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Probability Theory as Logic: Data Assimilation for Multiple Source Reconstruction

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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.

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Notes

  1. To allow for the presence of localized sources with time-varying emission rates, it is straightforward to deal with this case by simply selecting the source atoms in the structure-based representation of Eq. 5 to have the following form: \(Q_k \delta({\mathbf{x}}-{\mathbf{x}}_{s,k})\left[{\mathcal H}(t-T_b^k)-{\mathcal H}(t-T_e^k)\right]\) (\(k=1,2,\ldots, N_{\rm s}\)) where \({\mathcal H}(\cdot)\) denotes the Heaviside unit step function; and, T b and T e are the activation and deactivation times of the source atom, respectively.

  2. Actually, the source–receptor relationship considered explicitly here is a special case of Eq. 1 involving the steady continuous emission of a contaminant from a sustained source into a statistically stationary atmosphere.

  3. The binomial distribution B(p *;N s,min, N s,max) has a probability distribution function defined as follows:

    $$ p(N_{\rm s}|I)={\frac{(N_{{\rm s},{\rm max}}-N_{{\rm s},{\rm min}})!}{(N_{\rm s}-N_{{\rm s},{\rm min}})!(N_{{\rm s},{\rm max}}-N_{\rm s})!}} p^{*(N_{\rm s}-N_{{\rm s},{\rm min}})} (1-p^*)^{N_{{\rm s},{\rm max}}-N_{\rm s}}, $$

    for \(N_{\rm s} = N_{\rm s,min},N_{\rm s,min}+1, \ldots , N_{\rm s,max}.\) Note that in this definition, the standard form of the binomial distribution has been offset by the minimum number of source atoms N s,min. If the expected degree of complexity in the source molecule is unknown a priori, then the assignment of the maximum number of source atoms can be omitted if the binomial distribution for p(N s|I) is replaced by a discrete probability distribution having support over the natural numbers \({{\mathbb{N}}}\) (e.g., Poisson distribution, geometric distribution).

  4. More specifically, a Bernoulli-uniform mixture model for Q k has the following form:

    $$ p(Q_k|I) = (1-\gamma)\delta(Q_k) + \gamma {{\mathbb{I}}}_{(0,Q_{\rm max})}(Q_k)\big/Q_{\rm max}, $$

    (\(k=1,2,\ldots,N_{\rm s}\)) and \({{\mathbb{I}}_A(x)}\) is the indicator function for set A, with \({{\mathbb{I}}_A(x) = 1}\) if \(x \in A\) and \({{\mathbb{I}}_A(x) = 0}\) if \(x \not\in A. \)

  5. The forward increments of the marked particle position and velocity are defined as \(\hbox{d}{\mathbf{X}}(t) \equiv {\mathbf{X}}(t+\hbox{d}t)-{\mathbf{X}}(t)\) and \(\hbox{d}{\mathbf{U}}(t) \equiv {\mathbf{U}}(t+\hbox{d}t)-{\mathbf{U}}(t), \) respectively, with dt > 0.

  6. A Wiener process W i is a stochastic process which has independent increments with \({\hbox{d}W_i(t+\hbox{d}t) \equiv W_i(t+\hbox{d}t) - W_i(t) \sim {\mathcal{N}}(0,\hbox{d}t)}\) where \({{\mathcal{N}}(\mu,\sigma^2)}\) denotes a Gaussian distribution with mean μ and variance σ2.

  7. In this paper, we use the term “sampler” to refer to any algorithm that has been designed to draw samples from the posterior PDF of the source parameters.

  8. A Markov chain is a discrete-time random process with the Markov property (viz, the next state in the chain depends only on the current state and not on any of the past states).

  9. The prior distribution \(p(\Uptheta|I)\) is composed of standard probability distributions for which independent sampling is easy.

  10. This simply implies that \(\{\Uptheta_k(\lambda)\}_{k=1}^{N_{\rm mem}}\) can be interpreted as an ensemble of source molecules drawn from \(p_\lambda(\Uptheta|{\mathbf{D}},I).\)

  11. The finite value of N s,max = 8 chosen here assumes that the user knows a priori that the unknown source distribution consists of a (small) number of discrete point sources. Furthermore, the choice of p * = 1/7 for p(N s|I) implies that the expected a priori number of sources in the domain is \(\langle N_{\rm s} \rangle = N_{\rm s,min} + (N_{\rm s,max}-N_{\rm s,min})p^* = 2. \)

  12. The marginal posterior distribution \(p(N_{\rm s}|{\mathbf{D}},I), \) which is required for the inference of the number of sources, N s, is estimated as follows:

    $$ {\hat p}(N_{\rm s}|{{\mathbf{D}}},I)=\frac{1}{N_*}\#\left\{t: N_{\rm s}^{(t)} = N_{\rm s}\right\}=\frac{1}{N_*}\sum_{t=1}^{N_*} {\mathbb{I}}_{N_{\rm s}}(N_{\rm s}^{(t)}), $$

    where N * is the number of samples (source molecules) drawn from the Markov chain. Note that this estimate counts the number of source molecules with exactly N s source atoms and normalizes the count by N *.

  13. The algorithm was applied with other values for \(p^* \in (0,1)\) with essentially no change in the results with respect to both the best estimates of the source parameters and the uncertainty quantification for these estimates.

  14. A p% highest posterior density (HPD) interval encloses a source parameter with p% probability, and is constructed so that the lower and upper bounds of the specified interval are such that the probability density function within the interval is everywhere larger than outside it. This interval can be used as a uncertainty specification for a source parameter.

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Acknowledgments

This work has been partially supported by Chemical Biological Radiological-Nuclear and Explosives Research and Technology Initiative (CRTI) program under project number CRTI-07-0196TD.

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    Eugene Yee

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Yee, E. Probability Theory as Logic: Data Assimilation for Multiple Source Reconstruction. Pure Appl. Geophys. 169, 499–517 (2012). https://doi.org/10.1007/s00024-011-0384-1

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