基于模式和广义射线理论研究水平层状介质中面波模式的接触现象

doi:10.12196/j.issn.1000-3274.2023.04.006

摘要/Abstract

摘要： 在近地表面波勘探中, 尤其对于典型沉积盆地的探测, 基于观测数据得到的频率-相速度域的能谱中, 通常可以观察到基阶和一阶模式的模式接触现象(Mode-kissing)。基于模式理论, 频率域中面波模式对应由频散方程的根给出的一系列极点。基于广义射线理论, 时空域的多径, 造成了频率域中的多模。本文结合广义射线理论和简正模理论研究广义射线理论中极和极两个表面波极点贡献对频散曲线的影响, 解释实际观测的模式接触现象。基于两层介质模型, 分析下层半空间S波速度β⁽²⁾变化时, 随极点在复射线参数平面变化, 引起的基阶和一阶模式的频散曲线、本征位移和质点偏振的变化特征。研究发现, 模式接触现象出现在复射线参数平面内, 极越过1/β⁽²⁾表示的支点, 基本进入简正模区域时。与该极点对应的一阶露能模式, 其地表质点轨迹为顺进椭圆, 但本征位移随深度的变换表现出经典表面波特征, 能量主要集中在地表。

关键词: 广义射线, 本征值, 频散曲线, Cagniard-de Hoop方法, 广义反、透射系数, 自由表面波, 能陷波

Abstract: In near-surface surface wave exploration, especially in typical sedimentary basins, mode-kissing of the fundamental and first-order modes is often observed for the dispersion image in the frequency-phase velocity domain. In the mode theory of waves, surface wave mode in the frequency domain is associated with a series of poles determined by the dispersion equation. In the generalized ray theory, the multipath in the spatial-temporal domain gives rise to the multimode in the frequency domain. Combined with the generalized theory and the mode theory, the influence of the $\bar{P}$ pole and the $\bar{S}$ pole on dispersion curves is studied to explain the mode-kissing phenomenon. For a two-layer model, the characteristics of the dispersion curves, eigen displacement, and polarization of the particle, which varies with the moving of $\bar{P}$ pole in the complex ray parameter plane, are studied by changing the S wave velocity β⁽²⁾ of the bottom half space, both for the fundamental and the first overtone. It is found that mode-kissing appears when the $\bar{P}$ pole passes through the branch point 1/β⁽²⁾ in the complex ray parameter plane and just enter the area belongs to the normal mode. The particle motion of the first leaky mode corresponding to the $\bar{P}$ pole is a prograde ellipse, but the eigen displacement shows the characteristic of the classical surface wave, i.e., the displacement mainly concentrates near the surface and decays rapidly with depth.

Key words: Generalized ray, Eigenvalue, Dispersion curve, Cagniard-de Hoop method, Generalized reflection-transmission coefficient, Free surface wave, Trapped wave

中图分类号:

P315.3⁺1

王少曈, 鲁来玉. 基于模式和广义射线理论研究水平层状介质中面波模式的接触现象[J]. 地震, 2023, 43(4): 76-100.

WANG Shao-tong, LU Lai-yu. Study on the Mode-kissing of Surface Waves Based on the Theory of the Mode and Generalized Ray[J]. EARTHQUAKE, 2023, 43(4): 76-100.

参考文献

[1] Gilbert F, Knopoff L. The directivity problem for a buried line source[J]. Geophysics, 1961, 26(5): 626-634.
[2] Helmberger D V. The crust-mantle transition in the Bering Sea[J]. Bulletin of the Seismological Society of America, 1968, 58(1): 179-214.
[3] Wigginst R A, Helmberger D V. Synthetic seismogram computation by expansion in generalized rays[J]. Geophysical Journal International, 1974, 37(1): 73-90.
[4] Harkrider D G. Surface waves in multilayered elastic media I. Rayleigh and Love waves from buried sources in a multilayered elastic half-space[J]. Bulletin of the Seismological Society of America, 1964, 54(2): 627-679.
[5] Kerry N J. Synthesis of seismic surface waves[J]. Geophysical Journal International, 1981, 64(2): 425-446.
[6] Haskell N A. The dispersion of surface waves on multilayered media[J]. Bulletin of the Seismological Society of America, 1953, 43: 17-34.
[7] Lu L, Zhang B. Analysis of dispersion curves of Rayleigh waves in the frequency wavenumber domain[J]. Canadian Geotechnical Journal, 2004, 41(4): 583-598.
[8] Kennett B L N. Seismic wave propagation in stratified media[M]. Canberra: Cambridge University Press, 1983.
[9] Chen X. Seismogram synthesis in multi-layered half-space Part I. Theoretical formulations[J]. Earthquake Research in China, 1999, 13(2): 149-174.
[10] von der Pol B, Bremmer H. XIII. The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to badiotelegraphy and the theory of the rainbow.—Part I[J]. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1937, 24(159): 141-176.
[11] Cagniard L. Réflexion et réfraction des ondes séismiques progressives[M]. Paris: Gauthier-Villars, 1939.
Cagniard L. Reflecion and refraction of progressive seismic waves[M]. Paris: Gauthier-Villars, 1939 (in French).
[12] Cagniard L. Reflection and refraction of progressive seismic waves[M]. New York: McGraw-Hill, 1962.
[13] De Hoop A T. A modification of Cagniard's method for solving seismic pulse problems[J]. Applied Scientific Research, Section B, 1960, 8(1): 349-356.
[14] Spencer T W. The method of generalized reflection and transmission coefficients[J]. Geophysics, 1960, 25(3): 625-641.
[15] Rosenbaum J H. The long-time response of a layered elastic medium to explosive sound[J]. Journal of Geophysical Research, 1960, 65(5): 1577-1613.
[16] Gilbert F. Propagation of transient leaking modes in a stratified elastic waveguide[J]. Reviews of Geophysics, 1964, 2(1): 123-153.
[17] Haddon R A W. Computation of synthetic seismograms in layered earth models using leaking modes[J]. Bulletin of the Seismological Society of America, 1984, 74(4): 1225-1248.
[18]Aki K, Richards P G. Quantitative seismology (2nd ed.)[M]. Sausalito: University Science Books, 2002.
[19] Ben-Menahem A. Mode-ray duality[J]. Bulletin of the Seismological Society of America, 1964, 54(5A): 1315-1321.
[20] Dahlen F A, Tromp J. Theoretical global seismology[M]. Princeton: Princeton University Press, 1998.
[21] 张碧星, 鲁来玉. 层状半空间中导波的传播[J]. 声学学报, 2002, 27(4): 295-304.
ZHANG Bi-xing, LU Lai-yu. Propagation of guided waves in stratified half-space[J]. Acta Acustica, 2002, 27(4): 295-304 (in Chinese).
[22] 张碧星, 肖柏勋, 杨文杰, 等. 瑞利波勘探中“之”形频散曲线的形成机理及反演研究[J]. 地球物理学报, 2000, 43(4): 557-567.
ZHANG Bi-xing, XIAO Bai-xun, YANG Wen-jie, et al. Mechanism of “zigzag” dispersion curves in Rayleigh exploration and its inversion study[J]. Chinese Journal of Geophysics, 2000, 43(4): 557-567 (in Chinese).
[23] Gao L, Xia J, Pan Y. Misidentification caused by leaky surface wave in high-frequency surface wave method[J]. Geophysical Journal International, 2014, 199(3): 1452-1462.
[24] Xia J, Xu Y, Luo Y, et al. Advantages of using multichannel analysis of love waves (MALW) to estimate near-surface shear-wave velocity[J]. Surveys in Geophysics, 2012, 33(5): 841-860.
[25] Gao L, Xia J, Pan Y, et al. Reason and condition for mode kissing in MASW method[J]. Pure and Applied Geophysics, 2016, 173: 1627-1638.
[26] Qin T, Lu L, Ding Z, et al. High-resolution 3D shallow S wave velocity structure of Tongzhou, subcenter of Beijing, inferred from multimode Rayleigh waves by beamforming seismic noise at a dense array[J]. Journal of Geophysical Research: Solid Earth, 2022, 127(5): e2021JB023689.
[27] 张碧星, 鲁来玉, 鲍光淑. 瑞利波勘探中“之”字形频散曲线研究[J]. 地球物理学报, 2002, 45(2): 263-274.
ZHANG Bi-xing, LU Lai-yu, BAO Guang-shu. A Study on “zigzag” dispersion curves in Rayleigh wave exploration[J]. Chinese Journal of Geophysics, 2002, 45(2): 263-274 (in Chinese).
[28] 鲁来玉, 张碧星, 汪承灏. 基于瑞利波高阶模式反演的实验研究[J]. 地球物理学报, 2006, 49(4): 1082-1091.
LU Lai-yu, ZHANG Bi-xing, WANG Cheng-hao. Experiment and inversion studies on Rayleigh wave considering higher modes [J]. Chinese Journal of Geophysics, 2006, 49(4): 1082-1091 (in Chinese).
[29] Sezawa K, Kanai K. Discontinuity in dispersion curves of Rayleigh-waves[J]. Proceedings of the Imperial Academy, 1935, 11(1): 13-14.
[30] Boaga J, Cassiani G, Strobbia C L, et al. Mode misidentification in Rayleigh waves: Ellipticity as a cause and a cure[J]. Geophysics, 2013, 78(4): EN17-EN28.
[31] Boaga J, Vignoli G, Deiana R, et al. The influence of subsoil structure and acquisition parameters in MASW mode mis-identification[J]. Journal of Environmental and Engineering Geophysics, 2014, 19(2): 87-99.
[32] 鲁来玉. 分层介质半空间瑞利波模式分析和介质参数反演[D]. 北京: 中国科学院声学研究所, 2004.
LU Lai-yu. Analysis of half-space Rayleigh wave patterns in layered media and inversion of media parameters[D]. Beijing: Institute of Acoustics, Chinese Academy of Sciences, 2004 (in Chinese).
[33] Foti S. Multistation methods for geotechnical characterization using surface waves[M]. Turin: Politecnico di Torino, 2000.
[34] 吴勃. 面波和地球自由振荡的本征模理论[D]. 合肥: 中国科学技术大学, 2016.
WU Bo. The theory of normal modes of surface waves and Earth's free oscillations [D]. Hefei: University of Science and Technology of China, 2016 (in Chinese).
[35] 刘雪峰, 凡友华, 陈晓非. Rayleigh 波频散曲线“交叉”及多模式耦合作用研究[J]. 地球物理学报, 2009 (9): 2302-2309.
LIU Xue-feng, FAN You-hua, CHEN Xiao-fei. A study of “crossover” and multi-mode coupling of Rayleigh wave dispersion curve[J]. Chinese Journal of Geophysics, 2009 (9): 2302-2309 (in Chinese).
[36]Gilbert F, Laster S J. Excitation and propagation of pulses on an interface[J]. Bulletin of the Seismological Society of America, 1962, 52(2): 299-319.
[37]Tanimoto T, Rivera L. Prograde Rayleigh wave particle motion[J]. Geophysical Journal International, 2005, 162(2): 399-405.
[38]Malischewsky Auning P G, Lomnitz C, Wuttke F, et al. Prograde Rayleigh-wave motion in the valley of Mexico[J]. Geofísica internacional, 2006, 45(3): 149-162.
[39] Malischewsky P G, Scherbaum F, Lomnitz C, et al. The domain of existence of prograde Rayleigh-wave particle motion for simple models[J]. Wave Motion, 2008, 45(4): 556-564.
[40]Gribler G, Liberty L M, Mikesell T D, et al. Isolating retrograde and prograde Rayleigh-wave modes using a polarity mute[J]. Geophysics, 2016, 81(5): V379-V385.
[41] Gribler G, Mikesell T D. Methods to isolate retrograde and prograde Rayleigh-wave signals[J]. Geophysical Journal International, 2019, 219(2): 975-994.
[42] Ma Y, Clayton R W, Li D. Higher-mode ambient-noise Rayleigh waves in sedimentary basins[J]. Geophysical Journal International, 2016, 206(3): 1634-1644.
[43] Hu S, Luo S, Yao H. The frequency-Bessel spectrograms of multicomponent cross-correlation functions from seismic ambient noise[J]. Journal of Geophysical Research: Solid Earth, 2020, 125(8): e2020JB019630.
[44] Ji Z, Wang B, Yang W, et al. Observation of higher-mode surface waves from an active source in the Hutubi Basin, Xinjiang, China[J]. Bulletin of the Seismological Society of America, 2021, 111(3): 1181-1198.
[45] Laster S J, Foreman J G, Linville A F. Theoretical investigation of modal seismograms for a layer over a half-space[J]. Geophysics, 1965, 30(4): 571-596.
[46]Cochran M D, Woeber A F, De Bremaecker J C. Body waves as normal and leaking modes, 3. Pseudo modes and partial derivatives on the (+-) sheet[J]. Reviews of Geophysics, 1970, 8(2): 321-357.
[47] Dainty A M. Leaking modes in a crust with a surface layer[J]. Bulletin of the Seismological Society of America, 1971, 61(1): 93-107.
[48] Watson T H. A real frequency, complex wave-number analysis of leaking modes[J]. Bulletin of the Seismological Society of America, 1972, 62(1): 369-384.
[49] Shi C, Ren H, Li Z, et al. Calculation of normal and leaky modes for horizontal stratified models based on a semi-analytical spectral element method[J]. Geophysical Journal International, 2022, 230(3): 1928-1947.
[50] Alsop L E. The leaky-mode period equation—a plane-wave approach[J]. Bulletin of the Seismological Society of America, 1970, 60(6): 1989-1998.
[51] Oliver J, Major M. Leaking modes and the PL phase[J]. Bulletin of the Seismological Society of America, 1960, 50(2): 165-180.
[52] Ewing W M, Jardetzky W S, Press F, et al. Elastic waves in layered media[M]. New York: McGraw-Hill, 1957.
[53] Brekhovskikh L M. Waves in layered media (2^nd ed.)[M]. New York: Academic Press, 1980.
[54] Thomson W T. Transmission of elastic waves through a stratified solid medium[J]. Journal of Applied Physics, 1950, 21(2): 89-93.
[55] Chen X. A systematic and efficient method of computing normal modes for multilayered half-space[J]. Geophysical Journal International, 1993, 115(2): 391-409.
[56] 何耀锋, 陈蔚天, 陈晓非. 利用广义反射-透射系数方法求解含低速层水平层状介质模型中面波频散曲线问题[J]. 地球物理学报, 2006, 49(4): 1074-1081.
HE Yao-feng, CHEN Wei-tian, CHEN Xiao-fei. Normal mode computation by the generalized reflection-transmission coefficient method in planar layered half space[J]. Chinese Journal of Geophysics, 2006, 49(4): 1074-1081 (in Chinese).
[57] Mellman G R, Helmberger D V. High-frequency attenuation by a thin high-velocity layer[J]. Bulletin of the Seismological Society of America, 1974, 64(5): 1383-1388.
[58] Phinney R A. Leaking modes in the crustal waveguide: 1. The oceanic PL wave[J]. Journal of Geophysical Research, 1961, 66(5): 1445-1469.
[59] Chapman C. Lamb's problem and comments on the paper ‘on leaking modes' by Usha Gupta[J]. Pure and Applied Geophysics, 1972, 94: 233-247.
[60] Roth M, Holliger K. The non-geometric S wave in high-resolution seismic data: observations and modelling[J]. Geophysical Journal International, 2000, 140(1): F5-F11.
[61] Gilbert F, Laster S J. Experimental investigation of PL modes in a single layer[J]. Bulletin of the Seismological Society of America, 1962, 52(1): 59-66.
[62] Oliver J. On the long period character of shear waves[J]. Bulletin of the Seismological Society of America, 1961, 51(1): 1-12.
[63] Oliver J. Propagation of PL waves across the United States[J]. Bulletin of the Seismological Society of America, 1964, 54(1): 151-160.
[64] Su S S, Dorman J. The use of leaking modes in seismogram interpretation and in studies of crust-mantle structure[J]. Bulletin of the Seismological Society of America, 1965, 55(6): 989-1021.
[65] Poupinet G, Wright C. The generation and properties of shear-coupled PL waves[J]. Bulletin of the Seismological Society of America, 1972, 62(6): 1699-1710.
[66] Baag C E, Langston C A. Shear-coupled PL[J]. Geophysical Journal International, 1985, 80(2): 363-385.
[67] Pettersen . Observations of shear-coupled PL waves from shallow earthquakes in the Sumatra and Java subduction zones[J]. Geophysical Journal International, 2007, 168(1): 299-314.
[68] Furumura T, Kennett B L N. Regional distance PL phase in the crustal waveguide—An analog to the teleseismic W phase in the upper-mantle waveguide[J]. Journal of Geophysical Research: Solid Earth, 2018, 123(5): 4007-4024.
[69] García-Jerez A, Sánchez-Sesma F J. Slowly-attenuating P-SV leaky waves in a layered elastic halfspace. Effects on the coherences of diffuse wavefields[J]. Wave Motion, 2015, 54: 43-57.
[70] Fuchs K, Mller G. Computation of synthetic seismograms with the reflectivity method and comparison with observations[J]. Geophysical Journal International, 1971, 23(4): 417-433.
[71] 李正波. 频率贝塞尔变换法提取地震记录中的频散信息[D]. 合肥: 中国科学技术大学, 2020.
LI Zheng-bo. Frequency-bessel transform method for extracting dispersion information from seismic records[D]. Hefei: University of Science and Technology of China, 2020 (in Chinese).
[72] Gilbert F, Laster S J, Backus M M, et al. Observation of pulses on an interface[J]. Bulletin of the Seismological Society of America, 1962, 52(4): 847-868.