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