Geophysical inversion with a neighbourhood algorithm - I. Searching a parameter space

This paper presents a new derivative-free search method for finding models of acceptable data fit in a multidimensional parameter space. It falls into the same class of method as simulated annealing and genetic algorithms, which are commonly used for global optimization problems. The objective here is to find an ensemble of models that preferentially sample the good data-fitting regions of parameter space, rather than seeking a single optimal model. (A related paper deals with the quantitative appraisal of the ensemble.) The new search algorithm makes use of the geometrical constructs known as Voronoi cells to derive the search in parameter space. These are nearest neighbour regions defined under a suitable distance norm. The algorithm is conceptually simple, requires just two 'tuning parameters', and makes use of only the rank of a data fit criterion rather than the numerical value. In this way all difficulties associated with the scaling of a data misfit function are avoided, and any combination of data fit criteria can be used. It is also shown how Voronoi cells can be used to enhance any existing direct search algorithm, by intermittently replacing the forward modelling calculations with nearest neighbour calculations. The new direct search algorithm is illustrated with an application to a synthetic problem involving the inversion of receiver functions for crustal seismic structure. This is known to be a non-linear problem, where linearized inversion techniques suffer from a strong dependence on the starting solution. It is shown that the new algorithm produces a sophisticated type of 'self-adaptive' search behaviour, which to our knowledge has not been demonstrated in any previous technique of this kind.