Geoid Model Research Articles

Coherence of Bangui Magnetic Anomaly with Topographic and Gravity Contrasts across Central African Republic

The interactions between the geophysical processes and geodynamics of the lithosphere play a crucial role in the geologic structure of the Earth’s crust. The Bangui magnetic anomaly is a notable feature in the lithospheric structure of the Central African Republic (CAR) resulting from a complex tectonic evolution. This study reports on the coherence in the geophysical data and magnetic anomaly field analysed from a series of maps. The data used here include raster grids on free-air altimetric gravity, magnetic EMAG2 maps, geoid EGM2008 model and topographic SRTM/ETOPO1 relief. The data were processed to analyse the correspondence between the geophysical and geologic setting in the CAR region. Histogram equalization of the topographic grids was implemented by partition of the raster grids into equal-area patches of data ranged by the segments with relative highs and lows of the relief. The original data were compared with the equalized, normalized and quadratic models. The scripts used for cartographic data processing are presented and commented. The consistency and equalization of topography, gravity and geoid data were based using GMT modules ‘grdfft’ and ‘grdhisteq’ modules. Using GMT scripts for mapping the geophysical and gravity data over CAR shows an advanced approach to multi-source data visualization to reveal the relationships in the geophysical and topographic processes in central Africa. The results highlighted the correlation between the distribution of rocks with high magnetism in the central part of the Bangui anomaly, and distribution of granites, greenstone belts, and metamorphosed basalts as rock exposure. The correspondence between the negative Bouguer anomaly (<−80 mGal), low geoid values (<−12 m) and the extent of the magnetic anomaly with extreme negative values ranging from −1000 to −200 nT is identified. The integration of the multi-source data provides new insights into the analysis of crustal thicknesses and the average density of the Earth in CAR, as well as the magnitude of the magnetic fields with notable deviations caused by the magnetic flux density in the Bangui area related to the distribution of mineral resources in CAR.

Determination of Accurate Dynamic Topography for the Baltic Sea Using Satellite Altimetry and a Marine Geoid Model

Accurate determination of dynamic topography (DT) is expected to quantify a realistic sea surface with respect to its vertical datum and in identifying sub-mesoscale features of ocean dynamics. This study explores a method that derives DT by using satellite altimetry (SA) in conjunction with a high-resolution marine geoid model. To assess the method, DT was computed using along-track SA from Sentinel- 3A (S3A), Sentinel-3B (S3B), and Jason-3 (JA3), then compared with DT derived from a tide-gauge-corrected hydrodynamic model (HDM) for the period 2017–2019 over the Baltic Sea. Comparison of SA-derived DT and corrected HDM showed average discrepancies in the range of ±20 cm, with root mean square errors of 9 cm (for S3B) and 6 cm (for S3A and JA6) and a standard deviation between 2 and 16 cm. Inter-comparisons between data sources and multi-mission SA over the Baltic Sea also potentially identified certain persistent and semi-persistent problematic areas that are either associated with deficiencies in the geoid, tide gauge, HDM, and SA or a combination of all of these. In addition, it was observed that SA data have the potential to show a more realistic (detailed) variation of DT compared to HDM, which tended to generate only a smooth (low-pass) surface and underestimate DT.

A detailed quasigeoid model of the Hong Kong territories computed by applying a finite-element method of solving the oblique derivative boundary-value problem

Abstract New gravity and precise levelling measurements have been performed throughout the Hong Kong territories to modernize a vertical geodetic datum that is currently realized by heights of levelling benchmarks defined in the Hong Kong Principal Datum (HKPD). Modernization of the HKPD involved delivering various products, including new detailed geoid and quasigeoid models and newly determined orthometric and normal heights of levelling benchmarks. In this study, we present the result of gravimetric quasigeoid modelling. The method used to compute a detailed gravimetric quasigeoid model is based on the finite-element method to solve the geodetic boundary-value problem with oblique derivative boundary conditions considered directly at computational nodes on the discretized Earth’s topography. The result of a gravimetric quasigeoid modelling shows a good agreement with a geometric quasigeoid model at the Global Navigation Satellite System (GNSS)-levelling benchmarks. The standard deviation of differences between the gravimetric and geometric quasigeoid heights of ±3.3 cm is compatible with the expected accuracy of gravity, levelling, and GNSS measurements.

Revisiting the determination of Mount Olympus Height (Greece)

The present study deals with the determination of Mount Olympus summit (Mytikas), exploiting modern observations such as Global Navigation Satellite Systems (GNSS) and existing geodetic information. The last official Olympus height determination goes back 102 years by the Swiss surveyor M. Kurz. Since then, only unofficial measurement campaigns have taken place. There is a variety of released heights, which range from 2917 to 2919 meters. In September and October of 2022, we conducted a GNSS campaign, focusing on the area of highest Olympus peaks. Employing precise GNSS measurements (accuracy of 1–3 cm), in conjunction with height information from Greece’s National Triangulation Network (5–10 cm accuracy) and an appropriate, recent geoid model (5–6 cm accuracy), we estimate through the so-called GNSS-levelling, the height of Olympus to 2917.727 m with respect to Greece’s officially accepted mean sea level and 2918.390 m with respect to the global vertical datum. Our estimation of Olympus highest peak shows remarkable consistency at the level of 12.8 cm to that of M. Kurz in 1921.

Methodology for local correction of the heights of global geoid models to improve the accuracy of GNSS leveling

At present, one of the methods used to determine the height of points on the Earth's surface is Global Navigation Satellite System (GNSS) leveling. It is possible to determine the orthometric or normal height by this method only if there is a geoid or quasi-geoid height model available. This paper proposes the methodology for local correction of the heights of high-order global geoid models such as EGM08, EIGEN-6C4, GECO, and XGM2019e_2159. This methodology was tested in different areas of the research field, covering various relief forms. The dependence of the change in corrected height accuracy on the input data was analyzed, and the correction was also conducted for model heights in three tidal systems: “tide free”, “mean tide”, and “zero tide”. The results show that the heights of EIGEN-6C4 model can be corrected with an accuracy of up to 1 cm for flat and foothill terrains with the dimensionality of 1°×1°,2°×2°,and3°×3°. The EGM08 model presents an almost identical result. The EIGEN-6C4 model is best suited for mountainous relief and provides an accuracy of 1.5 cm on the 1°×1° area. The height correction accuracy of GECO and XGM2019e_2159 models is slightly poor, which has fuzziness in terms of numerical fluctuation.

Application of GOCE Satellite Gravimetric Data For Mineral Exploration

Gravity is directly proportional to density thereby making possible the use of measurable gravity in mapping the density variations in the earth’s interior. The use of density as a means of mineral exploration is cost-effective for wider area coverage. GOCE gravity data, collected by ESA’s GOCE satellite at 255km altitude and 10km intervals leading to the acquisition of multiple gravity points over the globe made possible the application of gravimetry in mineral exploration. Gravity disturbance data sourced from BGI- was corrected with WGM2012 corrections computed using EGM2008 geoid and ETOPO1 models, with reference gravity computed using the Somigliana formula. The gravity disturbances were mapped over Zambia using Surfer and Q-GIS, with over 600 control points of known mineral occurrences plotted together with other surface features like roads, rivers, railways, etc. By relating the control points to the varying gravity disturbances using the triple integral principle, a cautious analysis led to the geological classification of the gravity disturbances which essentially involved mapping predominant mineral occurrences across different parts of Zambia. During ground truthing, it was observed that the results within a particular area of interest on the classified map, and those obtained using four (4) different metal detectors as well as another remote sensing method tallied. The metal detectors used were the GR-100MINI, AKS, AKS plus 3D, and Garrett Ace 400, each with its own characteristics. From the map outputs, the results showed that GOCE data can be used for geological classification and delineation of terrain types. The delineation of terrain types on the classified map output matched that on existing geological maps and also offered the delineation of sub-terrain types.

Gravimetric geoid model determination of Central Ukraine area using combination of LSC and truncated SHA methods

Gravimetric geoid model determination of Central Ukraine area using combination of LSC and truncated SHA methods

An assessment of the implications of gravity reduction methods on local geoid modelling over Ado Ekiti

There are several gravity reduction approaches that are available for use in geodetic applications. More so, with the rapid rate at which gravimetric geoid models are being incorporated into Global Navigation Satellite System (GNSS) solutions, the need for precise regional models is becoming increasingly relevant. The two chosen gravity reduction approaches (Bouguer and Residual Terrain Model (RTM)) were used to reduce gravity anomalies over Ado town. The reduced anomalies were then used to compute a local geoid for the study area using the conventional Stokes integral in the Remove Compute Restore (RCR) technique. Comparison of the computed geoid by both methods with GNSS Levelling data at selected locations show that the RTM reduced anomalies produce a better geoid model than the Bouguer reduced anomalies over the study area with a RMSE of 83.2cm and 83.5cm respectively.

Determination of the height reference surface for the Republic of Albania by using global geopotential models

Abstract For the development of various geodetic tasks within a state, determining the Height Reference Surface by the geoid model is extremely important. Considering this, one of the main task of geodesy is to determine the geoid, which is defined as an equipotential surface of the Earth’s gravity field, as a result, it corresponds on average to the sea level. The aim of this study is to analyze the best-fitting geoid model for the territory of the Republic of Albania. In this study, DFHRS (Digital Finite Element Height Reference Surface) method was used (www.dfhbf.de), developed by Reiner Jäger [Jäger R. State of the art and present developments of a general approach for GPS-based height determination. East Africa: University of Applied Sciences, Faculty Geoinformationswesen, Department of Surveying and Geomatics. Paper Presented at the First Workshop on GPS and Mathematical Geodesy in Tanzania (Kilimanjaro Expedition 1999); 1999] to determine the most suitable geoid model for the territory of Albania. This approach allows the conversion of ellipsoidal heights determined by GNSS into the standard heights, which refer to the height reference surface (HRS) of an orthometric. The DFHRS is defined as continuous HRS in arbitrarily large areas by bivariate polynomials over an irregular grid [Jäger R, Schneid S. Online and postprocessed GPS heighting based on the concept of a digital height reference surface (DFHRS), in vertical reference systems. In: IAG Symposium. Cartagena, Colombia, Heidelberg: Springer; 2001, vol 124:203–8 pp]. The DFHRS approach uses a wide range of input data (Geometric and Physical) and in our case, there were 151 GPS/levelling height data as well as physical derivatives from different global geopotential models. The main focus of this study is placed on the calculation of the most suitable geoid model for the territory of Albania using global geopotential models (EGM96, EGM2008, EIGEN04, EIGEN6C4 and European Gravimetric Geoid Model 1997 (EGG97)). After analyzing the results and comparing the models among themselves, the Albanian DFHRS-EIGEN6C4 model was selected as the most suitable model for the territory of Albania.

Accuracy assessment of the SRTM2gravity high-resolution topographic gravity model in geoid computation

The accuracy of the SRTM2gravity high-resolution topographic gravity model is tested in this study. Complete Bouguer anomaly datasets calculated with classical formula in planar approximation and using the SRTM2gravity model in spherical approximation were compared in the study area. Two geoid models computed using free-air anomalies, which were restored from each Bouguer anomaly datasets, were compared to each other and they were validated at GPS/leveling points. In the mountainous part of the area, two geoid models differ by up to 4 cm. Following GPS/leveling validations, the geoid models provided nearly the same accuracy (∼4.2 cm). In conclusion, the accuracy of the SRTM2gravity model data was found sufficient to use in gravity reduction and it is recommended as a ready-to-use product in the area for gravity interpolation/gridding and geoid determination purposes.

Iterative data assimilation approach for the refinement of marine geoid models using sea surface height and dynamic topography datasets

Iterative data assimilation approach for the refinement of marine geoid models using sea surface height and dynamic topography datasets

Assessments of recent Global Geopotential Models based on GPS/levelling and gravity data along coastal zones of Egypt

Abstract 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 GGMS: 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.

Determining the new height of Mount Qomolangma based on the International Height Reference System

ABSTRACT To achieve the goal that China and Nepal jointly announce the new height of Mount Qomolangma (MQ), the campaign of Qomolangma Height Survey (QHS) was carried out from 2019 to 2020. A high precision geoid model realizing the common height datum for both sides is necessary for determining the unique height of MQ. However, high altitude and rugged topography make it extremely difficult to conduct terrestrial gravity measurements in this region, the accuracy of geoid model is restricted by terrestrial gravity data gaps. In the campaign of 2020 QHS, the airborne gravity survey was carried out over MQ and its surrounding areas, the airborne gravity data covering an area of 12,700 km2 were successfully collected. For the first time, the high precision observations of terrestrial gravity and BeiDou Navigation Satellite System (BDS) at the peak of MQ were collected. These datasets pave the way for the precise determination of the orthometric height of MQ. According to the definition of the International Height Reference System (IHRS), we developed the IHRS-based gravimetric quasigeoid model by combining the airborne and terrestrial gravity data. Validations against highly accurate GNSS leveling data at 61 benchmarks demonstrate that the accuracy of the quasigeoid model is 3.8 cm, and the addition of airborne gravity data improves the accuracy by 51.3%. Based on the IHRS, the final orthometric height of the snow surface of the peak of MQ is determined to be 8848.86 m.

GOCE Downward Continuation to the Earth’s Surface and Improvements to Local Geoid Modeling by FFT and LSC

One of the main applications of the gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite data is their combination with local gravity anomalies for geoid and gravity field modeling purposes. The aim of the present paper was the determination of an improved geoid model for the wider Hellenic area, using original GOCE SGG data filtered to retain only useful signals inside the measurement bandwidth (MBW) of the satellite. The filtered SGGs, originally at the satellite altitude, were projected to a mean orbit (MO) and then downward continued to the Earth’s surface (ES) in order to be combined with local gravity anomalies. For the projection to an MO, grids of disturbing gravity gradients from a global geopotential model (GGM) were used, computed per 1 km from the maximum satellite altitude to that of the MO. The downward continuation process was then undertaken using an iterative Monte Carlo (MC) simulated annealing method with GGM gravity anomalies on the ES used as ground truth data. The final geoid model over the wider Hellenic area was estimated, employing the remove–compute–restore method and both Fast Fourier Transform (FFT) and Least Squares Collocation (LSC). Gravity-only, GOCE-only and combined models using local gravity and GOCE data were determined and evaluation of the results was carried out against available GNSS/levelling data in the study area. From the results achieved, it was concluded that even when FFT is used, so that a combined grid of local gravity and GOCE data is used, improvements to the differences regarding GNSS/levelling data by 14.53% to 27.78% can be achieved. The geoid determination with LSC was focused on three different areas over Greece, with different characteristics in the topography and gravity variability. From these results, improvements from 14.73%, for the well-surveyed local data of Thessaly, to 32.88%, over the mountainous area of Pindos, and 57.10% for the island of Crete for 57.10% were found.

Assessing the performance of multi-GNSS precise point positioning technique on the geoid model validation

This study aims to investigate the comparative performance of PPP technique in evaluating the precision of three distinct geoid models and to quantify the contribution of GLONASS constellation to GPS as well as the effect of observation length. We use recently and concurrently measured GNSS and levelling data crossing rough topography with steep geoid slope in Türkiye. The RMSE statistics about the mean of the geometric minus gravimetric geoid heights reveal that at least 4-hours of static data should be collected regardless of the constellation. We find out encouraging results with 8-hours of GPS + GLONASS solutions having RSME values of 3.0, 5.0 and 21.5 cm in validating national, globally combined and satellite-only geoid models, respectively. The multi-GNSS PPP processing with a relatively long observation period yields comparable results to relative positioning though the latter still performs slightly better.

A precise geoid model of Bosnia and Herzegovina by the KTH method and its validation

We compute the first gravimetric geoid model for Bosnia and Herzegovina (BHG2022) based upon the KTH method and additive corrections. The BHG2022 is computed with the help of 34,820 terrestrial gravity points and a digital elevation model produced by the Shuttle Radar Topography Mission. The gravity data is gridded onto a 0.02 arc-degree resolution via the nearest neighbour interpolation method. ITU_GGC16 and ITG-Grace2010 models are utilised to provide long wavelengths of gravity field up to 280 and 180 maximum degree/order in the geoid computation, respectively. On the basis of an external evaluation by 609 Global Navigation Satellite System (GNSS)-levelling points homogeneously distributed over the country, the accuracy of the BHG2022 is estimated to be 5.65 cm absolutely. This is the most precise geoid model ever seen in this territory. The result highlights the significant effect of all gravity data obtained from various sources on the accuracy of the geoid model.

Performance Comparison of Deterministic and Stochastic Modifications in Stokes’s and Hotine’s Formulas: The Case of Jilin Province, China

The high-precision regional geoid model provides important fundamental geospatial information for developing and applying many disciplines. Deterministic and stochastic modifications are applied to Stokes’s and Hotine’s formulas of geoid modeling to reduce errors. Based on the Experimental Geopotential Model 2019 (XGM2019), this paper used Stokes’s and Hotine’s formulas to analyze the variation of global root mean square error (RMSE) with modification parameters for two deterministic (Wong and Gore; and Vaníček and Kleusberg) and three stochastic modifications (biased, unbiased, and optimum). Taking the quasigeoid refinement of Jilin Province as an example, the global RMSE, approximate geoid undulation, and additive corrections were calculated. The parameter analysis and the global RMSE calculation showed that the variation of the modification limits and the terrestrial gravity data error variance had a centimeter-level effect on the global RMSE. In contrast, the impact of the integration radius was relatively small. The stochastic modifications were better than the deterministic ones in calculating the global RMSE. The global RMSE of Hotine’s formula was smaller than that of Stokes’s, and its unbiased and optimum modifications reached the minimum value of 12.9 mm. The validation of XGM2019 and the refined quasigeoid based on the high accuracy GPS/leveling points showed that the standard deviation (STD) of XGM2019 was 5.8 cm in Jilin Province, and the refined optimal quasigeoid model was 2.9 cm. Stokes’s and Hotine’s formulas provided the same accuracy in the study area. In the western plain area, the accuracy of the deterministic modifications was 2.0 cm, which was about 0.4 cm higher than that of the stochastic modifications. In the eastern mountainous area, the stochastic modifications were better than the deterministic ones, and the accuracy was about 3.2 cm. Stokes’s and Hotine’s formulas based on deterministic and stochastic modifications significantly improve the accuracy of the XGM2019. The deterministic and stochastic modifications show millimeter-level differences in plain and mountainous areas.

Методика преобразования трехмерных положений пунктов между геоцентрическими и референцными системами координат для региональных территорий

A technique was developed for converting three-dimensional positions of points for regional territories between geocentric and reference coordinate systems (RCS) by matching the Helmert parameters with the geoid height model for the reference coordinate system and the three-dimensional positions of points in both SC. The technique is based on replacing the quasigeoid height model to the EGM2008 geoid height model transformed for reference coordinate systems. For regional territories, the accuracy of determining the Helmert parameters and the accuracy of transformations of coordinates and heights of points at the level of accuracy of the positions of points in the general and reference coordinate systems is ensured. Examples of calculations of the agreed Helmert parameters with the transformed elevation model of the EGM2008 geoid between the reference coordinate system SK-95 and WGS-84, SK-42, GSK-2011 for the territory of the Novosibirsk region are given.

Методика формирования цифровых моделей высот геоида для референцных систем координат

A technique was developed for generating digital models of geoid heights for reference coordinate systems using the Trimble Bussines Center software (TBC software). The source models can be the publicly available high-precision global geoid elevation models EGM2008, EIGEN6C4 and other models whose files with the *.ggf extension are compatible with the TBC software. The technique is based on the possibility of exporting orthometric heights from the TBC software in ASCII format to a user-specified area on the surface of the global WGS-84 ellipsoid for a uniform grid of nodes in lati-tude and longitude. The resulting data table is the starting point for calculating geoid heights for reference coordinate systems, which are determined by the parameters of the corresponding reference ellipsoids and Helmert parameters - spatial position, orientation and linear scale relative to the global coordinate system, according to Standard. The initial data are sufficient to calculate the digital model of the geoid heights for an arbitrary reference coordinate system. The expediency of the developed technique lies in the possibility of its use for a much more accurate calculation of the Helmert parameters for regional and other territories compared to the accepted standard values and, accordingly, a more accurate transformation of the three-dimensional positions of points between the general earth and reference coordinate systems (these issues will be discussed in a subsequent publication by the author). Approbation of the technique was carried out for several regional territories: Novosibirsk and Moscow regions, the territory of Western Siberia, the Republic of Belarus in the SK42 and SK95 coordinate systems, and the results are presented graphically in the form of geoid height maps.

Evaluation of several geoid models for GNSS height transformation in Turkiye

Evaluation of several geoid models for GNSS height transformation in Turkiye