Fracture density reconstruction using direct sampling multiple-point statistics and extreme value theory

Improving ensemble forecasts is a complex process which involves proper scores such as the continuous ranked probability score (CRPS). A homogeneous Gaussian (hoG) model is introduced in order to better understand the characteristics of the CRPS. An analytical formula is derived for the expected CRPS of an ensemble in the hoG model. The score is a function of the variance of the error of the ensemble mean, the mean error of the ensemble mean and the ensemble variance. The hoG model also provides a score decomposition into reliability and resolution components. We examine whether the hoG model provides a useful approximation of the CRPS when applied to operational ECMWF medium‐range ensemble forecasts. The hoG approximation describes the spatial variations of the CRPS well while moderately overestimating the mean score. Seasonal averages over large domains are within 10% of the actual CRPS. Furthermore, the ability to approximate score changes is evaluated by (a) comparing raw ensemble forecasts with postprocessed ensemble forecasts, and (b) by examining score changes associated with a recent upgrade of the IFS. Overall, the hoG approximation predicts the actual CRPS changes well. One of the main anticipated applications of the hoG approximation are new diagnostics in verification software used by NWP developers routinely. The purpose of the diagnostics is to help developers explain impacts of forecast system changes on the CRPS in terms of the changes in mean error, changes in error variance and changes in ensemble variance. The diagnostics require little additional computational resources compared to the alternative of verifying postprocessed versions of the ensemble forecasts. Therefore, it will be feasible to apply the diagnostics easily to all variables that are examined as part of the model development process.

Ensemble prediction system (EPS) meteorological forecasts are increasingly being used as input to hydrological models in order to extend flood‐warning lead‐times and to improve forecasts and the knowledge of their uncertainty. Probabilistic skill scores classically used in meteorology are usually also utilized for assessing the quality of hydrological ensemble forecasts. However, the different river discharge magnitudes can make difficult the interpretation and comparisons of these scores, as it is for the continuous rank probability score (CRPS). In this letter, a novel ‘Reduction’ CRPS (RCRPS), which takes into account the different river discharge magnitudes, is proposed, and its usefulness is exhibited. Copyright © 2013 Royal Meteorological Society

Abstract. Stream water temperatures influence water quality, with effects on aquatic biodiversity, drinking-water provision, electricity production, agriculture, and recreation. Therefore, stakeholders would benefit from an operational forecasting service that would support timely action. Deep-learning models are well-suited to providing probabilistic forecasts at individual stations of a monitoring network. Here, we train and evaluate several state-of-the-art models using 10 years of data from 54 stations across Switzerland. Static catchment features, time of the year, meteorological observations from the past 64 d, and their ensemble forecasts for the following 32 d are included as predictors in the models to estimate daily maximum water temperature over the next 32 d. Results show that the temporal fusion transformer (TFT) model performs best, with a continuous rank probability score (CRPS) of 0.70 °C averaged over all lead times, stations, and 90 forecasts distributed over 1 year. The TFT is followed by the recurrent neural network encoder–decoder, with a CRPS of 0.74 °C, and the neural hierarchical interpolation for time series, with a CRPS of 0.75 °C. These deep-learning models outperform other simpler models trained at each station: random forest (CRPS = 0.80 °C), multi-layer perceptron neural network (CRPS = 0.81 °C), and autoregressive linear model (CRPS = 0.96 °C). The average CRPS of the TFT degrades from 0.38 °C at lead a time of 1 d to 0.90 °C at a lead time of 32 d, largely driven by the uncertainty of the meteorological ensemble forecasts. In addition, TFT water temperature predictions at new and ungauged stations outperform those from the other models. When analyzing the importance of model inputs, we find a dominant role of observed water temperature and future air temperature, while including precipitation and time of the year further improves predictive skill. Operational probabilistic forecasts of daily maximum water temperature are generated twice per week with our TFT model and are publicly available at https://www.drought.ch/de/impakt-vorhersagen-malefix/wassertemperatur-prognosen/ (last access: 20 March 2025). Overall, this study provides insights into the extended-range predictability of stream water temperature and into the applicability of deep-learning models in hydrology.

Indian summer monsoon governs agricultural productivity, water resources, and economic activity affecting 1.4 billion people, yet accurate prediction remains challenging due to complex non-linear dynamics involving atmosphere-ocean interactions, land surface processes, and stochastic forcing. Traditional numerical weather prediction models solve deterministic partial differential equations but struggle capturing uncertainty quantification and sub-grid scale processes. Machine learning approaches demonstrate skill learning patterns from historical data but lack physical constraints and interpretability. This research develops hybrid framework integrating stochastic differential equations representing atmospheric dynamics with deep learning capturing non-linear relationships, enabling probabilistic monsoon forecasting with improved accuracy and uncertainty quantification. The research formulated Indian monsoon system as coupled stochastic differential equation system: dX_t = μ(X_t, θ)dt + σ(X_t, θ)dW_t, where X_t represents state variables (rainfall, temperature, humidity, pressure), μ denotes drift coefficient encoding deterministic dynamics, σ represents diffusion coefficient capturing stochastic fluctuations, W_t denotes Wiener process, and θ parameters learned from data. Long Short-Term Memory neural networks parameterized drift and diffusion coefficients enabling flexible representation of complex non-linear relationships while preserving stochastic process structure. Training utilized 40 years of daily rainfall data (1980-2019) from 64 India Meteorological Department stations across eight monsoon regions, supplemented by atmospheric reanalysis including temperature, humidity, wind, and sea surface temperature. Numerical integration employed Euler-Maruyama scheme with adaptive time stepping ensuring stability and accuracy. Model evaluation compared hybrid stochastic-machine learning approach against traditional numerical prediction, pure stochastic models, and pure machine learning on 2017-2019 validation period. Performance metrics included root mean square error, mean absolute error, R-squared, continuous ranked probability score, and skill scores assessing probabilistic forecast quality. Results demonstrated hybrid model achieved RMSE of 2.1 mm daily rainfall compared to 3.2 mm for pure stochastic models and 5.4 mm for deterministic numerical predictions, representing 34% accuracy improvement. R-squared reached 0.967 indicating excellent predictive skill. Probabilistic forecasts provided well-calibrated uncertainty estimates with continuous ranked probability score of 1.8 mm outperforming deterministic predictions. Spatial analysis revealed consistent performance across regions with coastal areas (R²=0.967) and northeast India (R²=0.956) achieving highest accuracy, while northwest plains showed moderate performance (R²=0.912) due to continental influences. Temporal validation demonstrated robust monsoon onset prediction with 89.3% accuracy detecting rainfall commencement within ±3 days. Computational efficiency analysis revealed hybrid model required 2.3 hours training on GPU hardware compared to 48 hours for traditional numerical models, enabling operational forecasting. The research establishes mathematical framework for physics-informed machine learning combining stochastic calculus with deep learning, providing interpretable probabilistic forecasts supporting agricultural planning, water resource management, and disaster preparedness across India's monsoon-dependent economy.

SummaryIn Bayesian inference, predictive distributions are typically in the form of samples generated via Markov chain Monte Carlo or related algorithms. In this paper, we conduct a systematic analysis of how to make and evaluate probabilistic forecasts from such simulation output. Based on proper scoring rules, we develop a notion of consistency that allows to assess the adequacy of methods for estimating the stationary distribution underlying the simulation output. We then provide asymptotic results that account for the salient features of Bayesian posterior simulators and derive conditions under which choices from the literature satisfy our notion of consistency. Importantly, these conditions depend on the scoring rule being used, such that the choices of approximation method and scoring rule are intertwined. While the logarithmic rule requires fairly stringent conditions, the continuous ranked probability score yields consistent approximations under minimal assumptions. These results are illustrated in a simulation study and an economic data example. Overall, mixture‐of‐parameters approximations that exploit the parametric structure of Bayesian models perform particularly well. Under the continuous ranked probability score, the empirical distribution function is a simple and appealing alternative option.

DDOS-attacks detection using an efficient measurement-based statistical mechanism

From optimal supply decisions to anticipatory control systems, wind-based energy applications rely heavily upon accurate, local, short-term forecasts of future wind speed. Recent studies have shown continuous ranked probability score (CRPS) minimizing models with Gaussian assumptions to be effective for well-researched sites where those assumptions are appropriate. We consider the more general case where Gaussianity is not assumed and access to historical data may be constrained. Deriving a CRPS expression for a minimum Extreme Value distribution, we use it to propose a site-adaptive Weibull-based CRPS-minimizing model, which is tested and shown to perform better than both deterministic and probabilistic reference models on a ground-based array of weather observation sites in northern Japan.

In numerical weather prediction (NWP), the uncertainty about the future state of the atmosphere is described by a set of forecasts (called an ensemble). All ensembles have deficiencies that can be corrected via statistical post-processing methods. Several ensembles, based on different NWP models, exist and may be corrected using different statistical methods. These raw or post-processed ensembles can thus be combined. The theory of prediction with expert advice allows us to build combination algorithms with theoretical guarantees on the forecast performance. We adapt this theory to the case of probabilistic forecasts issued as stepwise cumulative distribution functions, computed from raw and post-processed ensembles. The theory is applied to combine wind speed ensemble forecasts. The second goal of this study is to explore the use of two forecast performance criteria: the continuous ranked probability score (CRPS) and the Jolliffe–Primo test. The usual way to build skilful probabilistic forecasts is to minimize the CRPS. Minimizing the CRPS may not produce reliable forecasts according to the Jolliffe–Primo test. The Jolliffe–Primo test generally selects reliable forecasts, but could lead to issuing suboptimal forecasts in terms of CRPS. We propose to use both criteria to achieve reliable and skilful probabilistic forecasts.

We present an ensemble prediction system using a Deep Learning Weather Prediction (DLWP) model that recursively predicts six key atmospheric variables with six‐hour time resolution. This computationally efficient model uses convolutional neural networks (CNNs) on a cubed sphere grid to produce global forecasts. The trained model requires just three minutes on a single GPU to produce a 320‐member set of six‐week forecasts at 1.4° resolution. Ensemble spread is primarily produced by randomizing the CNN training process to create a set of 32 DLWP models with slightly different learned weights. Although our DLWP model does not forecast precipitation, it does forecast total column water vapor and gives a reasonable 4.5‐day deterministic forecast of Hurricane Irma. In addition to simulating mid‐latitude weather systems, it spontaneously generates tropical cyclones in a one‐year free‐running simulation. Averaged globally and over a two‐year test set, the ensemble mean RMSE retains skill relative to climatology beyond two‐weeks, with anomaly correlation coefficients remaining above 0.6 through six days. Our primary application is to subseasonal‐to‐seasonal (S2S) forecasting at lead times from two to six weeks. Current forecast systems have low skill in predicting one‐ or 2‐week‐average weather patterns at S2S time scales. The continuous ranked probability score (CRPS) and the ranked probability skill score (RPSS) show that the DLWP ensemble is only modestly inferior in performance to the European Center for Medium Range Weather Forecasts (ECMWF) S2S ensemble over land at lead times of 4 and 5–6 weeks. At shorter lead times, the ECMWF ensemble performs better than DLWP.

Density regression characterizes the conditional density of the response variable given the covariates, and provides much more information than the commonly used conditional mean or quantile regression. However, it is often computationally prohibitive in applications with massive datasets, especially when there are multiple covariates. In this article, we develop a new data reduction approach for the density regression problem using conditional support points. After obtaining the representative data, we exploit the penalized likelihood method as the downstream estimation strategy. Based on the connections among the continuous ranked probability score, the energy distance, the L 2 discrepancy and the symmetrized Kullback–Leibler distance, we investigate the distributional convergence of the representative points and establish the rate of convergence of the density regression estimator. The usefulness of the methodology is illustrated by modeling the conditional distribution of power output given multivariate environmental factors using a large scale wind turbine dataset.

Additive dynamic models for correcting numerical model outputs

This study used a quantile regression model and a non-homogeneous regression model to calibrate probabilistic forecasts of wind speed. These techniques were applied to the forecasts of wind speed over Pyeongchang area using 51-member European Centre for Medium-Range Weather Forecast (ECMWF). Reliability analysis was carried out by using rank histogram to identify the statistical consistency of ensemble forecasts and corresponding observations. The performances were evaluated by rank histogram, mean absolute error, root mean square error and continuous ranked probability score. The results showed that the forecasts of quantile regression and non-homogeneous regression models performed better than the raw ensemble forecasts. However, the differences of prediction skills between quantile regression and nonhomogeneous regression models were insignificant.

Investment decisions for offshore wind-to-hydrogen (W2H) projects are often framed as “better forecasts reduce uncertainty,” but it is less clear when higher-fidelity scenario modelling meaningfully changes a financing decision versus merely narrowing outcome ranges. We address this question using a decision-coupled evaluation that scores forecast skill on propagated economic distributions and links it directly to financeability metrics.Using 61 years of ERA5 wind data at 150 m hub height, we generate 1000 synthetic 23-year hourly wind scenarios per method and propagate them through a techno-economic model of a 375 MW offshore W2H project (development in 2024, operation in 2026-2050, base hydrogen price €8/kg, discount rate 7%). We compare three probabilistic scenario generators: historical bootstrapping, parametric Weibull fitting, and a calibrated probabilistic long short-term memory (LSTM) sequence model (used as a benchmark rather than architectural novelty).We evaluate (a) continuous ranked probability score (CRPS) of levelized cost of hydrogen (LCOH), net present value (NPV), and internal rate of return (IRR), (b) decision bandwidths W(Y) = P95(Y) – P5(Y), (c) threshold-crossing probabilities Pr(NPV>0) and Pr(IRR>10%), and (d) a local elasticity E(Y) = dW(Y)/dCRPS that maps marginal forecast skill to risk-band compression. Finally, we run a financing price sweep to identify the minimum hydrogen offtake price that achieves a 90% probability target for NPV > 0 and the joint target NPV > 0, IRR > 10%.Results show that improved scenario modelling can substantially reduce economic distribution error and compress risk bands: the LSTM lowers CRPS by 30% for LCOH and NPV and by 25% for IRR versus the best bootstrap/Weibull configurations. However, under base assumptions the financeability thresholds are nearly invariant across methods: the 90%-target required hydrogen price is €7.76-7.78/kg for Pr(NPV>0) and €9.16-9.18/kg for Pr(NPV>0 and IRR>10%), with cross-method spread below €0.02/kg indicates a threshold-saturated regime where better modelling mainly narrows uncertainty rather than shifting the decision boundary. Sensitivity analysis indicates decision value is highest in moderate-margin regimes (roughly €5.5-8/kg) and diminishes at high profitability where models converge.This work reframes “better scenarios” into an investment-relevant diagnostic: use elasticity and threshold behaviour to identify when modelling improvements will shift financeability versus only compress risk bands, supporting more defensible screening and policy design.

Providing useful inflow forecasts of the Manantali dam is critical for zonal consumption and agricultural water supply, power production, flood and drought control and management (Shin et al., Meteorol Appl 27:e1827, 2019). Probabilistic approaches through ensemble forecasting systems are often used to provide more rational and useful hydrological information. This paper aims at implementing an ensemble forecasting system at the Senegal River upper the Manantali dam. Rainfall ensemble is obtained through harmonic analysis and an ARIMA stochastic process. Cyclical errors that are within rainfall cyclical behavior from the stochastic modeling are settled and processed using multivariate statistic tools to dress a rainfall ensemble forecast. The rainfall ensemble is used as input to run the HBV-light to product streamflow ensemble forecasts. A number of 61 forecasted rainfall time series are then used to run already calibrated hydrological model to produce hydrological ensemble forecasts called raw ensemble. In addition, the affine kernel dressing method is applied to the raw ensemble to obtain another ensemble. Both ensembles are evaluated using on the one hand deterministic verifications such the linear correlation, the mean error, the mean absolute error and the root-mean-squared error, and on the other hand, probabilistic scores (Brier score, rank probability score and continuous rank probability score) and diagrams (attribute diagram and relative operating characteristics curve). Results are satisfactory as at deterministic than probabilistic scale, particularly considering reliability, resolution and skill of the systems. For both ensembles, correlation between the averages of the members and corresponding observations is about 0.871. In addition, the dressing method globally improved the performances of ensemble forecasting system. Thus, both schemes system can help decision maker of the Manantali dam in water resources management.

Accurate prediction of the freezing level (FZL) is essential for hydrometeorological forecasting systems, with direct implications for runoff generation and reservoir management. In this study, we develop a deep learning–based postprocessing framework using the Unet convolutional neural network (CNN) architecture to refine the FZL forecasts from the West Weather Research and Forecasting (West-WRF) model. The proposed framework leverages reforecast data from West-WRF and FZL estimates from the California-Nevada River Forecast Center (CNRFC) to develop Unet models over the northern Sierra Nevada watersheds, such as the hydrologically critical Yuba-Feather watershed. We introduce two Unet model variants, Unet-log and Unet-GMM, that employ specialized loss functions beyond the standard benchmarks to enhance forecast skill. Unet-log utilizes the Log-Cosh cosine of Error, and Unet-GMM uses Gaussian Mixture Model loss functions, to enhance FZL forecasts. Results show that Unet-based postprocessing reduces centered root-mean-squared errors by up to 20% and increases forecast–observation correlation by about 10% compared to raw West-WRF. Evaluation using the continuous ranked probability score (CRPS) for Unet-GMM further demonstrates consistent improvements across lead times. While performance fluctuates with forecast horizon, storm variability, and diurnal forcing, Unet-GMM and Unet-log consistently outperform the baseline. The models capture the spatiotemporal variability of the FZL across different elevations, mitigating biases from the West-WRF model. This novel deep learning–based postprocessing approach demonstrates a promising pathway for integrating machine learning into hydrometeorological forecasting and decision support within the Forecast Informed Reservoir Operations (FIRO) framework.