Hydrologists have long modelled rainfall with discrete or continous time models based on point processes. In a first part, we show that most of the desired phenomenological properties of rainfall models are captured by critical Hawkes processes. Viewing this approach as a microscopic modelling, we zoom out in a second part our data to build a macroscopic model of aggregated rainfall. On several macroscopic data sets, we empirically establish that rainfall behaves like a rough fractional process with Hurst parameter close to 0.1; we further rigorously analyse the compatibility of this our approach across time scales, implying a heavy-tailed behaviour for Hawkes rainfall models which we observe in practice. As a consequence, an unexpected analogy with the theory of rough volatility seems to emerge for rainfall modelling. We discuss the consequences of these findings from a statistical point of view, in particular how it advocates for the need of better tools for analysing nonstationary or nearly stationary data.

Joint work with Thomas Deschatre (EDF Labs) and Mathieu Rosenbaum (Paris Dauphine-PSL).