Coping with model error in variational data assimilation using optimal mass transport

Classical variational data assimilation methods address the problem of optimally combining model predictions with observations in the presence of zero-mean Gaussian random errors. However, in many natural systems, uncertainty in model structure and/or model parameters often results in systematic errors or biases. Prior knowledge about such systematic model error for parametric removal is not always feasible in practice, limiting the efficient use of observations for improved prediction. The main contribution of this work is to advocate the relevance of transportation metrics for quantifying nonrandom model error in variational data assimilation for nonnegative natural states and fluxes. Transportation metrics (also known as Wasserstein metrics) originate in the theory of Optimal Mass Transport (OMT) and provide a nonparametric way to compare distributions which is natural in the sense that it penalizes mismatch in the values and relative position of "masses'' in the two distributions. We demonstrate the promise of the proposed methodology using 1-D and 2-D advection-diffusion dynamics with systematic error in the velocity and diffusivity parameters. Moreover, we combine this methodology with additional regularization functionals, such as the l(1)-norm of the state in a properly chosen domain, to incorporate both model error and potential prior information in the presence of sparsity or sharp fronts in the underlying state of interest.