Geometric Geoid Models Research Articles

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

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

On the construction of gravimetric geoid model on the Lviv region area

Baltic Height System 1977, currently used in Ukraine, the starting point of which is the zero of the Kronstadttide-gauge, isobsoletedue to the great distance from the zero-point of the reference height and the difficulty of adapting satellite methods of geodesy. For the successful modernization of the height system of Ukraine, it is necessary to integrate it into the United European Leveling Network (UELN). For the full functioning of any modern height system, namely to determine the gravity-depend heights by satellite methods, it is necessary to operate with a high-precision geoid model. Therefore, an important task is construction of a high-precision regional model of the geoid on the territory of our state. The rear many methods of constructing a model of the regional Earth’s gravitational field, including the geoid model, each of which has its advantages and disadvantages. The purpose of this article is to test the STHA-method for calculating the model of the regional gravitational field, in particular the gravimetric model of the geoid, on the territory of Lviv region and to assess its accuracy. Free air gravity anomalies ∆g from WGM2012 provided by the International Gravimetric Bureau (BGI) were used as initial data. The gravimetric STHA-model of the geoid was calculated with in the procedure «Remove-Compute-Restore» up to 8 degrees/order. To assess the accuracy of the model, it was compared with 213 points of GNSS leveling, as well as with the model EGM2008 up to 360 degrees/order. There are always differences between geometric and gravimetric geoid models duet or and ommeasurement errors, in consistencies in datums, different geodynamic effects etc. Respectively the parameters of the transition between gravimetric and geometric models of the geoid on the territory of Lviv region were also found. The proposed method can be used to build a high-precision model of the geoid for the entire territory of Ukraine with its subsequent coordination with the model of the European geoid EGG2015.

Least Square Modification of Stokes Formulae with Additive Corrections Estimator for Klang Valley Geoid Modeling

Klang Valley is a fast-growing area and its development shall be equivalent with precise measurements for a precise vertical reference. Thus, existing vertical reference with 3 centimetres (cm) is inadequate and processed with complicated remove-compute-restore (RCR) procedure. Apart from this, areas such as Klang Valley should better than one (1) centimetre level vertical reference. Meanwhile processing method for vertical reference should be simplified and easy tasking. Because of that, methodology for this study is by employing the least squares modification of Stokes formula with additive corrections (KTH). This approach fully uses anomalies rather than residuals which it is more complicated. At the same time, the additive corrections estimator introduced combining the direct and indirect computations method. Datasets used in this study were refined rigorously prior to the gridding scheme in cross validation, free air anomalies, as well as anomaly correction. The KTHKVGM2020 gravimetric and geometric geoid models are evaluated from the reference position using GNSS levelling. It found that KTHKVGM2020 Geoid model is better than one (1) centimetre for Klang Valley area with efficiency processing method. Therefore, the study is an essential in future to develop high-precision geoid model with efficient methods particular for urban and rapidly developing areas.

Establishment of local geometric geoid model for Busoga, Uganda

The importance of the local geoid model for the computation of accurate geoid heights, as well as orthometric heights used for engineering constructions, necessitated its establishment in areas, regions or countries. Consequently, this study establishes the local geometric geoid model of Busoga, Uganda, using the geometric method. A total of 26 points were used in the study, 20 points for the development of the model and 6 test points. GNSS observations were acquired with Trimble GNSS dual-frequency receivers and processed with Bernese (V5.2) and Spectra Precision Survey Office (v4.1) software to obtain the coordinates and ellipsoidal heights of the points. Differences between the existing orthometric and ellipsoidal heights were computed to obtain the geoid heights. The Least squares adjustment technique was applied to determine the fit, as well as the Bicubic and Multiquadratic models’ parameters. The Root Mean Squares Error (RMSE) index was used to compute the accuracy of the models. The geoid models were compared with their RMSE, as well as accuracy to determine which of them is more suitable for application in the study area. The comparison result shows that the Multiquadratic geoid model is more suitable for implementation in the study area. A Microsoft Excel program was developed for the application of the model in the study area.

Establishment of local geometric geoid model for Busoga, Uganda

The importance of the local geoid model for the computation of accurate geoid heights, as well as orthometric heights used for engineering constructions, necessitated its establishment in areas, regions or countries. Consequently, this study establishes the local geometric geoid model of Busoga, Uganda, using the geometric method. A total of 26 points were used in the study, 20 points for the development of the model and 6 test points. GNSS observations were acquired with Trimble GNSS dual-frequency receivers and processed with Bernese (V5.2) and Spectra Precision Survey Office (v4.1) software to obtain the coordinates and ellipsoidal heights of the points. Differences between the existing orthometric and ellipsoidal heights were computed to obtain the geoid heights. The Least squares adjustment technique was applied to determine the fit, as well as the Bicubic and Multiquadratic models’ parameters. The Root Mean Squares Error (RMSE) index was used to compute the accuracy of the models. The geoid models were compared with their RMSE, as well as accuracy to determine which of them is more suitable for application in the study area. The comparison result shows that the Multiquadratic geoid model is more suitable for implementation in the study area. A Microsoft Excel program was developed for the application of the model in the study area.

IMPROVING THE LOCAL GEOMETRIC GEOID MODEL OF FCT ABUJA ACCURACY BY FITTING A HIGHER ORDER/DEGREE POLYNOMIAL SURFACE

The improvement of the accuracy of a local geometric geoid model using the same data set (geoid heights) requires the fitting of a higher degree polynomial surface to the data set. Consequently, this paper presents improving the local geometric geoid model of FCT, Abuja accuracy by fitting a higher order polynomial surface. A fifth degree polynomial surface was fit to the existing geoid heights of 24 points used previously for the determination of the geometric geoid model of the study area to improve its accuracy. The least squares adjustment technique was applied to compute the model parameters, as well as the fit. The RMSE index was applied to compute the accuracy of the model. The computed accuracy (0.081m) of the model was compared with those of the previously determined geoid models (Multiquadratic, 0.110m and Bicubic, 0.136m models) of the study area to determine which of the models best fit the study area, as well as has the highest resolution. The comparison result shows that the fifth degree polynomial surface best fit the study area.

COMPARATIVE ANALYSIS OF THREE PLANE GEOMETRIC GEOID SURFACES FOR ORTHOMETRIC HEIGHT MODELLING IN KAMPALA, UGANDA

The conversion of theoretical, as well as geometric heights to practical heights requires the application of geoidal undulations from a geoid model. The various global geopotential models that are readily available for application in any part of the world do not best-fit regions, as well as countries. As a result, there is a need to determine the local geoid models of local areas, regions and countries. This study determines the local geoid model of Kampala in Uganda for orthometric heights computation by comparing three plane geometric geoid surfaces. A total of 19 points were used in the study. The least squares adjustment technique was applied to compute the models’ parameters. Microsoft Excel programs were developed for the application of the models in the study area. The Root Mean Square Index was applied to compute the accuracy of the models. The three geometric geoid models were compared using their accuracy to determine which of them is most suitable for application in the study area. The comparison results show that the three models can be applied in the study area with more reliability, with greater confidence in model 2.

The Implication of Significant Number in Geodetic Computations: Geometric Geoid Model of FCT Abuja as a Case Study

The position occupied by the geoid in the computation of orthometric height (H) from GNSS observations is crucial and hence requires the geoid model parameters obtained with the application of least squares technique, be approximated to a reasonable number of significant figures in its (geoid) modelling to enable a high-resolution geometric geoid, be determined. As a result, this study presents the implication of significant number in geodetic computations using geometric geoid model of FCT, Abuja as a case study. The number of significant figures of the model parameters was varied, starting from 6 to 22 significant figures to enable the Root Mean Square Errors (RMSEs), as well as the accuracy of the model, be obtained at the varied numbers of significant figures. The obtained results show that the highest accuracy of the model is obtained if the model parameters are approximated to a reasonable number of significant figures. Keywords: geometric, geoid, height, significant, figure, geodetic, computation, modelling DOI: 10.7176/JSTR/6-08-09

Relationship between Polynomial Geometric Surfaces Terms and Observation Points Numbers and Effect in the Accuracy of Geometric Geoid Models

The application of the geometric method of local geoid model determination which requires the fitting of geometric surfaces to known geoid heights to enable geoid heights of new points to be interpolated involves the use of least squares technique for computation of the models' parameters. The selection of polynomial geometric surfaces depends on the size of the study area, the variation of the geoid heights and the number of measurement points. The accuracy of the geometric geoid model increases as the number of observation points approximates the number of geometric surface terms. But in most cases, the number of observation points is not considered. To this effect, this paper presents the relationship between geometric surfaces terms and observation points numbers and effect in the accuracy of geometric geoid models. A total of 23 points of known local gravimetric geoid heights were used. Two polynomial geometric (third and fifth degrees) surfaces were fitted to the geoid heights at various observation point numbers and compared to determine the relationship between the number of model terms and that of observation points and effect in the accuracy of the models. Least squares adjustment technique was applied to obtain the model parameters. The differences between the models and the known geoid heights of the points were computed and used to obtain the RMSEs as well as the accuracy of the models. The obtained results showed that the accuracy of the polynomial geometric geoid models tends to the highest as the number of measurement points approximates the number of the model terms and in a unique solution where the number of observation points is equal to the number of the polynomial geometric model terms, the model accuracy is highest. The paper recommends that the geometric method of local geoid model determination should be strictly applied in small areas. Where the method will be applied in considerable large areas, higher degrees polynomial geometric surfaces with a larger number of terms approximating the number of observation points should be applied. This will enable a proper fit of the polynomial surface to the known geoid heights, as well as high accuracy to be obtained.

Spatial Distribution of Survey Controls and Effect on Accuracy of Geometric Geoid Models (Multi-quadratic and Bicubic) in FCT, Abuja

Spatially distributed studies may follow two broad approaches: points that are randomly located and points that are clustered or clumped together. They are discussed within a discipline described as geo-statistics. Geo-statistics is applied in order to interpolate geoid undulation (N) and hence generate through a geoid model orthometric heights from scattered/random or closely/clustered located controls. Control points are coordinated points within a primary/secondary geodetic survey network. Kriging method was adopted to produce topographical maps of the both scattered and closely scenarios. Accuracy computed revealed that standard deviation (σ) of multiquadratic and bicubic models in the study area are respectively 11cm and 14cm in lopsided control study area while over clustered distributed located controls are respectively 12cm and 15cm. Standard deviation with the lowest values among the determined geometric geoid models is at all times preferable scattered/random than closely/clumped scenario. This implies that the multiquadratic models can be applied across the entire study area with high accuracy/reliability irrespective of spatial distribution of the points. Hence, the accuracy of the models are better when the total number of points distributed within the entire study area was used than when a limited number of points within a particular part of the study area was used.

Spatial Distribution of Survey Controls and Effect on Accuracy of Geometric Geoid Models (Multi-quadratic and Bicubic) in FCT, Abuja

Spatial Distribution of Survey Controls and Effect on Accuracy of Geometric Geoid Models (Multi-quadratic and Bicubic) in FCT, Abuja

Local Geometric Geoid Models Parameters and Accuracy Determination Using Least Squares Technique

Local Geometric Geoid Models Parameters and Accuracy Determination Using Least Squares Technique

Geostatistical prediction of a local geometric geoid - kriging and cokriging with the use of EGM2008 geopotential model

We investigate prediction abilities of different variants of kriging and different combinations of data in a local geometric (GNSS/leveling based) geoid modeling. In order to generate local geoid models, we have used GNSS/leveling data and EGM2008 geopotential model. EGM2008 has been used twofold. Firstly, it was used as a basic long wave-length trend to be removed from geoid undulation data to generate a residual field of geoid heights modeled later by kriging (remove-restore technique). Secondly, EGM2008-based undulations were used as a secondary variable in a cokriging prediction procedure (as pseudo-observations). Besides the use of EGM2008, the kriging-based local geometric geoid models were generated only on the basis of raw undulations data. Kriging itself was used in two variants, i.e. ordinary kriging and universal kriging for univariate and bivariate cases (cokriging). The quality of kriging-based prediction for all its variants and all data combinations have been investigated on one fixed validation dataset consisting of 86 points and three training data sets characterized by a different density of sampling. Results of this study indicate that incorporation of EGM08 as a long wave-length trend in kriging prediction procedure outperforms cokriging strategy based on incorporation of EGM08 as a secondary spatially correlated variable.

IMPACT OF DATUM TRANSFORMATION ON LOCAL VARIATIONS OF GEOMETRIC GEOID IN NIGER

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.

USING PARTICLE SWARM OPTIMIZATION TO ESTABLISH A LOCAL GEOMETRIC GEOID MODEL

There exist a number of methods for approximating the local geoid surface and studies carried out to determine a local geoid. In this study, performance of geoid by PSO method in modeling local geoid was presented and analyzed. The ellipsoidal heights (h), derived from GPS observations, and known orthometric heights from first-order benchmarks were first used to create local geometric geoid model, then the PSO method was used to convert ellipsoidal heights into orthometric heights (H). The resulting values were used to compare between the spirit leveling and GPS methods. The adopted PSO method can improve the fitting of local geometric geoid by quadratic surface fitting method, which agrees with the known orthometric heights within ±1.02cm

Using a least squares support vector machine to estimate a local geometric geoid model

In this study, test-region global positioning system (GPS) control points exhibiting known first-order orthometric heights were employed to obtain the points of plane coordinates and ellipsoidal heights by using the real-time GPS kinematic measurement method. Plane-fitting, second-order curve-surface fitting, back-propagation (BP) neural networks, and least-squares support vector machine (LS-SVM) calculation methods were employed. The study includes a discussion on data integrity and localization, changing reference-point quantities and distributions to obtain an optimal solution. Furthermore, the LS-SVM was combined with local geoidal-undulation models that were established by researching and analyzing3 kernel functions. The results indicated that the overall precision of the local geometric geoidal-undulation values calculated using the radial basis function (RBF) and third-order polynomial kernel function was optimal and the root mean square error (RMSE) was approximately ± 1.5 cm. These findings demonstrated that the LS-SVM provides a rapid and practical method for determining orthometric heights and should serve as a valuable academic reference regarding local geoid models.

Orthometric Height Improvement in Tainan City using RTK GPS and Local Geoid Corrector Surface Models

AbstractVarious corrector surface models are proposed to mitigate systematic height errors to fit the orthometric heights determined by combining the Tainan City real-time kinematic (RTK) global positioning system (GPS) network and the local geoid model data to the published orthometric heights. Several data sets for Tainan City were tested and analyzed. Two geometric geoid models and one gravimetric-geometric geoid model were generated using the GPS and leveling data. Consequently, three types of orthometric heights were determined (Models I, II, and III). The selection of the optimal corrector surface model for different models was based on a series of statistical tests. The test results show that (1) the selection of the optimal corrector surface model is highly related to the geoid model generating method, hence the optimal corrector surface models are a fifth-degree polynomial for Models I and II and a seven-parameter similarity transformation for Model III; and (2) the determined orthometric height ...

A new height datum for Iran based on the combination of gravimetric and geometric geoid models

A new geoid model for Iran (IRG04) was computed based on the least squares modification of the Stokes formula. IRG04 was derived from the most recent gravity anomaly database, SRTM high resolution Digital Elevation Model (DEM) and GRACE GGM02 global geopotential model. In order to define a new height datum for Iran, we attempted to combine this high resolution gravimetric geoid model with GPS/levelling data using the corrective surface approach. The corrective surface was constructed from 224 GPS/levelling points and then evaluated with 35 independent points. Different interpolation techniques were tested for the creation of the corrective surface; among them the Kriging method was selected as it gave the smallest RMS and ‘noise level’ at the comparisons with GPS/levelling data. The RMS fit of the new combined geoid model versus the independent GPS/levelling data is 0.09 m, it is near four times better compared to the original gravimetric geoid model. The combined model should be more convenient and useful in definition of the new height reference surface, specifically in engineering and GPS/levelling projects.

Comparisons of GPS-Derived Orthometric Heights Using Local Geometric Geoid Models

The purpose of the research was to compare measuring orthometric heights using differential leveling, static global positioning observations, and real-time kinematic (RTK) Global Positioning System (GPS) observations in the vicinity of the University of Conn. campus. Height observations were recorded at temporary and permanent monuments using differential leveling, static GPS, and RTK. Heights at permanent monuments with published elevations were observed using each method and served as control and to check the quality of all measurements. Three-dimensional reference ellipsoid coordinates derived from GPS observations and published orthometric heights were used to create local geoid models, which were then used to convert ellipsoid heights observed with the static GPS and RTK to orthometric heights. The resulting values were used to compare between the measuring methods. In general, static GPS performed better than RTK. Several polynomial surface models of different orders were fitted to the geoid heights, from a constant up to a quadratic. No meaningful distinctions could be drawn between the models indicating that, for a small geographical area, shifting the national geoid model, GEOID03, by a constant vertical amount performs as well as the more complicated approaches. Experience showed that multiple, independent RTK observations on the same day are not sufficient to ensure freedom from systematic errors.