【摘要】 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.更多还原

【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.更多还原