Sampling and reconstruction of seismic wavefields in the curvelet domain

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 ofEarth, Ocean and Atmospheric Sciences, Department ofGraduat