复杂地表起伏多重变加密网格有限差分法波动方程数值模拟

Numerical Simulation of Wave Equation Using Finite Difference Method with Rugged Variable Refined Multigrid in Complex Surface

常规变加密网格有限差分波动方程数值模拟方法采用水平分层加密网格,该网格剖分策略在适应地形起伏和近地表速度结构变化特征方面效果较差。针对该问题,提出一种起伏多重变加密网格有限差分波动方程数值模拟方法。该方法根据地形起伏和近地表低速层到高速层的速度分布特征进行网格加密;采用不同网格中的变系数差分格式离散声波方程,在保证波场模拟精度的同时兼顾计算效率;同时,为了进一步保障地表附近波场模拟的精度,地表附近最细网格中的差分格式不做降阶处理,针对位于地表以上的虚像点的波场值,提出一种融入自由地表边界条件的法向虚像外推法。算例分析验证该算法对速度模型不同区域进行的网格多次加密显著提高了计算效率,以黄土塬实际模型为例,耗时为常规1 m×1 m网格耗时的43.3%,并可达到和细网格基本一致的模拟精度,模拟误差控制在10-12范围内,同时表现出很好的近地表散射压制和边界吸收效果,且算法能稳定地适应实际复杂地表介质。

The horizontal layered refined grid is adopted in the conventional finite difference method of variable refined grid for the wave equation based numerical simulation.But the method cannot well adapt to the characteristics of rugged topography and the variation of velocity structure in near-surface region.To solve this problem,a wave equation based numerical simulation using finite difference method with rugged variable refined multigrid is proposed in this paper.This method encrypts the grid based on rugged topography and velocity distribution characteristics of low-velocity layer to high-speed layer near the surface and adopts the variable-coefficient difference schemes in different meshes to discretize the acoustic wave equation,in order to ensure the accuracy of wave field simulation and calculation efficiency.Furthermore,in order to further guarantee the precision of wave field simulation near the surface,the difference schemes in the smallest grids near the earth's surface are not processed by order reduction.For the wave field of the virtual ghost points above the surface,a normal virtual ghost extrapolation method incorporating free surface boundary conditions is proposed.The numerical evaluation and example show that the algorithm significantly improves the computational efficiency by encrypting the grids in different regions of the velocity model.Taking the actual model of loess plateau as an example,the ratio of time consumption to conventional 1 m×1 m grid is 43.3%and the simulation accuracy is basically consistent with the fine grid,the simulation error is controlled within 10-12.The algorithm shows good near-surface scattering suppression and boundary absorption effect,and can stably adapt to the actual complex near-surface media.