UNIVERSITY OF NAIROBI

COLLEGE OF BIOLOGICAL AND

DEPARTMENT OF GEOLOGY

SGL 413: PROJECT IN GEOLOGY

WAVE-NUMBER APPLICATION

KWEYU BONFACE

A research project report submitted for the purpose of examination in accordance with the

requirement for a partial fullfilment of a Bachelor’s of Science Degree in Geology.

UNIVERSITY OF NAIROBI

BIOLOGICAL AND PHYSICAL SCIENCES

DEPARTMENT OF GEOLOGY

SGL 413: PROJECT IN GEOLOGY

NUMBER FILTERING OF POTENTIAL DATAAPPLICATION IN ANALYSIS OF GRAVITY MAPS

Presented by:

KWEYU BONFACE ASHIKOMERA

I13/2380/2006

Supervisor

PROF. J.O BARONGO

Project Co-ordinator

DR. C. M. GICHABA

A research project report submitted for the purpose of examination in accordance with the

for a partial fullfilment of a Bachelor’s of Science Degree in Geology.

JUNE, 2010

PHYSICAL SCIENCES

FILTERING OF POTENTIAL DATA: GRAVITY MAPS

A research project report submitted for the purpose of examination in accordance with the

for a partial fullfilment of a Bachelor’s of Science Degree in Geology.

ii

DEDICATION

This piece of work is dedicated to my dear parents, brothers and sisters for their profound

support and to my beloved sweetheart for her encouragement during the process of compiling

this report.

iii

DECLARATION

I declare that this is my own original work and that any information borrowed and included into

the project has been acknowledged and references made.

Name: KWEYU BONFACE ASHIKOMERA

Registration No: I13/2380/2006

Signature……………………………… Date………………………

APPROVAL

I approve that the presented research project report satisfies the requirements for the purpose of

examination.

SUPERVISOR:

PROF. J.O. BARONGO

Signature……………………………… Date………………………

Department of Geology

PROJECT CO-ORDINATOR:

DR. C. M. GICHABA

Signature……………………………… Date………………………

Department of Geology

iv

ACKNOWLEDGEMENT

I thank Professor Justus Barongo for his spirited support guidance and his resourcefulness in

ideas towards this project. I also thank Doctor Gichaba for his instructions and modeling up of

this project to the required shape. I also acknowledge the financial assistance from my parents,

brothers and sisters and the mutual support from friends and classmates at times of hardship in

pursuit of this project.

v

DEFINITION OF TERMS AND ABBREVIATIONS

Regional- The effects of deep masses

Noise- The effects of shallow masses usually of short wavelength.

Residual-the gravity field after the regional and the noise have been removed.

Threshold/cut-off frequency-The frequency above/below which the output frequency is 70.7%

Surfer-A program used to carry out digitization process.

FFT- Fast Fourier Transform

vi

ABSTRACT

Filtering of potential data is an analytical data processing method that involves separation of

noise, regionals and residuals which are all termed as anomalies, from crude gravity data

collected and presented in gravity (Bouguer) maps. Various methods were used to filter this

crude data. These methods included upward or downward continuation of gravity data where the

field over an arbitrary higher or lower surface respectively was determined if the field was

known completely over another surface with no matter where located between the two surfaces.

Vertical derivatives were also used for data filtering where second vertical derivatives were

traditionally used to enhance local anomalies obscured by broader regional trends and aided in

the definition of the edges of source bodies. The second vertical derivatives enhance near-surface

effects at the expense of deeper anomalies hence they are a measure of curvature where large

curvatures are associated with shallow anomalies. They are evaluated by averaging over circles

of different radii. Derivatives may enhance (magnify) noise during the calculations due to

uncertainties in the measured field. Low-pass filters allow easy passage of low frequency signals,

barring and attenuating high frequency signals where as high-pass filters give easy passage for

high frequency signals and attenuating low frequency signals. In spite of their conceptual

simplicity, both of these filters work rather poorly in some applications because abrupt and

discontinuous change at threshold wavelength (λc) could result in undesirable effects during

inverse transformation and thus introducing artifacts into the data and it is often not possible to

truly separate two sets of anomalies. This is because most real source bodies will produce

anomalies that have contributions from all wavelengths in the spectrum.

vii

TABLE OF CONTENTSDEDICATION................................................................................................................................................................... ii

DECLARATION AND RECOMMENDATION..................................................................................................................... iii

ACKNOWLEDGEMENT .................................................................................................................................................. iv

DEFINITION OF TERMS AND ABBREVIATIONS ...............................................................................................................v

ABSTRACT ..................................................................................................................................................................... vi

TABLE OF CONTENTS ............................................................................................................................................ vii

LIST OF FIGURES ......................................................................................................................................................... viii

1.0 CHAPTER ONE ..........................................................................................................................................................1

1.1 INTRODUCTION........................................................................................................................................................1

1.2 PROBLEM STATEMENT ............................................................................................................................................3

1.3 OBJECTIVES OF THE STUDY......................................................................................................................................3

1.4 SIGNIFICANCE/ JUSTIFICATION OF THE STUDY........................................................................................................3

2.0 CHAPTER TWO .........................................................................................................................................................4

2.1 LITERATURE REVIEW................................................................................................................................................4

3.0 CHAPTER THREE.......................................................................................................................................................6

3.1 MATERIALS AND METHODOLOGY ...........................................................................................................................6

3.1.1 Materials...............................................................................................................................................................6

3.1.2 Methodology ........................................................................................................................................................6

3.2 Filter designs............................................................................................................................................................7

4.0 CHAPTER FOUR ........................................................................................................................................................8

4.1 THEORY ....................................................................................................................................................................8

4.1.1 Low-pass filters .....................................................................................................................................................9

4.1.2 High-pass (short-wavelength) filters ....................................................................................................................9

4.1.3 Upward continuation of gravity data..................................................................................................................10

4.1.4 The Vertical derivatives ......................................................................................................................................10

4.2 Wave-number Filtering..........................................................................................................................................11

4.3 Wave-number Filtering Program ...........................................................................................................................12

4.4 RESULTS (Upward continuation)............................................................................................................................15

4.5 LIMITING ERRORS ..................................................................................................................................................18

4.5.1 Gibb’s Phenomenon...........................................................................................................................................19

3.5.2 Leakage...............................................................................................................................................................19

4.6 CORRECTION OF GIBB’S PHENOMENON AND LEAKAGE......................................................................................19

4.7 DISCUSSIONS .........................................................................................................................................................20

CONCLUSIONS .............................................................................................................................................................21

REFERENCES.................................................................................................................................................................22

viii

LIST OF FIGURES

Figure 1: Complete Bouguer gravity anomalies map with contour interval of 10 mGal ...............................................6Figure 2: Flow chart for the wave-number filtering method: continuation of △g0 to an elevation............................14Figure 3: Original Bouguer anomaly map with gravity scale. Contour interval is 10mGal ........................................15Figure 4: Gravity map after data continuation of 3m above ground surface ...............................................................16Figure 5: Gravity map after data continuation 5m above ground surface....................................................................17Figure 6: Gravity map after data continuation 10m above the ground surface. ...........................................................18

1

1.0 CHAPTER ONE

1.1 INTRODUCTIONGravity data analysis has been aided greatly over the past decade by the widespread use of digital

computers and by the availability of continent-wide gravity and topographic data bases in

computer compatible form. There now exist reliable and efficient gridding and contouring

algorithms which open door to many sorts of experimentations in the enhancement and display

of data.

In most areas where gravity surveys are carried out there are deep-seated structural features

causing variations in gravity over the surface which are much larger in areal extent than the

structures ordinarily of interest in prospecting. For example, in the Rio Grande Valley, there is an

increase in regional gravity of about 1 milligal per mile toward the Gulf of Mexico as one moves

eastward from a point 7 miles inland.

Regional gradients such as these often distort or obscure the effects of structures like salt domes

or buried ridges that might result in oil traps. For this reason we must subtract out the regional

effects in order to isolate more clearly the structural features in which we are more interested.

There are several methods of removing regional he gravity so as to leave the residuals as they are

called. Some are graphical others are analytical.

In graphical approach, the regional effect must be estimated from the contours of observed

gravity; regional contours are interpolated more or less arbitrarily, being superimposed over the

original contour lines. The regional field is then subtracted from the observed gravity field.

With analytical methods of determining residual gravity, routine numerical operations on the

observed data make it possible to isolate anomalies of economic interest without having to

exercise any great amount of judgment. Such techniques generally require that gravity values be

spaced in a regular array. Analytical approaches in common use include the determination of

second derivatives, for which several standard computational formulas are available, analytical

continuation, which transforms the gravity field as measured at the surface to the field that would

be observed on the horizontal plane buried at some specified depth or elevated to a higher level.

High-pass (with similar effect as vertical derivatives) and low-pass (with effects related to

2

upward continuation) methods are used. High-pass filters allow easy passage of higher frequency

signals above set threshold frequency and attenuate frequencies below this threshold frequency.

On the other hand, low-pass filters allow easy passage of low frequency signals and attenuating

those above the threshold frequency.

Filtering techniques, in some cases originally designed for processing or image enhancement

have been successfully applied to gravity data to reveal features and relations that were not

obviously original contoured presentations of the data. Needless to say, all such features require

careful checking to verify that they truly exist in the original data and that they are not simply

artifacts introduced by the filtering or the enhancement process. It is also important to note that

in the bid to discuss filtering methods, Fourier transform techniques form the basis of the design

and use of filters.

Fourier transform is a powerful tool for analyzing a function of time f(t). Efficient computer

algorithms exist to calculate Fourier transforms and their inverse rapidly, and complicated filter

operations in time domain, for example can become simple multiplications in the Fourier

domain. The Fourier transform equation is as follows:

F(kx ,ky )=∬ ( , ) (∞−∞ kx x+ky y)dx dy - i ∬ ( , ) (∞−∞ kx x+ky y)dx dy

Where kx and ky are called wave numbers with respect to x and y axes, respectively; and have

units of radians per kilometer if x and y are measured in kilometers. Wave numbers are the

spatial counterparts of frequency (measured in radians per second) and are inversely proportional

to wavelengths λx and λy ;

kx = and ky=

3

1.2 PROBLEM STATEMENT

In ideal wavelength data filtering the gravity data are gridded and transformed into the Fourier

transform domain and the wavelengths (usually very long ones) corresponding to regional trends

are removed by an appropriate filter, for instance a high-pass filter that leaves the

shorter-wavelength residual gravity anomalies. The fact that components of the regional field

often cover a wide range of wavelengths makes it difficult to accomplish the separation

satisfactorily in this way. In some cases, the short-wavelength components of the regional field

cannot be removed without removing significant portions of the residual. Moreover, shallow

features to be enhanced will have long-wavelength components of considerable significance in

their spectra that produce an overlap with regional spectra that limits the usefulness of the

technique. Thus, it’s not as simple as it might seem to design a spatial filter that will be effective

in separating residual from regional fields

1.3 OBJECTIVES OF THE STUDYThe main objective of this research is to learn how to use various mentioned filtering methods to

filter potential data. Alongside this are:

To be able to design 2-d filters for low pass and high pass filters, upward continuation and

vertical derivative.

To learn the various practical applications of the filters in real life situation.

1.4 SIGNIFICANCE/ JUSTIFICATION OF THE STUDYGravity data processing and interpretation play a pivotal role in oil and mineral resource

utilization. For instance, in the use of gravity techniques of geophysical prospecting, different

rocks have different gravity. It is therefore very important to understand this property for you to

be able to recognize the rock stratigraphy, vital for determining the depth to which a given ore

body, water body or oil occurs. Therefore for us to be able to meet all these requirements, gravity

reduct methods and data processing techniques involving filtering methods to give accurate or

near accurate effects of the buried anomalies (residuals) is inevitable. This study will therefore

provide important knowledge useful in both graphical and analytical interpretation techniques.

4

2.0 CHAPTER TWO2.1 LITERATURE REVIEWGravity data interpretation is the key to potential data filtering and processing techniques. This

started with very basic gravity reduct methods such as latitude, terrain, free-air, tides and

Bouguer corrections. When the corrected gravity values reduced by methods such as those

outlined are plotted on a map and contoured, the resulting picture will seldom give much usable

information on the subsurface geology until it is analysed by suitable interpretation techniques.

Direct computation of residuals, the simplest analytical approach was described by Griffin

(1949). It involves the averaging of gravity values along the periphery of a circle or regular

polyhedron with its centre at the point for which the residual is computed. The average value

around the circle is simply the arithmetic mean of a finite number of equally spaced points about

its circumference. The residual value is simply the observed value at the centre minus this

average. The principal problem with this method is in the choice of a radius. This must be large

enough that the circle will lie entirely outside the anomaly but not so large as to include

irregularities from other sources.

Elkins in his paper discusses the second vertical derivative where he points out that the

suitability of this method is where the spacing of the gravity readings is close and where the

precision is high. On maps of observed gravity such anomalies may be obscured or hidden

altogether by regional trends as well as by the effects of other small features in close proximity.

As the vertical rate of change of the change of gravity with depth, the second vertical derivative

magnifies the gravity effect of smaller and shallower depths. Thus the geologic structures of

greatest interest in oil and mineral exploration are emphasized at the expense of large regional

structures.

In a classic paper published in 1949, Peters, using the process of downward continuation of

gravity data showed how a potential field (in his paper, the magnetic field) measured at the

earth’s surface can be analytically projected upward or downward, i.e. mathematically

transformed to what it would be if it could be measured either above the earth’s surface or along

a horizontal plane inside the earth. Downward continuation has particular applicability to gravity

interpretation. A deeply buried source of limited size will not yield a gravity effect that stands

5

out prominently above background. The computation procedure of this method is similar to that

used in second derivative calculations, a series of circles with appropriately spaced windows

being superimposed over a grid of observed gravity values. A variation upon the technique

outlined by Peters has been proposed by Trejo.

However, among those concerned with analytical procedures, controversies have developed as to

the best coefficients for derivative and continuation calculations. Actually there is no best choice

of coefficients, the selection depending on the problem at hand. It is also important in choosing

techniques for determining residual gravity to take into account the precision of the original data.

Errors in the instrumental readings or, more significantly, very local sources of anomaly could

result in misleading residual and derivative maps if appropriate ring diameters and weighting

coefficients are not used. Grant has developed a statistical technique, analogous to least squares,

which smoothes out local disturbances and observational errors in the course of extracting

residual gravity.

All analytical methods involving interpolation of grid values from contour maps have one

disadvantage in common, namely, namely that the mode of contouring itself influences the final

maps (such as the second derivative maps) very strongly. As Romberg remarked, the second

derivative map may be more an interpretation of the contouring than the gravity when the

separation of data points is greater than the grid spacing. The removal of regional effects is one

of the two most important problems in gravity interpretation. The other problem is the deduction

from the residual field of geological information about its source.

6

3.0 CHAPTER THREE3.1 MATERIALS AND METHODOLOGY3.1.1 MaterialsThe material used in this project is a complete Bouguer map derived from regional gravity data

over eastern Sierra Nevada of California with a contour interval of 10 mGal as shown;

3.1.2 MethodologyBouguer maps are obtained from the results of field surveys, for instance gravity surveys where

data obtained by use of gravity meters (gravimeters) is computed and all corrections taken care

of. This results to gravity map that is then digitized for use in the filtering process. Various

digitization methods used include the use of surfer program, Arc View and the Arc GIS.

There are different types of filtering processes that are done using various kinds of filters.

Among them and here discussed include 2-dimension filters for upward continuation of gravity

data, vertical derivatives, low-pass filters and high-pass filters.

Figure 1: Complete Bouguer gravity anomalies map with contour interval of 10 mGal

7

These four methods can be classified into two basing on their related effects they have on data

i.e. upward continuation is a form of low-pass filter whereas vertical derivative is a form of

high-pass filter.

3.2 Filter designs Filter design is a highly developed art that is intended to convey a concept on how filters operate

in the Fourier domain. For example, if geologic information is available to suggest that

interfering sources are at rather different depths, or that they posses rather different frequency

attributes, then it may be possible to use the spectral characteristics of the gravity field to design

a filter best suited to the separation task at hand. Some such filters have been called matched

filters and sometimes provide both the best way of separating anomalies and a means of telling

from the spectrum how inherently separable a set anomalies may be.

Most filters fall into two categories;

i. Filters relying on Fourier transform of the field (spatial domain)

ii. Convolution methods (space domain)

Fourier transform techniques involve converting the data set into frequency domain, operating on

this data set in some way and then returning it in the space domain. This allows different spatial

frequencies (wavelengths) of data to be highlighted or suppressed.

8

4.0 CHAPTER FOUR4.1 THEORYFiltering of potential data is a key role in potential data processing and interpretation. Its success

therefore depends on the clarity of collected data and the accuracy of the instruments used.

Potential data filtering involves the removal of noise and the separation of regionals and

residuals.

Bouger maps show horizontal differences in the acceleration of gravity and thus only horizontal

changes produce anomalies. Purely vertical changes in density produce the same effect

everywhere and so no anomalies result. The gravity field is a superposition of anomalies

resulting from density changes (anomalous masses), at various depths. The smoothness (apparent

wavelength) of anomalies is generally roughly proportional to the lateral density changes. The

effects of shallow masses usually called near surface noise are usually of short wavelength. They

can be removed by largely filtering out (smoothing) short wavelength anomalies. The effects of

deep masses are called the regional. The gravity field after near-surface noise and the regional

have been removed is called the residual which presumably represents effects of the intermediate

zone of interest.

Wavelength filtering methods for the separation of residuals from regionals are based on the

degree of smoothness (or wavelength=) of anomalies. Filtering can also be done by transforming

map data to wave-number-wave-number domain using a two dimensional Fourier transform,

removing certain wave-number components (filtering) and the doing of an inverse

transformation to reconstitute the map but with certain wavelengths removed. What are removed

as the small wave-numbers (large wavelengths) of the regional, so that wave-number component

involved in the inverse transform are the large ones which correspond to the short wavelengths

of the residuals.

Various wavelength filtering methods used are as listed below.

Upward continuation

Vertical derivatives

Low-pass filters

High-pass filters

9

4.1.1 Low-pass filtersA low-pass filter is a circuit offering easy passage to low-frequency signals and difficult passage

to high-frequency signals. These higher frequency signals usually above the cut-off frequency

(threshold frequency) will have their amplitudes reduced, a process known as attenuation. The

cut-off frequency for a low-pass filter is that frequency at which the output (load) voltage equals

70.7% of the input (source) voltage.

In the Fourier domain, a convenient place for the application of filters that are designed to

eliminate certain undesirable wavelengths, if wavelengths less than λ= λc are deemed

undesirable in f1(x, y), we should define a filter F2(kx,ky ) such that;

F2(kx,ky ) = ⎷( + ) > ℎ

�

This type of filter is called a low-pass or long-wavelength filter because all wave-numbers less

than (or wavelength greater than λc ) are passed by the filter without modification.

4.1.2 High-pass (short-wavelength) filters A high-pass filter is a signal processing filter that passes high frequencies well but attenuates

(i.e., reduces the amplitude of) frequencies lower than the cut-off frequency. The actual amount

of attenuation for each frequency is a design parameter of the filter. It is sometimes called a

low-cut filter and the terms bass-cut filter or rumble filter are also used in audio applications. As

with low-pass filters, high-pass filters have a rated cut-off frequency above which the output

frequency increases above 70.7% above the input voltage. This filter rejects wavelengths longer

than the threshold (cut-off) frequency. For example;

F2(kx,ky ) = ⎷( + ) < ℎ

�

10

4.1.3 Upward continuation of gravity dataWhen gravity data measured on one surface is transformed to some higher surface, the process is

referred to as upward continuation. Upward continuation is therefore a filter operation that tends

to smooth the original data by attenuation of short wavelength anomalies relative to their longer

wavelength counterparts which is a logical consequence of the attenuation of anomaly amplitude

with increasing distance from the source. The filter (upward continuation), F2( kx,ky ) is

elegantly simple for the special case of data measured on a flat surface.

F2 ( k x ,k y ) = e-kz where k=⎷(k + k ) and z (>0) is the distance of upward

continuation. Upward continuation requires no knowledge about the actual shape of the masses

or their density distributions. From the above equation, all wave-numbers are attenuated by e-kz

as we move away from the source. There is also a rapid attenuation of highest wave-numbers

(shortest wavelengths).the operations can be carried out in space domain using the convolution

algorithm or the Schwarz-Christoffel transformation which deals with two dimensional sources

only. The Fourier domain may be employed for calculations from a horizontal surface.

Remarks;

Upward continuation resembles a very smooth low-pass filter. Like standard low-pass filters,

upward continuation often provides perspective concerning the large regional sources beneath a

study area, but its simple physical interpretation (how data would look if measured on the higher

surface) sometimes offers a definite advantage over the application of low-pass filter with less

obvious physical significances.

4.1.4 The Vertical derivativesThe definition of the edges of source bodies has been aided by the use of second vertical

derivatives to enhance the local anomalies obscured by broader regional trends. Geologic

features like salt domes which have a limited lateral extent and occurring shallowly will typically

have a gravity anomaly with greater curvature than the regional field probably originating from

deeper sources on which it is superimposed. The second vertical derivative gives a measure of

the difference of the gravity value at a point relative to its values at neighboring points.

The second vertical derivative is therefore greater over a localized feature than over the more

smoothly varying regional trend. Vertical derivatives enhance anomalies caused by small

features and suppress longer wavelength regional trends and as a result, they are regarded as

11

high-pass filters. The nth vertical derivative of a gravity field can be expressed as a very simple

form in the Fourier transform domain as;

F2 (kx , ky ) = (-k)n where k=⎷(k + k )

As with upward continuation, no information is required about the sources to calculate a vertical

derivative. It also greatly enhances short wavelength noise and other defects in the data. This

method has a great advantage in that the quality of a data set can sometimes be quickly spotted

by the application of derivative filters to a grid from the data. However, higher derivatives than

the second are not used. Each differentiation enlarges the imperfections of the field data and

these are in general too large to permit the use of still higher derivatives.

4.2 Wave-number FilteringWave-number filtering refers to the isolating or enhancing of data in the wave-number or

frequency domain. To perform wave-number filtering it is necessary to convert anomalies in the

gravity field, represented by a data matrix along an X,Y coordinate system, to a two-dimensional

set of amplitudes over a range of frequencies or wave-numbers. This is done with the Fourier

integral. The Fourier integral can be used to transform a data set in the space domain to the

frequency or wave-number domain. Once in the wave-number domain, the proper filter can be

applied. The filtered data in the wave-number domain can then be transformed back into the

space domain in the same manner using the inverse of the Fourier integral.

It is desirable to filter the data in a way that will isolate or enhance certain features. One may wish

to see anomalies of only a certain size or of a certain wavelength. The words frequency,

wavelength and wave-number are often incorrectly used interchangeably. Frequency refers to

period, usually dealing with time, as in cycles per second. It may also deal with space, as in cycles

per kilometer. Wavelength is the inverse, seconds per cycle or kilometers per cycle. The term

wave-number is in reference to the mathematics involved in the computer program. After

converting a data set from the space domain to the frequency domain, one has a set of values or

amplitudes at discrete intervals. The first wave-number is the frequency associated with the first

amplitude. The range of frequencies that can be observed is 0 to 1/(2 times the station spacing).

12

The number of wave-numbers in this range depends on how much data is available. Once the

wave-number spectrum of the data set is known, the wave-numbers that would most likely contain

the information of interest can be isolated by cutting out other wave-numbers. One way to achieve

this is multiplying the spectrum by a box function. The wave-numbers to be saved are multiplied

by 1 and the wave-numbers to be deleted are multiplied by 0. When the data set is transformed

back into the space domain, only anomalies of a certain size remain. Alternately, certain

wave-numbers can be multiplied by factors greater than 1 and other wave-numbers could be

multiplied by factors less than 1. This would enhance certain anomalies without actually cutting or

deleting any of the spectra.

4.3 Wave-number Filtering Program

This program performs various types of filtering on a matrix that has been prepared by SETUP.

This program, also written in VAX/VMS Fortran-77, performs the following steps:

1. The selected filter is created.

2. Each row of the input matrix (the results from SETUP) is read in and multiplied by the

corresponding row in the filter matrix.

3. The resulting matrix is transformed into the spatial domain by the Fast Fourier Transform.

4. If the original data was extended to four times its original size to limit edge effects, only

one quadrant of the matrix is now written to a direct access output file.

5. A summary file containing all parameters used in the filtering of the data is written. This

file will have the same name as the output data file, except with a FIL extension.

Four types of filters are available. They are High-pass, Low-pass, Upward/Downward

continuation, and Nth order vertical or horizontal derivative.

The program will ask the user to input the following.

1. Name of file from SETUP.

2. Name for output file. After filtering, this file will be contoured and plotted.

3. Which type of filtering is desired.

If Nth order derivative filtering is selected:

4. Order of the spatial derivative. Usually one performs first and second derivatives.

5. Direction in which to calculate the derivative. The user can select vertical, horizontal in the

X direction, or horizontal in the Y direction.

If upward or downward continuation filtering is selected:

6. Distance upward to continue the data. If downward con

number. Distance must be in the same units as given for the station spacing in SETUP.

For all filters, the program will be asked for:

7. Whether to add the mean that was removed in SETUP back to the data set. This has

meaning to derivative filters and little use in the other

This program filters a data set of 128 by 64 extended to 256 by 128 by SETUP in less than one

minute of CPU time on the VAX 11

The method used to filter gravity data using the Fou

in figure 2. When one begins with the original gravity data, taking Fourier transform of

, the gravity data in the wave

filtered gravity in the wave-number domain. Taking the inverse Fourier transform of

final answer, , the filtered gravity data.

13

Order of the spatial derivative. Usually one performs first and second derivatives.

Direction in which to calculate the derivative. The user can select vertical, horizontal in the

X direction, or horizontal in the Y direction.

If upward or downward continuation filtering is selected:

Distance upward to continue the data. If downward continuation is desired, enter a negative

number. Distance must be in the same units as given for the station spacing in SETUP.

For all filters, the program will be asked for:

Whether to add the mean that was removed in SETUP back to the data set. This has

meaning to derivative filters and little use in the other filters.

This program filters a data set of 128 by 64 extended to 256 by 128 by SETUP in less than one

minute of CPU time on the VAX 11-750.

The method used to filter gravity data using the Fourier Transform can be seen diagrammatically

. When one begins with the original gravity data, taking Fourier transform of

, the gravity data in the wave-number domain. Multiplying by a filter gives

number domain. Taking the inverse Fourier transform of

, the filtered gravity data.

Order of the spatial derivative. Usually one performs first and second derivatives.

Direction in which to calculate the derivative. The user can select vertical, horizontal in the

tinuation is desired, enter a negative

number. Distance must be in the same units as given for the station spacing in SETUP.

Whether to add the mean that was removed in SETUP back to the data set. This has no

This program filters a data set of 128 by 64 extended to 256 by 128 by SETUP in less than one

rier Transform can be seen diagrammatically

. When one begins with the original gravity data, taking Fourier transform of it results in

by a filter gives , the

number domain. Taking the inverse Fourier transform of gives the

CONVOLUTION IN THE SPATIAL DOMAIN IS EQUIVALENT TO

MULTIPLICATION IN

SPATIAL DOMAIN

* *

FOURIER

×

FREQUENCY DOMAIN

Figure 2: Flow chart for the wave-number filtering method: continuation of

GRAVITY AT

LEVEL 0

GRAVITYAT

LEVEL 0

CONTINUATION

CONTINUATION

14

CONVOLUTION IN THE SPATIAL DOMAIN IS EQUIVALENT TO

MULTIPLICATION IN THE FREQUENCY DOMAIN

=

TRANSFORM

=

number filtering method: continuation of △g0 to an elevation.

GRAVITYAT

LEVEL Z

GRAVITYAT

LEVEL Z

UPWARDCONTINUATION

FILTER

UPWARDCONTINUATION

FILTER

to an elevation.

15

4.4 RESULTS (Upward continuation)Upward continuation of the data was carried out three times and every level to which the data

was continued has its effects of different magnitude. Continuation was done at 3 meters, 5 meters

and 10 meters above the original surface where the original data of the map was taken.

Figure 3: Original Bouguer anomaly map with gravity scale. Contour interval is 10mGal

Figure 3The map shows a general decrease in gravity from south-west corner towards the

north-eastern corner. This gives an insight on the geology of the area where in general, the areas

with higher gravity have high density rocks as compared to areas with low gravity indicating

zones of low density rocks.

16

Figure 4: Gravity map after data continuation of 3m above ground surface

The map has been filtered by upward continuation of 3 meters above the surface. This has caused

the removal of some higher frequencies especially around the south-eastern corner where some

vibrations were initially evident. The kriging effect is seen to start occurring around the edges of

the gravity map due to the data gaps at the edges.

17

Figure 5: Gravity map after data continuation 5m above ground surface

In the above diagram, the original gravity data was upward continued 5 meters above the surface

and the new data used to plot the new map. It is clearly evident that the contours are beginning to

widen up resulting in a smooth map. The edge effect is becoming even much more pronounced

as continuation is done at higher levels.

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Figure 6: Gravity map after data continuation 10m above the ground surface.

The map has been upward continued 10m above the ground surface where the very high

frequencies initially experienced on the eastern side of the map have been considerably removed

but the edge effects have been heightened.

4.5 LIMITING ERRORSEarly methods of filtering potential field data were only approximations of the discrete

convolution filtering equations. The amount of calculations required for discrete convolution

made these equations difficult to use. Approximating the equations allowed calculations to be

performed much more quickly, but since the approximation methods did not truly represent the

equations, they produced a large number of errors. By using the Fast Fourier Transform (FFT)

algorithm, the filtering equations can be used directly. Those errors due to approximating the

filtering equations and performing convolution on a limited data set have been eliminated. Any

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remaining sources of error lie only within the FFT algorithm. By understanding these sources of

error, their effect can be limited.

4.5.1 Gibb’s Phenomenon

Gibb's phenomenon is an oscillation that appears in the data set after the Fourier transform due to

breaks or discontinuities in the data set. Gibb's phenomenon is one characteristic of the FFT

algorithm that causes error. Any large jump in value from one data point to the next causes a

discontinuity and will result in oscillations to appear when the data is transformed from one

domain to the other. Jumps in value may be real, such as due to an unusually high gradient in the

gravity field, but will still cause ripples to appear in the filtered data set. These ripples will be

greatest at the point of discontinuity and taper off to both sides. If the discontinuity is small, the

oscillations due to Gibb's phenomenon may be tolerable, they may not even be apparent when the

data set is contoured, but at times they can cause a great amount of error.

3.5.2 Leakage

Leakage refers to the data's amplitude on one side of a matrix affecting the data's amplitude on the

other side. If a data set is continuous and infinitely long, it can be transformed from the space

domain to the frequency domain by multiplying it by the Fourier integral. The FFT assumes a data

set of limited length represents a summation of an infinitely long series of sine and cosine waves.

The finite data set is assumed to be one fundamental period of the infinite series. That is, the data

set repeats over and over, making it infinitely long. As a result, one end of the data set is placed

against the other end. During filtering, this may cause information from each edge of the map to

affect data on the opposite edge. This is called leakage (Dobrin, 1976, Reed, 1980).

4.6 CORRECTION OF GIBB’S PHENOMENON AND LEAKAGE

Leakage and Gibb's phenomenon are limited by ensuring the discontinuity from one side of the

data set to the other is eliminated by extending the data set and filling the extended region with a

reflection of the data set.

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One method of limiting edge effects is to taper the edges of the data set. This is done by

multiplying the data set by a window so that each end of the data set tapers to zero. One such

window recommended to reduce leakage is the Hanning window, a type of cosine taper (Brigham,

1974, Blackman and Tukey, 1969). The Hanning window tapers both ends of the data set to zero,

eliminating any discontinuities from one side of the data to the other.

Another method to limit leakage and Gibb's phenomenon is to add a buffer of data to both ends of

the data set so that any errors will happen away from the original data set (Tsay, 1975). A The

errors are almost zero all the way across, but the curve does oscillate. Gibb's phenomenon remains.

With some data sets, the error due to Gibb's phenomenon is greater than the error due to leakage.

4.7 DISCUSSIONSThe Fourier Transforms allows an easy method to calculate the upward or downward

continuation of a gravity field to any other level. However to use the results of the continuation

filter one must understand what continuation can show. Upward continuation of a field to an

elevation z shows what the field would look like if it were to be measured at that elevation z.

In regards to determining structure, the gravity field at the surface would be highly affected by

small near surface bodies, disguising the gravity effect from deeply buried bodies.

The gravity field a great distance above the surface would be less influenced by small bodies,

only showing the gravity due to large deep seated structure. As the gravity filed is continued

upward, only the most regional trends would remain. In the frequency domain, when the data is

multiplied by the filter local density anomalies tend to be emphasized over regional features.

Elkins (1951) also states the second derivative will show the smaller shallower geologic

anomalies at the expense of larger regional features.

Upward continuation suppresses the high frequency anomalies, so the data set is filtered by

progressively higher upward continuation filters. The data set was upward continued to 3, 5 and 10

meters (figures 4, 5, 6). As can be seen with each progressive upward continuation, more of the

smaller features have disappeared leaving only larger trends. The 3 meter map shows little change

from the map of the original data. Only noise and very small features have disappeared. Once the

data have been upward continued to this level, it is questionable whether the map represents any

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aspect of the geology. Too many of the wave-numbers have been suppressed. Perhaps this map

may show a regional trend that indicates the continent thickening from south-west to north-east,

but nothing more. Most of the wave-numbers have been suppressed, such as at elevations greater

than 5 meters, the data remain stable, but little information can be seen.

Wave-number filtering method has helped in the analysis of gravity data of various parts including

the southern California. Many contour maps were created which highlighted various aspects of the

gravity field. These maps gave new insight on the size, shape and magnitude of structural features

in the region. From this analysis boundaries of similar gravity anomalies were determined. To

understand better the tectonics of the region, these boundaries are now compared to other types of

geological data.

CONCLUSIONSThe contour map of the original data is very complex with many anomalies of different size and

amplitude. In this study, large scale features are of the most interest. It is desirable to see the

effects of faults, to distinguish the boundaries of various terrains, and perhaps even to see lower

crust - upper mantle structure. The map's complexity makes analysis difficult with small near

surface features hiding the relationship of deeper features. To suppress these small wavelength -

high wave-number anomalies and enhance long wavelength features one should perform a

combination filtering methods such as low-pass band-pass filtering and upward continuation

filtering. To separate and delineate anomalies one should perform first and second vertical

derivative filtering. First and second horizontal derivative filtering would show where the gravity

field is affected by faults and boundaries. High pass band-pass filtering would remove any

regional trend. Multiple band-pass filtering would show the location of anomalies of

corresponding sizes. Anomalies of specific orientations can be found by performing numerous

strike-pass filtering.

Errors that occur in the whole process result not only from the complex equations that define the

filter operations. The errors begin right from the field where the raw data is being taken and

recorded. It is therefore very important that much care should be taken right from the initial

stages of data collection. The errors also may arise from the reading errors and the accuracy of

the gravimeters .

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