近似解析解方法在简正模计算中的应用

Approximate Analytical Method to the Computation of Normal Modes

利用合理的参数近似,导出分层、球形、弹性各向同性和含自重的地球变形的常系数微分方程组,从而给出矩阵形式的解析解。同时,采用一种高效稳定的传播矩阵方法,计算地球简正模的本征周期和本征函数。为了验证方法的正确性,首先将本征周期与均匀球模型的解析解进行比较,结果表明,球型和环型简正模若要实现10^(-2)s的本征周期精度,模型的分层厚度分别为50 m和0.1 m;本征函数与解析解一致,且收敛性较好;当地球模型的层厚足够小时,近似解析解可认为是真实解。最后,采用近似解析解方法计算PREM模型的本征周期,并对Mineos的计算结果进行精度检验,发现环型简正模和球型简正模的本征周期精度分别为10^(-3)s和10^(-2)s,可以满足当前观测的需求。

With appropriate parameter approximations,this paper derives a system of constant-coefficient differential equations for modeling the deformation of a spherical,layered,elastic,isotropic,and self-gravitating Earth.The system has analytical solutions in matrix form.We use an efficient and stable propagation matrix method to compute the eigenperiods and eigenfunctions of the normal modes.For validation,we compare the eigenperiods to the analytical solutions for a homogeneous model,indicating that layer thicknesses of 50 meters and 0.1 meters are required to achieve 10^(-2)second accuracy for spheroidal and toroidal normal modes respectively.Additionally,we demonstrate excellent convergence between the eigenfunctions and analytical solutions.The solutions are indeed the true ones when the thickness of the layers of the Earth model is small enough.Finally,we verify the eigenperiod results for the PREM model using the Mineos software.We find that the eigenperiod accuracies are 10^(-3)second and 10^(-2)second for the toroidal and spheroidal normal modes,respectively,which is sufficient for current observational requirements.