Blind Categorical Deconvolution in Two-Level Hidden Markov Models

Abstract

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.

Variational Bayesian inversion (VBI) of quasi-localized seismic attributes for the spatial distribution of geological facies

“…Another important consideration in probabilistic inversion is the ability to draw stochastic realizations (samples) from the posterior distribution. Methods which are based on McMC algorithm (e.g., Larsen et al 2006;Hammer & Tjelmeland, 2011;Rimstad & Omre, 2013;Lindberg & Omre, 2014 are computationally demanding as they generate samples of the model from full joint posterior distributions and estimate marginal posterior distributions from these realizations. However, the comparative advantage of such methods is that they provide samples that may be used to perform any desired inference that cannot be performed directly from the marginal posterior distributions alone.…”

“…Section 4.2 provides a description of how the localized likelihoods assumption is relaxed. Examples of previous research on Bayesian inversion methods in which likelihoods are not (fully) localized include Lindberg & Omre (2014, Grana et al (2017) and : they used a convolved two-level, 1D hidden Markov model for inversion of categorical variables (such as lithology-fluid classes) represented as the bottom hidden-layer of the model, continuous system response variables (such as reflection coefficients) represented as the middle hidden-layer, and the measured convolved data represented in the observation layer. The advantages of our new approach are that it is multi-dimensional and that it allows for joint estimation of model parameters and the spatial distribution of geological facies.…”

Bayesian Inversion of Convolved Hidden Markov Models With Applications in Reservoir Prediction

“…Focus is on the class of switching state-space models where the likelihood function for the observations vary according to an unobservable categorical random process, see [11] and references therein. Switching state-space models have numerous applications in, for example, econometrics [12], signal processing [4], speech recognition [22] and blind deconvolution [19].…”

Interpreting the Subsurface Lithofacies at High Lithological Resolution by Integrating Information From Well‐Log Data and Rock‐Core Digital Images

“…However, well-log data can be contaminated by white or heteroscedastic noises from multiple sources such as depth-dependent conditions (i.e., temperature or confining pressure changes; Masoudi et al, 2017). Furthermore, well-log signals are originated from integrated petrophysical responses over a finite interval (convoluted signal), which makes it difficult to identify detailed types of lithofacies (i.e., lithofacies at a high lithological resolution) from well-log data (Theys, 1991;Lindberg & Omre, 2014). However, a detailed subsurface heterogeneity in lithofacies can significantly affect the practical issues of the subsurface applications (e.g., the vertical/areal sweep efficiency of an oil recovery process and the migration of leaked CO 2 ) by controlling the physical mechanisms of fluids in porous media (Alusta et al, 2011;Frampton et al, 2009;Gershenzon et al, 2014;Krishnamurthy et al, 2017;Kumar et al, 2005).…”