Data Reconstruction via Sparse Bi-focal Transformation

In most reconstruction algorithms information about the subsurface can not be utilised, 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 in a prior velocity model. A sparseness constraint 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. In addition, we introduce a smart choice of focal operators, based on data-driven operator updating where we follow closely the reflection events in the data. This allows an even stronger compression of the data in the focal domain.