article.bib

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@ARTICLE{BergFriedlander:2008,
  author = {E. 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},
  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.},
  doi = {10.1137/080714488},
  file = {890:http\://link.aip.org/link/?SCE/31/890:PDF},
  keywords = {basis pursuit, convex program, duality, Newton{\textquoteright}s method,
	one-norm regularization, projected gradient, root-finding, sparse
	solutions},
  publisher = {SIAM}
}

@ARTICLE{vandenberg08gsv,
  author = {E. van den Berg and Mark Schmidt and Michael P. Friedlander and K.
	Murphy},
  title = {Group sparsity via linear-time projection},
  year = {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.},
  keywords = {SLIM},
  pdf = {http://www.optimization-online.org/DB_FILE/2008/07/2056.pdf},
  url = {http://www.optimization-online.org/DB_FILE/2008/07/2056.pdf}
}

@ARTICLE{bernabe04pas,
  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},
  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.},
  doi = {10.1029/2004JB00311},
  keywords = {permeability, porosity, SLIM},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/bernabe04pas.pdf}
}

@ARTICLE{Erlangga07oam,
  author = {Yogi A. Erlangga and R. Nabben},
  title = {On multilevel projection Krylov method for the Helmholtz equation
	preconditioned by shifted Laplacian},
  journal = {Elec. Trans. Numer. Anal.},
  year = {2007},
  volume = {submitted},
  abstract = {In [Erlangga and Nabben, SIAM J. Sci. Comput. (2007), to appear],
	a multilevel Krylov method is proposed to solve linear systems with
	symmetric and nonsymmetric matrix of coefficients. This multilevel
	method is developed based on shifting (or pro jecting) some small
	eigen- values to the largest eigenvalue, leading to a more favorable
	spectrum for convergence acceleration of a Krylov subspace method.
	Such a pro jection is insensitive with respect to the approximation
	of the small eigenvalues to be pro jected, which for a particular
	choice of deflation subspaces is equivalent to solving a coarse-grid
	problem analogue to multigrid. Different from multigrid, in the multilevel
	Krylov method, however, the coarse-grid problem is solved by a Krylov
	method, whose convergence rate is further accelerated by applying
	pro jection to the coarse-grid system. A recursive application of
	pro jection and coarse-grid solve by a Krylov iterative method then
	leads to the multilevel Krylov method. The method has been successfully
	applied to 2D convection-diffusion problems for which a standard
	multigrid method fails to converge. In this paper, we extend this
	multilevel Krylov method to indefinite linear systems arising from
	a discretization of the Helmholtz equation, preconditioned by shifted
	Laplacian as introduced by [Erlangga, Oosterlee and Vuik, SIAM J.
	Sci. Comput. 27(2006), pp. 1471{\textendash}1492]. Since in this
	case pro jection must be applied to the preconditioned system AM
	- 1 , the coarse-grid matrices are approximated by a product of some
	low dimension matrices associated with A and M . Within the Krylov
	iteration and pro jection step in each coarse-grid solve, a multigrid
	iteration is used to approximately invert the preconditioner. Hence,
	a multigrid-multilevel Krylov method results. Numerical results are
	given for high wavenumbers and show the effectiveness of the method
	for solving Helmholtz problems. Not only can the convergence be made
	almost independent of grid size h, but also only mildly independent
	of wavenumber k},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/erlangga08oam.pdf}
}

@ARTICLE{fomel07rce,
  author = {S. Fomel and Gilles Hennenfent},
  title = {Reproducible computational experiments using {SC}ons},
  journal = {IEEE International Conference on Acoustics, Speech and Signal Processing
	(ICASSP)},
  year = {2007},
  volume = {4},
  pages = {IV-1257-IV-1260},
  month = {April},
  abstract = {SCons (from software construction) is a well-known open-source program
	designed primarily for building software. In this paper, we describe
	our method of extending SCons for managing data processing flows
	and reproducible computational experiments. We demonstrate our usage
	of SCons with a simple example.},
  bdsk-url-1 = {http://dx.doi.org/10.1109/ICASSP.2007.367305},
  bdsk-url-2 = {http://lcav.epfl.ch/reproducible_research/ICASSP07/FomelH07.pdf},
  bdsk-url-3 = {http://www.ee.columbia.edu/~dpwe/LabROSA/proceeds/icassp/2007/pdfs/0401257.pdf},
  date-added = {2008-05-22 11:42:36 -0700},
  date-modified = {2008-08-14 13:52:16 -0700},
  doi = {10.1109/ICASSP.2007.367305},
  issn = {1520-6149},
  keywords = {SLIM},
  pdf = {http://lcav.epfl.ch/reproducible_research/ICASSP07/FomelH07.pdf},
  url = {http://dx.doi.org/10.1109/ICASSP.2007.367305}
}

@ARTICLE{friedlander07dtd,
  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},
  doi = {10.1214/009053607000000479},
  keywords = {dantzig, SLIM, statistics},
  url = {http://www.cs.ubc.ca/~mpf/downloads/FriedlanderSaunders08.pdf}
}

@ARTICLE{friedlander07ero,
  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},
  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.},
  doi = {10.1137/060675320},
  keywords = {SLIM},
  url = {http://www.cs.ubc.ca/~mpf/index.php?q=cpreg.pdf}
}

@ARTICLE{hennenfent08nii,
  author = {Gilles Hennenfent and E. 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 = {July-August},
  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},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/hennenfent07pareto/paper_html}
}

@ARTICLE{hennenfent10nct,
  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},
  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.},
  doi = {10.1190/1.3494032},
  keywords = {curvelet transforms, data acquisition, geophysical techniques, seismology},
  publisher = {SEG},
  url = {http://link.aip.org/link/?GPY/75/WB203/1}
}

@ARTICLE{hennenfent:WB203,
  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},
  abstract = {Seismic data are typically irregularly sampled along spatial axes.
	This irregular sampling may adversely a?ect some key steps, e.g.,
	multiple prediction/attenuation or imaging, in the processing workßow.
	To overcome this problem almost every large dataset is regularized
	and/or interpolated. Our contribution is twofold. Firstly, we extend
	our earlier work on the nonequispaced fast discrete curvelet transform
	(NFDCT) and introduce a second generation of the transform. This
	new generation di?ers from the previous one by the approach taken
	to compute accurate curvelet coe?cients from irregularly sampled
	data. The Þrst generation relies on accurate Fourier coe?cients obtained
	by an ?2 -regularized inversion of the nonequispaced fast Fourier
	transform, while the second is based on a direct, ?1 -regularized
	inversion of the operator that links curvelet coe?cients to irregular
	data. Also, by construction, the NFDCT second generation is lossless,
	unlike the NFDCT Þrst generation. This property is particularly attractive
	for processing irregularly sampled seismic data in the curvelet domain
	and bringing them back to their irregular recording locations with
	high Þdelity. Secondly, we combine the NFDCT second generation 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 is, in our example, 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 no obvious acquisition
	imprint; a result very competitive with results from other reconstruction
	methods overall.},
  doi = {10.1190/1.3494032},
  keywords = {curvelet transforms; data acquisition; geophysical techniques; seismology},
  publisher = {SEG},
  url = {http://link.aip.org/link/?GPY/75/WB203/1}
}

@ARTICLE{hennenfent08sdw,
  author = {Gilles Hennenfent and Felix J. Herrmann},
  title = {Simply denoise: wavefield reconstruction via jittered undersampling},
  journal = {Geophysics},
  year = {2008},
  volume = {73},
  number = {3},
  month = {May-June},
  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.},
  doi = {10.1190/1.2841038},
  keywords = {sampling, SLIM},
  publisher = {SEG},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/hennenfent07jitter/paper_html}
}

@ARTICLE{hennenfent06sdw,
  author = {Gilles Hennenfent and Felix J. Herrmann},
  title = {Seismic denoising with non-uniformly sampled curvelets},
  journal = {Computing in Science and Engineering},
  year = {2006},
  volume = {8},
  number = {3},
  month = {May-June},
  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.},
  doi = {10.1109/MCSE.2006.49},
  keywords = {curvelet transform, SLIM},
  publisher = {IEEE},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/hennenfent06cdw.pdf}
}

@ARTICLE{herrmann10rsg,
  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},
  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.},
  doi = {10.1190/1.3506147},
  keywords = {data acquisition, geophysical techniques, Nyquist criterion, sampling
	methods, seismology},
  publisher = {SEG},
  url = {http://link.aip.org/link/?GPY/75/WB173/1}
}

@ARTICLE{herrmann05sdb,
  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 = {January},
  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},
  publisher = {IOS Press},
  url = {http://slim.eos.ubc.ca/~felix/public/RobinsonSub.pdf}
}

@ARTICLE{Herrmann2005Spec,
  author = {Felix J. Herrmann},
  title = {Seismic deconvolution by atomic decomposition: a parametric approach
	with sparseness constraints},
  journal = {Integr. Computer-Aided Eng.},
  year = {2005},
  volume = {12},
  pages = {69-91},
  number = {1},
  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.},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/herrmann05sdb.pdf}
}

@ARTICLE{herrmann04ssa,
  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},
  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.},
  doi = {10.1111/j.1365-246X.2004.02464.x},
  keywords = {percolation, SLIM},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/herrmann04ssa.pdf}
}

@ARTICLE{herrmann07nlp,
  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},
  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.},
  doi = {10.1111/j.1365-246X.2007.03360.x},
  keywords = {signal separation, SLIM},
  url = {http://onlinelibrary.wiley.com/doi/10.1111/j.1365-246X.2007.03360.x/abstract;jsessionid=956C674A489BB3A13D2D9D7F87D68FE7.d02t03}
}

@ARTICLE{HerrmannM2007,
  author = {Felix J. Herrmann and U. Boeniger and D. J. Verschuur},
  title = {Nonlinear primary-multiple separation with directional curvelet frames},
  year = {2007},
  volume = {170},
  pages = {781-799},
  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. The last 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. [...] 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.},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/herrmann07npm.pdf}
}

@ARTICLE{herrmann09cbm,
  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},
  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},
  url = {http://slim.eos.ubc.ca/Publications/Public/TechReports/herrmann08cmp-r.pdf}
}

@ARTICLE{herrmann09csf,
  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},
  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},
  url = {http://slim.eos.ubc.ca/Publications/Public/TechReports/herrmann08csf-r.pdf}
}

@ARTICLE{herrmann08nps,
  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 = {April},
  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.},
  doi = {10.1111/j.1365-246X.2007.03698.x},
  keywords = {curvelet transform, reconstruction, SLIM},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/herrmann07nps.pdf}
}

@ARTICLE{herrmann08sac,
  author = {Felix J. Herrmann and Peyman P. Moghaddam and C. C. 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 = {March},
  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.},
  doi = {10.1016/j.acha.2007.06.007},
  keywords = {curvelet transform, imaging, SLIM}
}

@ARTICLE{HerrmannACHA07,
  author = {Felix J. Herrmann and Peyman P. Moghaddam and C. C. Stolk},
  title = {Sparsity- and continuity-promoting seismic image recovery with curvelet
	frames},
  year = {2007},
  note = {Accepted for publication in the Journal of Applied and Computational
	Harmonic Analysis},
  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 expansion, 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 application
	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 operator. 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 $\ell_1$-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$^\prime$ salt model.},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/HerrmannACHA07.pdf}
}

@ARTICLE{herrmann08cbs,
  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},
  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},
  doi = {10.1190/1.2799517},
  keywords = {curvelet transform, SLIM},
  publisher = {SEG}
}

@ARTICLE{Herrmann2007Let,
  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 = {2007},
  volume = {73},
  pages = {A1-A5},
  number = {1},
  abstract = {I In this letter, the solutions to three seismic processing problems
	are presented that exploit the multiscale and multi-angular properties
	of the curvelet transform. Data regularization, multiple removal,
	and restoration of migration amplitudes are all formulated in terms
	of a sparsity promoting program that employs the high degree of sparsity
	attained by curvelets on seismic data and images. For each problem
	the same nonlinear program is solved, simultaneously minimizing the
	data misfit and the one norm ($\ell_1$) on the desired curvelet-domain
	solution. Parsimony of curvelets on seismic wavefields and images
	of the sedimentary crust with wavefront-like features underlies the
	successful solution of these problems and is a clear indication of
	the broad applicability of this transform in exploration seismology.},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/herrmann08cbs.pdf}
}

@ARTICLE{herrmann08acd,
  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},
  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},
  doi = {10.1190/1.2904986},
  keywords = {SLIM},
  publisher = {SEG},
  url = {http://slim.eos.ubc.ca/Publications/Private/Journals/herrmann08match/paper_html/paper.html}
}

@ARTICLE{VanLeeuwen2010,
  author = {Tristan van Leeuwen and Felix J. Herrmann},
  title = {Seismic waveform inversion by stochastic optimization},
  journal = {International Journal of Geophysics},
  year = {2011},
  volume = {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. A batch in the current setting represents a single
	random superposition of sources. The ultimate goal of such an approach
	is to dramatically reduce the number of shots that need to be modeled.
	Assuming that the computational costs grow linearly with the number
	of shots, this promises a significant speed-up. We introduce the
	stochasticity in the waveform inversion problem in a rigorous way
	via a technique called \emph{randomized trace estimation} and review
	some theoretical results that underlie recent developments in the
	use of stochastic methods for waveform inversion. We present some
	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. This leads to an order of magnitude speed up for waveform
	inversion.},
  url = {http://www.hindawi.com/journals/ijgp/2011/689041/cta/}
}

@ARTICLE{lin07cwe,
  author = {Tim T.Y. Lin and Felix J. Herrmann},
  title = {Compressed wavefield extrapolation},
  journal = {Geophysics},
  year = {2007},
  volume = {72},
  pages = {SM77-SM93},
  number = {5},
  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},
  doi = {10.1190/1.2750716},
  keywords = {SLIM, wave propagation},
  publisher = {SEG},
  url = {http://slim.eos.ubc.ca/Publications/Public/Journals/lin07ce.pdf}
}

@ARTICLE{saab08srb,
  author = {Rayan Saab and Ozgur Yilmaz},
  title = {Sparse Recovery by Non-Convex Optimization - Instance Optimality},
  journal = {Applied and Computational Harmonic Analysis},
  year = {2009},
  abstract = {In this note, we address the theoretical properties of $Î_p$, 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 $Î_1$, based on $\ell^1$
	minimization, to $Î p$ 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 $Î_1$ the decoders
	$Î_p$ 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 $Î_1$, the decoders $Î_p$ 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},
  url = {http://dx.doi.org/10.1016/j.acha.2009.08.002}
}

@ARTICLE{wang08bws,
  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},
  number = {5},
  month = {September-Octobe},
  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.},
  doi = {10.1190/1.2952571},
  keywords = {curvelet transform, SLIM},
  url = {http://slim.eos.ubc.ca/Publications/Private/Journals/wang08bayes/paper_html/paper.html}
}