Iterative finite-difference solution analysis of acoustic wave equation in the Laplace-Fourier domain

We have investigated numerical characteristics of iterative solutions to the acoustic wave equation in the Laplace-Fourier (LF) domain. We transformed the time-domain acoustic wave equation into the LF domain; the transformed equation was discretized with finite differences and was solved with iterative methods. Finite-difference modeling experiments demonstrate that iterative methods require an infinitesimal stopping tolerance to accurately compute the pressure field especially at long offsets. To understand the requirement for such infinitesimal tolerance values, we analyzed the evolution of intermediate solution vectors, residual vectors, and search direction vectors during the iteration. The analysis showed that the requirement arises from the fact that in the solution space, the amplitude of the pressure field varies more than sixty orders of magnitude on the common log scale. Accordingly, we propose a rule of thumb for choosing a proper stopping tolerance Value. We also examined numerical dispersion errors in terms of the grid sampling resolutions per skin depth and wavelength. We found that despite the similarity of the form of the acoustic wave and electromagnetic diffusion equations, the former is different from the latter due to the fact that in the LF domain, the skin depth of the acoustic wave equation is decoupled from its wavelength. This aspect requires that in the LF domain, its grid size be determined by considering the minimum grid sampling resolutions based not only the wavelength but also the skin depth.