Three‐dimensional acoustic modeling by the Fourier method

Abstract

A three‐dimensional forward modeling algorithm, allowing arbitrary density and arbitrary wave propagation velocity in lateral and vertical directions, directly solves the acoustic wave equation through spatial and temporal discretization. Spatial partial differentiation is performed in the Fourier domain. Time stepping is performed with a second‐order differencing operator. Modeling includes an optional free surface above the spatial grid. An absorbing boundary is applied on the lateral and bottom edges of the spatial grid. Three‐dimensional forward modeling represents a challenge for computer technology. Computation of meaningfully sized models requires extensive calculations and large three‐dimensional data sets which must be retrieved and restored during the computation of each time step. The computational feasibility of the Fourier method is demonstrated by implementation on the multiprocessor CRAY X‐MP computer system using the large secondary memory of the solid‐state storage device (SSD). Calculations use vectorization and parallel processing architecture. The similarity of numerical and analytical results indicates sufficient accuracy for many applications.

Support minimized inversion of acoustic and elastic wave scattering

“…Although the choice of an efficient forward process was secondary to this work, an understanding of the forward process is of primary importance in any inversion effort. Approximate models provide adequate solution with much less computation [33,18,5,19,29,4,45,44,3]. However, they do not provide solutions that satisfy the physics governing the measurement process.…”

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.…”

Hybrid Pseudospectral/Finite‐Difference Numerical Simulation and Stability Analysis of Pure P‐Wave Propagation in Vti Media

“…The pure P-wave equations usually involve fractional pseudo-differential operators that make the numerical implementation difficult when using the finite-difference (FD) method alone. The most natural choice to solve these equations is pseudospectral (PS) method (Kosloff and Baysal, 1982;Reshef et al, 1988aReshef et al, , 1988b. Besides, some other modified spectral methods are used to solve those special pure P-wave equations (Etgen and Brandsberg-Dahl, 2009;Crawley et al, 2010, Pestana andStoffa, 2010).…”