Description: The ocean load in glacial isostatic adjustment (GIA) modelling is represented by the so-called sea level equation (SLE). The SLE describes the mass redistribution of water between ice sheets and oceans on a deforming Earth. Despite various teams independently investigating GIA, there has been no systematic intercomparison among the numerical solvers of the SLE through which the methods may be validated. The goal of this paper is to present a series of synthetic examples designed for testing and comparing the numerical implementations of the SLE in GIA modelling. The 10 numerical codes tested combine various temporal and spatial parametrizations. The time-domain or Laplace-domain discretizations are used to solve the SLE through time, while spherical harmonics, finite differences or finite elements parametrize the GIA-related field variables spatially. The surface ice-water load and solid Earth’s topography are represented spatially either on an equiangular grid, a Gauss–Legendre or an equiarea grid with icosahedron-shaped spherical pixels. Comparisons are made in a series of five benchmark examples with an increasing degree of complexity. Due to the complexity of the SLE, there is no analytical solution to it. The accuracy of the numerical implementations is therefore assessed by the differences of the individual solutions with respect to a reference solution. While the benchmark study does not result in GIA predictions for a realistic loading scenario, we establish a set of agreed-upon results that can be extended in the future by including more complex case studies, such as solutions with realistic loading scenarios, the rotational feedback in the linear-momentum equation, and by considering a 3-D viscosity structure of the Earth’s mantle. The test computations performed so far show very good agreement between the individual results and their ability to capture the main features of sea-surface variation and the surface vertical displacement. The differences found can often be attributed to the different approximations inherent in the various algorithms. This shows the accuracy that can be expected from different implementations of the SLE, which helps to assess differences noted in the literature between predictions for realistic loading cases.

Description: We apply synthetic aperture radar data and geophysical modeling to assess ground deformation changes at the 284 km2 large Toktogul Reservoir in Kyrgyzstan, Central Asia, which is used for hydropower generation and irrigation. The reservoir’s water level is prone to significant changes during the year, but also shows inter-annual variations due to overall water recession or accumulation. We use Envisat ASAR data to analyse the ground deformation during a time of exaggerated use of water between 2004 – 2009 (net water level drop of 60 m / 13.5 km3 ) and Sentinel-1 data to derive the ground deformation during a time of overall water level increase between 2014 – 2016 (net water level plus of 51 m / 11.2 km3 ). The deformation pattern was measured by generating an interferometric time-series using the Small BAseline Subset (SBAS) approach. After removing heavily impacting atmospheric effects by applying the elevation dependent powerlaw approach, results show that both sensors are able to image related uplift and subsidence signals in the order of approximately 1 mm per 1 m water level change for the investigated time periods. Moreover, time-series results from Sentinel-1 also resolve intra-annual changes induced by 40 m periodical water level changes. Reasons for this superior behaviour of Sentinel-1 data are a short temporal baseline of 12 days and a small orbital tube, which both lead to a higher temporal sampling compared to the Envisat setting and at the same time to a better correlation of points within the interferograms. The derived spatial pattern of land-deformation rate is validated against modeling of the elastic deformation, based on a Love-number approach. The load forcing due to lake-level changes is derived from satellite-based radar altimetry.

Description: The influence of the elastic Earth properties on seasonal or shorter periodic surface deformations due to atmospheric surface pressure and terrestrial water storage variations is usually modeled by applying a local half-space model or an one dimensional spherical Earth model like PREM from which a unique set of elastic load Love numbers, or alternatively, elastic Green's functions are derived. The first model is valid only if load and observer almost coincide, the second model considers only the response of an average Earth structure. However, for surface loads with horizontal scales less than 2500 km2, as for instance, for strong localized hydrological signals associated with heavy precipitation events and river floods, the Earth elastic response becomes very sensitive to inhomogeneities in the Earth crustal structure. We derive a set of local Green's functions defined globally on a 1° × 1° grid for the 3-layer crustal structure TEA12. Local Green's functions show standard deviations of ±12% in the vertical and ±21% in the horizontal directions for distances in the range from 0.1° to 0.5°. By means of Green's function scatter plots, we analyze the dependence of the load response to various crustal rocks and layer thicknesses. The application of local Green's functions instead of a mean global Green's function introduces a variability of 0.5 − 1.0 mm into the hydrological loading displacements, both in vertical and in horizontal directions. Maximum changes due to the local crustal structures are from −25% to +26% in the vertical and −91% to +55% in the horizontal displacements. In addition, the horizontal displacement can change its direction significantly. The lateral deviations in surface deformation due to local crustal elastic properties are found to be much larger than the differences between various commonly used one-dimensional Earth models.

Description: In this study, we propose a statistical method to validate sea-level reconstructions using geological records known as sea-level indicators (SLIs). SLIs are often the only available data to retrace late-glacial relative sea level (RSL). Determining the RSL from SLI height is not straight forward, the elevation at which an SLI was found usually does not represent the past RSL. In contrast, it has to be related to past RSL by investigating sample’s type, habitat and deposition conditions. For instance, water distribution at which a specific specimen is found today can be related to the indicator's depositional height range. Furthermore, the precision of dating varies between geological samples, and, in case of radiocarbon dating, the age has to be calibrated using a non-linear calibration curve. To avoid an a-priori assumption like normal-distributed uncertainties, we define likelihood functions which take into account the indicative meaning’s available error information and calibration statistics represented by joint probabilities. For this conceptional study, we restrict ourselves to one type of indicators, shallow-water shells, which are usually considered as low-grade samples giving only a lower limit of former sea level, as the depth range in which they live spreads over several tens of meters, and does not follow a normal distribution. The presented method is aimed to serve as a strategy for glacial isostatic adjustment reconstructions, in this case for the German Paleo-Climate Modelling Initiative PalMod (https://www.palmod.de/en) and by extending it to other SLI types.

Description: Advancements in the Global Geodetic Observing System (GGOS) have enabled us to investigate the effects of lateral heterogeneities in the internal Earth structure on long-term surface deformations caused by the Glacial Isostatic Adjustment (GIA). Many theories have been developed so far to consider such effects based on analytical and numerical approaches, and 3D viscosity distributions have been inferred. On the other hand, fewer studies have been conducted to assess the effects of lateral heterogeneities on short-term, elastic deformations excited by surface fluids, with 1D laterally homogeneous theories being frequently used. In this paper, we show that a spectral finite-element method is applicable to calculate the elastic deformation of an axisymmetric spherical Earth. We demonstrate the effects of laterally heterogeneous moduli with horizontal scales of several hundred kilometers in the upper mantle on the vertical response to a relatively large-scale surface load. We found that errors due to adopting a 1D Green’s function based on a local structure could amount to 2–3% when estimating the displacement outside the heterogeneity. Moreover, we confirmed that the mode coupling between higher-degree spherical harmonics needs to be considered for simulating smaller-scale heterogeneities, which agreed with results of previous studies.

Description: Glacial-isotactic adjustment (GIA) is one of the key processes considering relative sea-level (RSL) and paleo-topography during the last glacial cycle. Especially in former ice-covered regions the subsidence of the solid Earth due to ice loads can reach more than 500 m and contributes to the stability of ice-sheets (e.g. position of grounding line and ice-sheet elevation), whereas at the coasts of the world oceans the deformation is governed by global RSL fall of more than 100 m. Because the viscoelastic response of the solid Earth is governed by its viscosity structure, the effect of lateral viscosity variations on deformations due to GIA has to be estimated. The importance was already shown for the differences in earth structure below the glacial ice sheets of Fennoscandia and Laurentide, as well as for a number of peripheral and far-field regions. One open question arises: Can the 3D earth properly be parameterized by locally optimized 1D earth structures? In this study, we apply a 3D Earth structure which we derived from seismic tomography and further geodynamic constraints as an a priori estimation of the Earth viscosity distribution. Applying a standard glaciation history, we compare the response characteristics of 1D and 3D earth parameterizations and discuss the limits of optimized 1D earth parametrizations. We will focus on reconstructions of RSL during the last deglaciation in view of sea level index points which are generally used for validating the GIA process.

Description: The elastic deformation of the Earth’s surface due to atmospheric surface pressure, terrestrial water storage, and ocean bottom pressure on seasonal or shorter time scales is usually represented by a set of elastic load Love numbers or the corresponding Green’s function, determined from a radial Earth structure like PREM. Thereby, the influence of local deviations of the Earth’s crustal and mantle properties is assumed to be negligible. However, local Green’s functions derived individually for 1◦ grid cells from the 3D crustal structure model CRUST1 show large variations for in particular smaller distance angles. The loading response due to small-scale surface loads extending over less than 2500km2 significantly depends on the heterogeneous shallow structure of the Earth. In this contribution, we discuss the influence of lateral variations in the crust and mantle structure on atmospheric, hydrologic, and oceanic surface loads with regard to their spatial scales and distribution. Non-tidal atmospheric loading is calculated from an atmospheric surface pressure time series covering four decades (1976 - 2015) based on 3-hourly atmospheric data of ECMWF that has been homogenized by mapping surface pressure to a common reference orography. Hydrological loading is calculated for daily terrestrial water storage from LSDM over the same time period, where the surface water compartment is mapped from the 0.5◦ model resolution to a 0.125◦ GIS-based river network. Ocean tidal loading is exemplarily calculated based on the FES2014 ocean tidal model (0.0625◦ ). Especially along the coasts of the oceans; in regions with steep orographic gradients; and in areas with thick crustal layers or sediments we will show the significant influence of the Earth’s structure on small-scale deformation features caused by surface loads.