• ISSN 2097-1893
  • CN 10-1855/P

黏弹地球地震变形理论研究进展和展望

唐河 孙文科

引用本文: 唐河,孙文科. 黏弹地球地震变形理论研究进展和展望. 地球与行星物理论评,2021,52(1):11-26
Tang H, Sun W K. Progress and prospect of deformation theory in the viscoelastic earth. Reviews of Geophysics and Planetary Physics, 2021, 52(1):11-26

黏弹地球地震变形理论研究进展和展望

doi: 10.19975/j.dqyxx.2020-005
基金项目: 国家自然科学基金资助项目(41774088,41974093,41331066,41474059)
详细信息
    通讯作者:

    孙文科(1956-),男,教授,博士生导师,主要从事地震位错理论及应用、重力变化的观测与解释、重力卫星GRACE的应用研究. E-mail:sunw@ucas.ac.cn

  • 中图分类号: P315.01, P315.4, P315.2

Progress and prospect of deformation theory in the viscoelastic earth

Funds: Supported by the National Natural Science Foundation of China (Grant No. 41774088, 41974093, 41331066, 41474059)
  • 摘要:Love(1911)研究了自重球体的弹性变形后,基于不同的黏弹性地球模型,许多科学家都对地震变形问题进行了深入研究,主要发展了基于半无限空间和球形地球模型的黏弹地球地震变形理论. 地震变形问题通常经积分变换、基函数展开等技术处理后,简化为求解满足特定震源和地表边界条件的常微分方程组问题. 针对这一独特的数学物理边值问题,本文以全解析、半解析和数值积分解等求解形式概述了近几十年发展的基于规则几何形态地球模型的黏弹地球地震变形理论,并讨论了各种方法的特点. 此外,针对三维地球模型,本文也简单回顾了目前的研究进展和存在问题. 总之,本文综述了过去半个多世纪以来的黏弹地球地震变形理论的发展历史、研究现状和最新进展,并讨论目前存在的问题和未来的发展趋势.

     

  • 图  1  分层球形地球模型中黏弹地震变形转化为复域等效弹性变形(一致性原理)示意图. (a)为Burgers体黏弹性变形. (b)为复域内等效弹性变形. Burgers体由Maxwell体和Kelvin体串联得到,弹簧原件表示弹性变形,阻尼器表示黏性变形,$\rho (r)$为密度,$\lambda {\rm{(}}r{\rm{)}}$$\mu (r)$为Lamé参数,$\eta (r)$为黏度,$\lambda {\rm{(}}s{\rm{)}}$$\mu (s)$为复域Lamé参数,它们均是半径r的函数,下标“K”和“M”分别表示Kelvin体和Maxwell体. 地球模型中:内核为弹性固体,外核为液态,地壳和地幔为黏弹性介质

    Figure  1.  The diagram of transforming a viscoelastic seismic deformation problem into an equivalent elastic deformation problem in complex domain (the correspondence principle) in a layered spherical Earth model. (a) the viscoelastic deformation of the Burgers body. (b) the equivalent elastic deformation in complex domain. The Burgers body is obtained by a series connection of a Maxwell body and a Kelvin body. Spring element represents elastic deformation; damper represents viscous deformation; $\rho (r)$ is density, $\lambda {\rm{(}}r{\rm{)}}$ and $\mu (r)$ are Lamé parameters, $\eta (r)$ is viscosity; $\lambda {\rm{(}}s{\rm{)}}$ and $\mu (s)$ is the complex Lamé parameters. Subscripts "K" and "M" represent Kelvin body and Maxwell body, respectively. All parameters are functions of radius r. In the Earth model: the inner core is elastic solid; the outer core is liquid; the crust and mantle are viscoelastic media

    图  2  由复Love数或Green函数,经逆Laplace变换计算时间域内Love数或Green函数的方法示意图. 负半实轴上的灰色点表示奇异点,“…”表示省略的奇异点.(a)中的黑色圆圈表示用Bromwich回路积分计算该奇异点处的留数,向上的有向直线表示逆Laplace积分定义式,对应路径${\rm{Re}} {\rm{(}}s{\rm{) = }}c$;(b)中阴影中的五角星表示Post-Widder方法计算点;(c)中有向矩形回路表示回路积分路径;(d)中半圆形表示积分核近似方法的路径,其内虚线表示纵向积分的实际路径,黑色点表示采样位置,$a$为引入参数,$t$为时间

    Figure  2.  The method of calculating Love numbers or Green function in time domain by inverse Laplace transform from complex Love numbers or Green function. The grey points on the negative semireal axis denote singular points. “…” denotes omitted singular points. The black circle in (a) represents the residue at the singular point calculated by Bromwich contour integral. The upward directed straight line ${\rm{Re}} {\rm{(}}s{\rm{) = }}c$ represents the inverse Laplace transform by the definition. In (b) the pentagram in the shadow represents the calculation point of the Post-Widder method. The directed rectangular in (c) represents the contour integration path. The semicircle in (d) represents the path of the integral kernel approximation method. Here the dotted line inside (d) represents the actual path of the vertical integration, the black dot represents the sampling position, $a$ is the introduced parameter, and $t$ is the time

  • [1] Aki K, Richards P G. Quantitative Seismology: Theory and Methods[M]. W. H. Freeman, 1980.
    [2] Bekaert D P S, Segall P, Wright T J, et al. A network inversionfilter combining GNSS and InSAR for tectonic slip modeling [J]. Journal of Geophysical Research-Solid Earth, 2016, 121(3): 2069-2086. doi: 10.1002/2015JB012638
    [3] Bonafede M, Ferrari C. Analytical models of deformation and residual gravity changes due to a Mogi source in a viscoelastic medium [J]. Tectonophysics, 2009, 471(1-2): 4-13. doi: 10.1016/j.tecto.2008.10.006
    [4] Broerse T, Riva R, Simons W, et al. Postseismic GRACE and GPS observations indicate a rheology contrast above and below the Sumatra slab [J]. Journal of Geophysical Research-Solid Earth, 2015, 120(7): 5343-5361. doi: 10.1002/2015JB011951
    [5] Cambiotti G. Joint estimate of the coseismic 2011 Tohoku earthquake fault slip and post-seismic viscoelastic relaxation by GRACE data inversion[J]. Geophysical Journal International, 2020, 220(2): 1012-1022.
    [6] Cambiotti G, Barletta V R, Bordoni A, et al. A comparative analysis of the solutions for a Maxwell Earth: the role of the advection and buoyancy force[J]. Geophysical Journal International, 2009, 176(3): 995-1006. doi: 10.1111/j.1365-246X.2008.04034.x
    [7] Cambiotti G, Klemann V, Sabadini R. Compressible viscoelastodynamics of a spherical body at long timescales and its isostatic equilibrium [J]. Geophysical Journal International, 2013, 193(3): 1071-1082. doi: 10.1093/gji/ggt026
    [8] Cambiotti G, Sabadini R. The compressional and compositional stratifications in Maxwell earth models: the gravitational overturning and the long-period tangential flux [J]. Geophysical Journal International, 2010a, 180(2): 475-500. doi: 10.1111/j.1365-246X.2009.04434.x
    [9] Cambiotti G, Sabadini R, Bordoni A. The continuous relaxation spectrum of Maxwell Earth models: a new method [J]. Egu General Assembly, 2010b, 12:9288.
    [10] Cambiotti G, Sabadini R, Yuen D A. Time-dependent geoid anomalies at subduction zones due to the seismic cycle[J]. Geophysical Journal International, 2018, 212(1): 139-150. doi: 10.1093/gji/ggx421
    [11] Cannelli V, Melini D, Piersanti A, et al. Application of the Post-Widder Laplace inversion algorithm to postseismic rebound models [J]. Nuovo Cimento Della Societa Italiana Di Fisica C-Colloquia on Physics, 2009, 32(2): 197-200.
    [12] Chao B F, Ding H. Spherical harmonic stacking for the singlets of Earth's normal modes of free oscillation [J]. Geophysical Ressearch Letters, 2014, 41(15): 5428-5435. doi: 10.1002/2014GL060700
    [13] Chao B F, Gross R S. Changes in the Earth's rotation and low-degree gravitational field induced by earthquakes [J]. Geophysical Journal International, 1987, 91(3): 569-596. doi: 10.1111/j.1365-246X.1987.tb01659.x
    [14] Chen W, Ray J, Li J C, et al. Polar motion excitations for an Earth model with frequency-dependent responses: 1. A refined theory with insight into the Earth's rheology and core-mantle coupling [J]. Journal of Geophysical Research: Solid Earth, 2013, 118(9): 4975-4994. doi: 10.1002/jgrb.50314
    [15] Chen W, Shen W B, Han J C, et al. Free wobble of the triaxial earth: Theory and comparisons with international earth rotation service (IERS) data [J]. Surveys in Geophysics, 2009, 30(1): 39-49. doi: 10.1007/s10712-009-9057-3
    [16] 程威,胡小刚. 南极上地幔结构异常对长周期自由振荡的影响 [J]. 地球物理学报, 2018, 61(8): 3211-3218. doi: 10.6038/cjg2018K0057

    Cheng W, Hu X G. The effect of upper mantle structure on long period Earth's free oscillation at the South Polar[J]. Chinese Journal of Geophysics, 61(8): 3211-3218 (in Chinese). doi: 10.6038/cjg2018K0057
    [17] Dahlen F, Tromp J. Theoretical Global Seismology[M]. New Jersey: Princeton University Press, 1998.
    [18] Davis J L, Elósegui P, Mitrovica J X, et al. Climate‐driven deformation of the solid Earth from GRACE and GPS [J]. Geophysical Ressearch Letters, 2004, 31(24): 24605-24608. doi: 10.1029/2004GL021435
    [19] Ding H. Attenuation and excitation of the ∼6 year oscillation in the length-of-day variation [J]. Earth and Planetary Science Letters, 2019, 507(131-139. doi: 10.1016/j.jpgl.2018.12.003
    [20] Ding H, Chao B F. Data stacking methods for isolation of normal-mode singlets of Earth's free oscillation: Extensions, comparisons, and applications [J]. Journal of Geophysical Research: Solid Earth, 2015, 120(7): 5034-5050. doi: 10.1002/2015JB012025
    [21] Dziewonski A M, Anderson D L. Preliminary reference Earth model [J]. Physics of the Earth and Planetary Interiors, 1981, 25(4): 297-356. doi: 10.1016/0031-9201(81)90046-7
    [22] Fang M, Hager B H. A singularity-free approach to post glacial rebound calculations [J]. Geophysical Ressearch Letters, 1994, 21(19): 2131-2134. doi: 10.1029/94GL01886
    [23] Fang M, Hager B H. The singularity mystery associated with a radially continuous Maxwell viscoelastic structure [J]. Geophysical Journal International, 1995, 123(3): 849-865. doi: 10.1111/j.1365-246X.1995.tb06894.x
    [24] Farrell W E. Deformation of the Earth by surface loads [J]. Reviews of Geophysics & Space Physics, 1972, 10(3): 761-797.
    [25] Feng M X, Bie L D, Rietbrock A. Probing the rheology of continental faults: decade of post-seismic InSAR time-series following the 1997 Manyi (Tibet) earthquake [J]. Geophysical Journal International, 2018, 215(1): 600-613. doi: 10.1093/gji/ggy300
    [26] Fernández J, Rundle J B. Gravity changes and deformation due to a magmatic intrusion in a two-layered crustal model [J]. Journal of Geophysical Research: Solid Earth, 1994, 99(B2): 2737-2746. doi: 10.1029/93JB02449
    [27] Fernández J, Yu T T, Rundle J B. Deformation produced by a rectangular dipping fault in a viscoelastic-gravitational layered earth model. Part I: Thrust fault—FLTGRV and FLTGRH FORTRAN programs [J]. Computers & Geosciences, 1996, 22(7): 735-750.
    [28] Fu G Y, Sun W K. Effects of spatial distribution of fault slip on calculating co-seismic displacement: Case studies of the Chi-Chi earthquake (MW7.6) and the Kunlun earthquake (MW7.8) [J]. Geophysical Ressearch Letters, 2004, 31(21):L21601.
    [29] Fu G Y, Sun W K. Surface coseismic gravity changes caused by dislocations in a 3-D heterogeneous earth [J]. Geophysical Journal International, 2008, 172(2): 479-503. doi: 10.1111/j.1365-246X.2007.03684.x
    [30] Fu G Y, Sun W K, Fukuda Y, et al. Effects of Earth’s curvature and radial heterogeneity in dislocation studies: Case studies of the 2008 Wenchuan earthquake and the 2004 Sumatra earthquake [J]. Earthquake Science, 2010a, 23(4): 301-308. doi: 10.1007/s11589-010-0727-5
    [31] Fu G Y, Sun W K, Fukuda Y C, et al. Coseismic displacements caused by point dislocations in a three-dimensional heterogeneous, spherical earth model [J]. Geophysical Journal International, 2010b, 183(2): 706-726. doi: 10.1111/j.1365-246X.2010.04757.x
    [32] Gaver D P. Observing stochastic processes and approximate transform inversion [J]. Operations Research, 1966, 14(3): 444-459. doi: 10.1287/opre.14.3.444
    [33] Gharti H N, Langer L, Tromp J. Spectral-infinite-element simulations of coseismic and post-earthquake deformation [J]. Geophysical Journal International, 2019a, 216(2): 1364-1393. doi: 10.1093/gji/ggy495
    [34] Gharti H N, Langer L, Tromp J. Spectral-infinite-element simulations of earthquake-induced gravity perturbations [J]. Geophysical Journal International, 2019b, 217(1): 451-468. doi: 10.1093/gji/ggz028
    [35] Gilbert F. Excitation of the normal modes of the earth by earthquake sources [J]. Geophysical Journal International, 1971, 22(2): 223-226. doi: 10.1111/j.1365-246X.1971.tb03593.x
    [36] Gilbert F, Backus G. A computational proplem encountered in a study of the earth's normal modes[C]// Proceedings of the December 9-11, 1968, fall joint computer conference, part II on-AFIPS '68 (Fall, part II). San Francisco, California; ACM. 1968: 1273-1277.
    [37] Han S C, Sauber J, Pollitz F. Postseismic gravity change after the 2006-2007 great earthquake doublet and constraints on the asthenosphere structure in the central Kuril Islands [J]. Geophysical Ressearch Letters, 2016, 43(7): 3169-3177. doi: 10.1002/2016GL068167
    [38] Hanyk L, Matyska C, Yuen D A. Initial-value approach for viscoelastic responses of the earth’s mantle. In: Wu Patrick (ed) Dynamics of the Ice Age Earth: A Modern Perspective[M]. Trans Tech Publication Ltd, Switzerland, 1998: 135-154.
    [39] Hanyk L, Moser J, Yuen D A, et al. Time-domain approach for the transient responses in stratified viscoelastic earth models [J]. Geophysical Ressearch Letters, 1995, 22(10): 1285-1288. doi: 10.1029/95GL01087
    [40] Hanyk L. Viscoelastic response of the Earth: Initial-value approach[D]. Charles University, 1999.
    [41] Hanyk L, Yuen D A, Matyska C. Initial-value and modal approaches for transient viscoelastic responses with complex viscosity profiles [J]. Geophysical Journal International, 1996, 127(2): 348-362. doi: 10.1111/j.1365-246X.1996.tb04725.x
    [42] Hashima A, Takada Y, Fukahata Y, et al. General expressions for internal deformation due to a moment tensor in an elastic/viscoelastic multilayered half-space [J]. Geophysical Journal International, 2008, 175(3): 992-1012. doi: 10.1111/j.1365-246X.2008.03837.x
    [43] Haskell N A. The dispersion of surface waves in multilayered media [J]. Bullseismol Socam, 1951, 43:1-2.
    [44] Heki K, Miyazaki S, Tsuji H. Silent fault slip following an interplate thrust earthquake at the Japan Trench[J]. Nature, 1997, 386(6625): 595-598. doi: 10.1038/386595a0
    [45] Hetland E A, Hager B H. Postseismic and interseismic displacements near a strike‐slip fault: A two‐dimensional theory for general linear viscoelastic rheologies [J]. Journal of Geophysical Research: Solid Earth, 2005, 110:B10401. doi: 10.1029/2005JB003689
    [46] Hu Y. Three-dimensional viscoelastic finite element model for postseismic deformation of the great 1960 Chile earthquake [J]. Journal of Geophysical Research, 2004, 109:B12403. doi: 10.1029/2004JB003163
    [47] 胡小刚,薛秀秀, 郝晓光. 利用地球简正模耦合研究上地幔过渡区方位各向异性 [J]. 地球物理学报, 2012, 55(6): 1903-1911. doi: 10.6038/j.issn.0001-5733.2012.06.011

    Hu X G, Xue X X, Hao X G. Study of azimuthal anisotropy in the transition zone of the Earth’s upper mantle with the coupling of normal modes [J]. Chinese Journal of Geophysics. 2012, 55(6): 1903-1911 (in Chinese). doi: 10.6038/j.issn.0001-5733.2012.06.011
    [48] Imanishi Y, Sato T, Higashi T, et al. A network of superconducting gravimeters detects submicrogal coseismic gravity changes [J]. Science, 2004, 306(5695): 476-478. doi: 10.1126/science.1101875
    [49] 江颖,徐建桥, 孙和平. 深内部地球结构对内核平动振荡本征周期的影响[J]. 地球物理学报, 2014, 57(4): 1041-1048. doi: 10.6038/cjg20140403

    Jiang Y, Xu J Q, Sun H P. The influence of deep interior structure on the eigenperiod of inner core’s translational oscillations[J]. Chinese Journal of Geophysics. 2014, 57(4): 1041-1048 (in Chinese). doi: 10.6038/cjg20140403
    [50] Jonsson S, Segall P, Pedersen R, et al. Post-earthquake ground movements correlated to pore-pressure transients [J]. Nature, 2003, 424(6945): 179-183. doi: 10.1038/nature01776
    [51] Kanamori H, Anderson D L. Importance of physical dispersion in surface wave and free oscillation problems: Review [J]. Reviews of Geophysics, 1977, 15(1): 105-112. doi: 10.1029/RG015i001p00105
    [52] Langer L, Gharti H N, Tromp J. Impact of topography and three-dimensional heterogeneity on coseismic deformation [J]. Geophysical Journal International, 2019, 217(2): 866-878. doi: 10.1093/gji/ggz060
    [53] Liu T, Fu G Y, She Y W, et al. Green’s functions for post-seismic strain changes in a realistic earth model and their application to the Tohoku-Oki MW 9.0 earthquake [J]. Pure Appl Geophys, 2018, 176(9): 3929-3949.
    [54] Longman I M. A Green's function for determining the deformation of the Earth under surface mass loads: 1. Theory [J]. Journal of Geophysical Research, 1962, 67(2): 845-850. doi: 10.1029/JZ067i002p00845
    [55] Longman I M. A Green's function for determining the deformation of the Earth under surface mass loads: 2. Computations and numerical results [J]. Journal of Geophysical Research, 1963, 68(2): 485-496. doi: 10.1029/JZ068i002p00485
    [56] Love A E H. Some Problems of Geodynamics[M]. Cambridge: Cambridge University Press, 1911.
    [57] 鹿璐, 李小凡, 陈楠, 等. 核幔边界地幔侧不同尺度横向非均匀对地球自由振荡的影响 [J]. 地球物理学进展, 2017, 32(04): 1465-1473. doi: 10.6038/pg20170407

    Lu L, Li X F, Chen N, et al. Influence of different scales of lateral inhomogeneous structures on the mantle side of the CMB on the Earth's free oscillation [J]. Progress in Geophysics, 32(4): 1465-1473 (in Chinese). doi: 10.6038/pg20170407
    [58] Matsu'ura M, Tanimoto T. Quasi-static deformations due to an inclined, rectangular fault in a viscoelastic half-space [J]. Journal of Physics of the Earth, 1980, 28(1): 103-118. doi: 10.4294/jpe1952.28.103
    [59] Mcconnell R K. Isostatic adjustment in a layered earth [J]. Journal of Geophysical Research, 1965, 70(20): 5171-5188. doi: 10.1029/JZ070i020p05171
    [60] Melini D, Cannelli V, Piersanti A, et al. Post-seismic rebound of a spherical Earth: new insights from the application of the Post-Widder inversion formula [J]. Geophysical Journal International, 2008, 174(2): 672-695. doi: 10.1111/j.1365-246X.2008.03847.x
    [61] Metois M, Socquet A, Vigny C. Interseismic coupling, segmentation and mechanical behavior of the central Chile subduction zone [J]. Journal of Geophysical Research-Solid Earth, 2012, 117(B3): 406-421.
    [62] Molodenskiy S M. The influence of horizontal inhomogeneities in the mantle on the amplitude of tidal oscillations [J]. Physics of the Solid Earth, 1977, 13(2): 77-80.
    [63] Molodenskiy S M. The effect of lateral heterogeneities upon the tides [J]. BIM Fevrier, 1980, 80:4833-4850.
    [64] Moore J D, Yu H, Tang C H, et al. Imaging the distribution of transient viscosity after the 2016 MW7.1 Kumamoto earthquake [J]. Science, 2017, 356(6334): 163-167. doi: 10.1126/science.aal3422
    [65] Okada Y. Surface deformation due to shear and tensile faults in a half-space [J]. Bulletin of the Seismological Society of America, 1985, 75(4): 1135-1154.
    [66] Okada Y. Internal deformation due to shear and tensile faults in a half-space [J]. Bulletin of the Seismological Society of America,1992, 82(2): 1018-1040.
    [67] Okubo S. Asymptotic solutions to the static deformation of the Earth - I. Spheroidal mode [J]. Geophysical Journal International, 1988, 92(1): 39–51. doi: 10.1111/j.1365-246X.1988.tb01119.x
    [68] Okubo S. Potential and gravity changes raised by point dislocations [J]. Geophysical Journal International, 1991, 105(3): 573-586. doi: 10.1111/j.1365-246X.1991.tb00797.x
    [69] Okubo S. Gravity and potential changes due to shear and tensile faults in a half-space [J]. Journal of Geophysical Research: Solid Earth, 1992, 97(B5): 7137-7144. doi: 10.1029/92JB00178
    [70] Piersanti A, Spada G, Sabadini R. Global postseismic rebound of a viscoelastic Earth: Theory for finite faults and application to the 1964 Alaska earthquake [J]. Journal of Geophysical Research-Solid Earth, 1997, 102(B1): 477-492. doi: 10.1029/96JB01909
    [71] Piersanti A, Spada G, Sabadini R, et al. Global post-seismic deformation [J]. Geophysical Journal International, 1995, 120(3): 544-566. doi: 10.1111/j.1365-246X.1995.tb01838.x
    [72] Pollitz F F. Postseismic relaxation theory on the spherical earth [J]. Bulletin of the Seismological Society of America, 1992, 82(1): 422-453.
    [73] Pollitz F F. Gravitational viscoelastic postseismic relaxation on a layered spherical Earth [J]. Journal of Geophysical Research-Solid Earth, 1997, 102(B8): 17921-17941. doi: 10.1029/97JB01277
    [74] Pollitz F F. Transient rheology of the uppermost mantle beneath the Mojave Desert, California [J]. Earth and Planetary Science Letters, 2003a, 215(1-2): 89-104. doi: 10.1016/S0012-821X(03)00432-1
    [75] Pollitz F F. Post-seismic relaxation theory on a laterally heterogeneous viscoelastic model [J]. Geophysical Journal International, 2003b, 155(1): 57-78. doi: 10.1046/j.1365-246X.2003.01980.x
    [76] Pollitz F F, Burgmann R, Banerjee P. Post-seismic relaxation following the great 2004 Sumatra-Andaman earthquake on a compressible self-gravitating Earth [J]. Geophysical Journal International, 2006, 167(1): 397-420. doi: 10.1111/j.1365-246X.2006.03018.x
    [77] Post E L. Generalized differentiation [J]. Transactions of the American Mathematical Society, 1930, 32(1-4): 723-781.
    [78] Qiu Q, Moore J D P, Barbot S, et al. Transient rheology of the Sumatran mantle wedge revealed by a decade of great earthquakes [J]. Nature Communications, 2018, 9(1): 995-1007. doi: 10.1038/s41467-018-03298-6
    [79] Rodkin M V, Kaftan V I. Post-seismic relaxation from geodetic and seismic data [J]. Geodesy and Geodynamics, 2017, 8(1): 13-16. doi: 10.1016/j.geog.2017.01.001
    [80] Rogister Y, Rochester M G. Normal-mode theory of a rotating earth model using a Lagrangian perturbation of a spherical model of reference [J]. Geophysical Journal International, 2004, 159(3): 874-908. doi: 10.1111/j.1365-246X.2004.02447.x
    [81] Rundle J B. Static elastic-gravitational deformation of a layered half-space by point couple sources [J]. Journal of Geophysical Research, 1980, 85(Nb10): 5355-5363. doi: 10.1029/JB085iB10p05355
    [82] Rundle J B. Viscoelastic-gravitational deformation by a rectangular thrust-fault in a layered earth [J]. Journal of Geophysical Research, 1982, 87(Nb9): 7787-7796. doi: 10.1029/JB087iB09p07787
    [83] Sabadini R, Yuen D A, Boschi E. The effects of post-seismic motions on the moment of inertia of a stratified viscoelastic earth with an asthenosphere[J]. Geophysical Journal of the Royal Astronomical Society, 1984, 79(3): 727-745. doi: 10.1111/j.1365-246X.1984.tb02865.x
    [84] 单新建, 屈春燕, 龚文瑜, 等. 2017年8月8日四川九寨沟7.0级地震InSAR同震形变场及断层滑动分布反演 [J]. 地球物理学报, 2017, 60(12): 4527-4536. doi: 10.6038/cjg20171201

    Shan X J, Qu C Y, Gong W Y, et al.Coseismic deformation field of the Jiuzhaigou MS7.0 earthquake from Sentinel-1A InSAR data and fault slip inversion. Chinese Journal of Geophysics,60(12): 4527-4536 (in Chinese). doi: 10.6038/cjg20171201
    [85] Spada G, Boschi L. Using the Post-Widder formula to compute the Earth's viscoelastic Love numbers [J]. Geophysical Journal International, 2006, 166(1): 309-321. doi: 10.1111/j.1365-246X.2006.02995.x
    [86] Steketee J A. On Volterra's dislocations in a semi-infinite elastic medium [J]. Canadian Journal of Physics, 1958a, 36(2): 192-205. doi: 10.1139/p58-024
    [87] Steketee J A. Some geophysical application of the elasticity theory of dislocations [J]. Canadian Journal of Physics, 1958b, 36(9): 1168-1198. doi: 10.1139/p58-123
    [88] Suito H, Freymueller J T. A viscoelastic and afterslip postseismic deformation model for the 1964 Alaska earthquake [J]. Journal of Geophysical Research-Solid Earth, 2009, 114(B11): 404-426.
    [89] Sun T H Z, Wang K L. Viscoelastic relaxation following subduction earthquakes and its effects on afterslip determination [J]. Journal of Geophysical Research-Solid Earth, 2015, 120(2): 1329-1344. doi: 10.1002/2014JB011707
    [90] Sun W K. Asymptotic theory for calculating deformations caused by dislocations buried in a spherical earth: geoid change [J]. Journal of Geodesy, 2003, 77(7-8): 381-387. doi: 10.1007/s00190-003-0335-4
    [91] Sun W K. Asymptotic solution of static displacements caused by dislocations in a spherically symmetric Earth [J]. Journal of Geophysical Research-Solid Earth, 2004a, 109(B5): 402-419.
    [92] Sun W K. Short Note: Asymptotic theory for calculating deformations caused by dislocations buried in a spherical earth — gravity change [J]. Journal of Geodesy, 2004b, 78(1): 76-81.
    [93] 孙文科. 地震位错理论[M].北京: 科学出版社, 2012a.

    Sun W K. Theory on Earthquake Dislocation[M]. Beijing: Science Press, 2012a (in Chinese).
    [94] 孙文科. 地震位错理论在地震学研究中的作用与存在的问题[J]. 国际地震动态, 2012b, 000(6): 17.

    Sun W K. The theory of earthquake dislocation in seismic research functions and problems [J]. International Earthquake Dynamics.2012b, 000(6): 17 (in Chinese).
    [95] Sun W K, Dong J. Geo-center movement caused by huge earthquakes [J]. Journal of Geodynamics, 2014, 76(1):67-73.
    [96] Sun W K, Okubo S. Surface potential and gravity changes due to internal dislocations in a spherical earth-I. Theory for a point dislocation[J]. Geophysical Journal International, 1993, 114(3): 569-592. doi: 10.1111/j.1365-246X.1993.tb06988.x
    [97] Sun W K, Okubo S. Surface potential and gravity changes due to internal dislocations in a spherical earth-II. Application to a finite fault [J]. Geophysical Journal International, 2002, 132(1): 79-88. doi: 10.1046/j.1365-246x.1998.00400.x
    [98] Sun W K, Zhou X. Coseismic deflection change of the vertical caused by the 2011 Tohoku-Oki earthquake (MW 9.0) [J]. Geophysical Journal International, 2012, 189(2): 937-955. doi: 10.1111/j.1365-246X.2012.05434.x
    [99] Takeuchi H, Saito M. Seismic surface waves [J]. Methods in Computational Physics Advances in Research & Applications, 1972, 11(1): 217-295.
    [100] Tanaka Y, Hasegawa T, Tsuruoka H, et al. Spectral-finite element approach to post-seismic relaxation in a spherical compressible Earth: application to gravity changes due to the 2004 Sumatra–Andaman earthquake [J]. Geophysical Journal International, 2015, 200(1): 299-321. doi: 10.1093/gji/ggu391
    [101] Tanaka Y, Klemann V, Fleming K, et al. Spectral finite element approach to postseismic deformation in a viscoelastic self-gravitating spherical Earth [J]. Geophysical Journal International, 2009, 176(3): 715-739. doi: 10.1111/j.1365-246X.2008.04015.x
    [102] Tanaka Y, Okuno J, Okubo S. A new method for the computation of global viscoelastic post-seismic deformation in a realistic earth model (I) - vertical displacement and gravity variation [J]. Geophysical Journal International, 2006, 164(2): 273-289. doi: 10.1111/j.1365-246X.2005.02821.x
    [103] Tanaka Y, Okuno J, Okubo S. A new method for the computation of global viscoelastic post-seismic deformation in a realistic earth model (II)-horizontal displacement [J]. Geophysical Journal International, 2007, 170(3): 1031-1052. doi: 10.1111/j.1365-246X.2007.03486.x
    [104] Tang H, Sun W K. Asymptotic expressions for changes in the surface co-seismic strain on a homogeneous sphere [J]. Geophysical Journal International, 2017, 209(1): 202-225.
    [105] Tang H, Sun W K. Asymptotic co- and post-seismic displacements in a homogeneous Maxwell sphere [J]. Geophysical Journal International, 2018a, 214(1): 731-750. doi: 10.1093/gji/ggy174
    [106] Tang H, Sun W K. Closed-form expressions of seismic deformation in a homogeneous Maxwell Earth model [J]. Journal of Geophysical Research-Solid Earth, 2018b, 123(7): 6033-6051. doi: 10.1029/2018JB015594
    [107] Tang H, Sun W K. New method for computing postseismic deformations in a realistic gravitational viscoelastic Earth model [J]. Journal of Geophysical Research-Solid Earth, 2019, 124(5): 5060-5080. doi: 10.1029/2019JB017368
    [108] Tang H, Dong J, Sun W K. An approximate method to simulate post-seismic deformations in a realistic earth model[M]. International Association of Geodesy Symposia, 2020a.
    [109] Tang H, Zhang L, Chang L, et al. Optimized approximate inverse Laplace transform for geo-deformation computation in viscoelastic Earth model [J]. Geophysical Journal International, 2020b, 223(1): 444–453. doi: 10.1093/gji/ggaa322
    [110] Thomson W T. Transmission of elastic waves through a stratified solid medium [J]. Journal of Applied Physics, 1950, 21(2): 89-93. doi: 10.1063/1.1699629
    [111] Valsa J, Brancik L. Approximate formulae for numerical inversion of Laplace transforms [J]. International Journal of Numerical Modelling-Electronic Networks Devices and Fields, 1998, 11(3): 153-166. doi: 10.1002/(SICI)1099-1204(199805/06)11:3<153::AID-JNM299>3.0.CO;2-C
    [112] Vermeersen L L A, Sabadini R. A new class of stratified viscoelastic models by analytical techniques [J]. Geophysical Journal International, 1997, 129(3): 531-570. doi: 10.1111/j.1365-246X.1997.tb04492.x
    [113] Vermeersen L L A, Sabadini R, Spada G. Analytical visco-elastic relaxation models [J]. Geophysical Ressearch Letters, 1996, 23(7): 697-700. doi: 10.1029/96GL00620
    [114] Wang H S. Surface vertical displacements, potential perturbations andgravity changes of a viscoelastic earth model induced by internal point dislocations[J]. Geophysical Journal International, 1999a, 137(2): 429-440.
    [115] Wang H, Wu P. Effects of lateral variations in lithospheric thickness and mantle viscosity on glacially induced surface motion on a spherical, self-gravitating Maxwell Earth [J]. Earth and Planetary Science Letters, 2006a, 244(3-4): 576-589. doi: 10.1016/j.jpgl.2006.02.026
    [116] Wang H S, Wu P. Effects of lateral variations in lithospheric thickness and mantle viscosity on glacially induced relative sea levels and long wavelength gravity field in a spherical, self-gravitating Maxwell Earth [J]. Earth & Planetary Ence Letters, 2006b, 249(3-4): 368-383.
    [117] Wang H S, Wu P. Analytical approach for the toroidal relaxation of viscoelastic earth [J]. Geophysical Journal International, 2006c, 167(1): 1-19. doi: 10.1111/j.1365-246X.2006.02980.x
    [118] 汪汉胜, Patrick Wu, Hugo Schotman, 等. 横向非均匀地球负荷问题的CLFE有限元算法的有效性 [J]. 地球物理学报, 2006, 49(6): 1657-1664. doi: 10.3321/j.issn:0001-5733.2006.06.012

    Wang H S, Wu P, Schotman H, et al. Validation of the coupled Laplace-Finite-Element method for the loading problem of a laterally heterogeneous sphencal Earth[J]. Chinese Journal of Geophysics, 2006, 49(6): 1657-1664 (in Chinese). doi: 10.3321/j.issn:0001-5733.2006.06.012
    [119] Wang K. The seismogenic zone of subduction thrust faults—17.Elastic and viscoelastic models of crustal deformation in subduction earthquake cycles[M].Dixon T, Moore J C(ed.). New York: Columbia University Press, 2007a: 540-575.
    [120] Wang K, Hu Y, He J. Deformation cycles of subduction earthquakes in a viscoelastic Earth [J]. Nature, 2012a, 484(7394): 327-332. doi: 10.1038/nature11032
    [121] Wang L, Shum C K, Simons F J, et al. Coseismic and postseismic deformation of the 2011 Tohoku-Oki earthquake constrained by GRACE gravimetry [J]. Geophysical Ressearch Letters, 2012b, 39(7): L07301.
    [122] 王启欣, 江在森, 武艳强, 等. 不同模型下地震位错理论的对比及其应用进展综述 [J]. 地震学报, 2015, (4): 690-704. doi: 10.11939/jass.2015.04.014

    Wang Q X, Jiang Z S, Wu Y Q, et al. A review on comparison and progress in applications of earthquake dislocation theories based on different models[J]. Acta Seismologica Sinica, 2015, 37(4): 690-704 (in Chinese). doi: 10.11939/jass.2015.04.014
    [123] Wang R, Wang H S. A fast converging and anti-aliasing algorithm for Green's functions in terms of spherical or cylindrical harmonics [J]. Geophysical Journal International, 2007, 170(1): 239-248. doi: 10.1111/j.1365-246X.2007.03385.x
    [124] Wang R J. A simple orthonormalization method for stable and efficient computation of Green's functions [J]. Bulletin of the Seismological Society of America, 1999b, 89(3): 733-741.
    [125] Wang R J. The dislocation theory: a consistent way for including the gravity effect in (visco)elastic plane-earth models [J]. Geophysical Journal International, 2005a, 161(1): 191-196. doi: 10.1111/j.1365-246X.2005.02614.x
    [126] Wang R J. On the singularity problem of the elastic-gravitational dislocation theory applied to plane-Earth models [J]. Geophysical Ressearch Letters, 2005b, 32(6):L06307
    [127] Wang R J, Lorenzo-Martín F, Roth F. PSGRN/PSCMP—a new code for calculating co- and post-seismic deformation, geoid and gravity changes based on the viscoelastic-gravitational dislocation theory [J]. Computers & Geosciences, 2006, 32(4): 527-541.
    [128] Wang W Z, Wang H, Liu Y C, et al. A comparative study of the methods for calculation of surface elastic deformation [C]//Proceedings of the Institution of Mechanical Engineers Part J-Journal of Engineering Tribology, 2003, 217(J2): 145-153.
    [129] Wen Y M, Li Z H, Xu C J, et al. Postseismic motion after the 2001 MW 7.8 Kokoxili earthquake in Tibet observed by InSAR time series [J]. Journal of Geophysical Research: Solid Earth, 2012, 117(B8): 405-409.
    [130] Widder D V. The inversion of the laplace integral and the related moment problem [J]. Transactions of the American Mathematical Society, 1934, 36(1-4): 107-200.
    [131] Wong M C, Wu P. Using commercial finite-element packages for the study of Glacial Isostatic Adjustment on a compressible self-gravitating spherical earth – 1: Harmonic loads [J]. Geophysical Journal International, 2019, 217(3): 1798-1820. doi: 10.1093/gji/ggz108
    [132] Wu P. Using commercial finite element packages for the study of earth deformations, sea levels and the state of stress [J]. Geophysical Journal International, 2004, 158(2): 401-408. doi: 10.1111/j.1365-246X.2004.02338.x
    [133] Xu C J, Gong Z, Niu J M. Recent developments in seismological geodesy [J]. Geodesy and Geodynamics, 2016, 7(3): 157-164. doi: 10.1016/j.geog.2016.04.009
    [134] Xu C Y, Sun W K. Earthquake-origin expansion of the Earth inferred from a spherical-Earth elastic dislocation theory [J]. Geophysical Journal International, 2014, 199(3): 1655-1661. doi: 10.1093/gji/ggu364
    [135] Yang H Y, Tromp J. Synthetic free-oscillation spectra: an appraisal of various mode-coupling methods [J]. Geophysical Journal International, 2015a, 203(2): 1179-1192. doi: 10.1093/gji/ggv349
    [136] Yang J Y, Zhou X, Yi S, et al. Determining dislocation love numbers using GRACE satellite mission gravity data [J]. Geophysical Journal International, 2015b, 203(1): 257-269. doi: 10.1093/gji/ggv265
    [137] Yu T T, Rundle J B, Fernández J. Deformation produced by a rectangular dipping fault in a viscoelastic-gravitational layered earth model. Part II: Strike-slip fault—STRGRV and STRGRH FORTRAN programs [J]. Computers & Geosciences, 1996, 22(7): 751-764.
    [138] Zhang G Q, Shen W B, Xu C Y, et al. Coseismic gravity and displacement signatures induced by the 2013 Okhotsk MW8.3 earthquake [J]. Sensors, 2016, 16(9):1410. doi: 10.3390/s16091410
    [139] 钟敏, 罗少聪, 申文斌, 等. 地球主惯性矩的精密确定及三轴分层地球自转动力学研究 [J]. 科技资讯, 2016, 14(18): 181-181. doi: 10.3969/j.issn.1672-3791.2016.18.102

    Zhong M, Luo S C, Shen W B, et al. Precise determination of the Earth's principal moment intertia and study of tri-axial layered earth rotation dynamics[J]. Science & Technology Information, 2016, 14 (18): 181-181 (in Chinese). doi: 10.3969/j.issn.1672-3791.2016.18.102
    [140] Zhou J, Pan E, Bevis M. A point dislocation in a layered, transversely isotropic and self-gravitating Earth. Part I: analytical dislocation Love numbers [J]. Geophysical Journal International, 2019a, 217(3): 1681-1705. doi: 10.1093/gji/ggz110
    [141] Zhou J, Pan E, Bevis M. A point dislocation in a layered, transversely isotropic and self-gravitating Earth — Part II: accurate Green's functions [J]. Geophysical Journal International, 2019b, 219(3): 1717-1728. doi: 10.1093/gji/ggz392
    [142] Zhou J, Pan E, Bevis M. A point dislocation in a layered, transversely isotropic and self-gravitating Earth – Part III: Internal deformation [J]. Geophysical Journal International, 2020, (3):1681-1705.
    [143] Zhou X, Cambiotti G, Sun W, et al. The coseismic slip distribution of a shallow subduction fault constrained by prior information: the example of 2011 Tohoku (MW 9.0) megathrust earthquake [J]. Geophysical Journal International, 2014, 199(2): 981-995. doi: 10.1093/gji/ggu310
    [144] Zhou X, Sun W K, Zhao B, et al. Geodetic observations detecting coseismic displacements and gravity changes caused by the MW= 9.0 Tohoku-Oki earthquake [J]. Journal of Geophysical Research: Solid Earth, 2012, 117: B05408.
  • 加载中
图(2)
计量
  • 文章访问数:  2066
  • HTML全文浏览量:  900
  • PDF下载量:  304
  • 被引次数: 0
出版历程
  • 收稿日期:  2020-07-02
  • 录用日期:  2020-08-05
  • 网络出版日期:  2021-09-13
  • 刊出日期:  2021-01-01

目录

    /

    返回文章
    返回