A 3D Helmholtz iterative solver with a semi-analytical preconditioner for acoustic wavefield modeling in seismic exploration problems

An approach to the iterative solution of the 3D acoustic wave equation in a frequency domain is proposed, substantiated, and verified numerically. 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 can be efficiently evaluated via 2D FFT in x and y directions followed by the solution of a number of ordinary differential equations in z directions. While solving ODE's. it is possible to treat the 1D velocity function as a piecewise constant one and to search for the exact solution as a superposition of upgoing and downgoing waves. This approach allows one not to use explicit finite-difference approximations of derivatives at all. The method has excellent dispersion properties in both lateral and vertical directions.