In this paper, we present a nonlinear curvelet-based sparsity-promoting formulation for three problems related to seismic noise, namely the ’good’, corresponding to noise generated by random sampling; the ’bad’, corresponding to coherent noise for which (inaccurate) predictions exist and the ’ugly’ for which no predictions exist. We will show that the compressive capabilities of curvelets on seismic data and images can be used to tackle these three categories of noise-related problems.

The surface-related multiple elimination (SRME) method has proven to be successful on a large number of data cases. Most of the applications are still 2D, as the full 3D implementation is still expensive and under development. However, the earth is a 3D medium, such that 3D effects are difficult to avoid. Most of the 3D effects come from diffractive structures, whereas the specular reflections normally have less of a 3D behavior. By separating the seismic data in a specular reflecting and a diffractive part, multiple prediction can be carried out with these different subsets of the input data, resulting in several categories of predicted multiples. Because each category of predicted multiples can be subtracted from the input data with different adaptation filters, a more flexible SRME procedure is obtained. Based on some initial results from a Gulf of Mexico dataset, the potential of this approach is investigated.

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Incomplete data represents a major challenge for a successful prediction and subsequent removal of multiples. In this paper, a new method will be represented that tackles this challenge in a two-step approach. During the first step, the recenly developed curvelet-based recovery by sparsity-promoting inversion (CRSI) is applied to the data, followed by a prediction of the primaries. During the second high-resolution step, the estimated primaries are used to improve the frequency content of the recovered data by combining the focal transform, defined in terms of the estimated primaries, with the curvelet transform. This focused curvelet transform leads to an improved recovery, which can subsequently be used as input for a second stage of multiple prediction and primary-multiple separation.

n this paper, two different applications of phase transitions to exploration seismology will be discussed. The first application concerns a phase diagram ruling the recovery conditions for seismic data volumes from incomplete and noisy data while the second phase transition describes the behavior of bi-compositional mixtures as a function of the volume fraction. In both cases, the phase transitions are the result of randomness in large system of equations in combination with nonlinearity. The seismic recovery problem from incomplete data involves the inversion of a rectangular matrix. Recent results from the field of "compressive sensing" provide the conditions for a successful recovery of functions that are sparse in some basis (wavelet) or frame (curvelet) representation, by means of a sparsity ($\ell_1$-norm) promoting nonlinear program. The conditions for a successful recovery depend on a certain randomness of the matrix and on two parameters that express the matrix' aspect ratio and the ratio of the number of nonzero entries in the coefficient vector for the sparse signal representation over the number of measurements. It appears that the ensemble average for the success rate for the recovery of the sparse transformed data vector by a nonlinear sparsity promoting program, can be described by a phase transition, demarcating the regions for the two ratios for which recovery of the sparse entries is likely to be successful or likely to fail. Consistent with other phase transition phenomena, the larger the system the sharper the transition. The randomness in this example is related to the construction of the matrix, which for the recovery of spike trains corresponds to the randomly restricted Fourier matrix. It is shown, that these ideas can be extended to the curvelet recovery by sparsity-promoting inversion (CRSI) . The second application of phase transitions in exploration seismology concerns the upscaling problem. To counter the intrinsic smoothing of singularities by conventional equivalent medium upscaling theory, a percolation-based nonlinear switch model is proposed. In this model, the transport properties of bi-compositional mixture models for rocks undergo a sudden change in the macroscopic transport properties as soon as the volume fraction of the stronger material reaches a critical point. At this critical point, the stronger material forms a connected cluster, which leads to the creation of a cusp-like singularity in the elastic moduli, which in turn give rise to specular reflections. In this model, the reflectivity is no longer explicitly due to singularities in the rocks composition. Instead, singularities are created whenever the volume fraction exceeds the critical point. We will show that this concept can be used for a singularity-preserved lithological upscaling.

L'inversion des formes d'ondes (FWI) est une procédure d'imagerie sismique pour imager le sous-sol de la Terre. FWI est résolue comme un problème d'optimisation. En fonction du contenu en fréquence des données, la fonction objective de FWI peut être fortement non linéaire. Pour des données associées des réflexions, ce problème empêche notamment les méthodes basées sur le gradient de retrouver les grandes longueurs d'onde du modèle de vitesse. Dans cette thèse, nous proposons une variante de FWI basée sur la séparation des champs d'ondes, typiquement en champs montants et descendants, pour atténuer la non-linéarité du problème. Il consiste à décomposer le gradient de FWI en une partie de courte longueur d'onde et une partie de grande longueur d'onde après décomposition des champs d'ondes. L'inversion est effectuée d'une manière alternée entre ces deux parties. Nous appliquons cette méthode à plusieurs études de cas et montrons que la nouvelle approche est plus robuste en particulier pour la construction du modèle de grande longueur d'onde. / Full Waveform Inversion (FWI) is a seismic imaging procedure to image the subsurface of the Earth. FWI is resolved as an optimization problem . Depending on the frequency content of the data, the objective function of FWI may be highly nonlinear. If a data set mainly contains reflections, this problem particularly prevents the gradient-based methods from recovering the long wavelengths of the velocity model.In this thesis, I propose a variant of FWI based on the wavefield separation, typically between up- and down- going waves, to mitigate the nonlinearity of the problem. The new method consists of decomposing the gradient of FWI into a short-wavelength part and a long-wavelength part after wavefield decomposition. The inversion is performed in an alternating fashion between these two parts. We apply this method to several case studies and show that the new method is more robust especially for constructing the long-wavelength model.

In this paper, the property of Curvelet transforms for preconditioning the migration and normal operators is investigated. These operators belong to the class of Fourier integral operators and pseudo-differential operators, respectively. The effect of this preconditioner is shown in term of improvement of sparsity, convergence rate, number of iteration for the Krylov-subspace solver and clustering of singular(eigen) values. The migration operator, which we employed in this work is the common-offset Kirchoff-Born migration.

We propose an efficient iterative data interpolation method using continuity along reflectors in seismic images via curvelet and discrete cosine transforms. The curvelet transform is a new multiscale transform that provides sparse representations for images that comprise smooth objects separated by piece-wise smooth discontinuities (e.g. seismic images). The advantage of using curvelets is that these frames are sparse for high-frequency caustic-free solutions of the wave-equation. Since we are dealing with less than ideal data (e.g. bandwidth-limited), we compliment the curvelet frames with the discrete cosine transform. The latter is motivated by the successful data continuation with the discrete Fourier transform. By choosing generic basis functions we circumvent the necessity to make parametric assumptions (e.g. through linear/parabolic Radon or demigration) regarding the shape of events in seismic data. Synthetic and real data examples demonstrate that our algorithm provides interpolated traces that accurately reproduce the wavelet shape as well as the AVO behavior along events in shot gathers.

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In this abstract, we present a nonlinear curvelet-based sparsity promoting formulation of a seismic processing flow, consisting of the following steps: seismic data regularization and the restoration of migration amplitudes. We show that the curvelet’s wavefront detection capability and invariance under the migration-demigration operator lead to a formulation that is stable under noise and missing data.