Block-Krylov methods for Multi-Dimensional Deconvolution

We address the estimation of the impulse response at a reflector by deconvolving the up- and downgoing waves. Deconvolution in the time domain can be written as a linear system with multiple right-hand sides in the frequency domain. A straightforward way of solving these systems is by applying an iterative method like LSQR. However, these solvers are dependent on the right-hand side and for every right-hand side we have to use LSQR again. This can be a costly process. We propose to solve the linear systems using block Krylov methods. We show that these methods give comparable accuracy compared to standard Krylov methods, but at a much lower computational cost. This is due to the fact that block methods are able to exploit similarities in the data and are able to constructs solutions in a much richer subspace. We also show that it is hard to solve the MDD problem in the frequency domain alone, and that additional optimization in the time domain is most likely required.