1. Reflection Seismic Data
Stochastic Geometry, Spatial Statistics and their Applications
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
International Workshop
February 14-17, 2007
Schloss Reisensburg, Germany

2. OUTLINE Background and Motivation Point Processes
Description of the Data Sets
Data Analysis
Results
Discussion and Future Work
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

3. BACKGROUND
Slave-Northern Cordillera
Lithospheric Evolution Experiment
Courtesy: Kevin Hall
Courtesy: Elissa Lynn
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

4. BACKGROUND
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

5. BACKGROUND Reflection Seismic Experiment
Courtesy: Arie van der Velden
Stochastic Geometry, Spatial Statistics and their Applications
(Adapted from Cook et al., The Southern Appalachians and the Growth of Continents, Scientific American, 243, (1980))
Schloss Reisensburg, Germany
February 14-17, 2007

6. BACKGROUND
(Vasudevan et al., Adaptation of seismic skeletonization for other geoscience applications, Geophysical Journal International, 161, (2005)
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

7. BACKGROUND Seismic Data Processing
(courtesy: Arie van der Velden)
Stochastic Geometry, Spatial Statistics and their Applications
(Adapted from Cook et al., The Southern Appalachians and the Growth of Continents, Scientific American, 243, (1980))
Schloss Reisensburg, Germany
February 14-17, 2007

8. BACKGROUND Slave Northern Cordillera Lithospheric Evolution
INTERPRETED REFLECTION PROFILE OF LINE 1
Study area
Reflection profile of 720 km in length and 110 km in depth
(Cook et al., Frozen subduction on Canada’s Northwest Territories: Lithorpobe deep lithospheric reflection profiling of the
Western Canadian Shield, Tectonics, 18(1), 1-24 (1999)
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

9. MOTIVATION Seismic interpretation of binary images
Understand geological processes
Geometrical patterns and structure
Pattern recognition tools, classical statistics tools
NEW
Extracting and analyzing the spatial point patterns
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

10. SPATIAL POINT PROCESSES
Model Descriptions
Poisson point process Matern hard core point process
Window Size
100x100
l=0.01
l=0.01; D=10
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

11. SPATIAL POINT PROCESSES
Model Descriptions
Matern cluster point process
lp=0.003, lc=0.1, R=10
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

12. SPATIAL POINT PROCESSES
Construction principle
Matern hard core point process
Matern cluster point process
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

13. { { SPATIAL POINT PROCESSES
Theoretical pair correlation function (Matern cluster) {
gMC(r ) = 1 +
Theoretical L-function (Matern cluster)
2 + + {
KMC(r ) = 1
where
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

14. SPATIAL POINT PROCESSES
Point process characteristics
Matern cluster point process
Pair correlation function, gMC (r)
L-function, LMC(r) - r
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

15. POINT PROCESS CHARACTERISTICS
Intensity Measure
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

16. POINT PROCESS CHARACTERISTICS
Pair correlation function
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

17. POINT PROCESS CHARACTERISTICS
Pair correlation function
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

18. POINT PROCESS CHARACTERISTICS
Pair correlation function
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

19. POINT PROCESS CHARACTERISTICS
L-function
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

20. POINT PROCESS CHARACTERISTICS
L-function
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

21. POINT PROCESS CHARACTERISTICS
L-function
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

22. DESCRIPTION OF THE DATA SETS
Region 1
Region 2
Region 1
Region 2
Stochastic Geometry, Spatial Statistics and their Applications
(Cook et al., Tectonics, 18(1),1-24 (1999))
Schloss Reisensburg, Germany
February 14-17, 2007

23. DESCRIPTION OF THE DATA SETS
FORT SIMPSON BASIN
Region 1
Region 2
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
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

24. DATA ANALYSIS Segments used for point pattern analysis REGION 1
CF,M,CF CF
CF,M,CF
CF : Coherency-filtered
M : Migrated
REGION 1
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

25. DATA ANALYSIS Segments used for point pattern analysis REGION 2
CF,M,CF CF
CF,M,CF
CF : Coherency-filtered
M : Migrated
REGION 2
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

26. DATA ANALYSIS Generation of points from seismic binary images
Seismic bitmap
Object (Coherency-filtered
segment)
Point
(Centre of gravity of the object)
Point pattern
Stochastic Geometry, Spatial Statistics and their Applications
(Beil et al., Journal of Microscopy, 220, 84-95(2005))
Schloss Reisensburg, Germany
February 14-17, 2007

27. RESULTS Point patterns built by the centers of gravity of the objects
CF,M,CF CF
CF,M,CF
REGION 1
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

28. RESULTS Point patterns built by the centers of gravity of the objects
CF,M,CF CF
CF,M,CF
REGION 2
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

29. RESULTS Angular distribution of point pairsCF
CF,M,CF
CF,M,CF
ISOTROPY
TEST
REGION 1 CF
CF,M,CF
CF,M,CF
REGION 2
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

30. RESULTS Estimated pair correlation functions
( Bandwidth h=0.15l-1/2 ) CF
CF,M,CF ^ CF
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

31. RESULTS Estimated functions L(r)-r^
Estimated functions L(r)-r
Region 1 ^
Region 2 CF
CF,M,CF CF
CF,M,CF
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

32. RESULTS Monte Carlo tests on Complete Spatial Randomness
Distance value, d
a, b are the width and length of the window; 5% significance level
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

33. RESULTS Monte Carlo tests on Complete Spatial Randomness Region 1CF
ESTIMATED PAIR CORRELATION
FUNCTION
ESTIMATED L-FUNCTION
Rank = 98
Reject null-hypothesis
Rank = 100
Reject null-hypothesis
5% significance level
Region 1
5% significance level
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

34. RESULTS Monte Carlo tests on Complete Spatial Randomness Region 1
CF,M,CF
ESTIMATED PAIR CORRELATION
FUNCTION
ESTIMATED L-FUNCTION
Rank=100
Reject null-hypothesis
Rank=100
Reject null-hypothesis
5% significance level
Region 1
5% significance level
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

35. RESULTS Monte Carlo tests on Complete Spatial Randomness Region 1
CF,M,CF
ESTIMATED L-FUNCTION
ESTIMATED PAIR CORRELATION
FUNCTION
Rank 100
Reject null-hypothesis
Rank 100
Reject null-hypothesis
5% significance level
5% significance level
Region 1
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

36. RESULTS Monte Carlo tests on Complete Spatial Randomness Region 2CF
ESTIMATED PAIR CORRELATION
FUNCTION
ESTIMATED L-FUNCTION
Rank=100
Reject null-hypothesis
Rank=90
Not reject null-hypothesis
5% significance level
5% significance level
Region 2
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

37. RESULTS Monte Carlo tests on Complete Spatial Randomness Region 2
CF,M,CF
ESTIMATED PAIR CORRELATION
FUNCTION
ESTIMATED L-FUNCTION
Rank=100
Reject null-hypothesis
Rank=98
Reject null-hypothesis
5% significance level
5% significance level
Region 2
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

38. RESULTS Monte Carlo tests on Complete Spatial Randomness Region 2
CF,M,CF
ESTIMATED PAIR CORRELATION
FUNCTION
ESTIMATED L-FUNCTION
Rank 100
Reject null-hypothesis
Rank 99
Reject null-hypothesis
5% significance level
5% significance level
Region 2
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

39. RESULTS Monte Carlo tests on Complete Spatial Randomness
Image Function Rank Reject null-hypothesis
CF data, region g(r) 100 Y
L(r) Y
CF, M, CF data, region 1a g(r) 100 Y
L(r) Y
CF, M, CF data, region 1b g(r) 100 Y
L(r) Y
CF data, region g(r) 100 Y
L(r) N
CF, M, CF data, region 2a g(r) 100 Y
CF, M, CF data, region 2b g(r) 100 Y
L(r) Y
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

40. DISCUSSION AND FUTURE WORK
The point patterns built by the centres of gravity are not completely
randomly distributed.
The two regions picked for study show marked differences in spatial
point pattern characteristics.
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
Schloss Reisensburg, Germany
February 14-17, 2007

41. DISCUSSION AND FUTURE WORK
DEFINING A SINGLE STATISTICAL MEASURE
“L-function attribute”
1. L-function attribute Y
Sum of the squares of the difference between the estimated L-function and the CSR result over r for a given window, W. W
A moving window procedure with
an overlap between windows X
W: A window of point patterns
Colour-coding the attribute map for
analysis and interpretation
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

42. DISCUSSION AND FUTURE WORK
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007

43. DISCUSSION AND FUTURE WORK
Preliminary results of spatial point pattern analysis of deep crustal
reflection seismic data look promising
Additional studies on point process models such as Matern cluster
point 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
Schloss Reisensburg, Germany
February 14-17, 2007

44. ACKNOWLEDGEMENTS
Natural Sciences and Engineering Research Council of Canada
DFG-Graduiertenkolleg 1100 (S. Eckel)
Peter Ehlers, University of Calgary
Freddie Yau, Mathematics and Statistics, University of Calgary
Stochastic Geometry, Spatial Statistics and their Applications
Schloss Reisensburg, Germany
February 14-17, 2007