@article{Tarantola1982Bayesian, author = {Tarantola, Albert and Valette, Bernard}, title = {Inverse Problems = Quest for Information}, journal = {Journal of Geophysics}, volume = {50}, pages = {159-170}, year = {1982}, } @article{VirieuxOverview2009, author = {Virieux, J. and Operto, S.}, title = {An overview of full-waveform inversion in exploration geophysics}, journal = {GEOPHYSICS}, volume = {74}, number = {6}, pages = {WCC1-WCC26}, year = {2009}, doi = {10.1190/1.3238367}, URL = { http://dx.doi.org/10.1190/1.3238367 }, eprint = { http://dx.doi.org/10.1190/1.3238367} } @article {Tarantola1982FWI, author = {Tarantola, Albert and Valette, Bernard}, title = {Generalized nonlinear inverse problems solved using the least squares criterion}, journal = {Reviews of Geophysics}, volume = {20}, number = {2}, issn = {1944-9208}, url = {http://dx.doi.org/10.1029/RG020i002p00219}, doi = {10.1029/RG020i002p00219}, pages = {219--232}, year = {1982}, } @conference {aravkin2012EAGErobust, title = {Source estimation for frequency-domain {FWI} with robust penalties}, booktitle = {EAGE}, year = {2012}, month = {06}, abstract = {Source estimation is an essential component of full waveform inversion. In the standard frequency domain formulation, there is closed form solution for the the optimal source weights, which can thus be cheaply estimated on the fly. A growing body of work underscores the importance of robust modeling for data with large outliers or artifacts that are not captured by the forward model. Effectively, the least-squares penalty on the residual is replaced by a robust penalty, such as Huber, Hybrid {\textquoteleft}1-{\textquoteleft}2 or Student{\textquoteright}s t. As we will demonstrate, it is essential to use the same robust penalty for source estimation. In this abstract, we present a general approach to robust waveform inversion with robust source estimation. In this general formulation, there is no closed form solution for the optimal source weights so we need to solve a scalar optimization problem to obtain these weights. We can efficiently solve this optimization problem with a Newton-like method in a few iterations. The computational cost involved is of the same order as the usual least-squares source estimation procedure. We show numerical examples illustrating robust source estimation and robust waveform inversion on synthetic data with outliers.}, keywords = {EAGE}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/aravkin2012EAGErobust/aravkin2012EAGErobust.pdf}, presentation = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/aravkin2012EAGErobust/aravkin2012EAGErobust_pres.pdf}, url2 = {http://earthdoc.eage.org/publication/publicationdetails/?publication=59196}, author = {Aleksandr Y. Aravkin and Tristan van Leeuwen and Henri Calandra and Felix J. Herrmann} } @article {vanLeeuwen2014GEOPcav, title = {Comment on: {\textquotedblleft}Application of the variable projection scheme for frequency-domain full-waveform inversion{\textquotedblright} (M. Li, J. Rickett, and A. Abubakar, Geophysics, 78, no. 6, R249{\textendash}R257)}, journal = {Geophysics}, volume = {79}, number = {3}, year = {2014}, note = {(discussion by Tristan van Leeuwen, Aleksandr Y. Aravkin, and Felix J. Herrmann)}, month = {05}, pages = {X11-X17}, keywords = {variable projection, waveform inversion}, doi = {10.1190/geo2013-0466.1}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2014/vanLeeuwen2014GEOPcav/vanLeeuwen2014GEOPcav.pdf}, author = {Tristan van Leeuwen and Aleksandr Y. Aravkin and Felix J. Herrmann} } @article{MaokunLi2013srcest, author = {Li, M. and Rickett, J. and Abubakar, A.}, title = {Application of the variable projection scheme for frequency-domain full-waveform inversion}, journal = {GEOPHYSICS}, volume = {78}, number = {6}, pages = {R249-R257}, year = {2013}, doi = {10.1190/geo2012-0351.1}, URL = { http://dx.doi.org/10.1190/geo2012-0351.1 }, eprint = { http://dx.doi.org/10.1190/geo2012-0351.1 } } @ARTICLE{vanLeeuwen2013GJImlm, author = {Tristan van Leeuwen and Felix J. Herrmann}, title = {Mitigating local minima in full-waveform inversion by expanding the search space}, year = {2013}, month = {10}, volume = {195}, pages = {661-667}, journal = {Geophysical Journal International}, abstract = {Wave equation based inversions, such as full-waveform inversion and reverse-time migration, are challenging because of their computational costs, memory requirements and reliance on accurate initial models. To confront these issues, we propose a novel formulation of wave equation based inversion based on a penalty method. In this formulation, the objective function consists of a data-misfit term and a penalty term, which measures how accurately the wavefields satisfy the wave equation. This new approach is a major departure from current formulations where forward and adjoint wavefields, which both satisfy the wave equation, are correlated to compute updates for the unknown model parameters. Instead, we carry out the inversions over two alternating steps during which we first estimate the wavefield everywhere, given the current model parameters, source and observed data, followed by a second step during which we update the model parameters, given the estimate for the wavefield everywhere and the source. Because the inversion involves both the synthetic wavefields and the medium parameters, its search space is enlarged so that it suffers less from local minima. Compared to other formulations that extend the search space of wave equation based inversion, our method differs in several aspects, namely (i) it avoids storage and updates of the synthetic wavefields because we calculate these explicitly by finding solutions that obey the wave equation and fit the observed data and (ii) no adjoint wavefields are required to update the model, instead our updates are calculated from these solutions directly, which leads to significant computational savings. We demonstrate the validity of our approach by carefully selected examples and discuss possible extensions and future research.}, url1 = {http://gji.oxfordjournals.org/content/195/1/661}, url2 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/GeophysicalJournalInternational/2013/vanLeeuwen2013GJImlm/vanLeeuwen2013mlm.pdf}, doi = {10.1093/gji/ggt258}, eprint = {http://gji.oxfordjournals.org/content/early/2013/07/30/gji.ggt258.full.pdf+html} } @UNPUBLISHED{vanLeeuwen2013Penalty2, author = {Tristan van Leeuwen and Felix J. Herrmann}, title = {A penalty method for PDE-constrained optimization}, year = {2013}, institution = {UBC}, abstract = {We present a method for solving PDE constrained optimization problems based on a penalty formulation. This method aims to combine advantages of both full-space and reduced methods by exploiting a large search-space (consisting of both control and state variables) while allowing for an efficient implementation that avoids storing and updating the state-variables. This leads to a method that has roughly the same per-iteration complexity as conventional reduced approaches while dening an objective that is less non-linear in the control variable by implicitly relaxing the constraint. We apply the method to a seismic inverse problem where it leads to a particularly ecient implementation when compared to a conventional reduced approach as it avoids the use of adjoint state-variables. Numerical examples illustrate the approach and suggest that the proposed formulation can indeed mitigate some of the well-known problems with local minima in the seismic inverse problem.}, keywords = {waveform inversion, optimization, private}, month = {Apr}, url = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2013/vanLeeuwen2013Penalty2/vanLeeuwen2013Penalty2.pdf} } @conference {leeuwen2014EAGEntf, title = {A new take on FWI: Wavefield Reconstruction Inversion}, booktitle = {EAGE}, year = {2014}, month = {01}, publisher = {UBC}, organization = {UBC}, abstract = {We discuss a recently proposed novel method for waveform inversion: Wavefield Reconstruction Inversion (WRI). As opposed to conventional FWI {\textendash} which attempts to minimize the error between observed and predicted data obtained by solving a wave equation {\textendash} WRI reconstructs a wave-field from the data and extracts a model-update from this wavefield by minimizing the wave-equation residual. The method does not require explicit computation of an adjoint wavefield as all the necessary information is contained in the reconstructed wavefield. We show how the corresponding model updates can be interpreted physically analogously to the conventional imaging-condition-based approach.}, keywords = {EAGE, Full-waveform inversion, Optimization, penalty method, Wavefield Reconstruction Inversion}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/leeuwen2014EAGEntf.pdf}, author = {Tristan van Leeuwen and Felix J. Herrmann and Bas Peters} } @conference {peters2014EAGEweb, title = {Wave-equation based inversion with the penalty method: adjoint-state versus wavefield-reconstruction inversion}, booktitle = {EAGE}, year = {2014}, month = {06}, abstract = {In this paper we make a comparison between wave-equation based inversions based on the adjoint-state and penalty methods. While the adjoint-state method involves the minimization of a data-misfit and exact solutions of the wave-equation for the current velocity model, the penalty-method aims to first find a wavefield that jointly fits the data and honours the physics, in a least-squares sense. Given this reconstructed wavefield, which is a proxy for the true wavefield in the true model, we calculate updates for the velocity model. Aside from being less nonlinear{\textendash}the acoustic wave equation is linear in the wavefield and model parameters but not in both{\textendash}the inversion is carried out over a solution space that includes both the model and the wavefield. This larger search space allows the algortihm to circumnavigate local minima, very much in the same way as recently proposed model extentions try to acomplish. We include examples for low frequencies, where we compare full-waveform inversion results for both methods, for good and bad starting models, and for high frequencies where we compare reverse-time migration with linearized imaging based on wavefield-reconstruction inversion. The examples confirm the expected benefits of the proposed method.}, keywords = {EAGE, Full-waveform inversion, Imaging, Optimization, penalty method}, doi = {10.3997/2214-4609.20140704}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/peters2014EAGEweb/peters2014EAGEweb.pdf}, presentation = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/peters2014EAGEweb/peters2014EAGEweb_pres.pdf}, author = {Bas Peters and Felix J. Herrmann and Tristan van Leeuwen} } @article{TristanFWI2013, author = {van Leeuwen, Tristan and Herrmann, Felix J.}, title = {Fast {W}aveform {I}nversion without {S}ource-{E}ncoding}, journal = {Geophysical Prospecting}, volume = {61}, publisher = {Blackwell Publishing Ltd}, issn = {1365-2478}, url = {http://dx.doi.org/10.1111/j.1365-2478.2012.01096.x}, pages = {10--19}, keywords = {Full-waveform inversion, Source-encoding, Stochastic optimization}, year = {2013}, } @article{MartinMcMC2012, author = {Martin, J. and Wilcox, L. and Burstedde, C. and Ghattas, O.}, title = {A {S}tochastic {N}ewton {MCMC} {M}ethod for {L}arge-Scale {S}tatistical {I}nverse {P}roblems with {A}pplication to {S}eismic {I}nversion}, journal = {SIAM Journal on Scientific Computing}, volume = {34}, number = {3}, pages = {A1460-A1487}, year = {2012}, URL = {http://epubs.siam.org/doi/abs/10.1137/110845598}, eprint = {http://epubs.siam.org/doi/pdf/10.1137/110845598} } @article{Overview2009, author = {Virieux, J. and Operto, S.}, title = {{An} {O}verview of {F}ull-waveform {I}nversion in {E}xploration {G}eophysics}, journal = {GEOPHYSICS}, volume = {74}, number = {6}, pages = {WCC1-WCC26}, year = {2009}, URL = {http://library.seg.org/doi/abs/10.1190/1.3238367}, eprint = {http://library.seg.org/doi/pdf/10.1190/1.3238367} } @article{AVO2003, author = {Buland, A. and Omre, H.}, title = {Bayesian linearized AVO inversion}, journal = {GEOPHYSICS}, volume = {68}, number = {1}, pages = {185-198}, year = {2003}, URL = {http://geophysics.geoscienceworld.org/content/68/1/185.short#cited-by}, } @article{GhattasMcMC2013, author = {Bui-Thanh, T. and Ghattas, O. and Martin, J. and Stadler, G.}, title = {A {C}omputational {F}ramework for {I}nfinite-{D}imensional {B}ayesian {I}nverse {P}roblems {P}art {I}: {T}he {L}inearized {C}ase, with {A}pplication to {G}lobal {S}eismic {I}nversion}, journal = {SIAM Journal on Scientific Computing}, volume = {35}, number = {6}, pages = {A2494-A2523}, year = {2013}, URL = {http://epubs.siam.org/doi/abs/10.1137/12089586X}, eprint = {http://epubs.siam.org/doi/pdf/10.1137/12089586X} } @article{MichaelSO2012, author = {Friedlander, M. and Schmidt, M.}, title = {{H}ybrid {D}eterministic-{S}tochastic {M}ethods for {D}ata {F}itting}, journal = {SIAM Journal on Scientific Computing}, volume = {34}, number = {3}, pages = {A1380-A1405}, year = {2012}, URL = {http://epubs.siam.org/doi/abs/10.1137/110830629}, eprint = {http://epubs.siam.org/doi/pdf/10.1137/110830629} } @article{Jo1996FD, author = {Jo, C., Shin, C. and Suh, J.}, title = {{A}n {O}ptimal 9-{P}oint, {F}inite-{D}ifference, {F}requency-{S}pace, 2-D {S}calar {W}ave {E}xtrapolator}, journal = {GEOPHYSICS}, volume = {61}, number = {2}, pages = {529-537}, year = {1996}, doi = {10.1190/1.1443979}, URL = {http://library.seg.org/doi/abs/10.1190/1.1443979}, eprint = {http://library.seg.org/doi/pdf/10.1190/1.1443979} } @article{Choi01102008, author = {Choi, Yunseok and Min, Dong-Joo and Shin, Changsoo}, title = {{F}requency-Domain {E}lastic {F}ull {W}aveform {I}nversion {U}sing the {N}ew {P}seudo-{H}essian {M}atrix: {E}xperience of {E}lastic {M}armousi-2 {S}ynthetic {D}ata}, volume = {98}, number = {5}, pages = {2402-2415}, year = {2008}, doi = {10.1785/0120070179}, abstract ={A proper scaling method allows us to find better solutions in waveform inversion, and it can also provide better images in true-amplitude migration methods based on a least-squares method. For scaling the gradient of a misfit function, we define a new pseudo-Hessian matrix by combining the conventional pseudo-Hessian matrix with amplitude fields. Because the conventional pseudo-Hessian matrix is assumed to neglect the zero-lag autocorrelation terms of impulse responses in the approximate Hessian matrix of the Gauss–Newton method, it has certain limitations in scaling the gradient of a misfit function relative to the approximate Hessian matrix. To overcome these limitations, we introduce amplitude fields to the conventional pseudo-Hessian matrix, and the new pseudo-Hessian matrix is applied to the frequency-domain elastic full waveform inversion. This waveform inversion algorithm follows the conventional procedures of waveform inversion using the backpropagation algorithm. A conjugate-gradient method is employed to derive an optimized search direction, and a backpropagation algorithm is used to calculate the gradient of the misfit function. The source wavelet is also estimated simultaneously with elastic parameters. The new pseudo-Hessian matrix can be calculated without the extra computational costs required by the conventional pseudo-Hessian matrix, because the amplitude fields can be readily extracted from forward modeling. Synthetic experiments show that the new pseudo-Hessian matrix provides better results than the conventional pseudo-Hessian matrix, and thus, we believe that the new pseudo-Hessian matrix is an alternative to the approximate Hessian matrix of the Gauss–Newton method in waveform inversion.}, URL = {http://www.bssaonline.org/content/98/5/2402.abstract}, eprint = {http://www.bssaonline.org/content/98/5/2402.full.pdf+html}, journal = {Bulletin of the Seismological Society of America} } @book{kaipio:2004, author = {Kaipio, Jari and Somersalo, Erkki}, day = {01}, edition = {1}, howpublished = {Hardcover}, isbn = {0387220739}, keywords = {bayes, inverse\_problems}, month = dec, posted-at = {2009-04-21 12:23:18}, priority = {2}, publisher = {Springer}, title = {{{S}tatistical and {C}omputational {I}nverse {P}roblems (Applied Mathematical Sciences) (v. 160)}}, url = {http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20\&path=ASIN/0387220739}, year = {2004} } @conference {fang2014EAGEfuq, title = {{F}ast {U}ncertainty {Q}uantification for 2D {F}ull-waveform {I}nversion with {R}andomized {S}ource {S}ubsampling}, booktitle = {76th EAGE}, year = {2014}, month = {01}, publisher = {}, organization = {}, keywords = {EAGE, FWI, Markov chain Monte Carlo, randomized sources subsampling, Uncertainty quantification}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2014/fang2014EAGEfuq.pdf}, author = {Zhilong Fang and Curt Da Silva and Felix J. Herrmann} } @techreport{tristan20133DFWI, author = {Tristan van Leeuwen and Felix J. Herrmann}, title = {3{D} {F}REQUENCY-{D}OMAIN {S}EISMIC {I}NVERSION WITH {C}ONTROLLED {S}LOPPINESS.}, year = {2013}, month = {04}, publisher = {UBC}, organization = {UBC}, keywords = {Optimization, private, waveform inversion}, url = {https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnx0cmlzdGFudmFubGVldXdlbnxneDoyYTEzM2YwZjBjMGFkNzg2} } @article {Li11TRfrfwi, title = {{F}ast {R}andomized {F}ull-waveform {I}nversion with {C}ompressive {S}ensing}, journal = {Geophysics}, volume = {77}, number = {3}, year = {2012}, month = {05}, pages = {A13-A17}, address = {University of British Columbia, Vancouver}, keywords = {Compressive Sensing, FWI, Optimization, SLIM}, doi = {10.1190/geo2011-0410.1}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2012/Li11TRfrfwi/Li11TRfrfwi.pdf}, author = {Xiang Li and Aleksandr Y. Aravkin and Tristan van Leeuwen and Felix J. Herrmann} } @article {Aravkin11TRridr, title = {{R}obust {I}nversion, {D}imensionality {R}eduction, and {R}andomized {S}ampling}, journal = {Mathematical Programming}, volume = {134}, number = {1}, year = {2012}, month = {8}, pages = {101-125}, address = {University of British Columbia, Vancouver}, keywords = {FWI, Inverse problems, Optimization, Robust estimation, Seismic inversion, Stochastic optimization}, doi = {10.1007/s10107-012-0571-6}, url = {http://www.springerlink.com/content/35rwr101h5736340/}, url2 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/MathematicalProgramming/aravkin2012MPrid/aravkin2012MPrid.pdf}, author = {Aleksandr Y. Aravkin and Michael P. Friedlander and Felix J. Herrmann and Tristan van Leeuwen} } @article {aravkin2012IPNuisance, title = {{E}stimating {N}uisance {P}arameters in {I}nverse {P}roblems}, journal = {Inverse Problems}, volume = {28}, number = {11}, year = {2012}, month = {10}, keywords = {full waveform inversion, students t, variance}, doi = {10.1088/0266-5611/28/11/115016}, url = {http://arxiv.org/abs/1206.6532}, url2 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/InverseProblems/2012/aravkin2012IPNuisance/aravkin2012IPNuisance.pdf}, author = {Aleksandr Y. Aravkin and Tristan van Leeuwen} } @conference {vanleeuwen2012EAGEcarpcg, title = {{P}reconditioning the {H}elmholtz {E}quation via {R}ow-projections}, booktitle = {EAGE}, year = {2012}, month = {06}, publisher = {EAGE}, organization = {EAGE}, keywords = {EAGE, Helmholtz equation, precondition}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/vanleeuwen2012EAGEcarpcg/vanleeuwen2012EAGEcarpcg.pdf}, presentation = {https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2012/vanleeuwen2012EAGEcarpcg/vanleeuwen2012EAGEcarpcg_pres.pdf}, url2 = {http://earthdoc.eage.org/publication/publicationdetails/?publication=58891}, author = {Tristan van Leeuwen and Dan Gordon and Rachel Gordon and Felix J. Herrmann} } @unpublished {zfang2014SEGsqn, title = {A stochastic quasi-Newton {McMC} method for uncertainty quantification of full-waveform inversion}, year = {2014}, month = {04}, publisher = {UBC}, organization = {UBC}, abstract = {In this work we propose a stochastic quasi-Newton Markov chain Monte Carlo (McMC) method to quantify the uncertainty of full-waveform inversion (FWI). We formulate the uncertainty quantification problem in the framework of the Bayesian inference, which formulates the posterior probability as the conditional probability of the model given the observed data. The Metropolis-Hasting algorithm is used to generate samples satisfying the posterior probability density function (pdf) to quantify the uncertainty. However it suffers from the challenge to construct a proposal distribution that simultaneously provides a good representation of the true posterior pdf and is easy to manipulate. To address this challenge, we propose a stochastic quasi-Newton McMC method, which relies on the fact that the Hessian of the deterministic problem is equivalent to the inverse of the covariance matrix of the posterior pdf. The l-BFGS (limited-memory Broyden{\textendash}Fletcher{\textendash}Goldfarb{\textendash}Shanno) Hessian is used to approximate the inverse of the covariance matrix efficiently, and the randomized source sub-sampling strategy is used to reduce the computational cost of evaluating the posterior pdf and constructing the l-BFGS Hessian. Numerical experiments show the capability of this stochastic quasi-Newton McMC method to quantify the uncertainty of FWI with a considerable low cost.}, keywords = {FWI, McMC, private, quasi-Newton, Uncertainty quantification}, url = {https://www.slim.eos.ubc.ca/Publications/Private/Conferences/SEG/2014/zfang2014SEGsqn/zfang2014SEGsqn.html}, author = {Zhilong Fang, Felix J. Herrmann and Chia Ying Lee} } @unpublished {vanLeeuwen2014pmpde, title = {A penalty method for {PDE}-constrained optimization ({CONFIDENTIAL})}, year = {2014}, note = {Patent filed on April 22, 2014. PCT International Application.}, month = {04}, publisher = {UBC}, organization = {UBC}, abstract = {The invention relates to a partial-differential-equation (PDE) constrained optimization method and especially to a partial-differential-equation (PDE) constrained optimization method for geophysical prospecting.}, keywords = {FWI, Optimization, patent, private}, url = {https://www.slim.eos.ubc.ca/Publications/Private/Submitted/2014/vanLeeuwen2014pmpde/vanLeeuwen2014pmpde.pdf}, author = {Tristan van Leeuwen and Felix J. Herrmann} }