  This paper presents a new variational data assimilation (VDA) approach for
the formal treatment of bias in both model outputs and observations. This
approach relies on the Wasserstein metric stemming from the theory of optimal
mass transport to penalize the distance between the probability histograms of
the analysis state and an a priori reference dataset, which is likely to be
more uncertain but less biased than both model and observations. Unlike
previous bias-aware VDA approaches, the new Wasserstein metric VDA (WM-VDA)
dynamically treats systematic biases of unknown magnitude and sign in both
model and observations through assimilation of the reference data in the
probability domain and can fully recover the probability histogram of the
analysis state. The performance of WM-VDA is compared with the classic
three-dimensional VDA (3D-Var) scheme on first-order linear dynamics and the
chaotic Lorenz attractor. Under positive systematic biases in both model and
observations, we consistently demonstrate a significant reduction in the
forecast bias and unbiased root mean squared error.
