基于优化有限差分和混合吸收边界条件的三维VTI介质声波和弹性波数值模拟

传统有限差分系数是通过泰勒级数展开求取的，这样导致所计算的频散曲线在大波数区域会产生较强的数值误差.针对二阶空间偏导数的显式有限差分离散，本文发展了一种新的优化差分系数方法：首先将泰勒级数展开与多点采样方法结合应用于空间频散关系，基于最大范数建立直观有效的优化目标函数，采用Remez算法求解该目标函数，从而获得最优化差分系数.利用优化有限差分方法求解三维垂直对称轴横向各向同性（VTI）介质中的声波和弹性波方程.另外，本文将二维混合吸收边界条件推广到三维VTI介质中，用于吸收人工截断边界反射；基于各向异性特征，合理调整了边界区域的速度值来提高吸收效果.考虑到三维情况下计算效率的问题，本文波场外推过程中采用图形处理器（GPU）取代传统的中央处理器（CPU）.数值精度分析表明，相比较于传统的泰勒级数展开方法，优化有限差分方法在大波数区域对频散误差的压制效果更明显.在三维均匀和修改的Hess VTI模型中的数值模拟实验证明了本文方法具有更高的精度与效率，混合吸收边界条件在三维VTI介质中具有良好的边界吸收效果.

3D acoustic and elastic VTI modeling with optimal finite-difference schemes and hybrid absorbing boundary conditions

The coefficients of the conventional finite-difference (FD) stencil are commonly determined by the Taylor-series expansion method (TEM), which can produce major numerical dispersion errors at large wavenumbers. To solve this problem, we develop a novel optimal explicit FD method (FDM) for the second-order spatial derivative. We first combine the TEM and multiple sampling to the spatial dispersion relation, and then construct an intuitive and effective objective function using the maximum-norm method (MNM) and solve it by the Remez iteration algorithm (RIA). The maximum-norm (MN)-based FDM is adopted to solve the 3D acoustic and elastic vertical transversely isotropic (VTI) wave equations. Moreover, we extend the 2D hybrid absorbing boundary conditions (HABCs) to 3D acoustic and elastic VTI media, and improve the absorbing effectiveness through reasonably adjusting wavefield velocities in the absorption areas based on anisotropy properties. To improve computational efficiency for a 3D case, we utilize graphic processing unit (GPU) instead of traditional central processing unit (CPU) architecture. Numerical accuracy analyses indicate that the optimal scheme has better tolerance to the numerical dispersion at large wavenumbers than the conventional TEM. Numerical modeling experiments for 3D homogeneous and modified Hess VTI models demonstrate that the proposed schemes have greater accuracy and efficiency than the conventional schemes and achieve significant absorbing effects.