Regularized framework for inverse problems in continuous atmospheric emissions: An application to the Fukushima accident

The inverse modeling technique has been widely adopted to estimate atmospheric emissions, which aims to complement the subjective inference and provides rare retrieval when unavailable source information. The inversion generally requires the environmental observations and the source-receptor relationship constructed by an air dispersion model. But these two kinds of input lead to an ill-posed inverse problem in continuous atmospheric emissions. For the observations, the measurement network cannot capture all information on a specific emission progress, because of the nature of spatial sparse and limited temporal collections. Besides, there are inevitable model-observation discrepancies introduced by the discretization and imperfect parameters in the physical model and the diagnostic meteorology model. In this dilemma, the estimated atmospheric emissions are featured with discontinuous elements such as temporal gaps, artificial oscillations, and negative values, which are biased from the continuous emission progress in the real world.

This paper describes a regularized inversion framework to objectively address these artifacts and promote the continuity of emissions. This framework consists of the joint estimation model and the total variation (TV) regularization to handle the model-observation discrepancies and the insufficient observations respectively. The former implements site-by-site corrections by adding a diagonal matrix to the residual term of the inversion, and thus reduces the oscillations. The latter enhances a prior with the piecewise-constant feature by the L1-norm of the gradient of the emission vector, and therefore recovers the missing emissions. An adaptive parameterization scheme is tailored for the TV regularization to correct negative values.

The proposed method has been applied to the Fukushima accident to estimate the lasting emissions of ¹³⁷Cs, facing the observations with nearly half temporal incomplete of the estimation period and unavoidable deviations introduced by the atmospheric dispersion model. The results produce a discrete emission profile that accurately approximates the continuous emission progress, which better matches the recognized one by expert judgments than nine published estimates, with a Pearson’s correlation coefficient of 0.92 and an index of agreement of 0.82. The estimated profile agrees with the timing of on-site gamma dose rate peaks as well. The evaluation was also conducted with respect to atmospheric simulations, providing significantly improved air concentrations and depositions, with the ten-factor agreement (FAC10) values of 0.56 and 0.99 respectively. The uncertainty analysis with respect to the regularization parameters shows a limited variation range of the estimation error (median value below 15.04%), demonstrating the potential for operational inversions with automatic parameterization.