seismic data interpolation problem forward & ÒclassicalÓ ... · seismic laboratory for...

Felix Herrmann, Ph.D.Assistant professorDepartment of Earth & Ocean Sciences

Ozgur Yilmaz, Ph.D.Assistant professorDepartment of Mathematics

Michael Friedlander, Ph.D.Assistant professorDepartment of Computer Science

NSERC CRD site visitFebruary 10th, 2006University of British Columbia

Contact:Felix J. [email protected]

Website:slim.eos.ubc.ca/NSERC2006

DNOISEDynamic Nonlinear Optimization for Imaging in Seismic Exploration

SEG international exposition & 76th annual meetingSPNA 2: topics in seismic processing IWednesday, October 4, 2006

Gilles [email protected]://wigner.eos.ubc.ca/~hegilles

Felix [email protected]://slim.eos.ubc.ca

Seismic Laboratory for Imaging & ModelingDepartment of Earth & Ocean SciencesThe University of British Columbia

Application of stable signal recovery to seismic data interpolation

1

Seismic Laboratory for Imaging and Modeling

Motivation

! improve

– multiple prediction & removal

– aliased ground roll removal

– imaging

! reduce acquisition cost & time

– acquire less data

2


Approach

! exploit geometry of seismic data

– high dimensional

• typically 5D - i.e. time ! source

location ! receiver location

– very strong geometrical structure (i.e.

wavefronts)

! provide sampling criteria for seismic data

– how well can one expect to recover

seismic data given an acquisition

geometry? (interpolation of vintage

survey)

– what is the ‘optimal’ acquisition

geometry in order to recover seismic

data within a given accuracy? (sparse

sampling scheme)

3


Agenda

! seismic data interpolation problem

– forward & “classical” inverse problem

! Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI)

– compressibility as a prior

– curvelets

! synthetic and real data examples

4


Seismic data interpolation problem

5


Forward & “classical” inverse problem

! (severely) underdetermined system of linear equations

– infinitely many solutions

! among “classical” approaches

– minimize energy (i.e. quadratic constraint)

signal =

picking

matrix

y

f0

P + n noise

ideal data

f̃= argminf

1

2!y"Pf!22! "# $

data misfit

+! !Lf!22! "# $

energy constraint

6

Released to public domain under Creative Commons license type BY (https://creativecommons.org/licenses/by/4.0).Copyright (c) 2006 SLIM group @ The University of British Columbia.


Compressibility as a prior

! what is compressibility?

– generalization of sparsity

– x is compressible if its sorted entries decay sufficiently fast

– compressible signals have small l1 norm

!x!1 :=

!

i

|x|(i)

7



! what is compressibility?

– generalization of sparsity

– x is compressible if its sorted entries decay sufficiently fast

– compressible signals have small l1 norm

!x!1 :=

!

i

|x|(i)

8



! why use sparsity/compressibility?

– powerful property (i.e. extra piece of information about the signal)

Idea of promoting sparsity for geophysical problems is commonly attributed to Claerbout and Muir in 1973 and was further developed e.g. by Oldenburg who proposed to deconvolve seismic traces for reflectivity as sparse spike trains.

9


Compressible representations

! seek

– simplicity

• signal f0 is built as a linear combination of few atoms from dictionary D

– expressiveness

• each selected atom significantly contributes to the construction of f0 (i.e.

energy of the signal f0 is concentrated in few significant coefficients)

coefficientssignal

x0

= D

dictionary

f0

10


Representations for seismic data

Transform Underlying assumption

FK plane waves

linear/parabolic Radon transform linear/parabolic events

wavelet transform point-like events (1D singularities)

curvelet transform curve-like events (2D singularities)

! curvelet transform

– multi-scale: tiling of the FK domain into

dyadic coronae

– multi-directional: coronae sub-

partitioned into angular wedges, # of

angle doubles every other scale

– anisotropic: parabolic scaling principle

– local

k1

k2angular

wedge2j

2j/2

11


2D curvelets

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4

12


0

0.5

1.0

1.5

2.0

Tim

e (

s)

-2000 0 2000Offset (m)

0

0.5

1.0

1.5

2.0

Tim

e (

s)

-2000 0 2000Offset (m)

Significant

curvelet coefficientCurvelet

coefficient~0

13

Sparsity-promoting inversion*

! reformulation of the problem

! Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI)

– look for the sparsest/most compressible,

physical solution

signal =y + n noise

curvelet representation

of ideal data

PCH

x0

KEY POINT OF THE RECOVERY

(P0)

!

""""#

""""$

x̃= arg

sparsity constraint% &' (

minx

!x!0 s.t. !y"PCHx!2 # !

f̃= CH x̃

(P1)

!

"

#

"

$

x̃= argminx !Wx!1 s.t. !y"PCHx!2 # !

f̃= CH x̃

(P0)

!

""""#

""""$

x̃= arg

sparsity constraint% &' (

minx

!x!0 s.t.

data misfit% &' (

!y"PCHx!2 # !

f̃= CH x̃* inspired by Stable Signal Recovery (SSR) theory by E. Candès, J. Romberg, T. Tao & Fourier Reconstruction with Sparse Inversion (FRSI) by P. Zwartjes

14


Sampling & aliasing

15


From aliasing to noise

16


Examples

! synthetic: Delphi dataset

– uplift from image to volume (time ! source location ! receiver location)

interpolation

– influence of missing data structure

! real: ExxonMobil test dataset

– challenging land data

• ground roll

– slow (i.e. weak minimum velocity constraint)

– strong (~ 30 dB stronger than signal)

• underlying de-aliasing problem

17


Model

shot of study

spatial sampling: 12.5 m

18


shot of study

avg. spatial sampling: 62.5 m

Data20% tracesremaining

19


t

xs

xr

data

imageinterpolation

t

xr

t

xs

xr

volumeinterpolation

t

xr

windowing

t

xr

windowing

comparison

20


40% tracesremaining

image interpolation withno velocity constraint

volume interpolation withno velocity constraint

data

11.67 dB 16.99 dB

20% tracesremaining

image interpolation withno velocity constraint

volume interpolation withno velocity constraint

data

3.90 dB 9.26 dB



SNR = 20 ! log10

!

"model"2

"reconstruction error"2

"

21


40% tracesremaining

difference (image) difference (volume)data

20% tracesremaining

difference (image) difference (volume)data



11.67 dB 16.99 dB

3.90 dB 9.26 dB

22




23


Interpolated result


(no minimum velocity constraint)

SNR = 16.92 dB

24


Model


25


Difference



SNR = 16.92 dB

26




27


Interpolated result



SNR = 9.26 dB

28


Model


29


Difference



SNR = 9.26 dB

30


Experiment

comparison

1

2 3

4

5

1. remove random

receiver positions

2. increase spatial

sampling

3. interpolate

4. decrease spatial

sampling

5. apply basic FK filter

31


avg. spatial sampling: 10 m

32


avg. spatial sampling: 10 m

33


34


SNR = 7.79 dB

35


SNR = 7.79 dB

36


(basic FK filtering )

37


Experiment

1 2 3

1. increase spatial

sampling

2. interpolate

3. apply basic FK filter

38


39


40


Conclusions

! curvelets exploit the very strong geometrical structure of seismic data

! compressibility is a powerful property (i.e. extra piece of information about the signal) that offers striking benefits

! randomness in the structure of missing data significantly helps recovery

! Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI) performs well

– synthetic: Delphi dataset

• from 62.5 m to 12.5 m

• significant uplift from image to volume interpolation

• significant influence of the structure of missing data

– real: ExxonMobil test dataset

• from 10 m to 2.5 m

• CRSI interpolates both signal & noise (i.e. ground roll)

41


Acknowledgments

! SLIM team members with a special thanks to H. Modzelewski, S. Ross-Ross, and D. Thomson for their great help with SLIMpy

! ExxonMobil Upstream Research Company and in particular the Subsurface Imaging Division with a special thanks to R. Neelamani, W. Ross, J. Reilly, and M. Johnson for exciting discussions and for the test dataset

! E. Verschuur for the Delphi dataset

! E. Candès, L. Demanet, and L. Ying for CurveLab

! S. Fomel and the developers of Madagascar

This presentation was carried out as part of the SINBAD project with financial support, secured through ITF, from the following organizations: BG, BP, Chevron, ExxonMobil, and Shell. SINBAD is part of the collaborative research & development (CRD) grant number 334810-05 funded by the Natural Science and Engineering Research Council (NSERC).

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seismic data interpolation problem forward & ÒclassicalÓ ... · seismic laboratory for...

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