各向同性和TI弹性波方程高精度有限差分数值解法新方法研究

With the development of seismic exploration from P-wave exploration to elastic wave exploration, more attentions have been gradually paid to forward modeling and reverse-time migration, one of whose key issues is numerically solving wave equations. Finite-difference (FD) methods have been widely applied in solving wave equations and high-order FD methods are commonly used to improve the accuracy of numerical solutions. Present high-order FD methods based on time-space domain dispersion relations can obtain high-accuracy numerical solutions when applying to the acoustic wave equation, but not high when applying to the elastic wave equations because the methods can not simultaneously meet accuracy requirements of P-wave and S-wave propagations. This project will study new high-accuracy high-order FD methods to numerically solve 2D and 3D isotropic elastic wave equations and transversely isotropic (TI) elastic wave equations respectively. The new methods will be implemented through designing several FD operators to meet the accuracy requirements of P-wave and S-wave propagations and thus suppress anisotropic dispersions. This project plans to develop new FD methods with (2M)th-order accuracy along all propagation directions for 2D and 3D isotropic elastic wave equations, new FD methods significantly suppressing anisotropic dispersions for 2D and 3D TI elastic wave equations, absorption boundary conditions and optimized FD methods improving the computational efficiency. Numerical modeling and analysis will be performed for simple and complex isotropic and TI elastic models to demonstrate validities and advantages of the proposed methods.

随着地震勘探由纵波向弹性波勘探发展，基于弹性波方程的正演和逆时偏移逐渐受到关注，其关键问题之一是弹性波方程数值求解。有限差分法是波动方程求解中常用的一种方法，高阶差分是提高精度的有效方法之一。目前的基于时空域频散关系的高阶差分方法应用于声波方程求解中能获得高精度数值解，但应用于弹性波方程求解中无法同时满足纵波和横波传播精度要求，精度不高。本项目针对二维、三维各向同性和TI(Transverse Isotropy，横向各向同性)弹性波方程，研究高精度高阶差分新方法，通过设计几个差分算子求解波动方程，来同时满足纵波和横波传播精度要求，并压制频散各向异性。经过研究，形成各个传播方向均具有2M阶精度的二维和三维各向同性弹性波方程有限差分数值解法、压制频散各向异性的二维和三维TI弹性波方程有限差分数值解法、针对性的吸收边界条件以及提高计算效率的差分算子优化方法，并对简单和复杂模型进行数值模拟与分析。

波动方程数值解广泛应用于地震正演、偏移和全波形反演中。由于有限差分法具有占有内存小、计算量小和易于实现等优点，因而成为最为常用的地震波动方程求解方法。如何提高波动方程有限差分法的精度和效率是近年来研究的重点。随着地震勘探由纵波向弹性波勘探发展，基于弹性波方程的正演和逆时偏移逐渐受到关注。本项目主要研究各向同性和TI弹性波方程高精度有限差分数值求解新方法。针对二维和三维各向同性弹性波方程，分别形成了基于菱形和椎体模板的空间2M阶、时间2N阶精度有限差分数值解法，当M = N，任意方向可以达到2M阶精度；针对二维和三维VTI弹性波方程，形成了基于Remez算法优化空间差分系数来压制数值频散的有限差分数值解法；针对TTI弹性波方程，形成了基于优化差分系数的显式及隐式旋转交错网格有限差分数值解法；形成了针对三维各向同性和VTI弹性波方程的混合吸收边界条件；形成的方法具有精度高、计算效率高、稳定性好等特点，应用于各向同性及各向异性复杂模型正演模拟并取得良好效果；达到项目研究目标要求。另外，在各向同性和各向异性声波波动方程数值求解和吸收边界条件、声波和弹性波逆时偏移及波形反演研究中取得一些进展，形成声波方程2M阶空间精度2N阶时间精度有限差分方法、声波方程时空域无网格优化有限差分正演方法、改进的混合吸收边界条件、弹性波最小二乘逆时偏移方法和弹性波全波形反演方法等成果。本项目研究成果，对于提高地震波动方程数值模拟、逆时偏移和全波形反演的精度、效率，具有重要应用前景。