Gravity field model calculated by using the GOCE data

In this contribution, we describe the global GOCE-only gravity field model ITG-Goce02 derived from 7.5 months of gradiometer and orbit data. This model represents an alternative to the official ESA products as it is computed completely independently, using a different processing strategy and a separate software package. Our model is derived using the short arc approach, which allows a very effective decorrelation of the highly correlated GOCE gradiometer and orbit data noise by introducing a full empirical covariance matrix for each arc, and gives the possibility to downweight ‘bad’ arcs. For the processing of the orbit data we rely on the integral equation approach instead of the energy integral method, which has been applied in several other GOCE models. An evaluation against high-resolution global gravity field models shows very similar differences of our model compared to the official GOCE results published by ESA (release 2), especially to the model derived by the time-wise approach. This conclusion is confirmed by comparison of the GOCE models to GPS/levelling and altimetry data.

We present a method with which we determined the local lunar gravity field model over the South Pole‐Aitken (SPA) basin on the farside of the Moon by estimating adjustments to a global lunar gravity field model using SELENE tracking data. Our adjustments are expressed in localized functions concentrated over the SPA region in a spherical cap with a radius of 45° centered at (191.1°E, 53.2°S), and the resolution is equivalent to a 150th degree and order spherical harmonics expansion. The new solution over SPA was used in several applications of geophysical analysis. It shows an increased correlation with high‐resolution lunar topography in the frequency band l = 40–70, and admittance values are slightly different and more leveled when compared to other, global gravity field models using the same data. The adjustments expressed in free‐air anomalies and differences in Bouguer anomalies between the local solution and the a priori global solution correlate with topographic surface features. The Moho structure beneath the SPA basin is slightly modified in our solution, most notably at the southern rim of the Apollo basin and around the Zeeman crater.

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.

Three GOCE-based gravity field solutions have been computed by ESA’s high-level processing facility and were released to the user community. All models are accompanied by variance-covariance information resulting either from the least squares procedure or a Monte-Carlo approach. In order to obtain independent external quality parameters and to assess the current performance of these models, a set of independent tests based on satellite orbit determination and geoid comparisons is applied. Both test methods can be regarded as complementary because they either investigate the performance in the long wavelength spectral domain (orbit determination) or in the spatial domain (geoid comparisons). The test procedure was applied to the three GOCE gravity field solutions and to a number of selected pre-launch models for comparison. Orbit determination results suggest, that a pure GOCE gravity field model does not outperform the multi-year GRACE gravity field solutions. This was expected as GOCE is designed to improve the determination of the medium to high frequencies of the Earth gravity field (in the range of degree and order 50 to 200). Nevertheless, in case of an optimal combination of GOCE and GRACE data, orbit determination results should not deteriorate. So this validation procedure can also be used for testing the optimality of the approach adopted for producing combined GOCE and GRACE models. Results from geoid comparisons indicate that with the 2 months of GOCE data a significant improvement in the determination of the spherical harmonic spectrum of the global gravity field between degree 50 and 200 can be reached. Even though the ultimate mission goal has not yet been reached, especially due to the limited time span of used GOCE data (only 2 months), it was found that existing satellite-only gravity field models, which are based on 7 years of GRACE data, can already be enhanced in terms of spatial resolution. It is expected that with the accumulation of more GOCE data the gravity field model resolution and quality can be further enhanced, and the GOCE mission goal of 1–2 cm geoid accuracy with 100 km spatial resolution can be achieved.

GOCE will be the first satellite ever to measure the second derivatives of the Earth’s gravitational potential in space. With these measurements it is possible to derive a high accuracy and resolution gravitational field if systematic errors have been removed to the extent possible from the data and the accuracy of the gravity gradients has been assessed. It is therefore necessary to understand the instrument characteristics and to setup a valid calibration model. The calibration parameters of this model could be determined by using GOCE data themselves or by using independent gravity field information. Also the accuracy or error assessment relies on either GOCE or independent data. We will demonstrate how state-of-the-art global gravity field models, terrestrial gravity data and observations at satellite track crossovers can be used for calibration/validation. In addition we will show how high quality terrestrial data could play a role in error assessment.

\(W_{0}\) is defined as the potential value of a particular level surface of the Earth’s gravity field called the geoid. Since the most accepted definition of the geoid is understood to be the equipotential surface that coincides with the worldwide mean ocean surface, a usual approximation of \(W_{0}\) is the averaged potential value \(W_{\mathrm{S}}\) at the mean sea surface. In this way, the value of \(W_{0}\) depends not only on the Earth’s gravity field modelling, but also on the conventions defining the mean sea surface. \(W_{0}\) computations performed since 2005 demonstrate that current published estimations differ by up to \(-2.6~\hbox {m}^{2}~\hbox {s}^{-2}\) (corresponding to a level difference of about 27 cm), which could be caused by the differences in the treatment of the input data. The main objective of this study is to perform a new \(W_{0}\) estimation relying on the newest gravity field and sea surface models and applying standardised data and procedures. This also includes a detailed description of the processing procedure to ensure the reproducibility of the results. The following aspects are analysed in this paper: (1) sensitivity of the \(W_{0}\) estimation to the Earth’s gravity field model (especially omission and commission errors and time-dependent Earth’s gravity field changes); (2) sensitivity of the \(W_{0}\) estimation to the mean sea surface model (e.g., geographical coverage, time-dependent sea surface variations, accuracy of the mean sea surface heights); (3) dependence of the \(W_{0}\) empirical estimation on the tide system; and (4) weighted computation of the \(W_{0}\) value based on the input data quality. Main conclusions indicate that the satellite-only component \((n = 200)\) of a static (quasi-stationary) global gravity model is sufficient for the computation of \(W_{0}\). This model should, however, be based on a combination of, at least, satellite laser ranging (SLR), GRACE and GOCE data. The mean sea surface modelling should be based on mean sea surface heights referring to a certain epoch and derived from a standardised multi-mission cross-calibration of several satellite altimeters. We suggest that the uncertainties caused by geographically correlated errors, including shallow waters in coastal areas and sea water ice content at polar regions should be considered in the computation of \(W_{0}\) by means of a weighed adjustment using the inverse of the input data variances as a weighting factor. This weighting factor should also include the improvement provided by SLR, GRACE and GOCE to the gravity field modelling. As a reference parameter, \(W_{0}\) should be time-independent (i.e., quasi-stationary) and it should remain fixed for a long-term period (e.g., 20 years). However, it should have a clear relationship with the mean sea surface level (as this is the convention for the realisation of the geoid). According to this, a suitable recommendation is to adopt a potential value obtained for a certain epoch as the reference value \(W_{0}\) and to monitor the changes of the mean potential value at the sea surface \(W_{\mathrm{S}}\). When large differences appear between \(W_{0}\) and \(W_{\mathrm{S}}\) (e.g., \({>}\pm 2\) m\(^{2}\) s\(^{-2})\), the adopted \(W_{0}\) may be replaced by an updated (best estimate) value. In this paper, the potential value obtained for the epoch 2010.0 (62,636,853.4 m\(^{2}\) s\(^{-2})\) is recommended as the present best estimate for the \(W_{0}\) value. It differs \(-2.6~\hbox {m}^{2}~\hbox {s}^{-2}\) from the so-called IERS \(W_{0}\) value (62,636,856.0 m\(^{2}\) s\(^{-2})\), which corresponds to the best estimate available in 1998.

<p>We proposed an alternative method to determine the height of Mount Everest (HME) based on the shallow layer method (SLM), which was put forward by Shen (2006). We use the precise external global Earth gravity field model (i.e., EGM2008 and EIGEN-6C4 models) as input information, and the digital topographic model (i.e., DTM2006.0) and crust models (i.e., CRUST2.0 and CRUST1.0 models) to construct the shallow layer model. There are four combined strategies：(1) EGM2008 and CRUST1.0 models, (2) EGM2008 and CRUST2.0 models, (3) EIGEN-6C4 and CRUST1.0 models, and (4) EIGEN-6C4 and CRUST2.0 models, respectively. We calculate the HME by two approaches: first approach, the HME is directly calculated by combining the geoid undulation (N) and geodetic height (h); second approach, we calculate the HME by the segment summation approach (SSA) using the gravity field inside the shallow layer determined by the SLM. Numerical results show that for four combined strategies, the differences between our results and the authoritatively released value 8848.86 m by the Chinese and Nepalese governments on December 8, 2020 are 0.448 m, -0.009 m, -0.295 m, and -0.741 m using first approach and 0.539 m, 0.083 m, -0.214 m, and -0.647 m using second approach. The combined calculation of the HME by the terrain model and gravity field model is more accurate than that by the gravity field model alone. This study is supported by the National Natural Science Foundations of China (NSFC) under Grants 42030105, 41721003, 41804012, 41631072, 41874023, Space Station Project (2020)228.</p>

Global static gravity field models are used for various applications such as determination of geoid height (i.e., vertical datum), satellite orbits, and ocean circulation models. Such static gravity models are obtained by combination of various satellite gravimetry data such as GRACE and GOCE, altimetry measurements over the oceans and terrestrial gravity measurements. These datasets have different error characteristics as well as spatial and spectral contents, which makes the data combination in model production very challenging. Therefore, it is very important to validate the global static gravity field models using independent observations not used in those models. Laser ranging interferometer (LRI) measurements acquired by the GRACE-FO show extraordinary sensitivity to small-scale static gravity signals down to 100 km. We use one year of GRACE LRI observations in 2019 to validate six state-of-the-art global static gravity models: GGM05C, GGM05S, ITSG-Grace2018s, GOCO-TIM-R06e, EIGEN-6C4, and GOCO06s. Our comprehensive analysis leads to detection of the spectral (spherical harmonic) band and spatial regions where these models suffer from errors. We discuss inefficacies of each model as well as their potential causes. Our unique study helps inform the model developers about the spectral band and spatial regions where their models need improvement.

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.

The performance of Gravity field and steady-state Ocean Circulation Explorer (GOCE) global gravity field models (GGMs), at the end of GOCE mission covering 42 months, is evaluated using geoid undulations and free-air gravity anomalies over Japan, including six sub-regions (Hokkaido, north Honshu, central Honshu, west Honshu, Shikoku and Kyushu). Seventeen GOCE-based GGMs are evaluated and compared with EGM2008. The evaluations are carried out at 150, 180, 210, 240 and 270 spherical harmonics degrees. Results show that EGM2008 performs better than GOCE and related GGMs in Japan and three sub-regions (Hokkaido, central Honshu and Kyushu). However, GOCE and related GGMs perform better than EGM2008 in north Honshu, west Honshu and Shikoku up to degree 240. This means that GOCE data can improve geoid model over half of Japan. The improvement is only evident between degrees 150 and 240 beyond which EGM2008 performs better than GOCE GGMs in all the six regions. In general, the latest GOCE GGMs (releases 4 and 5) perform better than the earlier GOCE GGMs (releases 1, 2 and 3), indicating the contribution of data collected by GOCE in the last months before the mission ended on 11 November 2013. The results indicate that a more accurate geoid model over Japan is achievable, based on a combination of GOCE, EGM2008 and terrestrial gravity data sets.Graphical abstractStandard deviations of the differences between observed and GGMs implied (a) free-air gravity anomalies over Japan, (b) geoid undulations over Japan. n represents the spherical harmonic degrees

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/.

To understand many geophysical phenomena, including ocean circulation, tectonic plate movement, and the Earth’s interior, it is important to study the Earth’s gravity field. During the past three decades, satellites have measured the Earth’s static gravity. As a result, high-resolution global gravity field models have been available. This paper aims to examine the appropriateness of the satellite derived gravity field models for the offshore and coastal regions of Sri Lanka.While these eight gravity field models with ID 155-EIGEN-6S4 V2, 168-Tongji-Grace 02K, 171-GO-CONS-GCF2-TIM-R6, 174-ITSG-GRACE 2018S, 178-Tongji-GMMG2021S, 148-EIGN-6C4, 152-GECO, 167-SGG-UGM-1, and 177-SGG-UGM2 were used for the study. The mean difference value, standard deviation value, and Root Mean Square values modeled by each of the model between the observed gravity data and the BGI observed gravity data. The results indicated that 171-GO-CONS-GCF2-TIM-R6 and 177-SGG-UGM2 is the more suited model for the coastal and offshore region of Sri Lanka. The study used the SARAL-Altika satellite Altimetry data to investigate the relationship between Mean see surface height and observed gravity. No significant relationship between Mean sea surface height and marine gravity is indicated by. In addition, the influence of gravity model type on each of the LEO satellite orbit predictions was also studied. The findings of the research demonstrate that the optimal type of gravity model applied for LEO satellite orbit prediction depends on a short-term or long-term predictions. The models JGM3, EGM2008, and GL04C are more appropriate for short-term predictions, while the models JGM3, EGM96, and EIGEN2 are the best for long-term predictions. The gravity order and permanent tides also have to be taken into account for the orbital prediction.In conclusion, this research provides valuable insights into the suitability of various satellite-derived gravity field models for Sri Lanka's coastal and offshore regions. The findings also emphasize the need to consider the impact of gravity models on LEO satellite orbit prediction, particularly for new applications such as LEO navigation, which require real-time precise orbits.

A earth gravity field recovery from CHAMP kinematic orbits using energy conservation approach is presented. A method combining the Newton's numerical differential formula and the remove-restore procedure for determining kinematic velocities is developed. Three earth gravity field models with 50 degree and order are recovered using CHAMP kinematic orbits provided by Technical University of Munich (TUM). The comparisons of degree variance and geoid difference with GRIM5-C1, EIGEN-1S and EIGEN-2 indicate that the recovered models yield a better fit to EIGEN-2. The assessments using Arctic gravity anomalies data show that these models have nearly the same quality of EIGEN-1S and EIGEN-2, and perform better than GRIM5-C1.

Summary The estimation of topographic gravity field models has attracted significant interest in recent years due to its growing relevance in Earth sciences. In this study, we present a robust methodology for the computation and comprehensive validation of global, complete spherical Bouguer and isostatic gravity anomalies that are essential for accurately interpreting subsurface mass distributions therefore geological structures. We synthesize these crucial gravitational functionals by leveraging spherical harmonic coefficients from high-resolution global gravity field models and various topographic/topographic-isostatic gravity field models. Our findings underscore the critical role of comprehensive terrain corrections in deriving physically meaningful, complete Bouguer gravity fields. The calculated global anomalies demonstrate strong coherence with established benchmark datasets, such as the World Gravity Map 2012. Residual differences are primarily attributed to variations in input Digital Terrain Models. Comparisons with regional Bouguer datasets reveal systematic biases that are largely explained by differing terrain correction methodologies. After removing this effect, there is a high level of consistency between the calculated global and published regional datasets, highlighting the utility of our global solutions, particularly in regions with sparse terrestrial data. Furthermore, the globally computed isostatic gravity anomalies exhibit significant agreement with both external global and diverse regional datasets, notably without the large systematic biases observed in Bouguer comparisons. This agreement reflects the effectiveness of the combined topographic and isostatic corrections in capturing Earth’s mass balance. This research provides valuable tools for new studies in the geoscience community by offering globally consistent and complete Bouguer and isostatic gravity field anomalies that have been rigorously validated for the ICGEM service.

In 2014, the Jeddah Municipality made a call for an estimate of a centimetric precision geoid model to be used for engineering and surveying applications, because the regional geoid model available at that time did not reach a sufficient precision. A project was set up to this end and dedicated sets of gravity and Global Positioning System (GPS)/levelling data were acquired in the framework of this project. In this paper, a thorough analysis of these newly acquired data and of the last available Global Gravity Field Models (GGMs) has been done in order to obtain a geoid undulation estimate with the prescribed precision. In the framework of the Remove–Compute–Restore (RCR) approach, the collocation method was used to obtain the height anomaly estimation that was then converted to geoid undulation. The remove and restore steps of the RCR approach were based on GGMs, derived from the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) and Gravity Recovery and Climate Experiment (GRACE) dedicated gravity satellite missions, which were used to improve the long wavelength components of the Earth’s gravity field. Furthermore, two different quasi-geoid collocation estimates were computed, based on gravity data only and on gravity plus GPS/levelling data (the so-called hybrid estimate). The best solutions were obtained with the hybrid geoid estimate. This was tested by comparison with an independent set of GPS/levelling geoid undulations that were not included in the computed solutions. By these tests, the precision of the hybrid geoid is estimated to be 3.7 cm. This precision proved to be better, by a factor of two, than the corresponding one estimated from the pure gravimetric geoid. This project has been also useful to verify the importance and reliability of GGMs developed from the last satellite gravity missions (GOCE and GRACE) that have significantly improved our knowledge of the long wavelength components of the Earth’s gravity field, especially in areas with poor coverage of terrestrial gravity data. In fact, the geoid models based on satellite-only GGMs proved to have a better performance, despite the lower spatial resolution with respect to high-resolution models (i.e., Earth Gravitational Model 2008 (EGM2008)).