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

Utilizing the adopted average topographic density of 2670 kg/m^(3)in the reduction of gravity anomalies introduces errors attributed to topographic density variations,which consequently affect geoid modeling accuracy.Furthermore,the mean gravity along the plumbline within the topography in the definition of Helmert orthometric heights is computed approximately by applying the Poincar e-Prey gravity reduction where the topographic density variations are disregarded.The Helmert orthometric heights of benchmarks are then affected by errors.These errors could be random or systematic depending on the specific geological setting of the region where the leveling network is physically established and/or the geoid model is determined.An example of systematic errors in orthometric heights can be given for large regions characterized by sediment or volcanic deposits,the density of which is substantially lower than the adopted topographic density used in Helmert's definition of heights.The same applies to geoid modeling errors.In this study,we investigate these errors in the Hong Kong territory,where topographic density is about 20%lower than the density of 2670 kg/m^(3).We use the digital rock density model to estimate the effect of topographic density variations on the geoid and orthometric heights.Our results show that this effect on the geoid and Helmert orthometric heights reach maxima of about 2.1 and 0.5 cm,respectively.Both results provide clear evidence that rock density models are essential in physical geodesy applications involving gravimetric geoid modeling and orthometric height determination despite some criticism that could be raised regarding the reliability of these density models.However,in regions dominated by sedimentary and igneous rocks,the geological information is essential in these applications because topographic densities are substantially lower than the average density of 2670 kg/m^(3),thus introducing large systematic errors in geoid and orthometric heights.

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.

Modeling a Local Geoid: Application in Thies, Senegal

Many applications in geodesy, hydrography and engineering require determining heights linked to the geoid. Direct leveling, which is the traditional method of obtaining these elevations, is slow, time consuming and expensive. The contribution of space techniques can make it possible to overcome these constraints provided that we have a precision geoid model compatible with that obtained by the GNSS method. There are today relatively precise regional geoid models, at least outside of mountain ranges, in all developed countries, which is not yet the case in developing countries like Senegal. An alternative is to use local models restricted to a small area. Thus, this study aims to produce a geoid model by combining multi-source data for the city of Thies intended mainly to support leveling operations by GNSS. To achieve this objective, direct precision leveling and GNSS leveling (static mode) were carried out covering the study area. The reference points used are, among others, those of the RRS04 (Reference Network of Senegal 2004) and the NGAO53 (General Leveling of West Africa 1953). Additionally, gravimetric measurements were conducted using the Sensor Play-Data Recorder application. The calculation of the model was carried out by the SRBF (Spherical Radial Basis Function) method using the PAGravf4.5 software. The SRBF method uses EGM08 to first calculate height and gravity anomalies. These are then compared with the raw data in order to determine the residuals which will allow the model to be refined. In order to validate our model, control points (GNSS/leveled) were chosen based on a homogeneous geographical distribution in the area in order to evaluate their altitude. An accuracy of less than 2 cm was obtained. Comparing our model with the existing local model GGSV12v1 shows that our model is more accurate.

Establishment of a Geometric Geoid Model and Evaluation of the EGM2008 and EIGEN-6CA Models over the Dakar-Thies-Mbour Triangle in Senegal

High-accuracy geoid determination is an essential goal that many groups of scientists and countries are striving to achieve. Techniques for determining geoid models have evolved over time. Unfortunately, this all-important determination requires relatively substantial technical and financial resources, depending on the type of geoid to be determined. This situation justifies the inadequacy, and sometimes absence, of accurate geoid models in many countries, despite the new challenges of altimetric positioning using space or satellite positioning techniques. This study focuses on the establishment of a geometric geoid model using simplistic techniques that are accessible and applicable in restricted or wide areas, with or without gravimetric data. The study was applied to the Dakar-Thiès-Mbour triangle, the two regions in the extreme west of Senegal that are home to the most infrastructure projects with the highest socio-economic stakes, as well as mines currently being exploited, and therefore the highest stakes in terms of positioning. This study also enabled us to assess the accuracy of a number of global field models in Senegal, which are used by some professionals for altimetric positioning using Global Positioning Satellite Systems (GNSS) in the absence of a local geoid model. The estimated geoid model is based on the determination of undulation at various sample points in the study area. To this end, a campaign of GNSS observations and direct levelling was carried out on the various points spread across the study area. These measurements were then used to determine the undulation at each point. Bilinear interpolation was used to deduce the undulations throughout the study area, based on the altimeter conversion grid. This grid was evaluated using GPS/level control points.

Geoid interpolation in Gongzhuling City and the 1cm quasi geoid refinement

The refinement of the 1cm geoid holds significant importance in engineering applications,where the accuracy of the geoid is frequently constrained by its resolution.However,there has been limited exploration into the specific relationship between geoid resolution and accuracy.This article aims to address this gap by thoroughly examining said relationship.This study employs the CapRCR modification to calculate the geoid of Gongzhuling City.The findings indicate that the accuracy can be enhanced by 2%to 9%through encryption of the geoid resolution from 2.5′×2.5′to 1.5′×1.5′.Furthermore,this improvement can be augmented by 15%to 21%through encryption of the gravity anomaly to the same resolution.The accuracy of the geoid exhibits a linear relationship with the resolution of gravity anomalies.The theoretical accuracy of the geoid,excluding integration errors,is determined to be 1.21 cm.In engineering applications,opting for a resolution of 30''×30''can result in the geoid containing only an integration error of 2 mm.However,to attain an accuracy of 1 cm,it becomes imperative to further mitigate data errors.Consequently,the final refined quasi-geoid accuracy is established at 0.56 mm.

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.

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.

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.

Models for Fitting Gravimetric Geoids and GPS Results

The aim of this investigation is to study how to use a gravimetric (quasi) geoid for levelling by GPS data in an optimal way.The advent of precise geodetic GPS has made the use of a technique possible,which might be called GPS_ gravimetric geoid determination.In this approach,GPS heights above the reference ellipsoid are determined for points whose levelled (orthometric) height H is above sea level people have already surveyed;for these points,we thus have the values of the geoid undulation N .These values are then used to constrain the geoid undulations N′ obtained from the gravimetric solution.

Determination of Geoid over South China Sea and Philippine Sea from Multi-satellite Altimetry Data

A new computational procedure for derivation of marine geoid on a 2.5′×2.5′grid in a non-tidal system over the South China Sea and the Philippine Sea from multi-satellite altimeter sea surface heights is discussed. Single-and dual-satellite crossovers were performed, and components of deflections of the vertical were determined at the crossover positions using Sand-well's computational theory, and gridded onto a 2.5′×2.5′resolution grid by employing the Shepard's interpolation procedure. 2.5′×2.5′grid of EGM96-derived components of deflections of the vertical and geoid heights were then used as reference global geopotential model quantities in a remove-restore procedure to implement the Molodensky-like formula via 1D-FFT technique to predict the geoid heights over the South China Sea and the Philippine Sea from the gridded altimeter-derived components of deflec-tions of the vertical. Statistical comparisons between the altimeter-and the EGM96- derived geoid heights showed that there was a root-mean-square agreement of ±0.35 m between them in a region of less tectonically active geological structures. However, over areas of tectonically active structures such as the Philippine trench, differences of about -19.9 m were obtained.

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.