Rajiv Kumar WEMVA Workshop · Rajiv Kumar Affordable omnidirectional subsurface extended image...
Transcript of Rajiv Kumar WEMVA Workshop · Rajiv Kumar Affordable omnidirectional subsurface extended image...
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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.
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University of British ColumbiaSLIM
Tristan van Leeuwen and Felix J. Herrmann
Affordable omnidirectional subsurface extended image volumes
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Why we need image gathers ?Full subsurface offset volumes allow us to conduct
• AVA analysis
• Geological dip estimation
• Velocity analysis in complex geological enviornment
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• 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
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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
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grid
poin
tssources
4D image volume as matrix
nx x nz
Extended images
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example for one layer
Extended images
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diagonal
Extended images
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Impediments
Prohibitively expensive to compute and store
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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
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Motivation
Can we avoid computing all of the wavefields if not forming the full image volumes?
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Can we gleans information from the full image volume without requiring a-priori knowledge of the geology?
Motivation
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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]
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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
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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
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Computation computation of an image point gather
time (s) memory (MB)
conventional 23.6 103
ours 2.02 0.03
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University of British ColumbiaSLIM
Application to WEMVA
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WEMVA [ conventional method]
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h E
.⇤
.⇤ stand for element-wise multiplication
Biondo & Symes, ’04 , Symes 2008, Sava & Vasconcelos, ’11
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⇤matrix-matrix multiplication
Focusing [ propose method]
Ediag(x) ⇡ diag(x)E
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⇤
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diag(x) E
⇤⇡
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Focusingwhere represents horizontal, vertical or all offset.x
horizontal offsethorizontal +vertical
offset
all offsets
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Why do we need all offsets
x (m)
z (m
)
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Yang and Sava, 2015, van Leeuwen et. al. 2015
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Common image gathersHorizontal reflector
z − zo (km)
z (k
m)
−0.4 −0.2 0 0.2 0.4
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1.8x − xo (km)
z (k
m)
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z (m
)
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x (km)
z −
z o (km
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0
0.5x (km)
x −
x o (km
)0.5 1 1.5
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Common image gathersVertical reflector
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Common image point gathers
x − xo (km)
z −
z o (km
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z −
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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?
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Randomized probing [reflection]
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true model initial model
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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
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α
||E
dia
g(x
) −
dia
g(x
)E||
F2
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α
||E
dia
g(x
) −
dia
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)E||
F2
K = 10 K = 80
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Lens Model x (m)
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True model Initial model WEMVA
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Lens Model [image gathers]
offset(km)
z(k
m)
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True model Initial model WEMVA
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Least-squares RTM Images
True model Initial model WEMVA
offset(km)
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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
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University of British ColumbiaSLIM
Image gathers w/ surface-related multiplesNing Tu
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Why need multiples ?
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S R4R1 R2 R3
subsurface)reflector
water)surface
R1 R2 R3 R4S
subsurface)reflector
water)surface
illumination by primaries
illumination by multiples
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Least-‐squares imaging: primary-‐only shot gather Least-‐squares imaging: multiple-‐only shot gather
Tu and Herrmann, 2015
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• Leverage benefits of SRME
- highly accurate data-driven multiple prediction
• All in one go method
- we combine SRME within the extended imaging condition
Motivation
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Verschuur et. al. ’92
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Extended imaging with multiples
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eE = EW = H�1PTs (Q� P )P ⇤PrH
�1W
: areal source
: total upgoing wavefieldP
(Q� P )
where
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Least-square extended imaging
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where
Kumar et. al. ’14
minimizeeE
1
2kF( eE)� Pk2F ,
F( eE) = PrH�1EWTH�1PT
s (Q� P ),
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Velocity model
True model Initial model
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Least-squares RTM images
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Primary only Primary + multiplesw/o areal sources
Primary + multipleswith areal sources
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Primary only40
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1000
Primary + multipleswith areal sources
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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.
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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/