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## SSG 4.189 DYNAMIC THEORIES OF
DEFORMATION AND GRAVITY FIELDS

**Detlef
Wolf, GeoForschungsZentrum Potsdam, Germany**

**1. Scientific program
**

SSG
4.189 `Dynamic theories of deformation and gravity fields' was
established in response to the continuing need to develop improved
dynamical models for the interpretation of time-dependent deformation
and gravity fields as better data become available from GPS, VLBI and
absolute gravity measurements or are expected from the satellite
gravity missions CHAMP and GRACE. Whereas the development of improved
theoretical models for the different types of forcing responsible for
the deformation and gravity fields is defined as the principal
activity of SSG 4.189, a substantial portion of its research during
the period 1999-2001 also involved the application of existing theory.

**2. Regular members
**

H. Abd-Elmotaal (University of Minia, Egypt)

J.-P. Boy (Louis Pasteur University, France)

V. Dehant (Royal Observatory of Belgium, Belgium)

R. Eanes (University of Texas, USA)

J. Engels (University of Stuttgart, Germany)

J. Fernández (Ciudad University, Spain)

G. Kaufmann (University of Göttingen, Germany)

Z. Martinec (Charles University, Czech Rebublic)

G. Milne (University of Durham, UK)

H.-P. Plag (Norwegian Mapping Authority, Norway)

G. Spada (University of Urbino, Italy)

W. Sun (University of Tokyo, Japan)

P. Varga (Geodetic and Geophysical Research Institute, Hungary)

K.
Wieczerkowski (GeoForschungsZentrum, Potsdam, Germany)

D.
Wolf (GeoForschungsZentrum Potsdam, Germany)

P. Wu (University of Calgary, Canada)

**3. Associate members
**

P. Gegout (Louis Pasteur University, France)

E. Grafarend (University of Stuttgart, Germany)

J. Hinderer (Louis Pasteur University, France)

L. Sjöberg (Royal Institute of Technology, Sweden)

**4. Scientific results
**

4.1. Fundamental theory

Sun and Sjöberg (1999a) revisited the classical problem of surface
loading of a radially symmetric elastic body and studied the radial
dependence of the load Love numbers and the Green functions for
displacement, potential and gravity perturbations. Grafarend (2000)
computed the gravity field of an arbitrary deformable body under the
assumption that the topographic surface, the interfaces and the
internal mass distribution vary over time. Grafarend
et al. (2000) studied the relationship between the incremental
Cartesian moments of the mass density, the incremental moments of
inertia and the incremental gravitational potential coefficients for
an arbitrary deformable body. As excitation, they considered tidal
forcing, normal and tangential surface forcings and rotational
variations. Dehant et al. (1999) calculated tidal Love numbers for rotating
aspherical earth models. In addition to elastic earth models, they
also investigated effects caused by assuming an inelastic convecting
mantle.

In two papers, the problem of load-induced, viscoelastic perturbations of
a compressible earth initially in hydrostatic equilibrium was
considered. Whereas Wolf and Kaufmann (2000) were concerned with the
plane-earth approximation of the problem, Martinec et al. (2001)
considered the generalized problem for a spherical earth consisting of
compositionally homogeneous shells. The density stratification was
given by Darwin's law, which can be shown to satisfy the field
equations governing the initial state. In another study, a systematic
comparison between the solutions for load-induced perturbations of
spherical, incompressible earth models with Maxwell or Burgers
rheology was carried out (Göbell et al., 1999).

Attempts were also made to obtain solutions of the field equations for
2-D and 3-D incompressible viscoelastic earth models. Whereas Kaufmann
and Wolf (1999) obtained an approximate analytical solution for a 2-D
plane earth, Martinec and Wolf (1999) derived the exact analytical
solution for two axially nested spheres. The analytical solutions are
required to test more general numerical solutions for arbitrary 2-D or
3-D viscoelastic earth models (Martinec, 1999, 2000).

4.2. Glacial loading

Th oma and Wolf (1999) interpreted a subset of the glacial-isostatic
adjustment data available for Fennoscandia in terms of 1-D earth
models and proposed improved bounds for the viscosity stratification.
An alternative approach was followed by Wieczerkowski et al. (1999),
who employed formal inverse theory to infer the viscosity
stratification below Fennoscandia. More recently, Milne et al. (2001)
considered GPS data from Fennoscandia. They showed that lithosphere
thicknesses and asthenospere viscosities inferred from this type of
data are consistent with those obtained using relative sea-level data.
Kaufmann and Amelung (2000) used subsidence data from the artifical
Lake Mead, Nevada, to infer the viscosity stratification in this
region and found very low viscosity values. Thoma and Wolf (2001)
interpreted land uplift induced by the recent melting of the Vatnajökull
ice cap, Iceland, and found anomalously low values for the lithosphere
thickness and asthenosphere viscosity in this region. Kaufmann and
Lambeck (2000) interpreted convectively supported geoid perturbations
as well as glacially induced changes of sea level, rotation and the
gravity field and inferred global average values of the upper- and
lower-mantle viscosities.

Wu (1999) raised the question of whether relative sea-level changes in
Hudson Bay and along the Atlantic coast of North America can also be
explained in terms of the glacial-isostatic adjustment of a flat earth
with non-Newtonian rheology. His results show that reconciling all
sea-level data is difficult for non-Newtonian rheologies.
Subsequently, Wu (2001) incorporated tectonic stress and found that
this modification makes the assumption of a non-Newtonian rheology
more reasonable. Giunchi and Spada (2000) developed a spherical earth
model with non-Newtonian rheology and concluded that, in this case,
the long-wavelength signatures of glacial-isostatic adjustment become
largely insensitive to the viscosity of the lower mantle.

Wu et al. (1999)
discussed the question of whether deglaciation-induced stresses are
sufficiently strong to have triggered paleo-earthquakes in
Fennoscandia. They found that glacial-isostatic adjustment is probably
the cause of the large postglacial faults observed but is unlikely to
be responsible for the current seismicity in this region. Wu and
Johnston (2000) studied a similar problem for North America and
concluded that stresses are sufficiently strong for triggering
earthquakes at locations not too far from the former ice-sheet margin.
Klemann and Wolf (1999) investigated the consequences of a ductile
layer inside an otherwise elastic lithosphere for glacial-isostatic
adjustment. Their results show that the stress pattern is
significantly affected by the presence of a ductile layer.

Milne et al. (1999)
developed an improved method of accounting for the influx of ocean
water to once ice-covered marine regions after melting and analyzed
the implications of this effect for the interpretation of glacial-isostatic
adjustment. Kaufmann (2000) predicted glacially induced variations of
the gravity field due to Late-Pleistocene and present-day changes in
glaciation and discussed the question of their detectability by the
satellite missions CHAMP and GRACE.

Kaufmann et al. (2000)
revisited the issue of glacial-isostatic adjustment in Fennoscandia.
In particular, they investigated whether lateral variations in
lithosphere thickness and viscosity may be resolved from the
observational record. Their results show that predictions of relative
sea level, uplift rates and gravity anomalies differ significantly if
lateral variations are taken into account.

4.3. Surface, internal and tidal loading

Abd-Elmotaal (1999a) calculated Moho depths for a test area in Austria
using the Vening Meinesz and the Airy-Heiskanen isostatic models and
compared the results with seismic Moho depths. Abd-Elmotaal (1999b,
2000) reviewed the inverse Vening Meinesz isostatic problem defined as
finding the Moho depth for which the isostatic gravity anomalies
become zero. Sun and Sjöberg (1999b) calculated global geoid
perturbations on the assumption that the topographic loads are
compensated by elastic deformation only. They found positive
correlations between the calculated and observed perturbations,
although large differences remained for long wavelengths due to the
neglect of dynamic processes. Dehant
et al. (2000)
computed the response of Mars to nutational, tidal and loading
excitation and studied the influence of the planet's assumed material
properties on its response. Arnoso et
al. (2001) analyzed tidal gravity observations from
Lanzarote, Canary Islands, and discussed whether they can be used to
resolve structural details of the upper crust below the island. Neumeyer
et al. (1999) and Hagedoorn et al. (2000)
investigated the total atmospheric contribution to gravity
perturbations using an elastic earth model. They found that their
results are superior to those obtained by simply using empirical
relationships between pressure and gravity changes.

4.4. Seismo-volcanic forcing

Nostro et al. (1999) compared spherical and flat earth models for
computing co- and postseismic deformations in order to assess in which
cases the neglect of sphericity and self-gravitation is justified. In
a related study, Boschi et al. (2000) calculated the global
deformation caused by a shear dislocation located in the mantle.
Important points of their study were the consideration of sources
below the lithosphere and effects due the presence of a low-viscosity
asthenosphere. Folch et al. (2000)
considered the viscoelastic deformation caused by an inflated magma
chamber and investigated the errors introduced by neglecting the
finite dimension of the chamber or the topography of the region. In a
related study, Fernández et al. (2001) interpreted deformation and
gravity change data from Long Valley Caldera, California. They showed
that incorrect interpretations may result if only one type of data is
used.

**5. Other activities
**

The research carried out in SSG 4.189 was reported by several of its
members and invited guests during the 7th International Winter Seminar
on Geodynamics on `Viscoelastic Theories in Geodynamics' held in
Sopron, Hungary, February 19-23, 2001. The meeting was financially
supported by the Hungarian Academy of Science.

**6. Selected publications of members: 1999-2001
**

Abd-Elmotaal, H., 1999a.

Inverse Vening Meinesz isostatic problem: theory and practice.

Boll. Geod. Sci. Aff., 58, 53-70.

Abd-Elmotaal,
H., 1999b.

Moho depths versus gravity anomalies.

Surv. Rev., 35, 175-186.

Abd-Elmotaal, H., 2000.

Vening Meinesz inverse isostatic problem with local and global Bouguer
anomalies.

J. Geod., 74, 390-398.

Arnoso, J., Fernández, J., Vieira, R., 2001.

Interpretation of tidal gravity anomalies in Lanzarote, Canary
Islands.

J. Geodyn., 31, 341-354.

Boschi, L., Piersanti, A., Spada, G., 2000.

Global postseismic deformation: deep earthquakes,

J. Geophys. Res., 105, 631-652.

Dehant,
V., Defraigne, P., Wahr, J.M., 1999.

Tides for a convective earth.

J. Geophys. Res., 104, 1035-1058.

Dehant V., Defraigne P., Van Hoolst T., 2000.

Computation of Mars' transfer function for nutation tides and surface
loading.

Phys. Earth Planet. Inter., 117, 385-395.

Fernández,
J., Charco, M., Tiampo, K. F., Jentzsch, G., Rundle, J. B., 2001.

Joint interpretation of displacements and gravity changes in volcanic

areas. A test example: Long Valley Caldera, California.

Geophys. Res. Lett., 28, 1063-1066.

Folch,
A., Fernández, J., Rundle, J.B., Martí, J., 2000.

Ground deformation in a viscoelastic medium composed of a layer

overlying a half space. A comparison between point and extended
sources.

Geophys. J. Int., 140, 37-50.

Giunchi, C., Spada, G., 2000.

Postglacial rebound in a non-Newtonian spherical earth.

Geophys. Res. Lett., 27, 2065-2068.

Göbell,
S., Thoma, M., Wolf, D., Grafarend, E.W., 1999.

Berechnung auflastinduzierter Vertikalverschiebungen, Geoidhöhen und

Freiluft-Schwereanomalien für ein selbstgravitierendes, sphärisches
Erdmodell

und unterschiedliche Rheologien.

Sci. Techn. Rep. GFZ Potsdam, STR99/24, 76 pp.

Grafarend, E.W., 2000.

The time-varying gravitational potential field of a massive,
deformable body.

Stud. Geophys. Geod., 44, 364-373

Grafarend, E.W., Engels, J., Varga, P., 2000.

The temporal variation of the spherical and Cartesian multipoles of
the

gravity field: the generalized MacCullagh representation.

J. Geod., 74, 519-530.

Hagedoorn, J.M., Wolf, D., Neumeyer, J., 2000.

Modellierung von atmosphärischen Einflüssen auf hochgenaue
Schweremessungen

mit Hilfe elastischer Erdmodelle.

Sci. Techn. Rep. GFZ Potsdam, STR00/15, 87 pp.

Kaufmann,
G., 2000.

Ice-ocean mass balance during the Late Pleistocene glacial cycles in view

of CHAMP and GRACE satellite missions.

Geophys. J. Int., 143, 142-156.

Kaufmann,
G., Amelung, F., 2000.

Reservoir-induced deformation and continental rheology in vicinity of
Lake Mead, Nevada.

J. Geophys. Res., 105, 16341-16358.

Kaufmann,
G., Lambeck, K., 2000.

Mantle dynamics, postglacial rebound and the radial viscosity profile.

Phys. Earth Planet. Inter., 121, 301-327.

Kaufmann,
G., Wolf, D., 1999.

Effects of lateral viscosity variations on postglacial rebound: an
analytical approach.

Geophys. J. Int., 137, 489-500.

Kaufmann,
G., Wu, P., Li, G., 2000.

Glacial isostatic adjustment in Fennoscandia for a laterally
heterogeneous earth.

Geophys. J. Int., 143, 262-273.

Klemann,
V., Wolf, D., 1999.

Implications of a ductile crustal layer for the deformation caused by
the Fennoscandian ice sheet.

Geophys. J. Int., 139, 216-226.

Martinec, Z., 1999.

Spectral, initial value approach for viscoelastic relaxation of a
spherical earth

with a three-dimensional viscosity - I. Theory.

Geophys. J. Int., 137, 469-488.

Martinec, Z., 2000.

Spectral-finite element approach to three-dimensional viscoelastic

relaxation in a spherical earth.

Geophys. J. Int., 142, 117-141.

Martinec, Z., Wolf, D., 1999.

Gravitational viscoelastic relaxation of eccentrically nested spheres.

Geophys. J. Int., 138, 45-66.

Martinec, Z., Thoma, M., Wolf, D., 2000.

Material versus local incompressibility and its influence on glacial-isostatic
adjustment.

Geophys. J. Int., 143, 1-25.

Milne, G.A., Mitrovica, J.X., Davis, J.L., 1999.

Near-field hydro-isostasy: the implementation of a revised sea-level
equation.

Geophys. J. Int., 139, 464 -483.

Milne, G.A., Davis, J.L., Mitrovica, J.X., Scherneck, H.-G., Johansson,
J.M.,

Vermeer, M., Koivula, H., 2001.

Space-geodetic constraints on glacial isostatic adjustment in
Fennoscandia.

Science, 291, 2381-2385.

Neumeyer, J., Barthelmes, F., Wolf, D., 1999.

Estimates of environmental effects in superconducting gravimeter data,

Bull. Inf. Mare 'es Terr., 131, 10153-10159.

Nostro, C., Piersanti, A., Antonioli, A., Spada, G., 1999.

Spherical versus flat models of coseismic and postseismic
deformations.

J. Geophys. Res., 104, 13115-13134.

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.

Thoma, M., Wolf, D., 1999.

Bestimmung der Mantelviskosität aus Beobachtungen der Landhebung und
Schwere in Fennoskandien,

Sci. Techn. Rep. GFZ Potsdam, STR99/02, 101 pp.

Thoma, M., Wolf, D., 2001.

Inverting land uplift near Vatnajökull, Iceland, in terms of
lithosphere

thickness and viscosity stratification.

in: Gravity, Geoid and Geodynamics 2000, Springer, Berlin, in press.

Wieczerkowski, K., Mitrovica, J., Wolf, D., 1999.

A revised relaxation-time spectrum for Fennoscandia,

Geophys. J. Int., 139, 69-86.

Wolf,
D., Kaufmann, G., 2000.

Effects due to compressional and compositional density stratification
on

load-induced Maxwell-viscoelastic perturbation.

Geophys. J. Int., 140, 51-62.

Wu, P., 1999.

Modeling postglacial sea-levels with power law rheology and realistic

ice model in the absence of ambient tectonic stress.

Geophys. J. Int., 139, 691-702.

Wu, P., 2001.

Postglacial induced surface motion and gravity in Laurentia for
uniform mantle

with power-law rheology and ambient tectonic stress.

Earth Planet. Sci. Lett., 186, 427-435.

Wu, P., Johnston, P., 2000.

Can deglaciation trigger earthquakes in N. America?

Geophys. Res. Lett., 27, 1323-1326.

Wu, P., Johnston, P., Lambeck, K., 1999.

Postglacial rebound and fault instability in Fennoscandia.

Geophys. J. Int., 139, 657-670.

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