球坐标系下谱元法三维地震波场模拟

谱元法已成为区域性乃至大陆性尺度地震波场模拟的重要工具.对于区域或大陆尺度层析成像而言，地球曲率不可忽略，此时模拟地震波传播采用球坐标系更为合适.本文从球坐标系下弹性波动方程弱形式出发，基于球坐标系变分原理给出了球坐标系下求解三维地震波方程的谱元法.另一方面，计算Fréchet敏感核是进行全波形反演的关键，本文借助伴随原理，推导了全波走时层析成像三维Fréchet敏感核表达式.为了验证球坐标系下谱元法的精度，我们将数值模拟结果与normal mode方法得到的解析解在1-D PREM模型下进行了对比.同时，我们将此方法应用到华北克拉通区域，以期获得地球内部结构精确成像.基于3-D全球径向各向异性地幔模型S362ANI和3-D地壳模型Crust1.0，我们建立了华北克拉通初始3-D背景模型，并将数值模拟结果与实际观测台站记录波形资料进行对比分析，利用互相关方法提取走时残差，最后给出了Fréchet敏感核在3-D空间中的分布，这些工作为下一步开展球坐标系下三维大尺度全波形反演奠定了基础.

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 Fréchet kernel is critical to the full waveform inversion. With the help of the adjoint method, we derive the expression of 3-D Fréchet 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 Crust1.0, 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 Fréchet kernel, which lays a foundation for the next large full-waveform inversion in the spherical coordinate system.