A Helmholtz Iterative Solver with Semi–analytical Preconditioner for Seismic Imaging and Inversion

We propose a preconditioned iterative method for solving the Helmholtz equation in heterogeneous media. Our method is based on Krylov type linear solvers, similarly to several other iterative solver approaches. The distinctive feature of our method is the use of a right preconditioner, obtained as the solution of the Helmholtz equation in a 1D medium, where velocities vary only with depth. This preconditioner improves the convergence of the iterative solver for the initial heterogenous medium, and it is computed efficiently via FFT along the horizontal direction(s), followed by the numerical solution of a system of ordinary linear differential equations. We illustrate the properties of our method using a complex velocity model, and demonstrate in particular, propagation of signals without dispersion in the horizontal direction, and, a fast convergence rate for a wide band of temporal frequencies (from 2 Hz to 70 Hz).