Maslov渐近理论与辛几何算法
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摘要
为了克服地震层析成像中的焦散问题，本文系统地研究了Ｍａｓｌｏｖ渐近理论与辛几何算法，同时提出了一种基于辛几何算法计算Ｍａｓｌｏｖ波场的数值计算方法，并就其中射线追踪这一重要环节，利用辛算法和非辛Ｒｕｎｇｅ－Ｋｕｔｔａ方法进行了数值模拟．计算结果表明，这两类算法在精度上相差无几，但辛算法的速度要快；在Ｈａｍｉｌｔｏｎ量保持方面，辛算法具有非辛Ｒｕｎｇｅ－Ｋｕｔｔａ方法无可比拟的优越性．
In order to overcome the caustic problem in seismic tomography, we systematically study the Maslov asymptotic theory and symplectic algorithm. A numerical method for evaluating Maslov wave field, which is based on symplectic algorithm, is presented. In addition, we perform numerical simulation of ray tracing using symplectic and nonsymplectic algorithm. The result shows that the precision of both is almost the same, however, the symplectic algorithm is faster. Furthermore, in keeping Hamitonlian quantities for a long time, the symplectic algorithm has an incomparable superiority over nonsymplectic algorithm.
In order to overcome the caustic problem in seismic tomography, we systematically study the Maslov asymptotic theory and symplectic algorithm. A numerical method for evaluating Maslov wave field, which is based on symplectic algorithm, is presented. In addition, we perform numerical simulation of ray tracing using symplectic and nonsymplectic algorithm. The result shows that the precision of both is almost the same, however, the symplectic algorithm is faster. Furthermore, in keeping Hamitonlian quantities for a long time, the symplectic algorithm has an incomparable superiority over nonsymplectic algorithm.
引文
1) QIN Meng-Zhao,ZHANG Mei-Qing.Symplectic Runge-Kutta schemes for Hamilton systems.Beijing:Computing Center,Chinese Academy of Sciences,1998.
2) QIN Meng-Zhao,Zhu Wen-Jie,ZHANG Mei-Qing.Construction of a three-stage difference scheme for ordinary differntial equations.Beijing:Computing Center,Chinese Academy of Sciences,1990.
[1]尹峰,李幼铭.反演二维非均匀介质背景中速度扰动的波动方程层析成像方法见:杨文采等编,应用地震层析成像,北京:地质出版社.1993
YIN Feng, LI You-Ming. Wave equation tomography by inversion of velocity disturbance in twodemensional inhomogeneous media. In: YANG Wen-Cai ed. Applied Seismic Tomography. Beijing:Geological Publishing House, 1993.
[ 2 ] George A Mcmechan, Walter D Mooney. Asymptotic ray theory and synthetic seismograms for laterallyvarying structures: theory and application to the imperial valley, California. Bull. Seism. Soc. Am. 1980,70: 2021 -2035.
[ 3 ] C H. Chapman, K Drummond, Body-wave seismograms in inhomogenous media using Maslov asympotictheory. Bull. Seism Soc. Am., 1982, 72:277-317.
[ 4 ] G D Spence, K P Whittall, R M. Clowes. Pratical synthetic seismograms laterallyvaring media calculatedby asympoticray theory. Bull. Seism. Soc. Am. 1984, 74: 1209-1223.
[ 5 ] C H. Chapman. Ray theory and its extensions: WKBJ and Maslov seismograms. J. Geophs., 1985, 58:27-43.
[ 6 ] V P. Maslov. Semi-classical Approximation in Quantum Mechanics. Hingham, Mass: D. Reidel, PublishingCompany, 1981.
[ 7 ] Ziolkowski R W, Deschamps G A. Asymptotic evaluation of high-frequency fields near a caustic: anintrodtction to Maslov method. Rodio Sci., 1984, 19: 1001-1025.
[ 8 ] V I Arnold. Mathematical methods of classical mechanics. Springer-Verlag, 1978.
[9] Feng Knag. On difference scheme and symplectic geometry, proceedings of the 1984 Beijing symposiumon differential geometry and differential equations, ed. Feng kang, Beijing: Science press, 1985, 42-58.
[10] Qin Meng-Zhao, Wang Dao-Liu and Zhang Mei-Qing. Explicit symplectic difference schemes for separableHamiliton system. JCM 1991, 9(3): 211-221.
[11] Ruth R D. A canonical integration technique. IEEE Trans. on Nuclear Seiences,1983, NS-30.2669-2671.
[12] M Suzuki. General theory of higher-order decomposition of exponential operators and symplectic integrators.Phys. Lett. A., 1922, 165:387-395.
[13] Haruo Yoshida. Construction of higher order sympletic integrators. Phys. Lett. A., 1990, 150: 262-269.
[14] Qin Meng-Zhao and Zhu Wen-Jie. Construction of Higher Order Symplectic schemes by composition.computing, 1991, 47: 309-321.
[15]高亮,李幼铭,陈旭荣,等,地震射线的辛几何算法尝试,地球物理学报,2000,43(3)402-410
GAO Liang, LI You-Ming, CHEN Xu-Rong, et al., An attempt to seismic ray tracing with symplectic algorithm. Chinese J. of Geophys. (in Chinese), 2000, 43(3):402-410.
2) QIN Meng-Zhao,Zhu Wen-Jie,ZHANG Mei-Qing.Construction of a three-stage difference scheme for ordinary differntial equations.Beijing:Computing Center,Chinese Academy of Sciences,1990.
[1]尹峰,李幼铭.反演二维非均匀介质背景中速度扰动的波动方程层析成像方法见:杨文采等编,应用地震层析成像,北京:地质出版社.1993
YIN Feng, LI You-Ming. Wave equation tomography by inversion of velocity disturbance in twodemensional inhomogeneous media. In: YANG Wen-Cai ed. Applied Seismic Tomography. Beijing:Geological Publishing House, 1993.
[ 2 ] George A Mcmechan, Walter D Mooney. Asymptotic ray theory and synthetic seismograms for laterallyvarying structures: theory and application to the imperial valley, California. Bull. Seism. Soc. Am. 1980,70: 2021 -2035.
[ 3 ] C H. Chapman, K Drummond, Body-wave seismograms in inhomogenous media using Maslov asympotictheory. Bull. Seism Soc. Am., 1982, 72:277-317.
[ 4 ] G D Spence, K P Whittall, R M. Clowes. Pratical synthetic seismograms laterallyvaring media calculatedby asympoticray theory. Bull. Seism. Soc. Am. 1984, 74: 1209-1223.
[ 5 ] C H. Chapman. Ray theory and its extensions: WKBJ and Maslov seismograms. J. Geophs., 1985, 58:27-43.
[ 6 ] V P. Maslov. Semi-classical Approximation in Quantum Mechanics. Hingham, Mass: D. Reidel, PublishingCompany, 1981.
[ 7 ] Ziolkowski R W, Deschamps G A. Asymptotic evaluation of high-frequency fields near a caustic: anintrodtction to Maslov method. Rodio Sci., 1984, 19: 1001-1025.
[ 8 ] V I Arnold. Mathematical methods of classical mechanics. Springer-Verlag, 1978.
[9] Feng Knag. On difference scheme and symplectic geometry, proceedings of the 1984 Beijing symposiumon differential geometry and differential equations, ed. Feng kang, Beijing: Science press, 1985, 42-58.
[10] Qin Meng-Zhao, Wang Dao-Liu and Zhang Mei-Qing. Explicit symplectic difference schemes for separableHamiliton system. JCM 1991, 9(3): 211-221.
[11] Ruth R D. A canonical integration technique. IEEE Trans. on Nuclear Seiences,1983, NS-30.2669-2671.
[12] M Suzuki. General theory of higher-order decomposition of exponential operators and symplectic integrators.Phys. Lett. A., 1922, 165:387-395.
[13] Haruo Yoshida. Construction of higher order sympletic integrators. Phys. Lett. A., 1990, 150: 262-269.
[14] Qin Meng-Zhao and Zhu Wen-Jie. Construction of Higher Order Symplectic schemes by composition.computing, 1991, 47: 309-321.
[15]高亮,李幼铭,陈旭荣,等,地震射线的辛几何算法尝试,地球物理学报,2000,43(3)402-410
GAO Liang, LI You-Ming, CHEN Xu-Rong, et al., An attempt to seismic ray tracing with symplectic algorithm. Chinese J. of Geophys. (in Chinese), 2000, 43(3):402-410.
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