wavefield-denoising and source encoding · wavefield-denoising and source encoding...
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University of British ColumbiaSLIM
Wavefield-denoising and source encodingRongrong Wang, Ozgur Yilmaz, and Felix Herrmann
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.
Outline
Wavefield Reconstruction Inversion (WRI)-denoising version
2
Source encoding using Wavefields SVD
Conclusion
• Fourier, Curvelet domain denoising [Sacchi et al., 1998] [Hennenfent, and Herrmann 2006]
• f-‐x Singular Spectrum Analysis [Trickett 2002] [Oropeza and Sacchi, 2011]
• Mid-‐point offset domain denoising [Kumar et al., 2013]
Denoising techniques for incoherent noise
3
Issues : the denoising result is not obvious for FWI.
Minimizing data misfit
where,
Full-waveform inversion
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[J. Virieux and S. Operto, 2009 ]
[Tarantola, 1984]
minimize
X
i,j
kP⌦iui,j � di,jk22
subject to Ai,j(m)ui,j = qi,j ,
m - squared slowness
di,j - observed data of the ith shot at the lth frequency
qi,j - the ith shot at the lth frequency
Ai,j - Helmholtz operator of qi,j
⌦i - receivers of the ith source
P⌦i - restriction to the receiver locations
Wavefield-Reconstruction Inversion (WRI)
Unconstained formulation:
where,
Eliminating u by variable projection
So
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[van Leeuwen, T and Herrmann, F J , 2013]
[Peters, B, Herrmann, F J and van Leeuwen, T and , 2014]
(u, m) = minu,m
g�(u,m)
g�(u,m) =X
i,j
�2kP⌦iui,j � di,jk22 + kAi,j(m)ui,j � qi,jk22
g�(m) = g�(m, u(m))u(m) = minu
g�(m,u)
[Y. Aravkin, A. and van Leeuwen, T 2013]
rg�
(m) = rx
g�
(m, u)
minm,u
g�(m,u) = minm
g�(m)
⌘ rmkA(m)u� qk2F
Alternating projection method
Fix , solve from
Fix , update by first or second order methods
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Ai,j(mk�1)
�P⌦i
�ui,j =
qi,j
�di,j
�
mk�1
mkuk
uk
mk = mk�1 � ↵H�1rkA(m)uk � qk2F
Wavefield denoising, interpolation and extrapolation
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Look at the first update u1
, which is a solution to
Ai,j(m0
)
�P⌦i
�ui,j =
qi,j
�di,j
�
u1
can be used as a
• denoiser: P⌦iui,j ⇡ di,j is smoothed data.
• interpolation/extrapolation method: dext
i,j ⌘ P⌦
ui,j , where ⌦ = [⌦i
Denoising effects
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The denoising e↵ect depends on �
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receivers
h=1h=.01h=.002
Interpolation
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0 50 100 150 200 250 300 350−50
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observed noisy data with missing receivers
Interpolation
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0 50 100 150 200 250 300 350-50
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0
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observedtrueλ=0.01
0 50 100 150 200 250 300 350-50
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0
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observedtrueλ=0.002
Interpolation
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0 50 100 150 200 250 300 350-50
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0
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observedtrueλ=0.0003
0 50 100 150 200 250 300 350-50
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0
10
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observedtrueλ=0.0001
wrong direction
Extrapolation
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0 50 100 150 200 250 300 350 400−100
−50
0
50
data from true modeldata from initial modelextrapolated
0 50 100 150 200 250 300 350 4000
2
4
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Source location : 167
Receivers of this source: 117 to 217
Union of all receivers ⌦ = [⌦i: 0 to 384
Error
Receivers
real part of data
Pros and Cons
Pros: provided that a good and are used, the output
Cons
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• is sparse in curvelet domain• has fast decaying singular vectors in the mid-‐point offset domain
• is of low rank in the SSA analysis• has a consistent frequency as the current frequency slice• is not very sensitive to
�
�
�• no good strategy to choose , even when the noise level is known
• one may need a different for each frequency/source• large computational complexity
u1
m0
m0
The denoising formulationSuppose the noise level is roughly known. We propose to
solve
The problem intrinsically decouples for all the , so we solve
each independently.
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min
ukA(m0)u� qk2F
subject to kP⌦iui,j � di,jk2 ✏i,j , 8i, j.
i, j
ui,j
The method of multipliers
The method of multipliers solves general problems of the form
The augmented Lagrangian of the above system
Alternatively update x and p
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min
x
f(x) subject to g
i
(x) 2 C
i
C
i
are convex,f ,g
i
are di↵erential with Lipschitz continuous gradients
L(x, p) = f(x) +X
i
1
2�ikDCi(pi + �ihi(x)k22 �
1
2�ikpik22
x
k+1 = argminx
L(x, pk)
p
k+1i
= D
�iCi(pk
i
+ �
i
h
i
(xk+1)) where DC(p) = p�⇧Cp
Implementation detailsLet
Alternatively update
.
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L(ui,j ,p) = kA(m)ui,j � qi,jk2 +1
2�kD✏i,j (p+ �(P⌦iui,j � di,j)k22 �
1
2�kpk22
pk+1i,j = D✏i,j (p
ki,j + �(P⌦iu
ki,j � di,j))
where D✏(x) =
✓max
⇢0, 1� ✏
kxk2
�◆x
uk+1i,j = min
vL(v,pk
i,j) rvL(v,pki,j) = AT (m) (A(m)v � qi,j) +
1
2PT⌦i
·D✏(pki,j + �(P⌦iu
ki,j � di,j))
LBFGS:
Denoising result
is the best for WRI that we found manually.
The denoising algorithm automatically gives a similar output.
170 10 20 30 40 50 60 70
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λ=0.0003WRI-denoise
� = 0.0003 �
Marmousi
True Initial
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km/s km/s
Marmousi
• 2160m ✕ 8060m• 12m ✕ 12m grid• 223 uniformly spaced source and receivers at the surface• 1D initial guess• 3Hz-‐ 12Hz• WRI: 10 iterations for each frequency • FWI: stops when converges to local minimum• Same SNR for each frequency
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Marmousi
True Initial
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Clean data
FWI inversion WRI inversion
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Noisy data
FWI, SNR=6.8 WRI, SNR=4.8
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Souce encoding/data compression
[Habashy, T., ABubakar, A, Pan, G, and Belani, A. 2011] proposed to use SVD for data compression.
The simultaneous sources corresponding to is
So, it’s also a source encoding technique that speeds up the implementation.
23
dcompr
= dobs
Vr
,dobs
= U⌃V T ⇡ Ur
⌃r
V T
r
dcompr
qcompr
⌘ qVr
SVD on the WavefieldRecall in WRI
Observe
where
24
rg�
(m) = rx
g�
(m, u) ⌘ rmkA(m)u� qk2F
kA(m)u� qk2F ⇡ kA(m)uVr � qVrk2F
u = U ⌃V T
SVD on the WavefieldRecall in WRI
Observe
where
25
rg�
(m) = rx
g�
(m, u) ⌘ rmkA(m)u� qk2F
kA(m)u� qk2F ⇡ kA(m)uVr � qVrk2F
u = U ⌃V T
Source encoding
Simulation Result: clean dataUse 25 out of 223 sources, data compression rate 9:1. Inversion method WRI
26x
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Encoding with Encoding withVr Vr
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Clean Data
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Use 25 out of 223 sources, data compression rate 9:1. Inversion method FWI
Encoding with Encoding withVr Vr
km/s km/s
Denoising effect: white Gaussian noise (4.8 SNR)
28x
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Use all data Use 1/3 of the data dobs
Vr
km/s km/s
Denoising effect: erratic noise
Use all data Use 1/3 of the data
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dobs
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Clean data: Towing in a row
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Use all data Use SVD, compression rate 3:1 Offsets: 0-‐3.5km, 1 moving sources, 101 receivers,
km/s km/s
Noisy, towing in a row (SNR=6)
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Offsets: 0-‐3.5km, 1 moving sources, 101 receivers, Use all data Use SVD, compression rate 3:1
km/s km/s
Conclusion
We proposed two Wavefield based denoising methods that can be used to remove both white and erratic noise.
The wavefield SVD approach is also a good compression technique, is useful to set source locations that best illuminate the subsurface.
We showed the denoising version of WRI is effective and fast, which opens the door for another inversion technique.
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This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada via the Collaborative Research and Development Grant DNOISEII (375142-‐-‐08). This research was carried out as part of the SINBAD II project which is supported by the following organizations: BG Group, BGP, CGG, Chevron, ConocoPhillips, DownUnder GeoSolutions, Hess, Petrobras, PGS, Schlumberger, Sub Salt Solutions and Woodside
Acknowledgements
Thank you!