2.4.2 Free network adjustment with minimum trace and minimum solution vector

If there is no or insufficient datum information, the solution of the normal equation can also be attained by a purposeful adding of conditions into the normal equation. In comparison to the automatic removal of rank defects corresponding to Section 2.4.1, this is the obviously better and securer method. It is especially recommended for engineering applications.

The consequences are on the one hand the minimum standard deviations of the adjusted results, and on the other hand the minimum solution vector of the corrections to the unknowns.

The conditions to be added are bundled in a matrix B appended to the normal equation matrix. The necessary theoretical background is given in textbooks and papers. See e.g. I. Illner/1983/. The solution of the resulting negative definite normal equation is described in G. Funcke/1986/.

If not all points are included in the Matrix B, only a part of the traces will be minimised. The points included in the Matrix are called datum points. Around the centre of these points the standard deviation of the adjusted results is a minimum. The centre point and therefore the datum points should be chosen always in that way, that the standard deviations will be minimised in that part of the network, which is mainly interesting.

The network position is defined exclusively by the position of the approximations of the datum points. The solution vector x is only minimised in relation to these approximations. If the approximations are not identical with the coordinates of the datum points this should be considered.

If the demanded network positioning cannot be obtained by the selection of the approximation coordinates, an additional 3-D transformation will be necessary (Program HEL3D).

The type and the size of the introduced matrix are defined by type and number of the free parameters. BINGO offers a selection of several useful combinations of these free datum parameters (see also Section 6.3).