Frozen Gaussian approximation for 3-D elastic wave equation and seismic tomography

Abstract

The purpose of this work is to generalize the frozen Gaussian approximation (FGA) theory to solve the 3-D elastic wave equation and use it as the forward modeling tool for seismic tomography with high-frequency data. FGA has been previously developed and verified as an efficient solver for high-frequency acoustic wave propagation (P-wave).The main contribution of this paper consists of three aspects: 1. We derive the FGA formulation for the 3-D elastic wave equation. Rather than standard ray-based methods (e.g. geometric optics and Gaussian beam method), the derivation requires to do asymptotic expansion in the week sense (integral form) so that one is able to perform integration by parts. Compared to the FGA theory for acoustic wave equation, the calculations in the derivation are much more technically involved due to the existence of both P-and Swaves, and the coupling of the polarized directions for SH-and SV-waves. In particular, we obtain the diabatic coupling terms for SH-and SV-waves, with the form closely connecting to the concept of Berry phase that is intensively studied in quantum mechanics and topology (Chern number). The accuracy and parallelizability of the FGA algorithm is illustrated by comparing to the spectral element method for 3-D elastic wave equa-arXiv:1810.06760v2 [physics.geo-ph] 21 Nov 2018 2 J. C. Hateley, L. Chai, P. Tong, X. Yang tion in homogeneous media; 2. We derive the interface conditions of FGA for 3-D elastic wave equation based on an Eulerian formulation and the Snell's law. We verify these conditions by simulating high-frequency elastic wave propagation in a 1-D layered Earth model. In this example, we also show that it is natural to apply the FGA algorithm to geometries with non-Cartesian coordinates; 3. We apply the developed FGA algorithm for 3-D seismic wave-equation-based traveltime tomography and full waveform inversion, respectively.

Convergence of the Frozen Gaussian Approximation for High Frequency Elastic Waves

“…This paper is to analytical show that the first ordered frozen Gaussian approximation for the elastic wave equation derived and reported in [5] is indeed the same order for strict hyperbolic systems, as proven in [9] and verified numerical in [7,2,3,5]. The proof will use all the same machinery from [9], with a new derivation and new error estimates for the evolution equation.…”

Seismic Tomography with Random Batch Gradient Reconstruction

“…To use RBM in seismic tomography, natural choices for computing synthetic seismograms are numerical methods of particle type, e.g., generalized ray theory [11,32], Kirchhoff migration [8,15], Gaussian beam migration [12,13,21,9,7,22], and frozen Gaussian approximation (FGA) [4,5,10,3]. Here for the sake of convenience, we use FGA to compute wave equations, which do not need to solve ray paths by shooting to reach the receivers, and can provide accurate solutions in the presence of caustics and multipathing, with no requirement on tuning beam width parameters to achieve a good resolution [2,12,6,23,19,33].…”

“…When the earthquake is modeled by a point source, one can choose s(t, x) = f (t)δ d (x − x s ) with f (t) as the source time function at x s with compact support on [0, ∞), and δ d as the Dirac delta function. Remark that the formulation here can be easily generalized to elastic wave propagation as in, e.g., [31,10], and we focus on seismic tomography using the propagation of P-wave for the sake of simplicity.…”

“…(10) [4,5,10]. In (9) we assume the same beam number N for both forward and adjoint simulations and N does not change during the iteration procedure (8).…”

“…As it defined in equations ( 9)-( 12), the kernel K can be seen as a summation of Ġj Ġ † l 's. Each G j or G † l is a Gaussian function whose parameters are given by a set of ODEs (10). By the smooth dependence on the initial condition and parameters for solution of ODEs, one can deduce that Ġj and Ġ † l 's are smooth in X and thus K is Lipschitz continuous in X.…”