ALADIN-HUNEPS Ensemble Prediction System

• ALADIN-HUNEPS is a limited-area ensemble prediction system that quantifies forecast uncertainty using high-resolution NWP and perturbation-based ensemble members.
• It applies advanced statistical post-processing techniques, including Bayesian Model Averaging and EMOS, to correct biases and under-dispersion in raw forecasts.
• The system’s design supports regime-switching and joint multivariate calibration, enhancing the reliability of wind speed and temperature predictions.

The ALADIN-HUNEPS Ensemble Prediction System (EPS) is the operational, limited-area ensemble prediction framework of the Hungarian Meteorological Service (HMS) designed to quantify forecast uncertainty for regional weather variables, primarily wind speed and temperature, over Hungary and surrounding Central European domains. ALADIN-HUNEPS leverages high-resolution numerical weather prediction (NWP) by introducing stochasticity through initial condition perturbations and applies advanced statistical post-processing, most notably Bayesian Model Averaging (BMA) and Ensemble Model Output Statistics (EMOS), to address systematic biases and under-dispersion in raw ensembles. The system has been both the basis and the testbed for significant developments in probabilistic weather forecasting, including joint multivariate calibration and regime-switching statistical models.

1. System Construction and Ensemble Design

ALADIN-HUNEPS constructs its ensemble by launching a high-resolution, limited-area NWP model multiple times per forecast cycle. The operational ensemble comprises 11 members: one control forecast obtained from the unperturbed analysis and 10 perturbed members derived by explicit modification of the initial conditions. Perturbations are computed as five distinct perturbation vectors; for each, the positive and negative variants are formed by addition and subtraction from the control state, resulting in five odd-numbered and five even-numbered members. This induces a natural statistical grouping of members into "control," "odd" (positive perturbation), and "even" (negative perturbation) subgroups, which are considered "exchangeable" within group (statistically indistinguishable under the system’s design).

The ensemble architecture is summarized as:

Group	Member Index	Description
Control	0	Unperturbed analysis
Odd members	1, 3, 5, 7, 9	Positive perturbations
Even members	2, 4, 6, 8, 10	Negative perturbations

This structure allows for targeted statistical calibration methods that respect the underlying ensemble generation mechanism.

2. Statistical Properties and Deficiencies of Raw Ensembles

Like most limited-area ensembles, raw forecasts from ALADIN-HUNEPS are characteristically under-dispersive: the spread among members, as measured by verification rank histograms and coverage of central prediction intervals, is systematically too narrow compared to the true forecast uncertainty. For example, raw ensemble quantiles might bracket the verifying observation only about 46–70% of the time, rather than at the nominal 83.33% expected from an 11-member ensemble. "Subgroup" exchangeability, evidenced by member construction, implies potential group-specific biases or spreads.

Systematic under-dispersion in raw ensemble output leads to mis-calibrated probabilistic interpretations and unreliable quantifications of risk, particularly for rare or extreme events—motivating the need for statistical post-processing.

3. Bayesian Model Averaging for Ensemble Calibration

BMA is central to HMS's post-processing strategy. In the BMA framework, each ensemble member is associated with a conditional predictive probability density function (PDF), parameterized to account for bias and variance:

• Wind Speed: Component PDFs are modeled as either gamma densities (historically) or, in later work, as truncated normal densities strictly supported on the nonnegative real line. For a two-group model (control/others), the general BMA predictive density is:

$$p(x \mid f_c, f_{\ell,1}, \ldots, f_{\ell,10}; \theta) = \omega \, g(x; f_c, b_0, b_1, c_0, c_1) + \frac{1-\omega}{10} \sum_{j=1}^{10} g(x; f_{\ell,j}, b_0, b_1, c_0, c_1)$$

where $g(x; f, b_0, b_1, c_0, c_1)$ is a gamma or truncated normal PDF with mean and standard deviation linked linearly to the forecast $f$.

• Temperature: BMA applies normal PDFs per member/group. For the two-group case:

$$p(x \mid f_c, f_{\ell,1}, \ldots, f_{\ell,10}) = \omega \, \mathcal{N}(x; b_{c,0} + b_{c,1} f_c, \sigma^2) + \frac{1-\omega}{10} \sum_{j=1}^{10} \mathcal{N}(x; b_{\ell,0} + b_{\ell,1} f_{\ell,j}, \sigma^2)$$

• Three-Group Model: Calibration is refined by distinguishing odd and even-numbered perturbed members, leading to three weights (for control, positive, and negative perturbations) subject to normalization constraints.

BMA model parameters—including groupwise weights, bias corrections, and spread parameters—are estimated by maximizing the likelihood over a fixed-length training period (typically 28–40 days), often via the Expectation–Maximization (EM) algorithm. For group-specific parameters, member exchangeability justifies parameter parsimony.

BMA directly corrects for bias, inflates forecast dispersion to match empirical variability, and yields predictive densities from which both point (mean, median) and probabilistic (intervals, quantiles) forecasts are derived.

4. Ensemble Model Output Statistics (EMOS) and Regime-Switching Extensions

EMOS provides an alternative single-distribution approach whereby a parametric PDF (normal for temperature; truncated normal, gamma, or log-normal for wind speed) is fit, with its parameters modeled as functions of the ensemble forecast vector or statistics (mean, variance). For wind speed, the predictive mean and variance ($m, v$) for a log-normal EMOS model are: $$m = \alpha_{0} + \sum_{j=1}^{M} \alpha_{j} f_{j}, \quad v = \beta_{0} + \beta_{1} S^{2}$$ where $S^2$ is the ensemble variance.

A key innovation is the regime-switching model, in which the choice between truncated normal and log-normal (or GEV) predictive distribution depends on the ensemble median forecast $f_{\text{med}}$ and a threshold $\theta$. For instance:

• If $f_{\text{med}} < \theta$: use truncated normal (TN).
• If $f_{\text{med}} \geq \theta$: use log-normal (LN).

Model parameters and thresholds are optimized based on proper scoring rules, e.g., mean Continuous Ranked Probability Score (CRPS), over rolling training periods.

5. Joint Multivariate Calibration: Bivariate BMA and EMOS

The more advanced stages of ALADIN-HUNEPS ensemble post-processing employ bivariate (wind speed, temperature) calibration to capture inter-variable dependencies. Both bivariate BMA and bivariate EMOS employ truncated bivariate normal distributions:

• Marginal for wind speed is truncated at zero.
• Temperature remains unbounded.
• Location parameter is an affine function of the ensemble forecast vector, and a shared covariance matrix allows explicit modeling of wind–temperature correlation.

The predictive density is

$$g(\mathbf{x} \mid \boldsymbol{\mu}, \Sigma) = \frac{|\Sigma|^{-1/2}}{2\pi \Phi(\mu_W / \sigma_W)} \exp\left[ -\frac{1}{2}(\mathbf{x} - \boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right] I_{x_W \geq 0}$$

where $\boldsymbol{\mu}$ and $\Sigma$ are estimated groupwise or globally, and $\Phi$ is the standard normal CDF.

Multivariate EMOS, by modeling the location as a groupwise affine function and the covariance as a function of the ensemble, can achieve calibration comparable to bivariate BMA with reduced computational cost, due to the absence of mixture-weight estimation.

6. Verification, Performance, and Operational Impact

Quantitative verification of both raw and post-processed ALADIN-HUNEPS ensembles utilizes probabilistic (CRPS, energy score, reliability index) and deterministic (MAE, RMSE) scores, as well as PIT/rank histograms and prediction interval coverages.

Empirical results consistently demonstrate that:

• BMA and EMOS markedly improve probabilistic calibration, as measured by PIT histogram uniformity, interval coverage (66.7%, 83.33%, 90%), and lower CRPS values relative to raw ensemble output.
• Point forecast error (e.g., MAE, RMSE of mean or median) is modestly reduced, with primary gains in uncertainty quantification.
• Bivariate models better capture inter-variable dependence (e.g., wind–temperature correlation).
• Regime-switching EMOS (TN-LN) achieves lower CRPS and better interval coverage than TN or GEV-based models alone, while avoiding allocation of probability to negative wind speeds.

Table: Comparative Performance of Calibration Models (ALADIN-HUNEPS; based on (Baran et al., 2014, Baran et al., 2013))

Model	CRPS (↓)	MAE / RMSE (↓)	Central Interval Coverage (%)	Computational Cost
Raw Ensemble	High	Moderate	Low (46–70)	Low
BMA (Gamma/TN)	Low	Low	≈ Nominal (e.g., 83.3)	Moderate–High
EMOS (TN/LN)	Low	Low	≈ Nominal	Low–Moderate
Regime-switch EMOS (TN-LN)	Very Low	Very Low	≈ Nominal	Low–Moderate
Bivariate BMA	Very Low	Very Low	≈ Nominal (joint)	Very High
Bivariate EMOS	Very Low	Very Low	≈ Nominal (joint)	Moderate

7. Extensions, Operational Considerations, and Future Directions

ALADIN-HUNEPS has provided the operational and scientific basis for methodological advances in local, spatially interpolated EMOS calibration—employing geostatistical kriging with local mean wind speed as a covariate, as well as for the integration of deep learning (e.g., conditional GAN-based ensemble spread emulation (Brecht et al., 2022)) to reduce computational burdens associated with running large ensembles.

Challenges persist in sustaining calibration in rapidly changing regimes, optimum selection of training period lengths (balancing adaptation and statistical stability), and jointly modeling additional variables or lead times. Recently, regime-switching and bivariate/multivariate frameworks have significantly advanced joint uncertainty quantification.

Continued research focuses on improving the computational efficiency of calibration algorithms (especially EM approaches), refining grouping structure, tuning regime-switching thresholds, and expanding post-processing to spatially continuous gridded domains for high-resolution probabilistic prediction.

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In summary, the ALADIN-HUNEPS EPS has driven substantial developments in both the science and operational implementation of statistical post-processing for regional ensemble forecasting, with a continuous interplay between diagnostics, probabilistic calibration, and advanced modeling frameworks resulting in empirically verifiable gains in reliability and forecast value.