University  of  British  ColumbiaSLIM

Rajiv Kumar

Affordable omnidirectional subsurface extended image volumes

Released to public domain under Creative Commons license type BY (https://creativecommons.org/licenses/by/4.0).Copyright (c) 2015 SLIM group @ The University of British Columbia.

University  of  British  ColumbiaSLIM

Tristan van Leeuwen and Felix J. Herrmann

Affordable omnidirectional subsurface extended image volumes

Why we need image gathers ?Full subsurface offset volumes allow us to conduct

• AVA analysis

• Geological dip estimation

• Velocity analysis in complex geological enviornment

• Given two-way wave equations, source and receiver wavefields are defined as

where discretization of the Helmoltz operator

source

data matrix

samples the wavefield at the source and receiver positions

squared-slowness

Extended images

H(m)U = PTs Q

H(m)⇤V = PTr D

H(m) :

Q :

D :

Ps, Pr :

m :

Forward  propagation

Backward  propagation

Extended images• Organize wavefields in monochromatic data matrices where each column

represents a common shot gather

• Express image volume tensor for single frequency as a matrix

E = UV ⇤

E = H�1PTs QD⇤PrH

�1

grid

poin

tssources

4D image volume as matrix

nx  x  nz

Extended images

example for one layer

Extended images

500 1000 1500 2000 2500

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1000

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2500

full matrixx [m]

z [m

]

800 900 1000 1100 1200

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600

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1000

one columnx [m]

z [m

]

800 900 1000 1100 1200

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diagonal

Extended images

Impediments

Prohibitively expensive to compute and store

Current Workflow• Compute all the source and receiver wavefields

• Severely subsample the image volume in the subsurface coordinates

• Allowed limited interactions in predefined directions• horizontal or vertical based upon geology of interest

Biondi and Symes, 2004, Sava and Biondi, 2004, Sava and Vasconcelos, 2011, Yang and Sava, 2015

Motivation

Can we avoid computing all of the wavefields if not forming the full image volumes?

Can we gleans information from the full image volume without requiring a-priori knowledge of the geology?

Motivation

Extended images• Probe volume with tall matrix

where represents single scattering points

van Leeuwen 2012eE = EW = H�1PTs QD⇤PrH

�1W

W = [w1, . . . ,wl]

wi = [0, . . . , 0, 1, 0, . . . , 0]

Computation

• mat-vec with extended image :

• (one subsurface source)

• (surface source function)

• (one surface source)

14

eE = EW = H�1PTs QD⇤PrH

�1w

ed = PrH�1w

ey = QD⇤ed

eE = H�1PTs ey

Computation computation of an image point gather

# of PDE solves “flops for correlations”

conventional 2Ns Ns x Nh

ours 2Nx Ns x Nr

Ns - # of sources Nr - # of receiversNh - # of subsurface offsetsNx - # of sample points

15

Computation computation of an image point gather

time (s) memory (MB)

conventional 23.6 103

ours 2.02 0.03

16

University  of  British  ColumbiaSLIM

Application to WEMVA

WEMVA [ conventional method]

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25000

0.2

0.4

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1

h E

.⇤

.⇤ stand for element-wise multiplication

Biondo & Symes, ’04 , Symes 2008, Sava & Vasconcelos, ’11

⇤matrix-matrix multiplication

Focusing [ propose method]

Ediag(x) ⇡ diag(x)E

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2500

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E diag(x)

⇤

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1000

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2500500 1000 1500 2000 2500

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1000

1500

2000

2500

diag(x) E

⇤⇡

Focusingwhere represents horizontal, vertical or all offset.x

horizontal offsethorizontal +vertical

offset

all offsets

Why do we need all offsets

x (m)

z (m

)

0 1000 2000 3000 4000 5000 6000

0

1000

2000

Yang and Sava, 2015, van Leeuwen et. al. 2015

Common image gathersHorizontal reflector

z − zo (km)

z (k

m)

−0.4 −0.2 0 0.2 0.4

1.3

1.4

1.5

1.6

1.7

1.8x − xo (km)

z (k

m)

−1 −0.5 0 0.5 1

1.4

1.5

1.6

1.7

1.8

x (m)

z (m

)

0 1000 2000 3000 4000 5000 6000

0

1000

2000

x (km)

z −

z o (km

)

0.5 1 1.5

−0.5

0

0.5x (km)

x −

x o (km

)0.5 1 1.5

−1

−0.5

0

0.5

1

x (m)

z (m

)

0 1000 2000 3000 4000 5000 6000

0

1000

2000

Common image gathersVertical reflector

Common image point gathers

x − xo (km)

z −

z o (km

)

−1 −0.5 0 0.5 1

−0.5

0

0.5x − xo (km)

z −

z o (km

)

−1 −0.5 0 0.5 1

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

x (m)

z (m

)

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0

1000

2000

Fast WEMVA w/ randomized probing• Measure the error in some norm

• The Frobenius norm can be estimated via randomized trace estimation :

where

Avron and Toledo, 2011

||A||2F = trace(ATA)

KX

i=1

wiwTi ⇡ I

⇡KX

i=1

wTi A

TAwi =KX

i=1

||Awi||22

minm

||E(m)diag(x)� diag(x)E(m)||2?

Randomized probing [reflection]

x(m)z(

m)

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0

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x(m)

z(m

)

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true model initial model

Randomized probing [reflection]

•Exact • Error bar of approximated objective function (different color represents different random realization)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.5

2

2.5

3

3.5

4

x 105

α

||E

dia

g(x

) −

dia

g(x

)E||

F2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 21.5

2

2.5

3

3.5

4

x 105

α

||E

dia

g(x

) −

dia

g(x

)E||

F2

K  =  10 K  =  80

Lens Model x (m)

z (

m)

0 500 1000 1500 2000 2500 3000

0

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1000

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True model Initial model WEMVA

x (m)

z (m

)

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z (m

)

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x (m)

z (m

)

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Lens Model [image gathers]

offset(km)

z(k

m)

−1.5 −1 −0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

offset(km)

z(k

m)

−1.5 −1 −0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

offset(km)

z(km

)

−1.5 −1 −0.5 0 0.5 1 1.5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

True model Initial model WEMVA

Least-squares RTM Images

True model Initial model WEMVA

offset(km)

z(k

m)

1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

offset(km)

z(k

m)

1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

offset(km)

z(k

m)

1 1.2 1.4 1.6 1.8 2

0

0.5

1

1.5

Conclusions• probings allows us to get offset information for all sub-surface

direction

• Prior knowledge of geology is not required

• randomized trace estimation allows us to compute WEMVA objective cheaply

University  of  British  ColumbiaSLIM

Image gathers w/ surface-related multiplesNing Tu

Why need multiples ?

33

S R4R1 R2 R3

subsurface)reflector

water)surface

R1 R2 R3 R4S

subsurface)reflector

water)surface

illumination  by  primaries

illumination  by  multiples

34

Least-­‐squares  imaging:    primary-­‐only  shot  gather Least-­‐squares  imaging:  multiple-­‐only  shot  gather

Tu and Herrmann, 2015

• Leverage benefits of SRME

- highly accurate data-driven multiple prediction

• All in one go method

- we combine SRME within the extended imaging condition

Motivation

35

Verschuur et. al. ’92

Extended imaging with multiples

36

eE = EW = H�1PTs (Q� P )P ⇤PrH

�1W

                               :  areal  source  

                               :  total  upgoing  wavefieldP

(Q� P )

where

Least-square extended imaging

37

where

Kumar et. al. ’14

minimizeeE

1

2kF( eE)� Pk2F ,

F( eE) = PrH�1EWTH�1PT

s (Q� P ),

Velocity model

True  model Initial  model

38

x (m)

z (m

)

0 500 1000 1500 2000

0

200

400

600

800

1000 1700

1800

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2000

2100

2200

2300

x (m)

z (m

)

0 500 1000 1500 2000

0

200

400

600

800

1000 1700

1800

1900

2000

2100

2200

2300

39

Least-squares RTM images

z (

m)

x (m)0 500 1000 1500 2000

0

200

400

600

800

1000

z (

m)

x (m)0 500 1000 1500 2000

0

200

400

600

800

1000

z (

m)

x (m)0 500 1000 1500 2000

0

200

400

600

800

1000

Primary only Primary + multiplesw/o areal sources

Primary + multipleswith areal sources

Primary only40

z (m

)

x − xo (m)

−400 −200 0 200 400

0

200

400

600

800

1000

z (m

)

x − xo (m)

−400 −200 0 200 400

0

200

400

600

800

1000

Primary + multiplesw/o areal sources

z (

m)

x − xo (m)

−400 −200 0 200 400

0

200

400

600

800

1000

Primary + multipleswith areal sources

Conclusions

41

Multiples provide extra illumination that can complement primaries.

Multiples can be used with primaries to form subsurface image gathers via least-squares inversion.

Acknowledgements

This  work  was  in  part  financially  supported  by  the  Natural  Sciences  and  Engineering  Research  Council  of  Canada  Discovery  Grant  (22R81254)  and  the  Collaborative  Research  and  Development  Grant  DNOISE  II  (375142-­‐08).  This  research  was  carried  out  as  part  of  the  SINBAD  II  project  with  support  from  the  following  organizations:  BG  Group,  BGP,  BP,  CGG,  Chevron,  ConocoPhillips,  ION,  Petrobras,  PGS,  Statoil,  Total  SA,  WesternGeco,  and  Woodside.

Thank  you  for  your  attention  !  https://www.slim.eos.ubc.ca/