% This file was created with JabRef 2.8.1. % Encoding: MacRoman @ARTICLE{moghaddam2013GEOPnoa, author = {Peyman P. Moghaddam and Henk Keers and Felix J. Herrmann and Wim A. Mulder}, title = {A new optimization approach for source-encoding full-waveform inversion}, year = {2013}, month = {may}, journal = {Geophysics}, volume = {78}, number = {3}, pages = {R125-R132}, abstract = {Waveform inversion is the method of choice for determining highly heterogeneous subsurface structure. However, conventional waveform inversion requires that the wavefield for each source is computed separately. This makes it very expensive for realistic 3D seismic surveys. Source-encoding waveform inversion, in which the sources are modelled simultaneously, is considerably faster than conventional waveform inversion but suffers from artifacts. These artifacts can partly be removed by assigning random weights to the source wavefields. We found that the misfit function, and therefore also its gradient, for source-encoding waveform inversion is an unbiased random estimation of the misfit function used in conventional waveform inversion. We found a new method of source-encoding waveform inversion which takes into account the random nature of the gradients used in the optimization. In this new method, the gradient at each iteration is a weighted average of past gradients such that the most recent gradients have the largest weights with exponential decay. This way we damped the random fluctuations of the gradient by incorporating information from the previous iterations. We compare this new method with existing source-encoding waveform inversion methods as well as conventional waveform inversion and found that the model misfit reduction is faster and smoother than those of existing source-encoding waveform inversion methods, and it approaches the model misfit reduction obtained in conventional waveform inversion.}, keywords = {Geophysics, FWI, optimization, source encoding}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2013/moghaddam2013GEOPnoa/moghaddam2013GEOPnoa.pdf} } @ARTICLE{vanderneut12irs, author = {Joost {van der Neut} and Felix J. Herrmann}, title = {Interferometric redatuming by sparse inversion}, year = {2012}, journal = {Geophysical Journal International}, volume = {192}, pages = {666-670}, abstract = {Assuming that exact transmission responses are known between the surface and a particular depth level in the subsurface, seismic sources can be effectively mapped to that level by a process called interferometric redatuming. After redatuming, the obtained wavefields can be used for imaging below this particular depth level. Interferometric redatuming consists of two steps, namely (i) the decomposition of the observed wavefields into up- and down-going constituents and (ii) a multidimensional deconvolution of the up- and downgoing wavefields. While this method works in theory, sensitivity to noise and artifacts due to incomplete acquisition call for a different formulation. In this letter, we demonstrate the benefits of formulating the two steps that undergird interferometric redatuming in terms of a transform-domain sparsity-promoting program. By exploiting compressibility of seismic wavefields in the curvelet domain, we not only become robust with respect to noise but we are also able to remove certain artifacts while preserving the frequency content. These improvements lead to a better image of the target from the redatumed data.}, keywords = {Controlled source seismology, Interferometry, Inverse theory}, month = {02/2013}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysical Journal International/2013/vanderneut12irs/vanderneut12irs.pdf} } @ARTICLE{berg2008SJSCpareto, author = {Ewout {van den Berg} and Michael P. Friedlander}, title = {Probing the Pareto frontier for basis pursuit solutions}, journal = {SIAM Journal on Scientific Computing}, year = {2008}, volume = {31}, pages = {890-912}, number = {2}, month = {01/2008}, abstract = {The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously differentiable over all points of interest, and show that it gives an explicit relationship to two other optimization problems closely related to BPDN. We describe a root-finding algorithm for finding arbitrary points on this curve; the algorithm is suitable for problems that are large scale and for those that are in the complex domain. At each iteration, a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint. Only matrix-vector operations are required. The primal-dual solution of this problem gives function and derivative information needed for the root-finding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.}, keywords = {basis pursuit, convex program, duality, Newton{\textquoteright}s method, one-norm regularization, projected gradient, root-finding, sparse solutions,Optimization}, optdoi = {10.1137/080714488}, publisher = {SIAM}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/vanderberg08ptp.pdf} } @ARTICLE{vandenberg08gsv, author = {Ewout {van den Berg} and Mark Schmidt and Michael P. Friedlander and K. Murphy}, title = {Group sparsity via linear-time projection}, year = {2008}, number = {TR-2008-09}, month = {06/2008}, abstract = {We present an efficient spectral projected-gradient algorithm for optimization subject to a group one-norm constraint. Our approach is based on a novel linear-time algorithm for Euclidean projection onto the one- and group one-norm constraints. Numerical experiments on large data sets suggest that the proposed method is substantially more efficient and scalable than existing methods.}, institution = {UBC - Department of Computer Science}, keywords = {SLIM,Optimization}, url = {http://www.optimization-online.org/DB_FILE/2008/07/2056.pdf} } @ARTICLE{aravkin2012IPNuisance, author = {Aleksandr Y. Aravkin and Tristan {van Leeuwen}}, title = {Estimating Nuisance Parameters in Inverse Problems}, journal = {Inverse Problems}, year = {2012}, volume = {28}, number = {11}, month = {10/2012}, abstract = {Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large- scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.}, keywords = {full waveform inversion, students t, variance}, optdoi = {10.1088/0266-5611/28/11/115016}, url1 = {http://arxiv.org/abs/1206.6532}, url2 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/InverseProblems/2012/aravkin2012IPNuisance/aravkin2012IPNuisance.pdf}, url3 = {http://iopscience.iop.org/0266-5611/28/11/115016/} } @ARTICLE{Aravkin11TRridr, author = {Aleksandr Y. Aravkin and Michael P. Friedlander and Felix J. Herrmann and Tristan {van Leeuwen}}, title = {Robust inversion, dimensionality reduction, and randomized sampling}, journal = {Mathematical Programming}, year = {2012}, volume = {134}, pages = {101-125}, number = {1}, month = {07/2012}, abstract = {We consider a class of inverse problems in which the forward model is the solution operator to linear ODEs or PDEs. This class admits several dimensionality-reduction techniques based on data averaging or sampling, which are especially useful for large-scale problems. We survey these approaches and their connection to stochastic optimization. The data-averaging approach is only viable, however, for a least-squares misfit, which is sensitive to outliers in the data and artifacts unexplained by the forward model. This motivates us to propose a robust formulation based on the Student’s t-distribution of the error. We demonstrate how the corresponding penalty function, together with the sampling approach, can obtain good results for a large-scale seismic inverse problem with 50 % corrupted data.}, keywords = {Inverse problems, Seismic inversion, Stochastic optimization, Robust estimation, Optimization, FWI}, optdoi = {10.1007/s10107-012-0571-6}, url = {http://www.springerlink.com/content/35rwr101h5736340/} } @ARTICLE{bernabe2004JGRpas, author = {Y. Bernab{\'e} and U. Mok and B. Evans and Felix J. Herrmann}, title = {Permeability and storativity of binary mixtures of high-and low-porosity materials}, journal = {Journal of Geophysical Research}, year = {2004}, volume = {109}, pages = {B12207}, month = {10/2004}, abstract = {As a first step toward determining the mixing laws for the transport properties of rocks, we prepared binary mixtures of high- and low-permeability materials by isostatically hot-pressing mixtures of fine powders of calcite and quartz. The resulting rocks were marbles containing varying concentrations of dispersed quartz grains. Pores were present throughout the rock, but the largest ones were preferentially associated with the quartz particles, leading us to characterize the material as being composed of two phases, one with high permeability and the second with low permeability. We measured the permeability and storativity of these materials using the oscillating flow technique, while systematically varying the effective pressure and the period and amplitude of the input fluid oscillation. Control measurements performed using the steady state flow and pulse decay techniques agreed well with the oscillating flow tests. The hydraulic properties of the marbles were highly sensitive to the volume fraction of the high-permeability phase (directly related to the quartz content). Below a critical quartz content, slightly less than 20 wt \%, the high-permeability volume elements were disconnected, and the overall permeability was low. Above the critical quartz content the high-permeability volume elements formed throughgoing paths, and permeability increased sharply. We numerically simulated fluid flow through binary materials and found that permeability approximately obeys a percolation-based mixing law, consistent with the measured permeability of the calcite-quartz aggregates.}, keywords = {permeability, porosity, SLIM, Modeling}, optdoi = {10.1029/2004JB00311}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Journal of Geophysical Research/2004/bernabe04JGRpas/bernabe04JGRpas.pdf} } @ARTICLE{mansour2012IEEETITrcs, author = {Michael P. Friedlander and Hassan Mansour and Rayan Saab and Ozgur Yilmaz}, title = {Recovering compressively sampled signals using partial support information}, journal = {IEEE Trans. on Information Theory}, year = {2012}, volume = {58}, pages = {1122-1134}, number = {2}, month = {02/2012}, abstract = {We study recovery conditions of weighted $\ell_1$ minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that if at least 50\% of the (partial) support information is accurate, then weighted $\ell_1$ minimization is stable and robust under weaker sufficient conditions than the analogous conditions for standard $\ell_1$ minimization. Moreover, weighted $\ell_1$ minimization provides better upper bounds on the reconstruction error in terms of the measurement noise and the compressibility of the signal to be recovered. We illustrate our results with extensive numerical experiments on synthetic data and real audio and video signals.}, address = {University of British Columbia, Vancouver}, institution = {Department of Computer Science}, keywords = {Compressive Sensing}, optdoi = {10.1109/TIT.2011.2167214}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/IEEETransInformationTheory/2012/mansour2012IEEETITrcs/mansour2012IEEETITrcs.pdf} } @ARTICLE{friedlander2007TASdtd, author = {Michael P. Friedlander and M. A. Saunders}, title = {Discussion: The Dantzig Selector: Statistical estimation when p is much larger then n}, journal = {The Annals of Statistics}, year = {2007}, volume = {35}, pages = {2385-2391}, number = {6}, month = {03/2007}, keywords = {dantzig, SLIM, statistics}, optdoi = {10.1214/009053607000000479}, url = {http://www.cs.ubc.ca/~mpf/2007-discussion-of-the-dantzig-selector.html} } @ARTICLE{Friedlander11TRhdm, author = {Michael P. Friedlander and Mark Schmidt}, title = {Hybrid deterministic-stochastic methods for data fitting}, journal = {SIAM Journal on Scientific Computing}, year = {2012}, volume = {34}, pages = {A1380-A1405}, number = {3}, month = {01/2012}, abstract = {Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms (both deterministic and randomized) offer inexpensive iterations by sampling only subsets of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate of convergence analysis and numerical experiments illustrate the potential for the approach.}, keywords = {Optimization}, optdoi = {10.1137/110830629}, publisher = {Department of Computer Science}, url = {http://www.cs.ubc.ca/~mpf/2011-hybrid-for-data-fitting.html} } @ARTICLE{friedlander2011CoRRhybrid, author = {Michael P. Friedlander and Mark Schmidt}, title = {Hybrid Deterministic-Stochastic Methods for Data Fitting}, journal = {CoRR }, year = {2011}, month = {04/2011}, abstract = {Many structured data-fitting applications require the solution of an optimization problem involving a sum over a potentially large number of measurements. Incremental gradient algorithms (both deterministic and randomized) offer inexpensive iterations by sampling only subsets of the terms in the sum. These methods can make great progress initially, but often slow as they approach a solution. In contrast, full gradient methods achieve steady convergence at the expense of evaluating the full objective and gradient on each iteration. We explore hybrid methods that exhibit the benefits of both approaches. Rate of convergence analysis and numerical experiments illustrate the potential for the approach.}, keywords = {Optimization}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/CoRR/2011/friedlander11hybrid.pdf} } @ARTICLE{friedlander2007SJOero, author = {Michael P. Friedlander and P. Tseng}, title = {Exact Regularization of Convex Programs}, journal = {SIAM J. Optim}, year = {2007}, volume = {18}, pages = {1326-1350}, number = {4}, month = {05/2007}, abstract = {The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a Lagrange multiplier. Moreover, the regularization parameter threshold is inversely related to the Lagrange multiplier. We use this result to generalize an exact regularization result of Ferris and Mangasarian [Appl. Math. Optim., 23(1991), pp. 266{\textendash}273] involving a linearized selection problem. We also use it to derive necessary and sufficient conditions for exact penalization, similar to those obtained by Bertsekas [Math. Programming, 9(1975), pp. 87{\textendash}99] and by Bertsekas, Nedi , Ozdaglar [Convex Analysis and Optimization, Athena Scientific, Belmont, MA, 2003]. When the regularization is not exact, we derive error bounds on the distance from the regularized solution to the original solution set. We also show that existence of a {\textquoteleft}{\textquoteleft}weak sharp minimum{\textquoteright}{\textquoteright} is in some sense close to being necessary for exact regularization. We illustrate the main result with numerical experiments on the l1 regularization of benchmark (degenerate) linear programs and semidefinite/second-order cone programs. The experiments demonstrate the usefulness of l1 regularization in finding sparse solutions.}, keywords = {SLIM,Optimization}, optdoi = {10.1137/060675320}, url = {http://www.cs.ubc.ca/~mpf/2007-exact-regularization.html} } @ARTICLE{haber10TRemp, author = {Eldad Haber and Matthias Chung and Felix J. Herrmann}, title = {An effective method for parameter estimation with PDE constraints with multiple right hand sides}, journal = {SIAM Journal on Optimization}, year = {2012}, volume = {22}, number = {3}, month = {07/2012}, abstract = {Often, parameter estimation problems of parameter-dependent PDEs involve multiple right-hand sides. The computational cost and memory requirements of such problems increase linearly with the number of right-hand sides. For many applications this is the main bottleneck of the computation. In this paper we show that problems with multiple right-hand sides can be reformulated as stochastic programming problems by combining the right-hand sides into a few „simultaneous” sources. This effectively reduces the cost of the forward problem and results in problems that are much cheaper to solve. We discuss two solution methodologies: namely sample average approximation and stochastic approximation. To illustrate the effectiveness of our approach we present two model problems, direct current resistivity and seismic tomography.}, keywords = {SLIM,Full-waveform inversion,Optimization}, url = {http://dx.doi.org/10.1137/11081126X} } @ARTICLE{hennenfent2008GEOPnii, author = {Gilles Hennenfent and Ewout {van den Berg} and Michael P. Friedlander and Felix J. Herrmann}, title = {New insights into one-norm solvers from the {P}areto curve}, journal = {Geophysics}, year = {2008}, volume = {73}, number = {4}, month = {07/2008}, abstract = {Geophysical inverse problems typically involve a trade off between data misfit and some prior. Pareto curves trace the optimal trade off between these two competing aims. These curves are commonly used in problems with two-norm priors where they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one norm, Pareto curves remain relatively unexplored. We show how these curves lead to new insights in one-norm regularization. First, we confirm the theoretical properties of smoothness and convexity of these curves from a stylized and a geophysical example. Second, we exploit these crucial properties to approximate the Pareto curve for a large-scale problem. Third, we show how Pareto curves provide an objective criterion to gauge how different one-norm solvers advance towards the solution.}, keywords = {Pareto, SLIM, Geophysics,Optimization,Acquisition,Processing}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2008/hennenfent08GEOnii/hennenfent08GEOnii.pdf } } @ARTICLE{hennenfent2010GEOPnct, author = {Gilles Hennenfent and Lloyd Fenelon and Felix J. Herrmann}, title = {Nonequispaced curvelet transform for seismic data reconstruction: A sparsity-promoting approach}, journal = {Geophysics}, year = {2010}, volume = {75}, pages = {WB203-WB210}, number = {6}, month = {12/2010}, abstract = {We extend our earlier work on the nonequispaced fast discrete curvelet transform (NFDCT) and introduce a second generation of the transform. This new generation differs from the previous one by the approach taken to compute accurate curvelet coefficients from irregularly sampled data. The first generation relies on accurate Fourier coefficients obtained by an l2-regularized inversion of the nonequispaced fast Fourier transform (FFT) whereas the second is based on a direct l1-regularized inversion of the operator that links curvelet coefficients to irregular data. Also, by construction the second generation NFDCT is lossless unlike the first generation NFDCT. This property is particularly attractive for processing irregularly sampled seismic data in the curvelet domain and bringing them back to their irregular record-ing locations with high fidelity. Secondly, we combine the second generation NFDCT with the standard fast discrete curvelet transform (FDCT) to form a new curvelet-based method, coined nonequispaced curvelet reconstruction with sparsity-promoting inversion (NCRSI) for the regularization and interpolation of irregularly sampled data. We demonstrate that for a pure regularization problem the reconstruction is very accurate. The signal-to-reconstruction error ratio in our example is above 40 dB. We also conduct combined interpolation and regularization experiments. The reconstructions for synthetic data are accurate, particularly when the recording locations are optimally jittered. The reconstruction in our real data example shows amplitudes along the main wavefronts smoothly varying with limited acquisition imprint.}, keywords = {curvelet transforms, data acquisition, geophysical techniques, seismology,SLIM,Processing}, optdoi = {10.1190/1.3494032}, publisher = {SEG}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2010/hennenfent2010GEOPnct/hennenfent2010GEOPnct.pdf} } @ARTICLE{hennenfent2008GEOPsdw, author = {Gilles Hennenfent and Felix J. Herrmann}, title = {Simply denoise: wavefield reconstruction via jittered undersampling}, journal = {Geophysics}, year = {2008}, volume = {73}, pages = {V19-V28}, number = {3}, month = {05/2008}, abstract = {In this paper, we present a new discrete undersampling scheme designed to favor wavefield reconstruction by sparsity-promoting inversion with transform elements that are localized in the Fourier domain. Our work is motivated by empirical observations in the seismic community, corroborated by recent results from compressive sampling, which indicate favorable (wavefield) reconstructions from random as opposed to regular undersampling. As predicted by theory, random undersampling renders coherent aliases into harmless incoherent random noise, effectively turning the interpolation problem into a much simpler denoising problem. A practical requirement of wavefield reconstruction with localized sparsifying transforms is the control on the maximum gap size. Unfortunately, random undersampling does not provide such a control and the main purpose of this paper is to introduce a sampling scheme, coined jittered undersampling, that shares the benefits of random sampling, while offering control on the maximum gap size. Our contribution of jittered sub-Nyquist sampling proves to be key in the formulation of a versatile wavefield sparsity-promoting recovery scheme that follows the principles of compressive sampling. After studying the behavior of the jittered undersampling scheme in the Fourier domain, its performance is studied for curvelet recovery by sparsity-promoting inversion (CRSI). Our findings on synthetic and real seismic data indicate an improvement of several decibels over recovery from regularly-undersampled data for the same amount of data collected.}, html_version = { https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2008/hennenfent08GEOsdw/paper_html/paper.html }, keywords = {sampling, Geophysics,SLIM,Acquisition,Processing,Optimization,Compressive Sensing}, optdoi = {10.1190/1.2841038}, publisher = {SEG}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2008/hennenfent08GEOsdw/hennenfent08GEOsdw.pdf} } @ARTICLE{hennenfent2006CiSEsdn, author = {Gilles Hennenfent and Felix J. Herrmann}, title = {Seismic Denoising with Nonuniformly Sampled Curvelets}, journal = {Computing in Science \& Engineering}, year = {2006}, volume = {8 Issue:3}, pages = {16 - 25}, month = {05/2006}, abstract = {The authors present an extension of the fast discrete curvelet transform (FDCT) to nonuniformly sampled data. This extension not only restores curvelet compression rates for nonuniformly sampled data but also removes noise and maps the data to a regular grid.}, keywords = {CiSE,Processing}, optdoi = {10.1109/MCSE.2006.49}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/CiSE/2006/hennenfent06CiSEsdn/hennenfent06CiSEsdn.pdf } } @ARTICLE{herrmann2012IIsi, author = {Felix J. Herrmann}, title = {Seismic advances}, journal = {International Innovation}, year = {2012}, pages = {46-49}, month = {12/2012}, abstract = {Current seismic exploration techniques are hampered by bottlenecks in data sampling and processing due to challenges in data collection, demand for more data and the increasing need to study highly complex geological settings. Professor Felix J. Herrmann's group is developing novel techniques to overcome these barriers which could greatly benefit the hydrocarbon industry.}, keywords = {Seismic exploration techniques, compressive sensing, wave-equation-based data mining, dynamic nonlinear optimization}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/InternationalInnovation/2012/herrmann2012IIsi/herrmann2012IIsi.pdf} } @ARTICLE{herrmann2010GEOPrsg, author = {Felix J. Herrmann}, title = {Randomized sampling and sparsity: Getting more information from fewer samples}, journal = {Geophysics}, year = {2010}, volume = {75}, pages = {WB173-WB187}, number = {6}, month = {12/2010}, abstract = {Many seismic exploration techniques rely on the collection of massive data volumes that are subsequently mined for information during processing. Although this approach has been extremely successful in the past, current efforts toward higher-resolution images in increasingly complicated regions of the earth continue to reveal fundamental shortcomings in our workflows. Chiefly among these is the so-called {\textquotedblleft}curse of dimensionality{\textquotedblright} exemplified by Nyquist{\textquoteright}s sampling criterion, which disproportionately strains current acquisition and processing systems as the size and desired resolution of our survey areas continue to increase. We offer an alternative sampling method leveraging recent insights from compressive sensing toward seismic acquisition and processing for data that are traditionally considered to be undersampled. The main outcome of this approach is a new technology where acquisition and processing related costs are no longer determined by overly stringent sampling criteria, such as Nyquist. At the heart of our approach lies randomized incoherent sampling that breaks subsampling related interferences by turning them into harmless noise, which we subsequently remove by promoting transform-domain sparsity. Now, costs no longer grow significantly with resolution and dimensionality of the survey area, but instead depend only on transform-domain sparsity. Our contribution is twofold. First, we demonstrate by means of carefully designed numerical experiments that compressive sensing can successfully be adapted to seismic exploration. Second, we show that accurate recovery can be accomplished for compressively sampled data volumes sizes that exceed the size of conventional transform-domain data volumes by only a small factor. Because compressive sensing combines transformation and encoding by a single linear encoding step, this technology is directly applicable to acquisition and to dimensionality reduction during processing. In either case, sampling, storage, and processing costs scale with transform-domain sparsity. We illustrate this principle by means of number of case studies.}, keywords = {data acquisition, geophysical techniques, Nyquist criterion, sampling methods, seismology,SLIM,Acquisition,Compressive Sensing,Optimization}, optdoi = {10.1190/1.3506147}, publisher = {SEG}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2010/herrmann2010GEOPrsg/herrmann2010GEOPrsg.pdf} } @ARTICLE{herrmann2005ICAEsdb, author = {Felix J. Herrmann}, title = {Seismic deconvolution by atomic decomposition: a parametric approach with sparseness constraints}, journal = {Integrated Computer-Aided Engineering}, year = {2005}, volume = {12}, pages = {69-90}, number = {1}, month = {01/2005}, abstract = {In this paper an alternative approach to the blind seismic deconvolution problem is presented that aims for two goals namely recovering the location and relative strength of seismic reflectors, possibly with super-localization, as well as obtaining detailed parametric characterizations for the reflectors. We hope to accomplish these goals by decomposing seismic data into a redundant dictionary of parameterized waveforms designed to closely match the properties of reflection events associated with sedimentary records. In particular, our method allows for highly intermittent non-Gaussian records yielding a reflectivity that can no longer be described by a stationary random process or by a spike train. Instead, we propose a reflector parameterization that not only recovers the reflector{\textquoteright}s location and relative strength but which also captures reflector attributes such as its local scaling, sharpness and instantaneous phase-delay. The first set of parameters delineates the stratigraphy whereas the second provides information on the lithology. As a consequence of the redundant parameterization, finding the matching waveforms from the dictionary involves the solution of an ill-posed problem. Two complementary sparseness-imposing methods Matching and Basis Pursuit are compared for our dictionary and applied to seismic data.}, address = {Amsterdam, The Netherlands, The Netherlands}, issn = {1069-2509}, keywords = {deconvolution, SLIM,Processing,Modelling}, publisher = {IOS Press}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/IntegratedComputerAidedEngineering/2005/herrmann2005ICAEsdb/herrmann2005ICAEsdb.pdf} } @ARTICLE{herrmann2004GJIssa, author = {Felix J. Herrmann and Y. Bernab\'e }, title = {Seismic singularities at upper-mantle phase transitions: a site percolation model}, journal = {Geophysical Journal International}, year = {2004}, volume = {159}, pages = {949-960}, number = {3}, month = {12/2004}, abstract = {Mineralogical phase transitions are usually invoked to account for the sharpness of globally observed upper-mantle seismic discontinuities. We propose a percolation-based model for the elastic properties of the phase mixture in the coexistence regions associated with these transitions. The major consequence of the model is that the elastic moduli (but not the density) display a singularity at the percolation threshold of the high-pressure phase. This model not only explains the sharp but continuous change in seismic velocities across the phase transition, but also predicts its abruptness and scale invariance, which are characterized by a non-integral scale exponent. Using the receiver-function approach and new, powerful signal-processing techniques, we quantitatively determine the singularity exponent from recordings of converted seismic waves at two Australian stations (CAN and WRAB). Using the estimated values, we construct velocity{\textendash}depth profiles across the singularities and verify that the calculated converted waveforms match the observations under CAN. Finally, we point out a series of additional predictions that may provide new insights into the physics and fine structure of the upper-mantle transition zone.}, keywords = {percolation, SLIM,Modelling}, optdoi = {10.1111/j.1365-246X.2004.02464.x}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysical%20Journal%20International/2004/herrmann2004GJIssa/herrmann2004GJIssa.pdf} } @ARTICLE{herrmann2007GJInlp, author = {Felix J. Herrmann and U. Boeniger and D. J. Verschuur}, title = {Non-linear primary-multiple separation with directional curvelet frames}, journal = {Geophysical Journal International}, year = {2007}, volume = {170}, pages = {781-799}, number = {2}, month = {08/2007}, abstract = {Predictive multiple suppression methods consist of two main steps: a prediction step, during which multiples are predicted from seismic data, and a primary-multiple separation step, during which the predicted multiples are {\textquoteright}matched{\textquoteright} with the true multiples in the data and subsequently removed. This second separation step, which we will call the estimation step, is crucial in practice: an incorrect separation will cause residual multiple energy in the result or may lead to a distortion of the primaries, or both. To reduce these adverse effects, a new transformed-domain method is proposed where primaries and multiples are separated rather than matched. This separation is carried out on the basis of differences in the multiscale and multidirectional characteristics of these two signal components. Our method uses the curvelet transform, which maps multidimensional data volumes into almost orthogonal localized multidimensional prototype waveforms that vary in directional and spatio-temporal content. Primaries-only and multiples-only signal components are recovered from the total data volume by a non-linear optimization scheme that is stable under noisy input data. During the optimization, the two signal components are separated by enhancing sparseness (through weighted l1-norms) in the transformed domain subject to fitting the observed data as the sum of the separated components to within a user-defined tolerance level. Whenever, during the optimization, the estimates for the primaries in the transformed domain correlate with the predictions for the multiples, the recovery of the coefficients for the estimated primaries will be suppressed while for regions where the correlation is small the method seeks the sparsest set of coefficients that represent the estimation for the primaries. Our algorithm does not seek a matched filter and as such it differs fundamentally from traditional adaptive subtraction methods. The method derives its stability from the sparseness obtained by a non-parametric (i.e. not depending on a parametrized physical model) multiscale and multidirectional overcomplete signal representation. This sparsity serves as prior information and allows for a Bayesian interpretation of our method during which the log-likelihood function is minimized while the two signal components are assumed to be given by a superposition of prototype waveforms, drawn independently from a probability function that is weighted by the predicted primaries and multiples. In this paper, the predictions are based on the data-driven surface-related multiple elimination method. Synthetic and field data examples show a clean separation leading to a considerable improvement in multiple suppression compared to the conventional method of adaptive matched filtering. This improved separation translates into an improved stack.}, keywords = {signal separation, SLIM,Processing}, optdoi = {10.1111/j.1365-246X.2007.03360.x}, url = { https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysical Journal International/2007/herrmann07nlp/herrmann07nlp.pdf } } @ARTICLE{herrmann2009GEOPcbm, author = {Felix J. Herrmann and Cody R. Brown and Yogi A. Erlangga and Peyman P. Moghaddam}, title = {Curvelet-based migration preconditioning and scaling}, journal = {Geophysics}, year = {2009}, volume = {74}, pages = {A41}, month = {09/2009}, abstract = {The extremely large size of typical seismic imaging problems has been one of the major stumbling blocks for iterative techniques to attain accurate migration amplitudes. These iterative methods are important because they complement theoretical approaches that are hampered by difficulties to control problems such as finite-acquisition aperture, source-receiver frequency response, and directivity. To solve these problems, we apply preconditioning, which significantly improves convergence of least-squares migration. We discuss different levels of preconditioning that range from corrections for the order of the migration operator to corrections for spherical spreading, and position and reflector-dip dependent amplitude errors. While the first two corrections correspond to simple scalings in the Fourier and physical domain, the third correction requires phase-space (space spanned by location and dip) scaling, which we carry out with curvelets. We show that our combined preconditioner leads to a significant improvement of the convergence of least-squares {\textquoteleft}wave-equation{\textquoteright} migration on a line from the SEG AA{\textquoteright} salt model.}, keywords = {migration,SLIM,Imaging}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/herrmann08cmp-r/herrmann08cmp-r.pdf} } @ARTICLE{herrmann2009GEOPcsf, author = {Felix J. Herrmann and Yogi A. Erlangga and Tim T.Y. Lin}, title = {Compressive simultaneous full-waveform simulation}, journal = {Geophysics}, year = {2009}, volume = {74}, pages = {A35}, month = {08/2008}, abstract = {The fact that computational complexity of wavefield simulation is proportional to the size of the discretized model and acquisition geometry, and not to the complexity of the simulated wavefield, is a major impediment within seismic imaging. By turning simulation into a compressive sensing problem{\textendash}-where simulated data is recovered from a relatively small number of independent simultaneous sources{\textendash}-we remove this impediment by showing that compressively sampling a simulation is equivalent to compressively sampling the sources, followed by solving a reduced system. As in compressive sensing, this allows for a reduction in sampling rate and hence in simulation costs. We demonstrate this principle for the time-harmonic Helmholtz solver. The solution is computed by inverting the reduced system, followed by a recovery of the full wavefield with a sparsity promoting program. Depending on the wavefield{\textquoteright}s sparsity, this approach can lead to significant cost reductions, in particular when combined with the implicit preconditioned Helmholtz solver, which is known to converge even for decreasing mesh sizes and increasing angular frequencies. These properties make our scheme a viable alternative to explicit time-domain finite-differences.}, keywords = {full-waveform,SLIM,Modelling,Compressive Sensing}, url = {https://www.slim.eos.ubc.ca/Publications/Public/TechReport/2009/herrmann2009GEOPcsf/herrmann2009GEOPcsf.pdf} } @ARTICLE{Herrmann11TRfcd, author = {Felix J. Herrmann and Michael P. Friedlander and Ozgur Yilmaz}, title = {Fighting the Curse of Dimensionality: Compressive Sensing in Exploration Seismology}, journal = {Signal Processing Magazine, IEEE}, year = {2012}, volume = {29}, pages = {88-100}, number = {3}, month = {05/12}, abstract = {Many seismic exploration techniques rely on the collection of massive data volumes that are mined for information during processing. This approach has been extremely successful, but current efforts toward higher resolution images in increasingly complicated regions of Earth continue to reveal fundamental shortcomings in our typical workflows. The "curse" of dimensionality is the main roadblock and is exemplified by Nyquist's sampling criterion, which disproportionately strains current acquisition and processing systems as the size and desired resolution of our survey areas continues to increase.}, issn = {1053-5888}, keywords = {Earth, Nyquist sampling criterion;dimensionality curse, higher-resolution images, massive data volumes,seismic exploration techniques, strains current acquisition system,strains current processing system, geographic information systems,seismology}, optdoi = {10.1109/MSP.2012.2185859}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/IEEESignalProcessingMagazine/2012/Herrmann11TRfcd/Herrmann11TRfcd.pdf} } @ARTICLE{herrmann2008GJInps, author = {Felix J. Herrmann and Gilles Hennenfent}, title = {Non-parametric seismic data recovery with curvelet frames}, journal = {Geophysical Journal International}, year = {2008}, volume = {173}, pages = {233-248}, month = {04/2011}, abstract = {Seismic data recovery from data with missing traces on otherwise regular acquisition grids forms a crucial step in the seismic processing flow. For instance, unsuccessful recovery leads to imaging artifacts and to erroneous predictions for the multiples, adversely affecting the performance of multiple elimination. A non-parametric transform-based recovery method is presented that exploits the compression of seismic data volumes by recently developed curvelet frames. The elements of this transform are multidimensional and directional and locally resem- ble wavefronts present in the data, which leads to a compressible representation for seismic data. This compression enables us to formulate a new curvelet-based seismic data recovery algorithm through sparsity-promoting inversion. The concept of sparsity-promoting inversion is in itself not new to geophysics. However, the recent insights from the field of {\textquoteleft}compressed sensing{\textquoteright} are new since they clearly identify the three main ingredients that go into a successful formulation of a re- covery problem, namely a sparsifying transform, a sampling strategy that subdues coherent aliases and a sparsity-promoting program that recovers the largest entries of the curvelet-domain vector while explaining the measurements. These concepts are illustrated with a stylized experiment that stresses the importance of the degree of compression by the sparsifying transform. With these findings, a curvelet-based recovery algorithms is developed, which recovers seismic wavefields from seismic data volumes with large percentages of traces missing. During this construction, we benefit from the main three ingredients of compressive sampling, namely the curvelet compression of seismic data, the existence of a favorable sam- pling scheme and the formulation of a large-scale sparsity-promoting solver based on a cooling method. The recovery performs well on synthetic as well as real data and performs better by virtue of the sparsifying property of curvelets. Our results are applicable to other areas such as global seismology.}, keywords = {curvelet transform, reconstruction, SLIM,Acquisition}, optdoi = {10.1111/j.1365-246X.2007.03698.x}, url = { https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysical Journal International/2008/herrmann08nps/herrmann08nps.pdf } } @ARTICLE{herrmann11GPelsqIm, author = {Felix J. Herrmann and Xiang Li}, title = {Efficient least-squares imaging with sparsity promotion and compressive sensing}, journal = {Geophysical Prospecting}, year = {2012}, volume = {60}, pages = {696-712}, number = {4}, month = {07/2012}, abstract = {Seismic imaging is a linearized inversion problem relying on the minimization of a least-squares misfit functional as a function of the medium perturbation. The success of this procedure hinges on our ability to handle large systems of equations---whose size grows exponentially with the demand for higher resolution images in more and more complicated areas---and our ability to invert these systems given a limited amount of computational resources. To overcome this ``curse of dimensionality'' in problem size and computational complexity, we propose a combination of randomized dimensionality-reduction and divide-and-conquer techniques. This approach allows us to take advantage of sophisticated sparsity-promoting solvers that work on a series of smaller subproblems each involving a small randomized subset of data. These subsets correspond to artificial simultaneous-source experiments made of random superpositions of sequential-source experiments. By changing these subsets after each subproblem is solved, we are able to attain an inversion quality that is competitive while requiring fewer computational, and possibly, fewer acquisition resources. Application of this concept to a controlled series of experiments showed the validity of our approach and the relationship between its efficiency---by reducing the number of sources and hence the number of wave-equation solves---and the image quality. Application of our dimensionality-reduction methodology with sparsity promotion to a complicated synthetic with well-log constrained structure also yields excellent results underlining the importance of sparsity promotion.}, address = {University of British Columbia, Vancouver}, keywords = {SLIM,Imaging,Optimization,Compressive Sensing}, optdoi = {10.1111/j.1365-2478.2011.01041.x}, url1 = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/GeophysicalProspecting/2012/herrmann11GPelsqIm/herrmann11GPelsqIm.pdf}, url2 = {http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2478.2011.01041.x/full} } @ARTICLE{herrmann2008ACHAsac, author = {Felix J. Herrmann and Peyman P. Moghaddam and Chris Stolk}, title = {Sparsity- and continuity-promoting seismic image recovery with curvelet frames}, journal = {Applied and Computational Harmonic Analysis}, year = {2008}, volume = {24}, pages = {150-173}, number = {2}, month = {03/2011}, abstract = {A nonlinear singularity-preserving solution to seismic image recovery with sparseness and continuity constraints is proposed. We observe that curvelets, as a directional frame expan- sion, lead to sparsity of seismic images and exhibit invariance under the normal operator of the linearized imaging problem. Based on this observation we derive a method for stable recovery of the migration amplitudes from noisy data. The method corrects the amplitudes during a post-processing step after migration, such that the main additional cost is one ap- plication of the normal operator, i.e. a modeling followed by a migration. Asymptotically this normal operator corresponds to a pseudodifferential operator, for which a convenient diagonal approximation in the curvelet domain is derived, including a bound for its error and a method for the estimation of the diagonal from a compound operator consisting of discrete implementations for the scattering operator and its adjoint the migration opera- tor. The solution is formulated as a nonlinear optimization problem where sparsity in the curvelet domain as well as continuity along the imaged reflectors are jointly promoted. To enhance sparsity, the l1 -norm on the curvelet coefficients is minimized, while continuity is promoted by minimizing an anisotropic diffusion norm on the image. The performance of the recovery scheme is evaluated with a time-reversed {\textquoteright}wave-equation{\textquoteright} migration code on synthetic datasets, including the complex SEG/EAGE AA salt model.}, keywords = {curvelet transform, imaging, SLIM,Imaging,Processing}, optdoi = {10.1016/j.acha.2007.06.007}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/ACHA/2008/herrmann2008ACHAsac/herrmann2008ACHAsac.pdf} } @ARTICLE{herrmann2008GEOPcbs, author = {Felix J. Herrmann and Deli Wang and Gilles Hennenfent and Peyman P. Moghaddam}, title = {Curvelet-based seismic data processing: a multiscale and nonlinear approach}, journal = {Geophysics}, year = {2008}, volume = {73}, pages = {A1-A5}, number = {1}, month = {03/2008}, abstract = {Mitigating missing data, multiples, and erroneous migration amplitudes are key factors that determine image quality. Curvelets, little {\textquoteleft}{\textquoteleft}plane waves,{\textquoteright}{\textquoteright} complete with oscillations in one direction and smoothness in the other directions, sparsify a property we leverage explicitly with sparsity promotion. With this principle, we recover seismic data with high fidelity from a small subset (20\%) of randomly selected traces. Similarly, sparsity leads to a natural decorrelation and hence to a robust curvelet-domain primary-multiple separation for North Sea data. Finally, sparsity helps to recover migration amplitudes from noisy data. With these examples, we show that exploiting the curvelet{\textquoteright}s ability to sparsify wavefrontlike features is powerful, and our results are a clear indication of the broad applicability of this transform to exploration seismology. {\copyright}2008 Society of Exploration Geophysicists}, keywords = {curvelet transform, SLIM,Acquisition,Processing}, optdoi = {10.1190/1.2799517}, publisher = {SEG}, url = { https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2008/herrmann08GEOcbs/herrmann08GEOcbs.pdf } } @ARTICLE{herrmann2008GEOPacd, author = {Felix J. Herrmann and Deli Wang and D. J. Verschuur}, title = {Adaptive curvelet-domain primary-multiple separation}, journal = {Geophysics}, year = {2008}, volume = {73}, pages = {A17-A21}, number = {3}, month = {08/2008}, abstract = {In many exploration areas, successful separation of primaries and multiples greatly determines the quality of seismic imaging. Despite major advances made by surface-related multiple elimination (SRME), amplitude errors in the predicted multiples remain a problem. When these errors vary for each type of multiple in different ways (as a function of offset, time, and dip), they pose a serious challenge for conventional least-squares matching and for the recently introduced separation by curvelet-domain thresholding. We propose a data-adaptive method that corrects amplitude errors, which vary smoothly as a function of location, scale (frequency band), and angle. With this method, the amplitudes can be corrected by an elementwise curvelet-domain scaling of the predicted multiples. We show that this scaling leads to successful estimation of primaries, despite amplitude, sign, timing, and phase errors in the predicted multiples. Our results on synthetic and real data show distinct improvements over conventional least-squares matching in terms of better suppression of multiple energy and high-frequency clutter and better recovery of estimated primaries. {\copyright}2008 Society of Exploration Geophysicists}, keywords = {Geophysics, SLIM,Processing}, optdoi = {10.1190/1.2904986}, publisher = {SEG}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2008/herrmann08GEOacd/herrmann08GEOacd.pdf } } @ARTICLE{herrmann2011RECORDERcsse1, author = {Felix J. Herrmann and Haneet Wason and Tim T.Y. Lin}, title = {Compressive sensing in seismic exploration: an outlook on a new paradigm}, journal = {CSEG Recorder}, year = {2011}, volume = {36}, pages = {19-33}, number = {4}, month = {04/2011}, abstract = {Many seismic exploration techniques rely on the collection of massive data volumes that are subsequently mined for information during processing. While this approach has been extremely successful in the past, current efforts toward higher resolution images in increasingly complicated regions of the Earth continue to reveal fundamental shortcomings in our workflows. Chiefly amongst these is the so-called “curse of dimensionality” exemplified by Nyquist's sampling criterion, which disproportionately strains current acquisition and processing systems as the size and desired resolution of our survey areas continues to increase. We offer an alternative sampling method leveraging recent insights from compressive sensing towards seismic acquisition and processing for data that, from a traditional point of view, are considered to be undersampled. The main outcome of this approach is a new technology where acquisition and processing related costs are decoupled the stringent Nyquist sampling criterion. At the heart of our approach lies randomized incoherent sampling that breaks subsampling-related interferences by turning them into harmless noise, which we subsequently remove by promoting sparsity in a transform-domain. Acquisition schemes designed to fit into this regime no longer grow significantly in cost with increasing resolution and dimensionality of the survey area, but instead its cost ideally only depends on transform-domain sparsity of the expected data. Our contribution is split into two part.}, url = {http://www.cseg.ca/publications/recorder/2011/04apr/Apr2011-Compressive-sensing-in-seismic-expl.pdf} } @ARTICLE{kumar2010TNPecr, author = {Vishal Kumar and Jounada Oueity and Ron Clowes and Felix J. Herrmann}, title = {Enhancing Crustal Reflection Data through Curvelet Denoising}, journal = {Technophysics}, year = {2011}, volume = {508}, pages = {106-116}, number = {1-4}, month = {07/2011}, abstract = {Suppression of incoherent noise, which is present in the seismic signal and may often lead to ambiguous interpretation, is a key step in processing associated with crustal reflection data. In this paper, we make use of the parsimonious representation of seismic data in the curvelet domain to perform the noise attenuation while preserving the coherent energy and its amplitude information. Curvelets are a recently developed mathematical transform that has as one of its properties minimal overlap between seismic signal and noise in the transform domain, thereby facilitating signal-noise separation. The problem is cast as an inverse problem and the results are obtained by updating the solution at each iteration. We demonstrate the effectiveness of this procedure at removing noise on both synthetic shot gathers and a synthetic stacked seismic section. We then apply curvelet denoising to deep crustal seismic reflection data where the signal-to-noise ratio is low. The reflection data were recorded along Lithoprobe's SNORCLE Line 1 across Paleoproterozoic-Archean domains in Canada's Northwest Territories. After initial processing, we apply the iterative curvelet denoising to both pre-stack shot gathers and post-stack data. Ground roll, random noise and much of the anomalous vertical energy is removed from the pre-stack shot gathers, to the extent that crustal reflections, including those from the Moho, are clearly seen on individual gathers. Denoised stacked data show a series of dipping reflections in the lower crust that extend into the Moho. The Moho itself is relatively flat and characterized by a sharp, narrow band of reflections. Comparing the results for the stacked data with those from F-X deconvolution, curvelet denoising outperforms the latter by attenuating incoherent noise with minimal harm to the signal. Because curvelet denoising retains amplitude information, it provides opportunities for further studies of seismic sections through attribute analyses. Curvelet denoising provides an important new tool in the processing toolbox for crustal seismic reflection data.}, keywords = {SLIM,Processing}, optdoi = {10.1016/j.tecto.2010.07.01}, url = {http://www.sciencedirect.com/science/article/pii/S0040195110003227} } @ARTICLE{vanLeeuwen2010IJGswi, author = {Tristan van Leeuwen and Aleksandr Y. Aravkin and Felix J. Herrmann}, title = {Seismic waveform inversion by stochastic optimization}, journal = {International Journal of Geophysics}, year = {2011}, volume = {2011}, month = {12/2011}, abstract = {We explore the use of stochastic optimization methods for seismic waveform inversion. The basic principle of such methods is to randomly draw a batch of realizations of a given misfit function and goes back to the 1950s. The ultimate goal of such an approach is to dramatically reduce the computational cost involved in evaluating the misfit. Following earlier work, we introduce the stochasticity in waveform inversion problem in a rigorous way via a technique called randomized trace estimation. We then review theoretical results that underlie recent developments in the use of stochastic methods for waveform inversion. We present numerical experiments to illustrate the behavior of different types of stochastic optimization methods and investigate the sensitivity to the batch size and the noise level in the data. We find that it is possible to reproduce results that are qualitatively similar to the solution of the full problem with modest batch sizes, even on noisy data. Each iteration of the corresponding stochastic methods requires an order of magnitude fewer PDE solves than a comparable deterministic method applied to the full problem, which may lead to an order of magnitude speedup for waveform inversion in practice.}, keywords = {SLIM,Full-waveform inversion,Optimization}, notes = {Article ID: 689041, 18pages}, optdoi = {10.1155/2011/689041}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/InternationJournalOfGeophysics/2011/vanLeeuwen10IJGswi/vanLeeuwen10IJGswi.pdf} } @ARTICLE{VanLeeuwen11TRfwiwse, author = {Tristan van Leeuwen and Felix J. Herrmann}, title = {Fast waveform inversion without source encoding}, journal = {Geophysical Prospecting}, year = {2012}, month = {07/2012}, abstract = {Randomized source encoding has recently been proposed as a way to dramatically reduce the costs of full waveform inversion. The main idea is to replace all sequential sources by a small number of simultaneous sources. This introduces random crosstalk in the model updates and special stochastic optimization strategies are required to deal with this. Two problems arise with this approach: i) source encoding can only be applied to fixed-spread acquisition setups, and ii) stochastic optimization methods tend to converge very slowly, relying on averaging to get rid of the cross-talk. Although the slow convergence is partly offset by the low iteration cost, we show that conventional optimization strategies are bound to outperform stochastic methods in the long run. In this paper we argue that we don¬øt need randomized source encoding to reap the benefits of stochastic optimization and we review an optimization strategy that combines the benefits of both conventional and stochastic optimization. The method uses a gradually increasing batch of sources. Thus, iterations are very cheap initially and this allows the method to make fast progress in the beginning. As the batch size grows, the method behaves like conventional optimization, allowing for fast convergence. Numerical examples suggest that the stochastic and hybrid method perform equally well with and without source encoding and that the hybrid method outperforms both conventional and stochastic optimization. The method does not rely on source encoding techniques and can thus be applied to non fixed-spread data.}, keywords = {SLIM,Full-waveform inversion,Optimization}, optdoi = {10.1111/j.1365-2478.2012.01096.x}, optvolume = {Online publication}, url = {http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2478.2012.01096.x/abstract} } @ARTICLE{Li11TRfrfwi, author = {Xiang Li and Aleksandr Y. Aravkin and Tristan van Leeuwen and Felix J. Herrmann}, title = {Fast randomized full-waveform inversion with compressive sensing}, journal = {Geophysics}, year = {2012}, volume = {77}, pages = {A13-A17}, number = {3}, month = {05/2012}, abstract = { Wave-equation based seismic inversion can be formulated as a nonlinear inverse problem where the medium properties are obtained via minimization of a least- squares misfit functional. The demand for higher resolution models in more geologically complex areas drives the need to develop techniques that explore the special structure of full-waveform inversion to reduce the computational burden and to regularize the inverse problem. We meet these goals by using ideas from compressive sensing and stochastic optimization to design a novel Gauss-Newton method, where the updates are computed from random subsets of the data via curvelet-domain sparsity promotion. Application of this idea to a realistic synthetic shows improved results compared to quasi-Newton methods, which require passes through all data. Two different subset sampling strategies are considered: randomized source encoding, and drawing sequential shots firing at random source locations from marine data with missing near and far offsets. In both cases, we obtain excellent inversion results compared to conventional methods at reduced computational costs. }, keywords = {SLIM,Full-waveform inversion,Compressive Sensing,Optimization}, optdoi = {10.1190/geo2011-0410.1}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2012/Li11TRfrfwi/Li11TRfrfwi.pdf} } @ARTICLE{lin2007GEOPcwe, author = {Tim T.Y. Lin and Felix J. Herrmann}, title = {Compressed wavefield extrapolation}, journal = {Geophysics}, year = {2007}, volume = {72}, pages = {SM77-SM93}, number = {5}, month = {08/2007}, abstract = {An explicit algorithm for the extrapolation of one-way wavefields is proposed that combines recent developments in information theory and theoretical signal processing with the physics of wave propagation. Because of excessive memory requirements, explicit formulations for wave propagation have proven to be a challenge in 3D. By using ideas from compressed sensing, we are able to formulate the (inverse) wavefield extrapolation problem on small subsets of the data volume, thereby reducing the size of the operators. Compressed sensing entails a new paradigm for signal recovery that provides conditions under which signals can be recovered from incomplete samplings by nonlinear recovery methods that promote sparsity of the to-be-recovered signal. According to this theory, signals can be successfully recovered when the measurement basis is incoherent with the representa-tion in which the wavefield is sparse. In this new approach, the eigenfunctions of the Helmholtz operator are recognized as a basis that is incoherent with curvelets that are known to compress seismic wavefields. By casting the wavefield extrapolation problem in this framework, wavefields can be successfully extrapolated in the modal domain, despite evanescent wave modes. The degree to which the wavefield can be recovered depends on the number of missing (evanescent) wavemodes and on the complexity of the wavefield. A proof of principle for the compressed sensing method is given for inverse wavefield extrapolation in 2D, together with a pathway to 3D during which the multiscale and multiangular properties of curvelets, in relation to the Helmholz operator, are exploited. The results show that our method is stable, has reduced dip limitations, and handles evanescent waves in inverse extrapolation. {\copyright}2007 Society of Exploration Geophysicists}, keywords = {SLIM, wave propagation,Modelling}, optdoi = {10.1190/1.2750716}, publisher = {SEG}, url = { https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2007/lin07cwe/lin07cwe.pdf } } @ARTICLE{Mansour11TRssma, author = {Hassan Mansour and Haneet Wason and Tim T.Y. Lin and Felix J. Herrmann}, title = {Randomized marine acquisition with compressive sampling matrices}, journal = {Geophysical Prospecting}, year = {2012}, volume = {60}, pages = {648-662}, number = {4}, month = {07/2012}, abstract = {Seismic data acquisition in marine environments is a costly process that calls for the adoption of simultaneous-source or randomized acquisition - an emerging technology that is stimulating both geophysical research and commercial efforts. Simultaneous marine acquisition calls for the development of a new set of design principles and post-processing tools. In this paper, we discuss the properties of a specific class of randomized simultaneous acquisition matrices and demonstrate that sparsity-promoting recovery improves the quality of reconstructed seismic data volumes. We propose a practical randomized marine acquisition scheme where the sequential sources fire airguns at only randomly time-dithered instances. We demonstrate that the recovery using sparse approximation from random time-dithering with a single source approaches the recovery from simultaneous-source acquisition with multiple sources. Established findings from the field of compressive sensing indicate that the choice of the sparsifying transform that is incoherent with the compressive sampling matrix can significantly impact the reconstruction quality. Leveraging these findings, we then demonstrate that the compressive sampling matrix resulting from our proposed sampling scheme is incoherent with the curvelet transform. The combined measurement matrix exhibits better isometry properties than other transform bases such as a non-localized multidimensional Fourier transform. We illustrate our results with simulations of ‘ideal’ simultaneous-source marine acquisition, which dithers both in time and space, compared with periodic and randomized time-dithering.}, keywords = {Curvelet transform,Fourier,Marine acquisition}, optdoi = {10.1111/j.1365-2478.2012.01075.x}, url = {http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2478.2012.01075.x/abstract} } @ARTICLE{saab2008ACHAsrb, author = {Rayan Saab and Ozgur Yilmaz}, title = {Sparse Recovery by Non-Convex Optimization - Instance Optimality}, journal = {Applied and Computational Harmonic Analysis}, year = {2010}, volume = {29}, pages = {30-48}, number = {1}, month = {07/2010}, abstract = {In this note, we address the theoretical properties of $Œî_p$, a class of compressed sensing decoders that rely on $l^p$ minimization with $p {\i}n (0, 1)$ to recover estimates of sparse and compressible signals from incomplete and inaccurate measurements. In particular, we extend the results of Cand{\textquoteleft}es, Romberg and Tao [3] and Wojtaszczyk [30] regarding the decoder $Œî_1$, based on $\ell^1$ minimization, to $Œî p$ with $p {\i}n (0, 1)$. Our results are two-fold. First, we show that under certain sufficient conditions that are weaker than the analogous sufficient conditions for $Œî_1$ the decoders $Œî_p$ are robust to noise and stable in the sense that they are $(2, p)$ instance optimal. Second, we extend the results of Wojtaszczyk to show that, like $Œî_1$, the decoders $Œî_p$ are (2, 2) instance optimal in probability provided the measurement matrix is drawn from an appropriate distribution. While the extension of the results of [3] to the setting where $p {\i}n (0, 1)$ is straightforward, the extension of the instance optimality in probability result of [30] is non-trivial. In particular, we need to prove that the $LQ_1$ property, introduced in [30], and shown to hold for Gaussian matrices and matrices whose columns are drawn uniformly from the sphere, generalizes to an $LQ_p$ property for the same classes of matrices. Our proof is based on a result by Gordon and Kalton [18] about the Banach-Mazur distances of p-convex bodies to their convex hulls.}, keywords = {non-convex,Compressive Sensing}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/ACHA/2010/saab2008ACHAsrb/saab2008ACHAsrb.pdf} } @ARTICLE{wang2008GEOPbws, author = {Deli Wang and Rayan Saab and Ozgur Yilmaz and Felix J. Herrmann}, title = {Bayesian wavefield separation by transform-domain sparsity promotion}, journal = {Geophysics}, year = {2008}, volume = {73}, pages = {1-6}, number = {5}, month = {07/2008}, abstract = {Successful removal of coherent noise sources greatly determines the quality of seismic imag- ing. Ma jor advances were made in this direction, e.g., Surface-Related Multiple Elimination (SRME) and interferometric ground-roll removal. Still, moderate phase, timing, amplitude errors and clutter in the predicted signal components can be detrimental. Adopting a Bayesian approach along with the assumption of approximate curvelet-domain independence of the to-be-separated signal components, we construct an iterative algorithm that takes the predictions produced by for example SRME as input and separates these components in a robust fashion. In addition, the proposed algorithm controls the energy mismatch between the separated and predicted components. Such a control, which was lacking in earlier curvelet-domain formulations, produces improved results for primary-multiple separation on both synthetic and real data.}, keywords = {curvelet transform, SLIM, Geophysics,Processing,Optimization}, optdoi = {10.1190/1.2952571}, url = {https://www.slim.eos.ubc.ca/Publications/Public/Journals/Geophysics/2008/wang08GEObws/wang08GEObws.pdf } }