Sequential Markov random fields and Markov mesh random fields for modelling of geological structures
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We have been given a two-dimensional image of a geologicalstructure. This structure is used to construct a three-dimensionalstatistical model, to be used as prior knowledge in the analysisof seismic data. We consider two classes of discrete latticemodels for which efficient simulation is possible; sequentialMarkov random field (sMRF) and Markov mesh random field (MMRF). Wefirst explore models from these two classes in two dimensions,using the maximum likelihood estimator (MLE). The results indicatethat a larger neighbourhood should be considered for all themodels. We also develop a second estimator, which is designed tomatch the model with the observation with respect to a set ofspecified functions. This estimator is only considered for the sMRFmodel, since that model proved to be flexible enough to givesatisfying results. Due to time limitation of this thesis, wecould not wait for the optimization of the estimator to converge.Thus, we can not evaluate this estimator. Finally, we extractuseful information from the two-dimensional models and specify asMRF model in three dimensions. Parameter estimation for this modelneeds approximative techniques, since we only have givenobservations in two dimensions. Such techniques have not beeninvestigated in this report, however, we have adjusted theparameters manually and observed that the model is very flexibleand might give very satisfying results.