Datum transformation

A datum transformation involves the change of the horizontal datum which is often accompanied with a change of map projection. This is the case when the source projection is based upon a different horizontal datum than the target projection. If the difference in horizontal datums is ignored, there will not be a perfect match between adjacent maps of neighbouring countries or between overlaid maps originating from different projections. It may lead to differences of several hundreds of metres in the resulting coordinates. Therefore, spatial data with different underlying horizontal datums may require datum transformation.

Suppose we wish to transform spatial data from the UTM projection to the Dutch RD system, and suppose that the data in the UTM system are related to the European Datum 1950 (ED50), while the Dutch RD system is based on the Amersfoort datum. To achieve a perfect match, in this example the change of map projection should be combined with a datum transformation step.

The inverse equation of projection A is used first to take us from the map coordinates (x, y) of projection A to the geographic coordinates (ϕ, λ, h) for datum A. A height coordinate (h or H) may be added to the (x, y) map coordinates. Next, the datum transformation takes us from these coordinates to the geographic coordinates (ϕ, λ, h) for datum B. Finally, the forward equation of projection B takes us from the geographic coordinates (ϕ, λ, h) for datum B to the map coordinates (x′, y′) of projection B.

Mathematically, a datum transformation is feasible via the geocentric coordinates (x, y, z) or directly by relating the geographic coordinates of both datum systems. The latter relates the ellipsoidal latitude (ϕ) and longitude (λ), and possibly also the ellipsoidal height (h), of both datum systems⁽¹⁾.

Geographic coordinates (ϕ, λ, h) can be transformed into geocentric coordinates (x, y, z), and vice versa. The datum transformation via the geocentric coordinates implies a 3D similarity transformation. This is essentially a transformation between two orthogonal 3D Cartesian spatial reference frames together with some elementary tools from adjustment theory. The transformation is usually expressed with seven parameters: three rotation angles (α, β, γ), three origin shifts (X₀, Y ₀, Z₀) and a scale factor (s). The inputs are the coordinates of points in datum A and coordinates of the same points in datum B. The output are estimates of the seven transformation parameters and a measure of the likely error of the estimate.

Datum transformation parameters have to be estimated on the basis of a set of selected points whose coordinates are known in both datum systems. If the coordinates of these points are not correct—often the case for points measured on a local datum system—the estimated parameters may be inaccurate and hence the datum transformation will be inaccurate.

Inaccuracies often occur when we transform coordinates from a local horizontal datum to a global geocentric datum. The coordinates in the local horizontal datum may be distorted by several tens of metres because of the inherent inaccuracies of the measurements used in the triangulation network. These inherent inaccuracies are also responsible for another complication: the transformation parameters are not unique. Their estimation depends on the particular choice of common points and whether all seven transformation parameters, or only some of them, are estimated.

The example in Table 1 illustrates the transformation of the Cartesian coordinates of a point in the state of Baden-Württemberg, Germany, from ITRF to Cartesian coordinates in the Potsdam Datum. Sets of numerical values for the transformation parameters are available from three organizations:

1. The federal mapping organization of Germany (labelled “National set” in Table 1 provided a set calculated using common points distributed throughout Germany. This set contains all seven parameters and is valid for whole Germany.

2. The mapping organization of Baden-Württemberg (labelled “Provincial set” in Table 1 provided a set calculated using common points distributed throughout the state of Baden-Württemberg. This set contains all seven parameters and is valid only within the state borders.

3. The National Imagery and Mapping Agency (NIMA) of the U.S.A. (labelled “NIMA set” in Table 1 provided a set calculated using common points distributed throughout Germany and based on the ITRF. This set contains a coordinate shift only (no rotations, and scale equals unity). This set is valid for whole Germany.

The three sets of transformation parameters vary by several tens of metres, for reasons already mentioned. The sets of transformation parameters were used to transform the ITRF cartesian coordinates of a point in the state of Baden-Württemberg. Its ITRF (X, Y, Z) coordinates are:

(4,156,939.96 m, 671,428.74 m, 4,774,958.21 m).

The three sets of transformed coordinates for the Potsdam datum are given in Table 2.

The three sets of transformed coordinates differ by only a few metres from each other. In a different country, the level of agreement could be a within centimetres, but it can be up to tens of metres of each other, depending upon the quality of implementation of the local horizontal datum.