DFHRS-based computation of quasi-geoid of Latvia.

Janpaule, Inese ; Jager, Reiner ; Younis, Ghadi 等

Introduction

The objective of this work was to calculate the QGgeoid height reference surface for Latvia. The computation was performed by applying the DFHRS software of Karlsruhe University of Applied Sciences.

The obtained geoid height reference surface was then compared with Latvian gravimetric geoid model LV'98 which is broadly used by land surveyors for more than 10 years. It is based on gravimetric measurements, data digitized from gravimetric maps and satellite altimetry data over Baltic Sea, its computation was performed by applying GRAVSOFT software. The estimated accuracy of LV'98 geoid model is 6-8 cm (Kaminskis 2010).

Comparisons were made also between the geoid height reference surface obtained by using DFHRS software and EGG97, EGM2008, Eigen5c, Eigen6c, GOCE GO_CONS_GFC_2_DIR_R3 models, and between LV'98 and above mentioned regional and global models. Appropriate transformation was applied because the global models are not fitted to national height system.

1. DFHRS software

The DFHBF (Digitale Finite-Element Hohen-Bezugs-Flache) or DFHRS (Digital Finite-Element Height Reference Surface) software (www.dffibf.de), developed at the Karlsruhe University of Applied Sciences, Faculty of Geomatics (Jager 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 (Jager 1999):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where f is a design matrix and [p.sub.k] is a polynomial parameter matrix. The corresponding polynomial coefficients are introduced as [a.sub.ij,k] over m meshes (i = 0,l; j = 0,l; k = 1,m).

The surface between two neighboring meshes c and d should be continuous, so the continuity conditions are considered between elements to have a continuous surface with [C.sub.1], [C.sub.1] and [C.sub.2] continuity levels at the border line. The continuity level type [C.sub.0] implies equal functional values along each common mesh border. Level Q implies equal tangential planes and level [C.sub.2] implies equal curvature along the common borders of the QGeoid HRS model. Each group of meshes forms a patch. To reduce the effect of medium or long wave length systematic shape deflections in the QGeoid height observations N and the vertical deflections ([xi],[eta]) from QGeoid or geopotential models (GPM), these observations are subdivides in a number of patches. Patches are formed depending on distribution of fitting points, each patch should contain at least 4 fitting points, which are necessary for a datum parameter estimation. The patches are related to individual sets of datum parameters, in order to model the long-wavelength systematic errors from QGeoid or GMP.

The calculated continuous polynomial parameters are stored together with the mesh topology information in a DFHRS-DB database. The DFHRS computed height N is provided by means of this database, which contains the QGeoid HRS parameters together with mesh and patch design information.

The mathematical model for computation of DFHRS-DB parameters is described by (Jager et al. 2012).

2. Computation process

As an input data (N, [xi], [eta]) the European Gravimetric Geoid Model 1997 (EGG97) was used. EGG97 is based on high-resolution gravity and terrain data available in 1997 as well as the global geopotential model EGM96 (Denker, Torge 1998). The geoid undulations and vertical deflections for 25 points in each mesh from EGG97 model were introduced as observations in the DFHBF approach and software.

102 GPS/levelling points (B,L,h | H)--serving both as itting points for the QGeoid surface as well as the datum parameter-estimation of the patches--with known ellipsoidal GNSS heights h and normal heights H were introduced. Most of the itting points are located within territory of Latvia, 3 points in Estonia and 17 points in Lithuania.

The designed mesh size is 5x5 km, so the total number of meshes is 4601 in the computation area. The patch size varies depending on distribution of reference points (from 80x40 km to 150x100 km). In Fig. 1 the thin blue lines represent meshes, the thick blue lines--the patch borders and the green triangles--the fitting points.

[FIGURE 1 OMITTED]

DFHRS software provides internal quality control of the fitting points. The functional models mean an over-determined system of equations with respect to the parameters, and require parameter estimations for redundant equation systems. The DFHRS software solves the respective parameter estimation by a Least Squares Adjustment. Based on Least Squares Adjustment the quality control and the quality indicators are set up as statistical tests (data snooping) for all single observations in the DFHBF-software.

The computation of Latvia QGeoid HRS was repeated three times, removing fitting points with detected residuals exceeding 3 cm after first and second computation. Most of the removed fitting points are located in North-West part of Latvia, in Kurzeme region (Fig. 2).

3rd order polynomials for the HRS meshes parameterization were used on all computations. Both geoid undulations (115025) and vertical deflections (230050) from EGG97 model were included as mesh observations. The number of continuity conditions was 65227, the number of unknown polynomial parameters was 46010 and the redundancy number was 364311.

[FIGURE 2 OMITTED]

3. QGeoid computation results

The resulting QGeoid model is presented on Fig. 3. QGeoid heights in the territory of Latvia vary from 19.20 m in North-East part to 24.50 m in South-West part.

For the quality control of obtained QGeoid HRS result, it was tested using the same 102 GPS/levelling points which were used during DFHRS computation. The same test was performed for LV'98 QGeoid model, 2[sigma] (96 GPS/levelling points) results are shown in Table 1. Residual RMS of 1.6 cm for Latvian QGeoid HRS was obtained. The same RMS appears for LV'98 QGeoid model, however the minimum and maximum values are larger than those in DFHRS solution.

For comparison between LV'98 model and DFHRS solution (Fig. 4), minimum difference is -25.7 cm, maximum: 21.1 cm, RMS: 5.8 cm.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

4. Comparison with EGG97 and Global Geopotential Models

Because the global models are not fitted to national height system, following transformation was performed:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where point position is represented by geographical latitude and longitude (B , L), h is ellipsoidal height, H is orthometric height, [N.sub.G] is geoid height, N is normal radius of curvature, W = a / N, [e.sup.2] = 2f--[f.sup.2] (a = main axis f = flattening of reference ellipsoid GRS80), three translations (u--[u.sub.G], v--[v.sub.G], w--[w.sub.G]), two rotations ([e.sub.x]--[e.sub.x,G], [e.sub.y]--[e.sub.y,G]) in horizontal plane and scale difference ([[DELTA].sub.m], [[DELTA].sub.mG]) are present (Jager 1999).

For the transformation of European Gravimetric Geoid Model 1997 and global geopotential models to national height system, 13 itting points evenly distributed within territory of Latvia were introduced.

After the fitting of EGG97 and GGM's, the comparisons between EGG97, EGM2008 (Pavlis et al. 2008),

Eigen5c (Forste et al. 2008), Eigen6c (ICGEM), GOCE GO_CONS_GFC_2_DIR_R3 (Brunisma et al. 2010) models and LV'98, as well DFHRS solution, were made. The results are shown in Table 2.

The comparison with global models shows that LV'98 model and DFHBF solution has the best agreement with EGG97 and EGM2008-RMS is 4-6 cm. Because EGG97 model was implemented in DFHRS solution, its RMS is 1.7 cm. GO_CONS_GFC_2_DIR_R3 model contain satellite only information, derived from the tracking of artiicial Earth satellites--Goce, Grace and Lageos. This satellite only model has the worst agreement with both LV'98 and DFHRS solution. Some of the comparisons are shown in Figs. 5 and 6.

Similar statistic results were obtained using the same 102 GPS/levelling points which were used during DFHRS computation for datum deinition. 2[sigma] (96 GPS/ levelling points) results are shown in Table 3.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

5. Quality control based on independent measurements

Table 4 shows three levelling control points, which were not used for the computation of Latvian QGeoid HRS. An example of quality control of Moldova QGeoid HRS is presented at (Jager 2010). The GNSS control measurements were performed on 14 December 2012. by geodesists of Latvia University of Agriculture, Faculty of Rural Engineering, as a static 4 hour measurements. Hleveuing values in BAS-77 (Baltic Height System 1977) are obtained from Latvian Geospatial Information agency database. Point No 1636 is located in North-West part of Latvia, point No 1155 is located in West central part and point No 1727 is located in central part of Latvia. Table 4 gives an overview of the results of GPS control measurement comparison of [H.sub.levelling] and [h.sub.ellipsoidal] with [N.sub.DFHRS]. Differences between [H.sub.levelling] and [H.sub.GNSS] = [h.sub.ellipsoidal]--[N.sub.DFHRS] are between 0 cm to 2.6 cm.

[FIGURE 7 OMITTED]

6. Detection of height deformations in the area of Riga using the DFHRS database

11 levelling control points, which were not used for the computation of Latvian QGeoid HRS, were selected in Riga city territory (Fig. 7). These points are considered as the most stable Riga city levelling network points (ground benchmarks), [H.sub.levelling] (1977) values are taken from 2nd order levelling network catalogue of years 1975-1977, [h.sub.ellipsoidal] values were obtained during RTK measurements in 2010 (Balodis et al. 2011).

Table 5 gives an overview on the sites of the GNSS RTK measurement and levelling catalogue value comparison to DFHRS computed QGeoid HRS values. The mean difference of this comparison reaches -4.9 cm. However, since 1977, Riga levelling network has not been controlled. According the investigation of (Silabriedis 2012) the Riga leveling network has experienced significant deformations. Table 5 shows that height differences between [H.sub.GNSS] = [h.sub.ellipsoidal]--[N.sub.DFHRS] and [H.sub.levelling] (1977) reflect the height deformations of 5 cm subsidence since years 1975-1977.

Conclusions

QGeoid height reference surface for Latvia of RMS 1.6 cm was obtained using DFHBF software. In case of poor coverage of fitting point data it is possible to change input parameters (mesh and patch size) of DFHBF software to obtain better accuracy. High accuracy geoid height reference surface can be achieved by minimum number of observations (102 fitting points). The DFHBF software version 5 (www.dfhbf.de) will also be able to handle terrestrial gravity observations in the integrated Least-Squares-Adjustment approach, which can--in opposite to the Stokes approach--be controlled each by data-snooping. The comparisons with Global Geopotential Models and fitting point quality assessment show that the high resolution and integrated EGM2008 has the agreement with both LV'98 and new DFHBF based solution.

Currently the project of digital zenith camera and its control software for vertical deflection measurements is under development at the University of Latvia, Institute of Geodesy and Geoinformation (Abele et al. 2012). After obtaining first vertical deflection measurements, these will be used for an update of the new Latvian QGeoid model solution, using again the DFHBF software.

doi: 10.3846/20296991.2013.788827

Acknowledgement

This research was funded by ERDF, project Nr 2010/0207/2DP/2.1.1.1.0/APIA/VIAA/077 and FP7 project "FOTONIKA-LV--FP7-REGPOT-CT-2011285912". Authors wish to thank Prof. J. Balodis for his advice and support during this research.

References

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Inese Janpaule (1), Reiner Jager (2), Ghadi Younis (3), Janis Kaminskis (4), Ansis Zarins (5)

(1, 4, 5) Institute of Geodesy and Geoinformation,University of Latvia, Raina bulv. 19, LV-1586 Riga, Latvia

(2) Faculty of Informationmanagement and Media, Institute of Geomatics, University of Applied Sciences, Moltkestrasse 30, 76133 Karlsruhe, Germany

(3) Palestine Polytechnic University, College of Engineering and Technology, Civil and Architectural Engineering, Wade Alhareya, C Building, P.O.Box: 198, Hebron, Palestine

E-mails: (1) inesej@inbox.lv (corresponding author); (2) reiner.jaeger@web.de; (3) ghadi@engineer.com; (4) janis.kaminskis@rtu.lv; 5ansiszx@inbox.lv

Received 07 February 2013; accepted 26 February 2013

Inese JANPAULE. Ph.D. student at the Institute of Geodesy and Geoinformation, University of Latvia, Raina bulv. 19, LV-1586 Riga, Latvia. Ph +371 67034435, e-mail: inesej@inbox.lv.

Graduate of Riga Technical University (Mg. sc. ing. 2008).

Research interests: GNSS, land surveying, Bernese GNSS software applications.

Reiner JAGER. Prof. Dr, Ing. Course Director of International Study programme Geomatics, Institute of Geomatics, Faculty of Informationmanagement and Media, University of Applied Sciences, Moltkestrasse 30, 76133 Karlsruhe, Germany. Ph +49 0721 925 2620, e-mail: reiner.jaeger@web.de. (Dr phys. 1988).

Research interests: satellite geodesy, mathematical geodesy, physical geodesy, adjustment, sotware development, GNSS/ MEMS multisensor navigation, physical geodesy.

Ghadi YOUNIS. Lecturer at Palestine Polytechnic University, College of Engineering and Technology, Civil and Architectural Engineering Department, Wade Alhareya C Building, P.O.Box: 198, Hebron, Palestine. Ph +972 2 2233050, e-mail: ghadi@engineer.com.

Graduate of University of Applied Sciences (Mg. sc. 2006), Ph.D. studies at TU-Darmstadt (2008-2013).

Research interests: satellite geodesy, mathematical geodesy, gravity ield modeling (q/geoid), sotware development.

Janis KAMINSKIS. Associate professor and leading researcher of Institute of Geodesy and Geoinformation, University of Latvia, Raina bulv. 19, LV-1586 Riga, Latvia. Ph +371 27476220, e-mail: janis.kaminskis@gmail.com. (Dr phys. 2010).

Research interests: fundamental and satellite geodesy, geodynamics, GNSS networks and application, gravity ield modeling (q/geoid) and astronomy.

Ansis ZARINS. Leading researcher at the Institute of Geodesy and Geoinformation, University of Latvia, Raina bulv. 19, LV-1586 Riga, Latvia (Ph +371 67034435), e-mail: ansiszx@inbox.lv. (Dr phys. 1988).

Research interests: control and data processing systems for satellite observation and astrometric instruments.

Table 1. Fitting point statistics (2s-95% of
fitting points)

 Min (m) Max (m) RMSE (m)

DFHRS -0,036 0,034 0,016
LV'98 -0,042 0,040 0,016

Table 2. Comparison with EGG97 and Global Geopotential Models

 Min (m) Max (m) Average (m) RMS (m)

DFHRS-EGG97 -0,064 0,063 -0,013 0,017
DFHRS-EGM2008 -0,278 0,179 -0,046 0,052
DFHRS-Eigen6c -0,233 0,188 0,005 0,064
DFHRS-Eigen5c -0,206 0,672 0,130 0,115
DFHRS- GO_CONS_GFC_2 -0,460 1,030 0,381 0,266
_DIR_R3

LV'98-EGG97 -0,257 0,211 0,000 0,058
LV'98-EGM2008 -0,138 0,134 -0,033 0,044
LV'98-Eigen6c -0,203 0,201 0,018 0,063
LV'98-Eigen5c -0,211 0,480 0,142 0,123
LV'98- GO_CONS_GFC_2 -0,461 1,077 0,393 0,272
_DIR_R3

Table 3. Fitting point statistics for EGG97 and Global
Geopotential Models (2s-95% of fitting points)

 Min (m) Max (m) RMS (m)

EGG97 -0,078 0,074 0,037
EGM2008 -0,085 0,079 0,042
Eigen6c -0,151 0,125 0,055
Eigen5c -0,162 0,317 0,153
GO_CONS_GFC_2_DIR_R3 -0,662 0,745 0,346

Table 4. 1st order leveling benchmarks for quality control

1st order levelling Year of Latitude Longitude
benchmark name, No levelling

Mersrags, 1636 2007 57,38516 22,97266
Vane, 1155 2007 56,94707 22,44686
Eleja, 1727 2005 56,41508 23,69842

1st order levelling [h.sub.ellipsoidal] [H.sub.levelling]
benchmark name, No (m) (m)

Mersrags, 1636 27,620 6,890
Vane, 1155 116,975 94,480
Eleja, 1727 55,359 32,380

1st order levelling [N.sub.DFHRS] Difference to
benchmark name, No (m) [N.sub.DFHRS]
 (m)
Mersrags, 1636 20,704 0,026
Vane, 1155 22,495 0,000
Eleja, 1727 23,001 -0,022

Table 5. Levelling benchmark measurement comparison to
DFHRS computed QGeoid HRS values

Point No Latitude Longitude [h.sub.ellipsoidal]
 (m)

5715 56,90561 24,21107 32,890
1193 56,86987 24,27417 31,844
915 56,94540 24,01676 32,804
3389 56,89100 24,07539 33,490
8540 57,03301 24,13018 30,232
37 57,00825 24,24445 27,707
938 57,00033 24,27682 34,027
3336 56,94287 24,24295 30,359
T8 56,94285 24,13518 25,691
0895 56,95117 24,00392 29,418
868 56,98750 24,22432 29,513

Point No [H.sub.levelling(1977)] [N.sub.DFHRS] Difference to
 (m) (m) [N.sub.DFHRS]
 (m)

5715 11,811 21,116 -0,037
1193 10,566 21,258 0,020
915 11,823 21,036 -0,055
3389 12,314 21,213 -0,038
8540 9,623 20,682 -0,072
37 7,042 20,728 -0,063
938 13,328 20,756 -0,058
3336 9,439 20,976 -0,056
T8 4,746 20,997 -0,052
0895 8,415 21,021 -0,017
868 8,736 20,814 -0,037
 Mean -0,0489
 Systematic
 Difference