引用本文: 王静, 刘洋, 周泓宇. 2023. 时间-空间高阶精度矩形交错网格隐式有限差分声波正演模拟. 地球物理学报, 66(1): 368-382, doi: 10.6038/cjg2022P0778

有限差分方法因其操作简单、计算消耗低而成为地震勘探领域中最为常用的数值模拟方法之一, 然而用离散的显式差分算子数值逼近地震波动方程中的连续导数容易导致数值频散, 并且基于正方形网格离散形式的有限差分方法对不同地质模型的适应性较低.针对一阶变密度声波方程的数值模拟, 本文发展了一种适用于矩形网格离散形式的时间高阶空间隐式有限差分格式, 可以有效压制时间和空间频散, 同时灵活的网格剖分增强了其应用的广泛性.基于本文矩形交错网格时间高阶空间隐式有限差分格式的时空域频散关系和变量替换的思想, 首先采用泰勒级数展开方法求解不同方向的非轴上时间差分系数及轴上空间差分系数, 使本文差分格式可以获得任意偶数阶时间和空间精度.为了进一步提高本文差分格式在更大波数区域的空间模拟精度, 我们采用线性优化方法来求取新的轴上空间差分系数用于一阶变密度声波方程的波场迭代求解中.频散、稳定性分析及数值模拟算例表明: 相比于传统十字形空间域隐式有限差分格式, 本文矩形交错网格时间高阶空间隐式有限差分格式在精度、稳定性和效率方面均具有优势.

矩形交错网格 隐式有限差分 线性优化算法 时间、空间高阶精度 Abstract:

The Finite-Difference (FD) method has become one of the most popular numerical simulation methods for seismic exploration due to its easy implementation and low computational consumption. However, using discrete difference explicit operators to numerically approximate the continuous derivatives of the seismic wave equation will lead to numerical dispersion easily. Besides, the extensive application is hindered by the square-grid discretization. For the numerical simulation of the first-order variable-density acoustic wave equation, we develop a more flexible rectangular-grid discretization-based temporal high-order and spatial implicit FD scheme, which can effectively suppress the temporal and spatial dispersions. Based on the time-space domain dispersion relation of our proposed new FD scheme, also combined with a variable substitution idea, we first use the Taylor series expansion method to solve the off-axial temporal and axial spatial FD coefficients in different directions to obtain arbitrary even-order temporal and spatial accuracy. In order to further improve the spatial accuracy of our proposed FD scheme in a large wavenumber region, a linear optimization method is adopted to obtain the optimized axial spatial FD coefficients of the first-order variable-density acoustic wave equation. Dispersion, stability analysis and numerical examples demonstrate that compared with the conventional cross-stencil-based FD scheme in the space domain, the newly proposed temporal high-order and spatial implicit FD scheme has obvious advantages in terms of simulation accuracy, stability and efficiency.

Key words: Rectangular-grid discretization Implicit finite-difference Linear optimization strategy Temporal and spatial high accuracy 陈东, 梁文全, 辛维等. 2016. 适用于声波方程数值模拟的时间-空间域隐式有限差分算子优化方法. 地球物理学报, 59(4): 1491-1497, doi: 10.6038/cjg20160429 . http://www.geophy.cn/article/doi/10.6038/cjg20160429 董良国, 马在田, 曹景忠等. 2000. 一阶弹性波方程交错网格高阶差分解法. 地球物理学报, 43(3): 411-419. doi: 10.3321/j.issn:0001-5733.2000.03.015 http://www.geophy.cn/article/id/cjg_3734 胡自多, 刘威, 雍学善等. 2021. 三维波动方程时空域混合网格有限差分数值模拟方法. 地球物理学报, 64(8): 2809-2828, doi: 10.6038/cjg2021O0296 . http://www.geophy.cn/article/doi/10.6038/cjg2021O0296 梁文全, 杨长春, 王彦飞等. 2013. 用于声波方程数值模拟的时间-空间域有限差分系数确定新方法. 地球物理学报, 56(10): 3497-3506, doi: 10.6038/cjg20131024 . http://www.geophy.cn/article/doi/10.6038/cjg20131024 Figure 1.

Proposed rectangular staggered-grid discretization based temporal high-order and spatial implicit FD stencil for approximating ∂/∂ _x

Figure 2.

Variations of the dispersion curves with M and N for two proposed rectangular staggered-grid based temporal high-order and spatial implicit FD methods

Figure 3.

Variations of the dispersion curves with time step τ for three rectangular staggered-grid based implicit FD methods

Figure 4.

Stability factor curves of three rectangular staggered-grid based implicit FD methods

Figure 5.

Snapshots at 1.9 s and the corresponding magnified images of different rectangular staggered-grid based implicit FD methods

Figure 6.

Velocity (a) and density (b) of the 2D SEG/EAGE salt model

Figure 7.

Snapshots at 2.3 s of different rectangular staggered-grid based implicit FD methods for SEG/EAGE salt model

Figure 8.

Seismograms computed by three rectangular staggered-grid based implicit FD methods for SEG/EAGE salt model

Figure 9.

Waveforms computed by three rectangular staggered-grid based implicit FD methods for SEG/EAGE salt model