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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.

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.

Conventionally, the determination of height that has physical meaning for a location or area can be determined geodetically by measuring the height difference relative to a reference point using the spirit-leveling method. This type of height, orthometric height, is essential to explain any physical phenomena, e.g., determining which direction fluid flows, which is used in many scientific and engineering applications. The coverage area is one limitation of conventional leveling, using waterpass precise leveling. The wider the coverage area, the more time and cost-consuming it will be. Another alternative to determine the height at a particular location is using the Global Navigation Satellite System – Leveling (GNSS-Leveling) method, which can be used to resolve the drawbacks of the previous method. GNSS-Leveling method is able to provide an efficient height determination solution if an accurate geoid undulation model and Mean Dynamic Topography (MDT) model are available over the area of interest. MDT is the height deviation between the geoid surface and the mean sea surface (MSS). Information related to geoid undulation and MDT is important to ensure any GNSS-Leveling measurement referring to the local Mean Sea Level (MSL) surface, which is widely needed in various applications. Applying the EGM2008 geoid model and altimetric-derived MDT solution to GNSS-Leveling measurements in the Jakarta area, the average orthometric height difference relative to the reference value is estimated to be about 5 cm. This shows that GNSS-Leveling is a promising solution when combined with geoid and MDT models.

Korean geoid models have been continuously developed for more than 20 years. However, the precision of previous models was approximately 8–15 cm according to evaluations based on newly obtained Global Navigation Satellite System (GNSS)/Leveling data because of irregular distribution and low precision of the gravity and GNSS/Leveling data. Therefore, in 2008, NGII began to obtain new terrestrial gravity and GNSS/Leveling data and collected more than 12,000 points of gravity data and 4492 points of GNSS/Leveling data by the end of 2017. As a result, the newest model, Korean National Geoid 2018 (KNGeoid18), achieved a degree of fit (DOF) of 2.3 cm. The precision was significantly improved compared to previous models including KNGeoid14, but precision in the mountainous areas remained still lower than that of the plain areas. Also, inconsistent differences between the GNSS/Leveling data and KNGeoid18 remained as a problem that should be solved. Through KNGeoid project, the NGII observed a positive effect of supplementing fundamental data, so NGII is obtaining new terrestrial gravity data. Regarding GNSS/Leveling data, the installation of 3D control points was completed in 2019 and the adjustment of GNSS and leveling data is ongoing to allow for the application of verified and adjusted GNSS/Leveling data in the future. Therefore, a new geoid model will be developed until the end of 2023 by applying new terrestrial gravity and GNSS/Leveling data. Overall, it is expected that the precision will be improved to approximately 1–1.5 cm, except in the mountainous areas, owing to new data gathering efforts and the maintenance of the fundamental data.

This study presents a methodology for constructing a quasi-geoid model with millimeter-level accuracy over the Shangyu area in China, following the guidelines of the International Association of Geodesy Joint Working Group 2.2.2, known as “The 1 cm geoid experiment”. Our approach combines two steps to ensure exceptional accuracy. First, we employ Molodensky’s theory to model the gravity field, accounting for non-level surfaces and considering complex terrain effects. Through an exhaustive analysis of these influential factors, we implement a comprehensive suite of applicable formulae within Molodensky’s series solution, enabling a thorough assessment of their impacts on height anomalies within the gravimetric quasi-geoid model. Second, we utilize a hybrid method that involves a multi-surface function using the least-squares method and a robust estimation technique. This approach enables the interpolation of quasi-geoid heights by incorporating ellipsoidal and leveling normal heights, as well as gravimetric quasi-geoid data. Through a numerical example, we demonstrate the efficiency of our solution concept, achieving an accuracy of 0.79 cm compared to independent global navigation satellite system (GNSS)/leveling measurements. By developing this methodology, our study contributes to the advancement of geodesy research and provides a valuable methodology for creating highly precise quasi-geoid models in geodetic applications.

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

The orthometric height has an essential role in a variety of civil engineering projects and it is defined as the length of the curved plumbline from a point (on the earth surface) to its intersection with the geoid surface. Leveling process is considered as the most accurate technique for obtaining these heights. However, regardless of its potentials, it is tedious, costly, and time consuming. Recently many organizations and research centers have developed multi Global Geopotential Models (GGMs) depending on several types of available gravity and height datasets to estimate orthometric heights from GNSS measurements. In this study, we present an evaluation and assessment of the accuracy of five of recent and popular GGM : XGM2016, XGM2019e, EIGEN-6C4, GO_CONS_GCF_2_TIM_R6e, and EGM2008 using actual 145 GNSS/leveling points and 96 terrestrial gravity points. The goal of this research is to find the best fit model along the study area located along the coastal zones of Egypt with distances of about 1,970 km for further determination of geoid modeling at regional scale. The selection of these areas basically was due to their developmental, urban, and economical importance and their continuous need for protection works to fight against the coastal erosion caused by climate change and global warming. The results indicated that for geoid undulation, GO_CONS_GCF_2_TIM_R6e model is the best fit GGM for the estimation of geoid model along Mediterranean Sea coastal line, while XGM2019e_2159 model is the best suitable for coastal line of the Red Sea. And regarding the gravity anomalies, the most reliable GGMs for this study area are XGM2019e_2159 and EIGEN-6C4 for Bouguer and free-air gravity anomaly, respectively.

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.

This special issue (SI) includes papers related to some recent efforts on geoid modeling in the Asia-Pacific region. In total, twelve papers were submitted to this SI, covering geoid models in Australia, mainland China, India, Indonesia, South Korea, Malaysia, Nepal, the Philippines, Taiwan, and Thailand. The methods for geoid modeling are rather diversified, with different considerations in gravity data processing and terrain effects. It is suggested that a mechanism for gravity data sharing should be developed and software packages can be freely distributed to geoid modelers. Observed GNSS/leveling along a route over varying terrains across Taiwan are released for testing geoid modeling methods and for accuracy assessments.

The accurate estimation of the zero-height geopotential level in a local vertical datum (LVD) is critical for linking traditional height reference systems to a global height system. In this paper, we investigate the theoretical and practical challenges involved in determining the offset between the North American vertical datum (NAVD) and the global vertical datum (GVD). Drawing on the classical theory of the vertical system in physical geodesy, we define the vertical datum offset and derive rigorous formulas for its calculation. We examine various factors that affect the determination of the offset, including the global gravitational models (GGMs), geodetic reference system, tide system, tilt error, and omission error. Using terrestrial gravity data and gravity anomalies from multiple GGMs in conjunction with Global Navigation Satellite System (GNSS) and orthometric heights, we estimate the vertical offset between the NAVD and GVD. Our results indicate that the geopotential difference approach and the geodetic boundary value problem (GBVP) approach yield consistent results. When the normal gravity geopotential of the geodetic reference system is selected as the gravity geopotential of the global height datum, the NAVD is approximately 0.04 m higher than the GVD relative to the GRS80 ellipsoid, and 0.97 cm higher than the GVD relative to the WGS84 ellipsoid. When the Gauss–Listing geopotential value is chosen as the gravity geopotential of the global height datum, the NAVD is roughly 1.45 m higher than the GVD relative to the GRS80 ellipsoid, and approximately 0.52 m higher than the GVD relative to the WGS84 ellipsoid.