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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 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.

This study presents a solution of the ‘1 cm Geoid Experiment’ (Colorado Experiment) using spherical radial basis functions (SRBFs). As the only group using SRBFs among the fourteen participated institutions from all over the world, we highlight the methodology of SRBFs in this paper. Detailed explanations are given regarding the settings of the four most important factors that influence the performance of SRBFs in gravity field modeling, namely (1) the choosing bandwidth, (2) the locations of the SRBFs, (3) the type of the SRBFs as well as (4) the extensions of the data zone for reducing the edge effect. Two types of basis functions covering the same spectral range are used for the terrestrial and the airborne measurements, respectively. The non-smoothing Shannon function is applied to the terrestrial data to avoid the loss of spectral information. The cubic polynomial (CuP) function which has smoothing features is applied to the airborne data as a low-pass filter for filtering the high-frequency noise. Although the idea of combining different SRBFs for different observations was proven in theory to be possible, it is applied to real data for the first time, in this study. The RMS error of our height anomaly result along the GSVS17 benchmarks w.r.t the validation data (which is the mean results of the other contributions in the ‘Colorado Experiment’) drops by 5% when combining the Shannon function for the terrestrial data and the CuP function for the airborne data, compared to those obtained by using the Shannon function for both the two data sets. This improvement indicates the validity and benefits of using different SRBFs for different observation types. Global gravity model (GGM), topographic model, the terrestrial gravity data, as well as the airborne gravity data are combined, and the contribution of each data set to the final solution is discussed. By adding the terrestrial data to the GGM and the topographic model, the RMS error of the height anomaly result w.r.t the validation data drops from 4 to 1.8 cm, and it is further reduced to 1 cm by including the airborne data. Comparisons with the mean results of all the contributions show that our height anomaly and geoid height solutions at the GSVS17 benchmarks have an RMS error of 1.0 cm and 1.3 cm, respectively; and our height anomaly results give an RMS value of 1.6 cm in the whole study area, which are all the smallest among the participants.

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.

As humanity aspires to explore the solar system and investigate distant worlds such as the Moon, Mars, and beyond, there is a growing need to estimate and model the rate of clocks on these celestial bodies and compare them with the rate of standard clocks on Earth. According to Einstein’s theory of relativity, the rate of a standard clock is influenced by the gravitational potential at its location and its relative motion. A convenient choice of local reference frames allows for the comparison of local time variations of clocks due to gravitational and kinematic effects. We estimate the rate of clocks on the Moon using a locally freely falling reference frame coincident with the center of mass of the Earth–Moon system. A clock near the Moon’s selenoid ticks faster than one near the Earth’s geoid, accumulating an extra 56.02 μs day−1 over the duration of a lunar orbit. This formalism is then used to compute the clock rates at Earth–Moon Lagrange points. Accurate estimation of the rate differences of coordinate times across celestial bodies and their intercomparisons using clocks on board orbiters at Lagrange points as time transfer links is crucial for establishing reliable communications infrastructure. This understanding also underpins precise navigation in cislunar space and on celestial bodies’ surfaces, thus playing a pivotal role in ensuring the interoperability of various position, navigation, and timing systems spanning from Earth to the Moon and to the farthest regions of the inner solar system.

Upper Cretaceous to Miocene continental volcanism in NE Brazil spans 350 km in a N–S direction and 60 km in width, forming the Macau-Queimadas alignment (MQA). This study combines fieldwork, petrography, geochemistry, and Sr–Nd–Pb isotopes to explore its origin and evolution. The MQA consists of volcanic and hypabyssal mafic rocks intruding Cretaceous and Precambrian basement rocks, divided into two groups: (i) alkaline (foidite to trachy-basalt); and (ii) subalkaline (basalt and basaltic andesite). Both are sodic and LREE-enriched, with distinct La/Yb ratios. The alkaline group reflects an asthenospheric source (Nd model age of 1.1–0.4 Ga), while the subalkaline group incorporates an older lithospheric component (Nd model age of 2.1–1.2 Ga). These magmas originated from picritic parental melts, with < 15% melting for the alkaline group and ~ 25–30% melting for the subalkaline group, derived from spinel- to garnet-bearing peridotite. Differentiated series formed by successive small melt volumes, with some samples undergoing crustal fractional crystallization of clinopyroxene + olivine + plagioclase (alkaline group), and clinopyroxene + orthopyroxene + Ca-plagioclase (subalkaline group). The persistence of basaltic magmatism over ~ 90 Myr indicates sustained upper mantle melting. The alignment of volcanics, its association with a positive geoid anomaly, and its parallelism with the Mid-Atlantic Ridge suggest the MQA may represent an aborted ridge that never progressed to an oceanic stage.

The thermochemical structure of the subcontinental mantle holds information on its origin and evolution that can inform energy and mineral exploration strategies, natural hazard mitigation and evolutionary models of Earth. However, imaging the fine-scale thermochemical structure of continental lithosphere remains a major challenge. Here we combine multiple land and satellite datasets via thermodynamically constrained inversions to obtain a high-resolution thermochemical model of central and southern Africa. Results reveal diverse structures and compositions for cratons, indicating distinct evolutions and responses to geodynamic processes. While much of the Kaapvaal lithosphere retained its cratonic features, the western Angolan–Kasai Shield and the Rehoboth Block have lost their cratonic keels. The lithosphere of the Congo Craton has been affected by metasomatism, increasing its density and inducing its conspicuous low-topography, geoid and magnetic anomalies. Our results reconcile mantle structure with the causes and location of volcanism within and around the Tanzanian Craton, whereas the absence of volcanism towards the north is due to local asthenospheric downwellings, not to a previously proposed lithospheric root connecting with the Congo Craton. Our study offers improved integration of mantle structure, magmatism and the evolution and destruction of cratonic lithosphere, and lays the groundwork for future lithospheric evolutionary models and exploration frameworks for Earth and other terrestrial planets. Cratons in central and southern Africa exhibit diverse structures, compositions and responses to geodynamic settings, according to a high-resolution thermochemical regional model constructed from land- and satellite-based geophysical observations.

This study assesses the effect of the UNB Topographical Density Model on the accuracy of geoid determination in Sarajevo, Bosnia & Herzegovina. Using the KTH method, 1020 gravimetric geoid models were developed, incorporating both constant and variable density values, simple and complete Bouguer anomalies. The study found that the model computed by the UNB Topographical Density Model and complete Bouguer anomalies achieved the highest precision, with an RMSE of 1.33 cm. The final geoid model was adjusted to the old vertical datum (Trieste height), resulting in an RMSE of 3.44 cm when tested with static GNSS points. These findings underscore the importance of incorporating variable density models for improving geoid accuracy and suggest further refinement using local geological data could enhance precision.