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

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

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

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

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, 156-168 (1980)) Schloss Reisensburg, Germany February 14-17, 2007

BACKGROUND (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 Schloss Reisensburg, Germany February 14-17, 2007

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, 156-168 (1980)) Schloss Reisensburg, Germany February 14-17, 2007

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

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

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

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

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

{ { 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

RESULTS Angular distribution of point pairs CF 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

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

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

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

RESULTS Monte Carlo tests on Complete Spatial Randomness Region 1 CF 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

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

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

RESULTS Monte Carlo tests on Complete Spatial Randomness Region 2 CF 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

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

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

RESULTS 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 CF, M, CF data, region 2b g(r) 100 Y L(r) 99 Y Stochastic Geometry, Spatial Statistics and their Applications Schloss Reisensburg, Germany February 14-17, 2007

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

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

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

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

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