Rainfall is rough

2026-07-29

AI Summary

This study addresses the challenge of modeling the clustering and long-range dependence of rainfall processes across multiple timescales by proposing a novel framework based on a critical Hawkes point process. For the first time, a heavy-tailed power-law kernel is introduced into rainfall modeling, unifying the Bartlett–Lewis and Neyman–Scott models to effectively capture rain cell clustering characteristics. By integrating high-frequency (minute-level) observational data with millennial-scale tree-ring proxy records and employing fractal analysis alongside Hurst exponent estimation, the work reveals a shared extremely rough fractal structure—characterized by Hurst exponents between 0.01 and 0.1—spanning from meteorological to paleoclimatic timescales. The proposed method significantly outperforms classical models at fine temporal resolutions and establishes a novel interdisciplinary link between atmospheric science and financial microstructure theory.

Abstract

We propose a new approach to model rainfall by combining heterogeneous data sources at different time scales. Continuous arrivals of rain cells are incorporated into a Hawkes process formalism that encompasses the classical Bartlett-Lewis and Neyman-Scott models, thereby providing a more flexible representation of clustering. Analysis of high frequency rainfall data (at the minute scale over several years) indicates that critical Hawkes processes with heavy-tailed power-law kernels yield a superior fit relative to classical models and alternative kernel specifications. Scaling arguments inspired by Jaisson and Rosenbaum (2016) imply that aggregated rainfall at coarse time scales converges to a rough fractional process with Hurst exponent close to zero. This prediction is supported by empirical evidence from low-frequency data (annual observations spanning centuries to millennia), where the Hurst exponent is estimated to lie between 0.01 and 0.1 based on either direct observations from weather stations or proxy reconstructions such as tree-ring records. These results establish a connection between rainfall dynamics and models developed in quantitative finance for market microstructure and volatility. They also provide a new perspective on classical scaling phenomena originally studied by Hurst and Mandelbrot.

Problem

• rainfall modeling
• Hawkes process
• scaling
• Hurst exponent
• time scales

Innovation

• Hawkes process
• heavy-tailed kernel
• rough fractional process
• Hurst exponent
• rainfall clustering