Description: In this study, a new method for computing the sensitivity of the glacial isostatic adjustment (GIA) forward solution with respect to the Earth's mantle viscosity, the so-called the forward sensitivity method (FSM), and a method for computing the gradient of data misfit with respect to viscosity parameters, the so-called adjoint-state method (ASM), are presented. These advanced formal methods complement each other in the inverse modelling of GIA-related observations. When solving this inverse problem, the first step is to calculate the forward sensitivities by the FSM and use them to fix the model parameters that do not affect the forward model solution, as well as identifying and removing redundant parts of the inferred viscosity structure. Once the viscosity model is optimized in view of the forward sensitivities, the minimization of the data misfit with respect to the viscosity parameters can be carried out by a gradient technique which makes use of the ASM. The aim is this paper is to derive the FSM and ASM in the forms that are closely associated with the forward solver of GIA developed by Martinec. Since this method is based on a continuous form of the forward model equations, which are then discretized by spectral and finite elements, we first derive the continuous forms of the FSM and ASM and then discretize them by the spectral and finite elements used in the discretization of the forward model equations. The advantage of this approach is that all three methods (forward, FSM and ASM) have the same matrix of equations and use the same methodology for the implementation of the time evolution of stresses. The only difference between the forward method and the FSM and ASM is that the different numerical differencing schemes for the time evolution of the Maxwell and generalized Maxwell viscous stresses are applied in the respective methods. However, it requires only a little extra computational time for carrying out the FSM and ASM numerically. An straightforward approach to compute the gradient of the data misfit is the brute-force method, whereby the partial derivatives of the misfit with respect to model parameters are approximated by the centred difference of two forward model runs. Although the brute-force method is useful for computing the gradient of the data misfit with respect to a small number of model parameters, it becomes expensive for a viscosity model with a large number of parameters. The ASM offers an efficient alternative for computing the gradient of the misfit since the computational time of the ASM is independent of the number of viscosity parameters. The ASM is thus highly efficient for calculating the gradient of the misfit for models with large numbers of parameters. However, the forward-model solution for each time step must be stored, hence the memory demands scale linearly with the number of time steps. This is the main drawback of the ASM.

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: Two regions in West Antarctica have received increased attention over the last years due to their potential sensitivity to climate change: the Antarctic Peninsula, which currently experiences warming at rates greater than the global average (Vaughan, 2006), ongoing ice shelf disintegration and subsequent glacier acceleration (Scambos et al., 2004); and the Amundsen Sea Sector (AS), where ice velocities, ice discharge and glacial imbalances are extreme compared to the rest of the continent (Rignot et al., 2008). The satellite gravimetry mission GRACE reveals interannual variations in ice mass related to varying snowfall, which is strongly influenced by the global climate phenomenon El Nino (Sasgen et al., 2010).

Description: The Greenland and Antarctic ice sheets have been reported to be losing mass at accelerating rates1, 2. If sustained, this accelerating mass loss will result in a global mean sea-level rise by the year 2100 that is approximately 43 cm greater than if a linear trend is assumed2. However, at present there is no scientific consensus on whether these reported accelerations result from variability inherent to the ice-sheet–climate system, or reflect long-term changes and thus permit extrapolation to the future3. Here we compare mass loss trends and accelerations in satellite data collected between January 2003 and September 2012 from the Gravity Recovery and Climate Experiment to long-term mass balance time series from a regional surface mass balance model forced by re-analysis data. We find that the record length of spaceborne gravity observations is too short at present to meaningfully separate long-term accelerations from short-term ice sheet variability. We also find that the detection threshold of mass loss acceleration depends on record length: to detect an acceleration at an accuracy within ±10 Gt yr−2, a period of 10 years or more of observations is required for Antarctica and about 20 years for Greenland. Therefore, climate variability adds uncertainty to extrapolations of future mass loss and sea-level rise, underscoring the need for continuous long-term satellite monitoring.