**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.

**Membership**

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., D*ehant, 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.