Global Gravity Field Models Research Articles

3D density structure of upper mantle beneath the Antarctic plate: The influence of Moho depth

In this study, we explore the impact of using different Moho surfaces on the reconstruction of the upper mantle geophysical parameters. The study area is the subsurface of the Antarctica continent. Using the optimization program of Sequential Integrated Inversion (SII) and the gravity anomalies synthetized by a global Gravity Field Model (GFM), we reconstructed the upper mantle density and the related 3D distribution of the ρ-vSV couplings down to the depth of 400 km, with a lateral resolution of 0.5° × 0.5°. Here, we present the results obtained for four models, built with four different Moho surfaces. A correlation analysis showed that different Moho structures affect the optimization of the intensity of the anomalies and the ρ-vSV couplings. The possibility of having both a density model and a 3D distribution of the ρ-vSV couplings enables us to highlight several significant features for all models that are not disclosed by seismic tomography. Among them, a trend of positive, presumably compositional anomalies suggests the contribution of the Lambert Rift System (LRS) to the Gamburstev Mountains (GSM) uplift, which in turn may have influenced the formation of the Maud Subglacial Basin (MSB). A continuous low-density anomaly and positive ρ-vSV phase coupling anomaly, extending from the northwestern side of the Transantarctic Mountains (TAM) to Victoria Land, support the thermal buoyancy force as the causative element of the formation of the TAM. A circumscribed negative density anomaly extending up to depth of 365 km, which is associated to a negative variation in the angular coefficients of the ρ-vSV couplings, indicates the presence of an active magmatic system in the upper mantle or a Cenozoic mantle plume beneath the region of Mary Byrd Land (MBL).

Analysis of high-resolution global gravity field models for the estimation of International Height Reference System (IHRS) coordinates in Argentina

Abstract Following the definition and realization of the International Height Reference System (IHRS), the vertical coordinate of a given point at the Earth’s surface can be obtained from the computation of the geopotential value from a harmonic expansion of a Global Gravity Model of High-Resolution (GGM-HR) or based on the computation of a local or regional pure gravimetric geoid or quasigeoid. Therefore, an evaluation of the accuracy of GGMs-HR and the geoid model available is needed in order to assess its capability to infer IHRS coordinates. In this study, different GGMs-HR are evaluated against 2287 benchmarks in Argentina. Moreover, the most recent geoid model of Argentina is also evaluated. Geoid undulations at these benchmarks are obtained based on ellipsoidal and orthometric heights in the local vertical datum. Results suggest that among the evaluated GGMs-HR, XGM2019e provides the best agreement with the observed geoid heights, but none of them is accurate enough in order to infer vertical coordinates in the IHRS. Similar conclusions are obtained for the local geoid model for Argentina demonstrating the necessity for a more precise geoid model, following the standards and recommendations given for the IHRS.

A Review of Marine Gravity Field Recovery from Satellite Altimetry

Marine gravity field recovery relies heavily on satellite altimetry. Thanks to the evolution of altimetry missions and the improvements in altimeter data processing methods, the marine gravity field model has been prominently enhanced in accuracy and resolution. However, high-accuracy and high-resolution gravity field recovery from satellite altimeter data remains particularly challenging. We provide an overview of advances in satellite altimetry for marine gravity field recovery, focusing on the impact factors and available models of altimetric gravity field construction. Firstly, the evolution of altimetry missions and the contribution to gravity field recovery are reviewed, from the existing altimetry missions to the future altimetry missions. Secondly, because the methods of altimeter data processing are of great significance when obtaining high-quality sea surface height observations, these improved methods are summarized and analyzed, especially for coastal altimetry. In addition, the problems to be resolved in altimeter data processing are highlighted. Thirdly, the characteristics of gravity recovery methods are analyzed, including the inverse Stokes formula, the inverse Vening Meinesz formula, Laplace’s equation, and least squares collocation. Furthermore, the latest global marine gravity field models are introduced, including the use of altimeter data and processing methods. The performance of the available global gravity field model is also evaluated by shipboard gravity measurements. The root mean square of difference between the available global marine gravity model and shipboard gravity from the National Centers for Environmental Information is approximately 5.10 mGal in the low-middle latitude regions, which is better than the result in high-latitude regions. In coastal areas, the accuracy of models still needs to be further improved, particularly within 40 km from the coastline. Meanwhile, the SDUST2021GRA model derived from the Shandong University of Science and Technology team also exhibited an exciting performance. Finally, the future challenges for marine gravity field recovery from satellite altimetry are discussed.

Global Gravity Field Model from Taiji-1 Observations

Taiji-1 is the first technology demonstration satellite of the Taiji program of China’s space-borne gravitational wave antenna. After the demonstration of the key individual technologies, Taiji-1 continues collecting the data of the precision orbit determinations, satellite attitudes, and non-conservative forces exerted on the S/C. Therefore, during its free-fall, Taiji-1 can be viewed as operating in the high-low satellite-to-satellite tracking mode of a gravity recovery mission. In this work, we have selected and analyzed the one month data from Taiji-1’s observations, and developed the techniques to resolve the long term interruptions and disturbances in the data due to the scheduled technology demonstration experiments. The first global gravity model TJGM-r1911, that independently derived from China’s own satellite mission, is successfully built from Taiji-1’s observations. Compared with gravity models from CHAMP and other satellite gravity missions, the accuracy discrepancies exist, which is mainly caused by the data discontinuity problem. As the extended free-falling phase been approved, Taiji-1 could serve as a gravity recovery mission for China since 2022 and it will provide us the independent measurement of both the static and the monthly time-variable global gravity field.

Определение гравитационных аномалий в акватории Вьетнама и на прилегающей территории по данным спутниковой альтиметрии CryoSat-2

The purpose of this study is to determine marine gravity anomalies with high accuracy using the CryoSat-2 satellite altimetry data in the Middle Vietnamese Sea and the adjacent territory. For this, the mentioned data and the process of determining gravity anomalies from the CryoSat-2 data were analyzed. The record from 818 GNSS-leveling points in Vietnam were used to evaluate and select the best global gravity field model for use in the calculation process. The findings from 31 gauge tidal were also used to assess and select the best mean dynamic topography model for the study area. Marine gravity anomalies in the study area were calculated using data from the CryoSat-2 satellite altimeter at 72 483 measurement points. The calculated results are compared with the ship-derived gravity anomalies at 1025 measurement points for assessment. The research results show that of the Earth Geopotential Model EIGEN6C4 and the Mean Dynamic Topography Model DTU15MDT are the most suitable for the research area. The accuracy of satellite-derived gravity anomalies, calculated from the CryoSat-2 satellite altimetry data in the study area, reaches ±1,18 mGal. Compared to the available results in the study area, this accuracy is the highest.

Research progress of ocean satellite altimetry and its recovery of global marine gravity field and seafloor topography model

Research progress of ocean satellite altimetry and its recovery of global marine gravity field and seafloor topography model

Analysis of data on pure and mixed gravity anomalies obtained from the results of the space gravity missions GRACE, GOCE

The article analyzes the data of pure and mixed gravity anomalies on the territory of the Novosibirsk region, obtained from the results of the space gravimetric missions GRACE, GOCE. Gravimetric and geodetic measurements carried out in the study area made it possible to obtain mixed and simulate pure gravity anomalies. The information obtained was the starting point for the analysis of gravity anomalies calculated from the coefficients of the global geopotential models XGM2019e_2159, GOCO01S. The results of the study led to the conclusion that when calculating mixed gravity anomalies according to the combined geopotential model XGM2019e_2159 and pure gravity anomalies according to the GOCO01S satellite model data, equal root-mean-square errors were obtained (for mixed anomalies - 3.45 mGal, for pure - 3.44 mGal). Therefore, modern combined global gravity field models can be used to obtain pure anomalies on the earth's surface.

The Refined Gravity Field Models for Height System Unification in China

A unified height datum is essential for global geographic information resource construction, ecological environment protection, and scientific research. The goal of this paper is to derive the geopotential value for the Chinese height datum (CNHD) in order to realize the height datum unification in China. The estimation of height datum geopotential value usually depends on high-precision global gravity field models (GFMs). The satellite gravity missions of the Gravity Recovery and Climate Experiment (GRACE) and Gravity field and steady-state Ocean Circulation Exploration (GOCE) provide high-accuracy, medium–long-wavelength gravity field spectra, but satellite-only GFMs are limited to medium–long wavelengths, which will involve omission errors. To compensate for the omission errors in satellite-only GFMs, a spectral expansion approach is used to obtain the refined gravity field models using the EGM2008 (Earth Gravitational Model 2008) and residual terrain model (RTM) technique. The refined GFMs are evaluated by using high-quality GNSS/leveling data, the results show that the quasi-geoid accuracy of the refined DIR_R6_EGM2008_RTM model in China has optimal accuracy and, compared with the EGM2008 model and the DIR_R6 model, this refined model in China is improved by 9.6 cm and 21.8 cm, and the improvement ranges are 35.7% and 55.8%, respectively. Finally, the geopotential value of the Chinese height datum is estimated to be equal to 62,636,853.29 m2s−2 with respect to the global reference level defined by W0 = 62,636,853.4 m2s−2 by utilizing the refined DIR_R6_EGM2008_RTM model and 1908 high-quality GNSS/leveling datapoints.

Assessment of high-degree reference models and Recent Goce/Grace Global Geopotential Models over Sudan based on the GPS/Leveling data

Nowadays, Global Geopotential Models (GGMs) can be used as a reference to develop more detailed regional/local geoids, or they can be used to provide geoid heights on their own. Since 2000, several GGMs have been released, and they are mainly derived from satellite gravity measurements, satellite-only models, terrestrial gravimetry, altimeter-derived gravity data in marine areas, and airborne gravity data. With a precise geoid model, ellipsoidal heights obtained from GPS can be converted to orthometric heights, which is reasonably quite needed in Geodesy, Civil Engineering, etc. These heights reflect changes in topography as well as local variations in gravity. This paper evaluates some of the latest releases of high degree reference models and the satellite-only global gravity field model over Sudan using 19 GPS/Leveling stations. We have been selected 6 GGMs based on Gravity field Goce and Grace, and they released in 2020, 2019, 2014, 2008, and 1996 as shown in the International Centre for Global Earth Models website (ICGEM). The accuracy evaluation of the GGM models have been discussed, the accurate GGMs over Sudan are XGM2019e_2159 and GOCO05s, which have indicated -0.019 and 0.046 meters, respectively. The evaluation results produce valuable information to academia and geoid modeling research topics in Sudan, which shows the precise model from the selected GGMs in Sudan by using the available GPS/Leveling data.

Global Gravity Field Modeling of Small Bodies with Heterogeneous Mass Distributions

This paper develops a global and general characterization of the gravity field around small celestial bodies, which commonly have heterogeneous mass distributions. The availability of such a characterization is important in providing guidance for close-proximity exploration missions, rover deployments, and dynamic studies. The determination of the gravitational potential is formulated as a boundary value problem for Laplace’s equation, which can be solved using the boundary element method. To address the near-singularity arising from the fundamental solution, a semi-analytic form of the boundary integral with an adaptive collocation strategy is derived, and the virtual boundary method (VBM) is incorporated. The existence and uniqueness of the analytic continuation of gravitational potential inside the small body are proved by investigating the convergence of arbitrary derivatives on the boundary, which serves as the necessary condition of the application of VBM. Furthermore, the well-posed nature of the boundary integral equation expressed as a single-layer representation on the virtual boundary is proved using the Lax–Milgram theorem, which guarantees the robustness of VBM. Simulations conducted on comet 67P/Churyumov–Gerasimenko and asteroids (216) Kleopatra and (433) Eros demonstrate the ability of the proposed characterization to provide a global high-fidelity gravity field.

Artefacts in gravity field modelling

Artefacts in applications of the global gravity field and topographic models based on satellite and terrestrial data of diverse kinds are studied. Various types of artefacts are presented and analysed with the aim to understand how the artefacts originated, how to reduce them (if feasible) and to avoid misinterpretations in geoscience. We work with the gravity aspects (gravity anomalies, the Marussi tensor of the second derivatives of disturbing potential, gravity invariants and their specific ratio, strike angles, and virtual deformations), and with surface or bedrock topography. Examples for the Earth, the Moon, and Mars are presented.

Gravity for Lithosphere Architecture Determination and Analysis: the Central Eastern Mediterranean case study

ABSTRACTThe Central‐Eastern Mediterranean region is known to be a complex area due to the interaction of four tectonic plates, namely Arabia, Africa, Anatolia and Eurasia, and by the presence of an ancient oceanic crust in the Herodotus and Ionian Basins. The analysis of the available literature highlights that the distribution of the freely available geophysical data (i.e. seismic, gravity and magnetic observations) is quite disparate. In this framework, high‐resolution global gravity field models, such as XGM2019e, mainly based (offshore) on satellite data, can be profitably used as a uniform dataset to study, in a coherent way, large regions.In the current work, we exploit the XGM2019e model, together with a set of a priori information, derived mainly from geophysical data retrieved from the literature, to study the structure of the crust in the Central‐Eastern Mediterranean area. The study is organized in three different phases: in the first one, we enhanced the map of geological crustal provinces by means of an automatic Bayesian classification algorithm applied to the second radial derivatives of the gravitational potential. In the second phase, using as observation a grid of gravity anomalies, we applied a full three‐dimensional inversion procedure (always based on a Bayesian paradigm) to estimate the mass density variations and the geometries of the main geological units in the whole study area. Finally, in the third phase, we performed a refined three‐dimensional local inversion on the Cyprus area to improve the modelling of the shallowest layers. The main results of this study, carried out in the framework of the European Space Agency GIADA project, are freely available, upon request, at https://www.g‐red.eu/geophysics/.

Mapping the Moho depth and ocean-continent transition in the South China Sea using gravity inversion

• The lithosphere thermal gravity anomaly in the South China Sea is calculated. • The depth of the Moho surface in the South China Sea is obtained by inversion in spherical coordinates with the thermal gravity anomaly correction. • We delineate the ocean-continent transition (OCT) of the northern South China Sea. In this paper, the depth of the Moho surface in the South China Sea is investigated by the gravity inversion method under the spatial-domain spherical coordinate system based on the global gravity field model WGM2012. The gravity data are corrected for the lithospheric thermal gravity anomaly due to the large negative residual thermal anomaly caused by lithospheric extension in the ocean and continental margins. The calculation results show that the thermal anomaly is mainly concentrated near the residual oceanic ridges in the South China Sea and that the density anomalies caused by the anomaly show a “fusiform” variation with depth; i.e., the anomaly is larger in the middle of the lithosphere and smaller at the top and bottom. The Moho inversion results indicate that the depth of the Moho surface in the South China Sea ranges from 7 to 32 km with an overall northeastward wedge-shaped distribution and is characterized by a shallow basin and a deep perimeter. In addition, a depression of the Moho surface remains under the remnant oceanic ridges, with the depth increasing by 2–5 km, and the depth of the Moho surface on both sides of the basin changes rapidly. Further, we determined the location of the OCT in the South China Sea based on the Moho depth mapped by gravity inversion and combined it with the results of seismic reflection profiles and other relevant geophysical data; its width is approximately 200 km in the east and 70 km in the west.

World ocean annual mean absolute geostrophic velocity on marine geoid of EIGEN‐6C4 from WOA13

AbstractThis data set was established from two open sources: (a) geoid undulation (N) from EIGEN‐6C4, which is a static global combined gravity field model up to degree and order 2,190, and (b) ‘World ocean geostrophic velocity inverted from World Ocean Atlas 2013 with the P‐vector method’ (NCEI accession 0,121,576). With the given non‐positive values of N in the oceans, absolute geostrophic currents (u, v) are easily obtained on N with 1°×1° resolution from the second data set except the equatorial zone (5oS – 5oN) due to the non‐existence of the geostrophic balance. Altogether, the data set contains 15,868 (u, v) data pairs. The data set shows that the hypothetical situation that there would be no flow on the geoid does not exist.

Evaluation of global gravity field models using shipborne free-air gravity anomalies over the Gulf of Guinea, Central Africa

In this work, the principal aim is to evaluate some recent GGMs on the Gulf of Guinea region using shipborne free-air gravity data. Two types of GGMs are evaluated in this study: combined models (EGM2008, EIGEN-6C4, GECO, SGG-UGM-1, GOCO05C and XGM2016) and satellite-only models (DIR_R5, ITU_GGC16, SPW_R5, TIM_R5, GGM05G and NULP_02S). The comparisons between these GGMs and the preliminary filtered shipborne gravity data were performed before and after the Spectral Enhancement Method (SEM). The statistical results have shown that the combined models have a better performance than the satellite-only models. Before and after SEM, we note that the EGM2008 model presents the best statistical results compared to the shipborne gravity anomalies (RMS and SD values respectively changes from 3.87 to 3.55 mGal and from 3.57 to 3.32 mGal). However, the DIR_R5 model is the best of the satellite-only GGMs of this study.

Determination of the geopotential value on the permanent GNSS stations in Vietnam based on the Geodetic Boundary Value Problem approach

SUMMARY In the realisation of the International Height Reference System, the determination of the geopotential value and its variations in time plays an important role. In this study, the geodetic boundary value problem approach is applied for direct determination of the gravity potential value using a GOCE global gravity field model enhanced with terrestrial gravity data. This determination is carried out on the Global Navigation Satellite System-Continuously Operating Reference Stations (GNSS-CORS) stations towards the realisation of the International Height Reference System in Vietnam. First, the effects of the GOCE global gravity field model omission error, the indirect bias term on the disturbing potential and the systematic cumulative errors in levelling data are estimated. These errors affect the estimated geopotential value. The results calculated on the GNSS/levelling points show that the effect of the GOCE DIR-R5 (up to degree/order 260) omission error on the offset potential value is quite significant. This effect was eliminated using high-resolution terrestrial gravity data using the remove-compute-restore technique. The indirect bias term on the disturbing potential can be safely neglected by using a GOCE global gravity field model for degrees higher than 60 for this study region. The systematic cumulative errors in levelling data can be modelled and removed using a third-order polynomial model. Then, the mean zero-height gravity potential of the Vietnam local vertical datum is estimated equal to ${\rm{W}}_0^{{\rm{LVD}}}$ = 62 636 846.69 m2 s–2 with standard deviation of 0.70 m2 s–2 based on the proposed approach. Finally, the geodetic boundary value problem approach was used to determine the geopotential on the surface of three GNSS-CORS stations in Vietnam. Based on time-series of the vertical component derived from the GNSS observations as well as InSAR data, temporal variations in geopotential are also estimated on these permanent GNSS stations. The purpose is to monitor deformation of the vertical datum. The results indicate that the geopotential value needs to be monitored and determined with the time-dependent component on the three Vietnamese permanent GNSS stations for a vertical datum. These stations may contribute to increase the density of reference points in the International Terrestrial Reference Frame, which is being researched and implemented by the International Association of Geodesy.

Joint analysis of selected GRACE monthly spherical harmonic solutions and monthly MASCON solutions

Study presented in this paper is focused on comparison and statistical assessment of differences between the selected Level 2 products of the satellite mission Gravity Recovery and Climate Experiment (GRACE). Global monthly gravity field models in terms of spherical harmonic coefficients produced by three institutes of GRACE Science Data System are compared with the partially independent MASCON global gravity field model. Detailed comparison and statistical analysis of differences is performed in 5 selected river basins: Amazon, Congo, Danube, Yenisei and Lena. For each spherical harmonic solution, 8 different filtrations available at International Center for Global Gravity Field Models (ICGEM) are tested over the time span from April 2002 to July 2016. Fischer test at two significance levels 10% and 5% has been performed in order to qualify the statistical significance between the particular solutions.

Quantifying the individual contributions of GOCE gradients for regional quasi-geoid modeling in a relatively rough topography

Gravity field and steady-state Ocean Circulation Explorer (GOCE) is the first satellite mission that measures the gravity gradients (GGs) in space with primary objective of determining the gravity field of the Earth with maximum spatial resolution and accuracy. GOCE GGs have been used for the computation of global gravity field models (GGMs) represented by the surface spherical harmonics for about 10 years. However, they may potentially improve the regional quasi-geoid solutions computed from terrestrial gravity and a Global Geopotential Model (GGM) particularly in mountainous areas with rough gravity field. This study exploits the full signal content in the GOCE Level-2 GGs given in Local North-Oriented Frame for improved regional quasi-geoid modeling in Auvergne test area located in France. The least squares collocation method is used in single-input and single-output (SISO) mode where the input is the residual terrestrial gravity data corrected for GGM contribution and residual terrain modeling effects, and also in multiple-input and single-output mode (MISO) where the terrestrial data input is augmented with the four precise GOCE GGs or their combinations. The single outputs for both modes are the quasi-geoid heights at 75 GPS collocated leveling benchmarks distributed in the central area and at 7857 Digital Elevation Model (DEM) points covering the whole area. The combination of the terrestrial gravity with the GOCE along-track gradient, having a lower noise level than the others, gives the best results when a single gradient is used. The further incorporation of the off-diagonal gradient to the along-track gradient results in the best MISO solution showing improvements with respect to the SISO solution only at GPS/leveling benchmarks near a large data gap. Intercomparison of the best MISO and SISO solutions at the DEM points suggests substantial quasi-geoid differences up to 13.6 cm in areas of sparse or no gravity data near the Alps where these results could not be assessed due to the lack of GPS-leveling data.

The earth\u2019s gravity field recovery using the third invariant of the gravity gradient tensor from GOCE

Due to the independence of the gradiometer instrument’s orientation in space, the second invariant I_2 of gravity gradients in combination with individual gravity gradients are demonstrated to be valid for gravity field determination. In this contribution, we develop a novel gravity field model named I3GG, which is built mainly based on three novel elements: (1) proposing to utilize the third invariant I_3 of the gravity field and steady-state ocean circulation explorer (GOCE) gravity gradient tensor, instead of using the I_2, similar to the previous studies; (2) applying an alternative two-dimensional fast fourier transform (2D FFT) method; (3) showing the advantages of I_3 over I_2 in the effect of measurement noise from the theoretical and practical computations. For the purpose of implementing the linearization of the third invariant, this study employs the theory of boundary value problems with sphere approximation at an accuracy level of O(J_2^2cdot T_{ij}). In order to efficiently solve the boundary value problems, we proposed an alternative method of 2D FFT, which uses the coherent sampling theory to obtain the relationship between the 2D FFT and the third invariant measurements and uses the pseudo-inverse via QR factorization to transform the 2D Fourier coefficients to spherical harmonic ones. Based on the GOCE gravity gradient data of the nominal mission phase, a novel global gravity field model (I3GG) is derived up to maximum degree/order 240, corresponding to a spatial resolution of 83 km at the equator. Moreover, in order to investigate the differences of gravity field determination between I_3 with I_2, we applied the same processing strategy on the second invariant measurements of the GOCE mission and we obtained another gravity field model (I2GG) with a maximum degree of 220, which is 20 degrees lower than that of I3GG. The root-mean-square (RMS) values of geoid differences indicates that the effects of measurement noise of I3GG is about 20% lower than that on I2GG when compared to the gravity field model EGM2008 (Earth Gravitational Model 2008) or EIGEN-5C (EIGEN: European Improved Gravity model of the Earth by New techniques). Then the accuracy of I3GG is evaluated independently by comparison the RMS differences between Global Navigation Satellite System (GNSS)/leveling data and the model-derived geoid heights. Meanwhile, the re-calibrated GOCE data released in 2018 is also dealt with and the corresponding result also shows the similar characteristics.

GOCO06s – a satellite-only global gravity field model

Abstract. GOCO06s is the latest satellite-only global gravity field model computed by the GOCO (Gravity Observation Combination) project. It is based on over a billion observations acquired over 15 years from 19 satellites with different complementary observation principles. This combination of different measurement techniques is key in providing consistently high accuracy and best possible spatial resolution of the Earth's gravity field. The motivation for the new release was the availability of reprocessed observation data for the Gravity Recovery and Climate Experiment (GRACE) and Gravity field and steady-state Ocean Circulation Explorer (GOCE), updated background models, and substantial improvements in the processing chains of the individual contributions. Due to the long observation period, the model consists not only of a static gravity field, but comprises additionally modeled temporal variations. These are represented by time-variable spherical harmonic coefficients, using a deterministic model for a regularized trend and annual oscillation. The main focus within the GOCO combination process is on the proper handling of the stochastic behavior of the input data. Appropriate noise modeling for the observations used results in realistic accuracy information for the derived gravity field solution. This accuracy information, represented by the full variance–covariance matrix, is extremely useful for further combination with, for example, terrestrial gravity data and is published together with the solution. The primary model data consisting of potential coefficients representing Earth's static gravity field, together with secular and annual variations, are available on the International Centre for Global Earth Models (http://icgem.gfz-potsdam.de/, last access: 11 June 2020). This data set is identified with the following DOI: https://doi.org/10.5880/ICGEM.2019.002 (Kvas et al., 2019b). Supplementary material consisting of the full variance–covariance matrix of the static potential coefficients and estimated co-seismic mass changes is available at https://ifg.tugraz.at/GOCO (last access: 11 June 2020).