THEORETICAL PROBLEMS OF DISCRETE GAUSSIAN ANALYSIS OF GEOMAGNETIC FIELD

Starting from the fact that Gaussian (spherical harmonic) analysis of geomagnetic field is a 'three-dimensional problem', some theoretical problems of Gaussian analysis arising from the discretization observational points are studied in this paper. Firstly, the orthogonality of discrete harmonic functions is discussed. Secondly, the basic relationship for estimating the errors of Gaussian coefficients to be determined in Gaussian analysis of discrete observational points is obtained by using perturbation method, and from this relationship the various factors influencing the errors of Gaussian coefficients to be determined and their relationship are studied. Finally, the investigation of the independence of Gaussian coefficients in discrete Gaussian analysis points out that if the distribution of observational points and the choice of weights make the orthogonality relations not satisfied, 'three-dimensional significance' of Gaussian analysis will be impaired.     

APPLICATIONS OF ORTHOGONALITY THEORY OF DISCRETE NORMAL MODES TO THE IDENTIFICATION OF FREE OSCILLATION EIGENFREQUENCIES AND DETERMINA TIONS OF SPECTRAL PEAK POSITIONS AND SPECTRAL LINE WIDTHS

Owing to the loss of the orthogonality relations of discrete normal modes of free oscillations of the Earth in practical analysis, not only stacking method fails to give the desired results, but also stripping method may lead to large errors and even mis-identifications for certain normal modes. For this reason, a method in which the data from many earthquakes are used and appropriately weighted in order to make the orthogonality relations of discrete normal modes approximately satisfied is put forward. In addition, the conditions under which the orthogonality relations of discrete normal modes hold exactly within the range of a given degree and order have been found.     

地球磁场的离散高斯分析理论

本文从地磁场的高斯(球谐)分析是“三维问题”出发,研究了高斯分析由于测点离散化而产生的一系列理论问题。包括:1.离散球谐函数的正交性;2.高斯分析在测点离散条件下的误差分析,给出了估计高斯系数误差的基本关系式,并由此分析了影响待求高斯系数误差的各种因素;3.测点离散条件下高斯系数的独立性,指出测点的分布和权重选取不满足离散球谐函数正交关系将影响高斯分析的“三维意义”。     

地球自由振荡离散简正模正交理论在特征频率的分辨和谱峰位置及谱线宽度的确定中的应用

本文指出由于实际分析中地球自由振荡离散简正模正交关系不成立,不仅“叠加”(stacking)方法往往不能奏效,而且“剥离”(StriPping)方法可能对某些简正模将引进较大的误差,甚至发生分辨的错误。为此提出了一种利用从多个震源获得的资料和适当加权以使离散简正模正交关系近似成立的“叠加”方法。此外本文还给出了理论上在一定阶次范围内离散简正模正交关系严格成立的条件。