@conference {witte2016SEGpve,
	title = {Phase velocity error minimizing scheme for the anisotropic pure P-wave equation},
	booktitle = {SEG Technical Program Expanded Abstracts},
	year = {2016},
	note = {(SEG, Dallas)},
	month = {10},
	pages = {452-457},
	abstract = {Pure P-wave equations for acoustic modeling in transverse isotropic media are derived by approximating the exact pure Pwave dispersion relation. In this work, we present an alternative approach to the approximate dispersion relation of Etgen and Brandsberg-Dahl, in which we approximate the exact dispersion relation through a polynomial expansion and determine its coefficients by solving a linear least squares problem that minimizes the phase velocity error over the entire range of phase angles. The coefficients are also optimized over a pre-defined range of Thomsen parameters, so that the phase error is small for models with spatially varying anisotropy. Phase velocity error analysis shows that the optimized pure P-wave equation is up to one order of magnitude more accurate than other popular pure P-wave equations, even for highly non-elliptic anisotropy. The optimized equation can be easily turned into a time-domain forward modeling scheme and comparisons of the modeled waveforms with analytical travel times once more illustrate its high accuracy. We also provide an efficient implementation of our approach for 3D tilted TI media that limits the count of fast Fourier transforms per time step to a number that is comparable to other pure P-wave equations.},
	keywords = {anisotropy, least squares, Modelling, SEG, TTI},
	doi = {10.1190/segam2016-13844850.1},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/witte2016SEGpve/witte2016SEGpve.html},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/witte2016SEGpve/witte2016SEGpve_poster.pdf},
	author = {Philipp A. Witte and Christiaan C. Stolk and Felix J. Herrmann}
}
@presentation {witte2016SINBADFpve,
	title = {Phase velocity error minimizing scheme for the anisotropic pure P-wave equation},
	year = {2016},
	publisher = {SINBAD},
	abstract = {Pure P-wave equations for acoustic modeling in transverse isotropic media are derived by approximating the exact pure P-wave dispersion relation. In this work, we present an alternative to the common Taylor expansion based equations, in which we approximate the exact dispersion relation through a polynomial expansion and determine its coefficients by solving a least squares problem that minimizes the phase velocity error over the complete range of phase angles. We show that this approach is up to an order of magnitude more accurate than other pure P-wave equations at a comparable computational cost.},
	keywords = {Presentation, SINBAD, SINBADFALL2016, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/witte2016SINBADFpve/witte2016SINBADFpve.pdf},
	url2 = {https://slim.gatech.edu/Publications/Public/Conferences/SINBAD/2016/Fall/witte2016SINBADFpve/witte2016SINBADFpve.mov},
	author = {Philipp A. Witte and Christiaan C. Stolk and Felix J. Herrmann}
}
@article {herrmann2008ACHAsac,
	title = {Sparsity- and continuity-promoting seismic image recovery with curvelet frames},
	journal = {Applied and Computational Harmonic Analysis},
	volume = {24},
	number = {2},
	year = {2008},
	month = {03},
	pages = {150-173},
	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 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 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 salt model.},
	keywords = {curvelet transform, Imaging, Processing, SLIM},
	doi = {10.1016/j.acha.2007.06.007},
	url = {https://slim.gatech.edu/Publications/Public/Journals/ACHA/2008/herrmann2008ACHAsac/herrmann2008ACHAsac.pdf},
	author = {Felix J. Herrmann and Peyman P. Moghaddam and Christiaan C. Stolk}
}
@conference {moghaddam2007CSEGmar,
	title = {Migration amplitude recovery using curvelets},
	booktitle = {CSEG Annual Conference Proceedings},
	year = {2007},
	month = {05},
	publisher = {CSEG},
	organization = {CSEG},
	abstract = {In this paper, we recover the amplitude of a seismic image by approximating the normal operator and subsequently inverting it. Normal operator (migration followed by modeling) is an example of pseudo-differential. curvelets are proven to be invariant under the action of pseudo-differential operators under certain conditions. Subsequently, curvelets are forming as eigen-vectors for such an operator. We propose a seismic amplitude recovery method that employs an eigen-value decomposition for normal operator using curvelets as eigen-vectors and to be estimated eigenvalues. A post-stack reverse-time, wave-equation migration is used for evaluation of the proposed method.},
	keywords = {SLIM},
	url = {http://cseg.ca/assets/files/resources/abstracts/2007/168S0131.pdf},
	author = {Peyman P. Moghaddam and Felix J. Herrmann and Christiaan C. Stolk}
}
@conference {moghaddam2007SEGrsi,
	title = {Robust seismic-images amplitude recovery using curvelets},
	booktitle = {SEG Technical Program Expanded Abstracts},
	volume = {26},
	year = {2007},
	pages = {2225-2229},
	publisher = {SEG},
	organization = {SEG},
	abstract = {In this paper, we recover the amplitude of a seismic image by approximating the normal (demigration-migration) operator. In this approximation, we make use of the property that curvelets remain invariant under the action of the normal operator. We propose a seismic amplitude recovery method that employs an eigenvalue like decomposition for the normal operator using curvelets as eigen-vectors. Subsequently, we propose an approximate non-linear singularity-preserving solution to the least-squares seismic imaging problem with sparseness in the curvelet domain and spatial continuity constraints. Our method is tested with a reverse-time {\textquoteleft}wave-equation{\textquoteright} migration code simulating the acoustic wave equation on the SEG-AA salt model. \copyright2007 Society of Exploration Geophysicists},
	keywords = {Presentation, SEG, SLIM},
	doi = {10.1190/1.2792928},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2007/moghaddam07SEGrsi/moghaddam07SEGrsi.pdf},
	presentation = {https://slim.gatech.edu/Publications/Public/Conferences/SEG/2007/moghaddam07SEGrsi/moghaddam07SEGrsi_pres.pdf},
	author = {Peyman P. Moghaddam and Felix J. Herrmann and Christiaan C. Stolk}
}
@conference {moghaddam2007EAGEsar,
	title = {Seismic amplitude recovery with curvelets},
	booktitle = {EAGE Annual Conference Proceedings},
	year = {2007},
	month = {06},
	abstract = {A non-linear singularity-preserving solution to the least-squares seismic imaging problem with sparseness and continuity constraints is proposed. The applied formalism explores curvelets as a directional frame that, by their sparsity on the image, and their invariance under the imaging operators, allows for a stable recovery of the amplitudes. Our method is based on the estimation of the normal operator in the form of an {\textquoteright}eigenvalue{\textquoteright} decompsoition with curvelets as the {\textquoteright}eigenvectors{\textquoteright}. Subsequently, we propose an inversion method that derives from estimation of the normal operator and is formulated as a convex optimization problem. Sparsity in the curvelet domain as well as continuity along the reflectors in the image domain are promoted as part of this optimization. Our method is tested with a reverse-time {\textquoteright}wave-equation{\textquoteright} migration code simulating the acoustic wave equation.},
	keywords = {EAGE, SLIM},
	url = {https://slim.gatech.edu/Publications/Public/Conferences/EAGE/2007/moghaddam07EAGEsar/moghaddam07EAGEsar.pdf},
	url2 = {http://earthdoc.eage.org/publication/publicationdetails/?publication=6935},
	author = {Peyman P. Moghaddam and Felix J. Herrmann and Christiaan C. Stolk}
}
@conference {moghaddam2007CSEGsac,
	title = {Sparsity and continuity enhancing seismic imaging},
	booktitle = {CSEG Annual Conference Proceedings},
	year = {2007},
	month = {05},
	publisher = {CSEG},
	organization = {CSEG},
	abstract = {A non-linear singularity-preserving solution to the least-squares seismic imaging problem with sparseness and continuity constraints is proposed. The applied formalism explores curvelets as a directional frame that, by their sparsity on the image, and their invariance under the imaging operators, allows for a stable recovery of the amplitudes. Our method is based on the estimation of the normal operator in the form of an {\textquoteright}eigenvalue{\textquoteright} decompsoition with curvelets as the eigenvectors{\textquoteright}. Subsequently, we propose an inversion method that derives from estimation of the normal operator and is formulated as a convex optimization problem. Sparsity in the curvelet domain as well as continuity along the reflectors in the image domain are promoted as part of this optimization. Our method is tested with a reverse-time {\textquoteright}wave-equation{\textquoteright} migration code simulating the acoustic wave equation.},
	keywords = {SLIM},
	url = {http://cseg.ca/assets/files/resources/abstracts/2007/091S0130.pdf},
	author = {Peyman P. Moghaddam and Felix J. Herrmann and Christiaan C. Stolk}
}
