@article {ghadermarzy2013ncs,
	title = {Non-convex compressed sensing using partial support information},
	journal = {Journal of Sampling Theory in Signal and Image Processing},
	volume = {13},
	number = {3},
	year = {2014},
	pages = {249-270},
	abstract = {In this paper we address the recovery conditions of weighted $\ell_p$ minimization for signal reconstruction from compressed sensing measurements when partial support in- formation is available. We show that weighted $\ell_p$ minimization with 0 < p < 1 is stable and robust under weaker sufficient conditions compared to weighted $\ell_1$ minimization. Moreover, the sufficient recovery conditions of weighted $\ell_p$ are weaker than those of regular $\ell_p$ minimization if at least 50\% of the support estimate is accurate. We also review some algorithms which exist to solve the non-convex $\ell_p$ problem and illustrate our results with numerical experiments.},
	keywords = {compressed sensing, nonconvex optimization, sparse reconstruction, weighted $\ell_p$},
	url = {http://arxiv.org/abs/1311.3773},
	author = {Navid Ghadermarzy and Hassan Mansour and Ozgur Yilmaz}
}
@article {moghaddam2013GEOPnoa,
	title = {A new optimization approach for source-encoding full-waveform inversion},
	journal = {Geophysics},
	volume = {78},
	number = {3},
	year = {2013},
	month = {05},
	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 = {FWI, Geophysics, Optimization, source encoding},
	doi = {10.1190/GEO2012-0090.1},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2013/moghaddam2013GEOPnoa/moghaddam2013GEOPnoa.pdf},
	author = {Peyman P. Moghaddam and Henk Keers and Felix J. Herrmann and Wim A. Mulder}
}
@article {hennenfent2010GEOPnct,
	title = {Nonequispaced curvelet transform for seismic data reconstruction: a sparsity-promoting approach},
	journal = {Geophysics},
	volume = {75},
	number = {6},
	year = {2010},
	month = {12},
	pages = {WB203-WB210},
	publisher = {SEG},
	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, Processing, seismology, SLIM},
	doi = {10.1190/1.3494032},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2010/hennenfent2010GEOPnct/hennenfent2010GEOPnct.pdf},
	author = {Gilles Hennenfent and Lloyd Fenelon and Felix J. Herrmann}
}
@article {hennenfent2008GEOPnii,
	title = {New insights into one-norm solvers from the Pareto curve},
	journal = {Geophysics},
	volume = {73},
	number = {4},
	year = {2008},
	month = {07},
	pages = {A23-A26},
	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 = {Acquisition, Geophysics, Optimization, Pareto, Processing, SLIM},
	doi = {10.1190/1.2944169},
	url = {https://slim.gatech.edu/Publications/Public/Journals/Geophysics/2008/hennenfent08GEOnii/hennenfent08GEOnii.pdf},
	author = {Gilles Hennenfent and Ewout van den Berg and Michael P. Friedlander and Felix J. Herrmann}
}
@article {herrmann2008GJInps,
	title = {Non-parametric seismic data recovery with curvelet frames},
	journal = {Geophysical Journal International},
	volume = {173},
	year = {2008},
	month = {04},
	pages = {233-248},
	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 resemble 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 recovery 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 sampling 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 = {Acquisition, curvelet transform, reconstruction, SLIM},
	doi = {10.1111/j.1365-246X.2007.03698.x},
	url = {https://slim.gatech.edu/Publications/Public/Journals/GeophysicalJournalInternational/2008/herrmann2008GJInps.pdf},
	author = {Felix J. Herrmann and Gilles Hennenfent}
}
@article {herrmann2007GJInlp,
	title = {Non-linear primary-multiple separation with directional curvelet frames},
	journal = {Geophysical Journal International},
	volume = {170},
	number = {2},
	year = {2007},
	month = {08},
	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. 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 = {Processing, signal separation, SLIM},
	doi = {10.1111/j.1365-246X.2007.03360.x},
	url = {https://slim.gatech.edu/Publications/Public/Journals/GeophysicalJournalInternational/2007/herrmann2007GJInlp/herrmann2007GJInlp.pdf},
	author = {Felix J. Herrmann and Urs Boeniger and D. J. Verschuur}
}
