Wavefield reconstruction is a crucial step in the seismic processing flow. For instance, unsuccessful interpolation leads to erroneous multiple predictions that adversely affect the performance of multiple elimination, and to imaging artifacts. We present a new non-parametric transform-based reconstruction method that exploits the compression of seismic data b the recently developed curvelet transform. The elements of this transform, called curvelets, are multi-dimensional, multi-scale, and multi-directional. They 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 (CRSI). 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 reconstruction problem, namely a sparsifying transform, a sub-Nyquist sampling strategy that subdues coherent aliases in the sparsifying domain, and a data-consistent sparsity-promoting program. After a brief overview of the curvelet transform and our seismic-oriented extension to the fast discrete curvelet transform, we detail the CRSI formulation and illustrate its performance on synthetic and read datasets. Then, we introduce a sub-Nyquist sampling scheme, termed jittered undersampling, and show that, for the same amount of data acquired, jittered data are best interpolated using CRSI compared to regular or random undersampled data. We also discuss the large-scale one-norm solver involved in CRSI. Finally, we extend CRSI formulation to other geophysical applications and present results on multiple removal and migration-amplitude recovery.

Wavefield reconstruction is a crucial step in the seismic processing flow. For instance, unsuccessful interpolation leads to erroneous multiple predictions that adversely affect the performance of multiple elimination, and to imaging artifacts. We present a new non-parametric transform-based reconstruction method that exploits the compression of seismic data b the recently developed curvelet transform. The elements of this transform, called curvelets, are multi-dimensional, multi-scale, and multi-directional. They 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 (CRSI). 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 reconstruction problem, namely a sparsifying transform, a sub-Nyquist sampling strategy that subdues coherent aliases in the sparsifying domain, and a data-consistent sparsity-promoting program. After a brief overview of the curvelet transform and our seismic-oriented extension to the fast discrete curvelet transform, we detail the CRSI formulation and illustrate its performance on synthetic and read datasets. Then, we introduce a sub-Nyquist sampling scheme, termed jittered undersampling, and show that, for the same amount of data acquired, jittered data are best interpolated using CRSI compared to regular or random undersampled data. We also discuss the large-scale one-norm solver involved in CRSI. Finally, we extend CRSI formulation to other geophysical applications and present results on multiple removal and migration-amplitude recovery.

Wavefield reconstruction is a crucial step in the seismic processing flow. For instance, unsuccessful interpolation leads to erroneous multiple predictions that adversely affect the performance of multiple elimination, and to imaging artifacts. We present a new non-parametric transform-based reconstruction method that exploits the compression of seismic data b the recently developed curvelet transform. The elements of this transform, called curvelets, are multi-dimensional, multi-scale, and multi-directional. They 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 (CRSI). 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 reconstruction problem, namely a sparsifying transform, a sub-Nyquist sampling strategy that subdues coherent aliases in the sparsifying domain, and a data-consistent sparsity-promoting program. After a brief overview of the curvelet transform and our seismic-oriented extension to the fast discrete curvelet transform, we detail the CRSI formulation and illustrate its performance on synthetic and read datasets. Then, we introduce a sub-Nyquist sampling scheme, termed jittered undersampling, and show that, for the same amount of data acquired, jittered data are best interpolated using CRSI compared to regular or random undersampled data. We also discuss the large-scale one-norm solver involved in CRSI. Finally, we extend CRSI formulation to other geophysical applications and present results on multiple removal and migration-amplitude recovery. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

Predictive multiple suppression methods consist of two main steps: a prediction step, in which multiples are predicted from the seismic data, and a subtraction step, in which the predicted multiples are matched with the true multiples in the data. The last step appears crucial in practice: an incorrect adaptive subtraction method will cause multiples to be sub-optimally subtracted or primaries being distorted, or both. Therefore, we propose a new domain for separation of primaries and multiples via the Curvelet transform. This transform maps the data into almost orthogonal localized events with a directional and spatialtemporal component. The multiples are suppressed by thresholding the input data at those Curvelet components where the predicted multiples have large amplitudes. In this way the more traditional filtering of predicted multiples to fit the input data is avoided. An initial field data example shows a considerable improvement in multiple suppression.

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.