A Helmholtz iterative solver with semianalytical preconditioner for the frequency‐domain full‐waveform inversion

Abstract

Summary A preconditioned iterative method for solving the Helmholtz equation in heterogeneous media is proposed. We consider it as a tool for frequency-domain fullwaveform inversion. Our method is based on a Krylov-type linear solver, similar to several other iterative approaches. The distinctive feature is the use of a right preconditioner, obtained as a solution of the complex damped Helmholtz equation in a 1D medium, where velocities vary only with depth. As a result, a matrix-by-vector multiplication of the preconditioned system may be efficiently evaluated using a fast 2D-Fourier transform in horizontal space coordinates, followed by the solution of a system of ordinary differential equations in the vertical space coordinate. To solve these equations, we treat the 1D background velocity as piecewise constant and search for the exact solution as a superposition of upgoing and downgoing waves. In our approach, we do not approximate derivatives by finite differences. The method has good dispersion properties in both lateral and vertical directions. We illustrate the properties of our method using a realistic 2D velocity model, and demonstrate propagation of waves without visible dispersion with fast convergence rates for a wide band of temporal frequencies. Finally, results of 2D fullwaveform inversion using the proposed forward modeling engine are presented for single-offset VSP data.