Seismic data interpolation problem Forward & ÒclassicalÓ ... · Seismic Laboratory for Imaging...
Transcript of Seismic data interpolation problem Forward & ÒclassicalÓ ... · Seismic Laboratory for Imaging...
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
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Seismic Laboratory for Imaging and Modeling
Motivation
! improve
– multiple prediction & removal
– aliased ground roll removal
– imaging
! reduce acquisition cost & time
– acquire less data
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Seismic Laboratory for Imaging and Modeling
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)
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Seismic Laboratory for Imaging and Modeling
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
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Seismic Laboratory for Imaging and Modeling
Seismic data interpolation problem
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Seismic Laboratory for Imaging and Modeling
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
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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.
Seismic Laboratory for Imaging and Modeling
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)
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Seismic Laboratory for Imaging and Modeling
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)
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Seismic Laboratory for Imaging and Modeling
Compressibility as a prior
! 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.
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Seismic Laboratory for Imaging and Modeling
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
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Seismic Laboratory for Imaging and Modeling
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
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Seismic Laboratory for Imaging and Modeling
2D curvelets
-0.4
-0.2
0
0.2
0.4
-0.4 -0.2 0 0.2 0.4
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Seismic Laboratory for Imaging and Modeling
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
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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
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Seismic Laboratory for Imaging and Modeling
Sampling & aliasing
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Seismic Laboratory for Imaging and Modeling
From aliasing to noise
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Seismic Laboratory for Imaging and Modeling
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
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Seismic Laboratory for Imaging and Modeling
Model
shot of study
spatial sampling: 12.5 m
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Seismic Laboratory for Imaging and Modeling
shot of study
avg. spatial sampling: 62.5 m
Data20% tracesremaining
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Seismic Laboratory for Imaging and Modeling
t
xs
xr
data
imageinterpolation
t
xr
t
xs
xr
volumeinterpolation
t
xr
windowing
t
xr
windowing
comparison
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Seismic Laboratory for Imaging and Modeling
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
avg. spatial sampling: 31.25 m
avg. spatial sampling: 62.5 m
SNR = 20 ! log10
!
"model"2
"reconstruction error"2
"
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Seismic Laboratory for Imaging and Modeling
40% tracesremaining
difference (image) difference (volume)data
20% tracesremaining
difference (image) difference (volume)data
avg. spatial sampling: 31.25 m
avg. spatial sampling: 62.5 m
11.67 dB 16.99 dB
3.90 dB 9.26 dB
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Seismic Laboratory for Imaging and Modeling
avg. spatial sampling: 62.5 m
Data20% tracesremaining
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Seismic Laboratory for Imaging and Modeling
Interpolated result
spatial sampling: 12.5 m
(no minimum velocity constraint)
SNR = 16.92 dB
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Seismic Laboratory for Imaging and Modeling
Model
spatial sampling: 12.5 m
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Seismic Laboratory for Imaging and Modeling
Difference
spatial sampling: 12.5 m
(no minimum velocity constraint)
SNR = 16.92 dB
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Seismic Laboratory for Imaging and Modeling
avg. spatial sampling: 62.5 m
Data20% tracesremaining
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Seismic Laboratory for Imaging and Modeling
Interpolated result
spatial sampling: 12.5 m
(no minimum velocity constraint)
SNR = 9.26 dB
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Seismic Laboratory for Imaging and Modeling
Model
spatial sampling: 12.5 m
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Seismic Laboratory for Imaging and Modeling
Difference
spatial sampling: 12.5 m
(no minimum velocity constraint)
SNR = 9.26 dB
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Seismic Laboratory for Imaging and Modeling
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
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Seismic Laboratory for Imaging and Modeling
avg. spatial sampling: 10 m
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Seismic Laboratory for Imaging and Modeling
avg. spatial sampling: 10 m
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Seismic Laboratory for Imaging and Modeling
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Seismic Laboratory for Imaging and Modeling
SNR = 7.79 dB
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Seismic Laboratory for Imaging and Modeling
SNR = 7.79 dB
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Seismic Laboratory for Imaging and Modeling
(basic FK filtering )
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Seismic Laboratory for Imaging and Modeling
Experiment
1 2 3
1. increase spatial
sampling
2. interpolate
3. apply basic FK filter
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Seismic Laboratory for Imaging and Modeling
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Seismic Laboratory for Imaging and Modeling
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Seismic Laboratory for Imaging and Modeling
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)
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Seismic Laboratory for Imaging and Modeling
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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