Pseudospectral picture of the seismic algorithm for Surface Multiple Attenuation

Given a two dimensional signal, e.g. surface recorded 2-D wavefield representing marine seismic data, we consider the task of removing interference due to multiple reflection (the so-called multiple events). This problem has been successfully modeled in the previous work by the third author as a matrix optimization problem. Seen from an abstract perspective, this optimization problem is related to the mathematical notion of the pseudo-spectrum of a matrix. Pseudospectrum is sometimes also called the spectral portrait of a matrix and it presents a powerful visualization tool for analyzing the spectral properties of a nonhermitian matrix. Furthermore, there are several freely available visualization tools to plot the pseudospectrum, most notably Tom Wright's eigtool. The original mathematical problem can be seen as a quest for a minimum of the pseudo-spectrum for a certain nonhermitian matrix pair in the Frobenius matrix norm. The main contribution in the original algorithm was the use of matrix Eigenvalue Decomposition in the preprocessing stage of the algorithm to reduce the computational cost of the inner optimization loop to O(n
²
). Here n is the dimension of the matrix (e.g. 2-D signal). Motivated by the work of Trefethen and Embree we present several alternatives to this algorithm which also result with inner loop procedures having O(n
²
) cost. Such algorithms are based on a Lanczos type singular value algorithms and solve the optimization problem with regard to the spectral norm. We further discuss issues related to the numerical stability of the complete algorithm with regard to the choice of the used matrix factorization in the preprocessing module. Notably, we consider the cost and the numerical stability for the use of Schur, Hessenberg and Eigenvalue matrix decompositions in this context. Let us close by noting that the structure of the computational problem is such that it falls into the class of “embarrassingly parallel” algorithms. This will also be exploited in the presented work.