Complex functions, such as the output of computer simulators, can be difficult to optimize. The task becomes even more difficult when only some of the function evaluations return real numbers and others simply fail to return a value. We combine statistical emulation, classification, sequential design, and optimization with an asymmetric entropy measure to solve the thorny problem of finding an optimum along a constraint boundary. This approach is demonstrated on simulated examples and a real problem in groundwater remediation.
A convolved two-level hidden Markov model is defined as an observed top level representing convolutions of an unobserved middle level of responses to an unobserved bottom level containing a Markov chain of categorical classes. The associated model parameters include a Markov chain transition matrix, response levels and variances, a convolutional kernel, and an observation error variance. The convolutional kernel and the error variance are defined to be unknown. Focus is on the joint assessment of the unknown model parameters and the sequence of categorical classes given the observed top level. This is termed blind categorical deconvolution and is cast in a Bayesian inversion setting. An approximate posterior model based on an approximate likelihood model in factorizable form is defined. The approximate model,
including the likelihoods for the unknown model parameters, can be exactly assessed by a recursive algorithm. A sequence of approximations is defined such that tradeoffs between accuracy and computational demands can be made. The model parameters are assessed by approximate maximum-likelihood estimation, whereas the inversion is represented by the approximate posterior model. A limited empirical study demonstrates that reliable model parameter assessments and inversions can be made from the approximate model. An example of blind seismic deconvolution is also presented and discussed.Index Terms-Bayesian inversion, blind deconvolution, forward-backward (FB) algorithm, hidden Markov models (HMMs), maximum approximate likelihood estimation.
Log-facies classification methods aim to estimate a profile of facies at the well location based on the values of rock properties measured or computed in welllog analysis. Statistical methods generally provide the most likely classification of lithological facies along the borehole by maximizing a function that describes the likelihood of a set of rock samples belonging to a certain facies. However, most of the available methods classify each sample in the well log independently and do not account for the vertical distribution of the facies profile. In this work, a classification method based on hidden Markov models is proposed, a stochastic method that accounts for the probability of transitions from one facies to another one. Differently from other available methods where the model parameters are assessed using nearby fields or analogs, the unknown parameters are estimated using a statistical algorithm called the Expectation-Maximization algorithm. The method is applied to two different datasets: a clastic reservoir in the North Sea where four litho-fluid facies are identified and an unconventional reservoir in North America where four lithological facies are defined. The results of the applications show the added value of the introduction of a vertical continuity model in the facies classification and the ability of the proposed method of inferring model parameters such as facies transition probabilities and facies posterior distributions. The application also includes a sensitivity analysis and a comparison to other statistical methods.
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