Optimized Mass and Stiffness Matrices for Accurate and Efficient 3D Acoustic Wave Modeling

Abstract

Abstract Numerical modeling of the 3D acoustic wave equation regularly faces a trade‐off between accuracy and efficiency. Usually, an improved computational efficiency leads to a compromised numerical accuracy, in which the numerical dispersion error increases. Such a trade‐off is commonly reported for the finite difference methods, and the consistent, lumped, eclectic finite element matrices, and the high‐order spectral element methods. Therefore, we present a methodology that simultaneously optimizes both the accuracy and the computational efficiency of the finite element method for modeling the 3D acoustic wave equation in the frequency domain. The numerical dispersion objective function is derived and minimized for a tri‐linear interpolation function of a cube element in terms of the mass and the stiffness matrices. With four grid points per wavelength sampling, for less than 0.5% error in velocities, the optimized modeled solutions outperform the results from extant approaches. Numerical experiments with the homogeneous, 3D unbounded 2‐layer and the 3D Society of Exploration Geophysicists salt velocity models reveal that the optimized accuracy is persistent across all source‐receiver offsets, a performance observably absent from other finite element matrices. The optimized matrices are not only useful for numerical modeling, but are also readily available for accurate computation of the sensitivity matrices of the density and the bulk modulus parameters for the multi‐parameterization inversion.