Applica'on  of  a  convex  phase  retrieval  method  to  blind  seismic  deconvolu'on  

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Number of traces used

Aver

age

SNR

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d = 1e−16d = 1e−14d = 1e−12d = 1e−10d = 1e−8d = 1e−6d = 1e−4true

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1

Ernie  Esser  and  Felix    J.  Herrmann  

1

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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50

100

150

200

250 −0.03

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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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S

data

mis

fit

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trueestimated

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frequency

abs

valu

e of

fft

true magnitude responseraw datadenoised raw datafrom estimated source wavelet

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frequency

abs

valu

e of

fft

true magnitude responseraw datadenoised raw datafrom estimated source wavelet

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trueestimated

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S=50S=90S=130S=170S=220true

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S=50S=90S=130S=170S=220

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S=100S=150S=200S=250S=300

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S=100S=150S=200S=250S=300true

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.