Estonia – Geographical Studies 9: 99-106 (2004)

Edited by KAARE, T. & PUNNING, J-M.

ISSN 1406-6092


Datum transformation parameters between the historical and modern Estonian geodetic networks

Gábor TIMÁR1,*, Raivo AUNAP2, Gábor MOLNÁR1


1Space Research Group, Dept. of Geophysics, Eötvös University of Budapest, Hungary

2Dept. of Geography, University of Tartu, Estonia



            In the 19th and 20th century the history of Estonia was very unsettled and as a mirror of this, the geodetic network has been changed a few times. Starting with C. Tenner’s and W. Struve’s triangulation in 1811/1816-1819, Estonia has been covered by several geodetic networks (Treikelder, 2000). During the independence between the two World Wars, Estonia developed its own geodetic network, re-equalised in 1937. Skipping here the short period of the first Soviet invasion and the German occupation – and the geodetic works of the German Army Grid as well –, the long-lasting Soviet occupation resulted a new geodetic system. This was the Russian Pulkovo 1942, which was widely used not only in the former Soviet Union but also in the countries of the later dissolved Warsaw Pact. After re-gaining its independence, Estonia developed a new three-dimensional geocentred geodetic system EUREF-EST92 and its realisation EUREF-EST97 derived by GPS measurements (Randjärv, 2002, pp. 13-15).

            In this paper the geodetic datum parameters are presented between the world’s quasi-standard, the geocentred WGS84 datum and some older datums, and the Eesti Vabariigi Triangulatsioon as well. These data set enables to fit the old Estonian maps to the modern ones in GIS packages without using selected ground control points but using instead the indicated grid values and crosshairs only. Errors are described where possible.


The concept of geodetic datums and datum transformations

            As the Earth’s surface is not a perfect ellipsoid but a complex mathematical shape called geoid, the ellipsoid-based triangulation and geodetic systems have to approximate the geoid with an ellipsoid, and this approximation is valid with low error only in the surveyed area.

            After choosing an ellipsoid (fixing its semimajor axis and flattening), the coordinates of several basepoints are determined by triangulation. In some points of this network (the so-called ’Laplace-points’) the astronomical coordinates are also measured directly. Because of the shape difference between the geoid and the ellipsoid, the astronomical and the triangulation coordinates do not match. These differences are spread on the survey area by the equalisation process.

            The result of the geodetic equalisation is an ellipsoid with the pre-defined semimajor axis and flattening with a relative displacement and orientation. Relative means here that the geometric centre of the ellipsoid is not at the Earth’s mass centre and the network North is not exactly the true North. If the same area is surveyed repeatedly using different ellipsoids with different shape (as has been done in Estonia, too), the geometric centres of the different ellipsoids are different, as well as the orientation of the network Norths.

            Using the GPS technique, we are able to determine our position relative to the real mass centre of the Earth. The World  Geodetic System 1984 (WGS84; DMA 1986) is a true Earth centred ellipsoid via the orbit elements of the GPS satellites. That’s why all local datums are parametrised with their transformation parameters to the global WGS84 datum.

            These transformations can define the ellipsoid’s displacement only (simple ellipsoid shifts, 3-parameter or Molodensky-Badekas [MB] models) or the displacement, the orientation and the scale factor, too (shifts with rotation, 7-parameter or Bursa-Wolf models). In the next two chapters, both transformation types are described. For both, it is necessary to use the geocentric coordinates (Cartesian coordinates with the zero in the center of the ellipsoid, X-axis pointing to the crossing of the Equator and the null meridian, Y-axis pointing ro the crossing of the Equator and the Eastern meridian of 90 degrees, Z-axis pointing to the North Pole):





where  the radius of curvature in prime vertical; a and e are the semimajor axis and the eccentricity of the ellipsoid; Φ, Λ and h are the latitude, longitude and ellipsoidal height of the point; X, Y and Z are its geocentric coordinates. For the reverse (geocentric to geodetic) conversion, the exact solution of Borkowski (1989) and the simpler but more approximative one of Bowring (1976) can be used. Both datum transformation methods are interpreted on geocentric coordinates.


The Bursa-Wolf (7-parameter) transformation model

            For the transformation of the geocentric coordinates of a given point on a certain datum to the ones on another datum, Bursa (1962) and Wolf (1963) suggested a simplified form of the three-dimensional Helmert-transformation. As it is a simplification, the Bursa-Wolf method can be used in case of very small (several arc second order) rotations, as follows:




where X’, Y’ és Z’ are the geocetric coordinates on the target datum, dX, dY and dZ are the shifting, εX, εY and εZ are the rotation parameters, κ is the scale factor.

            It has to be underlined that the sign convention of the rotation parameters can be of two kinds: the one represented in this equation is the coordinate frame rotation convention, mostly used in the USA and Australia (and therefore, by the common GIS packages, too). The other one, the position vector rotation convention (used in Western Europe and also proposed by the ISO19111 draft) is very similar but all rotation parameters have a reversed sign in the (4) equation and in the parameter set. For converting a parameter set from one rotation convention to the other one, the only thing to do is to change the signs of the rotation parameters.




The Molodensky-Badekas (3-parameter) transformation model

            If we suppose that the source and the target datums (ellipsoids) have parallel axes, Equation (4) will be simplified to a simple three-dimensional shift (Badekas, 1969) as








Moreover, in this case it is not necessary to deal with the geocetric coordinates. Using the so-called abridged Molodensky-formulas (Molodenskiy et al., 1960; DMA, 1990), the transformation can be made as follows:






where  the radius of curvature in prime meridian; ΔΦ” and ΔΛ” are the latitude- and longitude difference between the geodetic coordinates on the source and target datums, Δh is the difference between the ellipsoidal heights, a and f are the semi-major axis and the flattening of the source datum, da and df are the differences between the semi-major axes and flattenings of the source and target datums. N and e are described after Equation (3). If ellipsoidal heights are not presented, they can be computed using global or local geoid models, or Equation (9) can be simply skipped.


The used datasets and methods

            For the determination of the parameters of the Estonian 1937 datum, coordinates of several geodetic stations throughout Estonia were provided by the National Land Survey. After the evaluation, 74 of them occurred identical both in the old (1937) and new (1992) systems.

            For each points, the L-EST97 coordinates were presented along with the old Estonian Lambert coordinates of the respective zone. The older coordinates are fit to a system with zero False Eastings and False Northings, instead of the known cartographic description (Jürgenson, 1997). Geodetic coordinates of the points on both datums were computed using the projection equation, based on the work of Snyder (1987).

            No elevation data were presented at the basepoints. In horizontal investigations it is not necessary, so no altitude reference was used in the computation of the horizontally best fitting MB parameters. In computing the MB parameters using the fundamental point of a datum, the WGS84/GRS80 ellipsoid height was estimated using the EGM96 global geoid model (NIMA, 1997) while the ellipsoidal height based on the local datum was set to zero. In computing the Bursa-Wolf parameters, the elevation data of all points on the local datum were set to zero while the ellipsoid heights of the points were set to 17 m as a countrywide average of geoid undulation on the WGS84/GRS80 ellipsoid.

            For the parameters of the Triangulation of the coast of the Baltic Sea, the Triangulation of Finland and St. Petersburg Territory (of Russia), the Baltic Sea Triangulation and the Russian Pulkovo 1942 datum, data of the respective fundamental points given by Treikelder (2000) were used.


Parameters of some older systems (1838-1915)

            In case of the older system, simple MB type parameter sets can be computed using the known coordinates of the respective basepoints on the old system and on a new one (S-42 or WGS84) as well. If the coordinates of the orientation points are known also on a modern datum, a seven-parameter solution can be reached – but in this case these data were not available.

            The fundamental point of the Triangulation of the coast of the Baltic Sea (1829-1838) is the Observatory of Tallinn and its old and also its WGS84 coordinates are described by Treikelder (2000). The computed three-parameter datum transformation description is given in Table 1, column 2, along with the parameters of the Walbeck ellipsoid. Without information on the zero meridian of Tallinn observatory, the longitude of the fundamental point has been set to L=24°47’32.55”, its approximate GRS-80 value calculated by Treikelder (2000).

            The fundamental point of the Triangulation of Finland and St. Petersburg Territory (1891-1903) and the Baltic Sea Triangulation (1910-1915) is the dome hall centre of the Pulkovo Observatory. The coordinates of the same point on the S-42 datum is known as well (Treikelder, 2000). Transformation parameters between these datums and S-42 and WGS84 are given in Table 1. Having no more basepoint data, we did not compute the errors of these parameter sets. In case of the older Baltic Sea datum, the Pulkovo zero meridian value was set to L=30°19’42.09”, its S-42 value.




CBS--> GRS80

FSP1903 --> S-42

BST1915 --> S-42

S-42 --> GRS80

FSP1903 --> GRS80

BST1915 --> GRS80

dX (m)







dY (m)







dZ (m)














a (m)














Table 1. Molodensky-Badekas datum transformation parameters between some historical Estonian and modern datums. The ellipsoid and its semi-major axis and inverse flattening values of the first datums are also indicated. Abbreviations: CBS1838 – Triangulation of the coast of the Baltic Sea; FSP1903 – Triangulation of Finland and St. Petersburg Territory; BST1915 – Baltic Sea Triangulation. Note that in case of CBS1838 and FSP1903 the Tallinn and Pulkovo zero meridians have been set to the modern longitude of those points.


The Estonian 1937 system

A new geodetic basis was adjusted for the Historical Lambert projection as a result of triangulation on the Bessel1841 ellipsoid, conducted by Ottomar Douglas. The Varesmäe first-order triangulation point was selected as the fundamental point of the datum. Its coordinates:

Φ = 59° 18’ 34.465”

Λ = 26° 33’ 41.441”


EESTI37--> GRS80

1. 7-params

(all points)

2. 7-params

(52 points)

3. 3-params

(DMA, 1986)

4. 3-params


5. 3-params

(best fit)

dX (m)






dY (m)






dZ (m)






κ (ppm)






εX (arc sec)






εY (arc sec)






εZ (arc sec)






average error






max. error






Table 2. Datum transformation parameter sets between the Eesti Vabariigi triangulatsioon (EESTI37) and the GRS80 (practically identical to WGS84).


            Evalutaing the results, at first sight it can be surprising that any 3-parameter solution can be better than a 7-parameter one. However, these sets were computed applying different methods: 7-parameter type ones are optimized to the best three-dimensional fit and the ’best fit’ 3-parameter set was optimised for the best horizontal fit. The errors indicated in Table 2 are horizontal ones.

            Nevertheless in GIS applications it is suggested that the transformation No. 4 in Table 2 be used. As its error is low enough for GIS usage, it is valid also in the three-dimensional space. In spite of its lower average horizontal error, the three-dimensioal validity is definitely not true for the transformation No. 5; esimating the ellipsoidal heights, this one results in an error of several-metres.


The Pulkovo 1942 (S-42) system

            The Bursa-Wolf type transformation parameter set between S-42 and WGS84 valid for Estonia is provided by Aunap (2001). For the parametrising of the older systems, a simple datum shift parameter set was computed. This set is primarily valid only in Estonia but as Pulkovo is not too far from this region, the set has been assumed to be valid also for that point. The 3-parameter set is given in Table 1, Column 5. The errors of this set is not estimated as we had no more basepoint data, just the coordinates of the fundamental point.


Using the datum parameters in the GIS and GPS practice

            The correct usage of the datum parameters – along with the map projections – enables to georeference the historical maps and to fit them to modern ones without ground control points. For the practical use, the most important capability is to fit the old maps from the 1930s to the modern ones automatically, using only the map grid lines. This could be a useful tool for the geographic reaseach of the change of natural and built environment.

            Most GPS receivers enable to set ‘user datums’ based on the 3-parameter datum transformation parameters. For the EESTI37 and S-42 systems it may have practical importance to set them into GPS receivers for the fieldwork.



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