UBC Faculty Research and Publications

Compressed wavefield extrapolation with curvelets Lin, Tim T. Y.; Herrmann, Felix J. 2007

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
lin07SEG.pdf [ 2.56MB ]
Metadata
JSON: 1.0107427.json
JSON-LD: 1.0107427+ld.json
RDF/XML (Pretty): 1.0107427.xml
RDF/JSON: 1.0107427+rdf.json
Turtle: 1.0107427+rdf-turtle.txt
N-Triples: 1.0107427+rdf-ntriples.txt
Original Record: 1.0107427 +original-record.json
Full Text
1.0107427.txt
Citation
1.0107427.ris

Full Text

Compressed Wavefield Extrapolation with CurveletsTim T.Y. Lin and Felix J. HerrmannUniversity of British ColumbiaSEG 2007San Antonio, Sept 25Introduction? Concerned with explicit forms of wavefield propagator       of the linearized forward model? Would like to find explicit       suitable for wave-equation migration:? simultaneously operates on sets of traces ? fully incorporates velocity information of medium? no parabolic approximationss+W- R+ W+P-WWIntroduction? Goal: employ the complete 1-Way Helmholtz operator for? Problem: computation & storage complexity? creating and storing      is trivial? however        is not trivial to compute and store WIntroduction? In this case        is computed by eigenvalue decomposition? requires, per frequency:? 1 eigenvalue problem (O(n4))? 2 full matrix-vector for eigenspace transform (O(n2))WIntroduction? Band-diagonalization techniques like parabolic approximation trades for speed with approximations? Is there another way?Our approach? Consider a related, but simpler problem: shifting (or translating) signal? operator is ?     is differential operatorOur approach? Computation requires similar approach to? However, for    ,               , so computation trivial with FFTOur approach? Suppose FFT does not exist yet......Our approach? suppose some nodes didn?t finish their jobs......Our approach? mathematically, the system is incomplete? evidently some information of original         is invariably lost. Or is it? =Compressed Sensing? states that given system of the formmeasuredsignal =sparse representation of original datalinear model of restricted measurement process(measurement basis)Compressed Sensing? states that given system of the form? can exactly ?recover? x from y by solving L1 problemmeasuredsignal =sparse representation of original datalinear model of restricted measurement process(measurement basis)restrictedsamplingsignal in time domainrestrictedsamplingsignal in time domain signal in Fourier domainFrestrictedsamplingsignal in time domain signal in Fourier domainrestricted signal in Fourier domain(real)Frestrictedsamplingsignal in time domain signal in Fourier domainrestricted signal in Fourier domain(real)recovered signal in time domainFL1Compressed Sensing? x has to be sparse? A has to be Fourier transform? Compressed sensing theory gives us strict bounds on regions of recoverability? Enables deliberate incomplete computationsCompressed Sensing ?Computation?? if we ?shift? s(k) with             , what happens when we recover s(x) using s?(k)?= =Compressed Sensing ?Computation?? if we ?shift? s(k) with             , what happens when we recover s(x) using s?(k)?? Answer: we recover a shifted s(x)!= =signal in space domainsignal in space domainF L1L1incomplete signal in Fourier domainincomplete and shifted signal in Fourier domain shifted signal in space domainsignal in space domainCompressed SensingCompressed ProcessingFsignal in space domainsignal in space domainFL1shifted signal in Fourier domainincomplete and shifted signal in Fourier domain shifted signal in space domainStraightforward ComputationCompressed ProcessingFshifted signal in space domainFCompressed Sensing ?Computation?? In a nutshell:? Trades the cost of L1 solvers for a compressed operator that is cheaper to compute, store, and synthesize? L1 solver research is currently a hot topic in applied mathematicsCompressed Wavefield Extrapolation? Recall the similarity between        andCompressed Wavefield Extrapolation? Structure of ? analytically? discretelyCompressed Wavefield extrapolationAsymptotically identical to the Cosine transformeigenfunctions of        at 30 Hz for constant velocity mediumCompressed Wavefield extrapolationeigenfunctions of        at 30 Hz for Marmousi velocity mediumCompressed Wavefield extrapolationfairly close to the Cosine transformeigenfunctions of        at 30 Hz for Marmousi velocity mediumwavefield in space-time domainL1back-extrapolated wavefield in H2 domainStraightforward 1-Way inverse Wavefield ExtrapolationCompressed 1-Way Wavefield ExtrapolationLback-extrapolated to impulse source in space-time domainwavefield in space-time domainback-extrapolated to impulse source in space-time domainincomplete back-extrapolated wavefield in H2 domainLTLTCompressed wavefield extrapolationsimple 1-D space/time propagation example with point scatterspropagated 1.5km downCompressed wavefield extrapolationrecovered though L1 inversonsimple 1-D space/time propagation example with point scattersRestricted L transform to ~0.01 of original coefficientsSparsity through curvelets? for extrapolation to reflectivity, we first transform signal into a sparsifies reflectivity? we know reflectivity are sparse in curveletsExample (Canadian overthrust)original reflectivity downward extrapolated 50minverse extrapolated explicitlyExample (Canadian overthrust)inverse extrapolated explicitly inverse extrapolated with compressed computation~15% coefficients usedDiscussions? Bottom line: synthesis, operation, and storage cost savings versus L1-solver cost? require good sparsity-promoting basis (ie Curvelets)? potential to apply same technique to a variety of different operatorsConclusions? 1) Take linear operator with suitable structure for compressed sensing, having a diagonalizing basis which is incoherent with the signal basis? 2) Compressed sensing theory tells us how much computation we can throw away while still recovering full signal with L1 solver? 3) Then we can take advantage of results in compressed sampling for compressed computation? Take home point:? Exploit compressed sensing theory for gains in scientific computationStill awake?? Check-out the full paper at:Lin, T.T.Y. and F. Herrmann, 2007, Compressed wavefield extrapolation: Geophysics, 72, SM77-SM93Compressed wavefield extrapolation? Randomly subsample in the Modal domain? Recover by norm-one minimization? Capitalize on ? the incoherence between modal functions and impulse sources? reduced explicit matrix sizeCompressed wavefield extrapolation with curvelets? Original and reconstructed signals remain in the curvelet domain? Original curvelet transform must be done outside of the algorithm

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
Germany 4 0
United States 4 0
Japan 4 0
Canada 1 0
China 1 0
City Views Downloads
Unknown 4 0
Ashburn 4 0
Tokyo 4 0
Ottawa 1 0
Shenzhen 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

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>
                        
                    
IIIF logo 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.340.1-0107427/manifest

Comment

Related Items

Admin Tools

To re-ingest this item use button below, on average re-ingesting will take 5 minutes per item.

Reingest

To clear this item from the cache, please use the button below;

Clear Item cache