Forward modeling by a Fourier method

TLDR

The authors present a Fourier or pseudospectral forward‑modeling algorithm for solving the two‑dimensional acoustic wave equation. The method computes spatial derivatives via FFT on a numerical grid, applies second‑order differencing for time derivatives, and is validated on a single‑layer analytic problem and a wedge physical model. The scheme achieves the same accuracy with fewer grid points than finite‑difference methods, proving more efficient—particularly in 3‑D—and its results agree with analytic and physical models while elucidating time‑section events.

Abstract

A Fourier or pseudospectral forward‐modeling algorithm for solving the two‐dimensional acoustic wave equation is presented. The method utilizes a spatial numerical grid to calculate spatial derivatives by the fast Fourier transform. Time derivatives which appear in the wave equation are calculated by second‐order differencing. The scheme requires fewer grid points than finite‐difference methods to achieve the same accuracy. It is therefore believed that the Fourier method will prove more efficient than finite‐difference methods, especially when dealing with three‐dimensional models. The Fourier forward‐modeling method was tested against two problems, a single‐layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling. The numerical results agreed with both the analytic and physical model results. Furthermore, the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account.