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Because optimal transport (OT) acts as displacement interpolation in physical space rather than as interpolation in value space, it can avoid double-penalty errors generated by mislocations of geophysical fields. As such, it provides a very attractive metric for non-negative, sharp field comparison – the Wasserstein distance – which could further be used in data assimilation (DA) for the geosciences. However, the algorithmic and numerical implementations of such a distance are not straightforward. Moreover, its theoretical formulation within typical DA problems faces conceptual challenges, resulting in scarce contributions on the topic in the literature. We formulate the problem in a way that offers a unified view with respect to both classical DA and OT. The resulting OTDA framework accounts for both the classical source of prior errors, background and observation, and a Wasserstein barycentre in between states which are pre-images of the background state and observation vector. We show that the hybrid OTDA analysis can be decomposed as a simpler OTDA problem involving a single Wasserstein distance, followed by a Wasserstein barycentre problem that ignores the prior errors and can be seen as a McCann interpolant. We also propose a less enlightening but straightforward solution to the full OTDA problem, which includes the derivation of its analysis error covariance matrix. Thanks to these theoretical developments, we are able to extend the classical 3D-Var/BLUE (best linear unbiased estimator) paradigm at the core of most classical DA schemes. The resulting formalism is very flexible and can account for sparse, noisy observations and non-Gaussian error statistics. It is illustrated by simple one- and two-dimensional examples that show the richness of the new types of analysis offered by this unification.

In the next few years, numerous satellites with high-resolution instruments dedicated to the imaging of atmospheric gaseous compounds will be launched, to finely monitor emissions of greenhouse gases and pollutants. Processing the resulting images of plumes from cities and industrial plants to infer the emissions of these sources can be challenging. In particular traditional atmospheric inversion techniques, relying on objective comparisons to simulations with atmospheric chemistry transport models, may poorly fit the observed plume due to modelling errors rather than due to uncertainties in the emissions. The present article discusses how these images can be adequately compared to simulated concentrations to limit the weight of modelling errors due to the meteorology used to analyse the images. For such comparisons, the usual pixel-wise ℒ2 norm may not be suitable, since it does not linearly penalise a displacement between two identical plumes. By definition, such a metric considers a displacement as an accumulation of significant local amplitude discrepancies. This is the so-called double penalty issue. To avoid this issue, we propose three solutions: (i) compensate for position error, due to a displacement, before the local comparison; (ii) use non-local metrics of density distribution comparison; and (iii) use a combination of the first two solutions. All the metrics are evaluated using first a catalogue of analytical plumes and then more realistic plumes simulated with a mesoscale Eulerian atmospheric transport model, with an emphasis on the sensitivity of the metrics to position error and the concentration values within the plumes. As expected, the metrics with the upstream correction are found to be less sensitive to position error in both analytical and realistic conditions. Furthermore, in realistic cases, we evaluate the weight of changes in the norm and the direction of the four-dimensional wind fields in our metric values. This comparison highlights the link between differences in the synoptic-scale winds direction and position error. Hence the contribution of the latter to our new metrics is reduced, thus limiting misinterpretation. Furthermore, the new metrics also avoid the double penalty issue.