Post on 11-Oct-2020
Applica'on of a convex phase retrieval method to blind seismic deconvolu'on
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d = 1e−16d = 1e−14d = 1e−12d = 1e−10d = 1e−8d = 1e−6d = 1e−4true
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Ernie Esser and Felix J. Herrmann
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M
MX
j=1
| ˆfj |2 =
1
M
MX
j=1
|w|2|uj |2 ⇡ c|w|2 for some c
Data Model
Es'ma'ng autocorrela'on of source wavelet from data
Assuming unknown reflec'vity is white,
Source wavelet Sparse reflec'vity series
Convolu'on Noisy data fj = w ⇤ uj + ⌘j , j = 1, ...,M
w ⇤ uj
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Assume and set kwk = 1 b = denoise
PMj=1 | ˆfj |2
mean(
PMj=1 | ˆfj |2)
!⇡ |w|2
Rela've error versus SNR and M k|wtrue|�pbkp
N
Ricker wavelet Minimum phase approxima'on |w|
Reconstruc'ng minimum phase wavelet
In principle, the minimum phase wavelet can be computed from b by applying the Discrete Hilbert Transform to . [White and O’Brien 1974], [Claerbout 1976], [Lines and Ulrych 1977]
This is not well posed if b has too many zeros, and the result using shiXed data can be sensi've to d.
Minimum phase wavelet es'mated using
pb+ d
log(
pb)
Regulariza'on
Directly encourage energy to be concentrated at the beginning by adding a weighted l2 penalty [Lamoureux and Margrave 2007] to a nonconvex model for phase retrieval
minw
NX
n=1
n2w2n s.t. k|w|2 � bk ✏ and kwk = 1
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LiXing
Convex semidefinite relaxa'on
minW⌫0,tr(W )=1
tr(CW ) s.t. kA(W )� bk ✏
w
Let F be the DFT matrix and define a linear operator so that
W = wwT
A(W ) = diag(FWF ⇤) A(wwT ) = |w|2
wwTLiX to a posi've semidefinite matrix W and solve a convex problem for W [Candes, Strohmer and Voroninski 2013]
Douglas Rachford method
where C is diagonal with Cnn = n2
Diagonal of C
Solve for W by itera'ng V k+1 = ⇧kA(·)�bk✏(2W
k � V k)�W k + V k
W k+1 = ⇧4(V k+1 � ↵C) ,
where and is the orthogonal projec'on onto symmetric posi've definite matrices with trace equal to one. [Demanet and Hand 2012]
↵ > 0 ⇧4
Seismic Laboratory for Imaging and Modeling
Recovering wavelet
The trace penalty usually encourages W to be rank one. Otherwise recover w as its top normalized eigenvector. Results for .01 and .1 are visually similar. ✏ =
Impulsive w is recovered well from noisy data (SNR = 10)
Less impulsive w is shiXed leX but its magnitude response s'll matches the data
Acknowledgements
max(|w|� 10
�6, 0) + d
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true magnitude responseraw datadenoised raw datafrom estimated source wavelet
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true magnitude responseraw datadenoised raw datafrom estimated source wavelet
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S=50S=90S=130S=170S=220true
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Future Work
Alterna'ng Minimiza'on
w uj
Let where . With a good es'mate of S and an ini'al guess for z, recover w by solving
w = Xz =
z0
�
A good ini'al guess for z is the top normalized eigenvector of [Netrapalli, Jain and Sanghavi 2013].
real(XTF ⇤diag(b)FX)
Alternate and sk+1 =FXzk
|FXzk|zk+1 =
1
Nreal(XTF ⇤diag(
pb)sk+1)
Recovered w given different support es'mates Ini'al guesses
It may be possible to automa'cally es'mate the length of the support of w by using the alterna'ng minimiza'on strategy to compute w for many choices of S and choosing S to be where the data misfit begins to level off.
Data misfit versus S
Use the assump'on that is sparse to recover and/or correct w by alterna'ng minimiza'on techniques using an l1 penalty [Ulrych and Sacchi [2006], nonconvex sparsity penal'es such as l1/l2 [Krishnan, Tay and Fergus 2011] or even liXing w and together [Ahmed, Recht and Romberg 2012].
uj uj
uj
Also consider generalizing the model to include mul'ples [Lin and Herrmann 2013].
This work was partly supported by the NSERC Discovery Grant (22R81254) and the Collabora've Research and Development Grant DNOISE II (375142-‐08).
This research was carried out as part of the SINBAD II project with support from
minz2RS ,s2CN
1
2kFXz � diag(
pb)sk2 s.t. |sj | = 1
X =
IS0
�
Conclusions • LiXing is a viable approach for recovering an impulsive source wavelet from an es'mate of its magnitude response. The proposed convex model is robust to noise and independent of the ini'al guess. • Alterna'ng minimiza'on and minimum phase es'ma'on are faster alterna'ves that are, however, sensi've to the ini'al guess and processing of the data.
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.