Geophysical inversion with a neighbourhood algorithmöII. Appraising the ensemble

Monte Carlo direct search methods, such as genetic algorithms, simulated annealing, etc., are often used to explore a ¢nite-dimensional parameter space. They require the solving of the forward problem many times, that is, making predictions of observables from an earth model. The resulting ensemble of earth models represents all `information' collected in the search process. Search techniques have been the subject of much study in geophysics; less attention is given to the appraisal of the ensemble. Often inferences are based on only a small subset of the ensemble, and sometimes a single member. This paper presents a new approach to the appraisal problem. To our knowledge this is the ¢rst time the general case has been addressed, that is, how to infer information from a complete ensemble, previously generated by any search method. The essence of the new approach is to use the information in the available ensemble to guide a resampling of the parameter space. This requires no further solving of the forward problem, but from the new `resampled' ensemble we are able to obtain measures of resolution and trade-o¡ in the model parameters, or any combinations of them. The new ensemble inference algorithm is illustrated on a highly non-linear waveform inversion problem. It is shown how the computation time and memory requirements scale with the dimension of the parameter space and size of the ensemble. The method is highly parallel, and may easily be distributed across several computers. Since little is assumed about the initial ensemble of earth models, the technique is applicable to a wide variety of situations. For example, it may be applied to perform `error analysis' using the ensemble generated by a genetic algorithm, or any other direct search method.