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 shalysand reservoirs with statistical attributes
• multidimensional scaling (MDS) and
• kernel principal component analysis (KPCA)
Workflow
Markov Chains
Rock Physics
Thin sand-shale
sequences
Sand/shale model
sand, shaly sand sandy-shale, shale
Total porosity
0.4
Interpretation
•Net-to-gross ratios
•Saturations
0.6
300
0.4
400
0.2
500
600
200
400
Seismogram #
Second principal component
200
600
0.45
0.2
-0.2
0.5
400
600
200
400
Seismogram #
600
0
Sh
S
Sh
S
0.35
0.3
-0.6
Second principal component
200
S
-0.4
0.25
-0.2
0
0.2
0.4
First principal component
0.6
Net-to-gross ratios
0.5
0.5
1
Sh
0.4
0
0.45
0.4
0
-0.5
-0.5
0.35
0.3
0.25
0
First principal component
Attributes
MDS/KPCA
0.5
Seismic Responses
S-sh
Sh-s
0.1
0
0.2
0.4
0.6
0.8
Volume fraction of clay
3.5
1
clay
1
Sh-s
3
S
0.5
S-sh
2.5
Sh
2
0.1
0.4
-0.8
-0.4
K(xi,xj)
Seismogram #
Seismogram #
0.8
P-wave velocity (km/s)
Net-to-gross ratios
0.5
0.6
100
Sh
0.2
0
K(xi,xj)
1
S
0.3
0.2
0.3
Total porosity
0
Markov chain for lithologies
Discrete states: sand, shaly sand, sandy shale, shale
Transition probability matrix:
100 m
Properties from rock physics
Sand
Shaly-sand
Dvorkin and Gutierrez (2001)
Sandy-Shale
Shale
Properties from rock physics
Total porosity
0.4
Sh
0.2
P-wave velocity (km/s)
S-sh
Sh-s
0.1
0
Marion (1990) and Yin (1992)
S
0.3
0
0.2
0.4
0.6
0.8
Volume fraction of clay
3.5
1
clay
1
Sh-s
3
S
0.5
S-sh
2.5
Sh
2
0.1
0.2
Total porosity
0
0.3
Generate Seismic Response
0
Full waveform, normallyincident, reflected
seismograms are simulated
using the Kennett
algorithm (Kennett, 1983)
with a 30-Hz, zero-phase
Ricker wavelet
500
1000
1500
2000
2500
-0.1 0 0.1
Generate Seismic Response
0
Multiple realizations (Monte Carlo simulation)
0
0
500
0
0
500
500
500
500
1000
1000
1000
1000
1500
1500
1500
1500
1500
1000
2000
2000
2000
2500
-0.1 0 0.1
2500
2500 -0.1 0 0.1
-0.1 0 0.1
2000
2000
2500
-0.1 0 0.1
2500
-0.1 0 0.1
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
Kernel principal component
analysis (KPCA)
Perform linear PCA
1
2
1
Map from 2D to 3D
2-D
3-D
Linearly separable
Distance Functions
1/ r

r
d ( A, B)    | ak  bk | 
 k 1

n
General Minkowski metric
r = 2: Euclidean distance
1/ r

r
   k 
 k 1

n
 k  ak  bk
Dynamic Distance Function
1/ r

r
DPF     k 
  k  m 
Li et al. (2003)
Dynamic Partial Function
 k  ak  bk
m : set of smallest m ’s from {1,…,n}
The features for measuring similarity depend on
the objects being compared pairwise
Dynamic Similarity Kernel function
Dynamic similarity
kernel (Yan et al., 2006)
 DPF 2 ( xi , x j )
k ( xi , x j )  e
1
r


DPF ( A, B)     kr  ,
  k  m 
 k  ak  bk
and Δm = {the smallest
m δ’s of (δ1,…, δn)}
(Li et al., 2003)
2
Kernel functions
 xi  x j
Gaussian kernel
k ( xi , x j )  e
2
2 2
 DPF 2 ( xi , x j )
Dynamic similarity kernel
(Yan et al., 2006)
k ( xi , x j )  e
2
1
r


DPF ( A, B)     kr  ,
  k  m 
(Li et al., 2003)
 k  ak  bk and Δm = {the smallest
m δ’s of (δ1,…, δn)}
1
Inverse multi-quadric kernel
k ( xi , x j ) 
Polynomial kernel
k ( xi , x j )  ( xi  x j  c) d
c  xi  x j
2
2
2500
-0.1
Projections of seismograms
0
0.1
2000
0
0
1500
1000
500
1500
0
600
2000
Measure of
dissimilarity among
seismograms
2500
-0.1
0
2500
0.1
-0.1
400
500
1500
2000
300
Seismogram #
1000
1000
0
0.1
K(xi,xj)
MDS/KPCA
1
200
100
0.8
200
0.6
200
400
300
Seismogram #
600
0.4
400
Dissimilarity
matrix
500
or kernel
matrix
600
200
400
Seismogram #
0.2
600
Second principal component
100
500
Seismogram #
500
Net-to-gross ratios
0.5
0.6
0.4
0.45
0.2
0.4
0
-0.2
0.35
-0.4
0.3
-0.6
-0.8
-0.4
0.25
-0.2
0
0.2
0.4
First principal component
0.6
Configuration of
points color-coded
by net-to-gross
ratios or other
properties
• Results from MDS and KPCA: projections of input
seismograms onto selected principal components
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)
3
1
2
0
-1
-2
1
0
-1
-2
-3
-2
Classification
success rate
-1
0
1
First coordinate
2
-3
Net-to-gross ratios
0.5
3
Second coordinate
2
Second coordinate
Second coordinate
Net-to-gross ratios (MDS)
-2
-1
0
1
First coordinate
2
2
0.45
1
0.4
0
0.35
-1
0.3
-2
-2
0.25
-1
0
1
First coordinate
Classical MDS
Metric MDS
Non-metric MDS
56%
74%
73%
2
Net-to-gross ratios (KPCA)
0.6
0.2
0
-0.2
-0.4
-0.6
-0.8
-0.4
-0.2
0
0.2
0.4
First principal component
0.4
0.2
0
-0.2
-0.4
0.6
-0.5
0
0.5
First principal component
Second principal component
0.4
2
0.5
Second principal component
Second principal component
Second principal component
0.6
0
-0.5
-0.5
0
First principal component
Net-to-gross ratios
0.5
1
0.45
0
0.4
-1
0.35
-2
0.3
-3
-4
0.5-2
0.25
0
2
First principal component
4
Kernel
Gaussian
Dynamic
similarity
Inverse multiquadric
Polynomial
Classification
success rate
81%
90%
79%
59%
Classification of 3 NTG classes
Stratified 10-fold cross validation
Saturations (KPCA)
(A)
(C)
(B)
.05 .05 .85
.05 .05 .85
.05 .05 .05
.05 .05 .05
05 .05 .45
.05 .05 .45
.05 .05 .45
.05 .05 .45
Different transition matrices
Same nominal NTG
05 .05 .05
.05 .05 .05
.05 .05 .85
.05 .05 .85
Sh
S
Sh
S
More blocky sands
Sh
S
Saturations (KPCA)
Dynamic similarity kernel
(A)
(B)
0.8
sw=0.1
sw=0.5
sw=1
0.6
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
1
Brine sand
0.4
0.2
0
-0.2
-0.4
-0.6
-0.5
0
0.5
First principal component
0.6
Second principal component
0.4
0.8
0.8
Second principal component
Second principal component
0.6
(C)
-0.8
-1
0.2
0
-0.2
-0.4
-0.6
Oil sand
-0.5
0
0.5
First principal component
0.4
1
-0.8
-1
-0.5
0
0.5
First principal component
1
Saturations (KPCA)
0.4
(B)
0.8
sw=0.1
sw=0.5
sw=1
0
-0.2
-0.4
-0.6
-0.8
-1
0.6
0.6
0.2
0.4
0.2
0
-0.2
-0.4
Kernel
1
-0.8
-1
-0.5
0
0.5
First principal component
Gaussian
A
Classification 62%
success rate
B
73%
0.4
0.2
0
-0.2
-0.4
-0.6
-0.6
-0.5
0
0.5
First principal component
(C)
0.8
Second principal component
Second principal component
0.6
(A)
Second principal component
0.8
1
-0.8
-1
Dynamic
similarity
C
A
B
-0.5
0
0.5
First principal component
1
Inverse multiquadric
C
A
61% 88% 87% 84% 64%
3 saturation classes; stratified 10-fold cross validation
B
C
67%
60%
Polynomial
A
B
C
65% 65% 52%
Selecting components (KPCA)
0.6
Sixth principal component
0.2
Coordinate Value
0.1
0
-0.1
-0.2
-0.3
1
2
3
4
5 6 7
Coordinate
sw=0.1
sw=0.5
sw=1
8 9 10
Use 1st and 2nd components: success rate = 60%
Use 1st and 6th components: success rate =73%
Use the first 10 components: success rate = 82%
0.4
Parallel
coordinates plot
0.2
0
-0.2
-0.4
-0.4 -0.2
0
0
First principal com
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
0.6
Time
0.45
0.5
0.55
0.6
5
10
X
well
15
20
??
unknown
Second
principal component
2
Coordinate
0.4
0.4
??
0.2
0
-0.2
-0.4
-0.5
0
0.5
First principal component
Coordinate 1
3.5
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
N/G
0.8
1
Acknowledgements
• Stanford Rock Physics and Borehole
Geophysics project (SRB)and the Stanford
Center for Reservoir Forecasting (SCRF)