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.

翻译：从概率空间的视角探索集合预报及其组合。将集合预报视为离散概率分布，多模型集成（MMEs）可重新表述为这些分布的重心。重心基于给定距离定义，其中基于L2距离的重心被证明等价于池化方法。随后，该重心方法被扩展到分布空间中具有有趣特性的不同距离：Wasserstein距离。重心方法的另一有趣特性是可对集合赋予不同权重，从而自然构建加权MME。作为概念验证，将L2重心和Wasserstein重心应用于S2S数据库中两个模型的组合，即欧洲中期天气预报中心（ECMWF）模型和美国国家环境预测中心（NCEP）模型。通过七个冬季欧洲地区周平均2米气温预报，评估两种（加权）MME的性能。基于两个指标（CRPS和预报技巧比例）优化重心中模型的权重，这些权重对两种基于重心的MME的技巧具有重要影响。尽管ECMWF模型整体表现优于NCEP，但重心集成通常能超越两者。然而，最佳MME方法及权重均取决于评估指标。这些结果构成该方法在扩展到更多模型组合前的首次有前景的实现。