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Staudinger Martin

A Cost Orientated Approach to Geodetic Network Optimisation
by Martin Staudinger

Surveyors have, among other things, the task to determine information about the spatial position of different objects in the world. The objects are usually divided into individual points, and these points brought in connection to a spatial reference system. Usually, this is done by means of a coordinate system, describing every point by one, two, or three coordinates. In many cases, the appropriate measuring- and calculation-method for that purpose is a geodetic network.
The theory of geodetic networks connects mathematic-geometrical and statistical concepts. Starting with a set of redundant measurements, we get the appropriate position information as well as specifications concerning their quality (accuracy and reliability). The method of least squares adjustment has been well-known for now almost two hundred years.
Since then, surveyors always tried to establish their measurements in the required quality but with as little cost as possible. The method of least squares makes it possible to estimate the quality of a geodetic network before the actual field work (a priori adjustment). Thus, the network design can be optimised regarding accuracy and reliability. For an optimisation of the network costs, an appropriate cost function was missing so far.
In this thesis, we show that it is possible to find a cost function which, at least, indicates the cost differences between different network variants. Among a number of simulated network variants, we can find out the one which will presumably cause the smallest cost. The individual network versions differ according to the type and number of geodetic observations. Both the spatial conditions (costs for accessing the points and travelling between them) and the actual activities, which are undertaken by the field crew, are considered. In previous work, the spatial positions of the points were not considered; thus all observations of a special type (all distance observations, e.g.) were regarded as causing the same costs and only their number was optimised.
For the optimisation, we used the method of simulated annealing. Simulated annealing is a heuristic procedure. It is suitable to solve combinatorial optimisation problems which would otherwise not be solvable in polynomial computation time. The algorithm produces sub-optimal solutions and is based on an improvement of a simple local search-algorithm: A known solution is improved by searching for a further solution in the neighbourhood of the first one. The new solution is accepted with a certain probability even if the new value of the objective function (in our case: the cost of the network) is worse than the last one. By this means it is prevented to get trapped in a local optimal solution. The procedure was tested by some examples. The result is that
the found cost function is able to indicate cost differences between different network versions
the cost estimation can easily be integrated in the usual network adjustment
thus, the network-design can be optimised regarding its costs.

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