Description: The problem of compressibility in modelling of viscoelastic deformations of planetary bodies is still a topic under discussion. Studies facing this topic discuss the error when considering a stratification of layers with constant material parameters. But homogeneous compressible layers imply that the initial state is not stable. So, any perturbation method applied to this type of state results in an ill-posed problem, evident in a denumerable infinite set of modes in the spectral representation of the solution. Based on the analytic solution of Cambiotti and Sabadini, we discuss any violation from the stable Adams–Williamson condition to result in unphysical behaviour where we concentrate here on the consequences for the horizontal displacement and deformation within the mantle due to surface loading. This focus motivates to revisit the Longman paradox, which discusses the boundary conditions for a compressible fluid core.

Description: Global navigation satellite systems (GNSSs) have revealed that a mega-thrust earthquake that occurs in an island-arc trench system causes post-seismic crustal deformation. Such crustal deformation data have been interpreted by combining three mechanisms: afterslip, poroelastic rebound and viscoelastic relaxation. It is seismologically important to determine the contribution of each mechanism because it provides frictional properties between the plate boundaries and viscosity estimates in the asthenosphere which are necessary to evaluate the stress behaviour during earthquake cycles. However, the observation sites of GNSS are mostly deployed over land and can detect only a small part of the large-scale deformation, which precludes a clear separation of the mechanisms. To extend the spatial coverage of the deformation area, recent studies started to use satellite gravity data that can detect long-wavelength deformations over the ocean. To date, compared with theoretical models for calculating the post-seismic crustal deformation, a few models have been proposed to interpret the corresponding gravity variations. Previous approaches have adopted approximations for the effects of compressibility, sphericity and self-gravitation when computing gravity changes. In this study, a new spectral-finite element approach is presented to consider the effects of material compressibility for Burgers viscoelastic earth model with a laterally heterogeneous viscosity distribution. After the basic principles are explained, it is applied to the 2004 Sumatra–Andaman earthquake. For this event, post-seismic deformation mechanisms are still a controversial topic. Using the developed approach, it is shown that the spatial patterns of gravity change generated by the above three mechanisms clearly differ from one another. A comparison of the theoretical simulation results with the satellite gravity data obtained from the Gravity Recovery and Climate Experiment reveals that both afterslip and viscoelastic relaxation are occurring. Considering the spatial patterns in satellite gravity fields is an effective method for investigating post-seismic deformation mechanisms.

Description: The ability of any satellite gravity mission concept to monitor mass transport processes in the Earth system is typically tested well ahead of its implementation by means of various simulation studies. Those studies often extend from the simulation of realistic orbits and instrumental data all the way down to the retrieval of global gravity field solution time-series. Basic requirement for all these simulations are realistic representations of the spatio-temporal mass variability in the different sub-systems of the Earth, as a source model for the orbit computations. For such simulations, a suitable source model is required to represent (i) high-frequency (i.e., subdaily to weekly) mass variability in the atmosphere and oceans, in order to realistically include the effects of temporal aliasing due to non-tidal high-frequency mass variability into the retrieved gravity fields. In parallel, (ii) low-frequency (i.e., monthly to interannual) variability needs to be modelled with realistic amplitudes, particularly at small spatial scales, in order to assess to what extent a new mission concept might provide further insight into physical processes currently not observable. The new source model documented here attempts to fulfil both requirements: Based on ECMWF’s recent atmospheric reanalysis ERA-Interim and corresponding simulations from numerical models of the other Earth system components, it offers spherical harmonic coefficients of the time-variable global gravity field due to mass variability in atmosphere, oceans, the terrestrial hydrosphere including the ice-sheets and glaciers, as well as the solid Earth. Simulated features range from sub-daily to multiyear periods with a spatial resolution of spherical harmonics degree and order 180 over a period of 12 years. In addition to the source model, a de-aliasing model for atmospheric and oceanic high-frequency variability with augmented systematic and random noise is required for a realistic simulation of the gravity field retrieval process, whose necessary error characteristics are discussed. The documentation of the updated ESA Earth System Model (updated ESM) for gravity mission simulation studies is organized as follows: The characteristics of the updated ESM along with some basic validation is presented in Volume 1. A detailed comparison to the original ESA ESM (Gruber et al., 2011) is provided in Volume 2, while Volume 3 contains the description of a strategy to derive realistic errors for the de-aliasing model of high-frequency mass variability in atmosphere and ocean.

Description: We present regional-scale mass balances for 25 drainage basins of the Antarctic Ice Sheet (AIS) from satellite observations of the Gravity and Climate Experiment (GRACE) for the years 2002–2011. Satellite gravimetry estimates of the AIS mass balance are strongly influenced by mass movement in the Earth interior caused by ice advance and retreat during the last glacial cycle. Here, we develop an improved glacial-isostatic adjustment (GIA) estimate for Antarctica using newly available GPS uplift rates, allowing us to more accurately separate GIA-induced trends in the GRACE gravity fields from those caused by current imbalances of the AIS. Our revised GIA estimate is considerably lower than previous predictions, yielding an (upper) estimate of apparent mass change of 48 ± 18 Gt yr−1. Therefore, our AIS mass balance of −103 ± 23 Gt yr−1 is considerably less negative than previous GRACE estimates. The Northern Antarctic Peninsula and the Amundsen Sea Sector exhibit the largest mass loss (−25 ± 6 Gt yr−1 and −126 ± 11 Gt yr−1, respectively). In contrast, East Antarctica exhibits a slightly positive mass balance (19 ± 16 Gt yr−1), which is, however, mostly the consequence of compensating mass anomalies in Dronning Maud and Enderby Land (positive) and Wilkes and George V Land (negative) due to interannual accumulation variations. In total, 7% of the area constitute more than half of the AIS imbalance (53%), contributing −151 ± 9 Gt yr−1 to global mean sea-level change. Most of this imbalance is caused by long-term ice-dynamic speed up expected to prevail in the future.

Description: The amount of water that was stored in the ice sheets during the last glacial cycle is reflected as a global sea level drop of more than 120 m around 20,000 yr before present. This drop is evident in geological samples, the sea-level indicators (SLIs), which are analysed in regions far away from the glaciation regions. The analysis of this type of geological data allows the reconstruction of sea level variations during the glacial cycle and, so, to assess the mass transport between ice-sheets and ocean. Nevertheless, these farfield SLIs show a significant deviation of up to 20 m with respect to the eustatic sea level, which represents the global average of relative sea level. The largest variations appear for coastal sites, where the ocean floor is unloaded due to the reduced water column whereas the proximate land area is not affected. This mechanism, discussed as hydro-isostatic contribution in literature, results in a flexure of the lithosphere along all coastlines and, so, depends on the shape of the coastline but also on the rheological structure of the respective area.We assess this variabiltity of sea level change at typical SLI locations for different earth models and glaciation histories and show a large sensitivity of the predicted deviations from eustatic sea level to the considered earth model, whereas the influence of the considered glaciation history is rather small. This confirms a general strategy to model the hydro-isostatic contribution adopting a spherical model which is parameterized by the regional tectonic setting.

Description: Understanding the response of the Earth to the waxing and waning ice sheets is crucial in various contexts, ranging from the interpretation of modern satellite geodetic measurements to the projections of future sea level trends in response to climate change. All the processes accompanying Glacial Isostatic Adjustment (GIA) can be described solving the so-called Sea Level Equation (SLE), an integral equation that accounts for the interactions between the ice sheets, the solid Earth, and the oceans. Modern approaches to the SLE are based on various techniques that range from purely analytical formulations to fully numerical methods. Here we present the results of a benchmark exercise of independently developed codes designed to solve the SLE. The study involves predictions of current sea level changes due to present-day ice mass loss. In spite of the differences in the methods employed, the comparison shows that a significant number of GIA modellers can reproduce their sea-level computations within 2% for well defined, large-scale present-day ice mass changes. Smaller and more detailed loads need further and dedicated benchmarking and high resolution computation. This study shows how the details of the implementation and the inputs specifications are an important, and often underappreciated, aspect. Hence this represents a step toward the assessment of reliability of sea level projections obtained with benchmarked SLE codes.