A conditional time-invariant framework for probabilistic storm surge forecasting

Forecast uncertainty is essential for operational decision-making in storm surge forecasting. Current probabilistic operational systems typically estimate uncertainty by ensembling deterministic numerical models forced by meteorological ensemble members. While this approach captures atmospheric uncertainty, it neglects uncertainty in the oceanic response and surge propagation, leading to forecasts that are commonly underdispersed.

We present a framework that explicitly accounts for these missing sources of uncertainty, including those arising from waves and other coupled processes. Building on the deterministic tidal response method of Munk and Cartwright, we model storm surges as conditional time-invariant stochastic processes. These processes are defined by distributions over nonlinear impulse-response functions that map gravitational, meteorological, and other forcings to total sea level. The response distributions can be flexibly conditioned on recent observations, such as in-situ gauge data, while remaining robust when such data are unavailable.

To learn these processes, we introduce Mixture Density Neural Processes (MDNPs), a Bayesian neural architecture that combines the expressiveness of neural networks with the stochastic function modeling capabilities of Gaussian processes. The models are trained on in-situ observational data at locations used for operational decision-making but, due to the conditional time-invariant formulation, do not require such data to generate subsequent forecasts.

We demonstrate state-of-the-art performance against operational baselines from the UK, the Netherlands, and the United States. We further show how MDNPs can be coupled with traditional numerical models to improve both forecast accuracy and uncertainty calibration. Currently being trialed in UK and Dutch operational systems, the approach maintains high performance during extreme events, producing calibrated forecasts even for previously unseen peaks. We attribute this robustness on tail events to the impulse-response formulation and discuss the broader applicability of the framework to multiscale and compound coastal hazards.