statistical analysis of spatial point patterns: reflection seismic data k. vasudevan 1, s. eckel 2,...
TRANSCRIPT
Statistical Analysis of Spatial Point Patterns:
Reflection Seismic Data
K. Vasudevan1, S. Eckel2, F. Fleischer2, V. Schmidt2 and F.A. Cook1
1 Department of Geology and Geophysics, University of Calgary
2 Institute of Stochastics, Ulm University
Stochastic Geometry, Spatial Statistics and their Applications
International WorkshopFebruary 14-17, 2007
Schloss Reisensburg, Germany
OUTLINE
Stochastic Geometry, Spatial Statistics and their Applications
• Background and Motivation
• Point Processes
• Description of the Data Sets
• Data Analysis
• Results
• Discussion and Future Work
Schloss Reisensburg, Germany February 14-17, 2007
BACKGROUND
Stochastic Geometry, Spatial Statistics and their ApplicationsFebruary 14-17, 2007Schloss Reisensburg, Germany
Courtesy: Elissa Lynn
Courtesy: Kevin Hall
Slave-Northern Cordillera Lithospheric Evolution Experiment
BACKGROUND
Stochastic Geometry, Spatial Statistics and their ApplicationsSchloss Reisensburg, Germany February 14-17, 2007
Stochastic Geometry, Spatial Statistics and their Applications
BACKGROUND
Reflection Seismic Experiment
(Adapted from Cook et al., The Southern Appalachians and the Growth of Continents, Scientific American, 243, 156-168 (1980))
Courtesy: Arie van der Velden
Schloss Reisensburg, Germany February 14-17, 2007
Stochastic Geometry, Spatial Statistics and their Applications
BACKGROUND
February 14-17, 2007Schloss Reisensburg, Germany
(Vasudevan et al., Adaptation of seismic skeletonization for other geoscience applications, Geophysical Journal International, 161,975-993 (2005)
Stochastic Geometry, Spatial Statistics and their Applications
BACKGROUND
Seismic Data Processing
(Adapted from Cook et al., The Southern Appalachians and the Growth of Continents, Scientific American, 243, 156-168 (1980))
(courtesy: Arie van der Velden)
Schloss Reisensburg, Germany February 14-17, 2007
Stochastic Geometry, Spatial Statistics and their Applications
BACKGROUND
(Cook et al., Frozen subduction on Canada’s Northwest Territories: Lithorpobe deep lithospheric reflection profiling of theWestern Canadian Shield, Tectonics, 18(1), 1-24 (1999)
February 14-17, 2007Schloss Reisensburg, Germany
Slave Northern Cordillera Lithospheric Evolution
INTERPRETED REFLECTION PROFILE OF LINE 1
Reflection profile of 720 km in length and 110 km in depth
Study area
Stochastic Geometry, Spatial Statistics and their Applications
MOTIVATION
Seismic interpretation of binary images
Geometrical patterns and structure Pattern recognition tools, classical statistics tools
Understand geological processes
NEW
Extracting and analyzing the spatial point patterns
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
SPATIAL POINT PROCESSES
Model Descriptions
Poisson point process Matern hard core point process
=0.01 =0.01; D=10
Window Size
100x100
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
SPATIAL POINT PROCESSES
Model Descriptions
Matern cluster point process
p=0.003, c=0.1, R=10
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
SPATIAL POINT PROCESSES
Construction principleMatern hard core point process Matern cluster point
process
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
SPATIAL POINT PROCESSES
February 14-17, 2007Schloss Reisensburg, Germany
gMC(r ) = 1 +
2
2
22 41
22arccos
2
R
r
R
r
R
r
R p
0{
KMC(r ) =
p
r
12
zzz arcsin2arccos)48[(
1 2
+
])1(6)1(4 232 zzzz
1
{2 +
where origin. the aroundball the of radiusthe isRR
rz ;
2
Theoretical pair correlation function (Matern cluster)
Theoretical L-function (Matern cluster)
Stochastic Geometry, Spatial Statistics and their Applications
SPATIAL POINT PROCESSES
Point process characteristics
Matern cluster point process
Pair correlation function, gMC (r)
L-function, LMC(r) - r
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
POINT PROCESS CHARACTERISTICS
Intensity Measure
22
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)1)()((ˆ
:
||
)(ˆ
.
:#)(
W
WXWX
W
WX
W
XXWXnWX nn
2
2
for estimator An
: measure, intensitythefor estimator An
R a window
in located of pointsofNumber
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
POINT PROCESS CHARACTERISTICS
Pair correlation function
kernel ikov Epanechn
where
,n functioncorrelatio pair thefor estimator An
)(14
3)(
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2
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)2(
2
)2(
,
xIh
x
hxk
WW
XXrk
rr
rrg
hhh
jiWXX XX
jih
ji ji
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
POINT PROCESS CHARACTERISTICS
pairs pointofRepulsion
. distance, with pairs pointof Clustering
1)(
1)(
1)(
rg
rrg
rgPoisson
Pair correlation function
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
POINT PROCESS CHARACTERISTICS
February 14-17, 2007Schloss Reisensburg, Germany
Pair correlation function
Stochastic Geometry, Spatial Statistics and their Applications
POINT PROCESS CHARACTERISTICS
L-function
jiWXX XX
jirb
ji jiWW
XXIrk
rkrK
rKrLL
,,
),0(
2
||
)()(
ˆ)(
)(ˆ
)(ˆ)(ˆ
and
where
function, thefor estimator An
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
POINT PROCESS CHARACTERISTICS
L-function
r
rrL
r
rrL
rrLPoisson
distance, with pairs
pointofRepulsion slope negative
distance, with pairs
pointof Clustering slope positive
ndomness)Spatial Ra (Complete
)(
)(
0)(
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
POINT PROCESS CHARACTERISTICS
February 14-17, 2007Schloss Reisensburg, Germany
L-function
Stochastic Geometry, Spatial Statistics and their Applications
DESCRIPTION OF THE DATA SETS
February 14-17, 2007Schloss Reisensburg, Germany
Region 1
Region 1
Region 2
Region 2
(Cook et al., Tectonics, 18(1),1-24 (1999))
Stochastic Geometry, Spatial Statistics and their Applications
DESCRIPTION OF THE DATA SETS
February 14-17, 2007Schloss Reisensburg, Germany
FORT SIMPSON BASIN
Buried Proterozoic basin
Layering typical of sedimentary basins
Pattern recognition methods to characterize the layering
Objects denoted by black linear and/or curvilinear segments: coherency segments of the data Starting point for point pattern analysis
Region 1
Region 2
Stochastic Geometry, Spatial Statistics and their Applications
DATA ANALYSIS
REGION 1
Segments used for point pattern analysis
February 14-17, 2007Schloss Reisensburg, Germany
CF
CF,M,CF
CF,M,CF
CF : Coherency-filtered
M : Migrated
Stochastic Geometry, Spatial Statistics and their Applications
DATA ANALYSIS
REGION 2
Segments used for point pattern analysis
February 14-17, 2007Schloss Reisensburg, Germany
CF
CF,M,CF
CF,M,CF
CF : Coherency-filtered
M : Migrated
Stochastic Geometry, Spatial Statistics and their Applications
DATA ANALYSIS
February 14-17, 2007Schloss Reisensburg, Germany
Object (Coherency-filtered segment)
(Centre of gravity of the object)Point
Point pattern
Seismic bitmap
Generation of points from seismic binary images
(Beil et al., Journal of Microscopy, 220, 84-95(2005))
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Point patterns built by the centers of gravity of the objects
REGION 100391.0ˆ
00330.0ˆ
00400.0ˆ
c)
b)
a)
February 14-17, 2007Schloss Reisensburg, Germany
CF
CF,M,CF
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Point patterns built by the centers of gravity of the objects
REGION 200665.0ˆ
00690.0ˆ
00869.0ˆ
f)
e)
d)
February 14-17, 2007Schloss Reisensburg, Germany
CF
CF,M,CF
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Angular distribution of point pairs
February 14-17, 2007Schloss Reisensburg, Germany
ISOTROPYTEST
REGION 1
REGION 2
CF
CF
CF,M,CF CF,M,CF
CF,M,CF CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Estimated pair correlation functions ( Bandwidth h=0.15-1/2 )
February 14-17, 2007Schloss Reisensburg, Germany
^
CF
CF,M,CF
CF
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Estimated functions L(r)-r
^
^
February 14-17, 2007Schloss Reisensburg, Germany
CF
CF,M,CF
CF
CF,M,CF
Region 1
Region 2
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
February 14-17, 2007Schloss Reisensburg, Germany
Monte Carlo tests on Complete Spatial Randomness
},min{
2
1
0
2))()(ˆ(
ba
rltheoretica rLrLd
a, b are the width and length of the window; 5% significance level
Distance value, d
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Monte Carlo tests on Complete Spatial Randomness
Region 1
February 14-17, 2007Schloss Reisensburg, Germany
ESTIMATED PAIR CORRELATION FUNCTION
ESTIMATED L-FUNCTION
5% significance level
5% significance level
Rank = 100Reject null-hypothesis
Rank = 98Reject null-hypothesis
CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Monte Carlo tests on Complete Spatial Randomness
Region 1
February 14-17, 2007Schloss Reisensburg, Germany
ESTIMATED PAIR CORRELATIONFUNCTION
ESTIMATED L-FUNCTION
5% significance level
5% significance level
Rank=100Reject null-hypothesis
Rank=100Reject null-hypothesis
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Monte Carlo tests on Complete Spatial Randomness
Region 1
February 14-17, 2007Schloss Reisensburg, Germany
ESTIMATED PAIR CORRELATIONFUNCTION
ESTIMATED L-FUNCTION
5% significance level
5% significance level
Rank 100Reject null-hypothesis
Rank 100Reject null-hypothesis
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Monte Carlo tests on Complete Spatial Randomness
Region 2
February 14-17, 2007Schloss Reisensburg, Germany
ESTIMATED PAIR CORRELATIONFUNCTION
ESTIMATED L-FUNCTION
5% significance level 5% significance level
Rank=100Reject null-hypothesis
Rank=90Not reject null-hypothesis
CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Monte Carlo tests on Complete Spatial Randomness
Region 2
February 14-17, 2007Schloss Reisensburg, Germany
ESTIMATED PAIR CORRELATIONFUNCTION
ESTIMATED L-FUNCTION
5% significance level 5% significance level
Rank=100Reject null-hypothesis
Rank=98Reject null-hypothesis
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
Monte Carlo tests on Complete Spatial Randomness
Region 2
February 14-17, 2007Schloss Reisensburg, Germany
ESTIMATED PAIR CORRELATIONFUNCTION
ESTIMATED L-FUNCTION
5% significance level 5% significance level
Rank 100Reject null-hypothesis
Rank 99Reject null-hypothesis
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
RESULTS
February 14-17, 2007Schloss Reisensburg, Germany
Monte Carlo tests on Complete Spatial Randomness
Image Function Rank Reject null-hypothesis
CF data, region 1 g(r) 100 Y L(r) 98 Y
CF, M, CF data, region 1a g(r) 100 Y L(r) 100 Y
CF, M, CF data, region 1b g(r) 100 Y L(r) 100 Y
CF data, region 2 g(r) 100 Y L(r) 90 N
CF, M, CF data, region 2a g(r) 100 Y L(r) 98 Y
CF, M, CF data, region 2b g(r) 100 Y L(r) 99 Y
Stochastic Geometry, Spatial Statistics and their Applications
DISCUSSION AND FUTURE WORK
February 14-17, 2007Schloss Reisensburg, Germany
1. The point patterns built by the centres of gravity are not completely randomly distributed.
2. The two regions picked for study show marked differences in spatial point pattern characteristics.
3. The intensity, pair correlation function, and L-function show similar characteristics for the same region with different processing schemes.
4. The clustering effects for small point pair distances are stronger for region 1 than for region 2.
Stochastic Geometry, Spatial Statistics and their Applications
DISCUSSION AND FUTURE WORK
February 14-17, 2007Schloss Reisensburg, Germany
DEFINING A SINGLE STATISTICAL MEASURE“L-function attribute”
X
YW
W: A window of point patterns
1. L-function attribute
Sum of the squares of the difference between the estimated L-function and the CSR result over r for a given window, W.
2. A moving window procedure with an overlap between windows
3. Colour-coding the attribute map for analysis and interpretation
Stochastic Geometry, Spatial Statistics and their Applications
DISCUSSION AND FUTURE WORK
February 14-17, 2007Schloss Reisensburg, Germany
Stochastic Geometry, Spatial Statistics and their Applications
DISCUSSION AND FUTURE WORK
February 14-17, 2007Schloss Reisensburg, Germany
Preliminary results of spatial point pattern analysis of deep crustalreflection seismic data look promising
Additional studies on point process models such as Matern clusterpoint process model
Examining anisotropy in point patterns and introducing new model descriptions
Additional studies on attributes based on point process characteristics of spatial point patterns
Investigating other extraction procedures for point patterns and other point process characteristics
Stochastic Geometry, Spatial Statistics and their Applications
ACKNOWLEDGEMENTS
Natural Sciences and Engineering Research Council of Canada
DFG-Graduiertenkolleg 1100 (S. Eckel)
February 14-17, 2007Schloss Reisensburg, Germany
Peter Ehlers, University of CalgaryFreddie Yau, Mathematics and Statistics, University of Calgary