We present an analytical form of the layer propagator matrix for the response of a locally incompressible, layered, linear-viscoelastic sphere to an external load assuming that the initial density ...stratification ϱ0(r) within each layer is parametrized by Darwin's law. From this, we show that the relaxation of a sphere consisting of locally incompressible layers is governed by a discrete set of viscous modes. The explicit dependence of the layer propagator matrix on the Laplace transform variable allows us to determine the amplitudes of the viscous modes analytically. Employing Darwin's parametrization, we construct three simplified earth models with different initial density gradients that are used to compare the effects of the local incompressibility constraint, div (ϱ0u)=0, and the material incompressibility constraint, div u=0, on viscoelastic relaxation. We show that a locally incompressible earth model relaxes faster than a materially incompressible model. This is a consequence of the fact that the perturbations of the initial density are zero during viscoelastic relaxation of a locally incompressible medium, so that there are no internal buoyancy forces associated with the continuous radial density gradients, only the buoyancy forces generated by internal density discontinuities. On the other hand, slowly decaying internal buoyancy forces in a materially incompressible earth model cause it to reach the hydrostatic equilibrium after a considerably longer time than a locally incompressible model. It is important to note that the approximation of local incompressibility provides a solution for a compressible earth model that is superior to the conventional solutions for a compressible earth with homogeneous layers because it is based on an initial state that is consistent with the assumption of compressibility.
Spherical harmonic coefficients of the Earth's external gravity field represent important characteristics of the Earth. Satellite tracking data, land-borne gravity observations and altimeter data ...must be combined to determine these coefficients with a sufficient reliability and resolution. This altimetry-gravimetry problem is governed by the Laplace equation in the space outside the Earth with mixed types of boundary condition. It has been proved that the classical least-squares solution of the continuous altimetry-gravimetry problem is not stable. On the contrary, this paper demonstrates that the least-squares solution of the discrete altimetry-gravimetry problem does not fail and is stable provided that the boundary functionals are discretized in an equal angular grid, the cut-off degree of a global gravity model is not greater than 500, and the covariance matrices of the boundary data are set up in accordance with today's accuracy of geodetic measurements. Moreover, provided that the number of observations is greater than the cut-off degree of the potential series, the approximation error of the least-squares solution does not depend on the number of observations. This paper also treats the numerical stability of the discrete altimetry-gravimetry problem overdetermined by extra gravity data measured by ships over a part of the sea surface. Overlapping boundary data do not have, however, a significant impact on the conditionality of the matrix of normal equations, due to the low accuracy of sea-borne gravity observations. It should be emphasized that the leading motivation of the paper is to show that the classical least-squares solution to the discrete altimetry-gravimetry problem provides us with a reliable tool for constructing global gravity models. The method suggested here avoids converting the altimeter data to gravity disturbances—the preparatory step often made in the construction of a global gravity model—because such a transformation is ill-posed.
We infer regional mass changes in Antarctica using ca. 4 years of Gravity Recovery and Climate Experiment (GRACE) level 2 data. We decompose the time series of the Stokes coefficients into their ...linear as well as annual and semi-annual components by a least-squares adjustment and apply a statistical reliability test to the Stokes potential-coefficients' linear temporal trends. Mass changes in three regions of Antarctica that display prominent geoid-height change are determined by adjusting predictions of glacier melting at the tip of the Antarctic Peninsula and in the Amundsen Sea Sector, and of the glacial-isostatic adjustment (GIA) over the Ronne Ice Shelf. We use the GFZ RL04, CNES RL01C, JPL RL04 and CSR RL04 potential-coefficient releases, and show that, although all data sets consistently reflect the prominent mass changes, differences in the mass-change estimates are considerably larger than the uncertainties estimated by the propagation of the GRACE errors. We then use the bootstrapping method based on the four releases and six time intervals, each with 3.5 years of data, to quantify the variability of the mean mass-change estimates. We find 95% of our estimates to lie within 0.08 and 0.09 mm/a equivalent sea-level (ESL) change for the Antarctic Peninsula and within 0.18 and 0.20 mm/a ESL for the Amundsen Sea Sector. Forward modelling of the GIA over the Ronne Ice Shelf region suggests that the Antarctic continent was covered by 8.4 to 9.4 m ESL of additional ice during the Last-Glacial Maximum (ca. 22 to 15 ka BP). With regards to the mantle-viscosity values and the glacial history used, this value is considered as a minimum estimate. The mass-change estimates derived from all GRACE releases and time intervals lie within ca. 20% (Amundsen Sea Sector), 30% (Antarctic Peninsula) and 50% (Ronne Ice Shelf region) of the bootstrap-estimated mean, demonstrating the reliability of results obtained using GRACE observations.
We introduce a time-domain method to solve the problem of geomagnetic induction in a heterogeneous Earth excited by variations of the ionospheric and magnetospheric currents with arbitrary ...spatiotemporal characteristics. The problem is formulated in the integral sense. The Galerkin discretization is based on the vector spherical harmonic functions and piecewise linear finite elements in the angular and radial directions, respectively. A semi-implicit time integration scheme is employed. The method is validated against semi-analytical (SA) solutions for conductivity models consisting of multiple eccentrically nested homogeneous spheres.
We have constructed Green's function to Stokes's boundary-value problem with the gravity data distributed over an ellipsoid of revolution. We show that the problem has a unique solution provided that ...the first eccentricity e0of the ellipsoid of revolution is less than 0·65041. The ellipsoidal Stokes function describing the effect of ellipticity of the boundary is expressed in theE-approximation as a finite sum of elementary functions which describe analytically the behaviour of the ellipsoidal Stokes function at the singular point ψ = 0. We prove that the degree of singularity of the ellipsoidal Stokes function in the vicinity of its singular point is the same as that of the spherical Stokes function.
We revisit the results of the ISMIP-HEINO benchmark by first analyzing the differences in various model outputs using a wavelet-based spectral technique. Second, the ISMIP-HEINO benchmark experiments ...are recomputed with a novel numerical ice-sheet model based on the SIA-I algorithm that enables both the shallow-ice and a higher-order approximation of the ice-flow equations to be performed. To assess the significance of the higher-order approximation in the ISMIP-HEINO experiment, a numerical sensitivity study for the shallow-ice approximation (SIA) simulations is also carried out. A high sensitivity of the SIA model response to surface temperature perturbations is found. We conclude that the variations in ISMIP-HEINO results are due to the differences in (1) simulated basal temperatures and (2) numerical treatment of the basal sliding condition.
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