Uncertainty quantification of pollutant source retrieval: comparison of Bayesian methods with application to the Chernobyl and Fukushima-Daiichi accidental releases of radionuclides
Résumé
Inverse modelling of the emissions of atmospheric species and pollutants has significantly
progressed over the past 15 years. However, in spite of seemingly reliable estimates, the
retrievals are rarely accompanied by an objective estimate of their uncertainty, except when
Gaussian statistics are assumed for the errors, which is often an unrealistic assumption.
Here, we assess rigorous techniques meant to compute this uncertainty in the context of the
inverse modelling of the time emission rates – the so-called source term – of a point-wise
atmospheric tracer. Log-normal statistics are used for the positive source term prior and
possibly the observation errors; this precludes simple Gaussian statistics-based solutions.
Firstly, through the so-called empirical Bayesian approach, parameters of the error
statistics – the hyperparameters – are first estimated by maximizing their likelihood via
an expectation–maximization algorithm. This enables a robust estimation of a source
term. Then, the uncertainties attached to the retrieved source rates and total emission are
estimated using four Monte Carlo techniques: (i) an importance sampling based on a Laplace
proposal, (ii) a naive randomize-then-optimize (RTO) sampling approach, (iii) an unbiased
RTO sampling approach, and (iv) a basic Markov chain Monte Carlo (MCMC) simulation.
Secondly, these methods are compared to a more thorough hierarchical Bayesian approach,
using an MCMC based on a transdimensional representation of the source term to reduce
the computational cost.
Those methods, and improvements thereof, are applied to the estimation of the atmospheric
caesium-137 source terms from the Chernobyl nuclear power plant accident in April and
May 1986 and Fukushima Daiichi nuclear power plant accident in March 2011. This study
provides the first consistent and rigorous quantification of the uncertainty of these best
estimates.