﻿ 基于MATLAB的简正模正常谱峰分裂计算
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 大地测量与地球动力学  2019, Vol. 39 Issue (6): 583-586  DOI: 10.14075/j.jgg.2019.06.007

### 引用本文

ZHANG Lingyun, SUN Heping, XU Jianqiao, et al. Computation of Normal Splitting of Normal Modes Based on MATLAB[J]. Journal of Geodesy and Geodynamics, 2019, 39(6): 583-586.

### Foundation support

National Key Basic Research Program of China, No. 2014CB845902; National Natural Science Foundation of China, No. 41474062, 41604070.

### 第一作者简介

ZHANG Lingyun, PhD candidate, majors in Earth's free oscillations, surface wave, seismometer and SG data processing, E-mail: zly198712280572@sina.com.

### 文章历史

1. 中国科学院测量与地球物理研究所大地测量与地球动力学国家重点实验室，武汉市徐东大街340号，430077;
2. 中国科学院大学，北京市玉泉路19号甲，100049

1 基本方法

 $\begin{array}{l} a = \frac{1}{3}[1 - l(l + 1)\chi ]{\left( {\frac{\mathit{\Omega }}{{{\omega _0}}}} \right)^2} + \\ \;\;\;\;\;\;\frac{1}{2}\left( {v w_0^{ - 2} - \tau } \right) + {\alpha _2}{\left( {\frac{\mathit{\Omega }}{{{\omega _0}}}} \right)^2}\\ b = \chi \left( {\frac{\mathit{\Omega }}{{{\omega _0}}}} \right)\\ c = - \frac{3}{2}l(l + 1)\left[ {vw_0^{ - 2} - \tau } \right] + {\gamma _2}{\left( {\frac{\mathit{\Omega }}{{{\omega _0}}}} \right)^2} \end{array}$ (1)

 $\begin{array}{l}{\varepsilon^{1 \mathrm{st}}=-3 f_{2} / 2} \\ {\varepsilon^{2 \mathrm{nd}}=-3 f_{2} / 2-5 f_{4} / 8-3 f_{2}^{2} / 8}\end{array}$ (2)

 $\begin{array}{l}{J_{2}=-f_{2}-\overline{m} / 3-11 f_{2}^{2} / 7-\overline{m} f_{2} / 7} \\ {J_{4}=-f_{4}-36 f_{2}^{2} / 35-6 \overline{m} f_{2} / 7}\end{array}$ (3)

2 计算结果

 图 1 PREM模型一阶和二阶椭率随半径的变化 Fig. 1 The first and second order ellipticity of PREM varied with radius

 图 2 基于PREM模型0S2一阶和二阶单谱频率 Fig. 2 The first and second order singlet frequencies of starting model PREM

 $s=\sigma_{l-2}^{m}+\tau_{l-2}^{m}+\sigma_{l}^{m}+\tau_{l+1}^{m}+\sigma_{l+2}^{m}, |m| \leqslant l-2$ (4)
 $s = \tau _{l - 1}^{l - 1} + \sigma _l^{l - 1} + \tau _{l + 1}^{l - 1} + \sigma _{l + 2}^{l - 1}, \;|m| \le l - 1$ (5)
 $s = \sigma _l^l + \tau _{l + 1}^l + \sigma _{l + 2}^l, \;|m| = l$ (6)

3 结语

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Computation of Normal Splitting of Normal Modes Based on MATLAB
ZHANG Lingyun1,2     SUN Heping1,2     XU Jianqiao1     Deng Mingli1
1. State Key Laboratory of Geodesy and Earth's Dynamics, Institute of Geodesy and Geophysics, CAS, 340 Xudong Road, Wuhan 430077, China;
2. University of Chinese Academy of Sciences, A19 Yuquan Road, Beijing 100049, China
Abstract: In this paper, an interface is built to compute the normal modes splitting parameters, based on the MINEOS software, ode45 toolbox of MATLAB, and interpolation function. The hydrochloric coefficient of J2 and J4 are used to compute the first and second order term of Earth's ellipticity. The splitting parameters and singlets frequencies of 0S2 are generated based on the PREM and 1066A. It is proven to be correct when compared with the results of Dahlen. Our research shows that the second order ellipticity has a little effect on the splitting parameters, but almost no influence on the singlet frequencies. Finally, the coupling matrix and coupling strength between 0S2 and selected modes are given. This lays a good base to use the splitting function to inverse the interior Earth structure.
Key words: normal modes; second order ellipticity; normal splitting parameters; singlet; coupling strength