The H-factor in regional geoid modelling: a case study with different scenarios

New‐generation synthetic aperture radar altimetry, as implemented on CryoSat‐2, observes sea surface heights in coastal areas that were previously not monitored by conventional altimetry. Therefore, CryoSat‐2 is expected to improve the coastal mean dynamic topography (MDT). However, the MDT remains highly reliant on the geoid. Using new regional geoid models as well as CryoSat‐2 data, we determine three geodetic coastal MDT models in Norway and validate them against independent tide‐gauge observations and the operational coastal ocean model NorKyst800. The CryoSat‐2 MDTs agree on the ∼3–5 cm level with both tide‐gauge geodetic and ocean MDTs along the Norwegian coast. In addition, we compute geostrophic surface currents to help identifying errors in the geoid models. We find that even though the regional geoid models are all based on the latest satellite gravity data as provided by GOCE, the resulting circulation patterns differ. We demonstrate that some of these differences are due to erroneous or lack of marine gravity data. This suggests that there is significant MDT signal at spatial scales beyond GOCE, and that the geodetic approach to MDT determination benefits from the additional terrestrial gravity information provided by a regional geoid model. We also find that the border of the geographical mode mask of CryoSat‐2 coincides with the Norwegian Coastal Current, making it challenging to distinguish between artifacts in the CryoSat‐2 observations during mode switch and ocean signal.

In mountainous regions with scarce gravity data, gravimetric geoid determination is a difficult task that needs special attention to obtain reliable results satisfying the demands, e.g., of engineering applications. The present study investigates a procedure for combining a suitable global geopotential model and available terrestrial data in order to obtain a precise regional geoid model for Konya Closed Basin (KCB). The KCB is located in the central part of Turkey, where a very limited amount of terrestrial gravity data is available. Various data sources, such as the Turkish digital elevation model with 3 ″ × 3″ resolution, a recently published satellite-only global geopotential model from the Gravity Recovery and Climate Experiment satellite (GRACE) and the ground gravity observations, are combined in the least-squares sense by the modified Stokes’ formula. The new gravimetric geoid model is compared with Global Positioning System (GPS)/levelling at the control points, resulting in the Root Mean Square Error (RMS) differences of ±6.4 cm and 1.7 ppm in the absolute and relative senses, respectively. This regional geoid model appears to be more accurate than the Earth Gravitational Model 2008, which is the best global model over the target area, with the RMS differences of ±8.6 cm and 1.8 ppm in the absolute and relative senses, respectively. These results show that the accuracy of a regional gravimetric model can be augmented by the combination of a global geopotential model and local terrestrial data in mountainous areas even though the quality and resolution of the primary terrestrial data are not satisfactory to the geoid modelling procedure.

High precision geoid determination is a challenging task at the national scale. Many efforts have been conducted to determine precise geoid, locally or globally. Geoid models have different precision depending on the type of information and the strategy employed when calculating the models. This contribution addresses the challenging problem of combining different regional and global geoid models, possibly combined with the geometric geoid derived from GNSS/leveling observations. The ultimate goal of this combination is to improve the precision of the combined model. We employ fitting an appropriate geometric surface to the geoid heights and estimating its (co)variance components. The proposed functional model uses the least squares 2D bi-cubic spline approximation (LS-BICSA) theory, which approximates the geoid model using a 2D spline surface fitted to an arbitrary set of data points in the region. The spline surface consists of third- order polynomial pieces that are smoothly connected together, imposing some continuity conditions at their boundaries. In addition, the least-squares variance component estimation (LS- VCE) is used to estimate precise weights and correlation among different models. We apply this strategy to the combined adjustment of the high-degree global gravitational model EIGEN-6C4, the regional geoid model IRG2016, and the Iranian geometric geoid derived from GNSS/leveling data. The accuracy of the constructed surface is investigated with five randomly selected subsamples of check points. The optimal combination of the two geoid models along with the GNSS/leveling data shows a reduction of 21 mm (~20%) in the RMSE values of discrepancies at the check points.

Taiwan’s terrain is predominantly mountainous, with approximately 73% of its total land area classified as sloped terrain. The region is characterized by complex geological conditions, short yet steep and fast-flowing rivers, and a high susceptibility to natural disasters. During the summer and autumn seasons, typhoons and heavy rainfall frequently trigger landslides in mountainous areas, posing significant risks to infrastructure and communities. While numerous studies have investigated landslide susceptibility in this region, few have examined the impact of different geoid models on these analyses. In Taiwan, geoid models are periodically updated, and these changes can influence key analytical factors in landslide susceptibility assessments, potentially affecting the outcomes. This study utilizes high-resolution digital elevation models (DEMs) based on various global geoid models, such as EGM96 and EGM2008, as well as regional geoid models like TWGEOID2014, TWGEOID2023, and TWGEOID2024, to assess their influence on landslide susceptibility. This study focuses on the mountainous areas of central Taiwan, which also exhibit the largest differences in geoid models. Logistic regression analysis is performed using IBM SPSS statistics software, incorporating terrain factors such as aspect, slope, curvature, relief, and roughness to evaluate landslide susceptibility. Landslide susceptibility maps and receiver operating characteristic (ROC) curves are generated for each geoid model and compared to assess their differences. The findings of this research aim to improve the precision of disaster prediction and provide valuable insights for disaster prevention efforts, soil and water conservation, and integrated risk management strategies. Additionally, this study highlights the importance of geoid model selection in geospatial analyses and its broader implications for environmental and engineering applications.

Data requirements for the determination of a sub-centimetre geoid

The continuous efforts on establishment and modernization of the geodetic control in Turkey include a number of regional geoid models that have been determined since 1976. The recently released gravimetric Geoid of Turkey, TG03, is used in geodetic applications where GPS-heights need to be converted to the local vertical datum. To reach a regional geoid model with improved accuracy, the selection of the appropriate global geopotential model is of primary importance. This study assesses the performance of a number of recent satellite-only and combined global geopotential models (GGMs) derived from CHAMP and GRACE missions’ data in comparison to the older EGM96 model, which is the underlying reference model for TG03. In this respect, gravity anomalies and geoid heights from the global geopotential models were compared with terrestrial gravity data and low-pass filtered GPS/levelling data, respectively. Also, five new gravimetric geoid models, computed by the Fast Fourier Transform technique using terrestrial gravity data and the geopotential models, were validated at the GPS/levelling benchmarks. The findings were also compared with the validation results of the TG03 model.

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.

We tested EGM2008 on GPS/leveling data from Scandinavia and adjacent areas. EGM2008 performs at the same level as the best regional geoid model, NKG2004. However, the direct evaluation of EGM2008 is difficult in Greenland because no leveling data are available. Nevertheless, we show on 78 GPS-MSS data that EGM2008 also performs at the same level as the best regional geoid model GOCINA04.KeywordsGeopotential modelsEGM2008Leveling

Based on the success of the satellite mission GOCE in providing information on the global gravity field with high quality and spectral resolution, the realization of the 1 cm-geoid is at reach, leading to an increased interest in regional geoid modeling. It is therefore necessary to review theoretical and numerical aspects of regional geoid modeling, including availability of adequate data. In this study, we deal with the latter aspect, specifically the representation error implied by the available gravity data.

Geoid undulation model is typically used for the conversion between GNSS obtained ellipsoidal heights and their orthometric counterparts. In this study, an adaptive approach for optimizing regional geoid model based on global geoid model and local GNSS/leveling measurements is presented. A rigorous sequential least squares approach coupled with an adjustable criterion is used for identifying appropriate observables so that an optimized model that fulfills a predefined quality level can be automatically produced. It has been demonstrated that a city scale geoid undulation model with a quality level around ± 14 can be obtained by applying the proposed approach.

Geoid models have important applications in geosciences as well as engineering, for example, for the conversion from ellipsoidal heights observed by GNSS techniques to orthometric heights. To meet the user’s demands, the International Service for the Geoid (ISG, https://www.isgeoid.polimi.it/) provides access to a repository of local, regional, and continental geoid models through its website. Among hundreds of worldwide models, there are many covering countries in the Asia–Pacific area. The focus of this study is about this region, performing a series of analyses to assess the geoid models stored in the ISG repository through some relative comparisons. In particular, three kinds of analyses are performed with the purpose of: (a) investigating the evolution in time of a geoid series referring to the same country, (b) comparing the information provided by local and regional geoid models on overlapped areas, and (c) assessing the agreement between local and global models. These analyses are firstly performed on sample models, providing a detailed description, and then applied to all Asia–Pacific geoid models currently stored in the ISG repository, providing summary statistics.

The aim of this research is the optimal determination of the regional geoid model of Iran based on radial basis functions (RBFs). In this case, the type and number of RBFs, their horizontal positions, depths, and unknown coefficients must be properly determined. The quality of calculations strongly depends on the correct choice of these unknown parameters. Given the precise geocentric position of any point on the Earth’s surface with the beginning of the global navigation satellite system (GNSS), the surface gravity disturbances were used to calculate the height anomaly according to Molodensky’s theory. The residual surface gravity disturbances derived by subtracting the global gravitational model EIGEN-6C4 up to degree and order 360 were applied to determine the unknown RBF parameters using the stabilized orthogonal matching pursuit (SOMP) algorithm. Based on this iterative sparse approach, non-zero components of unknown RBF parameters having the maximum recoverable energy for the desired signal were found at each iteration. The SOMP algorithm was applied for optimal determination of the proper basis functions since each unknown RBF coefficient is related to a specific basis function. Only the RBFs representing the best solution to the problem were selected at each iteration, then several new RBFs were added at suitable positions to enhance the calculation result. The new RBF-based regional geoid model entitled IRG2016 was calculated by applying the geoid-to-quasigeoid corrections to the height anomaly. The IRG2016 was fitted to 1288 GNSS/levelling control points over Iran, by applying the polynomial corrector surface. Relying on this new strategy, the calculated height reference surface shows an RMS value of approximately 0.23 m for the difference in geoidal height at the independent control points, which is comparable with the last Stokes-based geoid model.

Regional geoid models are based on the combination of satellite-only gravity field information and terrestrial data. Satellite information is conveniently provided in terms of spherical harmonic global potential models. Terrestrial information is mostly provided in terms of point or block mean values of gravity in the region of interest. Combination of the two sources of information in the overlapping spectral band is either based on deterministic or on stochastic considerations. We have tested different schemas for weighting satellite and terrestrial information and compared the results to GNSS-levelling data in Norway. The results provide implications for the quality of terrestrial data in the study area and for regional geoid modeling based on GOCE satellite models in general.

We constructed a regional geoid model and compared it with the EGM2008 global gravity-field model to evaluate the performance of EGM2008 for Korea. By using the Remove–Compute–Restore (RCR) technique, we developed an improved gravimetric geoid model JNUGEOID2010 on a 1 × 1′ grid by combining the EIGEN-GL04C global geopotential model (GGM), 8316 land-gravity data, an altimetry-derived marine gravity model DNSC08, and the Shuttle Radar Topography Mission (SRTM) digital elevation model (DEM). The computations were done using a two-dimensional four-band spherical fast Fourier transform (FFT) with 100% zero padding. In addition, we compared the Korean national gravimetric geoid model KGEOID98 to EGM2008. For the comparison, geometric geoid heights derived from 735 GPS-levelling stations were used. The smallest standard deviation of geoid height differences is ± 0.163 m for EGM2008 to degree and order 2190. Thus, we conclude that the new model EGM2008 is superior to the two regional gravimetric geoid models, JNUGEOID2010 and KGEOID98, for Korea. Key words: Regional geoid model, EGM2008, Global geopotential models, GPS-levelling, geoid height, Korea.

This study investigates the contribution of global geopotential models which are calculated with GOCE satellite mission data to the improvement of gravimetric geoid models in Turkey. In this context, direct (DIR), time-wise (TIM), space-wise (SPW), and GOCO satellite-only model series were considered. The research was carried out in two parts. The first part includes the validation of models in each series at 100 homogeneously distributed GNSS/leveling stations over the country utilizing spectrally enhanced geoid heights to determine the best performing model and its optimal expansion degree. According to obtained statistics, the TIM-R6 model was selected as the best model with an optimal expansion degree of 204. In the second part, the TIM-R6 model up to 204 degree/order was linearly blended with EGM2008 to obtain an improved version up to 360 degree/order of expansion. To clarify the contribution of the linearly blended model to the improvement of the regional geoid model, the gravimetric geoid models were computed adopting TIM-R6 up to 204 degree/order and its improved version up to 360 degree/order as reference models. To further emphasize the contribution of the GOCE mission’s data, the gravimetric geoid computations were repeated relying on EGM2008 up to 204 and 360 degrees of expansions, since EGM2008 does not contain GOCE data. In addition, we computed gravimetric geoids based on another combined model that includes GOCE mission data, the EIGEN-6C4 model. The calculated regional geoids were compared to each other and validated using GNSS/leveling data set. The obtained results revealed a ∼23% improvement in regional geoid model accuracy when the blended GOCE-based geopotential model was used as a reference. In addition, the results of this study presented the significance of GOCE contribution to mapping the gravity field in Turkey. The best accuracy obtained from this study was 7.7 cm for the Turkey geoid.