Gravimetric Geoid

The determination of gravimetric geoid models using gravity anomalies requires the computation of normal gravity on specific ellipsoids. For global models, the International Gravity Formula is applied to the WGS 84 ellipsoid, while local models use the ellipsoid adopted for geodetic computation in the region of study. In countries that use the Clarke 1880 ellipsoid, a theoretical gravity model is computed on this surface to establish a precise local geoid model. This paper formulates a local theoretical gravity model for calculation of theoretical gravity values on the Clarke 1880 spheroid using the Somiglinana's closed formula for normal gravity, angular velocity, gravitational constant, and the Clarke 1880 spheroid parameters. The model is presented in two forms (A and B) and compared to ensure accuracy. Both forms of the model are suitable for accurate normal gravity computation on the Clarke 1880 ellipsoid.

This research investigates the performance of six satellite-only GOCE-based Global Geopotential Models (GGMs) released between 2018 and 2023. They include the WHU-SWPU-GOGR2022S, GOSG2S, Tongji-GMMG2021S, GO_CONS_GCF_2_TIM_R6, IGGT_R1C, and GOSG01S models. The assessment has been carried out using 221 first-order observed gravity stations in Egypt where 16 of them are of absolute gravity measurements. In addition, the assessment comprises the World Gravity Map 2012 (WGM 2012) as another unique open-source of gravity information worldwide. The accomplished findings have demonstrated that the WHU-SWPU-GOGR2022S model produces the least standard deviation of ±11.272 mGal over the available absolute gravity stations. On relative gravity points, the GOSG2S is the accurate GGM a with standard deviation equals ±15.480 mGal. Even though the WGM 2012 is not a GOCE-based GGM it has produced standard deviations of ±7.598 and ± 12.296 mGal at absolute and relative gravity locations respectively. Thus based on the available datasets and attained results, it can be concluded that the WGM 2012 and the WHU-SWPU-GOGR2022S GGM precisely describe the Earth's gravitational field over Egypt in particular. Furthermore, it is recommended to utilize those two sources of free-air gravity anomalies in geodetic activities chiefly for local geoid modelling.

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2007Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2007Bu çalışmada, GPS tekniğinden elde edilen elipsoidal yüksekliklerin bölgesel düşey datumdaki ortometrik yüksekliklere dönüştürülmesinde lokal geoit modeli çözümlerinin kullanımı araştırılmıştır. Araştırma iki kısımdan oluşmaktadır. Birinci kısımda Türkiye ulusal geoitleri TG99A ve TG03 ile Avrupa gravimetrik geoidi EGG97 beş adet lokal bölgedeki GPS/nivelman verileri kullanılarak test edilmiştir. Daha sonra bu modeller GPS/nivelman noktalarında geoit yükseklik farkları kullanılarak hesaplanan yüzey modelleri ile ulusal düşey datuma uydurulmuş ve bağımsız GPS/nivelman verileri yardımıyla sonuçlar test edilmiştir. Bölgesel geoit modelleri ile gerçekleştirilen bu testlerden başka lokal GPS/nivelman geoit modelleri hesaplanmış doğrulukları irdelenmiştir. Test edilen ve yeni hesaplanan geoit modellerinin doğrulukları karşılaşt...

Historical gravity observations have been utilized to compute the gravimetric geoid for the kingdom of Saudi Arabia. Using the "remove and restore" techniques, the global geoid model (EGM2008) heights and free air anomaly have been computed using Moledensky formula. The gravimetric geoid model covers about 70 percent of the kingdom. Secondly, GNSS (GPS) space based observations along with the Bench Mark observations were also used to compute the geoid. Determination of GNSS/Benchmark geoid based on the vertical reference network of Saudi Arabia, which was built in early 1970's as first order vertical control network by spirit leveling based on tidal gauges along the Arabian Gulf and Red Sea. More than 5000 GPS/BM observations using the remove and restore techniques. The trend correction was applied using the least square method, and the high was implemented for high frequency correction. Analysis of the two geoids reveals some geoid biases, which is possibly due to th...

Validation of recent GOCE/GRACE geopotential models over Khartoum state - SudanThis paper evaluates a number of latest releases of GOCE/GRACE global geopotential models (GGMs) using the GPS-levelling geometric geoid heights, terrestrial gravity data and existing local gravimetric models. We investigate each global model at every 5 degree of spherical harmonics. Our analysis shows that the satellite-only models derived by space-wise and time-wise approaches (SPW_R1, SPW_R2 TIM_R1 and TIM_R2), GOCO01S together with EGM08 (combined model) are very distinct and consistent to the local data, which guarantees one of them to be selected as the best of candidate models and then to be utilized in our further geoid studies. One of Satellite-only models will be employed for acquiring the long wavelength geoid component which is one of major steps in the geoid determination. EGM08 will be used to compensate and restore the missing gravity data points in the un-surveyed parts within the target a...

GPS (Global Positioning System) is a passive, all-weather satellite-based navigation and positioning system, which is designed to provide precise three dimensional position and velocity, as well as time information on a continuous worldwide basis. The ability of GPS for providing high-precision vertical component of relative coordinates could be exploited for monitoring, and hence mapping, the sea level. In this case, the local nature of GPS measurements could nicely supplement the temporal and spatial resolution of satellite altimetry measurements which has a more or less global nature. This paper would describe and discuss the concept and method of monitoring sea level using GPS, and the technical and operational aspects that should be taken into account in implementing this technique. The paper would also discuss the prospect of using this GPS-based approach for monitoring sea level in Indonesia, along with its potential related applications. The paper would be sum up with some concluding remarks and recommendations. I.

In this study, a thorough validation of the recently released Global Geopotential Models (GGMs) has been carried out by comparing all models with both the free-air (FA) gravity anomaly observations and GPS/Levelling data in Kuwait. First, the gravity functional in terms of FA gravity anomalies and geoid heights that derived from GGMs and compared with the ground-based data before applying the Spectral Enhancement Method (SEM). The findings indicate that the combined GGMs, generally, fit the ground data better than the satellite-only GGMs due to the existing spectral gaps. Second, the FA gravity anomalies and geoid heights, obtained from GGMs at maximum Spherical Harmonics (SH) degree/order (d/o) for the combined models and from spherical harmonics d/o ranging from 100 to their maximum d/o, with a step of 10 d/o for the satellite only GGMs, were compared with the corresponding ones from terrestrial data using the SEM. For the FA gravity signal, the EIGEN-6C4 manifested the best results of combined GGMs with differences characterised with a mean of 0.967 mGal and a Standard Deviation (STD) of 2.845 mGal, while TIM_R5 at SH (d/o) 200 showed the best results with a mean of satellite only GGMs with a mean if 1.008 mGal and a STD of 2.657 mGal. In the case of the evaluation of the geoidal height with GPS/levelling, XGM2016 manifested the best results of combined GGMs with differences characterised with a STD of 0.234 m. For the satellite only GGMs, the best results have been obtained from SPW_R5 at SH d/o 250 with a STD 0.207 m.

Digital Elevation Model (DEM) is crucial to a wide range of surveying and civil engineering applications worldwide. Some of the DEMs such as ASTER, SRTM1 and SRTM3 are freely available open source products. In order to evaluate the three DEMs, the contribution of EGM96 are removed and all DEMs heights are becoming ellipsoidal height. This step was done to avoid the errors occurred due to EGM96. 601 points of observed ellipsoidal heights compared with the three DEMs, the results show that the SRTM1 is the most accurate one, that produces mean height difference and standard deviations equal 2.89 and ±8.65 m respectively. In order to increase the accuracy of SRTM1 in EGYPT, a precise Global Geopotential Model (GGM) is needed to convert the SRTM1 ellipsoidal height to orthometric height, so that, we quantify the precision of most-recent released GGM (five models). The results show that, the GECO model is the best fit global models over Egypt, which produces a standard deviation of geoid undulation differences equals ±0.42 m over observed 17 HARN GPS/leveling stations. To confirm an enhanced DEM in EGYPT, the two orthometric height models (SRTM1 ellipsoidal height + EGM96) and (SRTM1 ellipsoidal height + GECO) are assessment with 17 GPS/leveling stations and 112 orthometric height stations, the results show that the estimated height differences between the SRTM1 before improvements and the enhanced model are at rate of 0.44 m and 0.06 m respectively.

According to the wide spread use of satellitebased positioning techniques, especially Global Navigation Satellite Systems (GNSS), a greater attention has been paid to the precise determination of geoid models. As it is known, leveling measurements require high cost and long time in observation process that make it not convenient for the practical geodetic purposes. Thus obtaining the orthometric heights by GNSS is the most conventional way of determining these heights. Verifying this goal was the main objective behind the current research. The current research introduces a numerical solution of geoid modeling by applying a surface fitting for a few sparse data points of geoid undulation using minimum curvature surface (MCS). The MCS is presented for deriving a system of linear equations from boundary integral equations. To emphasize the precise applicability of the MCS as a tool for modeling the geoid in an area using GPS/leveling data, a comparison study between EGM2008 and MCS geoid models, is performed. The obtained results showed that MCS technique is a precise tool for determining the geoid in Egypt either on regional and/ or local scale with law distortion at check points.

The regional gravimetric geoid solved using boundary-value problems of the potential theory is usually determined in two computational steps: (1) downward continuing ground gravity data onto the geoid using inverse Poisson's integral equation in a mass-free space and (2) evaluating geoidal heights by applying Stokes integral to downward continued gravity. In this contribution, the two integration steps are combined in one step and the so-called one-step integration method in spherical approximation is implemented to compute the regional gravimetric geoid model. Advantages of using the one-step integration method instead of the two integration steps include less computational cost, more stable numerical computation and better utilization of input ground gravity data (reduced in each integration step to avoid edge effects). A discrete form of the onestep integral equation is used to convert mean values of ground gravity anomalies into mean values of geoidal heights. To evaluate mean values of the integral kernel in the vicinity of the computation point, a fast and numerically accurate analytical formula is proposed using planar approximation. The proposed formula is tested to determine the regional gravimetric geoid of the Auvergne test area, France. Results show a good agreement of the estimated geoid with geoidal heights estimated at GNSS-levelling reference points, with the standard deviation for the difference of 3.3 cm. Considering the uncertainty of geoidal heights derived at the GNSS/levelling reference points, one can conclude the geoid models computed by the one-step and two-step integration methods have negligible differences. Thus, the one-step method can be recommended for regional geoid modelling with its methodological and numerical advantages.

The Italian official height system is defined through a high precision levelling network established and maintained by the Istituto Geografico Militare - IGM. During the last 20 years, IGM has performed levelling campaigns on almost the whole peninsular area of Italy with the aim of both densifying the existing network and updating the reference heights. This paper reports about the procedure applied to correct the levelling observations for the gravity effects and the assessment on the results. The needed gravity values were predicted from the Italian gravity dataset (IGD), and both from EGM2008 and XGM2019e high resolution global gravity models. A new formulation of the normal correction as well as the standard orthometric correction were applied. The IGD derived corrections proved to be effective by reducing the misclosure error of critical loops below the tolerance level. Gravity data derived from EGM2008 and XGM2019e proved to be too poor for the correction purposes, as it was ...

The GOCE satellite mission is one of the main achievements of the satellite geodesy for the Earth's gravitational field recovery. Three different approaches have been developed for the estimation of harmonic coefficients from gradiometry data measured on board of GOCE-satellite. In this paper a special version of the space-wise method based on the second method of Neumann for fast determination of the harmonic coefficients C nm , S nm of the Earth's gravitational potential is given based on the radial gravity gradients of the EGG_TRF_2 product, except of two polar gaps filled by radial gradients from the EGM2008 gravity model. In the pre-processing stage GOCE-based second degree radial derivatives were averaged to the regular grid through Kalman static filter with additional Gaussean smoothing of residual radial derivatives. All computations are made by iterations. As the first step the determination of the preliminary NULP-01S model up to degree/order 220 derived from the Gaussean grid of the GOCE radial derivatives with respect to the WGS-84 reference field was developed based only one of the radial gradients EGG_TRF_2 in the EFRF-frame. In the second iteration the same algorithm is applied to build the NULP-02S gravity field model up to degree/order 250 using the same Gaussean grid with respect to the NULP-01S reference field. The NULP-02S model was verified by means of applying various approaches for the construction of the gridded gravity anomalies and geoid heights in the Black Sea area using processing of datasets from six altimetry satellite missions. Comparison of different models with GNSS-levelling data in the USA area demonstrates the independent verification of achieved accuracy of the constructed NULP-02S Earth's gravity field model.

The gravity field is important to determine the size and shape of the earth, to determine the heights, to investigate the structure of the crust, and to explore the mineral wealth. A lot of gravity observations have been taken in Egypt. Those gravity observations are taken over a long-time span with different gravimeters, different references, and different processing methodologies. Those huge amounts of data need to be filtered and unified before using them in precise geoid determination. In the advent of satellite missions for gravity field determination and the yielded global gravity models with good resolution and precision, filtering the terrestrial observed gravity and improving the global gravity models should be done. In this research, the recent Global Geopotential Models (GGMs) will be used in filtering the terrestrial gravity anomalies, and then the reliable terrestrial anomalies will be used in improving the global model gravity anomalies. Four recent GGMs (EGM2008, XGM2019e_2159, GOCO06s, and SGG-UGM 2) are validated using terrestrial gravity anomalies to determine which model best fits the Earth's external gravity field in Egypt. Thousands of terrestrial gravity anomalies are collected as different data sets, so one of the aims of this research is to unify them using the common points among them. Finally, gravity anomalies are used to obtain their corresponding undulation values using a simple model. The obtained results of this study are recommended when collecting all the gravity observations taken all over Egypt to compute precise gravimetric geoid.

, GETECH Ltd. at the University of Leeds, is acknowledged for providing gravity data to the thesis research. Also, Steve Kenyon is thanked for providing the SRTM data. I acknowledge the International Gravimetric Bureau (BGI) for providing a part of the data used in this thesis. I am grateful to my parents Öznur and Mustafa Özöner, and my sister Özge Özöner, for their continuous support, generosity and encouragement during such a long and difficult period. They always did their best to keep me motivated and made me feel that I am very precious for them. They have always believed that I could succeed and this has made me stronger. Never ending thanks are given to them. My parents-in-law Fatmana and Mehmet Erol gave me their support and encouraged me all the time during this study, I am grateful to this. Last, but with great importance, I would like to thank my husband, Serdar Erol, who made this stressful period bearable with his understanding, kindness and patience. His endless support, help, encouragement and love made this thesis possible. I always feel the luckiest person because I share my life with him.

The last geopotential model EGM2008 has been compared with gravity and GPS/leveling data over the Central Mediterranean area. In this area, the gravity field has sharp variations due to strong topography/bathymetry signals and to relevant geophysical features. More than 300.000 gravity values covering the whole area have been considered in the comparisons. These data were used in the last Italian geoid computation and have been validated for datum consistency and for outliers. Further comparisons have been carried out using GPS/leveling data which are homogeneously distributed over the Italian Peninsula. About 1000 available values have been used in the comparison. The results show that, in this area, EGM2008 fits gravity and GPS/leveling data better than other existing global models. The geopotential model performances have been also compared with those of the last Italian geoid estimate, the ITALGEO05 geoid which has been computed in 2005. The statistics of the residuals with GPS/leveling data show that EGM2008 and ITALGEO05 have comparable accuracies, even though EGM2008 is slightly better.

In this paper, some features of the local disturbing potential model developed by the GGI method (based on Geophysical Gravity Inversion) were analyzed. The model was developed for the area of the Western Carpathians covering the Polish-Slovak border. A detailed assessment of the model's property was made regarding the accuracy of the disturbing potential values (height anomalies), gravity values, complete Bouguer anomalies (CBA), and differences between geoid undulations and height anomalies (N − ζ). Obtained accuracies of the GGI quasigeoid model (in terms of standard deviation of the residuals to the reference quasigeoid models) were at the level of ±2.2 cm for Poland and ±0.9 cm for the Slovak area. In terms of gravity, there was shown dependence of the accuracy of the GGI model on the digital elevation model (DEM) resolution, the point height, the density of gravity data used, and used reference density of topography model. The best obtained results of gravity prediction were characterized by an error of approximately 1 mGal. The GGI approach were compared with classical gravity prediction methods (using CBA and topographic-isostatic anomalies supported by Kriging prediction), getting very similar results. On the basis of the GGI model, CBA and differences (N − ζ) were also determined. The strong dependence of resolution of the CBA model obtained by GGI approach, on the size of the constant density zones, has been demonstrated. This significantly reduces the quality of such a model. The crucial importance of the topographic masses density model for both determined values (CBA and (N − ζ)) was also indicated. Therefore, for determining these quantities, all available information on topographic mass densities should be used in modelling.

In this paper, some features of the local disturbing potential model developed by the GGI method (based on Geophysical Gravity Inversion) were analyzed. The model was developed for the area of the Western Carpathians covering the Polish-Slovak border. A detailed assessment of the model's property was made regarding the accuracy of the disturbing potential values (height anomalies), gravity values, complete Bouguer anomalies (CBA), and differences between geoid undulations and height anomalies (N − ζ). Obtained accuracies of the GGI quasigeoid model (in terms of standard deviation of the residuals to the reference quasigeoid models) were at the level of ±2.2 cm for Poland and ±0.9 cm for the Slovak area. In terms of gravity, there was shown dependence of the accuracy of the GGI model on the digital elevation model (DEM) resolution, the point height, the density of gravity data used, and used reference density of topography model. The best obtained results of gravity prediction were characterized by an error of approximately 1 mGal. The GGI approach were compared with classical gravity prediction methods (using CBA and topographic-isostatic anomalies supported by Kriging prediction), getting very similar results. On the basis of the GGI model, CBA and differences (N − ζ) were also determined. The strong dependence of resolution of the CBA model obtained by GGI approach, on the size of the constant density zones, has been demonstrated. This significantly reduces the quality of such a model. The crucial importance of the topographic masses density model for both determined values (CBA and (N − ζ)) was also indicated. Therefore, for determining these quantities, all available information on topographic mass densities should be used in modelling.

Terrestrial measurements can provide accurate gravity data, but it is costly and time-consuming for large and remote area. Airborne gravity measurements have actually been carried out in Indonesia since 2008 by Technical University of Denmark (DTU) in collaboration with the Geospatial Information Agency (BIG). Purpose of the project was to develop a geoid model used for converting elevations from GPS/GNSS measurements that refers to ellipsoid to orthometric elevations that refer to sea level. The data can actually be explored so that it can be used for geophysical and other geoscience purposes, but the data must be carefully treated and extracted into observational gravity data. This study aims to improve the accuracy of gravity airborne data to produce an accurate complete Bouguer anomaly map. The data used in this study were airborne gravity data over Province of Southeast Sulawesi collected on September 29, 2008 to October 1, 2008.  Variation in flight height at the time of conse...

Several post-processing methods have been developed over the last years in order to take into consideration topography within satellite-based solar radiation maps using digital elevation models (DEM). If the main part of these procedures is to estimate the obstructed horizon around each DEM point of a given region so as to consider terrain-based shading effects, the size of the area can also limit this implementation. That is why we have developed a new efficient horizon model based on the DEM retrieved during the Shuttle Radar Topography Mission (SRTM). In order to be usable at any world location with the same expected accuracy, this model is only derived from mathematical statements without any kind of empirical approximation. Validation against in situ horizons and comparison with some other models have finally shown this one presents both better accuracy (RMSE of 1.555 • against 1.712 • or more) and lower computation time (at least 4 times faster). Furthermore, in the case of very large areas, we propose an optimization procedure allowing the user to knowingly alter the modeling error in order to reduce processing time. Finally, using in situ data, we have also developed a method for predicting the repercussion of the original SRTM DEM error on the final horizon precision.

One of the main challenges in physical geodesy today is achieving the 1-centimetre gravimetric quasi-geoid model, since a model of such accuracy could be used in the definition and realisation of the height reference frame. One of the main obstacles in this challenge is the lack of terrestrial gravity data or its uneven distribution. Therefore, the main question arises: what density of gravity points is necessary in order to obtain a gravimetric quasi-geoid model with an accuracy of 1 centimetre or even better? In this simulated study, the results show a trend of decreasing RMS related to the sparser dataset used in computation, leading to the conclusion that the determination of the sub-centimetre quasi-geoid model is no longer a theory, but can be achieved with the exact density of terrestrial gravity data: gravity observation stations have to be distributed no farther than 500 m from one another. When dealing with a very rough topography, the distance should be even shorter.

The heights determined by  Global Positioning System (GPS) refer to the ellipsoid called the World Geodetic System 1984 (WGS84).  Global Geopotential Models (GGMs) that are available on GNSS commercial software are generally used to transform  ellipsoidal heights to orthometric heights. In this study, the geoid heights of GPS/Levelling were computed to evaluate the accuracy of the geoid heights obtained from two GGMs, namely, the Earth Gravitational Model 96 (EGM96) and the Earth Gravitational Model 08 (EGM08). Seventeen (17) GPS/Levelling stations of the High Accuracy Reference Network (HARN) over Egypt were used for this purpose. The standard deviations for the differences between the geoid heights obtained through GPS/Levelling and those obtained from EGM96 and EGM08 were determined as  ± 1.212 m and ± 0.543 m, respectively. This research confirms that the geoid heights obtained from EGM08 are closer to the geoid heights determined using GPS/Levelling over Egypt.

The development of a high-resolution and high accuracy geoid model is becoming nowadays a fundamental component of any modern geodetic infrastructure. The Kingdom of Saudi Arabia (KSA) has devoted the last decade a significant number of resources and manpower to collect high-quality land and airborne gravity data as well as GNSS/Levelling observations to create a state-of-the-art geoid model as a fundamental part of the Saudi Arabia National Spatial Reference System (SANSRS). In that frame, this work focuses on the collected gravity, terrain, and GNSS/Levelling data for the area under study, and their pre-processing in terms of horizontal, vertical and gravity reference system homogenization, blunder detection and removal. Given the availability of these data the latest gravimetric geoid model for the KSA is developed.The gravity data pre-processing relied on the available metadata to collect information about the horizontal, vertical and gravity reference system. Hence, all this in...

Nowadays the Global Positioning System (GPS) is one of the most favorite techniques in practical geodesy. A major dilemma in GPS surveying lies in its ellipsoidal-based type of heights, while in engineering practice orthometric heights are usually utilized. Thus, it is important to convert GPS heights into orthometric heights through applying an accurate geoid model. The objectives of this paper are to model a local geoid in the study area using GPS/levelling technique, and to evaluate the performance of several Global Geopotential Models (GGMs) particularly the OUS-91A, EGM96 and EGM2008 in the study area, which is located in the northern Egypt at Rosetta zone area. The accomplished results show that the EGM 2008 represents the most precise global geopotential model to be used for geoid determination in Egypt. Furthermore, the achievable accuracy of local geoid determination in the study area after using regression method models is ranges between 0.059 meter to -0.083 meter, with a...

The Italian official height system is defined through a high precision levelling network established and maintained by the Istituto Geografico Militare - IGM. During the last 20 years, IGM has performed levelling campaigns on almost the whole peninsular area of Italy with the aim of both densifying the existing network and updating the reference heights. This paper reports about the procedure applied to correct the levelling observations for the gravity effects and the assessment on the results. The needed gravity values were predicted from the Italian gravity dataset (IGD), and both from EGM2008 and XGM2019e high resolution global gravity models. A new formulation of the normal correction as well as the standard orthometric correction were applied. The IGD derived corrections proved to be effective by reducing the misclosure error of critical loops below the tolerance level. Gravity data derived from EGM2008 and XGM2019e proved to be too poor for the correction purposes, as it was ...

In this study, a comparison was held among the Stokes, Hotine and Deflection-geoid formulae for gravimetric geoid modelling in Egypt, regarding the resulting statistical geoid features and accuracies. For this purpose, relevant 2' x 2' grids of gravity anomalies, graviv disturbances and vertical deflection components were input, yielding 2' x 2' detailed geoid models. The comparison showed that the Stokes and Hofine methods yielded practically the same geoid features and accuracy. The Deflection-geoid technique showed different geoid features and significalrtly better accuracy. The comparison of the three geoid solutions at independent GPSLeveling control, showed that the mean and standard deviation of discrepancies, pertaining to the Deflection-geoid formula, are less by about 36% and 8%, respectively, compared to those relevant to the other two formulae. Therefore, the vertical deflections are recommended as homogeneous input data lype in gravimetric geoid detehination. cjy; u'-$ (Deflection-geoid) J (Hotine) J (Stokes) &= & ~JUI ,spj-1 3

The development of a high-resolution and high accuracy geoid model is becoming nowadays a fundamental component of any modern geodetic infrastructure. The Kingdom of Saudi Arabia (KSA) has devoted the last decade a significant number of resources and manpower to collect high-quality land and airborne gravity data as well as GNSS/Levelling observations to create a state-of-the-art geoid model as a fundamental part of the Saudi Arabia National Spatial Reference System (SANSRS). In that frame, this work focuses on the collected gravity, terrain, and GNSS/Levelling data for the area under study, and their pre-processing in terms of horizontal, vertical and gravity reference system homogenization, blunder detection and removal. Given the availability of these data the latest gravimetric geoid model for the KSA is developed.The gravity data pre-processing relied on the available metadata to collect information about the horizontal, vertical and gravity reference system. Hence, all this in...

Each gravity observation technique has different parameters and contributes to different pieces of the gravity spectrum. This means that no one gravity dataset is able to model the Earth’s gravity field completely and the best gravity map is one derived from many sources. Therefore, one of the challenges in gravity field modelling is combining multiple types of heterogeneous gravity datasets.  The aim of this study is to determine the optimal method to produce a single gravity map of the Canterbury case study area, for the purposes of use in geoid modelling.  This objective is realised through the identification and application of a four-step integration process: purpose, data, combination and assessment. This includes the evaluation of three integration methods: natural neighbour, ordinary kriging and least squares collocation.  As geoid modelling requires the combination of gravity datasets collected at various altitudes, it is beneficial to be able to combine the dataset using an...

The Earth's potential information is important for exploration of the Earth's gravity field. The techniques of measuring the Earth's gravity using the terrestrial and ship borne technique are time consuming and have limitation on the vast area. With the space-based measuring technique, these limitations can be overcome. The satellite gravity missions such as Challenging Mini-satellite Payload (CHAMP), Gravity Recovery and Climate Experiment (GRACE), and Gravity-Field and Steady-State Ocean Circulation Explorer Mission (GOCE) has introduced a better way in providing the information on the Earth's gravity field. From these satellite gravity missions, the Global Geopotential Models (GGMs) has been produced from the spherical harmonics coefficient data type. The information of the gravity anomaly can be used to predict the bathymetry because the gravity anomaly and bathymetry have relationships between each other. There are many GGMs that have been published and each of the models gives a different value of the Earth's gravity field information. Therefore, this study is conducted to assess the most reliable GGM for the Malaysian Seas. This study covered the area of the marine area on the South China Sea at Sabah extent. Seven GGMs have been selected from the three satellite gravity missions. The gravity anomalies derived from the GGMs are compared with the airborne gravity anomaly, in order to figure out the correlation (R 2) and the root mean square error (RMSE) of the data. From these assessments, the most suitable GGMs for the study area is GOCE model, GO_CONS_GCF_2_TIMR4 with the R 2 and RMSE value of 0.7899 and 9.886 mGal, respectively. This selected model will be used in the estimating the bathymetry for Malaysian Seas in future.

The main goal of the study is the calculation of a high resolution dataset that models the geoid for Greece using several kind of data collected by the Hellenic Military Geographical Service (HMGS). In situ gravity measurements and GPS/levelling on triangulation points plays a central role in the formation of covariance and cross-covariance functions used for the calculation of high frequency residuals. Also global models such as EGM2008 and EIGEN-6C4 contribute in the analysis of collected data and in the removal of low frequencies. A database of older gravity measurements completes and guarantees the data coverage of the whole region leading to a high resolution exported product. The whole project is based on the Remove-Compute-Restore (RCR) technique and the Least Squares Collocation (LSC) method is used at its core during the computation of the residual geoid height. In order to fulfill the RCR technique topographic corrections have been calculated on each measured point and the indirect effect has been computed for the total region. Rasters of the above have been extracted for visualization and analysis. The final product has been transformed through a parametric model for orthometric height adaption. Several scripts have been developed in Matlab and Python for the reckonings as no commercial or scientific software was used. Data combination and visualization in raster format has been made using the ESRI ArcGIS software. The study concludes to three different beta geoid height models depending on the RCR usage for further discussion and which will be evaluated in the light of new data collection.

The aim of this study is to analyze the best fitting geoid model for an arbitrary selected area of investigation. The Digital Finite Element Height Reference Surface (DFHRS) method, developed by Hochschule Karlsruhe has been chosen to define the height reference surface for the territory of Kosovo, which should be defined as national vertical datum. This approach allows the conversion of ellipsoidal heights determined by GPS into the standard heights, which refer to the height reference surface (HRS) of an orthometric, or normal surface system. The DFHRS is defined as continues HRS in arbitrary large areas by bivariate polynomials over an irregular grid (Jager, Schneid 2001). The DFHRS approach uses wide range of the input data (Geometric and Physical) and in our case there were 30 GPS/leveling height data as well as physical derivatives from different global geopotential models. The proposed approach has been successfully applied and results are compared to actual normal heights an...

The purpose of this paper is to determine a high-precision the Republic of Albania geoid model
in order to convert the ellipsoidal heights into orthometric heights. The Digital Finite-Element Height
Reference Surface method was used to solve this problem. The following models were chosen as the Republic
of Albania global geopotential models: EGM96, EGG97 and EGM2008. Calculations of mentioned models
were performed for 151 points of the Republic of Albania territory using the Digital Finite-Element Height
Reference Surface method to determine the height reference surface. The minimum, maximum, mean values,
standard deviation and mean quadratic error of the models in question were identified. The results showed that
EGM96 had a standard deviation of 8 cm, EGG97 had a standard deviation of 4.8 cm and EGM2008 had a
standard deviation of 4.2 cm as the best model. Thus, it was proved that EGM2008 has topographic distribution
and fits well in the extension of the territory of the Republic of Albania.

The existing Bouguer anomaly map, which covers the territory of the Republic of Slovenia is a few decades old. Since then, quite a few new gravimetric measurements (data) for the territory of Slovenia as well as high quality digital terrain models that are needed for creating such a map have been made available. The methodology and standards for creating gravity anomaly maps are also changing. Thus, the national Bouguer anomaly map was updated. There were some gross errors detected in the set of old gravimetric data. Additionally, the influence of new updated gravimetric data was analyzed. The comparison of the various maps and the analysis of the influence of input gravimetric data indicates that the new gravimetric data of Slovenia has a significant influence on the creation of the gravimetric anomaly maps for Slovenia (even over 30 mGals at some points).

This paper presents an attempt to consider whether it is possible to determine a geoid at the centimetre level in the territory of Egypt based on recently available global and local gravity field data. The paper has two main objectives. Firstly, the paper overviews previously published geoid solutions, while the second objective investigates the performance of the recent global geopotential models (GGM) in Egypt. The existing geoid solutions have illustrated that there is an insufficient distribution of data which is sampled inconsistently. At this time, data deficiency still exists, and to overcome it, we have selected a "data window" and applied the Least Square Collocation (LSC) technique. The outcome from LSC was interesting and acceptable, and we obtained a "sample" geoid that has a standard deviation of 11 cm for the external control points.

Local gravimetric geoid determination was carried out in the test region of Soskut. In this solution 1610 point gravity data as well as a high resolution DTM were used. The latest Hungarian gravimetric geoid solution of the BUTE was chosen as a reference for the local geoid determination. From the point gravity anomalies and the elevation data a surface density model has been derived. Afterwards the surface density data were used for the calculation of the terrain-corrections and the indirect effect. The calculations were made with constant density values too. The difference between the two solutions reached the maximum of 2 mm.

Vertical deflection can be determined by geometrical and physical measurement. In geometrical way, vertical deflection is obtained by comparing astronomical coordinate and geodetical coordinate. In physical way, vertical deflection can be computed from gravity measurement. In the past, vertical deflection was computed from gravity anomaly data. Gravity anomaly data measurement is difficult because it need reduction of gravity from surface of the earth to the geoid using orthometric height from spirit level measurement. In modern era, gravity anomaly data may be replaced by gravity disturbance data whose only required gravity and GNSS (Global Navigation Satellite System) measurement. This research aims to determine vertical deflection in Semarang City from terrestrial gravity disturbance data. The gravity data were measured in March of 2016. Formula of Vening Meinesz that usually used for vertical deflection was replaced by new formula that generated from derivation of function of Ho...

Terrestrial measurements can provide accurate gravity data, but it is costly and time-consuming for large and remote area. Airborne gravity measurements have actually been carried out in Indonesia since 2008 by Technical University of Denmark (DTU) in collaboration with the Geospatial Information Agency (BIG). Purpose of the project was to develop a geoid model used for converting elevations from GPS/GNSS measurements that refers to ellipsoid to orthometric elevations that refer to sea level. The data can actually be explored so that it can be used for geophysical and other geoscience purposes, but the data must be carefully treated and extracted into observational gravity data. This study aims to improve the accuracy of gravity airborne data to produce an accurate complete Bouguer anomaly map. The data used in this study were airborne gravity data over Province of Southeast Sulawesi collected on September 29, 2008 to October 1, 2008.  Variation in flight height at the time of conse...

Original scientific paper Modern scientific research is largely focused on interdisciplinary aspect of individual technical disciplines in research and scientific approach in discovering new insights. In the field of geo-sciences, there is a wide range of mutual methods of measuring physical parameters needed for the determination of geologic and structural composition of the underground area, the density of surface layers, the Earth's gravity field, i.e. local gravity changes. This paper explains the influences of topography of the Earth masses in the determination of vertical deflections needed for the transformation of the heights by means of trigonometric levelling methods in the real gravity field acceleration.

The virtual point-mass method has been widely used in dealing with the approximation of the local gravity field which is a difficult problem in internal currently. In this paper, the approximation theory of point-mass model is briefly introduced, and the characteristics of the elements in the coefficient matrix for the model construction are analyzed by numerical calculation. The observations of gravity anomaly is simulated from EGM2008 with degree and order 720 and the approximated region is 32~34 N o o and 103~105 E o o. A four-tier point-mass model which is on the base of the geopotential model with degree and order 36 from low frequency to high frequency is applied to approximate the local earth's gravity field. The results of the experiments show that the truncation error of gravity disturbance created by using the point-mass model is less than 2 mGal on the radial direction and there is an optimal truncation error for some certain spectrum gravity field in the space.

The Global Positioning System (GPS) has been extensively utilized in surveying and mapping applications worldwide and in Egypt, which results in a crucial need for precise geoid models. Traditionally, global and local gravimetric quasi-geoid models are determined and, then, fitted to GPS/leveling data. In the context of precise geoid determination, a quasi-geoid determination has to be converted to a geoid model prior to fitting process. Neglecting such a transforming process leads to the absorption of these effects in the fitting residuals that, hence, may produce misleading results and interpretations. The geoid/quasi-geoid separations in Egypt have been computed and found to range from-0.074 m to 0.367 m, with an average of 0.060 m. Additionally, when the quasi-geoid to geoid conversion is performed, it has been found that the undulation of the EIGEN-CG01C global quasi-geoid model when compared to true undulations of HARN stations, have been decreased, in the mean sense, from 0.52 m to 0.46 m. That means an average improvement of accuracy of about 11%. Based on the obtained results, it can be concluded that the conversion of the quasi-geoid to the geoid is quite significant and it should be a regular step in precise geoid determination in Egypt.

Validation of recent GOCE/GRACE geopotential models over Khartoum state - SudanThis paper evaluates a number of latest releases of GOCE/GRACE global geopotential models (GGMs) using the GPS-levelling geometric geoid heights, terrestrial gravity data and existing local gravimetric models. We investigate each global model at every 5 degree of spherical harmonics. Our analysis shows that the satellite-only models derived by space-wise and time-wise approaches (SPW_R1, SPW_R2 TIM_R1 and TIM_R2), GOCO01S together with EGM08 (combined model) are very distinct and consistent to the local data, which guarantees one of them to be selected as the best of candidate models and then to be utilized in our further geoid studies. One of Satellite-only models will be employed for acquiring the long wavelength geoid component which is one of major steps in the geoid determination. EGM08 will be used to compensate and restore the missing gravity data points in the un-surveyed parts within the target a...

Poisson Downward Continuation Solution by the Jacobi MethodDownward continuation is a continuing problem in geodesy and geophysics. Inversion of the discrete form of the Poisson integration process provides a numerical solution to the problem, but because the B matrix that defines the discrete Poisson integration is not always well conditioned the solution may be noisy in situations where the discretization step is small and in areas containing large heights. We provide two remedies, both in the context of the Jacobi iterative solution to the Poisson downward continuation problem. First, we suggest testing according to the upward continued result from each solution, rather then testing between successive solutions on the geoid, so that choice of a tolerance for the convergence of the iterative method is more meaningful and intuitive. Second, we show how a tolerance that reflects the conditioning of the B matrix can regularize the solution, and suggest an approximate way of choosing ...

This work is licensed under CC BY 4.0 The description of the earth's gravity field is usually expressed in terms of spherical harmonic coefficients, derived from global geopotential models. These coefficients may be used to evaluate such quantities as geoid undulations, gravity anomalies, gravity disturbances, deflection of the vertical, etc. To accomplish this, a global reference normal ellipsoid, such as WGS84 and GRS80, is required to provide the computing reference surface. These global ellipsoids, however, may not always provide the best fit of the local geoid and may provide results that are aliased. In this study, a regional or localized geocentric level ellipsoid is used alongside the EGM2008 to compute gravity field functionals in the state of Johor. Residual gravity field quantities are then computed using GNSS-levelled and raw gravity data, and the results are compared with both the WGS84 and the GRS80 equipotential surfaces. It is demonstrated that regional level ellipsoids may be used to compute gravity field functionals with a better fit, provided the zero-degree spherical harmonic is considered. The resulting residual quantities are smaller when compared with those obtained with global ellipsoids. It is expected that when the removecompute-restore method is employed with such residuals, the numerical quadrature of the Stoke's integral may be evaluated on reduced gravity anomalies that are smoother compared to when global equipotential surfaces are used

This work is licensed under CC BY 4.0 The description of the earth's gravity field is usually expressed in terms of spherical harmonic coefficients, derived from global geopotential models. These coefficients may be used to evaluate such quantities as geoid undulations, gravity anomalies, gravity disturbances, deflection of the vertical, etc. To accomplish this, a global reference normal ellipsoid, such as WGS84 and GRS80, is required to provide the computing reference surface. These global ellipsoids, however, may not always provide the best fit of the local geoid and may provide results that are aliased. In this study, a regional or localized geocentric level ellipsoid is used alongside the EGM2008 to compute gravity field functionals in the state of Johor. Residual gravity field quantities are then computed using GNSS-levelled and raw gravity data, and the results are compared with both the WGS84 and the GRS80 equipotential surfaces. It is demonstrated that regional level ellipsoids may be used to compute gravity field functionals with a better fit, provided the zero-degree spherical harmonic is considered. The resulting residual quantities are smaller when compared with those obtained with global ellipsoids. It is expected that when the removecompute-restore method is employed with such residuals, the numerical quadrature of the Stoke's integral may be evaluated on reduced gravity anomalies that are smoother compared to when global equipotential surfaces are used

A high-resolution gravimetric quasigeoid model for Vietnam and its surrounding areas was determined based on new gravity data. A set of 29,121 land gravity measurements was used in combination with fill-in data where no gravity data existed. Global Gravity field Models plus Residual Terrain Model effects and gravity field derived from altimetry satellites were used to provide the fill-in information over land and marine areas. A mixed model up to degree/order 719 was used for the removal of the long and medium wavelengths and the calculation of the quasigeoid restore effects. The residual height anomalies have been determined employing the Stokes integral using the Fast Fourier Transform approach and deterministic kernel modification proposed by Wong-Gore, as well as by means of Least-Squares Collocation. The accuracy of the resulting quasigeoid models was evaluated by comparing with height anomalies derived from 812 co-located GNSS/levelling points. Results are very similar; both local quasigeoid models have a standard deviation of 9.7 cm and 50 cm in mean bias when compared to the GNSS/levelling points. This new local quasigeoid model for Vietnam represents a significant improvement over the global models EIGEN-6C4 and EGM2008, which have standard deviations of 19.2 and 29.1 cm, respectively, for this region.

In the course of the IAG African Geoid Project, it is needed to study the impact of the lakes on the gravity reduction and geoidal heights. The aim of this paper is to study the impact of the water in Nasser Lake on gravity reduction and geoidal heights for Egypt. The determination of the gravimetric geoid is based on the well-known remove-restore technique. The problem of the lakes occurs because the popular programs widely used in practice (e.g., TC-program (Forsberg, 1984)) assume that all positive elevations are filled with rock topography, and all negative elevations are filled with ocean water. This is, however, not true for the case of Nasser Lake, which lies completely above sea level, at about 180 m elevation, with a water depth of about 20 m. The paper presents an approach on estimating the impact of Nasser Lake on gravity reduction and geoidal heights using TC-program with some tricky cases. The results show that the impact of Nasser Lake on both gravity anomalies and geoid undulation is limited to the area of the lake. The impact of Nasser Lake on the gravity anomalies is in the order of sub mgal, while the impact of Nasser lake on the geoid undulation is significant and reaches few centimeters.

Validation of recent GOCE/GRACE geopotential models over Khartoum state - SudanThis paper evaluates a number of latest releases of GOCE/GRACE global geopotential models (GGMs) using the GPS-levelling geometric geoid heights, terrestrial gravity data and existing local gravimetric models. We investigate each global model at every 5 degree of spherical harmonics. Our analysis shows that the satellite-only models derived by space-wise and time-wise approaches (SPW_R1, SPW_R2 TIM_R1 and TIM_R2), GOCO01S together with EGM08 (combined model) are very distinct and consistent to the local data, which guarantees one of them to be selected as the best of candidate models and then to be utilized in our further geoid studies. One of Satellite-only models will be employed for acquiring the long wavelength geoid component which is one of major steps in the geoid determination. EGM08 will be used to compensate and restore the missing gravity data points in the un-surveyed parts within the target a...

Engineering activities and development plans rely chiefly on the availability of precise spatial data, which in turn depends the accessibility of an accurate Geodetic Data Infrastructure (GDI) as reference frames for all geospatial measurement GDI is considered a key component in the undergoing National Spatial Data Infrastructure (NSDI) for Egypt. The current study proposes a structure of a GDI in Egypt that has multi database combining several types of geodetic datasets within a Geographic Information Systems (GIS) environment. The study develops such a GDI for the Nile delta region, as a case study, containing the available heterogeneous geodetic datasets. The problems and advantag recommendations are presented, too, for constructing an integrated GDI over a national scale in Egypt.

The project of modeling a gravimetric quasigeoid for Romania is carried out in stages, based on the gravimetric measurements in the area of each county, in the gravimetric points of the 0, 1st and 2nd order gravimetric network, in the checkpoints points (with GNSS/levelling data) and also in the new designed points. The remove-compute-restore technique was used to remove the long-wavelength component from Global Geopotential Model (GGM) and effect of the shortwavelength signal by applying terrain corrections, to compute residual geoid heights, and to restore the effect of the GGM and topography. As a preliminary analysis for the development and evaluation of a precise quasigeoid for Romania, different gravimetric geoid solutions were computed using two gravity reduction techniques (Helmert’s second condensation method and the Residual Terrain Model method), and two methodologies to compute residual geoid heights and residual height anomalies (Stokes' integral and Least Squares C...