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University of British ColumbiaSLIM
Ning Tu and Felix Herrmann*
Fast linearized inversion with surface-related multiples with source estimation
Released to public domain under Creative Commons license type BY (https://creativecommons.org/licenses/by/4.0).Copyright (c) 2014 SLIM group @ The University of British Columbia.
Motivation
• make use of primaries and multiples simultaneously‣ higher SNR from primaries and extra illumination from multiples
• eliminate acausal artifacts from multiples by inversion‣ cross-‐correlation of wrong orders of up-‐/down-‐going wavefields
• look for a computationally efficient approach‣ dense matrix products in multiple prediction by SRME or EPSI‣ expensive simulation in iterative inversion
Berkhout 1993; GuiNon 2002; Muijs et al., 2007; Verschuur 2011; Long et al. 2013; Wong et al., 2014
primary illuminated region
Illustration: primary wave propagation
RTM image of primaries: 1 source
multiple illuminated region
Illustration: surface-multiples propagation[Each receiver serves as a virtual secondary source]
much smaller incident anglescompared to primaries
RTM image of multiples: 1 source
Motivation
• make use of primaries and multiples simultaneously‣ higher SNR from primaries and extra illumination from multiples
• eliminate acausal artifacts from multiples by inversion‣ cross-‐correlation of wrong orders of up-‐/down-‐going wavefields
• look for a computationally efficient approach‣ dense matrix products in multiple prediction by SRME or EPSI‣ expensive simulation in iterative inversion
Muijs et al., 2007; Lin et al., 2010; Liu et al., 2011; Whitmore et al., 2010; Verschuur et al., 2011; Wong et al., 2012
Inversion of multiples: 1 source, 60 iterations[~120X the simulation cost of the 1 source RTM image]
Motivation
• make use of primaries and multiples simultaneously‣ higher SNR from primaries and extra illumination from multiples
• eliminate acausal artifacts from multiples by inversion‣ cross-‐correlation of wrong orders of up-‐/down-‐going wavefields
• look for a computationally efficient approach‣ dense matrix products in multiple prediction by SRME or EPSI‣ expensive simulation in iterative inversion
Herrmann and Li, 2012; Tu and Herrmann, 2012 a,b
Solutions
• Model all orders of surface-‐related multiples using two-‐way wave-‐equations based on the SRME relation
• Use primaries by matching them by on-‐the-‐fly source estimation
• Eliminate expensive dense matrix products in multiple prediction
• Fast least-‐squares inversion with a cost of a single RTM of all the data
Caveat: internal multiples are not modelled in multiple prediction by the SRME relation
Method: the SRME relation
Verschuur et al., 1992
: total up-‐going wavefield
: surface-‐free dipole Green’s function
: point-‐source wavefield
: surface reflectivity
: frequency index
P
R
X
S
i 1 · · ·nf
⇡ �I
= wiI
Pi = Xi(Si +RiPi)
Eliminating dense matrix-matrix products[SRME relation & wave-equation solver]
Tu and Herrmann, 2012 a,b
Linearized modelling of the surface-‐free Green’s function:
Xi ⇡ rFi[m0, �m, I]
= !2DrHi[m0]�1Hi[m0]
�1diag(�m)D⇤sI
.= rFi[m0, �m](D⇤
sI)
rFi
m0
�m
I
: linearized modelling: background model: model perturbations: impulsive source array
Dr
Hi
D⇤s
: data restriction at receivers: time-‐harmonic Helmholtz operator: source injection operator
Eliminating dense matrix-matrix products[SRME relation & wave-equation solver]
Tu and Herrmann, 2012 a,b
Combined with free-‐surface physics:
Pi ⇡ rFi[m0, �m, I](Si �Pi)
= rFi[m0, �m](D⇤sI)(Si �Pi)
= rFi[m0, �m](D⇤s(Si �Pi))
.= rFi[m0,Si �Pi].
Dense matrix-‐matrix products
Wave-‐equation solves with total downgoingdata injected as “areal” source
Compressive imaging w/ sparsity promotionHerrmann and Li, 2012; Tu and Herrmann, 2012 a,b; Tu et al., 2013 a
� ˜m = C
Hargmin
x
kxk1
subject to
X
i2⌦
kpi�rFi[m0,Si �Pi]C
Hxk22 �2
: curvelet transform: wavefields with subsampled source experiments, i.e., , : randomized frequency subset: misfit tolerance
C·⌦
�
pi= vec(Pi)Pi = PiE
Rerandomization:‣ SPGl1 solves this problem via a series of LASSO subproblems. ‣ Draws independent and for each subproblem.‣ Faster progress to solution, higher robustness to linearization errors.
⌦ E
Synthetic data examples
• model cropped from the sedimentary part of the Sigsbee 2B model, 3.8km deep, 6km wide, 7.62m grid spacing
• 15Hz Ricker wavelet, 8.184s recording time, 311 freq. samples• 261 co-‐located sources/receivers, fixed spread, 22.86m spacing, 7.62m deep• data modelled using iWave with free-‐surface BC.• 31 (~10%) frequencies & 26 (~10%) simultaneous sources used for fast inversion• run for 50 iterations, simulation cost ~1 RTM with all sources and frequencies• caveat: true wavelet used for joint inversion of primaries and multiples• comparison between RTM (adjoint migration) w. multiples and fast inversion w. multiples
multiples
primaries
A shot-gather of total data
RTM image, with total downgoing data as areal source
Fast inversion w/ sparsity promotion, with total downgoing data as areal source, simulation cost ~1 RTM of all data
Source estimation[50% simulation overhead incurred w/ variable projection]
Aravkin and van Leeuwen, 2012; Aravkin et al., 2013; Tu et al., 2013b
Given an estimate for , source wavelet has a closed-‐form solution:
Now solve (remember ):
x
Si = wiI
wi(x) =< rF[m0, I]CH
x, p
i+rF[m0,Pi]C
Hx >
k < rF[m0, I]CHxk22
� ˜m = C
Hargmin
x
kxk1
subject to
X
i2⌦
kpi�rFi[m0, wi(x)I�Pi]C
Hxk22 �2
Pseudo-codeInput:total upgoing wavefield P
i
, background velocity model m0, tolerance � = 0,iteration limit k
max
Initialization:iteration index k 0, subproblem index l 0,x
l
0, wi
= 1 for all i 21, · · · , n
f
while k < kmax
do⌦
l
,El
new independent draw, Pi
= Pi
El
, I = IEl
, pi
= vec(Pi
)
⌧l
determine from ⌧l�1 and � by root finding on the Pareto curve
xl
(argmin
x
Pi2⌦lkp
i
�rFi
[m0, wi
(x)I�Pi
]CHxk22subject to kxk1 ⌧
l
//warm start
with xl�1, solved in k
l
iterations, in each iteration, compute
wi
(x) =<rF[m0,I]C
Hx, p
i+rF[m0,Pi]C
Hx>
k<rF[m0,I]CHxk2
2
k k + kl
, l l + 1
end whileOutput: Model perturbation estimate �m = CHx
Fast inversion w/ sparsity promotion w. source estimation, w/ total downgoing data as areal source, simulation cost ~1.5 RTM of all the data
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Iterations
Nor
mal
ized
cro
ss−c
orre
latio
n
Blue: true sourceBlack: source estimation
Model error decrease[measured by normalized cross-correlation]
Field data example
• North Sea data set, courtesy of PGS• water depth about 90m (two-‐way travel time of sea bottom reflection ~1.2s)• dual-‐sensor streamer data• after up / down data decomposition to remove receiver ghosts• 401 co-‐located sources / receivers, 12.5m spacing• data after 75Hz low-‐pass-‐filter to remove receiver aliasing• the imaging procedure uses data up to 48Hz with a 6.25m grid spacing• 30 simultaneous source, 30 randomized frequencies, 30 iterations so the simulation cost of the inversion of total data with source estimation is ~1 RTM with all the data
• Robust EPSI algorithm used to separate primaries and multiples, and to get a source estimate to obtain conventional RTM images
Lin and Herrmann 2013
List of examples
• conventional RTM of total data and primaries(i.e., before and after de-‐multiple via Robust EPSI)‣ EPSI output source wavelet‣ high-‐pass filtered to remove low-‐frequency artifacts‣ depth weighted to boost amplitudes of the deeper part
• difference between the two images‣ artifacts of multiples in the image
• fast inversion of multiples‣ turn multiples from noise to signal
• fast inversion of total data with source estimation‣ more coherent reflector events via joint imaging of primaries and multiples
Lateral distance (m)
Dep
th (m
)
0 1000 2000 3000 4000 5000
0
500
1000
1500
2000
2500
30001600
1800
2000
2200
2400
2600
2800
3000
3200
Migration velocity model converted from NMO velocity (m/s)
Lateral distance (m)
Dep
th (m
)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
2500
3000
Conventional RTM of primaries
Lateral distance (m)
Dep
th (m
)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
2500
3000
Conventional RTM of total data
Lateral distance (m)
Dep
th (m
)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
2500
3000
ghost image of the sea bottom
artifacts from multiples
Artifacts from multiples
Lateral distance (m)
Dep
th (m
)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
2500
3000
Inversion of multiples
true reflectors also imaged by the primaries
Lateral distance (m)
Dep
th (m
)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
2500
3000
Conventional RTM of primaries
true reflectors also imaged by the primaries
Lateral distance (m)
Dep
th (m
)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
2500
3000
Inversion of multiples
reduced artifacts from multiples
Lateral distance (m)
Dep
th (m
)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0
500
1000
1500
2000
2500
3000
Inversion of total data
Conclusions
Primaries and multiples can be jointly imaged by iterative inversion‣ acausal artifacts in RTM of multiples largely removed ‣ high-‐fidelity on-‐the-‐fly source estimation
Inversion can be carried out efficiently by‣ eliminating dense matrix products in multiple prediction‣ reducing simulation cost using compressive imaging
Inversion can also lead to higher spatial resolution‣ by properly inverting the source wavelet
ReferencesBerkhout, A. J., 1993. Migration of multiple reflections, in SEG Technical Program Expanded Abstracts, vol. 12, pp. 1022–1025, SEG. Guitton, A., 2002. Shot-profile migration of multiple reflections, in SEG Technical Program Expanded Abstracts, vol. 21, pp. 1296–1299.Muijs, R., Robertsson, J. O. A., & Holliger, K., 2007. Prestack depth migration of primary and surface-related multiple reflections: Part i—imaging, Geophysics, 72, S59–S69.Verschuur, D. J., 2011. Seismic migration of blended shot records with surface-related multiple scattering, Geophysics, 76(1), A7–A13.Long, A. S., Lu, S., Whitmore, D., LeGleut, H., Jones, R., Chemingui, N., & Farouki, M., 2013. Mitigation of the 3d cross-line acquisition footprint using separated wavefield imaging of dual- sensor streamer seismic, in 75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013, EAGE.Wong, M., Biondi, B., & Ronen, S., 2014. Imaging with multiples using least-squares reverse time migration, The Leading Edge, 33(9), 970–976.Lin, T. T., Tu, N., & Herrmann, F. J., 2010. Sparsity-promoting migration from surface-related multiples, in SEG Technical Program Expanded Abstracts, vol. 29, pp. 3333–3337, SEG.Liu, Y., Chang, X., Jin, D., He, R., Sun, H., & Zheng, Y., 2011. Reverse time migration of multiples for subsalt imaging, Geophysics, 76(5), WB209–WB216.Whitmore, N. D., Valenciano, A. A., & Sollner, W., 2010. Imaging of primaries and multiples using a dual-sensor towed streamer, in SEG Technical Program Expanded Abstracts, vol. 29, pp. 3187–3192.Wong, M., Biondi, B., & Ronen, S., 2012. Imaging with multiples using linearized full-wave inversion, in SEG Technical Program Expanded Abstracts, pp. 1–5, Chapter 706, SEG.
References (cont.)Herrmann, F. J. & Li, X., 2012. Efficient least-squares imaging with sparsity promotion and compressive sensing, Geophysical Prospecting, 60(4), 696–712.Tu, N. & Herrmann, F. J., 2012a. Least-squares migration of full wavefield with source encoding, in 74th EAGE Conference & Exhibition incorporating SPE EUROPEC 2012 EAGE .Tu, N. & Herrmann, F. J., 2012b. Imaging with multiples accelerated by message passing, in SEG Technical Program Expanded Abstracts, pp. 1–6, Chapter 682, SEG.Verschuur, D. J., Berkhout, A. J., & Wapenaar, C. P. A., 1992. Adaptive surface-related multiple elimination, Geophysics, 57(9), 1166–1177.Tu, N., Li, X., & Herrmann, F. J., 2013a. Controlling linearization errors in l1 regularized inversion by rerandomization, in SEG Technical Program Expanded Abstracts, pp. 4640–4644, SEG.Aravkin, A. Y. & van Leeuwen, T., 2012. Estimating nuisance parameters in inverse problems, Inverse Problems, 28(11).Aravkin, A. Y., van Leeuwen, T., & Tu, N., 2013. Sparse seismic imaging using variable projection, in Acoustics, Speech and Signal Processing (ICASSP), 2013 IEEE International Conference on, pp. 2065–2069.Tu, N., Aravkin, A. Y., van Leeuwen, T., & Herrmann, F. J., 2013b. Fast least-squares migration with multiples and source estimation, in 75th EAGE Conference & Exhibition incorporating SPE EUROPEC 2013, EAGE.Lin, T. T. & Herrmann, F. J., 2013. Robust estimation of primaries by sparse inversion via one-norm minimization, Geophysics, 78(3), R133–R150.
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, Subsalt Solutions Ltd., Total SA, WesternGeco, and Woodside.
Many thanks to the authors of iWave, SPGl1, SLIM FD modelling engine, the Sigsbee 2B model, and to PGS for providing the Nelson field data.Thanks to Dr. Eric Verschuur and Dr. Joost van der Neut for beneficial discussions.
You are invited to check out our website: www.slim.eos.ubc.ca
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