Using multidimensional scaling and kernel principal component analysis to interpret seismic signatures of thin shaly-sand reservoirs Piyapa Dejtrakulwong1, Tapan Mukerji2, and Gary Mavko1 1Stanford Rock Physics Laboratory (SRB), Department of Geophysics, 2Stanford Center for Reservoir Forecasting (SCRF), Department of Energy Resources and Engineering, Stanford University

Motivation Limitation in seismic resolvability Interpretations of the sub-resolution layers Goal: To investigate seismic signatures of thin shaly-sand reservoirs with statistical attributes multidimensional scaling (MDS) and kernel principal component analysis (KPCA)

Thin sand-shale sequences Workflow Markov Chains Rock Physics Sand/shale model Thin sand-shale sequences Interpretation Net-to-gross ratios Saturations Attributes MDS/KPCA Seismic Responses

Markov chain for lithologies Discrete states: sand, shaly sand, sandy shale, shale Transition probability matrix: 100 m

Properties from rock physics Sand Shaly-sand Sandy-Shale Shale Dvorkin and Gutierrez (2001)

Properties from rock physics Marion (1990) and Yin (1992)

Generate Seismic Response Full waveform, normally-incident, reflected seismograms are simulated using the Kennett algorithm (Kennett, 1983) with a 30-Hz, zero-phase Ricker wavelet

Generate Seismic Response Multiple realizations (Monte Carlo simulation)

Multidimensional scaling (MDS) transforms the dissimilarity matrix into points in lower dimensional (Euclidean) space configures points such that their Euclidean distances (dij) in the space match the original dissimilarity (δij ) of the objects as much as possible

Multidimensional scaling (MDS) Atlanta Chicago Denver Houston Los Angeles Miami New York San Francisco Seattle Washington DC Kruskal and Wish 1978

Kernel principal component analysis (KPCA) Perform linear PCA 1 2 1 Map from 2D to 3D 2-D 3-D Linearly separable

Distance Functions General Minkowski metric r = 2: Euclidean distance

Dynamic Distance Function Li et al. (2003) Dynamic Partial Function Dm : set of smallest m d’s from {d1,…,dn} The features for measuring similarity depend on the objects being compared pairwise

Dynamic Similarity Kernel function Dynamic similarity kernel (Yan et al., 2006) (Li et al., 2003) and Δm = {the smallest m δ’s of (δ1,…, δn)}

Kernel functions Gaussian kernel Dynamic similarity kernel (Yan et al., 2006) Inverse multi-quadric kernel Polynomial kernel (Li et al., 2003) and Δm = {the smallest m δ’s of (δ1,…, δn)}

Projections of seismograms MDS/KPCA Results from MDS and KPCA: projections of input seismograms onto selected principal components Measure of dissimilarity among seismograms Dissimilarity matrix or kernel matrix Configuration of points color-coded by net-to-gross ratios or other properties

Investigate net-to-gross ratios and saturations Effect of net-to-gross ratios: we study a set of aggrading-type transition matrices with various net-to-gross ratios. (Sw=0.1 for sand layers and 1 for the others) Effect of saturations: we generate sequences from the same transition matrix but now vary saturation (in the sand layers only)

Net-to-gross ratios (MDS) Classical MDS Metric MDS Non-metric MDS Classification success rate 56% 74% 73%

Net-to-gross ratios (KPCA) Kernel Gaussian Dynamic similarity Inverse multi-quadric Polynomial Classification success rate 81% 90% 79% 59% Classification of 3 NTG classes Stratified 10-fold cross validation

Saturations (MDS) (A) (B) (C) Non-metric MDS Classical MDS Metric MDS Classification success rate 65% 66% 53% 67% 57% 69% 58%

Saturations (KPCA) Different transition matrices Same nominal NTG More blocky sands

Saturations (KPCA) Dynamic similarity kernel (A) (B) (C) Brine sand Oil sand

Inverse multi-quadric Saturations (KPCA) (A) (B) (C) Kernel Gaussian Dynamic similarity Inverse multi-quadric Polynomial A B C Classification success rate 62% 73% 61% 88% 87% 84% 64% 67% 60% 65% 52% 3 saturation classes; stratified 10-fold cross validation

Selecting components (KPCA) Parallel coordinates plot Use 1st and 2nd components: success rate = 60% Use 1st and 6th components: success rate =73% Use the first 10 components: success rate = 82%

Conclusions Dynamic Similarity Kernel (DSK) best differentiates both the net-to-gross classes and the saturation classes. The features for measuring similarity depend on the objects being compared Increasing coordinates improves classification. In addition a subset of most relevant coordinates for the property of interest can also be chosen. Similar workflow using MDS and KPCA can be applied to real seismic data to characterize thin shaly-sand reservoirs.

Interpreting seismic signatures ?? Time Coordinate 2 X ?? Coordinate 1 well unknown N/G

Acknowledgements Stanford Rock Physics and Borehole Geophysics project (SRB)and the Stanford Center for Reservoir Forecasting (SCRF)