2. Mathematical functions
One important task of the surveyor is the determination of three-dimensional coordinates of heterogeneous objects. In order to check the measurements, more observations are made than there are unknowns. The resulting system of redundant observations can in practice only be solved by an optimisation process.
Every optimisation is uniquely defined by edge conditions and a target function. Edge conditions here are the geometric relationships between the observations (paths, angles, photo measurements, etc.) and the unknowns (points, photo orientations, etc.) described in the design matrix A. They are discussed later in the Section 2.1 Functional Model. For a specific data set they cannot be exchanged.
Selected target functions for geodetic applications might be e.g. Minimax, L1-Norm or L2-Norm. This selection might have a high influence to the adjusted result. In BINGO this equation system will be solved by adjustment according to the method of least squares (L2-norm). Section 2.2 Stochastical Model describes the intricacies of this application.
Because of the non-linear approach this adjustment is always an iterative process.
A more detailed description of the mathematical functions, the mode of operation of the adjustment algorithm, and practical examples, may be seen in /Kruck 1983/.
Functional and stochastic model are strictly separate from each other. The only connection is the standard deviation of the weight unit s0 as will become clear further on in this chapter. An important consequence of this fact is that e.g. standard deviations of individual points cannot say anything at all about their true accuracy, if measuring errors still exist in the data set. A measuring error at an individual point affects the standard deviation of this point not directly but only via the global factor s0 for the entire net.