Forward modeling by a Fourier method

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.

Simulation of seismic waves at the earth's crust (brittle–ductile transition) based on the Burgers model

“…The numerical solution is based on a Taylor expansion of this operator up to the fourth order, and the Runge-Kutta algorithm is used (Jain, 1984;Carcione, 2007). The spatial derivatives are calculated with the Fourier pseudospectral method (Kosloff and Baysal, 1982;Carcione, 2007). This method consists of a spatial discretization and calculation of spatial derivatives using the fast Fourier transform.…”

Seismic Waves - Research and Analysis

“…Other spatial interpolations are possible. Previous discrete expressions are based on Lagrange interpolations while other interpolations are possible such as Chebychev or Laguerre polynomial or Fourier interpolations (Kosloff & Baysal, 1982;Kosloff et al, 1990;Mikhailenko et al, 2003). Interpolation basis could be local (Lagrange) or global(Fourier) ones based on equally spaced nodes or judiciously distributed nodes for keeping interpolation errors as small as possible: this will have a dramatic impact on the accuracy of the numerical estimation of the derivative and, therefore, on the resolution of partial differential equations.…”

3-D areal shot-source modeling

“…Another approach to modeling is pseudospectral method which uses Fourier transforms and requires fewer grid points per wavelength than other numerical modeling methods such as the finite-difference and finite-element methods to achieve the same accuracy. The characteristics of pseudospectral algorithms are described by references [1,2,3]. As the required storage extremely exceeded the capacity of the central memory at that time, the algorithm called for retrieving and restoring 3-D data sets in each of time steps.…”