一种基于部分分式展开的隐式有限差分波动方程正演方法

Implicit fnite difference wave equation forward modeling based on partial fraction expansion

网格离散造成的数值频散是限制有限差分数值模拟精度的关键问题。隐式高阶有限差分算子随着差分阶数的增加能迅速地逼近理论精度,但是涉及带状矩阵的求解导致计算效率较低。本文基于平面波理论,推导一种在波数域部分分式展开的低阶隐式差分算子(PFIFD)来逼近理论算子精度。这种算子将常规的多对角矩阵表示成不同方向上多级并联的三对角线性矩阵,简化了稀疏线性方程组求解的复杂程度。同时利用追赶法求解三对角矩阵线性方程组,提高了隐式差分正演中计算空间偏导数方法的计算效率。频散分析和数值模拟结果表明,与其它方法比较,在保持相当的计算效率下,本文提出的方法能够更好地提高数值模拟的精度。

Numerical dispersion caused by mesh discretization is a critical factor affecting fnite difference numerical simulation accuracy.High-order implicit fnite difference operator may be used to solve this problem because the convergence rate increases with the order,but the effciency is low because of the computation of the band matrix.We derived a low-order implicit difference operator with partial fraction expansion in the wave number domain from the plane wave theory to improve the accuracy of numerical simulation.Using this operator,the routine multi-diagonal matrix is rewritten as multi-parallel linear tri-diagonal matrices in different directions to simplify the solution of the sparse linear equations.We adopted the chasing method to solve the linear equations composed of tri-diagonal matrices.This improves the computational effciency of the space partial derivative in the implicit difference forward modeling process.Dispersion analysis and numerical simulations show that our method yields excellent results with high accuracy and acceptable computational effciency.