Bayesian inversion of seismic attributes for geological facies using a Hidden Markov Model

Abstract

Markov chain Monte-Carlo (McMC) sampling generates correlated random samples such that their distribution would converge to the true distribution only as the number of samples tends to infinity. In practice, McMC is found to be slow to converge, convergence is not guaranteed to be achieved in finite time, and detection of convergence requires the use of subjective criteria. Although McMC has been used for decades as the algorithm of choice for inference in complex probability distributions, there is a need to seek alternative approaches, particularly in high dimensional problems. Walker & Curtis (2014) developed a method for Bayesian inversion of two-dimensional spatial data using an exact sampling alternative to McMC which always draws independent samples of the target distribution. Their method thus obviates the need for convergence and removes the concomitant bias exhibited by finite sample sets. Their algorithm is nevertheless computationally intensive and requires large memory. We propose a more efficient method for Bayesian inversion of categorical variables, such as geological facies that requires no sampling at all. The method is based on a 2D Hidden Markov Model (2D-HMM) over a grid of cells where observations represent localized data constraining each cell. The data in our example application are seismic attributes such as P-and S-wave impedances and rock density; our categorical variables are the hidden states and represent the geological rock types in each cell-facies of distinct subsets of lithology and fluid combinations such as shale, brine-sand and gas-sand. The observations at each location are assumed to be generated from a random function of the hidden state (facies) at that location, and to be distributed according to a certain probability distribution that is independent of hidden states at other locations-an assumption referred to as localized likelihoods. The hidden state (facies) at a location cannot be determined solely by the observation at that location as it also depends on prior information concerning the spatial distribution of other hidden states elsewhere. The prior information is included in the inversion in the form of a training image which represents a conceptual depiction of local geologies that might be expected, but other forms of prior information can be used in the method as desired. The method provides direct estimates of posterior marginal probability distributions over each

Rapid Discriminative Variational Bayesian Inversion of Geophysical Data for the Spatial Distribution of Geological Properties

“…Removing the assumption of localized likelihoods and conditional independence of data means that our method should be able to account for any correlations present in the data due to spatial blurring of data or due to correlated noise, as long as we can model some salient characteristics (or features) such as the spatial correlation of noise. In order to test this, and to benchmark the current method against previous research, we used the same test Earth model as used in the previous research of Walker and Curtis () and Nawaz and Curtis (, ). Here for the first time, we demonstrate that the new method is capable of inverting seismic attributes for facies with reasonable accuracy even in the presence of strongly correlated noise.…”

“…In that method every sample is an independent sample from the posterior probability distribution leading to far greater information content in any fixed set of samples. Using alternative strategies for such spatial models, Nawaz and Curtis (, ) developed inversion methods which avoid sampling entirely by computing the posterior distribution using numerical optimization. Our current method follows the latter philosophy and allows probabilistic inversion to be performed while avoiding McMC.…”

“…For example, the McMC‐based method of Larsen et al () performs a trace‐by‐trace inversion of recorded seismic waveforms (traces) and assumes that each trace is conditionally independent of others, given the geology. Walker and Curtis () and Nawaz and Curtis (, ) perform a multidimensional inversion (with 2‐D examples) but still assume conditional independence of data given the geology.…”

“…To limit the analytical and computational complexity of modeling the joint distribution over all of the observed and hidden variables, previous research in geostatistical inversion assumed that the likelihoods are localized (or quasi-localized) and that the observed data are independent and identically distributed (e.g., Caers et al, 2006;Grana, 2018;Hoffman & Caers, 2007;Larsen et al, 2006;Nawaz & Curtis, 2017;Shahraeeni & Curtis, 2011;Shahraeeni et al, 2012;Ulvmoen & Omre, 2010;Walker & Curtis, 2014a). The localized-likelihood assumption models the probabilistic relationship between the data observed at a location and the model parameters at the same location independently from other locations.…”

Neural Network Travel-Time Tomography

“…These methods have been used at a global scale to invert surface wave velocities for global crustal thicknesses and seismic velocities (Meier et al, 2007a,b) and for water content in the mantle transition zone (Meier et al, 2009), at a reservoir scale to infer petrophysical parameters from velocities (Shahraeeni and Curtis, 2011;Shahraeeni et al, 2012), for earthquake source parameter estimation (Käufl et al, 2014(Käufl et al, , 2015 and to assess the uncertainty in model parameters of the Earth's global average (1-dimensional) radial velocity structure from P-wave travel time curves (De Wit et al, 2013). They have also been used in conjunction with Markov random fields and other statistical and graphical models to solve geophysical inverse problems with spatially sophisticated prior information (Nawaz and Curtis, 2017. They have been used in conjunction with seismic gradiometry to perform near-real time 3D surface wave tomography (Cao et al, submitted).…”