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Due to the high cost and time constraints, geoid and gravity surveys utilizing airborne and shipborne surveys can only be implemented in marine areas with limited coverage. Satellite altimeter has become an essential tool for global geoid and gravity field recovery. With approximately 60% observation coverage of the Earth's surface for ocean height, researchers can replace conventional marine geoid models by conducting faster surveys over massive areas at a minimal cost. This study attempts to develop a marine geoid model from multi-mission satellite altimetry along-track data using the Least Squares Modification of Stokes with Additive Corrections. The gravity anomaly has been computed using Gravity Software, and the planar Fast Fourier Transformation method has been applied. The evaluation, selection, blunder detection, combination, and re-gridding of the altimetry-derived gravity anomalies and Global Geopotential Model data are demonstrated. The cross-validation approach has been employed in the cleaning and quality control the data using the Kriging interpolation method. The optimal condition modification parameters of the spherical cap, terrestrial gravity data error, and correlation length have also been applied. Then, the additive corrections based on Downward Continuation, Atmospheric Effects, and Ellipsoidal corrections were combined with the estimated geoid to provide a precise marine geoid over the Malaysian seas. The findings have been evaluated with the MyGeoid17 model from the Department of Survey and Mapping Malaysia. The validation yields satisfactory results with accuracy up to a centimetre level. Hence, the marine geoid model can be utilized for orthometric height determination in marine areas over the Malaysian seas. [ABSTRACT FROM AUTHOR]

Satellite altimetry is a vital tool for global ocean observation, providing critical insights into ocean gravity and its gradient. Over the past six years, satellite data from various space agencies have nearly tripled, facilitating the development of high-precision ocean gravity anomaly and ocean vertical gradient of gravity anomaly (VGGA) models. This study constructs a global ocean VGGA model named SDUST2023VGGA using multi-directional mean sea surface data. To address computational resource limitations, the global ocean is divided into ten sub-regions. In each sub-region, the DTU21 Mean Sea Surface (MSS) model and the CNES-CLS22 Mean Dynamic Topography (MDT) model are used to derive the geoid. To mitigate the influence of long-wavelength signals on the calculations, the study subtracts the long-wavelength geoid derived from the XGM2019e gravity field model from the original geoid, resulting in a residual geoid (short-wavelength). To ensure the accuracy of the VGGA calculations, a weighted least-squares method is employed using residual geoid data from a 17′ × 17′ area surrounding the computation point. This approach effectively accounts for the actual ocean environment, thereby enhancing the precision of the calculation results. After combining the VGGA models for all sub-regions, the model's reliability is validated against the SIO V32.1 VGGA (named curv) model. The comparison between the VGGA and the SIO V32.1 model shows a mean of -0.08 Eötvös (E) and an RMS of 8.50 E, indicating a high degree of consistency across the global scale. Analysis of the differences reveals that the advanced data processing and modeling strategies employed in the DTU21 MSS model enable SDUST2023VGGA to maintain stable performance across varying ocean depths, unaffected by ocean dynamics. The effective use of multi-directional mean sea surface data allows for the detailed capture of ocean gravity field information embedded in the MSS model. Analysis across diverse ocean regions demonstrates that the SDUST2023VGGA model successfully reveals the internal structure and mass distribution of the seafloor. The SDUST2023VGGA dataset is freely available at https://doi.org/10.5281/zenodo.14177000 (Zhou et al., 2024). [ABSTRACT FROM AUTHOR]

This study conducts an evaluation of the performance of seven ultra-high-degree Global Geopotential Models (GGMs) across Egypt, utilizing GNSS/leveling data as a basis for assessment. The models under investigation include SGG-UGM-2, XGM2019e_2159, SGG-UGM-1, GECO, EIGEN-6C4, EGM2008, and GGMplus. The evaluation procedure comprises three principal steps: an absolute assessment, the implementation of Residual Terrain Modelling (RTM), and the fitting of GGMs to GNSS/leveling data. Among the models assessed, GECO exhibited the highest performance in the absolute assessment, achieving a standard deviation (STD) of 0.310 m, while SGG-UGM-1 recorded the largest STD at 0.353 m. Given Egypt’s predominantly flat topography, the application of RTM yielded only modest benefits. Nonetheless, all models demonstrated significant enhancements in accuracy through the fitting process, with EIGEN-6C4 emerging as the most successful model, attaining an STD of 0.116 m in external assessment. The accuracy improvements following the fitting procedure ranged from 49 % to 63 % across all models assessed. [ABSTRACT FROM AUTHOR]

The planform is a defining feature of mantle convection. It can be gleaned from the stratigraphic record by mapping the continent-scale distribution of hiatus and no hiatus surfaces serving as a proxy for high and low dynamic topography. We carry this out for all continents apart from Antarctica for eight geological series since the Upper Jurassic, showing that: (i) the planform as indicated by our maps contains wavelengths of the order of 1000 km, smaller than the convective scales implied by the geoid. (ii) The planform changes on timescales of geological series (10–20 Myrs), smaller than the mantle transit time. (iii) Flood basalt eruptions are frequently preceded by hiatus surfaces. (iv) Some hiatus surfaces are not linked to any known plume, potentially reflecting the lateral transport of material in the asthenosphere. Our results reveal the importance of mantle viscosity stratification in shaping the convective planform and the resulting dynamic topography. Geodynamic Earth models should aim to reproduce the global characteristics of our maps, as well as specific regional events identified in this work. Finally, we separate the effects of sea-level variation from regional changes in base level induced by dynamic topography by contrasting the stratigraphic evolution of different regions. [ABSTRACT FROM AUTHOR]

A large number of fully normalized associated Legendre function (fnALF) calculations are required to compute Earth's gravity field elements using ultra high-order gravity field coefficient models. In the surveying and mapping industry, researchers typically rely on CPU-based systems for these calculations, which leads to limitations in execution speed and power efficiency. Although modern CPUs improve instruction execution efficiency through instruction-level parallelism, the constraints of a shared memory architecture impose further limitations on the execution speed and power efficiency. This results in exponential increases in computation time as demand rises alongside high power consumption. In this article, we present a new computational implementation of an fnALF based on the ZYNQ platform. We design a task-parallel "pipeline" architecture which converts the original serial logic into a more efficient hardware implementation, and we utilize a redundant calculation layer to handle repetitive coefficient computations separately. The experimental results demonstrate that our system achieved accurate and rapid calculations. Under the only one geocentric residual latitude condition, we measured the computation times for spherical harmonic coefficient degrees of 360, 720, and 1080 to be 0.155922 s, 0.520950 s, and 1.401609 s, respectively. In the case of the multiple geocentric residual latitudes condition, our design generally yielded efficiency gains of over three times those of MATLAB R2020b implementation. Additionally, our calculated results were used to determine the geoid height in the field with an error of less than ± 0.1 m , confirming the reliability of our computations. [ABSTRACT FROM AUTHOR]

In the current most accepted global geopotential model, EGM2008, there are often data gaps in the source data used to compute the model over inland water. As a result, EGM2008 may be less reliable over lakes. Satellite altimetry has the potential to estimate gravity anomalies and update EGM2008 over lakes. Here, we evaluate the first attempt to extract gravity anomalies from ICESat-2 laser altimetry over several medium (100–1000 km 2 ) and large (>1000 km 2 ) lakes and compare them with conventional radar altimetry from CryoSat-2 and SARAL to investigate the performance of ICESat-2 for gravity determination over lakes. Aerial gravimetry from the GRAV-D project over the United States are utilized as the best estimate of the gravity field over the lakes. Gravity determination from altimetry is done using Fast Fourier Techniques (FFT) within a remove-restore geoid-to-gravity approach. The resulting altimetry derived gravity anomalies are then compared to the EGM2008 geoid over each lake with respect to GRAV-D. 18 lakes with area ranging from 108 km 2 to 82,220 km 2 across the United States were considered. Overall, gravity determination from ICESat-2 provides more reliable estimates than the other two radar altimetry missions. For all considered lakes, the performance of ICESat-2, measured in terms of standard deviation with GRAV-D, is comparable or better than the EGM2008 field over the same lake. Lake Pend Orielle is the best performing case, in which the standard deviation of the ICESat-2 derived gravity field is 2.14 mGal and the standard deviation of the EGM2008 gravity field is 2.66 mGal with respect to the GRAV-D measurements. Over Lake Tahoe, which is surrounded by mountainous terrain, ICESat-2 performs comparably to EGM2008 and captures clear gravity signal related to the lake's bathymetry, whereas CryoSat-2 produces very unstable results. The method presented here for deriving gravity anomalies from altimetry applied to ICESat-2 laser altimetry data produces results that validate in supplement to the GRAV-D project over medium to large lakes in the United States. [ABSTRACT FROM AUTHOR]

Antarctica is renowned for its ancient cratons, difficult‐to‐observe sutures and active continental rifts. Detailed lithospheric structure and strength estimates are crucial for understanding the potential distribution, long‐term geological evolution, and deformation patterns of this continent. The lithospheric structure of the Antarctic continent is investigated based on joint modeling of elevation and geoid data with the incorporation of seismic data and thermal constraints. Moreover, these results are used to infer the yield strength envelopes across Antarctica. The laterally variable lithospheric strength is finally generated by vertically integrating the above envelopes. Our results show that the variations in the depth of the lithosphere‐asthenosphere boundary (LAB) are analogous to the variations in the integrated lithospheric strength; these parameters vary from 60 to 220 km and from 0.5 × 1013 Pa m to 4.5 × 1013 Pa m, respectively. Several regions of East Antarctica exhibit thick and strong lithospheric mantle, suggesting the presence of scattered cratonic blocks. In contrast, a thin and weak lithosphere is observed in the tectonically active West Antarctica and East Antarctic orogenic belts. Generally, our new LAB model correlates well with previous estimates, but there is a significant inconsistency beneath the Gamburtsev Subglacial Mountains, where a shallow LAB (∼120 km), high Moho temperature, and low strength (<1.0 × 1013 Pa m) are estimated. We speculate that this anomaly may be similar to the thinned lithosphere in Dronning Maud Land and reflects the removal of the mantle lithosphere during or after the orogenesis associated with the assembly of the Gondwana supercontinent. Plain Language Summary: Knowing lithospheric structure and strength helps us understand tectonic deformation and evolutionary process in Antarctica. However, most previous models provide limited spatial resolution beneath this region, impacting subsequent geological interpretation. In this paper, we use geophysical data to yield high‐resolution models which provide new insights into the remarkable lateral variations in lithospheric thickness and strength for the Antarctic continent and reveal more structural details. In East Antarctica, we find that a few cratons (old, cold and rigid portion of the continental lithosphere) have high values of lithospheric strength, but a few of them are very weak, suggesting these cratons have been modified by late geological events and have lost their original signature. One of the collisional orogens in the East Antarctic interior exhibits strongly variable lithospheric strength, which probably represents the distinct lithospheric deformation along its strike. These findings advance our understanding of the tectonic evolution of the Antarctic lithosphere. Key Points: We present new maps of lithospheric thickness and strength across the Antarctic continentVariations in lithospheric strength reflect the potential distributions of cratons and collisional orogens in the East Antarctic interiorThin lithosphere of the Gamburtsev Subglacial Mountains, implying possible orogenic‐root delamination, is presented for the first time [ABSTRACT FROM AUTHOR]

Astrogeodetic deflections of the vertical (DoVs) are close indicators of the slope of the geoid. Thus, DoVs observed along horizontal profiles may be integrated to create geoid undulation profiles. In this study, we collected DoV data in the Eastern Swiss Alps using a Swiss Digital Zenith Camera, the COmpact DIgital Astrometric Camera (CODIAC), and two total station-based QDaedalus systems. In the mountainous terrain of the Eastern Swiss Alps, the geoid profile was established at 15 benchmarks over a two-week period in June 2021. The elevation along the profile ranges from 1185 to 1800 m, with benchmark spacing ranging from 0.55 km to 2.10 km. The DoV, gravity, GNSS, and levelling measurements were conducted on these 15 benchmarks. The collected gravity data were primarily used for corrections of the DoV-based geoid profiles, accounting for variations in station height and the geoid-quasigeoid separation. The GNSS/levelling and DoV data were both used to compute geoid heights. These geoid heights are compared with the Swiss Geoid Model 2004 (CHGeo2004) and two global gravity field models (EGM2008 and XGM2019e). Our study demonstrates that absolute geoid heights derived from GNSS/leveling data achieve centimeter-level accuracy, underscoring the precision of this method. Comparisons with CHGeo2004 predictions reveal a strong correlation, closely aligning with both GNSS/leveling and DoV-derived results. Additionally, the differential geoid height analysis highlights localized variations in the geoid surface, further validating the robustness of CHGeo2004 in capturing fine-scale geoid heights. These findings confirm the reliability of both absolute and differential geoid height calculations for precise geoid modeling in complex mountainous terrains. [ABSTRACT FROM AUTHOR]

The geoid serves as a critical reference surface for precise mapping applications, particularly in the context of satellite-based positioning systems like the Global Navigation Satellite System (GNSS). While GNSS offers efficient positioning solutions, it relies on an ellipsoidal surface that lacks physical meaning for vertical reference, highlighting the need for accurate geoid height models. A precise geoid model is essential for converting geodetic heights into orthometric heights, which are crucial for practical applications. This study investigates potential discrepancies among geoid models derived from different methods, focusing on the Stokes-Helmert (SH), remove-compute-restore (RCR), and Kungl Tekniska Högskolan (KTH) methods. The primary differences among these methods lie in their approaches to modifying the Stokes formula and their reduction schemes. Conducted in the central part of Java Island, Indonesia, this study uses terrestrial gravity observations to model the geoid and GNSS/leveling data for validating the geoid models. The RCR method demonstrated the highest accuracy, with an RMS error of 8.4 cm, outperforming the KTH method (9.2 cm) and the SH method (10.7 cm). Discrepancies between SH and RCR models were less pronounced, with differences around 30 cm, compared to over 1 meter between KTH and the other methods. The comparison with the global EGM2008 model showed that the gravimetric geoid models were more accurate, with RMS differences reaching up to 10 cm, primarily due to systematic differences with the EGM2008 model. Statistical analysis using t-tests with 95% confidence intervals indicated that the differences among SH, RCR, and KTH methodologies were not statistically significant. Despite the RCR method's apparent superior performance, these differences did not achieve statistical significance. The study notes the limitations of using a relatively limited terrestrial gravity dataset and emphasizes the need for incorporating additional gravity data, such as recent airborne gravity datasets, to improve geoid model performance. Future research should also aim for denser GNSS/leveling observations with stricter measurement requirements to provide a more robust absolute assessment of gravimetric geoid models. [ABSTRACT FROM AUTHOR]

Satellite altimetry data, with its increasing density and quality, has become the primary source for marine deflection of the vertical (DOV) and gravity anomaly modelling. Limited by orbital inclinations, the precision of the meridian component of the gridded deflection of the vertical (GDOV) calculated by traditional altimetry satellites is significantly better than that of the prime vertical component, and the excessive precision difference between these two components restricts the inversion precision of marine gravity anomaly model. The study of cross-track deflection of the vertical (CTDOV) is enabled by the multibeam synchronous observation mode of the new laser altimetry satellite, Ice, Cloud and Land Elevation Satellite-2 (ICESat-2). Based on the remove-restore method, residual geoid gradients are first calculated in this paper using three approaches: along-track (A-T), cross-track (C-T) and an integration of along-track and cross-track. Vertical deflections are then computed on a 1′ × 1′ grid using the least squares collocation (LSC) method, and the precision is verified against the SIO V32.1_DOV model. An optimized combination is proposed to address the issue of precision differences between the meridian and prime vertical components, and to enhance the precision of DOV inversion. A new DOV combination is formed by combining the meridian component from along-track deflection of the vertical (ATDOV) with the prime vertical component from cross-track deflection of the vertical (CTDOV) based on the remove-restore method. The Philippine Sea (0°–35°N, 120°–150°E) is selected as the test area to verify the feasibility of the optimized combination. The results indicate that the optimized combination of the meridian and prime vertical components achieved test precision of 2.63 and 3.33 μrad, respectively, when compared against the SIO V32.1_DOV model. The precision gap between the components is effectively narrowed by this approach, which maintains the precision of the meridian component and enhances that of the prime vertical component, thereby achieving optimal inversion precision for gravity anomalies. [ABSTRACT FROM AUTHOR]

The development of global geoid models became feasible following the launch of specialised satellite missions. Today, the root mean square deviation of the heights in global models of high degree & order varies from centimetres to decimetres across different countries. In countries where the accuracy of such models is lower, there is potential to enhance their precision by applying specific corrections. This study presents a novel methodology for locally modelling the height errors of high degree & order global geoid models using levelling sub-benchmarks for GNSS stations. The methodology is based on a combination of optimal interpolation methods, filtering, and the concept of data weighting by gravity anomaly differences. The methodology is aimed at creating a hybrid model that aligns with the local characteristics of the geoid (or quasi-geoid) derived from the traditional levelling network. The advantage of this methodology lies in its ability to reduce the residual height errors of the EGM2008 and EIGEN6C4 models to less than 1 cm when using only four control points. Such results exceed the initial values of the systematic height errors of these models by 90–96 %. For the GECO model, the residual errors are around 2 cm, while for the XGM2019e_2159 model, they reach 3 cm. These results indicate that this methodology can be applied to all global models of high degree & order, although its effectiveness may vary depending on the specifics of a particular model. [ABSTRACT FROM AUTHOR]

Utilizing the adopted average topographic density of 2670 kg/m3 in the reduction of gravity anomalies introduces errors attributed to topographic density variations, which consequently affect geoid modeling accuracy. Furthermore, the mean gravity along the plumbline within the topography in the definition of Helmert orthometric heights is computed approximately by applying the Poincaré-Prey gravity reduction where the topographic density variations are disregarded. The Helmert orthometric heights of benchmarks are then affected by errors. These errors could be random or systematic depending on the specific geological setting of the region where the leveling network is physically established and/or the geoid model is determined. An example of systematic errors in orthometric heights can be given for large regions characterized by sediment or volcanic deposits, the density of which is substantially lower than the adopted topographic density used in Helmert's definition of heights. The same applies to geoid modeling errors. In this study, we investigate these errors in the Hong Kong territory, where topographic density is about 20% lower than the density of 2670 kg/m3. We use the digital rock density model to estimate the effect of topographic density variations on the geoid and orthometric heights. Our results show that this effect on the geoid and Helmert orthometric heights reach maxima of about 2.1 and 0.5 cm, respectively. Both results provide clear evidence that rock density models are essential in physical geodesy applications involving gravimetric geoid modeling and orthometric height determination despite some criticism that could be raised regarding the reliability of these density models. However, in regions dominated by sedimentary and igneous rocks, the geological information is essential in these applications because topographic densities are substantially lower than the average density of 2670 kg/m3, thus introducing large systematic errors in geoid and orthometric heights.

To achieve the goal that China and Nepal jointly announce the new height of Mount Qomolangma (MQ), the campaign of Qomolangma Height Survey (QHS) was carried out from 2019 to 2020. A high precision geoid model realizing the common height datum for both sides is necessary for determining the unique height of MQ. However, high altitude and rugged topography make it extremely difficult to conduct terrestrial gravity measurements in this region, the accuracy of geoid model is restricted by terrestrial gravity data gaps. In the campaign of 2020 QHS, the airborne gravity survey was carried out over MQ and its surrounding areas, the airborne gravity data covering an area of 12,700 km2 were successfully collected. For the first time, the high precision observations of terrestrial gravity and BeiDou Navigation Satellite System (BDS) at the peak of MQ were collected. These datasets pave the way for the precise determination of the orthometric height of MQ. According to the definition of the International Height Reference System (IHRS), we developed the IHRS-based gravimetric quasigeoid model by combining the airborne and terrestrial gravity data. Validations against highly accurate GNSS leveling data at 61 benchmarks demonstrate that the accuracy of the quasigeoid model is 3.8 cm, and the addition of airborne gravity data improves the accuracy by 51.3%. Based on the IHRS, the final orthometric height of the snow surface of the peak of MQ is determined to be 8848.86 m.

Accurate geoid modelling in marine areas requires the integration of gravity data from multiple sources including shipborne gravity measurements, global geopotential models, and satellite altimetry-derived gravity data. This study aims to develop homogenized gravity data for the coastal region of Nigeria to improve geoid modelling accuracy. Residual linear drifts in the shipborne gravity dataset from the Bureau Gravimétrique International (BGI) were corrected using crossover adjustments for each survey leg. We eliminated gross errors for each survey leg by using the 2-sigma method. Outliers in the historical shipborne gravity data were identified and removed using the leave-one-out cross-validation technique, resulting in a refined shipborne gravity dataset. The refined shipborne data were compared with the gravity data predicted by DTU21GRA, SSv29.1, SGG-UGM-2, XGM2019e_2159, GECO, EIGEN-6C4, and EGM2008. Our findings show that DTU21GRA outperformed the other models in the same region when compared with shipborne gravity data. The refined shipborne gravity data were merged with the DTU21GRA data using Least-Squares Collocation (LSC) to create a combined gravity dataset. The results of comparison between the complete refined shipborne gravity data and DTU21GRA before and after the integration process, shows that both the mean offset and the SD values decreased from 0.43 to −0.02 mGal and 3.14 to 2.69 mGal, respectively, which reveal an improvement in the final combined data. The geoid model constructed using the combined gravity data before and after the integration process showed an improvement in the SD values, decreasing from 0.023 m to 0.016 m when evaluated against the CNES-CLS22 MDT. [ABSTRACT FROM AUTHOR]

The isle of Gavdos, and its wider area, is one of the few places worldwide where the calibration and validation of altimetric satellites has been carried out during the last, more than, two decades using dedicated techniques at sea and on land. The sea-surface calibration employed for the determination of the bias in the satellite altimeter's sea-surface height relies on the use of a gravimetric geoid in collocation with data from tide gauges, permanent global navigation satellite system (GNSS) receivers, as well as meteorological and oceanographic sensors. Hence, a high-accuracy and high-resolution gravimetric geoid model in the vicinity of Gavdos and its surrounding area is of vital importance. The existence of such a geoid model resides in the availability of reliable, in terms of accuracy, and dense, in terms of spatial resolution, gravity data. The isle of Gavdos presents varying topographic characteristics with heights larger than 400 m within small spatial distances of ~7 km. The small size of the island and the significant bathymetric variations in its surrounding marine regions make the determination of the gravity field and the geoid a challenging task. Given the above, the objective of the present work was two-fold. First, to collect new land gravity data over the isle of Gavdos in order to complete the existing database and cover parts of the island where voids existed. Relative gravity campaigns have been designed to cover as homogenously as possible the entire island of Gavdos and especially areas where the topographic gradient is large. The second focus was on the determination of a high-resolution, 1 ′ × 1 ′ , and high-accuracy gravimetric geoid for the wider Gavdos area, which will support activities on the determination of the absolute altimetric bias. The relative gravity campaigns have been designed and carried out employing a CG5 relative gravity meter along with geodetic grade GNSS receivers to determine the geodetic position of the acquired observations. Geoid determination has been based on the newly acquired and historical gravity data, GNSS/Leveling observations, and topography and bathymetry databases for the region. The modeling was based on the well-known remove–compute–restore (RCR) method, employing least-squares collocation (LSC) and fast Fourier transform (FFT) methods for the evaluation of the Stokes' integral. Modeling of the long wavelength contribution has been based on EIGEN6c4 and XGM2019e global geopotential models (GGMs), while for the contribution of the topography, the residual terrain model correction has been employed using both the classical, space domain, and spectral approaches. From the results achieved, the final geoid model accuracy reached the ±1–3 cm level, while in terms of the absolute differences to the GNSS/Leveling data per baseline length, 28.4% of the differences were below the 1 c m S i j   [ k m ] level and 55.2% below the 2 c m S i j   [ k m ] . The latter improved drastically to 52.8% and 81.1%, respectively, after deterministic fit to GNSS/Leveling data, while in terms of the relative differences, the final geoid reaches relative uncertainties of 11.58 ppm (±1.2 cm) for baselines as short as 0–10 km, which improves to 10.63 ppm (±1.1 cm) after the fit. [ABSTRACT FROM AUTHOR]

The objective of this research is to assess temporal variations of orthometric/normal height (ΔH/ΔH*) at proposed International Height Reference Frame (IHRF) sites. GRACE/GRACE-FO-based Global Geopotential Models were utilised to determine temporal variations of geoid/quasigeoid height (ΔN/Δζ) and time-dependent changes of ellipsoidal height (Δh) induced from the temporal mass loading at those sites. ΔH/ΔH* at each site were determined by combining ΔN/Δζ and Δh. The results reveal that ΔH/ΔH* at proposed IHRF sites cannot be considered negligible as peak-to-peak variation of ΔH/ΔH* can reach 81 mm, and secular variations of ΔH/ΔH* range from approximately –2 mm/year to 9 mm/year. [ABSTRACT FROM AUTHOR]

One of the critical factors influencing the accuracy of a local geoid model is the quality of the digital elevation model (DEM). A properly selected high-resolution DEM can significantly mitigate errors in geoid modeling, gravity anomaly processing, and topography and downward continuation correction. Purpose. Evaluating the accuracy of five global DEMs obtained from open sources to identify the most suitable model for creating a local geoid. Methodology. The vertical accuracy of the DEMs was assessed by comparing the heights between the DEM and control points across different types of terrain. The reference values are based on 344 ground benchmarks, where GNSS observations were performed with subsequent adjustment of coordinates and heights. The accuracy analysis involved calculating statistical indicators of the height differences between the GNSS data and the DEM data. Findings. The standard deviation assessment showed favorable values for the COPERNICUS and ALOS DEMs, followed by SRTM, ASTER, and ETOPO. In the mean absolute error calculations for mountainous areas, the ALOS model performed best, followed by COPERNICUS, SRTM, ASTER, and ETOPO. For other types of terrain, COPERNICUS demonstrated the best results in mean absolute error. Originality. This study distinguishes itself through the incorporation of advanced high-resolution DEMs, such as GLO30, providing a modern and thorough evaluation of DEM accuracy specifically for Kazakhstan. What is new is a detailed analysis of the impact of terrain features (plain, hilly, mountainous) on modeling accuracy. This approach advances beyond previous assessments, delivering new and significant insights into the performance of contemporary DEMs. Practice value. The practical value of the results obtained consists in issuing recommendations regarding the possibility of using the studied DEM for the regions of Kazakhstan which differ among themselves in terms of landscape characteristics. The findings indicate that COPERNICUS and ALOS DEMs are highly suitable for precise geoid modeling in southern Kazakhstan. These models can significantly improve the accuracy of local geoid models, benefiting applications in geospatial science and engineering. [ABSTRACT FROM AUTHOR]

A levelling network was readjusted and a new geoid model compiled within the framework of geodetic vertical datum modernization at the Hong Kong territories. To accomplish all project objectives, the quasigeoid model has to be determined too. A quasigeoid model can be obtained from existing geoid model by applying the geoid-to-quasigeoid separation. The geoid-to-quasigeoid separation was traditionally computed as a function of the simple planar Bouguer gravity anomaly, while disregarding terrain geometry, topographic density variations, and vertical gravity changes due to mass density heterogeneities below the geoid surface. We applied this approximate method because orthometric heights of levelling benchmarks in Hong Kong were determined only approximately according to Helmert's theory of orthometric heights. Considering a further improvement of the accuracy of orthometric heights by applying advanced numerical procedures, we determined the geoid-to-quasigeoid separation by applying an accurate method. The comparison of the accurately and approximately computed values of the geoid-to-quasigeoid separation revealed significant differences between them. The approximate values are all negative and reach -2.8 cm, whereas values from the accurate method vary between -4.1 and + 0.2 cm. In addition, we assessed the effect of anomalous topographic density on the geoid-to-quasigeoid separation by employing a newly developed digital rock density model. According to our estimates the effect of anomalous topographic density reaches a maximum value of 1.6 cm, reflecting a predominant presence of light volcanic rocks and sedimentary deposits at the Hong Kong territories. Our numerical findings indicate that the conversion between geoid and quasigeoid models should be done accurately, even in regions with a moderately elevated topography. [ABSTRACT FROM AUTHOR]

This research investigates precision enhancement in regional quasi-geoid refinement through ADS80 tri-linear array scanning stereoscopic imagery for aerial triangulation coupled with GNSS gravity-potential modeling. By acquiring stereoscopic imagery and analyzing triangulation accuracy using an ADS80 camera, we performed this study over the Qinghai–Tibet Plateau's elevated, desolate terrain, collecting 593 GNSS points following high-precision stereoscopic imagery modeling. By utilizing 12 gravity satellite models, we computed geoid heights and China's 1985 Yellow Sea elevations for 28 benchmarks and GNSS points, thereby refining the Qinghai Province Quasi-Geoid Model (QPQM) using geometric techniques. The findings reveal that POS-assisted ADS80 stereoscopic imagery yields high-precision triangulation with maximal horizontal and elevation accuracies of 0.083/0.116 cm and 0.053/0.09 cm, respectively, across five control point arrangements. The RMSE of normal heights for 1985, processed via these GNSS points, achieved decimeter precision. By applying error corrections from benchmarks to the 1985 elevation data from gravity satellites and performing weighted averaging, the precision of EGM2008, SGG-UGM-2, and SGG-UGM-1 models improved to 8.61 cm, 9.09 cm, and 9.38 cm, respectively, surpassing the QPQM by 9.22 cm to 9.99 cm. This research demonstrates that the proposed methods can significantly enhance the precision of regional quasi-geoid surfaces. Additionally, these methods offer a novel approach for rapidly establishing regional quasi-geoid models in the uninhabited areas of the Qinghai–Tibet Plateau. [ABSTRACT FROM AUTHOR]

We analyze the feasibility of so-called radial surface currents as derived from Sentinel-1 Wave mode Doppler shift observations to support the geodetic estimation of the mean dynamic topography (MDT). Broadly defined as subtracting the geoid from the mean sea surface, this separation problem suffers from data inhomogeneities and usually requires strong prior assumptions about the MDT's smoothness. Recent advancements in calibration and current retrieval methodology for Sentinel-1 data give reason to assess potential gains from this observation type, which can be linked to the gradient of the MDT under geostrophic approximation. Due to current lack of long time-series, we synthesize 10 years of observations from daily surface currents grids and include these in a parametric least-squares adjustment of the MDT. Our results utilizing the synthetic data in place of smoothness constraints are promising, showing 2 cm RMS agreement with a state-of-the-art MDT model. [ABSTRACT FROM AUTHOR]

With the progress in Global Navigation Satellite Systems (GNSS) technology, accurate geoid modelling has started to play an essential role in geodetic applications such as establishing height datum as a continuous surface model and related vertical control for infrastructure projects. Thus, numerous geoid modelling methods have been offered since 1990's, each of them has its own algorithm and approximation theories. Classical Stokes-Helmert is one of the most well-known methods all over the world by geodetic communities. However, a user-friendly software package of the method is not publicly accessible on the Internet. Therefore, a compact and user-friendly software package "CSHSOFT" is developed and presented for scholars in this field. A fractionated programming strategy has been treated to build individual components striving high accuracy and computational efficiency for geoid heights. Subsequently, the CSHSOFT is simply tested to construct a geoid model in the mountainous area in Auvergne test-bed where several geoid modelling techniques are implemented. Afterward, the new geoid model of the region is externally evaluated by GNSS-levelling data in terms of rigorous orthometric heights. The fitting statistics of 2.75 cm and 0.36 ppm in absolute and relative height differences fairly indicate that the CSHSOFT is a vigorous tool for gravimetric geoid modelling, and can be comfortably employed for geoscientific and technical studies. [ABSTRACT FROM AUTHOR]

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To achieve the goal that China and Nepal jointly announce the new height of Mount Qomolangma (MQ), the campaign of Qomolangma Height Survey (QHS) was carried out from 2019 to 2020. A high precision geoid model realizing the common height datum for both sides is necessary for determining the unique height of MQ. However, high altitude and rugged topography make it extremely difficult to conduct terrestrial gravity measurements in this region, the accuracy of geoid model is restricted by terrestrial gravity data gaps. In the campaign of 2020 QHS, the airborne gravity survey was carried out over MQ and its surrounding areas, the airborne gravity data covering an area of 12,700 km2 were successfully collected. For the first time, the high precision observations of terrestrial gravity and BeiDou Navigation Satellite System (BDS) at the peak of MQ were collected. These datasets pave the way for the precise determination of the orthometric height of MQ. According to the definition of the International Height Reference System (IHRS), we developed the IHRS-based gravimetric quasigeoid model by combining the airborne and terrestrial gravity data. Validations against highly accurate GNSS leveling data at 61 benchmarks demonstrate that the accuracy of the quasigeoid model is 3.8 cm, and the addition of airborne gravity data improves the accuracy by 51.3%. Based on the IHRS, the final orthometric height of the snow surface of the peak of MQ is determined to be 8848.86 m. [ABSTRACT FROM AUTHOR]

A precise gravimetric geoid model is computed utilising Stokes’s formula, supposing an absence of topography above the geoid. Subsequently, the geoid model undergoes a simple correction for topographic masses, the constant density is taken as 2670 kg/m3. Notably, the true density of topographical mass deviates by approximately ±20% from this constant value. Recently, the University of New Brunswick in Canada released a global topographical density model at a 30 arc-second resolution. This paper investigates the impact of incorporating this model on the precision of the gravimetric geoid within a mountainous region in the Colorado test region. Numerical findings reveal that variations in geoid undulation attributable to this model can extend to a few decimetres, a discrepancy that cannot be neglected in geoid modelling with one-centimetre precision. It is therefore recommended that the considerable impact of topographic density fluctuations on geoid determination be taken into account, particularly in mountainous regions. [ABSTRACT FROM AUTHOR]

Indonesia released a new regional geoid model in 2020—the Indonesian Geoid 2020 (INAGEOID2020). It covers the Indonesian region with a spatial resolution of 0.01 × 0.01 degree with the unit in meters. The model was generated through a series of data and computations. Three components of gravity data, i.e., the observed free-air anomaly, the long-wave from the global geoid model, and the short-wave from the terrain model, were employed. The computation was performed using the Remove-Compute-Restore technique with the Fast Fourier Transformation approach. The output was then fitted to the geoid at tide stations by adding a fitting plane to the geoid model. The fitting plane was constructed based on the difference between the geoid model and each tide gauge benchmark. The final geoid model was evaluated by comparing the model with the reference data. Based on quality metrics, the accuracy of INAGEOID2020 varied between 6 cm to 29 cm. Any interested user can use this gridded geoid model to convert geodetic to orthometric heights and vice versa. [ABSTRACT FROM AUTHOR]

In this study, a tailored gravity-field model is developed to fit and recover local terrestrial gravity data by integrating gravity from global gravity-field models, residual gravity derived from topographic data and observed terrestrial gravity over two study sites in Africa (Cameroon and South Africa). During the modelling phase, two-thirds of the terrestrial gravity data is utilised, reserving the remaining one-third for validation purposes. Additionally, an independent validation is conducted by comparing computed quasigeoid models (derived from tailored gravity data) with height anomalies from GPS/levelling data over the two study sites. The accuracy of the tailored gravity model in reproducing observed gravity data is noteworthy, with a ±8.9 mGal accuracy for the study site in South Africa at 2867 test points and a ±10.4 mGal accuracy for the study site in Cameroon at 637 test points. Comparing height anomalies from GPS/levelling with the SATGQG quasigeoid model (developed from tailored gravity data) and the recent CDSM09A quasigeoid model at 11 GPS/levelling data points reveals comparable accuracies of ±0.10 m and ±0.05 m, for SATGQG and CDSM09A, respectively for the site in South Africa. For the Cameroon site, the differences between height anomalies from GPS/levelling and the CTGQG quasigeoid model (developed from tailored gravity data), along with the recent CGM20 quasigeoid model at 38 GPS/levelling data points, show practically equal accuracies of ±0.15 m for CTGQG and ±0.11 m for CGM20. These findings underscore the potential of tailored gravity-field model in developing accurate quasigeoid models, particularly in regions with limited gravity data coverage. This approach holds promise for gravity recovery and precise geoid modelling in developing countries and regions with insufficient coverage of terrestrial gravity data. [ABSTRACT FROM AUTHOR]

Sri Lanka is still in need of a well-defined local geoid model. This geoidal void has made present-day Global Navigation Satellite System (GNSS) surveys heavily dependent on Global Geopotential Models (GGMs) for height determination. Further, in many instances, the accuracy of GGMs have shown drawbacks in elevation determination over Sri Lanka. Therefore, the study focused on developing a hybrid geoid model (HGM) for Sri Lanka by integrating the available GGMs. Five high-resolution (2190°) GGMs; EGM2008, EIGEN-6C4, GECO, XGM2019e-2159, and SGG-UGM2 were employed to extract GGM-derived geoid undulation for 21 Fundamental Benchmarks (FBMs). The residuals (geoid height deviation) were calculated relative to the observed geoid undulation using GNSS/leveling on the FBMs. The data set was clustered based on topography, and residuals were adjusted using weighted least squares adjustment (LSA). The uneven distribution of the FBMs promotes topography-based clustering. EIGEN-6C4 is found to be the robust GGM for Sri Lanka to develop a hybrid approach, with a 0.001 m RMS value of estimated residuals in LSA. The resulting HGM was interpolated at 1 arc-second grid resolution (30 m × 30 m) using the Inverse Distance Weighted Interpolation. Regression lines were generated for the interpolated HGM with respect to the interpolated observed geoid undulation for 9 transects along the parallel passing through Mount Pedro and for the 16 transects along the meridian. The coefficient of determination on both lines is 0.999. HGM generated by EIGEN-6C4 has shown reliable RMS gradient and intercept values of 8.860078 × 10−9 and 0.0039239, respectively, in first-order polynomial fitting. [ABSTRACT FROM AUTHOR]

Following the Regulation of the Head of the Geospatial Information Agency (BIG) No. 13 of 2021, geoid is used as the Vertical Geospatial Reference System in Indonesia. Applications using the geoid as an ideal reference require a much higher accuracy and resolution than the geoid obtained from models derived solely from satellite data. The Indonesian Geospatial Information Agency considers the geoid ideal if it has reached an accuracy of better than 15 cm. Recent studies have combined satellite and other gravimetric data to produce a combined geoid model with increased resolution. Gravimetric data obtained from measurements close to the Earth's surface, such as airborne and terrestrial gravity data, are particularly attractive because the high-frequency portion of the signal is more apparent and can contribute to the medium to high frequencies of the gravity field. This study models the geoid over Bali Island by combining satellite, airborne and terrestrial gravity data. Calculations were performed using Least Square Collocation (LSC) and Remove-Compute-Restore (RCR) techniques. The gravimetric geoid model was tested against the geometric geoid profile calculated from a GNSS/Levelling survey. The geoid, calculated by combining the GOCO06S satellite gravity model, the GGMplus gravity model and airborne gravity data at an average flight altitude of 4100 m produces a standard deviation of 14.46 cm along the 125 km validation path. After also adding terrestrial gravity data, the standard deviation increased to 16.37 cm. By comparison, the results of the validation of the geoid model from GOCO06S and INAGEOIDV2 with geometric geoids have standard deviation values of 79.56 cm and 16.40 cm, respectively. However, the results of the statistical tests are strongly influenced by the data quality used as validation, in this case, GNSS/Levelling. It is shown that the GNSS/Levelling data over Bali contains significant errors, which have been reduced based on an analysis of geometric vertical deflections. A geometric geoid profile with higher accuracy is required to test the accuracy of the gravimetric geoid models more reliably. [ABSTRACT FROM AUTHOR]

Global geopotential models (GGMs) are mainly used in the remove-compute-restore (RCR) technique applied to gravity field modeling such as geoid determination and height datum unification. The increase in the number and quality of gravity data has led the developers of GGMs to produce models with higher resolution and accuracy. Basically, the long-wavelength coefficients of the gravity field are computed based on satellite data, while the medium- and short-wavelength coefficients are calculated based on terrestrial (land and sea) data. One of the main challenges regarding the evaluation of high-degree GGMs is to compute the associated Legendre functions of the first kind based on the usual recursive formulas. Since most computational softwares use the double-precision format by default, an important question is whether this level of precision is sufficient to numerically evaluate the associated Legendre functions of the first kind? To answer this question, the computation of the associated Legendre functions of the first kind in different degrees and latitudes is studied based on MATLAB software, which uses the double-precision format by default. From the numerical results, we find that the calculation of associated Legendre functions of the first kind up to degree of 2190 (the highest degree of existing GGMs), does not have sufficient accuracy at latitudes between 56°20′ and 78°33′ where the most critical state occurs at the latitude 60°. We also find that the accuracy of the calculation of associated Legendre functions at the latitude 60° (the most critical state) significantly decreases for the degrees higher than 2029. These results imply that the usual computational softwares based on the double-precision format are not suitable for calculating the associated Legendre functions in all degrees and latitudes. This is due to the fact that if we consider the associated Legendre functions of the first kind in the form of a matrix with the dimensions corresponding to the degree and order of the functions, as the degree increases, the numbers on the main diagonal approach to the number 10-308 and thus they are considered zero. In the recursive method, the entries below the main diagonal are calculated from the entries on the main diagonal. Since the entries below the main diagonal become very large as they move away from the main diameter, any error in computing the main diagonal entries leads to a large error in computing the entries below the main diagonal. In this paper, we also study the challenges of using the associated Legendre functions of the first kind in the production of gravity field functionals based on a GGM, utilizing MATLAB software. The results show that the gravity potential computation up to degree of 2190 suffers from very large computational errors at latitudes between 57°32′ and. 60°13′. We observe that the safe degrees for the gravity potential computation in all latitudes are degrees less than 2065. The critical latitudes and degrees for the gravity calculation are somewhat different. The results indicate that the gravity computation up to degree of 2190 leads to very large errors at latitudes between 57°41′ and 60°163′. In addition, the maximum degree of expansion that grants sufficient accuracy for the calculation of gravity for all latitudes is estimated to be 2071. Therefore, since the usual computational software based on the doubleprecision format is not suitable for evaluating the current high-degree GGMs, in this research, a new proposal based on the use of the “long double-precision” format is presented and evaluated. Based on our evaluations, the use of the long double-precision format throughout the computational procedure provides sufficient accuracy to compute the gravity field functionals based on the current high-degree GGMs. [ABSTRACT FROM AUTHOR]

Orthometric heights are vital for fields such as topographic mapping, construction, engineering, and geophysical studies. This study examines the performance of five Global Geopotential Models (GGMs) and Global Navigation Satellite System (GNSS) data in deriving orthometric heights to determine the optimum model. Data from 21 ground control points were collected using South GNSS receivers, processed for ellipsoidal height, and compared with geoid undulations from five GGMs (EGM 2008, EIGEN-6C4, GECO, SGG-UGM-2, and XGM 2019e) to determine the orthometric heights. The derived heights were validated against existing orthometric data. The geoid models showed a high correlation (0.999), with EGM 2008, GECO, SGG-UGM-2, and XGM 2019e displaying similar standard deviations (0.06-0.07 m). EIGEN-6C4, however, was an outlier, with a higher standard deviation (SD = 4.8750 m) and lower minimum value (Min = 2.467m). Root Mean Square Errors (RMSEs) ranged between 10.92 m and 11.27 m for all models. Statistical tests, including t-tests and ANOVA, showed no significant difference between most models at a 95% confidence level, though GECO and XGM 2019e had a small, statistically significant difference. XGM 2019e was identified as the most precise model, showing a near-perfect correlation with other models with the least standard deviation.

Accurate leveling is critical to every hydrogeological investigation. Areas of study in need of data point leveling include flooding, design of drainage infrastructure, mapping of drainage basins, solute/contaminant transport, groundwater recharge and discharge zones. The objective of this paper is therefore to employ the use of geoid method in the correction of Global Positioning System (GPS) derived elevation values to delineate the natural topography vis-à-vis the mean sea level at B-dere and Ejama-Ebubu in Rivers State Nigeria. The data revealed that B-dere study site located about 19 km southeast of Ejama-Ebubu was found to be below sea level as all five sampling points returned elevation values varying between -6.4 m-msl and -3.5 m-msl. Four sampling points located northwest of Ejama-Ebubu study site out of the thirteen positions occupied also yielded elevation values differing between - 4.2 m-msl and -1.1 m-msl. These areas are topographic troughs and subject to flooding - a realization impossible to reach using the GPS elevation reading directly since results are always positive. [ABSTRACT FROM AUTHOR]

Geodesy can provide valuable information on marine current estimation based on the combination of gravity and altimetry. Gravity is standardly used to estimate the geoid undulation, i.e. the height of the geoid over a given reference ellipsoid. As it is well known, the geoid undulation over the oceans is closely related to the Mean Sea Surface (MSS) with discrepancies that can reach 1–2 m at global scale. By satellite altimetry, one can get the MSS and then estimate the Mean Dynamic Topography (MDT) as the difference between the MSS and the geoid undulation. As the MDT is related to the ocean circulation, information on the ocean circulation to be compared with oceanographic estimates can be provided using these geodetic measurements. In this context, the GeoMed2 project aims at estimating a high-accuracy and high-resolution geoid model for the Mediterranean Sea based on land and marine gravity data and on recent Global Geopotential Models. In this paper, the processing methodology based on the well-known remove–compute–restore approach for the determination of the geoid in the Mediterranean area is presented. In a pre-processing step, all available gravity observations for the wider Mediterranean basin have been collected, validated, homogenized, and unified in terms of their horizontal and gravity system. In this way, a reliable gravity database to be used for the determination of the geoid has been prepared. This data set has been used in computing a gravimetric geoid estimate based on which the MDT over the Mediterranean Sea was obtained. The results of this computation were then revised, commented and compared with other existing MDT solutions. By these comparisons, it can be concluded that the geodetic computed MDT is not yet satisfactory since it is too noisy. This is possibly due to some inconsistencies still present in the gravity data used for estimating the geoid undulation and to the adopted MSS which seems to be too smooth over the Mediterranean area. [ABSTRACT FROM AUTHOR]

Land and sea vertical datums have traditionally been separate reference levels, due to the different methodologies and observations used in their derivations. However, the need for a seamless connection of these datums has become important due to a wide range of applications in the coastal zone. This study evaluates existing methods, oceanographic and geodetic models and observations for the development of an operational model of coastal heights called AUSHYDROID. We use a complex study area in north west Australia to gain a new insight into how accurately the separation between the lowest astronomical tide (LAT) and a reference ellipsoid can be estimated from existing models, which could form the basis of a national AUSHYDROID model for Australia. Our results are the first attempt to use this combination of existing data in this location, suggesting that ellipsoidal heights of LAT can be estimated to an accuracy of ∼0.2 m at the coast, with the combination of DTU18MSS and FES2014b having the lowest RMS of 0.125 m. However, in some complex coastline areas such as bays and estuaries, the differences increase to >0.5 m so that additional tide gauge observations with GNSS levelling connections and improved models are needed in these regions. [ABSTRACT FROM AUTHOR]

The wide-swath altimeter satellite Surface Water and Ocean Topography (SWOT) will provide high spatiotemporal resolution sea surface heights (SSHs), which is crucial for studying the impact of observation errors on marine gravity recovery. This study uses simulated SWOT data to derive deflection of the vertical (DOV) and gravity anomalies in the northern South China Sea. We quantified the impact of SWOT errors on DOV and gravity anomalies, and analysed the contributions from different directions of geoid gradient. The results show that the geoid gradient in the cross-track direction significantly improves gravity field recovery by enhancing the precision of east component of DOV. For one-cycle SWOT observations, phase errors emerge as the most impactful error affecting both DOV and gravity anomalies, followed by random errors. 2-D Gaussian filtering and the tilt correction proposed in this study could effectively mitigate their impact. Using the corrected data for DOV computation, the precision in the east and north components improves by 75.32 and 46.80 per cent, respectively, while enhancing the accuracy of the gravity field by 70.23 per cent. For 17-cycle data, phase errors and random errors remain the predominant factors affecting DOV and gravity anomalies, but their impact diminishes with an increase in SWOT observations. Our results indicate that marine gravity accuracy improves by approximately 70 per cent compared to a single cycle. Whether for single-cycle or multicycle data, the impact of phase errors is roughly twice that of random errors. These data processing strategies can serve as valuable references for wide-swath altimeter data processing, aiming to advance the precision and resolution of marine gravity field recovery. [ABSTRACT FROM AUTHOR]

We investigate the geodynamic and melting history of Mars using 3D spherical shell models of mantle convection, constrained by the recent InSight mission results. The Martian mantle must have produced sufficient melt to emplace the Tharsis rise by the end of the Noachian–requiring on the order of 1–3 × 109 km3 of melt after accounting for limited (∼10%) melt extraction. Thereafter, melting declined; however, abundant evidence for limited geologically recent volcanism necessitates some present‐day melt even in the cool mantle inferred from InSight data. We test models with two mantle activation energies and a range of crustal Heat Producing Element (HPE) enrichment factors and initial core‐mantle boundary temperatures. We also test the effect of including a hemispheric (spherical harmonic degree‐1) step in lithospheric thickness to model the Martian dichotomy. We find that a higher activation energy (350 kJ mol−1) rheology produces present‐day geotherms consistent with InSight results, and crustal HPE enrichment factors of 5–10‐times produce localized melting near or up to present‐day. The 10‐times crustal HPE enrichment is consistent with both InSight and geochemical results and also produces present‐day geoid power spectra consistent with Mars. However, calculations that match the present‐day geoid power spectra require more than 60% melt extraction to produce the Tharsis swell. The addition of a degree‐1 hemispheric dichotomy, as an equatorial step in lithospheric thickness, does not significantly improve upon melt production or the geoid. Plain Language Summary: Mars' mantle needed to produce an extremely high volume of melt by ∼3.7 billion years ago in order to build the immense volcanic plateau of Tharsis. There is also abundant evidence for small volumes of geologically recent volcanism, yet the InSight mission results are consistent with a relatively cool present‐day mantle. We use 3D numerical models of the Martian mantle to determine what properties can produce a melting history and present interior temperatures consistent with InSight results and Mars' volcanic history. We test sets of models with two different sensitivities of the mantle viscosity to changes in temperature (i.e., activation energy), and a range of enrichment factors in Heat Producing Elements in the crust. We also test the effect of including a simplified version of the Martian hemispheric dichotomy. Our models with the higher activation energy and 10‐times crustal enrichment in HPEs (consistent with Mars' crustal composition) produce melt near the present‐day as well as temperature profiles consistent with InSight. However, it is very difficult to produce sufficient melt to form Tharsis in such a cool mantle, without assuming most of the melt produced in the mantle reaches the surface. Addition the simplified dichotomy does not significantly improve our results. Key Points: Our results favor a higher activation energy mantle rheology and 10‐times crustal heat producing element enrichment factorEven with the cool mantle consistent with InSight results, we obtain models having melt up to present‐dayProducing sufficient melt to form Tharsis requires a degree of melt extraction 6‐times as efficient as Earth's mid‐ocean ridges [ABSTRACT FROM AUTHOR]

One of the most challenging environments for accurate geoid models is in high, rugged mountain areas. Orthometric heights derived from GNSS and a geoid model can easily have errors at the decimeter level. To investigate the effect of geoid model variability on the elevations of peaks in high, rugged mountain areas, this paper is focused on the “Fourteeners” of Colorado, USA (a group of about 60 peaks that are above 14,000 feet = 4267.2 m). Airborne LiDAR data are used to determine geometric (ellipsoidal) heights, which first requires removing a hybrid geoid model, as the LiDAR data is originally provided as orthometric heights. We quantify a significant improvement when using these derived ellipsoidal heights compared with the original orthometric heights: from ± 0.074 to ± 0.054 m (RMSE), an improvement of 28%. Next, a mean geoid model is determined with a relative accuracy of ± 0.06 to 0.08 m and used as a “stand in” realization of the future, official geopotential datum of the USA, NAPGD2022. Using the LiDAR ellipsoidal heights and geoid model, elevations (and uncertainties) for each of the Fourteener summits are determined and found to be, on average, 1.6 m lower than currently published values. This is a much larger change than the 0.5 m decrease expected from the new datum shift alone. The bulk of the difference is due to the original treatments of the vertical angle, triangulation data. A reanalysis of 32 of the 60 peaks shows that the historic data were indeed too high by about 1.0 m or more. Ultimately, no peak falls below the 14,000-foot level nor are any peaks elevated above this level. [ABSTRACT FROM AUTHOR]

Physical heights is one of the most important topics in physical geodesy. Their original concept, introduced in the 19-th century, defined physical heights as lengths of plumblines of the Earth's gravity field between the geoid and points of interest. There are orthometric heights of surface points, that have been traditionally estimated by spirit levelling and measured gravity; however, the knowledge of the density distribution of topographic masses (masses between the geoid and Earth's surface) is required that significantly affects their determinability. This was also the main reason why a new type of physical heights was proposed in the mid of the 20-th century. Normal heights approximate orthometric heights in a sense that the Earth's gravity field is replaced by the normal gravity field, an analytic model based on the theory of an equipotential ellipsoid. This height system has been introduced since that time in different countries in Europe and beyond. Contrary to the classical height system based on orthometric heights, its counterpart based on normal heights may have slightly different definitions. Moreover, normal heights are often defined as heights of points above the quasigeoid. This contribution reviews alternative definitions of normal heights and respective height systems. It is argued that both orthometric and normal heights refer to the geoid. In the case physical heights are estimated by satellite positioning, normal heights must be computed through the height anomaly estimated at each point of interest, whether it is below, at or above the Earth's surface. On the contrary, orthometric heights of all points along the same plumbline, be it below, at or above the Earth's surface, are estimated by introducing one value of the geoid height. Normal heights of surface points can be estimated by spirit levelling easier than orthometric heights as no topographic mass density hypothesis is required; however, one has to keep in mind the gravity field approximation used both for their definition and realization. [ABSTRACT FROM AUTHOR]

The marine geoid can be determined through the utilization of the marine gravity anomaly model. The accuracy of retrieving marine gravity anomaly model with altimetry data has been improved due to the abundance and high quality of available altimetry data. The SDUST2022 GRA marine gravity anomaly model is the global marine gravity anomaly model constructed by integrating multi-source altimeter satellites, including ICEsat-2 laser altimeter satellite for the first time. The model optimizes the deflection of the vertical accuracy imbalance. The Stokes formula is employed in this study to construct the geoid model, while the two-dimensional fast Fourier transform convolution in planar coordinate is utilized to enhance computational efficiency. Additionally, the remove-computer-restore method is applied during the calculation process. Firstly, the accuracy of the algorithm is verified using the 2190-degree XGM2019e_2159 gravity field model and the 2159-degree EGM2008 gravity field model. Subsequently, the XGM2019e_2159 is removed from the SDUST2022 GRA marine gravity anomaly model, and the residual geoid height is computed utilizing the Stokes formula. Then, the geoid height of the corresponding reference field is restored. Finally, a geoid model for China Sea and its adjacent ocean is constructed. The results showed that the geoid of the China Sea and its adjacent ocean gradually increases from west to east, and the longwave and shortwave characteristics of the seafloor can be clearly seen. The reliability of the model established in this paper was verified by the least-squares collocation method, and the geoid model calculated by the two methods had a good consistency. The mean sea surface model was introduced to calculate the mean dynamic topography. According to the mean dynamic topography, the Kuroshio Extension and the mesoscale eddy phenomenon caused by the instability of background current could be clearly seen. Compared with the mean dynamic topography of DTU22MDT, the difference between the two mean dynamic topography was basically within the range of centimeters. This showed that the mean dynamic topography of the China Sea and its adjacent ocean calculated in this paper was reliable, that is, the geoid of the China Sea and its adjacent ocean constructed in this paper was reliable.

Astrogeodetic deflections of the vertical (DoVs) are close indicators of the slope of the geoid. Thus, DoVs observed along horizontal profiles may be integrated to create geoid undulation profiles. In this study, we collected DoV data in the Eastern Swiss Alps using a Swiss Digital Zenith Camera, the COmpact DIgital Astrometric Camera (CODIAC), and two total station-based QDaedalus systems. In the mountainous terrain of the Eastern Swiss Alps, the geoid profile was established at 15 benchmarks over a two-week period in June 2021. The elevation along the profile ranges from 1185 to 1800 m, with benchmark spacing ranging from 0.55 km to 2.10 km. The DoV, gravity, GNSS, and levelling measurements were conducted on these 15 benchmarks. The collected gravity data were primarily used for corrections of the DoV-based geoid profiles, accounting for variations in station height and the geoid-quasigeoid separation. The GNSS/levelling and DoV data were both used to compute geoid heights. These geoid heights are compared with the Swiss Geoid Model 2004 (CHGeo2004) and two global gravity field models (EGM2008 and XGM2019e). Our study demonstrates that absolute geoid heights derived from GNSS/leveling data achieve centimeter-level accuracy, underscoring the precision of this method. Comparisons with CHGeo2004 predictions reveal a strong correlation, closely aligning with both GNSS/leveling and DoV-derived results. Additionally, the differential geoid height analysis highlights localized variations in the geoid surface, further validating the robustness of CHGeo2004 in capturing fine-scale geoid heights. These findings confirm the reliability of both absolute and differential geoid height calculations for precise geoid modeling in complex mountainous terrains.

The determination of the geoid–quasigeoid separation (GQS) is most often based on the use of Bouguer gravity anomalies or disturbances with additional corrections, which allow for the determination of so-called complete or accurate GQS values. This study presents analyses related to an attempt to determine accurate GQS values using the GGI approach (based on the geophysical gravity inversion technique). This approach allows for the modeling of various parameters of the gravity field, and it also enables the determination of the GQS or geoid undulations. Such capabilities of the method have not yet been tested. In this study, the details of the GGI solution in terms of determining the GQS and the first results from tests performed in the area of the Colorado 1 cm geoid computation experiment are presented. The GQS values determined by the GGI approach were compared with the reference values determined previously using the complete classical approach. The differences between the compared values were small, with a standard deviation of 0.007 m, and the maximum differences reached 0.075 m. The analyses also revealed the significant impact of changes in the density of topographic masses on both the geoid undulations and GQS values determined using the GGI approach. [ABSTRACT FROM AUTHOR]

Objectives: The determination of the orthometric height of Mount Qomolangma requires the prior calculation of geoid height, which is determinined by the quasi-geoid (height anomaly) and the height difference between the quasi-geoid and geoid. Methods: First, based on Molodensky principle and remove-restore technology, a 1.5'×1.5' gravity quasi-geoid model in Mount Qomolangma region can be established by using a terrain model, an earth gravity field model, and the fusion data of ground and airborne gravity. Then, system errors can be eliminated by correcting the gravity quasi-geoid model with GNSS-leveling points. The quasi-geoid based on the National Vertical Datum 1985 can be obtained. Finally, the height difference between the quasi-geoid and geoid can be calculated by using the remove-restore method by gravity at the summit of Mount Qomolangma. Results: The results show that the EIGEN-6C4 earth gravity field model performs best in the region of Mount Qomolangma compared to EGM2008 and XGM2019e_2159. The precision of the gravity geoid model is 7.8 cm based solely on the EIGEN-6C4 and ground gravity data, while it is 4.8 cm based on the EIGEN-6C4 and the fusion data of ground and airborne gravity. The accuracy of model is improved by approximately 40% than using ground gravity data only. At the summit of Mount Qomolangma, the height anomaly is -25.586 m, the height difference between the quasi-geoid and geoid is -1.216 m, and the geoid height is (-26.802±0.06) m in the National Height Datum 1985. Conclusions: Airborne gravity can effectively provide data for the region where ground gravity data is sparse or non-existent. By incorporating airborne gravity, the density of gravity data at the summit of Mount Qomolangma has been greatly improved, leading to an increased accuracy of the quasi-geoid model. The availability of richer data, more accurate models, and more advanced calculation methods in 2020 resulted in more reliable results compared to the 2005 measurement. [ABSTRACT FROM AUTHOR]

The Stokes operator is used to transform gravity anomaly into geoid, and the remove-compute-restore (RCR) technique and Kungliga Tekniska Högskolan (KTH) method are commonly employed in calculating the geoid. The KTH method estimates the geoid by (least-squares modification of Stokes formula) LSMSF, while the traditional RCR technique maintains its unbiasedness mathematically. This study modifies the RCR technique by adding a set of adjustment parameters, and proposes a general model of RCR modification to obtain the optimal solution. This model is unbiased when taking the adjustment parameters as zeros, which explains its generalization, and the equivalence between it and LSMSF is illustrated. The additive corrections for the RCR modification are re-derived. The test in Northeast China indicated that the proposed model can reach high accuracy at small integration radius, and refine the quasi-geoid with the best precision comparing with other methods. [ABSTRACT FROM AUTHOR]

The novelty and relevance of scientific solutions lie in the necessity to establish a national reference system, namely the determination of parameters for a regional ellipsoid. The methodology of such scientific research involves the task of defining a regional ellipsoid, which practically reduces to finding corrections to the adopted global ellipsoid GRS80. The regional ellipsoid for the territory of Ukraine should best represent the geoid of this region. In other words, the geoid heights relative to the regional ellipsoid within the territory of Ukraine should be minimized as much as possible. The purpose of the article is to obtain the parameters of the regional ellipsoid for the territory of Ukraine and investigate the effectiveness of such a reference system in solving certain practical and scientific geodetic tasks. Thus, based on the results obtained from the conducted research, the following observations can be made. Determining all five parameters of the regional ellipsoid is strongly influenced by the functional dependence of these parameters on each other. This dependence (correlation) is well demonstrated by the values of mean square errors, which are proportional to the obtained parameters and even exceed them. The highest correlation arises between the correction to the major axis of the ellipsoid and the linear elements of its center displacement within the Earth's body. Taking these observations into account, it can be concluded that the simultaneous determination of all five parameters using the least squares method for the territory of Ukraine does not yield the expected good results. This is evident from the calculations of geoid heights presented in the form of a spheroidal trapezoid describing the territory of Ukraine. In contrast, research on determining only three parameters of ellipsoid displacement at given dimensions provides the opportunity to reasonably fit a regional ellipsoid that best represents the geoid constructed on the territory of Ukraine. The solution to these problems has demonstrated very different results obtained under the same data for the same territory. This leads us to the necessity of conducting additional research to obtain correct solutions for so-called ill-conditioned problems [ABSTRACT FROM AUTHOR]

This article investigates the role of different approaches and interpolation methods in gridding terrestrial gravity anomalies. In this regard, first of all, simple and complete Bouguer anomalies are considered in gravity data gridding. In the comparison results of gridding these two Bouguer anomaly datasets, the effect of the high-frequency contribution of topographic gravitation (by means of the terrain correction) is clarified. After that, the role of the used interpolation algorithm on the resulting grid of mean gravity anomalies and hence on the geoid modeling accuracy is inspected. For this purpose, four different interpolation methods including geostatistical Kriging, nearest neighbor, inverse distance to a power (IDP), and artificial neural networks (ANNs) are applied. Here, the IDP and nearest neighbor methods represent simple-structured algorithms among the interpolation methods tested in this study. The ANN method, on the other hand, is preferred as a complex, optimization-based soft computing method that has been applied in recent years. In addition, the geostatistical Kriging method is one of the conventional methods that is mostly applied for gridding gravity data in geodesy and geophysics. The calculated gravity anomalies in grids are employed in high-resolution geoid model computations using the least squares modifications of Stokes formula with additive corrections (LSMSA) technique. The investigations are carried out using the test datasets of Auvergne, France that are provided by the International Service for the Geoid for scientific research. It is concluded that the interpolation algorithms affect the gravity gridding results and hence the geoid model determination. The ANN method does not provide superior results compared to the conventional algorithms in gravity gridding. The geoid model with 4.1 cm accuracy is computed in the test area. [ABSTRACT FROM AUTHOR]

The technology for determining a point's coordinates on the earth's surface using the global navigation satellite system (GNSS) is becoming the norm along with ground-based methods. In this case, determining coordinates does not cause any particular difficulties. However, to identify normal heights using this technology with a given accuracy, special research is required. The fact is that satellite determinations of geodetic heights (h) over an ellipsoid surface differ from ground-based measurements of normal height (HN) over a quasi-geoid surface by a certain value called quasi-geoid height or height anomaly (ζ). In relation to determining heights of a certain territory, the concept of geoid height (N) is usually operated when dealing with a geoid model. In this work, geodetic and normal heights are determined for five control points in three different regions in Lebanon, where measurements are carried out using GNSS technology and geometric levelling. The obtained quasi-geoid heights are compared with geoid heights derived from the global Earth model EGM2008. The results obtained showed that, in the absence of gravimetric data, the combination of global Earth model data, geometric levelling for selected areas, and satellite determinations allows for the creation of a highly accurate altitude network for mountainous areas. [ABSTRACT FROM AUTHOR]

The National Geospatial Policy (NGP) and guidelines of India mention that the threshold value for vertical or elevation shall be 3 m. However, the terms 'height', 'elevation' or 'vertical' alone are not sufficiently self-explanatory. Therefore, this article provides an overview of India's height systems, vertical datums and vertical reference surfaces. The ellipsoidal heights obtained from GNSS have been discussed briefly, but the main focus remains on the physical height, commonly known as heights above mean sea level. This is because only the latter is used for large infrastructural projects and contouring in topographical maps. The geoid, a geopotential surface by definition, is the best candidate for a vertical reference surface. Some countries also employ quasigeoid, but India has always pursued geoid. Developing a geoid model is also one of the milestones for 2025 in the NGP. Geoid modelling has been studied in India for over a century but has never been discussed in detail. This article comprehensively discusses all the pertinent information on heights and vertical reference surfaces used in the country, which is important for various stakeholders and users of the NGP and guidelines. Some suggestions towards the successful implementation of the NGP in terms of maintaining consistency and avoiding duplication in densifying the national fundamental elevation dataset and a roadmap for developing the consistently precise national geoid model have also been provided for consideration by the national agencies and engineering surveyors. [ABSTRACT FROM AUTHOR]

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Bathymetry data over lake areas are not included in the current and previous NGS (National Geodetic Survey) geoid models. Lake surfaces are simply treated as land surfaces during the modeling regardless of the apparent density difference between water and rock, resulting in artificial masses that distort the model from the actual gravity field and the corresponding geoid surface. In this study, compiled high-resolution bathymetry data provided by National Centers for Environmental Information are used to identify the real volume of water bodies. Under the mass conservation principle, two strategies are deployed to properly account the water body bounded by the mean lake surface and the bathymetry indicated lake floor into the current NGS geoid modeling scheme, where the residual terrain modeling method is used to account for topographic effects. The first strategy condenses water bodies into equivalent rock masses, with the cost of changing the geometrical shape of the water body. The second one keeps the shape of the water body unchanged but replaces the water and rock densities inside each topographical column bounded by the geoid surface and the mean lake surface by an averaged density. Both strategies show up to 1-cm geoid changes when compared with the previous geoid model that does not consider bathymetric information. All three geoid models are evaluated by local GNSS/Leveling benchmarks and multi-year-multi-mission altimetry indicated mean lake surface heights. The results show that both strategies can improve the geoid model precision. And the second strategy yields more realistic results. [ABSTRACT FROM AUTHOR]