Geoid Model Research Articles

Precise Geoid Determination in the Eastern Swiss Alps Using Geodetic Astronomy and GNSS/Leveling Methods

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.

Assessment of digital elevation models accuracy for local geoid modeling

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.

Assessment of geoid models for geopotential values determination in Mexico'S continuous monitoring network

The requirements of the reference systems related to the vertical part are aimed at adopting a global height system that allows taking advantage of the development of satellite technology that provides terrestrial data with higher quality. Due to this need, the International Association of Geodesy (IAG) has proposed the realization and implementation of the International Height Reference System (IHRS), based on geopotential numbers obtained from the gravity potential difference at a point on the Earth's surface (W(P)) referred to the geoid (W0), whose value adopted by convention corresponds to 62636853.4 m2s-2. Nowadays, efforts are underway worldwide to materialize this system. Considering this, the present work entails the analysis of information gathered from SIRGAS continuous monitoring stations in Mexico: MTY2, IDGO, INEG, MERI, and ICAM, to estimate its gravity potential from sufficiently precise geocentric coordinates and absolute gravity values, which when combined with heights derived from geoid models, allows obtaining a modern height system such as the IHRS. The analysis aims to evaluate the feasibility of estimating the gravity potential at the stations, by using precise geocentric coordinates and absolute gravity values which when combined with high-resolution geoid models such as EGM2008, GGM10, EIGEN6C4, XGM2019e_2159, SGGUGM2, and the xGEOID20B, to implement a modern height reference system in Mexico; as well as to contribute to the International Height Reference Framework (IHRF), which in its latest version only includes the MERI station, located in Mérida, Yucatán. Therefore, we consider that for their characteristics the stations analyzed can be included to densify the frame in the implementation of the system. The results show the differences in gravity potential in the different regions of the country, reinforcing the need to consider more reference stations. Additionally, benchmark points close to the selected stations were analyzed to determine the reference potential value of the GGM10, with which it was also possible to analyze the behavior between the observed geoid and the different geoid models, toward the modernization of the official height system, which is based on orthometric heights called NAVD88, which shows us the differences between the geoid observed with the official height system and the analyzed models. Finally, the study examines the consistency of the calculated orthometric heights with geopotential numbers derived from the different geoid models.

Modelling geoid height errors for local areas based on data of global models

Abstract 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.

Drone‐towed electromagnetic and magnetic systems for subsurface characterization and archaeological prospecting

AbstractDrone‐based geophysical surveying is an emerging measuring platform. High time‐cost efficiency and flexibility to survey over inaccessible areas make drones attractive or sometimes the only feasible option to carry geophysical measurements. This study presents a new drone‐towed electromagnetic induction and magnetic gradient sensor system used for near‐surface characterization and areal mapping.The system uses various datasets to enhance processing and interpretation. The system includes; an electromagnetic induction instrument; magnetic sensors; GNSS‐IMU system; photogrammetry; Lidar model data; and geoid model data. Robust data processing and stochastic inversion subsurface characterization for archaeological prospecting with drone‐towed electromagnetic induction and magnetic gradient sensor systems. Robust statistical methods were used to process the data.We conducted the fieldwork at one of the ancient Viking settlements in Denmark. The surveyed area was approximately 100200 m. We then implemented and applied a one‐dimensional laterally constrained non‐linear stochastic inversion to image the subsurface electrical conductivity. The inversion results show a consistent conductive layer at 5–8 m depths, likely associated with the groundwater level. This conductive layer is disrupted under a prominent anomaly within a 2–4 m wide area. Our analysis showed that this conductivity disruption could be a flint mine extending 7 m deep. This anomaly also has a strong signature in magnetic gradient data.

Evaluation and homogenization of a marine gravity database from shipborne and satellite altimetry-derived gravity data over the coastal region of Nigeria

Abstract 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.

Local Gravity and Geoid Improvements around the Gavdos Satellite Altimetry Cal/Val Site

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 1cmSij [km] level and 55.2% below the 2cmSij [km]. 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.

Impact of the UNB topographical density model on precise geoid determination in the high mountainous region

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.

Geoid Undulation Model as Vertical Reference in Indonesia

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.

Evaluating 3-D positioning infrastructure quality and utilization: The potential improvement with multi-GNSS methods

This article evaluates the quality of the national 3-D positioning infrastructure using multi-criteria decision making (MCDM) to simulate the potential application of multi-GNSS method. The MCDM evaluation used coverage and availability of Indonesia Continuous Operating Reference System (INACORS) services, distribution of survey pillars, and accuracy of height determination using the Indonesian Geoid Model (INAGEOID). The term multi-GNSS method refers to the utilization of PPP method as a complement to the conventional differential GNSS method for the production of mapping control points. The results of this evaluation were complemented by a questionnaire analysis on the utilization of positioning infrastructure by respondents from various professional backgrounds. The MCDM evaluation results showed that Java had nearly 100% good or excellent 3-D positioning infrastructure quality. Other regions in Indonesia still had significant areas of average, fair, or even poor quality. The questionnaire results showed that many users have faced some problems in areas with fair or poor infrastructure quality. The application of multi-GNSS method can contribute to reduce up to half of the area with fair and poor positioning infrastructure quality.

Benefit of classical leveling for geoid-based vertical reference frames

Classically, vertical reference frames were realized as national or continent-wide networks of geopotential differences derived from geodetic leveling, i.e., from the combination of spirit leveling and gravimetry. Those networks are affected by systematic errors in leveling, leading to tilts in the order of decimeter to meter in larger networks. Today, there opens the possibility to establish a worldwide unified vertical reference frame based on a conventional (quasi)geoid model. Such a frame would be accessible through GNSS measurements, i.e., physical heights would be derived by the method of GNSS-leveling. The question arises, whether existing geodetic leveling data are abolished completely for the realization of vertical reference frames, are used for validation purposes only, or whether existing or future geodetic leveling data can still be of use for the realization of vertical reference frames. The question is mainly driven by the high quality of leveled potential differences over short distances. In the following we investigate two approaches for the combination of geopotential numbers from GNSS-leveling and potential differences from geodetic leveling. In the first approach, both data sets are combined in a common network adjustment leading to potential values at the benchmarks of the leveling network. In the second approach, potential differences from geodetic leveling are used as observable for regional gravity field modeling. This leads to a grid of geoid heights based on classical observables like gravity anomalies and now also on leveled potential differences. Based on synthetic data and a realistic stochastic model, we show that incorporating leveled potential differences improves the quality of a continent-wide network of GNSS-heights (approach 1) by about 40% and that formal and empirical errors of a regional geoid model (approach 2) are reduced by about 20% at leveling benchmarks. While these numbers strongly depend on the chosen stochastic model, the results show the benefit of using leveled potential differences for the realization of a modern geoid-based reference frame. Independent of the specific numbers of the improvement, an additional benefit is the consistency (within the error bounds of each observation type) of leveling data with vertical coordinates from GNSS and a conventional geoid model. Even though we focus on geodetic leveling, the methods proposed are independent of the specific technique used to observe potential (or equivalently height) differences and can thus be applied also to other techniques like chronometric or hydrodynamic leveling.

A conversion of the geoid to the quasigeoid at the Hong Kong territories

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.

Determination of the Geoid Heights of Awka and Environ from the Satellite Altimetry data using Broadview RADAR Altimeter Toolbox (BRAT) and Sentinel-3 Missions

Satellite altimetry has revolutionized our ability to measure Earth's surface with unprecedented accuracy, offering invaluable insights into various geophysical phenomena. This study presents the determination of geoid heights (Ns) of Awka and Environ utilizing the Broadview RADAR Altimeter Toolbox (BRAT) in conjunction with data from the Sentinel-3 missions. The geoid, a surface of constant gravitational potential representing mean sea level, is a fundamental reference surface for geodetic measurements and understanding Earth's gravity field. The methodology involves processing raw altimetry data acquired by the Sentinel-3 missions using BRAT, followed by precise corrections for various factors affecting the altimeter measurements, such as atmospheric delays, sea state bias, and orbit errors. Subsequently, the derived SSH data are combined with precise geoid models to compute the geoid heights at different spatial resolutions. The results demonstrate the effectiveness of the approach in determining geoid height (N) with high precision and spatial resolution, offering valuable contributions to geodetic research and applications. The utilization of Sentinel-3 data combined with BRAT facilitates robust and accurate geoid determination, which is essential for a wide range of geospatial applications, including oceanography, geophysics, and climate studies. The integration of BRAT and Sentinel-3 missions offers a powerful tool for geodetic research and applications, contributing to our understanding of Earth's dynamic processes and improving the accuracy of geospatial measurements.

Application of Geoid and Mean Dynamic Topography Models as an Alternative to the Mean Sea Level Approach in Determining Vertical Datum (Case Study: Java Island and Surrounding)

Abstract Indonesia’s archipelagic territory provides constraints for determining a seamless vertical datum throughout the region. The main constraint is how to connect the vertical datum between islands separated by sea. To define the physical height system of land areas, it usually refers to the Mean Sea Level (MSL), which has different values in each area, and the ideal MSL information is not yet available throughout regions with a minimum data duration of one cycle of 18.6 years. To overcome this constraint, in this research, another alternative method is used: applying a vertical datum that refers to a physical geoid reference surface, namely Mean Dynamic Topography (MDT) using Altimetry satellite data. The study area is Java Island and its surroundings, using vertical information from Geodetic Control Point stations, which also function as Tidal stations in the Indonesian Geospatial Reference System network. The range of MDT deviation values obtained from the Satellite Altimetry with Geodetic Ground Control data is ± 2 meters, which is a good value for coastal areas facing the open sea. This shows that the application of the geoid and MDT models can be used as an alternative, seamless MSL approach in the Java Island region and its surroundings.

A user-friendly software package for modelling gravimetric geoid by the classical Stokes-Helmert method

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.

The Importance of Ground Control Points In A Photogrammetric Workflow

Abstract This study investigates the optimal distribution and pattern of ground control points (GCPs) in aerial photogrammetric projects. Aerial triangulation (AT), also known as bundle adjustment, is the fundamental step in refining 3D reconstruction models and camera positions, thereby minimizing reprojection errors. The study utilizes data from a national project in Romania, employing high-resolution aerial images acquisition using photogrammetric sensors. The project has rigorous requirements of ground control points (GCP) placement and field measurements using GNSS and geometric leveling techniques. The study employs various scenarios, manipulating the number and distribution of GCPs, to assess their influence on planimetric and altimetric accuracy. Results indicate that the configuration and number of GCPs significantly affect the accuracy of photogrammetric products, such as dense image point clouds, digital surface models, and orthophotos. Moreover, the study underscores the importance of precise GCP determination methods, especially in regions lacking a precise gravimetric geoid model. In scenarios with inadequate GCP coverage the outcomes have inferior quality, emphasizing the critical role of GCPs in ensuring the quality of photogrammetric products. Overall, the research gives a clear view on the best placement patterns of GCPs and their influence on AT process evaluation performed in check points (CHKs).

Improving mean water lake surface elevation estimates using dense lidar measurements from the GEDI satellite mission

Lakes are an essential component of the global water cycle and are considered as sentinels of climate change. An accurate estimate of lake's water levels is necessary for reliable continuous monitoring of their water balance. Due to their huge number worldwide, satellite altimetry has been identified as an efficient tool for measuring their temporal variations of water level. This technique can reach an accuracy of a few centimeters for water level estimates over large lakes. Due to the coarse across-track spatial resolution of the radar altimetry missions, their spatial sampling only covers a small part of the lakes surface. To obtain an accurate estimate of the lake water level, it is necessary to apply a correction to account for the small-scale orthometric height undulations occurring on the lake area and not present in the current global geoid models. In this study, the high-density lidar measurements from GEDI are used to estimate the mean lake surface. From the 1,366,211 GEDI data over 1239 tracks acquired over more than 6000 km2 of the surface of Lake Issy-Kul (Kyrgyzstan, Central Asia) between May 2019 and mid-November 2021, 654,592 shots over 843 tracks were selected after a multi-step quality-check and processed to generate a mean lake surface of Lake Issyk-Kul. Comparison between the GEDI-based mean lake surface and GNSS kinematic surveys corrected from the lake level variations exhibit an RMSE of 0.065 m. The principal objective of performing such mean lake surface is to further use it in the framework of the calibration/validation of the Surface Water and Ocean Topography (SWOT) mission.

The effect of topographic density variations on the geoid and orthometric heights in Hong Kong

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.

Moving mountains: reevaluating the elevations of Colorado mountain summits using modern geodetic techniques

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.

Прогнозування потенційних похибок аномалій висот глобальних моделей геоїда для різних систем припливів

The article presents the method of predicting the errors of height anomalies obtained from high-order global geoid models (EGM_2008, EIGEN_6C4, GECO, XGM2019e2159, SGG_UGM_2) and taking into account different tide systems (free/zero/mean-tide). A correct understanding of how much the choice of the global model height tide system affects the accuracy of calculating the normal heights of the Earth’s physical surface is an important component for obtaining reliable results. The purpose of the work is to predict the potential errors of height anomalies for three main concepts of tidal corrections (tidal systems) used in the creation of global geoid models. Method. The essence of the prediction method is to find a connection between the height anomalies of the global geoid models and the PL-quasi-geoid2021 regional quasi-geoid model. On the basis of the established connection, the prediction of potential errors of height anomalies was made, taking into account the choice of one or another system of tides. The prediction of the errors of height anomalies is implemented on the basis of 200 GNSS stations located on the territory of Ukraine and in the border sector. To find the connection between the heights of the global geoid models and the PL-quasi-geoid2021 regional model, a territory was allocated in the western border sector measuring 48°-52° in latitude and 21°-25° in longitude. For 36 GNSS stations located in the border area, a scale coefficient between global and regional height anomalies was found. Results. In the border sector, the factual errors of height anomalies are estimated at the level of 0.01-0.03 m based on the standard deviation and at the level of the mean square deviation of 0.17–0.18 m, 0.15-0.16 m, and 0.08–0.09 m for the free, zero and mean-tide systems, respectively. The standard deviations of the predicted potential errors for the border sector were 0.02-0.03 m, and the mean square values were 0.17–0.18 m, 0.13 m and 0.06 m for similar tidal systems. The residual errors of height anomalies for the border sector amounted to 0.03 m according to the standard and mean square deviation. Assessment of accuracy prediction of potential errors of height anomalies of models EGM_2008, EIGEN_6C4, GECO, XGM2019e2159, SGG_UGM_2 carried out on the basis of 200 GNSS stations was 0.03 m by standard deviation and 0.15 m (free-tide), 0.13 m (zero-tide) and 0.06 m (mean-tide) by mean square value. Scientific novelty. First implemented was the prediction of potential errors of height anomalies of global geoid models, which was performed by the scale factor for GNSS stations located on the territory of Ukraine and taking into account different tidal systems. Practical significance. The predicted errors of height anomalies make it possible to calculate the normal heights of GNSS stations and points on the Earth’s physical surface with an accuracy of about 3 cm. The obtained results allow the use of height anomalies of global geoid models of high order for more accurate geodetics work.