4.5 Estimation of the transformation parameters

If we have a geodetic base-point set, containing the coordinates in two different datums, the transformation parameters between these datums can be estimated, according to both the AM and the BW methods.

The AM-parameters, the vector components between the geometric centers of the two datum ellipsoids, can be obtained easily. This calculation can be made even if the coordinates of just one common point (in most cases, the fundamental point) are known. It this case, we calculate the Cartesian coordinates of the point in both systems, using the sizes and figures of the ellipsoids and the geoid undulation values. Interpreting these two coordinate triplets as position vectors of the point in the two different systems, the desired parameters can be obtained as the components of the difference vector between them. First, the coordinates should be transformed to geocentric Cartesian ones:

(4.5.1)

first on the Datum 1 then on Datum 2. The first datum is usually a local one while the second is the WGS84. Then the parameters are:

(4.5.2)

In the Equation (4.5.1), h expresses the elevation above the ellipsoid (cf. Chapter 3). If we have the elevations above the sea level (above the geoid), we shall convert them all to ellipsoidal heights, using the geoid undulation values on local datum. If the elevations are unknown at all, they should be replaced simply by the geoid undulation values. If these values are unknown, use zero values. The geoid undulation values for the WGS84 (the second datum) can be obtained from a global geoid model, e.g. the EGM96. EGM96 data is available directly on the Internet, and free calculation programs are also available.

If we have more common points, we can repeat the above procedure for every point and the final parameters are provided by averaging.

Estimation of the BW-parameters are much more complicated. There are two approaches to do it. Usually, it is done by standard parameter estimation of the least square method. It is far beyond the goal of this handout to show the whole procedure, however it is worth to note that the method estimates the parameters simultaneously. This means that the parameters cannot be interpreted independently from each other – that’s why we can’t substitute the AM-type shift parameters to a BW parameter set, leaving the rotation and scale parameters untouched. In general, it is possible that the same transformation is described well by apparently very different BW parameter sets, and – contrary to the AM method – there is no easy way to show their similarity. However, there is another BW parameter estimation method, simply enough to explain, providing real, geometrically independent parameters, albeit its accuracy is a bit worse.

Let’s suppose that we shall derive parameters for a transformation between the WGS84 and a local datum with known fundamental point, whose coordinates are known on both the local datum and the WGS84. In the first step, we calculate the AM type shift parameters between the two systems, using Equation (4.5.2). In the following, we choose rotation and scale parameters for them, to improve the horizontal and spatial accuracy of the transformation.

First, we shall use the fact that the effect of the scale factor to the horizontal coordinates is much less than the effect of the rotation. Moreover, we shall realize that there is a connection between the three rotation parameters and the location of the fundamental point plus the observer azimuth at it (spherical case):

(4.5.3)

(4.5.4)

(4.5.5)

The inverse formulas for the ellipsoid:

(4.5.6)

(4.5.7)

(4.5.8)

The coordinates of the fundamental point are known. We also know that the rotation is around this point, by a single angle of α. We can estimate this angle α by calculations only if we know the azimuths from the fundamental point in both systems. However, the problem is reduced to a one-variable minimum search, even if we don’t know both azimuths. We shall seek the angle α, thus rotation parameters rx, ry and rz, which provides the best fit between the coordinates throughout the whole base-point set. This minimum search can be easily carried out by iteration, even in any spreadsheet software.

The scale factor reflects to the length measurement error at the baseline, or some minor mismatch in inappropriate using of length etalons. However, if we set the shift and rotation parameters, the scale can be estimated by another iteration step.

This method is slightly less accurate than the standard simultaneous parameter estimation, as the scale and the rotation is not fully independent from each other. However, the provided parameters can be interpreted separately geometrically. The standard estimation procedure is provided in the Appendix.