Geoid Model Research Articles

ГЕОДЕЗІЯ, КАРТОГРАФІЯ І АЕРОФОТОЗНІМАННЯ

A number of factors can significantly affect the accuracy of height determination when applying the GNSS leveling method. In general, it is possible to distinguish those related to the process of GNSS observations and their post-processing, and those related to the selection of the geoid/quasi-geoid height model. This work focuses on aspects of GNSS leveling accuracy when choosing global geoid models. In particular, to better ensure accuracy, it is important to understand the significance of the heights tidal system selection of global geoid models. The purpose of the work is to analyze the influence of different tide systems of global geoid models on the accuracy of height determination by the GNSS leveling method. This paper considers the heights of global geoid models EGM08, EIGEN-6C4, GECO, and XGM2019e_2159 of high degree and order calculated in the tide systems of “tide-free”, “mean-tide”, “zero-tide”. The analysis of the actual accuracy of the geoid heights was carried out on the basis of the standard and root mean square deviations of the heights differences of global geoid models in the corresponding tidal systems in relation to the GNSS leveling data. GNSS leveling data were obtained at 14 high-precision geometric leveling points of accuracy class 1-2, covering the central part of the Lviv region. Similarly, the accuracy of the geoid models was analyzed taking into account the differences of gravity anomalies concerning the high-resolution anomalies of the WGM2012 model. Data presenting differences of height and gravitational anomalies allowed us to correct the height of the models according to the weighted average principle. In addition, corresponding statistics were calculated for them. The conducted analysis shows that for the EGM08 model, the system of “mean-tide” is optimal with an accuracy assessment at the level of σ=2-3 cm and m=4 cm. For the EIGEN-6C4 model, it is best to use the “zero-tide” system which will ensure accuracy up to 4-5 cm. The accuracy of the EGM08 and EIGEN-6C4 models is confirmed by the statistical characteristics analysis results of the gravity anomaly differences. The GECO and XGM2019e_2159 models give ambiguous results within 3-9 cm by both parameters and in all tidal systems. Only after correction of the heights, their accuracy is 2-5 cm. Considering the optimal tidal system, the heights of the EGM08 and EIGEN-6C4 models can provide an accuracy of 1-3 cm after the correction by weighting coefficients.

Improvement of GOCE-Based Global Geopotential Models for Gravimetric Geoid Modeling in Turkey

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.

NW Pacific‐Panthalassa Intra‐Oceanic Subduction During Mesozoic Times From Mantle Convection and Geoid Models

AbstractPacific‐Panthalassa plate tectonics are the most challenging on Earth to reconstruct during the Mesozoic and Cenozoic eras due to extensive subduction, which has resulted in large (>9,000 km length) unconstrained gaps between the Pacific and Laurasia (now NE Asia) back to the Early Jurassic. We build four contrasted NW Pacific‐Panthalassa global plate reconstructions and assimilate their velocity fields into global geodynamic models. We compare our predicted present mantle structure, synthetic geoid and dynamic topography to Earth observations. P‐wave tomographic filtering of predicted mantle structures allows for more explicit comparisons to global tomography. Plate reconstructions that include intra‐oceanic subduction in NW Pacific‐Panthalassa fit better to the observed geoid and residual topography, challenging popular models of Andean‐style subduction along East Asia. Our geodynamic models predict significant SE‐ward lateral slab advections within the NW Pacific basin lower mantle (∼2,500 km from Mesozoic times to present) that would confound “vertical slab sinking”‐style restorations of imaged slabs and past subduction zone locations.

Determination of radii of curvature for high resolution geoid models using the harmonic synthesis algorithm

Different types of radii of curvature were assessed for the geoid based on the GECO geopotential model, up to degree and order 2190. The route values of gravity and the three horizontal gravity gradients were computed based on such geopotential model and the angular velocity of the Earth. The investigation was performed on coarse global grids and finer grids covering the Egyptian territory. Respective latitudinal and longitudinal profiles for the geoidal radii were extracted. Comparisons were held with the radii of curvature on the WGS-84 ellipsoid, and with the geoidal radii derived from other models of lower resolutions. Unlike the ellipsoid, the values of the geoidal radii exhibited a rather irregular behaviour that is far from any geographical symmetry. The principal radii of the geoid do not generally occur along the meridian and prime-vertical directions. Such irregularities were found to be more exaggerated with higher degrees. At all investigated resolution levels, the signs of the principal radii assured the convexity of the geoid surface. This enabled to define, compute and compare the Gaussian mean radii for the geoid. The local residual geoidal radii showed a decaying spectral tendency. Also, the results implied that the utilized algorithm proved to be convergent.

An assessment of the mass line and mass prism models in terrain correction for regional geoid modelling in low mountainous areas

Terrain correction is a very crucial step in gravity reduction and the reliability of the adopted topographic model plays a significant role in the overall geoid computation process especially in mountainous terrain. Given the cumbersome computation involved in the mass prism model, the mass line topographic model is often utilized in practical computation while implementing terrain correction for most gravity reduction schemes. In this study, an assessment is made of the accuracy of the mass line (ML) model viz a viz the mass prism (MP) model in a low-ranged mountainous area like Ado Ekiti township; with a view to determine the suitability of the continued use of ML in such regions. Both models were implemented in the spatial domain using MATLab codes written from the conventional formulae. Results obtained indicate that the minimum and maximum differences in computed Bouguer anomalies using ML and MP models are 0.0196mgals - 0.0610mgals. Consequently, the choice of model did not have significant effect on the computed geoid models as the derived geoid from both models produced the same RMSE of 83cm when compared with GNSS-Leveling geoid at validation points. The study concludes that for topographic ranges less than 300m, either of both topographic models could be used and similar level of accuracy will be obtained in the resulting geoid.

A crossvalidation-based comparison of kriging and IDW in local GNSS/levelling quasigeoid modelling

Abstract This study compares two interpolation methods in the problem of a local GNSS/levelling (quasi) geoid modelling. It uses raw data, no global geopotential model is involved. The methods differ as to the complexity of modelling procedure and theoretical background, they are ordinary kriging/least-squares collocation with constant trend and inverse distance weighting (IDW). The comparison itself was done through leave-one-out and random (Monte Carlo) cross-validation. Ordinary kriging and IDW performance was tested with a local (using limited number of data) and global (using all available data) neighbourhoods using various planar covariance function models in case of kriging and various exponents (power parameter) in case of IDW. For the study area both methods assure an overall accuracy level, measured by mean absolute error, root mean square error and median absolute error, of less than 1 cm. Although the method of IDW is much simpler, a suitably selected parameters (also trend removal) may contribute to differences between methods that are virtually negligible (fraction of a millimetre).

Analysis of high-resolution global gravity field models for the estimation of International Height Reference System (IHRS) coordinates in Argentina

Abstract Following the definition and realization of the International Height Reference System (IHRS), the vertical coordinate of a given point at the Earth’s surface can be obtained from the computation of the geopotential value from a harmonic expansion of a Global Gravity Model of High-Resolution (GGM-HR) or based on the computation of a local or regional pure gravimetric geoid or quasigeoid. Therefore, an evaluation of the accuracy of GGMs-HR and the geoid model available is needed in order to assess its capability to infer IHRS coordinates. In this study, different GGMs-HR are evaluated against 2287 benchmarks in Argentina. Moreover, the most recent geoid model of Argentina is also evaluated. Geoid undulations at these benchmarks are obtained based on ellipsoidal and orthometric heights in the local vertical datum. Results suggest that among the evaluated GGMs-HR, XGM2019e provides the best agreement with the observed geoid heights, but none of them is accurate enough in order to infer vertical coordinates in the IHRS. Similar conclusions are obtained for the local geoid model for Argentina demonstrating the necessity for a more precise geoid model, following the standards and recommendations given for the IHRS.

Comparison of different resolutions of six free online DEMs with GPS elevation data on a new 6th of October City, Egypt

The Digital Elevation Models (DEMs) are used to represent the topographic surface of the earth as it is used in many applications. Nowadays, many free DEMs are available online. This study aims to assess the accuracy of six free DEMs. This research uses the GPS observations to determine the elevation stations in the study area by using the Post-Processing Kinematic (PPK) technique and compares these elevations against the elevations computed at the same stations by using six free online Digital Elevation Models (DEMs): SRTM 1 Arc-Second, ASTER 1 Arc-Second, NASADEM 1 Arc-Second, SRTM 3 Arc-Second, GTOPO30, and ALOS PALSAR 12.5 m to find the best results from the free DEMs. The elevations of all stations of different data sources were calculated using the geoid model EGM96. The results indicate that the DEMs that give the best results in this study are obtained from NASADEM 1 Arc-Second, where (RMSE is 2.99 m, and the SD is 2.97 m), and SRTM 3 Arc-Second, where the (RMSE is 3.70 m, and the SD is 3.53 m).

Hybrid geoid model over peninsular Malaysia (PMHG2020) using two approaches

Hybrid geoid model over peninsular Malaysia (PMHG2020) using two approaches

Towards realistic dynamic topography from coast to offshore by incorporating hydrodynamic and geoid models

One of the major factors preventing utilization of realistic sea-level data is that various sources are referred to inconsistent vertical reference datum (VRD). This study presents a methodology for deriving instantaneous and realistic sea level data from the coastal to offshore areas of the Baltic Sea, by synergizing different sources of sea level data (tide-gauge (TG), hydrodynamic model (HDM) and satellite altimetry (SA)). The key component being the geoid that links the VRD of these different sources.HDMs are known to be a representable source of sea level data, however their VRD is often undisclosed. Therefore, they are often spatially and temporally biased with respect to in-situ data. This study demonstrates that by using geoid-referenced TG data that represents ‘realistic’ sea level it is possible to correct HDM, by deriving and applying a spatial bias correction. Three interpolation methods (linear, thin plate spline regression, and inverse distance weighted) are examined to derive the corrected HDM. The results showed an improvement with respect to TG data, and the annual mean dynamic topography of corrected HDM was improved by a factor of almost 1.5. Examination of the Sentinel-3A satellite along-track data also confirms the corrected HDM data to be more accurate. The methodology applied also identified problematic locations of HDM, SA and geoid data along with unreliable TG. This accuracy and conformity in sea level data are urgently required for a comprehensive understanding of climate change, marine engineering and navigation applications, which can be achieved by adapting the uniquely defined geoid for the vertical reference datum.

Compilation of the new detailed geoid model HKGEOID-2022 for the Hong Kong territories

Number of detailed geoid models have been developed to convert geodetic heights measured by the Global Navigation Satellite Systems (GNSS) to heights in the Hong Kong Principal Datum (HKPD). Although gravity measurements were used to compile these geoid models, heights of leveling benchmarks in HKPD were determined from precise spirit leveling measurements but without involving gravity data. To address these inconsistencies, the orthometric heights of HKPD leveling benchmarks were determined from leveling and gravity data. Moreover, the new geoid model HKGEOID-2022 was computed and fitted with the geometric geoid heights at GNSS-leveling benchmarks derived from newly determined orthometric heights. Numerical procedures used to prepare the HKGEOID-2022 geoid are discussed in this study. A gravimetric geoid was computed by using the KTH method. A systematic bias between the gravimetric and geometric geoid heights at GNSS-leveling benchmarks was modeled and reduced by applying a 7-parameter similarity transformation. The accuracy analysis revealed that the resulting detailed geoid model HKGEOID-2022 fits the geometric geoid heights with a standard deviation of ±2.2 cm. This accuracy is compatible with the estimated uncertainties of GNSS measurements as well as with the expected accuracy of a newly developed geoid model, both at the level of approximately ±1–2 cm.

Geoid model validation and topographic bias

Abstract Recently a number of geoid campaigns were performed to verify different types of geoid and quasigeoid modeling techniques. Typically, GNSS-leveling was employed as an independent method, but in some cases zenith camera astronomic deflection data were also used in astrogeodetic determinations of the geoid and/or quasigeoid. However, due to the uncertainty in the topographic density distribution data (and thereby in orthometric heights), we conclude that neither GNSS-leveling nor astrogeodetic techniques can reliably verify differences between gravimetric geoid models at several centimeter levels in rough mountainous regions. This is because much the same topographic data are used both in the gravimetric geoid models and in their verifications by geometric and/or astrogeodetic geoid models. On the contrary, this is not a problem in verifying gravimetric quasigeoid models, as they are independent of the topographic density distribution, and so is the related normal height used in GNSS-leveling.

Computation and Analysis of Geopotential Number in São Paulo, Brazil

In recent decades, important steps have been taken to implement the physical concepts of Geodesy in practice, con- cerning height systems. Despite the difficulties involving gravity field modeling, with the establishment of conventions, standards, and computation strategies, the realization of the International Height Reference System (IHRS) is well underway. For a global system, there are constraints for some countries, especially for those with sparse gravity data, mountain regions, and vast areas. In terms of methodology, the computation can be performed directly using the Global Geopotential Models (GGM), recovering existing geoid models, or determining pointwise the gravity potential using integral formulas. In general, the regional gravity modeling is given by numerical integration or least-squares collocation and more recently adopting the spherical radial basis functions. The first approach allows determining the earth’s gravity component at a specific point and adjusting the integral formula according to the gravity coverage. Since so far there is no common sense about the best methodology, computation strategies are been analyzed. In this con- text, the paper aims to contribute to IHRF, computing the geopotential number in the scope of IHRF, using numerical integration to solve the Geodetic Boundary Value Problem and an existing recent quasi-geoid model in four stations in São Paulo state, Brazil. The first approach was performed considering two cases: a radius of 210 km and 110 km of gravimetric data coverage and the Global Geopotential Model GOCO05S truncated at 100 and 200, respectively. The results between solutions have shown a maximum difference of 94 cm, and a minimum difference of 10 cm.

Towards a new height datum for Uganda

A new height datum for Uganda is computed using the corrective surface principle. It is based on a combination of the Uganda Gravimetric Quasigeoid model (UGQ) 2014 and GNSS/levelling. UGQ2014 was derived from the Uganda Gravimetric Geoid model (UGG) 2014, which was computed from sparse terrestrial gravity data from the International Gravimetric Bureau, the 3 arc second Shuttle Radar Topography Mission digital elevation model and the GOCE – only global geopotential model GO_CONS_GCF_2_TIM_R5. The corrective surface was constructed based on 21 discrete GNSS/levelling points and then evaluated with 4 independent points. Three interpolation techniques were tested for the creation of the corrective surface with the Kriging method giving the lowest standard deviation and noise level suggesting that it is the best method for interpolation. In absolute terms, the Root Mean Square of the fit between the known and computed normal-orthometric heights based on the new height datum is 11cm, which is about 5cm (31%) better than using UGQ2014 alone. For relative heights an average precision of 29 ppm is computed for all baselines tested. Both the absolute and relative tests show that the new height datum satisfies the precision and accuracy requirements of third order precise levelling. Therefore, UGQ2014C represents a significant step towards the determination of a precise new height datum for Uganda.

Аналіз сучасних моделей відлікових поверхонь для визначення висот методом GNSS-нівелювання

This paper discusses various sources of information related to the issue of determining heights by GNSS-leveling. Implementation of the GNSS leveling method requires the presence of geoid or quasi-geoid heights, which today can be obtained from the corresponding models. In recent decades, scientists from around the world have developed many global, regional and local models of geoid and quasigeoid. This has contributed to the emergence of numerous publications on GNSS leveling. The purpose of the work is to perform the analysis of modern models of reference surfaces on the basis of materials of scientific publications according to the criteria that have a significant impact on research in the field of determining the height by GNSS-leveling. Method. The paper considers 44 publications published in the period 2001–2021. Among the research in this direction, there are three types of work: 1) 13 publications on methods of constructing the models themselves; 2) 12 works on their accuracy analysis and 3) 19 works on “adjustment” of model heights. At the first stage of the research the analysis was performed according to criteria related to geoid and quasi-geoid surface modeling, including Stokes and Molodensky theory, mathematical methods of data analysis and processing, tidal systems, zero-order geoid undulation and scale levels of geoid and quasi-geoid models. In the second stage, the frequency of publications by years was analyzed and the activity of publications of existing geoid and quasigeoid models on the sample of countries, which was made on the basis of all selected works in this study, was established. At the third stage, a quantitative analysis of official published models of geoid and quasi-geoid on the frequency of publications for the study period. The ratio of the accuracy of the heights of global geoid models in relation to the magnitude of the degree and order of their calculation is established. Results. In 58 % of the analyzed works, the authors use Stokes’ theory in their research, and in 42 % – Molodensky’s theory. Among the mathematical methods of data analysis and processing, 27 % use the method of mean square collocation, 20 % use the least squares method, the “delete-restore” method and the method of modifying Stokes’ formula with least squares (or KTH-method). The Fast Fourier transform method was used in 13 % of the works. Publications on the creation of global models of the Earth mostly use the parameters of the tide free tidal system in their calculations – a total of 40 % works. No less important criterion (33 % of works) should be taken into account the parameter of undulation of the geoid of zero order (“zero degree term”). In total, 41 % of research is aimed at creating and analyzing quasi-geoid models on a regional scale. During the study period, most works were published in 2012 and 2018. Leading countries in the development of quasi-geoid models are Canada, Poland, Sweden and the United States, and global geoid models are Germany, the United States and China. For the period 2001–2021, 99 global geoid models of various degree and order were officially presented, among which GOCO, EIGEN and EGM series models are most often used for research. Also during this period, 177 quasi-geoid models were presented, the most of which were published in 2019. Based on these data, the relationship with the frequency of publications in 2008–2021 can be traced. The accuracy of global geoid models in relation to the degree and order of their calculation is systematic in the range of 0.52–1.92 m, 0.38–0.50 m, 0.23–0.38 m and 0.12–0,14 m for models 60-220, 220-260, 260-720 and 720-2190 degree and order respectively. Scientific novelty. It is shown that the analysis of modern models of reference surfaces on the basis of materials of scientific publications in the field of GNSS-leveling method allows establishing the existing advantages and disadvantages of research in this field. Practical significance. The data of such analysis can be used to solve key problems in determining the height of the GNSS-leveling method, which require further research and to find modernized solutions.

EGM08 중력 모델을 이용한 심층신경망 기반 지오이드 기복 예측 정확도 평가

The difference between mean sea level and ellipsoid height is defined as geoid relief, which is essential information when building barometric altimeter/GPS combined systems or terrain matching systems. Therefore, an accurate calculation of the geoid undulation is required. In this paper, we propose a deep neural network method for calculating geoid undulations in real time in an embedded system. Then, using the EGM08 geoid model of order 2160, training data at intervals of 0.001 degrees were generated, and the prediction model accuracy was evaluated for 4 cases according to the number of hidden layers. As the number of hidden layers increased, the prediction accuracy increased, and it was confirmed that the calculation time also increased proportionally.

Comparative analysis among Asia–Pacific geoid models stored at the ISG repository

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.

Investigation of the Geoid Model Accuracy Improvement in Turkey

This study investigated the accuracy improvement of the gravimetric geoid model in Turkey using densified grids of the gravity anomalies with the least-squares modification of the Stokes integral with additive corrections (LSMSA) method. In LSMSA computations of the geoid models, the surface gravity anomalies, which were derived from the complete Bouguer anomaly grid data set by forward modeling the topographic masses, were used. In the computations, in addition to the original 5′-resolution (∼9 km) Bouguer anomaly data set, its densified versions to 1′-resolution (∼2 km) also were used. The densification of the gravity grids was carried out using both planar- and spherical-type complete Bouguer anomalies. Thus, the effects of both gravity grid densification and the applied approximation for the Bouguer reduction process in gravity gridding on the geoid model accuracies were examined in order. The Bouguer gravity reductions were carried out using classical formulas for the planar approximation, whereas the gravimetric terrain corrections from the high-resolution SRTM2gravity global model were employed for the spherical approximation in numerical tests. The calculated geoid models were validated at 100 homogeneously distributed GPS/leveling (GPS means Global Positioning System) benchmarks over the country. In the tests, the geoid undulations derived from the models also were compared with the observations at six tide gauge stations along the coastlines. The latter provided an additional check for the assessment of the models’ compatibility at the coastal boundaries. In conclusion, the accuracy of the best geoid model solution (8.6 cm by means of the standard deviation of geoid undulation differences at GPS/leveling benchmarks) was achieved using 1′-resolution gravity anomalies restored from the spherical complete Bouguer anomalies in Turkey.

New Gravimetric Infrastructure in Southeast Brazil: From Absolute Gravity Network to a Geoid Model

New Gravimetric Infrastructure in Southeast Brazil: From Absolute Gravity Network to a Geoid Model

Geoid modeling with least squares modification of Hotine’s integral using gravity disturbances in Turkey

Geoid modeling with least squares modification of Hotine’s integral using gravity disturbances in Turkey