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The primary objective of the 1-cm geoid experiment in Colorado (USA) is to compare the numerous geoid computation methods used by different groups around the world. This is intended to lay the foundations for tuning computation methods to achieve the sought after 1-cm accuracy, and also evaluate how this accuracy may be robustly assessed. In this experiment, (quasi)geoid models were computed using the same input data provided by the US National Geodetic Survey (NGS), but using different methodologies. The rugged mountainous study area (730 km × 560 km) in Colorado was chosen so as to accentuate any differences between the methodologies, and to take advantage of newly collected GPS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17) which are now available to be used as an accurate and independent test dataset. Fourteen groups from fourteen countries submitted a gravimetric geoid and a quasigeoid model in a 1′ × 1′ grid for the study area, as well as geoid heights, height anomalies, and geopotential values at the 223 GSVS17 marks. This paper concentrates on the quasigeoid model comparison and evaluation, while the geopotential value investigations are presented as a separate paper (Sánchez et al. in J Geodesy 95(3):1. https://doi.org/10.1007/s00190-021-01481-0 , 2021). Three comparisons are performed: the area comparison to show the model precision, the comparison with the GSVS17 data to estimate the relative accuracy of the models, and the differential quasigeoid (slope) comparison with GSVS17 to assess the relative accuracy of the height anomalies at different baseline lengths. The results show that the precision of the 1′ × 1′ models over the complete area is about 2 cm, while the accuracy estimates along the GSVS17 profile range from 1.2 cm to 3.4 cm. Considering that the GSVS17 does not pass the roughest terrain, we estimate that the quasigeoid can be computed with an accuracy of ~ 2 cm in Colorado. The slope comparisons show that RMS values of the differences vary from 2 to 8 cm in all baseline lengths. Although the 2-cm precision and 2-cm relative accuracy have been estimated in such a rugged region, the experiment has not reached the 1-cm accuracy goal. At this point, the different accuracy estimates are not a proof of the superiority of one methodology over another because the model precision and accuracy of the GSVS17-derived height anomalies are at a similar level. It appears that the differences are not primarily caused by differences in theory, but that they originate mostly from numerical computations and/or data processing techniques. Consequently, recommendations to improve the model precision toward the 1-cm accuracy are also given in this paper.

The origin of the Earth's lowest geoid, the Indian Ocean geoid low (IOGL) has been controversial. The geoid predicted from present‐day tomography models has shown that mid to upper mantle hot anomalies are integral in generating the IOGL. Here we assimilate plate reconstruction in global mantle convection models starting from 140 Ma and show that sinking Tethyan slabs perturbed the African Large Low Shear Velocity province and generated plumes beneath the Indian Ocean, which led to the formation of this negative geoid anomaly. We also show that this low can be reproduced by surrounding mantle density anomalies, without having them present directly beneath the geoid low. We tune the density and viscosity of thermochemical piles at core‐mantle boundary, Clapeyron slope and density jump at 660 km discontinuity, and the strength of slabs, to control the rise of plumes, which in turn determine the shape and amplitude of the geoid low. Plain Language Summary The origin of the deepest geoid on Earth, the Indian Ocean geoid low (IOGL), is debated. Several competing hypotheses exist, amongst which, a recent study employing tomography models suggested that hot anomalies at mid to upper mantle depths are crucial in generating this elusive feature. Assimilating plate motion in global mantle convection models from the Mesozoic till the present day, we attempt to trace the formation of this geoid low. We show that flow induced by downwelling Tethys slabs perturbs the African Large Low Shear Velocity province and gives rise to plumes that reach the upper mantle. These plumes, along with the mantle structure in the vicinity of the geoid low, are responsible for the formation of this negative geoid anomaly. Exploring a wide model parameter space, such as the density and intrinsic viscosity of the thermochemical piles, Clapeyron slope and density jump at 660 km depth, strength of slabs, we show that plumes are integral in generating the IOGL. The contribution of lower mantle Tethys slabs is secondary but also necessary in generating this geoid low. Key Points Employing time dependent global mantle convection models since the Cretaceous we simulate the origin of the enigmatic Indian Ocean geoid low Plumes forming along the edges of the African Large Low Shear Velocity province (LLSVP) control the regional geoid in the Indian Ocean These plumes, in turn are generated by lower mantle Tethyan slabs that perturb the African LLSVP

The mean dynamic topography (MDT) is a key reference surface for altimetry. It is needed for the calculation of the ocean absolute dynamic topography, and under the geostrophic approximation, the estimation of surface currents. CNES-CLS mean dynamic topography (MDT) solutions are calculated by merging information from altimeter data, GRACE, and GOCE gravity field and oceanographic in situ measurements (drifting buoy velocities, hydrological profiles). The objective of this paper is to present the newly updated CNES-CLS18 MDT. The main improvement compared to the previous CNES-CLS13 solution is the use of updated input datasets: the GOCO05S geoid model is used based on the complete GOCE mission (November 2009–October 2013) and 10.5 years of GRACE data, together with all drifting buoy velocities (SVP-type and Argo floats) and hydrological profiles (CORA database) available from 1993 to 2017 (instead of 1993–2012). The new solution also benefits from improved data processing (in particular a new wind-driven current model has been developed to extract the geostrophic component from the buoy velocities) and methodology (in particular the computation of the medium-scale GOCE-based MDT first guess has been revised). An evaluation of the new solution compared to the previous version and to other existing MDT solutions show significant improvements in both strong currents and coastal areas.

The harmonic correction (HC) is one of the key quantities when using residual terrain modelling (RTM) for high-frequency gravity field modelling. In the RTM technique, high-frequency topographic gravitational signals are obtained through removing gravitational effects of a long-wavelength reference surface, e.g., MERIT2160. There might be points located below the reference surface. In such cases, the RTM gravity field is calculated in the non-harmonic condition, HC is therefore required. Over past decades, though various methods have been proposed to handle the HC issue for the RTM technique, most of them were focused on the HC for RTM gravity anomaly rather than for other gravity functionals, such as RTM geoid height. In practice, the HC for RTM geoid height was generally assumed to be negligible, but a detailed quantification was missing for present-day RTM computations. This might cause large errors in the regional geoid determination over rugged areas. In this study, we derive HC expressions for the RTM geoid height in the framework of the classical condensation method. The HC terms are derived under four different assumptions separately: residual masses approximated by an unlimited Bouguer plate, residual masses approximated by a limited Bouguer plate which overcomes the mass inconsistency effect, residual masses approximated by a Bouguer shell which overcomes the effect of planar approximation, and residual masses approximated by a limited Bouguer shell which overcomes the errors induced by both planar approximation and mass-inconsistency. The errors due to various approximations in HC terms are investigated through comparison among various terms. Besides, HC terms are computed using an expansion up to degree and order 2159. Our results show that HC for RTM geoid height is less 1 mm and could be ignored over ∼99% of continental areas, but be of great significance for regional geoid determination over mountain areas, e.g., more than 10 cm effect over very rugged areas. The validation through comparison with terrestrial measurements and a baseline solution of the RTM technique proves that the HC terms provided in this study can improve the accuracy of RTM geoid heights and are expected to be useful for applications of the RTM technique in regional and global gravity field modelling.

In traditional airborne gravimetry, the vertical component of the gravity vector is used as an approximation of the measured magnitude of the gravity vector, which enters the determination of the local geoid. In this study, a comprehensive computational scheme for determining the local geoid using three components of the airborne gravity vector is presented. Our approach extends the existing one-step method for local geoid modeling by incorporating the full gravity vector measured by airborne sensors as boundary values in the gravimetric boundary-value problem. We derive integral kernel functions along with far-zone contributions for the three components of the airborne gravity vector and apply deterministic modifications to them. To validate our derivations, we use a global geopotential model (GGM)-based airborne gravity vectors burdened with realistic colored noise at one of the most challenging test sites for geoid determination, the 1-cm geoid test area in Colorado (USA). Results of closed-loop tests confirm that applying all three components of the GGM-based airborne gravity vector improves the internal accuracy of the geoid by 50% compared to using only the vertical component. We further use real airborne gravity vectors observed at a test site in the same region and show that the STD of the estimated geoid heights evaluated against the reference geoidal heights along the Geoid Slope Validation Survey of 2017 (GSVS17) Line is 2.3 cm using the “traditional approach” and 1.3 cm including the horizontal components. This indicates a significant improvement in the external accuracy (~ 46%) of the geoid when the full gravity vector is used, without using other heterogeneous observations. Graphical Abstract

Earth's residual geoid is dominated by a degree-2 mode, with elevated regions above large low shear-wave velocity provinces on the core—mantle boundary beneath Africa and the Pacific. The edges of these deep mantle bodies, when projected radially to the Earth's surface, correlate with the reconstructed positions of large igneous provinces and kimberlites since Pangea formed about 320 million years ago. Using this surface-to-core—mantle boundary correlation to locate continents in longitude and a novel iterative approach for defining a paleomagnetic reference frame corrected for true polar wander, we have developed a model for absolute plate motion back to earliest Paleozoic time (540 Ma). For the Paleozoic, we have identified six phases of slow, oscillatory true polar wander during which the Earth's axis of minimum moment of inertia was similar to that of Mesozoic times. The rates of Paleozoic true polar wander (<1°/My) are compatible with those in the Mesozoic, but absolute plate velocities are, on average, twice as high. Our reconstructions generate geologically plausible scenarios, with large igneous provinces and kimberlites sourced from the margins of the large low shear-wave velocity provinces, as in Mesozoic and Cenozoic times. This absolute kinematic model suggests that a degree-2 convection mode within the Earth's mantle may have operated throughout the entire Phanerozoic.

Integration of land and marine vertical datums is an important aspect of geospatial reference systems. Therefore, this study has been conducted to identify an optimum approach to integrate the marine and land vertical datums. Two hybrid geoid models have been developed and fitted to the the land levelling datum at benchmark and to the tide gauge-benchmark station (TGBM). The differences between the two hybrid geoid models were computed to establish a vertical datum transformation model (VDT). Among the 305 GNSS-levelling points, 295 have been used in the hybridization process and 10 have been used for validation. Based on the comparison, the geoidal differences at the 10 points range from −7.2 to 7.0 cm while the mean and RMSE of differences are 1.3 cm and ± 4 cm, respectively. The second hybrid geoid, which was fitted to local MSL, was developed by directly adding to the offset between the gravimetric geoid and local MSL at nine TGBM stations. The result indicates that the offset derived at Tanjung Gelang is the optimum one with an RMSE of ± 0.045 m. The VDT model developed shows a transformation accuracy of approximately ± 4 cm.

This study aims to introduce the geoid modeling process in Sulawesi and demonstrate the practical issues faced. There are limitations of terrestrial gravity surveys in Sulawesi due to its complex geography, so airborne gravity surveys were conducted from 2008 to 2019 through a collaboration between the Badan Informasi Geospasial (BIG), the Technical University of Denmark (DTU), and the National Chiao Tung University (NCTU) gravity research team. The airborne gravity data currently cover almost the entire land area of Indonesia. The geoid modeling process involved refining the EGM08-derived geoid heights by incorporating downward-continued airborne gravity data and RTM-derived geoid effects, and adjusting the geometric geoid heights to accommodate diverse mean sea levels used in different GPS/leveling datasets. The impacts of different global gravitational models (GGMs), such as EIGEN-6C4, GECO, XGM2019e, and SGG-UGM-2, on geoid modeling were examined, and it revealed that differences arise from the different datasets used in the development of the GGM. This study revealed that airborne gravity data can significantly improve the accuracy of the geoid, achieving an impressive accuracy of approximately 0.04 to 0.05 m. In addition, the compatibility issue between gravity data and GGM in geoid modeling is highlighted. The strategy to unify the vertical datum between different GPS/leveling datasets for the validation of gravimetric geoid model was discussed in detail. Once the corresponding geopotential is determined, the conversion between the global vertical datum and the local vertical datum can be achieved. Accurate geoid is critical for infrastructure development, land-use planning, and resource management and play an integral role in supporting sustainable development goals (SDGs) by providing accurate spatial referencing, ensuring precise mapping, and offering location-based services.

The article presents the results of research on the development of methods for obtaining normal heights in Vietnam using the global geoid model EGM2008 and software for processing GNSS measurements. The errors between the normal heights computed by the global model of the EGM2008 geoid and the heights found by GNSS measurements and geometric levelling on the territory of Vietnam are determined. The localisation method for the global geoid model EGM2008 for Vietnam's territory is suggested, such a geoid model is practically formed as the EGM2008_ TN file. Tests of the developed technique and the localised model of the EGM2008__TN geoid were carried out, showing the possibility of its application to obtain normal heights from GNSS measurements in Vietnam with the 4th levelling accuracy class.

An accurate knowledge of the ocean mean dynamic topography (MDT) is mandatory for the optimal use of altimetric data, including their assimilation into operational ocean forecasting systems. A new global 1/4° resolution MDT was computed for the 1993–1999 time period with improved data and methodology compared to the previous RIO05 MDT field. First, a large‐scale MDT is obtained from the CLS01 altimetric Mean Sea Surface and a recent geoid model computed from 4.5 years of GRACE (Gravity Recovery and Climate Experiment) data. Altimetric sea level anomalies and in situ measurements are then combined to compute synthetic estimates of the MDT and the corresponding mean currents. While the RIO05 MDT was based on 10 years of in situ dynamic heights and drifting buoy velocities, the new field benefits from an enlarged data set of in situ measurements ranging from 1993 to 2008 and includes all hydrological profiles from the Argo array. Moreover, the processing of the in situ data has been updated. A new Ekman model was developed to extract the geostrophic velocity component from the drifting buoy measurements. The handling of hydrologic measurements has also been revisited. Compared to the previous RIO05 solution, the new global MDT resolves much stronger gradients in western boundary currents, with mean velocities being doubled in some places. Moreover, in comparison to several other recent MDT estimates, we find that the new CNES‐CLS09 MDT is in better agreement with independent in situ observations. Key Points Computation of a new mean dynamic topography Development of a new Ekman model Optimal filtering of MSS‐Geoid