Sect_11.2.4

11.2.4 GPS Network Transformations

THE SOLUTION PROCEDURE

The minimum number of observations required to solve for the parameters of a 3-D similarity transformation is seven (section 11.1.4). This condition could be satisfied by having three common points, each with their three coordinate components known in each net, leaving a redundancy of two. However it is desirable to have as many common stations as possible so as to ensure a reasonable "degree of freedom" for the Least Squares determination of the transformation parameters. At the same time, the analyst may wish to leave some common points out of the solution, to be used later as "check points".

There are a number of network adjustment programs that will estimate the parameters of a similarity transformation as a by-product of the integration of one network with another. There are also many commercial GPS software packages that have the capability of determining datum transformation parameters.

Functional Model

For the Bursa-Wolf formulation (eqn (11.1-1)) the functional model is:

(11.2-1)

Note that the scale parameter has been redefined to represent the change in scale between the two networks (expressed in "parts per million"). Each of the model equations involve a mixture of parameters and more than one observable. Hence neither the "parametric" nor the "condition" method of Least Squares adjustment is suitable (section 7.1.2). Rather, the "combined" method of Least Squares is used (see HARVEY, 1994, for details). A similar functional model can be defined for the Molodensky-Badekas form (eqn (11.1-2)). The partial derivatives for the design matrix are given in HARVEY (1994).

Network Geometry

The geometrical formulation of a similarity transformation represents a continuum. Replacing this continuum by a discrete set of common points may lead to errors of interpolation. Moreover, in the presence of observational and computational errors, the accuracy of the estimated parameters may vary considerably. This depends on the spatial distribution of the points used. Some guidelines are:

Points should not be distributed in a collinear fashion because components of rotations about axes parallel to the line of points cannot be determined.
For a stable solution it is important that the points are spatially well distributed. For example, a network with an uneven geographical spread of points will cause the solution to be biased towards the areas of higher point density. Points in areas of lower density will be disadvantaged and probably result in large corrections to their coordinates.
The common points should be mostly around the perimeter of the region in which the transformation parameters are to be used.
If distortions in scale and orientation are present in the geodetic network, then it may be better to solve for sets of local parameters rather than determining a single set of transformation parameters for the whole network.

Parameterisation

Although there are seven parameters in a 3-D similarity transformation model, there may be a choice between estimating more than seven parameters, or less.

If a network does contain distortions it is possible to add parameters representing systematic errors to the basic similarity transformation model. However, if the model contains too many parameters, the adjustment may lead to a poorly conditioned system of equations. Many parameters will usually fit the data better (that is, produce smaller residuals) than a few parameters, but the estimates of the parameters may not be reliable. Furthermore, the degrees of freedom of the adjustment will be reduced and statistical testing will be less effective. A careful investigation of the network should be made before using additional parameters to "soak up", for example, scale distortions due to ground height errors.

On the other hand, solving less than the full set of seven parameters can sometimes be justified. In general, all seven parameters are estimated and then tested using statistical tests to determine if a subset of parameters is significantly different to zero. Then another solution may be obtained in which these parameters are held fixed. It often happens that the scale factor is insignificant, or that some of the rotation angles are not separable from the translation parameter estimates.

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