VÚGTK

Proceedings of Research Works 1998

250 66 Zdiby 98, tel. 02/685 7375, fax: 02/685 7056, e-mail: odis@vugtk.cz
Sborník titulní strana


Jan Kostelecký1,2), Miloš Cimbálník 2), Jiří Provázek 3), Ladislav Zajíček 3)

REFERENCE COORDINATE SYSTEM S-JTSK/95 – ITS ESTABLISHMENT IN THE CZECH REPUBLIC

Proceedings of Research Works 1998. - Roč.44. - Zdiby : VÚGTK, 1998. - S.55-63. - ISBN 80-85881-10-1



Introduction

As the present information-oriented society expands to include the position-related dimension, GPS is being configured to serve nearly every application. It has proven to be a beneficial tool also for geodesy and surveying. Its use in the horizontal geodetic control in former Czechoslovakia was motivated by two requirements. The first is the improvement of the existing horizontal geodetic control, the second is the set up of a coordinate system which would enable a direct use of GPS in all fields of geodesy and of everyday surveying practice and the combination of GPS observations and conventional terrestrial survey data.

This new coordinate system, called S-JTSK/95, will be also in future represented by a set of coordinates of the monumented triangulation stations. Its ability to serve the conventional survey techniques will be ensured by a set of plane coordinates Y, X in the national "Křovák's" map projection accomplished by sea level heights referred to the "Baltic Vertical Datum after Adjustment". For the use of GPS technique geocentric coordinates in ETRS-89 will be available represented by the geodetic (ellipsoidal) coordinates and the ellipsoidal heights referred to the ellipsoid GRS80. The relation between the ellipsoidal and the plane coordinates will be clearly defined by a unique mathematical relation. Because the new system will be constituted by the GPS technology in the sense that this technology eliminates the deficiencies of the existing system S-JTSK (large - scale and local deformations), the new plane coordinates will, in general, differ from the old ones. The experiments show that the rms difference is 9 cm in each coordinate (the maximum is 45 cm). With respect to such a small difference practically all existing maps will keep being utilized after the new system is introduced. The realization of this system was described in detail e.g. in (Cimbálník, Kostelecký, 1994).

Since the "geocentric component" of the above mentioned system will be realized through the ETRF-89, this contribution brings a brief description of the approach and the activities towards its realization in the Czech Republic. The first stage of realization was also reported in (Kostelecký, Šimek, 1995).

1. Establishment of the Reference Frame ETRF-89 in the Czech Republic.
GPS Observation Campaigns 1991 - 1994

In correspondence with the conception of the Czech National Geodetic Service in the field of geodetic control a series of GPS - campaigns were realized in 1991 - 1994. The conception stemmed from the idea of hierarchic building based on the sequence of observation campaigns directed to a stepwise densification or supplementing the existing set of coordinates. The initial campaign was EUREF-CS/H- 91 in which 6 stations were observed in former Czechoslovakia, 3 of them in the present Czech Republic (Pecný, Přední Příčka and Kleť). The second (or the first densification) campaign CS-NULRAD-92 consisted in observations at 19 stations, 10 of them in the Czech Republic and the interconnection campaign CS-BRD-93 (6 stations in the Czech Republic) linked the zero-order GPS reference network to the German reference network DREF. In the large-scale densification campaign DOPNUL the ETRF-89 coordinates of 176 stations were determined. The DOPNUL stations are regularly distributed over the entire territory with the spacing 20 - 25 km. The campaigns are reported in detail in (Kostelecký, Šimek, 1995) and (Šimek, Kostelecký, 1994).

2. Realization of ETRS-89 in the Czech Republic - First Stage

The coordinates of six stations EUREF-CS/H-91 obtained from the final adjustment (Seeger et al., 1995)are underlying for the realization of ETRF-89 in the Czech Republic. A strategy for the setup of a reference frame represented by coordinates of 176 stations determined by an appropriate combination of the results of all relevant campaigns is described in (Kostelecký, Šimek, 1995). The resulting ellipsoidal coordinates are referred to the ellipsoid GRS80 in the reference frame ETRF-89, epoch 1989.0.

3. Realization of ETRS-89 in the Czech Republic - Second Stage.
Transformation of the Czech National Triangulation Network into the ETRF-89

Because all 176 stations observed in above mentioned campaigns are identical with triangulation stations, the ETRF-89 represented by these stations could be further densified using the results of classical terres- trial triangulation. On the basis of coordinates of 178 stations (176 in the Czech Republic and 2 in the Slovak Republic) available in both the ETRF-89 and S42/83, which is the best terrestrial reference system in the Czech Republic, the coordinates of about 29 thousands triangulation stations have been transformed from S42/83 into ETRF-89. Because the file of coordinates in S42/83 is located in the Military Topographic Institute at Dobruška, this step has been carried out in cooperation between civilian and military topographic services.

The transformation routine consists in the following steps:

  1. The identities of 178 stations were independently checked by two methods developed at the Department of Military Territorial Information of the Military Academy in Brno and in the Research Institute of Geodesy, Topography and Cartography (VUGTK) Zdiby. Both approaches are based on testing smoothness of the field of transformation residuals. On the basis of the evaluation of performed tests only 174 stations were taken for the determination of seven transformation parameters between S42/83 and ETRF-89.
  2. The parameters of the 7-parameter similarity transformation were determined. For the determination of cartesian coordinates in S42/83 the ellipsoidal heights above the Krassovskij ellipsoid were used. Using the derived transformation parameters the coordinates of all triangulation stations were transformed into the ETRF-89.
  3. The rms residual after transformation of 174 stations was 14 cm. The residuals are due to inaccuracy of S42/83 coordinates and also to the linearity of the transformation. The residuals had to be distributed to the "surrounding" stations so that the resulting coordinates of "identical" stations were exactly equal to those, determined from GPS and, consequently, the homogeneity of the network was preserved. For the distribution of residuals three, to some extent independent, methods have been developed. All of them are based on the determination of the residual shift by weighted mean of a certain number of residuals at "surrounding" identical stations
    Duj= ĺipiDuj
    ĺipi
    where u is one of the coordinates, i is the index of a surrounding station, j is the index of the determined station and the weight 1p=1/s or 2p=1/s2. In the approach of the VUGTK the weight 2p was used empirically determined for the area with the radius of 40 km taking the residuals preliminarily reduced by the conformal cuODIS transformation of the entire state area. This approach was used in direct transformation between ETRF-89 and S-JTSK (Czech cadastral coordinate system). The approaches of the Military Topographic Institute and of the Military Academy Brno are based on a combination of both types of weights and on the interpolation inside the triangles created by optimal triangulation of the area. Both methods differ from each other only in application of the extrapolation which was necessary in the border zones. After some testing performed in the Military Topographic Institute the method developed by the Military Academy was accepted. The rms difference in position between all three methods is 2-3 cm except for the border zones where the differences reach up to 10 cm.

The above mentioned approach was used for the transformation of all 29000 triangulation stations in the Czech Republic. Doing that, the system S42/83 was "corrected" by matching to the ETRF-89 which is by one order more accurate. The ETRF-89 coordinates of the triangulation stations thus are more homogenous, however, one showed keep in mind that their relative accuracy is still that of the classical terrestrial system S42/83.

4. Description of the coordinate system S-JTSK/95

4.1 Fundamental characteristics

System JTSK/95 was defined by 7 parameter Helmert's space transformation of the points in the ETRF- 89 (space X,Y,Z coordinates are derived from the ellipsoidal coordinates and heights related to the GRS80 geocentric ellipsoid) to old cadastral system S-JTSK (space X,Y,Z coordinates are derived from the ellipsoidal coordinates and heights related to the Bessel ellipsoid). Fixation of the Bessel ellipsoid with respect to absolute quasigeoid was determined using the condition

ĺdH 2 = min. ,


where H is a distance between Bessel ellipsoid and quasigeoid. Rectangular planar cartographic coordinates Y, X are computed using Křovák's cartographic projection applied onto ellipsoidal coordinates, geographic latitude B, and geographic longitude L on the Bessel ellipsoid.

Further approach between S-JTSK/95 and S-JTSK is realized by quadratic transformation (2 additional terms) applying the condition:

ĺ[(XS-JTSK/95 - XS-JTSK)2 + (YS-JTSK/95 - YS-JTSK)2 ] = min. ,


determined additional terms are finally add to Křovák's projection (see below), then we obtained "modified Křovák's projection". The defined system enables:
  1. to introduce the geocentric coordinate system, i.e. to allow a direct use of GPS-techniques;
  2. to define plane coordinates in the user map projection on the basis of geocentric coordinates X, Y, Z or B, L, H, i.e. to allow the use of classical geodetic measurements (angular and distance measurements using e.g. total stations);
  3. to allow the use of existing map in scale series from 1:1000 towards smaller scales; fulfilling this requirement the system meets the needs of both the engineering or cadastral works and data locating within GIS/LIS;
  4. to make an extensive use of the existing geodetic systems.
The system removes:
  1. Global scale deformation of the old S-JTSK.
  2. Local deformations of the old S-JTSK.
Reference frame is realized by marked stations with the set of coordinates
B, L, in the ETRF-89 (epoch 1989.0)
H(el) (ellipsoid GRS80)		 Y, X, planar in modif. Křovák's proj.
H(BpV)=(Baltic height system after evaluation)
The unambiguous relations between ellipsoidal coordinates B, L and planar coordinates Y, X exist, this relation is defined by the sequence of transformations
B, L, H(el)(GRS80)®[1]®B, L, H(el) (Bessel)®[2]®Y, X (planar), H(BpV) ,
inverse transformation is based on the sequence:
Y, X (planar), H(BpV)®[2']®B, L, H(el) (Bessel)®[1']®B, L, H(el)(GRS80).
Ellipsoidal heights H(el)(GRS80), H(el)(Bessel) and levelling heights H(BpV) are connected by the relation

H(el)(GRS80) = H(BpV) + height of quasigeoid above the ellipsoid GRS80,

H(el)(Bessel) = H(BpV) + height of quasigeoid above the Bessel ellipsoid.

The accuracy of the height transformation depends substantially on the accuracy of the quasigeoid realization.

4.2 Transformation [1] - transformation of the B, L, H(el) from the ellipsoid GRS80 to the B, L, H(el) onto the Bessel ellipsoid and transformation [1'] - transformation of the B, L, H(el) on the Bessel ellipsoid onto B, L, H(el) on the ellipsoid GRS80

Transformation [1] is realized by the sequence of steps:

B, L, H(el) GRS80 ® [1.1] ® x, y, zETRF89 ® [1.2] ® x, y, zS-JTSK/95 ® [1.1'] ® B, L, H(el) Bessel

Inverse transformation [1'] is created by the sequence:

B, L, H(el)Bessel ® [1'.1] ® x, y, zS-JTSK/95 ® [1'.2] ® x, y, zETRF89 ® [1'.1'] ® B, L, H(el)GRS80

Transformations [1.1] and [1'.1] are then defined by:

(4.1)

x = [N + Hel] cos B cos L,
y = [N + Hel] cos B sin L,
z = [N (1 - e2) + Hel] sin B,

where B is a geodetic latitude, L is a geodetic longitude, N is a radius of curvature in transversal direction, N = a/W , W is a first fundamental geodetic function:

(4.2)

W =

a is semimajor axis of the ellipsoid of reference, and e is eccentricity. Numerical values are:

for ellipsoid GRS80:  a = 6 378 137.0 m,  e2 = 0.00669 43800 22901,
for Bessel ellipsoid:  a = 6 377 397.155 m,  e2 = 0.00667 43722 3062.


For transformation [1.1'] and [1'.1'] the inverse formulas are used:

(4.3)

L = arctan ć
ç
č		 y
x		 ö
÷
ř

B0 = arctan é
ę
ë		z
(1 + e2
1 - e2		)ů
ú
ű

and then using iterative process for i = 1, 2 ....

(4.4)
Ni=a
,

Heli=
cos Bi-1		- Ni,

Bi = arctan é
ę
ë		z
(1 - Nie2
Ni + Heli		)-1ů
ú
ű		,
is valid.

Transformation [1.2] from the system "0" to the system "1" is given by:

(4.5)x1 = (1 + T4 . 10-6 ) [ x0 + T5 y0 - T6 z0 ] + T1 ,
y1 = (1 + T4 . 10-6 ) [ - T5 x0 + y0 + T7 z0 ] + T2 ,
z1 = (1 + T4 . 10-6 ) [ T6 x0 - T7 y0 + z0 ] + T3 .
Coefficients T1 , T2 ,..., T7 for transformation [1.2] from ETRF89 to S-JTSK/95 are based on the realized transformation between ETRF-89 and S-JTSK, and they are see (Cimbálník et al., 1996): T1 = -570.828 m, T2 = -85.677 m, T3 = -462.842 m ,T4 = -3.56231, T5 = 5.26108", T6 = 1.58672", T7 = 4.99840".

The coefficients of inverse transformation [1'.2] between S-JTSK/95 and ETRF89 differs slightly from T1,...,T7 due to:

  1. high correlations between parameters (parameters of transformation are determined from the territory of the Czech Republic only),
  2. equations (4.5) valid in the case of rotation higher than 1 arcsec only approximately, in our case the rotations are about 5". Coefficients T5,T6 and T7 then are not directly rotations, but they are the combinations of goniometric functions of rotations.
They are, see (Cimbálník et al., 1996): T1 = 570.838 m, T2 = 85.683 m, T3 = 462.847 m, T4 = 3.56102, T5 = -5.26111", T6 = -1.58671", T7 = -4.99845".

4.3 Transformations [2] and [2'] : Modified Křovák's projection

Modified Křovák's projection is defined as "original" Křovák's projection accompanied by small additional terms (determined on the basis of minimalization of the differences between coordinates of S-JTSK/95 and S-JTSK).

Basic formulas:

  1. Constants:
    0 = 49o 30', a = 6 377 397.155 m, e2 = 0.00667437223062
    a = [ 1 + (e2 cos4 0) / ( 1 - e2 ) ]1/2 ,
    UQ = 59o 42' 42.69689" ,
    U0 = arcsin ( a-1 sin 0 ),
    g( 0 ) = { (1 + e sin 0) / (1 - e sin 0) } ae/2 ,
    k = tan (U0 / 2 + 45o) . cotan ( 0 / 2 + 45o) . g( 0 ) ,
    k1 = 0.9999 ,
    N0 = a (1 - e2)1/2 / (1 - e2 sin2 0 ),
    S0 = 78o 30' ,
    n = sin S0 ,
    0 = k1 . N0 cotan S0 .

  2. Transformation [2]: B, L (Bessel) to Y, X planar:
    g(B) = { (1 + e sin B) / (1 - e sin B) } ae/2
    U = 2 {arctan [ k . tana ( / 2 + 45o ) . g(B)-1 ] - 45o} ,
    l' = L + 17o 40' ,
    DV = a . (42o 30' - l'),
    a' = 900 - UQ ,
    S = arcsin [ cos a' sin U + sin a' cos U cos DV ] ,
    D = arcsin [ cos U sin DV / cos S ] ,
    e= n . D ,
    = 0 tann(S0 / 2 + 45o) . cotann(S / 2 + 45o) ,
    Y' = sin e , X' = cos e ,
    and resulting Y, X are:
    Y = ( Y' - DY ) + 5 000 000.0 m , X = ( X' - DX ) + 5 000 000.0 m , where DY and DX will be defined later.

  3. Transformation [2'] for transformation Y, X planar to B, L (Bessel) :
    Y' = ( Y + DY ) - 5 000 000.0 m, X' = ( X + DX ) - 5 000 000.0 m,
    = ( Y'2 + X'2 )1/2
    e = arctan (Y / X) ,
    D = e / sin S0 ,
    S = 2 { arctan [ ( 0/ )1/n . tan (S0 / 2 + 45o) ] - 45o}
    U = arcsin [ cos a' sin S - sin a' cos S cos D ] ,
    DV = arcsin [ cos S sin D / cos U ] ,
    L = 24o 50' - DV / a ,

    B is determined by iterative process:
    Bi = 2 {arctan [ k(-1/a) tan1/a (U / 2 + 45o) . [ (1 + e sin Bi-1)/( 1 - e sin Bi-1 ) ]e/2 ] - 45o },
    when Bo = U , i = 1, 2, ....

  4. Additional terms

    Additional terms DY and DX are functions of Y' and X'; they are determined by means of the equations:

    DY = A2 + A3 . Yred + A4 . Xred + 2 A5 Yred . Xred + A6 (Xred2 - Yred2) ,
    DX = A1 + A3 . Xred - A4 . Yred + A5 (Xred2 - Yred2) - 2 A6 Yred . Xred     ,


    where for Yred , Xred :

    Yred = Y' - 654 000.0 m ,
    Xred = X' - 1 089 000.0 m ,


    is valid and coefficients A1, ..., A6 are:
    A1 = .5839284707E-01, A2 = .4718658410E-01, A3 = .8227606925E-07,
    A4 = -.3337763709E-06, A5 = .8850984442E-11, A6 = .1444547818E-11.

4.4 Quasigeoid heights

Due to the fact the new system S-JTSK/95 keeps spatial relations between geocentric ellipsoid GRS80 and Bessel ellipsoid, it is necessary to use ellipsoidal instead of sea level heights for transformation. For the computation of the ellipsoidal heights model of quasigeoid VÚGTK 1995 (solution of Research Institute of Geodesy, Topography and Cartography) is used. The heights of the quasigeoid above both used ellipsoids are content in this model.

4.5 Software products for the transformations

The FORTRAN77, 5.1 version procedures (subroutines) are prepared.

For transformation of B, L, H(el) in ETRF89 to Y, X in S-JTSK/95 the subroutine E89J95 is used.

For transformation of Y, X, H(BpV) in S-JTSK/95 to B, L, H(el) in ETRF89 the subroutine J95E89 can be used. Description of the subroutines (input and output parameters) are presented in (Cimbálník et al., 1996).

4.6 Model of transformation between S-JTSK and S-JTSK/95

In the case of application of the S-JTSK/95 to geodetic practice will be necessary to realize transforma tion between old S-JTSK to new S-JTSK/95 system for great amount of the data.

From this reason, the special software for transformation has been prepared in Surveying Office, Prague. Software is based on the interpolation of the coordinate differences between above mentioned systems in regular 1 km times 1 km grids. Accuracy of this interpolation is better than 2 cm in each coordinate.

Detail information about relations between S-JTSK and S-JTSK/95, testing of the accuracy of the realization of ETRF89 on the territory of the Czech Republic is presented in (Zajíček 1993a,b, 1994, 1995).

Conclusion

The S-JTSK/95 coordinates of the triangulation stations have been further used for practical geodetic and cadastral measurements. The ongoing best measurement show that even though the improved system is quite sufficient for cadastral works and it also makes possible to use GPS-observations in current surveying practice, the local discrepancies between precise measurements and the coordinates may reach up to 10 cm. Therefore, the third stage of the realization of an improved reference frame was conceived, consisting in its further densification by exclusively GPS-technique down to the average density of 10 km. For those who need the transformation between the S-JTSK and ETRF-89 on the decimetre/3 metres accuracy level (e.g. positioning in GIS, positioning of boreholes in geological prospection, flight control etc.) a software has been developed for direct transformation between ETRF-89 and the existing user system S-JTSK with an accuracy higher than 15 cm/3 m in all three coordinates, see (Cimbálník, Kostelecký, 1996).

The application of GPS in geodynamics is in the Czech Republic going in a separate way. With respect to higher claims to the stability and protection of geodynamical stations a geodynamical network has been designed the stations of which are not, but a few identical with those defining the ETRF-89.

REFERENCES

Cimbálník, M., Kostelecký, J. (1994): Realization of the geocentric and terrestrial systems in the Czech Republic. In: Report on the Symposium of the IAG Subcommis. EUREF held in Warsaw 8 - 11 June 1994. Veröffent. der Bayer. Kommission für die Intern. Erdmessung der BAW, Heft Nr. 54, München 1994, p. 228.

Cimbálník, M., Kostelecký, J., Provázek , J., Zajíček, L. (1996) Realization of the S-JTSK/95 (Transformation between geocentric and planar coordinates). Technical Report of the Research Institute of Geodesy, Topography and Cartography, Zdiby and Land Survey Office, Prague. (in Czech).

Cimbálník, M., Kostelecký, J. (1996): Direct transformation between the ETRS-89 and the Czech cadastral system S-JTSK. In: Report on the Symposium of the IAG Subcommis. EUREF held in Ankara, May 22 - 25, 1996. Veröffent. der Bayer. Kommission für die Intern. Erdmessung der BAW, Heft Nr. 57, München 1996.

Kostelecký, J., Šimek, J. (1995): EUREF and its Evolution in the Czech Republic. In: Re-port on the Symposium of the IAG Subcommis. EUREF held in Helsinki 3 - 6 May 1995. Veröffent. der Bayer. Kommission für die Intern. Erdmessung der BAW, Heft Nr. 56, München 1995, p. 149.

Seeger, H., Schlüter, W., Talich, M., Kenyeres, A., Arslan, E., Neumaier, P., Habrich, H. (1994): Results of the EUREF-CS/H'91 GPS Campaign, In: Report on the Symposium of the IAG Subcommis. EUREF held in Warsaw 8 - 11 June 1994. Veröffent. der Bayer. Kommission für die Intern. Erdmessung der BAW, Heft Nr. 54, München 1994, p. 87.

Šimek, J., Kostelecký, J. (1994): National Report of the Czech Republic. On the improvement of the national geodetic control for the Czech Republic. In: Report on the Symposium of the IAG Subcommis. EUREF held in Warsaw 8 - 11 June 1994. Veröffent. der Bayer. Kommission für die Intern. Erdmessung der BAW, Heft Nr. 54, München 1994, p. 264.

Zajíček, L. (1993a): Transformation of the S-JTSK to S-GS. Report of the Land Survey Institute Prague (in Czech).

Zajíček, L. (1993b): The Czech geocentric coordinate system and S-JTSK. Report of the Land Survey Institute Prague (in Czech).

Zajíček, L. (1994): The future of the geodetic control. Vojenský topografický obzor 3/93, p. 24 - 30, (in Czech).

Zajíček, L. (1995): Realization of the system JTSK/95 in the Czech Republic. Report of the Land Survey Office, Prague (in Czech).

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