求解弹性波解耦方程的一种优化拟解析方法

常规伪谱方法二阶时间差分格式时间精度较低，且对于大步长时间采样间隔，常规伪谱方法不稳定.拟解析方法对于速度变化剧烈的模型，在时间和空间上均有较大误差.本文提出了一种基于解耦的二阶位移弹性波方程波场模拟及矢量波场分解的优化拟解析方法，将归一化的拟拉普拉斯算子分别应用于P波和S波波场延拓，延拓矢量波场的同时，可分解并延拓纯纵波和纯横波波场.利用弹性波优化拟微分算子表示拟拉普拉斯算子，该拟微分算子不仅包括原始微分算子的谱估计，而且还包含一个时间补偿项，其可在波数-空间域精确地补偿波动方程在时间方向上采用二阶有限差分引起的误差.利用低秩分解近似求解弹性波优化拟微分算子，可有效提高计算效率.2D均匀模型、层状模型以及部分盐丘模型数值正演模拟结果表明：相比较于常规的伪谱法和拟解析法，本文方法在时间和空间上均有很高的精度，并且稳定性条件比较宽松.

Optimized pseudo-analytical method for decoupled elastic wave equations

A number of numerical methods have been used to solve seismic wave equations in various media. Among them, the most commonly used is the Finite Difference (FD) method which has advantages of easy implementation and high efficiency. However, it suffers from numerical dispersion and conditionally stable problems. Compared with the FD method, the spectral method has a high accuracy in space, providing a generally dispersion-free wavefield. The second-order time stepping scheme of the conventional pseudo-spectral method, however, may produce time stepping errors and instabilities at a larger time step. The pseudo-analytical method can be applied to solve these problems through using a pseudo-Laplacian operator with constant compensation velocity. While this method can handle mild velocity variations with several compensation velocities, it also causes significant errors in the case of high velocity variations. In this paper, we propose to simulate vector wavefileds and decompose them into pure wave models simultaneously by the optimized pseudo-analytical method based on the decoupled elastic wave equations. We approximate the normalized pseudo-Laplacian operator with two variable compensation velocities, one is applied to the P-wave model, the other is applied to the S-wave model, through low-rank decomposition. Simulation on 2-D synthetic models demonstrates that the proposed method has high accuracy both in time and space with a more relaxed stability condition compared with the conventional pseudo-spectral and pseudo-analytical methods.