Seismic imaging and processing with curvelets Felix J. Herrmann joint work with Deli Wang Combinations of parsimonious signal representations with nonlinear sparsity promoting programs hold the key to the next-generation of seismic data processing algorithms ... Since they allow for formulations that are stable w.r.t. noise incomplete data moderate phase rotations and amplitude errors Finding a sparse representation for seismic data & images is complicated because of wavefronts & reflectors are multiscale & multidirectional the presence of caustics, faults and pinchouts The curvelet transform 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) Properties curvelet transform: multiscale: tiling of the FK domain into dyadic coronae multi-directional: coronae subpartitioned into angular wedges, # of angle doubles every other scale anisotropic: parabolic scaling principle Rapid decay space Strictly localized in Fourier Frame with moderate redundancy (8 X in 2-D and 24 X in 3-D) k2 2j/2 angular wedge 2j fine scale data k1 coarse scale data 2-D curvelets curvelets are of rapid decay in space x-t curvelets are strictly localized in frequency f-k Oscillatory in one direction and smooth in the others! Obey parabolic scaling relation length ≈ width2 Curvelet tiling & seismic data Angular wedge Curvelet tiling # of angles doubles every other scale doubling! Real data frequency bands example Data is multiscale! 1 2 3 4 5 6 7 8 Single frequency band angular wedges 6 6th scale image Data is multidirectional! Seismic Laboratory for Imaging and Modeling Decomposition in angular wedges Wavefront detection 0 -2000 Offset (m) 0 2000 Time (s) 0.5 1.0 1.5 2.0 Significant curvelet coefficient Curvelet coefficient~0 Extenstion to 3-D Cartesian Fourier space [courtesy Demanet ‘05, Ying ‘05] Curvelets live in a wedge in the 3 D Fourier plane... 3-D curvelets Curvelets are oscillatory in one direction and smooth in the others. Coefficients Amplitude Decay In Transform Domains Fourier Wavelets Curvelets Partial Reconstruction Curvelets (1% largest coefficients) SNR = 6.0 dB Curvelet sparsity promotion Forward model Linear model for the measurements of a function m0: y = Km0 + n with y = data K = the modeling matrix m0 = the model vector n = noise inversion of K either ill-posed or underdetermined. seek a prior on m. Key idea ˜ = arg min x x x 1 sparsity enhancement s.t. Ax − y 2 ≤ data misfit When a traveler reaches a fork in the road, the l1 -norm tells him to take either one way or the other, but the l2 -norm instructs him to head off into the bushes. John F. Claerbout and Francis Muir, 1973 New field “compressive sampling”: D. Donoho, E. Candes et. al., M. Elad etc. Preceded by others in geophysics: M. Sacchi & T. Ulrych and co-workers etc. Inline 0 Linear quadratic (lsqr): • 2 s.t. Ax − y 2 ≤ model Gaussian Non-linear s.t. Ax − y 2 ≤ 0.5 model Cauchy (sparse) Problem: • 500 300 400 500 Inline Time (s) • 200 400 1.5 : 1 100 300 1.0 2.0 0 ˜ = arg min x x x 200 0.5 Time (s) ˜ = arg min x x x 100 1.0 data does not contain point scatterers 1.5 not sparse 2.0 Our contribution Inline 0 Model as superposition of little plane waves. Time (s) Compound modeling operator with curvelet synthesis: 0.5 1.0 1.5 K m0 ˜ m T → KC = ˜ C x → x0 T Exploit parsimoniousness of curvelets on seismic data & images ... 2.0 100 200 300 400 500 Sparsity-promoting program Problems boils down to solving for x0 signal y = A + n noise x0 curvelet representation of ideal data with P : ˜ = arg minx x x ˜ = CT x ˜ m 1 s.t. Ax − y 2 ≤ exploit sparsity in the curvelet domain as a prior find the sparsest set of curvelet coefficients that T ˜ match the data, i.e., y ≈ KC x invert an underdetermined system Solver Initialize: i = 0; x0 = 0; Choose: L, AT y while y − Axi 2 ∞ > > λ1 > λ2 > · · · do for l = 1 to L do xi+1 = Tλsi xi + AT y − Axi end for i = i + 1; end while f = CT xi . Applications Problems in seismic processing can be cast in to P stable under noise stable under missing data Obtain a formulation that explicitly exploits compression by curvelets is stable w.r.t. noise exploits the “invariance” of curvelets under imaging Applications include seismic data regularization primary-multiple separation seismic amplitude recovery Seismic data regularization joint work with Gilles Hennenfent Motivation Irregular sub-sampling incoherent noise Noisy because of irregular sampling ... Sparsity-promoting inversion* Reformulation of the problem signal y = H RC + n noise x0 curvelet representation of ideal data Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI) P : look for the sparsest/most compressible, physical solution KEY POINT OF THE RECOVERY data misfit sparsity constraint sparsity constraint = arg ˜ H H − PC minmin x 0x 0 s.t. s.t. y −yPC x 2x≤2!≤ ! x =x˜arg ˜0 )= arg minxx Wx x s.t. Ax − y 2 ≤ (P0 )(Px 1 T ˜ ˜H f = C x H ˜f = ˜fC= xC ˜ x˜ * inspired by Stable Signal Recovery (SSR) theory by E. Candès, J. Romberg, T. Tao, Compressed sensing by D. Donoho & Fourier Reconstruction with Sparse Inversion (FRSI) by P. Zwartjes Original data 80 % missing CRSI recovery with 3-D curvelets Primary multiple separation Joint work with Eric Verschuur, Deli Wang, Rayan Saab and Ozgur Yilmaz Motivation Primary-multiple separation step is crucial moderate prediction errors 3-D complexity & noise Inadequate separation leads to remnant multiple energy deterioration primary energy Introduce a transform-based technique stable insensitive to moderate shifts & phase rotations Exploit sparsity and parameterization transformed domain Move-out error Move-out error now two sparsityThe matrices y =problem Ax0 one + n,for each signal co Sparse signal model: y = Ax0 + n, with A = [A1 A2 ] and x0 = [x01 augmented synthesis and sparsity vectors index11 <->2primary 0 01 index 2 <-> multiple T x02 ] T A = [A A and ] and xvector, = [x respectivel x02 ] ynthesis matrix sparsity erved for primaries and multiples. The above s se weights drive the two signal components apart during the optimization ‡ accuracy . The solution The w-weighted optimization problem becomes to reasonable The weighted norm-one minx x w,1 Pw : sˆ1 = A1 xˆ 1 given: s˘2 [w1 , optimization problem: subject to y − Ax 2 ≤ε and sˆ2 = A2 xˆ 2 and w(y, s˘2 ) with T w2 ] the weighting vectors withT strictly positive weights defined in w := w1 , w2 T primaries T multiples. TheA estimates for the and multiples are computed from := C , C s.2During := thepredicted multiples minimizes Pw˘ optimization, the sparsity vector is recovered b ˘s1 ˘2 := S − S Solution cont’d The weights with ˘ 1 ≈ C˘s1 u ˘ 2 ≈ C˘s2 u √ w1 := max σ · 2 log N , C1 |˘ u1 | √ w2 := max σ · 2 log N , C2 |˘ u2 | during minimization signal components are driven apart curvelet compression helps separates on the basis of position, scale and direction Synthetic example total data SRME predicted multiples Synthetic example SRME predicted primaries curvelet-thresholded Synthetic example SRME predicted primaries estimated Real example total data SRME predicted multiples Real example curvelet thresholded curvelet estimated SRME predicted primaries curvelet estimated primaries Seismic amplitude recovery Joint work with Chris Stolk and Peyman Moghaddam Motivation Migration generally does not correctly recover the amplitudes. Least-squares migration is computationally unfeasible. Amplitude recovery (e.g. AGC) lacks robustness w.r.t. noise. Existing diagonal amplitude-recovery methods do not always correct for the order (1 - 2D) of the Hessian [see Symes ‘07] do not invert the scaling robustly Moreover, these (scaling) methods assume that there are no conflicting dips (conormal) in the model is infinite aperture are infinitely-high frequencies etc. Existing scaling methods Methods are based on a diagonal approximation of Ψ. Illumination-based normalization (Rickett ‘02) Amplitude preserved migration (Plessix & Mulder ‘04) Amplitude corrections (Guitton ‘04) Amplitude scaling (Symes ‘07) We are interested in an ‘Operator and image adaptive’ scaling method which estimates the action of Ψ from a reference vector close to the actual image assumes a smooth symbol of Ψ in space and angle does not require the reflectors to be conormal <=> allows for conflicting dips stably inverts the diagonal Our approach “Forward” model: y ≈ Ax0 + ε with y = A := C Γ T AA r K = K Km + ε T migrated data T T ≈ K Kr = the demigration operator = migrated noise. diagonal approximation of the demigration-migration operator costs one demigration-migration to estimate the diagonal weighting Solution Solve P: with minx J(x) subject to y − Ax 2 ˜ = (AH )† x ˜ m sparsity J(x) = α x 1 +β Λ 1/2 A H † continuity x p . ≤ Example SEGAA’ data: “broad-band” half-integrated wavelet [5-60 Hz] 324 shots, 176 receivers, shot at 48 m 5 s of data Modeling operator Reverse-time migration with optimal check pointing (Symes ‘07) 8000 time steps modeling 64, and migration 294 minutes on 68 CPU’s Scaling requires 1 extra migration-demigration Migrated data Amplitude-corrected & denoised migrated data Noise-free data Noisy data (3 dB) Data from migrated image Data from amplitude-corrected & denoised migrated image Nonlinear data Conclusions The combination of the parsimonious curvelet transform with nonlinear sparsity & continuity promoting program allowed us to... recover seismic data from large percentages missing traces separate primaries & multiples recover migration amplitudes This success is due to the curvelet’s ability to detect wavefronts <=> multi-D geometry differentiate w.r.t. positions, angle(s) and scale diagonalize the demigration-migration operator Because of their parsimoniousness on seismic data and images, curvelets open new perspectives on seismic processing ... Acknowledgments The authors of CurveLab (Demanet,Ying, Candes, Donoho) William Symes for the reverse-time migration code. These results were created with Madagascar developed by Sergey Fomel. 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 (334810-05) of F.J.H. This research was carried out as part of the SINBAD project with support, secured through ITF (the Industry Technology Facilitator), from the following organizations: BG Group, BP, Chevron,ExxonMobil and Shell.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Faculty Research and Publications /
- Seismic imaging and processing with curvelets
Open Collections
UBC Faculty Research and Publications
Seismic imaging and processing with curvelets Herrmann, Felix J. 2007
pdf
Page Metadata
Item Metadata
Title | Seismic imaging and processing with curvelets |
Creator |
Herrmann, Felix J. |
Contributor | University of British Columbia. Seismic Laboratory for Imaging and Modeling |
Publisher | European Association of Geoscientists & Engineers |
Date Issued | 2007 |
Extent | 11320469 bytes |
Subject |
curvelet transform nonlinear sparsity data recovery seismic demigration migration wavefront detection continuity scaling |
Genre |
Presentation |
Type |
Text Still Image |
File Format | application/pdf |
Language | eng |
Date Available | 2008-03-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | All rights reserved |
DOI | 10.14288/1.0102591 |
URI | http://hdl.handle.net/2429/582 |
Affiliation |
Science, Faculty of Earth and Ocean Sciences, Department of |
Citation | Herrmann, Felix J. 2007, Seismic imaging and processing with curvelets. 69th EAGE Conference & Exhibition. |
Peer Review Status | Unreviewed |
Scholarly Level | Faculty |
Copyright Holder | Herrmann, Felix J. |
Aggregated Source Repository | DSpace |
Download
- Media
- 52383-herrmann07EAGEIMPROC.pdf [ 10.8MB ]
- Metadata
- JSON: 52383-1.0102591.json
- JSON-LD: 52383-1.0102591-ld.json
- RDF/XML (Pretty): 52383-1.0102591-rdf.xml
- RDF/JSON: 52383-1.0102591-rdf.json
- Turtle: 52383-1.0102591-turtle.txt
- N-Triples: 52383-1.0102591-rdf-ntriples.txt
- Original Record: 52383-1.0102591-source.json
- Full Text
- 52383-1.0102591-fulltext.txt
- Citation
- 52383-1.0102591.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.52383.1-0102591/manifest