Sparse Linear Algebra and Geophysical Migration

The pre-stack depth migration of reflection seismic data can be expressed, with the framework of waveform inversion, as a linear least squares problem. While defining this operator precisely, additional main characteristics of the forward model, like its huge size, its sparsity and the composition with convolution are detailed. It ends up with a so-called discrete ill-posed problem, whose acceptable solutions have to undergo a regularization procedure. Direct and iterative methods have been implemented with specific attention to the convolution, and then applied on the same data set: a synthetic bidimensional profile of sensible dimensions with some added noise. The efficiency with regard to computational effort, storage requirements and regularizing effect is assessed. From the standpoint of the global inverse problem, the extra feature of providing a solution that can be differentiated with respect to a parameter such as background velocity is also discussed