Giorgio Spada

Global postseismic rebound of a viscoelastic Earth: Theory for finite faults and application to the 1964 Alaska earthquake

A spherically symmetric Earth model with viscoelastic rheology is used to study the postseismic rebound associated with finite lithospheric dislocations. We perform a systematic study of surface deformations due to sources characterized by two- and three-dimensional faults, modeled by a linear and planar distribution of point sources. Our approach is based on the normal mode technique for a layered Earth with linear viscoelastic rheology and allows for a self-consistent description of the time evolution of postseismic displacements due to strike- and dip-slip faults. As a case study, we compare the predicted horizontal displacements due to the 1964 Alaska earthquake with very long baseline interferometry (VLBI) baselines changes observed in this region in the period 1984–1989. Although subduction and postglacial isostatic adjustment are expected to contribute significantly to present-day velocities, our results show that the postseismic rebound due to the 1964 Alaska earthquake plays a relevant role in the changes of several VLBI baselines. The last section is devoted to the analysis of the evolution of other North American baselines. As suggested by recent investigations based on forward approaches, isostatic readjustment of the Earths crust in response to the melting of the last Pleistocene ice sheets is a major contributor to baseline variations in this area. Our results indicate that a correct interpretation of VLBI baseline changes in North America should account also for the effects of postseismic deformation.

Analytical visco‐elastic relaxation models

In the past, relaxation processes employing PREM or 1066B-stratified earth models have been solved numerically, either directly in the time domain (initial value models) or in the Laplace-transformed domain (normal mode models). Solving N-layer stratified models analytically has only been performed for small values of N. We present a brief outline of the analytical method of general radially stratified N-layer, self-gravitating, Maxwell rheological models. The analytical models allow for the relaxation times of the myriad of relaxation modes to be determined with very high accuracies, as the secular determinant from which they derive is an analytical function. We show by explicitly comparing results on a 5 and 30-layer incompressible Maxwell model with a convex mantle viscosity profile, that the differences in the results between the two models are small when the 5-layer model has the volume-averaged structural and rheological properties of the 30-layer model. The fact that in both models the fluid values of the Love numbers reach the same expected limits leaves no room for hypothetical so-called non-modal contributions. The application to compressible linear rheologies of this generalized analytical normal mode method is more involved, but straightforward.

Isostatic deformations and polar wander induced by redistribution of mass within the Earth

The effects at the surface of the Earth of time-dependent mantle mass anomalies are analyzed within the framework of a viscoelastic mantle with Maxwell rheology. The implications for the Earths rotation are developed using the linearized Liouville equations valid for small polar displacements. Our approach is appropriate for a simplified modeling of subduction. The displacement of the Earths axis of rotation, called true polar wander, is very sensitive to the viscosity profile of the mantle and to the nature of the 670-km seismic discontinuity. Phase change models generally yield a huge amount of polar wander, except for large viscosity increases. For chemically stratified models, true polar wander is drastically reduced as a consequence of dynamic compensation of the mass anomalies at the upper-lower mantle interface. When the source is embedded in the upper mantle in the proximity of the chemical density jump, the polar wander stops after a finite time controlled by the slowest isostatic mode.

Spherical versus flat models of coseismic and postseismic deformations

We perform an exhaustive study of coseismic and postseismic surface deformations induced by shear dislocations using flat and spherical Earth models. Our aim is to examine the effects of the spherical geometry, the vertical layering, and the self-gravitation on surface displacement field. For a 100 km long fault, spherical and flat models produce comparable coseismic deformations up to a distance of ∼300 km from the epicenter. This distance is sensibly reduced in the postseismic regime and when infinitely long strike-slip faults are considered. The differences between predictions based on flat and spherical models are due both to their global geometry and the effect of the gravity forces. Self-gravitation has a minor role with respect to that of sphericity for surface coseismic deformations, while in the postseismic regime its effects increase considerably. As a case study, we consider the coseismic and postseismic deformations due to the great 1960 Chilean earthquake. The results of the spherical stratified model differ sensibly from those of a flat uniform model. Moreover, within the framework of spherical Earth models, the rheological stratification plays a major role in determining the pattern of the displacement field. We show that the present-day rates of vertical and horizontal deformations are considerably large (∼10 -2 m yr -1 ) for an asthenospheric viscosity ranging from 10 19 to 10 20 Pa s. These rates, which could possibly be detected by geodetic investigations, are found to be also sensitive to the rheological properties of the mantle beneath the asthenosphere.

Paleofluvial Mega-Canyon Beneath the Central Greenland Ice Sheet

Ice Lubricant The Greenland and Antarctic ice sheets both possess hydrological systems that allow water accumulating from the melting of surface ice to be transported to the base of the ice sheet. If that water, when it reaches the ice-bedrock interface, is distributed over large areas, it will lubricate rapid ice sheet flow toward the sea. Bamber et al. (p. 997) report the existence of a large, 750-km-long subglacial canyon in northern Greenland, which may act as a channel for the transport of basal meltwater to the margin of the ice sheet and thus influence overall ice sheet dynamics. A large subglacial canyon extends for more than 750 kilometers from central Greenland to its northern margin. Subglacial topography plays an important role in modulating the distribution and flow of basal water. Where topography predates ice sheet inception, it can also reveal insights into former tectonic and geomorphological processes. Although such associations are known in Antarctica, little consideration has been given to them in Greenland, partly because much of the ice sheet bed is thought to be relatively flat and smooth. Here, we present evidence from ice-penetrating radar data for a 750-km-long subglacial canyon in northern Greenland that is likely to have influenced basal water flow from the ice sheet interior to the margin. We suggest that the mega-canyon predates ice sheet inception and will have influenced basal hydrology in Greenland over past glacial cycles.

Lateral viscosity variations and post‐glacial rebound: Effects on present‐day VLBI baseline deformations

By means of a finite element algorithm we investigate the effects of lateral viscosity variations upon the horizontal motions currently detected by VLBI techniques in Europe. Our axisymmetric flat models, appropriate to describe the rebound due to the melting of the Fennoscandia ice-sheet, are characterized by a layered compressible mantle with linear Maxwell rheology. We have first computed the time-evolution of theoretical baselines characterized by a simple geometry. In agreement with previous results, the rates of horizontal deformation have been found to be greatly sensitive to both lateral viscosity variations and deep mantle stratification, especially for baselines located in the vicinity of the ice sheet margin. To complete our study, we have compared our numerical results with the observed time-evolutions of relevant European baselines. Onsala-Wettzell and Onsala-Eflsberg are fitted by a laterally varying asthenosphere and sharp viscosity increase in the lower mantle; longer baselines, connecting these sites to Medicina (northern Italy), indicate that post glacial rebound is responsible only for a fraction of the VLBI observations, the residual being thus attributable to continental collision in the Mediterranean Sea.

Toroidal/poloidal partitioning of global post‐seismic deformation

An Earth model subject to dislocations is studied in order to investigate the toroidal/poloidal content of global post-seismic deformation. Differently from previous analyses, our approach allows us to deal with some of the main complexities of the real Earth, such as sphericity, self-gravitation, and rheological stratification. The time-dependent ratio between toroidal and poloidal displacements is evaluated in the near and far-field of both strike and dip-slip lithospheric sources characterized by a finite length. Our findings show that post-seismic deformation changes its nature from poloidal to toroidal (and viceversa) on time-scales of several decades and over distances comparable with the dimensions of the lithospheric plates. The effects on horizontal velocities in North America of both poloidal and toroidal components induced by the Alaska 1964 earthquake is comparable with the signatures due to postglacial rebound; these two geodynamical phenomena are currently detectable by variations along baselines connecting VLBI stations.

Long-term rotation and mantle dynamics of the Earth, Mars, and Venus

In response to internal and surface tectonic processes the terrestrial planets have the ability to displace the axis of rotation with respect to the mantle. This behavior is quantified by means of a nondimensional number, defined here as the rotational number Ro, that allows classification of the planets into two categories, the first containing Mars and the Earth, where true polar wander is a feasible mechanism, and the second, to which Venus belongs, where rotation equilibrium is attained by means of mega-wobbles driven by internal mass anomalies. The number Ro is related to the timescale characterizing the readjustment of the equatorial bulge during long-term polar motion and to the length of the sidereal day. If these two timescales are well separated, the planet experiences true polar wander. Nonlinear Liouville equations for stratified viscoelastic bodies with linear Maxwell rheology are solved in order to assess the relevance of surface and mantle processes in driving long-term rotation instabilities in the terrestrial planets. The amount of true polar wander estimated for the Earth and Mars is reproduced correctly by our modeling with mantle viscosities and lithospheric thickness consistent with other studies. The major difference between the Earth and Mars is the driving mechanism, subduction for the Earth and lithospheric loading for Mars. When mantle viscosities similar to those of the Earth are considered for Venus, the most updated estimate for the offset of about 0.5° between the rotation axis and the axis of maximum inertia is well reproduced during the mega-wobbles induced by internal mass redistribution. We show that the degree 2 topography of these three planets can be affected by their rotation, which is responsible for the dominance of the sectorial component on the Earth and Mars and for the zonal component on Venus.

Modeling Earth's post‐glacial rebound

Efforts to mathematically model the Earths post-glacial rebound, or, in general, long-term planetary-scale viscoelastic deformations, have been ongoing for several decades. Unfortunately, research in the post-glacial rebound community has not been characterized by much exchange of knowledge. Groups around the world have developed their code independently, sometimes with profoundly different approaches, occasionally leading to inconsistent results [e.g., Boschi et al., 1999]. Postglacial Rebound Calculator (TABOO) is a post-glacial rebound software that is being made freely available (through Samizdat Press at http://samizdat.mines.edu/taboo/) in the hope that it might become a common reference for all post-glacial rebound researchers. TABOO is portable and has been tested on Unix, Linux, and Windows systems; all it requires is a Fortran90 compiler supporting quadruple precision. The software is easy to use. It comes with a detailed guide that can work as a quick reference cookbook, and it is also accompanied by a textbook, The Theory Behind TABOO, collecting the most significant theoretical results from post-glacial rebound literature. TABOO is not a “black-box,” although it may easily be used as such. The entire source code is provided and should be easy to understand for intermediate-level Fortran programmers.

Global postseismic deformation: Deep earthquakes

We study the global viscoelastic deformations associated with a shear dislocation on a fault embedded in a viscoelastic mantle. To address this problem, we extend the theory of the quasi-static deformations of a Maxwell, spherical, N-layer incompressible Earth, previously limited to the modeling of the effects of earthquakes occurring only within the elastic lithosphere, Permanent postseismic deformations, which can be generated by lithospheric sources, cannot be sustained if the source region is viscoelastic; however, the transient response of the mantle strongly depends on the viscosity of the source region. We use the technique developed here to investigate thoroughly the quasi-static surface deformations induced by seismic events of variable depth. We show that owing to the combined effect of sphericity and viscoelastic mantle relaxation, the surface displacements do not systematically decrease as the depth of the source increases. Instead, with increasing source depth we predict an increasing efficiency of mantle relaxation in triggering postseismic deformations of large size. Most of our results are based on a simple four-layer model, with a 100-km-thick lithosphere, upper and lower mantle separated by the 670-km discontinuity, and a fluid core; the last part of this work is devoted to a study of the effects of a low-viscosity asthenosphere on the rates of deformation detected at the Earths surface. The findings reported here may be useful for the interpretation of the transient motions of the Earths surface in response to deep-focus earthquakes.