Report of IAG Special Study Group 4.176
for the period 1995 - 1999
Detlef Wolf
GeoForschungsZentrum Potsdam
Division 1: Kinematics and Dynamics of the Earth
Section 1.4: System Theory and Modelling
D-14473 Potsdam


Scientific program

SSG 4.176 Models of temporal variations of the gravity field was established in response to the growing need of developing geodynamical models for the interpretation of the time-dependent gravity field as better data are provided by superconducting and absolute gravimeters or are expected from planned satellite gravity missions. Whereas the principal activity of SSG 4.176 was defined to be the development of improved theoretical models for the individual types of forcing responsible for gravity variations, a substantial portion of its research during the period 1995-1999 also involved the application of existing theory. This included the calculation of other measures of deformation, such as displacement and stress. The broader scope of the research performed is also reflected by the large number of publications of members of SSG 4.176 not concerned with gravity variations. However, in accordance with the topic of SSG 4.176, this report concentrates on that part of the research which is concerned explicitly with gravity variations. Clearly, this restriction is somewhat arbitrary in view of the fact that the same theory governs gravity variations, displacements, stresses and other measures of deformation. Another restriction is that research predominantly concerned with tidal gravity variations was largely excluded. This reflects that research on earth tides is represented in the IAG structure separately.


The following scientists were regular members of SSG 4.176:

Veronique Dehant (Royal Observatory of Belgium, Brussels, Belgium)

Martin Ekman (National Land Survey of Sweden, Gävle, Sweden)

Johannes Engels (University of Stuttgart, Germany)

José Fernández (Ciudad University, Madrid, Spain)

Erik Grafarend (University of Stuttgart, Germany)

Paul Johnston (Australian National University, Canberra, Australia)

Xi-lin Li (Chinese Academy of Sciences, Wuchang, China)

James Merriam (University of Saskatchewan, Saskatoon, Canada)

Jerry Mitrovica (University of Toronto, Canada)

Shubei Okubo (University of Tokyo, Japan)

Lars Sjöberg (Royal Institute of Technology, Stockholm, Sweden)

Giorgio Spada (University of Urbino, Italy)

Leif Svensson (Lund Institute of Technology, Lund, Sweden)

Bert Vermeersen (Delft University of Technology, Delft, Netherlands)

Hans-Georg Wenzel (University of Hannover, Germany)

Detlef Wolf (GeoForschungsZentrum, Potsdam, Germany)

Scientific results

The research completed in SSG 4.176 may be caterogized as follows:

Theory: Fundamental theoretical studies are due Grafarend et al. (1997), who formulated a theory for the space-time gravitational field of a deformable earth without assumptions on the geometry and constitution of the earth model. Varga et al. (1999) computed in particular the variation of the degree-2 harmonic of the gravity field and pointed out an apparent discrepancy with observational results. The classical problem of gravitational-elastic perturbations of a spherically symmetric earth model was revisited by Sun & Sjöberg (1998, 1999a), who calculated the radial variations of the Love-Shida numbers for realistic parameter values. The modification of the relation between changes of gravity and inertia effected by the choice of the core-mantle interface conditions was investigated by Spada (1995). The theory of gravitational viscoelastodynamics was developed in systematic form by Wolf (1997, 1998). A number of papers are concerned with solutions of these field equations for special cases. Thus, Johnston et al. (1997) introduced into the theory the modifications required in the presence of phase boundaries. An analytic solution of the field equations for the case of an earth model composed of homogeneous incompressible shells was derived by Vermeersen & Sabadini (1997). Martinec & Wolf (1998) considered the same type of earth model and derived explicit expressions for the propagator matrix entering into the solution. The problem of solving the viscoelastic field equations in the case of compressibility was investigated by Vermeersen et al. (1996). A number of papers are concerned with the solution of the field equations on the assumptions of lateral variations of the viscosity. Thus, D’Agostino et al. (1997) employed a spectral method to solve the equations for a spherical earth in this more general case. Kaufmann & Wolf (1999) used the perturbation approach valid on the assumption of small variations of viscosity and derived analytical solutions for a number of simple 2-D plane earth models. Tromp & Mitroviva (1999a, b) developed a more general perturbation approach valid for a 3-D spherical earth. A special model consisting of two eccentrically nested spheres and designed for testing numerical codes valid for arbitrarily large 2-D variations of viscosity was developed by Martinec & Wolf (1999).

Topographic and glacial loading: Using the technique of mass condensation, the incremental gravity generated by topographic masses and their isostatic compensation was studied by Engels et al. (1996). The authors concluded that the observed geoid heights confirm that the earth's crust cannot be represented by a constant-density shell. In a related study, Sun & Sjöberg (1999b) calculated gravity changes generated by topographic loads on the assumption of a perfectly elastic earth model. In view of the observed geoid heights, they pointed out that dynamic processes must also be responsible for the anomalies. Ekman & Mäkinen (1996) analysed gravity variations in Fennoscandia and related them in terms of a simple flow model to glacial-isostatic adjustment. Johnston & Lambeck (1999) considered in particular the temporal variation of the degree-2 harmonic of the gravity field and investigated the sensitivity of the predictions on the details of the earth and load models. In similar studies, Vermeersen et al. (1997) and Milne et al. (1998) determined the influence of the viscosity stratification on the degree-2 harmonic. Wolf et al. (1997a, b) and Thoma & Wolf (1999) predicted deglaciation-induced gravity variations for Iceland and Fennoscandia, respectively, and suggested that the signals be observable after a period of several years.

Internal and tidal loading: The problem of calculating the gravity signatures associated with convective density inhomogeneities in a Newtonian-viscous compressible earth model with phase boundaries was studied by Defraigne et al. (1996). The same problem was considered for a viscoelastic earth by Mitrovica & Forte (1997), who also investigated the consistency of the earth's viscosity stratification inferred from dynamic geoid anomalies with that inferred from glacial-isostatic adjustment. The tidal loading problem was revisited for the case of a spherical elastic earth model in an initial state of hydrostatic equilibrium by Grafarend et al. (1996), who derived an integral relation between the Love-Shida numbers. Mathews et al. (1997) also calculated Love-Shida numbers, taking into account effects due to ellipticity, rotation and anelasticity. Later, Dehant et al. (1998, 1999) further generalized the problem and studied tidal loading for an aspherical earth model with an inelastic mantle and in a non-hydrostatic initial state. Wieczerkowski & Wolf (1998) were also concerned with tidal loading. The emphasis of their study was on assessing the modifications introduced by compressibility and by different types of viscoelasticity.

Seismotectonic forcing: In a number of papers, the gravity changes caused by various types of forcing associated with seismic, tectonic or volcanic activity are considered. Thus, Fernández et al. (1997a, b) and Yu et al. (1997) studied several types of faulting and volcanic intrusions and computed the associated deformation and gravity change for plane elastic or viscoelastic earth models. In similar studies, Piersanti et al. (1997) and Sun & Okubo (1998) employed spherical elastic or viscoelastic earth models to determine the co- or post-seismic deformation and gravity change on a global scale. Soldati & Spada (1999) considered in particular the earthquake-generated degree-2 harmonic of the gravitational field for a viscoelastic earth model and also studied the implications for the earth's rotation.

Miscellaneous: Brimich et al. (1995) investigated the effects produced by heat sources in layered elastic earth models. In a theoretical study, Degryse & Dehant (1995) readdressed the problem of computing the period of the Slichter modes of the inner core, which may be detectable in the records of superconducting gravimeters. Neumeyer et al. (1997) computed the direct attraction of the atmosphere and the secondary contributions due to the earth's deformation in response to the atmospheric loading. In a related study, Neumeyer et al. (1999) also estimated the effects due to rainfall and groundwater.

Other activities

The research carried out in SSG 4.176 was reported by its members in two meetings held in Walferdange, Luxembourg, during March 17-19, 1997 (9 presentations) and in Potsdam, Germany, during November 23-25, 1998 (15 presentations). Both meetings were financially supported by the IAG; for the first meeting, additional funding was provided by the European Centre for Geodynamics and Seismology. Abstracts of the presentations given were issued subsequently (Wolf, 1997, 1998). A further activity of SSG 4.176 was the compilation of a bibliography on the theory and modelling of temporal gravity variations for the period 1960-1999 (Wolf, 1999). The abstract volumes and the bibliography are available from the chairperson of SSG 4.176 upon request.

Selected publications of members: 1995-1999 Brimich, L., Fernández, J., Granell, R.D.R., Hvoždara, M., 1995. Some comments on the effects of earth models on ground deformation modelling. Stud. Geophys. Geod., 40, 14-24.

Defraigne, P., Dehant, V., Wahr, J.M., 1996. Internal loading of an inhomogeneous compressible earth with phase boundaries. Geophys. J. Int., 125, 173-192.

Degryse, K., Dehant, V., 1995. Analytical computation of modes for an earth with viscous boundary layers, and influence of viscosity on the non-ratating Slichter period. Man. Geod., 20, 498-514.

Dehant V., Defraigne P., Wahr J.M., 1998. Tides for an earth in a non-hydrostatic equilibrium, in Ducarme, B., Pâ quet, P., eds., Proc. 13th Int. Symp. Earth Tides, Brussels, pp. 261-263. Royal Observatory of Belgium, Brussels.

Dehant, V., Defraigne, P., Wahr, J.M., 1999. Tides for a convective earth. J. Geophys. Res., 104, 1035-1058.

D’Agostino, G., Spada, G., Sabadini, R., 1997. Postglacial rebound and lateral viscosity variations: a semi-analytical approach based on a spherical model with Maxwell rheology. Geophys. J. Int., 129, F9-F13.

Ekman, M., Mäkinen, J., 1996. Recent postglacial rebound, gravity change and mantle flow in Fennoscandia. Geophys. J. Int., 126, 229-234.

Engels, J., Grafarend, E.W., Sorcik, P., 1996. The gravitational field of topographic-isostatic masses and the hypothesis of mass condensation II - the topographic-isostatic geoid. Surv. Geophys., 17, 41-66.

Fernández, J., Rundle, J.B., Yu, T.-T., Carrasco, J.M., 1997a. Displacement and gravity changes due to different sources in layered media. Compt. Rend. J. Luxemb. Geodyn., 82, 55-60.

Fernández, J., Yu, T.-T., Rundle, J.B., 1997b. Geodetic signature produced by different sources in a gravitational layered earth model. Publ. Inst. Astron. Geod., 191, 1-32.

Grafarend, E., Engels, J., Varga, P., 1996. The gravitational potential of a deformable massive body generated by tidal and load potentials. Acta Geod. Geophys. Hung., 31, 283-292.

Grafarend, E., Engels, J., Varga, P., 1997. The spacetime gravitational field of a deformable body. J. Geod., 72, 11-30.

Johnston, P., Lambeck, K., 1999. Postglacial rebound and sea level contributions to changes in the geoid and the earth's rotation axis. Geophys. J. Int., 136, 537-558.

Johnston, P., Lambeck, K., Wolf, D., 1997. Material vs isobaric internal boundaries in the earth and their influence on postglacial rebound. Geophys. J. Int., 129, 252-268.

Kaufmann, G., Wolf, D., 1999. Effects of lateral viscosity variations on postglacial rebound: an analytical approach. Geophys. J. Int., 137, 489-500.

Martinec, Z., Wolf, D., 1998. Explicit form of the propagator matrix for a multi-layered, incompressible viscoelastic sphere. Sci. Techn. Rep. GFZ Potsdam, STR98/08, 1-13.

Martinec, Z., Wolf, D., 1999. Gravitational-viscoelastic relaxation of eccentrically nested spheres. Geophys. J. Int., 138, 45-66.

Mathews, P.M., Dehant, V., Gipson, J.M., 1997. Tidal station displacements. J. Geophys. Res., 102, 20469-20477.

Milne, G.A., Mitrovica, J.X., Forte, A.M., 1998. The sensitivity of glacial isostatic adjustment predictions to a low-viscosity layer at the base of the upper mantle. Earth Planet. Sci. Lett., 154, 265-278.

Mitrovica, J.X., Forte, A.M., 1997. Radial profile of mantle viscosity: results from the joint inversion of convection and postglacial rebound observables. J. Geophys. Res., 102, 2751-2769.

Neumeyer, J., Barthelmes, F., Wolf, D., 1997. Atmospheric pressure correction for gravity data using different methods, in Ducarme, B., Pâ quet, P., eds., Proc. 13th Int. Symp. Earth Tides, Brussels, pp. 431-438. Royal Observatory of Belgium, Brussels.

Neumeyer, J., Barthelmes, F., Wolf, D., 1999. Estimates of environmental effects in superconducting gravimeter data. Bull. Inf. Mareés Terr., 131, 10153-10159.

Piersanti, A., Spada, G., Sabadini, R., Bonafede, M., 1997. Global post-seismic deformation. Geophys. J. Int., 120, 544-566.

Soldati, G., Spada, G., 1999. Large earthquakes and earth rotation: the role of mantle relaxation. Geophys. Res. Lett., 26, 911-914.

Spada, G., 1995. Changes in the earth inertia tensor: the role of boundary conditions at the core-mantle interface. Geophys. Res. Lett., 22, 3557-3560.

Sun, W., Okubo, S., 1998. Surface potential and gravity changes due to internal dislocations in a spherical earth-II. Application to a finite fault. Geophys. J. Int., 132, 79-88.

Sun, W., Sjöberg, L.E., 1998. Gravitational potential changes of a spherically symmetric earth model caused by a surface load. Phys. Chem. Earth, 23, 47-52.

Sun, W., Sjöberg, L.E., 1999a. Gravitational potential changes of a spherically symmetric earth model caused by a surface load. Geophys. J. Int., 137, 449-468.

Sun, W., Sjöberg, L.E., 1999b. A new global topographic-isostatic model. Phys. Chem. Earth, 24, 27-32.

Sun, W., Okubo, S., Vaníèek, P., 1996. Global displacements caused by point dislocations in a realistic earth model. J. Geophys. Res., 101, 8561-8577.

Thoma, M., Wolf, D., 1999. Bestimmung der Mantelviskosität aus Beobachtungen der Landhebung und Schwere in Fennoskandien. Sci. Techn. Rep. GFZ Potsdam, STR 99/02, 1-101.

Tromp, J., Mitrovica, J.X., 1999a. Surface loading of a viscoelasic earth-I. General theory. Geophys. J. Int., 137, 847-855.

Tromp, J., Mitrovica, J.X., 1999b. Surface loading of a viscoelastic earth-II.~Spherical models. Geophys. J. Int., 137, 856-872.

Varga, P., Grafarend, E.W., Engels, J., 1999. Earth tide generated variations of second degree geopotential. Bull. Inf. Mareés Terr., 131, 10217-10223.

Vermeersen, L.L.A., Sabadini, R., 1997. A new class of stratified viscoelastic models by analytical techniques. Geophys. J. Int., 129, 531-570.

Vermeersen, L.L.A., Sabadini, R., Spada, G., 1996. Compressible rotational deformation. Geophys. J. Int., 126, 735-761.

Vermeersen, L.L.A., Fournier, A., Sabadini, R., 1997. Changes in rotation induced by Pleistocene ice masses with stratified analytical earth models. J. Geophys. Res., 102, 27689-27702.

Wieczerkowski, K., Wolf, D., 1998. Viscoelastic tidal dissipation in planetary models, in Ducarme, B., Pâ quet, P., eds., Proc. 13th Int. Symp. Earth Tides, Brussels, pp. 277-285. Royal Observatory of Belgium, Brussels.

Wolf, D., 1997. Gravitational viscoelastodynamics for a hydrostatic planet. Veröff. Deut. Geodät. Komm., Reihe C, 452, 1-96

Wolf, D., 1998. Gravitational-viscoelastic field theory, in Wu. P., ed., Dynamics of the Ice Age Earth: A Modern Perspective, pp. 55-86. Trans Tech Publications, Uetikon.

Wolf, D., Barthelmes, F., Sigmundsson, F., 1997a. Predictions of deformation and gravity variation caused by recent change of Vatnajökull ice cap, Iceland. Compt. Rend. J. Luxemb. Geodyn., 82, 36-42.

Wolf, D., Barthelmes, F., Sigmundsson, F., 1997b. Predictions of deformation and gravity change caused by recent melting of the Vatnajökull ice cap, Iceland, in Segawa, J., Fujimoto, H., Okubo, S., eds., Gravity, geoid and marine geodesy, pp. 311-319. Springer, Berlin.

Yu, T.-T., Fernández, J., Rundle. J.B., 1997. Subsurface deformation and geoid changes due to earthquakes. Publ. Inst. Astron. Geod., 191, 33-50.

Abstracts, bibliography

Wolf, D., ed., 1997. Special Study Group 4.176: Models of Temporal Variations of the Gravity Field, First Meeting, Walferdange, Luxembourg, 17-19 March 1997, Abstracts, GeoForschungsZentrum, Potsdam.

Wolf, D., ed., 1998. Special Study Group 4.176: Models of Temporal Variations of the Gravity Field, Second Meeting, Potsdam, Germany, 23-25 November 1998, Program and Abstracts, GeoForschungsZentrum, Potsdam.

Wolf, D., ed., 1999. Special Study Group 4.176: Models of Temporal Variations of the Gravity Field, Bibliography: 1960-1999, GeoForschungsZentrum, Potsdam.