The Refined Gravity Field Models for Height System Unification in China

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)).

The Earth’s time-variable gravity field is of great significance to study mass change within the Earth’s system. Since 2002, the NASA-DLR Gravity Recovery and Climate Experiment (GRACE) and its successor GRACE follow-on mission provide observations of monthly changes in the Earth gravity field with unprecedented accuracy and resolution by employing low-low satellite-to-satellite tracking (LLSST) measurements. In addition to LLSST, monthly gravity field models can be acquired from satellite laser ranging (SLR) and high-low satellite-to-satellite tracking (HLSST). The monthly gravity field solutions HLSST+SLR were derived by combining HLSST observations of low earth orbiting (LEO) satellites with SLR observations of geodetic satellites. Bandpass filtering was applied to the harmonic coefficients of HLSST+SLR solutions to reduce noise. In this study, we analyzed the performance of the monthly HLSST+SLR solutions in the spectral and spatial domains. The results show that: (1) the accuracies of HLSST+SLR solutions are comparable to those from GRACE for coefficients below degree 10, and significantly improved compared to those of SLR-only and HLSST-only solutions; (2) the effective spatial resolution could reach 1000 km, corresponding to the spherical harmonic coefficient degree 20, which is higher than that of the HLSST-only solutions. Compared with the GRACE solutions, the global mass redistribution features and magnitudes can be well identified from HLSST+SLR solutions at the spatial resolution of 1000 km, although with much noise. In the applications of regional mass recovery, the seasonal variations over the Amazon Basin and the long-term trend over Greenland derived from HLSST+SLR solutions truncated to degree 20 agree well with those from GRACE solutions without truncation, and the RMS of mass variations is 282 Gt over the Amazon Basin and 192 Gt in Greenland. We conclude that HLSST+SLR can be an alternative option to estimate temporal changes in the Earth gravity field, although with far less spatial resolution and lower accuracy than that offered by GRACE. This approach can monitor the large-scale mass transport during the data gaps between the GRACE and the GRACE follow-on missions.

Gravity Recovery and Climate Experiment (GRACE) satellite gravimetry can monitor surface mass variations with a spatial resolution of ~300km and a temporal resolution of 1 month. GRACE intersatellite geopotential difference (GPD) is a type of in-situ gravimetric observables with explicit geophysical interpretation, which can directly reflect surface mass variations. The intersatellite GPDs are estimated from GRACE Level-1B (RL03) data (including KBR range-rate measurements (KBR1B), satellite orbits (GNV1B), accelerometer measurements (ACC1B), and satellite attitude measurements (SCA1B)) based on an improved energy balance equation, and the orbit and background model errors are reduced using empirical parameters based on the remove-compute-restore (RCR) technique. Background models used for estimating GRACE intersatellite GPDs are consistent with GRACE Level-2 Processing Standards Document (for Level-2 Product RL06). GRACE intersatellite GPD data are processed over the period from April 2002 to July 2016 with a sampling rate of 5s along the satellite orbits (orbit height is ~450km), and stored on a monthly basis. There are total 155 months of data files over the period from April 2002 to July 2016, and the other 17 months of data are not available due to the instrument failure or other influencing factors. Each data file includes the positions of GRACE satellites A and B in Earth-centered Earth-fixed (ECEF) coordinate system, intersatellite geopotential differences after deducting the mean gravity field model GGM05C (i.e., the disturbance potential difference TAB=TB-TA), and the corresponding quality flags. GRACE intersatellite GPD data are validated through the estimation of monthly gravity field solutions and regional surface mass variations. The estimates inferred from GPD data are compared with GRACE official monthly solutions and mascon solutions, such as the products published by the Center for Space Research (CSR), Jet Propulsion Laboratory (JPL), German Research Center for Geosciences (GFZ), and Goddard Space Flight Center (GSFC). It is demonstrated that the differences between GPD-inferred results and the official released models are within the range of the differences among those official models.  GRACE intersatellite GPD dataset enriches the GRACE data products, and can be applied to directly invert the monthly gravity field models, global and regional surface mass variations, which is helpful to study the terrestrial water cycle, ice sheets and mountain glacier mass balance, sea level change, and ocean bottom pressure variations, etc.

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.

The results of this study suggest that a traditional height network based on fundamental benchmarks can be used meaningfully to assess the quality of a satellite‐derived geoid model. We compare the pre‐ Gravity Recovery and Climate Experiment (GRACE) OSGM02 and the recent OSGM05 geoid models with the datum surface inferred by the UK fundamental benchmark network combined with network monument ellipsoidal heights derived from GPS data. OSGM05 incorporates long‐wavelength gravity field data from GRACE to degree/order 90. Switching from OSGM02 to OSGM05, the mean difference between the leveling datum and the geoid model reduces from −0.129 m to −0.004 m, and RMS differences reduce from 0.110 m to 0.048 m. The mean tilt between the height datum and the geoid model in the north‐south direction reduces from 34 mm to 12 mm per degree of latitude. Hence we demonstrate a systematic convergence between the geoid models and the leveling datum surface in terms of mean offset, RMS scatter and tilt without any fitting of one surface to the other. The implication is that the leveling datum surface represented a more accurate model of the geoidal geometry compared to OSGM02 in terms of form (in both short‐ and long‐wavelength features) and surprisingly also in terms of its offset from the GRS80 ellipsoid. While the latter point can be no more than a coincidence, the former indicates that the UK leveling network, based on its fundamental benchmark network, can be used to test the accuracy of geoid model improvements derived from missions such as GRACE, with implications for the forthcoming Gravity Field and Steady State Ocean Circulation Explorer (GOCE) mission.

Thanks to the unprecedented success of Gravity Recovery and Climate Experiment (GRACE), its successive mission GRACE Follow-On (GFO) has been in orbit since May 2018 to continue measuring the Earth’s mass transport. In order to possibly enhance GFO in terms of mass transport estimates, four orbit configurations of future polar-type gravity mission (FPG) (with the same payload accuracy and orbit parameters as GRACE, but differing in orbit inclination) are investigated by full-scale simulations in both standalone and jointly with GFO. The results demonstrate that the retrograde orbit modes used in FPG are generally superior to prograde in terms of gravity field estimation in the case of a joint GFO configuration. Considering the FPG’s independent capability, the orbit configurations with 89- and 91-degree inclinations (namely FPG-89 and FPG-91) are further analyzed by joint GFO monthly gravity field models over the period of one-year. Our analyses show that the FPG-91 basically outperforms the FPG-89 in mass change estimates, especially at the medium- and low-latitude regions. Compared to GFO & FPG-89, about 22% noise reduction over the ocean area and 17% over land areas are achieved by the GFO & FPG-91 combined model. Therefore, the FPG-91 is worthy to be recommended for the further orbit design of FPGs.

The geoid-to-quasigeoid correction has been traditionally computed approximately as a function of the planar Bouguer gravity anomaly and the topographic height. Recent numerical studies based on newly developed theoretical models, however, indicate that the computation of this correction using the approximate formula yields large errors especially in mountainous regions with computation points at high elevations. In this study we investigate these approximation errors at the study area which comprises Himalayas and Tibet where this correction reaches global maxima. Since the GPS-leveling and terrestrial gravity datasets in this part of the world are not (freely) available, global gravitational models (GGMs) are used to compute this correction utilizing the expressions for a spherical harmonic analysis of the gravity field. The computation of this correction can be done using the GGM coefficients taken from the Earth Gravitational Model 2008 (EGM08) complete to degree 2160 of spherical harmonics. The recent studies based on a regional accuracy assessment of GGMs have shown that the combined GRACE/GOCE solutions provide a substantial improvement of the Earth’s gravity field at medium wavelengths of spherical harmonics compared to EGM08. We address this aspect in numerical analysis by comparing the gravity field quantities computed using the satellite-only combined GRACE/GOCE model GOCO02S against the EGM08 results. The numerical results reveal that errors in the geoid-to-quasigeoid correction computed using the approximate formula can reach as much as ~1.5 m. We also demonstrate that the expected improvement of the GOCO02S gravity field quantities at medium wavelengths (within the frequency band approximately between 100 and 250) compared to EGM08 is as much as ±60 mGal and ±0.2 m in terms of gravity anomalies and geoid/quasigeoid heights respectively.

The accuracy of Earth's gravitational field from the Gravity Recovery and Climate Experiment (GRACE) Follow‐On satellite mission is efficiently and rapidly estimated for the first time based on the analytic method. Firstly, the new single and combined analytic error models of cumulative geoid height influenced by four error sources including the intersatellite range‐rate of interferometric laser ranging system, orbital position and velocity of GPS receiver and nonconservative force of accelerometer from GRACE Follow‐On satellites are established using the power spectrum principle, respectively. Secondly, the dependability of analytic error model is validated according to the consistency of the matching accuracy indexes of GRACE key payloads from the single analytic error models and the GRACE Level 1B provided by the American Jet Propulsion Laboratory (JPL), and the conformity of GRACE cumulative geoid height errors from the combined analytic error model and the Earth's gravitational field model EIGEN‐GRACE02S released by the German GeoForschungsZentrum Potsdam (GFZ). Finally, the influences of different matching accuracy indexes of key payloads and orbital altitudes of GRACE Follow‐On satellites on the accuracies of Earth's gravitational field are demonstrated contrastively. At the 360 degree, cumulative geoid height error is 1.231×10−1 m using combined analytic error model based on orbital altitude 250 km, intersatellite range 50 km, intersatellite range‐rate error 1×10−9 m/s, orbital position error 3×10−5 m, orbital velocity error 3×10−8 m/s and nonconservative force error 3×10−13 m/s2. This work not only can provide the theoretical foundation and calculational guarantee for the efficient and rapid determination of the accuracies of current GRACE and future GRACE Follow‐On Earth's gravitational field, but also has some reference significance to the successful execution of the future Gravity Recovery and Interior Laboratory (GRAIL) lunar satellite gravity exploration mission.

The terrestrial water resources in Shaanxi–Gansu–Ningxia (SGN) region are relatively scarce, and its climate change is unstable. Research on the deformation driven by terrestrial water load is of great significance to the dynamic maintenance of reference station networks. In this paper, data derived from Gravity Recovery and Climate Experiment (GRACE) and Global Navigation Satellite System (GNSS) from 2010 to 2014 were combined to monitor the spatiotemporal characteristics of surface vertical deformation caused by terrestrial water load change. The single scale factor was calculated by comparing CPC, WGHM, and GLDAS hydrological model to restore filtering leakage signal. The singular spectrum analysis (SSA) method was used to extract the principal component of temporal vertical deformation, and its spatial distribution was analyzed. At the same time, in order to study the relationship between the terrestrial water load deformation from GRACE and that from GNSS, the first-order term correction, the Atmosphere and Ocean De-aliasing Level-1B product (GAC) correction, and the first-order load LOVE number correction for GRACE were adopted in this paper. In addition, a quantitative comparative analysis of both the monitoring results was carried out. The results show that the time-variable characteristics of surface vertical deformation characterized by the filtered three hydrological models were consistent with those of GRACE. The correlation coefficient and Nash–Sutcliffe efficiency coefficient (NSE) values were the highest in the GLDAS model and the GRACE model, respectively; the former index is 0.93, while the latter is 0.85. The crustal vertical deformation from terrestrial water load showed a declining rate from 2010 to 2014. Its spatial change rate showed an obvious ladder distribution, with the surface subsidence rate gradually decreasing from south to north. In addition, weighted root mean square (WRMS) contribution rate of the crustal vertical deformation resulting from GRACE with GAC correction between the different GNSS stations ranged from 18.52% to 54.82%. The correlation coefficient between them was close to 0.70. After deducting the mass load impact of GRACE only, the WRMS contribution rate of the corresponding stations decreased from −8.42% to 21.18%. The correlation coefficient between them reduced noticeably. Adding GAC back can increase the comparability with GRACE and GNSS in terms of monitoring the crustal vertical deformation. The annual amplitude and phase of surface vertical deformation resulting from GRACE with GAC correction were close to those of GNSS. The research results can help to explore the motion mechanism between water migration and surface deformation, which is of benefit in the protection of the water ecological environment in the region.

The monthly global gravity field solutions derived using the measurements from the GRACE (Gravity Recovery and Climate Experiment) satellites have been continuously improved by the processing centers. One of the improvements in the processing method is a more detailed calibration of the on-board accelerometers in the GRACE satellites. The accelerometer data calibration is usually restricted to the scale factors and biases. It has been assumed that the three different axes are perfectly orthogonal in the GRACE science reference frame. Recently, it was shown by Klinger and Mayer-Gurr (2016) that a fully-populated scale matrix considering the non-orthogonality of the axes and the misalignment of the GRACE science reference frame and the GRACE accelerometer frame improves the quality of the C20 coefficient in the GRACE monthly gravity field solutions. We investigate the effect of the more detailed calibration of the GRACE accelerometer data on the C20 coefficient in the case of the AIUB (Astronomical Institute of the University of Bern) processing method using the Celestial Mechanics Approach. We also investigate the effect of the new calibration parameters on the stochastic parameters in the Celestial Mechanics Approach.

To evaluate the performance of the global geopotential models (GGMs) in a more unbiased way, ground-based gravity and GNSS/levelling datasets are highly required. In this study, the eight latest releases of the satellite-only and combined GGMs are evaluated on the regional scale using the available terrestrial gravity and GNSS/Levelling data over Sudan, considering the spectral consistency issue by applying the spectral enhancement method (SEM). The evaluation process consists of three stages: firstly, the eight GGMs are evaluated globally with each other by using different degree variances in terms of geoid heights, gravity anomalies, and signal-to-noise ratio (SNR); secondly, the GGMs are compared against the Earth Gravitational Model 2008 (EGM2008) on a regional scale over Sudan; thirdly, apply the SEM strategy by incorporating high (SEM_WITHOUT_RTM technique) and very-high (SEM technique) frequencies of the gravity field spectrum from the EGM2008 and high-resolution residual terrain model (RTM), respectively. For reliable robustness of the latter evaluation process, three different DEMs are used, namely, SRTM30, ASTER30, and GTOPO30. Our findings on the evaluation process using SEM_WITHOUT_RTM technique show improved gravity anomalies solutions regarding differences of standard deviations (STD) from 19–20.7 mGal to about 14 mGal. When applying the SEM technique, more improvements are achieved, providing STD differences in gravity anomalies and geoid heights of about 12 mGal and 45 cm, respectively. Among the three applied DEMs, it has been found that despite the slight refinements, the ASTER30 and GTOPO30 models show better performance than the SRTM30 model.

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

Accurate determination of geocenter motion is essential not only for establishing a stable terrestrial reference frame, but also for deriving a complete picture of large‐scale global mass redistribution in the Earth system. For geophysical applications, reliable geocenter motions can be inferred from time‐variable gravity fields provided by the Gravity Recovery and Climate Experiment (GRACE) since 2002, but it is more challenging for the pre‐GRACE era where only low‐degree gravity fields are available from the Satellite Laser Ranging (SLR). In addition, geocenter motion estimates derived from SLR using the direct method lack the trend in a linear reference frame and are therefore generally not suitable for studying mass change rates. In this study, we derive the geocenter motion from low‐degree gravity fields up to degree and order 5 after properly addressing signal leakage. Using the leakage‐corrected land mass patterns combined with corresponding ocean mass fingerprints, we generate geocenter motion estimates and compare them with those derived from GRACE, geophysical models, and the SLR direct method between 2002 and 2020. The trends in our estimates are consistent with GRACE and models, with differences below 0.1∼0.2 mm/yr depending on the quality of the gravity field models, while the SLR direct estimates yield opposite trends, leading to significantly underestimated global ocean mass change rates. Our study provides the first promising solution to derive long‐term geocenter motion rates from low‐degree gravity fields, which can be used to track large‐scale mass change back to the 1990s.

Since few time‐variable gravity field models were developed before the Gravity Recovery and Climate Experiment (GRACE) era, one normally estimates the global mean ocean mass (GMOM) changes by summing the mass contributions. In this study, new monthly gravity field models named Tongji‐LEO2021 developed from Early Low Earth Orbit (LEO) satellites' observations are used to directly estimate GMOM changes and quantify the closure of global sea‐level budget with Altimetry and Steric observations between 1993 and 2004. The resulting GMOM changes from Tongji‐LEO2021 solutions are comparable to those from the Center for Space Research RL06 GRACE solutions and the IGG‐SLR‐HYBRID solutions, with the correlation coefficients of 0.91 and 0.88. Our results show that the LEO‐derived GMOM change rate is 0.88 ± 0.16 mm/year, slightly larger than 0.83 ± 0.16 mm/year from IGG‐SLR‐HYBRID, which are both close to Altimetry‐Steric estimate (0.87 ± 0.33 mm/year). Overall, Tongji‐LEO2021 solutions can estimate the GMOM change and understand the global sea‐level change before the GRACE era.

SUMMARY This study aims to provide valuable scientific insights into various estimation techniques of geocentre motion (GCM) from the perspective of signal analysis, thereby enhancing Gravity Recovery and Climate Experiment (GRACE) users' understanding and application of GCM. Initially, it utilizes the satellite laser ranging (SLR) technique with the network shift approach to estimate over 30 yr of weekly GCM time-series from 1994 to 2024. Subsequently, we employ two approaches to estimate three types of monthly GCM time-series spanning more than 20 yr from 2002 to 2023: combining GRACE data with an ocean bottom pressure model (GRACE-OBP approach), the fingerprint approach (FPA), and the fingerprint approach with satellite altimetry data (FPA-SA, up to 2022). The former is referred to as SLR-based GCM estimates, while the latter, which uses GRACE Earth's gravity field models, is termed GRACE-based GCM estimates. Furthermore, this study pioneers the use of multichannel singular spectrum analysis (MSSA) for GCM analysis, especially focusing on the latest GRACE-based GCM estimates from the GRACE-OBP and FPA/FPA-SA approaches, marking the first comprehensive analysis of GCM estimated by various techniques. The results show that MSSA can effectively extract common signals from the three components of the GCM time-series. The seasonal components extracted from GRACE-based GCM estimates using MSSA are consistent with those from SLR-based GCM estimates, although the former exhibit slightly larger amplitudes of the annual and semi-annual signals. After correcting the atmosphere-ocean dealiasing, the amplitudes of the SLR-based estimates correspondingly decrease, remaining slightly larger but becoming closer to those of the GRACE-based estimates. However, a periodic signal with an approximate 160-d period is detectable in all GRACE-based GCM estimates, but it is absent in SLR-based GCM estimates. Further investigation using MSSA into higher degree spherical harmonic (SH) coefficients of the Earth's gravity field models reveals that these SH coefficients contain a 160-d periodic signal. This finding suggests that the signal detected in GRACE-based GCM estimates originates from systematic errors in these SH coefficients, offering new insights for improving the accuracy of GRACE Earth's gravity field solutions. Additionally, GRACE-based GCM estimates show significant secular non-zero trends, notably larger than those in SLR-based GCM estimates, which are not expected to exhibit any trends. However, the reliance of GRACE-based GCM estimates on geophysical models (e.g. glacier melting, glacial isostatic adjustment and hydrological models) limits the accuracy of their trends, underscoring the need for further validation. Overall, this study highlights new challenges regarding the accuracy of GRACE-based GCM estimates and emphasizes the necessity for further validation in mass change research.