  Ensemble forecasts and their combination are explored from the perspective of
a probability space. Manipulating ensemble forecasts as discrete probability
distributions, multi-model ensembles (MMEs) are reformulated as barycenters of
these distributions. Barycenters are defined with respect to a given distance.
The barycenter with respect to the L2-distance is shown to be equivalent to the
pooling method. Then, the barycenter-based approach is extended to a different
distance with interesting properties in the distribution space: the Wasserstein
distance. Another interesting feature of the barycenter approach is the
possibility to give different weights to the ensembles and so to naturally
build weighted MME.
  As a proof of concept, the L2- and the Wasserstein-barycenters are applied to
combine two models from the S2S database, namely the European Centre
Medium-Range Weather Forecasts (ECMWF) and the National Centers for
Environmental Prediction (NCEP) models. The performance of the two (weighted-)
MMEs are evaluated for the prediction of weekly 2m-temperature over Europe for
seven winters. The weights given to the models in the barycenters are optimized
with respect to two metrics, the CRPS and the proportion of skilful forecasts.
These weights have an important impact on the skill of the two barycenter-based
MMEs. Although the ECMWF model has an overall better performance than NCEP, the
barycenter-ensembles are generally able to outperform both. However, the best
MME method, but also the weights, are dependent on the metric. These results
constitute a promising first implementation of this methodology before moving
to combination of more models.
