Rainfall is rough
How a self-exciting point-process model from stock-market microstructure, and a roughness idea from financial volatility, quietly bridge two rainfall traditions that never talked to each other — the minute-scale clustering of rain cells, and the multi-century scaling laws hydrologists have chased since Hurst.
The two rainfalls
Zoom into five-minute rain-gauge data and rainfall looks like a scatter of discrete, clustering events — a point process. Zoom out to centuries of tree rings and lake sediments and it looks like a continuous, self-similar, fractal curve. Hydrology has treated these as two separate research traditions for forty years. This paper connects them with an actual formula.
Random processes and events
A stochastic process is just a randomly-drawn path over time. A point process tracks discrete arrivals instead — and the entire richness of the model lives in one rule: the intensity, the instantaneous chance of the next arrival, given everything that happened so far. A plain Poisson process has a flat, memoryless intensity — no clustering at all.
The classic storm models
Bartlett–Lewis and Neyman–Scott (1987) are the textbook rainfall models: parent storm cells arrive plainly, and each parent spawns a batch of children nearby in time. Exactly two generations, by construction — no grandchildren, no children of children.
Hawkes: when rain begets rain
A Hawkes process (Hawkes, 1971 — originally for earthquake aftershocks) removes the generation cap. Every event, parent or child, can trigger its own children, who can trigger theirs, in an open-ended branching cascade. The same idea now runs finance order-book models, epidemiology, neuroscience, and criminology.
Kernels: short memory vs. long memory
The kernel is the "echo function" — how strong the triggering boost is right after an event, and how fast it fades. An exponential kernel fades fast, like a plucked string. A heavy-tailed power-law kernel fades slowly, leaving a long faint influence. Real rain-gauge data prefers the heavy tail, with a tail parameter α landing close to one-half.
Criticality: on the edge
The total accumulated echo from one event — call it the branching ratio — measures how many "children" one event produces on average. Near 0: clustering fizzles fast. Near 1: the process is teetering right at the edge between dying out and exploding, a regime called near-critical. High-frequency financial markets were shown to sit exactly here. This paper finds rain-cell clustering sits there too.
A hidden unity
Proposition 2.2 in the paper: a Hawkes process with a plain exponential kernel produces exactly the same means, variances, and correlations as the classical Bartlett–Lewis and Neyman–Scott models, under a matching parameter translation. The old models were never a rival framework — they were a narrow corner of this one, hiding in plain sight for forty years.
Testing at the minute scale
Fitted month-by-month to a 69-year Bochum, Germany record and four 20-year French Météo-France stations, the heavy-tailed Hawkes model beats the classical models and the plain-exponential Hawkes model almost everywhere, by information-criterion scoring that penalizes extra parameters.
Roughness and the Hurst exponent
The Hurst exponent H, between 0 and 1, measures how jagged a random path is at every scale at once. H = 0.5 is ordinary Brownian motion (a jittering pollen grain). H above 0.5 is smoother, more "persistent." H below 0.5 is rougher than Brownian motion — jagged at every zoom level — and since a 2018 finance paper gave it the name, that's simply called rough.
From clustering cells to a rough curve
Borrowing a 2015–2016 theorem built for financial order flow (Jaisson & Rosenbaum), the paper shows: a near-critical, heavy-tailed Hawkes process, rescaled over a long time horizon, converges to the running total of a rough fractional process — with H = α − 1/2. Real gauge data gives α ≈ 0.5–0.6, so the predicted H lands near 0 to 0.1 — before ever touching the multi-century data.
Testing across centuries and millennia
16 weather-station records (200+ years, mostly European) plus 5 paleoclimatic proxy archives — tree rings, lake sediment, pollen, spanning 372 to 3,512 years — independently estimate H. Despite wildly different measurement methods, continents, and eras, nearly all of them land in the predicted 0.01–0.1 band.
The Hurst–Mandelbrot paradox, resolved
Hurst (1951) found long rainfall/river records show strong "long memory," HHurst ≈ 0.7–0.9. Mandelbrot & Wallis (1968) modeled the cumulative total as a smooth, persistent fractional Brownian motion to explain it. This paper models the underlying rate directly and finds the opposite: H ≈ 0.05, rough. Same data family — two different objects, not a contradiction.
Rainfall's other language: multifractals
Schertzer & Lovejoy's multifractal cascades (since the 1980s) are the dominant scaling framework in operational hydrology. They look different from a single Hurst number — but the authors show the multifractal "conservation parameter," long found near zero for rainfall, is measuring nearly the same thing as this paper's H ≈ 0 once you compare the right statistics. In the small-H limit, rough processes and multifractal cascades converge mathematically.
Why this matters
Four things worth holding onto: this is a genuine transplant of finance-only mathematics into meteorology; it unifies two forty-year-old, previously disconnected rainfall literatures under one explicit formula; it's mechanistic — the large-scale exponent is derived from small-scale clustering physics, not just fitted; and it sits in the same "rough analysis" intellectual family as path signatures, related but distinct — one is about local statistical smoothness of a scalar path, the other about faithfully encoding a path's full geometric shape.
My own take
The most impressive fact here isn't the accuracy — it's the direction: two numbers estimated from minute-scale clustering, fed into a theorem borrowed from finance, generated a falsifiable prediction about millennia-old tree rings before checking. That's rare in a field where scaling exponents are usually fit after the fact, separately, at whatever scale the data happens to exist. Where I'd want more: the bridge relies on an asymptotic limit theorem, and real gauge records are only approximately near-critical (0.78–0.99, not exactly 1) — how fast the convergence to the rough limit actually kicks in isn't fully pinned down. And everything here is single-point, single-gauge; the natural next step — the one that would connect most directly to gridded PRISM-style work — is whether this same roughness signature holds up spatially, not just at one station through time.