目前球形分层大地格林函数在理论推导和数值计算方面还存在着较大的困难.首先,通过球形分层电磁理论推导无穷勒让德级数形式的格林函数,并提出勒让德级数权重函数的递推算法.在此基础上,提出求解球形分层格林函数的复镜像方法,将格林函数的无穷级数求和转化为复镜像位函数的叠加,并推导算法的误差上限计算公式,通过算例验证方法的准确性.针对地球尺度级别球形分层格林函数的数值奇异问题和极缓慢收敛特性,提出基于多精度算法的解决方案,进一步证明复镜像法在计算速度和精度上的优势.所提方法能够解决球形分层格林函数的理论和计算难题,为求解地球尺度级别球形分层格林函数提供有效解决方案.
At present,there are still some difficulties in the theoretical derivation and numerical calculation of the Green's function of the spherical layered earth.First,the Green's function in the form of infinite Legendre series is derived by the spherical layered electromagnetic theory,and the recursive algorithm of the weight function of Legendre series is proposed.On this basis,the complex image method for solving spherical layered Green's function is presented,which transforms the summation of infinite series of Green's function into the superposition of complex image potential function.Moreover,the upper error limit of the algorithm is derived,and the accuracy of the method is verified by examples.In order to solve the numerical singularity and the extremely slow convergence of the earth-scale spherical layered Green's function,a solution based on multiple precision algorithm is proposed,which further proves the advantages of the complex image method in computing speed and accuracy.The method in this paper solves the theoretical and computational problems of spherical layered Green's function,and can be used as an effective solution to the spherical layered Green's function at the earth scale.