基于广义正交多项式褶积微分算子的地震波场数值模拟方法

地震波场模拟方法研究对于与波动现象有关的地震学问题的重要性是不言而喻的.就目前现有的各种正演算法来说，精度较高的算法（如有限元法、谱元法、高阶有限差分法等），其计算速度较慢；计算速度较快的算法（如低阶有限差分法、付氏伪谱法等）计算精度却比较低.为了兼顾地震波场模拟的精度与速度，本文推出了一种快速的、高精度地震波场模拟方法（基于Forsyte广义正交多项式的褶积微分算子法），该方法是以计算数学中的Forsyte广义正交多项式插值函数为基础，构建一个新的褶积微分算子，并将该算子引入到地震波动方程的一阶速度-应力方程的空间微分运算中去，采用时间交错网格有限差分算子替代普通的差分算子以匹配高精度的褶积微分算子，从而构造一种全新的地震波场数值模拟方法.该方法同时具有广义正交多项式方法的高精度和短算子低阶有限差分算法的高速度.通过对算子长度的调节及算子系数的优化，可同时兼顾波场解的全局信息与局部信息.复杂非均匀介质模型中的波场数值模拟实验证实了该方法的可行性及优越性.

Seismic waves modeling by convolutional Forsyte polynomial differentiator method

Numerical modeling is important in studying a variety of problems in seismic wave propagation, many methods have been investigated for calculating the spatial terms in the wave equations, the most widely used methods are the classical finite difference (FD) method, the Fourier pseudo-spectral (FPS) method and Finite Element (FE) method. It is clear that each method has its own merits and drawbacks. For improving the precision and efficiency of seismic modeling, this paper develops a new modeling approach (Convolutional Forsyte Polynomial Differentiator Method) by using optimized convolutional operators for spatial differentiation and staggered-grid finite-difference for time differentiation in wave equation computation, a theoretical computation example of 2-D seismic wave field is given and the numerical results show that the algorithm can bring reliable outcomes with high precision and fast speed, and be adapted to complex inhomogeneous geological models and readily be extended to wave modeling for anisotropic media.