Geoid

The use of GPS for the estimation of orthometric heights in a given region, with the help of existing levelling data requires the determination of a local geoid and the bias between the local levelling and the one implicitly defined when the geoid is calculated which is generally based on the gravity anomalies data. The heights of new data can

The RTK networks are expanding as the favored GNSS solution. There are two main requirements needed in their operations, the first is the transformation of the observed coordinates into the local coordinate system, and the second is the value of the physical height of the observed point. These requirements need to be achieved in real time environment with minimal interaction from the user side. The RTCM 3.1 standard addressed these two problems in the development of its transformation messages to accomplish these two main tasks.
This paper explains the communication in the RTK network, which is the base of the client/server architecture for transformation messages and shows the different ways of communications between the client (rover) and the server (RTK control server).
The mathematical geodesy foundation in RTCM 3.1 transformation messages is presented, the conversion operations from geographic to geocentric and the transformation operations from geocentric to geocentric, and the similarity equation for seven parameters are also explained.
The paper provides the algorithms and data structures associated with the transformation messages, and how the transformation engine in the server side manipulates the received observed coordinates, and transforms it to the desired coordinate system.
The implementation of the physical height is explained, and how the transformation transmits the physical height (for height indicator =1) or the geoid height (for height indicator =2).
The different types of transformation messages are presented and their combination of use. The application of the mathematical geodesy algorithms to the observed coordinates is explained, and how the transformation parameters are computed and applied.
The implementation for some cases for realization of the transformation messages is presented for the server side. The manipulation of the client to the received transformation message is declared, and how the rover uses these data to present the observed data in the required coordinate system.

The Global Positioning System (GPS) satellite-based technology has been utilized extensively, in the last few years, for a wide range of Geomatics and Geographic Information Systems (GIS) applications within Makkah metropolitan area. One of the chief concerns dealing with GPS-based heights is their conversion to the Mean Sea Level (MSL) heights that normally used in surveying and mapping through a geoid model. Currently, there is no local geoid model covers Makkah administrative area. Global Geopotential Models (GGM) provide appropriate cost-effective transformation alternatives in such case. A number of the most-recent GGM have been obtained and utilized for representing the Earth's gravity field and converting GPS heights into MSL heights in Makkah city. Terrestrial first-order control points have been used to assess the external quality of GGM's results in order to choose the optimum GGM for geomatics application within the study area. The attained results showed that the EGM2008 model is the most precise GGM within Makkah metropolitan area, with an overall accuracy level about ± 0.16 meter. Moreover, a new modified geoid model was developed for Makkah city, that is based on integrating the original EGM2008 with precise local geodetic datasets. Accordingly, it is recommended to apply the modified version of EGM2008 geoid model in GPS, GIS, surveying, and geomatics projects in Makkah until a complete national Saudi geoid model is developed.

The use of GPS for the estimation of orthometric heights in a given region, with the help of existing levelling data requires the determination of a local geoid and the bias between the local levelling and the one implicitly defined when the geoid is calculated which is generally based on the gravity anomalies data. The heights of new data can be collected swiftly without using the orthometric heights from levelling; it is what one calls commonly levelling by GPS. In this framework, the Least Squares Collocation method (LSC) has been used to evaluate the quality of the available GPS-Levelling data, to determine a gravimetric geoid in the North region of Algeria and to estimate the constant datum bias. The data used in the setting of this study are: The geopotential model EGM96, a total number of 2534 gravity anomalies, as well as 43 GPS points connected to the geodetic network levelling present on the whole North part of Algerian.

In this study, we have conducted an investigation on the impact of the coordinates’ transformation on local variations of geometric geoid. The study area is limited by 1°43′12″ to 4°00′37″ East and 13°01′57″ to 14°31′20″ North in the southwest of the Niger Republic. We used 39 network GPS/levelling points established by the Japan International Cooperation Agency (JICA) and the National Geographic Institute of Niger (IGNN), including the DOPPLER point ANG302/no.65. Using other coordinates of point no. 65 provided by IGNN, we transformed the points into WGS84 and computed a new geometric geoid model. The comparison of the new model with EGM2008 geoid up to d/o 2160 gives the STD of 15 cm and the RMS of 16cm. Local variations of the geometric geoids, were compared to that of EGM2008 geoid. The comparison through basic statistics, trend lines and 3D overlaps, showed a similar trend between the geometric geoid from the transformed coordinates and that of EGM2008. On the contrary, the JICA-IGNN geometric geoid generated an opposite and exaggerated trend. The Jarque-Bera test confirms that the three samples follow a normal distribution at the significance level α = 5%. The equality of variances between EGM2008 and JICA-IGNN geoids has been rejected by the Fisher’s F-Test/two-tailed at α = 10%. However the test confirms the variances equality between EGM2008 and the transformed geometric geoid at α = 5% and α = 10%. The two-tailed Student’s T-Test at α = 5% also confirms the equality of means between EGM2008 geoid and transformed geometric geoid samples.

Determination of the ellipsoidal height by Global Navigation Satellite Systems (GNSS) is becoming better known and used for purposes of leveling with the aid of geoid models. However, the disadvantage of this method is the quality of the geoid models, which degrade heights and limit the application of the method. In order to provide better quality in transforming height using GNSS leveling, this research aims to develop a hybridization methodology of gravimetric geoid models EGM08, MAPGEO2015 and GEOIDSP2014 for the State of São Paulo, providing more consistent models with GNSS technology. Radial Basis Function (RBF) neural networks were used to obtain the corrector surface, based on differences between geoid model undulations and the undulations obtained by GNSS tracking in benchmarks. The experiments showed that the most suitable interpolation for correction modeling is the linear RBF. Checkpoints indicate that the geoid hybrid models feature root mean square deviation ± 0.107, ± 0.104 and ± 0.098 m, respectively. The results shows an improvement of 30 to 40% in consistencies compared with the gravimetric geoids, providing users with better quality in transformation of geometric to orthometric heights.

Due to current situation in Kosova with the national geoid, which is still under development, utilization of global EGM's is more than necessary. In this research, multiband raster datasets with 1km spatial resolution of existing EGM 2008, 1996, and 1984 have been developed. Maximal and minimal differences between the models, range of extreme values of geoid heights in three geoid models, mean heights and other data calculations, are given in this paper. Calculation have been performed with online geoid calculator GeoidEval, in grid of 10893 points. Based on developed multiband raster model dataset and calculated differences between the models, three maps for the territory of Kosova were compiled. Main purpose of performed research analyses and developed GIS datasets for geoid heights, is free and open for usage by the geo community in Kosova, till the establishing of state gravimetrical network and model.

The whole process of the approximation of
the anomalous gravity potential is reviewed from the
point of view of available geodetic data and their pri-
mary combination to define a mathematical standard
model, i.e. a suitable boundary value problem.
The construction of global models is then outlined,
showing where recent theorems of existence, unique-
ness and stability play a role. The local modelling
is also summarized, clarifying the conditions under
which it has meaning. Finally, some proposal to fol-
low a slightly different line in global modelling are put
forward, while open problems in local modelling that
need to be further studied are highlighted.

Abstract. Global gravity models are being developed according to new data sets available from satellite gravity
missions and terrestrial/marine gravity data which are provided by different countries. Some countries do
not provide all their available data and the global gravity models have many vague computational methods.
Therefore, the models need to be evaluated locally before using. It is generally understood that the accuracy
of global gravity models is enough for local (civil, mining, construction, etc.) projects, however, our results
in Iran show that the differences between synthesized values and observation data reach up to ~300 mGal
for gravity anomalies and ~2 m for geoid heights. Even by applying the residual topographical correction
to synthetized gravity anomalies, the differences are still notable. The accuracy of global gravity models for
predicting marine gravity anomalies is also investigated in Persian Gulf and the results show differences of
~140 mGal in coastal areas. The results of evaluating selected global gravity models in Iran indicate that the
EIGEN-6C4 achieves the lowest RMS for estimating the geoid heights. EGM08 predicts the closest results to
terrestrial gravity anomalies. DIR-R5 GOCE satellite-only model estimates the low-frequency part of gravity
field more accurately. The best prediction of marine gravity anomalies is also achieved by EGM08.

Geoid data are more sensitive to density distributions deep within the Earth, thus the data are useful for studying the internal processes of the Earth leading to formation of geological structures. In this paper, we present much improved version of high resolution (10 _ 10) geoid anomaly map of the northern Indian Ocean generated from the altimeter data obtained from Geodetic Missions of GEOSAT and ERS-1 along with ERS-2, TOPEX/POSIDEON and JASON satellites. The geoid map of the Indian Ocean is dominated by a significant low of _106 m south of Sri Lanka, named as the Indian Ocean Geoid Low (IOGL), whose origin is not clearly known yet. The residual geoid data are retrieved from the geoid data by removing the long-wavelength core–mantle density effects using recent spherical harmonic coefficients of Earth Gravity Model 2008 (EGM2008) up to degree and order 50 from the observed geoid data. The coefficients are
smoothly rolled off between degrees 30–70 in order to avoid artifacts related to the sharp truncation at degree 50. With this process we observed significant improvement in the residual geoid data when compared to the previous low-spatial resolution maps. The previous version was superposed by systematic broad regional highs and lows (like checker board) with amplitude up to ±12 m, though the trends of geoid in general match in both versions. These methodical artifacts in the previous version may have arisen due to the use of old Rapp’s geo-potential model coefficients, as well as sharp truncation of reference model at degree and order 50. Geoid anomalies are converted to free-air gravity anomalies and validated with cross-over corrected ship-borne gravity data of the Arabian Sea and Bay of Bengal. The present satellite derived gravity data matches well with the ship-borne data with Root Mean Square Error (RMSE) of 5.1–7.8 mGal, and this is found to be within the error limits when compared with other globally
available satellite data. Spectral analysis of ship-borne and satellite data suggested that the satellite gravity data have a resolution down to 16–18 km. Further, the geoid, residual geoid and gravity anomalies are integrated with seismic data along two profiles in the Bay of Bengal and Arabian Sea, and inferences have been made in terms of density distributions at different depths. The new residual geoid anomaly map shows excellent correlation with regional tectonic features such as Sunda subduction zone, volcanic traces (Chagos–Laccadive, Ninetyeast and 85_E ridges) and mid-ocean ridge systems (Central Indian and Carlsberg ridges).

Urban planning and hydraulic risk management are a worldwide necessity which is best achieved when natural and artificial elements located closely to watercourses are known in great detail. A geodatabase is a practical tool to store and manage such information. When working at small scales, however, any well established methodology exists to map the position and the height of the various elements with centimetric accuracy. For this purpose we propose a methodology that we tested on the Arno river (Italy) and its most urbanized tributaries, a demonstrative case of hydrological risk around large fluvial systems. We surveyed 116 km of river traits to collect GPS measurements and information about all the natural and artificial elements connected to hydraulic risk and fluvial dynamics. The mapped elements include (but are not limited to) buildings, assets, bridges, hydraulic works, weirs, drainage outlets, dikes, riverbanks, structural damages, fluvial bars and eroding banks. All these elements were mapped with high accuracy, in particular a local geoid model, related only to the study area, was developed to obtain orthometric heights affected with errors  0.05 m. Consequently a GIS geodatabase was built to visualize the spatial distribution of the mapped elements and to store a series of technical data, including the present preservation condition for man-made objects. The geodatabase provides an overview of the territories connected with the fluvial dynamics, highlighting that in the studied territory, the more is urbanized, the more it is exposed to hydraulic risk. In a similar context, the geodatabase itself represents a useful tool for the management of the hydrological risk and for hydraulic policy and urban planning.

Nowadays, Global Geopotential Models (GGMs) can be used as a reference to develop more detailed regional/local geoids, or they can be used to provide geoid heights on their own. Since 2000, several GGMs have been released, and they are mainly derived from satellite gravity measurements, satellite-only models, terrestrial gravimetry, altimeter-derived gravity data in marine areas, and airborne gravity data. With a precise geoid model, ellipsoidal heights obtained from GPS can be converted to orthometric heights, which is reasonably quite needed in Geodesy, Civil Engineering, etc. These heights reflect changes in topography as well as local variations in gravity. This paper evaluates some of the latest releases of high degree reference models and the satellite-only global gravity field model over Sudan using 19 GPS/Leveling stations. We have been selected 6 GGMs based on Gravity field Goce and Grace, and they released in 2020, 2019, 2014, 2008, and 1996 as shown in the International Centre for Global Earth Models website (ICGEM). The accuracy evaluation of the GGM models have been discussed, the accurate GGMs over Sudan are XGM2019e_2159 and GOCO05s, which have indicated -0.019 and 0.046 meters, respectively. The evaluation results produce valuable information to academia and geoid modeling research topics in Sudan, which shows the precise model from the selected GGMs in Sudan by using the available GPS/Leveling data.

This paper presents the first and second order gravimetric relative network planning for the territory of the Republic of Albania, as well as the calculation of free-air anomalies, Bouguer anomalies, and Bouguer gravity. The planning of the first and second order relative gravimetric network is done for the whole territory of the Republic of Albania, based on the three absolute gravimetric points. The planning of the first order network is done using the 1 point/1000km2 surface criteria, while the second order network planning is done again using the 1 point/100km2 surface criteria. The ArcGis software tested the best dot coverage of the network of triangles based on two criteria set out as above. The tests done on all three gravimetric absolute points shows that the best coverage of the whole territory, with first and second order points has absolute gravimetric points at Saranda station. So, taking this fact into consideration, a grid network has been built in ArcGis software based on regular triangles. As a result, 30 first-order relative and 289 second order relative points were obtained, for which the calculations of free-air anomalies, Bouger anomalies and Bouguer gravity using WGS84 parameters were performed in Excel. To enable the calculation of free-air anomalies, Bouguer anomalies, and Bouguer gravity, first must be calculated the normal ellipsoid gravity, then the gravity of height and finally the reduction of free-air. Based on the measured gravity (which in our case was obtained the measured gravity point at the Saranda station) by adding free-air reduction and then removing the normal gravity value, the free-air anomalies were calculated. Bouguer anomaly calculation first must be reduced to topography with the Bouguer plate, and then we remove the gravity of height at the gravity point P₀. Bouguer gravity is calculated from the measured gravity (which in our case is taken the measured gravity point at the Saranda station) by removing the Bouguer plate and adding free...

ABSTRACT
New additional Global Geopotential Models [GGMs] have now been released into the public domain,
those including data from the CHAMP, GREACE and GOCE dedicated satellite gravimetric mission. Those
satellite tracking data have resolved the long wave length component of the global gravity field
with rather very high accuracy [10].
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2013, Civil Engineering Research Magazine CERM, Vol. 35 .No. (3), Published by Faculty of
Engineering, Al-Azhar University, Cairo, Egypt.










Therefore, it is important to evaluate those new models over Egypt to determine which; of them is
the most appropriate GGM there. In this study a comparison of the performance of three of the GGMs
released between 1996 tell now (EGM96, EGM2008 and EIGEN-6C 2011) over Egypt is made. The gravity
anomalies computed from the models are compared with point free air gravity anomalies on land. The
results have indicated the outstanding performance of EGM2008 to the other examined GGMs
undoubtedly. EGM2008 has1.23 times better statistics than the EGM96 and 2.54 times than EIGEN-6C
2011, in terms of root mean square error [r.m.s.e].

We compile a new geoid model at the computation area of New Zealand and its continental shelf using the method developed at the Royal Institute of Technology (KTH) in Stockholm. This method utilizes the least-squares modification of the Stokes integral for the biased, unbiased, and optimum stochastic solutions. The modified Bruns-Stokes integral combines the regional terrestrial gravity data with a global geopotential model (GGM). Four additive corrections are calculated and applied to the approximate geoid heights in order to obtain the gravimetric geoid. These four additive corrections account for the combined direct and indirect effects of topography and atmosphere, the contribution of the downward continuation reduction, and the formulation of the Stokes problem in the spherical approximation. The gravimetric geoid model is computed using two heterogonous gravity data sets: the altimetry-derived gravity anomalies from the DNSC08 marine gravity database (offshore) and the ground ...

The selection of a global geopotential model (GGM) for modeling the long-wavelength for geoid computation is imperative not only because of the plethora of GGMs available but more importantly because it influences the accuracy of a geoid model. In this study, we propose using the Gaussian averaging function for selecting an optimal GGM and degree and order (d/o) for the remove-compute-restore technique as a replacement for the direct comparison of terrestrial gravity anomalies and GGM anomalies, because ground data and GGM have different frequencies. Overall, EGM2008 performed better than all the tested GGMs and at an optimal d/o of 222. We verified the results by computing geoid models using Heck and Grüninger's modification and validated them against GPS/trigonometric data. The results of the validation were consistent with those of the averaging process with EGM2008 giving the smallest standard deviation of 0.457 m at d/o 222, resulting in an 8% improvement over the previous geoid model. In addition, this geoid model, the Ghanaian Gravimetric Geoid 2017 (GGG 2017) may be used to replace second-order class II leveling, with an expected error of 6.8 mm/km for baselines ranging from 20 to 225 km.

An attempt has been made to evaluate the geoid-quasigeoid separation term over Pakistan by using solutions of terms involving first and second order terrain heights. The first term, involving the Bouguer anomaly, has a significant value and requires being incorporated in any ...

The conversion of geometric as well as ellipsoidal heights from GNSS observations to practical heights for engineering constructions has necessitated the determination of the local geoid model of areas. Benin City is a developing area which requires a local geoid model for conversion of geometric heights to orthometric heights for physical developments in the area. This paper is on the best local geoid model of Benin City, Nigeria by comparing three gravimetric-geometric geoid models of the study area. GNSS and gravimetric observations were carried out on 49 points to respectively obtain their coordinates and absolute gravity values. The theoretical gravity values of the points were computed on the Clarke 1880 ellipsoid, subtracted from the absolute gravity values and corrected for the air (free air) to obtain the free air gravity anomalies of the points. The computed free air gravity anomalies were applied to compute the geoid heights of the points using the integration of the modified Stokes integral. Three geometric geoid surfaces (plane, second degree and third degree surfaces) were fitted to the computed gravimetric geoid heights using the least squares technique to obtain the gravimetric-geometric geoid models of the study area. The RMSE of the three gravimetric-geometric geoid models were computed to determine their (the models) accuracy. The three gravimetric-geometric geoid models were compared using their accuracy to obtain the most suitable geoid model of the study area. The results of the comparison showed that the third degree gravimetric-geometric geoid model is most suitable for application in the study area. It is recommended that ellipsoidal heights obtained from GNSS observation within Benin City, Nigeria should be converted to orthometric heights using the third degree geoid model.