Appendix A. Appendix: procedures to estimate the datum transformation parameters

Table of Contents

Estimation of the abridging Molodensky-parameters, providing the best horizontal fit
Estimation of the Burša-Wolf parameters

The abridging Molodensky formulae describe the datum transformation simply by the components of the position vector pointing from the geometric center of the target datum ellipsoid to the one of the source datum ellipsoid. It does not take the orientation and scale differences into account. It is also referred to as three-parameter datum transformation. The neglected orientation and scale parameters are applied in the Burša-Wolf method; besides the three position shift parameters, uses three orientation and one scale parameters, too. Therefore it is also called as seven-parameter datum transformation method. The parameters of both transformations (as well as the ones of other procedures) are derived from coordinates of geodetic base points, whose coordinates are known in both the source and the target datums.

In this Appendix we show the estimation methods of

Estimation of the abridging Molodensky-parameters, providing the best horizontal fit

The procedure – verifying its name – provides direct connection between the geodetic coordinates and ellipsoidal heights in the source and the target datums. To estimate the shift parameters, we need base points, whose ellipsoidal coordinates are known on both datums. In the practice, usually the low-order geodetic base points are used as common points, whose coordinates are given in well-defined projection systems. The inverse projection parameters should be used to obtain the ellipsoidal coordinates.

The abridging Molodensky formulae are given in Equations (4.2.2), (4.2.3) and (4.2.4). Using the base point set with the coordinates both in the source and the target system, the differences between the observed and the calculated coordinates should be minimized, as follows:

(10.1)

where the ’S’ lower index indicates the source coordinates and the ’T’ indicates the target ones. These values can be calculated using the Molodensky formulae as a function of the geodetic coordinates. To get the optimum in a planar system instead of the geodetic system, the longitude difference is scaled by The condition of the minimum is that the partial derivative of the square sums of the differences in the first two equations, by the parameters, should be all zeroes.

Doing the partial derivations and using the value C=a·df+f·da, the Equation (10.1) can be expressed in the following matrix form:

(10.2)

where the elements of the (symmetric) matrix A and the vector b are:

 

(10.3)

In the Equation (10.3) all the coordinates, the M and N values are interpreted in the source system. This is an inhomogeneous linear equation system, whose solution is

(10.4)

where A-1 is the inverse of the matrix A. The solution vector x contains the dX, dY and dZ parameters. In the practice, the parameters can be easily determined by the Cramer rule.