地球自由振荡数据不仅可以对一维球对称模型提供基本的约束，而且对于约束三维密度结构、构建全球地幔模型有着独特的优势。为了将观测值与理论值进行比较以构建或精化地球的三维成像模型，一维地球模型本征模的精确计算对基于模态耦合理论合成的三维地球模型的自由振荡谱尤为重要。鉴于现有的计算地球本征模的程序在高频情况对诸如斯通利模等一些特殊振型的计算精度很低，本项目拟利用基于Langer近似得到的渐近解，将在水平层状均匀介质中已发展得很成熟的广义反透射系数算法应用于垂向非均匀球状介质，根据不同振型所具有的不同振荡特征，导出相应的特征方程，给出求解地球本征模的自适应算法，为一维地球模型的本征模数据（特别是现有程序所计算不准的那些振型）的生成提供一种替代性选择，进而为合成理论地震图或其他耦合计算提供更加精确的基函数，为更加真实地反演地球三维密度结构和认识地球深内部结构奠定重要基础。

The free-oscillation data not only places an essential constraint on the 1-D spherically symmetric earth model, but also has a unique advantage in constraining 3-D density structure and constructing global mantle model. In order to invert or refine 3-D tomographic models by comparing the observations with the theoretical values, the accurate computation of the normal modes of the 1-D Earth model would be quite crucial to the calculation of synthetic mode spectra for 3-D models based on mode-coupling theory. Due to the insufficient accuracy of the calculation of some special modes (e.g., Stoneley modes) at high frequencies by current codes, this project will utilize the asymptotic solution obtained based on Langer approximation, and derive the corresponding characteristic equations according to the different oscillating character of the various modes, after adopting the generalized reflection/transmission coefficient algorithm, which has been fully developed for horizontal stratified homogeneous media and would be applied here to vertically inhomogeneous spherical layers. Finally, an adaptive method of computing the Earth’s normal modes is presented, which may be used instead when one generates the normal modes of a 1-D Earth model, especially those calculated quite inaccurately by current codes. Consequently, it will provide more accurate basis functions for synthesizing theoretical seismograms or other coupling calculations and underlie the more accurate inversion of the 3-D density structure and the understanding of the more realistic Earth’s deep interior.