statistical analysis of spatial point patterns: reflection seismic data k. vasudevan 1, s. eckel 2,...

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


• 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


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



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)


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)



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)


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


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



SPATIAL POINT PROCESSES

Model Descriptions

Poisson point process Matern hard core point process

=0.01 =0.01; D=10

Window Size

100x100




Model Descriptions

Matern cluster point process

p=0.003, c=0.1, R=10




Construction principleMatern hard core point process Matern cluster point

process





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)



Point process characteristics

Matern cluster point process

Pair correlation function, gMC (r)

L-function, LMC(r) - r



POINT PROCESS CHARACTERISTICS

Intensity Measure

22

||

)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




Pair correlation function

kernel ikov Epanechn

where

,n functioncorrelatio pair thefor estimator An

)(14

3)(

,||

|)|(

2

1)(ˆ

ˆ)(ˆ

)(ˆ

,2

2

,

)2(

2

)2(

,

xIh

x

hxk

WW

XXrk

rr

rrg

hhh

jiWXX XX

jih

ji ji




pairs pointofRepulsion

. distance, with pairs pointof Clustering

1)(

1)(

1)(

rg

rrg

rgPoisson





L-function

jiWXX XX

jirb

ji jiWW

XXIrk

rkrK

rKrLL

,,

),0(

2

||

)()(

ˆ)(

)(ˆ

)(ˆ)(ˆ

and

where

function, thefor estimator An




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)(





L-function


DESCRIPTION OF THE DATA SETS


Region 1

Region 1

Region 2

Region 2

(Cook et al., Tectonics, 18(1),1-24 (1999))


DESCRIPTION OF THE DATA SETS


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


DATA ANALYSIS

REGION 1

Segments used for point pattern analysis


CF

CF,M,CF

CF,M,CF

CF : Coherency-filtered

M : Migrated


DATA ANALYSIS

REGION 2

Segments used for point pattern analysis


CF

CF,M,CF

CF,M,CF

CF : Coherency-filtered

M : Migrated


DATA ANALYSIS


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))


RESULTS

Point patterns built by the centers of gravity of the objects

REGION 100391.0ˆ

00330.0ˆ

00400.0ˆ

c)

b)

a)


CF

CF,M,CF

CF,M,CF


RESULTS

Point patterns built by the centers of gravity of the objects

REGION 200665.0ˆ

00690.0ˆ

00869.0ˆ

f)

e)

d)


CF

CF,M,CF

CF,M,CF


RESULTS

Angular distribution of point pairs


ISOTROPYTEST

REGION 1

REGION 2

CF

CF

CF,M,CF CF,M,CF

CF,M,CF CF,M,CF


RESULTS

Estimated pair correlation functions ( Bandwidth h=0.15-1/2 )


^

CF

CF,M,CF

CF

CF,M,CF


RESULTS

Estimated functions L(r)-r

^

^


CF

CF,M,CF

CF

CF,M,CF

Region 1

Region 2


RESULTS


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


RESULTS


Region 1


ESTIMATED PAIR CORRELATION FUNCTION

ESTIMATED L-FUNCTION

5% significance level


Rank = 100Reject null-hypothesis

Rank = 98Reject null-hypothesis

CF


RESULTS


Region 1


ESTIMATED PAIR CORRELATIONFUNCTION




Rank=100Reject null-hypothesis


CF,M,CF


RESULTS


Region 1






Rank 100Reject null-hypothesis


CF,M,CF


RESULTS


Region 2




5% significance level 5% significance level


Rank=90Not reject null-hypothesis

CF


RESULTS


Region 2







CF,M,CF


RESULTS



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


DISCUSSION AND FUTURE WORK


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.




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




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


ACKNOWLEDGEMENTS

Natural Sciences and Engineering Research Council of Canada

DFG-Graduiertenkolleg 1100 (S. Eckel)


Peter Ehlers, University of CalgaryFreddie Yau, Mathematics and Statistics, University of Calgary

statistical analysis of spatial point patterns: reflection seismic data k. vasudevan 1, s. eckel 2,...

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