University of British ColumbiaSLIM

Ben Bougher, SEG 2016

AVA analysis as a machine learning problem

Released to public domain under Creative Commons license type BY (https://creativecommons.org/licenses/by/4.0).Copyright (c) 2016 SLIM group @ The University of British Columbia.

Reflection seismology as an unsupervised learning problem

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Approaches: Physics  driven  (conventional)  Data  driven  (thesis  contributions)  

Motivation: Inability  to  discover  hydrocarbons  directly  from  seismic  data.  

Problem: Automatically  segment  potential  hydrocarbon  reserves  from  seismic  images.  

Angle domain common image gathers

8

θProblem:  Need  angle  dependent  reflectivity  responses  

Solution:  Angle  domain  common  image  gather  migration

x1

x

z1

dept

h

✓

refle

ctiv

ity

Earth  Model R(𝜃)  at  (x1,z1)

Scattering theory (Zoeppritz)

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θ1θ1

θ2

φ1

φ2

Vp1, Vs1, ρ1

Vp2, Vs2, ρ2

Rpp Rps

Tpp

Tps

2

664

RPP

RPS

TPP

TPS

3

775 =

2

6664

� sin ✓1 � cos�1 sin ✓2 cos�2cos ✓1 � sin�1 cos ✓2 � sin�2sin 2✓

VP1VS1

cos2�1⇢2VS22VP1⇢1V 2S1VP 2

cos 2✓1⇢2VS2VP1⇢1V 2S1

cos 2�2

�cos�2 VS1VP1 sin 2�1⇢2VP2⇢1VP1

⇢2VS2⇢1VP1

sin 2�2

3

7775

�1 2

664

sin ✓1

cos ✓1

sin 2✓1

cos 2�1

3

775

Problem: Relate angle dependent reflectivity to rock physics.

Assumption: Ray theory approximation.

Solution:

Problem: Non-linear, not useful for inversion.

R(✓) / VP , VS , ⇢

Scattering theory (Shuey)

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Limitations: Small  perturbations  over  a  background  trend  valid  <  30  degrees  

Benefits:  linear  for  i  and  g  invert  using  simple  least  squares  

0 10 20 30 40 50 60 70 80 90

theta [deg]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Rpp

Zoeppritz2-term Shuey

Rpp(✓) = i(�VP ,�⇢) + g(�VP ,�VS ,�⇢) sin2 ✓

R(𝜃)  at  (x1,z1)

Shuey  approximation

Shuey term inversion as a projection

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✓

Rpp

Rpp(✓) at x1, z1

✓

ADCIG at x1

θ

ADCIGs

Image  for  every  scattering  angle.  An  angle  gather  for  every  slice.  A  reflectivity  curve  for  every  point.

Shuey term inversion as a projection

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✓

Rpp

Rpp(✓) at x1, z1

Fit  the  reflectivity  curve  using  a  linear  combination  of  the  Shuey  vectors.  Each  reflectivity  curve  is  projected  down  to  two  coefficients.

Shuey term inversion as a projection

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i g

i,g at x1

i

g

Reduced  the  dimensionality  of  the  angle  gathers  to  two  coefficients.  Projection  coefficients  can  be  plotted  to  analyze  the  multivariate  relationships.

Relation to hydrocarbons

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�0.4 �0.2 0.0 0.2 0.4I

�0.4

�0.2

0.0

0.2

0.4

G

mudrock linesand - shalegas sand - shale

g =i

1 + k

h1� 4 hVSihVP i

⇣ 2m

+ khVSihVP i

⌘i

*m,  c,  k  are  geological  parameters  determined    empirically  from  well  logs/laboratory  measurements

Hydrocarbon  reserves  are  found  from  outliers  of  a  crossplot!

Brine-‐saturated  sands  and  shales  follow  a    mudrock  line:  

Hydrocarbon  saturated  sands  deviate  from  this  trend.

Reality bites

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5 10 15 20 25 30✓ [deg]

�1.6

�1.4

�1.2

�1.0

�0.8

�0.6

�0.4

�0.2

Rpp

⇥104

0 5 10 15 20 25 30 35 40✓ [deg]

0.00

0.05

0.10

0.15

0.20

0.25

Rpp

0 5 10 15 20 25 30✓ [deg]

0

1

2

3

4

5

6

Rpp

0 5 10 15 20 25 30✓ [deg]

0.0

0.1

0.2

0.3

0.4

0.5I

G

Problem:    Shuey  components  can’t  explain  the  features  in  real  data.

Solution:    Use  unsupervised  machine  learning  to  find  better  projections.

BG  dataset Penobscot  dataset Mobil  Viking  dataset

Unsupervised learning problem

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X 2 Rn⇥d

n  is  number  of  samples  in  the  image,    d  is  the  number  of  angles

Principle component analysis (PCA)

Eigendecomposition  of  the  covariance  matrix:  

Project  onto  the  eigenvectors  with  the  two  largest  eigenvalues.  Maximizes  the  variance  (a  measure  of  information).

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C = XTX =1

n

nX

i

xixTi

Marmousi II Earth model

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0 1000 2000 3000 4000x [m]

0

500

1000

1500

2000

z[m

]

vp

12001400160018002000220024002600

0 1000 2000 3000 4000x [m]

0

500

1000

1500

2000z

[m]

vs

0

150

300

450

600

750

900

1050

0 1000 2000 3000 4000x [m]

0

500

1000

1500

2000

z[m

]

rho

105012001350150016501800195021002250

0 1000 2000 3000 4000x[m]

0

500

1000

1500

2000

z[m

]

zero-o↵set rpp

�0.32�0.24�0.16�0.080.000.080.160.240.32

Specifically  made  for  testing  amplitude  vs.  offset  analysis  

Contains  gas-‐saturated  sand  embedded  in  shales  and    brine-‐sands.  

Seismic modeling

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0 500 1000 1500 2000 2500 3000 3500 4000x [m]

0

500

1000

1500

2000

z[m

]

0 5 10 15 20 25 30✓ [deg]

500 1000 1500 2000 2500 3000 3500 4000 4500 5000x [m]

0

500

1000

1500

2000

2500

z[m

]

�5 0 5 10 15 20 25 30✓ [deg]

Physically  consistent  with  the  Zoeppritz  equations

Migrated  visco-‐acoustic  survey

True  reflectivity Migrated  seismic

Kernel PCA

Problem:  Find  a  non-‐linear  projection  that  provides  better  discrimination  of  trends  and  outliers.  

Solution:  Use  the  “kernel-‐trick”  to  compute  PCA  in  a  high-‐dimensional  non-‐linear  feature  space  

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Kernel trickPCA  can  be  calculated  from  the  Gramian  inner  product  matrix:  

Replace                          with  a  kernel    

Example  c=2,  b=1

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hxi,xji (xi,xj)

�(x) = [1,p2x1,

p2x2, x

21, x

22,p2x1x2]

XXT =

0

BBB@

hx1,x1i hx1,x2i . . . hx1,xnihx2,x1i hx2,x2i . . . hx2,xni

......

. . ....

hxn,x1i hxn,x2i . . . hxn,xni

1

CCCA

(xi,xj) = (xTi xj + b)

c = h�(xi),�(xj)i

RecapSituation:  Exploring  seismic  data  for  anomalous  responses.  Problem:  Physical  model  can  not  explain  real  world  data.  Solution:  Learn  useful  projections  directly  from  the  data.  Assessment:  

• PCA  is  equivalent  for  physically  consistent  data,  but  more  robust  to  processing/acquisition  artifacts.  

• Kernel  PCA  makes  outliers  linearly  separable  from  the  background.  

Next:  Automatic  segmentation  (clustering)  

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Each  point  is  initially  considered  a  cluster.  Each  iteration  merges  the  closest  clusters  into  a  larger  cluster.  Builds  a  hierarchy  until  a  defined  number  of  classes  is  reached.

Hierarchical clustering

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9�0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

1

2

3 4

5

6

7

8

9

5 3 1 2 47 96 80

A

BB

C

Unclassified  data Classification  hierarchy

Results - PCA

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Clustered  physically  consistent  data Clustered  migrated  data

Results - Kernel PCA

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Clustered  physically  consistent  data Clustered  migrated  data

Summary

Successes:  Each  projection  could  segment  the  reservoir.  Kernel  PCA  provided  advantageous  multivariate  geometries  (linearly  separable).  

Challenges:  Clustering  is  highly  sensitive  to  user  chosen  parameters.  Kernel  PCA  is  computationally  expensive  and  lacks  interpretation.  

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Robust PCA

Problem:  Find  a  sparse  set  of  outlying  reflectivity  responses  against  a  background  trend.  Assumption:  The  background  trend  of  similar  curves  is  highly  redundant,  which  forms  a  low  rank  matrix.  Solution:  

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minL,S

kLk⇤ + �kSk1,1 s.t. L+ S = X

Results - physically consistent data

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0 1000 2000 3000 4000x [m]

0

500

1000

1500

z[m

]

Results - migrated seismic

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Summary

Successes:  Segmentation  of  reservoir  in  both  images.  Physically  interpretable  segmentation  without  clustering.  

Challenges:  Requires  tuning  of  one  optimization  trade  off  parameter.  Convergence  sensitive  to  the  rank  of  outliers,  not  well  understood.

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Epilogue

Outcome:  Generalized  a  conventional  analysis  approach  using  unsupervised  learning  models.  Successful  in  segmented  potential  hydrocarbon  reserves  from  seismic  data  Future:  More  data,  standardized  datasets  Quantitative  benchmarks

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References[1]Zoeppritz, K., 1919, VII b. Über Reflexion und Durchgang seismischer Wellen durch Unstetigkeitsflächen: Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1919, 66–84 [2]Shuey, R. T., 1985, A simplification of the Zoeppritz equations: Geophysics, 50, 609–614. [3]Gardner, G. H. F., L. W. Gardner, and A. R. Gregory, 1974, Formation velocity and density; the diagnostic basics for stratigraphic traps: Geophysics, 39, 770–780. [4]Castagna, J. P., M. L. Batzle, and R. L. Eastwood, 1985, Relationships between compressional-wave and shear-wave velocities in clastic silicate rocks: Geophysics, 50, 571–581. [5]Castagna, J. P., H. W. Swan, and D. J. Foster, 1998, Framework for AVO gradient and intercept interpretation: Geophysics, 63, 948–956. [6]Edgar, J., and J. Selvage, 2013, Dynamic Intercept-gradient Inversion

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