文章快速检索 高级检索
 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

 [1] Backus G, Gilbert F. The Rotational Splitting of the Free Oscillations of the Earth[J]. Proceedings of the National Academy of Sciences, 1961, 47(3): 362-371 DOI:10.1073/pnas.47.3.362 (0) [2] Dahlen F A, Tromp J. Theoretical Global Seismology[M]. Princeton: Princeton University Press, 1998 (0) [3] Dahlen F A. The Spectra of Unresolved Split Normal Mode Multiplets[J]. Geophysical Journal of the Royal Astronomical Society, 1979, 58(1): 1-33 DOI:10.1111/j.1365-246X.1979.tb01008.x (0) [4] Deuss A, Ritsema J, Heijst V H. A New Catalogue of Normal-Mode Splitting Function Measurements up to 10 MHz[J]. Geophysical Journal International, 2013, 193(2): 920-937 DOI:10.1093/gji/ggt010 (0) [5] Auer L, Boschi L, Becker T W, et al. Savani: A Variable Resolution Whole-Mantle Model of Anisotropic Shear Velocity Variations Based on Multiple Data Sets[J]. Journal of Geophysical Research: Solid Earth, 2014, 119(4): 3006-3034 DOI:10.1002/2013JB010773 (0) [6] Tanimoto T. The Backus-Gilbert Approach to the Three-Dimensional Structure in the Upper Mantle-I. Lateral Variation of Surface Wave Phase Velocity with Its Error and Resolution[J]. Geophysical Journal of the Royal Astronomical Society, 1985, 82(1): 105-123 DOI:10.1111/j.1365-246X.1985.tb05130.x (0) [7] Woodhouse J H, Girnius T P. Surface Waves and Free Oscillations in a Regionalized Earth Model[J]. Geophysical Journal of the Royal Astronomical Society, 1982, 68(3): 653-673 DOI:10.1111/j.1365-246X.1982.tb04921.x (0) [8] Ishii M, Tromp J. Normal-Mode and Free-Air Gravity Constraints on Lateral Variations in Velocity and Density of Earth's Mantle[J]. Science, 1999, 285(5431): 1231-1236 DOI:10.1126/science.285.5431.1231 (0) [9] Koelemeijer P, Ritsema J, Deuss A, et al. SP12RTS: A Degree-12 Model of Shear-and Compressional-Wave Velocity for Earth's Mantle[J]. Geophysical Journal International, 2015, 204(2): 1024-1039 (0) [10] Ritzwoller M, Masters G, Gilbert F. Observations of Anomalous Splitting and Their Interpretation in Terms of Aspherical Structure[J]. Journal of Geophysical Research: Solid Earth, 1986, 91(B10): 10203-10228 DOI:10.1029/JB091iB10p10203 (0) [11] Ritsema J, Deuss A, Heijst V H J, et al. S40RTS: A Degree-40 Shear-Velocity Model for the Mantle from New Rayleigh Wave Dispersion, Teleseismic Traveltime and Normal-Mode Splitting Function Measurements[J]. Geophysical Journal International, 2011, 184(3): 1223-1236 DOI:10.1111/gji.2011.184.issue-3 (0) [12] Stacey F D. Physics of the Earth[M]. New York: John Wiley, 1977 (0) [13] Yuan K, Beghein C. Seismic Anisotropy Changes across Upper Mantle Phase Transitions[J]. Earth and Planetary Science Letters, 2013, 374(14): 132-144 (0) [14] Woodhouse J H, Dziewonski A M. Mapping the Upper Mantle: Three-Dimensional Modeling of Earth Structure by Inversion of Seismic Waveforms[J]. Journal of Geophysical Research: Solid Earth, 1984, 89(B7): 5953-5986 DOI:10.1029/JB089iB07p05953 (0) [15] Woodhouse J H, Dahlen F A. The Effect of a General Aspherical Perturbation on the Free Oscillations of the Earth[J]. Geophysical Journal of the Royal Astronomical Society, 1978, 53(2): 335-354 DOI:10.1111/j.1365-246X.1978.tb03746.x (0) [16] Nakiboglu S M. Hydrostatic Figure and Related Properties of the Earth[J]. Geophysical Journal International, 1979, 57(3): 639-648 DOI:10.1111/gji.1979.57.issue-3 (0) [17] Chambat F, Ricard Y, Valette B. Flattening of the Earth: Further from Hydrostaticity than Previously Estimated[J]. Geophysical Journal International, 2010, 183(2): 727-732 DOI:10.1111/j.1365-246X.2010.04771.x (0) [18] Dahlen F A, Sailor R V. Rotational and Elliptical Splitting of the Free Oscillations of the Earth[J]. Geophysical Journal of the Royal Astronomical Society, 1979, 58(3): 609-623 DOI:10.1111/j.1365-246X.1979.tb04797.x (0) [19] Giardini D, Li X D, Woodhouse J H. Splitting Functions of Long-Period Normal Modes of the Earth[J]. Journal of Geophysical Research: Solid Earth, 1988, 93(B11): 13716-13742 DOI:10.1029/JB093iB11p13716 (0) [20] Rogister Y. Splitting of Seismic-Free Oscillations and of the Slichter Triplet Using the Normal Mode Theory of a Rotating, Ellipsoidal Earth[J]. Physics of the Earth and Planetary Interiors, 2003, 140(1-3): 169-182 DOI:10.1016/j.pepi.2003.08.002 (0) [21] Lognonné P. Normal Modes and Seismograms in an Anelastic Rotating Earth[J]. Journal of Geophysical Research: Solid Earth, 1991, 96(B12): 20309-20319 DOI:10.1029/91JB00420 (0)
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