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Computation of Green’s Functions for Ocean Tide Loading

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Abstract

The Earth deforms periodically due to the varying weight of the ocean tides. Classical terrestrial geodetic techniques such as gravimetry, strain and tilt observe this deformation clearly. Also space geodetic techniques have reached an accuracy level where this loading signal can no longer be ignored. We present here the basic physical assumptions that underlie, and the derivation of, the mathematical framework that is used nowadays to compute these deformations. Special attention has been paid to the definition of the boundary conditions, how to treat the fluid core and the peculiarities that surround the deformation of the solid Earth at degree one. Also a short explanation of the Longman-Paradox is given. Finally, we discuss numerical methods to solve the differential equations and how to form the Green's functions that describe the ocean tide loading due to a point load.

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Notes

  1. 1.

    Almost ignore; you need to assume that \(g(r)/r=\hbox{const}.\) in order to retain the structure of the analytical solution (Vermeersen et al. 1996).

References

  • Agnew D (2007) Earth tides. In: Herring T (eds) Treatise on geophysics: geodesy, Elsevier, Amsterdam, pp 163–195

    Chapter  Google Scholar 

  • Alterman Z, Jarosch H, Pekeris CL (1959) Oscillations of the Earth. Proc R Soc London, Ser A 252:80–95

    Article  Google Scholar 

  • Arfken G (1985) Mathematical methods for physicists. 3rd edn. Academic Press Inc, New York

    Google Scholar 

  • Backus G (1986) Poloidal and toroidal fields in geomagnetic field modeling. Rev Geophys 24:75–109

    Article  Google Scholar 

  • Blewitt G (2003) Self-consistency in reference frames, geocenter definition, and surface loading of the solid earth. J Geophys Res 108: . doi:10.1029/2002JB002082

    Google Scholar 

  • Bos MS, Baker TF, Røthing K, Plag HP (2002) Testing ocean tidemodels in the nordic seas with tidal gravity observations. Geophys J Int 150(3):687–694

    Article  Google Scholar 

  • Boyd JP (2000) Chebyshev and Fourier spectral methods. Dover Publications Inc, New York

    Google Scholar 

  • Chinnery MA (1975) The static deformation of an Earth with a fluid core: a physical approach. Geophys J R Astron Soc 42:461–475

    Article  Google Scholar 

  • Dahlen FA (1974) On the static deformation of an Earth model with a fluid core. Geophys J Int 36:461–485. doi:10.1111/j.1365-246X.1974.tb03649.x

    Article  Google Scholar 

  • Dahlen FA, Fels SB (1978) A physical explanation of the static core paradox. Geophys J R Astr Soc 55:317–331

    Article  Google Scholar 

  • Dahlen FA, Tromp J (1998) Theoretical global seismology. Princeton University Press, Princeton

    Google Scholar 

  • Dziewonski AM, Anderson DL (1981) Preliminary reference earth model. Phys Earth Planet Int 25:297–356

    Article  Google Scholar 

  • Farrell WE (1972) Deformation of the earth by surface loads. Rev Geophys Space Phys 10(3):761–797

    Article  Google Scholar 

  • Francis O, Mazzega P (1990) Global charts of ocean tide loading effects. J Geophys Res 95(C7):11,411–11,424

    Article  Google Scholar 

  • Gantmacher F (1950) The theory of matrices. vol 2, Chelsea Publishing Company, New York

    Google Scholar 

  • Gilbert F, Backus GE (1966) Propagatormatrices in elasticwave and vibration problems. Geophysics 31:326–332

    Article  Google Scholar 

  • Guo JY, Ning JS, Zhang FP (2001) Chebyshev-collocation method applied to solve ODEs in geophysics singular at the Earth center. Geophys Res Lett 28:3027–3030. doi:10.1029/2001GL012886

    Article  Google Scholar 

  • Guo JY, Li YB, Huang Y, Deng HT, Xu SQ, Ning JS (2004) Green’s function of the deformation of the Earth as a result of atmospheric loading. Geophys J Int 159:53–68. doi:10.1111/j.1365-246X.2004.02410.x

    Article  Google Scholar 

  • Haskell NA (1953) The dispersion of surface waves in multilayered media. Bull Seism Soc Am 43:17–24

    Google Scholar 

  • Hoskins LM (1910) The strain of a non-gravitating sphere of variable density. Trans Am Math Soc 11:494–504

    Article  Google Scholar 

  • Hoskins LM (1920) The strain of a gravitating sphere of variable density and elasticity. Trans Am Math Soc 21:1–43

    Article  Google Scholar 

  • Jekeli C (2007) Potential theory and static gravity field of the Earth. In: Herring T (ed) Treatise on Geophysics, vol 11. pp 11–42

    Google Scholar 

  • Jentzsch G (1997) Earth tides and ocean tidal loading. In: Wilhelm H, Zurm W, Gwenzel H (ed) Tidal phenomena. pp 145

    Google Scholar 

  • Kampfmann W, Berckhemer H (1985) High temperature experiments on the elastic and anelastic behavior of magmatic rocks. Phys Earth Planet Int 40:223–247

    Article  Google Scholar 

  • Kaula WM (1963) Elastic models of the mantle corresponding to variations in the external gravity field. J Geophys Res 68(17):4967–4978

    Google Scholar 

  • Knopoff L (1964) Q Rev Geophys 2:625–660

    Google Scholar 

  • Lamb H (1895) Hydrodynamics. Cambridge University Press, Cambridge

    Google Scholar 

  • Lambeck Cazenave A K, Balmino G (1974) Solid Earth and ocean tides estimated from satellite orbit analyses. Rev Geophys Space Phys 12:421–434

    Article  Google Scholar 

  • Longman IM (1962) A Green’s function for determining the deformation of the Earth under surface mass loads, 1. theory. J Geophys Res 67(2):845–850

    Article  Google Scholar 

  • Longman IM (1963) A Green’s function for determining the deformation of the Earth under surfacemass loads, 2. computations and numerical results. J Geophys Res 68(2):485–496

    Article  Google Scholar 

  • Love AEH (1911) Some problems of geodynamics. Dover Publications Inc, New York, 1967

    Google Scholar 

  • Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice Hall Inc, Englewood Cliffs

    Google Scholar 

  • Martinec Z (1989) Thomson–haskell matrix method for free spheroidal elastic oscillations. Geophys J Int 98:195–199. doi:10.1111/j.1365-246X.1989.tb05524.x

    Article  Google Scholar 

  • Merriam JB (1985) Toroidal love numbers and transverse stress at the Earth’s surface. J Geophys Res 90:7795–7802. doi:10.1029/JB090iB09p07795

    Article  Google Scholar 

  • Merriam JB (1986) Transverse stress Green’s functions. J Geophys Res 91:13903–13913. doi:10.1029/JB091iB14p13903

    Article  Google Scholar 

  • Müller G (1983) Generalised maxwell bodies and estimates of mantle viscosity. Royal Astron Soc Geophys J 87:1113–1141

    Article  Google Scholar 

  • Munk WH, MacDonald GJF (1960) The rotation of the Earth; a geophysical discussion. Cambridge [Eng.] University Press, Cambridge

    Google Scholar 

  • Nowick A, Berry B (1972) relaxation in crystalline solids. Academic, New York

    Google Scholar 

  • Okubo S (1988) Asymptotic solutions to the static deformation of the Earth-I. Spheroidal mode. Geophys J Int 92:39–51. doi:10.1111/j.1365-246X.1988.tb01119.x

    Article  Google Scholar 

  • Pekeris CL, Accad Y (1972) Dynamics of the liquid core of the Earth. R Soc London Philos Trans Ser A 273:237–260. doi:10.1098/rsta.1972.0093

    Article  Google Scholar 

  • Pekeris CL, Jarosch H (1958) The free oscillations of the Earth. In: Benioff H, Ewing M, Howell BF, Press F (eds) Contributions in geophysics in honor of Beno Gutenberg, Pergamon, New York, pp 171–192

    Google Scholar 

  • Phinney RA, Burridge R (1973) Representation of the elastic—gravitational excitation of a spherical Earth model by generalised spherical harmonics. Geophys J R Astron Soc 34:451–487

    Article  Google Scholar 

  • Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1988) Numerical recipes in C. Cambridge University Press, Cambridge

    Google Scholar 

  • Rothwell EJ (2008) Computation of the logarithm of Bessel functions of complex argument and fractional order. Commun Numer Methods Eng 24:237–249. doi:10.1002/cnm.972

    Article  Google Scholar 

  • Ruotsalainen H (2001) Modernizing the finnish long water-tube tiltmeter. J Geod Soc Jpn 47:28–33

    Google Scholar 

  • Scherneck H-G (1991) A parametrized solid Earth tide model and ocean tide loading effects for global geodetic baseline measurements. Geophys J Int 106:677–694

    Article  Google Scholar 

  • Slichter LB, Caputo M (1960) Deformation of an Earth model by surface pressures. J Geophys Res 65:4151. doi:10.1029/JZ065i012p04151

    Article  Google Scholar 

  • Sun W, Sjöerg LE (1999) Gravitational potential changes of a spherically symmetric Earth model caused by a surface load. Geophys J Int 137:449–468

    Article  Google Scholar 

  • Takeuchi H, Saito M (1972) Seismic surface waves. In: Bolt BA (eds) Seismology: surface waves and free oscillations, methods computer physics, Academic, San Diego, pp 217–295

    Google Scholar 

  • Thomson W, Tait PG (1867) A treatise on natural philosophy. Oxford Press, Oxford

    Google Scholar 

  • Vermeersen LLA, Sabadini R, Spada G (1996) Compressible rotational deformation. Geophys J Int 126:735–761. doi:10.1111/j.1365-246X.1996.tb04700.x

    Article  Google Scholar 

  • Wahr J, Sasao T (1981) A diurnal resonance in the ocean tide and in the Earth’s load response due to the resonant ‘free core nutation’. Geophys J R Astr Soc 64:747–765

    Article  Google Scholar 

  • Wu P, Peltier WR (1982) Viscous gravitational relaxation. Geophys J R Astr Soc 70:435–485

    Article  Google Scholar 

  • Wunsch C (1974) Simple models of the deformation of an Earth with a fluid core-I. Geophys J Int 39:413–419. doi:10.1111/j.1365-246X.1974.tb05464.x

    Article  Google Scholar 

  • Zschau J (1983) Rheology of the Earth’s mantle at tidal and Chandler wobble periods. In: Kuo J (ed) In: Proceedings of the 9th international symposium on Earth tides, E. Schweizerbart’sche Verlagsbuchhandlung (Nägele und Obermiller), Stuttgart, pp 605–629

    Google Scholar 

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Authors and Affiliations

  1. Centro Interdiscriplinar de Investigacão Marinha e Ambiental,  Universidate do Porto, Rua dos Bragas 289, 4050-123,  Porto,  Portugal

    M. S. Bos

  2. Chalmers University of Technology, Earth and Space Sciences, SE-412 96,  Gothenburg, Sweden

    H.-G. Scherneck

Authors
  1. M. S. Bos
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  2. H.-G. Scherneck
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Correspondence to M. S. Bos or H.-G. Scherneck .

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Editors and Affiliations

  1. Abt. 1 Geodäsie und Fernerkundung, GeoForschungsZentrum Potsdam, Telegrafenberg A 17, Potsdam, 14473, Brandenburg, Germany

    Guochang Xu

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Bos, M., Scherneck, HG. (2013). Computation of Green’s Functions for Ocean Tide Loading. In: Xu, G. (eds) Sciences of Geodesy - II. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28000-9_1

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