Surface vertical displacements, potential perturbations andgravity changes of a viscoelastic earth model induced by internal point dislocations

Using the viscoelastic correspondence principle, we utilize the surface coseismic spheroidal deformation fields (i.e. vertical displacements, potential perturbations and gravity changes) of SNREI earth models caused by four typical types of point dislocation, derived by Sun & Okubo (1993 ), to deduce the fundamental formulas for spheroidal fields relevant to viscoelastic earth models. In computations, we employ a strike-slip dislocation on a vertical plane buried at the bottom of the lithosphere to estimate the maximal viscous relaxation responses to this kind of source that possibly exist on the surface of the earth. We take the seismic moment as 10²²  N  m, which is characteristic of an average large earthquake. The numerical results demonstrate that, if we take the viscosity as 10¹⁹  Pa  s in the asthenosphere, and 10²¹  Pa  s in the other mantle layers, the rates of surface vertical displacements and gravity changes within about 2.5° for the 10 postseismic years are respectively 1.5–8.1  cm  yr⁻¹ and 4.0–14.9  μgal  yr⁻¹ : the viscous relaxation for this mantle viscosity profile proceeds much faster than for a constant mantle viscosity of 10²¹  Pa  s.     

Pleistocene deglaciation and the Earth's rotation: implications for mantle viscosity

Summary. Recent results from the analysis of postglacial rebound data suggest that the viscosity of the Earth's mantle increases through the transition region. Models which fit both relative sea-level and free air gravity data have viscosities which increase from a value near 10²² poise in the upper mantle beneath the lithosphere to a value of about 10²³ poise in the lower mantle. In this paper we analyse the effect of deglaciation upon the Earth's rotation and thereby show that the observed secular trend (polar wander) evident in the ILS—IPMS pole path, and measurements of the non-tidal acceleration of the length of day, are both consistent with the viscosity profile deduced from postglacial rebound. The two analyses are therefore mutually reinforcing.     

Small-scale instabilities below the cooling oceanic lithosphere

Interpretation of satellite altimetry data as well as ship bathymetry data revealed strongly elongated anomalies roughly perpendicular to the mid-ocean ridges in the Indian and east Pacific oceans. A spectral analysis of gravity altimetry data along profiles parallel to the East Pacific Rise indicated wavelengths of about 150–180  km close to the ridge and about 250  km further away. A simple model of Rayleigh–Taylor instabilities developing at the base of the cooling lithosphere is discussed and applied to the data. By considering thermal diffusion and comparing Rayleigh–Taylor growth rates to the velocity of the thermal front in the cooling lithosphere, we are able to explain the observed anomalies by instabilities developing below the lithosphere in a layer with a viscosity of about 10¹⁹  Pa  s above an asthenospheric layer with a viscosity reduction of 2–3 orders of magnitude.     

Deglaciation effects on the rotation of the Earth

Summary. The viscoelastic response of the Earth to the mass displacements caused by late Pleistocene deglaciation and concomitant sea level changes is shown to be capable of producing the secular motion of the Earth's rotation pole as deduced from astronomical observations. The calculations for a viscoelastic Earth yield a secular motion in the direction of 72° W meridian which is in excellent agreement with observed values. The average Newtonian viscosity and the relaxation time obtained from polar motion data are about (1.1 ± 0.6)10²³ poise (P) and 10⁴ (1 ± 0.5) yr. The non-tidal secular acceleration of the Earth can also be attributed to the viscoelastic response to deglaciation and results in an independent viscosity estimate of 1.6 × 10²³ P with upper and lower limits of 1.1 × 10²³ and 2.8 × 10²³ P. These values are in agreement with those based on the polar drift analysis and indicate an average mantle viscosity of 1–2 × 10²³ P.     

Subsurface mass redistributions at Mount Etna (Italy) during the 1995–1996 explosive activity detected by microgravity studies

Gravity changes are presented from a series of field microgravity surveys conducted at Mt Etna between August 1994 and November 1996, a period including the 1995–1996 explosive summit activity. Data were collected along a microgravity network of 69 stations at a monthly to annual sampling rate, depending on each subarray of the network.
  Results show that seasonal changes in water level within the volcano may induce gravity changes of up to 20  μgal on Etna's southern slope, and indicate that significant magma movement occurred within and below Etna's edifice between 1994 and 1996. In particular, between September 1994 and October 1995, a mass increase of 2 × 10¹⁰  kg occurred 2000  m beneath the summit craters. Between October 1995 and July 1996 this mass was lost, while another 2 × 10¹⁰  kg was injected at about 1000  m  a.s.l. into the 1989 fracture system. From the gravity data alone, it is not possible to distinguish whether the first shallow intrusion (1994–1995) was then injected laterally into the 1989 fracture, or summit activity was fed by the first shallow intrusion, while new magma entered the 1989 fracture system.
  While magma was being redistributed within the volcanic edifice, measurements along an E–W-trending profile on the southern slope of the volcano detected some 1.5 × 10¹¹  kg of magma accumulating 2–3  km below sea level between October 1995 and November 1996.     

Effect of viscosity structure on fault potential and stress orientations in eastern Canada

Previous investigations of the causal relationship between postglacial rebound and earthquakes in eastern Canada have focused on the mode of failure and the observed timing of the pulse of earthquake/faulting activity following deglaciation. In this study, the observational database has been extended to include observed orientations of the contemporary stress field and the rotation of stress since deglacial times. It is shown that many of these observations can be explained by a realistic ice history and a viscoelastic earth with a uniform 10²¹ Pa s mantle.
The effects of viscosity structure on the above predictions are also examined. It is shown that, since most of the above observations are found within the ice margin, they are not very sensitive to lithospheric thickness. Also, the inclusion of a 25 or 50 km ductile layer within the lithosphere will not decouple the seismogenic upper crust. High viscosity (10²² Pa s) in the lower mantle is rejected by the stress orientation and rotation observations. A low-viscosity (6 times 10²⁰Pa s) upper mantle with 1.6 times 10²¹ Pa s in the upper part of the lower mantle and 3 times 10²¹ Pa s in the lower part of the lower mantle below 1200 km depth has been found to give predictions that are in general agreement with the observations.     

Palaeointensity results from Ethiopian basalts: implications for the Oligocene geomagnetic field strength

From a large collection of Ethiopian flood basalts (~30  Myr old) sampled for magnetostratigraphy, ⁴⁰Ar/³⁹Ar geochronology and geochemical investigations, 47 samples were selected in order to test their suitability for Thellier palaeointensity experiments. Only 17 samples from eight individual flows yielded reliable palaeointensity estimates, with flow-mean virtual dipole moments ranging from 3.0 to 10.5 × 10²²  A  m² .
  A critical review of the Oligocene palaeointensity data set, including these new Ethiopian data, indicates an Oligocene mean virtual dipole moment of 5.1 ± 2.5 × 10²²  A  m² for the complete data set. After applying mild selection criteria, the reduced data set yields a mean value of only 4.6 ± 1.9 × 10²²  A  m² . This value is significantly lower than the present-day field strength but is higher than the Mesozoic dipole low mean field. This low Oligocene field might be in agreement with the high palaeosecular variation and rather high non-dipole field invoked around 30  Ma. However, the Oligocene data set is largely dependent on the palaeointensity determinations from Armenia, obtained mainly from baked contacts, which show amazingly low dispersion at both flow and between-flow levels. More data are needed to reduce the weight of these determinations on the mean value and avoid a possible bias.     

Glacial rebound of the British Isles—II. A high-resolution, high-precision model

Observations of ice movements across the British Isles and of sea-level changes around the shorelines during Late Devensian time (after about 25 000 yr BP) have been used to establish a high spatial and temporal resolution model for the rebound of Great Britain and associated sea-level change. The sea-level observations include sites within the margins of the former ice sheet as well as observations outside the glaciated regions such that it has been possible to separate unknown earth model parameters from some ice-sheet model parameters in the inversion of the glacio-hydro-isostatic equations. The mantle viscosity profile is approximated by a number of radially symmetric layers representing the lithosphere, the upper mantle as two layers from the base of the lithosphere to the phase transition boundary at 400 km, the transition zone down to 670 km depth, and the lower mantle. No evidence is found to support a strong layering in viscosity above 670 km other than the high-viscosity lithospheric layer. Models with a low-viscosity zone in the upper mantle or models with a marked higher viscosity in the transition zone are less satisfactory than models in which the viscosity is constant from the base of the lithosphere to the 670 km boundary. In contrast, a marked increase in viscosity is required across this latter boundary. The optimum effective parameters for the mantle beneath Great Britain are: a lithospheric thickness of about 65 km, a mantle viscosity above 670 km of about (4-5) 10²⁰ Pa s, and a viscosity below 670 km greater than 4 × 10²¹ Pa s.     

Deglacial land emergence and lateral upper-mantle heterogeneity in the Svalbard Archipelago—II. Extended results for high-resolution load models

In Paper I (Breuer & Wolf 1995), a preliminary interpretation of the postglacial land emergence observed at a restricted set of six locations in the Svalbard Archipelago was given. The study was based on a simple model of the Barents Sea ice sheet and suggested increases in lithosphere thickness and asthenosphere viscosity with increasing distance from the continental margin.
In the present paper, the newly developed high-resolution load model. BARENTS-2, and land-uplift observations from an extended set of 25 locations are used to study further the possibility of resolving lateral heterogeneity in the upper mantle below the northern Barents Sea. A comparison of the calculated and observed uplift values shows that the lithosphere thickness is not well resolved by the observations, although values above 110 km are most common for this parameter. In contrast to this, there are indications of a lateral variation of asthenosphere viscosity. Whereas values in the range 10¹⁸-10²⁰Pas are inferred for locations close to the continental margin, 10²⁰-10²¹ Pa s are suggested further away from the margin.
A study of the sensitivity of the values found for lithosphere thickness and asthenosphere viscosity to modifications of load model BARENTS-2 shows that such modifications can be largely accommodated by appropriate changes in lithosphere thickness, whereas the suggested lateral variation of asthenosphere viscosity is essentially unaffected. An estimate of the influence of the Fennoscandian. ice sheet leads to the conclusion that its neglect results in an underestimation of the thickness of the Barents Sea ice sheet by about 10 per cent.     

Stresses due to phase changes in subduction zones and an empirical equation of state for the mantle

Summary. An empirical equation of state, assuming mineralogical equilibrium, is developed for the top 700 km of the mantle. Assuming a uniform viscosity, this equation of state is used to show that the stresses due to the changes in phase induced in a descending lithospheric plate in a subduction zone are an order of magnitude larger than those due to the negative buoyancy of the slab in the asthenosphere. The stresses predicted are well within the power law creep region for likely mantle materials and so the effective viscosity will vary within the slab. Consequently the stresses will be smaller than those of 7.0 × 10⁸N/m² obtained here using uniform viscosities. These stresses are relatively compressional near the sides of the slab and tensional in the centre.     

The effects of post-seismic motions on the moment of inertia of a stratified viscoelastic earth with an asthenosphere

Summary. We give the analytical formulation for calculating the transient displacement of fields produced by earthquakes in a stratified, selfgravitating, incompressible, viscoelastic earth. We have evaluated the potential of viscous creep in the asthenosphere in exciting the Chandler wobble by a four-layer model consisting of an elastic lithosphere, a two-layer Maxwell viscoelastic mantle, and an inviscid core. The seismic source is modelled as an inhomogeneous boundary condition, which involves a jump condition of the displacement fields across the fault in the lithosphere. The response fields are derived from the solution of a two-point boundary value problem, using analytical propagator matrices in the Laplace-transformed domain. Transient flows produced by post-seismic rebound are found to be confined within the asthenosphere for local viscosity values less than 10²⁰P. The viscosity of the mantle below the low-viscosity channel is kept at 10²²P. For low-viscosity zones with widths greater than about 100 km and asthenospheric viscosities less than 10¹⁸P, we find that viscoelasticity can amplify the perturbations in the moment of inertia by a factor of 4–5 above the elastic contribution within the time span of the wobble period. We have carried out a comparative study on the changes of the inertia tensor from forcings due to surface loading and to faulting. In general the global responses from faulting are found to be much more sensitive to the viscosity structure of the asthenosphere than those produced from surface loading.     

Secular variation of the poloidal magnetic field at the core–mantle boundary

A region of enhanced conductivity at the base of the mantle is modelled by an infinitesimally thin sheet of uniform effective conductance adjacent to the core–mantle boundary. Currents induced in this sheet by the temporally varying magnetic field produced by the geodynamo give rise to a discontinuity in the horizontal components of the poloidal magnetic field on crossing the sheet, while the radial component is continuous across the sheet. Treating the rest of the mantle as an insulator, the horizontal components of the poloidal magnetic field and their secular variation at the top of the core are determined from geomagnetic field, secular variation and secular acceleration models. It is seen that for an assumed effective conductance of the sheet of 10⁸  S, which may be not unrealistic, the changes produced in the horizontal components of the poloidal field at the top of the core are usually ≤10 per cent, but corrections to the secular variation in these components at the top of the core are typically 40 per cent, which is greater than the differences that exist between different secular variation models for the same epoch. Given the assumption that all the conductivity of the mantle is concentrated into a thin shell, the present method is not restricted to a weakly conducting mantle. Results obtained are compared with perturbation solutions.     

Model of the geoelectrical structure of the mid- and lower mantle in the Europe–Asia region

The conductivity structure of the Earth's mantle was estimated using the induction method down to 2100  km depth for the Europe–Asia region. For this purpose, the responses obtained at seven geomagnetic observatories (IRT, KIV, MOS, NVS, HLP, WIT and NGK) were analysed, together with reliable published results for 11  yr variations. 1-D spherical modelling has shown that, beneath the mid-mantle conductive layer (600–800  km), the conductivity increases slowly from about 1  S  m⁻¹ at 1000  km depth to 10  S  m⁻¹ at 1900  km, while further down (1900–2100  km) this increase is faster. Published models of the lower mantle conductivity obtained using the secular, 30–60  yr variations were also considered, in order to estimate the conductivity at depths down to the core. The new regional model of the lower mantle conductivity does not contradict most modern geoelectrical sounding results. This model supports the idea that the mantle base, situated below 2100  km depth, has a very high conductivity.     

Iceocean mass balance during the Late Pleistocene glacial cycles in view of CHAMP and GRACE satellite missions

During the last glacial cycles, global sea level dropped several times by about 120 m and large ice sheets covered North America, northern Europe and Antarctica during the glacial stages. The changes in the iceocean mass balance have displaced mantle material mainly via viscous flow, and the perturbation of the equilibrium figure of the Earth by glacial isostatic adjustment is still observable today in timedependent changes of gravitational and rotational observations. Contemporary iceocean mass balance from volume changes of polar ice caps also contributes to secular variations of the Earth's gravitational field.
In the near future, several satellite gravity missions will significantly improve the accuracy of the observed timedependent gravitational field. In view of the expected improvements in the observations, we predict glacially induced perturbations of the gravitational field, induced by Late Pleistocene and contemporary ice volume changes, for a variety of radial mantle viscosity profiles. We assess the degree of uncertainty for the glacially induced contributions to gravitational and rotational parameters, both in the spectral and the spatial domain.
Predictions of power spectra for the glacially induced freeair gravity and geoid anomalies are about one order of magnitude lower than the observed values, and uncertainties arising from different plausible viscosity profiles are around 0.150.4 mGal and 0.21.5 m, respectively. Uncertainties from different ice models are of secondary importance for the predicted power spectra. Predicted secular changes in geoid anomalies in formerly glaciated areas are mainly controlled by the viscosity profile and contemporary ice volume changes. We also show that the simple threelayer viscosity profiles currently employed for the majority of postglacial rebound studies represent a limited subset for model predictions of the timedependent gravitational field.     

Core–mantle boundary deformations and J2 variations resulting from the 2004 Sumatra earthquake

The deformation at the core–mantle boundary produced by the 2004 Sumatra earthquake is investigated by means of a semi-analytic theoretical model of global coseismic and postseismic deformation, predicting a millimetric coseismic perturbation over a large portion of the core–mantle boundary. Spectral features of such deformations are analysed and discussed. The time-dependent postseismic evolution of the elliptical part of the gravity field ( J ₂) is also computed for different asthenosphere viscosity models. Our results show that, for asthenospheric viscosities smaller than 10¹⁸ Pa s, the postseismic J ₂ variation in the next years is expected to leave a detectable signal in geodetic observations.     

On the penetration of a hot diapir through a strongly temperature-dependent viscosity medium

Summary. The ascent of a hot spherical body through a fluid with a strongly temperature-dependent viscosity has been studied using an axisymmetric finite element method. Numerical solutions range over Peclet numbers of 10⁻¹– 10³ from constant viscosity up to viscosity variations of 10⁵. Both rigid and stress-free boundary conditions were applied at the surface of the sphere. The dependence of drag on viscosity variation was shown to have no dependence on the stress boundary condition except for a Stokes flow scaling factor. A Nusselt number parameterization based on the stress-free constant viscosity functional dependence on the Peclet number scaled by a parameter depending on the viscosity structure fits both stress-free and rigid boundary condition data above viscosity variations of 100. The temperature scale height was determined as a function of sphere radius. For the simple physical model studied in this paper pre-heating is required to reduce the ambient viscosity of the country rock to less than 10²² cm² s⁻¹ in order for a 10 km diapir to penetrate a distance of several radii.     

Combining geomorphological and geodetic data to determine postglacial tilting in Manitoba

Estimates of postglacial rebound in central North America from Laurentide deglaciation to the present time are uncertain as a result of lack of data from the continental interior. A more precise knowledge of postglacial tilt history will assist studies of the evolution of the major lakes in Manitoba and will facilitate the engineering and environmental management of the present-day hydrological system. This paper explores the benefits of combining geomorphological data with high-precision, real-time geodetic data (GPS positioning and absolute gravity) and lake-gauge tilt data now being collected for postglacial rebound studies in Manitoba and adjacent regions in the US. Presently-available data sets representing these data types are (1) tilting of the 9.5 kyr B.P. Campbell strand line south and west of Lake Winnipeg, (2) the rate of decrease in absolute gravity values measured from 1987 to 1995 at Churchill, Manitoba, and (3) the present-day regional tilt rate derived from water-level gauges in southern Manitoba lakes. These data are compared to theoretical predictions based on the published ICE-3G loading history and on a model of Earth rheology characterized by a 1066B density and elastic structure, an upper-mantle viscosity of 10 ²¹Pa s, a lower-mantle viscosity of 2 × 10 ²¹Pa s, and a lithosphere thickness of 120 km (Tushingham & Peltier, 1991). All three data types show disagreement in Manitoba with ICE-3G and the standard Earth model. ICE-4G does better but could not be investigated in any detail. The constraints on model parameters provided by the different data types were investigated by varying, one at a time, three key parameters, (1) the thickness of the lithosphere in excess of 120 km, (2) the lower mantle viscosity, and (3) the thickness of Laurentide ice over the Prairies, to obtain better fits to the data. The present data do not appear to constrain lithosphere thickness in excess of 120 km very well. While both the Campbell strand line data and the Churchill absolute gravity data are consistent with an increase in lower-mantle viscosity, the present-day, lake-gauge data are not. All three data types are consistent with a thinning of the Laurentide ice-sheet over the Prairies relative to the ICE-3G model. Simultaneous adjustment of model parameters with the advantage of anticipated new data in Manitoba and adjacent regions in the US will lead to better understanding of the trade-offs between Earth rheology and ice sheet history and hence to an improved Laurentide postglacial rebound model.     

Strain accumulation along an oblique plate boundary: the Reykjanes Peninsula, southwest Iceland

We use annual GPS observations on the Reykjanes Peninsula (RP) from 2000 to 2006 to generate maps of surface velocities and strain rates across the active plate boundary. We find that the surface deformation on the RP is consistent with oblique plate boundary motion on a regional scale, although considerable temporal and spatial strain rate variations are observed within the plate boundary zone. A small, but consistent increase in eastward velocity is observed at several stations on the southern part of the peninsula, compared to the 1993–1998 time period. The 2000–2006 velocities can be modelled by approximating the plate boundary as a series of vertical dislocations with left-lateral motion and opening. For the RP plate boundary we estimate left-lateral motion  18⁺⁴₋₃ mm yr⁻¹  and opening of  7⁺³₋₂ mm yr⁻¹  below a locking depth of  7⁺¹₋₂ km  . The resulting deep motion of  20⁺⁴₋₃ mm yr⁻¹  in the direction of  N(100⁺⁸₋₆)°E  agrees well with the predicted relative North America–Eurasia rate. We calculate the areal and shear strain rates using velocities from two periods: 1993–1998 and 2000–2006. The deep motion along the plate boundary results in left-lateral shear strain rates, which are perturbed by shallow deformation due to the 1994–1998 inflation and elevated seismicity in the Hengill–Hrómundartindur volcanic system, geothermal fluid extraction at the Svartsengi power plant, and possibly earthquake activity on the central part of the peninsula.     

Uplift and heat flow following the injection of magmas into the lithosphere

Summary. The thermal effect of a rapid injection of hot magmas into the lower part of the lithosphere is modelled as an increase in heat production through the invaded region. The change in surface heat flow and the uplift resulting from the thermal expansion are determined in three-dimensional axially symmetric geometry: they are expressed as the space time convolutions of a Green's function with the anomalous heat production.
The anomalies with shorter wavelength (compared to the lithospheric thickness) are attenuated. This filtering affects the surface uplift more than the heat flow anomaly; the attenuation effect is larger when only the lower part of the lithosphere is invaded.
The uplift time constant is of the same order as the heat conduction time if the lower lithosphere is invaded by magmas at a moderate rate (i.e. the rate of injection does not exceed the equivalent of 0.1 per cent of the lithospheric volume in 10⁶yr). Fifty per cent of the total uplift takes place in about 80 × 10⁶yr for a lithosphere 100 km thick. The uplift is slightly faster when the whole lithosphere is invaded. The heat flow anomaly is delayed when the lower part of the lithosphere is invaded.
The spatial extent and the timing of the uplift and heat flow anomalies are critical in determining the mechanism's feasibility. Magma injections explain rapid uplifts [> 100 m (10⁶ yr)⁻¹] only if the magma is supplied at a very high rate (i.e. at least 10 per cent of the lithosphere volume per 10⁶yr). It is a feasible mechanism for uplifts that occur over longer periods of time (≊ 30 × 10⁶yr) such as those that seem to have occurred when the African plate came to rest with respect to the mantle.     

Long-period geomagnetic variations and mantle conductivity: an inversion using Bailey's method

Summary. We have implemented an algorithm which is based on Bailey's solution of the inverse problem of electromagnetic induction in the Earth. The study was motivated by recent determinations of very long period data and also benefited from recent redeterminations of high frequency data. The algorithm has been successfully tested to provide reliable estimates of conductivity down to a depth of 2000 km, using synthetic data in the period range from 4 days to 11 years. Smooth data sets, which are required for the inversion, were constructed from various sources. At a given depth, the range of inverted models is less than one order of magnitude. Due to the lack of high frequency data, the conductivity of the upper 600 km of the mantle, which is found to be of the order of 10⁻¹Ω⁻¹ m⁻¹, may be overestimated. The algorithm performs well in the middle mantle, where conductivity rises steadily from 1 to 50 Ω⁻¹ m⁻¹. The lack of very low frequency data and limitations of the algorithm prevent one from obtaining meaningful estimates in the lower mantle. However, the study of the propagation of the late 1960s secular variation acceleration provides an estimate of the mean conductivity of the whole mantle. Thus, a complete mantle profile can be constructed. It is found that deep mantle conductivity probability does not exceed a few hundred Ω⁻¹ m⁻¹.