Estimation of the Burša-Wolf parameters

Here the goal is to provide such parameters of the BW-transformation (Equation 4.3.1) that provide minimum difference between the measured (given) coordinates of a point set and the calculated coordinates of the same set, deriving from the coordinates in the other datum by the BW-method. It is formulated as follows:

(10.5)

In Equation (10.5) the ~ sign refers to the measured data, while the lower index indicates the source (1) and the target (2) systems. The index i runs for the point set, the number of the point in this set is N, while the index j refers to the dimension – in case of planar coordinates, it is 2 while at spatial coordinates, j=3.

If we substitute the following variables in the Helmert transformation:

(10.6)

it appears as follows:

(10.7)

These equations are linear ones to the parameters to be estimated. Again, the least squares method can be applied for the estimation of the parameters. The minimum condition is:

(10.8)

The equation system (10.8) contains the measured coordinates of the identical points, X, Y, Z are their coordinates in the source and the target systems. The minimum condition is set for the square-sums of the minimums between the measured and the calculated coordinates. In other words, it is set for the squares of the metric distances. The condition is similar to minimization of the absolute value of the distance difference.

The condition of the minimum is that the partial derivatives according to the parameters (the dX, dY, dZ, A, B, C and D values) should be zeroes. These partial derivatives are:

(10.9)

In the Equation (10.9), the seven parameters can be moved before the summation in each row. After rearrangement, this can be written as an inhomogeneous linear equation system, similar to the form of Equation (10.2). Making the derivations,

(10.10)

occurs. The elements of the matrix A and the vector b – similarly to the solution of the abridging Molodensky method – show the sums of the values of all base points of the set. We shall use this simplification to avoid an equation image too complex and make it ready to print. Where the squares or the mixed products of the coordinates occur among the matrix or vector elements, the summarization should be made for them. Together with the omission of the sum sign, we also omit the index i.

The vector x, containing the estimated parameters, can be provided by the inverse of the matrix A, similarly to the Equation (10.4). Afterwards, according to the Equation (10.6), the κ scale factor and the α, β and γ angle values can be computed from the A, B, C and D values. For this procedure, we have to have at least three base points with their X, Y, Z coordinates both in the source and target datums.