Compressive sampling: a new paradigm for seismic data … · 2019. 11. 14. · Use recent insights...

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

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

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

ION Technical forum - Sprowston, UKTuesday 15th – Thursday 17th April

Compressive sampling: a new paradigm for seismic data acquisition and processing?

Wednesday, April 23, 2008

Released to public domain under Creative Commons license type BY (https://creativecommons.org/licenses/by/4.0).Copyright (c) 2008 SLIM group @ The University of British Columbia.

mailto:[email protected]


http://wigner.eos.ubc.ca/~hegilles






Seismic Laboratory for Imaging and Modeling

Motivation

Current state of affairs:– Seismic data processing firmly rooted in paradigm of regular Nyquist sampling

– Practitioners go all out to create regularly-sampled data volumes

– Preferred by Fourier-based processing flows

Recent theoretical & hardware developments– Alternative multiscale, localized & directional transform domains that compress

seismic data

– New nonlinear sampling theory that supersedes the overly pessimistic Nyquist sampling criterion

– New autonomous data acquisition devices that allow for more flexibility during acquisition

– New simultaneous & continuous recording

Recent successful application of directional transforms in seismic– wavefield separation

– wavefield matching

– image-amplitude recovery



Today’s agenda

Sparsity-promoting wavefield recovery– sparsifying transform

– favorable (random) acquisition

– nonlinear recovery by sparsity promotion

Seismic data processing with curvelets– primary-multiple separation

A look ahead ...– stable wavefield inversion

– multidimensional acquisition design


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



Sampling and reconstruction of seismic wavefields in the curvelet domain







Wavefield reconstruction methods

filter-based methods [Spitz’91, Fomel’00]

– convolve the incomplete data with an interpolating filter

wavefield-operator-based methods [Canning and Gardner’96, Biondi et al.’98, Stolt’02]

– explicitly include wave propagation

– require knowledge of velocity model

– computationally intensive

transform-based methods [Sacchi et al.’98, Trad et al.’03, Zwartjes and Sacchi’07]

– fastest approaches

– no explicit link with wave propagation

Performance of most aforementioned methods deteriorates for data with acquisition irregularities.



Key ideas

Use recent insights from the field of compressive sensing to

formulate a new wavefield reconstruction method that handles both regular and irregular acquisition geometries

– curvelet reconstruction with sparsity-promoting inversion (CRSI) [Herrmann and Hennenfent‘08]

develop a new random coarse sampling scheme that maximizes the performance of CRSI

– jittered undersampling scheme [Hennenfent and Herrmann‘08]

implement a new large-scale, one-norm solver– iterative soft thresholding with cooling (ISTc) [Herrmann and Hennenfent’08, Hennenfent et

al.‘08]

formalize nonlinear ad hoc methods – anti-leakage Fourier transform [Xu et. al. ‘05]



Problem statement

Consider the following (severely) underdetermined system of linear equations

Is it possible to recover x0 accurately from y?

unknown

data(measurements/observations)

x0

Ay=



Perfect recovery

conditions– A obeys the uniform uncertainty principle

– x0 is sufficiently sparse

procedure

performance– S-sparse vectors recovered from roughly on the order of S measurements (to within

constant and log factors)

minx

‖x‖1

︸︷︷︸

sparsity

s.t. Ax = y︸︷︷︸

perfect reconstruction

x0

Ay=

[Candès et al.‘06][Donoho‘06]



Simple example

x0

A

A := RFH

y=

Fourier coefficients(sparse)

with

Fouriertransform

restrictionoperator

signal



NAIVE sparsity-promoting recovery

inverseFourier

transform

detection +data-consistent

amplitude recovery

Fouriertransform

y

AH

=

A

y=detection

Ar data-consistent amplitude recovery

y

A†r

=

x0



Coarse sampling schemes

Fourier

transform

✓

✗

3-fold under-sampling

significant coefficients detected

ambiguity

few significant coefficients

Fourier

transform

Fourier

transform

[Hennenfent and Herrmann‘08]



Observations

Random undersampling breaks the constructive interferences, i.e. aliases

Turns alias into incoherent noise

Works by virtue of – incoherence (correlations) between the rows of the Dirac measurement basis

and the columns of the Fourier synthesis basis

– maximum spreading of Diracs in Fourier domain

– maximum leakage

– independence amongst columns of A, i.e., there exists a subset of columns of A that forms an orthonormal basis

According to theory of compressive sensing – recovery stable w.r.t. noise

– measurement & sparsity bases can be more general



Sparsity-promoting wavefield reconstruction

x0

Ay= with

sparsifying transformfor seismic data

restriction operator

A := RSH

[Sacchi et al.‘98][Xu et al.‘05]

[Zwartjes and Sacchi‘07][Herrmann and Hennenfent‘08]

complete wavefield (transform domain)

acquireddata

Interpolated data given by withf = SHx

x = arg minx

||x||1 s.t. y = Ax



Key elements

sparsifying transform– typically localized in the time-space domain to handle the complexity of

seismic data

advantageous coarse sampling– generates incoherent random undersampling “noise” in the sparsifying

domain

– does not create large gaps

• because of the limited spatiotemporal extent of transform elements used for the reconstruction

sparsity-promoting solver– requires few matrix-vector multiplications



Representations for seismic data

curvelet transform– multiscale: tiling of the FK domain into

dyadic coronae

– multidirectional: coronae sub-partitioned into angular wedges, # of angles doubles every other scale

– anisotropic: parabolic scaling principle

– local

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)

k1

k2angular

wedge2j

2j/2



2D discrete curvelets

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4

time

freq

uenc

y

spacewavenumber



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)

Significantcurvelet coefficient

Curveletcoefficient~0



3D discrete curvelets

~2 j

~2 j/2

(1,αl,βl)

ω1 ω2ω3

(a) (b)

Figure 3. 3D frequency tilings. (a) Schematic plot for the frequency tiling of continuous 3D curvelets. (b) Discrete

frequency tiling. ω1, ω2 and ω3 are three axes of the frequency cube. Smooth frequency window eUj,! extracts thefrequency content near the shaded wedge which has center slope (1, α!, β!).

This frame of discrete curvelets has all the required properties of the continuous curvelet transform in Section2. Figure 2(b) shows one typical curvelet in the spatial domain. To summarize, the algorithm of the 2D discretecurvelet transform is as follows:

1. Apply the 2D FFT and obtain Fourier samples f(ω1,ω2), −n/2 ≤ ω1,ω2 < n/2.

2. For each scale j and angle ", form the product Uj,!(ω1,ω2)f(ω1,ω2).

3. Wrap this product around the origin and obtain W(Uj,!f)(ω1,ω2), where the range for ω1 and ω2 is now−L1,j,!/2 ≤ ω1 < L1,j,!/2 and −L2,j,!/2 ≤ ω2 < L2,j,!/2. For j = j0 and je, no wrapping is required.

4. Apply a L1,j,!×L2,j,! inverse 2D FFT to each W(Uj,!f), hence collecting the discrete coefficients cD(j, ", k).

4. 3D DISCRETE CURVELET TRANSFORMThe 3D curvelet transform is expected to preserve the properties of the 2D transform. Most importantly, thefrequency support of a 3D curvelet shall be localized near a wedge which follows the parabolic scaling property.One can prove that this implies that the 3D curvelet frame is a sparse basis for representing functions with surface-like singularities (which is of codimension one in 3D) but otherwise smooth. For the continuous transform, wewindow the frequency content as follows. The radial window smoothly extracts the frequency near the dyadiccorona {2j−1 ≤ r ≤ 2j+1}, this is the same as the radial windowing used in 2D. For each scale j, the unit sphereS2 which represents all the directions in R3 is partitioned into O(2j/2 · 2j/2) = O(2j) smooth angular windows,each of which has a disk-like support with radius O(2−j/2), and the squares of which form a partition of unityon S2 (see Figure 3(a)).

Like the 2D discrete transform, the 3D discrete curvelet transform takes as input a 3D Cartesian grid of theform f(n1, n2, n3), 0 ≤ n1, n2, n3 < n, and outputs a collection of coefficients cD(j, l, k) defined by

cD(j, ", k) :=∑

n1,n2,n3

f(n1, n2, n3) ϕDj,!,k(n1, n2, n3)

where j, " ∈ Z and k = (k1, k2, k3).



2D nonequispaced fast discrete curvelets


time

space

data with acquisition irregularities

5

NFDCT on 2-D regular grid

256 by 256FDCT 0.85 s NFDCT 0.85 sIFDCT 0.82 s ANFDCT 0.85 sAccuracy 4.10−16 Accuracy 7.10−2



NFDCT on 2-D unstructured grid (1-D regular & 1-D un-structured)

ϕ grid Fourier grid TNFDCT TANFDCT

256 by 100 256 by 256 0.87 s 0.82 s256 by 200 256 by 256 0.88 s 0.82 s512 by 200 512 by 512 2.39 s 2.20 s512 by 300 512 by 512 2.25 s 2.30 s

!0.2 !0.15 !0.1 !0.05 0 0.05 0.1 0.15

160

180

200

220

240

260

280

300

320

340

360

Figure 1: Sample of an unequispaced curvelet

CONCLUSION

In this report, I presented the first 2-D fast discrete curvelet transform at non equi-spaced knots (NFDCT) along one axis and regular points along the other axis. Thetransform is an extension to the 2-D fast discrete curvelet transform (Candes et al.,2005a). I have presented an very efficient implementation based on the fast Fouriertransform at non equispaced knots (Kunis and Potts, 2002). The generalization of

space

time

“seismic” curvelet

time

space

processed

using

processed

using

fast discretecurvelet transform



Key elements


seismic data


domain




✓



Localized transform elements & gap size

v v

✓ ✗

x = arg minx

||x||1 s.t. y = Ax



Discrete random jittered undersampling

receiverpositions

receiverpositions

PDF

receiverpositions

PDF

receiverpositions

PDF


Typical spatial convolution kernel

Sampling schemeType

po

orl

yjit

tere

do

pti

mal

lyjit

tere

dra

nd

om

reg

ula

r



Key elements


seismic data


domain




✓

✓



quadratic programming [many references!]

basis pursuit denoise [Chen et al.’95]

LASSO [Tibshirani’96]

Approaches

BPσ : minx‖x‖1 s.t. ‖y −Ax‖2 ≤ σ

QPλ : minx

12‖y −Ax‖2

2 + λ‖x‖1

LSτ : minx

12‖y −Ax‖2

2 s.t. ‖x‖1 ≤ τ



One-norm solvers

iterative soft thresholding (IST)

iterative reweighted least-squares (IRLS)

spectral projected gradient for l1 (SPGl1)

iterative soft thresholding with cooling (ISTc)

Pareto curve++++

[Hennenfent et al.‘08]



Key elements


seismic data


domain




✓

✓

✓



Model



Regular 3-fold undersampling



CRSI from regular 3-fold undersampling

SNR = 20 × log10

(

‖model‖2

‖reconstruction error‖2

)

SNR = 6.92 dB



Random 3-fold undersampling



CRSI from random 3-fold undersampling

SNR = 20 × log10

(

‖model‖2

‖reconstruction error‖2

)

SNR = 9.72 dB



Optimally-jittered 3-fold undersampling



CRSI from opt.-jittered 3-fold undersampling

SNR = 10.42 dB



avg. spatial sampling: 10 m


Seismic Laboratory for Imaging and ModelingWednesday, April 23, 2008


SNR = 7.79 dB



(basic FK filtering )



Conclusions

new wavefield reconstruction method that handles both regular and irregular acquisition geometries

– curvelet reconstruction with sparsity-promoting inversion (CRSI) [Herrmann and Hennenfent‘08]

extension of the fast discrete curvelet transform to handle irregular seismic data

– nonequispaced fast discrete curvelet transform (NFDCT) [Hennenfent and Herrmann‘06]

new coarse sampling schemes that maximize performance of CRSI– jittered undersampling schemes [Hennenfent and Herrmann‘08]

new large-scale, one-norm solver– iterative soft thresholding with cooling (ISTc) [Herrmann and Hennenfent’08, Hennenfent et

al.‘08]



Opportunities

paradigm shift– from an assumption of band-limited to sparse representation for seismic data

– from linear to nonlinear wavefield sampling theory

design of advantageous coarse sampling schemes– same image quality at a lower acquisition cost

– better image quality at a given acquisition cost


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

P. Moghaddam, D. Wang, R. Saab, O. Yilmaz


D. J. Verschuur, Delphi

C. C. Stolk, Twente University


Other applications of curvelet-domain processing







Other applications

Curvelet-domain primary-multiple separation– sparsity promotion [Herrmann et. al. ’07, Saab ’07, Wang ‘08]

– primary-multiple matching [Herrmann et. al. ’08]

Curvelet-domain migration amplitude recovery– sparsity-promotion [Herrmann et. al. ’08b]

– image-remigrated-image matching [Herrmann et. al. ’08b]

– migration preconditioning



Primary-multiple separation

Motivation– residual multiple energy and inadvertent removal primaries are problematic

– Achilles’ heel is adaptive separation after prediction

– use curvelet-domain sparsimony and adaptivity

New curvelet-domain technology– uses non-agressive (SRME) prediction as input

– produces improved separation for primaries and multiples

Three stages– Single-term optimized SRME prediction for the multiples

– Curvelet-domain matching of predicted multiples with multiples in data

– Bayesian separation of matched multiples and primaries based on sparsity promotion



Total data Predicted multiplesTotal data Predicted multiples



SRME primaries Predicted multiplesOne-term SRME predicted primaries

Predicted multiples



Not scaled Bayesian Predicted multiplesBayesian estimate without matching

Predicted multiples



Difference between SRME and scaled Bayesian

Scaled Bayesian Difference between SRME and Bayesian with matching

Bayesian estimate with matching



SRME primaries Scaled BayesianOne-term SRME predicted primaries

Bayesian estimate with matching



Conclusions

Nyquist sampling criterion is too pessimistic for seismic data processing

– new acquisition design based on controlled randomness

– leverages recent developments in wireless acquisition systems

Application of curvelet-technology opens a tantalizing perspective of redesigning seismic processing flows via combination of

– sparsity promotion through norm-one optimization

– phase-space adaptation through curvelet matching

By no longer combating sampling irregularity but by embracing it we open the possibility to supersede Nyquist’s criterion and further push the envelope ...



Acknowledgments

SLIM team members– C. Brown, H. Modzelewski, and S. Ross-Ross for SLIMpy (slim.eos.ubc.ca/

SLIMpy)

D. J. Verschuur for the synthetic dataset

Norsk Hydro for the real dataset

E. J. Candès, L. Demanet, D. L. Donoho, and L. Ying for CurveLab (www.curvelet.org)

E. van den Berg and M. P. Friedlander for SPGL1 (www.cs.ubc.ca/labs/scl/spgl1) & Sparco (www.cs.ubc.ca/labs/scl/sparco)

S. Fomel, P. Sava, and the other developers of Madagascar (rsf.sourceforge.net)

This work 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).


http://slim.eos.ubc.ca/SLIMpy




http://www.curvelet.org

http://www.curvelet.org

http://www.cs.ubc.ca/labs/scl/spgl1




http://www.cs.ubc.ca/labs/scl/sparco

http://www.cs.ubc.ca/labs/scl/sparco

http://rsf.sourceforge.net

http://rsf.sourceforge.net


References

T. Lin and F. J. Herrmann. Compressed extrapolation. Geophysics, Volume 72, Issue 5, pp. SM77-SM93, September-October 2007

F.J. Herrmann and U. Boeniger and D.J. Verschuur. Nonlinear primary-multiple separation with directional curvelet frames. , Geophysical Journal International, Vol. 170, 781-799, 2007

Felix J. Herrmann, Deli Wang, Gilles Hennenfent and Peyman Moghaddam. Curvelet-based seismic data processing: a multiscale and nonlinear approach. Geophysics, Vol. 73, No. 1, pp. A1–A5, January-February 2008

F.J. Herrman, P.P. Moghaddam and C. C. Stolk. Sparsity- and continuity-promoting seismic image recovery with curvelet frames. Appl. Comput. Harmon. Anal. Vol 24/2, 150-173, 2008

F. J. Herrmann and G. Hennenfent. Non-parametric seismic data recovery with curvelet frames, Geophysical Journal International, 173, 233–248, 2008

F. J. Herrmann, D. Wang and D. J. Verschuur. Adaptive curvelet-domain primary-multiple separation, Geophysics, 73(3), May-June 2008

G. Hennenfent and F. J. Herrmann. Simply denoise: wavefield reconstruction via jittered undersampling. Geophysics, 73(3), May-June 2008

D. Wang, R. Saab, O. Yilmaz and F. J. Herrmann. Bayesian wavefield separation by transform-domain sparsity promotion. To appear in Geophysics. 2008


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Compressive sampling: a new paradigm for seismic data … · 2019. 11. 14. · Use recent insights...

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