Applications of multigrid processing featuring deconvolution

Abstract

The efficient solution of linear systems is an important part of many routines in geophysical data processing and inversion. One option for the solution of very large systems is the multigrid method. The algorithms are fast, robust, and able to solve linear and nonlinear systems at a fraction of the computer cost of other methods. Applications of multigrid processing are common in fluid dynamics but there has been little effort to apply the technique to seismic processing and inversion. Multigrid processing uses different grid sizes when forming iterative solutions to inverse problems. The process commences with an estimate on a coarse grid that is improved with one or more iterations of inversion. The data is then interpolated to a higher resolution grid and used as the input for another iterative solution. This sequence is repeated until the desired resolution and accuracy is obtained. Two examples are used to illustrate the properties and benefits of the method. A simple 2D solution to Laplace?s equation illustrates the poor low frequency convergence of a conventional Gauss-Seidel method and the superior low frequency convergence of the multigrid method. A 1D deconvolution example also illustrates the simplicity and rapid convergence of the method.