Bayesian Analysis of Correlated Evolution of Discrete Characters by Reversible‐Jump Markov Chain Monte Carlo

Upscaling log measurements acquired at high frequencies and correlating them with corresponding low-frequency values from surface seismic and vertical seismic profile data is a challenging task. We have applied a sampling technique called the reversible jump Markov chain Monte Carlo (RJMCMC) method to this problem. A key property of our approach is that it treats the number of unknowns itself as a parameter to be determined. Specifically, we have considered upscaling as an inverse problem in which we considered the number of coarse layers, layer boundary depths, and material properties as the unknowns. The method applies Bayesian inversion, with RJMCMC sampling and uses simulated annealing to guide the optimization. At each iteration, the algorithm will randomly move a boundary in the current model, add a new boundary, or delete an existing boundary. In each case, a random perturbation is applied to Backus-average values. We have developed examples showing that the mismatch between seismograms computed from the upscaled model and log velocities improves by 89% compared to the case in which the algorithm is allowed to move boundaries only. The layer boundary distributions after running the RJMCMC algorithm can represent sharp and gradual changes in lithology. The maximum deviation of upscaled velocities from Backus-average values is less than 10% with most of the values close to zero.

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Estimation of trace gas fluxes with objectively determined basis functions using reversible-jump Markov chain Monte Carlo

Geoscientific Model Development ◽

10.5194/gmd-9-3213-2016 ◽

2016 ◽

Vol 9 (9) ◽

pp. 3213-3229 ◽

Cited By ~ 22

Author(s):

Mark F. Lunt ◽

Matt Rigby ◽

Anita L. Ganesan ◽

Alistair J. Manning

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Parameter Space ◽

Spatial Domain ◽

Monte Carlo Algorithm ◽

Trace Gas ◽

Reversible Jump ◽

Pseudo Data ◽

Atmospheric Trace Gas

Abstract. Atmospheric trace gas inversions often attempt to attribute fluxes to a high-dimensional grid using observations. To make this problem computationally feasible, and to reduce the degree of under-determination, some form of dimension reduction is usually performed. Here, we present an objective method for reducing the spatial dimension of the parameter space in atmospheric trace gas inversions. In addition to solving for a set of unknowns that govern emissions of a trace gas, we set out a framework that considers the number of unknowns to itself be an unknown. We rely on the well-established reversible-jump Markov chain Monte Carlo algorithm to use the data to determine the dimension of the parameter space. This framework provides a single-step process that solves for both the resolution of the inversion grid, as well as the magnitude of fluxes from this grid. Therefore, the uncertainty that surrounds the choice of aggregation is accounted for in the posterior parameter distribution. The posterior distribution of this transdimensional Markov chain provides a naturally smoothed solution, formed from an ensemble of coarser partitions of the spatial domain. We describe the form of the reversible-jump algorithm and how it may be applied to trace gas inversions. We build the system into a hierarchical Bayesian framework in which other unknown factors, such as the magnitude of the model uncertainty, can also be explored. A pseudo-data example is used to show the usefulness of this approach when compared to a subjectively chosen partitioning of a spatial domain. An inversion using real data is also shown to illustrate the scales at which the data allow for methane emissions over north-west Europe to be resolved.

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Markov Chain Monte Carlo

Bayesian Models ◽

10.23943/princeton/9780691159287.003.0007 ◽

2015 ◽

Author(s):

N. Thompson Hobbs ◽

Mevin B. Hooten

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Bayesian Analysis ◽

Posterior Distribution ◽

Mcmc Algorithm ◽

Bayesian Analyses ◽

Seminal Paper ◽

Formal Treatment ◽

High Level

This chapter explains how to implement Bayesian analyses using the Markov chain Monte Carlo (MCMC) algorithm, a set of methods for Bayesian analysis made popular by the seminal paper of Gelfand and Smith (1990). It begins with an explanation of MCMC with a heuristic, high-level treatment of the algorithm, describing its operation in simple terms with a minimum of formalism. In this first part, the chapter explains the algorithm so that all readers can gain an intuitive understanding of how to find the posterior distribution by sampling from it. Next, the chapter offers a somewhat more formal treatment of how MCMC is implemented mathematically. Finally, this chapter discusses implementation of Bayesian models via two routes—by using software and by writing one's own algorithm.

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