High-resolution Calibrated Probabilistic Hourly Precipitation from a Deterministic Forecast

This paper describes an “Attention Residual U-Net” method for probabilistic quantitative precipitation forecasting (PQPF) that predicts the hourly probability of no precipitation plus the distribution of positive precipitation from a weighted mixture of two Gamma distributions. The neural network is trained on patches of numerical weather prediction (NWP) hourly precipitation from The Weather Company’s convection-permitting GRAF (Global high-Resolution Atmospheric Forecasting) model along with terrain information and column-average relative humidity from the National Oceanic and Atmospheric Administration’s (NOAA’s) Global Forecast System (GFS). The target data are NOAA’s Multi-Radar, Multi-Sensor (MRMS) gauge-corrected, quality controlled radar data sampled to the same grid as the GRAF data. The network outputs distributional parameters for each model grid point. Training uses negative log-likelihood as a proper scoring rule, with climatological initialization for stable convergence. Inference is performed as a single forward pass over the contiguous United States (CONUS) domain, with edge-replication padding to satisfy the network’s spatial-divisibility requirement. The subsequent forecasts are spatially detailed, highly reliable, and skillful with respect to climatology and a simpler reference forecast method. The method is particularly useful for estimating probabilities in regions with large terrain variation.

Improvements to AI Systems:

1. Hybrid Physics-Statistical Loss Function: Integrate the negative log-likelihood (NLL) proper scoring rule with a physics-based constraint (e.g., conservation of total precipitation mass or terrain-induced orographic enhancement) to reduce unphysical extremes and improve generalization in unobserved meteorological regimes.

2. Uncertainty-Aware Spatial Attention with Edge-Adaptive Padding: Replace fixed edge-replication padding with a learned, attention-driven padding mechanism that explicitly models boundary conditions, enabling the network to handle arbitrary domain shapes and non-divisible grids without performance degradation—useful for global or irregular regional forecasting.

3. Climatological Prior as a Learnable Embedding: Instead of only using climatological initialization for stable convergence, inject a learnable climatological embedding (e.g., a low-dimensional vector of local seasonal precipitation statistics) into the attention residual blocks, allowing the model to dynamically adjust its output distribution based on long-term local climate context—improving skill in data-sparse or extreme-event scenarios.

4. Multi-Scale Gamma Mixture with Temporal Recurrence: Extend the single-step mixture of two Gamma distributions to a recurrent architecture (e.g., a ConvLSTM or temporal attention) that outputs time-evolving mixture weights and parameters, enabling probabilistic forecasts over multiple lead times with consistent temporal correlations—critical for flash flood warnings.

5. Calibration-Aware Inference with Post-Hoc Isotonic Regression: After the single forward pass, apply a lightweight, grid-point-specific isotonic regression on the predicted probability of no precipitation to correct residual miscalibration, especially in regions with complex terrain, without retraining the network.

What the Improved AI System Can Do:

• Generate physically consistent, spatially detailed probabilistic precipitation forecasts (e.g., 0–6 hour lead times) that are reliable even in mountainous regions, with reduced bias in extreme rainfall events.

• Seamlessly forecast over any domain (CONUS, global, or irregular watersheds) without grid-divisibility constraints, enabling real-time operational use on arbitrary NWP grids.

• Provide calibrated, temporally coherent probability sequences (e.g., hourly probabilities of no rain and Gamma-distributed positive rain amounts) that outperform both climatology and simple reference methods, while also flagging high-uncertainty zones for human forecaster review.

• Adapt to local climate regimes (arid vs. humid) automatically, improving skill in regions with sparse radar coverage or rapid land-use changes.

• Run as a single-pass, edge-aware inference engine that can be deployed on limited hardware (e.g., for emergency response) while maintaining high spatial resolution and probabilistic reliability.

An ``Attention Residual U-Net'' method is described for probabilistic quantitative precipitation forecasting (PQPF) that predicts the hourly probability of no precipitation plus the distribution of positive precipitation from a weighted mixture of two Gamma distributions. The neural network is trained on patches of numerical weather prediction (NWP) hourly precipitation from The Weather Company's convection-permitting GRAF (Global high-Resolution Atmospheric Forecasting) model along with terrain information and column-average relative humidity from the National Oceanic and Atmospheric Administration's (NOAA's) Global Forecast System (GFS). The target data are NOAA's Multi-Radar, Multi-Sensor (MRMS) gauge-corrected, quality controlled radar data sampled to the same grid as the GRAF data. The network outputs distributional parameters for each model grid point. Training uses negative log-likelihood as a proper scoring rule, with climatological initialization for stable convergence. Inference is performed as a single forward pass over the contiguous United States (CONUS) domain, with edge-replication padding to satisfy the network's spatial-divisibility requirement. The subsequent forecasts are spatially detailed, highly reliable, and skillful with respect to climatology and a simpler reference forecast method. The method is particularly useful for estimating probabilities in regions with large terrain variation.