用于弹性波方程数值模拟的有限差分系数确定方法

有限差分方法是波场数值模拟的一个重要方法,交错网格差分格式比规则网格差分格式稳定性更好,但方法本身都存在因网格化而形成的数值频散效应,这会降低波场模拟的精度与分辨率.为了缓解有限差分算子的数值频散效应,精确求解空间偏导数,本文把求解波动方程的线性化方法推广到用于求解弹性波方程交错网格有限差分系数;同时应用最大最小准则作为模拟退火(SA)优化算法求解差分系数的数值频散误差判定标准来求解有限差分系数.通过上述两种方法,分别利用均匀各向同性介质和复杂构造模型进行了数值正演模拟和数值频散分析,并与传统泰勒展开算法、最小二乘算法进行比较,验证了线性化方法和模拟退火方法都能有效压制数值频散,并比较了各个算法的特点.

Methods to determine the finite difference coefficients for elastic wave equation modeling

Numerical simulation of the elastic wave equation with staggered-grid finite-difference algorithms is widely used to synthesize seismograms theoretically, and is also the basis of the reverse time migration. With some stability conditions, grid dispersion often appears because of the discretization of the time and the spatial derivatives in the wave equation. How to suppress the grid dispersion is therefore a key problem for finite-difference approaches. Different methods have been proposed to address this issue. The most commonly used methods are the high order Taylor expansion (TE) methods. In this paper, we extend the linear method for solving the acoustic wave equation to the staggered grid finite difference method for solving the elastic wave equation. We also apply the maximum-minimum criterion to measure the dispersion error when performing simulated annealing (SA) algorithm. Dispersion analysis and numerical simulation demonstrate that a linear method without iteration is nearly equal to the SA method and the least squares (LS) method in the space domain, and is better than the TE methods.For the finite difference coefficients obtained by the two methods, using homogeneous isotropic and complex structural model, we performed a numerical forward modeling and numerical dispersion analysis firstly, then compared it with the traditional Taylor expansion (TE) method and least squares(LS) method.Dispersion analysis and numerical simulation demonstrate the following conclusions: (1) With the increase of the length of the operator, various methods are able to maintain the dispersion relation in a larger wave number range. (2) The coefficients obtained by the TE method covers the minimal wave number range, coefficients from SA and LS method cover the maximal wave number range, the wave number range of linearization method is much larger than that of TE method, and is very close to that of the optimization method. (3)Although the wave number range of the linearization method is slightly less than the optimization method, but the dispersion error of this method is smaller in lower wave number (low frequency). Taking the spectrum of seismic source into account, the coefficient of linearization finite difference method is able to effectively reduce the numerical dispersion. (4) The linearization method can be used to solve finite difference coefficients directly. Its computational efficiency is much higher than the LS method and SA method.Comparison of the numerical simulation results from the uniform medium model and the complex structure model indicates that the linearization method, LS method and SA method can significantly suppress the numerical dispersion (seismograms almost coincide), thus the coefficients confirmed by the linearization finite difference method can be used to speed-stress staggered grid elastic wave modeling and time-reverse migration to improve the accuracy and efficiency of the calculation.