Extreme rainfall events have three dimensions: how intense, how widespread, and how long. Traditional extreme-value analysis handles intensity well — the theory of maxima is highly developed — but spatial extent and temporal duration require different mathematical tools. The intensity of the most extreme point tells you little about how many square kilometers were affected or how many hours the event lasted.

The researchers frame all three quantities through spatial geometry. Instead of treating rain as a time series at a point, they treat it as a random field whose extremes define geometric objects — regions in space where the field exceeds a threshold, whose area captures extent and whose temporal persistence captures duration. The geometric framework makes spatial extent and temporal duration natural consequences of the field’s structure rather than separate quantities requiring separate models.

The non-stationarity is handled through topographical and seasonal covariates — mountains and seasons modulate the extreme-value parameters, which is standard. What is less standard is the sampling approach: rather than extracting maxima over blocks or peaks over thresholds (the two classical approaches), they sample in a way that preserves the temporal ordering needed for duration estimation. The temporal dimension is not discarded during extreme-value extraction.

The through-claim: geometry converts a problem about intensity into a problem about shape. The same random field, analyzed geometrically, simultaneously answers questions about magnitude, area, and time — three quantities that appear to require three models but are connected through the spatial structure of the field.

(arXiv:2603.18149)