Critical decisions frequently rely on high-dimensional output from complex computer simulation models that show intricate crossvariable, spatial and temporal dependence structures, with weather and climate predictions being key examples. There is a strongly increasing recognition of the need for uncertainty quantification in such settings, for which we propose and review a general multi-stage procedure called ensemble copula coupling (ECC), proceeding as follows:1. Generate a raw ensemble, consisting of multiple runs of the computer model that differ in the inputs or model parameters in suitable ways.2. Apply statistical postprocessing techniques, such as Bayesian model averaging or nonhomogeneous regression, to correct for systematic errors in the raw ensemble, to obtain calibrated and sharp predictive distributions for each univariate output variable individually.3. Draw a sample from each postprocessed predictive distribution. 4. Rearrange the sampled values in the rank order structure of the raw ensemble to obtain the ECC postprocessed ensemble.The use of ensembles and statistical postprocessing have become routine in weather forecasting over the past decade. We show that seemingly unrelated, recent advances can be interpreted, fused and consolidated within the framework of ECC, the common thread being the adoption of the empirical copula of the raw ensemble. Depending on the use of Quantiles, Random draws or Transformations at the sampling stage, we distinguish the ECC-Q, ECC-R and ECC-T variants, respectively. We also describe relations to the Schaake shuffle and extant copula-based techniques. In a case study, the ECC approach is applied to predictions of temperature, pressure, precipitation and wind over Germany, based on the 50-member European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble.

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Contemporary weather forecasts are typically based on ensemble prediction systems, which consist of multiple runs of numerical weather prediction models that vary with respect to in the initial conditions and/or the the parameterization of the atmosphere. Ensemble forecasts are frequently biased and show dispersion errors and thus need to be statistically postprocessed. However, current postprocessing approaches are often univariate and apply to a single weather quantity at a single location and for a single prediction horizon only, thereby failing to account for potentially crucial dependence structures. Non-parametric multivariate postprocessing methods based on empirical copulas, such as ensemble copula coupling or the Schaake shuffle, can address this shortcoming. A specific implementation of the Schaake shuffle, called the SimSchaake approach, is introduced. The SimSchaake method aggregates univariately postprocessed ensemble forecasts using dependence patterns from past observations. Specifically, the observations are taken from historical dates at which the ensemble forecasts resembled the current ensemble prediction with respect to a specific similarity criterion. The SimSchaake ensemble outperforms all reference ensembles in an application to ensemble forecasts for surface temperature from the European Centre for Medium-Range Weather Forecasts.

Two seemingly unrelated postprocessing concepts retaining inter-variable, spatial and temporal rank dependence structures from raw ensemble forecasts are compared and unified. The first concept combines ensemble model output statistics (EMOS), which fits and adjusts a parametric probability density function, with the ensemble copula coupling (ECC) reordering notion. The second concept is the member-by-member postprocessing (MBMP), based on a direct parametric adjustment of the raw ensemble members. It is shown that MBMP can be interpreted as a specific EMOS-ECC variant. In an application to ensemble forecasts for temperature, EMOS-ECC and MBMP succeed in outperforming the raw ensemble and conserving correlation structures.