Bridging the Scaling Gap: A Review of Nonlinear Paradigms for the Estimation and Understanding of Extreme Rainfall from Heavy Storms

Short-duration extreme rainfall is a major trigger of flash floods and urban inundation, yet its quantification remains a profound challenge due to the scarcity of high-resolution observations. This review synthesizes how three central paradigms of nonlinear science, multifractal cascade theory, self-organized criticality (SOC) and chaos theory, provide critical insights and practical methodologies for bridging this observational gap. We examine how multifractal temporal downscaling leverages scale-invariance to derive sub-hourly rainfall statistics from coarser data. The SOC paradigm is discussed for its ability to explain the power-law statistics of rainfall extremes and cluster properties, offering a physical basis for estimating rare events. The role of chaos theory and its modern evolution into complex network analysis is explored for diagnosing predictability and spatiotemporal organization. By comparing and integrating these perspectives plus recent developments in stochastic hydrology, this review highlights their collective potential to advance the estimation, understanding, and prediction of short-duration extreme rainfall, ultimately informing improved risk assessment and climate resilience strategies.