Gravimetric Geoid Research Articles

Local Gravity and Geoid Improvements around the Gavdos Satellite Altimetry Cal/Val Site

The isle of Gavdos, and its wider area, is one of the few places worldwide where the calibration and validation of altimetric satellites has been carried out during the last, more than, two decades using dedicated techniques at sea and on land. The sea-surface calibration employed for the determination of the bias in the satellite altimeter’s sea-surface height relies on the use of a gravimetric geoid in collocation with data from tide gauges, permanent global navigation satellite system (GNSS) receivers, as well as meteorological and oceanographic sensors. Hence, a high-accuracy and high-resolution gravimetric geoid model in the vicinity of Gavdos and its surrounding area is of vital importance. The existence of such a geoid model resides in the availability of reliable, in terms of accuracy, and dense, in terms of spatial resolution, gravity data. The isle of Gavdos presents varying topographic characteristics with heights larger than 400 m within small spatial distances of ~7 km. The small size of the island and the significant bathymetric variations in its surrounding marine regions make the determination of the gravity field and the geoid a challenging task. Given the above, the objective of the present work was two-fold. First, to collect new land gravity data over the isle of Gavdos in order to complete the existing database and cover parts of the island where voids existed. Relative gravity campaigns have been designed to cover as homogenously as possible the entire island of Gavdos and especially areas where the topographic gradient is large. The second focus was on the determination of a high-resolution, 1′×1′, and high-accuracy gravimetric geoid for the wider Gavdos area, which will support activities on the determination of the absolute altimetric bias. The relative gravity campaigns have been designed and carried out employing a CG5 relative gravity meter along with geodetic grade GNSS receivers to determine the geodetic position of the acquired observations. Geoid determination has been based on the newly acquired and historical gravity data, GNSS/Leveling observations, and topography and bathymetry databases for the region. The modeling was based on the well-known remove–compute–restore (RCR) method, employing least-squares collocation (LSC) and fast Fourier transform (FFT) methods for the evaluation of the Stokes’ integral. Modeling of the long wavelength contribution has been based on EIGEN6c4 and XGM2019e global geopotential models (GGMs), while for the contribution of the topography, the residual terrain model correction has been employed using both the classical, space domain, and spectral approaches. From the results achieved, the final geoid model accuracy reached the ±1–3 cm level, while in terms of the absolute differences to the GNSS/Leveling data per baseline length, 28.4% of the differences were below the 1cmSij [km] level and 55.2% below the 2cmSij [km]. The latter improved drastically to 52.8% and 81.1%, respectively, after deterministic fit to GNSS/Leveling data, while in terms of the relative differences, the final geoid reaches relative uncertainties of 11.58 ppm (±1.2 cm) for baselines as short as 0–10 km, which improves to 10.63 ppm (±1.1 cm) after the fit.

Impact of the UNB topographical density model on precise geoid determination in the high mountainous region

A precise gravimetric geoid model is computed utilising Stokes’s formula, supposing an absence of topography above the geoid. Subsequently, the geoid model undergoes a simple correction for topographic masses, the constant density is taken as 2670 kg/m3. Notably, the true density of topographical mass deviates by approximately ±20% from this constant value. Recently, the University of New Brunswick in Canada released a global topographical density model at a 30 arc-second resolution. This paper investigates the impact of incorporating this model on the precision of the gravimetric geoid within a mountainous region in the Colorado test region. Numerical findings reveal that variations in geoid undulation attributable to this model can extend to a few decimetres, a discrepancy that cannot be neglected in geoid modelling with one-centimetre precision. It is therefore recommended that the considerable impact of topographic density fluctuations on geoid determination be taken into account, particularly in mountainous regions.

A user-friendly software package for modelling gravimetric geoid by the classical Stokes-Helmert method

With the progress in Global Navigation Satellite Systems (GNSS) technology, accurate geoid modelling has started to play an essential role in geodetic applications such as establishing height datum as a continuous surface model and related vertical control for infrastructure projects. Thus, numerous geoid modelling methods have been offered since 1990’s, each of them has its own algorithm and approximation theories. Classical Stokes-Helmert is one of the most well-known methods all over the world by geodetic communities. However, a user-friendly software package of the method is not publicly accessible on the Internet. Therefore, a compact and user-friendly software package “CSHSOFT” is developed and presented for scholars in this field. A fractionated programming strategy has been treated to build individual components striving high accuracy and computational efficiency for geoid heights. Subsequently, the CSHSOFT is simply tested to construct a geoid model in the mountainous area in Auvergne test-bed where several geoid modelling techniques are implemented. Afterward, the new geoid model of the region is externally evaluated by GNSS-levelling data in terms of rigorous orthometric heights. The fitting statistics of 2.75 cm and 0.36 ppm in absolute and relative height differences fairly indicate that the CSHSOFT is a vigorous tool for gravimetric geoid modelling, and can be comfortably employed for geoscientific and technical studies.

The Importance of Ground Control Points In A Photogrammetric Workflow

Abstract This study investigates the optimal distribution and pattern of ground control points (GCPs) in aerial photogrammetric projects. Aerial triangulation (AT), also known as bundle adjustment, is the fundamental step in refining 3D reconstruction models and camera positions, thereby minimizing reprojection errors. The study utilizes data from a national project in Romania, employing high-resolution aerial images acquisition using photogrammetric sensors. The project has rigorous requirements of ground control points (GCP) placement and field measurements using GNSS and geometric leveling techniques. The study employs various scenarios, manipulating the number and distribution of GCPs, to assess their influence on planimetric and altimetric accuracy. Results indicate that the configuration and number of GCPs significantly affect the accuracy of photogrammetric products, such as dense image point clouds, digital surface models, and orthophotos. Moreover, the study underscores the importance of precise GCP determination methods, especially in regions lacking a precise gravimetric geoid model. In scenarios with inadequate GCP coverage the outcomes have inferior quality, emphasizing the critical role of GCPs in ensuring the quality of photogrammetric products. Overall, the research gives a clear view on the best placement patterns of GCPs and their influence on AT process evaluation performed in check points (CHKs).

The effect of topographic density variations on the geoid and orthometric heights in Hong Kong

Utilizing the adopted average topographic density of 2670 kg/m3 in the reduction of gravity anomalies introduces errors attributed to topographic density variations, which consequently affect geoid modeling accuracy. Furthermore, the mean gravity along the plumbline within the topography in the definition of Helmert orthometric heights is computed approximately by applying the Poincaré-Prey gravity reduction where the topographic density variations are disregarded. The Helmert orthometric heights of benchmarks are then affected by errors. These errors could be random or systematic depending on the specific geological setting of the region where the leveling network is physically established and/or the geoid model is determined. An example of systematic errors in orthometric heights can be given for large regions characterized by sediment or volcanic deposits, the density of which is substantially lower than the adopted topographic density used in Helmert's definition of heights. The same applies to geoid modeling errors. In this study, we investigate these errors in the Hong Kong territory, where topographic density is about 20% lower than the density of 2670 kg/m3. We use the digital rock density model to estimate the effect of topographic density variations on the geoid and orthometric heights. Our results show that this effect on the geoid and Helmert orthometric heights reach maxima of about 2.1 and 0.5 cm, respectively. Both results provide clear evidence that rock density models are essential in physical geodesy applications involving gravimetric geoid modeling and orthometric height determination despite some criticism that could be raised regarding the reliability of these density models. However, in regions dominated by sedimentary and igneous rocks, the geological information is essential in these applications because topographic densities are substantially lower than the average density of 2670 kg/m3, thus introducing large systematic errors in geoid and orthometric heights.

Evaluating the accuracy of the digital elevation models (DEMs) within the territory of Bosnia and Herzegovina

This study focuses on evaluating the accuracy of Digital Elevation Models (DEMs) in Bosnia and Herzegovina, a region characterized by diverse topography. The accuracy of the gravimetric geoid model is directly correlated with the resolution and accuracy of the digital elevation model (DEM). To date, no research has been conducted on the accuracy of global and regional models for the area of Bosnia and Herzegovina, using ground control points. The analysis encompasses various aspects of DEMs, including interpolation methods (bicubic, B-spline, bilinear, and nearest neighbor), data sources, control points, and datum alignment. Both global and regional DEM models are integrated, introducing a native terrain model. Two sets of control points are utilized for accuracy assessment, along with datum alignment approach. The performance of the BIHDEM model is scrutinized, and a comparison with LIDAR data is conducted. Accuracy assessment involves calculating the Root Mean Square Error (RMSE) for all DEMs and interpolation methods. The study reveals that the bicubic interpolation method yields the best results. The BIHDEM model proves to be the most effective for SET1 (high precision leveling network), while COPERNICUS emerges as the optimal choice for SET2 (trigonometric network). Furthermore, an analysis of the elevation profiles obtained from LIDAR and DEM models highlights significant differences, with ASTER, MERIT, NASADEM and SRTM exhibiting the largest variations. In contrast, FABDEM and BIHDEM models demonstrate minimal differences. Mean values and RMSE further support the superiority of FABDEM and BIHDEM models, providing valuable insights for geospatial applications in the region.

GeoMed2,the geoid of the Mediterranean: work in progress

Geodesy can provide valuable information on marine current estimation based on the combination of gravity and altimetry. Gravity is standardly used to estimate the geoid undulation, i.e. the height of the geoid over a given reference ellipsoid. As it is well known, the geoid undulation over the oceans is closely related to the Mean Sea Surface (MSS) with discrepancies that can reach 1–2 m at global scale. By satellite altimetry, one can get the MSS and then estimate the Mean Dynamic Topography (MDT) as the difference between the MSS and the geoid undulation. As the MDT is related to the ocean circulation, information on the ocean circulation to be compared with oceanographic estimates can be provided using these geodetic measurements. In this context, the GeoMed2 project aims at estimating a high-accuracy and high-resolution geoid model for the Mediterranean Sea based on land and marine gravity data and on recent Global Geopotential Models. In this paper, the processing methodology based on the well-known remove–compute–restore approach for the determination of the geoid in the Mediterranean area is presented. In a pre-processing step, all available gravity observations for the wider Mediterranean basin have been collected, validated, homogenized, and unified in terms of their horizontal and gravity system. In this way, a reliable gravity database to be used for the determination of the geoid has been prepared. This data set has been used in computing a gravimetric geoid estimate based on which the MDT over the Mediterranean Sea was obtained. The results of this computation were then revised, commented and compared with other existing MDT solutions. By these comparisons, it can be concluded that the geodetic computed MDT is not yet satisfactory since it is too noisy. This is possibly due to some inconsistencies still present in the gravity data used for estimating the geoid undulation and to the adopted MSS which seems to be too smooth over the Mediterranean area.

Evaluating the Impact of the Recent Combined and Satellite-Only Global Geopotential Model on the Gravimetric Geoid Model

Geoid represents Earth’s surface, ocean, and gravitational field, which influence the elevations, shape, and mass distribution of the geopotential surface, a hypothetical surface that is perpendicular to the direction of gravity at every point. This geopotential surface serves as a reference for measuring elevations and is used as a fundamental reference surface for geodetic and surveying purposes. In this study, the Least Squares Modification of Stokes Formula (LSMS) with Additive Corrections (AC), also known as the KTH method, is used to generate a new gravimetric geoid model for Peninsular Malaysia. The KTH method was developed at the Royal Institute of Technology (KTH) in Stockholm-Sweden. The dataset used is the most recent global digital elevation model, Shuttle Radar Topography Mission (SRTM) 1 Arc-Second Global, generated by the National Aeronautics and Space Administration (NASA) and the National Imagery and Mapping Agency (NIMA). In addition to this elevation data, the dataset includes the Global Geopotential Model (GGM), which is composed of the XGM2016, XGM2019e, Tongji_GGMG2021S, and Tongji-Grace02k models. Furthermore, it incorporates sets of regional gravity data, including terrestrial gravity, airborne gravity, and marine gravity anomalies, all of which are derived from the Technical University of Denmark (DTU 21). The actual 45 Global Navigation Satellite System (GNSS)-levelling points data have been compared to the gravimetric geoid model developed in this study and the geoid acquired from Department of Survey and Mapping Malaysia (DSMM). According to the statistical results, NXGM2019e provides better accuracy, with the Root Mean Square Error (RMSE) geoid model errors of ±0.033 m, compared to the deviations in free-air anomalies, XGM2019e, which has the minimum RMSE of 10.291 mGal. Meanwhile, Tongji-GMMG2021S has the maximum RMSE of 14.792 mGal. The geoid is derived from the XGM2019e model and has maximum and minimum values of 0.032 m and 0.147 m, respectively, with mean residuals of 0.089 m. In conclusion, the XGM2019e has the potential to determine a precise local geoid model for Peninsular Malaysia

Two alternative techniques for fitting the gravimetric geoid for Egypt

In this paper, two proposed geoid fitting techniques for Egypt's gravimetrically determined geoid and Global Positioning System GPS/levelling-derived geoid are introduced. First, any errors in the available GPS stations are ruled out. These methods rely on the absolute geoid difference, which is the gravimetric geoid height minus the geoid obtained through comparing GPS and levelling. The suggested geoid fitting techniques use an optimization algorithm scheme to choose the minimum number of the best-suited GPS stations to be used for fitting the gravimetric geoid. The least-squares collocation method is used to determine each GPS point's impact on the remaining GPS points. The GPS stations with the least impact on the other points are used for external validations, till an acceptable limit of the influence of the GPS points on the remaining ones (when the lowest standard deviation of the differences between gravimetric and geometric geoids is achieved with minimum average or standard deviation is larger than the accuracy of GPS observations, or standard deviation is larger than the target accuracy for geoid; or until a maximum of 30% of the GPS/levellings are excluded). This method operates in a way that automatically selects the fewest GPS stations that are most suitable for usage in the geoid fitting procedure. The geoid quality is then checked externally using the remaining GPS stations. A kriging trend surface is then taken out of the absolute geoid difference to complete the fitting. The proposed geoid fitting techniques are compared with that using polynomial regression of different degrees. The results proved that the proposed techniques give extremely better results. The findings of this study affirm that the adoption of the proposed techniques yields a geoid with an external accuracy of approximately 19 cm for Egypt.

Computation of gravimetric geoid model using free air vertical gravity gradient anomaly in geoid-quasigeoid formula

Geoid-quasigeoid separation term is a crucial parameter that allows the conversion between different national height systems based on geoid and quasigeoid surfaces. The consideration of free-air Vertical Gravity Gradient Anomalies (VGGA) in the separation term formula is generally ignored and is not sufficiently investigated. However, this may be important, particularly for the mountainous areas. In this study, the impact of considering modelled VGGA in the special formulation of separation term is investigated. The study area is in mountainous western Turkey where modelled VGGA are previously validated with the homogenous and dense VGGA measurements (in total: 159). The separation term is calculated with and without considering the VGGA. Thus, gravimetric geoid models have been obtained with two different strategies. These gravimetric geoid models are compared with geometric geoid model calculated from recently measured GPS/levelling measurements. The results demonstrate that the consideration of modelled VGGA in the special formula of separation term changes the gravimetric geoid model with the statistics of min: −10.6 cm, max: 17 cm, mean: 0.2 cm, std.: 1.5 cm. This is quite significant for a goal of cm accurate geoid modelling. Especially at elevations higher than 1000 m, the use of modelled VGGA is found to be more effective on the gravimetric geoid.

Analysis of different combinations of gravity data types in gravimetric geoid determination over Bali

Abstract Following the Regulation of the Head of the Geospatial Information Agency (BIG) No. 13 of 2021, geoid is used as the Vertical Geospatial Reference System in Indonesia. Applications using the geoid as an ideal reference require a much higher accuracy and resolution than the geoid obtained from models derived solely from satellite data. The Indonesian Geospatial Information Agency considers the geoid ideal if it has reached an accuracy of better than 15 cm. Recent studies have combined satellite and other gravimetric data to produce a combined geoid model with increased resolution. Gravimetric data obtained from measurements close to the Earth’s surface, such as airborne and terrestrial gravity data, are particularly attractive because the high-frequency portion of the signal is more apparent and can contribute to the medium to high frequencies of the gravity field. This study models the geoid over Bali Island by combining satellite, airborne and terrestrial gravity data. Calculations were performed using Least Square Collocation (LSC) and Remove-Compute-Restore (RCR) techniques. The gravimetric geoid model was tested against the geometric geoid profile calculated from a GNSS/Levelling survey. The geoid, calculated by combining the GOCO06S satellite gravity model, the GGMplus gravity model and airborne gravity data at an average flight altitude of 4100 m produces a standard deviation of 14.46 cm along the 125 km validation path. After also adding terrestrial gravity data, the standard deviation increased to 16.37 cm. By comparison, the results of the validation of the geoid model from GOCO06S and INAGEOIDV2 with geometric geoids have standard deviation values of 79.56 cm and 16.40 cm, respectively. However, the results of the statistical tests are strongly influenced by the data quality used as validation, in this case, GNSS/Levelling. It is shown that the GNSS/Levelling data over Bali contains significant errors, which have been reduced based on an analysis of geometric vertical deflections. A geometric geoid profile with higher accuracy is required to test the accuracy of the gravimetric geoid models more reliably.

Validation of European Gravimetric Geoid models in context of realization of EVRS system in Poland

Validation of European Gravimetric Geoid models in context of realization of EVRS system in Poland

Gravimetric geoid for Egypt based on properly application of Helmet's method of condensations with window remove-restore technique

Gravimetric geoid for Egypt based on properly application of Helmet's method of condensations with window remove-restore technique

Determination of a gravimetric geoid model for Eastern Province in the Kingdom of Saudi Arabia

Abstract A high-resolution gravimetric geoid model was developed for the Eastern Province Kingdom of Saudi Arabia region. The model was created using various datasets, including 320,434 land gravity measurements, 5442 shipborne marine gravity data, the DTU17 altimetry gravity model, and the XGM2019e global geopotential model. The computation strategy employed for modeling the gravimetric geoid involved the Remove-Compute-Restore method with Residual Terrain Model reduction and the 1D-Fast Fourier Transform approach technique. Geoid heights were determined using the Stokes integral with the Wong-Gore modification. To assess the accuracy of the resulting geoid models, they were compared with 4401 GNSS/Levelling points. The geoid accuracy throughout the entire area is better than 1.6 cm in terms of standard deviation (STD) after fitting.

Investigating the Congruence between Gravimetric Geoid Models over India

A major motivation of precise geoid computation is to adopt it as a national vertical datum or as an alternative vertical reference surface to aid surveyors in calculating physical heights using the Global Navigation Satellite System (GNSS). There exist several methods of geoid computation, and only one particular method is generally used to calculate the national geoid model irrespective of the size and topographical landforms of the country. The validation of the developed geoid models is done with the complete GNSS-leveling data set. In such a case, it is hard to claim the consistency in the precision of the developed geoid model throughout the country. This study aims to identify the consistency of the geoid models over India computed using the three approaches primarily followed in Curtin University of Technology (CUT), the University of New Brunswick (UNB), and the Royal Institute of Technology (KTH). Three analyses have been done on the calculated geoid models: (1) clusterwise validation with the GNSS-leveling data, (2) intermodel comparison for the whole study area, and (3) intermodel comparison for Indian states and Union Territories (UT) only. The GNSS-leveling validation results show that the standard deviations of differences for all the methods are within a range of ±0.01 m with the exception of Uttar Pradesh West with the UNB method. However, inter-model comparison shows that the mean (meters) and standard deviation (meters) of the differences between the pairs (CUT-UNB), (CUT-KTH), and (KTH-UNB) are 0.241±0.854, −0.133±0.498, and 0.374±1.239, respectively, with maximum difference sometimes exceeding 5 m. There is only one UT and four states for which the mean value is within (−0.20 m, 0.20 m) and standard deviation ≤±0.05 m for all the three pairs. Therefore, the analysis shows that it is difficult to calculate a precise national geoid model using any one method alone and a strategy is required to merge various regional precise geoid models or methods to develop a consistently precise national geoid model.

Geoid model determination for the Hellenic area “Hellas Geoid 2023”

The latest geoid model "HELLAS GEOID 2023" (HG2023) derived by the Hellenic Military Geographical Service is the most comprehensive model for the entire Hellenic area. Long-term gravity data, orthometric and geometric heights, seabed topography and a high-resolution digital terrain model were implemented in the calculations. Data evaluation and their accuracy estimation were of major importance to ascertain compatibility among data sources. In this study, data from neighboring countries were used as they were essential for the completeness of the model developed. The technique remove–compute–restore was adopted for the separation of the gravity signal in the heterogeneous data utilizing EIGEN 6C4 (full degree and order 2190), since the global geodynamic model was found to fit best in the Greek region. Mean gravity residual contribution to the local geoid model was calculated using Stokes' theorem in the frequency domain and implementing the Fourier transform using the Wang and Core modification. The final surface of the resulting gravimetric geoid was adapted to the existing height system of the State utilizing the Least Squares Collocation method by fitting points of known orthometric and geometric heights, distributed throughout Greece. The external accuracy of the estimated geoid model was estimated at the level of 6 cm.

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.

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.