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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 paper studies the contribution of airborne gravity data to improvement of gravimetric geoid modelling across the mountainous area in Colorado, USA. First, airborne gravity data was processed, filtered, and downward-continued. Then, three gravity anomaly grids were prepared; the first grid only from the terrestrial gravity data, the second grid only from the downward-continued airborne gravity data, and the third grid from combined downward-continued airborne and terrestrial gravity data. Gravimetric geoid models with the three gravity anomaly grids were determined using the least-squares modification of Stokes’ formula with additive corrections (LSMSA) method. The absolute and relative accuracy of the computed gravimetric geoid models was estimated on GNSS/levelling points. Results exhibit the accuracy improved by 1.1 cm or 20% in terms of standard deviation when airborne and terrestrial gravity data was used for geoid computation, compared to the geoid model computed only from terrestrial gravity data. Finally, the spectral analysis of surface gravity anomaly grids and geoid models was performed, which provided insights into specific wavelength bands in which airborne gravity data contributed and improved the power spectrum.

In the summer of 2017, the National Geodetic Survey (NGS) conducted its third and final Geoid Slope Validation Survey in the rugged terrain of southern Colorado, USA. As in previous surveys, the intent is to acquire the most accurate and precise field observations to determine geoid slopes. In turn, these data can be used to quantify the accuracy of various geoid models as NGS looks ahead to creating a highly accurate gravimetric geoid model for use as a national vertical datum. Long period GPS sessions, spirit leveling, absolute gravity, and deflection of the vertical (DoV) observations were acquired along a 360 km line, ranging from 1900 to 3300 m in elevation, with a station spacing of approximately 1.6 km. Our absolute gravity and DoV datasets are unique in that they were collected at 222 field stations in highly mountainous terrain at an unprecedented observational accuracy of 10 µGal and 0.04″, respectively. Further, by employing tailored refraction corrections to the spirit leveling data, we improved the agreement between heights derived from the DoV and spirit leveling from ± 1.9 to ± 1.3 cm RMS, or by more than 30%, across the line. At all length scales, from 1.6 to 360 km, the agreement is better than 2 cm. Finally, as a description of the validation process, we compare the observations with recent NGS experimental geoid models. We find that typical agreement is at about 3–5 cm, with no single model being best at all length scales. The data from this project are freely available to the community and should serve as test beds for not only geoid modeling comparisons, but also the refinement of numerous field techniques.

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.

Numerical methods, like the finite element method (FEM) or finite volume method (FVM), are widely used to provide solutions in many boundary value problems. In previous studies, these numerical methods have also been applied in geodesy but demanded extensive computations because the upper boundary condition was usually set up at the satellite orbit level, hundreds of kilometers above the Earth. The relatively large distances between the lower boundary of the Earth's surface and the upper boundary exacerbate the computation loads because of the required discretization in between. Considering that many areas, such as the US, have uniformly distributed airborne gravity data just a few kilometers above the topography, we adapt the upper boundary from the satellite orbit level to the mean flight level of the airborne gravimetry. The significant decrease in the domain of solution dramatically reduces the large computation demand for FEM or FVM. This paper demonstrates the advantages of using FVM in the decreased domain in simulated and actual field cases in study areas of interest. In the simulated case, the FVM numerical results show that precision improvement of about an order of magnitude can be obtained when moving the upper boundary from 250 to 10 km, the upper altitude of the GRAV-D flights. A 2–3 cm level of accurate quasi-geoid model can be obtained for the actual datasets depending on different schemes used to model the topographic mass. In flat areas, the FVM solution can reach to about 1 cm precision, which is comparable with the counterparts from classical methods. The paper also demonstrates how to find the upper boundary if no airborne data are available. Finally, the numerical method provides a 3D discrete representation of the entire local gravity field instead of a surface solution, a (quasi) geoid model.

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 geoid and quasi-geoid serve as the reference surfaces of the orthometric and normal height systems, respectively. In order to improve the accuracy of the (quasi-) geoid determined by the Stokes integral with use of the Remove-Compute-Restore (RCR) technique, various modification methods for the spherical Stokes’ kernels, including the spheroidal, cosine-, power-, and Molodensky-modified kernels, are studied in this paper. In addition to the traditional Molodensky-modified Stokes’ kernel, a more effective Molodensky-modified Stokes’ kernel is put forward. A general formula for spectral decomposition of the Stokes integral in the RCR mode is derived, followed by the spectral analysis to reveal the transfer principles of gravity data when using different Stokes’ kernels. The spheroidal and modified Stokes integrals can cause spectral leakage phenomenon, and a method to eliminate spectral leakage is presented based on spectral analysis. The research indicates the low truncation degree of the spheroidal Stokes’ kernel and the low modification degrees of the modified Stokes’ kernel affect the accuracy of the (quasi-) geoid significantly. Quantitative methods for estimating the empirical values of the parameters of the low-degree spheroidal and modified Stokes’ kernels are proposed and the effectiveness of the methods is validated through numerical tests.

In order to satisfy the Stokes’ solution to the Robin’s problem, the masses above the geoid are removed using an appropriate reduction scheme. This requires heights of the gravity station and at the integration or running points to be known with reasonable accuracy. The height data is used to compute the topographic, atmospheric, and downward continuation effects—essential elements of any geoid model—as well as to estimate free-air gravity anomalies. The height of the gravity station is usually tied to the local vertical network using standard heighting methods, ideally spirit levelling, while a Digital Elevation Model (DEM) is used to derive the heights at the integration points. In most developing countries, the gravity station heights are either not available or are unreliable for geoid modelling applications, since they were mostly observed to cater for the needs of geophysical exploration. By the end of the day, the geodesist has to accommodate the height information in the integral equations for computing topographical effects using the data available in the country. In this study, the different options for height information are investigated using the data available in Auvergne, central France. Geoid models are computed using different height data combinations to represent different scenarios which exist in different countries. Spherical one-dimensional (1D) Fourier transform is used to evaluate the Stokes’ integral in the framework of the Remove-Compute-Restore (RCR) technique. Results show that orthometric heights derived from a high-resolution Global Geopotential Model (GGM) with ellipsoidal heights may be as good as, if not better than, spirit-levelled heights, when used at the gravity station.