/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

“Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks”の学習報告

Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.

/pdf/an-overview-of-full-waveform-inversion-in-exploration-59sei7fzvh.pdf

An overview of full-waveform inversion in exploration geophysics

This paper describes two digital implementations of a new mathematical transform, namely, the second generation curvelet transform in two and three dimensions. The first digital transformation is based on unequally spaced fast Fourier transforms, while the second is based on the wrapping of specially selected Fourier samples. The two implementations essentially differ by the choice of spatial grid used to translate curvelets at each scale and angle. Both digital transformations return a table of digital curvelet coefficients indexed by a scale parameter, an orientation parameter, and a spatial location parameter. And both implementations are fast in the sense that they run in O(n^2 log n) flops for n by n Cartesian arrays; in addition, they are also invertible, with rapid inversion algorithms of about the same complexity. Our digital transformations improve upon earlier implementations—based upon the first generation of curvelets—in the sense that they are conceptually simpler, faster, and far less redundant. The software CurveLab, which implements both transforms presented in this paper, is available at http://www.curvelet.org.

/pdf/fast-discrete-curvelet-transforms-1z0gmpoaao.pdf

Fast Discrete Curvelet Transforms

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

SUMMARY 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 artefacts 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 wave fronts 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 ‘compressed sensing’ 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 favourable 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 by virtue of the sparsifying property of curvelets. Our results are applicable to other areas such as global seismology.

/pdf/non-parametric-seismic-data-recovery-with-curvelet-frames-3bf5izdudw.pdf

Non-parametric seismic data recovery with curvelet frames

Wave-equation based inversions, such as full-waveform inversion, are challenging because of their computational costs, memory requirements, and reliance on accurate initial models. To confront these issues, we propose a novel formulation of full-waveform inversion based on a penalty method. In this formulation, the objective function consists of a data-misfit term and a penalty term which measures how accurately the wavefields satisfy the wave-equation. Because we carry out the inversion over a larger search space, including both the model and synthetic wavefields, our approach suffers less from local minima. Our main contribution is the development of an efficient optimization scheme that avoids having to store and update the wavefields by explicit elimination. Compared to existing optimization strategies for full-waveform inversion, our method differers in two main aspects; i) The wavefields are solved from an augmented wave-equation, where the solution is forced to solve the wave-equation and fit the observed data, ii) no adjoint wavefields are required to update the model, which leads to significant computational savings. We demonstrate the validity of our approach by carefully selected examples and discuss possible extensions and future research.

Mitigating local minima in full-waveform inversion by expanding the search space

We present a new, discrete undersampling scheme designed to favor wavefield reconstruction by sparsity-promoting inversion with transform elements localized in the Fourier domain. The work is motivated by empirical observations in the seismic community, corroborated by results from compressive sampling, that indicate favorable (wavefield) reconstructions from random rather than regular undersampling. Indeed, 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. Thus, we introduce a sampling scheme, termed jittered undersampling, that shares the benefits of random sampling and controls the maximum gap size. The contribution of jittered sub-Nyquist sampling is key in formu-lating a versatile wavefi...

Simply denoise: Wavefield reconstruction via jittered undersampling

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

Seismic denoising with nonuniformly sampled curvelets

Mitigating missing data, multiples, and erroneous migration amplitudes are key factors that determine image quality. Curvelets, little “plane waves,” 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'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.

Felix J. Herrmann

Papers

Non-parametric seismic data recovery with curvelet frames

Mitigating local minima in full-waveform inversion by expanding the search space

Simply denoise: Wavefield reconstruction via jittered undersampling

Seismic denoising with nonuniformly sampled curvelets

Curvelet-based seismic data processing : A multiscale and nonlinear approach