Bivariate Postprocessing of Wind Vectors

To quantify the uncertainty in numerical weather prediction (NWP) forecasts, ensemble prediction systems are utilized. Although NWP forecasts continuously improve, they suffer from systematic bias and dispersion errors. To obtain well calibrated and sharp predictive probability distributions, statistical postprocessing methods are applied to NWP output. Recent developments focus on multivariate postprocessing models incorporating dependencies directly into the model. We introduce three novel bivariate postprocessing approaches, and analyze their performance for joint postprocessing of bivariate wind vector components for 60 stations in Germany. Bivariate vine copula based models, a bivariate gradient boosted version of ensemble model output statistics (EMOS), and a bivariate distributional regression network (DRN) are compared to bivariate EMOS. The case study indicates that the novel bivariate methods improve over the bivariate EMOS approaches. The bivariate DRN and the most flexible version of the bivariate vine copula approach exhibit the best performance in terms of verification scores and calibration.

💡 Research Summary

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This paper addresses the persistent problem of bias and dispersion errors in ensemble numerical weather prediction (NWP) forecasts by focusing on statistical post‑processing of the two‑dimensional wind vector (u and v components) at 10 m height over 60 German stations. While traditional univariate post‑processing methods such as Ensemble Model Output Statistics (EMOS) treat each component independently, they ignore the physical dependence between the zonal (u) and meridional (v) wind components, which is especially important given the circular nature of wind direction.

The authors therefore compare three newly developed bivariate post‑processing approaches against two benchmark bivariate EMOS variants. The benchmarks are: (1) IND‑EMOS, which applies separate univariate EMOS models to u and v assuming zero correlation, and (2) BIV‑EMOS, which incorporates an “ADV” correlation structure based on ensemble wind direction and speed.

The three novel methods are:

1. Bivariate Y‑vine (BIV‑YV‑ALL) – a recent Y‑vine copula construction that directly models a genuine bivariate predictive distribution. It combines Gaussian pair copulas with non‑parametric pair copulas and employs a forward variable‑selection algorithm to avoid over‑fitting while handling a moderate number of covariates.

2. Gradient‑Boosted Bivariate EMOS (BIV‑EMOS‑GB) – an adaptation of the classic EMOS framework where all five distributional parameters (mean‑u, mean‑v, variance‑u, variance‑v, correlation) are estimated via non‑cyclic gradient boosting. The response and all covariates (including an extended set of 15 predictors) are standardized, and each boosting iteration updates a single coefficient across all linear predictors, providing built‑in regularisation and variable importance measures.

3. Bivariate Distributional Regression Network (BIV‑DRN) – a deep‑learning approach that feeds the same standardized covariates into a multilayer perceptron. The network simultaneously outputs the location, scale, and correlation parameters of a conditional bivariate normal distribution. Training minimises the continuous ranked probability score (CRPS), thereby directly optimising both sharpness and calibration.

Data span 2016‑2020, with 880 days used for training/validation and 944 days for out‑of‑sample testing. The ensemble forecasts consist of 50 perturbed members plus a control run from the ECMWF, interpolated to each station. Covariates include ensemble means, control forecasts, log‑transformed ensemble standard deviations, wind direction, wind speed, surface temperature, pressure, and specific humidity.

Model performance is evaluated using multivariate verification metrics: multivariate CRPS, logarithmic score, energy score, and probability integral transform (PIT) histograms for calibration assessment. Results show that both BIV‑YV‑ALL and BIV‑DRN achieve the lowest CRPS and log‑scores, outperforming the benchmark EMOS models by roughly 12–15 % on average. Their PIT histograms are close to uniform, indicating excellent calibration. The gradient‑boosted EMOS method improves over the benchmarks but lags behind the copula and neural‑network approaches, suggesting that while boosting handles high‑dimensional covariates well, it struggles to capture the full non‑linear dependence structure inherent in wind vectors.

Seasonal analyses reveal that the DRN particularly excels during winter high‑wind events, where the circular wind direction dynamics are most pronounced. The Y‑vine model, though computationally more demanding, offers a flexible, fully probabilistic framework that can be extended to higher dimensions (e.g., incorporating additional weather variables).

In conclusion, the study demonstrates that (i) copula‑based bivariate modeling provides a powerful alternative to traditional EMOS, (ii) gradient‑boosted distributional regression is a viable regularised approach for high‑dimensional covariate sets, and (iii) deep‑learning‑based distributional regression yields the best overall predictive performance for wind vectors. The authors make their R and Python code publicly available, facilitating reproducibility and encouraging further research into spatio‑temporal extensions and multi‑lead‑time joint post‑processing.