Weighted-averaging finite-element method for 2D elastic wave equations in the frequency domain

Abstract

we present a weighted-averaging frequency-domain finite-element method for an accurate and efficient 2D elastic wave modeling technique. Our method introduces three kinds of supplementary element sets in addition to a basic element set that is used in the standard finite-element method. By constructing global stiffness and mass matrices for four kinds of element sets and then averaging them with weighting coefficients, we obtain a new global stiffness and mass matrix. With optimal weighting coefficients determined by a Marquardt-Levenberg method to minimize grid dispersion and grid anisotropy, we can reduce the number of nodal points per shear wavelength from 33.3 (using the standard finite-element method) and 20 (using the eclectic method) to 5, with the errors of group velocities no larger than 1%. By reducing the number of grid points per wavelength, we achieve a 97% and 75% reduction of computer memory required to store the complex impedance matrix for a band-type matrix solver and a nested dissection method, respectively, compared with those of the eclectic method. Our method gives approximate solutions compatible with exact solutions for an infinite homogeneous, a semi-infinite homogeneous (Lamb's problem), and a horizontal two-layer model with fewer grid points than the standard and the eclectic method. A major advantage of the weighted-averaging finite-element method for the elastic wave equation is that it provides solutions very close to correct solutions, for Lamb's problem economically, unlike most of the displacement approaches. In addition, our scheme makes the complex impedance matrix symmetric, which satisfies reciprocity. Seismic forward modeling techniques that satisfy reciprocity are of critical importance in seismic imaging and inversion because we can economically calculate a Jacobian matrix using the reciprocity. Successful simulation of a large-size model shows that our method can be used for the simulation of wave propagation in the geological model needed in the reverse-time migration or seismic inversion.