Frequency Domain Iterative Solver for Elasticity without Use of Finite-difference Approximations

A preconditioned iterative method for solving frequency domain elastic wave equations is presented. Our method is based on Krylov type linear solvers. Its distinctive feature is the use of a right preconditioner, obtained as a solution of the damped elastic wave equations in a vertically heterogeneous medium. We represent the actual differential operator as a perturbation of the preconditioner. As a result, a matrix-by-vector multiplication of the preconditioned system is effectively evaluated via the fast Fourier transform in horizontal direction(s) followed by the solution of a number of systems of ordinary differential equations in the vertical direction. To solve these equations we introduce a piecewise constant 1D background medium and search for the exact solution in the 1D medium as a superposition of upgoing and downgoing P- and S- waves. The method has excellent dispersion properties because it does not use any finite-difference approximation of derivatives and converges reasonably fast.