Numerical modeling of the 3-D seismic wavefield with the spectral element method in spherical coordinates

The spectral element method has become an important tool for seismic wavefield simulation of regional and even continental scales. For regional or continental tomography, the curvature of the earth cannot be neglected, so it is more appropriate to use the spherical coordinate system to simulate the propagation of seismic waves. Starting from the weak form of elastic wave equation in spherical coordinates, the basic theory of the Legendre spectral element method in a spherical coordinate system is expounded based on variational principle of the spherical coordinate system. On the other hand, calculation of Frechet kernel is critical to the full waveform inversion. With the help of the adjoint method, we derive the expression of 3-D Frechet kernel. The numerical results obtained by the forward modeling, which are compared with the analytical solutions obtained by the normal mode method to verify the accuracy of the 1-D PREM model. Simultaneously, we apply this method to the North China Craton to create accurate imaging of the earth's interior structure. Based on the 3-D global radially anisotropic mantle model S362ANI with the 3-D crustal model Crustl. 0, the numerical simulation of seismic wave propagation in North China is carried out, and compared with the data recorded by observation stations. We obtain traveltime residuals from cross correlation and the spatial distribution of full wave traveltime 3-D Frechet kernel, which lays a foundation for the next large full-waveform inversion in the spherical coordinate system.