Massentransporte und Massenverteilungen im System Erde  
 SPP1257 / Projekte / VILMA 


SPP Project: VILMA

The topic of VILMA, Viscoelastic Lithosphere and Mantle model, is the reduction of global GRACE, GPS and altimetry data with respect to the glacial-isostatic adjustment applying a 3D viscoelastic earth model.



Temporal Earth-gravity variations recently derived from GRACE monthly solutions mainly result from the superposition of annual or semi-annual oceanic, atmospheric and hydrological mass movements as well as from the secular dynamics of the earth's interior. The separation of the individual contributions requires the application of advanced filtering and analysis techniques. This is exemplified in northern Europe, where the secular gravity variations are largely caused by the land uplift in response to the disappearance of the last Fennoscandian ice sheet at the end of the Pleistocene.

To model this glacial-isostatic adjustment, the earth's interior is represented as a viscoelastic fluid. In realistic computations, lateral variations of the lithosphere thickness and mantle viscosity related to the convective circulation in the earth's interior must be taken into account. For this purpose, a 3D viscoelastic earth model forced by the global deglaciation at the end of the Pleistocene is developed. This model can also be applied to regions with tectonic boundaries, such as Alaska, Antarctica, Iceland, Patagonia and Svalbard, where recent glacial changes are observed.


  • Validation of code against analytical 2D code
  • Development of tools, scripts for handling of the code
  • Parallelisation of code using (OpenMP)
  • Modification of formalism for time integration presented by Martinec (2000) to account for non-linear and anisotropic rheologies
  • Integration of non-linear sea-level equation according to Hagedoorn (2005) to model the effect of moving coast lines
  • Implementation of compressibility (Tanaka et al. 2010, submitted)
  • Stability problem of compressible continuum
  • Participation in COST action ES0701 benchmark of GIA codes (Spada et al. 2010, in prep.)
  • Implementation of stress-dependent viscosity
  • Cooperation with AGIA
  • SLI-data handling (Dransch et al. 2010, submitted)


2D version: (axial symmetry of viscoelastic structure and surface load) applied to Iceland plume, Patagonian slab and post-seismic adjustment

Hartmann et al., (2007): Improvement in fitting surface gravity and uplift in vicinity of Vatnajökull, Iceland.

Figure shows results of the gravity measuring campaigns in Iceland during 1991-2000 with error ellipses (grey shading) at different distances from the center of the Vatnajökull ice load and computed gravity-change rate as function of the plume viscosity for a laterally inhomogeneous earth model.

Klemann et al. (2007): Strong asymmetry of uplift and horizontal displacement east and west of South Patagonian Icefield due to material transport in asthenosphere. Slab is impeding material transport towards the Pacific Ocean.

Figure shows present-day displacement of crust and upper mantle due to glacial loading of South Patagonian Icefield located at 60° co-latitude.

3D version: (arbitrary lateral viscosity variations in mantle, no restrictions on surface-load geometry), presription of plate boundaries as 400 km-wide low-viscosity zones separating the elastic plates

Klemann et al. (2008): Analysis of kinematics

• Equipartioning of spheroidal and toroidal surface motion: Motion of plates due to GIA follows the same equipartitioning as motion of plates due to the mantle convection. Note that if a spherically symmetric structure of the Earth is considered there is no toroidal displacement induced by surface-mass loading. Thus, the toroidal component is induced by coupling mechanisms to the spheroidal excitation.

Figure shows degree variances of spheroidal-vertical (dotted), spheroidal-horizontal (solid) and toroidal-horizontal (dashed) displacements for 1D (spherically symmetric) and 3D earth model (consideration of plate boundaries and lithosphere thickness variations).

• Assessment of plate rotation vectors for 1D and 3D structure. Due to improved GPS analysis, predicted `GIA corrections' for plate motion are comparable or even larger than the accuracy of their determination due to geodetic methods (state of ITRF (Altamimi et al., 2007)).

Figure shows horizontal present-day GIA-induced surface velocities for rigid plate loations of North America considered in the ITRF2005 for the 1D earth model (left) and 3D earth model (right). Black arrows indicate the induced surface velocities and red arrows the velocities due to a rigid rotation with parameters given at the top of each plot.

GIA induced geocenter motion. It has been shown that the GIA induced contribution to the geocenter motion is most sensitive to lower-mantle viscosity. The amplitude due to GIA lies between 0.1 and 1 mm/yr depending on the lower-mantle viscosity structure, whereas the direction of the motion is pointing rather stable towards east of Hudson Bay, NA. Because the accuracy of the geodetically determined geocenter motion is only about 1 mm/yr, the interpretation of this signal in terms of GIA demands an improvement by one order of magnitude in the observation techniques.


• Interpretation of results in terms of dynamics of the Earth's interior

• Consideration of lateral viscosity variations derived from mantle tomography

• Implications for the interpretation of gravity variations

• Inversion strategy by assimilation of Sea-level indicators in order to get a consistent model

Figure shows expected lateral variability of viscosity in the lower mantle for a given scaling of shear-wave velocity variation to viscosity. Red bars denote the shear-wave velocity variation of an averaged tomographic model (Becker & Boschi, 2002).

List of publications (where VILMA is acknowledged)

  • Jacoby WR, Hartmann O, Wallner H, Smilde PL, Bürger S, Sjöberg LE, Erlingsson S, Wolf D, Klemann V, Sasgen I 2009. Temporal gravity variations near shrinking Vatnajökull icecap, Iceland. Pure Appl. Geophys., 166: 1283-1302.
  • Klemann V, Ivins ER, Martinec Z, Wolf D 2007. Models of active glacial isostasy roofing warm subduction: The case of the South Patagonian Icefield. J. Geophys. Res., 112: B09405.
  • Klemann V, Martinec Z, Ivins ER 2008. Glacial isostasy and plate motions. J. Geodyn., 46: 95-103.
  • Klemann V, Martinec Z 2009. Contribution of glacial-isostatic adjustment to the geocenter motion. Tectonophysics, online.
  • Sasgen I, Mulvaney R, Klemann V and Wolf D 2008. Glacial-isostatic adjustment and sea-level change near Berkner Island, Antarctica. Pure Appl. Geophys., submitted.
  • Tanaka Y, Klemann V and Okuno J, 2009. Application of a numerical inverse Laplace integration to surface loading in a viscoelastic compressible Earth model. Pure Appl. Geophys., 166: 1199-1216.
  • Tanaka Y, Klemann V, Fleming K and Martinec Z 2009. Spectral finite element approach to postseismic deformation in a viscoelastic self-gravitating spherical Earth. Geophys. J. Int., 176: 715-739.