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

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

Local geoid modeling in the central part of Java, Indonesia, using terrestrial-based gravity observations

The Global Navigation Satellite System(GNSS)positioning method has been significantly developed in geodetic surveying.However,the height obtained through GNSS observations is given in a geodetic height system that needs to be converted to orthometric height for engineering applications.Information on geoid height,which can be calculated using the global geopotential mode,is required to convert such GNSS observations into orthometric height.However,its accuracy is still insufficient for most engineering purposes.Therefore,a reliable geoid model is essential,especially in areas growing fast,e.g.,the central part of Java,Indonesia.In this study,we modeled the local geoid model in the central part of Java,Indonesia,using terrestrial-based gravity observations.The Stokes'formula with the second Helmert's condensation method under the Remove-Compute-Restore approach was implemented to model the geoid.The comparison between our best-performing geoid model and GNSS/leveling observations showed that the standard deviation of the geoid height differences was estimated to be 4.4 cm.This geoid result outperformed the commonly adopted global model of EGM2008 with the estimated standard deviation of geoid height differences of 10.7 cm.

Refining geoid and vertical gradient of gravity anomaly

We have derived and tested several relations between geoid （N） and quasi-geoid （~） with model validation. The elevation correction consists of the first-term （Bouguer anomaly） and second-term （vertical gradient of gravity anomaly）. The vertical gradient was obtained from direct measurement and terrain calcula- tion. The test results demonstrated that the precision of geoid can reach centimeter-level in mountains less than 5000 meters high.

Computing Local Geoid Model Using DTM and GPS Geodetic Points.Case Study:Mejez El Bab-Tunisia

Different methods have been deployed to compute the geoid, the altimetry reference for surveying applications. One of their main goals is to allow the use of GPS (Global Positioning System) or GNSS heights, which are related to an ellipsoid and therefore must be corrected. Some of these methods are accurate but quite heavy as developed by [1], but one of them is easy to use while giving very good results in a local system: some mm for a 10 × 10 km2 area developed by [2] [3]. In our study, we have used software called “Géoide Program”, previously used at the CERN in Switzerland and set up by [4], which they complete this software allowing a parameterization of general data to provide results in a general system. Then, tests have shown the way to optimize computations without any loss of accuracy. For our computations we use gridded of geodetic heights, from Lambert or WGS 84 datum’s, DTM (Digital Terrain Model) and leveled GPS points. To obtain these results, components of the vertical deflection are computed for every point on the grid, deduced from the attraction exerted by the mass Model. Then, geodetic heights are computed by an incremental way from an arbitrary reference. Once the calculation is performed, the geodetic height of any point located in the modelled area can be interpolated. The variations of parameters (mainly size and increments of the DTM and of the modeled area, and ground density) have shown that they do not play a significant role although DTM must be large enough to take into account an important area around a selected zone. However, the choice of the levelled GPS points is primordial. We have performed tests with real data concerning Mejez El Bab zone, in north of Tunisia. Nevertheless, for a few hundreds of square kilometers area, and just by using a DTM and a few levelled GPS points, this method provides results that look extremely promising, at least for surveying activities, as it shows a good possibility to use GPS for coarse precision levelling, and as DTM are now widely available in many countries.

Local geoid height approximation and interpolation using moving least squares approach

In this paper an introduction of the moving least squares approach is presented in the context of data approximation and interpolation problems in Geodesy.An application of this method is presented for geoid height approximation and interpolation using different polynomial basis functions for the approximant and interpolant,respectively,in a regular grid of geoid height data in the region 16.0417°≤φ≤47.9583°and 36.0417°≤λ≤69.9582°,with increment 0.0833°in both latitudal and longitudal directions.The results of approximation and interpolation are then compared with the geoid height data from GPS-Levelling approach.Using the standard deviation of the difference of the results,it is shown that the planar interpolant,with reciprocal of distance as weight function,is the best choice in this local approximation and interpolation problem.

Precise geoid computation using Stokes-Helmert’s scheme and strict integrals of topographic effects

The high-precision local geoid model was computed based on the improved Stokes-Helmert0 s boundary value problem and strict integrals of topographic effects. This proposed method involves three steps.First, the mathematical form of Stokes-Helmert0 s boundary value problem was derived, and strict computational formulas regarding topographic effects were provided to overcome the disadvantage of planar approximations. Second, a gravimetric geoid model was constructed using the proposed StokesHelmert0 s scheme with a heterogeneous data set. Third, a least squares adjustment method combined with a multi-surface function model was employed to remove the bias between the gravimetric geoid model and the GNSS/leveling data and to refine the final local geoid model. The accuracy of the final geoid model was evaluated using independent GNSS/leveling data. Numerical results show that an external precision of 1.45 cm is achievable.

Application of Geoid Anomalies to the Tectonic Research in the East Asian Continental Margin

In this paper, we calculated multi-scale residual geoid anomalies with the method of geoid separation processing, according to EGM2008 ultra-high order gravity field model, remove-restore technique and Stokes integral. The East Asian continental margin was selected as the study area. The residual geoid anomalies have been calculated by programming. On the basis of residual geoid anomalies at various orders, the interlayer geoid anomalies at different depths were calculated to depict the spatial distribution characteristics of the residual geoid. Finally, we conducted a detailed geophysical interpretation for the study area according to the geoid anomalies in combination with other geophysical datasets. Four conclusions can be outlined as follows: 1) it is impracticable that geoid anomalies are used in the interpretation of the shallow objects due to the influence of the terrain; 2) the anomalies of residual geoid can reflect the intensity of small-scale mantle convection in the asthenosphere; 3) the interlayer geoid anomalies can reflect the magmatic activities associated with the mantle convection and mantle plume in different scales; 4) the study of the geoid may provide an approach for the research of the subduction zone, mantle convection and mantle plume.

DISCUSSION ON SOME FFT PROBLEMS TO DETERMINE THE GEOID

The aim of this investigation is to study some FFT problems related to the application of FFT to gravity field convolution integrals.And the others,such as the effect of spectral leakage,edge effects,cyclic convolution and effect of padding,are also discussed.A numerical test for these problems is made.A large area of Western China selected for the test is located between 30°N～36°N and 96°E～102°E and includes 1 858 gravity observations on land.The results show that the removal of the bias in the residual gravity anomalies is important to avoid spectral leakage.One hundred percent zero padding is highly recommended for further research of the geoid to remove cyclic convolution errors and edge effects.1_D FFT is recommended for precise local geoid determination because it does not use kernel approximation.

A Local Geoid Determination Based on Ellipsoid Approximation in Western China

A new methodology for precise geoid determination with finest local details based on ellipsoidal approximation is presented.This methodology is formulated through the "fixed_free two_boundary value problem" based on the observable of the type modulus of gravity intensity,gravity acceleration and gravity potential at the GPS positioned stations,with support of the known geoid’s potential value,W

GPS-GRAVIMETRIC GEOID DETERMINATION IN EGYPT

The main objective of this study is to improve the geoid by GPS/leveling data in Egypt.Comparisons of the gravimetric geoid with GPS/leveling data have been performed.On the basis of a gravimetric geoid fitted to GPS/leveling by the least square method,a smoothed geoid was obtained.A high_resolution geoid in Egypt was computed with a 2.5′×2.5′ grid by combining the data set of 2 600 original point gravity values,30″×30″ resolution Digital Terrain Model (DTM) grid and the spherical harmonic model EGM96.The method of computation involved the strict evaluation of the Stokes integral with 1D_FFT.The standard deviation of the difference between the gravimetric and the GPS/leveling geoid heights is ±0.47 m.The standard deviation after fitting of the gravimetric geoid to the GPS/leveling points is better than ±13 cm.In the future we will try to improve our geoid results in Egypt by increasing the density of gravimetric coverage.

Influencing factors on the accuracy of local geoid model

Different modification methods and software programs were developed to obtain accurate local geoid models in the past two decades.The quantitative effect of the main factors on the accuracy of local geoid modeling is still ambiguous and has not been clearly diagnosed yet.This study presents efforts to find the most influential factors on the accuracy of the local geoid model,as well as the amount of each factor’s effect quantitatively.The methodology covers extracting the quantitative characteristics of 16 articles regarding local geoid models of different countries.The Statistical Package of Social Sciences(SPSS)software formulated a strong multiple regression model of correlation coefficient r = 0.999 with a high significance coefficient of determination R^2 = 0.997 and adjusted R^2 = 0,98 for the required effective factors.Then,factor analysis is utilized to extract the dominant factors which include:accuracy of gravity data(40%),the density of gravity data(25%)(total gravity factors is 65%),the Digital Elevation Model(DEM)resolution(16%),the accuracy of GPS/leveling points(10%)and the area of the terrain of the country/state under the study(9%).These results of this study will assist in developing more accurate local geoid models.

Rational formulation of modern geoid concept

Based on modern geometrical field theory, a rational interpretation of modern geodesic surface concept was formulated. The general equations for geodesic surface were given. The height concept and arc length concept were formulated strictly. Their intrinsic benefits and shortages of parameter-center reference and mass-center reference systems were studied in details. The research results show that the geodesic concept depends on the available technology. The technology-dependence feature of geodesy concept was expressed by related mathematic formulations. For the satellite-based observation system, the rationality of China-2000 Geoid Reference System was explained. Discussions about how to establish regional precision reference system were made based on general equations of arc length and space distance equations. A new correction item should be taken into consideration for arc length gauge calculation.

Development of Precise Geoid Model for the Establishment of Consistent Height System in Geoga Grand Bridge Construction Area

The national benchmarks on islands were mostly established by trigonometric leveling in Korea. This method results in inaccuracy, which is a serious problem in Geoga Grand Bridge construction work that tried to link the mainland and the islands. The Geoga Grand Bridge （Pusan-Geoje fixed link project） was selected as the study area, a huge construction work in Korea that will connect the mainland （Pnsan） and an island （Gecje island）. However, the orthometric heights issued at benchmarks （JINH and GOFJ） were not consistent, because they did not refer to the same zero point, which would make the linking of the sections problematic. This paper introduces the precise local geoidas a vertical datum for the construction area in order to establish a consistent height system. To determine the precise local geoid for the construction area, we firstly developed a precise gravimetric geoid for Korea and its adjoining seas as a whole. This gravimetric geoid was developed by use of all available gravity data, including surface and satellite data on land and on the ocean. The gravimetrie gecid was computed by spherical fast fourier transform with modified Stokes＇ kernels. The remove-restore technique was used to eliminate the terrain effects by use of the RTM reduction and to determine the residual geoid by combining the GGM02S/EGM96 geopotential model, free-air gravity anomalies and high-resolutinn DEM data. Finally, the gravimetric model was fitted to the geoid heights obtained from GPS and tide observations （Ncps/Tiae） by least square coUocatian, to provide the final GPS-consistent local precise geoid model. The post-fit error （std. dev. ） of the final geoid to the NetS/Tide derived from GPS and tide observations was ± 2.2 cm for the construction area. We solved the height inconsistency problem by calculating the orthometric height of the benchmarks and the cnntrol points using the final geoid model. Also, the highly accurate orthometric height was estimated through the GPS/leveling technique by applying the developed local precise geoid. Therefore, the precise local geoid is expected to improve the quality of the construction procedure of the Geoga Grand Bridge.

Calculation of geoid undulations and gravity anomaliesin the Northwest Pacific by using the Topex/Poseidonand Geosat altimeter data

The geoid undulations in the Northwest Pacific were calculated by usingtheTopex/Poseidon and Geosataltimeter data. Firstly the bias between two types of the altimeter data was removed and the geoid undulations in theNorthwest Pacific were acquired by a long wave bias diminishing model with a resolution of 30 km and precision of 14cm. Then an algorithm of inversion of gravity anomalies was derived , and the gravity anomalies in the East China Seawere calculated by using the algorithm and the geoid undulations. The rms of difference between the in situ measure-ments the gravity anomalies from altimeter data was 3 .8× 10-5 m/s2. A method to colculate the gravity anomaliesin a larger area was developed which combined gravity anomalies in four subregions overlapping each other into one data set in a larger region. The error analysys shaws that the model and result of the inversion of gravity anomalies were reliable.

Using satellite altimetry leveling to assess the marine geoid

Based on the concept of Global Position System(GPS)/leveling,the satellite altimetry leveling(SAL) is first proposed to evaluate the marine geoid.SAL is derived by the difference among the mean sea surface(MSS),mean dynamic ocean topography(MDT),and leveling origin.In this study,(1) the original satellite altimetry data are processed to infer the vertical deflection and gravity anomaly,(2) the Chinese coastal marine geoids(CMG) are determined by using the differe nt methods(including Molodensky,least square collocation,Stokes formula,and two-dimensional fast Fourier transformation(FFT) with the vertical deflection and gravity anomaly data),(3) CMG are evaluated by using the results from above different methods,the Gravity field and steady-state Ocean Circulation Explorer(GOCE) gravity potential model(GGPM),and SAL.The results show that(1) CMG from the Molodensky method has the highest precision by using vertical de flection data,(2) the accuracy of CMG indicate good consistency between the SAL and GGPM,(3) SAL can be used as a new method for assessing marine geoid.

Determination of Istanbul geoid using GNSS/levelling and valley cross levelling data

The Istanbul GPS Triangulation Network(IGTN) and the Istanbul Levelling Network(ILN),established in2006,provide data for the determination of a local GNSS/levelling geoid model.These networks’ measurements were done separately on both the Asian and European sides of the Bosphorus Strait in the vicinity of Istanbul.To connect these regions for those networks,a Valley Cross Levelling(VCL) data set,acquired in 1986 and 2004,was used.The use of this VCL data set was challenging in calculating the Istanbul geoid model,primarily because of its errors.In this study,this challenge was overcome through newly collected VCL data in 2010,allowing for the readjustment of the ILN and the newly collected VCL data set.The Istanbul geoid model was computed using soft computing techniques including the adaptive-network-based fuzzy inference system(ANFIS) and the artificial neural networks(ANNs).The resulting Istanbul GNSS/levelling geoid model is shown to be more reliable when compared with the model computed using conventional interpolation techniques.

Determination of Antarctic geoid by using global gravity field

With Chinese latest global gravity field model WDM94, the authors provide the geoid height and mean free air gravity anomaly of Antarctica (The range of latitude is from -60° to -90°). In order to conclude and analyze the characters of Antarctic geoid roundly, the authors collect the latest oversea global gravity field model OSU91 (to degree and order 360) and JGMOSU (to degree and order 360), get the corresponding geoid height and mean free air gravity anomaly. The results are compared with the results got from WDM94, thus we get the difference. The standard deviation of geoid height between WDM94 and OSU91 is ±1.90 m;the deviation of geoid between WDM94 and JGMOSU is ±2.09 m. The standard deviation of mean gravity anomaly are ±8.97 mGal and ±9.32 mGal respectively. [WT9.HZ]

Modelling Orthometric Heights from a Combination of Ellipsoidal Heights and Gravimetric Geoid Model in Rivers State, Nigeria

Many applications in geodesy, hydrography and engineering require geoid-related heights. Spirit leveling which is the traditional means of obtaining geoid- or mean sea level-related heights is slow, time-consuming and costly. Global Navigation Satellite Systems (GNSS) offer faster and relatively cheaper way of obtaining geoid-related heights when geoidal undulation is applied to ellipsoidal heights. However, difficulties involved in determining acceptable geoid height have seriously hampered the application of GNSS for leveling in Rivers State, thus necessitating the need to develop an acceptable geoid model which will serve as a means of conversion of GNSS-delivered ellipsoidal heights to their orthometric heights equivalent. In pursuance of this objective, a detailed gravimetric geoid has been evaluated for Rivers State, Nigeria. The computation of the geoid was carried out by the traditional remove-restore procedure. The Earth Geopotential Model 2008 (EGM08) was applied as the reference field for both the remove and restore parts of the procedures;spherical Fast Fourier Transform (FFT) was employed for the evaluation of the Molodenskii’s integral formula for the height anomaly, (ζ) to yield the quasi-geoid;while the Residual Terrain Modelling (RTM) was done by prism integration. The classical gravimetric geoid over Rivers State was obtained from the rigorously evaluated quasi-geoid by adding the quasi-geoid to geoid (N?- ζ) correction it. The minimum and maximum geoid height values are 18.599 m and 20.114 m respectively with standard deviation of 0.345 m across the study area. Comparison of the gravimetric geoidal heights with the GPS/Leveling-derived geoidal heights of 13 stations across Rivers State, Nigeria showed that the absolute agreement with respect to the GPS/leveling datum is generally better than 7 cm root mean squares (r.m.s) error. Results also showed that combining both GPS heights and the computed Rivers State geoid model can give orthometric heights accurate to 3 cm post-fit using a 4-parameter empirical model. The geoid model can thus serve as a good alternative to traditional leveling when used with GPS leveling, particularly for third order leveling in the study area.

Impact of the formation and ablation of Antarctic ice sheet on the global geoid and sea level

It is convenient to investigate the gravimetry using a harmonic spheric function for the description of the distribution and thickness of the Antarctic ice sheet. The gravitational theory and the Stokes' harmonic spheric function formula were used to determine the impact of the Antarctic ice cap on the global geoid. The Antarctic ice cap is formed from the condensation of seawater vapour whose mass is equal to a layer of seawater 59 m thick of covering the earth's surface, i.e. 2.7×10 19 kg. This will cause the global averaged geoid to decrease for around 23 m. The authors' computations show that the Antractic ice cap has a great impact on the global geoid, which increases (+) in some regions, but decreases (-) in other reigions. The geoid is +115 m, -37 m and +8 m at the South Pole, the 25°S parallel and the North Pole, respectively. If the Antarctic ice cap melts completely, on the rigid Earth's surface the seawater and geoid will return to its original position (and height) due to the balancing force of the fluid. Since the crust is almost in a state of isostasy, assuming that the crust is an elastic solid and the mantle is an incompressible fluid, the load of seawater will deflect the crust and drive the mantle material to flow. The material above the isostatic surface compensates mutually. If the densities of the mantle and seawater are 3270 kg/m 3 and 1030 kg/m 3, respectively, then the variation in the elevation of the continent is only 2.8 m with respect to the sea level after the Antarctic ice cap melts;it is not larger than that estimated by some people.It is worth noting that the above results were derived from an ideal Earth model. In the real Earth, the mantle and crust are visco elastic.

Determination of a Gravimetric Geoid Solution for Andalusia (South Spain)

The orthometric heights can be obtained without levelling by means of the ellipsoidal and geoidal heights. For engineering purposes, these orthometric heights must be determined with high accuracy. For this reason, the determination of a high-resolution geoid is necessary. In Andalusia (South Spain) a new geopotential model (EIGEN-GL04C) has been available since the publication of a more recent regional geoid. As a consequence, these new data bring about improvements that ought to be included in a new regional geoid of Andalusia. With this aim in mind, a new gravimetric geoid determination has been carried out, in which these new data have been included. Thus, a new geoid is provided as a data grid distributed for the South Spain area from 36 to 39 degrees of latitude and –7 to –1 degrees of longitude (extending to 3 × 6 degrees), in a 120 × 240 regular grid with a mesh size of 1.5’ × 1.5’ and 28800 points in the GRS80 reference system. This calculated geoid and previous geoids are compared to the geoid undulations obtained for 262 GPS/levelling points, distributed within the study area. The new geoid shows an improvement in accuracy and reliability, fitting the geoidal heights determined for these GPS-levelling points better than any previous geoid.