Novel MCMC methods for Bayesian inference of spatial parameter fields

2021 ◽  
Author(s):  
Sebastian Reuschen ◽  
Teng Xu ◽  
Fabian Jobst ◽  
Wolfgang Nowak
Keyword(s):  
Gaussian Random Fields ◽  
Mcmc Methods ◽  
Bayesian Inversion ◽  
Soil Parameters ◽  
Multiple Point ◽  
Test Problems ◽  
Parameter Distribution ◽  
Indirect Measurements ◽  
Spatial Parameter ◽  
Prior Models

<p>Geostatistical inference (or inversion) methods are commonly used to estimate the spatial distribution of heterogeneous soil properties (e.g., hydraulic conductivity) from indirect measurements (e.g., piezometric heads). One approach is to use Bayesian inversion to combine prior assumptions (prior models) with indirect measurements to predict soil parameters and their uncertainty, which can be expressed in form of a posterior parameter distribution. This approach is mathematically rigorous and elegant, but has a disadvantage. In realistic settings, analytical solutions do not exist, and numerical evaluation via Markov chain Monte Carlo (MCMC) methods can become computationally prohibitive. Especially when treating spatially distributed parameters for heterogeneous materials, constructing efficient MCMC methods is a major challenge.</p><p>Here, we present two novel MCMC methods that extend and combine existing MCMC algorithms to speed up convergence for spatial parameter fields. First, we present the<em> sequential pCN-MCMC</em>, which is a combination of the <em>sequential Gibbs sampler</em>, and the <em>pCN-MCMC</em>. This <em>sequential pCN-MCMC</em> is more efficient (faster convergence) than existing methods. It can be used for Bayesian inversion of multi-Gaussian prior models, often used in single-facies systems. Second, we present the <em>parallel-tempering sequential Gibbs MCMC</em>. This MCMC variant enables realistic inversion of multi-facies systems. By this, we mean systems with several facies in which we model the spatial position of facies (via training images and multiple point geostatistics) and the internal heterogeneity per facies (via multi-Gaussian fields). The proposed MCMC version is the first efficient method to find the posterior parameter distribution for such multi-facies systems with internal heterogeneities.</p><p>We demonstrate the applicability and efficiency of the two proposed methods on hydro-geological synthetic test problems and show that they outperform existing state of the art MCMC methods. With the two proposed MCMCs, we enable modellers to perform (1) faster Bayesian inversion of multi-Gaussian random fields for single-facies systems and (2) Bayesian inversion of more realistic fields for multi-facies systems with internal heterogeneity at affordable computational effort.</p>

Bayesian inversion and visualization of hierarchical geostatistical models

2020 ◽  
Author(s):  
Sebastian Reuschen ◽  
Teng Xu ◽  
Wolfgang Nowak
Keyword(s):  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Gaussian Model ◽  
Large Data ◽  
Bayesian Inversion ◽  
Soil Parameters ◽  
Multiple Point ◽  
Parameter Distribution ◽  
Small Uncertainty ◽  
Prior Models

<p>Geostatistical inversion methods estimate the spatial distribution of heterogeneous soil properties (here: hydraulic conductivity) from indirect information (here: piezometric heads). Bayesian inversion is a specific approach, where prior assumptions (or prior models) are combined with indirect measurements to predict soil parameters and their uncertainty in form of a posterior parameter distribution. Posterior distributions depend heavily on prior models, as prior models describe the spatial structure of heterogeneity. The most common prior is the stationary multi-Gaussian model, which expresses that close-by points are more correlated than distant points. This is a good assumption for single-facies systems. For multi-facies systems, multiple-point geostatistical (MPS) methods are widely used. However, these typically only distinguish between several facies and do not represent the internal heterogeneity inside each facies.</p><p>We combine these two approaches to a joint hierarchical model, which results in a multi-facies system with internal heterogeneity in each facies. Using this model, we propose a tailored Gibbs sampler, a kind of Markov Chain Monte Carlo (MCMC) method, to perform Bayesian inversion and sample from the resulting posterior parameter distribution. We test our method on a synthetic channelized flow scenario for different levels of data available: A highly informative setting (with many measurements) where we recover the synthetic truth with relatively small uncertainty invervals, and a weakly informative setting (with only a few measurements) where the synthetic truth cannot be recovered that clearly. Instead, we obtain a multi-modal posterior. We investigate the multi-modal posterior using a clustering algorithm. Clustering algorithms are a common machine learning approach to find structures in large data sets. Using this approach, we can split the multi-modal posterior into its modes and can assign probabilities to each mode. A visualization of this clustering and the according probabilities enables researchers and engineers to intuitively understand complex parameter distributions and their uncertainties.</p>

Bayesian petroelastic inversion with multiple prior models

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. M57-M71 ◽  
Author(s):  
Dario Grana
Keyword(s):  
Posterior Distribution ◽  
Prior Distribution ◽  
Transition Probabilities ◽  
Mathematical Formulation ◽  
Seismic Inversion ◽  
Seismic Facies ◽  
Bayesian Inversion ◽  
Prior Model ◽  
Facies Classification ◽  
Prior Models

Bayesian methods are commonly used for geophysical inverse problems, such as seismic and rock-physics inversion, for the prediction of petroelastic properties. Bayesian inversion is based on Bayes’ theorem and combines the information from a prior distribution and a likelihood function; in geophysical applications, the prior model generally includes the available geologic information about the model variables, whereas the likelihood includes the geophysical models that link the model to the data. The goal of Bayesian inversion is to estimate the posterior distribution of the model variables conditioned by the measured data. The focus is on the prior model and its parameters. Typically, the parameters of the prior distributions are assumed to be fixed, for example, the mean and standard deviation of the prior distribution of petroelastic properties in seismic inversion or the facies proportions and transition probabilities in facies classification. I have studied the posterior distribution of the model given the data in a Bayesian setting using multiple prior models. The posterior distribution is assessed by summing the contributions of all of the likelihood functions of the model given the data, using different sets of parameters, weighted by the probabilities of the parameters. I apply the mathematical formulation in different problems, including log-facies classification, seismic-facies classification, and petrophysical property prediction and using different methods for the prior model generation such as transition matrices, training images, and Gaussian mixture models with multiple modes. The results show that multiple prior models can match the data and that the uncertainty in the prior parameters should be accounted for in the posterior distribution of the reservoir properties.

Improving the Pattern Reproducibility of Multiple-Point-Based Prior Models Using Frequency Matching

2014 ◽  
Vol 47 (3) ◽  
pp. 317-343 ◽  
Author(s):  
Knud Skou Cordua ◽  
Thomas Mejer Hansen ◽  
Klaus Mosegaard
Keyword(s):  
Multiple Point ◽  
Frequency Matching ◽  
Prior Models

scholarly journals Complexity analysis of accelerated MCMC methods for Bayesian inversion

2013 ◽  
Vol 29 (8) ◽  
pp. 085010 ◽  
Author(s):  
Viet Ha Hoang ◽  
Christoph Schwab ◽  
Andrew M Stuart
Keyword(s):  
Complexity Analysis ◽  
Mcmc Methods ◽  
Bayesian Inversion

Indirect measurements in micro-space with the EM

1995 ◽  
Vol 53 ◽  
pp. 598-599
Author(s):  
Sterling P. Newberry
Keyword(s):  
Electron Microscope ◽  
Standard Model ◽  
Particle Physics ◽  
Information Storage ◽  
Storage Space ◽  
Indirect Measurements ◽  
Storage And Retrieval ◽  
Adequate Information ◽  
Compelling Reason ◽  
The Standard Model

At the 1958 meeting of our society, then known as EMSA, the author introduced the concept of microspace and suggested its use to provide adequate information storage space and the use of electron microscope techniques to provide storage and retrieval access. At this current meeting of MSA, he wishes to suggest an additional use of the power of the electron microscope.The author has been contemplating this new use for some time and would have suggested it in the EMSA fiftieth year commemorative volume, but for page limitations. There is compelling reason to put forth this suggestion today because problems have arisen in the “Standard Model” of particle physics and funds are being greatly reduced just as we need higher energy machines to resolve these problems. Therefore, any techniques which complement or augment what we can accomplish during this austerity period with the machines at hand is worth exploring.

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