The application op non-linear optimisation techniques in geophysics

Abstract

Non-linear optimisation techniques form an important subject in non-linear programming. They work by searching for an optimum of a function in the hyperspace of its variable parameters. The purpose of the present work is to test the applicability of the techniques to solving non-linear geophysical problems. A problem from each of the major branches of geophysics is considered. The problem of fitting continental edges is also considered. Direct search methods are slow but are robust and, therefore useful in the early stages of the search. Gradient methods are fast and are efficient in the proximity of the optimum. A gravity or magnetic anomaly due to a two-dimensional polygonal model has a unique solution in theory. In practice, ambiguity arises from the presence of several factors and takes the form of a scatter of local minima and elongated ‘valleys’, in the hyperspace. The solution becomes less ambiguous as the influence of these factors gets less and as more parameters in the model are specified. The techniques are used successfully to interpret two-dimensional gravity and magnetic anomalies. Their efficiency, and flexibility make it possible to tackle a wide range of gravity and magnetic problems. The required computer time can be reduced by careful programming. The techniques are useful in interpreting surface wave dispersion data; the large degree of ambiguity associated with the problem may be overcome by specifying several parameters. A fast curve matching process is deviced for interpreting apparent resistivity curves. The method of outputting the results reduces the effect of equivalence. A method of fitting continental edges, by minimising the gaps and overlaps between them, is given. The ambiguity in the precise position of the pole of rotation is illustrated using the same concept adopted in the gravity, magnetic and seismic problems.