Reliable probabilistic learning & accurate prediction - flooding in the Humber Estuary

The Humber estuary is one of the largest in the U.K. and crucially important to the British economy. It homes a large population and thousands of businesses. Additionally, its diverse geomorphologically-shaped landscape renders it a thriving area of conservation. The region however is at a worryingly high flood risk, owing to it being on low-lying tidal land that is exposed to the effects of storm surges even far inland. With climate change-induced rise in sea levels, risks of flooding are anticipated to only rise. Given this, it is imperative that we develop capacity for reliable and accurate predictions of indundation levels at any chosen location within the estuarine region, at a given set of hydrological parameters. This project offers such a capacity, following reliable and trustworthy probabilistic learning of the function that represents the relationship between water level at a chosen location and hydrological parameters, as well as the markers of said location. The most up-to-date work in this context includes prediction of water levels at distinct locations, given values of hydrological parameters, where said prediction is performed at one location independent of others within the wider area. This is noted to give rise to predictions for water levels at one location that are inconsistent with that at a neighbouring point. In other words, currently there exists no reliable projections of water levels corresponding to a given hydrological scenario, anywhere inside the Humber estuary. In this project, we will address this exact gap with closed-form and accurate predictions of the output (water level) of the learnt function, given the inputs that include five hydrological parameters, in addition to the longitude and latitude of a design location within the wider estarine region. These five hydrological variables include the flow rates in four major tributaries of the Humber, and the sea level. Since this function is an unknown - and an unknown is a random variable within the Bayesian approach - the aforesaid function is a (function-valued) random variable, that we will assign a probability distribution to. Now, probability distributions on a space of functions is given by stochastic processes (SPs). So, we will model the sought function with an adequately-chosen SP. In fact, we will choose this SP to be a Gaussian Process (GP) since GPs impose minimal constraints on their sample functions. The only constraint a GP imposes is that the joint probability of a finite number of realisations of any of its sample functions - where each realisation is made at a chosen input value - has a prescribed form. The shape of any of its sample function is however so generic in general, that it cannot be ascribed any parametric form. In fact, in this application, we anticipate this multivariate function to be differentially wriggly at different inputs, inspiring us to employ a mathematically-involved learning technique that captures such a challenging signature of the sought function. The distribution of the output predicted at a new set of inputs is also identifiable, and we will exploit this mathematical property to predict the output (namely, local water level), in a chosen location, when the hydrological input parameters are projected to attain chosen values. We retain the capacity for predicting the indundation level for the worst-case flooding scenarior, where the domain experts (such as our Project Partner) will define the "worst-case".