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Geoid-refinement approach as a synergetic combination of geoid information N G (B,L)' submitted to a datum change (N G ', datum 1 → → N G , correct datum 2) and the finite element model NFEM(p,x,y) as overlay (dotted). It is introduced to describe remaining systematics between N G and the true height reference surface HRS .
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... powerful synergy of both above approaches finally leads to the so called "geoid-refinement approach". It is used for the case that the available geoid information N G (B,L)' is to be refined by a finite element model NFEM(p,x,y), which is acting as additional overlay to improve the geoid model (see fig. 3). The geoid-refinement approach ...
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... The aim of this study is to achieve up to a 1 cm precise quasi-geoid model, but traditional relative gravimetry requires a lot of measurement points for this task. The alternative method of the DFHRS approach [6,7] was used, additionally using VDs, which were observed by the GGI developed Digital zenith Camera -VESTA (Vertical by STArs) [8]. In 1998 the first quasi-geoid model LV'98 [9] had been computed using Remove-Restore technique, implemented in GRAVSOFT software [10]. ...
Since the development of GNSS techniques, the determination of a precise quasi-geoid model has become even more actual. In terms of this project the staff of the Institute of Geodesy and Geoinformatics (GGI) has developed a new quasi-geoid model based on DFHRS (Digital Finite-element Height Reference Surface) approach additionally using astrogeodetic measurements – vertical deflections (VD), which can be observed by a Digital zenith camera. This paper evaluates a quasi-geoid model results based on vertical deflections, as a study area using the territory of Latvia: the standard deviation of the solution is equal to 0.006 m with observation residuals after the adjustment of minimum and maximum differences −0.012 and 0.012 accordingly. The standard deviation of quasi-geoid heights and h-H values from LGIA database is equal to 0.012 m with minimum and maximum differences −0.026 and 0.025 accordingly. The post-processed terrestrial VD observations have been compared to VD derivatives from EGM2008 and GGMplus geopotential models. The developed quasi-geoid has been compared to the national quasi-geoid model LV’14 and to the Scandinavian NKG2015.
... This means that an alternative method is required to create a geoid model as a height reference system. Jäger (1999) introduced the use of Finite Element Method (FEM) to define a Height Reference Surface (HRS) representing the Geoid/Quasi-Geoid undulations in a project called (Digital Finite Elements Height Reference Surface -DFHRS). The advantage of this concept is that it is possible to be applied in the areas with minimal number of leveling or GNSS field observations by the densification of global geoid models with high degree and order observations (Geoid Undulations and deflections of vertical) to define the surface and its change in directions (DFHRS, 2000). ...
... The basic principle of FEM to represent the local geoid of an area of interest is to divide the area into subareas (patches) (Jäger, 1999). The patches are used to fit the global geoid models (GGM) observations to the local datum. ...
... Referring to equation (7). The height fitting point observation equation located at the mesh (i) with an expected residual v reads (Jäger, 1999): ...
Nowadays, the use of modern and precise GNSS technologies for precise positioning is the most common tool for field surveyors. The output coordinate of GNSS are divided into geometric horizontal coordinates (latitude, longitude), or equivalently the mathematically transformed and projected coordinates (Easting, Northing), and the ellipsoidal Normal heights (h). The ellipsoidal heights (h) need to be transformed to match with properties and values of the engineering used physical/orthometric heights (H) that are typically produced using precise leveling. The transformation between both types of heights requires the availability of precise geoid model as a height reference surface (HRS). Typically, the modeling process of a Geoid requires dense networks of precise leveling, gravity and astronomical deflections of vertical. Here, the requirements of the availability of dense leveling and gravity networks for classical geoid modeling methods are overridden by the integration of the limited number of benchmarks and the freely available global geoid models (EGM2008, Eigen05c, EGM96 … etc.) is applied using finite elements method. Conceptually, the modeling area is divided into patches with dimensions (50-70km) to transform the global models' reference datum to fit to the local vertical datum. Afterward, each patch is then divided into smaller elements/meshes with the size of (5x5km) that are represented by 2 nd /3 rd order polynomial. To apply the least squares solution for the parameters of the polynomials, a combined system observation equations is applied using GNSS/Leveling and additionally Geoid heights and deflections of vertical by the global models for further observations and densification of the solution. To guarantee the continuity and the smoothness of the modeled surface, one least squares solution is applied for all element using zero, first and second-order continuity conditions. Finally, statistical analysis of the least squares solution and test points were used for the validation and accuracy assessment of the model. Residuals less than 3cm were obtained by the solution. Consistently, the accuracy of 1-3cm could be achieved using the test points.
... Other datum problem occurs when the orthometric heights (H ) are defined as the difference between the ellipsoidal height (h) and the geoid undulation (N ). The definition of the orthometric heights (H ), which follows the direction of the plumb line height and the ellipsoidal geometric/normal heights (h), does not fit, see Fig. 4 [11]. The solution of the heights datum problem can be solved using Molodensky three-dimensional coordinates' transformations. ...
In GNSS, 3D geometric position is calculated with high accuracy. This position is the ellipsoidal latitude, longitude and ellipsoidal height (λ,ϕ,h) with respect to WGS84 coordinates system. These coordinates are integrated with local horizontal/projected coordinates (X, Y) by mathematical coordinate’s transformations and map projections. The geometric ellipsoidal height (h) obtained by means of GNSS has to be integrated with the physical/orthometric heights (H) obtained by means of precise leveling. The physical surface defining the difference between both height systems is the geoid represented by the geoid undulation (N) at a given position. Different methods are used to build geoid models. Global models using terrestrial and satellite data are available to calculate the geoid heights as function of the earth potential (W). The most recent high degree and order models are EGM2008, Eigen05c, Eigen06c4, etc. (GFZ-Potsdam, ‘List of available global models. http://icgem.gfz-potsdam.de/ICGEM/modelstab.html, 2017). Some regional geoid models are available like the European Gravimetric Geoid (EGG97). These models mostly do not fit the local datum due to datum definition problems. Here, a group of precise height reference benchmarks measured with GNSS is used to fit the global models with the local vertical datum to define the local height reference system of Palestine. The accuracy of the different global geopotential models is evaluated before and after the application of the geoid fitting.
... Comparison with Latvian geoid model LV'98 was also performed and is shown in Fig. 1. The method of digital finite element height reference surface (DFHRS) was developed at the University of Applied Sciences in Karlsruhe, Germany [5]. This method is based on an adjustment of highly accurate global and gravimetric geoid models to the local height system represented by the set of GNSS/levelling data. ...
This paper discusses the research work done in Institute of Geodesy and Geoinformation, University of Latvia, and Department of Geomatics, Riga Technical Univesity, devoted to the geodynamics in Latvia: national geoid model computation, using different methods and data sets, in order to improve its precision; analysis of LatPos and EUPOS®-Riga GNSS permanent station observation data time series for time period of 5 years; development of digital zenith camera for vertical deflection determination.
... The DFHBF (Digitale Finite-Element Höhen-Bezugs-Fläche) or DFHRS (Digital Finite-Element Height Ref-erence Surface) software (www.dfhbf.de), developed at the Karlsruhe University of Applied Sciences, Faculty of Geomatics (Jäger 1999) was used for computation of QGeoid height reference surface (HRS) of Latvia. ...
... In the DFHRS concept the area is divided into smaller finite elements -meshes. The QGeoid HRS l is calculated by a polynomial of degree l in term of ( , x y ) coordinates in each mesh indicated by index k (Jäger 1999): ...
In geodesy, civil engineering and related fields high accuracy coordinate determination is needed, for that reason GNSS technologies plays important role. Transformation from GNSS derived ellipsoidal heights to orthometric or normal heights requires a high accuracy geoid or quasi-geoid model, respectively the accuracy of the currently used Latvian gravimetric quasi-geoid model LV'98 is 6–8 cm. The objective of this work was to calculate an improved quasi-geoid (QGeoid) for Latvia. The computation was performed by applying the DFHRS software. This paper discusses obtained geoid height reference surface, its comparisons to other geoid models, fitting point statistics and quality control based on independent measurements.
... A more general model, involving a stochastic signal term, has been introduced by Kotsakes and Sideris (1999), see also Fotopoulos (2005) and Grebenitcharsky et al. (2005). An alternative approach to the combination of geometric and gravimetric height anomalies with the goal to obtain an optimal transformation of GPS ellipsoidal heights into the national height system, has been introduced by Dinter et al. (1997) and refined by Jäger (1999, 2000) and Jäger and Schneid (2001). Least squares collocation (LSC) is also commonly used, but numerous other approaches have been trialled (e.g., Soltanpour et al. 2006; Lysaker et al. 2007). ...
... In practice, the " corrector surface " f is parameterized in different ways. For instance, de Min (1996) uses a bivariate linear algebraic polynomial in latitude and longitude for the area of the Netherlands; de Bruijne et al. (1997) use a linear combination of a bilinear polynomial and a trigonometric polynomial in latitude and longitude with 28 parameters for the North Sea region; Grebenitcharsky et al. (2005) use a four-parameter trigonometric model for Canada originally suggested by Heiskanen and Moritz (1967); Jäger (1999) uses a bivariate algebraic polynomial defined on triangular meshes for parts of Germany; Featherstone (2000) compares splines under tension and LSC; Nahavandchi and Soltanpour (2006) use, among others, cubic splines. The proper choice of a corrector surface, which is a model identification problem, is still an open issue. ...
We consider the problem of local (quasi-)geoid modelling from terrestrial gravity anomalies. Whereas this problem is uniquely solvable (up to spherical harmonic degree one) if gravity anomalies are globally available, the problem is non-unique if gravity anomalies are only available within a local area, which is the typical situation in local/regional gravity field modelling. We derive a mathematical description of the kernel of the gravity anomaly operator. The non-uniqueness can be removed using external height anomaly information, e.g., provided by GPS-levelling. The corresponding problem is formulated as a Cauchy problem for the Laplace equation. The existence and uniqueness of the solution of the Cauchy problem is guaranteed by the Cauchy–Kowalevskaya theorem. We propose several numerical procedures to compute the solution of the Cauchy problem from given differences between gravimetric and geometric height anomalies. We apply the numerical techniques to real data over the Netherlands and Germany. We show that we can compute a unique quasi-geoid from observed gravimetric and geometric height anomalies, which agree with the data within the expected noise level. We conclude that observed differences between gravimetric height anomalies and geometric height anomalies derived from GPS and levelling cannot only be attributed to systematic errors in the data sets, but are also caused by the intrinsic non-uniqueness of the problem of local quasi-geoid modelling from gravity anomalies. Hence, GPS-levelling data are necessary to get a unique solution, which also implies that they should not be used to validate local quasi-geoid solutions computed on the basis of gravity anomalies.
... Presently the computation of the preliminary height of the Uhuru Peak in the Tanzanian height system can not be performed on using the standard different GPS-height integration concepts [9] realized in the software HEIDI2. A sophisticated analysis and more reliable computation of the height of the Uhuru Peak using different height integration approaches will require a larger number of the identical points as these shown in the table below. ...
... At present the author of this paper is working on the evaluation of an optimum design for the position of additional identical height points H-tz [8], which will enable to compute a local geoid model in different ways. The first way is due to using the software HEIID2 and the second one to using the new DFHRS (Digital Finite Elements Height Reference Surface) software ( [9], [10]). ...
... In is to be noted that the accuracy of the ITRF-related heights H=h-N(B,L) is in the range of one meter only, because of a respective limitation of the EGM96 model geoid heights N=N(B,L) (see chap. 4).The transformation of the ellipsoidal ITRF height h to the local Tanzanian height datum H-tz, in a more accurate way, will be done using the software HEIDI2[9], as soon as some further identical heights H of the Tanzanian height system are available and observed by GPS according to the present planning and design studies[8]. ...
In the first part the contribution points out the present way of an acquisition of so called IGS- products from the internet servers of different IGS institutions. The respective IGS products - such as precise ephemeries, IGS-station coordinates, IGS station observations, ionosphere models etc. - are used together with the GPS-station observations on new points for the pre- cise DGPS positioning in the context with precise long baselines. The second part of the paper deals with the use of the IGS products, and gives an overview concerning the different processing strategies, which were applied for the processing of the GPS Kilimanjaro expedition 1999 data with the Bernese GPS software package. Additionally the GPS software SKI_PRO was used for a control of the GPS processing in the local area. In a third part of the contribution deals with the EGM96, which was used for the determina- tion of the geoid heights N in order to correct the ITRF-based ellipsoidal heights h of the points to receive the ITRF-related orthometric heights of the points. In the last part of the paper the final results, namely the ITRF-poisitions (B,L,h) of all pro- cessed points as well as the new ITRF-related orthometric heights H related to the EGM96 model are presented. Additionally the height Htz in the Tanzanian height system of Kiliman- jaro highest point, the Uhuru Peak, is evaluated using the given heights of some benchmarks observed with GPS. An outlook on future work concerning GPS-based height and also the methods of a combined GPS, levelling, geoid and gravity data based geoid determination is given.
For the development of various geodetic tasks
within a state, determining the Height Reference Surface by
the geoid model is extremely important. Considering this,
one of the main task of geodesy is to determine the geoid,
which is defined as an equipotential surface of the Earth’s
gravity field, as a result, it corresponds on average to the
sea level. The aim of this study is to analyze the best-fitting
geoid model for the territory of the Republic of Albania.
In this study, DFHRS (Digital Finite Element Height Reference
Surface) method was used (www.dfhbf.de), developed
by Reiner Jäger [Jäger R. State of the art and present
developments of a general approach for GPS-based height
determination. East Africa: University of Applied Sciences,
Faculty Geoinformationswesen, Department of Surveying
and Geomatics. Paper Presented at the First Workshop on
GPS and Mathematical Geodesy in Tanzania (Kilimanjaro
Expedition 1999); 1999] to determine the most suitable geoid
model for the territory of Albania. This approach allows the
conversion of ellipsoidal heights determined by GNSS into
the standard heights, which refer to the height reference
surface (HRS) of an orthometric. The DFHRS is defined as
continuous HRS in arbitrarily large areas by bivariate polynomials
over an irregular grid [Jäger R, Schneid S. Online
and postprocessed GPS heighting based on the concept of
a digital height reference surface (DFHRS), in vertical reference
systems. In: IAG Symposium. Cartagena, Colombia,
Heidelberg: Springer; 2001, vol 124:203–8 pp]. The DFHRS
approach uses a wide range of input data (Geometric and
*Corresponding author: Bashkim Idrizi, Department of Geodesy, University
of Pristina Faculty of Civil Engineering, Pristina, Kosovo,
E-mail: bashkim.idrizi@yahoo.com. https://orcid.org/0000-0002-1637-
7473
Fitore Bajrami Lubishtani, Department ofGeodesy,University of Pristina
Faculty of Civil Engineering, Pristina, Kosovo,
E-mail: fitore.bajrami@uni-pr.edu. https://orcid.org/0000-0002-4998-
8617
Milot Lubishtani, Private Sector, Ferizaj, Kosovo,
E-mail: milot.lubishtani1@hotmail.com. https://orcid.org/0000-0002-
4055-2438
Physical) and in our case, there were 151 GPS/levelling
height data as well as physical derivatives from different
global geopotential models. The main focus of this study is
placed on the calculation of the most suitable geoid model
for the territory of Albania using global geopotential models
(EGM96, EGM2008, EIGEN04, EIGEN6C4 and European
Gravimetric Geoid Model 1997 (EGG97)). After analyzing the
results and comparing the models among themselves, the
Albanian DFHRS-EIGEN6C4 model was selected as the most
suitable model for the territory of Albania.
GPS heighting remains an interesting issue to explore. The height derived from GPS
measurement (ellipsoidal height) is expected to be able to replace the high cost, time consuming,
and highway-dependency of conventional method of leveling. These two height data acquisition
methods are compared in this study, both on technical and non-technical aspects, in local network
as well as in wider regional network. Since the Indonesian geoid data quality is still limited, those
data will not be used in this comparative study.
Summary The DFHRS (Digital-Finite-Element-Height-Reference- Surface) research and development project is funded by the German Ministry of Education and Research. It aims at the conversion of ellipsoidal GPS-heights h in an on- line or post processed GPS-heighting into the standard heights H, which refer to the height reference surface (HRS) of an orthometric, NN- or normal standard height system. The DFHRS is mo delled as a continuous HRS in arbitrary large areas by bivariate polynomials over an irregular grid. Geoid information (geoid heights N, de- flections of the vertical ξ,η) provided with a HRS datum adoption parametrization and identical points (h,H) as observations in a least squares procedure enable the sta- tistically controlled DFHRS computation. Several geoid mo dels may be in troduced simultaneously and any geoid model may be splitted into different "geoid- patches" with individual datum-parameters and continu- ity requirements along the patch borders. The resulting DFHRS data-base provides a correction Δ=Δ(B,L,h) to transform an ellipsoidal GPS-height h di rectly and on- line into a standard height H. Examples for the compu- tation and use of DFHRS data-bases in DGPS-networks (e.g. SAPOS, Germany) are presented for different countries.
