Data Reconstruction via Sparse Double Focal Transformation: An Overview

In measurements for seismic exploration, the sampling of sources and receivers is usually not adequate to perform subsequent processing and imaging algorithms. Therefore, reconstruction of the seismic data to obtain aliasing-free, dense, and regularly sampled data is an important preprocessing step. In most reconstruction algorithms, information about the subsurface can not be utilized, even if such is available. Focal transformation is a way to effectively incorporate prior knowledge of the subsurface in seismic data reconstruction. The basis functions of this transformation are the focal operators. They can be understood as one-way propagation operators from certain effective depth levels to the measurement surface in a prior (approximate) velocity model. A sparseness constraint in the focal domain is used to penalize aliasing noise. By using several depth levels simultaneously, the data can be described with less parameters in the transform domain. This results in a better signal to noise separation and, therefore, improved reconstruction. The principles are described and some illustrations on synthetic seismic data demonstrate the virtues of the approach.