Accuracy of finite‐difference and finite‐element modeling of the scalar and elastic wave equations

Abstract

Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite‐difference and finite‐element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time‐domain and frequency‐domain techniques. In addition to versatility and ease of implementation, it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors and transmission coefficients. This paper shows that in homogeneous media the explicit finite‐element and finite‐difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson’s ratio less than 0.3. Finite‐elements are superior to finite‐differences when modeling elastic media with Poisson’s ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central‐differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency‐domain finite‐element solutions employing a weighted average of consistent and lumped masses yield the most accurate results, and they promise to be the most cost‐effective method for CDP, well log, and interactive modeling.

Inversion of Full Acoustic Wavefield in Local Helioseismology: A Study With Synthetic Data

“…Note that the impedance matrix (Marfurt 1984) is a very large and sparse matrix that links the seismic source terms to the seismic wavefield (Pratt 1999). We estimate the unknown physical properties (sound speed in our case) at the node points using 1. m sets of Fourier-transformed experimental observations, u d , recorded at a subset of the nodal points corresponding to the receiver locations, 2. an initial model from which synthetic data u s are generated.…”

Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

“…(2) contains the numerical dispersion error. Usually the analysis of the numerical dispersion error and its improvement for many space-discretization techniques such as the finite elements, spectral elements, isogeometric elements and others starts with the analysis and modifications of the elemental mass and stiffness matrices; see [10,1,2,7,8,15,16,17,18,22,11,14,20,21]. For example, one simple and effective finite-element technique for acoustic and elastic wave propagation problems is based on the calculation of the mass matrix M M M in Eq.…”

“…For example, one simple and effective finite-element technique for acoustic and elastic wave propagation problems is based on the calculation of the mass matrix M M M in Eq. (2) as a weighted average of the consistent and lumped mass matrices; see [15,16,17,18] and others. For the 1-D case and the linear finite elements, this approach reduces the error in the wave velocity for harmonic waves from the second order to the fourth order of accuracy.…”

Solving the 3D Acoustic Wave-Equation on Generalized Structured Meshes: A FDTD Approach

“…Examples of these types of approaches include the finite-element (Marfurt, 1984), spectral-element (Komatitsch and Vilotte, 1998) and discontinuous-Galerkin (Cockburn et al 2000) methods. By using meshes conformal to irregular surfaces, these approaches facilitate the numerical implementation of freesurface boundary conditions (Priolo et al, 1994) and are able to more accurately represent discontinuities across major interior lithological boundaries relative to finite-difference methods (Fornberg, 1988).…”