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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 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 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.

We describe the development of a hybrid geoid model for Peninsular Malaysia, based on two approaches. The first approach is utilising an ordinary method fitting the gravimetric geoid to the geometric undulation derived from GNSS-levelling data; the second approach directly fits the gravimetric geoid to the reference mean sea level derived from the tide measurements of Port Klang tide gauge station. The hybrid geoid model fitted to Port Klang (PMHGG2020_PK) is produced by adding an offset of 0.446 m to the gravimetric geoid, based on the comparison at the tide gauge benchmark. To calculate the gravimetric geoid, a new model for Peninsular Malaysia (PMGG2020) has been developed based on Least-Squares Modification of Stokes’ Formula with Additive correction (LSMSA). Three different sources of gravity data which are terrestrial, airborne, and satellite altimetry-derived gravity anomaly (DTU17) have been combined to construct the geoid model. The height information has been extracted from the newly released global digital elevation model, TanDEM-X DEM. GO_CONS_GCF_2_SPW_R4 model derived from GOCE data provides long-wavelengths gravity field up to maximum degree and order 130. The gravity datasets are gridded by 3D Least-Squares Collocation method. The PMGG2020 model is consistent with the geometric geoid heights from 173 GNSS-levelling measurements, with a standard deviation of ±5.8 cm. Evaluation of the hybrid geoid model constructed from the first approach shows a significant improvement over the two existing hybrid geoid models. The accuracy of ±4.6 cm has been achieved after evaluating by 20 GNSS-levelling points, externally. Hybrid geoid model fitted to Port Klang has also been evaluated via 173 GNSS-levelling points, and the result shows that 71% of the total data exhibit height differences lower than 10 cm. The overall results indicate that the hybrid geoid model developed in this study can be valuable as an alternative to the current modern height system in Peninsular Malaysia for surveying and mapping.

A geoid or quasigeoid model allows the integration of satellite measurements with ground levelling measurements in valid height systems. A precise quasigeoid model has been developed for the city of Krakow. One of the goals of the model construction was to provide a more detailed quasigeoid course than the one offered by the national model PL-geoid2011. Only four measurement points in the area of Kraków were used to build a national quasigeoid model. It can be assumed that due to the small number of points and their uneven distribution over the city area, the quasigeoid can be determined less accurately. It became the reason for developing a local quasigeoid model based on a larger number of evenly distributed points. The quasigeoid model was based on 66 evenly distributed points (from 2.5 km to 5.0 km apart) in the study area. The process of modelling the quasigeoid used height anomalies determined at these points on the basis of normal heights derived through levelling and ellipsoidal heights derived through GNSS surveys. Height anomalies coming from the global geopotential model EGM2008 served as a long-wavelength trend in those derived from surveys. Analyses showed that the developed height anomaly model fits the empirical data at the level of single millimetres – mean absolute difference 0.005 m. The developed local model QuasigeoidKR2019, similar to the national model PL-geoid2011, are models closely related to the reference and height systems in Poland. Such models are used to integrate GNSS and levelling observations. A comparison of the local QuasigeoidKR2019 and national PL-geoid2011 model was made for the reference frame PL-ETRF2000 and height datum PL-KRON86-NH. The comparison of the two models with respect to GNSS/levelling height anomalies shows a triple reduction in the values of individual quartiles and a mean absolute difference for the developed local model. These summary statistics clearly indicate that the accuracy of the local model for the city of Krakow is significantly higher than that of the national one.

This paper presents the results of research and analyses related to the development of a new quasigeoid model fitted to GNSS/levelling data for the area of Poland (PL-geoid2021). The model was determined employing two procedures based on the Geophysical Gravity data Inversion technique (GGI method): procedure A consisted of the determination of the gravimetric quasigeoid model in the first step and its subsequent fitting to GNSS/levelling data in the second step, and procedure B consisted of a one-step determination of the model fitted to GNSS/levelling data. Both models were developed using the global geopotential model SGG-UGM-2 and gravity data covering the area of Poland, and slightly extend beyond Poland's southern and northern borders. The average model was adopted as the final model. It was demonstrated that the accuracy of the gravimetric quasigeoid model had a very low dependence on the reference topographic mass density model used. On the basis of this model, the GNSS/levelling datasets were also assessed and outliers were identified. The estimated accuracy of the gravimetric model, determined based on four GNSS/levelling datasets, was in the range of ± 1.2 to ± 1.7 cm, in terms of the standard deviation of the differences between the measured and model-determined height anomalies. Due to partial lack of gravity data just beyond the Polish border, the edge effect was also analysed. The accuracy of the final quasigeoid model (estimated in the same way as the gravimetric model) ranges from ± 1.0 to ± 1.2 cm. It should be noted, however, that this assessment is not fully independent because three of the four sets of GNSS/levelling points used for it, were also used to build the final model.

We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant.

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.