摘要
为了改善Born散射级数解决地震强散射问题时的收敛性,将带有虚部分量的复波数格林函数引入到求解格林函数Lippmann–Schwinger(L-S)积分方程数值解的广义逐次超松弛迭代法中,弱化格林函数的奇异性。引入预条件算子降低系数矩阵的条件数,加速迭代级数的收敛速度,给出了复波数L-S方程的预条件广义逐次超松弛(preconditioned generalized successive over-relaxation,Pre-GSOR)迭代格式。通过数值分析和收敛性分析重新选取合适的衰减因子和预条件算子,得到了满足地震强散射条件的收敛Born级数,并将其用于地震强散射问题中数值格林函数的计算。数值结果表明:复波数L-S方程Pre-GSOR迭代法可以得到与实波数L-S方程直接法相匹配的数值模拟结果;复波数L-S方程Pre-GSOR迭代法系数矩阵条件数在高频时仅为原系数矩阵条件数的10%,相同迭代次数下归一化收敛残差可降低3个数量级以上,且对高频适应性强,可有效改善实波数L-S方程广义超松弛迭代法在强散射介质中的收敛停滞问题。
Born scattering series is often limited by weak scattering assumptions when solving strongly seismic scattering problems,resulting in slow convergence or divergence.A simple and effective way is to improve the iterative algorithm using numerical analysis.One such method is the generalized successive over-relaxation(GSOR)iterative method,which can be applied to solve the Lippmann-Schwinger(L-S)equation and obtain the desired convergent Born scattering series.However,in strongly heterogeneous media,the GSOR iterative method may also face the challenge of slow convergence speed while calculating the high-frequency Green’s function.In this paper,the complex wavenumber Green’s function is utilized with the GSOR iterative method to numerically solve the L-S equation of the Green’s function.The complex wavenumber has imaginary components that enable localizing the energy of the background Green’s function and exponential decay,reducing the singularity of the background Green’s function.To reduce the condition number of the coefficient matrix,we further introduce the preconditioning operator and provide a preconditioned generalized successive over-relaxation(Pre-GSOR)iteration format.The convergent iteration series is obtained by selecting an appropriate damping factor and preconditioning operator.Then it is used to calculate the numerical Green’s function in the seismic strongly scattering media.Numerical results indicate that the Pre-GSOR iteration method for the complex wavenumber L-S equations can produce simulation results consistent with those obtained by direct methods for real wavenumber L-S equations.The condition number of the coefficient matrix in the Pre-GSOR iterative method for the complex wavenumber L-S equation is only 10%of the original condition number at high frequencies.Under the same number of iterations,the normalized convergence residual obtained by this method can be reduced by more than three orders of magnitude.The new method exhibits lower convergence error,better convergence,and strong adaptability to high frequencies,effectively mitigating the convergence stagnation problem encountered by the generalized over-relaxation iterative method for the real wavenumber L-S equation in strongly scattering media.
作者
徐杨杨
商耀达
孙建国
Xu Yangyang;Shang Yaoda;Sun Jianguo(College of GeoExploration Science and Technology,Jilin University,Changchun 130026,China)
出处
《吉林大学学报（地球科学版）》
CAS
CSCD
北大核心
2024年第5期1696-1710,共15页
Journal of Jilin University：Earth Science Edition
基金
国家自然科学基金项目(41974135)。
关键词
地震散射波场
格林函数
广义超松弛迭代
预条件算子
seismic scattering wave field
Green’s function
generalized over-relaxation iteration
preconditioning operator
