Combining the multigrid and gradient methods to solve the seismic inversion problem

Abstract

Gradient methods are usually used to solve the seismic inversion problem. They converge rapidly for the high frequency components of the Earth model and slowly for the low frequency components. These iterative methods only ensure a solution corresponding to the local minimum closest to the initial starting point of the algorithm. Thus, they are very dependant on the initial guess for seismic inversion due to the high nonlinearity of the cost function for this problem. We propose to treat simultaneously high and low frequency components of the seismic inversion problem by combining a variation of the gradient method (the quasi-Newton method) with the multigrid method. The multigrid method helps the gradient method to find the global minimum regardless of the initial guess. In this paper, we illustrate the multigrid method of seismic inversion on a piece of the Marmousi data set and we demonstrate that the multigrid method succeeds where the quasi-Newton fails to invert the velocity model.