Uncertainty quantification of pollutant source retrieval: comparison of Bayesian methods with application to the Chernobyl and Fukushima Daiichi accidental releases of radionuclides

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.