A Helmholtz Iterative Solver without of Finite–difference Approximations

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 complex dumped Helmholtz equation in a 1D medium, where velocities vary only with depth. The actual Helmholtz operator is represented as a perturbation of the preconditioner. As a result, a matrix-by-vector multiplication of the preconditioned system may be effectively evaluated via 2D FFT in x and y directions followed by solution of a number of ordinary differential equations in z directions. To solve ODE’s we treat 1D background velocity as piecewise constant and search for exact solution as a superposition of upgoing and downgoing waves. We do not use explicit finite-difference approximations of derivatives. The method has excellent dispersion properties in both lateral and vertical directions. We illustrate the properties of our method using realistic 2D velocity model, and demonstrate in particular, propagation of signals without dispersion, and, a fast convergence rate for a wide band of temporal frequencies (from 2 Hz to 80 Hz).