Gilles Hennenfent

Simply denoise: Wavefield reconstruction via jittered undersampling

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...

Seismic denoising with nonuniformly sampled curvelets

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

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

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 curvelets ability to sparsify wavefrontlike features is powerful, and our results are a clear indication of the broad applicability of this transform to exploration seismology.

Nonequispaced curvelet transform for seismic data reconstruction: A sparsity-promoting approach

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 noneq...

New insights into one-norm solvers from the Pareto curve

Geophysical inverse problems typically involve a trade-off between data misfit and some prior model. Pareto curves trace the optimal trade-off between these two competing aims. These curves are used commonly in problems with two-norm priors in which they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one-norm prior, Pareto curves remain relatively unexplored. We show how these curves lead to new insights into one-norm regularization. First, we confirm 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 toward the solution.

Reproducible Computational Experiments using Scons

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.

Random Sampling: New Insights Into the Reconstruction of Coarsely-sampled Wavefields

In this paper, we turn the interpolation problem of coarsely-sampled data into a denoising problem. From this point of view, we illustrate the benefit of random sampling at sub-Nyquist rate over regular sampling at the same rate. We show that, using nonlinear sparsitypromoting optimization, coarse random sampling may actually lead to significantly better wavefield reconstruction than equivalent regularly sampled data.

Sparseness-constrained data continuation with frames: Applications to missing traces and aliased signals in 2/3-D

We present a robust iterative sparseness-constrained interpolation algorithm using 2-/3-D curvelet frames and Fourier-like transforms that exploits continuity along reflectors in seismic data. By choosing generic transforms, we circumvent the necessity to make parametric assumptions (e.g. through linear/parabolic Radon or demigration) regarding the shape of events in seismic data. Simulation and real data examples for data with moderately sized gaps demonstrate that our algorithm provides interpolated traces that accurately reproduce the wavelet shape as well as the AVO behavior. Our method also shows good results for de-aliasing judged by the behavior of the (f − k)-spectrum before and after regularization.

Application of Stable Signal Recovery to Seismic Data Interpolation

We propose a method for seismic data interpolation based on 1) the reformulation of the problem as a stable signal recovery problem and 2) the fact that seismic data is sparsely represented by curvelets. This method does not require information on the seismic velocities. Most importantly, this formulation potentially leads to an explicit recovery condition. We also propose a large-scale problem solver for the `1regularization minimization involved in the recovery and successfully illustrate the performance of our algorithm on 2D synthetic and real examples.

Algorithm 890: Sparco: A Testing Framework for Sparse Reconstruction

Sparco is a framework for testing and benchmarking algorithms for sparse reconstruction. It includes a large collection of sparse reconstruction problems drawn from the imaging, compressed sensing, and geophysics literature. Sparco is also a framework for implementing new test problems and can be used as a tool for reproducible research. Sparco is implemented entirely in Matlab, and is released as open-source software under the GNU Public License.