基于Remez迭代算法求解差分系数的双相介质自适应有限差分正演模拟

Adaptive finite difference forward modeling of two-phase media based on the Remez iterative algorithm for solving differential coefficients

利用传统有限差分方法对基于Biot理论的双相介质波动方程进行数值求解时,由于慢纵波的存在,数值频散效应较为明显,影响模拟精度.相对于声学近似方程及普通弹性波方程,Biot双相介质波动方程在同等数值求解算法和精度要求条件下,其地震波场正演模拟需要更多的计算时间.本文针对Biot一阶速度-应力方程组发展了一种变阶数优化有限差分数值模拟方法,旨在同时提高其正演模拟的精度和效率.首先结合交错网格差分格式推导Biot方程的数值频散关系式.然后基于Remez迭代算法求取一阶空间偏导数的优化差分系数,并用于Biot方程的交错网格有限差分数值模拟.在此基础上把三类波的平均频散误差参数限制在给定的频散误差阈值和频率范围内,此时优化有限差分算子的长度就能自适应非均匀双相介质模型中的不同速度区间.数值频散曲线分析表明:基于Remez迭代算法的优化有限差分方法相较传统泰勒级数展开方法在大波数范围对频散误差的压制效果更明显;可变阶数的优化有限差分方法能取得与固定阶数优化有限差分方法相近的模拟精度.在均匀介质和河道模型的数值模拟实验中将本文变阶数优化有限差分算法与传统泰勒展开算法、最小二乘优化算法进行比较,进一步证明其在复杂地下介质中的有效性和适用性.

When the finite difference method(FDM)is used to numerically solve the two-phase medium wave equation based on Biot theory,the numerical dispersion effect is obvious due to the existence of slow compressional waves(P-wave),which affects the accuracy of numerical simulation.Compared with the acoustic approximation equation and the ordinary elastic wave equation,the Biot two-phase medium wave equation requires more calculation time for the seismic wave-field forward modeling under the same numerical solution algorithm and accuracy requirements.To solve this problem,this work developed a variable-order optimal FDM(VOFDM)for Biot first-order velocity-stress equations to improve the accuracy and efficiency of the forward modeling.Firstly,the numerical dispersion relation of Biot equation is derived by combining the staggered-grid FD(SFD)scheme.Then,based on the Remez iterative(RI)algorithm,the optimal differential coefficients of the first-order spatial partial derivatives are obtained and used for the SFD of the Biot equation.On this basis,the average dispersion error parameters of three kinds of waves are limited to a given dispersion error threshold and frequency range,so that the length of the FD operator can be adaptive to different velocity intervals in the inhomogeneous two-phase media model.The numerical dispersion curve analysis shows that the optimized FDM based on the RI algorithm is more effective in suppressing the dispersion error in the large wave number range than the traditional Taylor-series expansion(TE)method.The VOFDM can achieve similar simulation accuracy with the fixed-order optimized FDM(FOFDM).The numerical simulation experiments on homogeneous medium and river models prove VOFDM′s effectiveness and applicability in complex underground media compared with the TE algorithm and least squares(LS)optimization algorithm.