Probability Theory as Logic: Data Assimilation for Multiple Source Reconstruction

Abstract

Probability theory as logic (or Bayesian probability theory) is a rational inferential methodology that provides a natural and logically consistent framework for source reconstruction. This methodology fully utilizes the information provided by a limited number of noisy concentration data obtained from a network of sensors and combines it in a consistent manner with the available prior knowledge (mathematical representation of relevant physical laws), hence providing a rigorous basis for the assimilation of this data into models of atmospheric dispersion for the purpose of contaminant source reconstruction. This paper addresses the application of this framework to the reconstruction of contaminant source distributions consisting of an unknown number of localized sources, using concentration measurements obtained from a sensor array. To this purpose, Bayesian probability theory is used to formulate the full joint posterior probability density function for the parameters of the unknown source distribution. A simulated annealing algorithm, applied in conjunction with a reversible-jump Markov chain Monte Carlo technique, is used to draw random samples of source distribution models from the posterior probability density function. The methodology is validated against a real (full-scale) atmospheric dispersion experiment involving a multiple point source release.

Locating and quantifying greenhouse gas emissions at a geological CO2storage site using atmospheric modeling and measurements

“…In our current formulation, we assume implicitly that the number of localized sources (Ns) is known a priori (viz., Ns is a fixed quantity that does not need to be estimated). The significantly more difficult problem (not considered herein) of the reconstruction of an a priori unknown number of localized sources was studied by Yee [2008Yee [ , 2010 as a generalized parameter estimation problem. Note that in this case, the source parameter vector q consists of the characteristics (e.g., location and emission rate) for each source as well as Ns, and the dimensionality of the source parameter space depends on Ns.…”

“…Note that in this case, the source parameter vector q consists of the characteristics (e.g., location and emission rate) for each source as well as Ns, and the dimensionality of the source parameter space depends on Ns. To address this problem, Yee [2008Yee [ , 2010 used a reversible jump Markov chain Monte Carlo algorithm to sample from the generalized parameter space, allowing changes in the dimensionality of the model (viz., changes in Ns). An alternative (and arguably simpler) approach to address the problem of an unknown number of sources is to treat the problem as a model selection problem rather than as a generalized parameter estimation problem.…”

Inverse Dispersion for an Unknown Number of Sources: Model Selection and Uncertainty Analysis

“…The problem of reconstruction of an unknown number of localized sources using a finite number of noisy concentration measurements is a significantly more difficult problem. A solution for this problem was ISRN Applied Mathematics 3 proposed by Yee 13,14 who approached the problem as a generalized parameter estimation problem in which the number of localized sources, N, in the unknown source distribution was included explicitly in the parameter vector θ, in addition to the usual parameters that characterize each localized source e.g., location, emission rate, source-on time, source-off time .…”

“…Furthermore, it was found that the RJMCMC algorithm sampled the parameter space of the unknown source distribution rather inefficiently. To overcome this technical difficulty, Yee 13,14 showed how the RJMCMC algorithm can be combined either with a form of parallel tempering based on a Metropolis-coupled MCMC algorithm 13 or with a simpler and computationally more efficient simulated annealing scheme 14 to improve significantly the sampling efficiency or, mixing rate of the Markov chain in the variable dimension parameter space.…”

“…In this paper, we address the inverse dispersion modeling problem for the difficult case of multiple sources when the number of sources is unknown a priori as a model selection problem, rather than as a generalized parameter estimation problem as described by Yee 13,14 . The model selection problem here is formulated in the Bayesian framework which involves the evaluation of model selection statistics that gauges rigorously the tradeoff between the goodness-of-fit of the source model structure to the concentration data and the complexity of the source model structure.…”