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PATTERN RECOGNITION TECHNIQUES FOR INTEGRATION OF GEOPHYSICS, REMOTE SENSING, GEOCHEMISTRY AND
GEOLOGY
A Thesis submitted for the degree of DOCTOR OF PHILOSOPHY
in the FACULTY OF SCIENCE OF
THE UNIVERSITY OF LONDON
by
JONG NAM PARK M.Sc (London)
Geophysics Section Royal School of Mines Imperial College of Science & Technology London SW7
APRIL 1983
ii
A B S T R A C T
An evaluation of pattern recognition techniques applied to
filtered multivariate data has been carried out on selected geo-
physical ,remote sensing, and geochemical test data from the Bodmin
Moor area, S.W. England. A total of 14 data sets was used.
Unsupervised classification methods, factor analysis and
cluster analysis, and supervised classification techniques,
empirical discirminant analysis and characteristic analysis, were
adopted. Their usefulness in geological mapping and outlining
potential mineralization has been compared.
Generally the supervised classification techniques proved
fast in classifying the multivariate data based on an initial template
of known data. The unsupervised classification techniques did not
require such initial information but needed more computing time.
It was found that regional lithological mapping could be
best achieved using transformed data, while the untransformed data
were more suited to defining potential mineralization zones. However,
the Bodmin Moor granite rock type was consistently identified by all of
the methods.
Much emphasis was put on feature extraction techniques,
a different technique being applied to each of the 14 data sets
depending on the characteristic property, either physical, chemical
or spectral.
During the course of the research, several new techniques of
data analysis have been developed, with associated computer programs,
especially data extrapolation in convolution filtering. This
technique has been effectively applied in geophysical data processing
in order to avoid loss of information at the edges of data sets.
Also, a new pseudogravity filter has been designed for use on equally
spaced data. This technique facilitates the interpretation of
complex subsurface geological features by comparing gravity data
with the pseudogravity transform of the corresponding magnetic data.
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ACKNOWLEDGEMENTS
I wish to express most gratitude to Dr D.R. Cowan for suggesting the research topic, and for his help, encouragement and advice throughout the research. My immense grati tude also goes to Dr R.J. Howarth, for providing useful computer programs on pattern recognition techniques and graphic plots, and for his help and advice particularly in the multivariate data analysis. Their cri t ical reading of the manuscript was also most grateful .
I am also very grateful to the British government for providing a grant for the duration of the research and in particular Miss Celia Wood of the British Council for her help.
I wish to thank the Korean Government, particularly, Professor B.K. Hyun, President of Korea Institute of Energy and Resources (KIER) and Dr J . H. Koo, Head of Geophysics Department of KIER, for allowing me to have leave of absence from KIER to undertake this research.
I am also much indebted to Professor R.G. Mason, Head of Geophysics Department of Imperial College (IC) for his help in obtaining the British Council grant and Dr Anna Thomas-Betts of IC for her help in providing the allocation of the computer units and also for useful discussions on certain mathematical aspects. Many thanks are also due to M Hale of IC for providing useful information on geology and geochemistry.
I am also very grateful to many people for helping in various ways during the research. Particular mention must be made of Dr R.B. Evans and Dr J. Hawkes from the Institute of Geological Sciences (IGS). Many thanks to Mr P. Jarvis of the Imperial College Computer Centre for his help in colour plotting and allowing access to his software program for Landsat MSS data manipulation. Immense thanks also goes to Mrs S. Cowan for the colour plot of the geo-chemical data over the Bodmin Moor area in Chapter
I also wish to thank T. Richardson for typing the thesis. Finally special thanks are due to my wife and family for
their patience, encouragement and great support throughout the period of the research.
This thesis is dedicated to the memory of my father W.K. Park who supported me in all aspects of my life and in particular my further education.
J.N. Park
V
LIST OF CONTENTS
Page
Abstract Acknowledgements iv List of Contents v
List of Figures ix List of Tables xiii List of Appendices xv Summary of Abbreviations xvi
CHAPTER 1: INTRODUCTION 1
CHAPTER 2: GEOLOGICAL SETTING 6
2.1 Geomorphology and Soil Types 6 2.2 Geology 7 2.2.1 The Geology of Cornwall 7
(a) Devonian and Carboniferous Rocks 9 (b) Grani tes and Minor Intrusions 9 (c) Mineralization 10 (d) The Structure 12
2.2.2 The Geology of the Bodmin Moor Area 15
(a) Devonian Rocks (b) Carboniferous Rocks (c) Devonian Volcanic Rocks and Greenstone 18 (d) Granites and Later Intrusive Rocks 19 (e) Elvans 2 0
(f) Aureole of Thermo-Metamorphism surrounding 20 the Granites
(g) Mineralization 21
CHAPTER 3: DATA PREPARATION AND QUALIFICATION ' 23 3.1 General Considerations in Data Preparation 23
and Qualification 3.2 Interpolation and Extrapolation 24 3.2.1 Interpolation 25 3.2.2 Extrapolation 26
(a) Extrapolation in Space Domain 26 (b) Extrapolation in Frequency Domain 27
3.3 Digitization of Two-Dimensional Data 28 3.3.1 Digitization of Geophysical Data 28
(a) Gravity Data 32 (b) Magnetic Data 32
3.3.2 Digitization of Landsat MSS Data 33 3.3.3 Digitization of Geochemical Data 35 3.4 Noise Evalaution 36 3.4.1 Estimates of Noise Contributions 36 3.4.2 Noise Filtering 39 3.5 Data Qualification 44 3.5.1 Stationarity 44 3.5.2 Normality 48 3.6 Discussions 49
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CHAPTER 4: FEATURE EXTRACTION 51
4 . 1 Introduction 51 4 .2 Geophysical Data Processing 52 4 . 2 . 1 Filter Operators for Analysis of 52
Potential Field Data 4 . 2 .2 Review of Filter Operators 55
(a) Derivatives 55 (b) Upward and Downward Continuations 56 (c) Lowpass and Highpass Filtering 58 (d) Reduction to the Pole 59 (e) Pseudogravity 60
4, .2. .3 Theoretical Background of Filter Operators 61 4, .2, .4 The Description of GEOPAK Program 64 4, .2, .5 Qualitative Interpretation of Potential Field 66
Data (a) Description of Filtered Gravity and Magnetic 66
Maps (b) First- and Second-Derivative Maps 68 (c) Upward and Downward continuation Maps 74 (d) Lowpass and Highpass Filtered Maps 78 (e) Reduction to the Pole Map 81 (f) Pseudogravity Map 82
4. .3 Landsat MSS Data Processing 84 4. .3. . 1 Introduciton 84 4. .3. .2 Basic Principles in Landsat MSS Data 89
Processing 4. .3. ,3 Feature Extraction and Interpretation 93
(a) Black-and-White MSS Images of Cornwall 93 (b) False-Colour Composite of Cornwall 100 (c) Colour-Ratio Composite of Cornwall 101 (d) Description of Surface Maps for Bodmin Moor 103
Area (e) False-Colour Composite of the Bodmin Moor Area 106 (f) Colour-Ratio Composite of the Bodmi'n Moor Area 108
4. 4 Geochemical Data Processing 1 11 4. 4. 1 Principles of Regional Geochemical Data 1 13
Processing (a) Analysis of the Regional Distribution Patterns 113
of Geochemical Elements (b) Detection of Geochemical Haloes using the 1 14
Probability Plot 4. 4. 2 Regional Distribution of Geochemical 1 15
Elements 4. 4. 3 Probability Analysis 1 18 4. 5 Conclusions 126
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CHAPTER 5: TREND SURFACE ANALYSIS 127
5.1 Introduciton 127 5.2 Mathematical Procedures 128 5.3 Applications to the Bodmin Moor Area Data 131 5.3.1 Geophysical Data 132
(a) Gravity Data 132 (b) Magnetic Data 132
5.3.2 Landsat MSS Data 136 5.3.3 Geochemical Data 142 5.3.4 Residuals from the Third-Degree Trend 144
Surface (a) Geophysical Data (Gravity & Magnetics) 144 (b) Landsat MSS Data 148 (c) Geochemical Data 149
5.4 Discussion 151
CHAPTER 6: SIMILARITY ANALYSIS 155
6.1 Introduction 155 6.2 Principles of the Procedures 155 6.2.1 Overall Similarity 155 6.2.2 Spatial Similarity 157 6.2.3 Coherence Analysis 159 6.3 Applications of Similarity Analysis 162 6.3.1 Overall Similarity 162 6.3.2 Similarity Map 171 6.3.3 Coherence Analysis 175 6.4 Discussion 181
CHAPTER 7: CLASSIFICATION AND IDENTIFICATION OF 185 MULTIVARIATE DATA
7.1 Introduction 185 7.2 Transformation of Data ' 186 7.3 Selection of Variables in Multivariate Data 193
Analysis 7.4 Unsupervised Classification 198 7.4.1 Factor Analysis 198
(a) Factor Analysis Procedures 198 (b) Applications of Factor Analysis 204
7.4.2 Cluster Analysis 228 (a) Cluster Analysis Procedures 228 (b) Testing the ISODATA Program 236 (c) Applications of Cluster Analysis 245
7.5 Supervised Classification 259 7.5.1 Discriminant Analysis 259
(a) Discriminant Analysis Procedures 259 (b) Selection of the Initial Condition for EDF 264
Program (c) Applications of EDF 269
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7.5.2 Characteristic Analysis 275 (a) Characteristic Analysis Procedures 275 (b) Application of Characteristic Analysis 279
7.6 Conclusions 282
CHAPTER 8 : CONCLUSIONS AND RECOMMENDATIONS 287
8.1 Some Concluding Remarks 287 8.2 Summary of Recommendations for Further Work 289
REFERENCES 292
IX
LIST OF FIGURES Page
1.1 A pattern recognition system 2
2.1 Geology of Cornwall (S.W. England) 8
2.2 The structural geology of S.W. England 13
2.3 Geology of the Bodmin Moor area 16
3.1 Contour map of raw gravity data in gravity units 30 (128 by 128 array size)
3.2 Contour maps of potential field data 31 (64 by 64 array size)
3.3 MSS Scanning Arrangement 34
3.4. One-dimensional representation of two-dimensional 37 power spectrum by radial averaging
3.5 Two-dimensional power spectra of the filtered 41 data in log scale
4.1 Contour maps of the filtered potential fields of the 67 Bodmin Moor area with a cutoff wavelength of 800m.
4.2 Derivative maps of the gravity data 70
4.3 Derivative maps of the magnetic data 72
4.4 Upward and downward continuation maps of the 75 gravity data
4.5 Upward and downward continuation maps of the magnetic 76 data
4.6 Upward continuation maps of the gravity and magnetic 77 data (h = 3)
4.7 Lowpass and highpass filtered maps of the gravity data 79
4.8 Lowpass and highpass filtered maps of the magnetic data 80
4.9 Reduction to the Pole map of the magnetic data 83
4.10 Pseudogravity map of the magnetic data 83
4.11 Contrast stretched black-and-white band images 94
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4.12 Frequency distributions of black-and-white 97 MSS band images
4.13 False Colour composite of MSS bands 4, 5 and 7. 102
4.14 Colour-composite of MSS band ratios, R5/4, R6/5 and 102 R7/6 in blue green and red, respectively
4.15 Surface maps of MSS bands 4 and 7 over the Bodmin 104 Moor area, Cornwall
4.16 Surface maps of MSS bands 4 and 7 over Bodmin Moor 105 granite
4.17 Colour representation of MSS bands 4, 5 and 7 by 107 normal slicing over the Bodmin Moor area and their false colour-composite
4.18 Colour representation of MSS band ratios R5/4, R6/5 and 110 R7/6 and their colour-composite
4.19 Geochemical data over Bodmin Moor area 116
4.20 Probability plots of 8 geochemical elements 120
4.21 Spatial distribution of geochemical haloes of Cu, Pb, 125 Sn and Zn
5.1 Comparison of trend surfaces of degree 1 for 14 133 variables in the Bodmin Moor area
5.2 Comparison of trend surfaces of degree 3 for 14 137 variables in the Bodmin Moor area
5.3 Contoured residual maps of the third degree poly- 145 nomial
5.4 Linear directions of the data and 'predominant1 153 direction of the linear trends
6.1 Dendrogram - clustering with absolute correlation 167 coefficients (untransformed data)
6.2 Dendrogram - clustering with absolute correlation 170 coefficients (transformed data)
6.3 Similarity maps 172
6.4 Coherence vs. Frequency - Bodmin Moor Area 176
6.5 Bandpass filtered maps of coherence region A 178
6.6 Bandpass filtered maps of coherence region B 180
6.7 Bandpass filtered maps of incoherent region C 182
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7.1 Histograms of the filtered data 189
7.2 Histograms of X transformed data except the gravity 191 data (a) in which case the arc sine transform was applied
7.3 Cattell's Scree Test for determining the correct 197 number of principal components calculated from 16 variables
7.4 Cattell's Scree Test for determining the correct 205 number of principal components of four different variable sets
7.5 FA (Factor analysis) of the untransformed 6-MSS data 208 set
7.6 FA of the transformed 6-MSS data set 209
7.7 FA of the untransformed 10-MSS variable set 212
7.8 FA of the transformed 10-MSS variable set 213
7.9 FA of the untransformed 8-geochemical element set 216
7.10 FA of the transformed 8-geochemical element set 220
7.11 FA of the untransformed 9-mixed variable set 224
7.12 FA of the transformed 9-mixed variable set 227
7.13 Flowchart for ISODATA 234
7.14 Results of clustering by ISODATA for 8-geochemical 247 element sets
7.15 Results of clustering by ISODATA for 9-mixed variable 252 sets
7.16 Results of clustering by ISODATA for 8-PCA Score sets 257
7.17 Two overlapped bivariate distribution showing the 261 effective classification by projecting onto the discriminant function line
7.18 Interpolated one-dimensional probability density 262 function for five training set samples with increasing values of smoothing parameter a.
7.19 Correct classification performance rates of the training 268 set for the untransformed and transformed variable sets
7.20 Results of classification by EDF for the 8-geochemical 270 element sets
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7.21 Results of classification by EDF for the 9-mixed 273 variable sets
7.22 Results of classification by EDF for the 8-PCA 274 score sets
7.23 Hypothetical data profile showing areas above local 277 inflexion points (second derivative negative) labelled
and other locations labelled '0'
7.24 Results of characteristic analysis for a variable set 281 of As, Cu, Pb, Sn and Zn
xiii
LIST OF TABLES Page
3.1 Noise filter coefficients 40
3.2 Test result of stationarity 47
4.1 Comparison of the number of multiplications 54 required between ordinary convolution, 8-folded convolution, discrete Fourier Transform and Fast Fourier Transform
4.3 Means and standard devl ations of 8 geochemical 119 elements from the Bodmin Moor area in lithologic units
4.4 Statistical results of probability analysis 123
5.1 Trend surface coefficients of degree 1 and 140 degree 3
5.2 Summary of the test values of goodness-of-fit and 141 F-test
6.1 Correlation matrix (untransformed data) 166
6.2 Correlation matrix (transformed data) 169
7.1 Estimation of mean X values using Dunlap and Duffy, 187 Skewness/Kurtosis 2/1, and maximum likelihood schemes
7.2 Statistical results of factor analysis for 6- 210 Landsat MSS variable sets
7.3 Statistical results of factor analysis for 214 10-Landsat MSS variable sets
7.4 Statistical results of factor analysis for 217 8-geochemical element sets
7.5 Statistical results of factor analysis for 225 9-mixed variable sets
7.6 An example of the input parameters for ISODATA 237
7.7 Test results of ISODATA of 8-geochemical element 239 sets by varying the spherical factor with the rest of the input parameters constant
7.8 Test results of ISODATA of 9-mixed variable sets by 240 varying the spherical factor with the rest of the input parameters constant
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7.9 Test of ISODATA of 8-PCA score sets by varying the 241 spherical factor with the rest of the input parameters constant
7.10 Test of ISODATA by varying the Euclidean distance 243 criterion (8 ) with the rest of the input parameters constant
7.11 Means and standard deviations of clustering for 248 8-geochemical element sets
7.12 Means and standard deviations of clustering for the 253 9-mixed variable sets
7.13 Eigenvectors of the first 8-principal components 256 of PCA with 16 variables
7.14 Means and standard deviations of clustering for the 258 8-PCA score sets
7.15 Characteristic weights of five elements (As, Cu, Pb, Sn 282 and Zn) for the model selected
XV
LIST OF APPENDICES Page
Appendix A New Extrapolation Techniques 315 A.1 Extrapolation in Frequency Domain using 315
Bicubic Spline Method A.2 Extrapolation in Space Domain 319
Appendix B Sampling of Landsat MSS Data for the 322 Bodmin Moor Area
Appendix C Calculation of Filter Operators 326 C.l Calculation of Continuations 326 C.2 Calculation of Reduction to the Pole 327 C.3 Calculation of First Vertical Derivative 328 C.4 Calculation of First Horizontal Derivative 329
in Gravity C.5 Calculation of Lowpass Filter 330 C.6 Derivation of Pseudogravity 331 C.7 First Horizontal Derivative of the 336
Magnetic Field C.8 Second Vertical Derivative 341
Appendix D Descriptions of GEOPAK Program 349
Appendix E Contrast Stretching using the Probability Denstiy 353 Function of Gaussian Distribution for PICPAC Gray Scale Plotting
Appendix F Trend Surface Analysis of First- and Third- 357 IXgree Polynomials
F.l Least Square Fit for a Flat Plane using 357 First-Degree Polynomials
F.2 Least Square Fit using Third-Degree 358 Polynomials
Appendix G Empirical Discriminant Function • 360
Microfiche Listing (rear folder)
GEOPAK
COHAN AND BPFILT
FACTOR
xv i
SUMMARY OF ABBREVIATIONS
BNG British National Grid
CCT Computer Compatible Tape
CHARAN CHARacteristic ANalysis
DFT Discrete Fourier Transform
EDT Empirical Discriminant Function
FA Factor Analysis
FFT Fast Fourier Transform
IGRF International Geomagnetic Reference Field
IGS the Institute of Geological Sciences
LDF Linear Discriminant Function
MSQ Mean SQuare Value
MSS MultiSpectral Scanner
PCA Principal Component Analysis
RSU Remote Sensing Unit
SLM Single Linkage Method
ULCC University of London Computer Centre
USGS United States Geological Survey
WPGM Weighted Pair Group average Method
1
CHAPTER ONE
INTRODUCTION
In the Earth Sciences there is a growing trend to change
from a univariate or bivariate data analysis to multivariate
data analysis in almost every aspect of the subject. This tendency
is due to several factors; firstly, in the study of Earth Sciences,
often no single variable is sufficient to extract useful geological
information from complex geological provinces or mineralized zones.
Often, a combination of several variables is more significant in
defining and delineating these areas. Secondly, modern methods in
Earth Sciences are providing a variety of data sets from the same
area, and thirdly, modern computing methods enable the manipulation
of complex data sets.
In this respect, pattern recognition techniques have been
successfully applied in various fields of multivariate data analysis.
Since the application of pattern recognition techniques to
Earth Sciences began in the 1960's, there has been increasing interest
in the utilization of the techniques for the extraction of
useful geological information, particularly for geological mapping and
evaluation of potential mineralization. In particular, the use of
Computer Compatible Tape (CCT) from Landsat Multi-Spectral Scanner
(MSS) data has increased the usefulness of the techniques in various
fields of study.
What then is pattern recognition and how is it useful in
Earth Sciences?
2
Pattern recognition has developed mostly as statistical
classification techniques (Fu, 1976). Tou (1974) has defined
pattern recognition techniques as "a process of categorization of input
data into identifiable classes via extraction of any features of
significance in the input data from a background of irrelevant details",
These techniques are involved with the extraction of features from
the input data and then classifying each feature into a class based
on certain criteria or a set of selected measurements extracted from
the input data. These criteria or selected measurements are
supposed to be invariant or less sensitive with respect to the commonly
encountered variations and distortions, and also supposed to contain
less redundancies. Under these conditions, Fu (1976) has noted that
pattern recognition can be considered as consisting of two subproblems.
1. What measurements should be taken from the input
patterns? Usually the decision of what to measure is
rather subjective and also depends on practical
considerations such as the availability of information or
the cost of measurements.
2. The second is the problem of classification based
on the measurements taken from the selected features.
A simplified block diagram of a pattern recognition system
is shown in Fig. 1.1.
Input Pat tern
Feature extraction
Class i-fication Dec is ion
Feature Measurements
Fig. 1.1: A pattern recognition system (After, Fu 1976)
3
There are two categories in pattern recognition classification -
supervised and unsupervised.
The former depends on prior information on the nature of classes
to be identified and establishes a classification of the samples based
on this information. Discriminant analysis or characteristic
analysis belongs to this category.
The unsupervised classifications are used when no prior
information is available on the nature of classes. Principle
component analysis or any cluster analysis techniques are in this
category.
These categories have been distinguished by Howarth (1973b)
as pattern classification and pattern analysis, respectively.
Applications of pattern recognition techniques to the Earth
Sciences and problems arising in their practical applications have
been fully discussed by Howarth (1973b, 1983).
The main objectives of the research involved in this thesis
are to investigate by computer methods any correlation between different
groups of data sets, and thereby to extract any features of geological
significance such as any potential mineralization or geological
variation.
This kind of computer-based analysis of a multivariate data
set is particularly useful because firstly computer analysis with more
data will provide unbiased and possibly more reliable results and also
subtle differences between data sets which are not very easy to
distinguish by other means may be readily distinguished in a reasonable
manner so that the result may provide insight into the geology.
Secondly full utilization of Landsat MSS data with its abundance and
repeatability of data would provide a cheap and fast way of data
analysis for preliminary reconnaissance surveying, particularly
4
where the area is not easily accessible and where any geological
or geochemical data are not readily available. Thirdly, the criteria
found in the chosen study area could be extrapolated to other areas
with similar geological conditions.
The research is involved with the integration of various
geophysical, geological and geochemical data bases with Landsat MSS
data for the chosen Bodmin Moor area of Cornwall, S.W. England, using
pattern recognition techniques as mentioned previously.
To achieve the research objectives, it is essential after
selecting good relevant data, to carefully make the digitization,
preprocessing (noise filtering) and data validation which are
essential to the data analysis.
Digitization of data was achieved by using either computer
interpolation techniques (gravity and geochemical data) or manually
(magnetic). For Landsat MSS data, an areal average method was used for
obtaining initial data for analysis and then the computer interpolation
technique was applied in order to project the data onto the same grid
points as used in the other methods.
In the noise filtering process, Landsat MSS and geochemical
data were filtered by a 3 by 3 spatial moving average method, and for
particular use in geophysical data, a 9 by 9 filter operator was
designed by using the concept of low-pass filtering.
The most important factors in data qualification, stationarity
and normality of data were also evaluated.
Various data sets were subject to analysis individually prior
to the multivariate data analysis using pattern recognition.
For geophysical data, various filter operators have been
designed for analysis to extract information due to regional or local
geological features. This includes vertical and horizontal
5
derivatives, upward and downward continuations, low- and high-pass
filtering and reduction to the pole of the magnetic data. Further
with the formula derived, pseudogravity filter operator for
2-dimensional data were then designed for data processing. This is of
great use to effectively analyse the coherence between gravity and
magnetic data, and thereby to extract spectral and further subsurface
structural features.
For Landsat MSS data, image enhancing techniques such as
contrast stretching using the Gaussian probability density function,
and ratioing techniques have been exercised and their colour composites
have been made for analysis.
The geochemical data were subject to level slicing by
concentration scale to examine the regional distribution of each
element and probability plots were applied to define threshold values
of anomalies and thus delineate spatial distribution of those anomalous oire&S.
Similarity analysis has been made of the data by using trend
surface analysis, analyses of correlation coefficients and similarity
map, and coherence analysis.
The information in this thesis will be presented in the
following sequence.
In Chapter 2, the general geomorphology and soil types,
regional geology and mineralization of the study area have been
reviewed. This is followed by data preparation and qualification in
Chapter 3. Chapter 4 deals with feature extraction of individual
data sets including geophysical, Landsat MSS and geochemical data.
The trend surface analysis is then described in Chapter 5 and
further similarity analysis in Chapter 6.
Finally various supervised and unsupervised pattern recognition
techniques have been applied for classification of geological signi-
fiance in Chapter 7.
6
CHAPTER TWO
GEOLOGICAL SETTING
2.1 Geomorphology and Soil Types
Cornwall can be conveniently divided into two major physio-
graphic upland units which are related to major rock types, the
Granites and the Slates. Extensive, gently sloping (influve) areas
lying from below 90m in Land1 s End to over 300m to the North Cornwall
characterize the land over Devonian and Carboniferous slates.
The granite upland rises from less than 200m in theLand's End area
to over 400m at Rough Tor on Bodmin Moor.
The climate is typical of southwestern Britain. Though very
mild onLand1s End and near the coast, conditions become more
climatically severe with increasing rainfall as the high Moor lands
are approached. Average annual rainfall ranges from over 1800mm
on the higher parts of the Bodmin Moor to below 1000mm on the coast
and along the lower parts of Land1sEnd.
The soils in the region are formed either directly in-situ
from local rocks, or by downslope movement of already weathered material
produced previously. Staines (1976, 1979) gives detailed description
of the soils in the region.
Much of the Devonian slate outcrop is dominated by soils belonging
to the Highweck series, of fine loamy typical brown earths in slaty
drift with depressions occupied by wetter soils. Close to the
granite contact are brown podzolic soils in the south but stagno-
podzolic soils in the north. On the volcanic outcrops in the north
brown podzolic soils dominate with humic gley soils in depressions.
The Upper Carboniferous outcrops in the north are occupied by stagno-
7
podzolic and surface-water gley soils.
In contrast, most of the higher, flatter ridges of the granitic
outcrop are occupied by stagnohumic gley soils, while the flanks are
occupied by stagnopodzols. Brown podzolic soils occur on the lower
formed granite land. On parts of Godolphin granite the granitic
head contains a significant proportion of the slate debris within it.
Depression sites such as broad valleys and valley heads are
occupied by humic gley and peat soils.
2.2 Geology
In the following description the regional geology of Cornwall
is abstracted mainly from Edmonds et al. (1975) and a detailed geology
of the Bodmin Moor area is reviewed from relevant Memoirs of the
Geological Survey of Great Britain Sheets and other published
references.
2.2.1 The Geology of Cornwall
The regional geological map of Cornwall is shown in Fig. 2.1.
The Pre-Cambrian igneous complex of the Lizard, which comprises
serpentinites, gabbro,hornblende schists and the Kennack gneiss and
granites, is known to contain the oldest rocks in this region. The next
oldest rocks are relatively small remanents of Ordovician quartzite
and Silurian limestone at Meneage and Nare Head. However, the region
is mainly dominated by intensely contorted thrust Devonian and
Carboniferous rocks (Killas) which are intruded by granite masses and
overlain in parts by relatively undisturbed Permian and later deposits.
Fig. 2.1 Geology of Cornwall (South-west England)
9
(a) Devonian and Carboniferous Rocks
The south-west peninsula of England is mainly composed of a broad
synclinal structure trending east-west. Devonian sediments occur
to the north and south, and the central area of the syncline is
occupied largely by Carboniferous rocks (Dearman, 1971).
The Devonian is divided into three groups; the Lower, the
Middle, and the Upper Devonian. It is mainly composed of slates, silt-
stones, conglomerate and some calcareous beds which vary in colour
from purple to green and grey. The Carboniferous rocks of south-west
England are divided into two series; a lower marine series of
evenly bedded shales and cherts with thin limestone, and an upper
series of a sequence of shales with turbidite sandstones and
siltstones (Reid, et al., 1911).
The Lower Carboniferous rocks occur in several tectonic slices
separated by thrusts and faults.
Extensive sheets of spillitic lava, tuffs and agglomerate from
volcanicity during the Lower Carboniferous times occur in northern
Cornwall and South Devon.
(b) Gran ite and Minor Intrusions
The oldest rocks of south-west England, apart from those of the
Lizard, are dark green dykes and sills called greenstones and they
include dolerite, gabbros and other basic igneous rocks. However,
the most widespread igneous rocks in Cornwall and Devon are granitic,
and they outcrop as cupolas from a single buried Airorican batholith.
The granites of south-west England occur as five great bosses
which trend WSW from Exeter to Land's End, and form the large topographic
features of Dartmoor, Bodmin Moor, St. Austell Moor, the Carnmenellis
and Land's End masses. Small bosses lie near the larger ones forming
10
Kit Hill and Hingston Down between Dartmoor and Bodmin Moor, Castle-an
Dinos and Belowda Beacon, north of St. Austell Moor and around
Carnmenellis Carn Brea, C a m Marth and Godolphin. Isolated from
Cornwall are the granite boss of the Scilly Isles and the underwater
HaigFras lying about 60 miles N.N.W. of the Scillies which may be part
of the same batholith displaced by faulting (Edmonds, et al., 1975).
Exposed contacts between granite and country rock (locally known as
killas) are always sharp, but often irregular and frequently complicated
by apophyses from the granites, or granite dykes cutting across the
contacts.
The Bouguer anomaly map of Bott, Day and Masson Smith (1958)
indicates that in general the granite contacts are steeper on the
south than on the north side. This is generally confirmed by the
aureoles (Edmonds, et al., 1975) and by gravity modelling of the granite
masses and drilling conducted by the Institute of Geological Science
(IGS) (Beer, Burley and Tombs (1975, Tombs 1977). Further confirmation
can be found in the 2-dimensional modelling conducted by Al-rawi (1980).
The granite is characterized by rugged tors dominating areas
of moorland or rough pasture. The rocks consist generally of quartz,
perthitic orthoclase crystals, plagioclase and brown mica (biotite).
Secondary white mica (muscovite) is locally present and consists of
tourmaline, zircon and apatite (Edmonds, et al., 1975).
(c) Mineralization
The mineralization in south-west England is widely recognised to
be related generally to the intrusion of the Amorican granites, while
later orogenic movements produced further mineralization. The
intrusion of quartz-porphyry, aplite and thin pegmatite dykes is also
associated with the late stages of granite emplacement. Hosking
11
(1949) suggested that the mineralization always occurs in close proximi
to the granite, but Dines (1956) noted that the relationship between
the mineral deposits to the granite bosses is not so close, based on
the study of lode distributions around the granite bosses.
A general fracture pattern, which eventually produced the
channels for the mineralizing fluids, probably developed at the
beginning of folding and granite intrusion. Later relaxation of the
pressure and the shrinkage caused by the cooling granite reactivated
the fractures. Local variations of stresses established distinct
local fracture patterns differing from the regional system.
The mineral content of the lodes varies both laterally and with
depth. Rich sections occur at changes of strike, intersections, and in
the steeply inclined sections. The lowest mineralization zone contains
dominant amounts of tin, while above it, is the copper rich zone, which
in turn is replaced upwards by a still more extensive lead rich zone.
Other elements present are zinc, arsenic, antimony, iron and manganese,
together with smaller amounts of tungsten, cobalt, nickel, uranium,
baryte and fluospar.
The gangue minerals associated with the ore-bodies in Cornwall
are mainly adularia, fluorite, white mica, tourmaline, chlorite,
hematite and quartz. Secondary alteration above the watertable
is widespread throughout the region. The upper parts of the lodes
consist mainly of gangue minerals with iron and manganese oxides,
forming the 'gossan1 or the so-called 'iron-hat' of the miner (Edmonds,
et al. , 1975).
There are currently five operating metal mines in S.W.England
and a number of developing prospects: Geever (Land's End), South Crofty
(Redruth), Wheal prosper and Mulberry (Lanivet), Wheal Concord
(Blackwater), and Hernerdon (Plymouth). All produce tin as the major
12
product with tungsten, copper, zinc, silver and gold as significant
associated products (Thorne, 1981).
(d) The Structure
During the Ordovician times the area was covered by a shallow
warm sea, deepening to the north towards the central Welsh trough.
The shallow sea became more extensive in Silurian times and stretched
from the Bristol Channel across eastern England.
Towards the end of the Silurian period the Caledonian Orogeny
uplifted the land to the north of the present peninsula and largely
produced a north-easterly structural trend. Later, the Amorican
orogeny encroached and produced easterly trending folds.
A Caledonoid zone in S.W. England is truncated south of a
line from Perranporth through Pentewan and the rest of the peninsula
is covered by the Amorican zone.
The Amorican orogeny, of late Carboniferous and early Permian
times, was characterized by north-south compressive forces which have
produced an east-west structural trend. Towards the end of
compression the Cornubian granite magmas, rising from great depths,
approached to within some hundreds of metres of the surface. Strata
were arched up over a number of cupolas and in places, as above the
Bodmin Moor granite, slices of pre-folded rock slid radially outwards
from the rising intrusion. Some of this low angle faulting took
place by reactivation along earlier thrust planes. As the granite
crystallised, and pressure relaxed, easterly trending normal faults
developed. North-westerly wrench faults may have been initiated by
earlier compressive forces, but many were reactivated in Tertiary times.
In general, the folds trend easterly but modification might have taken
place by Tertiary wrench faulting (Dearman,1964).
K E Y
HP Tertiary
New Red Rocks
Granite
Major fault lone
Major synclinal a m
Major anticlinal a m
Line ol confrontation between south-lacing and north-lacing lolds
Zone of intense detormation
Utiriltinil I'oini
Bude H'anmtt At<ntlit
Ciimhcuk/ Hitscv Ih'iicli• BosctsUe
FOLDS F A N N I N G O V E R S O U T H W A R D S I N T O
R E C U M B E N T F L E X U R E S P A R T L Y O B S C U R E C B Y \ L A T E Z I G Z A G F O L D S
LParfsfov
Tintsge I
^ T T e a r h . ' ' " ' R E C U M B E N t
V S O U T H - F A C I N G y
A v " " " (sr f- S & K ? & &
Portnrow ' o ^ ** T R U R O \ J P e n t e w a n
-f*c/ng f o l o s \ >
^Portnsdlcr
Doilnmn I'oinl Shirt Point
30 40
30
50
40
T~ 60 70
50 Miles
80 Kilometres
l.izonl I'oini
Fig. 2.2 The structural geology of S.W. England (After Edmonds, et al., 1975)
14
Open upright folds north of Bude fan out northwards
into northward-overturned folds and southwards into southward-
overturned folds. The latter become recumbent and isoclinial
towards Padstow, south of which the primary recumbent folds face
north.
The large low angle Rusey Fault marks a major tectonic break.
To the south of a line from Rusey Beach through Tremaine, deformation
is increased. Folding south of this belt is also recumbent but
south-facing with axes trending around E.N.E. and this pattern extends
to a line roughly from Padstow through the southern ends of the
Bodmin Moor and Dartmoor granites. At this line occurs a major
confirmation of south-facing folds to the north and north-facing folds to
the south within the St. Minver synclinorium. Possibly the two
opposing sets of folds are separated by a thrust or fault.
Immediately south of Padstow the folds are recumbent,
trending ENE, but south-dipping fold axial planes around Newquay, and
in a belt which extends eastwards, steepen southwards. A major
anticlinal axis, displaced locally by faults, runs eastwards from
Tremaine, north of Newquay, to Dartmouth.
The northern boundary of the Gamscatho Beds at Perranporth,
and all along the Perranporth-Pentewan line, was probably a slide on
which major north-facing recumbent folds moved in NNW directions
(Sanderson, 1971).
The Mylor Beds and Gamscatho Beds between Perranporth and
Penzance show recumbent first-phase folds trending ENE and facing north
or north-west, second-phase upright folds and third-phase flat-lying
folds.
The metamorphic rocks of the Lizard represent altered early
Devonian and older rocks. Movements at the end of Devonian times,
15
suggested by radiometric dating, may point to the thrusting north-
eastwards of a Lizard nappe.
The general structure of the region is shown in Fig. 2.2.
A general view of the structure is described by Dearman (1971)
and Edmonds et al. (1975) and a good description on structural
zonings of the Variscan fold in the region is given by Sanderson
and Dearman (1973). Geotectonic views of the Cornubian batholith
are mentioned by Bromley (1976) and Badham (1976), and Paleomagnetic
reconstruction for the upper paleozoic and the tectonic evolution
of the Variscan orogeny are given by Tarling (1979).
2.2.2 The Geology of the Bodmin Moor Area
The study area is in the British National Grid Square SX
and contained within British National Grid northing lines 9000-5190
and easting lines 20000-23810.
Fig. 2.3 shows a more detailed map of the area, centred on the
Bodmin Moor granite and surrounded by Devonian and Carboniferous sediments
with scattered minor complex intrusions. A part of the St. Austell
granite is also included in the south-west corner'of the area.
(a) Devonian Rocks
The oldest sediments in this region are of Lower Devonian
age; these are grouped into three categories; Dartmouth slates, Meadfoot
group, and Staddon grits. They consist mainly of slates, sandstones,
shales, conglomerates and some calcareous beds.
The Dartmouth slates are often siliceous and contain bands
and lenses of hard grit of quartzite. The rocks vary in colour from
purple to green and grey. In the Dartmouth slate the distinction
zz
Crackington Formation.
Crackington Form.* Lower C.
Lower Carboniferous.
Upper Devonian.
Upper and Middle D.
Middle Devonian.
Lower Devonian.
Lavas.
Elvan or quartz porphyry.
Diabase.(D).
Granite.
Picrite.
) CARBONIFEROUS
•DEVONIAN
/IGNEOUS
v.
>. '/Ay . -X' '.-y,^
Fig. 2.3 Geology of the Bodmin Moor area.
17
between cleavage and bedding is often obscure partly due to intense
thrusting causing the middle limbs of recumbent folds to be severed.
The Meadfoot group is distinguished by the presence of grey slates
with few fossils in their upper portion (Ussher, et al., 1907).
Surrounding the St. Austell granite, the Meadfoot group of slates contains
lenticules and beds of grit, often hard, compact, and evenly bedded
(Ussher,et al., 1909).
In the area between Bodmin and Grogley Downs the Staddon group
is encountered and the group consists of a number of thin bands
of grit or grey-wackes , hard and compact if fresh, dispersed
through a much thicker mass of killas of variable textures (Reid,et al,
1910).
To the north of the Staddon grit ridge in the Bodmin area are
the grey Middle Devonian slates, in which the only variations consist
of a few thin beds of limestone, some silty bands, and a few
intrusive sills (Ussher, et al., 1909). In the region between
the coast and Wadebridge, the slates are always grey and show well
the striking flatness of the cleavage planes. East of Wadebridge
the narrow Middle Devonian outcrop shows the usual EW strikes
but near the area where the granite outcrops the slate is disturbed
probably with much folding and faulting (Reid, et al., 1910).
The Upper Devonian rocks, unlike the Middle Devonian, are
very variable. In the lower part there are grey slates but higher
in the succession there occur purple and green slates, which are in
turn replaced by silty slates with thin bands of grit. The Upper
Devonian rocks are characterized by the presence of various belts
of contemporaneous lava and volcanic tuff (Reid, et al., 1910).
This group extends from near Tintagel Head, the north-west corner
of the study area passes north of Bodmin Moor granite to occupy a
18
large area to the East of the Bodmin Moor.
(b) Carboniferous Rocks
In the Boscastle district rocks termed Yeolmbridge Formation
assigned to the Lower Carboniferous, crop out in a belt trending
ESE from the coast near Boscastle. These rocks comprise mainly
fine-grianed, grey sericitic slates with scattered brown earthy
calcareous lenses and there appears to be a gradational downward
passage into possibly Upper Devonian slates. The slates are
commonly rust-spotted after pyrite, and aggregations of sericitic
and brown micaceous minerals show elongation due to tectonic stresses
(Mckeown, et al., 1973).
The Upper Carboniferous rocks are divided into two groups;
the Crackington Formation and the Bude Formation. However, the
description here is restricted to the Crackington Formation since
the Bude formation is outside the study area. The Crackington
Formation consists of dark grey cleaved shales with interbedded
siltstones, thin turbidite sandstones and a few thicker greyish
green sandstone units. To the east of Bodmin Moor there is a
transitional zone from the Lower Carboniferous to'Crackington
Formation (Mckeown, et al., 1973).
(c) Devonian Volcanic Rocks and Greenstones
There are various indications of contemporaneous volcanic
activity during Upper Devonian times. The volcanic rocks are splites,
schalsteins and tuffs. They are for the most part grey-green in colour.
The lavas are mainly tuffs with some vesicular, highly sheared and
occasionally porphyritic agglomerates.
19
Greenstones occur as sills or dykes, mainly intruded
into the lower beds of the Upper Devonian (Mckeown, et al. 1973, Reid
et al. 1910).
In the northern and eastern parts of the Bodmin Moor there
are many igneous rocks breaking through and penetrating the Upper
Devonian slates and often altering them to spicosites. These rocks
are mostly basic and ultrabasic. Several exposures of picrite are
flanked by long, wide, strap-shaped sills and possibly dykes of
diabase; partly contact-altered by the granite (Reid, et al., 1911).
(d) Granites and Later Intrusive Rocks
Except in the western and northern margins of the granite
moor, where the contacts are always marked by a steep slope, there is
generally no sharp ri se at the iunction with the slates. On the north
western side of the granite junction with the killas is a normal
intrusive one but towards the northern end it appears often to be a
fault. Over large areas the granite, whether fresh or decomposed,
shows little original variation either in composition or texture. In
hand specimen the granite is seen to be composed of orthoclase in
crystals about an inch long, quartz, in abundant grains of fair size,
a somewhat subordinate amount of plagioclase, and white and brown mica.
The Bodmin Moor granite is further linked to the other Cornish
granites by the occurrence within it of several good-sized
masses of finer granite, which are clearly intrusive. Schorl may be
an essential constituent of the intrusions as a whole. The
characteristic feature of this finer intrusive, is the almost complete
absence of the coarse grains of quartz, so abundent in the normal
rock (Reid, et al., 1910).
20
(e) Elvans
Numerous elvans or quartz-porphyry dykes, are associated with
the Bodmin Moor granite and they are most abundant toward the southern
and eastern end of the great intrusion, with a few extensions
westward toward Padstow and Wadebridge.
The elvans are slightly younger than the granite and the
edges of the elvans are always composed of fine-grained, felsic material,
and the centre is always coarser in texture than the edges.
The alteration appears to be connected to the infilling of
the tin-veins and is in all cases very pale in colour and almost
white. Two types of alteration can be recognised in the elvans
in this areas as in other districts of Cornwall; the first is
essentially the same as that which affects the granite when it is
converted into greisen. In this process the feldspar of the original
rock is replaced by aggregates of white mica and quartz. The second
type is tourmalinization (Reid, et al., 1910).
(f) Aureole of Thermo-metamorphism Surrounding the Granites
The Bodmin Moor granite is surrounded by an aureole of
thermometamorphosed rocks like those of other Cornish granites.
The surface width is approximately 600 to 1000 metres around the
Bodmin Moor.
The altered rocks are silvery-grey, or almost white but as the
proportion of chloritic material increases, the altered rocks become
darker and they also contain much more finely divided iron-ore
than those that are rich in white mica.
The unaltered killas is composed largely of minerals which contain
much water of hydration. The first effect of the granite intrusion
was to set free this water in the form of superheated steam, which
21
resulted in the metamorphic rocks within the aureole. The meta-
morphosed killas has almost always a foliated structure.
The eastern portion of the aureole of metamorphism presents
several points of special importance. In the first place, the rocks
in the eastern side of the granite were less affected by dynamic
action than those in the west, and in consequence they were less
schistose when metamorphosed by the intrusion. Further the junction
of the granite with the killas is often a fault, and the fault
trends, roughtly, east-west, the same as the mineral lodes (Reid,
et al., 1910).
The St. Austell granite and the smaller outlying granitic stocks
are also surrounded by well-marked aureole of contact meta-
morphism or pneumatolytisis (Ussher, et al., 1909).
(g) Mineralization
The principal mining areas in the area of study are situated
in the metamorphic aureoles of the great granite bosses as in other
areas of Cornwall.
In the Bodmin Moor area, the most important part of the area
for its mineral production has been the south-eastern part of the
granite. East-west striking lodes rich in tin and copper have been
extensively worked.
They also line up with similar striking lodes in the Kit Hill-,
which are mainly of tin and copper associated with lead, arsenic and
silver, etc. Most mines are rich in pyrites.
The oxides of iron also occur in the form of gossans,
particularly between St. Austell and Wadebridge to the west of the study
area, most iron lodes trend north-south and the lodes mainly
consist of red and brown hematites.
22
The deposits of manganese are usually associated with oxides
of iron; they appear to be connected with cherts in the Carboniferous
series (Reid, et al., 1911, Mckeown, et al., 1973).
Important lead zones have also been extensively worked at
Menheniod and Hendsfoot, and antimony was worked near Duloe and
Pillaton in the south of Bodmin Moor granite (Ussher, 1907).
Recently, some industries are more actively conducting some
exploration in the south-eastern margin of Bodmin Moor granite and near
the Kithill where a large number of east-west trending sulphide lodes
occur (Prest, 1982).
In St. Austell granite, the principal mining area is situated
in the metamorphic aureole on the southside of the granite. The
principal deposits occur in the killas and for the most parts are
associated with quartz-porphyry dykes (elvans). The lodes of
St. Austell and of the whole region in the north are mainly tin-bearing.
There also occur mineralized lodes younger than the main tin and
copper lodes. Some of these lodes contain minerals which are also
characteristic of tin-lodes, such as arsenic, copper, and small
quantities of uranium, cobalt, and nickel ores. These younger lodes
have been formed in fissures crosscutting the tin'or copper lodes,
and these younger lodes have a strike direction between north and
northwest, similar to the cross-courses.
St. Austell district is particularly important as a producer
of remarkably white and pure china clay, which is an alteration product
of the feldspathic constituents of the granite. While the eastern
part of the granite is comparatively free from this kind of
decomposition, the central and western portions show a widespread and
very complete kaolinization (Ussher, et al., 1909).
23
CHAPTER THREE
DATA PREPARATION AND QUALIFICATION
3.1 General Considerations in Data Preparation and Qualification
Bendat et al (1971) and Bath (1974) have described in
detail the data preparation procedures for digital processing of
data.
Data preparation is considered as one of the most arduous
tasks to be faced in any efforts to any data processing. This is largely
done by two steps; digitization and preprocessing.
Digitization is the process of creating machine readable
values for digital data processing, the sampling of which is usually
performed at equally spaced intervals. The problem is then
to determine an appropriate sampling interval,AS. On the one hand,
sampling at points which are too close together will yield correlated
and highly redundant data and thus unnecessarily increase the
labour and cost of calculations, whereas on the other hand sampling
at points which are too far apart will lead to confusion between high
and low frequency components in the original data', and lose the
resolution. The latter problem is known as 'aliasing'. In the space
domain, if the sampling rate is 1/AS samples /m where AS is the
spacing in meters, the maximum frequency we can pick up is f = 1/2AS.
Here f is so-called the Nyquist frequency or folding frequency.
Two practical methods exist for handling this aliasing problem
(Bendat et al, 1971). The first is to choose AS sufficiently small
so that it is physically unreasonable for data to exist above the
associated cutoff frequency f .
Generally it is a good rule to choose f to be one and a
24
half to two times greater than the maximum anticipated frequency
(Bendat et al. 1971, Agarwal 1968). The second method is to filter
the data prior to processing so that information above a maximum
frequency of interest is no longer contained in the filtered data.
The second method is preferred over the first in order to
save computing time and thus costs. However, the combined case
should be always considered between information of interest and
computing time.
After the data has been digitized and filtered, prior
to actual analysis, certain operations must be performed in order to
qualify the data, since the correct procedures for analysing
random data, as well as interpreting the analysed results, are
strongly influenced by certain characteristics which may or may not
be exhibited by the data.
The most important of these basic characteristics involved
with the study are the stationarity and the normality of the data.
Stationarity should be analysed because the analysis procedures
required for non-stationary data are generally more complicated
than those suitable for stationary data. The validity of an assumption
that the data have a Gaussian probability density' function should be
investigated as the normality assumption is vital to parametric
analysis of the data.
3.2 Interpolation and Extrapolation
An account of the interpolation technique used in the
gridding is described.
Serious problems with edge effects are usually encountered
when data processing either in space or in frequency domains. Thus,
25
an appropriate technique should be applied in order to reduce edge
effects which might lead to spurious results, in case of analysis in
frequency domain, or loss of information around edges by stripping
off half of the filter window around the border when convolving in
space domain. This can be done by applying an appropriate extra-
polation technique.
Extrapolation techniques both in space and in frequency
domain have been developed.
3.2.1 Interpolation
There are a variety of interpolation schemes, differing not
only on the principles but also sometimes on the properties of data.
In any case, the most important features are firstly to calculate
values as near as possible to the real values and secondly to introduce
minimum noise in the digitization process which can affect the computed
spectrum.
The gridding program applied in this study was originally
written by the USGS (Program W9322) and it utilizes the inverse-square
method. That is, the weighting factor applied is'inversely proportional
to the square of the distance. Mathematically, for a particular grid
point (i,j), the computed grid value Z(i,j) is
n 2
Z X(k) * l/dZ(k) Z ( i , j ) = — (3.1)
kS L l/d2(k)
t h
where x^k) t'"ie ^ originally measured value which is separated by
d(k) from the grid point (i,j). If the distance is in the grid unit,
the range of d(k) for a particular point (i,j) is
26
(i-1) <_ d(k) £ (i+1) in row
(j-1) _< d(k) £ (j + 1) in column
For those map points which are coincident to the data points,
the values themselves are assigned to the grid point. For missing points
which might occur particularly where the originally measured data th are sparsely distributed, either k local points or the 4-point
th boundary average method is applied. In the K local averaging method
for a missing point searches are made for calculated local grid
points by increasing the range of search-distance radially, and if
any calculated values are encountered, the average of those grid
values is taken for that missing point.
The 4-point boundary average method involves averaging
values at the nearest 4 points for the missing point value.
In this study, the latter method has been applied to obtaining
missing points in the gridding procedure because it is computationally
a little more efficient.
If the originally measured data is fairly uniformly
distributed, this is a very effective method particularly in potential
field data as the decay pattern of the field is generally inversely
proportional to the square of the distance. However, it has been
noted that if the data is sparse, it may introduce some high frequency
noise. This effect will be demonstrated in Section 3.3.
3.2.2 Extrapolation
(a) Extrapolation in Space Domain
In the space domain, information at edges can be lost up to
27
half of a filter window around the border of a map when convolving.
This can be avoided by generating data at ends of a profile or
around the border of a map. However, the problem with an extrapolation
is that in this case the information at edges is distorted so that
it might induce some spurious results.
A very reasonable way of extrapolation has been developed to
minimize the distortion, so that the information at edges might be
as useful as others (or at least no significant distortions at edges
in worst case ).
The technique uses the geometrical property of an exponential
function and the local background. This method is computationally
efficient.
(b) Extrapolation in Frequency Domain
The fundamental approach for reducing edge effects in
frequency domain is different from that of space domain, because of
differences in assumption of data structure. In the frequency domain
the data is assumed to be periodically continuous while in the space
domain it is not necessarily so.
The edge effects in the frequency domain-are due to the
truncation of data which results from violation of assumptions (Bath,
1974). This truncation of data introduces the so-called Gibb's
phenomenon where some unwanted oscillations are introduced into the data
in the processing. This phenomenon is larger at the edges of the data,
so that the processed results could be ambiguous at edges. Although
it is not presented here, the author has experienced serious edge
effects when Gaussian filtering was applied to the gravity data to
reduce the high frequency noise in the frequency domain. Another
problem is to avoid introducing any high frequency noise due to
28
overshooting when extrapolating, particularly with a short data
length.
The most common method may be the cosine tapering (Bath,
1974). Computationally a little more laborious, but one of the most
effective methods has been developed using Ku's (1977) algorithm
of simplified cubic spline method used in his interpolation scheme.
The so-called 'mean slope' method is advantageous compared to the
conventional cubic spline methods in that it attenuates overshooting
by interpolation so that high frequencies may not be introduced in
extrapolation even with very short data length.
Both extrapolation schemes either in space or in frequency
domains are described in detail in Appendix A.
3.3 Digitization of Two-dimensional Data
Considering the area being covered around the Bodmin Moor
and peculiarities of the research, a sampling interval of
300m would be required to make a 128 by 128 grid array and thus the 2
total area covered is about 1500Km . It is bounded by the British
National Grid coordinates 5190N - 9000N in the north-south and
20000E - 23810E in the east-west directions as mentioned in Section
2 . 2 . 2 .
Digitization was carried out either manually by projecting
a grid on the field contour maps and taking readings at regular interval
(magnetic), or automatically by using a computer program with irregular
data recorded on the magnetic tapes (gravity and geochemical data).
Landsat MSS data for the area were acquired by averaging for a certain
area and gridding using the computer program. Details of the method of
gridding each data set will be described in the next section.
29
Actually the first method is considered as the least
efficient way of performing conversion. This is usually done
in cases where no other alternatives exist. However, its accuracy,
when carefully done, compares favourably with other more sophisticated
methods. For example, the raw gravity in Fig. 3.2(a) obtained
by using the gridding program contains more high frequency noise
through the data than the raw magnetic map in Fig. 3.2(b) obtained
manually.
Besides noise being introduced in the digitizing process,
particularly when the gridding program is used for digitization, noise
also may be introduced in data acquisition procedures due to
geological, field disturbances, instrumental or cultural effects
and it may also be caused in data reduction procedures.
Aliasing was not examined by analysing the frequency
response of a profile but visually inspecting the contoured maps.
Since there is virtually no loss of resolution, as shown in
comparison between two sets of gravity maps in Fig. 3.1 with 128 by 128
grid data, and Fig. 3.2(a) with 64 by 64 grid data (which has been
taken for every second value in every second row), the data were
further reduced to a set of 64 by 64, to give a sampling interval of
600 meters. The advantage of reduced data size, is that the smaller
set is much more efficient to handle, particularly in multivariate data
analysis and yet it is still large enough for regional analysis of
multivariate data sets.
The noise in the data was reduced by applying appropriate
filtering techniques. This filtering aspect will be described in
the following sections.
30
3N*r(20oooe,qooo>4)
0 io 20 wm 1 I 1 ' S c a t e
Fig. 3.1 Contour map of Raw gravity data in gravity units. 128 by 128 array size.
31
(20QftOE,qoQON)
(a) Gravity
I 1 1 1 L
120000E ,<1ooon)
Fig. 3.2 Contour maps of potential field data (a) Gravity (b) Magnetics. 64 by 64 array size.
32
3.3.1 Digitization of Geophysical Data
(a) Gravity Data
The field data for the Bouguer gravity anomaly map was obtained,
reduced and compiled by the Institute of Geological Sciences (IGS).
Tombs (1977) has described the data reduction procedures in some detail. 3
A density of 2.67gr/cm was used for the reduction and the
Bouguer anomalies were calculated against the International Gravity
Formula 1967 and referred to the National Gravity Reference NET 1973.
For topography ranging within a radius of 50km (Hammer zone, A to M) 3
the data were terrain-corrected for an average density of 2.7gr/cm .
The terrain-corrected Bouguer anomaly values expressed in ten
times a gravity unit were dumped on the magnetic tape and gridded at
an interval of 300m. The density of field observations over the
Bodmin Moor area that was gridded is about 70 data points per 100 sq.km.
(b) Magnetic Data
The total magnetic field contour map was published by the
IGS in 1958 at 1/25,000 or 1/50,000 scales over the on-shore areas
of south-west England. The contour interval of this aeromagnetic map
is 10 gammas. This map was compiled from an airborne survey conducted
in 1957 by Hunting Survey Limited, under the contract to the Geological
Survey of Great Britain (now part of IGS) and the United Kingdom Atomic
Energy Resource Authority. The aeromagnetic survey was flown with a
flight line separation of 2km or closer, and with tie lines spaced
at about 10km. The mean terrain clearance was approximately 150m.
In the reduction of data , diurnal corrections were applied
for the data and then the backgrounds based on the IGRF (International
Geomagnetic Reference Field) was removed.
33
The contoured maps, at a scale of 1/25,000, or 1/50,000
where 1/25,000 scale of maps were not available, were obtained from
the IGS and digitized manually at every 300 meters grid intervals.
Some differences were observed between the two scales of maps, but
these were not significant and showed some consistencies, so that
reasonable adjustments could be made manually.
3.3.2 Digitization of Landsat MSS Data
The MSS in Landsat 1 and 2 is a four band scanner with six sensors
in each band. It operates by receiving the Earth's reflected solar
radiation in the spectral region from 0.5 to l.lym. The spectral
range of each band for the MSS used in the Landsat 1 and 2 are
Band 4 0.5 - 0.6)Jm
Band 5 0.6 - 0.7lJm
Band 6 0.7 - 0.8]Jm
Band 7 0.8 - 1.1pm
At the nominal orbital altitude and attitude, the MSS scans cross-
track swaths of 185km in width, simultaneously imaging six scan lines
across in each of the four spectral bands. The scan lines which are
projected on ground are approximately 185km long and in a west-
to-east direction. The arrangement of scanning is shown in
Fig. 3.3.
34
orrica
Fig. 3.3: MSS Scanning Arrangements (After European Space Agency Earthnet programme 1979).
The data obtained for this study was already radiometrically and
geometrically corrected and converted to a usable binary form on
Computer Compatible Tape (CCT), with the radiance values ranging
from 0 to 255 in all four bands. Each pixel of the data represents , r. ..m _,»m . . , approximately 57 x 79 in its area! coverage.
Further details on the Landsat MSS data acquisition procedure
are described in European Space Agency Earthnet programme (1979),
Siege I et al. (1980) and others.
The CCT tape for the south-west England produced at FUSINO
receiving station in Italy was purchased from the Remote Sensing Unit
(RSU) of the Space Department at the Royal Aircraft Establishment,
Farnborough. The CCT tape data was imaged by Landsat 1 on 24,4,1975.
35
The track number and frame number are 220 and 25 respectively, and
the centre location of latitude and longitude is (50.15N, 5.10W) in
decimal degrees of latitude and longitude.
The data over the Bodmin Moor area was obtained as follows. . lines pixels The data obtained was averaged for a point value by 8 x 10
which were then recorded on the DISC with X(E-W) and Y(N-S) coordinates
converted into the British National Grid Unit with respect to a
provisional origin. In order to project the Landsat data on the User's
grid", the coordinates were rescaled using a few reference points
on the map and readjusted against the gridding program mentioned
previously (Section 3.2.1) in order to obtain coincident data
sets on the Bodmin Moor area. Full detailed accounts are given in
Appendix B.
3.3.3 Digitization of Geochemical Data
The original geochemical data were obtained from the Wolfson
Geochemical Atlas, which was initially compiled by the Applied Geo-
chemistry Research Group at Imperial College for regional geochemical
analysis of stream sediment samples through Englahd and Wales. This
was to delineate potentially mineralized districts and also to provide
fundamental geochemical information related to the regional geology
(Webb, et al. 1978).
Digitization of the geochemical data was done by using the
gridding program to interpolate the data onto the User's grid
coordinates. Eight elements, Arsenic (As), Copper (Cu), Gallium (Ga),
" User's grid system on this thesis was devised by the author in order to standardise data from various sources for the purposes of this study. The origin is at (9000N, 20,000E) BNG.
36
Lithium (Li), Nickel (Ni), Lead (Pb), Tin (Sn) and Zinc (Zn), have
been chosen for the analysis since these elements are closely related
to local mineralization and geology (Howarth: Hale, personal
communication). The density of measured data over the area is about
40 samples per 100 sq.km.
3.4 Noise Evaluation
3.4.1 Estimates of Noise Contributions
The power density function yields information on the amplitude
of the dominant harmonics of which the data is composed (Agarwal, 1968).
This enables one to analyse noise levels in the data by using the
power spectrum of the data.
A study of power spectra was carried out for the raw and
filtered data sets using a two-dimensional Fast Fourier Transform
(FFT) program. For easy comparisons, the two-dimensional power spectra
were represented as one-dimensional spectra by averaging radially.
Fig. 3.4(a) to 3.4(n) show one-dimensional representation of power
spectra of the raw and filtered data. The extrapolation as mentioned
in 3.2.2(b) was applied prior to applying FFT for 'the analysis.
On the whole, the raw data show much higher power spectra
at shorter wavelengths than those of filtered ones. The raw gravity
data shows quite significant noise levels in Fig.3.4(a). This is
partly due to very near surface density variations, of little interest
to this study, and possibly occasional erorrs in the original
measurements or data reduction process. The gridding process may
also introduce some high frequency noise as mentioned before. The
contoured map of raw data in Fig. 3.2(a) clearly shows high frequency
effects. The energy spectrum of raw magnetic data in Fig. 3.4(b) also
represent those of raw data and dotted lines are of filtered data. (Nyquist frequencies indicated by t)
Fig. 3.4 continued
V.« i*. !• rfi 4. m •'.«« •'.«• i'm Ml Out OCT Cicus/tod Klin (1) Pb
(j) Li
i
(m) Sn
s
(k) Ni
39
shows some high energy levels at shorter wavelengths.
In the Landsat data the power spectra of raw data at shorter
wavelength are quite high compared to those of longer wavelengths,
which might depict the characteristic reflectivity of surface or
near surface features in the data.
The geochemical data have varying amounts of noise as
clearly shown in one-dimensional power spectra from Fig. 3.4(g) to
3.4(n).
Gilman et al. (1962) has noted in the study of power spectra
for time series of atmospheric and solar indices that the spectral
characteristics is subject to a general suppression of relative variance
at higher frequencies and consequent inflation at low frequencies, which
is called 'red noise'. Problems with discrete Fourier transforms
in estimating power spectra are also given by Cordell and Grauch (1982)
in their comparative study with integral Fourier transform. Thus, the
spectral representation may somehow be distorted. However, this
phenomenon is not considered to be serious in this study and has been
disregarded in the qualitative interpretation.
From the analysis of one-dimensional power spectra, it
can be concluded that the maximum powers are concentrated at longer
wavelengths, whereas at shorter wavelengths there are some spectral
highs which may be due to noise either of small scale geological
features of little interest to this study or digitization, etc.
3.4.2 Noise Filtering
Prior to the actual analysis of data, any high frequency
noise which may cause erroneous results should ideally be removed.
However, practically such a complete removal of unwanted noise is not
40
possible. Thus, some optimal procedures are applied to reduce the
noise, while preserving the signal.
Bee Bednar (1982) has noted in seismic data filtering that
weighting average is most effective when the additive noise and
signal component occupy different portion of the frequency spectrum.
Thus, for geophysical data, a 9 by 9 filter operator has been
designed using the principle of low-pass filtering as described in
Appendix C-5. This filter passes any frequencies lower than one and
a half of the Nyquist frequency. The filter operator so designed
is illustrated in Table 3.1.
TABLE 3.1: Noise filter coefficients (fourth quadrant)
0.05500 .04755 .02981 .01174 .00102 0.04755 .04109 .02568 .01006 .00087 0.02981 .02568 .01591 .00162 .00051 0.01174 .01006 .00612 .00227 .00018 0.00102 .00087 .00051 .00018 .00001
For Landsat MSS and geochemical data, a 3 by 3 box-car window
smoothing was applied to reduce the random noise in the data.
The calculated values might contain noise averaged over nine points,
but as assumed the error is random, the calculated errors tend to be
much smaller than the individual error. The effect of this procedure
through a frequency approach in a one-dimensional case is given by
Jenkins and Watts (1969, p.50).
The application of different noise filtering techniques is due
to their different physical properties, and perhaps due to the objective
of the study and the user's experience.
To analyse the noise levels of filtered data, a two-dimensional
power spectrum of data sets have been evaluated. The two-dimensional
power spectra of filtered data are shown in Fig. 3.5(a) to 3.5(n).
41
I
L
(a) Gravity
2
Mf QuCfcC 11$ .t ICIU'OOI" i 600" ' (b) Magnetic
Fig. 3.5 Two-dimensional power spectra of the filtered data in log scale (a) and (b) show gravity and magnetics, (c) to (f) are of Landsat MSS four bands, (g) to (m) are of eight geochemical elements of As, Cu, Ga, Li, Ni, Pb, Sn and Zn.
42 Fig. 3.5 continued
t n .
0 A
< ; J - <3 <
MfOtlCWCKS .IICKS/0«I« I»:i«»ail600ni (e) MSS Band 6
(g) As rxoucuciti .CTCifi/MT* mKivmitaoni
TO- : »
vjVi ^ • ^ u ° TO ii - o i
^ TO W
Er ' - TO
K-t:
• •taufxcifs .cicits/ooio i*it«von6ocmi
FXOuCkCltS .C»Cll5/0«1» imi«»«ll(00HI (f) MSS Band 7
MtOUtNCICI .CTCLCt/MT* INTIIVNll900A1 (h) Cu
'•lOutkCICS -CICKS/DO!" |NI(«vai I 600" I
(i) Ga (j) Li
Fig. 3.5 continued
IMMXIII ,CIClCt/0«t» |«t(a«*lllOWII (k) Ni
43
MfOuCXKI .C»CIIS/D««» l»»l«»KIMO"l
(1) Pb
(m) Sn (n) Zn
44
In this case extrapolation was not applied because it was unlikely
to significantly alter the general features of the spectrum. The
comparative noise levels between the raw and filtered data can be seen
more clearly in one-dimensional representation of two-dimensional
power spectra in Fig. 3.4(a) to 3.4(n).
All the two-dimensional power spectra show that the maximum
powers are concentrated at the centre and more or less circularly
symmetrically diminish to the high frequencies, except a few data
sets particularly including gravity and magnetics which have some
elongations to axial lines. Clearly noise levels have been
significantly reduced after the Nyquist frequency as indicated by
arrows in one-dimensional power spectra of filtered data.
Typical examples of filtered gravity and magnetic data are shown
in contoured maps of Fig. 4.1(a) and (b). Compared to the original
maps in Fig. 3.2(a) and (b), respectively, the filtered map eliminated
all high frequency noise and show smoother patterns than the original
maps (See Chapter 4).
3.5 Data Qualification
As mentioned in Section 3.1, two basic data qualifications
- stationarity and normality - have to be done for their validity in using
appropriate data processing, so that improvements for appropriate
statistical analysis can be achieved.
3.5.1 Stationarity
The stationarity of sampled random data depends upon the
physics of the phenomenon producing the data. Almost all physical
processes are more or less non-stationary. The degree of stationarity
45
may be a matter of gridding scale (Bath, 1974). In general we may
consider phenomenon with slow variation as stationary or quasi-
stationary, while it is customary to consider a rapidly varying
phenomenon such as seismic p-wave as transient and non-stationary.
There may be various evaluation schemes of stationarity
ranging from visual inspect of the data to detailed statistical tests
of appropriate data parameters.
Bendat et al (1971) has described an assumption of statinarity
can often be determined by a simple non-parametric test of sample
mean square values computed from the data. Mazzarella and Cesere (1980)
has written a program called MARKOV using Bendat's algorithm
for analysis of stationarity of a time series.
The mean square value MSQ is defined as,
MSQ = MEAN2 + DEVT2 (3.2)
where MEAN is the mean value and DEVT is the standard deviation
of the sample. Then the process of the test of stationarity of a random
sample in the program is summerized as follows:
1. Divide the sample in equal intervals (termed NA) of width (MS) where the data'in each interval may be considered independent.
2. Compute a mean square value (MSQ) for each interval and align these sample values in time sequence.
3. Test the sequence of mean square values for the presence of underlying trends or variations other than those due to expected sampling variation, using a nonparameteric test such as the RUNS test.
The Markov process was used by Palumbo and Mazzarella (1980) in a study of rainfall statistical prediction schemes of the atmosphere.
In this study, MARKOV has been modified to be used for
46
analysis of stationarity of the data line by line. Three different
kinds of tests have been conducted. The first is a test using
the filtered data itself and the second and third ones are tests
after regional trends have been removed. The two regional removal
techniques are 'Spencer's method of smoothing' and 'Double
Exponential filtering' (Davis, 1973, pp.226). The formulae used
are as follows:
1. Spencer's method of smoothing
? i = 350 ( 6 0 Yi + " ' W ^ i - l ' + 4 7 ( yi +2 + yi-2> +
33(yi+3+yi.3) • :8(yi+4yi_4) • fity^i-s' -
" 5 < yi+7 + yi-7 ) ~ 5 ( yi +8 + yi-8 ) " 3 ( yi + 9+ yi-9 ) " (yi+10+yi-10):>
(3.3)
2. Double exponential filtering
Y. = 0.3ly. + 0.16(y. +y. ,) + 0.08(y. +y. J + I J i l+l i-l i+2 i-2 0.04(y. +y. 0) + 0.02(y. ,+y. .) (3.4) J i+3 i-3 i+4 i-4
The result of the analysis is given in Table 3.2.
Tests for the original four Landsat MSS bands show that
about 30% of total lines are stationary at 0.05 per cent level,
but the rest of the data sets show none or very low rate of stationarity.
It is interesting to note that although the gravity and magnetic
data are usually considered to be at least quasi-stationary, none
of the lines satisfy the requirement of stationarity in the original
data. Geochemical data show also very low rate of stationarity in the
original data. These might be due to the regional trend of the data.
Thus, two kinds of regional removal have been applied further to
examine the stationarity; Spencer's method of smoothing and the
Double exponential filtering as mentioned before.
47
When Spencer's method of regional removal is applied, all
of the data sets generally increase the rate of stationarity with
maximum rate 75 per cent in the Landsat MSS band 6. In case of
the use of the Double exponential filter they show even higher
increase in their rates with maximum 100 per cent in the Landsat
MSS band 4. There are a few exceptions with geophysical data when
the rates are decreased compared to those of Spencer's method.
TABLE 3.2: Test result of stationarity. Row by row tests so that each variable consists of 64 test data profiles. Each variable was subjected to three different ways of testing: original, regional trend removed by Spencer's method and regional trend removed by the Double exponential filter (DEF). recorded numbers are those lines which satisfy the stationarity at 0.05 level of significance.
Regional trend Regional trend Variables original removed by removed by REMARKS
Spencer DEF
Gravity 0 36 16 number of Magnetic 0 12 10 degrees of MSS Band 4 18 25 64 freedom of MSS Band 5 23 27 61 the RUN test MSS Band 6 15 45 60 for these MSS Band 7 16 43 59 data = 15 As 7 12 30 Cu 0 16 26 Fe 6 29 50 Ga 3 33 45 Li 2 29 42 Ni 1 1 30 45 Pb 3 6 20 Sn 5 1 1 17 Zn 3 6 35
48
This may indicate that an appropriate regional removal should
be applied prior to exercising the test of stationarity. Howarth
has noted that it is possible to analyse the randomness of the data
by statistical tests of the trend surfaces. This account will be
described in Chapter 5.
3.5.3 Normality
The parametric statistical analysis assumes that the
distribution of samples is normal. Yet the results of an investigation
will be subject to considerable uncertainty if the samples are
drawn from a population whose characteristics are incompletely
understood (Howarth et al. 1979). Thus test of normality is essential
for appropriate statistical analyses.
Perhaps the most obvious way to test stationary random
data for normality is to measure the probability density function of
the data and compare it to the theoretical normal distribution. If
the sampling distribution of the probability density estimate is
known, various statistical tests for normality can be performed.
However, a knowledge of the sampling distribution of probability
density measurements requires frequency information for the data
which may be difficult to obtain in practical cases. In this case,
a nonparametric test is desirable.
One of the most convenient nonparametric tests for normality
is the chi-square goodness-of-fit test. Details of parametric
statistical tests of normality can be found in numerous statistical
references, for example, Davis (1973), Bendat et al. (1971) etc.
and various nonparametric tests are described by Siegel (1956) and
Bendat et al. (op. cit).
In this study, visual inspection of histogram of the
49
samples has been performed qualitatively to analyse the normality
of the data.
Histograms of various filtered data sets are shbwn in
Fig. 7.1 in Chapter 7. Histograms of four Landsat MSS data are
somewhat similar to patterns of normal distribution, except for long
tails toward low and high ends of data. The gravity data show the
typical pattern of bimodal distribution and the magnetic and all of
the geochemical data are positively skewed to some extent. As
expected, not only the geophysical and geochemical data but also the
four Landsat MSS data have been rejected at 5 percent of
significant level with nonparametric chi-square test of the normality.
Thus it is necessary to transform the data to near normal
distribution for approriate data analyses as otherwise the
analysed results will suffer from loss of reliability. The
transformation techniques will be described in Chapter 7. The
histograms of frequency distribution of filtered data are shown in
Fig. 7.1 in Section 7.2.
3.6 Discussions
Data preparation including digitization and noise filtering
may be one of the most difficult tasks in the data processing,
particularly when several different types of data are being
considered. Various different filtering techniques have to be
applied partly due to the different physical properties of the data
and depending on purpose of the study. This is because the noise
contents in the data are usually unknown except in a very few cases
where the input signal is correctly known. The concept of noise
is also partly subjective to the study involved depending on the type
50
of study.
In this study, the raw gravity and magnetic data were subject
to filtering with a specially designed filter operator in order to
reduce the noise, and all the other data were subject to box-car
smoothing filtering using a 3 by 3 window function.
Comparative plots of one-dimensional power spectra of the
raw and filtered data show significant reduction of energy at
shorter wavelengths, and thus random noise in the data has been
removed largely by filtering.
To increase the reliability of the result, it is essential
to perform certain operations in order to qualify the data for
appropriate data processing. These include tests of stationarity
of data for correct procedure of analysis and further normality
of data for an appropriate statistical analysis.
The filtered Landsat MSS show generally some degree of
stationarity, but the filtered geophysical and geochemical data show
none or a very low degree of stationarity. However, after removing
the regional trends by Spencer's method of smoothing or double exponential
filtering, the degrees of stationarity increase rapidly. Thus in
general it can be concluded that all the data are'at least quasi-station-
ary and also often stationary.
Except for Landsat MSS data, all the other data show significant
skewness, so that all the data including Landsat data were subject to
transformation of data to at least near-normality using the power
transform method for all the data except for gravity where an
arcsine transform method was applied.
51
CHAPTER FOUR
FEATURE EXTRACTION
4.1 Introduction
Feature extraction is the process of summarizing all of the
information and reducing it to a smaller, more manageable set, thus
eliminating as much of the redundant information as possible, while
retaining as much of the essential information as is required. There
are various techniques depending on the nature and properties of the
data set and the aim of study.
In this chapter, various feature extraction techniques have
been applied to sets of data individually including potential field
data, four Landsat MSS data, and eight geochemical elements of As,
Cu, Ga, Li, Ni, Pb, Sn, and Zn.
In geophysical data processing for gravity and magnetics, a
number of filtering techniques including derivatives, continuations,
high- and low-pass filtering, reduction to the pole, and pseudogravity
methods have been applied to extract any regional and local features
of geological interest.
Landsat MSS data were subject to image enhancement by contrast
stretching using the Gaussian probability density function and further
ratioing techniques in order to enhance features of the images.
For geochemical data, level slicing in concentration levels
(i.e. the slicing levels are determined by an exponential function; see
Section 4.4 for more details) was applied to examine the general
distribution patterns of each element. Probability analysis was also
made to determine threshold values to the geochemical halcfs for Cu, Pb,
Zn, and Sn.
Performing analysis of individual data sets is particularly important
52
prior to multivariate data analysis because firstly it extracts
certain types of anomalies and secondly properties of individual
data sets can be understood, so that the analysis and interpretation
of the multivariate data may become much easier.
4.2 Geophysical Data Processing
4.2.1 Filter Operators for Analysis of Potential Field Data
One of the most important problems in the interpretation of
potential field is the separation of the data giving information
characterizing different geological structures. This process is
called 'filtering', the procedure of which is to separate anomalies
of particular wavelength from the original data, so that the anomaly
features can be accentuated.
Various mathods were utilized to perform this separation
either in space or in the frequency domain, for example, vertical
derivatives, upward and downward continutations, lowpass and highpass
filtering, reduction to the pole, etc.
Analysis in the frequency domain involves the Fourier transform
of the data and removes those elements in the frequency domain corres-
ponding to unwanted effects, and applies an inverse Fourier transform
to obtain the resulting map in the space domain.
Alternatively a fully equivalent operation of convolution
filtering can be used to achieve the same objective without implementing
the Fourier analysis. The digital convolution of the filter with the
map is as follows
53
a-l b-l Z(x,y) = Z Z X(x - x,y - A) F(x,A) (4.1)
x=0 y=0
where X(x,y) is the input data, F(x,X) the lagged filter and Z(x,y)
is the filtered output. a and b are the filter windows in x and y
directions, respectively, and the filter windows are in most cases
equal.
Since the Fast Fourier Transform (FFT) was introduced (for
example, Cooley and Tukey 1965, Robinson 1967 etc), the frequency
analysis was preferred to the spatial analysis due to its computational
efficiency with readily available transfer functions. However, it has
some difficulty in computation of large data sets due to its requirement
of the double memory space. Further difficulty arises from well-known
'Gibb's phenomenon' which tends to introduce severe oscillations which
are particualrly high at edges of data as mentioned in Section 3.2.
Those problems in the frequency analysis can be avoided by
convolution in the space domain. However, convolution has two dis-
advantages; firstly much computational time is required and secondly
there occurs a loss of information at the margin of the data set when
convolving. Since most of the filter operators have the circularly
symmetrical property (i.e. in a discrete square grid the cirularly
symmetry is equivalent to 8-folded symmetry) except in a few cases
such as the reduction to the pole and pseudogravity, the computational
time with convolution can be considerably reduced by folding about the
orthogonal axes, so that for a moderate size of filter operator the
computation may be much less than that of frequency analysis. Comparative
numbers of multiplications involved between ordinary convolution, 8-
folded convolution, double Fourier transform (DFT), and FFT are given
in Table 4.1.
54
Table 4.1: Comparison of the number of multiplications required between ordinary convolution, 8-folded convolution, discrete Fourier Transform and Fast Fourier Transform.
filter size
Ordinary Convolution
8-folded Convolution
DFT FFT Remarks
5 x 5
7 x 7
9 x 9
11 x 11
13 x 13
15 x 15
17 x 17
21 x 21
25 x n'
49 x N
81 x N
121 x N'
169 x N'
225 x N'
289 x N'
441 x N'
6 x N'
10 x N'
15 x N
21 x N'
28 x N'
36 x N'
45 x N'
55 x N
.. Data 4N log_N 2 size N x N
2
In case of 64 x 64 data set, FFT requires at least 49 N
multiplications (this number includes multiplications with transfer
function and the inverse FFT) which is even greater than 8-folded
convolution with 17 x 17 filter size.
The other disadvantage in convolution, strip-off effect at edges,
can be solved if an appropriate extrapolation technique such as the
exponential extrapolation as mentioned in Section 3.2 is applied.
Therefore, in many cases such as vertical derivatives, continua-
tions, pass filters etc., spatial analysis is preferable to the
frequency analysis computationally, and in its usefulness of the processed
results if the filter operators are properly designed.
As noted by Mufti (1972% it is possible to design relatively
small operators which will give results practically equivalent to those
from much larger operators but with a greatly reduced number of numerical
operation. Thus, analysis in the space domain by convolution can be
55
used as it is computationally efficient and effective.
4.2.2 Review of Filter Operators
(a) Derivatives
The second vertical derivative filtering is often used to
enhance the character of maps from the potential field surveys. The
method has the property of amplifying small-scale fluctuations relative
to broad-scale anomalies, so that the resultant map may indicate
certain geological features of small scale not easily identifiable
on the original map. The theoretical second derivative is unsuitable
for use with practical data because of noise in the data. The presence
of noise can lead to divergence of the expressions for differentiation
of the map. This fact has led to the awareness that useful expressions
are with the theoretical equations with an appropriate weighting function.
Various different operators have been proposed by many authors for
computation of second vertical derivatives: The early works include
those by Peter (1949), Henderson and Zietz (1949), Elkins (1951),
Rosenbach (1953) and Nettleton (1954) etc. and later by Henderson (1960)
Paul (1961), Agarwal and Lai (1971), Agarwal and Lai (1972) etc.
Recently Gupta et al (1982) has successfully applied so called optimum
second vertical derivatives designed by analysing the power spectrum
of the gravity data in geological mapping and mineral exploration .
Frequency analysis of the filter operators (Me sko 1965, 1966, Danes and
Oncley 1962, Darby and Davies 1967, Zurflueh 1967, Fuller 1967,
Kanasewich and Agarwal 1970, Agarwal and Lai 1971 etc.) provides an
insight to understand the nature and purpose of differentiation when
applying these operators upon gridded data.
56
It may be clear from these analyses that there is no uniquely
preferable operator in practice. The choice of operators depends on the
depth or size of the structure of interst, and the probable random
noise present in the data. Agarwal and Lai (1971) have shown that
although the filter operator by Rosenbach (1953) is much nearer to the
theoretical response than that of Elkins (1951), it is not to be preferred
to the latter in some practical applications where the data contain high
frequency noise. This is because the Rosenbach filter operator tends
to emphasize the noise or smaller features of no interest and thus lead
to difficulties in the interpretation.
Agarwal and Lai (1972) suggested empirically that the maximum
frequency content of the anomaly is inversely proportional to the depth
of the structure. Mathematically, they represented this by expression:-
Maximum frequency * Depth = Constant
It is therefore obvious that in cases where we are interested in deeper
structures a lowpass filter such as Elkins (1951), or Agarwal and Lai
(1972) where a = 0.23 (see appendix C.8), would yield good results,
whereas such filters may not have much application for shallow structures.
(b) Upward and Downward Continuations
The process of potential field continuation above or below the
original datum can be used to map subsurface geological structures by
interpreting the continued field. <
Upward continuation is used in potential field data interpretation
as a filter to remove the effect of shallow sources relative to those
of basement (steenland and Brod 1960, Nettleton and Cannon 1962,
Kanasewich and Agarwal 1970).
57
Downward continuation can be used to sharpen the anomalous
sources and to separate sources with overlapping effects. It has much
the same general effect in localizing anomaly boundaries and trends
as a second derivatives or residual calculations (Nettleton, 1976).
Upward continuation is a straight forward operation, since the
surfaces are in field-free space. In downward continutation it is
more unstable because of the inherent uncertainty in the location and
size of the structures represented by the field at the datum plane
because we must assume that there are no sources between the levels
over which the continuation is being made.
Roy (1966, 1967) and Rudman, et al (1971) have examined using
model studies the reliability of the downward continuation to determine
the maximum possible depth and to outline the variation in the shape of
homogeneous sources at various depths. If continuation is carried out
to depths greater than the source, the continued field will begin to
oscillate: this oscillation could be a criterion of depth of the source.
The half maximum value at the oscillation level may be used as an
outline of the top of the source.
Peters (1949) and Henderson and Zietz (1949a) have designed
sets of coefficients for computation of the continued field. Henderson
(1960) has reviewed the effectiveness of various coefficient sets,
including comparison of actually measured and continued fields.
Fundamentals of continuation fields in the frequency domain
have been described by Henderson (1966), Nagi (1967) and Roy (1967) etc.
The computation of continued fields in the frequency domain can be
achieved by multiplying the Fourier transformed data with the transfer t 2 x TO z (u + u ) , , . . . , function, e , and applying an appropriate weight Wt in the
case of downward continuation. If Z is positive, downward continuation
58
is computed, while if Z is negative, upward continuation is computed.
In the above formula u and u are angular frequencies in x and y
directions, respectively. In case of upward continuation no weigth
is applied since the continued field is much smoother than the
original field (Fuller, 1967).
(c) Lowpass and Highpass Filtering
Two-dimensional wavelength filtering is a means of separating
anomalies of different wavelength from each other, so that regional
trends, or other local features that are not easily detectable on the
original map can be emphasized. Separation by filtering is based on
the assumption that the near surface, presumably higher wave number
components of the field, can be isolated from the lower wavenumber
trends caused by deeper sources.
The general theory for the design of two-dimensional filters
to be applied to regular grid data has been given by Darby and Davies
(1967), Fuller (1967) and Zurflueh (1967).
The basic procedure applied in these papers is to specify in
digital form the desired wavenumber response of th'e filter (lowpass,
highpass or bandpass). Filter coefficients to be applied in the space
domain are determined by taking the inverse Fourier transform of the
wavenumber response.
Baranov (1975) has applied the function sin (ttx/Q)/ttx in order
to design the lowpass filter operator in the space domain. Since the
convolution of any function F(x) with the above function has the effect
of replacing the function F(x) with an arbitrary spectrum by a function
f(x) whose spectrum does not exceed tt/Q, various lowpass filter operators
can be designed by varying Q, so that desired lowpass filtering maps
are obtained for interpretation.
59
Alternatively, one can also carry out the filtering operation
in the frequency domain and then obtain the inverse Fourier transform
of the resultant spectrum.
Polynomial surface fittings are also applied to problems which
can be analysed with two-dimensional wavelength filters. However,
there are several advantages in using wavenumber filters as pointed out
by Zurflueh (1967). First pseudo-anomalies are a severe problem with
polynomial surfaces, especially if the data have a large dynamic range.
Filters are more reliable in this respect. Secondly the goodness of
fit of the polynomial varies in different parts of a map and the fit
becomes unreliable at the edges. Wavelength filters yield uniform
results over the whole extent of a map. Thirdly, filter programs can
process larger amounts of data in one computer operation than is possible
with polynomial programs.
(d) Reduction to the Pole
Reduction to the pole method is a mapping procedure helpful
in the interpretation of magnetic anomalies. Its purpose is to convert
data which have been recorded in the inclined earth's magnetic field
to what the data would have looked like if the magnetic field had been
vertical for the causative body, thereby reducing the distortion in the
pattern of magnetic anomalies resulting from the dependence on the angle
of magnetic inclination. Thus the map of data reduced to the pole may
be interpreted more easily (Baranov and Nandy 1964, Bhattacharrya 1965,
Kanasewich and Agarwal 1970, Nettleton 1976).
Some examples of filter oporators for computation of the reduced
to the pole are given by Baranov and Nandy (1964) and Baranov (1975).
Ervin (1976) has developed an algorithm for the reduction to the
pole using a Fast Fourier Series by considering the inducing magnetic
60
field only. Bhattacharrya (1965) has described the filtering procedure
in the frequency domain, taking into account both the induced and
remanent components of magnetization. Kanasewich and Agarwal (1970)
has modified Bhattacharrya's method in order to take advantage of FFT.
Spector (1975) has also shown the transfer function of the reduction to
thepolewhich can be implemented with FFT.
Obviously, in computational effort, analysis of reduction to the
pole in the frequency domain may be preferred. However, as mentioned in
Section 3.2, it is preferable to analyse in the space domain in order to
avoid difficulties in the frequency analysis with requirement of large
computer memory space and also spurious oscillations in the processed
map by FFT.
(e) Pseudogravity
Baranov (1957) first introduced the pseudogravity formula using
thepoisson's relation of practical application to gravity and magnetic
data interpretation.
The pseudogravity is a process to convert the magnetic field
into a gravity field. The transform involved is the elimination of the
distortion due to the oblique angularity of the normal magnetic field,
so that the resulting anomalies will be located on the vertical line
above the disturbing magnetized bodies and do not depend on the inclination
of the magnetic field, nor on the direction of the magnetization. Thus,
the interpretation and all the subsequent computations become very simple.
However, as was noted by Affleck (Baranov, (1957)), the pseudo-
gravity anomalies represent the effects of the magnetic rocks only, while
the observed gravity anomalies represent the combined effects of all
rocks, magnetic and non-magnetic. Thus Aina (1979) has pointed out in
61
his review of the method that the full capabilities of Poisson's
equations in combined gravity and magnetic data interpretation have not
been fully realized, and he has also suggested that further development
should be made for special cases of situations whereby anomalies are
caused by arbitrary distribution of magnetization and density in irregular
shaped bodies.
The application of this method was made for the Fourier transform
approach by Kanasewich and Argarwal (1970), for the Matrix method by
Bott and Ingles (1972), and for the Hilbert transform method by Shuey
(1972).
Spector (1975) has also given the transfer function of the pseudo-
gravity which can be implemented into FFT in the frequency domain. So
far in the literature no description has been given of attempts at
interpretation of potential field data by analysis with filter operators
in the space domain on a regular grid.
4.2.3 Theoretical Background of Filter Operators
Baranov (1975) has described details of potential fields and
derived some filter operators such as vertical derivatives, continuations,
reduction to the pole and so on.
The theoretical background of potential fields is briefly
described and further developments to pseudogravity and first horizontal
derivatives for the magnetic field are described.
The Laplace equation holds for potential fields,
V2u = 0 (4.2)
It can be solved in the cartesian coordinates as,
62
, v -iax-iBy-yz u(x,y,z) = e 3 (4.3)
2 2 2 where y = a + 3 (4.3.1)
A more generalized solution of Eq (4.2) is given by Baranov
by multiplying by a constant as
u(x,y,z) = 4/
u(0)B) e-iax~^y-yz d a d e (4.4)
- 7 T
where u(a,3) is the Fourier transform of the function
u(x,y,0) = u(x,y), that is
u(a,3) = ^ ^ ( x . y ) dxdy (4.5)
Filter operators such as continuations, reduction to the pole, first
vertical derivative, lowpass filter and first horizontal derivative
for potential fields have been computed for this study using Baranov's
method derived from Eq(4.4). Pseudogravity and first horizontal deri-
vative operator for magnetic data have also been designed in this study
from the derived equations. The second vertical derivative operator
is readily available from many publications as detailed in Section
4.2.2a. In this study the second vertical derivative operator has been
computed by using Agarwal's method (1972).
For all filter operators so designed, Hann's shortening
operator (Blackman and Tukey, 1958) used by Fuller (1967) were applied
for the edge correction in order to prevent a sharp cut-off at the
edges by truncation of the operators, and finally a weighted normalization
designed by the author has been applied (i.e. the sum of filter
63
operators is equal to 1 except those for derivative operators in which
case it is equal to zero).
The formula for shortening operator is
S(k,n) =
+ cos 7r(k2 + nV/(x 2 + y V for | k| £ x
|n| < y
0 for |k| > x
|n| > y (4.6)
The newly developed normalization procedure is
W(k,n) = W(k,n,d) + |w(k,n,d)| (4.7)
S S where AA = 1 - £. E, W(k,n,d) in cases of continuations, pass filters, k=I n=l reduction to the pole and pseudogravity operators and AA = -
W(k,n,d) in cases of derivative operators , AB = I I |w(k ,n,d)| and k— i. n— i
S the filter size.
A comparative study of the shortening operators by Agarwal and
Singh (1977) has shown that Martin's shortening operator is best for
normalizing sets of filter coefficients. In many cases, however, the
shortening operator combining Hann's shortening and weighted
normalization seems to be much preferred to Marten's method of
shortening in the author's experience. This is because the shortening
operator designed by the author can achieve both sharpening at the
edges and normalization by weighting rather than the normalization
only produced by Martin's method which often encounters problems with
sudden edge cutoffs.
Detailed calculations of filter operators including the
64
filter coefficients used in the study are given in Appendix C and
the programs for computing filter operators are listed in Microfiche 2
in rearfolder.
4.2.2 The Description of GEOPAK Program
Spatial filtering is an areal analysis technique that can be
applied to any form of contourable geological information. The process
involves digitizing the input data, designing a filter and convolving
the filter with the data.
GEOPAK has been written initially by ROBINSON (1971) in order
to utilize spatial filtering techniques. This is an integrated
package of FORTRAN IV subroutines whose main features are those of the
convolution and transformation of the convolved data into alphanumeric
character sets to print out the map data on the lineprint output.
The program has been modified in order to enhance the
efficiency and effectiveness of the computing by adapting several
significant subroutines. These include
1. 8-folded convolution
2. extrapolation of the data
3. gray-scale line-print output, etc.
(1) 8-folded convolution; The convolution of the data in the space
domain is the operational Fourier transform of multiplication. The
process thus requires the multiplication of the corresponding map data
with a filter and adding them all to produce a resultant data set
representing the value at the centre of the filter window and this
process goes through the map until an entire new map is produced.
Spatial filtering permits a large map to be handled compared to the
analysis in the frequency domain.
65
The convolution is thus the main routine of the program, which
largely dictates the computing time. The initial program computes the
convolution by multiplying every filter coefficient with the corresponding
map data and adding the all resultants together, so that the computing
is rather slow.
Since most of the filter operators have the circularly symmetrical
property (this is equivalent to 8-folded symmetry in the gridded system),
it is possible to reduce the number of multiplications significantly
by folding about symmetrical axes. The comparative numbers of
multiplication are shown in Table 4.1.
(2) Extrapolation; One of the most serious problems in spatial
filtering with convolution is the loss of information when convolving
the data with a filter operator. The strip-off of data by convolution
is as much as a half of the filter window around the border of the map.
To avoid this loss of information at the edges of the data, a reliable
extrapolation technique was designed using the exponential function
(see Section 3.2 and Appendix A).
(3) The gray-scale line-print output; This program subroutine
initially written by Blenkenship (Gonzalez and Wintz, 1977) has been
modified to print out the map by darkness depending on the amplitude
of the values. Several slicing techniques including equal-interval,
log-level, power level and absorption level slicings are employed in
the routine in order to be suitable for a variety of data sets which
show different patterns of frequency distribution. For example, geo-
physical data are commonly applied with equal-level slicing, while
geochemical data may be adequately treated with power-level or
absorption level slicing.
The flow chart of the GEOPAK program used and its descriptions
are illustrated in Appendix D and the program list is on Micro-iche 1
66
in rearfolder.
4.2.5 Qualitative Interpretation of Potential Field Data
(a) Description of the filtered gravity and magnetic maps
The filtered maps of gravity and magnetic data are shown in
Fig. 4.1 (a) and (b), respectively. In Fig. 4.1 (a), the main features
of the Bouguer anomaly map are that large negative anomalies occur in
the centre and southwest corner of the map which correspond to the Bodmin
Moor and a part of the St. Austell granites. Further a small patch of
negative anomaly appears in the centre of the eastern margin, the locality
of which corresponds to the Kithill granite. All these negative anomalies
are attributed to the low density of the granitic masses and they do
appear to be connected in the subsurface with the general WSW-ENE trend,
as noted by many authors,for example, Bott, et al 1958.
Apart from the granite bosses, the Bouguer gravity anomaly is
gradually increasing toward the sediments surrounding the granites, and
the southern flank of the anomaly over the Bodmin Moor granite usually
shows steeper gradients than the northern part.
Another interesting feature is the EW trending high anomaly in
the southern border of the map. Similar features also appear in the
northeast and northwest corner of the map. These features are difficult
to interpret uniquely from the gravity data alone. However, considering
the gravity data with the magnetic data may indicate that the northern
and southern gravity anomalies are due to different causes; in the
south the anomaly might be due to the limit of the granite batholith
and a deepening of the sedimentary basin but in the north it might be
due to uprising of the basement with only a local thickening of the
sediments as a contributing factor in the distortion of the contours
67
(a) Bouguer gravity anomaly (in gravity units)
O to ao Km • * * * * Scale
Fig. 4.1v Contour maps of the fittered potential fields of the Bodmin area with a cutoff wave length of 800 m.
68
in the northeast of the map.
Some irregularities in the contour map, particularly around
the Bodmin Moor granite may be due to differences in subsurface relief
and density of overburdens possibly partly controlled by fracture
patterns.
The magnetic map in Fig. 4.1 (b) shows strong positive and
negative anomalies trending eastwest or WNW-ESE in the northeast
quadrant of the map. The anomalies trending WNW-ESE correspond
generally to areas in which basic intrusive trending in this direction
are widely distributed. The strong negative anomaly north of the
main granite is coupled with a high positive anomaly to the north
of the negative althou^iit is not included in the map area.
Al-rawi (1980) has noted by bandpass filtering of the regional
data that the large positive anomaly can be attributed to deeply buried
basic rocks and he suggests that the direction of the remanence
vector in the basic rocks is southerly dipping. Thus, the negative
anomaly might be partly complementary to the positive anomaly.
Elsewhere, there appear no particular features but the general
tendency is the gradual increase from south to north with a largely
east-west trend. The magnetic map shows generally no relation to
the granite bosses, but may reflect largely the trend of crystalline
or metamorphosed basement under the main granite, which may be getting
deeper to the south.
Areas of some distortion and steep gradient in the northeast
may be related to some local magnetic features as well as some structural
features such as folds and faults etc.
(b) First-and second-derivative maps
The gravity and magnetic derivative maps including first
69
vertical, second vertical, and first horizontal derivatives have been
calculated with the appropriate filter operators designed (see Appendix
C),by convolving them with data from the Bodmin Moor area using the
GEOPAK program package as mentioned before.
Fig. 4.2 (a) to (c) show the first, second vertical derivative
and first horizontal derivative maps of the gravity fields, respectively.
The first and second derivative maps show broadly similar features;
small patches of positive anomalies are distributed around the sediments
surrounding the granites, and some highs also occur on the granite area.
However, the first horizontal derivative map in Fig. 4.2 (c) shows
almost the mirror image of the vertical derivative maps. The positive
anomalies in the first horizontal derivative map are generally around
the boundary of the granites where the steepest gradients occur, whereas
the positive anomalies in the first and second derivative maps appear
to be outside the granite outcrop. The reason may be that the first
horizontal derivatives pick up maximum gradients of any anomalies,
while vertical derivatives outline local anomalies or edges of anomalous
bodies. Particularly anomalous areas of vertical derivatives might
correspond to areas of inhomogeneities in the shallow subsurface which
might be related to local structures such as small granitic subsurface
intrusions. These kinds of subsurface intrusions could be related to
local mineralizations.
On the other hand, all the magnetic derivative maps from Fig.
4.3 (a) to (c) show that the general anomaly patterns are very similar
to each other, except in the magnitude of anomalies. The greater
amplitude in the second vertical derivative map (Fig. 4.4 (b)) may
indicate that the successive vertical derivatives accentuate the
relative effect of the shallower features such as the near surface
70
liOoooE, Rooo N) • —
M
(a) First vertical derivative map \o 2o Km
Scale lao&ooe, qoooN)
(b) Second vertical derivative map
Fig. A.2 Derivative maps of the gravity data. (a) First vertical derivative, (b) Second vertical derivative and (c) First horizontal derivative map.
71
continued from Fig. 4.2
0 § ip £.0 Km Scale
72
0 10 oo Km 1 I I I L
Fig. 4.3 Derivative maps of (a) First vertical derivative and (c)
the magnetic data. derivative, (b) Second vertical First horizontal derivative map.
73
continued from Fig. 4.3
0 io 20 Km \ i i • • (20ODOE, qoooN.) Scale
N 4
(c) First horizontal derivative map
74
basic intrusions the same as those in the gravity maps.
Except in the north-eastern part of the map, there appear no
significant features. The strong positive anomaly trends in a WNW-ESE
direction and there are associated negative anomalies to the south of the
positive ones. Other patches of anomalies trend approximately east-
west in the middle of the eastern margin of the map. All these anomalies
have been shifted very little from the original magnetic map.
Elsewhere, weak anomaly patches bearing an east-west trend
appear on either side of the main anomaly in the north-west and north-
east corner of the map.
All of these anomalies correspond to those areas where basic
rocks have been geologically mapped, so that these anomalies would
appear to be caused by the shallow basic rocks which have a high magnetic
susceptibility contrast with the surrounding rocks. In both gravity and
magnetic derivatives, numerous interruption zones can be found in the
maps. They are often found to be associated with fractures, faults,
and other geological lineaments.
One must be cautious in applying derivative filtering that
the resultant maps of the derivative filtering do not necessarily
contain high and low anomalies which have structural significance but
may be the result of algebraic properties, particularly as might be
produced in the gravity derivative maps.
(c) Upward and downward continuation maps
The gravity and magnetic data in the Bodmin Moor area were
continued upward to 1200 metres and downward to 600 metres being
multiples of grid interval (600 m) .
The upward continuation map of the gravity field in Fig. 4.4
(a) shows smoothed anomaly features associated with the granites. The
75
o 10 20 Km
. 4.4 Upward and downward continuation maps of the gravity data. (a) upward (h = 2), (b) downward (h = 1) continuation maps, h is in terms of grid spacing (600 m). Contour interval 25 gravity unit.
76
£o Km
Fig. 4.5 Upward arid downward continuation maps of the magnetic data. (a) upward (h = 2) and (b) downward (h = 1) continuation maps, contour interval 25 gammas.
77
(a) Gravity |0 20 Km
S c a l e (IOOOOE^OOON)
V / (b) Magnetics
Fig. 4.6 Upward continuation maps of the gravity and magnetic data (h = 3). (a) Gravity (b) Magnetic.
78
contribution of filtering in the upward continuations is not very \
significant partly because in the filtering process of the raw data
only the high frequency components have been significantly reduced.
On the other hand, the downward continuation map in Fig. 4.4
(b) shows that this operation has sharpened the local inhomogeneities.
In Fig. 4.5 (a) the upward continued magnetic map also shows smoothed
anomalies, attenuating largely small anomaly features trending WNW-
ESE and showing the broad E-W anomaly trend associated largely with
basement structures.
The strong negative anomaly north of the Bodmin Moor granite
appears to indicate a magnetic body extending to depth. The downward
continuation map in Fig. 4.5 (b) certainly shows a sharpening of small
anomaly features and localizes them compared to the filtered map in
Fig. 4.1 (b).
This may be comparable to filtering features of the derivative
maps which also have the same general effect in localizing individual
anomalies.
A further upward continued map to 1800 metres (h = 3) in Fig.
4.6 (a) and (b) shows smoother features than the map at 1200 metres
(h = 2). Particularly the magnetic anomaly features trending NW-SE in
the northeast margin of the granite have nearly completely disappeared
in the filtered map to 1800 metres (h = 3). These upward continued
maps clearly indicate the causative basic rocks extend to a limited
depth only.
(d) Lowpass and highpass filtered maps
The lowpass filtered map has been calculated from the filter
operator designed to pass frequencies lower than 0.3 cycles/600 m.
This is equivalent to a cutoff wavelength of approximately 2000 m. The
79
(a) Lowpass filtered map o 11> ao Km t 1— J 1 1
Scale
Fig. 4.7 Lowpass and highpass filtered maps of the gravity data. (a) lowpass filtered and (b) highpass filtered maps. Cutoff wavelength is 2 km.
80
(a) Lowpass filtered map 0 >o 20 Kffl 1 I ! ! l
s c a l e
Fig. 4.8 Lowpass and highpass filtered maps of the magnetic data (a) lowpass filtered and (b) highpass filtered maps. Cutoff wavelength is 2 km.
81
highpass filtered map is conveniently called that because it is the
residuals of the lowpass filtering, and it must represent effectively
a high frequency range of the data as described in Section 4.2.2.
Fig. 4.7 (a) and (b) illustrates the lowpass and highpass
filtered maps of the gravity data, respectively. The lowpass filtered
map shows the regional trend, eliminating the small local features,
whereas the highpass filtered map delineates local anomalies associated
with shallow geological structures.
The highpass filtered map is much the same as those of vertical
derivatives (see Fig. 4.2 (a) and (b)) and its patterns are strikingly
inverse to those of the horizontal derivative map in Fig. 4.2 (c).
The lowpass filtered map of the magnetic data in Fig. 4.8 (a)
shows the broad features to be largely east-west trending, which
reflect the deep subsurface structural features possibly associated
with the American earth movement.
The highpass filtered map in Fig. 4.8 (a) shows much the same
patterns as the vertical derivative maps, emphasing and isolating
local anomalies more clearly. Henderson and Zietz (1949) and
Swartz (1954) have also described the similarity between residuals and
derivatives. The positive anomaly trending WNW-ESE appears to be due
to the effect associated with the shallow basic igneous rocks since
this anomaly is nearly attenuated in the lowpass filtered map, as
also described in the upward continuation map (see Fig. 4.6). However
the strong negative anomaly abruptly interrupted in the centre of the
northern margin of the map appears to be caused by a body extending
from near the surface to great depth as partly confirmed in the
upward continuation map.
(e) Reduction to the pole map
82
Fig. 4.9 shows the reduction to the pole map of the total
magnetic intensity field. This has been calculated by convolving
the magnetic data with the filter operator designed (see Appendix
C.2), by assuming that the magnetization vector of the total field
is the same as that of the inducing field (Inclination = 68°,
Declination = 10°W). Thus if the magnetic anomalies are mainly caused
by only induced magnetization the reduction to the pole map will produce
a symmetrical anomal located above the causative body.
The reduction to the pole map is very similar to the filtered
map (Fig. 4.1 (b)), but a careful examination shows a slight shift of
the anomaly patterns in the former map toward the north. This may
be expected if we consider the steep inclination and a shallow
declination applied in the design of the filter operator. According
to Cornwell (1967), mean values of 188° in declination, -13°in
inclination were calculated for the Exter lava which is of the
permian age. Although the ages may be different between the two lavas,
considering large difference from the present dipole field direction
and the high Q value (>1), the magnetization vector of the total field
for the areas of volcanic lavas might be much different from the current
inducing field vector. Al-rawi (1980) indicated this in his regional
study for the strong positive anomaly coupled with negative anomaly
in the north of the granite, so that those strong positive or negative
anomalies may not be correctly represented by the transformations.
(f) Pseudogravity map
Further transformation has been made to convert the magnetic
data into gravity. Fig. 4.10 illustrates the pseudogravity map
calculated from the 21 by 21 filter operator. The anomaly patterns
[•20ooo&,qooou)
o to 30 Km I 1 ! ! I
Scale
Pseudo-gravity map of the magnetic data
84
are much the same as the reduction to the pole map except the anomalies
are shifted very little toward the north. The smoothed feature of Fig.
4.10 may be due to the filter size (21 by 21), whose coefficients are
not alternating but this phenomenon does not arise in the case of
the reduction to the pole because they are of alternating signs.
4.3 Landsat MSS Data Processing
4.3.1 Introduction
Remote sensing plays an important role in geological exploration
because it provides a quick, economic, overview particularly in arid
and semi-arid areas where the terrain is inaccessible. Geological
applications of remote sensing are well described in numerous references,
for example, by Siegal et al (1980), Sabins (1978), Smith (1977), etc.
Though aerial photographs were recorded by Daguerre as early as
1839 (Reeves,et al,1975), the concept of the Earth Resources Technology
Satellite (ERTS; later Landsat) was developed in the late 1960's by
joint NASA/USGS studies, stimulated by the proven geological value of
orbital photographs from the Mercury and Gemini flights.
Landsat 1 was launched in 1972 and began returning imagery
every 18 days from both imaging systems: the Return Beam Vidicon (RBV)
television system and the Multi-Spectral Scanner (MSS). Land 2 and
3 (c) were launched in 1975 and in 1978, respectively. The images have
been used for a wide variety of applications. Landsat 4(d) is now
operational, after being successfully launched on schedule on July 16
1982.
In the beginning of the geological applications of Landsat images
85
it was a simple small-scale comparison with the small
geological map (Lowman, 1969). Structural sketch maps made from the
colour-composite have in many cases shown lineaments not included in
the geological map. The detection of many large unknown structures
gave indication of potential value of Landsat imagery for the study of
regional structures.
In this study, geological applications of remote sensing data
are two fold: regional lithologic mapping and mineral exploration.
Lithologic mapping from images can often be accomplished by
analysis and interpretation of the spectral and spatial information
within the images. The spectral and spatial information available
for lithologic identification is expressed by landform development,
drainage pattern and density, vegetation differences, and spectral
reflectivity, all integrated in the context of climatic effects (Abrams,
1980).
Contrast enhancement is one of the most widely used image
processing techniques for lithologic mapping. It is the process of
redistributing frequencies in an image to maximize the contrast.of
areas of interest. Details on various contrast st-retching methods
are described by Taranik (1978), Siegal and Gillespie (1980) and
others.
Drake (1975) and Houston et at (in Freden et al, 1974)
have shown that colour-composite mapping using bands 4, 5 and 7
made it possible to make many stratigraphic subdivisions.
Thillaigovindarajan et al (1979) have also noted that using the 2
I S Additive Viewer the false-colour composite was found to be the
best for interpretation of linear and circular features.
Rationing is another common processing technique for lithologic
mapping. This is a method of enhancing minor differences between
86
materials by defining the slope of the spectral curve between two
bands. Ratio images also tend to reduce the effects due to topography
and to emphasize the changes in brightness values in materials (Chavez,
1975).
However, at the same time, ratio images accentuate noise, making
interpretation more difficult. In addition, dissimilar materials
having similar spectral slopes but different albedos, which are easily
separable in standard images, may become inseparable in ratio images
(Siegal et al 1980). Ratio images have been used in many geological
investigations for lithologic mapping by Blodget et al (1975) and others.
Rowan et al (197 6) has noted that the R4/5, 5/6, and 6/7 composite
was found to provide the greatest amount of information for discriminating
rocks.
Newton (1974, 1981) has noted by comparative study of several
different types of aerial photography for geological interpretation
that colour and infrared photographs are only marginally better for
study of superficial deposits and even under relatively unfavourable
conditions, normal black-and-white photography can be of great value,
particularly in the early stages of mapping an unknown area.
However, in many cases, colour photographs provide greatly
increased information content over black-and-white images since the
human eye can discriminate many more shades of colour than it can
tones of gray.
Thorough study of visible and near-infrared spectra of minerals
and rocks are mentioned by Hunt et al (1971 a, b, c, d, e, 1972, 1973,
1974 (a) and (b), 1977) .
The application of Landsat imagery to exploration for minerals
and hydrocarbons is one of the most difficult to study because of the
proprietary nature of the information. Correlation of lineament
87
studies with field mapping, aeromagnetic and gravity surveys,
geochemical sampling and relations to known ore deposits have been
described by Raines (1976, 1977), Salas (1977), and Rowan et al
(1981). Economic applications have been summarized by Vincent
(1977) and many others.
For mineral exploration, there are three promising methods
of study: the regional study of linear features, analysis of areas
of alteration through the use of multispectral data, and the
application of geobotanical techniques.
The regional features in Landsat images may generally
indicate structural regions, representing the surface expression
of faults, fold, lithologic contacts, or other geological discontinuities.
Many authors (for example, Rowan et al 1975, Correa 1975, Baker 1977,
Carter and Rowan 1977, Mercanti 1977, and Rowan et al 1979) have noted
that the study of linear features has often led to the discovery of
ores. This may be because ore deposits are generally related to some
type of deformation of the lithosphere and u.ost theories of ore formation
and concentration comprise tectonic and deformational concepts (Siegal et al
1980). Fernandez, et al (1979) and others have noted that of the many Landsat
features the circular ones are most likely to be correlated with the
presence of mineral deposits.
Regarding the analysis of alteration zones, gossans may be
valuable indicators of mineral deposits that are concealed beneath
the weathered surface, although not all gossans are associated with
ore bodies. The colours of gossans contrast with those of adjacent
country rocks, but most gossans are relatively small so that they
are difficult to detect on Landsat images. Hydrothermal alteration
zones are more areally extensive, so that it is one of the most
useful indirect approaches to exploration for ore bodies that is
based on studies of gaugue minerals associated with ore deposits.
88
Rowan et al (1976, 1977) and Carter and Rowan (1979) have
noted that ratioing of MSS bands have proven to be the most
effective method to detect hydrothermal alteration.
Mineral concentrations at depth often accompanied by
secondary surface indicators, many of which (e.g. tonal anomalies
associated with geochemical or geobotanical stress-gossans or
alteration aureoles) can often be detected by computer-assisted
multispectral image analysis (Baker, 1977). Lyon (1975) has
also suggested that it may be possible to detect vegetation
anomalies associated with porphyry copper deposits by examining
spectral ratios on a pixel by pixed basis.
The principle applications of geobotanical techniques for
mineral exploration in remote sensing are largely based on three
factors;
1. observation of the distributions of indicator plants,
2. vegetation density changes, especially bare spots,
and 3. morphological changes in plants.
Detection by indicator plants with any remote sensing
technique may be difficult due to the small areal extent of most
patches of such species. The general environment of mineral
deposits are often unfavorable for vegetation growth so that grassy
clearings or stunted plant growth may result. Such clearings are
probably the most widely used geobotanical indicator in prospecting
(Lag and Bolvikan,1974). Some morpholgical and physiological
changes in plants, and toxicity symtoms due to concentration of
some elements may be detectable by remote sensing techniques due to
different reflectance caused by changing colours or changes in growth
89
form.
Gausman (1977) has described in some detail the properties
of the spectral reflectance of a leaf. However, the spectral
reflectance and emittance properties of a plant are much more
variable and more complex than that of a leaf. Colwell (1974)
has mentioned factors on the leaf reflectance properties to be
important when considering a vegetation canopy.
However, though geobotanical symtoms related to a mineral
deposit are in general obvious, there are subtle effects that can
only be seen on the Landsat image by the trained eye.
The unique advantages and characteristics of Landsat MSS
are its synoptic and repetitive character of space-acquired imagery.
The computer compatible digital format of the MSS images is further
advantageous compared to the photoimagery since the MSS images can
utilize its dynamic range of brightness values for further enhancement.
Ligget et al (1974) has shown that in reconnaissance
exploration for mineral, groundwater, and geothermal resources, and
in the study of geological hazards, Landsat MSS may provide cost
savings of about 10 to 1 compared with conventional methods.
Computer processing of Landsat MSS data for geologic applications
is well described by Gillespie (Siegal and Gillespie, 1980) and
Taranik (1978) etc.
4.3.2 Basic Principles in Landsat MSS Data Processing
Two aspects of the image processing were applied in this study.
Firstly, the regional study over the whole Cornwall area was conducted
for regional geological mapping such as lithologic mapping, detection
90
of linear features or any alteration zones which might be related
to any mineralization. This is based partly on the data produced 2 . . .
by using I S Additive Viewer set up at the Remote Sensing Unit m
Imperial College. Sampling over Cornwall was taken at every fourth
pixel in every third line since this is approximately a square
grid image and also fits within the limitation of the TV screen of
the system which holds a maximum of 512 x 512 array size at any one
time.
A false-colour composite of bands 4, 5 and 7 in blue, green
and red light, respectively, was made for the interpretation. Colour
-ratio composite techniques usually offer an efficient means for
combining stretched band ratio images for discrimination of rock types.
Discrimination is increased not only because information from several
ratio images is combined, but also the human eye is capable of
discriminating much better in colours than in shades of gray.
Since we can have 6 ratioed data from four spectral bands,
a total of 20 colour ratio combinations is possible if we are using
the colour additive viewer.
Rowan et al (1976) have noted that the most useful combination
for discriminating the main rock types and altered areas in exposed
regions, was the colour subtractive view of MSS R4/5, 5/6, and 6/7 in
cyan, yellow and magenta, respectively. However, facilities using
the diazo process to produce the colour subtractive view of composite
images were not available to the author. Therefore, an optimum
combination for geologic analysis of the study area was determined
using the colour additive viewer with the following ratio image
combinatiore: R5/4, R6/5 and R7/6 in blue, green and red, respectively.
All were subjected to histogram normalization.
To extract any linear features, individual black-and-white images
91
were produced on a pixel by pixel basis using a ULCC (University
of London Computer Centre) 64 level gray scale picture processing
package, PICPAC routine. In the gray level slicing process, a
contrast stretching technique was designed using a probability
density function of Gaussian distribution with a mean of zero and
a variance of 1.
The mathematical expression for the probability density
function of the Gaussian distribution with a mean of zero and a
variance of 1 is:
PDF = f(x) = - 00 < x < 00 (4.8) /2tt
Detailed procedures of the level slicing technique and program used
are given in Appendix D.
The second aspect is the processing of the digitized images
over the Bodmin Moor area to examine the regional spectral features
of the data. Again, in this study, a false-colour composite of MSS
bands 4, 5, and 7, and a colour composite of band ratios 5/4, 6/5 2
and 7/6 were enhanced by histogram normalization in the I S system
to produce colour additive pictures.
In order to form meaningful images for ratios between
Landsat bands, it is usually necessary to subtract a constant bias
value from each band (Chaven 1975 ) due
to direct scatter of sublight from the atmosphere into the sensor.
It depends mainly on the wavelength of the scattered light. Usually
11 for band 4, 5 for band 5 and 3 for band 6 are subtracted from
the recorded values. No atmospheric noise is considered for band 7.
However, in this study, this aspect was not considered.
Instead, the following procedures have been performed in the ratioing
92
since relative significance between two methods can be preserved
similarly.
To make the ratioing exact, the least square regression
line was calculated between two bands as follows:
y = a Q + a 1 x (4.9)
where y is the numerator and x the denominator in ratioing.a^ is the
intercept value at x = 0, and a^ the slope.
The calculated coefficients between two bands are given
in Table 4.2.
Table 4.2 Computed coefficients of linear trend between two variables
numerator denominator 0 b^(intercept values in x-axis)
Band 5
Band 6
Band 7
Band 6
Band 7
Band 7
Band 4
Band 4
Band 4
Band 5
Band 5
Band 6
-27.1280
5.1274
1.2360
41.4188
34.2470
-22.6880
1.7689
2.5926
2.4650
1.5808
1.5526
1.1648
-26.2012
-22.0578
In order to calculate the ratios, the intercept values (S q) were first
subtracted from the numerator (y) and then divided by the relative
denominator (x), that is
z i = " a o ) / x i ( 4 . 9 . 1 )
However, for ratio data of R6/5 and R7/5, the intercept values of
93
x-axis (b^) were subtracted from the denominators rather than subtracting
a^ from the numerators. In this way, the ratio can be performed in
the first quadrant with the least square linear regression line passing
through the origin, and the significance of the relative data are
preserved in this manner since the data are being standardized prior
to any multivariate data analysis.
4.3.3 Feature Extraction and Interpretation
Either black and white images of individual bands, colour-
composites of the original Cornwall region and digitized data over the
Bodmin Moor area, or their ratios, produced by photographic process
have been subjected to detailed interpretation. The photographic
processing of the images caused loss of some resolution.
Band 5 is by far the best for geological mapping and band
7 is optimum for recognizing contacts between land and water, vegetation
differences, and probably differences in soil moistures.
(a) Black and White MSS images of Cornwall
Gray scale images of the Cornwall area are enhanced by contrast
stretching with Gaussian distribution and are shown in Fig. 4.11 (a)
to (d). Frequency distributions of the original four bands are shown
in Fig. 4.12(a) and (d). From the histograms shown, spectral features
of Land in the higher range are well separated from those of water in
the lower range on bands 6 and 7; but the features is not so distinct
on bands 4 and 5. Particularly the spectral distribution pattern of
land in the near-infrared bands (bands 6 and 7) shows a near-normal
dis tribution.
Although distinctive geological features may not be seen
(a) Band 4
(b) Band 5
Fig.4.11 Contrast stretched black-and-white Band images over Cornwall. a) Band 4, b) Band 5, c) Band 6 , d) Band 7 and e) Lineaments stidied from Band 5.
Fig.4-11.continued.
Band 6
Band 7
96
Fig. 4.11. continued
(e) Lineaments studied from black-and-white MSS Band 5 images.
97
63 127 191 255
(a) Band 4
il G3 127 191 255
(b) Band 5
TOMiliTO G3 127
(c) Band 6 191 255
,l\llUuHHUl.... 63 127
(d) Band 7 191 2 5 5
Fig.4.12 Frequency distributions of black-and-white MSS Band images over Cornwall, a) Band 4,b) Band 5, c) Band 6 and d) Band 7.
98
in the images, they generally show some regional features which are
related to the regional geology.
Bands 4 and 5 in Fig. 4.11 (a) and (b), respectively, show
particularly high reflectance over the high Moor lands. Although
this feature does not appear in the Land's End and Carnmenellis granite
areas, careful investigation makes it possible to outline the granite
boundaries throughout the peninsula in all four bands.
Although it is not so clear, probably due to the photographic
process, dark colours in the coast areas and some river channels in
the northwest may be due to high spectral features of sanddunes. Some
areas along the southeast coast line are also dark, which is attributed
to high reflectance by cultural features and possibly sanddunes. As
noted by Davis, et al (1981), the high reflectance of sanddunes is due
to the lack of iron staining and. other dark minerals in the visible
bands. Although mineralized areas usually appear to be bright
(Rowan, et al 1976) it is not so apparent in the study area. This
may be due to the almost complete coverage of vegetation which may not
always show distinct reflectance properties from other features in the
black and white images. Also further detailed distinction between
different rock types is not possible because the vegetation may mask
subtle differences of the reflectances between the various rock types.
Numerous linear features are apparent through the area. Some
of them may be drainage patterns but many may represent geological
structures such as fractures, faults, or curved features representing
folds. Some of these lineaments correspond to some known major faults
and other could not be correlated with any of the geological features
in the existing geological map.
Between the images, band 5 is by far the best in the study of
99
lineaments and the next is band 7 in the study area. All interpreted
linear features are summarized in Fig. 4.11 (e).
The conspicuous northwest trending regional linear features
around the Bodmin Moor granite area marked well-known wrench-faults.
These lineaments are manifested in the space image by the tonal
differences in vegetation and alignment of subsequent stream segments
and topographic forms etc.
The contact line between Caledonian and Amorican zones is
also marked between Perranporth in the west and Pentowan in the east.
Lines of confrontation or oppositely facing folds may also be seen
in the images as in the north of Padstow.
Other features which are interesting to note are the parabolic
features cut by lineaments in areas between Bodmin Moor and Dartmoor
granites. These features appear to be in the south of the recumbent
fold belt on the structural geology map (see Fig. 2.2) but these may
be parts of many recumbent folds in the area.
The lineaments which cannot be identified with any of the
geologic features in the geologic map are possibly related to deeply
buried concealed structures.
In any case, such detailed remote sensing studies which also
involved correlation of geophysical data interpretation and also
checking these lineaments from the standpoint of their correspondence .
to concealed structures will be very important because the structural
control is very often the main factor in the location of mineral
deposits as described in Section 4.3.1.
Many lineaments in the area may be related to local mineral-
izations. Correlation with possible mineralization can also be
supported by available geochemical maps (Webb et al 1978) in the area.
100
However, the enhanced images do not provide any clear
indication of the presence of the mineral deposits. Thus, in all
cases in the study area, spectral tonal variations found to exist
in the Landsat imagery of heavily vegetated regions requires
detailed ground examination to determine whether they are the result
of geologic, simple botanical or other factors such as any cultural
features.
(b) False-colour composite of Cornwall
Fig. 4.13 is the false-colour composite of bands 4, 5 and 7
using blue, green and red filters, respectively. The map shows the
major physiographic and structural features actually visible on
it and a few well-known features such as the wrench faults around
the Bodmin Moor are also included. Many lineaments, expressing
faults or folds are obvious.
On the other hand, the Landsat images show albedo differences,
generally resulting from vegetation patterns, which may be of some
geologic value by directing attention to areas or features with
distinctive vegetation patterns.
In this colour composite, vegetation appears to be red
because of the high reflectivity of vigorous vegetation in MSS band 7
compared with MSS bands 4 and 5 (Rowan et al 1976).
In the study area, the colour composite may be most useful
for discrimination of vegetated areas. The darkest red, and therefore
the densest vegetation, occurs in areas surrounding the Bodmin Moor
granite. Some red tinges are apparent throughout the whole peninsula.
Particularly, Carnmenellis, Land's End and some part of Lizard complex
show uniform distribution of vegetation. Colour composites appear
to offer a little improvement over the stretched MSS images for
101
discriminating geological features. High moor lands of St. Austell,
Bodmin Moor and Dartmoor granites are white to greenish white in the
image, but this feature does not appear on the low lying Land's End
and Carnmenellis granite areas. Sanddunes appear to be white in the
NW and SE coasts and along some river channels. The greenish feature
in the middle of the Lizard complex may be due to sparse vegetation
or stunted growth in serpentine rocks. Cultural features in the municipal
areas are blue as in St. Austell and Plymouth because of the high
reflectivities of those features in the visible bands.
Although most of the metamorphic aureoles appear to be light
brown to reddish, the colour composite may not offer a reliable means
for detecting altered areas in the study area where the surface is
almost completely covered by cultivated vegetation. These do not show
distinct spectral features from others in the study area.
(c) Colour-ratio composite of Cornwall
The colour-ratio composite shown in Fig. 4.14 was analysed by
comparing with the regional geological map. The colour-ratio composite
image virtually offer little improvement over the false-colour composite
for discriminating any rock types.
Areas of high Moor granite and sanddune are blue, while the
cultural features appear to be dark blue. Another blue feature in the
Lizard complex corresponds largely to areas of granite and gneiss.
Vigorously vegetated areas appear to be bright orange in the ratio
composite. One interesting point to note is that surrounding the
Bodmin Moor and St. Austell granites and over Land's End and Carnmenellis
granites are marked dark orange distinctive from other vegetated areas
having bright orange colour. It is not so clear whether this feature
is directly related to reflective features from the altered areas or
102
Fig.4.13 False colour-composite of MSS Band 4,5 and 7 in blue,green and red,respectively.
Fig. 4. 14 Colour-composite of MSS Band ratios R5/4,R6/5, and R7/6 in blue,green and red,respectively.
103
not since the comparative colour density differences between altered
areas and vigorously vegetated areas are still too subtle to be clearly
differentiated. Areas of serpentine rocks are more apparent in the
ratio composite than in the false-colour composite.
Many linear features are apparent in both false-colour
composite and band ratio composite images. These kinds of composites
may be sometime advantageous in studying regional lineaments since they
remove small local variations. However, it will not be pursued further
here since it has been already described in the previous section.
(d) Description of surface maps for the Bodmin Moor area
Surface maps in the study area have been plotted for two
comparative bands in order to increase perspective views: bands 4 and
7.
In Fig. 4.15 (a) the high reflective feature of MSS band 4
is apparent over the high Moorland areas. Typical highs arise in many
parts of the St. Austell granite and a few in the Bodmin Moor granite.
These may be due to the high reflectivity of clay minerals in open-pit
mining areas in the visible spectral bands. However, the high
reflectance features do not appear in band 7 as shown in Fig. 4.15
(b). Instead, the reflectance features in these areas are lower
than the vegetated surroundings. The sudden drops in the NW, SE and
a little in the SW corner of the map reflect the low reflectivity of
the offshore water. Although details on every feature are not described
here, the general feature is a sort of reflection of morphological
features rather than any geological ones.
In general, on land areas in band 4, high reflective features
occur in high or dry areas while low ones occur in wet and thickly
vegetated areas.
104
I 1.641
(a) MSS BAND 4
( 1 . 64 1
Fig. 4.15 Surface maps of MSS bands 4 and 7 over Bodmin Moor area, Cornwal1. The geometrical coordinate (1,1) is oriented as the northwest corner.
105
I 9.411
I 34 . 4 11
MSS BAND 7 OVER THE BODMIN MOOR GRANITE
Fig. 4.16 Surface maps of MSS Bands 4 and 7 over Bodmin Moor granite. The geometrical coordinate (9.20) represents the northwest part of the map.
106
On the contrary, in band 7, thick and vigorous vegetation
shows highest reflective features while dry areas and water show low
reflective features.
To examine more detailed reflective features over the Bodmin
Moor granite area, the data was partitioned for the area as shown in
Fig. 4.16 (a) and (b).
The high reflectance in band 4 is evident, the peaks of which
largely correspond to high tors in the area and toward the boundary the
spectral feature is flattened. On the other hand, the reflectance
features in band 7 of Fig. 4.16 (b) are reversed to those in band 4.
The high reflective features in band 4 appear to be low in band 7, and
the flat feature in band 4 disappear in band 7 and it rather shows some
irregular highs. These may mean the spectral feature in band 7 may be
little affected by the topographic elevation. Instead, it may be
influenced by vegetation conditions and surface and near-surface
moisture contents, judging from the fact that the area is almost
covered by vegetation which varies according to elevation.
The high moor lands consist of dry and brown vegetation
overgrazed by cattle, while lower lands are mainly composed of vigorous
vegetation and/or high moisture content in the soils particularly in
lowland depressions.
(e) False-colour composite of Bodmin Moor area
Spatially averaged data in Fig. 4.17 (a) to (d) show regional
reflective features over the Bodmin Moor area. Bands 4 and 5 are closely
similar in spectral features as in Fig. 4.17 (a) and (b).
Regionally the highest reflectance features occur in the high
lands of Bodmin Moor and St. Austell granites. Additional highs occur
sporadically throughout the study area, which may correspond to areas
107
( a ) b a n d 4 ( b ) b a n d 5
->p Km sea l e
( c ) b a n d 7
y J
f P i t
H ( d ) c o m p o s i t e
F i g . 4 . 1 7 R e p r e s e n t a t i o n o f MSS B a n d s 4 , 5 a n d 7 b y n o r m a l s l i c i n g o v e r t h e B o d m i n M o o r a r e a a n d t h e i r f a l s e c o l o u r - c o m p o s i t e ( d ) ( a ) B a n d 4 , ( b ) B a n d 5 a n d ( c ) B a n d 7 .
*'Overlay' in the rear folder is approximately the regional geology of the Bodmin Moor area.
108
with less vegetated or dry surface. Sanddunes near the coast lines
or cultural features also appear to give high reflectance.
Low reflectance in the visible bands may be attributed to
spectral features of waterlogged or intensely vegetated lands.
On the contrary, band 7 generally reflects the opposite
features to either band 4 or 5, except perhaps areas with water and
wet land. The high reflective features of high moor lands show lower
reflectance than the surroundings in band 7. Areas of high reflectance
largely correspond to highly vegetated areas. Low reflectance
in band 7 is generally attributed to water, sparse or dry vegetation
and features of the larger towns, while high reflectance may be due
to vegetation, the density and vividness of which is proportional to
its magnitude of reflectance.
The colour composite in Fig. 4.17 (d) shows that high tors
and open-pit mining areas are white and high moor lands with dry vege-
tation, cultural or dry lower land areas appear to be light blue,
and areas of intense vegetation show reddish. Dark colour represents
areas of moisture.
Many of the altered areas in the aureoles' of the granites in
the south are greenish to orange, but much of the altered zones in the
north to northeast of the Bodmin Moor granite are reddish.
It is uncertain whether these features are due to spectral
features of altered areas or not in the study area because they may be
obscured by vegetation cover although vegetation in altered areas often
shows certain distinct spectral features which can be distinguished from
others.
(f) Colour-ratio composite of the Bodmin Moor area
The stretched colour-ratio composite of R5/4, R6/5 and R7/6
109
assigned by blue, green and red, respectively, are shown in Fig. 4.18
(a) to (d). Geologically, although almost no improvement is offered
by the ratio data compared to the false-colour composite of the original
baids some relative reflective features between bands can be seen in
the ratio composite.
In Fig. 4.17 (a), the high land of Bodmin Moor granite are
bright, but St. Austell granite and two spots in the Bodmin Moor granite:
one in the northwest and the other in the southern margin, appear to
be dark. This indicates that though dry vegetated lands have relatively
higher reflectance on band 5 than band 4, the reflectivity of the open-
pit mining regions is relatively higher in band 4 than band 5 spectral
regions.
In the lowlands of sediments dark colour may be due to
relatively low reflectance of band 5 due to strong chlorophyll absorption
at near 0.68 ym spectrum. The offshore area in the northwest corner
of the map is dark but the offshore area in the south appears to be
white. Though the reason is not clear, considering shallow and maybe
dry stream channels flow in the northwest while more active water
channels flow in the southern coast through populated areas, it might
be due to differences in reflective features of contaminated water by
suspended sediments in the south from un- or less-contaminated water
in the northwest, since there are strong absorptions in blue and green
bands from the suspended sediments in the water as noted by Specht et
al (1973). This feature does not appear in the MSS ratio R6/5 and
R7/6. Most of the drainage system on land areas may also appear to
be bright in the band ratio R5/4 probably due to suspended sediments
by strong current or contamination.
In Fig. 4.18 (b), the general features of R6/5 are that water
and granitic moor lands are dark while areas of intense vegetation
(c) R7/6 (d) Ratio composite
4 . 1 8 C o l o u r r e p r e s e n t a t i o n o f MSS B a n d r a t i o R 5 / 4 , R 6 / 5 a n d R 7 / 6 i n b l u e , g r e e n a n d r e d r e s p e c t i v e l y a n d t h e i r c o l o u r - c o m p o s i t e ( d ) . ( a ) R 5 / 4 , ( b ) R 6 / 5 , ( c ) R 7 / 6 .
0 Io jiQ KlT\ 1 1 1 1 i Sea I e
111
appear to be bright green, the degree of which may be proportional to
its brightness and density.
Band ratio R7/6 in Fig. 4.18(c) shows that townships and
high moorlands are dark while water and vegetated areas are bright.
Nowhere do the images show any direct relation of the
distinct leflective features of alteration zones or any mineralization
though the colour composite of the ratioed data shows some indication
of alteration around the metamorphic aureoles surrounding the granites.
The colour-ratio composite in Fig. 4.18 (d) shows a more
dynamic range of reflectance features. The high land of Bodmin Moor
granite and maybe dry killas are blue but the high moor land of open-
pit mining and township areas appear to be dark.
The offshore, water and densely vegetated areas are reddish
to brown. The greenish to yellowish colours may be due to different
density and brightness of vegetation, the order of which is from dense
and bright to less dense and less bright.
4.4 Geochemical Data Processing
Geochemical methods of prospecting have been proven in practice
with a number of mineral discoveries. Almost invariably, geochemical
methods have been used in conjunction with other geological or geophysical
exploration methods (Rose, et al, 1979).
The stream sediment sample values obtained from the study from
the Wolfson Atlas project (see Section 3.3.3) approximate to a composite
sample of those materials derived from the catchment area upstream
from the sample sites, so that in appropriate circumstances the patterns
of metal distribution in the rocks and/or soils may be reflected to a
degree in corresponding variations in the composition of the stream
1 12
sediment except in areas of contamination (Webb, et al.,1978).
In this section the geochemical data for the stream sediment
samples were analysed to investigate the regional distribution patterns
of each element in relation to geology, and possibly delineate those
areas with potential mineralization prior to the multivariate regional
geological mapping as a tool to assist detection of possible mineralized
areas.
Geochemical elements usually have bimodal or polymodal concentration
distributions. When potential mineralization is of concern,the sample
distribution may be treated simply as bimodal; background and anomaly.
However, the background and anomaly populations commonly overlap, so
that completely satisfactory discrimination of background and anomaly
samples is not possible.
There are various ways of separating anomalous samples from
background, depending on the amount of data, purpose of the study and
economic consequences of the selection, etc. (Rose,et aL, 1979).
Most cases are involved with threshold selection to define
anomalous samples from backgournd. Others may include the extraction
of a local anomaly by subtracting regional values 'from the sample
values or using a number of sophiscated multiple statistical treatments
such as discriminant analysis or factor analysis.
Webb et al (1978) and Mancey (1980) have noted that colour
subtractive views of percentile slicing of three geochemical elements
in cyan, yellow and magenta illustrate the regional geology most
effectively.
In this study, threshold methods (including representation by
the absorption-scale level slicing method to evaluate the general
distribution patterns of the eight elements (As, Cu, Ga, Li, Ni, Pb,
Sn, and Zn), and probability plots) have been applied further to evaluate
1 13
the data as a means of assessing the potential for mineralization.
4.4.1. Principles of Regional Geochemical Data Processing
(a) Analysis of the regional distribution patterns of geochemical
elements
To analyse the regional distribution patterns in stream
sediments of As, Cu, Ga, Li, Ni, Pb, Sn and Zn, the data were sliced
using absorption levels.
The absorption level slicing technique takes the form:
x = aeby (4.10)
where y is the slicing level value and x is the sample value, a is the
minimum of the logarithmic data values and b the difference between
the maximum and minimum logarithmic values divided by the number of
gray levels (GN).
In another expression Eq. (4.10) becomes
y = ( l n X ~ a) * GN (4.11) b'
where b = ^r, so that the gray level of each sample value is determined G N
directly from the above equation, or the gray level threshold values
for the data are determined first and the level values are applied to
the data for selected levels by iterative method.
This kind of slicing may be adequate for application to
the geochemical elements since the frequency distributions of most
geochemical elements are approximately log-normal. Thus, the distribution
pattern can be histogram-normalized which may give maximum discriminatory
114
power of the regional distribution.
(b) Detection of geochemical haloes using the probability plot
The probability plot has been effectively used as a means of
separating the anomalous samples from the background.
The method makes use of probability plot paper devised by
Hazen (1913). The cumulative percentage scale of this paper is
specially arranged, so that when the ordered concentration values
for any normally distributed population are plotted, the points all
fall close to a straight line. If a bimodal or polymodal distribution
are mixed with distributions which are themselves normally distributed ,
they will, when plotted, give a curve which is the resultant of two
or more straight lines. There is a point, or points, of inflexion
on this sigmoidal curve where the direction of the curve changes.
These points of inflexion indicate that there are two or more populations
involved.
The plotting procedures are as follows:
When the number of samples is small, the data are arranged in
ascending order of magnitude, each individual sample is plotted in
such a way that there are equal percentage intervals between each
sample. In general, if there are N samples, the first in the sequence
is plotted on the 'probability' line whose value is 50/N % and the
succeeding (N-l) samples at equal intervals of 100/N %. Further
details are described by Sinclair (1976).
These procedures were implemented in the interactive graphic
program called GIRAF initially written by Steven Earle, Department of
Geology, Imperial College. The mean and standard deviation of each
sub-population are calculated statistically. The Kolmogorov-Smirnov
statistical test of normality of the sample is also implemented in
115
the program.
4.4.2 Regional Distribution of Geochemical Elements
The geochemical maps sliced by absorption levels in Fig. 4.19 show the regional distribution patterns in stream sediments. Slicing levels, with their relative symbols, are listed in Table 4.2.
In addition, means and standard deviations were calculated for lithologic units (including Upper and Lower Carboniferous rocks, Upper, Middle and Lower Devonian sediments and granite areas) in order to provide information in the lithologic distribution of the elements. The statistics for anomalous zones defined from the probability plot by threshold values were separated for more reasonable lithologic estimation of each element. The calculated statistics are given in Table 4.3.
Considering the data as a whole, there is a broad similarity in the distribution of groups of elements although there are some variations in details.
Principal composite patterns of the regional geochemical distribution are as follows.
1. Generally most of metallic sulphide elements such as As, Cu, Pb, Zn and Ni tend to be low on the granite compared to the surrounding country rocks. These typical lows could be due to kaolinization (Hale, pers. comm.). These elements show also typical low values in the Upper Carboniferous rocks (except Ni), and in the lower Devonian sediments except in areas of contamination by drainage patterns.
Ga, Li and Sn are largely higher on the granite. They also appear to be high in the middle Devonian near the granite, which may
N MOOR AREA
117
be due to the secondary enrichment by drainage systems. Extensive
lateral distribution of high tin patterns in the eastern part of the
St. Austell granite has in part been related to reworking during the
pliocene marine transgression. This explanation is deduced from the
absence of supporting arsenic and/or copper values (Webb et al 1978).
As, Cu, Ni and Zn are uniformly low on the granites while
Pb shows some high on the granite.
2. As, Cu, Ni, Pb and Zn generally tend to concentrate in
a number of areas around the granitic aureoles. Again, there are
marked differences within the overall pattern in distribution and
metal concentration.
Peak values for As and Cu occur east, southeast and south
of Bodmin Moor and near the Kithill granites. Pb and Zn show
maximum concentration in southeast and northwest of the main granite,
and near the Kithill granite. Zn is concentrated east of the St.
Austell granite and shows moderate highs east and south of the Bodmin
Moor granite.
Except in areas of drainage contamination and possibly old
mining activities and smelting (particularly in the southeastern part
of the study area and east of the St. Austell granite), the distribution
of the elements associated with mineralization is in broad accord with
mineral zoning.
3. Ga and Li are characterized by their low distribution
patterns in the country rocks, particularly Carboniferous and Lower
Devonian sediments, but they show patterns in the Middle Devonian
rocks:as high as the granite areas. This may be attributable to
redistribution of those components by particularly abundant drainage
systems south and west of the main granite.
Sn peaks occur in most of the mineralized areas east and
118
south of the Bodmin Moor granite and over the St. Austell granite. As
Hosking et al (1965) pointed out in the study of sediments of the
rivers to the south of the St. Austell granite mass, the wide dispersion
of the high tin values may be derived from the slate rather than from
particular mineralized zones or veins.
Table 4.2 Slicing-level values for 8 geochemical elements used in colour plotting in Fig. 4.19.
element
colour level
slicing-level element
colour level As Cu Ga Li Ni Pb Sn Zi
1 Dark Blue 10-2 30-7 6-8 4-6 5-8 1-3 1-3 11-20
2 Blue 2-4 7-15 8-11 6-10 8-13 3-7 3-10 20-40
3 Cyan 4-6 15-33 11-15 10-15 13-21 7-16 10-30 40-75
4 Green 6-11 33-73 15-20 15-23 21-33 16-42 30-100 75-150
5 Yellow 11-21 73-160 20-27 23-36 33-53 42-100 100-300 150-250
6 Orange 21-39 160-360 27-37 36-56 53-86 100-270 300-900 250-500
7 - Lemon 39-71 360-800 37-50 56-87 86-D8 270-690 900-3000 500-1000
8 Red 71- 800- 50- 87- 138- 690- 3000- 1000-
units are ppm
4.4.3 Probability Analysis
Fig. 4.20 (a) to (h) shows the probability plots of the
regional geochemical elements. The plotted lines show assymmetrically
placed sigmoidal curves. There is a point of inflexion where the
direction of the curves change, as indicated by arrows. This point
suggests that there are two population involved; a population of small
concentrations, and mixed with them a small population of large
119
Table 4.3: Means and standard deviations of 8 geochemical elements from the Bodmin Moor area in lithologic units. Threshold values of geochemical anomalies were determined from the probability analysis in Section 4.4.3.
Upper C 510
Lower C 69
Upper D 1000
Middle D 672
Lower D 896
Crani te 713
Anomalous area
* 23.1 10.7 10.0 6.5 8.1 65.1
As** 3.71 7.59 7.18 7.69 5.58 6.0 21.11
1.1- 21 .0 7.3-35.0 1.5-36.9 1.7-36. 1.1-32.3 37.2-131.0
510 43 872 610 894 707 224
44.6 199.0 90.7 58.5 73.8 44.6 961.7
Cu 15.96 128.41 157.23 56.69 94.48 64.11 283.52
24.7-169 .5 38.3-55.86 14.6-571.8 21.9-560.6 12.5-546.1 3.6-565.2 576.2-1763
510 69 876 651 888 701' 165
16.0 16.4 20.1 22.3 14.7 23.4 44.5
Ca 2.34 3.40 2.95 2.16 3.84 3.22 9.47
9.6-22.5 9.4-23.6 10.8-27.7 13.6-28.3 6.8-31.0 13.7-30.8 31.2-68.4
510 69 1000 672 839 636 134
9.2 10.7 12.2 14.2 10.3 17.9 44.6
L i 2.04 2.38 2.82 2.74- 2.49 4.36 25.31
4.9-19.5 5.6-15.4 4.3-22.4 9.1-23.7 5.3-24.6 7.3-29.9 25.0-135.8
510 69 1000 672 829 462 318
62.2 68.3 60.2 61.7 55.1 26.2 117.9
Ni 12.47 10.07 13.47 16.53 15.70 14.84 31.14
31.1-91.0 44.2-87.2 13.9-91.8 11.7-88.8 5.7-89.2 6.3-64.3 92.5-221.3
507 53 969 667 885 , 713 66
45.6 171.3 105.1 96.0 78.8 45.2 668.5
Pb 34.38 89.18 92.32 73.01 66.29 33.57 327.67
16.3-282 .6 29.7-373.6 17.2-388.0 26.0-382.5 2.0-368.7 2.0-325.1 390.0-1750
510 60 877 653 895 709 156
24.9 882.4 212.8 464.6 453.6 408.9 3725.9
Sn 64.20 456.82 369.11 493.34 458.20 446.28 1425.20
0-937.9 46.7-1806.5 0-1815.9 5.3-1822.2 5.2-1793.1 3.9-1828.9 1836.7-8880.:
510 52 869 518 662 524 725
157.8 312.0 214.4 246.4 223.3 111.3 796.7
Zn 60.59 87.06 84.43 100.79 85.21 70.50 255.02
51.1-524 .0 173.8-511.2 61.5-524.6 63.4-528.9 11.5-529.8 11.3-518.7 537.9-1905.6
510 55 933 617 870 692 183
* Means minimum and Maximum
* * Standar deviat ions • • number of s neiples
un i t s are ppm
120
'10 70 30 40 50 SO 70 60 90 9S 99 99 C U M U L A T I V E P E R C E N T
-i 1 1 1—r
I 7 5 10 70 10 40 50 60 70 60 90 95 9e 99 C U M U L A T I V E PERCENT
77 !7 IC 57 I.C '7 60 90 95 96 99 C U M U L A T I V E P E R C E N T
1 7 5 10 75 JO 4C 50 6C >0 80 90 95 96 99 C U M U L A T I V E P E R C E N T
Fig.4-20 Probability plots of 8 geochemical elements: As,Cu,Ga,Li,Ni,Pb,Sn and Zn.
121
Fig.4-20 continued.
C U M U L A T I V E P E R C E N T C U M U L A T I V E PERCENT
C U M U L A T I V E P E R C E N T C U M U L A T I V E P E R C E N T
122
concentrations. The two straight lines represent an approximation to
these two populations. The actual values for these two sub-populations
are plotted as dots along the straight lines. The straight lines are
actually fitted from the dot points by multiplying the percentages for
the smallest individuals with the inverse of its proportion to the total.
For example, if the small sub-population is 95% of the total, then the
inverse '100/95' is multiplied by the percentages of concentrations
involved. Likewise, for the largest concentrations '100/5' is multi-
plied to the percentages of concentrations from the larger sub-population.
The dots along the sigmoidal curve are the resultant estimate of the
two straight sub-populations. The Kolmogorov-Smirnov non-parametric test
was applied here as a test of goodness-of-fit of the estimate curves to
the observed data.
The statistical results calculated are shown in Table 4.4. The
total means and even the means of sub-population 1 are much higher than
the world mean crustal concetrations except for Ga, Li, and Ni. Ga, Li
and Ni are lower in total means and means of sub-population 1, but the
means of sub-population 2 of Ga and Ni are higher than those from the
world statistics.
However, in the case of Ni, the anomalous values were encountered
in the lower range, which may represent a local geological feature that
the granites are low in nickel.
In all cases, Komogorov-Smirnov tests show that the estimated
values are not significantly different from the observed samples at a
confidence level of 90% as shown in Table 4.4.
To further analyses the spatial distributions of the anomalous
samples to the background, colour pictures have been produced for the
main sulphide elements (Cu, Pb, Sn and Zn) which are closely related
to local mineralization. This was done by using the Colour Additive
Table 4.4 Statistical results of probability analysis
. Sample . _ , . , sub-pop 1 sub-pop 2 K-S tesl Variable r Std.Dev. Threshold 77 .. — — - _r / — . nr.„ Mean . Mean Std. Percent Mean Std. Percent at 90%
Dev. Dev.
As 11 15.6 3.7 8.6 7.2 94.86 54.9 19.6 5.14 A 0.804
Cu 110 206.8 575 93.2 66.9 96.19 790 238 3.81 A 2.079
Ga 20 6.7 31 18.6 4.5 96.61 36.1 2.9 3.39 A 0.612
Li 15 11.9 25 13.8 4.8 95.58 40.8 12.0 4.42 A 0.989
Ni 55 23.3 28 18.1 7.1 16.15 60.0 14.1 83.85 A 0.225
Pb 110 152.3 390 107 71 96.61 478 150 3.39 A 1.706
Sn 920 1496.3 1830 349 325 83.64 3600 1250 16.36 A 1.313
Zn 230 167.8 533 206 95 95.73 671 138 4.27 A 0.732
A: not significantly different at the significance level of 90%
K-S: Kolmogorov-Smirnov statistics
units are ppm
124
Viewer at the Remote Sensing Unit in Imperial College. Threshold values
estimated from the probability plots are shown in Thble 4.4.
Arsenic has been excluded from this analysis because As may not
be a good pathfinder as noted by Shouls et al (1968) since it is
inconsistently associated with other elements in mineralization.
The patterns were produced with five levels as follows:
1. blue: background (in the probability plot, the background
is treated by the sample values below the point of arrow).
2. light green: less than -1 SD (Standard Deviation) of
sub-population 2.
3. yellow : between -1 SD and 0 SD.
4. orange : between 0 SD and +1 SD.
5. red : greater than 1 SD.
As shown in Fig. 4.21, the pictures indicate generally well defined
anomaly patterns which might be related to local mineralization.
Most copper concentrations occur in the east and southeast
of the Bodmin Moor granite and a small patch near the Kithill granite.
They are largely associated with anomalous tin concentration as well.
These may occur associated with local copper-tin mineralization. Part
of the concentration in the southeast of the Bodmin Moor granite may
have contaminated drainage.
Sn also shows an extension of its anomaly further to the
south of the main granite, on the St. Austell granite and in its eastern
part.
Pb and Zn are largely assocaited with each other southeast and
northwest of Bodmin Moor, and near the Kithill granite areas. A strong
Zn anomaly occurring in the eastern part of St. Austell granite may be
due to a combination of mineralization with some contamination of the
drainage sediments from local smelting.
125
N
M b \ * • -
i (a) Cu (b) Pb
(c) Sn (d) Zn
10 20 Km
Scdi\e
F i g . 4 . 2 1 S p a t i a l d i s t r i b u t i o n o f g e o c h e m i c a l h a l o e s o f C u , P b , S n a n d Z n .
126
4.5 Conclusions
A variety of data processing techniques chosen with reference
to the properties of the data, have been applied to the geophysical,
Landsat MSS and geochemical data, to extract regional and local
features of geological significance.
In geophysics, regional features such as magnetic basement features
and regional gravity fluctuations were defined by regional analysis
including low-pass filtering and upward continuation. Local anomaly
features of magnetic sources and density variations such as tertiary
undulation or subsurface granite cusps which could be related to local
mineralization were also enhanced by horizontal and vertical derivatives
and highpass filtering in order to visualize those features more clearly.
In Remote Sensing, contrast stretching and ratioing techniques,
and their colour-composites applied to Landsat MSS data from Cornwall
and the Bodmin Moor area enabled the extraction of various regional
features showing lithologic or structural patterns. In particular,
many known lineaments were confirmed and further linear features
(which are not on the geological map and are probably geologically
significant) were detected from the enhanced images.
In addition,level slicing for geochemical data by the absorption
scale employed in this study shows the regional distribution patterns
of geochemical elements in conjunction with geological units. Further,
probability plots were effectively applied to determine threshold
values and thereby spatial distribution of geochemically anomalous zones
has been effectively delineated and these may be related to local
mineralizations.
127
CHAPTER FIVE
TREND SURFACE ANALYSIS
5.1 INTRODUCTION
Trend surface analysis has been widely used in the Earth
Sciences for recognition and measurement of trends that can be
represented by lines or surfaces.
It is a multiple regression technique used to fit
statistical equations or models to geographically distributed
data in order to define trends on a map surface. A trend analysis
illustrate regional,and residuals from the trend surface, and this
type of analysis may provide valuable geological information about linear
structures and related tectonic features.
Henkel (1968) has reviewed somfe of the geophysical trend
analysis and noted that apart from their use for examination of
linear features in the data they can be employed for various
analytical procedures. Geophysical anomalies with certain trends
can be distinguished as anomaly patterns using gridding operators
with specific properties such as the directional filter as described
by Fuller (1967). Trend surface analysis can also indicate
regional effects and detect whether these have been removed from
the residual anomalies.
Agarwal (1968) applied an empirical method using cross
correlation coefficients in order to trace the trend due to various
geological factors. There are numerous applications of the trend
surface analysis in geological studies. Excellent review on
problems arising in the trend surface analysis and its application
is given by Howarth (1983).
128
Polynomials are perhaps the most widely used as can be seen
in most of the statistical references. Lower order trend surfaces
may depict the regional or large-scale features and their residuals
may reflect the local or small-scale geological features.
Other approximations can also be used for trend surface
analysis such as orthogonal polynomials (the Fourier and Chebyshev
series, etc.) or the moving average method, etc.
Comparison of advantages and disadvantages of different
methods cannot be directly evaluated because of the different purposes
for which they are employed. Criteria for selecting the most
appropriate method for a trend surface depends upon the geological
objectives. Some comparison between polynomial, moving averages, and
the Double Fourier Series methods for various analysis are given by
Davis (1973). Some details on the trend surface analysis with
polynomial fitting and double Fourier series can be found in
Harbaugh and Merriam (1968).
In my study, first- and third-degree trend surfaces have
been applied for analysis. This is because it can be anticipated
that a basic pattern of similarity between variables should lie in
the lower degree terms.
5.2 Mathematical Procedures
The general n-degree polynomial for two independent
variables, often geographic coordinates, is as follows:
129
Z = b. + ( b x + b Y) + (b X 2 + b.XY + b Y 2) + 0 1 2 3 4 5 + (b X° + b Xn~^Y + .... + b. Y n) (5.1) k k+1 k+n
where k = n(n+l)/2 and n is the degree of the polynomial and
b^ (i=0,1,2,...,k+n) the coefficients to be calculated.
In this study, first and third degree trend surfaces have been
analysed by solving equations of polynomials by means of the least
square criterion. The expressions for the least square fit
of the first- and third-degree polynomials are given in Appendix E.
A statistical test can be performed to indicate the reliability
of the fit of the calculated values to the original data. Usually
the "goodness-o'f-fit" depends on the degree or order of the
calculation.
The most commonly used measure*of reliability of the fitted
surfaces is the sum of squares test which has been widely used,
for example, by Merriam and Sneath (1966), and Howarth (1967), etc.
Howarth (1967) calculated the percentage sum of squares accounted
for by first-, second-, and third-degree trend surfaces fitted to
random data in order to measure the reliability in extracting regional
trends, and noted that if the sum of squares test produces values
that fall below 6.0, 12.0, and 16.2 per cent for the first-, second-,
and third-degree trend surfaces respectively, the distribution of
data points is not significantly different from random at the 0.05 level.
Nordiffe (1969) derived improved values for critical % sums of
squares. If F^ for level of significance Ot, r^ = K-l, r2=n-k-2,
where K is the number of coefficients in trend surface equation, n the
number of data points, then
130
F (k-1) % 'explained' = x ,00 (5.2)
a
e.g. for n = 500 minimum % accounted for by random trends will be
F = 1.8 and 4.3% at a = 0.01, and F = 1.2% and 3.4% at a = 0.05 a ' a for linear and cubic polynomials, respectively.
In this study, the sum of squares has been calculated to
indicate the goodness-of-fit of the trend surfaces as follows.
R _ SSR 2 SST
n n Z X" . - ( Z X
regression regression )2/n
n n Z XZ, - ( Z X . ,r/n . , observed . , observed i=l i=l
(5.3)
where n is the number of data.
Further the F-test value can also be obtained by
F = [ SSR/m SSD/(n-m-1) (5.4)
where m and (n-m-1) are degrees of freedom of regression and
deviation from polynomial regression, respectively. m is one
less than the number of coefficients for the surface being used.
SSR and SSD are the sums of squares due to regression and deviation,
respectively. SST is the total variation, that is,
SST = SSR + SSD (5.5)
The F-test provides a measure of the random effect of the
regression.
131
Further details of the statistics employed can be found in
Davis (1973). The program in his reference (p.332) has been
modified by the author to apply to regularly gridded data sets for
this study.
5.3 Application to the Bodmin Moor Area Data
First- and third-degree trend surface analysis have been
applied to the gravity and magnetic data, four Landsat MSS bands,
and eight geochemical elements in the Bodmin Moor area. The are of
geochemical elements^ As, Cu, Ga, Li, Ni, Pb, Sn and Zn.
The first-degree maps of the analysed data are shown in
Fig.5.1(a) to (n), and the third-degree maps are illustrated in
Fig. 5.2(a) to (n). The residual maps of the third-degree polynomial
were contoured and shown in Fig. 5.3(a') to (n).
The trend surface coefficients are tabulated in the Table 5.1
(a) for the first-degree and (b) for the third-degree. Statistical
data pertinent to application of analysis of variance to particular
trend-surface analysis have been calculated. These data include (a) sums
of squares apportioned among first- and third-degree regression
components, (b) sums of squares associated with deviations of regression
from the original components, and the number of degrees of freedom
associated with regressions and deviations. From these data, the
goodness-o-f-fit (R^) and F ratio is calculated as shown in Table 5.2,
and further the F ratio values are used to determine the significance
level, expressed as a percentage by reference to Tables of F.
132
5.3.1 Geophysical data
(a) Gravity data
The first-degree trend surface fitted to the gravity data reveals
a gradual increase toward the east-northeast direction perpendicular
to the north-northwest linear direction in Fig. 5.1(a).
The generally low fit of the first-degree trend surface, of
14.6 per cent of total sum of squares (Table 5.2) indicates that the
bulk of variation of the gravity is not properly accounted for by the
first-degree trend surface.
A third-degree trend surface fitted to the same data in
Fig. 5.2(a) resulted in a significantly improved fit, accounting for
81.6 percent of total sum of squares. A dominant feature is the
major trend of low gravity values connecting the Bodmin Moor and
St. Austell granite bosses and this trend turns nearly E-W in the
eastern part of the study area showing clearly the trend in low values
of the Cornubian granite batholith (see Section 4.2.5). Outside
of this gravity trend, the trend surface increases toward the
Upper Paleozoic sediments.
(b) Magnetic data
The first-degree trend surface of the magnetic data in
Fig. 5.1(b) shows nearly E-W trends with a gradual increase from
south to north. Quite a high fit of the first-degree trend surface,
of 68.8% of the total sum of squares may indicate that the linear
trend surface may depict the regional trend of the magnetic source
in this area. This trend which is different from that of the
gravity data corresponds well to the Amorican structural trend.
The third-degree trend surface shows some improvement in
fitting, of 79.5%. The general trend is very similar in features
133
(c) MSS Band 4 (d) MSS Band 5
Fig. 5.1: Comparison of trend surfaces of degree 1 for 14 variables in the Bodmin Moor area. The variables are of (a) Gravity, (b) Magnetic, (c) MSS Band 4, (d) MSS Band 5, (3) MSS Band 6 (f) MSS Band 7, (g) As, (h) Cu, (i) Ga, (j) Li, (k) Ni (1) Pb, (m) Sn and (n) Zn. The values on the maps increase in order of (3 2 1 & A B C).
Fig. 5.1 continued
~~~~.
(e) MSS Band 6
(g) As
Ii) Ga
"~W""J:~ .- . . - . = : .::::.. : )'
:§l 1'1 ~ :i' =- 'I ::;: ;··S ;;;;. 'Ill - ··n == .'I!! ~ 11':1: ". , = ··.t~: ~ .= ~;.,tt: ~ - .. ~. = .'''.: ,,. = ::,'B:~ =- ::.;::'1:~ ._" ~.
I I II illl
.....• I
il . I
I .!
(f) MSS Band 7
(h) eu
.:::::: : . -.. -: ==-:.=" ........-:=" .' .'
(j) Li
...... ~ .... =
-::
(m) Sn (n) Zn
136
to the first-degree ones, showing the general E-W trend. A broad
basin-like feature appears in the south of the map which may represent
a local basin of non-magnetic rocks. The location of this basin
is nearly coincident geographically with the central part of the
Trevone basin. The general increase toward the north may reflect
a shallower magnetic basement in a northern direction.
5.3.2 Landsat MSS data
The first-degree trend surfaces of all Landsat MSS bands
reveal a generally common NNW-SSE linear trend. Trends of MSS bands
4 and 5 increase towards SW and are the reverse of the trends of the gravity
data, which is largely attributed to the high reflectance over the
granite masses compared with the relatively low reflectance in
the surrounding sediments.
However, in MSS bands 6 and 7, the direction of increase of
the trends is opposite and almost corresponds to those of the gravity.
This may be due to the lack of dominant features over the granite
relative to higher reflectance of the green vegetations in the
surrounding sediments toward the infrared spectral bands as described
in Section 4.3.3.
The extremely low fit of the first-degree trend surfaces of
all bands (0.0315 ~ 0.0525) indicates that the linear trend may not be
reliable for analysis of linearity of the data. The third-degree
trend surfaces in the MSS bands in Fig. 5.2(c) to (f) have resulted
in a much improved fit for the various data sets ranging from 27.3
to 46.5% of the total.
The values of the third-degree trend surfaces of the four bands
are the reverse of those of the gravity data. Low value gravity trends
corresponds to high value MSS data trends. The higher trends
137
fdr .piiiiiiiiiiiiih!,-
III' 4
^ i l i l i l " 1 ? t* M H U U n t M * .
(b) Magnetic
(c) MSS Band 4 (d) MSS Band 5
Fig. 5.2: Comparison of trend surfaces of degree 3 for 14 variables in the Bodmin Moor area. The variables are of (a) Gravity, (b) Magnetic, (c) MSS Band 4, (d) MSS Band 5, (e) MSS Band 6, (f) MSS Band 7, (g) As; (h) Cu; (i )Ga ; (j ) Li ; (k ) Ni ; (1) Pb; (m) Sn and (n) Zn.
The value on the maps increase in order of (3 2 1 & A B C)
Fig. 5.2 continued
a m i i w i K a H f f i i i
4444***** nllilllll!:
(e) MSS Band 6 (f) MSS Band 7
N
» . i t u i i u n : : 1!!!. i:::..: 11 • 1111:':
(g) As (h) Cu
(i) Ga (j) Li
Fig. 5.2 continued 1 39
(m) Sn (n) Zn
i
140
Table 5.1 Trend surface coefficients of degree 1 and degree 3
\ \ e n i : 1" Ts
v a r i u b l e ^ r ^ ^ b 0 b l b 2
G r a v i l y - 5 1 1657 1 . 7 7 2 5 - 1 1428 M a g n e t i c 39 7 1 8 5 - 0 5 7 3 - 3 0 8 4 2 Band 4 33 0 8 1 1 - 0364 0 0 1 4 2 Band 5 30 5 4 7 6 - 0454 0 0 3 2 0 Band 6 87 0 4 4 3 0 1447 - 0 8 4 0 Band 7 78 4 3 7 1 0 1940 - 1 1 5 3 As 3 1989 0 2 0 1 7 0 0 5 0 0 Cu - 2 6 7 8 2 1 2 8 1 9 6 1 2 8 0 0 Ga 25 3184 - 1544 - 0 0 8 8 Li 16 . 7 5 9 9 - 1810 0 1246 Ni 55 8134 0 1375 - 1687 Pb 5 8 . 2 3 9 5 1 0 0 6 3 0 5 3 0 5 Sn 5 0 2 6269 - 1 8 4414 31 2 4 7 9 Zn 1 4 5 . 4777 0 3 9 4 5 2 0 6 4 5
(b) Degree 3
b o b l b 2 b 3 b 4 b 5 b 6 b_ b s b 9 v a r i a b l e ^ ^ G r a v i t y 274 . 3 9 8 1 - 1 6 . 3 2 5 8 - 1 3 . 6 1 4 6 0 . 4156 0 . 0 7 4 9 0 . 1 4 0 8 - . 0 0 2 2 - . 0 0 4 2 0 . 0 0 4 3 - . 0 0 0 6 Magnet i c 60 . 6937 - 3 . 2 1 1 1 - 2 . 6 0 9 6 0 . 1123 - . 0505 - . 0 3 2 9 - . 0 0 0 7 - . 0 0 0 7 0 . 0 0 1 0 0 . 0 0 0 6 Band 4 25 . 6 6 8 0 0 . 3744 0 . 2 8 9 3 - . 0 0 4 5 - . 0 1 1 0 - . 0 0 2 3 - . 0 0 0 0 0 . 0 0 0 2 - . 0 0 0 0 0 . 0 0 0 0 Band 5 11 . 7776 1 . 0 2 4 9 0 . 8 4 3 3 - . 0 1 5 7 - . 0 2 4 5 - . 0 1 0 2 0 . 0 0 0 0 0 . 0 0 0 3 - . 0 0 0 0 0 . 0 0 0 1 Band 6 34 . 3 8 6 6 3 . 1 4 2 9 1 . 1 0 0 0 - . 0 4 9 1 - . 0 5 3 7 0 . 0 2 6 4 0 . 0 0 0 2 0 . 0 0 0 4 0 . 0 0 0 1 - . 0 0 0 5 Band 7 18 . 9 5 7 0 3 . 5 5 9 6 1 1 1 4 9 - . 0 5 5 7 - . 0 5 7 6 0 . 0 3 4 8 0 . 0 0 0 3 0 . 0 0 0 4 0 . 0 0 0 1 - . 0 0 0 6 As 18 . 5 8 7 6 0 . 0 0 7 5 - 2 . 0 5 0 6 - . 0 1 4 5 0 . 0 4 6 0 0 . 0 6 6 0 0 . 0 0 0 1 0 . 0 0 0 1 - . 0 0 0 7 - . 0 0 0 6 Cu 346 . 0 5 1 6 - 2 8 8 7 9 7 - 2 2 2 9 9 5 0 . 8 7 7 1 0 . 5 6 3 3 0 . 6 6 0 5 - . 0 0 8 7 - . 0 0 0 3 - . 0 0 7 4 - . 0 0 5 7 Ga 20 . 0 5 0 5 0 1036 0 0 0 4 5 0 . 0 0 2 2 - . 0 0 5 0 0 . 0 0 2 1 - . 0 0 0 1 0 . 0004 - . 0 0 0 4 0 . 0 0 0 1 Li 1 . 3 3 6 9 0 . 3 0 6 6 0 . 7 6 7 4 0 . 0 1 2 5 - . 0 2 5 4 - . 0 1 2 3 - . 0 0 0 3 0 . 0 0 0 7 - . 0 0 0 5 0 . 0 0 0 3 Ni 99 . 3 7 4 6 - 2 . 5 1 5 8 - 2 3 1 0 2 0 . 0 3 5 6 0 . 0 4 9 3 0 . 0 5 3 3 0 . 0 0 0 0 - . 0 0 0 7 0 . 0 0 0 1 - . 0 0 0 6 Pb 558 . 8 3 3 6 - 2 9 . 1 1 8 8 - 2 4 . 5 8 5 7 0 . 5 4 8 7 0 . 6 4 8 7 0 . 5 1 4 5 - . 0 0 3 8 - . 0 0 2 9 - . 0 0 4 3 - . 0 0 4 3 Sn - 2 7 9 . 2 5 4 2 48 7 9 0 5 - 8 2 . 7 7 6 4 - 1 1 5 8 5 0 . 0 8 2 7 5 . 7 0 2 4 0 . 0 0 5 2 0 . 0 2 5 4 - . 0 4 8 6 - . 0 5 0 8 Zn 368 . 4 6 0 8 - 3 . 9 4 0 1 - 1 6 . 4 9 0 4 - . 2 5 6 8 0 . 5 5 2 8 0 . 4 0 3 2 0 . 0 0 4 2 - . 0 0 2 7 - . 0 0 5 1 - . 0 0 2 9
141
Table j.2 Summary of test values of goodness-of-fit and F-test
TOs- degree
variable^^
first degree third degree TOs- degree
variable^^ goodness-of-fit F-test goodness-of-fit F-test
Gravity 14.65 351.4 81.61 2015.2
Magnetic 68.80 4513.0 79.51 1761.9
Band 4' 5.25 113.5 27.35 170.9
Band 5 3.15 66.6 30.75 201.6
Band 6 3.53 74.9 45.78 383.3
Band 7 ' 4.66 100.1 46.51 394.8
As 6.07 132.3 29.03 185.7
Cu 7.65 169.6 23.34 138.3
Ga 18.36 460.2 51.44 480.9
Li 11.66 270.1 42.73 338.7
Ni 3.57 75.7 15.8 85.2
Pb 1.90 39.7 28.64 182.2
Sn 20.07 513.9 40.82 313.1
Zn 5.36 115.8 14.40 76.4
Those underlined satisfy random effect at significance level =0.05 according to the criteria by Nordiffe(1969).
142
in MSS data connect the Bodmin Moor and St. Austell granite areas.
Bands 4 and 5 show their maximum values over the St. Austell
granite with general high trends in the NE, which decrease towards
the NW, NE and SE boundaries of the map. However, although the
general high trends in bands 6 and 7 are similar to those of bands
4 and 5, the typical highest peak in the St. Austell area disappears
in bands 6 and 7, and the high trend extends toward the NE border
of the map and decreases towards NW and south of the map. This is
again due to relatively low reflectance of bands 6 and 7 over
sparsely vegetated high Moor lands and relatively high reflectance
of well developed vegetation in the low lands associated with the
sediments. The sudden decrease at the NW and SE boundaries along
the coast lines is certainly attributed to the relatively low
reflectance of water compared to land areas in all bands.
5.3.3 Geochemical data
Again, the first-degree trend surfaces fitted to the eight
geochemical elements show the broad regional trend of each data set.
As, Cu and Pb elements in Fig. 5.1(g), (h) and (1) are similar to
each other and generally increase towards the SE. This may indicate
that these three elements may be regionally associated with each other
throughout the area. The high common occurrence of these elements in
the mineralized zones in the east of Bodmin Moor, and to a lesser
extent near the east of St. Austell granite, might be the main contributing
factor to the close correlation between the three trends.
Ga in Fig. 5.1(i) increases toward the west with a linear
trend of nearly N-S, and Li in Fig. 5.1(j) has a NE trend with an
increasing gradient to the NW. The general linear trends of Ga
143
and Li may be due to their high occurrence in the granite and to some
degree the neighbouring sediments as mentioned in Section 4.4.2.
Ni and Sn in Fig. 5.1(k) and (m), respectively, appear
to be similar in their NW-SE regional trends, but their values are
increasing in opposite directions. Ni increases to NE but Sn
shows its increase to SW. Zn has a linear direction of ENE-WSW
and increase toward the SE of the map.
The low reliability of these trends is indicated by generally
low fits of their linear trend surfaces of the data, ranging from
1.9% in Pb to 20.0% in Sn.
The third-degree trend surfaces show an improvement when
fitted to the same data with much higher fits of the regression
to the total sum of squares ranging from 14.4% in Zn to 51.4% in Ga.
Third-degree trend surface of the eight elements are shown in
Fig.5.2(g) to (n). Again, As, Cu and Pb in Fig. 5.2(g), (h) and (1),
respectively, show similar regional trends to each other. They
increase towards the eastern boundary and NW corner of the map, with
a saddle ridge of low values passing NNE to WSW in the west of the map.
The high trends represent those areas of most mineralization in
the east of the Bodmin Moor granite extending to the Kithill granite,
and also in the northwest part of the map.
Li and Ga appear to be similar in their third-degree trends,
which are largely moderately high over the Bodmin Moor granite
and the highest trend appears on the St. Austell granite in the
southwest corner of the map.
The regional trend apparent on the map of Ni in Fig. 5.2(k)
shows a broad depression with a low trend over the Bodmin Moor granite
and particularly low values over the St. Austell granite areas.
Surrounding the granites the trend increases toward the map boundary
144
in the east, northeast and northwest.
Sn in Fig. 5.2(m) shows a highest trend on the St. Austell
granite and the high trend extend to ENE, generally following the
southern margin of the Bodmin Moor granite. It decreases to the north
and SE corner of the map. This is largely related to the high content
of Sn over the St. Austell granite area with its extensions to
the southern margin of the Bodmin Moor granite and eastwards to the
Kithill granite.
Zn in Fig. 5.2(n) appears with a high trend extending NE
in the south of the map which joins the eastern boundary of the
St. Austell granite and the east of the Bodmin Moor granite area where
a number of zones of sulphide mineralization occur. Another high
appears in the northwestern part of the map, which is largely attributed
to Pb-Zn mineralization to the east of Wadebridge.
5.3.4 Residuals from the third-degree trend surfaces
The spatial distribution of third-degree trend surface
residuals is shown in Fig. 5.3(a) to (n) in order to extract local
isolated features which might have significance in geology or local
mineralization. Furthermore, the residual maps may provide
information on associations in detailed variations between variables.
a. Geophysical data (Gravity and Magnetic)
Clusters of gravity residual values in Fig. 5.3(a) reflect
local variations in the original data. The negative clusters that
show in the middle and SW corner of the map depict the trend of low
gravity values over the granite masses, and positive trends occur
outside the granite margins.
145
(200ooE,qooaN)
(a) Gravity (b) Magnetic
(c) MSS Band 4 (d) MSS Band 5
Fig. 5.3 Contoured Residual Maps of the Third degree polynomial: (a) Gravity, (b) Magnetic, (c) MSS Band 4, (d) MSS Band 5, (e) MSS Band 6, (f) MSS Band 7, (g) As, (h) Cu, (i) Ga, (j) Li (k) Ni, (1) Pb, (m) Sn and (n) Zn.
. »P . Km Sea l <8
Fig. 5.1 continued
(e) MSS Band 4
(g) As (h) Cu
(i) Ga (j) Li
147
Fig.4-20 continued.
148
This may be due to the steep gradient of the Bouguer anomaly
along the margin of the granites due to the steep intrusive contacts
of the granites with the surrounding sediments. Those dense
basic intrusives around the granite margin might also contribute to
emphasize these features. Sharp decreases towards the edges of the maps,
particularly in NW and NE, may be due to unreliable edge effect typical
in the polynomial fittings as described in Section 4.2.2c. However,
the E-W basin-like trend in the southern margin of the map may be of
some structural significance. This has been partially described
in Section 4.2.5 and will be described further in Section 6.3.3.
The magnetic residuals in Fig. 5.3(b) are similar in their
features to the vertical derivatives and highpass filtering maps
(see Section 4.2.5b) except in some margins of the map. It illustrates
well the isolation of local anomalies as described in the vertical
derivative maps. The high reliability' of this result is indicated by
its high 'goodness-to-fit' of the third-degree trend surface to the
data (See Table (5.2.).
b. Landsat MSS data
The third-degree residuals of MSS bands 4 and 5 are shown
in Fig. 5.3(c) and (d), respectively. The positive clusters that
aggregate in the middle and southwest corner of the map both represent
high reflectance over the high granite Moor areas. They are largely
attributed to dry and brownish vegetation over high elevations possibly
due to cattle overgrazing and partly due to the open-pit mining of
Kaolin as described in Section 4.3.3. These aspects were observed
on a field trip through the area in April 1982.
149
In the flat and low lands of the map, the low reflectance
may be caused by cultivated green vegetation toward the end of
April when the image was taken, particularly in band 5 which has
a strong absorption around 0.68]Jm of spectral value.
Dr Hawkes of I.G.S. who accompanied me on the field trip
considers it is unlikely that there has been much change in land use
between April 1975 when the Landsat data were imaged and 1982 when
the field data were collected.
On the other hand, MSS bands 6 and 7 are similar to each
other in their residuals in Fig. 5.3(e) and (f), but different from
those of bands 4 and 5. There is a lack of high reflectance in the
residuals over the granite Moor areas as in bands 4 and 5, and they
show rather negative values over the granite areas while they show
some positive values outside the granite, particularly near the coast-
line. Strong negtive reflectivity of the off-shore areas are distinct
in Bands 6 and 7 and show the clear feature of the coastline. This
feature is not so sharp in Bands 4 and 5.
The reflective feature over the Kithill granite area is not
similar to those of other granite areas in all bands. It may be
due to different vegetation coverage in areas lower than high Moor
areas as described in Section 4.3.3. The vegetation over the Kithill
granite might have consisted of a green agricultural cover which is
similar to the vegetation in the cultivated lower flat areas.
c. Geochemical data
Residuals of the geochemical elements from Fig. 5.3(g)
to (n) may show more direct relationship to local mineralization
and geological features than any other variables. In elements with
common sulphide minerals high positive residuals occur around the
150
margins of granite masses. Particularly high residuals are distinct
in the east to southeast borders of the Bodmin Moor granite and
this high trend extends further to the south, following some drainage
patterns. The extension of these high residuals to the Kithill granite
area is also characteristic of most elements with common sulphide
minerals such as As, Cu, Pb, Zn and Sn. Around the south to
southwest border of the Bodmin Moore granite and further towards the
eastern border of the St. Austell granite where there had been some
mining activities for sulphides in the past, high As, Cu, and Zn
residuals similarly occur. Zn also shows some drainage pattern
in the southeast of the St. Austell granite which might also be
partly due to contamination by old mining and smelting. Between
Wadebridge and the Bodmin Moor granite in the northwestern part of the
map, high positive residuals of Pb and Zn occur. This may be related
to local Pb-Zn mineralizations. Some anomalous residual of As
also occur in this area.
A small patch of positive Zn residuals is also shown in the
north of the granite where many basic volcanic rocks occur.
In Fig. 5.3(m), positive Sn residuals occur mainly near the
southern and southeast margins of the Bodmin Moor granite and over the
St. Austell granite and its eastern part.
Ni appears as largely high residuals around the metamorphic
aureoles of the granites and as typically low residuals over the granite
masses. A typical high residual in the southeast of the St. Austell
granite may be due to both effects, mineralization and contamination
by mining activities, etc.
The elements, Ga and Li in Fig. 5.3(i) and (j), respectively,
show in general, positive residuals in the granitic areas and from
the Bodmin Moor granite it extends further to the northeast of the
151
map which might indicate a contribution from local volcanic rocks
in the area.
5.4 Discussion
Regional trends of the data over the Bodmin Moor area were
extracted by analysis of trend surfaces of first- and third-degree, and
local irregularities or isolated anomalies were calculated by
subtraction of the computed third-degree trend surfaces from the
original data.
Some statistics were computed to examine the 1goodness-of-fit'
of the computed regression to the original data, and further random
effects of the data were assessed using the calculated F-test values.
The degree of randomness of the data has, to a certain extent, the inverse
relationship with the degree of 1goodness-of-fit1.
At the 0.1% level of significance (a = 0.001), test statistics
of a F-distribution with degrees of freedom (df)* (2,4093) and
(9,4086) for the total number of 4096 data values have critical values
of F ~ 6.9 and ~3.0, respectively. The computed test values of all
fourteen variables for both cases of degrees 1 and 3 fall well within
the critical region, that means the computed test values are all much
greater than the critical values, so that the regression of data
are fitted well for first- and third-degree polynomials.
*F for [(k-1), (n-k-2)]df where k = no. of coefficients in equation
of trend surface and n = no. of data points.
152
In a different way of expression for assessing the randomness
of data, the regression effect is significantly different from the
random effect. Anything with % sums of squares accounted for >1%
is definitely'significant1 in a statistical sense, but will still not
necessarily be geologically very meaningful.
Although the comparative study of similarity among different
data may not be so reliable in the trend surface analysis, it
provides at least some insight into the basic patterns of similarity
between variables.
To facilitate the comparison, a diagram was made of the
lienar directions of the data, with indication of the predominant
directions of various trends in Fig. 5.A.
Apart from the magnetic data and Zn, all the linear trends
can be effectively divided into two groups.
Group A consists of gravity da'ta, four Landsat MSS bands, Li,
Ni and Sn, and trends largely in a northwest to southeast direction.
Among those in this group, MSS bands 4 and 5 Li and Sn show the same
increasing direction of the trends to SW, while gravity, Ni, MSS bands
6 and 7 increase to NE.
The similarities and dissimilarities of the regional trends
of the variables in this group may be understood by the fact that those
variables commonly show most dominant features over the main granite
area.
Group B consists of As, Cu, Pb and Ga, and trends in the range
between North and NE. Sulphide related elements As, Cu, and Pb are close
to each other and show their increases to SE but Ga increases opposite
to the others to NW.
153
Fig. 5.4 Linear directions of the data and predominant directions of the linear trends. Values in the parenthesis are 'goodness of fit'. Higher values are more 'important'.
154
Significant geological features in Group B are evident.
There is an anomalously high common occurrence of As,
Cu, and Pb in the Middle and Upper Devonian and Lower Carboniferous
rocks with the maximum values in the mineralized zones, particularly
in the eastern flank and southeastern part of the Bodmin Moor granite,
which indicates that these elements are most likely associated in
these areas. There is low occurrence of these elements within the
granite itself and in the Lower Devonian and Upper Carboniferous
rocks.
By contrast, Ga shows its high concentration within the
granites and surrounding superficial sediments where streams drain
from the granite, while Ga has a low concentration in the rest of
the sedimentary rocks.
The residuals derived from the third-degree trend surfaces
illustrate localized anomaly patterns which might be closely
related to specific mineral concentrations.
The author's study has shown that analysis of the third-degree
trend surfaces of the variable is a more reliable means of assessing
the regional trends and local anomaly patterns than the other methods
of trend surface analysis considered.
155
CHAPTER SIX
SIMILARITY ANALYSIS
6.1 INTRODUCTION
There are basically three different types of map comparison,
overall similarity, spatial similarity and spectral (or
structural) similarity. These can be applied to geological data,
similar types of maps or different types'of maps.
In this chapter, these three methods of similarity analysis
have been employed to investigate the inter-relationships between
map variables.
Correlation coefficients were calculated for overall similari
analysis, and for spatial similarity the similarity map of Davis
(1973, p.393-407) was used between gravity and pseudogravity, and
between some geochemical elements. Coherence analysis was used for
analysis of spectral similarities and also for the structural
similarities between gravity and pseudogravity transform of magnetic
data.
6.2 Principles of the procedure
6.2.1 Overall Similarity
The Pearson product-moment correlation coefficient has been
calculated in making comparisons between variables. The calculated
coefficients result in a measure of statistical overall correspondenc
between two variables with no consideration of the sample location at
all.
When the correlation coefficients are estimated from the m
156
sample data, the computed formula is as follows,
n £ (X -X.)(X. -X.)
Sii k=l l k 1 J k J
= — (6.1) ij S S n _ n _ 1 3 [ Z (X., -X.) Z (X., -X.)
k=1 lk 1
k=1 Jk J
i = 1,2,...,m
j = 1,2,...,m
where S. and S. are the variances of variables X. and X. respectively 1 J . 1 J and S.. the covariance between the two variables, n is the ij number of samples in each variable.
Because of variability of correlation estimates, it is
desirable to verify that a non-zero value of the sample correlation
coefficients reflects any existence of statistically significant
correlation between variables. This is accomplished by testing the
hypothesis that the population correlation coefficient p^j = 0.
The acceptance region for a test of the hypothesis of zero correlation
given by Bendat et al (1971) is,
/—t" i+R.. ^El l n _ i l < |Z« | (6.2)
ij 2
where Z is the standardized normal variable and n the number
of samples. Values outside the above interval would constitute
evidence of statistical correlation at the a level of significance.
For example, at the significance level of Ot = 0.01, the
acceptance region of the hypothesis is I R — I £ 0.0404 (6.2.1)
Thus, if the absolute value of a computed correlation coefficient
is greater than 0.0404 , the hypothesis is rejected at the significance
level of CL = 0.01, so that the two variables can be said to have
statistical correlations.
157
The overall similarity analysis is very effective to see
concisely the degree of association between variables since it
gives a single number between -1 and 1. As noted by Howarth (1979),
although it may be seriously affected by several factors such as
erroneous data values, scaling of the data and so on, the
Pearson correlation matrix will give much useful information on
the structure of interelement relationships for little computational
effort.
However, it' does not provide information on spatial
significance, so that it might be difficult to interpret it in relation
the geology. There are also many other ways of overall similarity
measurement such as distance coefficient or cosine theta coefficient,
etc. Some detailed accounts of these overall similarity measures
are described by Harbaugh and Merriam (1968), Davis (1973) and Sokal
and Sneath (1963).
6.2.2 Spatial Similarity
A possible way of comparison of two maps with relation to
the spatial distribution is the so-called 'similarity map1 (Davis,
1973).
This is similar to matching coefficients or cross-association
coefficients as described by Harbaugh and Merriam (1968), and
Z-trend maps by Robinson and Merriam (1971), in which the data are
not necessarily required to be numerical and the results are commonly
reduced to two categories.
Instead, a similarity map is a numerical application of
those techniques and there are no particular limits on the number of
categories.
158
This is a very effective way of spatial comparison between
different sets of the same type of variables, since it is more complex
if two different maps have been mapped separately for comparison
(Davis, 1973).
This technique can be equally applied to two different types
of variables since the problem inherent in different maps based
on different variables can be avoided if the two original maps are
converted to standardized form so that both variables can be
represented by common dimensionless digits.
The standardized form is
X.-X Z. = (6.3) l s
where Z^ and X^ are the standardized and measured data at point i,
respectively, and X and s the mean and standard deviation of the
variable.
Because the map variables are standardized, it is not really
necessary to compute a correlation to obtain a measure of similarity
between the two maps. Instead, the surfaces can be multiplied
together. If both surfaces deviate from the mean in the same direction,
their product will be positive, while if they deviate in opposite
directions their product will be negative.
Care must be taken in that the high correlation does not
always imply the same geological significance since both negative
or positive ones with the same magnitude always produce the same
results.
Although it may provide information of some structural or
other geological significance, as noted by Davis (1973), this method
may not give direct insight into structural features themselves.
Rather, it provides spatial distributions of similarities between
map variables which may provide further qualitative information of
159
structural or other geological significance, if the original maps
are consulted in its interpretation.
6.2.3 Coherency Analysis
Coherency analysis has been widely used for analysis of
spectral or structural similarity in the potential fields (Agarwal,
1968; Kanasewich and Agarwal 1970, QEB Inc. Lakewood Co. 1978 etc.).
This provides an excellent measure of similarities between waveforms when
random noise is absent (QEB Inc. Lakewood Co. 1978). The mathematical
formulation is as follows:
Consider two spatial functions f(x,y) and g(x,y) where x and
y are an orthogonal coordinate system, and f and g are map data
sampled at discrete spatial intervals Ax and Ay. Letting
and represent the power spectrum of f and g respectively,
and P_ (io ,03 ) the cross-power spectrum between f and g where 0) and f g x y x
U) are angular frequencies in x and y directions, the coherence
between f and g are as follows.
|Pf (o> >|2
Coh( f, g)= x y , r- (6.3) P.(0) ,03 ) P (03 ,0) ) f x y g x y
Theoretically, the above coherence function would be equal to
1, independent of frequency, while this is not so in practical applications
due to windowing and smoothing effects (Bath, 1974).
To calculate Coh(f,g), the power spectra of both f and g,
together with the crosspower spectra between f and g have to be
determined. These are calculated by a Fast Fourier transform
algorithm (as mentioned in Section 4).
The Fourier transform of a function f defined over the
160
x-y plane, is
^ -id) x — icu y F(u>x,u> ) = J J f(x,y)e X e y dxdy (6.5)
—CO —CO
In practice, we deal with a discrete sample function over
a finite areal extent.
For the discrete case, Equation (6.5) becomes
N-l M-l -27Ti(^ + JLL) F(u,v) = Z Z f(k,n)e AXAY (6.6)
k=0 n=0
where AX and AY are the sample intervals in the X and Y directions
respectively, and N and M represent the total number of samples in X
and Y. u and v are the frequency indices, and k and n the location
indices in the x and y directions respectively.
The power spectrum P^ is obtained by
P (u,v) = |F(u,v)|2 (6.7)
and the cross-power spectrum P^ by
P. (u,v) = F"(u,v) G(u,v) (6.8) f >g
where * indicates the complex conjugate function. G(u,v) is the
Fourier transform of function g.
Thus, for the practical discrete case Equation (6.4) can
be written as
161
Coh(f,g) =
u +Au v +Av
u=u v=v F*(u,v)G(u,v)
u +Au v +Av u +Au v +Av Z Z |F(u,v)|2 Z Z |G(u,v)|2
u=u v=v u=u v=v
(6.9)
where F and G are the corresponding Fourier transforms of f and g,
and Au and Av are the bandwidths over u and v respectively. Thus
Equation (6.9) is an estimator for the coherence over a band centred
at u = uj + Au/2 and v = v^ + Av/2
A plot of coherence coefficients as a function of wavenumbers
will illustrate those spectral regions which have degrees of association
between f and g. If the coherence is very low for a particular band of
wavenumbers, f and g are said to be incoherent in that wavenumber band.
If f and g are, respectively, the Bouguer gravity field and a
pseudogravitational transform of the magnetic field, the incoherence
may indicate that the magnetic and gravity data do not represent the
same source body. Or, it may also indicate that significant
remanent magnetization is present. Random noise may also contribute,
to a certain degree, to reduce the coherence. On the other hand,
if the coherency is very high for a particular band of wavenumbers, then
the causative bodies of magnetic and gravity may come from the same source
unit. Thus, Equation (6.4) is of great value in interpreting
the subsurface structures of an area. The main point in utilizing
Equation (6.4) is that it is assumed that the rock unit is both
anomalously dense and magnetic, and the direction of the rock
magnetization is uniform throughout the area and also the same as
that of the present day inducing field, or alternatively, we know already
162
the precise direction of the magnetic vector.
Where considerable remanent magnetization is present,
and where the direction of the remanent magnetization is unknown,
then the coherence between the gravity and transformed magnetic fields
will be low even if the causative body is the same.
A programe 'COHAN' was written by the author using Equation
(6.4) and Bartlett windowing was implemented in the smoothing process
for the coherence analysis and further for bandpass filtering
'BPFILT' was also written to delineate structural features in space
domain for those spectral regions applied for the analysis.
6.3 Applications of Similarity Analysis
6.3.1 Overall Similarity
The Pearson-product moment correlation coefficients were
calculated by using Equation (6.1) for the data over the Bodmin Moor
area. The calculated results are shown in Table 6.1. The coefficients
show the existence of inter-relationships between variables and their
relative strengths.
Taking the significance level of OL = 0.05 as the critical value
then lower values are an indication of non-association and the range
of the correlation coefficients (R) falling below 0.0306 are those
of non-association as noted in the previous Section. Thus, the results
have been classified into five groups according to the magnitude of
the coefficients.
a.very strong association |R| > .9
b.strong association .6 < |R| < .9
c.significant association .3 < |R| < .6
163
d. weak association .0306 _< |R) < .3
e. non-association |R| < .0306
Statistical results of inter-relationship are discussed on the basis
of their attitudes and patterns of the regional distribution as
described in Chapter 4 and Chapter 5.
(a) Very strong association
Only Landsat data fall in this group. The coefficients
between bands 4 and 5, between bands 6 and 7 show above 0.96 of R
which may indicate very similar reflectances of the near-surface
features between those two visible spectral bands and between the near-
infrared spectral bands as described in Section 4.3.3. Although it
is not shown here, bands 6 and 7 with band ratios R6/4, R7/4, R6/5 and
R7/5 are also very strongly associated. None of the ratio data show
the strong association with band 4 or 5.
(b) Strong association
As-Cu: A strong correlation between the two elements
may be due to their common occurrence in the soil particularly
associated with mineralized zone in the study area contributes to high
increase in the correlation coefficient between these two elements.
Ga-Li: The common high occurrence of these elements in
the granite and local stream draining from the granite near its
margin together with their low occurrence in the sediments, cause
the strong association between these two elements.
164
(c) Significant association
Gravity-Magnetic, B4, B5, Ga, Li, Ni, and Sn: The gravity
data shows some positive correlation with magnetic and nickel.
Particularly the positive correlation with nickel can be seen from the
fact that high nickel occurrence appears in the aureoles of the
granite and typical low occurrence within the granite itself.
Negative correlations with bands 4 and 5 are expected due
to high reflectances of those visible bands over the high Moor lands.
Ga, Li and Sn elements are also negatively correlated with the
gravity due to their high occurrences on or near the granites and
low occurrences toward the sediments.
Magnetic-Sn: They show a moderately negative correlation
between these two data sets. Generally this may be due to the high
occurrence of tin in the south where the magnetic trend decreases
gradually, and the low occurrence of tin in the north where in
general there is a regional high magnetic trend.
B4,B5,-B6,B7, Ga, Li, and Ni: Bands 4 and 5 show positively
moderate correlations with bands 6 and 7 as expected.
Positive correlation with Ga and Li are due to their common
high values over the granites and low values towards the surrounding
sediments.
Negative correlation with nickel may be due to the absence
of nickel in the granites and its concentration in the surrounding
aureoles possibly in the form of pentlandite and pyrrhotite.
As, Cu-Pb, Sn and Zn: their moderate positive association
can be explained partly by their common occurrences in the soils
and also their association in mineralization and surrounding anomalous
zones.
165
Ga,Li - Ni,Sn: Ga and Li show moderately negative correlations
with Ni due to their inverse relationships in the granites and
surrounding metamorphic aure oles, and positive correlations with Sn
are due to their high occurrence in the granites and their margins
and low occurrence in the sediments.
Zn-Ni, Pb: Zn shows moderate correlations with Ni and Pb.
Particularly the significant association with Pb may be in part
due to their common association in mineral occurrences.
(d) Non-association: The data sets below are statistically
non-correlated at 95% confidence level in the area.
Gravity - Zn Magnetic - B6, B7, Ga B5 - As B6 - Ga, Li, Ni, Sn, Zn B7 - Ni, Sn Cu - Ga
To display complex relationships among variables more
effectively, a cluster algorithm 'CLUSTER1 (Davis, 1973, p.467)
was used to construct a dendrogram of the correlation matrix.
CLUSTER uses weighted pair-group average clustering. The
method operates on an N x N similarity matrix. The algorithm pairs
those two individuals, i and j say, which have the highest similarity
and replaces columns (and rows) i and j by a single column with
arithmetic 'average' similarity coefficients. The process is
then repeated by pairing individuals and clusters of previously combined
individuals.
Details of the weighted pair-group method can be referred
to in Davis (1973), Sokal and Sneath (1963) and Parks (1966). The
process of averaging together members of a cluster and treating
TABLE 6.1: Correlation Matrix (untransformed data)
Gravity Magnetic B4 B5 B6 B7 As Cu
Gravity 1.0000
Magnetic .3914 1.0000
B4 -.3873 -.0956 1.0000
B5 -.4480 -.1627 .9679 1.0000
B6 -.2319 .0111 .5410 .5645 1.0000
B7 -.2039 .0253 .4424 .4680 .9911 1.0000
As -.2763 -.2275 -.0663 -.0254 .1458 .1586 1.0000
Cu -.1248 -.2151 -.1126 -.0930 .0375 .0493 .7554 1.0000
Ga -.5481 -.0043 .3917 .3572 .0211 -.0314 .0336 -.0187
Li -.5800 -.1853 .5125 .4857 .0031 -.0625 -.0599 -.0775
Ni .4533 .1876 -.3651 -.3728 -.0306 .0067 .0971 +.0696
Pb .1671 -.0864 -.1993 -.2240 -.1802 -.1754 .4009 .4207
Sn -.5222 -.3479 -.0974 -.0900 .0269 .0265 .3413 .3575
Zn .0158 -.2394 -.1578 -.1624 .0224 .0440 .4519 .3864
Ga Li Ni Pb Sn Zn
0000 8250 1.0000
3237 -.5024 1.0000
0729 -.2078 .2049
3294 .3545 -.2276
0830 -.1484 .3695
1.0000
-.0685 1.0000
.4838 .2010 1.0000
• 1762 •3306 •4850 .6394 • .7938 .9482 .0990 .2534 .4078 .5622 .7166 .8710 1.0254
Gray
Ga
Li
Ni
Mag.
Sn
As
Cu
Pb
Zn
B4
B5
B6
B7
Fig. 6.1
.0990 .2534 .4078 .5622 .7166 .8710 1.0254 .1762 .3306 .4850 .6394 .7938 .9482
Dendrogram - clustering with absolute correlation coefficients (untransformed data) Values along X-axis are similarities.
.5641
.8250
.4332
.2726
.3479
• 1841
• 7554 • 4150 .4838 .1333 • 9679 .5040 .9911
CTN
168
them as a single new number introduces distortion into the dendrogram
and this distortion becomes increasingly apparent as successive
levels of clusters are averaged together.
However, this is an efficient way to represent inter-
relationships between variables in a cogent tree-structural manner.
The analysed results are shown in the dendrogram of Figure 6.1.
Absolute values of all correlation coefficinets were considered in
the analysis since the similarity and dissimilarity are different
in verbal sense. If similarity values above about 0.25 are taken
to be significant in grouping, there are three groups apparent in
the dendrogram.
Group 1 consists of gravity, Ga, Li, Ni, Sn and magnetic
in their association with descending order of correlation.
Group 2 consists of mainly sulphide metallic elements
including As, Cu, Pb and Zn.
Group 3 are those from all MSS bands.
The groupings of similarities are somehow different from
those of linear trends in Section 6.4. This is because significance
of magnitude of all individual values, are taken into account in the
similarity analysis with correlation coefficients while only regional
aspects of the sampled data contribute to the trend surface
analysis which ignores detailed variations of individual values.
Mancey (1980) has noted that the inter-relationships can
be found better with the transformed data rather than analysis with
the untransformed data. Analysis made with the transformed data
in this study has shown similar results with the untransformed data.
However, certain differences have been found in clustering results
between these two. Grouping at the same level of similarity in the
TABLE 6.2: Correlation Matrix (transformed data)
Gravity Magnetic B4 B5 B6 B7 AS
Gravity 1.0000
Magnetic .3246 1.0000
B4 -.3929 -.0517 1.0000
B5 -.4528 -.1204 .9689 1.0000
B6 -.0431 .1239 .4291 .4142 1.0000
B7 .0497 .1567 .2029 .1810 .9563 1.0000
As -.2947 -.2071 -.0520 .0075 .0977 .0949 1.0000
Cu .0865 -.1472 -.2720 -.2825 -.0257 .0328 .6248
Ga -.5554 .1220 .3366 .3159 -.0263 -.1235 .1413.
Li -.6820 -.1260 .3592 .3850 -.0694 -.1738 .1109
Ni .4803 .1543 -.3401 -.3536 .0368 .1215 .1757
Pb .1587 -.1603 -.2954 -.2978 -.0881 -.0433 .5346
Sn -.6720 -.6240 .1207 .16059 -.0987 -.1353 .3154
Zn .1603 -.2363 -.2480 -.2486 .0060 .0674 .4943
Cu Ga Li Ni Pb Sn Zn
0000 0737
1622
3556
5611
2891
5718
0000 7834
2862
1421
,2656
,2080
0000 4338
2905
,4695
,2511
1.0000
.2897
-.2345
.5471
1.0000
.0598 1.0000
.5589 .2142 1.0000
o vo
• 1751 • 4759 • 6263 • 7767 • 9271 • 0999 • 2503 • 4007 .5511 .7015
I I • I I
• 8519 1.0023 Gray
. — Sn
.— Ga
Li
,— Ni
— B4
— B5
— As
Cu
— Pb
Zn
Mag
B6
B7
•0999 .2503 .4007 .5511 .7015 .8519 1.0023 •1751 .3255 .4759 .6263 .7767 .9271
• 6720
• 4931
• 7834 • 3587 .3315 • 9689
• 2367
• 6248 • 5404 • 5589 • 1721
• 1333
• 9563
o
Fig.- 6.2 Dendrogram - clustering with absolute correlation coefficients (transformed data) • Values along X-axis are similarities.
171
transformed data (R = .25), B4 and B5 are clustered into group 1
and B6 and B7 are independently clustered as a group 3. Magnetics
shows no association at all to any of the groups. Clustering
of sulphide metallic elements remains same as Group 2 in the untrans-
formed data.
Qualitatively, the result of transformed data seems to be
better in its clustering in this study. Correlation coefficients
for the transformed data and dendrogram of clustering result are
given in Table 6.2 and Fig. 6.2 respectively.
6.3.2 Similarity map
The application of 'similarity map' analysis was made to
following data; between gravity and pseudogravity, As and Cu, Cu and Sn,
and Pb and Zn.
Tov avoid overweighting by outliers of data, the data were
reset to ±3.291 when the standardized data were greater than
3.291 in absolute values. This ensures the data fall in the
confidence level of 99.9%
Maps of the surface-product of two standardized data are given
in Fig. 6.3(a) - (d). Areas of high coincidence of two surfaces are
clearly shown, as are areas where they depart markedly from one another.
In Fig. 6.2(a) the similarity map between gravity and pseudogravity
shows high coincidence in the northeastern part of the map which
might depict dense magnetic bodies. In the northwest there is
moderately high correspondence. Along the sub-surface trend of the
Cornubian batholith there also occur a moderately positive correlations.
Low correlations or negative correlations occur in the northern
part of the granite and toward the southeast of the granite masses.
172
(2£0OPE, qooo H)
N
(a) Gravity and pseudogravity
Scale
M
3.0 Km
(b) Cu and Sn
Fig. 6.3 Similarity maps. Hatched areas represent negative values whereas blank areas are positive.
173
F i g . 6 .3 cont inued
(c) Pb and Zn 3 io ao 1 i » • —•
S e a l e
174
These may mean that either gravity and magnetic anomalies come from
different sources, or remanent magnetization may exist so that the
transformation of the magnetic field into gravity may not be complete.
Particularly in the area where very high negative magnetic anomaly
appears, the low negative correspondence appears to be due to the
latter as already discussed in Section 4.2.5e.
High correspondences between Cu and Sn in Fig. 6.3(b) largely
occurs east and southeast of the Bodmin Moor granite and Kithill
granite, and the northeast margin of the St. Austell granite,
and negative correlations occur in the southern margin of the Bodmin
granite and east of the St. Austell granite where largely high Sn occurs
along the granite margins while copper values are low. Another
negative trends occurs in the southeast of the map where largely high
Cu occurs partly due to mineralization and possibly contamination by
mining activities and smelting.
Elsewhere, there appears largely low positive correspondence
throughout the map. In Fig. 6.3(c) high association of Pb and Zn
appears in three areas: one in the northwest of the granite, another
to the southeast of the granite and the third in the Kithill granite
area. Strong negative correlations occur in the northwest of the main
granite and in the southwest of the study area. The first one coincides
with high Pb mineralization and the last two areas correspond with
high Zn mineralization and possible areas of contamination.
Elsewhere, except along the southern and eastern part of
the Bodmin granite and northwest of the map, where largely low negative
association occur, Pb and Zn are largely positively associated,which
means that they are positively correlated as shown in correlation
coefficients in Tables 6.1 and 6.2, respectively, for the untrans-
formed and transformed data.
175
In Fig. 6.3(d), the inter-relationships between As and Cu are
largely positive except a few areas of weak negative association.
The strong positive association which appears in the southeast of the
Bodmin granite and . Jn the Kithill granite may reflect those
highly mineralized or partly contaminated area.
The low correlation on the southern margin of the granite is
noteworthy because the copper and tin mineralization occur in this
area but the arsenic shows low values in this region which may imply
that Arsenic should not be used as a pathfinder for mineralization
as it is inconsistently associated with other elements in mineralization
as noted by Shouls et al (1968).
6.3.3 Coherence Analysis
A contour map of coherence coefficients calculated by
Equation (6.9) for the Bodmin area is plotted in Fig. 6.4 as a function
of spatial frequency. Interpreted regional subdivisions are also
illustrated in the map.
A bandwidth of Q = 2, was applied for smoothing to compute
the power and cross-power estimates together with the Bartlett windowing.
Coherence contours show discrete regions of high and low
coherence on the map. There are four distinct regions labelled A, B,
C, and D.
Region D has been ignored since the spatial region of D may be
insignificant in terms of geological results.
To interpret the significance of the spectral regions A, B, and
C, two dimensional filters were designed which pass or reject
gravity and/or magnetic data within the frequency limits defined by
these regions. These filters were then applied to the data in order
176
Fig. 6.4 Coherence vs. Frequency - Bodmin Moor Area
177
to produce filtered maps of various coherent and incoherent regions.
Digital filters were designed in the two-dimensional
frequency domain by assigning a weighting value of 1.0 to
frequencies within the desired region of the coherence plot, while
a value of 0 was assigned to frequencies outside the desired
region.
A cosine taporing method was applied to the frequency weights
for the transition zone between the pass and reject bands, in order
to reduce spurious oscillation due to the sharp truncation o'f data.
The filter function was then multiplied by the Fourier transform of
the gravity and magnetic data, and the resulting product was inversely
transformed to yield a filtered version of the original data. The
filtered maps are shown in Fig. 6.5 to 6.7.
a. Region A
The coherence of this region is not very high, but it is
presented here to illustrate spatial features for those low frequency
regions with a moderate magnitude of the coherence.
In Fig. 6.5(a) the gravity map was bandpass filtered to show
the general low trend of the gravity cupolas interconnected in the
subsurface and high ridges along the northern margin which may
correspond to the ridge between the two basins of Bude and Trevone.
This also largely corresponds to the E-W trending magnetic high
to the north of the granite.
The inverse association between gravity and magnetic data
in the southern part of the map may be due to the following reasons
as partly described in Section 4.2.5(a).
It indicates that the causative anomalies of magnetic and
gravity fields do not come from the same source bodies; the gravity
178
( 2 0 0 o o E , q o o o N )
M
(a) Gravity Coherence Map CtooooEjloeoN)
(b) Magnetic Coherence Map o to so Km , • »
s cai e Fig. 6.5 Bandpass filtered maps of coherence region A
179
high of the E-W trend may be due to regional undulations of the main
granite and thickening of the sedimentary basin, or the limit of lateral
extension of the main granite underneath, while the magnetic anomaly
may come from magnetic basement of high susceptibility under the
batholith or sediment basin. The area may correspond
geographically to the central part of the Trevone basin.
b. Region B
The moderately high coherence in'this spectral region may be due
to the general E-W structural trends in the north of the Bodmin Moor
granite as shown in the bandpass filtered gravity and magnetic maps
in Fig. 6.6(a) and (b), respectively. Although small fluctuations
occur in the filtered gravity map in this spectral region, the general
high and low gravity trends corresponds to those of the magnetic data
in the north of the Bodmin Moor granite.
The curved shape with alternating signs in this region may
be due to several slices of rocks which have different magnetic
properties. This may be due to shallow basic rocks intruded into
zones of stratigraphical weakness which formed during the intrusion
of the granite bosses and were therefore aligned along the margin of
the granites.
Freshney and Taylor (1971) have noted that slices of Lower
Carboniferous rocks are inserted into Upper Devonian rocks as far
south as Tintagel. They have also noted that rocks in a zone of
isoclinal folds below the Rusey Thrust show the effect of low-grade
metamorphism with development of fine chlorite, particularly
in Devonian and fine pyrite and pyrrhotite spots. Thus the magnetic
patterns may be partly due to stratigraphic effects of Lower Carboni-
ferous and Upper Devonian rocks.
180
(lOOooE, qoooW?
M
(a) Gravity Coherence Map
(2dOOD*,qooou)
(b) Magnetic Coherence Map
o to 2.0 W -Jl
S c a l e
Fig. 6.6 Bandpass filtered maps of coherence region B
181
The middle and southern parts of the map do not show
distinctive features. These areas may not be correlated
significantly as shown in Similarity map in Fig. (6.3(a)).
c. Region C
The incoherent gravity and magnetic maps of Region C
of the coherence plot were obtained by applying the filtering mentioned
above. Fig. 6.7(a) and (b) show the results of bandpass filtered
gravity and magnetic maps, respectively.
A number of intermediate and short wavelength highs and lows
trending generally N-S are apparent. In Fig. 6.6(a) the alternating
low and high N-S trends through the map area may indicate
the subsurface undulation of the granite and sedimentary rocks
e.g. local thickening of the superficial deposits, probably local
sub-regional tectonic faulting contributes somehow to those fluctuations
and some discontinuities in the area.
On the other hand, the magnetic map of incoherence Region C in
Fig. 6.5(b) shows local small scale anomalies apparently different
from those of gravity and trending largely N-S in the north of the
map where a number of basic volcanic rocks are intruded into the
Devonian sediments.
6.4 Discussion
Analysis of various similarity measures shows general inter-
associations between map variables.
The correlation coefficients provide a measure of overall
similarity in a simple number which might be geologically informative.
They may also be of further value in multivariate data analysis.
182
(a) Gravity Coherence Map
(b) Magnetic Coherence Map P • . ^o Hm
S c a l e Fig. 6.7 Bandpass filtered maps of incoherent region C
183
For example, the extremely high correlation coefficients, such
as between Bands 4 and 5 or Bands 6 and 7, may be redundant in the
multivariate data analysis, so that one of the two variables may be
reasonably removed in the multivariate data analysis, in order
to reduce the number of data sets for efficient and effective
computation. Further consideration can be given to those variables
which are not correlated with any other variables at all, since these
variables may occur completely independently from others.
This aspect will be further discussed in Chapter 7.
The similarity map clearly shows spatial correspondence
between variables, so that associated mineralized zones or
any geologically significant aspects such as structural information or
geological provinces, could be analysed by interpreting those maps
with original data.
Coherence analysis between the' gravity and a pseudogravity
transform of the magnetic data is very useful to analyse spectral
correlations between them. Furthermore, bandpass filters designed
from the coherence map for particular spectral regions, such as high
coherence or incoherence have been effectively used to extract structural
features for those spectral regions, so that causative anomalies
for both fields have been more clearly identified. In particular,
stratigraphic features in the northern part of the study area may
be due to shallow basic intrusions occurring along the stratigraphic
beddings or zones of structural weakness developed during the
Amorican orogeny. Stratigraphic overlappings between Upper Devonian
and Lower Carboniferous sediments may have contributed to these
effects to a lesser extent due to their differences in susceptibility.
184
Many known and unknown linear features could be delineated
by bandpass filtering designed from the coherence analysis between the
gravity and pseudogravity transform of the magnetic data in the
study area because they can be emphasized by the application of
filtering for particular spectral region.
185
CHAPTER SEVEN
CLASSIFICATION AND IDENTIFICATION OF MULTIVARIATE DATA
7.1 INTRODUCTION
Four types of pattern recognition techniques have been
used in order to examine their applicability to various combinations
of multivariate data sets with the purpose of defining geological
provinces, or areas of possible mineralization.
The methods include Factor Analysis (FA), Cluster Analysis,
empirical discriminant analysis and characteristic analysis. As
already mentioned in Chapter 1, the first two methods require no
pre-determined or established training set for the analysis and they
attempt 'natural' grouping of the multivariate data sets using
certain similarity or dissimilarity criteria, (i.e. unsupervised
learning). On the other hand, the last two types belong to supervised
learning which requires certain prior known training sets in order to
classify remaining unknown samples. The procedures of each classi-
fication method, together with reviews of references, will be
described rn the following sections.
Parametric statistical analysis assumes that the frequency
distributions of multivariate data are normal, otherwise results
will be less confident. Thus, the data were transformed to near-
normality by using the power transform technique (Howarth and Earle,
1979) or an arc sine method where it is appropriate, and comparison
was made between the results from untransformed and transformed data
sets.
Details on the transformation techniques applied will be
described in the next section.
186
In addition, one of the most
data analysis - the selection of the
pattern recognition analysis is also
difficult tasks in multivariate
best sets of variables for
described.
7.2 Transformation of Data
Parametric statistical analysis assumes that the data is
normally distributed. In a real situation most of the data are
skewed to some extent, so that they do not conform to the normal
distribution.
As shown in Fig. 7.1(a) to (t), histograms of various
original data sets clearly show some degree of skewness, and particular
geochemical elements such as As, Cu and Pb are highly positively
skewed. Many transformation methods have been proposed for reducing
this skewness.
Various transformation techniques are described by Hoyle (1973)
Mancey and Howarth (1980) have shown that the power transform
suggested by Box and Cox is one of the most effective techniques for
de-skewing data. In this study, the power transform was applied
to all the data sets except gravity, in which an arc sine method
was applied for transformation of the data.
The generalized power transform is:
z = (xA-l)/A , X K 0 1 x > 0 (7.1)
In A , A = 0 J
where z is the set of transformed data, x the set of original data
and A the power coefficient.
Howarth and Earle (1979) have implemented various optimization
criteria in the computer program for computation of A in order to
187
either reduce the skewness to zero, or jointly reduce the skewness
to zero and kurtosis to three. Mancey and Howarth (1980) have also
shown its applicability to an even larger data set- by making a
subset of the data for computation of X. Mancey (1980) has also
discussed in detail the power transformation method.
In this study, a subset of data for estimation of X has
been made by taking values at every second column in every second row.
The mean X of computational results obtained using three
kinds of optimization technique (Dunlap and Duffy, Skewness/Kurtosis 2/1
and maximum likelihood schemes) are shown in Table 7.1
Table 7.1 Mean X values estimated using the Dunlap and Duffy,skewness/kurtosis 2/1, and maximum likelihood schemes
Variables mean X Variables mean X
gravity 1 , .2301 BR 7/6 9, .5013 magnetic 0, .2126 As .2543 Band 4 .3003 Cu .3677 Band 5 0. .5933 Ga .0687 Band 6 7. .1788 Li -1 , .077 Band 7 4, .8857 Ni .8893 BR 5/4 7. .4403 Pb . 1445 BR 6/4 5. .3950 Sn 0. .0611 BR 7/4 3. .8829 Zn 0. .0176 BR 6/5 4. .6374 Gravity SVD 1. .0482 BR 7/5 3. ,3534 Magnetic SVD 1 . . 1906
The transformation of the data sets have been performed using
mean X values obtained from three different optimization techniques,
mentioned above. This may be most reasonable method (Howarth, perc.
comm.) and also the X values calculated from the three methods are
usually close to each other.
188
Histograms of the power transformed data except for the gravity
data are shown in Fig. 7.2(b) to (t). Generally the X transform has
regularized the data. Particularly the data having approximately a
unimodal distribution have been improved significantly (which means
that even in the bimodal, the population of the anomalous pattern
is relatively small compared to the background population - these
are the most common cases in geochemical elements).
This could be beneficial in parametric statistical analysis.
The Landsat MSS bands and their ratios (Fig. 7.2(c) to (1)) have
been reasonably de-skewed by the power transform. However, though
in all cases the overall asymmetries have been de-skewed clearly,
certain data have not been much improved in terms of normality compared
to the original data.
This phenomenon is particularly significant with the
geophysical data including gravity and magnetic which have patterns
of strong bimodal distribution. Some geochemical elements (eg, Ni)
are also not much improved. These data show strong bimodal distribution
patterns.
It appears to the writer that in the case of univariate data
the power transform is most effective, while in the case of typical
bimodal or polymodal data sets it improves its overall skewness
but the effectiveness is decreased with the degree of polymodal
distribution.
Thus, the arc sine method has been used for the gravity of
typical bimodal distribution as follows.
-1 (X.-X . Z. = sin l min (7.3) i (X -X . max min
where X . and X m m i max are the minimum and maximum of sample values,
respectively.
u - n V n - i n . * -m.m ti.m nt.ii n..n m..i .^TS'TT^^riS^^R^^^^^^^'Ei:!. ij.'.T rt.it
a ) GRAVITY b ) MAGNETIC c ) MSS BAND 4
11.11 ll.ti 11.11 m.n m.m <!.•• 41..i h.m m!|< 71.11 m.'ll im.'m uitu tiltn d ) MSS BRNO 5 ) mss BAND 5
n . n <i.i> it .M
f ) MSS BAND 7
: . n I.M i. i i i . n i . i i
g ) M S S 3R5/4 • ••! •••< l-ll l.m 1.11 i.m i.m <.|l c.x l.ll l.ll h) MSS 3R5/4 i) MSS 3R7/4
Li :.»• i.ii t.ii i.it i.,, i.ii i. ii :.i« .i.n o.ii i.»I i.m i.ii <.io i.m i.ii i.II
j ) m5s br6/5 k) mss 3R7/5 1 )mSS BR7/6
Fig.7.1 Histograms of the filtered data.
190
Fig.7.1 continued.
to ii.m m.m o.'n 57T77 m ) 0 S ELEMENT
P ) LI ELEMENT
s) SN ELEMENT
11.11 11.11 im.m im.o iii.n in.n m.n nt.it iiu.h UJ i.m ii.ii to.ti ti.it ii.m <•.:<
N ) CU ELEMENT O ) GFI ELRMRN"
».i ' I.ti n.tt tt.tr M.n ti.il It.I I .»7i* iiJti tiTti u.'m H. 'n uJL • i.ii iii.ii -...11 im.ii itl.m <|,.l« in .01 111.11
q ) N f F ELEMENT r ) pR ELEMENT
-tn.it iii.m kii.ii mii.ii imi.m imi.m m i l .1,1.11 ,,:„ in.m mi.hut.m 1,77m t )ZN ELEMENT
191
v = — r ^ — * —i i r- - - i 1 •f.tt - I . * H . ' l -t.tt ••»• I.M l.tl t.M l.lt I H t.M I I . . I I I .M lf.lt f.Of |.M f. l l I . I t » . » 1.11 ?•>
a) GRRV1TY AS b)MRGNETJC c) *SS BOND 4
i.t? t.tt «.«» it.M i ? .u m.m it.K n H i i i i i i i i u i i m i i M n t m i n i M i i u i i i i i f n M i m i m . i l c i : i V i n » i f i i M i i » i M i n i t < i i i n r n n i i i t » i i i t i i . . i « . i . i
d ) MS5 SKND 5 e ) ^SS BOND 6 f ) » S S 3RND 7
I.M .... .... .... I . n „ . „ „..c „.i. „ . , , , „ . „ .„. ' „ V , . n i , . „.',, , . ' „ „;..
g ) M S S BR5/4- h ) MS5 BR6/4 ^ ^ M<55 bri/4
? ' : . ' ! « . « f. l . l l ..t.lt I I I . I I M l . I I IM I . l l i"..U 11.11 .11.11 111.11 11 f I .M 1.11.1. 1111..I t - l l t.ft t.M I . . I t.ll 0.1
j ) HS5 5R6/5 k)MSS KR7/5 1 ) MSS HR7/6
Fig 7 2 Histograms of ^transformed data except for the gravity data(a) in which case the arc sine transform was applied.
192
Fig.7.2 continued,
•Vii i.n i.n I .n t.41 I.u i.n *.«« i.k i.»» i.m in »." ».•• " :.»i '•,t *•** m ) AS ELEMENT
:••>« i.ii ».»« i p ) L ! E L E M E N T
S ) S N E L E M E N T
N ) CU ELEMENT O ) G A E L E M E N T
!M mi Mt c7»» '>••» <••>* M . i i t i . t * •».«; •»»•»»
UkL wl >:•< t.'n i-ii Tm
Q ) N I E L E M E N T r) P B E L E M E N T
JiiL I M it"
LU '„ .!„ .!„ .!.. •.«•r.. T ) Z N E L E M E N T
193
From the characteristic of the sine curve it is clear that the
middle part of the data is compressed while both ends of the
distribution are stretched, so that typical bimodal distribution
could be transformed near to normality as shown in Fig. 7.2(a).
As Prelat (1977) has noted, this transform also prevents the variance
from being a function of the mean as occurs in a binomial distribution.
7.3 Selectioft of Variables in Multivariate Data Analysis
Selection of an appropriate group of variables is one of the
most difficult tasks in multivariate data analysis in order to
effectively analyse the data for geological exploration.
If the number of samples is kept small, the collection may not
be representative of the truth, and if the number of variables is
small, the variation due to some of the geological processes might not
be represented. Therefore, statistical and mathematical approaches
that require the use of automated procedures on computers should be
applied to a large set of samples analysed for a large number of
variables.
However, more data will not always guarantee better results
and we also have to face the problem of computer capacity to
accommodate such a large number of variables from large samples, so
that an optimum number of variables should be chosen in order to be
efficient in computation and effectively solve the geological
problems.
Castillo-Muhoz (1973) has shown in the study of the overall
recognition success rate for the testing set with a linear discriminant
fucntion that the maximum recognition success rate for samples of
known type presented to the classifier as 'unknown1 (the testing set)
194
is attained (for his data) with particular combinations of either 6, 7,
or 8 variables. Thus, the use of a certain limited number of variables
can be more successful in its analysed results and efficient
in computation than a larger number of variables.
There have been some propositions for the selection of variables
as for example in Schultz and Goggans (1961) and Mucciardi et al.
(1971) etc.
In this study, two methods were used in the selection: trial by
intuition and experience, and principal component analysis.
The trial method may be entirely user-dependent although
the selection should be performed based on geological reasoning.
Knowledge of the characteristics of each variable and geological
information are essential to be effective in the selection. Analysis
of inter-relationships between variables such as correlation coefficients
may also provide information in order to reduce the number of variables
to a smaller size as described in Section 6.3.
Principal component analysis is a well-known procedure in
reducing the dimension to a smaller size. As will be described in the
following sections, this is a linear transformation of variables
which are uncorrelated with each other. Each principal component
score calculated by transformation, would constitute a certain
proportion of the total variance (represented in its eigenvalue),
so that those scores for which variance becomes an insignificant
proportion of the total variance, could be removed from the further
analysis with minimum loss of information since these insignificant
scores probably consist of noise contributions as noted by Mancey (1980)
and Siegal (1980), etc. Thus, the reduced numbers of component
scores chosen can be used further in the cluster analysis or any dis-
criminant analysis. In addition, if any training set is well
established in the study area, then, a method called 'BAKWRD'
195
developed by Howarth (1973a) would be very effective and efficient
in the selection of variables. This is a sequential backward method
in order to find the best set of variables for effective discrimination.
It is suboptional in the sense that the overall best combination out
of all possible combinations may not be found.
As noted by Howarth (1973a) exhaustive examination of all
possible combinations is not practicable for more than about 7
variables. In most cases, setting-up an appropriate training set itself
is difficult in practice, so that this method has not been used in the
selection scheme of variable sets in this study.
In areas of known geology and mineralization, trial by
ISODATA (a cluster analysis program which will be described later
in this Chapter) could be very effective in the selection. As mentioned
above, the practical use of this method for thorough examination is
limited by the number of possible combinations of variables
{m(m-l)/2} where m is the total number of variables to be analysed.
Where any prior information is not available, the most
reasonable way of selection may be to apply principal components
analysis in order to reduce to a smaller manageable size of scores
which may contain most of the information in the multivariate data.
There appear to be no objective methods of determining what
combinations of the data will give the best result in relation to
geology or mineralization. Thus, the complication of variable selection
schemes leads the potential user to an individual optimum method
either by intuition or a statistical method.
Pattern recognition techniques, therefore, have been used in
this study to cluster three different kinds of untransformed and
transformed data chosen by trial using information available (such as:
correlation coefficients; examination of eigenvector matrix; and
196
comparison of the results from ISODATA analysis to the known geological
information) and two untransformed and transformed data sets
derived from the principal components analysis.
The three sets of data are as follows:
1. 6 Landsat MSS data set: B4, B5, B7, R5/4, R6/5 and R7/6.
2. 8 geochemical element set: As, Cu, Ga, Li, Ni, Pb, Sn and
Zn.
3. 9 mixed variable set: Gravity, Magnetic, B5, R5/4, R7/6, Cu,
Ni, Sn and Zn.
The two data sets computed from the PCA are the first 8 PCA scores
derived from the following 16 variables: Gravity, Magnetic, B4, B5,
B7, R5/4, R6/5, R7/6, As, Cu, Ga, Li, Ni, Pb, Sn and Zn.
PCA is calculated so that the first component scores have the
largest variance of the total, the second component scores have the
second largest variance of the total, and so on. Thus, the first few
component scores contain most of the information of the multivariate
data set, while the component scores towards the end are most likely to
contain the noise contribution of the data.
Therefore, by choosing the first few component scores and
ignoring the rest of the component scores, we represent all information
in it effectively for further easier manipulation of the data.
Joreskog et al. (1976) have noted by an empirical test that the Scree
Test proposed by Cattell (1966) has been found to be the most useful.
In this Scree Test, if the magnitude of the eigenvalues are
plotted against the successive component number, there is a point on the
slope of this plot where an exponential decay portion in the beginning
and the consistent linear portion of the later components come together.
This point divides the exponential part in which the major information
is contained (and thus these scores are to be retained for further
197
20
<u 10 3 .—i CO > C Q) Ml •H W
\ * \ Exponential \ portion
\ . Linear portion
V — .
principal components number 11 12 3 16
20
s? w (1) 3 i CO > C 0) M •H W
10.
\ ^ Exponential \ portion
\ Linear portion
Figure 7.3:
I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 principal components number
Cattell's Scree Test for determining the correct number of principal components calculated from 16 variables: Gravity, Magnetics, B4, B5, B7, R5/4, R6/5, R7/6, As, Cu, Ga, Li, Ni, Pb, Sn and Zn.
198
analysis) and the latter, linear, part that contains noise which can
be ignored in the further analysis. The 8 PCA scores chosen are
based on the criteria as shown in fig. 7.3.
As noted by Davis (1973), since the discriminant analysis is so
clo sely related to multiple regression, most of the procedures
for selecting the most effective set of predictors can also be used
to find the most effective set of discriminators. In this sense, the
'Ridge regression' technique presented by Howarth in the recent
Departmental Seminar (November 1982)seems to be very effective since
the aim of this technique is to find best predictor elements which
are stable and have high discriminating power. However, this technique
was not attempted, in this study, because of shortage of time.
Mathematical procedures for ridge regression are described by Jones
(1972) and Turner (1980).
7.4 Unsupervised Classification
7.4.1 Factor analysis
(a) Factor analysis procedures
Usually the multivariate data show intercorrelation to some
extent between variables so that the axes of the probability density
function are not statistically orthogonal.
Factor analysis (FA) or principal components analysis (PCA)
is a multivariate statistical technique that generates new variables whose
probability density function have orthogonal axes (i.e. uncorrelated
variables). Although the two techniques have much in common and
are often referred to in the literature as the same thing, there are
fundamental differences between them as described by Howarth
and Sinding-La'rsen (Howarth, 1983, Chapter 6). In PCA each
component is determined so as to maximize the common
199
variance (C ) that is common to all the variables,whereas in FA a
given number of 'factors' (less than the number of variables) are defined
so as to account maximally for the common intercorrelation between
the variables. PCA is thus variance-oriented whereas FA is
correlation-oriented.
The mathematical bases of the PCA and FA are described in many
references,e.g. Cooley and Lohnes (1971), Davis (1973) and Chien (1978),
etc. Comprehensive descriptions of the two methods are also given by
Howarth (1983).
When we are considering heterogeneous data sets prior to any
actual computation, we have to normalize the data set by standardization
in order to give each variable equal weighting. The standardization
procedure is given in Equation (6.3) for a single variable. In
PCA, the correlations between variables which are represented by the
covariance matrix have to be computed. • When the correlation
coefficients R. . are estimated from the sample data, the computational ij formula for the case of m variables is given in Equation (6.1) of
Section 6.2.
To generate independent variables when R ^ is not zero, one must
find a transform that transforms matrix R to R' so that for all i \ j,
R L j = 0. To do this, one must compute the eigenvalues of R. The
eigenvalues are actually the variances for each factor of the transformed
scores associated with R' and the eigenvalues are the solutions
to the characteristic polynomial of R. The characteristic equation
of R is a polynomial of degree m obtained by expanding the determinant
200
C =
Rn~xi R 21
31
ml
R 12
R22" X2
"32
m2
13
R23
R33~ X3
1m 2m 3m
R - • • • R -X mJ mm m
= 0
(7.4)
Here the eigenvalues X^ are the solutions to the equations. With
the known eigenvalues, a column matrix of the eigenvectors V can be
determined as follows:
V RV = R1 (7.5)
where V is the transpose of V and R1 is the matrix having the
eigenvalues (X.,X.,X0,...,X ) as diagonal elements. I 2 J m
The matrix formed from the column vectors (normalized to
unit length) defines the linear transformation that is applied to
the m-variables to produce the principal component scores.
The eigenvalues are ranked in descending order^, so that the
first transformed map will have the largest variance, and the next
will have the second largest variance and so on.
Some of the components scores may be contributing little to
the overall variance, and fewer principal components, say P variables,
than m may adequately represent the data. The P derived variables
that may retain as much of the variance of the original data as
possible are the first principal components.
In FA, the correlations among the m variables are assumed to be
accountable by a model in which the variance for any variable is
201
distributed between a number (p) of common factors and anumber
of 'unique factors', the remaining being attributed to random error
(Howarth, 1983). Thus,the factor model X^ may be expressed as
P X. = Z z. f + e. (7.6) l . lr r i r= 1
t h where fris the r common factor, p the specified number of factors
and z. the factor loading of X. on factor f and e. the random lr l r i
variation unique to the factor associated with X^.
In psychometrics, most problems are concerned with identifying
the common factors. However, in geological factor study, factor scores may
be more important because it retains geometrical information with the
significance of analysed results.
The variance for each variable has been reduced to unity by
normalisation. Therefore, the sum of squares of the factor loading P 2 of the common factors Z (Jl) provides a measure of the degree of
i= 1
fit of the samples to the common factors. This is referred to as
'communalities'.
The magnitude of the communalities is dependent upon the
number of factors that are retained. For this reason, care must be
taken in choosing the number of factors.
In the true factor analysis model, it is assumed that the
number of common factors is related to a number of known underlying
causative influences. The number of these is postulated beforehand
and used to define the factor solution. In geological work this is
not usually possible and the PCA solution, truncated to K(<m) components
is usually taken as the initial solution. This may be 'improved'
upon, in the sense that a more easily interpretable solution is obtained,
by subsequent (orthogonal or oblique) rotation to a final set of
202
variables, generally referred to in the geological literature as 'factors'.
The Scree test for analysis of information content for the
components has been described in the previous section. However, it is
not practically possible to display effectively all information for
analysis if the number of components containing information is larger.
Some workers recommend retaining all factors which have eigen-
values greater than one. Another approach may be to use only two or
three factors, because this is the maximum number that can conveniently
be displayed as scatter diagrams. Particularly when colour-composite
plotting techniques are used, choosing three factor scores is the most
effective in order to represent the resultant data by means of colour
additive or colour subtracting views as noted byHowarthet al ( 1976).
Any number larger than this increases the dimensionality of the
problem.
To ease the interpretation problems of the factor scores, a
further rotation of the factor can be applied. In this study, the
Varimax scheme of Kaiser has been used, taking the first three
principal components as the starting point.
The Varimax criterion involves maximization of the variance
of the loadings on the factors by obliquely rotating the principal
components axes but still preserving orthonormality between components
(Trochimczyk and Chayes, 1978) and thereby adjust the factor loadings
so that they are either near ±1 or near zero. Consequently for each
factor there will be a few significantly high loadings and many
insignificant loadings and the interpretation in terms of the original
variables is more easily apparent.
Factor analysis based on PCA in this way has certain important
properties which are essential in multivariate data analysis as noted
by Howarth (Pers. Comm.). First, it is repeatable ; second, data
203
compression can be achieved effectively; and third, interpretation of the
resulting map may be simplified.
In this study, the FACTOR program described in Davis (1973,
p519) has been significantly modified to apply it to the regularly
gridded multivariate data sets. The analysis has been made on a line
by line basis, keeping individual data files separate in order to
facilitate easy manipulation - this means that the multi-data files
individually recorded on a magnetic tape can be copied on the disc
when the job is run.' A line-by-line basis analysis could reduce the
dimension of the multivariate data set significantly so that the Central
Memory space is considerably reduced to a manageable size for the job.
The modified program is listed on Microfiche 3.
One restriction is that since the maximum number of files
allowed in a program run is limited by 50, the individual data files
may be limited to about 15 files in a run, so that for more than 15
individual data files the program might have to be modified.
204
(b) Applications of the factor analysis
PCA was first performed on the normalized variance-covariance
matrices (i.e.: correlation matrix) of a total of 8 data sets from
untransformed and transformed 6 Landsat MSS, 10 Landsat MSS, 8
geochemical elements and 9-mixed variables, the statistical results
being given in Table 7.2 to 7.5. Standardized data were reset to
±3.291, if the standardized data were larger than 3.291 in absolute
value, in order to avoid overweighting by extreme data values.
In all cases, the first threfe principal components were taken
for further Varimax rotation to yield the FA scores. The reason
for this is that not only do the first three components represent most
of the total variation in this study, but it is also most convenient 2 . . . .
to apply to the 'I S1 colour additive viewer at the Remote Sensing Unit
in the College to produce a colour-composite picture. However,
there is inevitably some loss of information judging from the Scree
Test criterion as shown in Fig. 7.4, in using only 3 components.
Statistical results, given in Table 7.2 to 7.5, show percentage
contributions of each component to the total variance and their
cumulative percentage.
The three factor scores retained from FA were stretched from 0 to 255 and then subjected to histogram normalization for effective
2 viewing on the I S viewer, as described in Section 4.3.1. The resultant
maps are shown in Fig. 7.3 to 7.10. In the maps, white and/or bright
colours represent high values and dark colours represent low values of
untransformed (U)
30
20
10
transformed (T)
1 2 3 4 5 6 7 1 2 3 4 5 6 7 principal component numbers
(c) 8-geochemical element sets
3
2C
1G (U)
30
20
10 (T)
50
40
30
20
10
60
50
40
30
20
(U)
50
40
30
20
101
(T)
> \ 2 3 4 5 6 2 3 4 5 6 (a) 6 Landsat MSS variable sets
50
40
30
20
(T)
3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10/1 2 3 4 5 6 7 (b)loLandsat MSS variable sets
N) o Cn
8 9 10 1 2 3 4 5 6 7 8 9 1 (d) 9-mixed variable sets
Fig. 7.4 Cattell's Scree test for determining the correct number of principal components of four different variable sets: 6MSS data, 10 MSS data, 8-geochemical data and 9-mixed variable data sets
206
the component. Detailed discussions on the resulting map will be
described in the following sections.
(b.l) Six Landsat MSS data
Fig. 7.5(a) to (d) and 7.6(a) to (d) are the results of FA for
the untransformed and transformed 6 Landsat data sets, respectively.
From the statistical results in Table 7.2, the first three
factor scores of the FA constitute about 95% of the total variance
in both untransformed and transformed data sets and thus the first
three factor scores represent effectively most of the total variance,
though there still occurs some loss of information according to
the Scree Test criterion as shown in Fig. 7.4(a).
(b.1.1) Untransformed data
The first factor score is very highly correlated with brightness"
in the visible bands 4 and 5, and band ratio R7/6, and to a lesser
extent, with band 7. This factor is responsible for 51.1% of the total
variance. It is apparent that this factor is a measure of brightness
in the visible bands. High negative correlation with R7/6 may indicate
that this ratio may reflect inversely similar features to those of bands
4 and 5 as shown in Fig. 4.18(c) in chapter 4 and further the
similarity may be seen from high negative correlation coefficients
(-.7493 and -.7310) with bands 4 and 5, respectively.
Thus bright areas may represent high Moorlands, or dry or
township areas while dark zones may indicate those areas with wet or
thick vegetation. Sanddunes in the northwest and southeast coast lines
show up as bright areas.
"'brightness1 referred in this section means the intensity of the value.
207
The second factor score appears to be correlated with
brightness in the band 7 and band ratio R6/5, and to a lesser extent,
with band 4 and 5. Considering high correlations between R6/5 and two
infrared bands 6 and 7 (0.9390 and 0.9656, respectively), this
factor is a measure of brightness in the infrared bands.
Thus, bright areas may correspond to areas of green vegetation,
while dark zones may represent areas of water or dry vegetation.
Township areas and sanddunes are also dark in the image.
The third factor score is very highly 'correlated with band
ratio R5/4 and to a lesser extent, with band 5, and thus the factor score
map may be comparable to the R5/4 picture as shown in Fig. 4.18(a) in
Chapter 4. Therefore, high values which produce bright areas of this factor
are represented by dry vegetation or contaminated waters while dark
spots indicate areas of bare soils (or kaolin), green vegetation or
uncontaminated water.
The composite picture in Fig. 7.5(d) shows effectively the
combination of the three factors and is responsible for 95.9%
of the total variance. The general features are more detailed than the
false-colour composite in Fig. 4.17 in Chapter 4.
The Bodmin Moor granite area appears to be blue except for
two spots in the northwest and middle south which are greenish yellow.
St. Austell granite area also appears to be greenish-yellow. This
feature represents areas of open-pit mining for kaolin in this region.
Township areas and sanddunes along the southeast and northwest coastlines,
and dry and sparsely vegetated lowlands, appear to be green. Red tints
represent thick vegetation.. Water or water-logged areas appear to be
black and blue.
Reddish orange to magenta occur around the granite margin
where most of the mineralization in this region is located. However,
direct relationships between those two features are not so distinctive
208
( c ) F a c t o r 2 3 0 4 - , % - ( d ) C o m p o s i t e 9 6 . 0 %
F i g . 7 . 5 : FA o f t h e u n t r a n s f o r m e d 6 - M S S v a r i a b l e s e t : MSS b a n d s 4 , 5 , 7 , R 5 / 4 , R 6 / 5 a n d R 7 / 6
6 t 'Q ao Km
sea 1 e
209
( b ) F a c t o r 2 3 2 . 9 %
( c ) F a c t o r 3 1 4 . 9 % ( d ) C o m p o s i t e 9 4 . 8 %
3.0 K m
Sea 1 e
F i g u r e 7 . 6 FA o f t h e t r a n s f o r m e d 6 MSS v a r i a b l e s e t : MSS b a n d s 4 , 5 , 7 , R 5 / 4 , R 6 / 5 a n d R 7 / 6 .
210
Table 7.2 Statistical results of Factor Analysis for 6-landsat MSS data sets:B4,B5,B7,R5/4,R6/5 and R7/6.
1. Untrasformed data, a) Eigenvalues with their associated percentages.
Factor Eigenvalue Percentage variance
Cumulative per-centage variance
1 3.0645 51.07 51.07 2 1.8261 30.45 81.51 3 0.8674 14.46 95.97 4 0.2349 3.92 99.88 5 0.0045 0.07 99.95 6 0.0026 0.04 100
b) Rotated factor matrix ^factor variancte^ 1 2 3
B4 0.9532 0.2342 0.0196 B5 0.9207 0.2279 0.2596 B7 0.2259 0.9596 0.1606 R5/4 0.1220 0.1423 0.9817 R6/5 -.0425 0.9842 0.0668 R7/6 -.9057 0.1855 -.0282
2. Transformed data, a) Eigenvalues with their associated percentages.
Factor Eigenvalue Percentage variance
Cumulative per-centage variance
1 2.8181 46.97, 46.97 2 1.9746 32.91 70.88 3 0.8930 14.88 94.76 4 0.2616 4.36 99.12 5 0.0493 0.82 99.94 6 0.0034 0.06 100
b) Rotated factor matrix *^\factor variabTe^
1 2 3
B4 0.9730 0.1096 0.0497 B5 0.9386 0.0846 0.2806 B7 0.0993 0.9819 0.0317 R5/4 0.1693 -.0312 0.9838 R6/5 -.1265 0.9670 -.0689 R7/6 -.8754 0.2455 -.0523
211
in the study area.
(b.1.2) Transformed data
Transformed data in Fig. 7.6(a) to (d) are almost the same
as those of the untransformed data in Fig. 7.5(a) to (d),
respectively. The first three factors represent 94.9% of the total
variance.
It can be expected from the fact that though the transformation
of the data changes values completely different from the original
ones, the frequency distributions of the transformed data are very
similar to the untransformed data (see Section 7.2.2, Fig. 7.1 and
7.2).
Comparison of the statistical results of eigenvalues and rotated
factor matrices in Table 7.2(a) and (b> clearly indicates the
similarities between the two analysis.
In order to illustrate how effectively the chosen six variable
sets represent the multivariate Landsat MSS data, FA has been applied
to the total 10 variables including four MSS bands and the whole combination
of 6 ratio data from the four bands.
The results of the computed factor scores for the untransformed
and transformed data sets are given in Fig. 7.7(a) to (d) and
Fig. 7.8(a) to (d), and the statistical results are summarized in
Table 7.3.
The resultant features are strikingly similar to each other.
The first three factors scores represent 97.3 and 95.7% of
the total variances for the untransformed and transformed data sets,
respectively.
Therefore, the chosen 6 variable sets represent the total
Landsat MSS data effectively and they are thus computationally
212
( a ) F a c t o r 1 6 1 . 9 % ( b ) F a c t o r 2 2 6 . 5 %
( c ) F a c t o r 3 ( d ) C o m p o s i t e 9 7 . 3 %
10 _i i — i 1. ao Km
scale
F i g u r e 7 . 7 : FA o f t h e u n t r a n s f o r m e d 1 0 - M S S v a r i a b l e s e t : MSS b a n d s 4 , 5 , 6 , 7 , R 5 / 4 , R 6 / 4 , R 7 / 4 , R 6 / 5 , R 7 / 5 a n d R 7 / 6
213
( c ) F a c t o r 3 9 . 1 % ( d ) C o m p o s i t e 9 3 . 7 %
( a ) F a c t o r 1 3 6 . 7 % ( b ) F a c t o r 2 2 9 . 9 %
F i g u r e 7 . 8 FA o f t h e t r a n s f o r m e d 1 0 - M S S v a r i a b l e s e t : M S S b a n d s 4 , 3 , 6 , 7 , R 5 / 4 , R 6 / 4 , R 7 / 4 , R 6 / 5 , R 7 / 5 a n d R 7 / 6
o io io Km J 1 ! S i
Scale
214 Table 7.3 Statistical results of Factor Analysis for 10-Landsat MSS data sets:B4,B5,B6,B7,R5/4,6/4,7/4, 6/5,7/5,and 7/6.
1. Untransformed data. a) Eigenvalues with their associated percentages.
Factor Eigenvalue Percentage variance Cumulativ centage v 1 6.1899 61.90 61.90 2 2.6488 26.49 88.39 3 0.8903 8.90 97.29 4 0.2512 2.51 99.80 5 0.0138 0.14 99.94 6 0.0028 0.03 99.97 7 0.0020 0.02 99.99 8 0.0011 0.01 99.999 9 0.0001 0.00 99.999
10 0.0000 0.00 100
LrSSnce
b) Rotate4 factor matrix ^-nLactor varialDle--. 1 2 3
B4 B5 B6 B7 R5/4 R6/4 R7/4 R6/5 R7/5 R7/6
0.2050 0.2088 0.9195 0.9580 0.1782 0.9854 0.9897 0.9947 0.9965 0.1757
-.9602 -.9316 -.3640 -.2469 -.1404 -.0120 0.0011 0.0259 0.0291 0,8961
-.0032 0.2372 0.1200 0.1215 0.9735 0.1442 0.1442 0.0304 0.0497 -.0182
Transformed data. Eigenvalues with their associated percentages. Factor Eigenvalue Percentag varianc e Cumulative per-e centage variance
1 2 3 4 5 6 7 8 9 10
5.67 2.99 0.91 0.28 0.12 0.01 0.01 0.00 0.00 0.00
56.74 29.87 9.08 2.85 1.20 0.11 0.10 0.03 0.01 0.00
56.74 86.61 95.69 98.54 99.74 99.85 99.95 99.98 99.99 100
Rotated factor matrix. --—doctor variaBT^- 1 2 3
B4 B5 B6 B7 R5/4 R6/4 R7/4 R6/5 R7/5 R7/6
0.0799 0.0623 0.8972 0.9717 -.0131 0.9812 0.9846 0.9788 0.9816 0.2358
-.9764 -.9470 -.3750 -.1272 -.1911 0.1302 0.1374 0.1156 0.1286 0.8648
0.0221 0.2547 0.0436 0.0068 0.9808 0.0308 0.0356 -.0813 -.0525 -.0444
215
efficient by reducing the dimension of multivariate data analysis
by removing those variables with high correlations as redundancy.
(b.2) Eight geochemical element sets
The results of FA for the untransformed and transformed eight
geochemical element sets are shown in Fig. 7.9(a) to (d) and
7.10(a) to (d), respectively, and statistical results are given in
Table 7.4.
The eigenvalue of each factor in the table shows its relative
magnitude to the total variance. This relative significance is
more clearly shown in the percentage representation of each variance
to the total. In the case of the untransformed data, the
contribution of the first three factor scores to the total variance
are 34.0%, 29.2%, and 11.3%, and thus the first three factor
scores represent 74% of the total variance.
On the other hand, in the case of the transformed data the
contribution of the first factor to the total variance increases to 38.7%,
but those of the second and third factors decrease slightly to
28.4% and 9.7% compared to those of the untransformed data.- Thus,
the overall contribution of the first three factors to the total
variance is an increase by 2.2% in case of the transformed data.
This could imply that the results of the transformed data set would
contain more information at least for the first three FA scores.
(b.2.1) Untransformed data
The characteristics of the first three factor scores which have
the largest portion of the total variance are as follows.
Factor 1 is very highly correlated with As, Cu and Sn and to a
lesser extent, with Zn and Pb. This is interpreted as old mining
regions in the study area. Thus, the factor may indicate Sn haloes
216
( c ) F a c t o r 3 1 1 . 3 % ( d ) C o m p o s i t e 7 4 . 6 %
F i g u r e 7 . 9 FA o f t h e u n t r a n s f o r m e d 8 - g e o c h e m i c a l e l e m e n t s e t : A s , C u , G a , L i , N i , P b , S n , a n d Z n
0 io 30 Km 1 I ! i 1 s^aie
217
Table 7.4 Statistical results of Factor Analysis for 8-geochemical element sets; As,Cu,Ga,Li,Ni,Pb,Sn and Zn.
1, Untransformed data. a) Eigenvalues with their associated percentages. Factor Eigenvalue Percentage Cumulative per-
variance centage variance 1 2.7227 34.03 34.03 2 2.3383 29.23 63.26 3 0.9047 11.31 74.57
0.7707 9.63 84.20 5 0.5333 6.67 90.87 6 0.3602 4.50 95:37 7 0.2343 2.93 98.30 8 0.1358 1.70 100
b) 'Rotated factor matrix. ^\£actor 1 2 3 variably As 0.8163 -.0365 0.3531 Cu 0.8481 -.0863 0.2879 Ga O.OI76 0.9422 0.0150 Li 0.0263 O.936O -.1834 Ni -.1804 -.4422 0.6039 Pb 0.2060 -.0436 0.7746 Sn O.6859 O.378O -.1705 Zn 0.3179 -.0178 0.7643
2. Transformed data, a) Eigenvalues with their associated percentages. Factor Eigenvalue Percentage Cumulative per-
variance centage variance 1 3.0957 38.70 38.70 2 2.2751 28.44 67.14 3 0.7753 9.69 76.83 4 O.6723 8.40 85.23 5 0.4423 5.53 90.76 6 0.3420 4.28 95.03 7 0.2467 3.08 93.12 8 0.1506 1.88 100
b) Rotated factor matrix. ^^actor 3 ^^actor 1 2 3 variabit^^ As 0.8008 0.2279 -.1359 Cu 0.8436 -.0848 -.1029 Ga -.0438 0.9578 -.0684 Li -.1126 0.8619 -.3701 Ni 0.5031 -.2010 0.6731 Pb 0.76 2 -.1096 0.0761 Sn 0.3263 0.2224 -.8167 Zn 0.8200 -.1893 0.0941
218
mainly associated with copper and arsenic most probably related
to mineralization in this region.
The brightest colour appears to be in the eastern flank
and southeast of the main granite. Relatively high concentrations
also occur near the Kithill granite, southern margin of the
Bodmin Moor granite and northeast of the St. Austell granite.
Most of metamorphic aureole zones appear to be dark, which
may be due to weakly negative association of Ni to component 1 and
general low association of other elements. Weak association of Pb
and Zn elements to the component resulted in dark colours for areas
associated only with Pb-Zn haloes.
Factor 2 shows its strongest association with Ga and Li and
to a lesser extent, with Sn positively but with Ni negatively. Thus,
this factor may reflect areas of granites and associated kaolization.
Prominent highs occur in the St. Austell and Bodmin Moor granite
and along its southern margins where drainage patterns flow down from
the granite inland to the surrounding sediments.
Factor 3 is mainly associated with Ni, Pb, and Zn, and to a
lesser extent, with As and Cu. Therefore, this factor may reflect
areas of Pb-Zn and Ni haloes in the study area, which are not necessarily
directly related to the local mineralization since contamination by old
mining activities, etc. might have occurred around the region.
The brightest areas are near the Kithill grnaite, northwest
and southeast of the main grtanite, and the eastern part of
the St. Austell granite, and to a lesser extent, along the metamorphic
aureoles of the Bodmin Moor granite. The granitic areas are typically
low in this component scores.
The composite picture in Fig. 7.9(d) representing 74.6%
of the total variance shows the combination of the three factors to
represent in an effective manner most of the information of the
219
data set.
Though the geological provinces are not clearly detailed, the main
granitic areas and different types and intensity of mineralized
zones are well defined. Upper and Middle Devonian sediments appear
to be the same, largely greenish colour. These rock units have been
differentiated from Carboniferous and Lower Devonian sediments which
mark the similar feature of dark colour.
Magenta and bluish colours outline the granite areas. Reddish
lemon colours in the eastern flank of the Bodmin Moor 'and north of the
St. Austell granite mark areas of tin mineralization associated with
copper and arsenic. Orange colours in the southeast of Bodmin Moor
and near the Kithill granites mark again areas of tin haloes associated
with most of the metallic sulphide elements but they are weaker than the
previous ones and sometimes contamination by drainage systems is
apparent. Magenta in the south of the main granite represents the tin
high associated with high As and Cu.
Bluish green in the south, southeast and northwest of the main
granite is mainly associated with Pb-Zn haloes probably related to the local
mineralization in this area. The widespread red colour in the east
of the St. Austell granite may be due to high Sn which may be attributable
to either contamination by drainage in this region or derived by local
sediments of shale rich in tin.
(b.2.2) Transformed data
Fig. 7.10(a) to (d) shows the characteristic features of the
first three factors and their composite obtained from the transformed
8-geochemical elements of As, Cu, Ga, Li, Ni, Pb, Sn and Zn. Compared
to the untransformed pictures, some distinct features are evident in the
transformed data though the general features are similar to each other.
220
( c ) F a c t o r 3 9 . 7 % ( d ) C o m p o s i t e 7 6 . 8 %
F i g u r e 7 . 1 0 FA o f t h e t r a n s f o r m e d 8 - g e o c h e m i c a 1 e l e m e n t s e t : A s , C u , G a , L i , N i , P b , Sn a n d Z n o io oo km j i i i i
S c a l e
221
Factor 1 is highly correlated with As, Cu, Pb and Zn, and to
a lesser extent, with Sn and Ni. This factor generally represents
patterns related to the local sulphide mineralization and
contamination by drainage or smelting. Thus, this factor is an
indication of geochemical haloes of sulphide elements around the
granites.
The brightest and thus the highest values appear in the south-
east of Bodmin Moor and around the Kithill granite. Slightly less
dominant values occur around the aureoles of the granites: east,
south, northwest and north of the Bodmin Moor granite and another
one northeast of the St. Austell granite.
Factor 2 is very similar to that of the untransformed data.
It is strongly correlated with Ga and Li and marginally correlated
with Sn, As and Ni, so that granitic features and their flanks are
dominant in the picture.
Factor 3 shows its strong association with Sn, and to a lesser
extent, Li in a negative manner and further with Ni in a positive
manner, so that Sn concentrations in and near the granites appear
to be dark while granitic aureoles where Ni enrichment occurs, are
bright in the picture.
The composite picture in Fig. 7.10(d) shows combined geological
features better compared to the untransformed data, while localization
of anomaly patterns are more diffuse than the untransformed data.
Carboniferous sediments have been clearly differentiated
from the lower Devonian rocks which have similar features to the
Carboniferous rock units in the untransformed data. It may be difficult
to say that the Upper Devonian rocks are markedly different from
the Middle Devonian but they show somewhat different features to the
transformed data.
222
Granitic areas are blue to black. The Bodmin Moor granite
has been delineated very well but some confusion still arises with
the St. Austell granite mainly due to heavy alteration by secondary
enrichment.
The anomalous bulge on the southern part of the Bodmin Moor
granite in both untransformed and transformed data sets may be due to
the secondary enrichment by drainage flowing down from the granite upland
to the lower sedimentary areas in this region.
Mineralized zones are a bit diffuse compared to the untransformed
data but they are regionally comparable with them.
The main geochemical haloes of sulphide elements are marked by
red to magenta, but Pb-Zn haloes in the northwest of the map appears
to be yellowish white. Areas of nickel enrichment are bright
green.
(b.3) Nine-mixed variable sets
Fig. 7.11(a) to (d) and 7.12(a) to (d) show the results of FA
for the untransformed and transformed 9-mixed variable sets of Gravity,
Magnetic, B5, R5/4, R7/6, Cu, Ni, Sn and Zn, respectively. Statistical
results are given in Table 7.5.
For the first three factors, the untransformed and transformed
data sets show 65 and 71% of the total variance, respectively, so
that the transformed data shows higher percentage of the total variance
(by 6%) th an the untransformed data. This again indicates that the
result obtained from the transformed data set is at least statistically
more significant than that from the untransformed data set as in the
analysis of 8-geochemical element sets. Detailed descriptions for
each factor follow in the next section.
223
(b.3.1) Untransformed data
The general features of the three main FA factors for the
untransformed data set in Fig. 7.11 are as follows.
Factor 1 is highly associated with Landsat MSS data, i.e.
B5, R5/4 in a negative manner and R7/6 in a positive manner, and to
a lesser extent with gravity and magnetics, so that Factor 1 could be
a measure of brightness of Landsat MSS data. Thus, though there are
some variations in detail, the general features are a reflection
of band 5 as shown in Fig. 4.17 in Chapter 4.
Water or water-logged areas and densely vegetated areas are
bright while dry and brown high Moorland, dry lower land with less
vegetation or township areas are dark blue. Sanddunes in the northwest
and southeast coast lines appear to be dark.
Factor 2 is mainly positively associated with Zn, Cu, Ni and, to
a lesser extent, with Sn positively but with magnetic data negatively, so
that this factor may define the geochemical haloes. Granitic areas appear
to be low in this factor. Negative correlation of the magnetic data in
this component may indicate that most of geochemical haloes in the
study area appear to be in areas of negative magnetic anomalies.
Factor 3 is mainly correlated with gravity and Ni, and to a
lesser extent, with magnetic in a positive manner but Sn in a negative
manner. Weak negative correlations occur also with Cu and B5.
Dark features mark the Cornubian batholith trend, while the
bright colour are mainly in the Lower Carboniferous and Upper Devonian
sedimentary areas. Thus bright colours may represent areas of Ni
concentration associated with high Bouguer gravity anomalies and
possibly high magnetic values.
The colour-composite of the three factors in Fig. 7.11(d),
representing 65% of the total information, show largely the regional
224
( c ) F a c t o r 3 1 2 . 6 % ( d ) C o m p o s i t e 6 5 . 2 %
F i g u r e 7 . 1 1 FA o f t h e u n t r a n s f o r m e d 9 - m i x e d v a r i a b l e s e t : G r a v i t y , M a g n e t i c , B 5 , R 5 / 4 , R 7 / 6 , C u , N i , S n a n d Z n
0 io 56 Km 1 I i S I S e a l e
225 Table 7*5 Statist ical results of Factor Analysis for 9-mixed. variable
sets: Gravity fMagnetic,B5»R5A»R7/6fCufNifSn and Zn.
1. Untransformed data. a) Eigenvalues-with their associated percentages.
Factor Eigenvalues Percentage variance
Cumulative per-centage variance
1 2.7527 30.58 30.58 2 1.9800 22.00 52.58 3 1.1361 12.62 65.21 4 O.947O 10.52 75.73 5 0.6755 7.51 83.24 6 0.5828 6.48 89.71 7 0.4274 ^.75 94.46 8 0.3133 3.48 97.9^ 9 0.1852 2.06 100
b) Rotated factor matrix. "^\factor 1 2 3
Gravity O.3532 -.0128 0.7757 Magnetic 0.2977 -.4246 0.4889 B5 -.8709 -.1915 -.2056 R5A -.6367 0.1845 -.0863 R7/6 0.7735 0.1945 0.0380 Cu 0.1000 0.6487 -.3101 Ni 0.1928 0.5184 0.6570 Sh 0.1019 O.2721 -.8021 Zn 0.0436 0.8458 O.O519
2. Transformed data.
a) Eigenvalues with their associated percentages.
Factor Eigenvalue Percentage variance
Cumulative per-centage variance
1 3.0404 33.78 33.78 2 2.1957 24.40 58.18 3 1.2101 13.^5 71.63 4 0.8240 9.16 80.78 5 0.6187 6.87 87.66 6 0.4456 4.95 92.61 7 O.3O8O 3.42 96.03 8 0.2084 2.32 98.34 9 0.1490 1.66 100
b) Rotated factor matrix. —factor variable—- 1 2 3 Gravity 0.7531 -.3181 -.28 56 Magnetic 0.7445 0.1949 -.1122 B5 -.1368 0.2509 0.8848 R5A -.2416 0.0195 0.5796 R7/6 0.0011 -.0952 -.8637 Cu -.2437 -.73^6 -.1990 Ni 0.4056 -.7667 -.1087 Sn -.9215 -.1267 O.O733 Zn -.1435 -.8823 -.0692
226
geological features.
The granitic areas are marked by dark and blue colours except
areas of contamination by drainage pattern and old mining. In the
northern part of the Bodmin Moor granite where a part of the granite
is marked by a greenish hue different from other granitic areas, may
be due to the secondary geological environment. This area largely
corresponds to a depression, so that geochemical elements can be
enriched due to a poor drainage system.
Most of the Upper Devonian appears to be similar to the
Carboniferous rock unit in the north.
The Middle Devonian may not be defined very well and is marked
largely by a greenish and dark orange. The Lower Devonian generally
appears to be dark due to the generally low content of most geochemical
elements used in the study. Blue features in or near the granites
may be related to the drainage patterns. Offshore areas are marked by
blue colours except coastal areas of contaminated streams drainage to
the sea, particularly in the southeast corner of the study area.
Mineralized zones are again indicated by varying degrees of
colour from red to yellow, and occur in the east and southern margin
of the Bodmin Moor, around the Kithill and St. Austell granite areas.
(b.3.2) Transformed data
The results of FA for the transformed 9-mixed data set are
shown in Fig. 7.12(a) to (d). Though the information in the three main
factors is equivalent to the untransformed data set, marked differences
are that factor 1 in the untransformed data corresponds to factor 3
in the transformed data set in a negative mode, factor 3 in the
untransformed data corresponds to factor 1 in the transformed data,
and factor 2 in both data sets are of the opposite signs.
227
PCA MXD9 (B R C : T)
0 1 0 2 0 Km 1 1 1 1 1 Sea I e
F i g u r e 7 . 1 2 FA o f t h e t r a n s f o r m e d 9 - m i x e d v a r i a b l e s e t : G r a v i t y , M a g n e t i c , B 5 , R 5 / 4 , R 7 / 6 , C u , N i , Sn a n d Z n
m e a n s t h e v a l u e s a r e r e v e r s e d .
228
Their information contents to the total variances are also different
(see Table 7.5). However, for easy comparison, the pictures of
each factor for both data sets have been made in the same fashion.
Regionally transformed pictures are similar to the corresponding
pictures in the untransformed data though the percentaged eigenvalues
significantly different compared to the total (for example, factor
1 in the untransformed data is 30.5%, while the corresponding factor 3
in the transformed data is 13.5% of the total variance). However,
there are marked differences in details between the two corresponding
pictures. Since quantitative comparison is not possible and the
regional aspects of both pictures are similar to each other, further
description will be omitted except for one point.
The geochemical haloes defined in the transformed data show again
a more diffuse pattern than the untransformed data, and the transformed
9-mixed data set for geological mapping is marginally better than the
untransformed 9-mixed data set in this study. This may be partly
due to histogram normalization applied for all FA scores in colour 2
processing using the 'I S' colour additive viewer.
However, in case of the 8-geochemical data sets, geological
mapping with the transformed data set is much better than the untrans-
formed data.
7.4.2 Cluster Analysis
(a) Cluster analysis procedures
Cluster analysis has been widely used by Numerical Taxonomists
as a classification tecnique and is increasingly employed in the field
of geological sciences, especially for geochemical applications
(Rhodes 1969, Obial 1970, Obial and James 1973, Castillo-Munoz 1973,
229
Crisp 1974, and Mancey 1980, etc).
This is a statistical treatment of data that is able to elaborate
on elemental and sample associations. The purpose of cluster analysis is
to divide the set of m-variables into subsets or clusters, those
objects in the same cluster being close or similar in some sense, and those
in different clusters being distant or dissimilar.
Cormack (1971) has defined three different types of clustering
procedure as
1. Hierarchical classification - the classes are them-selves grouped in a repetitive process at different levels to form a dendrogram.
2. Partitioning - the classes are mutually exclusive, thus forming the subset of the original data.
3. Clumping - the classes may overlap, the classes are regarded as different types of class.
Clustering is considered to be -partitioning the data set
into 'natural' and 'homogeneous' groups and finding the 'most
representative elements'. The 'natural' concept of clustering is that
there should be parts of the space in which points are very dense,
separated by parts with low density (Carmichael, 1968). Two most
important aspects must be taken into consideration in order to satisfy
this condition, namely, the choice of similarity measure and the way
of grouping of samples.
Several measures of similarity have been described in the
literature. They fall into two main categories: those using numerical
data such as the correlation coefficient, distance coefficient, or
cosine data coefficient; and those using non-numerical data coded into
two- or multi-scale forms, such as matching and association coefficients.
Details of computational procedures for these measures are given by
Sokal and Sneath (1963), Harbaugh and Merriam (1968), Fu (1976), etc.
230
The most widely used coefficients are the distance coefficients.
The commonest measure in the distance coefficients is the Euclidean
distance d.. which is expressed as
where X., and X., denote the Kth variable measured an object i and j, lk jk
respectively.
As expected, a low distance indicates the two objects are .
similar, whereas a large distance indicates dissimilarity.
The Euclidean distance measure has the property of giving
extra weight to outlying values of a variable. This is partly
overcome by scaling (Cormack, 1971). Several other different forms
of distance measure: are summarized by Cormack (1971) in his excellent
review on clustering methods.
In this study, the data for the analysis have beeen transformed
in order to give each variable equal weighting by standardization
and absolute values larger than 3.291 reset to 3.291 in order to
avoid extra weight to outliers, so that the data lies within 99.9%
confidence level.
Similarity measures with properties like correlation coefficients
are often attempted. The mathematical formula to calculate the
correlation coefficient generally used for clustering is given by
Equation 6.1 in Chapter 6. Some arguments against using it are when
circular features occur, because it can give R.. < R., when variables i j lk
E. and E. are more similar than E. and E, (Eades, 1965). Ball (1965) l j i k
also describes that the covariance matrix suffers from being sensitive
to noise and to the distribution of the data set. Sokal and Sneath
(1963) note that in the large data sets used in computing the
(7.6)
231
correlation coefficients reduce the sensitivity to a directional
property.
As also noted by Marriott (1971) in his examination of the
classification method, the method minimizing the determinant of the
within-group dispersion matrix is scale-dependent, and it may be
rather sensitive to the inclusion of highly correlated variables, and
the existence of genuine multimodality can be masked by the inclusion
of irrelevant variables, especially if the modes are ellipsoidal rather
than spherical.
Most techniques for grouping have developed as algorithms
without formal basis. A formal approach would set up a criterion
to be optimized over the set of partitions of the data set.
Cormack (1971) has summarized three types of groupings of samples:
(i) agglomerative: successive groupings of the individual sample into groups
(ii) divisive: partitioning of the data set into smaller groups
(iii) clustering: finding partitions in the data set with properties approximating to some desired criterion.
Agglomerative methods have been widely used in geochemical
applications as well as in Numerical Taxonomy (Crisp 1974, Sneath
and Sokal (1973). The divisive method has not been much used because
this type of process suffers from inherent difficulties if early
misclassifications are used for later subdivision (Crisp, 1974).
Gower (1967) has suggested that considerable distortion may be
caused by the dendrograms presented in the grouping process.
In agglomerative methods, there are three kinds of this
technique: a weighted pair group average method (WPGM), an
unweighted pair group average method (UPGM), and a single linkage
232
method (SLM), are in common use as criteria of entry of the samples
into a cluster. Obial and James (1973) suggest UPGM is most suitable
for geochemical data. Details of comparison of the three methods
are described by Obial and James (1973), and full descriptions of these
methods may be found in Sneath and Sokal (1973).
Some restrictions in choosing the grouping technique are
imposed by the nature of the data and its scientific purpose in classi-
fying. Cormack (1971) has stated that the single linkage is the only
hierarchical clustering method which satisfies both the stability
requirement and certain other sample requirements about the way in which
a classification should represent the mutual dissimilarities of
objects. However, Jardine (Cormack, 1971) argues that it is nevertheless
a method of classification which may produce inhomogeneous clusters
what Cormack referred to as 'the chain effect'.
Several dozen different methods of classification have been
developed so far. Overall reviews are given by Cormack (Op. Cit.).
Comparative reviews on some different clustering techniques are
given by Dubes and Jain (1976) and Gower (1967). Practical problems
in cluster analysis are described by Marriott (1971). Many clustering
techniques have been introduced to solve the problems of pattern
analysis and pattern classification.
Sammon (1970) has introduced on-line interactive graphic systems.
The dynamic clustering method developed by Diday (1973) is a non-
hierarchical classification procedure which can help the geologists in
exploration work.
In this study, a clustering method called 'ISODATA' initially
developed by Ball and Hall (1966) was used for classification.
The version used for this study has been modified from
the original ISODATA algorithm by David Wolf of the Stanford Research
233
Institute, and adopted for use in the Imperial College Computer
Centre CDC 6400 Computer by Richard Howarth (Crisp, 19V4). A
simplified flowchart is given in Fig. 7.13.
The ISODATA method is one of the most famous of the square-error
clustering methods (dubes and Jain, 1976). This method allows the
comparison of individual samples with a series of k cluster group means,
which are tentatively computed based on the distance threshold, from the
overall mean, and thus the samples are allocated to the nearest
cluster centre, and the means are recomputed for the new clusters.
This process is iteratively applied until the means represent the clusters
with a specified error-of-fit. Refinement of the basic process
allows for clusters to be joined or split depending on local conditions
and user's thresholds.
The repartitioning of the samples about the means is based
on the maximum variance criterion such -that the sum of square distances
of the features from their cluster centres is minimized. Thus,
for a fixed number of clusters the program minimizes the total squared
error, E
2 K 2 E = z e, (7.7)
k=l k
where e. 2 = Z Z (X. .-C, . )T(X. .-C, .) (7.7.1) k icr kJ kJ j=i i£ck i=l,2,...,n
and C is the kth cluster centre where C = (C C ,...,C n and K K K I KZ Km
m are numbers of measurements and variables, respectively. The performance of the algorithm is highly dependent upon
a set of control parameters.
234
Fig.7.13 Flowchart for ISODATA (After Dubes and Jain,1976)
235
The initial clustering is controlled by a user supplied
threshold, the "spherical factor". A new cluster centre is formed if
the Euclidean distance of a sample from the overall mean is greater
than the spherical factor multiplied by the overall standard
deviation or root mean square average distance between clusters.
Iteration of the basic process allows small clusters to be
discarded, and clusters to be fused into larger clusters
(lumped) or broken down into small clusters (split) as required by
the user. For each iteration, the user control options are the
maximum number of clusters that can be lumped in iteration (NCLST), the
threshold distance for lumping or splitting (THETA C), the number of
discards (THETA N) and an option whether lumping or splitting is
performed in the iteration.
Thus the algorithm consists of a basic partitioning process
that is iteratively used with the initial, lumping or splitting
routines to describe the data.
Since the ISODATA computes the similarity by means of the
Euclidean distance of each sample to the cluster centres, the
algorithm does not suffer from the distortions inherent in other
multivariate techniques based on the computation of a similarity
measure of a covariance matrix. It is also capable of handling larger
data sets and yet allows the detail of each data point to be considered,
a feature which is lost when a similarity matrix is used for the
analysis. The maximum likelihood estimation of the means of a mixture
of Gaussian distributions makes the technique a very powerful tool
in the analysis of mixed populations (Crisp, 1974).
The basic output of ISODATA program is a listing of cluster
centres, a table of distances between them, the squared-error for each
cluster, a nd the membership of each cluster. Several other statistics
236
are also computed.
Further detailed description on the method can be found in
references by Ball and Hall (1966) and Crisp (1974).
The central processing time used for the analysis is linearly
proportional to the number of measurements, the number of variables
and the number of iterations. CPU time for 10 iterations with 5
partitions of data per iteration in the analysis of a 9-variable set,
each of which consists of 1024 measurement values, was typically 280 seconds
on Cyber 174.
(b) Testing the ISODATA program
Crisp (1974) has tested the program using a standard set of
well-known multivariate data which consists of 150 samples with four
variables containing typically 3 species of iris. One of the three
species (category 1) is distinct from the other two species, which
are moderately well sorted into clusters but are not linearly separable.
He has found from the test that the clustering is most sensitive to
changes in the spherical factor. He has also noted that the distinct
category 1 is perfectly identified both at low and high spherical
factors, and the threshold for forming 50 clusters is shown where
the error is minimized with the iris test data set. The remainder
of the controls had very little effect on the basic clusterings.
In this study, in order to understand how the user parameters
affect the clustering in practical application of large data sets
four actual data sets (untransformed and transformed 8-geochemical
elements and 9-mixed variables) were used to test the algorithm for
the two most important cases.
237
1. Test of ISODATA by varying the spherical factor with the rest
of the control parameters fixed.
2. Test of ISODATA by varying the Euclidean distance threshold
with the rest of the control parameters fixed.
In both cases, 10 to 12 iterations were made with a fixed number of
partitions per iteration (5) and the maximum number of "lumpings" per
iteration (9). The numbers of discarded samples were kept the same in
all cases as follows: In the first 3 iterations no discards were
applied, in the next 3 iterations 5 discards were used, and for the
rest of the iterations 10 discards were used in the analysis. By
gradually increasing the discards in the following iterations, we may
effectively discard small clusters while keeping the maximum number
of discards around 5% or less to the total. In this way, it may
be more efficient computationally in practical cases where a larger number
of measurement values are dealt with and it may be more effective
in the analysed result in practical cases if we ignore fugitive
clusters, which have insignificant proportion of the total measurements,
since we can usually expect that those discards are from either
abnormally high or low values. An example of the input controls for
the run of the program is shown in Table 7.6
Table 7.6 An example of the input parameters for ISODATA
80.10 5102412 554 CLUSTER ANALYSIS OF 8 GEOCHEM.VARI.(TRN) (11F10.4)
00000000000 9 3.0 0LUMP 9 3.0 0LUMP 9 3.0 0LUMP 9 3.0 5LUMP 9 3.0 5LUMP 9 3.0 5LUMP 9 3.010LUMP 9 3.010LUMP 9 3.010LUMP 9 3.010LUMP 9 3.010LUMP
238
The results of the first case where the spherical factor was varied,
are shown in Table 7.7(a) and (b), 7.8(a) and (b), and
7.9(a) to (d) for the untransformed and transformed 8-geochemical
elements, 9-mixed variable sets and 8-PCA score sets respectively.
The results of the second case where the Euclidean distance
threshold was varied for the untransformed 8-geochemical elements and
transformed 9-mixed variable sets are shown in Table 7.10(a) and
(b), respectively.
From the analysis, it was noted that the spherical factor is
almost the only one sensitive to the results of clustering if numbers of
pattern classes and discards are kept the same.
The criterion for the assessment of the results in this study
is the error percentage of the clustering with consideration of the
numbers of pattern classes and discards.
From Tables 7.7 to 7.9, the minimum error occurs where the
clustering constitutes around 40 pattern classes in this case, and it
is gradually increased with increasing the number of iterations. Thus,
partitioning the data around 40 cluster centres in this case may be
the very distinct natural clusters to be identified, based on the
criterion of error of fit. However, this may not be so meaningful
in regional assessment of geological applications since 40 classes in
this study still make complications in interpretation. The most useful
one for the more meaningful interpretation in the study, was found
to be when the number of clusters is about 8 to 12.
For both untransformed and transformed 8-geochemical elements,
the best optimum spherical factor is 0.4 and the next is 0.6, but
in case of 9-mixed variable set the best optimum spherical factor
is 0.6 followed by 0.8. This phenomenon is varied in case of 8-PCA
score sets. It appears that the optimum spherical factor may vary
Table 7.7 Test results of ISODATA of 8-geochemical data sets "by varying the spherical factor with the rest of the input parameters constant(thetac=3.)
a) Untransformed 5-geochcmlc.nl data.
srtuf. iteration
0.1 0 . 2 0.1 0 . 6 0.8 1.0
1 * * * 5 0 / 0 15.83
50/0 15.51
50/0 11.13
5 0 / 0 11.13
50/0 11.19
50/0 15.16
2 16/0 13.50
13/0 13.80
16/0 10.73
16/0 11.88
11/0 12.61
31/0 15.90
3 ' "3/0 12.55
10/0 12.90
11/0 11.16
12/0 13.58
12/0 12.51
26/0 18.06
1 38/0 12.11
31/10 13.03
35/8 12.08
21/50 11.15
18/61 11.59
18/12 22.17
5 31/10 12.9"
30/10 13.70
31/8 13.36
17/50 16.26
11/61 13.19
11/12 25.86
6 31/10 13.39
26/10 11.18
29/8 13.91
11/50 19.79
10/61 21.16
11/12 28.99
7 21/16 13.18
Zl/16 15.87
20/56 13.21
8/77 23.57
5/71 30.81
6/25 36.01
8 19/16 11.91
18/16 16.88
16/56 15.71
5/77 32.32
1/71 37.72
5/25 39.71
9 16/16 16.72
16/16 I 8 . 5 6
12/56 19.8"
1/77 37.99
3/71 12.17
1/25 H . 3 5
10 13/"6 19.99
13/16 2 0 . 7 0
10/56 22.33
3/77 12.13
2/71 52.29
1/25 H.32
11 10/16 21.37
10/16 26.81
8/56 26.03
2/77 51.25
1/71 72.06
1/25 11.32
12 8/16 27.61
8/16 30.66
6/56 36.38
1/77 70.79
1/71 72.06
1/25 I I . 3 2
b) Transformed 8-geochcmical data.
sph.f. iteration
0 . 1 0 . 2 0 . 1 0 . 6 0 . 8 1 . 0
+ * * * 50/0
2 2 . 6 1 50/0 22.53
5 0 / 0 15.77
5 0 / 0 13.72
1 1 / 0 13.79
2 1 / 0 2 0 . H
2 1 6 / 0 1 7 . 7 3
1 6 / 0 20.3?
1 2 / 0 1 5 . 1 0
1 2 / 0 1 1 . 5 2
33/0 15.11
17/0 25.31
n 1 1 / 0 1 6 . 5 1
1 0 / 0 1 8 . 1 1
37/0 15.93
37/0 15.19
31/0 1 8 . 1 6
13/0 29.51
A- 31.11 1 6 . 8 6
2 8 / 2 0 17.69
3 0 / 1 2 1 6 . 6 5
27/8 1 8 . 0 0
2 0 / 1 6 2 0 . 9 1
9/0 35.68
c J 2 8 / 1 1 17.37
23/20 1 8 . 9 6
2 5 / 1 2 1 8 . 1 0
2 2 / 8 19.93
15/16 21.19
6 / 0 13.83
6 2 1 / 1 1 2 0 . 2 6
1 9 / 2 0 2 0 . 5 2
2 2 / 1 2 19.56
17/8 2 3 . 1 2
1 2 / 1 6 28.05
5/0 5 1 . 1 8
7 16/23 23.17
11/20 2 1 . 2 3 •
15/19 23.79
1 2 / 8 27.31
7/22 37.21
1/0 5 5 . 1 0
8 12/23 2 7 . 0 2
1 0 / 2 0 3 0 . 2 9
11/19 28.97
9/8 32.36
5/22 13.27
3 / 0 7 3 . 2 2
c 8/23 35.83
7 / 2 0 37.81
7/19 38.89
6 / 8 1 3 . 0 2
1 / 2 2 5 0 . 6 1
2 / 0 7 6 . 2 2
10 6/23 41.36
6 / 2 0 1 1 . 8 6
5/19 19.OI
3/8 50.31
3/22 6 9 . 2 2
2 / 0 7 6 . 2 1
11 5/23 4 8 . 5 8
5/2C 1 8 . 1 2
4/19 53.60
1 / 8 51.67
3 / 2 2 6 9 . 2 2
2 / 0 7 6 . 2 1
12 1/23 5 l . 1 i
1 / 2 0 52.03
3/19 7 0 . 6 6
3/8 72.33
3/22 6 9 . 2 2
2 / 0 7 6 . 2 1
ho to VO
* number of patterns/number of discards
* * error percentage
Table 7.8 Test results of ISODATA of 9-mixed variable sets by varying the spherical factor with the rest of input parameters constant (thetac=3.)•
a) Untransformed 9-mixed variable sot. b) Transformed 9-mixed variable set.
sph.f.
iteration 0.1 0.2 0.4 0.6 0.8 1.0 sph.f.
iteration 0.1 0.2 0.4 0.6 0.8 1.0
1 * * *
50/0 23.96
50/0 21.19
50/0 1 6 . 7 0
•50/0 14.34
50/0 13.07
33/0 17.47
1 * * * 50/0 23.94
50/0 25.27
50/0 20.60
5 0 / 0 16.14
50/0 14.80
31/0 19.40
n / 46/0 20.48
45/0 18.58
4 5 / 0 1 6 . 0 6
45/0 13.97
46/0 13.55
28/0 19.85
46/0 21.64
48/0 22.89
45/O 18.72
4 5 / 0 16,04
43/O 15.73
26/0 21.04
3 41/0 19.56
<K)/0 13.12
'K)/0 15.57
41/0 14 .79
40/0 15.75
25/0 2 1 . 9 8
3 41/0 20.90
45/0 22.01
40/0 18.46
40/0 16.87
39/0 16.51
20/0 24.16
4 27/25 19.70
31/20 18.15
29/23 15.93
2e/29 15.74
25/21 18.15
16/12 23.18
4 25/36 20.79
27/35 20.67
26/33 19.01
32/19 17.27
30/14 17.70
15/7 26.91
5 ?3/25 2 0 . 2 3
29/20 18.31
25/23 17.50
24/29 17.73
22/21 19.48
14/12 25.30 _ .
5 22/36 21.49
23/35 2 0 . e 6
23/33 19.79
26/19 18.46
26/14 19.16
12/7 .30.44
6 2 0 / 2 5 2 0 . 9 6
26/20 18.75
20/23 19.44
19/29 20.78
16/21 2 3 . 1 8
11/12 30.98
6 18/36 23.22
21/35 2 1 . 2 9
19/33 2 2 . 0 9
22/19 21.12
21/14 22.04
8/7 36.78
7 15/y* 22.65
21/35 19.72
14/51 2 1 . 5 2
15/48 21.20
11/29 27.44
9/20 33.25
7 13/46 25.35
1 6 / 4 4 2 3 . 2 3
16/33 24.08
14/46 24.22
17/24 23.27
6/7 46.56
8 1 2 / 3 4 2 5 . 9 5
18/35 2 0 . 8 9
1 2 / 5 1 24.68
13/48 23.57
9/29 32.74
3/20 42.66
8 10/46 30,22
1 2 / 4 4 27.46
12/33 28.89
10/<*6 28.23
14/24 26.91
6/7 46.55
9 1 0 / 3 4 31.64
15/35 2 2 . 9 4
9/51 31.22
10/48 20.17
8/29 42.26
6/20 45.71
9 6/46 33.97
9/44 32.35
9/33 33.33
8/46 32.99
11/24 28.87
6/7 46.55
10 9 / 3 4 34.18 13/35 25.05
8/51 40.47
9/48 31.44
6/29 51.50
5/20 48.85
10 6/46 43.67
7/44 40.24
8/33 37.36
7/46 38.19
8/24 34.e2
6/7 46.55
11 8/34 43.14
9/35 32.86
7/51 43.41
8/43 40.97
5/29 48.20
4/20 51.57
11 5/46 48.80
5/44 49.21
7/33 40.20
- 6/24 44.44
6/7 46.55
12 7/34 41.46
8/35 35.61
6/51 39.49
7/48 •43.91
4/29 59.72
3/20 59.88
12 5/46 46.04
5/44 46.31
6/33 44.76
- 5/24 48.89
6/7 46.55
* number of patterns/number of discards
* * error percentage
Table 7.9 Test results of ISODATA of 8-PGA score sets by varying the spherical factor with keeping the rest of input parameters constant*
a) Untransformed 8-PCA score set(thetac-3.;
sph.f.
iteration 0.2 0.4 0.6 0.8
1 * 50/0 50/0 50/0 50/0 * * 20.18 18.37 16.46 15.95
2 45/0 4 3 / 0 44/0 't4/0 18.32 17.12 16.84 16.70
3 41/0 40/0 39/0 39/0 17.58 17.30 18.02 17.60
4 32/24 31/19 28/14 28/22 17.75 17.89 19.42 19.53
5 27/24 28/19 24/14 23/22 19.21 18.58 21.24 22.86
6 23/24 24/19 18/14 20/22 21.33 20.67 25.99 24.85
7 15/44 l9/?8 16/14 12/59 24.74 24.17 27.27 30.76
e 13/44 16/28 14/14 9/59 26.86 2 6 . 5 6 30.82 38.74
9 10/44 14/28 11/14 8/59 36.86 28.39 37.90 46.37
10 8/44 12/28 9/14 7/59 45.17 36.73 48.84 51.70
11 7/44 9/28 8/14 6/59 49-55 43-57 53.42 59.17
12 6/44 8/28 7/14 5/59 54.58 50.38 57.87 6 8 . 1 3
b) Transformed 8-PCA score 3et(thetac-3.) sph.f.
iteration 0.2 0.4 0.6 0.8
1 * * * 49/0 24.33
49/0 25.15
50/0 20.85
50/0 18.15
2 45/0 2 2 . 2 3
46/0 23.20
4 7 / O 20.21
44/0 19.10
3 41/0 22.15
4 3 / O 22.06
4 3 / O 20.84
38/0 20.46
4 31/18 2 2 . 6 7
31/23 2 2 . 2 7
35/13 21.53
28/14 23.51
5 27/18 23.57
27/23 23.77
31/13 22.61
24/14 26.04
6 24/18 24.87
22/23 26.91
26/13 24.96
18/14 31.97
7 19/28 28.06
16/31" 30.00
19/21 28.77
12/23 38.89
8 16/28 31.71
13/31 34.08
15/21 33.29
10/23 44.27
9 12/28 40.52
10/31 41.89
12/21 39-57
8/23 55.89
10 10/28 43.00
8/31 49.41
8/21 54.92
6 / 2 3 6 3 . 2 8
11 7/28 57.31
6/31 58.97
7/21 57.48
5/23 70.02
12 6/28 60.23
5/31 67.41
6/21 59.13
4 / 2 3 73.49
* * *
number of patterns/number of discards error percentage
Table 7.9 continued.
c) Untransformed 8-PCA score set(thetac-5«) sph.f.
iteration 0.2 0.4 0.6 0.8
1 * 5 0 / 0 50/0 50/0 50/0 * * 20.18 18.37 16.46 15.95
2 4 5 / 0 43/0 44/0 44/0 18.30 17.12 16.84 16.70
3 41/0 40/0 39/0 39/0 17.58 17.30 18.02 17.60
4 32/24 31/19 28/14 28/22 17.75 17.89 19.42 19.53
5 27/24 28/19 24/14 23/22 19.21 18.58 21.24 22.86
6 23/24 24/19 18/14 20/22 21.33 20.67 25.99 24.85
7 15/44 19/28 16/14 12/59 24.74 24.17 27.27 30.76
6 13/44 16/28 14/14 9/59 26.86 2 6 . 5 6 30.82 38.74
9 10/44 14/28 11/14 6/59 36.86 28.39 37.90 53.06
10 8/44 12/28 8/14 4/59 45.17 36.73 49.86 64.44
11 6/44 8/28 6/14 3/59 55.69 51.55 60.74 72.54
12 3/44 5/28 3/14 2/59 74.09 66.41 79.15 79.70
* number of patterns/number of discards
* * error percentage
d) Transformed 8-PCA score set (thetac-5.)
sph.f.
iteration 0.2 0.4 0.6 0.8
1 * * * 49/0 24.33
49/0 25.15
50/0 20.85
50/0 18.15
2 4 5 / 0 2 2 . 2 3
46/0 23.20
47/0 20.21
44/0 19.10
3 41/0 22.15
43/0 22.06
43/0 20.84
38/0 20.46-
4 31/18 22.67
31/23 22.27
35/13 21.53
28/14 23.51
5 27/18 23.15
27/23 23.77
31/13 22.61
24/14 26.04
6 24/18 24.87
22/23 26.91
26/13 24.96
18/14 31.97
7 19/28 28.06
16/31 30.00
19/21 28.77
12/23 38.89
8 16/28 31.71
13/31 3 4 . O 8
15/21 33.29
10/23 44.27
9 12/28 40.52
10/31 41.89
12/21 39.57
8/23 55.89
10 10/28 43.OO
8/31 49.41
8/21 54.92
5/23 64.67
11 7/28 57.31
6/31 58.97
5/21 65.07
3/23 78.22
12 4/28 70.34
3/31 76.48
3/21 78.57
2/23 86.41
Table 7.10 Test results of ISODATA by varying the criterion(thetac) with the rest of the constant.
Euclidean distance input parameters
a) Untransformed 8-geochemical data (spherical factor=0.4)
thetac 0.1 0.5 1.0 3.0 5.0 8.0 iteration
0.5 5.0
1 * 5 0 / 0 5 0 / 0 5 0 / 0 50/0 50/0 50/0 * * 11.13 11.13 11.13 11.13 11.13 11.13
2 , 5 0 / 0 5 0 / 0 46/0 46/0 46/0 46/0 10.'47 10.47 10.73 10.73 10.73 10.73
3 5 0 / 0 5 0 / 0 41/0 41/0 41/0 41/0 10.39 10.39 11.46 11.46 11.46 11.46
4 48/8 48/8 35/8 35/8 35/8 35/8 10.29 10.29 12.08 12.08 12.08 12.08
5 48/8 48/8 33/8 31/8 31/8 31/8 10.29 10.29 12.81 13.36 13.36 13.36
6 48/8 48/8 32/8 29/8 29/8 29/8 10.29 10.29 12.88 13.94 13.94 13.94
7 37/99 37/99 25/56 20/56 20/56 20/56 8.10 8.10 11.66 13.24 13.24 13.24
8 37/99 37/99 2 5/56 16/56 1 6 / 5 6 16/56 8.10 8.10 11.58 15.71 15.71 15.71
9 37/99 37/99 25/56 12/56 12/56 12/56 8.10 8.10 11.54 19.84 19.84 19.84
10 - - 24/56 10/56 10/56 10/56 1 1 . 6 9 22.83 22.83 22.83
11 _ - 2 3 / 5 6 8 / 5 6 8/56 8/56 11.93 26.08 26.08 26.08
12 - - 23/56 6/56 5/56 5/56 11.91 36.38 40.01 40.01
b) Transformed 9-mixed variable set (spherical factor-0.6) thetac
Iteration 0.5 1 . 0 3.0 5.0 8.0 11.0 12.0
1 * * * 50/0 50/0 16.14 16.14
5 0 / 0 16.14
50/0 16.14
50/0 16.14
50/0 16.14
5O/O 16.14
2 50/0 15.41
4 9 / 0 15.48
45/0 16.04
45/0 16.04
45/0 16.04
W o 16.04
45/O 16.04
3 50/0 15.19
49/0 15.26
W o 16.87 w°
16.87
40/0 16.87 w°
16.87
40/0 16.87
u 44/21 14.89
4 3 / 2 2 14.94
32/19 17.27
3 2 / 1 9 17.27
32/19 17.27
32/19 17.27
32/19 17.27
5 44/21 14.89
43/22 14.94
23/19 19.46
28/19 18.46
28/19 18.46
28/19 18.46
28/19 18.46
6 44/21 14.89
43/22 14.94
22/19 21.12
22/19 21.12
22/19 21.12
22/19 21.12
22/19 21.12
7 33/108 13.81
32/109 13.85
l'i/46 24.22
14/46 24.32
14/46 24.32
14/46 24.32
14/46 24.32
8 33/103 13.81
3 2 / 1 0 9 13.85
10/46 22.23
10/46 28.23
10/46 28.23
10/46 28.23
10/46 28.23
9 33/108 13.81
32/109 13.35
3/46 32.99
8/46 32.99
8/46 32.99
8/46 32.99
8/46 32.99
10 33/103 13.61
3 2 / 1 0 9 13.85
7/46 - 33.19
5/46 44.70
5/46 44.70 44.70
5/46 44.70 ho -O U>
* number of patterns/ number of discards * * error percentage
244
vto determine-the oft'-«num sf her? cal faetor farevery M a from data set to data setj so that it might be necessary/whenever a
different variable set is applied for the analysis, but it is noted in
this study that both transformed cases remain the same as the
untransformed cases.
The Euclidean distance threshold may not affect the clustering
results provided the number of pattern classes is kept the same and
the number of discards remains same. However, the merging process
is considerably slower if the distance criterion is small.
In some cases of low Euclidean distance threshold, it may
never converge to form a small number of clusters, which may
be meaningful for interpretation. This has to be avoided in practical
computations. This phenomenon is shown in Table 7.9 where & equals
to 0.1, 0.5 and 1.0.
Otherwise, above a certain value (in this study ^ = 3), the
reuslts are virtually identical as far 'as the numbers of pattern
classes and discards remain the same.
It is also significant to note that the performance of
clustering is remarkably stable though sometimes there may be a small
improvement by iteration as indicated by vertical dark lines for the
same number of clusters with the same number of discards in the Tables.
Thus, the most efficient and effective procedures of the application
of the algorithm drawn from the analysis may be as follows.
1. Determine the optimum number of partition per
iteration (3-5) and the maximum number of clusters per iteration (in
this study it was 9).
2. Determine the discards in each iteration; the total number
of discards may be kept around 5% of the total measurements or less.
This can be done as follows. In the first few iterations, no
discards may be applied and the number of discards is increased
245
gradually in the following iterations.
3. Find the Euclidean distance threshold large enough to
form meaningful clusters by a computationally reasonable number of
iterations (in case of this study $ = 3 or larger).
4. Find the optimum spherical factor by varying the value from
0.1 to about 1.0 with the rest of the user's parameters fixed.
Thorough examination of all parameters was impracticable with the
test data sets used in the study because of the computation cost
and time. This is also partly because those parameters (the number '
of partitions per iteration and the maximum number of lumpings per
iteration) only make effects marginal to the final results, as also
noted by Crisp (1974). However, further analysis for varying degrees
of variable sets may be useful and thus provide means of efficient and
effective use of the sophisticated ISODATA program.
(c) Applications of cluster analysis
In this study the clusters designation by letters are different
for all the untransformed and transformed data sets.
(c.l) 8-geochemical element sets
The results of both untransformed and transformed 8-geochemical
element sets are shown in Fig. 7.14(a) and (b), respectively.
Though Crisp (1974) has noted that the untransformed data shows
much faster convergence to a smaller number of clusters and allows
high rate of discarding than the transformed data, this is not unique
in the case of this study. From Tables 7.7(a) and (b) to 7.9(a) and
(d) for the test results of ISODATA obtained by varying the spherical
factor and keeping the rest of the input control options constant, it
appears that the convergence rates of the untransformed data are
246
not always faster than the transformed data, and even the number of
discards of the untransformed data are not uniquely high in the case
of untransformed data with the same initial condition. Thus,
apart from the initial condition of the program run, not only are
their convergence rates but also the discards may be dependent upon
the .d Ts puersior\ patterns-: vn<thfo-'ettch var? able between variables
so that no uniqueness may be drawn for general cases.
The clusters produced show a marked consistency to changes in
spherical factors, as was also noted by Crisp (1974).
The means and standard deviations of 12 classes produced by
the run with iteration 9 for the untransformed data set is shown in
Table 7.11(a) and those of 10 clusters with the same iteration for the
transformed data set is shown in Table 7.11(b).
The clustering results may not be so meaningful with relation
to known geology as in the FA. This may be expected because (1) the
stream sediment sample values themselves are the resultant of regional
redistribution of upstream geology, and (2) the sampling process also
introduces smoothing effects with adjacent values,
and (3) the noise filtering process will merge the data to further distort
the geochemical composition on the spot.
However, bearing in mind the secondary environments together
with those factors mentioned above, they are similar to results obtained
from other methods, and thus indicate a consistency in the results.
Some regional characteristics are evident in relation to the
known geology or geochemical haloes in both untransformed and trans-
formed data sets, which might be enough to justify the use of the
method in classification.
The Bodmin Moor granitic area is well defined except
the northern part of the granite body but there are some confusions
247
(a) untransformed data
— l i t h o l o g i c a l boundary
Figure 7.14 Results of clustering by ISODATA for 8-geochemical element sets: As, Cu, Ga, Li, Ni, Pb, Sn and Zn
Table 7.11 Means and standard deviations of clustering for the 8-geochemical data sets.
a) Untransformed data (iteration 9,12 patterns,discards 56) b) Transformed data (iteration 9, 10 patterns, discards 3*0 variable
cluster no As Cu Ca Li Ni Pb Sn Zn no of
samples variable
cluster no. As Cu Ca Li Ni Pb Sn Zn no. of
samples A * * * -.46
. 2 1 -.31
. 1 2 -•94
.39 -.49
.14 0.01
.45 -.31
.28 -.48
.26 -.34
.31 246 A * * *
0.11 .61
0.06 .55
0.69 .43
0.99 .52
-1.30 .48
-.39 .60
1.10 .42
-.79 .46
64
3 - . 2 1 .34
-.28 .13
0.10 .33 - . 2 2
.24 O .63
.51 -.10
.60 -.44
.34 -.14
.3" 355 B 0 . 2 0
.49 0.41
.41 0.21
.43 -.22
.38 0.67
.47 1.30
.50 -.61
.44 0.59
.72 130
C O .23 .36
0.90 .72
-.83 .44
-M .12
0.26 .48
0.85 .44
-.33 .23
0.64 .35
35 C -1.10 .84
-.39 .44
-.65 .61
-1.00 . 5 4
0.34 .44
-.96 .39
-1.30 .50
-.53 .37
123
0 -.03 .20
-.25 .11
0.Z7 .16
-.13 .10
O .92 .44
0.55 .53
-.42 .27
2.10 .70
28 D -.06 .69
-.39 .28
0.30 .39
0 . 2 5 . 5 2
0.52 .61
-.45 .45
-.39 .71
-.12 -.38
201
2 -.50 .05
-.30 .08
1.50 .92
1.50 .72
-1.40 .73
-.53 .03
2.10 .40
2.90 .65
13 E -1.40 .56
-.38 .44
2.80 .51
2.20 .29
-2.00 .40
-2.00 .75
1.10 .51
-2.00 .88
22
F -.28 .30
-.36 •23
0.68 .47
0.70 . 6 0
-1.60 .44
-.39 .19
-.16 .47
-.84 .13
130 F 1.30 .53
1.10 .85
0.37 .40
0.44 .40
0.16 .73
0.34 .55
1.10 .40
0.67 .44
132
G 0.39 . 6 9
0.32 •55
0.02 .36
-.26 . 2 1
0.14 .34
2.70 .52
-.47 . 1 8
0.73 .40
19 G 1.50 .53
1.70 •3.6
-.58 .80
-.53 .70
0.61 . 6 9
1.70 .61
0.67 .43
1.50 .64
65
H 0.10 .53
-.02 .45
-.02 .59
-.05 • 35
-.46 .72
-.39 .20
2.20 .71
-.07 .57
72 H -.51 .61
-.17 .58
-1.30 .61
-.82 .47
0 . 0 3 .54
0.09 .55
0.19 .65
0.08 .39
147
I -.57 ..07
-.33 .03
3.00 .53
3.00 .4 7
-1.80 .39
-.16 .03
1.50 1.10
-.94 .26
24 I -.64 .31
-.08 .52
1.50 .73
1.70 .34
-.61 1.50
-.91 .28
1.50 .08
2 . 1 0 . 6 9
19
•J 1.20 .53
1.60 .97
0.12 .29
-.05 .13
0.34 .57
0.17 .37
0.74 .54
0.46 .40
21 J -.32
.67 -1.60
.79 0.86
.44 1.30
.46 -1.80
• 35 -.39
.53 0.19
.'•5 -1.50
.40 87
X 3.30 .07
2.00 .77
-.80 .30
-.46 . 1 1
0.71 .41
2.00 .54
0.64 .74
3.00 .32
12
. 1 3.20 .21
3.20 .16
0.39 .20
0.15 .09
-.70 .55
0.1b .20
0 . 9 3 .26
0.26 .24
" l3
ho -f> 00
* Arithnatic moans in uni ts of standard deviation from the sample means. * * Standard deviation
249
in the St. Austell granite. This may be due to effects of heavy
secondary enrichment which occurred throughout the area either by
drainage or old mining activities as mentioned previously.
As in the FA, the anomalous bulge to the south of the Bodmin Moor
granite and the indentation to the north of the Bodmin Moor granite
are evident in the clustering.
The untransformed data in Fig. 7.14(a) shows a broad
distinction between lithologic units. The general features of the
clustering are that Carboniferous and Lower Devonian rocks are
assigned as the same cluster A, Upper and Middle Devonian rocks are
clustered into a group assigned to as cluster B. Bodmin Moor granite
is largely well defined as a cluster (F) but the St. Austell
granite area is defined as clusters E, F, I and H.
Clusters D, E, G, J, K and L largely represent areas of high
geochemical haloes of metallic elements. Cluster D is marked by high Zn,
cluster E is characterized by markedly high Sn and Zn, and cluster G
represents areas of high Pb and moderately high Zn. Cluster K shows a
prominent high in most metallic sulphide elements including As, Cu, Pb
and Zn, which in this study area might be most closely associated
with areas of high metallic concentration near the Kithill granite.
Cluster L shows exceptionally high values only with As and Cu and
a moderate high Sn, and cluster J is characterized by moderately high
values in As and Cu. Cluster H represents areas of high Sn content,
and cluster C shows a moderately high value in most sulphide elements,
which may be due to contamination by drainage, etc. Discards are
mostly related to anomalous areas of either very high or very low values
in the geochemical elements.
In the transformed data, the lithologic units are generally better
defined with the same number of classes compared to the untransformed
250
data. The Carboniferous rocks (cluster C) have been differentiated
from the Lower Devonian rocks (cluster H) although some mis-
classifications still arise probably due to similar geological environments
between them. Marked lows in Ga, Li and Sn contents are characteristics
of the cluster C and H, and most of the metallic elements are prominently
low in these clusters as well.
The Bodmin Moor granite area is also differentiated into two
distinct groups : Cluster J of low Sn content and cluster A of high
Sn content. In the St. Austell granite area more localization of
pattern classes have been produced. Clusters A, E, and I are characterized
by high Ga and Li but are low in most metallic elements except cluster
I with dominant high Zn. Cluster F shows marked high in As and Cu,
and to a lesser extent, in Pb and Zn, so that this cluster is the most
representative of high sulphide elements in the St. Austell granite
in this study area.
Upper and Middle Devonian rocks are still smeared into a group
on the map but there occur two distinct groups: Cluster B is moderately
high in most sulphide elements but cluster D is generally much lower
in these components than the statistical means in the study area
(See Table 4.2) except nickel. Thus, cluster B is more likely associated
with some sort of mineralization in these rock units than cluster D.
Various differential clusters of geochemical haloes in
the untransformed data are collectively classified as cluster F and G of
much broader features in the transformed data than the untransformed
data. Cluster G shows marked highs in As, Cu, Pb, and
Zn while cluster F shows marked high in As and Cu but moderate high in
Pb and Zn contents. Discards in the transformed data are only related
to the very low contents of geochemical elements.
From the characteristic features of the maps, the regional
251
geology may be better defined and delineated by the transformed data,
while areas of geochemical haloes of metallic elements are characterized
by more diffuse patterns in the transformed than the untransformed
data.
(c.2) 9-mixed variable sets
The results of clustering from both untransformed and transformed
9-mixed variable sets are shown in Fig. 7.15(a) and (b), respectively.
The statistical means and standard deviations' of the clusters
for the untransformed and transformed data are also illustrated
in Table 7.12(a) and (b), respectively.
Generally the results show, regional geological features,
as well as mineralization patterns in both cases, although outlining
pattern classes are different from those of 8-geochemical element
sets in Fig. 7.14(a) and (b). Again, in the analysis the transformed data
is better in defining the regional geology but it shows a more diffuse
pattern in delineating areas of mineralized zones compared to the
untransformed data.
The Bodmin Moor granite is outlined by clusters F and H in
the untransformed map and clusters F and J in the transformed map.
Again, the anomalous bulge to the south of the main granite is evident
in both data sets. Depression in the northern part of the granite has
been differentiated from the granite in the transformed data but not
into the same cluster as the surrounding sediment as in the 8-geochemical
element sets. Of particular interest is the cluster F in the untrans-
formed data and J in the transformed data which largely correspond to
the Bodmin Moor granite. These characteristics might be due to the
dominant features of the MSS bands and gravity data over the granite area.
The St. Austell granite area is also clustered as different from the
252
lithological boundary
Figure 7.15 Results of clustering by ISODATA for 9-mixed variable sets: Gravity, Magnetic, B5, R5/4, R7/6, Cu, Ni, Sn and Zn
Table 7.12 Means and standard deviations of clustering for the 9-mixed variable sets.
aj Untransformed data ( iteration 9, 10 patterns, discards 18 ) b) Transformed data ( iteration 6j 10 patterns« discards 16 )
variable Cravity Magnetic B5 R5/1 R7/6 Cu Ni Sn Zn no. of
variable
cluster no« Gravity Magnetic »5 R5/1 R7/6 Cvl Hi En Zn no. of
samples cluster no. samples
variable
cluster no« no. of samples
A * * * 0 . 1 3
.69 -.52
.52 0 . 0 2
.37 0 . 2 8
.58 0 . 0 0
. 6 3 -.27
. 2 0 0 . 1 2
.59 -.31
. 1 1 -.05
.31 330
A * * * -.23
O ? - 0 2
'65 ".it
0 0 0 . 2 3
<59 0.21
<51 -.15
<36 0 . 6 2
<55 0.2S
.60 0 . 1 6
.67 197
-2.60 -.62 -.16 B 1.10 1.50 - 2 . 9 0 -2.60 2 . 6 0 0.71 0.73 -.97 0 . 1 1 22
3 1 . 2 0 1 . 5 0 -2.60 - 3 . 0 0 1 . 9 0 -.09 0.73 -.62 -.16 22 . 0 2 .21 . 0 7 .25 . 6 0 . 0 0 2 .001 .001 .001 22
.02 . 2 8 .05 . 2 1 .38 .001 .001 . 0 0 0 .001 0 . 9 3
.61
.05 .38 C 0 . 9 3
.61 1.10 -.26 -.51 0 . 3 2 -.19 0.10 -1.20 - . 3 1 207
C 0.97 1.60 -.32 -.13 0.39 -.31 O . 3 8 -.62 -.39 201 0 . 9 3
.61 .19 .27 .51 . 5 5 .10 • 51 .18 . 5 1 207
.51 .59 .25 .51 . 1 6 .08 .51 . 0 2 .33 D 0.82 -1.20 -.06
. 5 1 .51 .59 .25 .51 .51 .33 D 0.82 -1.20 - 2 . 3 0 -.06 1.80 0.21 -.18 -.16 0 . 1 7 130
D 1.10 -1.10 -1.90 - . 0 1 1 . 1 0 0 . 0 3 - . 2 1 -.55 -.19 36 • 39 .17 .05 .55 .85 .87 •51 .70 . 7 1 130
.16 .18 .83 .53 .83 .58 .37 .07 • 35 E 1.20 .17
-1.20 .16
- 2 . 3 0 - . 0 7 1.70 O . 2 3 -.17 -.17 0 . 1 6 31 E 0.26 -.17 0 . 0 5 0 . 0 0 - . 0 2 0.35 0.75 -.23 1.60 33
1.20 .17
-1.20 .16 .86 .51 .06 .86 .52 .69 . 7 2
31
.60 .62 .32 .11 .56 .76 .83 .15 .77 F -1.20 .18
-.35 -.05 0.06 0.21 -.25 -1.30 0.93 - . 9 1 97 F -1.20 0.01 1.60 1 . 1 0 -1.10 - . 1 2 -1 . 3 0 -.13 -.81 88
-1.20 .18 .31 ..31 . 6 0 • 53 •70 .56 .53 .61
97
.36 .33 .81 . 6 3 . 6 1 .05 .79 .20 . 2 1 C -1.60 -.55 -1.80 -2.10 -.39 -1.80 1.20 -1.80 22 C -1.10 -.71 0.33 - . 9 3 -.29 -.29 - 1 . 1 0 2.20 2.90 12 .25 .10 .80 • 51 .65 .19 .11 .17 .82
22
M .06 .59 . 5 6 .37 .08 .70 • 37 .65 H -.31 O . I 9 1.30 1.80 - 1 . 0 0 - . 1 6 O . 1 7 -.73 -.23 • 51
1 1 r H -1.10 -.15 -.06 0.16 0.32 -.16 -.93 1.30 -.39 111
• 51 . 1 0 .03 .78 . 5 1 .38 . 1 1 • 56 -.23
• 51 1 1
M .21 .32 . 5 6 . 1 8 .31 .83 1.20 .53 I -.13 -.66 - . 0 3 0.16 0 . 0 9 1.50 0.21 1 . 0 0 1.10 162. 2.10 -7.70 -1.70 2.20 -.85 13
•73 .37 .36 •59 . 5 5 .65 .71 . 1 1 .67 162.
I -1.50 -.59 2.10 - 1 . 9 0 -7.70 -.35 -1.70 2.20 -.85 13 •59 . 5 5 .65 .71 .67
.18 .07 .80 .62 .72 .06 . 1 0 .50 .18 J - 1 . 1 0 0.002 1.30 1.30 -1.10 -1.70 -1.80 0.12 -1.50 . 1 6
66
-.63 .23
2.80 .65
17 .26 .21 .71 .71 . 6 3 .88 .39 . 1 2
-1.50 . 1 6 ro
J -.71 .52
-.63 .23
- . 1 1 .32
0.18 .17
0.25 . 1 1
2.80 .65
0.12 .37
1.20 .89
0.97 1.10
17 . 6 3 .39 . 1 2
-1.50 . 1 6 on
u>
* Arithmatic means in units of standard deviation from the sample meana. * * standard deviations.
254
Bodmin Moor granite, showing different mean intensity of variables,
as in the 8-geochemical element sets. In both granite areas, tin
rich zones are differentiated from the other granite areas, as a group
H in the untransformed data and group F and I in the transformed
data.
Upper Devonian and Carboniferous rocks are generally grouped
into a cluster C except in the area east of the main granite in both
untransformed and transformed data sets. The reason for this exception
is not clear in the east of the main granite area unless it is due to
the solifluxion process or secondary effect mainly due to basic
rocks distributed in this area.
The Middle and Lower Devonian rocks are assigned as a cluster
A in the untransformed data but they are largely classified as
different clusters in the transformed data.
A seemingly odd cluster boundary between the Middle and Lower
Devonian, and a large cluster C associated with Carboniferous and
a large part of the Upper Devonian sediments in the north, might be due
to the dominant physical properties of the geophysical data which may
reflect the subsurface features which are irrelevant to the surface
geologic distinction, e.g. a dense or magnetic rock at depth.
Areas of main geochemical haloes have been grouped as two
different clusters, J and E, in the untransformed data but these are
•largely defined as a cluster I in the transformed data. In the untransformed
data, group J represents areas of main sulphide mineralized zones while
group E is largely coincident with areas of Pb-Zn anomalies.
The regional outlining of offshore areas is evident in both
untransformed and transformed data in the northwest and southeast corners
of the map.
Clustering of 9-mixed data sets seem to be much more stable than
255
those of 8-geochemical data sets though some odd outlining of geological
boundaries are recognized. This account will be further discussed
in the training set analysis in the following section.
Again in this analysis, the transformed data may be better
in defining the regional geological features while anomalous areas
may be more localized by the untransformed data.
(c.3) 8-PCA score sets
The multivariate data sets of 8-PCA scores derived from
16 variables have been further applied to analyses by the clustering
method and EDF (Empirical Discriminant Function). The proportion
of information for each component is already shown in Fig. 7.3
in Section 7.3. The eigenvector matrices of 8-principal components
for the untransformed and transformed data are illustrated in Table
7.13(a) and (b), respectively.
The analysed results of clustering for the untransformed and
transformed data sets are shown in Fig. 7.16(a) and (b), and
statistical results of clusterings are given in Table 7.14(a) and (b),
respectively.
The regional features of clustering are largely comparable to the
9-mixed variable sets in Fig. 7.15(a) and (b). However, detailed definition
of lithologic classification might be much better in the 8-PCA
data sets.
The Bodmin Moor granite is much better outlined in the 8-PCA
data sets than any other multivariate data sets. Lower Devonian
sediments may also be better defined in the 8TPCA data sets than any
other variable sets.
Between untransformed data sets from 9 -mixed and 8-PCA data the
8-PCA score data set is also better in defining anomalous areas but this
256
Table 7.13 Eigenvectors of the first 8 principal components of PCA with 16 variables; Gravity, Magnetic, B4, B5,B7,R5/4,R6/5,R7/6, As,Cu, Ga, Li, Ni, Pb, Sn and Zn.
(a) Untransformed data. l c. 3 6 5 6 7 fi
1 . 1 4 6 1 . 2 0 3 0 . 0 9 3 7 . 2 9 7 6 . 0 1 7 2 - . 0 6 1 1 - . 2 3 8 3 - . 0 6 7 5 ; . 1 1 1 1 . 1 2 0 0 . 1 2 0 1 . 5 2 5 1 - . 3 7 8 0 . 2 7 6 6 - . 2 6 7 6 1 - . < . 0 > 3 . 1 6 6 0 . 3 2 3 5 . 0 3 9 6 . 0 1 6 6 - . 1 9 5 6 - . 0 2 1 0 <i - . < > 1 2 0 . o m . 1 9 9 6 . 2 6 5 5 - . 1 0 3 6 - . 0 1 0 6 - . 0 3 6 7 - . 0 6 6 0 •> - . 1 4 7 0 . 5 5 6 7 - . 0 7 7 6 . 2 1 3 6 . 0 2 8 8 - . 0 5 0 7 . 1 7 7 5 r- - • 1 4 7 7 - . 1 2 0 2 . 2 5 2 0 - . 1 6 6 1 - . 5 5 5 6 - . 0 7 3 1 . 5 5 6 8 - . 1 5 5 9 7 - . n ? T , - . 1 0 6 ( 1 . 5 6 6 8 - . 1 5 7 0 . 2 8 2 5 . 0 5 3 6 - . 0 7 1 0 . 1 r? 2 4
n . 3 l ' . o - . 0 1 0 : 1 . 0 1 0 2 - . 6 9 2 3 . 1 6 0 2 . 0 1 2 3 . 1 6 9 1 . 2 5 7 2
c . 0 0 3 ' . - . 0 0 7 1 - . 0 1 7 5 . 0 5 0 3 . 0 6 8 6 - . 3 2 6 3 . 1 0 9 7 - . 1 7 6 9 1 0 . 0 3 0 6 - . < • 7 2 3 - . 0 0 0 2 . 0 0 6 1 . 0 6 0 0 - . 6 3 8 7 - . 1 2 8 2 - . 2 2 6 2 1 1 - . 1 1 1 0 - . 0 0 7 7 - . 2 6 2 6 - . 0 0 6 3 . 6 0 8 7 . 0 6 6 1 . 6 1 8 0 . 0 3 5 9 1 ? - . 3 7 0 7 • O 3 6 4 - . 2 7 6 6 - . 0 6 0 5 . 2 1 5 5 . 1 1 8 1 . 1 6 5 6 . 0 8 3 9 1 3 . 2 " 3 0 - . U R 0 6 . 1 6 9 7 . 2 2 3 0 . 1 2 3 9 . 6 6 9 1 . 2 5 5 2 - . 5 2 6 0 1 6 . 1 0 3 2 - . 2 8 6 6 - . 1 6 9 7 . 6 5 1 2 . 0 0 8 9 - . 1 1 6 6 . 1 6 1 3 . 5 6 3 0 I f . - . 1 - » 1 6 -.2991 - . 2 0 3 1 - . 3 2 2 6 . 1 6 9 3 . 1 5 0 2 - . 6 1 2 8 - . 3 1 6 7 ' 6 . 1 1 0 6 - . 3 9 0 - . a 2 9 7 . 2 3 6 9 . 0 5 3 6 . 5 3 5 2 . 0 6 0 0 . IOCJ
(b) Transformed data.
i 3 6 5 6 7 e 1 . 3 4 0 0 . 2 A 1 1 - . 0 1 7 9 - . 2 7 6 5 . 1 2 5 9 - . 0 7 7 0 - . 0 1 7 0 . 0 5 3 6 5 . 0 C 2 H . 3 2 2 5 . 0 0 7 6 . 1 7 2 9 . 5 2 6 0 . 2 9 9 9 - . C 3 0 2 - . 2 5 6 3 3 - . 3 5 1 0 . 0 5 4 4 . 3 1 7 1 - . 2 6 9 5 . 1 6 7 6 - . 2 2 3 5 - . 0 3 6 6 • 0C 1 5 t - . 3 A <3 0 . 0 1 0 9 . 3 1 6 0 - . 2 8 1 3 . 0 5 1 6 - . 0 3 1 0 . 0 0 1 3 -.02ce «. . 0 7 1 0 . 0 7 2 5 . 6 1 8 6 . 2 3 6 2 - . 1 3 0 3 * - . 0 6 3 6 - . 0 8 9 1 . 1 6 2 6 f -. PM - . 1 1 0 8 . 1 2 0 6 - . 2 29e - . 3 2 1 8 . 7 6 6 1 . 1 3 9 8 - . 0 3 5 2 7 . 1 M 0 . 0 6 6 5 . 5 6 7 8 . 3 1 9 3 - . 1 3 0 3 - . 0 7 3 3 . 0 0 7 1 . 0 3 9 3 E . 3 0 2 3 . 0 0 1 7 - . 1 1 2 7 . 6 5 2 1 - . 2 6 9 0 . 1 6 6 6 . 0 1 5 3 . 0 6 5 7 c . 0 3 0 0 - . 5 0 6 . 1 7 6 0 . 0 2 6 9 . 1 6 8 0 . 3 1 0 7 - . 2 1 1 1 - . 1 9 6 6 ; r . 1 < m - . 3 9 7 6 . 0 7 0 2 . 0 0 5 1 . 1 9 8 0 - . 1 3 8 7 - . 1 6 6 0 - . 5 9 7 3 1 1 - . 2"60 - . 0 9 5 2 - . 0 6 9 6 . 3 6 3 8 . 6 6 0 1 . 0 6 5 6 . 1 5 3 8 . 1 6 6 0 1 ? - • ? 7 0 - . 1 0 0 0 - . 1 0 1 6 . 3 2 1 6 . 1 8 1 6 . 0 6 9 6 . 1 5 8 6 . 2 6 6 0 1 ? . 2 O 5 0 - . 0 7 6 1 . 1 3 1 5 - . 1 6 1 1 . 2 1 6 5 . 0 6 0 8 . 6 9 6 2 . 0 1 3 6 1 4 . 2 9 6 6 - . 3 6 3 7 - . 0 0 7 2 - . 1 5 2 7 . 2 0 6 1 . 0 7 5 5 - . 6 6 8 1 . 5 6 6 0 1 5 - . 1 " ' , 1 - . 6 0 0 6 - . 0 6 6 3 . 1 6 6 6 - . 2 7 8 7 - . 2 9 3 9 . 1 7 9 6 - . 1 7 6 0 : > . 2 2 6 0 - . 3 6 6 0 . 1 1 6 6 - . 1 6 0 6 . 0 7 8 1 - . 1 2 8 6 . 3 3 5 8 • 2 7 C 5
257
— - — lithological boundary
(b) transformed data
Figure 7.16 Results of clustering by ISODATA for 8-PCA score sets
Table 7.14 Means and standard deviations
a) Untransformed data ( iteration 10, 12 patterns, discards 28, error 3 6 . 7 ) variable
cluster no. F1 F2 F3 F4 F5 F6 F7 F8 no. of
samples
A * -.42 -.79 -.31 -1.40 0 . 0 7 0.33 -.97 -.49 87 * * .36 .61 .28 .51 .38 .64 •97 .91
B 2.40 1.20 -3.30 0.52 1.30 - . 6 7 0.46 0.19 22 .09 .04 .00 .20 .11 .16 .33 .20
C 0.57 0.74 0.75 0.14 1.10 -.57 0 . 1 7 -.35 211 .32 .32 .30 .48 .48 .49 .71 .61
D 0.15 0.01 0.43 0.02 -.53 0.70 -.08 0.01 333 .29 .43 .30 . 6 9 .64 .58 .99 .71
E -1.20 0.41 -.04 -.35 -.66 -.65 0.85 0.64 148 .68 .30 .37 1.00 .61 .33 .57 1.10
F 0.39 -1.10 0,20 1 . 6 0 0.01 O . 5 8 0 . 5 8 2.00 39... • .24 .71 .19 .46 .71 .91 .62 .88
G -.44 -2.70 -.34 -.75 0.35 - 2 . 3 0 -.46 -1.10 27 .24 .42 .22 .61 .47 .66 .64 .57
H 0 . 3 1 -2,20 0.03 0.77 -.11 0.08 0.13 -.41 49 .31 .85 .29 • 85 .56 .67 . 6 9 .91
I 1.80 0.68 -2.30 -.51 -2.20 -.32 -.24 -.22 32 .48 .53 .69 .81 .50 .31 .82 .69
J -.79 - . 6 9 -1.80 -.48 1.10 2.90 -.05 -.38 16 .82 .47 .88 .92 .70 .50 1.30 1.50
K -2.10 O . 3 8 -1.70 1.50 0.41 -.68 0 . 9 6 13 .68 .42 .52 .68 .34 .70 .50 .77
L -2.80 0.72 -1.80 1.60 1.20 0.58 -2.10 -.43 19 .81 .44 .46 .63 .76 • 63 .77 •58
* Arithmatic means in un i t s of standard dev ia t ion from the sample means. Standard dev ia t ions .
clustering for the 8-PGA score sets.
b) Transformed data ( iteration 9, 12 patterns, discards 28, error 40.5 ) variable
cluster no. F1 F2 F3 F4 F5 F6 F7 F8 no. of
samples
A * * * -.16
.43 -1.20
.57 0.14
.48 0.67
.70 -.33
.51 0.01
.58 -.24
.36 -1.10
.70 130
B 1.90 . 1 0
0.08 .02
-3.20 .07
1.20 .24
2.00 .16
-.07 .36
-.67 .05
0.61 . 1 1
22
C 0.43 .41
-.86 .77
0.75 -.11 .75
0.39 .64
-.04 .51
-.02 .99
0.81 .79
180
D 0.39 .47
0.79 .79
0 . 3 4 . 6 0
0.30 .64
0.24 .65
0.29 .67
0.61 .77
-.31 .65
264
E -1.80 .59
0.27 .69
-.04 .28
1.30 .56
0.05 .53
-1.60 .52
0.03 -.95 .66
15
F -1.50 .69
0.13 .35
0.15 .49
-.89 .65
0.23 •53
1.80 .57
0 . 0 0 .76
-.20 . 6 0
68
G O .29 .y*
0.18 .65.
-.00 .58
-1.20 .78
-1.20 .67
-.76 .63
-.08 .65
0 . 2 9 .86
141
H - . 9 3 . 4 9
-.61 .41
-.81 .59
.29
.74 0.46
.65 -1.90
.61 1.70
.83 0 . 7 6
.72 22
I 0.13 .71
O . 2 3 .47
-.42 • 93
-1.10 .66
2 . 1 0 .72
-.44 .76
-.85 .42
-.04 .46
27
J -3.00 .38
0.34 .56
-.58 .23
0 . 0 9 .49
1 . 6 0 .56
-3.00 .33
0.06 .17
-1.00 . 6 0
18
X 1.40 .36
-.16 .73
-2.90 .47
- . 6 9 •55
-1 . 3 0 .64
0.39 .71
-.55 .32
-.28 .83
30
I -1.40 .67
0.41 .40
-.45 .36
0.97 .76
-.61 .57
O .56 .63
-1,00 .49
1.00 . 9 1
79
259
may be not always so in the case of transformed data sets although
lithologic classifications are generally better in the case of the
transformed 8-PCA score data.
However, though more definitive information could be drawn
from the data sets obtained by the 8-PCA component scores than any
other data sets chosen largely by intuition, there are certain
disadvantages in using PCA component score sets for futher classification
analysis since it is very difficult to interpret the pattern classes
in terms of geologic features of individual data sets, for example,
physical, chemical or spectral.
7.5 Supervised Classification
7.5.1 Discriminant analysis
(a) Discriminant analysis procedure's
Discriminant analysis has been applied to a wide variety of
research and problems of prediction as a statistical technique for
examining differences between two or more groups of measurements with
respect to several variables simultaneously.
In Earth Sciences particularly in geochemistry, discriminant
analysis is one of the most widely used multivariate procedures
because it is a powerful statistical tool and it provides an
additional link between univariate and multivariate statistics
(Davis, 1973).
It is a classification tool for use when prior information is
available on the nature of classes which we are interested in
distinguishing between. A training set consisting of a typical example
of these classes is used to find a decision rule which discriminates
well between these classes on the basis of the observed measurements, and
260
this rule is then applied to classify the remainder of the data.
The simplest of the discriminant function is a Linear
Discriminant Function (LDF). It transforms an original set of
measurements on a sample into a single discriminant score. That
score represents the sample's position along a line defined by the
LDF. The basis of the LDF is that of a search for a linear surface
in the hyperplane that effectively separates the classes. In the case
of two groups based on the presence or absence of ore mineralization, LDF
thus consists of finding a transform which gives a maximum ratio of the
between groups variance to the within-groups variance.
If we regard our two groups as consisting of two clusters
of points in multivariate space, we must search for the one orientation
along which the two clusters have the greatest separation, while
simultaneously each cluster has the least inflation. Fig. 7.17 shows
how the group A and B are adequately separated by the LDF.
A good description on the LDF is given by Harbaugh and Merriam
(1968), Rhodes (1969) and Davis (1973).
Howarth (1973) has suggested that the LDF may not be capable
of separating classes that are nonlinear. He used the Empirical
Discriminant Function (EDF) as an alternative to predict both the
stratigraphic affinity and occurrence of mineralization.
This method is the exponential form of the polynomial
method of Specht (1967). It is based on the nonparametric estimation
of a probability density function for each category to be classfied,
so that the Bayes decision rule may be implemented. A smoothing function
is applied so that a density function is estimated from the training
samples.
The effect of smoothing by varying the smoothing parameter Q
is shown in Fig. 7.18. As O increases, the five distinct modes of the
261
Figure 7.17 Two overlapped bivariate distributions showing the effective classification by projecting onto the discriminant function line (After Davis, 1973)
262
Xo= |oo...o. ..»•[
Figure 7.18 Interpolated one-dimensional probability density function for set of five training samples with increasing values of smoothing parameter O. (After Howarth, 1971a)
263
training samples are gradually smoothed out. A detailed account of the
mathematics involved is given by Specht (1967), Howarth (1971a, 1971b,
1973a) and Castillo-Muhoz (1973).
Ideally, the classification should be performed in such ways
that the classes for the classification are mutually exclusive
and exhaustive, and the variable set chosen should allow the perfect
assignment of unknown data to one of the classes. Hence the choice
of the variable set becomes an important consideration as already
discussed previously. Howarth (1973b) has reviewed the various
methods available, and he finds that the stepwise feature selection
techniques provide the best overall success rates in the classification.
As already described in Section 7.3, these techniques (BAKWRD)
consist of an iteration of the classification with the variable
causing the greatest improvement in the overall success rate being
added to or deleted from the variable sret. The problems of testing
the classifier performance is the large number of possible combinations
of the variate set, thati,s,{m(m-1 )/2} where m is the total number of
variables to be analysed. This point has been discussed already in
Section 7.3 and Howarth (1973b) has noted that the optimum method is
training with successive eliminations (also known as
11 jackknife"). These procedures are implemented in the EDF package
program developed by Howarth (1973a) and used by Castillo-Munoz (1973).
The mathematical procedures of EDF described by Howarth (1973a)
are given in Appendix G.
In this study, the EDF has been used to classify those areas
most closely related to the training set chosen by means of geological
and geochemical information and results of the cluster analysis. Further
details on the training set selection schemes will be described in
the next section.
264
(b) Selection of the initial condition
The control parameters required by the EDF computer program
(Howarth, 1971b) are as follows: (1) the number of variables to be
considered, (2) the number of pattern clusters into which the data
are to be grouped, (3) the a priori probability of the occurrence
of each of the classes specified, (4) the smoothing parameter
for the estimation of the probability density function for each variable
in each class, (5) whether the data are to be left untransformed or
log-transformed prior to exercising the discriminant analysis,
(6) a threshold value that an unassigned sample is to be assigned to in
one of the given categories, and (7) the data matrix coordinates
of one or more rectangular training areas for each class.
The program will then classify all the samples on the basis of
relative probability that an unassigned sample resembles each of the
defined classes. If the absolute probability that an input sample belongs
to one of the given categories falls below the threshold value, the
sample is then assigned to the 'unknown' class (Howarth, 1971b).
Since the input data are from different sources, provisions for log
transformation (base 10) in the program were not used. Instead, the input
data, have been provided from either untransformed or
data transformed by the power transformation technique or arc sine
transformation technique (see Section 7.2). There are options in
the program for assigning different a priori probabilities for the
occurrence of each class, or for giving a different penalty for making
an incorrect classification and these were arbitrarily made equal for
all classes, since it is not possible to determine them prior to the
analysis.
Above all, the most important initial step is the training set
selection and choice of the smoothing parameter(cr). The training set
265
samples are to be chosen in such ways that an artificial grouping of
the results should be avoided as far as possible.
Thus the training areas for the classes of the multivariate
data sets were selected by considering the results of the cluster
analysis by ISODATA, and the geological map was used to select
lithologically homogeneous geologic formations as far as possible.
Geochemical homogeneity was also taken into account by referring to the
regional distribution patterns of geochemical elements produced as
in Fig. 4.18 in Section 4.4. In particular, clustering results by
ISODATA played an important role since when only the lithologic
homogeneity was considered the EDF results were dubious.
In this study, seven training classes for the classification
were selected to represent the major lithological classes, including
a class for the potential mineralization. Further, a training class
representing the offshore area was also- selected for the 9-mixed
variable sets and 8-PCA score sets. Upper and Lower Carboniferous sediments
were grouped together into a class since they are generally intercollated
with each other by folding so that it might be difficult to
separate them in the regional sense. Upper and Middle Devonian strata
generally show geochemical homogeneity but they were separated as two
different groups in order to see what might be expected to be
produced. The Lower Devonian was assigned as a class though in some
occasions it shows geochemical homogeneity with Carboniferous rock
units and Middle and Upper Devonian rocks.
The Bodmin Moor and St. Austell granites are grouped as
different clusters because they are somehow different in geochemical
composition, and this was confirmed in the cluster analysis by
ISODATA which produced the granites as different clusters in the higher
number of pattern classes, though they were merged into one cluster
266
in the lower number of pattern classes. Selection of a class for
mineralized zones has been made by examining spatial information of
geochemical haloes of metallic sulphide elements as described in
Section 4.4 and the results of ISODATA analysis. Part of the
Crackington Formation in the east central part of the study area
(transitional zone) and basic intrusives in the northeast of the Bodmin
Moor granite were ignored since regionally these minor groups could be
merged into the neighbouring major groups.
However carefully they are chosen, in practical applications the
training sets selected by the investigator will always be too small
to reliably be extrapolated to new data sets (Nagy, 1968,
Howarth 1971a).
After having selected an appropriate training set in the study
area, the success of the method depends on a correct choice of the
smoothing parameter ( a ) . Since it is td some extent, data-dependent
(Howarth, 1971a), an optimum value must be determined prior to
performing the classification with EDF. Alternatively, the smoothing
option with successive increments is desirable to classify with the
data successively and compare the results with known geologic
information. However, analysis in this way may not be so desirable compu-
tationally and in its effectiveness since much effort has to be expended
in order to calculate for varying degrees of the smoothing parameter
and it also requires a complete knowledge of geology in the study
area for correct assessment of classification results obtained by
varying smoothing parameters.
Howarth (1973a) has presented a computer program BAKWRD
initially written for feature selection schemes. The program uses a
method of evaluating the probable classification success of the
training set originally proposed by Kanal and Chandrasekaran (1968).
267
In this study, BAKWRD was used to obtain a classification
performance table showing the true and assigned classifications of the
training set samples.
Fig. 7.19 shows for varying smoothing parameters the correct
performance rates of each class of the training set samples chosen
from untransformed and transformed 8-geochemical, 9-mixed variable sets
and 8-PCA score sets.
From the figures, the correct classification rates are high when
the smoothing parameter is between 0.1 and 2.0, but the rates decrease
gradually towards higher smoothing parameters. If the smoothing parameter
is very low (a=0.01), there is no misclassification at all but many
are unclassified. This indicates that certain optimum smoothing
parameters must be applied in general for the data set used in this
study. The plotted values of the correct classification rates at
Q = 0.01 on the figure are unclassified rates rather than misclassified
rates.
From the analysis, the transformed data sets are more stable
and higher in correct classification rates than the untransformed
data in all cases. Comparing results between the three different variable
sets, the 8-PCA score set is the highest and 9-mixed variable set
is second. Training set samples from the transformed 8-PCA score
data set show 100% correct classification rate bewteen Q = 0.1 and
Q = 2.0, and even at O = 5.0 the performance rate is over 90%. Except
in the untransformed 8-geochemical element set, they show 100%
correct classification rate between O = 0.1 and a = 1.0 and even the
untransformed 8-geochemical element is 100% between O = 0.1 and O = 0.5.
Assuming the training set samples chosen for the analysis are
adequate, optimum smoothing parameters for the data sets chosen for the
study lie between 0.1 and 0.5 (this tendency has been generally noticed
100 •*
90 J
80 b
70
60
50
40
30 .
20 .
10 .
untransformed 8-geochemical » transformed 8-geochemical + untransformed 9-mixed * transformed 9-mixed A untransformed 8-PCA n transformed 8-PCA
i i 1 0 smoothing parameter
Figure 7.19 Correct classification performance rates of the training set for the untransformed and transformed variable sets
r o 00
269
in the test with other trianing set samples). Thus the smoothing
parameters O = 0.2 or 0.5 were chosen as optimum values for the analysis
of EDF in this study. Indeed, comparison of the results of EDF
analysis with O = 0.2 and (7=0.5 shows marked resemblances between
each other and some differences noted are generally marginal.
The performance rates of each class in the training set samples
for the optimum smoothing parameters (CT = 0.2 and 0.5) analysed by
BAKWRD program are 100% for all data sets as shown in Fig. 7.18.
(c) Applications of EDF
(c.l) 8-geochemical element sets
The results of classification using the chosen 7 classes of the
training set are shown in Fig. 7.20(a) and (b) for the untransformed
and transformed data, respectively. Even with the small training set samples
(12.5% of the total samples) the lithologic units are largely well
classified in the analysis though there are some variations in
details. The classification reuslts are almost comparable to
the reuslts of clustering by ISODATA as shown in Fig. 7.14(a) and
(b). Classification of granitic areas, Carboniferous and Lower Devonian
rocks, Upper and Middle Devonian rocks in both analyses provide
general resemblances between Fig. 7.14 and 7.20.
The Bodmin Moor granite which is assigned as clusters E and F,
is probably the most successfully classified. This may be attributable
to the uniform regional lithologic features of the units and no serious
effect of secondary environment. However, the granite clusters are in
general spread over into the Devonian sediments, partiuclarly in
the south of the granite. An exception is in the northern part of the
granite where a portion of the granite area is assigned as Devonian
270
(a) untransformed data
— lithological boundary
t BY JINC COOHOlNOJtS (b) transformed data
Figure 7.20 Results of classification by EDF for the 8-geochemical sets: As, Cu, Ga, Li, Ni, Pb, Sn and Zn
271
or Carboniferous sediments. As these features have also been
recognized in results of the cluster analysis of the same data sets and
FA, the spreading to the south is attributed to the drianage system
flowing down from granitic uplands into Devonian lowlands
to the south, and the granite area assigned as Devonian or Carboniferous
sediments in the north corresponds to a poorly drained topographical
depression where thick oxide coating might have altered the chemical
characteristics of the soil.
Cluster F,'classified as the St. Austell granite in the southwest
corner of the study area, is also well represented except in a few
marginal areas where there might be distortion of data by secondary
effects such as drainage or mining activities and also by digitizing
and smoothing in the data preparation processes.
The Carboniferous rock unit has been generally well classified,
but the Upper and Middle Devonian rocks are intermixed together
which could be expected since these two rock units are similar in
geochemical composition. Parts of Lower Devonian rock (Cluster D)
have been assigned as cluster A of Carboniferous rock, which would indicate
there are some geochemical similarities between the two rock units.
Group G assigned as potential mineralization zones has been
differentiated in the east of the main granite and near Kithill and partly
in the southwestern part of the map between Bodmin Moor and St. Austell
granites.
Results of the classification for the transformed data in
Fig. 7.20(b) are similar in features to the untransformed data except
for the drainage pattern in the southeastern part of the map, and a slightly
more diffuse pattern in defining potential mineralized zones in
the east and south of the main granite and the northeast margin of
St. Austell grnaite, which are, to some extent, comparable to the
clu stering results of the same data in Fig. 7.14(b). Therefore, further
272
detailed descriptions will be omitted for the analysed results.
(c.2) 9-mixed variable sets
The results of classification by EDF analysis for the 9-mixed
variable sets in Fig. 7.21(a) and (b) generally show similar features
comparable to the results of clustering by ISODATA.
Though the training set samples chosen are small, EDF
analysis might be as stable as the sophisticated ISODATA program in
its performance of classification as long as the training set samples
are chosen carefully.
(c.3) 8-PCA score sets
The classification results of 8-PCA score sets by EDF are shown
in Fig. 7.22(a) and (b) for the untransformed and transformed data,
respectively.
Though regional features might be comparable to the clustering
results in Fig. 7.16, some distinctions in classification appear between
both results.
The Carboniferous and Lower Devonian sediments have been generally
differentiated from the Upper Devonian in the EDF analysis. In the
clustering analysis in Fig. 7.16, the Carboniferous and part of the
Upper Devonian rocks have been clustered as the same group, and Middle
and Lower Devonian sediments have been grouped into a single cluster.
The depression in the northern part of the Bodmin Moor
granite is again classified into the Devonian sediments, the feature
of which does not appear in the clustering analysis of the same data sets
as in Fig. 7.16.
There also occur some differences in classification of
273
(a) untransformed data
lithological boundary
M
*Voo 17.00 14.00 10.00 71.00 tostinc cooooinoies
(b) transformed data
Figure 7.21 Results of classification by EDF for the 9-mixed variable sets: Gravity, Magnetics, B5, R5/4, R7/6, Cu, Ni, Sn, Zn
274
lithological boundary
Figure 7.22 Results of classification by EDF for the 8-PCA score sets
275
anomalous units though their localities appear to be generally in the
same region.
Again, in these analyses, the transformed data set shows
more diffuse patterns in defining mineralized zones although the litho-
logical definitions have been classified in a similar way.
As in the cluster analysis, significant misclassifications also
arise in classifying different sedimentary rock units in the EDF analysis,
which could mean that they are, to a certain extent, geologically
similar and those' facies might be similar physically, chemically
or spectrally unless much distortion occurred in the preprocessing
of the data.
7.5.2 Characteristic Analysis (CHARAN)
The applications of characteristic analysis in this study is
mainly introductory in its application to the data. This is mainly due
to the limited time for the research.
(a) Characteristic analysis procedures
Characteristic analysis is a multivariate technique that has
been used to attempt the regional assessment of exploration targets
for a variety of types of deposit (Botbol, 1970, 1971, Botbol, et al.
1977, 1978, Sinding-Larsen et al. 1981, McCammon et al. 1981). It was
originally developed as a method for integrating regionalized multi-
variate data in geology, geochemistry and geophysics (McCammon et al.
1981).
There are basically three concepts involved with the method; data
transformation, 'favourable' model formulation and regional cell
evaluation.
276
The CHARAN program initially written by Botbol, Sinding-Larsen
and co-workers has been modified to use in this study.
1. Data transformation
The initial step in the CHARAN program is to transform the
data into binary form by assigning the value of 1 meaning favourable
or the value of 0, meaning unfavourable or unevaluated.
The manner in which this transform is performed depends on the
nature of the exploration model and the nature of data. For geochemical
data, favourability may be defined by local anomalies calculated
from second derivative surfaces, highpass filtering and so on.
Reduction of local anomalies by the second derivative method is
illustrated in Fig. 7.23.
Where a cell or location has a negative second derivative for
a particular variable, it is labelled '!' and is of interest because
the values within the cell are higher than the values in the neighbouring
cells. The 'O's represent all other data which may have no potential
values of interest. For those variables with known negative anomaly
representation, simple reversal of binary notation may be applied.
For geophysical data, on the other hand, favourability may be
determined on the basis of regional gradients, local increases or decreases,
and recognition of any special features such as any lineaments, etc.
For geological map data, the presence of a particular rock type
or any structural linear features could be a criterion to be considered.
This type of binary representation yields maps that indicate only
those areas that have values of major interest in exploration.
The latest version of the characteristic analysis program
facilitates the dynamic range of data values, by coding the input in
ternary form (1, 0, and -1) (McCammon et al. 1981). Here the value of
. l . l . l . t . o . o . o . o . l . t . l . l . l . t . o . l . t . t . t . o . o , ' . '
Figure 7.23 Hypothetical data profile showing areas above local inflection points (second derivative negative) labelled '1' and other locations labelled '0' (After Botbal et al. 1977) ^
278
1 means favourable as before, the value of 0 meaning unevaluated, and the
value of -1 meaning unfavourable. Adaption of logical combinations of
transformed variables make it most powerful to utilize the evaluation
of dynamic range of models.
2. 'Favourable' model formulation
After having generated a binary or ternary array for each
variable, the next step is to determine the relative weights of each
variable for region cell evaluation from a model.
The favourability of a given cell is defined as a weighted
linear combination of either the binary or ternary transformed variables,
as follows,
f = a.X + a X + ... + a X (7.8) 11 2 2 m m
where a^ and X , (i=l ,2, . . . ,m) represent the weights and m transformed
variables, respectively.
The weights, a^, in Equation (7.8) are determined by 'a product
matrix' defined by a selected model set of the transformed variables.
Mathematically,
(X'X)a = Xa (7.9)
where A is the largest eigenvalue of (X'X). X is the n x m matrix
of m variables for n selected cells that comprise the model.
The a^'s are the elements of the eigenvector a_ associated with
X and are scaled such that f in Equation (7.8) lies between 0 and 1
in case of binary input but -1 and 1 for ternary data set.
The model cells may be selected in areas of known geology or
deposits, depending on the nature of study, i.e. mineralized model or
lithologic model. Generally the model should be generalized in its appli-
279
cation not by the unique features of deposits or lithology in an area, but
by inclusion of regional cells that do not contain particular
deposits or lithology.
3. Region-cell evaluation
After the weights of the variables, which comprise a model,
have been calculated by Equation (7.9), the degree of association
for unknown region cells outside the model areas is thus calculated
by multiplying the binary or ternary vector that represents the
transformed variables of each region cell with the characteristic
vector of the model calculated as in Equation (7.8).
(b) Application of characteristic analysis
In this study, the original version of characteristic analysis
utilizing the binary form of the input variables was applied to test its
applicability to the data from the Bodmin Moor area.
Five sulphide geochemical elements of As, Cu, Pb, Sn and Zn were
selected for consideration and their model cells were chosen for use in
order to analyse for areas of potential mineralization.
Eight model cells were selected on the basis of known geochemical
haloes related to the known mineral lodes or veins in the eastern
part of the Bodmin Moor and near the Kithill granites. The model was
characterized with respect to the elements used. Table 7.15 shows the
characteristic weights for the five elements derived from the product
matrix of the model. Pb is the most strongly weighted component
of the model vector, followed by As and Sn. The ratio of the eigenvalue
(the characteristic root in the table) and the total number of I's in
the model cells indicates the degree of anomaly overlap.
Low overlap means low dependence of cells within the model, which
280
reflects the combination of cells and/or variables which does not
produce a diagnostic result, while high overlap indicates that the
model chosen shows a strong dependence adequate enough to produce the
diagnostic results.
In the case of this study, the model has a high degree of
similarity (about 73%).
The results of the analysis with this model are shown
in Fig. 7.24. Five classes were chosen to illustrate the degrees
of association between the region cells and the model.
There occur five most prominent non-model anomalies on the
map. All occur as individual mineral belts(?) which may or may not
be genetically related, but they are largely aggregated with moderate
degrees of association around them.
However, none of the prominent anomalies are associated with the
known mineral lodes or veins. This may indicate that the manner
in which the data transformation is performed in the program may not
be adequate for this kind of regional analysis.
It seems to the author that in any future analysis with more
time available it might be advisable to examine if the data trans-
formation for geochemical elements should be performed by other methods
such as highpass filtering, etc.
Obviously full utilization of the new version of characteristic
analysis with logical combinations would facilitate its dynamic
range of application to the assessment of the regional geology or
mineralization.
281
Figure 7.24 Results of characteristic analysis for a variable set of As, Cu, Pb, Sn and Zn. Rectangulars are the selected model cells. Degrees of association is indicated by symbols in the increasing order as blank, ',-,+,0
282
Table 7.15 Characteristic weights of five elements (As, Cu, Pb, Sn and Zn) for the model selected as indicated by two rectangulars in Fig. 7.24
Element Weights Remarks
As .47 * = 17.49 Cu .42 ** = 24 Pb .55 Sn .47 Zn .27
* The characteristic root ** Total sum of l's in the model cells
7.6 Conclusions
Four kinds of supervised and unsupervised pattern recognition
techniques have been applied to various sets of the multivariate data
from the Bodmin Moor area. The usefulness of each kind of technique
has been shown in use for regional assessment of geology or potential
mineralization.
In all cases, the classification of the Bodmin Moor granite
is the most successful. The regional geological mapping of sedimentary
rock units is in general confused, but the data sets which are most
appropriate are the 8-geochemical ones followed by 8-PCA score sets.
This might be due to similarities between different sediments,
particularly near their boundaries which would be due to secondary effects
such as contamination or smoothing, etc. Alternatively, this might be
due to inappropriate combinations of the multivariate data (particularly
of gravity and magnetics) for the classification, though the gravity data
might contribute significantly in classifying the granitic areas. In
283
this study area, these potential fields usually represent subsurface
geological features which might be unrelated to the surface lithologic
features.
However, the training set test of BAKWRD for varying the
smoothing parameter (a) has shown that the 8-PCA score sets are
most stable, followed by the 9-mixed data sets and the last by the 8-
geochemical element sets. This indicates that an appropriate
combination of data sets having various properties, for example, physical,
chemical or spectral, would constitute more stable data sets and thus
draw better results. In addition, in all cases, the transformed data
are more stable than the untransformed data, indicating favourability
of the transformed data over the untransformed data in the application
of pattern recognition techniques.
Thus, in this study, regional lithologic mapping of the
transformed data is better than the untransformed data, but at the expense
of diffuse patterns for localizing mineralized zones, whereas the
untransformed data are better in defining potential mineralization which
largely corresponds to areas of known mineral lodes or veins.
Dicards occurred both at typically low and high values in both
untransformed and transformed data in this study although many workers
(for example, Crisp 1974, Castillo-Munoz 1973, and Mancey 1980, etc.)
have noted in the analysis with geochemical data sets that the discards
might occur usually at high anomalous values of geochemical elements.
From the analyses of the techniques, the following general
conclusions may be drawn.
1. The factor analysis method is by far the most capable
of handling as large a number of data measurements as may be required,
since the dimension of the program depends on the input data structure
which can be modified as required. However, the display of major
284
component scores and the choice of combination of the major components
are not immediately possible and are more complicated than other
classification methods. The interpretation may also be somehow
subjective.
2. The cluster analysis by ISODATA is, to some extent,
subjective because of various initial options which would affect the
results. However, this method is much more conclusive (at least
from the statistical point of view), since the error percentage
of the clustering results together with various statistics useful
for interpretation, are calculated though the error rate is much
dependent on the number of discards.
Immediate recognition of patterns from the line-printer output
would add•to its usefulness for rapid and effective analysis.
3. Both 'EDF and CHARAN' supervised classification
techniques applied are very subjective since the classification results are
entirely dependent upon the user's input controls and no unique results
can be drawn.
However, these classification methods would be most usefully
applied to areas where the training set for regional geology or
particular mineralization is well-known and thus this is used further to
classify unknown areas by comparison.
While there are complications in setting-up the user's initial
conditions, their simplicity and nonparametric nature could make them
most appropriate in solving complex geological problems arising in
pattern recognition.
285
4. Transformation of the data may be useful at least for
statistical requirements. However, in this study, unique advantages of
transformed data over the untransformed data are not significant,
though the regional geological mapping seems to be generally better
with the transformed data but at the expense of diffuse patterns of
mineralized zones.
Problems with the transformed data might lie in the fact that
transformation of individual variables to near-normality may not
guarantee that the data set drawn from the transformed data are truly
Gaussian, as noted by Mancey (1980).
However, tests of correct classification rate with the
training set samples indicate that the transformed data are more stable
than the untransformed data. The test results also show that 8-PCA
score sets are the most stable compared to the smoothing parameter Q,
followed by the 9-mixed variable sets, and the least stable one is
the 8-geochemical data set.
This would indicate that combining the geophysical, remote
sensing and geochemical data might constitute a better multivariate
data set than any set of data drawn from any single set of properties
(for example, physical, chemical or spectral) of the region of interest.
5. Though the usefulness of combining various data having
different properties has been noted from the study conducted, odd
lithologic outlining of the 9-mixed variable sets indicates that some
data sets such as gravity and magnetics may not be adequately applied
in the analysis, since they usually show subsurface geological features
which might be different from the surface or near-surface geological
features.
286
6. None of the pattern recognition techniques applied in
this study are to be uniquely favoured over other methods. They are
largely complementary to each other. Particularly when EDF is used
for classification, it is almost essential to perform the cluster
analysis or some other unsupervised classification techniques to ensure
that the training set for the EDF constitutes 'natural' groupings.
287
CHAPTER EIGHT
CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions
Data preparation
Careful data preparation is essential in multivariate analysis;
otherwise, the results of analysis may be misleading. Data prepara-
tion included digitization, noise filtering and data qualification.
1.1 Digitization of data was achieved using either computer inter-
polation technique ( gravity and geochemical data ) or manually(magnetic).
For Landsat MSS data, an areal average method was used for obtaining
i n i t i a l data for regional analysis and then the computer interpolation
technique was applied in order to project the data onto the same grid
points as used in the other methods.
1.2 Spectral analysis of the raw data indicated that the power at
shorter wavelengths appears to be due to random noises. Thus, the
noise in the data has been reduced by either a specially designed
f i l t e r operator ( gravity and magnetics) or box-car smopthing f i l ter ing
using a 3 ty 3 window function ( Landsat MSS and geochemical data ).
Comparative plots of one-dimensional power spectra of the raw
and filtered data show significant reduction of energy at shorter wave-
lengths, and thus the random noise in the data has been considerably
reduced ( See Fig. )•
1.3 For data qualification, two important factors, stationarity
and normality, were evaluated for the f i l tered data •
The Landsat MSS data show some degree of stationarity, but
the geophysical and geochemical data show at bestveiytaudegrees of
stationa x i t y . However,", after removing the regional trends by
either Spencer's method or Double exponential f i l te r ing ( Davis,1973,
p226 ) f the degrees of stationarity increase rapidly, so that in general
I
288
i t can "be concluded that a l l the data are at least quasi-stationary
( See Table 3.2 ) .
Test of normality by visual inspection of histogram shows
significant skewness except Landsat MSS data, in which cases they
show, in general, symmetrical patterns.
Feature extraction
Prior to multivariate data analysis, feature extraction tech-
niques were needed, the type of technique being applied to each of the
14- data sets being determined by the physical, chemical or spectral
characteristic property. This enabled emphasis to be placed upon
regional or local features of geological significance or potential
mineralization.
2.1 In geophysics, regional features ,for example, due to magnetic
basement and/or regional gravity trends were defined by regional analysis
including lowpass f i l ter ing and upward continuation. Sources of
magnetic and gravity anomalies usu&hjdifferent from each other in this
study area. For example, the regional magnetic trend reflects base-
ment features while the regional gravity trend shows regional features
of the granite cupolas and adiacent sediment basins. Local anomaly
features of small scale magnetic and gravity sources ( e.g. perhaps
subsurface granite cusps ) were enhanced by vertical and horizontal
derivatives and highpass f i l ter ing in order to visualise those features
more clearly.
2.2 In remote sensing, feature enhancement techniques including
contrast stretching and ratioing methods, and derived colour-composites
applied to Landsat MSS data from Cornwall and the Bodmin Moor granite
area enabled the extraction of various regional features showing l i tho-
logical or stricturS-l patterns. I n particular, the granite cupolas
and many known lineaments ( e.g. wrench faults, the contact line between
289
Caledonian an ( l Amorican zones, and l ine of confrontation in the north
of Badstow, etc.) were confirmed and further linear features ( which
are not on the geological map hut are probably geologically s ignif icant)
were detected from the enhanced images.
This work for the study of lineaments shows that black-and-
white band 5 w as the best, followed by band 7> and for l i tholog ica l
mappings, the colour-composite of band ratios provide the best.
2.3 For geochemical data, level s l ic ing by concentration scale
employed shows the regional distribution patterns of geochemical elements
in relation to geology. Metallic sulphide elements including As,Cu,
Pb,Zn and Ni tend to be low in the granites compared to the surrounding
country rocks; this could be due to kaolization. These elements (
except Ni ) show also typical ly low values in the Upper Carboniferous
and Lower Devonian sediments except in areas of contamination by drainage,
etc. The metallic sulphide elements tend to concentrate in a number
of areas around the granitic aureoles.
Ga, Li, and Sn show high concentrations on the granite and i t s
margin, which may be due to the secondary enrichment by drainage systems.
They are characterized by their low values in the country rocks, parti*
cularly in the Carboniferous and Lower Devonian sediments.
Spatial distributions of the anomaly patterns of metallic
elements ( Cu, Pb, Sn and Zn ) were delineated by the probability analy-
s i s and these may be related to the local mineralizations. Sn haloes
appear in the east, southeast and along the southern border of the
Bodmin Moor granite and in the K i t h i l l granites. Also Sn haloes
appear in most of the St.Austell granite and in the northeast to east
of the St.Austell granite. The east and southeast of the Bodmin Moor
granite and the K i th i l l gimite appear to be associated with Cu haloes.
Pb and Zn haloes are generally associated in the southeast and
northwest of the Bodmin Moor granite and in the K i t h i l l granite. Zn
haloes occurring in the east of the St.Austell granite may be mainly due
290
to contamination by old mining, etc.
Trend surface analysis
Regional trends of the data sets over the Bodmin Moor area were
extracted by analysis of f i r s t - and third- degree trend surfaces, and
local irregularit ies or isolated anomalies were calculated from the
third-degree trend surface residuals.
3.1 The re l i ab i l i t i e s of the first-degree trend surfaces are generally
low in the analysis. However, two distinct groups can be found.
Group 1 consists of gravity, four Landsat MSS bands, Li, Ni and Sn; i t
trends approximately in a NW to SE direction. These variables common-
ly show most dominant features over the granite areas; bands k and Hi
and Sn show high occurrences while bands6 and 7> and Ni are typical ly
low on the granites.
Group 2 consists of As, Cu, Pb and Ga; i t trends between north
and NE. The signif icant geological features are high common occurrences
of As, Cu and Pb in the Upper and Middle Devonian rocks; the maximum
values are associated with the mineralized zones, particularly in the
east and southeast of the Bodmin Moor and in the K i th i l l granites.
There i s a low occurrence of these elements within the granite cupolas
and in the Lower Devonian and Upper Carboniferous rocks. Zn element
belongs to this category.
By contrast, Ga shows i t s high levels of occurrence within the
granites and surrounding superficial sediments where streams drain down
from the granites, while low concentrations occur in the rest of the
sediments.
3.2 The residuals derived from the third-degree trend surfaces
i l lustrate localized anomaly patterns which might be related to specific
lithology or mineral concentrations. However, detailed f ie ld checking
i s needed in order to identify these anomalies. This i s partly due
291
to low significances of the trend surface analysis, particularly near
the edge of the data.
3*3 In the analysis, the third-degree trend surfaces provide more
reliable means of assessing the regional trends and local anomaly patterns.
Similarity analysis
Three different types of similarity analyses were made to establish
inter-associationships between variables: correlation coefficients,
similarity map and coherence analysis.
4.1 The correlation coefficients which provide means of evaluating
overall similarity are computationally simple and perhaps geologically
informative (See Table 6.1 & 6.2 for the untransformed and transformed
data, respectively ). Hierachical clustering of the correlation
matrix worked better with the transformed data and the results show
distinct three groups. Group 1 consists of gravity, Sn, Ga, Li, Ni,
bands 4 and 5 their association with the descending order of correlation.
*£he common features of this group are dominant features over the granites.
Group 2 is mainly of metallic sulphide elements including As, Cu, Pb and
Zn. Group 3 is composed of bands 6 and 7 which show typically high
reflectance features over the vegetation cover.
The groupings of similarities are different from those of linear
trends. This is because only regional aspects of the sample data
contribute to the trend surface analysis which ignores detailed variations
of individual values.
4.2 The similarity maps show spatial correspondence between variables,
so that associated mineralized zones or any geologically significant
features such as structural information or geological provinces, could
be analysed.
A strong association between gravity and pseudogravity appears
in the northeastern part of the study area which may depict dense magnetic
291-1
sources. The Cornubian batholith trend shows a relatively low but
positive association.
High associations between Sn and Cu, and As and Cu, commonly
occur in the east and southeast of the Bodmin Moor and in the K i th i l l
granites. Another high spatial correlation between Sn and Cu appears
in the northeast of the St.Austell granite. Pb and Zn show their
high associations in the southeast and northwest of the Bodmin Moor and
in the K i th i l l granites. The local i t ies of these correspond to areas
of known mineral lodes except for some drainage patterns, particularly
in the southeast of the main granite. Strong negative associations
between Pb and Zn in the east of the St.Austell granite correspond to
high Zn haloes probably contaminated by old mining act iv it ies,etc.
4-.3 The coherence between gravity and pseudogravity transform of
the magnetic data was in general low. The remanent magnetization of
the magnetic source, bodies might contribute to the low coherence.
Bandpass f i l ter ings designed for particularspectral regions
such as high coherence or incoherence show structural features for those
spectral regions. In particular, stratigraphic features in the north-
ern part of the study area ( See Fig. 6.6 ), derived from the bandpass
f i l ter ing of the high coherence spectral region, might be due to shallow
dense basic rocks occurring along the stratigraphic beddings or zones of
weakness developed during the Amorican orogeny. Stratigraphic over-
lappings between Upper Devonian and Lower Carboniferous rocks might have
contributed to these effects to a lesser extent due to their differences
in magnetic susceptibi l i t ies.
Pattern recognition thechniques
Pattern recognition techniques provide much more powerful means
of extracting regional geology as well as anomaly patterns by combining
various data sets than any feature extraction techniques for individual
data set.
291-2
The resear-ch was concerned with classification mapping of regional'
multivariate data which were derived from geophysical, remote sensing
and geochemical data against the known geology. The study has shown
how various pattern recognition techniques (FA, ISODATA, EDF and CHARAN)
based on the multivariate data can be used to solve geological problems.
Multivariate analysis requires that the set of samples be suffi-
ciently large to be representative and this set of samples should be >
analysed for a large enough number of variables. Selection of the
ootimum number of variables to solve the problem being tackled is crucial.
In this study two methods of selection were used (a) trial by
intuition and experiences and (b) principal components analysis.
The first method utilized all available information on geology, geophysics,
geochemistry including correlation coefficients and the eigenvector matrix
and three different bands of untransformed and transformed data were used.
1. 6 Landsat MSS data set : E^, E5, B7, R6/5, R7/6
2. 3 geochemical element set : As, Cu, Ga, Li, Ni, Fb, Sn, Zn
3. 9 mixed variable set : gravity, magnetic, B5» R R 7 / 6 , Cu, Ni,
Sn, Zn
The second method utilized two untransformed and transformed data sets
from the first 8 PCA scores derived from the 16 variables, gravity,magnetic,
B^, B5, B7» R5A» R6/5» R7/6, As, Cu, Ga, Li, Ni, Pb, Sn, Zn
5.1 In this work, it was found that the regional lithological
mapping could be best achieved using transformed data, while untrans-
formed data were more suited to defining potential mineralization zones.
However, the Bodmin Moor granite rock unit was consistently identified
by all of the methods.
291-3
5.2 The boundaries of the sedimentary rock units in the regional
geological map are not clearly defined. The most appropriate data
set in this study are the transformed 8-geochemical one, followed by the
transformed 8-PCA score set. The transformed 8-geochemical data set
was in general better in defining the regional l ithological units,
particularly Carboniferous and Lower Devonian rocks. The Lower
Devonian rocks was similarly well-defined in the analysis of the trans-
formed 8-PCA data set.
The relatively poor definition of the sedimentary rock units
could be due to the mixture of transported material with local material
, or smoothing effect in the digitization,etc. Alternatively, this
might be due to inappropriate combinations of the multivariate data
(particularly of gravity and magnetics) for the classification, though
the gravity data might contribute significantly in classifying the granite
areas. In this study area, these potential f ields usually represent
subsurface geological features which might be irrelevant to the surface
l ithological features.
5.3 Comparative contribution of each variable to the multivariate
data was derived on the basis of eigenvalues(Fig.7.3) and eigenvectors
(Table 7.13) obtained by PCA of 16 data sets(gravity,magnetic,B^,B5,B7,
R5/^,R6/5,R7/6,As,Cu,Ga,Li,Ni,Fb,Sn and Zn). The overall contribution
of the f i r s t 8-PCA scores i s set arbitrary to be 1.
From Table 8.1, columns of "original* shows the nature and
magnitude of the contribution of each variable of the multivariate data
(i.e. positive or negative and its weight). Columns of "absolute"
illustrate the significance of the contribution by individual data sets
of the multivariate data (i.e. overall contribution of each variable to
the multivariate data set) and it is indicated by the descending "order".
291-4
Table 8.1
variably
untransformed data transformed data
variably * original * * absolute **order original absolute order Gravity 1.1092 .0670 6 .0905 .0631 8
Magnetic .1005 .0721 3 .1225 .0611 12
Band 4 -.0470 .0673 5 -.0548 .0 675 2
Band 5 -.0578 .0657 9 -.0635 .0615 11
Band 7 .0114 .0596 12 .0927 .0529 16
R5/4 -.0512 .0677 4 -.0429 .0617 9
H 6/5 .0327 .053^ 13 .1095 .0592 13 R7/6 .0452 .0657 9 .0786 .0588 14
As -.0614 .0423 16 -.0252 .0564 15 Cu -.0693 .0506 14 -.0186 .0634 7 Ga -.0436 .0656 11 -.0185 .0647 6
Li -.0729 .0659 8 -.0577 .0 675 2
Ni .0979 .0726 2 .0923 .0616 10 Pb .0378 .0 660 7 -.0039 .0654 5 Sn -.1162 .0728 1 -.1171 .0682 1
Zn .0458 .0459 15 .0119 .0669 4
* analysed results from the orig inal eigenvector matrix
* * analysed results from the absolute values of eigenvector matrix
* * * the order of overall contribution derived from the absolute matrix
In the untransformed data(in descending order),gravity,magnetic
and Ni are strong positive contributors, while Sn and to a lesser extent,
Id show strong negative contribution. In the transformed data ,
magnetic, R6/5, Band 7> Ni, gravity and R7/6 are strong positive
contributors, while again Sn and to a lesser extent, Li show strong
negative contribution.
In the analysis of the overall contribution by the absolute
eigenvector matrices, the most signif icant element i s Sn followed by
Ni, magnetic and R5/4, etc. and the least s ignif icant one i s As in the
untransformed data. In the transformed data, again Sn i s the most
s ignif icant element followed by Band 4, L» and so on. THe least
contributor i s Band 7«
291-5 However, the ratios of the least contributors to the highest
contributors are O.58 and O.78, in the untransformed and transformed
data, respectively. This indicates that even the least contributors
in this analysis are s t i l l s ignif icant in multivariate data analysis.
I t was also found that each of the pattern recognition techniques
had i t s own specific advantages in i t s method of application ( as
described in Section 7*6 ) • As a result, the techniques may be
regarded as complementary to each other. Particularly when EDF or any
supervised class i f icat ion technique i s used, i t i s almost essential to
perform the cluster analysis or any other unsupervised class if icat ion
methods to ensure that the training set for the supervised classif ication
constitutes 'natural' and 'homogeneous' groupings.
5.5 I t was indicated that combining geophysical, remote sensing and
geochemical data might constitute better multivariate data set than any
set of data drawn from any single set of properties,either • -physical,
chemical or spectral, of the region of interest, since the training set
test by BAKWRD for varying smoothing parameter (-6") has shown that 8-PGA
score sets are the most stable, followed by 9~roixed data sets and the
l a s t by 8-geochemical data sets. In addition, in a l l cases, the
transformed data are more stable than the untransformed data, indicating
favourability of the transformed data over the untransformed data in the
application of pattern reconition techniques.,
5.6 In the analysis of ISODATA and EDF, discards occurred both at
typical ly low and high values in both untransformed and transformed data.
5.7 I t i s important to note that the conclusions in this section
are tentative since the data used for some of the analyses were not as
good as orig inal ly expected.
<91-6
8.2 Summary of Recommendations for further work
Feature extraction
1.1 As pointed out in Section 4.2.2 , analysis of reduction to the pole and pseudogravity in the frequency domain is preferable to analysis in the space domain.
Regional analysis including lowpass filtering and upward conti-
nuation could also be more conveniently and effectively achieved by
analysis in the frequency domain if edge effects be avoided by an
appropriate extrapolation. This is because for regional study in the
space domain, it is needed to design a large filter operator for effective
analysis, which results in a large amount of computing time.
1.2 In geologically virgin areas, the most effective way of mineral
exploration seems to be using the abundant Landsat data ( particularly
Landsat D if it is commercially available ) for the preliminary regional
study to outline lithological boundaries, structural features, hydrothermal
alteration zones or any botanical symptoms such as clearing or any toxic
effects of vegetation by metal concentrations. This is followed by
ground checking to delimit interesting areas of economic potential for
further effective detailed survey, by either geophysically, geochemically
or combining them together with geological mapping. Thus, Landsat
MSS data would play a very important role in the future study, particularly
in remote areas where the geology is not well-known.
Pattern recognition techniques
2.1 What is optimum and effective methods in applying pattern recogni-
tion techniques for geologically unknown areas ?
(l) Probably the most important thing in the pattern recognition is
selection of variables to a smaller manageable size which can be
conveniently handled by various pattern reconition techniques.
There are two common ways of doing this.
291-7
The f i r s t method i s to study the correlation coefficients in order to
find interrelationships "between the variables. I f any strong
association i s found , then only one of the variables i s selected
and the others are removed as redundancy. Complete independent
variables( the correlations to others are very low ) may also be
removed since these variables may not be representative of the local
geologyfetc. Further analysis of eigen-values and eigenvector
matrix would indicate how: much each of the variables contributes to
the multivariate data set. I f needed, further reduction can be
made by removing those variables whose contribution to the major
components scores are low.
Alternatively, PCA can be applied to yield the principal components
scores, each of which constitutes a certain proportion of the total
variance ( represented in i t s eigenvalue ) . Those components
scores whose variance become insignif icant proportion of the total
variance, could be removed fromthe analysis with minimum loss of
information since these components scores probably consist of noise
contributions as noted by Mancey(l980),
(2) Analysing the frequency distribution of each variable :
The frequency distribution i s supposed to be normally distributed.
I f i t i s not normally distributed , then a transformation technique i s .
applied to transform the data near to- normal.
(3) Applying appropriate unsupervised class if icat ion techniques ( such
as clustering by ISODATA ) : I f the ISODATA package i s used, then
the data set i s tested for various options for optimum results ( as
described in detail in Section 7«4.2(b)) .
(4) Supervised class if icat ion techniques : I f any training set i s
obtained with the unsupervised class i f icat ion , further study with
the supervised class i f icat ion methods(e.g. EDF ) can be pursued for
comparison.
291-1694
2.2 The use of on-line interactive graphic systems to handle large
data sets could improve the efficiency and effectiveness of pattern
recognition analysis methods.
2.3 In pattern reconition studies, i t i s better to combine data sets
having different properties, (e.g. physical, chemical or spectral ), than
to use data sets derived from a single property. This i s because any
s ingle property may not always present enough c lass i f icat ion of the
regional geology or mineralization. For example, Landsat MSS data
used in this study did not provide information on potential mineralization.
2.4 One of the most important but d i f f i cu l t tasks in multivariate
data analysis i s to select an appropriate variable data set for the
best discriminating power as described before. Thus, the use of
more powerful means of setting-up the multivariate data set i s desirable.
The Ridge Regression" technique might be one of the available methods
which could be more fu l l y used.
2 . 5 I t should be noted that in some cases regional trends in the
data might have to be removed in the preparation procedures, since they
might only ref lect deep-seated basement features having l i t t l e relevance
to surface or near-surface geological targets. I n these instances,
the regional variations may res t r i c t the effective dynamic range of the
relevant data.
For th i s reason, shallow re s i s t i v i t y and radiometric data, which
ref lect only immediate surface or near-surface physical features, would
be preferable as geophysical data on which to carry out multivariate
analysis, i f geological mapping or perhaps evaluation of potential
mineralization i s required. In remote sensing, as Lyon (1976) and
Kahle et a l . (1979) pointed out, the thermal infrared bands (7-15Am)
may offer a better basis than the v i s ib le bands for geological studies
since many rock-forming minerals including s i l i ca tes , carbonates, and
sulphates, etc. show much stronger absorption peaks in the infrared
2.91-9
spectral range than in the v is ib le bands, where the distinctions are
not so well made. Therefore, the recently launched Landsat D(4),
which has their detection, would be expected to contribute greatly in
the future study of pattern recognition f ie lds .
The addition of the 1.6 /im and 2.2^um bands in i t would also
allow discrimination between unaltered and hydrothermally altered rocks
better than the visible bands as noted by Pbdwysocki et a l . (1979).
In addition, the HCMM (heat capacity mapping mission) data, covering
v is ib le and near-infrared wavelengths, 0.55 to 1.1 jxm. and the thermal
infrared, 10.5 to 12.5/Am, are also expected to be of geological value
in rock discrimination.
2 £ The latest version of the characteristic analysis (NCHARAN)
allows a more definitive means of data transformation and manipulation
by log ica l processes, this technique could be applied more widely in
the future to geological data studies.
2 y There i s no absolute guarantee that even though the individual
data set may be transformed to the near-normal distribution, the overall
multivariate data derived from such individual data sets would constitute
multivariate normal distribution patterns. I t might, therefore, be
worthwhile to apply transformation of the multivariate data set collect-
ively using suitable transformation techniques such as the power trans?-
form, after the data have been transformed into a single dimension or
made dimensionless by standardization.
2.8 Finally, i t i s recommended that this research be continued in a
variety of geological environments to provide more examples and definite
conclusions. This should be extended to allow the application of
pattern recognition techniques over areas of relatively unknown and
complex geology, possibly to locate new mineral deposits.
292
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APPENDIX A
NEW EXTRAPOLATION TECHNIQUES
A.1 Extrapolation in Frequency Domain Using Bicubic Spline Method
As mentioned in Section 3.2.2(b), in practical application of
frequency analysis, truncation of data to a limited number always intro-
duces some oscillations throughout the data, particularly at the
edges of area of data coverage. Therefore, some technique must be
applied to avoid the problem which otherwise would lead to spurious
results.
Extrapolation of data at both ends of a profile or around the
border of a map to be analysed is a method of reducing such edge
effects.
Sato (1954) first attempted to find a solution by adding a
certain number of zeros at ends of the original array. Recently Tsay
(1975) has considered two methods with greater success compared to
Sato's method. Tsay's first method is similar, at least in idea, to
Sato's except in adding an array of certain constants rather than
the zeros in Sato's method. Tsay's second method called Cosine Series
method uses the symmetry of data by adding an array T(kAX), k=-2n,
-2n+1,...,-1, to the left side of the original data to be continuous
and letting
T(-AX) = T(AX)
T(-2AX) = T(2AX)
T(-2nAX) = T(2nAX)
Tsay's first method can be very successful when the data at the edges
316
change smoothly around the background level as demonstrated by his
experiments with the upward continuation.
However, it seems to have difficulties if the data at edges
change rapidly with the sudden truncation often encountered in this
study, for example, the gravity data in the northern part of the
St. Austell granite which is in the middle of the western border of
the study area, and also the magnetic data with strong negative values
in the middle of the northern boundary area. The problem is the
violation of continuity which is required in Fourier analysis.
This may cause some oscillations in the data. In the case of Cosine
Series method, this leads to a large increase in the array size and
thus increases memory requirements and time for two-dimensional array, e.g.
100 * 100 input array will produce 200 * 200 array which can be
computationally prohibitive for practical purposes.
Since the frequency analysis assumes the data to be infinitely
continuous, we need some extrapolation of data in such a way that two
points at the edges should be coincident at zero level to minimize fictitious
oscillations and also no high frequency signal should be introduced by
extrapolation.
The most common way may be the cosine tapering. A more sophisti-
cated method can be the application of the bicubic spline method used
by Ku (1977) in his interpolation.
As shown in Fig. A.1, if we have a data profile with its length
L which has to be extrapolated by extending to lengths and at
both ends in order to make the length of data to the power of two,
which is often required for the computational convenience.
317
yl
Fig. A.l: The original profile and its extrapolations
Taking two values at each end and using the datum value (this is the
background which must be removed prior to any actual data processing,
so that the datum could be always zero), we can draw a profile which
is continuous after extrapolation as shown in Fig. A.2.
datum
yn-l n
Fig. A.2: Continuity of a profile after extrapolation
Following Ku, the simplified version of the cubic polynomial
f(x) = C_ + C. (x-x. ) + C (x-x. J 2 + C/x-x. ,)3 U I i-l L l-l 3 l-l X . , < X < X . 1-1 1 (A.l)
We can approximate the function, in order to derive extrapolating
values at edges.
318
Letting
SL(i-l) = yi-l i-2 x. - x. l-l i-2
SL(i) yi " yi-l x. - x . . 1 1-1
(A.2)
C (i) = tan(0.5(tan 1(SL(i-l)) + tan ](SL(i))))
for i = 2,3, ... ,n (A.3)
and C(1) = c (2)
C (n+1) = C (n).
The remaining coefficients C^ and C^ in Equation (A.l) can then
be obtained by the following relation:
' h 2 h 3 ^
2h. 3hT l I
r c 2u) i
C3(i)
y. - y. - h.C.(i) I l-l l 1 C^i+1) - C (i)
(A.4)
where h. = x. - x. i=l,2,...,n. l l l-l
Let us consider one edge of a profile
SL(1) = yn " yn-l x - x n n-1 V yn-l
if we express the extrapolation interval in terms of unit length,
but at the end of a profile, we force SL(2) = 0, in order to satisfy
the boundary condition at the end of extrapolation (the boundary value
and its slope are zero).
Hence, C^(x'), (^(x1) and C^(x') are readily obtained using
Equation (A.3), Equation (A.4) and the relation in Equation (A.2).
C Q can be obtained using the datum value. As mentioned above,
319
it should be zero in order to minimize fictitious oscillations
called Gibb's phenomenon in the frequency analysis.
A. 2 Extrapolation in Space Domain
As mentioned in Section 3.3.2, in order to avoid the loss of
information at the ends when convolving, it is necessary to extrapolate
the data at ends of a profile by one half of the filter window.
Since the harmonic field can be expressed usually by the
exponential function (see Equation 4.2), it might be reasonable to
consider that the adjacent values in potential fields have exponential
relationships. This justifies the use of the exponential function
for extrapolation. One further thing to be considered is that values
to be extrapolated should.be bounded to certain limits, for example,
maximum and minimum of the data, local highs and lows, or possibly some
portions of the differences of the last two values at the end of the
profile.
The mathematical procedure is as follows:
We may generalize the extrapolation function as
y = a*ebx + c (A. 5)
Rewriting Equation (A.5),
Y = a* eb x (A.6)
where Y = y - c.
Differentiating Equation (A.6),
Y' = ab*ebx = bY (A.7)
Equation (A.7) means that the slope of the function is equal
320
to the original value multiplied by a constant b. Thus the extra-
polation function has the property of the constant rate of slope
changing. This establishes the geometric property which is to be
applied to the limits of the function.
Let the difference at a particular two adjacent values be
equal to d=(y n_ ry n) and
then,
yn-l " yn =
yn ~ V H yn+l yn+2
= b*d
= b2*d (A.8)
' _ = ,N yn+N-l yn+N
where N is the number of points to be extrapolated.
Summing Equation (-A.8) at both sides and letting N tend to
infinity, then the sum S is
S = lim(y - y M ) = d(l+b+b2 + ... + b N) (A.9) yn-l •'n+N
rt-*»
Since the extrapolated function has to be asymptotic to
certain limits, the sum of Equation (A.9) should be a constant. To
satisfy this condition, |b| should be less than 1.
Therefore, S = d -rXr- (A. 10) 1 -b
Letting S = d + rd, which means the maximum amplitude of
extrapolation is some multiple (r) of d, then
d + rd = d -b
b = (A.11) 1+r
321
Accordingly, for different values of r, the extrapolation
values are readily determined.
We evaluate Equation (A.11) further in practical cases.
(i) If r = 1, then we obtain b = y from Equation (A.11). This
means that if we extrapolate the function with a maximum change the same
as the amplitude d, the rate of slope changing should be y, so that every
point to be extrapolated changes half of the difference of the
previous two points in the equal grid system.
(ii) If r = 0, then b = 0 which means no slope changing. This
is equivalent to Tsay's first method which adopts the extrapolation
by adding a certain constant at the ends.
(iii) If r = oo, then b = 1, which means a constant slope. As
mentioned above, in order for the extrapolation function to be bounded,
b should be less than 1, .so that this case is unacceptable.
From the above examples, as long as the curve does not reverse
the trend of the end two values, b must be bounded from 0 to less than
1, while r varies from 0 to a certain constant.
Therefore, by examining the data set the value r may be
reasonably determined for practical applications.
In the#author's experience, r should be less than or equal
to 2 and if the differences between the last two values at the end of
a profile are large, then r = 0.5, might be reasonable. In general
case, it is more reasonable to apply r = 1.
Direct use of an exponential function for extrapolation has
some complications in practical computing due to its property of
divergence at one end. This complication could be removed if we apply
the geometrical property of the exponential fucntion as mentioned
above, so that the computation can be very effective.
322
APPENDIX B
SAMPLING OF LANSAT MSS DATA FOR THE BODMIN MOOR AREA
The orientation of Lansat frame does not coincide to the
UTM (Universal Transverse Mercator) grid which has been used for
the British National Grid (BNG)). Fig. B.1 is a diagram showing the
relationship between the Landsat frame orientation and UTM grid
systems. In the figure, the angle -9- includes the convergence of
the meridians (i.e. the angle at the frame centre between true north
and grid north) and satellite nominal heading.
GN
O x
0 image frame centre GN direction of UTM grid north OX direction of across-track scan motion OY direction of along-track satellite motion
Fig. B.1: Frame orientation
The orientation of frame for the CCT tape image of the Bodmin Moor
area is 0.2800308466 in radians (= 16 degrees) with respect to the
UTM grid.
There is no unique way of sampling of Landsat data in the user's
323
grid system correctly, so that some approximation has to be made
by the trial and error method.
In this study, original data large enough to cover the study
area for gridding have been obtained by averaging 8 by 10 pixels.
At the same time, x and y coordinates with respect to a provisional
origin are recorded in the user's grid system by approximating each
pixel size 57 by 79 square metres and finally rescaling the coordinate
values using the distance between some reference points so as to
apply the gridding package mentioned in Chapter 3.1 for the study area.
The detailed gridding procedures are as follows.
(1) Find the approximate location of the study area in the CCT tape
and outline an area large enough to cover the study area. This may
be done by trials using the grey scale plotting program package
GEOPAK (See Chapter 4).
(2) Take a provisional origin 0'(0,0) at the top left corner
and perform sampling by averaging 8 by 10 pixels in order to approximate
an area of 600 by 600 square metres. At the same time, x and y
coordinate values of each sample are computed with respect to the
origin in the orthogonal axes of the BNG and recorded. This has been
done as follows.
Assuming a pixel size of 57 and 79 in metres in E-W and N-S
directions respectively, then the grid in a profile between adjacent
sample points is Ax = 57* 1 Ocos(-90 and Ay = 57*10sin(-90 , so that the
coordinates of x and y are shifted by Ax and Ay for the immediate next
sampling point in the profile. For the next profile, the origin is moved
by (-79*8 sin(&), 79*8cos($0) and for the rest of the data in
the profile it moves (Ax,Ay) with respect to the previous point as
before, and this procedure is continued for the rest of the data.
324
(3) Print the data with the gray scale map on the line printer
particularly for MSS band 7 since this band shows most clearly shore
lines etc. due to its relative high reflectance of land to water
From the gray scale map and topographic map, find a few control
points by approximation, so that the coordinates of the control points
can be estimated in the gray scale map, in order to calculate the
distances of x and y coordinates separately between two control
points for scaling. The origin of the user's grid system may also
be estimated (X ,Y ).
(4) Compute the scale factor since the distance AB may not be
same in both grid systems. For example, from Fig. B2 assuming the
distance AB is approximated 1 1 . and 1 „, 1 _ in x and y directions xl y 1 x2 y2
both user's grid and gray scale systems respectively, the scale factor
in the x direction is S. = 1 ,/l Likewise the scale factor in fx xl x2 the y direction is S. = 1 ,/l fy yl y2 t
(5) These are used in the gridding program; subtract xq»Y q from
recorded x and y coordinates and scale the coordinate values by
multiplying the scale factors so as to apply the gridding package for
the data of the Bodmin Moor area.
325
Although it may not represent those values in the user's grid
coordinates exactly, the regional Landsat MSS data for a partiuclar
area can be obtained reasonably by the procedures mentioned above.
The correct projection of the grid point values is mainly dependent
upon the accuracy of the control points chosen in the both grid
systems for scaling and estimation of the grid origin 0(X ,Y ).
326
APPENDIX C
CALCULATION OF FILTER OPERATORS
Baranov (1975) has described most of the filter functions
except for the pseudogravity and the horizontal derivative of the
magnetic data which have been derived by the author as part of his research,
The second vertical derivative operator has been computed by
Agarwal's method (1972). Using these filter functions all filter
coefficients have been computed for the analysis of the potential
fields.
C. 1 Calculation of continuations
From Equation (4.5) if we let the x and y coordinates in
discrete form in terms of the grid interval be k and n respectively,
the spectral function can be written as
U = H U(k,n)e k n
kai + n8i ( C . l )
Then, the downward or upward continued at (x,y) to the altitude z
can be expressed, from Equation (4.4-), as
U(x,y,z) = g Z U(k,n) — ]— n 4tt
e _ Y Z + ( k - x ) a i + ( n - y ) e i . d a d 8
-tt (C.2)
Letting x = y = 0, then
U(0,0,z) = £ Z U(k,n) — ^ 4tt
e-YZ+kai+n8idad3 (C. 3)
where U(k,n) = U(k,n,0).
We then have U(0,0,z) = £ Z U(k,n) C(k,n)
where
327
C(k,n) = 4tt"
e-Yz+kai+n8i d a d 3 (C.4)
-tt
Hence C(k,n) represents the filter function of continuations.
If the sign of z is positive, an upward continuation coefficient
is computed, while if the sign is negative, a downward operator is
computed.
C.2 Calculation of reduction to the pole
From harmonic function of Equation (4.4-), the total magnetic
field is expressed as
T(x,y,z) = -X ff T(a,3) e-Yz-xai.-y2i dadf3 (Q 5 ) 4tt J.
where T(a,3) = H T(k,n) e k a i + n 8 1 (C.5.1) k n
The expression for the field at point (x,y,z) reduced to the pole in
the northern hemisphere can be derived from the potential field
Equation (4.4-) by integrating the total magnetic field in the upward
directions of the magnetization vector and inducing field as follows,
Tn(x,y,z) = 1
4tt T(a,S)e-zy-xai-Si £ dadB (C.6) AB
where A = A o Y + A.a. + A„3-3 li 2 I B = V-y + V a. + Vi. 3 1 l 2 l
and a n d 3 r e d-"-rect:'-on c o s i n e s of ^ (unit vector
of inducing field and V (magnetization vector) respectively.
Substituting T*(a,3) by the discrete form of Equation (C.5.1)
and letting x = y = z = 0, then
328
T0(0) = Z T(k,n) n 4tt'
kai+ngi y , J 0 e -j—- dad 3 AB (C.7)
Z T(k,n)C(k,n)
Thus, the filter coefficient C(k,n) = 1 kai+n8i y 477
-it < a,3 < it
AB dad 3
(C. 8)
C.3 Calculation of first vertical derivative
From harmonic function of Equation (4.4-), we obtain the field as
g(X,Y,Z) = 4tt
where g(a,3) £ Z g(k,n)e k n
S(o.B) e-yz-xai-y6i dad0
kai+n3i
(C. 9)
(C.9)'
The vertical derivative is then
ll = _L 4tt
g(a,3)e-yz"xai-yeiY dad3 (C.10)
Replacing g(a,3) in discrete form and letting x = y = z = 0,
then
If - E g s(k'n) 7 2 4tt
kai+n3i , JQ e y dad3
Z g(k,n) C(k,n) n
where C(k,n) = , 2 4tt .. kai+n3i , ,0 e y d a d3
(C.ll)
(C.12)
-7t < a, 3 < 7t
329
C.4 Calculation of horizontal derivative in gravity
In Equation (C.9) if we integrate over z from zero to infinity,
the result is a new representation for the potential;
U(x,y,z) = - 1 4tt'
g(a,3) e ' y - dadB (C.13)
Thus the horizontal derivatives in x and y directions are
9u 9x
9u 9y
4tt
4tt'
f n v -Yz-xai-y3i ia , | g(a,3) e ' — dadS
g(as3)iYz-xCli-y3i dad3
(C.14)
The derivatives at origin (x=y=z=0) are
9u 9 x J 0 4TT2
' 3u 1 9y 0 . 2 4TT
|(a,3) y dad3
g(a,3) y dad3
(C.15)
Therefore,
9u 9x
' 9u 9y
Z g(k,n) C (k,n) n x
Z g(k,n) C (k,n) n y
(C.16)
where
Cx(k,n) = 4TT
C (k,n) = y • 2 4tt
ekai+n3i ai Y
kai+n3i 3i ,.fl e — dad 3 Y
(C.17)
330
C.5 Calculation of lowpass filters
Baranov (1975) has described that the convolution expressed
by the formula
f ( x) = F(x) (C. 18) 7TX
has the effect of replacing the function F(x) with an arbitrary
spectrum by a function f(x) whose spectrum does not exceed tt/Q in
absolute value.
The discrete form of the 2-dimensional case f(x,y) at a single
point x = y = 0 is
£(0.0) - 2 2 F(kq,nq) sin,(l,kq/Q) * " ° < W Q > . kTT nTT k n
= Z Z F(kq,nq) * C(k,n) (C.19) k n
where the lowpass filter coefficient
C(k,n) = sin(fTkq/Q) « ,in(imq/Q) ( , 5 TTk nn
and q is the grid spacing.
In the study, Q = lOq has been chosen for calculation of the
lowpass filter operator. This is equivalent to cut-off frequency
about 0.3 cycles/data interval (600m). For highpass filtering, no
calculation of the filter operator has been made. Instead, the residuals
of lowpass filtering have been computed to represent the highpass
filtering because the residual should preserve all spectra higher than
tt/Q in the data.
331
C.6 Derivation of pseudogravity
The general expressions of the gravimetric potential and
magnetic potential are
U = G f a - dv (C.21) Jv y
V = J JV(—)dv (C.22) v ^
respectively, where G is the gravitational constant, O the density
contrast and J the magnetization vector.
From the above two equations, we can write the following
relation:
-JVU = GQV (C.23)
Letting v the unit vector .of magnetization and |j|= Ga, Equation (C.23)
then becomes
-v VU = V. (C.24)
If T^ is defined as a component of the magnetic field in a
direction ofunit vector X
TX(M) = - X VV(M) = - ^ (C. 25)
Combining Equation (C.24) and Equation (C.25)
332
Fig. C.l Geometry of the Magnetic Field
In the meantime, the gravity field g^ is
3u 80 ~ 9z (C.27)
Differentiating Equation (C.27) with respect to X and V, and
Equation (C.26) with respect to z and equating both, then
d2g 0 dXdV
8T; IT (C.28)
From the harmonic function of Equation (C.5),
V M ) - - T 4tt
T(a,3)e" Y z- x a i- y 3 i dadB (C.29)
If we assume the directions of vectors X and V to be
downward as in common in the northern hemisphere, Equation (C.28) must
333
be integrated in directions opposite to the vectors since the half
lines in Fig. (C.l) do not cut any magnetic sources.
It will be defined by the references s and t as
OQ = OM + MQ = OM + Xs + Vt (C.30)
If (x,y,z) are the coordinates of M, those of the variable points
Q will be
x + X s + Vjt, y + X^s + v2t, z + X^s + v3t (C.31)
Therefore, T T
!o(x,y;z) - / " / " A J T f(«.6)e- y z- x a i- y 6 i -As-Bt 4tt 0 0 -tt
where A = X.a. + X0B. + X_y li 2 I 3 B = v.a. + V-6. + v y Ii 2 i 3
TdadBldsdt
(C.32)
(C.32.1)
Since |e ^Sds = J- and 0 0
Equation (C.32) becomes
- uu f -Bt 1 e dt -
g0(x,y,z) = — 2 47T
T(a )6)e- Y Z" x a i- y 6 ii dadg A D
(C.33)
The spectral function T(a,8) is given in Equation (C.5.1). Therefore,
the pseudogravity at x = y = z = 0 is
g0(0) = £ Z T(k,n) 1 k n 4tt'
kai + n3i Y „ ,o a I d a d 3
= g Z T(k,n) C(k,n) (C.34)
where the pseudogravity filter operator
1 C(k,n) = 4tt'
kai + n3i Y .jo AB d0tdB
- I T < a, 3 < TT (C.35)
334
Practical calculation of pseudogravity
The .pseudogravity filter operator using Equation (C.35) can be
easily obtained by applying the same notation as Baranov (1975)
used in computation of the reduction to the pole operator. „/ r\ \ kOti + n(3i Y Letting F(a,3) = e , then mtegrati AB on
Equation (C.35) becomes
1 C(k,n) = 4tt
F(a,B) + F(-a,B) + F(a,-B) + F(-a,-3)]dad3
0 < a,3 < it (C.36)
This shows that the function C(k,n) is real.
Letting again G(a,3) = Re[(F(a,3) + F(a,-B)
C(k,n) = 1 27T
G(a,3) dad3
(C.37)
(C.38)
Referring Fig. C.2 /rr jx
6 ty
C(k,n) = 1 2tt
2tt
2TT
da 0 0 rTT r 3
d3 0 0
tt >a da
0 0
G(a,3)d3
G(a,B)da
[G(a,3)+G(3,a)]d3
-7T IT a
(C.39) Fig. C.2: Reduction of the region of integration to a triangle
Letting 3 = Otu, the variable u then varies from 0 to 1 and
Equation (C.39) becomes
335
/•TT
C(k,n) = 2TT
z-1 ada
0 0 1 /-TT
du 2TT
[G(a,au)+G(au,a)]du
[G(a,au)+G(au,a)]ada (C.40)
In Equation (C.37) the two terms differ only in the sign of u,
thus Equation (C.40) can be written as /•l /-7T
C(k,n) = 1 2tt
du
0 0 (k+nu)a.
F(a,au) = e
Re[F(a,au)+F(au,a)]ada
i /1+u2 P1 Qli a D11D12
(C.41)
(C.42)
F(au,a) = e ( k u + n ) a i / H u 2 P2 Q2i
a D D 21 22
(C.43)
where
12
A32(l-ru2) + U 1 + X 2 u ) 2
2 2 2 V 3 (1+u ) + (v +\>2u)
A3v3 (1+U2) - (A^A
X1 D21 D 22
[(A]+A2u)V3 + (V]+V2)A3] /1+U' A32(1+U2) + (Aju+ A2)2
2 2 2 V3 (1+u ) + (v]u+v2) A3V3(1+U2) -(A1U+A2)(V1U+V2) [(AiU+A2)v3 + (vlU+v2)A3] /l+u'
(C.43)'
In each of Equation (C.42) and Equation (C.43) appears an
exponential factor e xai with T = T.J = k + nu in Equation (C.42)
T = T 2 = ku + n in Equation (C.43) /•TT
Then 2TT
xai da = cf(-r) + i sf(x) (C.44)
336
where Cf(x) =
and Sf(t) =
sin TTT 2tt2T l-cos TTT 2TT 2T
(C.44•0
when TTT < 0.001, Cf(x) (1 — 2TT 6
Sf ( T ) = ± (X - J L . ) ( C . 4 4 . 2 )
where X = TTT.
C.7 First horizontal derivative of the magnetic field
As mentioned before (see Equation (C.25)), the magnetic field,
is related to the potential field according to the following relationship,
X - - « (C.45)
taken in a given direction A.
Finding the partial derivatives in x and y directions we have
magnetic field components
T = - — x 8x
T y 9y
(C.46)
are Thus, the horizontal derivatives in each direction x and y
m i _ _ 92V x V 1
dx , . (C.46.1) T • -
337
As shown in (C.6), Equation (C.45) leads to the solution:
oo V = J T(x+A|t, y+X2t, z+X3t)dt (C.47)
where K x + A ^ , y+A2t, z+X3t) = — -4TT
n\ -Yz-xai-y(3i -At, ,n T(a,3)e ' y dad3
(C.47.1)
and A is the same conventional defined in (C.32.1)
the spatial function T(a,3) is
T(a,3) = Z T(k,n)e n kai+n3i (C.48)
Substituting Equation (C.48) in Equation (C.47.1)and integrating
over t, then we obtain expressions for the coefficients to calculate
the derivatives at origin ( x = y = z = 0 ) .
VOc.n) 4TT
T '(k,n) = Y 4TT
T(k.n) e k a i + n e i a 2 dad3/A
T(k,n) a 2 d a d 3 / A
(C.49)
-TT < a , 3 < tt
Kanasewich and Agarwal (1970) has shown the horizontal
gradient along any declination D to be
T'(H) = T ' sinD + T 'cos D x y
Thus, the horizontal derivative is
1
(£.50)
T'(H) = 4tt'
T(k,n) e k a i + n 8 1 ( ai sinD+3icosD)dad3/A
Therefore, T'(H) = Z Z T(k,n) C(k,n) k n
(C.51)
(C.52)
338
where C(k,n) = ekai+nBi (iasinD+iBcosD)dadB/A (C.53) 4tt'
-7t < a , b < it .
Practical calculation of the horizontal derivative
Again, using the same notation as before, the practical
computation of the filter operator can be carried out without
difficulties. kai+nBi 2 Letting C (k,n) = — X t _ JL
and
4tt J.
1 C (k,n) = _ y / 2 J 4tt
a /A dadB
kai+nBi n2
(C.54)
B /A dadB (C.55)
Let us consider Cx(k,n). First the region of integration can be
reduced to the square 0 < a, B < 7t.
•kai+nBi a^ Letting F(a,B) = e
Whence the integral C (k,n) becomes
C (k,n) = -hr x . 2 4tt [F(a,B)+F(xx,-B)+F(-a,B)+F(a,-B)JdadB
0 < a,B < 7T.
Since the above function is real, we will set again
G(a,B) = Re[F(a,3)+F(a,-B)] (C.56)
Thus, the integration Equation (C.54) is
Cx(k,n) = 2tt
G(a,B)dadB (C.57)
The small square region of integration will be divided into two
triangles by the bisector B = a, letting 3 = au.
339
We can derive the integration as follows: rl\ /-I
Cx(k,n) = - L 2tt J
ada [G(a,au)+G(au,a)]du
2TT'
0 0 RL /-TT
du
0 0
[G(a,au)[G(au,a)]ada
If we integrate from u = -1 to 1, we get, by referring
Equation (C.57)
and
,.1 C (k,n) = — ^
2tt du Re[F(a,au)+F(au,oO jada
-1
2 \ (k+nu)ai (-(X +X u)i + X0(/l+u )a F(a,au) = e 1 2 3
(X /l+u + (Xj+X u) (k+nu)aif . v e (P-Q]i)a
D 1 1
(C.58)
(C.59)
2 2 2 where D^ = X^ (l+u ) + (X^+X^)
P = X3/l+u2
Q1 = (X1+X2u)
In the same way,
F(au,a) =e (ku+n)ai a (X3/l+u + X1ui+X2i)
(C.60)
(ku+n) = (P-Q2i)a
22
2 2 2 where D 2 2 = X3 (l+u ) + (X^u+X^
P = X3 /l+u2
Q2 = A]U + X2
340
In Equation (C.59) and Equation (C.60), appears an exponential TOti
factor e with T = k+nu in Equation (C.59)
T = t 2= ku+n in Equation (C.60)
The integration over a in Equation (C.58) is then reduced aTT
1 2,2
TOti ? e a da = C f ( x ) + i S f ( x )
2 3 2 where Cf(x) = sinTTx/(2x) + costtx/(ttx ) - s in(7TT)/(T TT )
2 3 2 S f ( x ) = - COSTTX/2 + sinTTX/ (x TT) + (cosTTX-1 ) / ( X TT )
Thus Cx(k,n) = J $(u)du (C.61) -1
P C f ( x ) + Q ] S f ( x ) P 2 C f ( x ) + Q 2 S f ( x ) where $(u) = +
11 22
likewise, referring to Equation (C.56) Cx(k,n) = C^(n,k)
C (k,n) = y -1
where
$1(u)du (C.62)
R.u 2 Cf(x)+u 2 S .S f (x ) R_u 2 C f ( x ) + S.u 2 S f ( x ) V » > - , „ •
= u $(u)
Finally,
C(k,n) = C (k,n) sinD + C (k,n)cosD (C.63) x y
341
C.8 Second vertical derivative
The basic computational principle of the second vertical
derivative applied in this study starts from the 2-dimensional
Fourier transform expressed as follows. oo
G(u,v) = |J g(x,y)e-2,,i(ku+nv)dkdn (C.64) —CO
Grau (1966) has shown that if G(u,v) is circularly symmetrical,
Equation (C.64) can be written as oo
G(R) = 27T J rg(r) JQ(27TrR)dr (C.65) 0
where u = R cosOt
v = R sina
x = r cos3
y = r sin3
and OL and 3 are angles of radial vectors with respect to positive u
and x axes in frequency and space domains, respectively. J ^ ^ h t R ) is the
Bessel function of zero order defined as
R 2TT „ . „ T fo I 27TirR COSU , f„ rr\ J (2nrR) = — — e du (C.66) o 2TT J
0 Agarwal and Lai (1972) have used this expression to design second
vertical derivative filter operators using the above notation. The
discrete form of Equation (C.65) as shown by Agarwal and Lai
is M
G(p) = 4y V w(r ) J Cpr ) (c.67) 2 (__, n 0 K n S r=0
where s is the grid spacing.
342
In case of second vertical derivative, G(p) is
G(p) = p2 exp(-Ap2) (C.68 )
2 2
where p = /u +v is the radial frequency, and u and v angular
frequencies in x and y directions, respectively. X is the
weighting factor.
Agarwal and Lai (op.cit.) have computed for the case of
X = 0.1, 0.15, 0.20 and 0.23 using Rational Approximation (Agarwal and
Lai, 1971). As described by Agarwal, the application of appropriate
X depends on the feature of interest and noise levels in the data.
For example, if X is smaller, it might be good for low noise level
of data to extract smaller features, while if X is larger it can be
applied to the data with some noise level for larger features.
Most of derived second vertical derivative filter operators
are bound between X = 0.1 and 0.3. The filter operator of Rosenbach
(1953) might be equivalent to that of X = 0.1 and the filter operator
of Elkin (1951) to that of X = 0.23.
The author has calculated the derivative coefficients
for X = 0.1 to X = 0.23. Both coefficients by Agarwal et al and the
author are listed in the Table C.l
Small differences in both (i) and (ii) in Table C.l may be
due to different method of calculation of the Bessel function of the first
kind (Jq(x))given in Equation (C.65) The author has used NAG Library
subroutine in the Imperial College collection and Agarwal and Lai has
used the Rational Approximation Method. Details of the Rational
Approximation Method can be found in Agarwal and Lai (1971).
Although the NAG routine is not listed here, the general
mathematical approximation is described.
Table C.l: Weights for the different ring averages in various second vertical derivatives.
(i) Derived wei ghts
radii R n A = = 0.05 0
Smoothing Parameter .10 0.15 0.20 0 .23
0 +4 .81396 +2. 72876 +1.68768 +1.12599 +0. 90856 S -6 .19426 -1 . 27348 +0.45395 +0.99468 + 1 . 09571
s/2 + 1 .86564 -1 . 64542 -2.15992 -1.87427 -1 . 64024 s/4 -1 .00698 +0. 44528 +0.27892 +0.02197 -0. 04013 S/5 +0 .55962 -0. 27584 -0.28208 -0.27377 -0. 31599 S/8 -0 .05443 +0. 02908 +0.02698 +0.00960 -0. 00216 s/ io +0 .01040 -0. 00568 -0.00552 -0.00419 -0. 00575
(ii) Agarwal and Lai's
radii R n A = = 0.05 0
Smoothing Parameter .10 0.15 0.20 0 .23
0 +4. .81777 +2 .72648 +1.68540 + 1 .12441 +0 .90749 S -6. .21198 -1 .26288 +0.46459 + 1.00206 + 1 .10074
S/2 + 1 . .891 18 -1 .66072 -2.17527 -1.88491 -1 .64754 S/4 -1 . .03145 +0 .46357 +0.29728 +0.03473 -0 .03128 S/5 +0. .58084 -0 .29126 -0.29486 -0.28267 -0 .32221 S/8 -0. .05735 +0 .03085 +0.02875 +0.01084 -0 .00126 s/ io +0. .01100 -0 .00604 -0.00589 -0.00446 -0 .00594
344
*NAG subroutine for approximation of the Bessel function of the
first kind Jn(x)•
The routine is based on the three Chebyshev expansions;
since jq (-x ) = j^Cx), so the approximation need only consider x > 0
(1) for 0 < x < 8
r ' 2
J-(x) = ) a T (t) , with t = 2(|-) - 1 (C.69 ) 0 L. r r 8 r=0
(2) for x > 8
j (x) = / T O / P ( x) cos(x - y) - Q(x) sin(x - y) 0 / n x ^ O 4 4 (C.70 ) i
where Pn(x) = ) b T (t) 0 L, r r r=0 2
with t = 2(-) - 1 x and Q (x) = - V c T (t) o x (_ r r
r=0
(3) for x near zero, Jq(x) ^ L (C.71 )
345 Table C.2 Filter Operators.
First Vertical Derivative - . 0 0 0 1 0 - . 0 0 0 2 6 - . 0 0 0 6 3 - . 0 0 0 4 0 . 0 0 5 4 6 - . 0 0 0 4 0 - . 0 0 0 6 3 - . 0 0 0 2 6 . 0 0 0 1 0 - . 0 0 0 2 6 - . 0 0 0 8 4 - . 0 0 1 6 4 - . 0 0 4 6 1 - . 0 2 5 0 3 - . 0 0 4 6 1 - . 0 0 1 6 4 - . 0 0 0 8 4 - . 0 0 0 2 6 - . 0 0 0 6 3 - . 0 0 1 6 4 - . 0 0 5 4 8 - . 0 0 7 9 1 . 0 3 7 8 3 - . 0 0 7 9 1 - . 0 0 5 4 8 - . 0 0 1 6 4 - . 0 0 0 6 3 - . 0 0 0 4 0 - . 0 0 4 6 1 - . 0 0 7 9 1 - . 0 7 7 8 1 - . 4 5 0 5 6 - . 0 7 7 8 1 - . 0 0 7 9 1 - . 0 0 4 6 1 - . 0 0 0 4 0
. 0 0 5 4 6 - . 0 2 5 0 3 . 0 3 7 8 3 - . 4 5 0 5 6 2 . 1 8 9 7 5 - . 4 5 0 5 6 . 0 3 7 8 3 - . 0 2 5 0 3 . 0 0 5 4 6 - . 0 0 0 4 0 - . 0 0 4 6 1 - . 0 0 7 9 1 - . 0 7 7 8 1 - . 4 5 0 5 6 - . 0 7 7 8 1 - . 0 0 7 9 1 - . 0 0 4 6 1 - . 0 0 0 4 0 - . 0 0 0 6 3 - . 0 0 1 6 4 - . 0 0 5 4 8 - . 0 0 7 9 1 . 0 3 7 8 3 - . 0 0 7 9 1 - . 0 0 5 4 8 - . 0 0 1 6 4 - . 0 0 0 6 3 - . 0 0 0 2 6 - . 0 0 0 8 4 - . 0 0 1 6 4 - . 0 0 4 6 1 - . 0 2 5 0 3 - . 0 0 4 6 1 - . 0 0 1 6 4 - . 0 0 0 8 4 - . 0 0 0 2 6 - . 0 0 0 1 0 - . 0 0 0 2 6 - . 0 0 0 6 3 - . 0 0 0 4 0 . 0 0 5 4 6 - . 0 0 0 4 0 - . 0 0 0 6 3 - . 0 0 0 2 6 . 0 0 0 1 0
Second vertical derivative (<<=0 .1; normalized) 0 0 - . 0 0 1 1 9 0 - . 0 0 1 1 9 0 0 0 . 0 2 1 6 3 - . 0 9 4 4 3 . 4 7 8 2 3 - . 0 9 4 4 3 . 0 2 1 6 3 0 - . 0 0 1 1 9 - . 0 9 4 4 3 - . 7 1 9 7 7 - . 6 0 3 6 1 - . 7 1 9 7 7 - . 0 9 4 4 3 - . 0 0 1 1 9 0 . 4 7 8 2 3 - . 6 0 3 6 1 4 . 0 5 9 0 5 - . 6 0 3 6 1 . 4 7 8 2 3 0 - . 0 0 1 1 9 - . 0 9 4 4 3 - . 7 1 9 7 7 - . 6 0 3 6 1 - . 7 1 9 7 7 - . 0 9 4 4 3 - . 0 0 1 1 9 0 . 0 2 1 6 3 - . 0 9 4 4 3 . 4 7 8 2 3 - . 0 9 4 4 3 . 0 2 1 6 3 0 0 0 - . 0 0 1 1 9 0 - . 0 0 1 1 9 0 0
First horizontal derivative(gravity) - . 0 0 0 1 7 - . 0 0 0 4 1 - . 0 0 0 8 6 - . 0 0 0 4 6 . 0 4 2 0 5 - . 0 0 0 4 6 - . 0 0 0 8 6 - . 0 0 0 4 1 - . 0 0 0 1 7 - . 0 0 0 4 1 - . 0 0 1 1 4 - . 0 0 1 8 6 - . 0 0 4 2 0 - . 0 1 7 0 7 - . 0 0 4 2 0 - . 0 0 1 8 6 - . 0 0 1 1 4 - . 0 0 0 4 1 - . 0 0 0 8 6 - . 0 0 1 8 6 - . 0 0 4 9 8 - . 0 0 5 4 0 . 1 4 5 5 4 - . 0 0 5 4 0 - . 0 0 4 9 8 - . 0 0 1 8 6 - . 0 0 0 8 6 - . 0 0 0 4 6 - . 0 0 4 2 0 - . 0 0 5 4 0 - . 0 3 5 3 9 . 1 0 2 4 6 - . 0 3 5 3 9 - . 0 0 5 4 0 - . 0 0 4 2 0 - . 0 0 0 4 6
. 0 4 2 0 5 - . 0 1 7 0 7 . 1 4 5 5 4 - . 1 0 2 4 6 0 - . 1 0 2 4 6 . 1 4 5 5 4 - . 0 1 7 0 7 . 0 4 2 0 5 - . 0 0 0 4 6 - . 0 0 4 2 0 - . 0 0 5 4 0 - . 0 3 5 3 9 . 1 0 2 4 6 - . 0 3 5 3 9 - . 0 0 5 4 0 - . 0 0 4 2 0 - . 0 0 0 4 6 - . 0 0 0 8 6 - . 0 0 1 8 6 - . 0 0 4 9 8 - . 0 0 5 4 0 . 1 4 5 5 4 - . 0 0 5 4 0 - . 0 0 4 9 8 - . 0 0 1 8 6 - . 0 0 0 8 6 - . 0 0 0 4 1 - . 0 0 1 1 4 - . 0 0 1 8 6 - . 0 0 4 2 0 - , . 0 1 7 0 7 - . 0 0 4 2 0 - . 0 0 1 8 6 - . 0 0 1 1 4 - . 0 0 0 4 1 - . 0 0 0 1 7 - . 0 0 0 4 1 - . 0 0 0 8 6 - . 0 0 0 4 6 . 0 4 2 0 5 - . 0 0 0 4 6 - . 0 0 0 8 6 - . 0 0 0 4 1 - . 0 0 0 1 7
First horizontal derivative(magnetic;D=350* 1=68° ) . 0 0 1 0 8 , .00035 - . 0 0 0 5 2 - . 0 0 3 8 1 - . 0 0 4 5 4 - . 0 0 5 0 1 - . 0 0 2 0 7 - . 0 0 0 3 8 . 0 0 1 3 3 . 0 0 0 0 9 ,V0663 - . 0 2 4 8 0 - . 0 1 0 3 2 - . 0 0 0 6 7 . 0 0 0 9 9 . 0 1 2 5 2 . 0 0 8 4 0 . 0 0 0 6 0 - . 0 0 3 3 3 . 0 1 4 1 4 . 0 2 6 9 8 - . 0 0 2 1 4 - . 0 6 7 0 1 - . 0 3 3 3 2 . 0 3 5 9 5 - . 0 1 0 1 1 - . 0 0 2 0 4 . 0 1 4 7 2 - . 0 3 8 3 4 . 0 1 8 4 4 . 0 8 4 8 5 - . 1 3 1 5 7 . 1 1 9 0 8 . 0 2 8 0 2 - . 0 4 5 7 1 . 0 1 4 0 1 - . 0 1 9 2 4 . 0 1 5 5 4 - . 0 8 4 6 8 - . 1 2 3 6 8 . 5 3 9 9 1 - . 2 0 2 2 1 - . 1 1 4 5 4 . 0 1 2 8 9 - . 0 2 1 0 7 . 0 1 4 5 1 - . 0 4 0 3 9 . 0 1 9 5 3 . 0 8 3 1 0 - . 1 2 1 4 5 . 1 0 6 0 1 . 0 3 4 1 6 - . 0 4 7 8 2 . 0 1 4 5 3 - . 0 0 2 6 5 . 0 0 8 1 5 . 0 2 8 8 6 - . 0 0 7 6 5 - . 0 6 3 2 2 - . 0 3 1 3 5 . 0 3 0 9 2 - . 0 1 6 5 4 - . 0 0 2 0 8 . 0 0 0 0 3 . 0 0 7 1 0 - . 0 2 2 0 9 - . 0 0 9 1 3 - . 0 0 0 2 8 . 0 0 1 5 4 . 0 1 6 2 4 . 0 0 7 3 8 . 0 0 0 7 9 . 0 0 1 1 6 . 0 0 0 1 1 - . 0 0 0 2 4 - . 0 0 4 2 3 - . 0 0 4 3 2 - . 0 0 4 4 1 - . 0 0 2 5 8 - . 0 0 0 3 0 . 0 0 1 1 8
Lowpass filter (Q=10) . 0 0 0 5 6 . 0 0 1 4 0 . 0 0 2 5 1 . 0 0 3 6 2 . 0 0 4 4 4 . 0 0 4 7 4 . 0 0 4 4 4 . 0 0 3 6 2 . 0 0 2 5 1 . 0 0 1 4 0 . 0 0 0 5 6 . 0 0 2 5 1 . 0 0 1 4 0 . 0 0 2 9 5 . 0 0 4 8 4 . 0 0 6 6 7 . 0 0 8 0 1 . 0 0 8 4 9 . 0 0 8 0 1 . 0 0 6 6 7 . 0 0 4 8 4 . 0 0 2 9 5 . 0 0 1 4 0 . 0 0 2 5 1 . 0 0 4 8 4 . 0 0 7 5 9 . 0 1 0 2 0 . 0 1 2 0 8 . 0 1 2 7 7 . 0 1 2 0 8 . 0 1 0 2 0 . 0 0 7 5 9 . 00484 . 0 0 2 5 1 . 0 0 3 6 2 . 0 0 6 6 7 . 0 1 0 2 0 . 0 1 3 5 2 . 0 1 5 8 9 . 0 1 6 7 5 . 0 1 5 8 9 . 0 1 3 5 2 . 0 1 0 2 0 . 0 0 6 6 7 . 0 0 3 6 2 , 00444 . 0 0 8 0 1 . 0 1 2 0 8 . 0 1 5 0 9 . 0 1 8 5 9 . 0 1 9 5 8 . 0 1 0 5 9 . 0 1 5 8 9 . 0 1 2 0 8 ,00801 . 0 0 4 4 4 , 00474 . 0 0 8 4 9 . 0 1 2 7 7 . 0 1 6 7 5 . 0 1 9 5 8 . 0 2 0 6 0 . 0 1 9 5 8 . 0 1 6 7 5 . O l 2 7 7 ,00849 . 0 0 4 7 4 . O l 2 7 7 ,00444 . 0 0 8 0 1 . 0 1 2 0 8 . 0 1 5 8 9 . 0 1 8 5 9 . 0 1 9 5 8 . 0 1 8 5 9 . 0 1 5 8 9 . 0 1 2 0 8 00801 . 0 0 4 4 4 0 0 3 6 2 . 0 0 6 6 7 . 0 1 0 2 0 . 0 1 3 5 2 . 0 1 5 8 9 . 0 1 6 7 5 . 0 1 5 8 9 . 0 1 3 5 2 . 0 1 0 2 0 0 0 6 6 7 . 0 0 3 6 2 00251 . 0 0 4 8 4 . 0 0 7 5 9 . 0 1 0 2 0 . 0 1 2 0 8 . 0 1 2 7 7 . 0 1 2 0 8 . 0 1 0 2 0 . 0 0 7 5 9 0 0 4 8 4 . 0 0 2 5 1 0 0 1 4 0 . 0 0 2 9 5 . 0 0 4 8 4 . 0 0 6 6 7 . 0 0 8 0 1 . 0 0 8 4 9 . 0 0 8 0 1 . 0 0 6 6 7 . 0 0 4 8 4 0 0 2 9 5 . 0 0 1 4 0 0 0 0 5 6 . 0 0 1 4 0 . 0 0 2 5 1 . 0 0 3 6 2 . 0 0 4 4 4 . 0 0 4 7 4 . 0 0 4 4 4 . 0 0 3 6 2 . 0 0 2 5 1 0 0 1 4 0 . 0 0 0 5 6
346
Upward continuation (h=2) 00010 . 0 0029 . 0 0058 . 00096 . 00129 . 00144 . 00129 . 0 0096 . 00058 00029 . 00010 000 29 . 00071 . 00142 . 00239 . 00332 . 00371 . 00332 . 00239 . 0 0142 00071 . 00029 00050 . 00142 . 00298 . 00540 . 00817 . 00952 . 00017 . 00540 . 0 0298 001 42 . 00050 00 096 . 0 0239 . 00540 . 01107 . 01901 . 02344 . 01901 . 0 1 1 0 7 . 00540 00239 . 0 0096 00 I 29 . 0 0332 . 0 0817 .01901 . 03916 . 05310 . 03916 . 01901 . 00817 00332 . 0 0129 001 44 . 00371 . 00952 . 0234 4 . 05318 .07611 . 05318 . 02344 . 00952 00371 . 00144 00129 . 00332 . 0 0817 .01901 . 03916 . 05318 . 0 3916 . 01901 . 00817 00332 . 0 0129 00096 . 0 0239 . 00540 . 01107 . 01901 . 02344 . 01901 . 0 1 1 0 7 . 00540 00239 . 0 0096 00050 . 00142 . 0 0298 . 00540 . 00817 . 00952 . 00817 . 0 0540 . 0 0 2 9 8 00142 . 00058 00029 . 00071 . 0 0142 . 00239 . 00332 . 00371 . 00332 . 0 0 2 3 9 . 0 0142 00071 . 0 0029 00010 . 0 0029 . 0 0058 . 00096 . 00129 . 00144 . 00129 . 0 0 0 9 6 . 0 0 0 5 8 00029 . 00010
Upward continuation (h=3) 00007 . 00020 . 00039 . 00065 . 0 0092 . 0 0114 . 00122 . 0 0114 . 00092 00065 . 00039 . 00020 . 00007 00020 . 00045 . 00086 . 00140 . 00201 . 00252 . 00272 . 00252 . 00201 00140 . 00086 . 00045 . 00020 00039 . 00086 . 00162 . 00271 . 0 0403 . 0 0 5 1 9 . 00567 . 0 0 5 1 9 . 00403 00271 . 00162 . 00086 . 00039 00065 . 00140 . 00271 . 00477 . 0 0 7 4 9 . 0 1 0 1 5 . 01132 . 0 1 0 1 5 . 00749 00477 . 00271 . 00140 . 00065 00092 . 00201 . 00403 . 0 0749 . 0 1 2 6 7 . 0 1 8 3 6 . 02106 . 0 1 8 3 6 . 0 1267 00749 . 00403 . 00201 . 00092 00114 . 00252 . 00519 . 0 1015 . 0 1 8 3 6 . 02851 . 03373 . 02851 . 01836 01015 . 00519 . 00252 . 00114 00122 . 00272 . 0 0567 . 01132 . 0 2106 . 0 3373 . 04050 . 03373 . 02106 01132 . 00567 . 00272 . 00122 00114 . 00252 . 00519 . 01015 . 0 1836 . 02851 . 03373 . 02851 . 01836 01015 . 00519 . 00252 . 00114 00092 . 00201 . 00403 . 00749 . 0 1 2 6 7 . 0 1836 . 02106 . 0 1836 . 01267 00749 . 00403 . 00201 . 00092 00065 . 00140 . 00271 . 0 0477 . 0 0 7 4 9 . 0 1015 . 01132 . 0 1015 . 00749 00477 . 00271 . 00140 . 00065 00039 . 00086 . 00162 . 00271 . 0 0403 . 0 0 5 1 9 . 00567 . 0 0519 . 0 0403 00271 . 00162 . 00086 . 00039 00020 . 00045 . 00086 . 00140 . 00201 . 0 0252 . 00272 . 00252 . 00201 00140 . 00086 . 00045 . 00020
00007 . 00020 . 00039 . 0 0065 . 0 0092 . 0 0114 . 00122 . 00114 . 00092 00065 . 00039 . 00020 . 0 0007
Downward continuation (h=-l) . 00107 .00444 . 01252 . 05022 . 24866 . 05022 . 01252 . 00444 . 00107
.00444 ,01179 ,03900 . 12843 ,68208 .12843 ,03900 , 01179 ,00444
,01252 .03900 .09982 .37977 .76083-.37977 . 09982 . 03900 . 01252
,05022 ,24866 ,05022 . 01252 - . 0 0 4 4 4 . 0 0107
. 12843 - , , 68208 . 12843 - . 0 3 9 0 0 . 0 1179 - . 0 0 4 4 4
.37977 1. , 76083 . 37977 . 09982 - . 0 3 9 0 0 . 01252
1, . 20809- 5 < . 63189 1, . 20809 - . 3 7 9 7 7 . 1 2843 - . 0 5 0 2 2
•5, . 631891 5 < .59474-•5 < . 63189 1 . 7 6 0 8 3 - . 6 8 2 0 8 . 24866
1, . 20809- 5 i , 63189 1 . 2 0809 - . 3 7 9 7 7 . 1 2843 - . 0 5 0 2 2
. 37977 1, . 76083 - . 3 7977 . 09982 - . 0 3 9 0 0 . 01252
. 12843 _ , , 68208 . 1 2843 - . 0 3 9 0 0 . 0 1179 - . 0 0 4 4 4
- . 05022 . 24866 - . 0 5022 . 01252 - . 0 0 4 4 4 . 00107
347
Reduction to the pole(21*21:D=350°, 1=68° )
. 0 0 0 0 1 _ . 0 0 0 0 1 _ . 0 0 0 0 4 . 0 0 0 0 4 . 0 0 0 0 9 - . 0 0 0 0 7 - . 0 0 0 1 7 - . 0 0 0 0 9 - . 0 0 0 2 5 . 0 0 0 0 4 - , . 0 0 1 1 5 - . 0 0 0 7 0 . 0 0 0 0 2 - . 0 0 0 1 1 . 0 0 0 0 5 - . 0 0 0 0 1 . 0 0 0 0 6 . 0 0 0 0 3
. 0 0 0 0 4 . 0 0 0 0 2 . 0 0 0 0 1 . 0 0 0 0 1 - - . 0 0 0 0 4 - . 0 0 0 0 5 - . 0 0 0 1 1 - . 0 0 0 1 0 - . 0 0 0 2 1 . 0 0 0 1 5 - . 0 0 0 3 3 . 0 0 0 1 2
. 0 0 0 5 5 . 0 0 1 3 3 . 0 0 0 7 3 - . 0 0 0 2 6 . 0 0 0 0 8 - . 0 0 0 0 5 . 0 0 0 1 2 . 0 0 0 0 5 . 0 0 0 1 1
. 0 0 0 0 6 . 0 0 0 0 6 . 0 0 0 0 3
. 0 0 0 0 4 - . 0 0 0 0 5 - . 0 0 0 1 2 - . 0 0 0 1 1 - . 0 0 0 2 5 - . 0 0 0 2 0 - . 0 0 0 4 2 - . 0 0 0 2 6 - . 0 0 0 6 3 - . 0 0 0 0 1 - . 0 0 2 4 9 - . 0 0 1 5 4 . 0 0 0 0 6 - . 0 0 0 1 8 . 0 0 0 2 0 . 0 0 0 0 8 . 0 0 0 2 4 . 0 0 0 1 4
. 0 0 0 1 7 . 0 0 0 0 9 . 0 0 0 0 7 . 0 0 0 0 4 - . 0 0 0 1 1 - . 0 0 0 1 1 - . 0 0 0 2 6 - . 0 0 0 2 3 - . 0 0 0 4 9 . 0 0 0 3 7 - . 0 0 0 7 9 - . 0 0 0 3 8 - . 0 0 1 2 6 . 0 0 2 4 4 . 0 0 1 4 4 - . 0 0 0 4 7 . 0 0 0 2 9 . 0 0 0 0 7 . 0 0 0 4 4 . 0 0 0 2 7 . 0 0 0 3 7
. 0 0 0 2 2 . 0 0 0 2 1 . 0 0 0 1 0 - . 0 0 0 1 0 - . 0 0 0 0 9 - . 0 0 0 2 5 - . 0 0 0 2 3 - . 0 0 0 5 2 - . 0 0 0 4 4 - . 0 0 0 9 4 - . 0 0 0 6 7 - . 0 0 1 4 7 - . 0 0 0 2 6 - . 0 0 4 8 8 - . 0 0 3 0 8 . 0 0 0 2 5 - . 0 0 0 0 8 . 0 0 0 7 5 . 0 0 0 4 9 . 0 0 0 7 5 . 0 0 0 4 7
. 0 0 0 4 8 . 0 0 0 2 7 . 0 0 0 2 2 - . 0 0 0 0 7 - . 0 0 0 2 2 - . 0 0 0 1 9 - . 0 0 0 5 0 . 0 0 0 4 4 - . 0 0 1 0 0 - . 0 0 0 8 4 - . 0 0 1 8 0 - . 0 0 1 1 4 - . 0 0 2 9 8 . 0 0 3 9 7 . 0 0 2 7 7 - . 0 0 0 5 5 . 0 0 1 2 2 . 0 0 0 8 3 . 0 0 1 4 4 . 0 0 0 9 4 . 0 0 1 0 0
. 0 0 0 5 8 . 0 0 0 5 1 . 0 0 0 2 6 - . 0 0 0 1 8 - . 0 0 0 1 4 - . 0 0 0 4 4 - . 0 0 0 3 6 - . 0 0 0 9 6 - . 0 0 0 8 2 - . 0 0 1 9 5 - . 0 0 1 6 2 . 0 0 3 6 2 - . 0 0 1 4 0 - . 0 0 9 8 3 - . 0 0 6 1 1 . 0 0 1 5 8 . 0 0 1 3 1 . 0 0 2 8 1 . 0 0 1 8 7 . 0 0 2 0 2 . 0 0 1 18
. 0 0 1 0 7 . 0 0 0 5 6 . 0 0 0 4 6 - . 0 0 0 0 8 - . 0 0 0 3 7 - . 0 0 0 2 3 - . 0 0 0 8 5 . 0 0 0 6 3 - . 0 0 1 8 5 - . 0 0 1 5 7 - . 0 0 4 0 9 . 0 0 3 4 5 - . 0 0 8 6 9 . 0 0 5 3 0 . 0 0 7 1 0 . 0 0 2 0 2 . 0 0 5 9 4 . 0 0 3 9 3 . 0 0 4 1 5 . 0 0 2 3 3 . 0 0 2 1 4 . 0 0 1 0 9 . 0 0 0 9 6 . 0 0 0 4 3 . 0 0 0 3 2 - . 0 0 0 0 8 - . 0 0 0 7 6 - . 0 0 0 3 0 - . 0 0 1 6 8 - . 0 0 0 9 9 - . 0 0 3 8 6 - . 0 0 3 2 5 - . 0 1 0 3 9 - . 0 0 8 1 6 - . 0 2 7 0 0 - . 0 1 0 4 4 . 0 1 4 7 5 . 0 0 9 3 4 . 0 0 9 3 3 . 0 0 4 5 6 . 0 0 4 2 9 . 0 0 1 9 2 . 0 0 1 8 9 . 0 0 0 7 5 . 0 0 0 7 6
. 0 0 0 3 5 - . 0 0 1 0 9 . 0 0 0 6 5 - . 0 0 2 3 3 . 0 0 0 9 2 - . 0 0 4 8 8 . 0 0 0 6 4 - . 0 1 1 9 8 - . 0 0 4 1 6 . 0 5 4 2 3 - . 0 1 0 9 4 . 0 8 6 4 1 . 0 3 0 5 1 . 0 2 5 8 1 . 0 0 7 5 7 . 0 0 9 7 0 . 0 0 2 1 3 . 0 0 4 2 4
. 0 0 0 5 7 . 0 0 1 8 5 . 0 0 0 1 1 . 0 0 5 6 8 - . 0 0 8 6 3 . 0 1 1 8 5 - . 0 1 7 1 3 . 0 2 1 9 7 - . 0 3 2 2 9 . 0 3 9 8 4 - . 0 6 6 4 8 . 0 8 1 2 4 - . 2 5 7 5 1 . 9 4 7 4 4 . 4 0 1 8 5 - . 0 4 8 5 8 . 0 8 7 0 6 - . 0 3 1 5 5 . 0 3 9 1 4 - . 0 1 8 6 7 . 0 1 9 9 9 - . 0 1 0 3 9 . 0 0 9 8 5 - . 0 0 5 0 7 - . 0 0 0 1 9 - . 0 0 0 3 1 - . 0 0 0 4 9 - . 0 0 0 7 9 - . 0 0 1 1 8 - . 0 0 2 0 6 . 0 0 3 1 6 - . 0 0 6 5 3 - . 0 1 1 9 8 - . 0 3 8 8 5 - . 1 0 8 3 4 . 0 0 2 0 4 . 0 2 1 4 2 . 0 1 1 7 6 . 0 0 8 7 3 . 0 0 4 8 3 . 0 0 3 6 2 . 0 0 2 0 4
. 0 0 1 5 4 . 0 0 0 8 4 . 0 0 0 6 0 - . 0 0 0 1 2 - . 0 0 0 3 6 - . 0 0 0 3 4 - . 0 0 0 8 6 - . 0 0 0 8 9 - . 0 0 2 0 0 - . 0 0 2 3 9 - . 0 0 5 1 3 . 0 0 7 2 5 - . 0 1 4 1 1 . 0 0 5 1 2 - . 0 0 3 1 2 . 0 0 0 0 0 . 0 0 4 9 1 . 0 0 3 4 9 . 0 0 3 4 9 . 0 0 2 0 4 . 0 0 1 8 0
. 0 0 0 9 7 . 0 0 0 8 2 . 0 0 0 3 9 - . 0 0 0 1 9 - . 0 0 0 1 9 - . 0 0 0 4 8 - . 0 0 0 4 8 . 0 0 1 0 9 - . 0 0 1 1 7 - . 0 0 2 4 2 - . 0 0 2 7 8 . 0 0 5 3 6 . 0 0 5 5 5 - . 0 1 9 3 3 - . 0 0 8 3 8 - . 0 0 0 6 1 - . 0 0 0 0 4 . 0 0 1 8 3 . 0 0 1 2 5 . 0 0 1 5 3 . 0 0 0 8 9 . 0 0 0 8 6 . 0 0 0 4 5 . 0 0 0 3 9 - . 0 0 0 1 0 ... . 0 0 0 2 5 - . 0 0 0 2 6 - . 0 0 0 5 9 . 0 0 0 6 1 - . 0 0 1 2 5 - . 0 0 1 3 0 - . 0 0 2 4 8 - . 0 0 2 3 8 . 0 0 4 1 9 . 0 0 5 2 3 . 0 0 0 1 3 . 0 0 2 2 5 . 0 0 0 0 2 - . 0 0 0 0 3 . 0 0 0 8 1 . 0 0 0 5 1 . 0 0 0 A) . 0 0 0 3 9 . 0 0 0 4 0 . 0 0 0 2 0 . 0 0 0 1 2 - . 0 0 0 1 3 - . 0 0 0 3 1 - . 0 0 0 3 2 . 0 0 0 6 6 . 0 0 0 6 6 . 0 0 1 2 7 - . 0 0 1 1 9 . 0 0 2 1 6 . 0 0 1 4 4 . 0 0 8 1 2 - . 0 0 4 0 1 . 0 0 0 6 9 , 0 0 0 9 3 . 0 0 0 0 7 - . 0 0 0 0 2 . 0 0 0 3 9 . 0 0 0 2 2 , 0 0 0 3 2 . 0 0 0 1 7 . 0 0 0 1 7 . 0 0 0 0 6 . 0 0 0 1 5 - . 0 0 0 1 6 . 0 0 0 3 4 . 0 0 0 3 4 . 0 0 0 6 7 - . 0 0 0 6 2 . 0 0 1 1 5 . 0 0 0 8 5 . 0 0 1 7 3 . 0 0 3 3 1 . 0 0 0 6 7 , 0 0 1 2 0 , 0 0 0 2 8 . 0 0 0 4 1 . 0 0 0 0 7 . 0 0 0 0 1 . 0 0 0 1 9 . 0 0 0 0 9 . 0 0 0 1 4 . 0 0 0 0 6 . 0 0 0 0 6 . 0 0 0 0 7 - . 0 0 0 1 6 . 0 0 0 1 7 . 0 0 0 3 5 . 0 0 0 3 2 . 0 0 0 6 1 . 0 0 0 4 9 . 0 0 0 9 5
- , , 0 0 0 4 3 . 0 0 4 0 0 - . 0 0 2 0 3 , 0 0 0 3 0 , 0 0 0 5 8 - . 0 0 0 1 2 - . 0 0 0 1 9 . 0 0 0 0 5 . 0 0 0 0 1 , 00008 . 0 0 0 0 3 . 0 0 0 0 5 , 0 0 0 0 3 . 0 0 0 0 7 - . 0 0 0 0 8 , 0 0 0 1 7 , 0 0 0 1 6 • . 0 0 0 3 2 . 0 0 0 2 7 - . 00051 . 0 0 0 3 1 , 00077 , 0 0 1 9 0 . 0 0 0 5 4 , 0 0 0 5 7 , 0 0 0 1 5 . 0 0 0 2 7 . 0 0 0 0 4 . .'10008 . 000'>3 , 00000 . 0 0 0 0 3 . 0 0 0 0 1 , 0 0 0 0 2 , 0 0 0 0 3 - . 0 0 0 0 7 , 0 0 0 0 7 , 0 0 0 1 5 - . 0 0 0 1 4 - . 0 0 0 2 7 - . 0 0 0 1 9 . 0 0 0 4 1 , 00012 , 0 0 1 9 3 - . 0 0 0 9 8 ,00011 , 0 0 0 2 8 . 0 0 0 0 7 . 0 0 0 1 2 . 00001 . 0 0 0 0 3 00001 , 0 0 0 0 0 . 0 0 0 0 1 ,00001 , 0 0 0 0 2 - . 0 0 0 0 3 , 0 0 0 0 6 , 0 0 0 0 6 ~ . 0 0 0 1 3 . 0 0 0 1 0 . 0 0 0 2 1 . o o o t o
- . 0 0 0 3 3 ,00094 . 0 0 0 3 1 ,00024 , 0 0 0 0 6 - . 0 0 0 1 3 . 0 0 0 0 3 . 0 0 0 0 5 . 0 0 0 0 0 00001 ,00000 . 0 0 0 0 0
348
Pseudogravity (21*21: D=350°, 1=68° ) . 0 0002 . 00004 - . 00000 . 00012 - . 0 0 0 0 0 .00024 - . 00000 . 00039 - . 0 0 0 0 0
. 00054 . 00053 . 00060 - . 0 0 0 0 0 . 0 0063 - . 00000 . 00051 - . 00000 . 0 0 0 3 ?
. 00000 . 00013 . 00006
. 00004 . 00007 . 00012 . 00018 . 00026 . 00035 . 00045 . 00056 . 0 0 0 6 8
. 00077 . 00096 . 00100 . 00097 . 0 0097 . 00090 . 00081 . 00067 . 0 0 0 5 3
. 00038 . 00025 . 00014 - . 0 0000 . 00011 . 00018 . 00026 - . 0 0 0 0 0 . 00048 - . 00000 . 00078 - . 0 0 0 0 0
. 00111 . 00112 . 00128 - . 0 0 0 0 0 . 0 0140 - . 0 0 0 0 0 . 00120 - . 0 0 0 0 0 . 00081
. 00061 .00041 - . 00000
. 00010 . 00016 . 00024 . 00034 . 0 0048 . 00062 . 00081 .00101 . 0 0 1 2 6
. 00148 . 00190 . 00202 . 00200 . 00202 . 00190 . 00172 . 00145 . 0 0 1 1 8
. 00089 . 00062 . 00040
. 00000 . 00021 - . 00000 . 00044 . 0 0059 . 00079 - . 0 0 0 0 0 . 00131 - . 0 0 0 0 0
. 00201 . 00210 . 00249 - . 0 0 0 0 0 . 00281 - . 0 0 0 0 0 . 00237 .00201 .00161 - . 00000 . 00087 - . 00000
. 0 0017 . 00025 . 00037 . 00052 . 00071 .00094 . 00125 . 00160 . 0 0210
. 00260 . 00356 . 00390 . 00393 . 0 0396 . 00365 . 00322 . 00267 . 0 0 2 1 3
. 00160 . 00115 . 00077 - . 00000 . 00030 - . 00000 . 00061 - . 0 0 0 0 0 .00111 . 00145 . 00197 - . 0 0 0 0 0
. 00350 . 00388 .00491 - . 0 0 0 0 0 . 0 0548 . 00499 . 00422 - . 0 0 0 0 0 . 0 0 2 6 8
. 00000 . 00143 - . 00000
. 00022 . 00033 . 00049 . 00065 . 00092 . 00121 . 00169 . 00224 . 0 0 3 2 5
. 00449 . 00718 . 00840 . 00829 . 00780 . 00660 . 00542 . 00423 . 0 0 3 2 5
. 00239 . 00171 .00114
. 00000 . 00037 . 00000 . 00073 - . 0 0 0 0 0 . 00136 - . 0 0 0 0 0 . 00262 . 0 0371
. 00625 . 00846 . 01225 . 01304 . 0 1 0 6 9 - . 0 0 0 0 0 . 00650 - . 0 0 0 0 0 . 0 0 3 6 9
. 00000 . 00190 - . 00000
. 00029 . 00031 . 00061 . 00062 . 00116 . 00116 . 00217 . 00234 . 00481
. 00730 . 02527 . 03028 . 01929 . 0 1436 . 00967 . 0 0757 . 00526 . 00411
. 00280 . 00209 . 00132
. 00050 . 00001 . 00105 . 00003 . 0 0196 . 00008 . 00362 . 00030 . 0 0 7 8 3
.00294 . 08180 . 05639 . 01821 . 0 1777 . 00817 . 00891 . 00438 . 0 0 4 7 9
. 00233 . 00243 . 00111
. 00026 . 00034 . 00055 . 00067 . 00104 . 00122 . 00192 . 00237 . 0 0 4 0 9
. 00653 . 01676 . 02215 . 01636 . 01234 . 00888 . 00684 . 00498 . 0 0 3 8 0
. 00270 . 0 0196 . 00129 - . 00000 . 00034 - . 0 0000 . 0 0067 - . 0 0 0 0 0 . 00122 - . 0 0 0 0 0 . 00225 . 0 0 3 2 3
. 00489 . 00760 . 00968 . 00990 . 00864 - .00000 . 00560 - . 0 0 0 0 0 . 0 0330
. 00000 . 00174 . 00000
.00021 . 00031 . 00044 . 00061 . 00082 . 00110 . 00144 . 00194 . 0 0 2 6 0
. 00358 . 0 0457 . 00577 . 00625 . 0 0593 . 00524 . 0 0 4 3 8 . 00354 . 0 0 2 7 5
. 00207 . 00149 . 00101 - . 00000 . 00027 - . 00000 . 0 0053 - . 0 0 0 0 0 . 00095 . 00125 . 00160 . 0 0000
. 00265 . 00351 . 00394 - . 0 0 0 0 0 . 00414 . 00380 . 00332 - . 0 0 0 0 0 . 0 0 2 2 0
. 00000 . 00121 . 00000
. 00015 . 00023 . 00033 . 00046 . 00061 . 00080 . 0 0103 . 00132 . 0 0 1 6 3
. 00203 . 00224 . 00266 . 0 0293 . 00292 . 00276 . 00246 . 00209 . 0 0169
. 00130 . 00094 .00064 - . 00000 . 00018 . 00000 . 0 0037 . 00050 . 00065 - . 0 0 0 0 0 . 00104 - . 0 0 0 0 0
.00151 . 00190 .00202 - . 0 0 0 0 0 . 0 0209 - . 0 0 0 0 0 . 00180 . 00154 . 0 0 1 2 6
.00000 . 00070 - , , 00000 *
. 00008 . 00014 .00021 . 00029 . 0 0039 . 00052 . 00065 .00081 . 0 0 0 9 7
.00115 . 00119 ,00137 . 0 0149 . 0 0 1 4 8 . 00142 . 00128 . 00110 . 0 0090 ,00069 . 00049 ,00032 .00000 . 00010 .00015 . 00022 - . 0 0 0 0 0 . 00039 - . 0 0 0 0 0 . 00060 - . 0 0 0 0 0 .00083 .00101 ,00105 - . 0 0 0 0 0 . 0 0105 . 00000 . 00089 . 00000 . 0 0 0 6 ? .00046 . 00032 ,00000 ,00003 . 00006 .00010 . 0 0015 . 00021 . 00028 . 00035 .0004 4 . 000V, 1 ,00060 . 00060 ,00068 . 00073 . 00071 . 00067 . 00060 . 00050 . 0 0040 ,00029 . 00019 ,00011 ,00001 . 00003 ,00000 . 00010 . 00000 . 00019 - . 0 0 0 0 0 . 00030 . 0 0 0 0 0 .00040 . 00048 ,0004V - . 0 0 0 0 0 . 0 0 0 4 7 . 00000 . 00038 .OOOOO . 00024 ,00000 . 00010 ,00004
349
APPENDIX D GEOPAK Program
Fig. D.l Flow chart of the GEOPAK
350
Descriptions of GEOPAK
1. IN; 5 Alphanumeric characters corresponding one of
those in the right of Fig. D.l.
2. ARGS: Argument numbers, consisting maximum 5 data.
3. Subroutine : APLOT
a routine to change the map data into alphanumeric data
sets. L = ARGS(l) output file recorded in alphanumeric
characters.
4. Subroutine CARDIN:
a routine to read in the data input
ARGS(1) = input file.
Three more input cards are required for the numbers of row and
column and title.
5. Subroutine CPLOT:
Contour plotting routine
ARGS(l) = 0 Read in options from input card otherwise, compute from the ARGS
ARGS(2) = number of contours
ARGS(3) = the increment of X multiplied by 100
ARGS(4) = the increment of Y multiplied by 100.
6. Subroutine CONVOL:
8-folded convolution routine with ARGS(l) = output file and ARGS(2) \ 2
Subroutine CONALL:
ordinary convolution routine with ARGS(l) = output file and ARGS(2) = 2.
351
7. Subroutine EXPOL:
a routine to extrapolate the data around the border of a map
ARGS(1) = the amplitude of maximum extrapolation in terms of differences at edge two points
8. FILT 6:
a routine to read in the filter coefficients
ARGS(l) = filter coefficient input file number
ARGS(2) = filter size
9. Subroutine GETNUM:
a routine to read in a data from a processing file
ARGS(l) = the number of first row to read
ARGS(2) = the number of last row to read
ARGS(3) = the number of first column to read
ARGS(4) = the number of last column to read
ARGS(5) = input file number
10. Subroutine GPLOT:
a gray scale plotting routine
ARGS(l) = 1 = equal-integral slicing 2 = power slicing 3 = log-slicing 4 = Exponential slicing
ARGS(2) = 0 An option to print higher values darker
ARGS(3) \ 0 An option to print lower values darker
11. HALT9: Stop execution to end
12 LIMIO: a routine to read in level values and corresponding character sets to change map data into alphanumeric characters
13: Subroutine PRYNT:
a line-print output routine
ARGS(1) = numbers of output printing
352
Subroutine PUTNUM:
a routine to write the map data in a file. The usage of ARGS is same as subroutine GETNUM
SIZE 3:
a routine to read in numbers of rows and columns
ARGS(l) = numbers of rows
ARGS(2) = numbers of columns
Subroutine SIMI:
a routine to compute the similarity map between two maps.
This routine requires to read two input data.
ARGS(I) = 0 no standardization of data input otherwise standardize the data
ARGS(2) = output file number
SYMB 6: to read in a character set to change the map data into alphanumeric data
ZMNX7: a routine to find the maximum and minimum of the data
Subroutine NORM:
A routine to reset a data into a certain range
ARGS(l) = lower limit of the range
ARGS(2) = upper limit of the range
ARGS(3) = ouput file number
ARGS(4) = 2 Apply A power transform and reset - otherwise just reset
ARGS(5) = A value multiplied by 100, in case of applying
A transform of the data
353
APPENDIX E
CONTRAST STRETCHING USING THE PROBABILITY DENSITY FUNCTION OF GAUSSIAN DISTRIBUTION FOR PICPAC GRAY SCALE PLOTTING
The Landsat MSS data are recorded in the range from 0 to
255 and the PICPAC plotting package in ULCC utilizes a maximum
of 64 gray levels, so that the Landsat data shculd be rescaled to
lie between 0 and 63 by using image enhancing techniques such as
contrast stretching.
In the procedures, a probability function of Gaussian distribution
has been applied in image enhancing in order to plot with at ULCC
gray scale mapping package called PICPAC.
Detailed procedures are as follows:
(1) Evaluate the frequency statistics of each band from 0 to 255
and establish its cumulative frequency distribution.
(2) Calculate 64 levels of theoretical Gaussian frequency
statistics with mean zero and variance 1 using Equation 4 in Section 4.
Assign 2.5% of total frequencies at each end value and establish
theoretical cumulative frequency statistics.
(3) Compute 64 gray level values either by fitting 2-points
linear interpolation or by fitting a Chebyshev polynomial of second
order for 3 points partial interpolation as shown in the figure (
(Anderson & Houseman, 1942).
354
Fig. E.l
*DECK HISTOGRAM STRETCHING PROGRAM HSTST (INPUT »OUTPUT»TAPE5=INPUT ..TAPE6=OUTPUT 1 .TAPE11.TAPE12.TAPE13.TAPE14)
C C A PACKAGE OF FORTRAN 10 PROGRAM TO CONDUCT THE GRAY-LEVEL C SLICING FOR EITHER EQUAL FREQUENCY DISTRIBUTION QR NORMAL C DISTRIBUTION FUNCTION WITH MEAN ZERO AND VARIANCE 1. C C THE PROGRAM INCLUDES TWO METHODS J 2-POINT LINEAR INTER-C POLATION AND PARTIAL INTERPOLATION WITH SECOND-ORDER C CHEBYSHEV POLYNOMIAL. IN CASE OF SOLVING THE CHEDY-C SHEV POLYNOMIAL . NEWTON-RAPHSON ITERATION TECHNIQUE C IS APPLIED . C C INPUT CONTROL CARDS i C IOPT(l ) = 0 STOP C = 1 ANALYSIS OF EQUAL FREQUENCY DISTRIBUTION C = 2 NORMAL DISTRIBUTION C IOPT< 2) = 1 FOR SOLVING BY 2-POINT LINEAR INTERPOLATION C = 2 FOR SOLVING BY CHEBYSHEV POLYNOMIAL C IOPT(3) = LU INPUT FILE NUMBER C C N NO. OF INPUT GRAY LEVELS C NL NO. OF OUTPUT GRAY LEVFI.S c DIMENSION X (65)» Y(65).Z(65 >»F(65). FT(65) DIMENSION FF<258).ZIN(256).IOPT(3) DATA SUMF/O.0/ DATA N f NL/256 » 64/ c READ (5.5) <IOPT(I> . 1 = 1.3) 5 FORMAT(312) IA = IOPT(I) IF(IA.LE.0 ) STOP IB = IOPT(2) LU = IOPT < 3) NL1 = NL+1 PI2 = SQRT(2.*3.141592) C c rf.ad in frequenty data
rewind i i) READ < LI). I'V, (?TN< I . J-.-i . n . 10 FORMAT(1 OLA.0) r C '"QMPUT F THE CUMULiVIIVF E RED''I ni. v lUNi'MON Of- . 1 NPU 1 C
ff <1) = 0.0 do 15 1 = 1.n
15 ff(i+1 ) = ff(i)+zin(i ) ff(n + 2 > ^ ff(ne1 ) C sum = ff(nf1) C go to (100.200). ia
c r c0mpu1f the fqi'ai distribution function 0
10a 'qntinuf a l a hi im / el. dat (nl"'
C CONHMPUTE THE THEORETICAL CUMULATIVE EQUAL DISTRIBUTION C FUNCTION
DO 105 1 = 1 . NL. 1 105 F ( I ) ••= AL
FT(1 ) = 0 . 0 DO 110 1=2.NL1
110 F T ( I ) = F T ( I - l ) + F<I ) FT(NLl) = SUM GO TO ( 1 2 0 . 1 3 0 ) . I D
C COMPUTE GRAY LEVELS BY USING 2-POINT LINEAR INTERPOLATION 120 CONTINUE
CALL LINT2P(Y » FF.FT.NL » N) GO TO 300
C COMPUTE GRAY LEVELS BY USING CHEBYSHEV POLYNOMIAL 130 CONTINUE
CALL CHED2D(Y.FF.FT.N.NL) GO TO 300
C COMPUTE THE NORMAL DISTRIBUTION FUNCTION C 200 CONTINUE
DX = "J . 92/FLOAT (NL-2 ) XX = - 1 . 9 6 + DX/2 . DO 220 1=3 .NI.. F ( I ) = E X P ( - X X * X X / 2 . ) / P I 2 SUMF = SUMFEF(I) XX = XX I DX
220 CONTINUE C C ACCEPT 2 . 5 PERCENT OF TOTAL FREQUENCY FOR EACH EDGE LEVEL C
CC = 5 . * S U M F / 1 9 0 . F(1) = 0.0 F ( 2 ) = CC F(NL1) = CC SUMF = SUMF 1-2 . *CC
C COMPUTE THEORETICAL FREQUENCIES AT EVERY LEVEL DO 225 1=2.NL1
225 F ( I ) = F(I>*SUM/SUMF C C COMPUTE THE THEORETICAL CUMULATIVE FREQUENCY FUNCTION
F T ( 1 ) ' 0 . 0 DO 230 I>2.NL1
230 FT ( T ) = L ) ( I 1 ) F F ( I ) (I GO TO ( 2 4 0 . 2 5 0 ) . I P
C COMPUTE GRAY LEVELS BY USING 2-POINT INTERPOLATION 240 CONTINUE
CALI... I... I. N I 21•' i T . FF . F T « NL. . N ) GO TO 300 ^
C C COMPUTE GRAY LEVELS BY USING CHEBYSHEV POLYNOMIAL ^
250 CONTINUE CALL CHED2D(Y.EF.FT.N.NI.)
300 CONTINUE C GENERATE VALUES FOR X-AXIS FOR GRAPHS
DO 310 1=1.NL1 310 X( l> = 1 -1
WRITE(6 .320) 320 FORMAT -3X. •ORDER' ? 5X- " LEVEL " . 10X. "FREQUENCY" )
DO 325 1=1»NLI IT = 1-2
325 WFv'I TE ( 6 i 330 ) I I i Y ( I ) . F ( I ) 330 F0RMAT(5X»I3 .F0 .1rHXfFlO.n
HO 340 1 = 1 »NL1 IY = I N T ( Y ( I > + 0 . 5 )
340 Y ( I ) = IY c. C COMPUTE NO. OF FREQUENCIES AI EACH LEVEL
DO 350 1 = 1»NL1 350 Z <I) = 0 . 0
DO 360 1=1»NL 11 = Y ( I ) +1 12 = Y ( I +1 ) DO 355 J =11r12
355 Z ( I+1 ) = Z C1 + 1) + Z I N ( J ) 360 CONTINUE
CALL GRAFt C(X » F » NL1) CALL GRAF IC < X . Z • Nl. I ) END SIJDROUTINE LIN T 2F" ( Y r F F' r F f t Ml. t N ) DIMENSION Y<65)»FT(65)»FF(25G^ C J = 2 Y(1) = 0.0 DO 70 1 = 1»NL 10 CONTINUE i f t j . g t . n ) go ro go I F ( F T ( I > . E O . F F ( J ) ) GO TO 50 IF (FT ( I > . GT . FF( J - 1 ) . AND . F'T < I ) . I.T. F'F ( J > > GO TO 60 I F ( F T ( I ) . L T . F F ( J - l ) ) GO TO 40 J = J+l GO TO 10
40 J = J - l GO TO 10
50 Y ( I ) = FLOAT(J-l) J = J + l GO TO 70
60 A = F F ( J ) - F F ( J - l ) B = FF(J ) -A*FLOAT(J) Y ( I ) = ( F T ( I ) - B ) / A - 1 . J = J+l
70 CONTINUE 80 CONTINUE
Y(NIF1) = 255 . RETURN END SUBROUUNF CHflLMH V.FF.I l .N.NI i DIMENSION ( v .<>5 ) , I F i 25G > » F T « 65 > » E1< 3 ) r E2 < 3 > DATA £ 1 / 1 . , 0 . » 1 .0 . DATA E2/1 . 0 » - 2 . r 1 . <
Y(1 ) = 0.0 DO 70 I =2 ? Nt CON TINUL l F ( J . G T . N ) GO TO GO I F ( F T ( I ) , L T . F F ( J - l ) ) GO TO 45 I F ( F T ( I ) . E O . F F ( J - l ) ) GO TO 40 IF ( F" T ( I > . G1 .F F ( J - 1 ) . AND . FT ( T ) . I I . FT (..(>> GO TO .1 = JF 1 i.O 10 (5 Y( n = JL0AT< J-l >
J = JT1 GO TO 70
45 J = J - l GO TO 35
50 CONTINUE k - J - i
SM =• 0 . 0 SE1 = 0 . 0 SE2 = 0 . 0 DO 60 L = 1f 3 I.L. - N+L-l SM = SM + FFU.L) SFI1 = SF. 1 +FF ( LL ) *E 1 ( L ) SE2 = SE2+FF(LL)*E2(L>
60 CONTINUE AO = SM/3. Al SE1/2. A2 - S E 2 / 6 . D = FLOATYK > C = FT( 1 )
65 CONTINUE BJ = B-FLUAIhJ) H = ((.: • ( AO + AI *BJ+A2#3. #BJ*B.J-2 . *A?> >/<Al#B+A2*6 . *B.J> B = D + H IF ( U . o r . 0 . I ) GO TO 65 Y ( I > = B - l . J = J+l
70 CONTINUE 80 CONTINUE
Y(N L +1 ) ; 2 5 5 . RETURN END
U) Ui ON
357
APPENDIX F
TREND SURFACE ANALYSIS OF FIRST- AND THIRD-DEGREE POLYNOMIALS
F.l Least square fit for a flat plane using first-degree polynomial
As the simplest case of trend surface analysis, the first-degree
trend surface which is equivalent to a flat plane, is represented by
Z = b + b X + b2Y (F.l)
where b^, b^, b^ are the unknown coefficients to be determined, and
X and Y are the values for x- and y-coordinates, respectively.
By applying the least square solution for the equation, and
solving for the unknown coefficients, we can obtain the mathematical
expression of the trend surface.
The mathematical procedure is as follows: n 2
S = I (b +b X. + b Y.- Z.) (F.2) . , O l i 2 1 i i= 1 where n is the number of data values. If S is to be minimized, it is
^ 9 S 9S 9S re necessary that -ttt— = = -^r— = 0. (r .3) 0 3bl 2
Differentiating (F.3) against the unknown coefficients, b^, b^ and
b^, and setting equal to zero, we then obtain the three normal
equations to find the solution.
bQn + b] X + b2 Y = Z
bQ X + b] X 2 + b2 XY = XZ
bQ Y + b] XY + b2 Y 2 = YZ J
(F . 4)
In matrix form
358
n ZX ZY ZX 2 EXZ ZXY ZY ZXY 2 ZY
> ' zz "
b , = ZXZ
y , 2YZ ,
(F.5)
In more simple matrix algebra terms
A] [B] = [c; (F.6)
where [A] is the XY matrix, [B] the column b^b^b2 vector and [C]
the column vector containing Z.
Equation (E.6) can be solved in the usual fashion as
[B] = [A] ^C] (F.7)
where [A] ^ is the inverse of the matrix [A]
F.2 Least square fit using third-degree polynomial
The third-degree polynomial is
Z 3 = bn + (b X+b Y) + (b X2+b,XY+b Y 2) + (b,X3+b X2+b_XY2+b_Y3) 0 1 2 3 4 5 6 7 8 9 (F.8)
By the same manner as in the first-degree polynomial, solving for
the coefficients using the least square criteria, we obtain the normal
equation as follows:
f Zn ZX ZY . . EYJ 1 ZX 2 ZX ZXY . . 3 . . ZXY ZY ZZXY 2 ZY . . 4 . . ZY
3 ^ ZY ZXY3 ZY4 . . . . ZY6 ,
r ZZ
ZXZ
ZYZ
3 ZY Z
(F.9)
359
In matrix algebra terms;
[A] [B] = [C] (F.10)
The solution is, as before, by multiplying the inverse of matrix A
[B] = [A]"1 [C] . (F.ll)
360
APPENDIX G
EMPIRICAL DISCRIMINANT FUNCTION (EDF)
Let us assume that for each class , j = l,2,...,k, we
have measurements on a m-feature vector X = {X ,X , . . . ,X }, and I 2 m
that the a priori probabilities h , j = l,2,...,k, of occurrence
of each class are known. Let the multivariate probability density
function for the jth category be f(X), that is, the probability
that X belongs to category j. These functions may be of any form
provided that they are everywhere non-negative integrable, and that
their integrals over all space equal unity. The classifier must
perform the classification on the basis of this information with a
minimum of misrecognition.
Defining a decision function d(X), where d(X) = d^ means that
X is assigned to , let e^ be the loss (penalty) incurred if
d(X) = d., when X is a member of fi.. It is assumed that the loss J i
is zero for a correct decision. The problem is now to choose a
decision criterion such that the average loss over all class is
minimized. Fu (1968) has shown that under the optimal rule, in the
sense of minimizing the average loss, Z I L ( X ) is smaller than
under any other decision rule. Using a symmetrical loss function
d(X) = d.; e. = 0 l I d(X) = d., j \ i; e. = const. J i
when X is a member of , then the Bayes decision rule is to assign
X to the category for which h e f (X) is a minimum. r r r
361
While it may be possible to estimate the a priori probabilities
and loss-function values to the correct order of magnitude for each
category., it may be impossible to know the probability density
func t ions.
Again following Specht, we will estimate the probability
densities for each category as a sum of exponentials based on the
set of training samples, which have a positive probability of occurrence,
and we will assume that samples not in the training set but near
a given sample point (in m space) will have about the same
probability of occurrence as the training samples.
Assuming that the estimated probability density function
for a category is smooth and continuous, and that the first partial
derivatives are small, Specht proposed that an interpolation
function g(X,X.) be found such that 1
f(X) = - Z g(X,X.) m . 1 l
where m is the number of training patterns available and g(X,X^) is
the contribution of the ith training patterns to the estimated density.
If it is assumed that each training pattern contributes independently
to the overall density distribution, and that g(X,X^) is a function
of the Euclidean distance of X from the ith pattern point in m
space, following Specht we write
-(X-X.)'(X-X.) i l g(X,X.) = (2TT)m/2am
exp
where a is a 'smoothing parameter1. The estimated
denstiy function for the ft.th category is then
362
k f0 (X) = L — . t" V e
(2TT)mA2a2 k A, 1= 1
where X .. is the ith training pattern from category ft.
The probability density function (for each category of the
training set) obtained with the increase of the smoothing parameter
may in some cases overlap other functions along the concentration
scale (Fig.l). Extrapolation of the data can take place in both
directions along the concentration scale. If a relatively large
smoothing parameter has to be used in order to extrapolate satisfactorily
between data points, points on the ends of the different populations
(e.g ft and ft , Fig. 1) could be assigned to rather different categories.
f r f ) f A ( X )
2 Concentration .scole
Figure 1 Extrapolation effect of two unimodal probability density functions for categories ft^ and ft^ with increase of CJ(q <Q_).
MICROFICHE 1
GEOPAK
MICROFICHE 2
COHAN and BPFILT
MICROFICHE 3
FACTOR
Overlay showing geology and drainage
C:Carboniferous ;UD:Upper Devonian MD:Middle Devonian LD:Lower Devonian BG:Bodmin Moor Granite SG:St.Austell Granite i
Overlay showing geology and drainage