Ensemble Model Output Statistics (EMOS)

• EMOS is a statistical framework that transforms raw ensemble forecasts into calibrated probabilistic predictions by linking distribution parameters to ensemble-derived statistics.
• It addresses issues like bias and underdispersion inherent in numerical weather prediction outputs, significantly improving forecast reliability across diverse weather variables.
• EMOS employs optimization of scoring rules such as the CRPS for parameter estimation and extends to multivariate, spatial, and hybrid modeling strategies for operational use.

Ensemble Model Output Statistics (EMOS) is a statistical postprocessing framework developed to calibrate ensemble weather forecasts by fitting a single parametric probability distribution whose parameters are linked to the ensemble output. EMOS addresses deficiencies inherent to raw ensemble outputs from numerical weather prediction (NWP) systems, such as bias and underdispersion, and provides a unified approach for generating sharp and reliable probabilistic forecasts across a spectrum of weather variables and model architectures.

1. Foundation and General Workflow

At its core, EMOS transforms raw ensemble forecasts into a full predictive probability distribution for a weather variable of interest. For a variable $Y$ and ensemble forecasts $f_1, \dots, f_M$, EMOS posits that $Y$ follows a parametric distribution—e.g., normal, truncated normal, log-normal, gamma, generalized extreme value (GEV), or logistic—with parameters expressed as functions (typically affine) of statistics derived from the ensemble (means, variances, etc.).

A generic univariate EMOS scheme for temperature, assumed normal, is: $$Y \mid \{f_1, \dots, f_M\} \sim \mathcal{N}\left(a_0 + a_1 f_1 + \cdots + a_M f_M,\; b_0 + b_1 S^2\right)$$ where $S^2 = \frac{1}{M-1} \sum_{k=1}^M (f_k - \bar{f})^2$. For wind speed, which is nonnegative and skewed, a truncated normal or log-normal predictive law is often used.

Parameters are fitted by optimizing proper scoring rules—most commonly, the continuous ranked probability score (CRPS)—on a rolling training window of historical forecast–observation pairs. This approach ensures both calibration and sharpness of the forecast distribution. For ensemble types with exchangeable members, groupings are respected to reduce parameter multiplicity.

2. Parametric Distributions and Extensions

EMOS is flexible with respect to the choice of predictive distribution, enabling specialized handling of the physical and statistical characteristics of different meteorological variables:

• Temperature: Normal law. Parameters linearly or affinely linked to ensemble mean and spread (Baran et al., 2013).
• Wind Speed: Truncated normal, log-normal, or GEV. For heavy-tailed behavior (high wind events), log-normal or GEV family may be used, with further hybridization in regime-switching or mixture models (Baran et al., 2014, Baran et al., 2015, Baran et al., 2020).
• Precipitation: Censored and shifted gamma (CSG), or censored GEV/logistic distributions, allow for discrete-continuous modeling accounting for zero-inflation and positive skewness (Scheuerer, 2013, Baran et al., 2015, Szabó et al., 2022, Friedli et al., 2019).
• Wind Vectors and Multivariate Quantities: Bivariate EMOS models encode the full dependence structure by modeling bias-corrected means, variances, and direction-dependent correlations (e.g., via trigonometric functions for wind vector components) (Schuhen et al., 2012, Baran et al., 2015).

Advanced EMOS schemes include conditional or nonlinear models, regime-switching (selecting different families based on covariate thresholds), mixtures (weighted sums of parametric distributions), and extensions grounded in time series theory (e.g., embedding autoregressive or GARCH processes in the predictive mean and/or variance) (Jobst et al., 1 Feb 2024).

3. Parameter Estimation Methodologies

Parameter estimation in EMOS typically proceeds via optimization of a proper scoring rule—most commonly CRPS, which rewards both calibration and sharpness: $$\text{CRPS}(F, y) = \int_{-\infty}^\infty \left(F(t) - \mathbb{I}_{[y,\infty)}(t)\right)^2 dt$$ CRPS allows closed-form or numerically efficient solutions for many EMOS distributions, enabling practical, operational deployment even at high spatial and temporal resolutions (Scheuerer, 2013, Baran et al., 2015, Szabó et al., 2022).

Alternative estimation based on maximum likelihood (minimizing mean logarithmic score) is sometimes preferred for computational reasons. CRPS optimization is generally more robust in forecast verification applications because it simultaneously accounts for the entire forecast distribution.

For models with high spatial or parametric complexity, estimation can be made "regional" (training on all sites), "local" (separate parameters for each site/time), or "semi-local" (grouping sites with similar climatology or forecast error characteristics via clustering or distance metrics). The semi-local paradigm efficiently balances estimation variance and specificity (Lerch et al., 2015, Möller et al., 2015, Díaz et al., 2018, Friedli et al., 2019).

4. Multivariate, Mixture, and Combination EMOS

The EMOS framework extends naturally to multivariate settings to capture cross-variable dependence. Bivariate (and, in principle, higher-dimensional) EMOS models use multivariate normal or truncated distributions, with parameters (means, variances, covariance/correlation) linked via affine functions and possibly conditioned on physical predictors (e.g., wind direction) (Schuhen et al., 2012, Baran et al., 2015).

Mixture EMOS models for wind speed (e.g., TN–LN mixtures) convexly combine two distributions, with the weights estimated to optimize forecast skill as measured by scoring rules like CRPS. This avoids the instability and interpretability issues of regime-switching approaches and provides smoother adaptation across regimes (Baran et al., 2014, Baran et al., 2015, Baran et al., 2016). More generally, forecast combinations via linear pooling, spread adjustment, or beta transformations allow EMOS to borrow strength from multiple parametric families (Baran et al., 2016).

Ensemble Copula Coupling (ECC) and copula-based EMOS extensions are used to restore/highlight joint dependence structures lost during univariate postprocessing (Schuhen et al., 2012, Baran et al., 2015, Friedli et al., 2019).

5. Spatial, Temporal, and Adaptive Strategies

Recent EMOS advances introduce spatial adaptivity by treating bias coefficients as realizations of spatial random fields (notably, Gaussian Markov random fields via SPDEs), solved via integrated nested Laplace approximations (INLA). This “MEMOS” approach yields spatially adaptive, Bayesian estimates that maintain calibration and sharpness at arbitrary grid locations, even with sparse observational data (Möller et al., 2015).

Time series–based extensions integrate seasonality, trend, and temporal autocorrelation or heteroscedasticity into EMOS. Seasonal patterns in location and scale can be encoded via finite Fourier series, while forecast error time series may be modeled using AR or GARCH structures to relax independence assumptions and exploit persistence (Jobst et al., 1 Feb 2024).

Conditioning on additional predictors (e.g., diurnal cycle, wind direction, planetary boundary layer height) or including real-time observations (for gap-filling between synoptic runs) further increases local adaptability, sharpness, and skill, especially in operational settings for renewable energy forecasting (Casciaro et al., 2021, Casciaro et al., 2022).

6. Empirical Performance and Case Studies

Empirical validation of EMOS and its variants consistently demonstrates substantial improvements over raw ensembles and climatological reference models:

• Substantial reduction in energy score, CRPS, MAE, and RMSE across variables and regions.
• Calibration improvement (more uniform rank/PIT histograms), eliminating underdispersion prevalent in raw NWP ensembles.
• Ability to achieve performance comparable to advanced nonparametric postprocessing approaches (e.g., quantile mapping) but with shorter training requirements and greater computational efficiency (Szabó et al., 2022).
• Extensions with neighborhood information and topographic/seasonal covariates further enhance performance for precipitation (Scheuerer, 2013, Friedli et al., 2019).
• Mixture and combination EMOS models consistently outperform single-family or regime-switching approaches, especially for heavy-tailed distributions such as wind speed, while avoiding physically implausible predictions (e.g., negative wind speeds) (Baran et al., 2015, Baran et al., 2020).
• In joint wind vector prediction, explicit modeling of the covariance as a function of mean wind direction leads to significantly sharper, better calibrated forecasts (Schuhen et al., 2012).

7. Operational and Research Implications

EMOS has become a central tool for uncertainty quantification and calibration in operational NWP workflows. Its flexibility to accommodate various predictive distributions and its extensibility to multivariate and spatial-temporal contexts make it suitable for a wide range of meteorological applications (temperature, wind, precipitation, wind power, and renewable integration). It is computationally tractable at high resolution and can be adapted to multi-model and dual-resolution ensemble scenarios.

Ongoing development focuses on integrating EMOS with machine learning for higher-dimensional, nonlinear calibration, hybridization with copula methods for spatial consistency, adaptation to novel ensemble designs (e.g., dual resolution, multi-physics), and seamless inclusion of new observational and ancillary variables, particularly for high-impact and renewable energy forecasting (Schulz et al., 2021, Casciaro et al., 2021, Casciaro et al., 2022).

In summary, Ensemble Model Output Statistics unifies statistical postprocessing for ensemble forecasts under a single, theoretically grounded distributional regression approach. EMOS acts as both a robust correction tool for systematic deficiencies in ensemble forecasts and a platform for continuous methodological innovation in probabilistic meteorological prediction and beyond.