Mixed memory, (non) Hurst effect, and maximum entropy of rainfall in the tropical Andes

Diverse linear and nonlinear statistical parameters of rainfall under aggregation in time and the kind of temporal memory are investigated. Data sets from the Andes of Colombia at different resolutions (15 min and 1-h), and record lengths (21 months and 8–40 years) are used. A mixture of two timescales is found in the autocorrelation and autoinformation functions, with short-term memory holding for time lags less than 15–30 min, and long-term memory onwards. Consistently, rainfall variance exhibits different temporal scaling regimes separated at 15–30 min and 24 h. Tests for the Hurst effect evidence the frailty of the R/S approach in discerning the kind of memory in high resolution rainfall, whereas rigorous statistical tests for short-memory processes do reject the existence of the Hurst effect.Rainfall information entropy grows as a power law of aggregation time, S(T) ∼ Tβ with 〈β〉 = 0.51, up to a timescale, TMaxEnt (70–202 h), at which entropy saturates, with β = 0 onwards. Maximum entropy is reached through a dynamic Generalized Pareto distribution, consistently with the maximum information-entropy principle for heavy-tailed random variables, and with its asymptotically infinitely divisible property. The dynamics towards the limit distribution is quantified. Tsallis q-entropies also exhibit power laws with T, such that Sq(T) ∼ Tβ(q), with β(q) ⩽ 0 for q ⩽ 0, and β(q) ≃ 0.5 for q ⩾ 1. No clear patterns are found in the geographic distribution within and among the statistical parameters studied, confirming the strong variability of tropical Andean rainfall.

Research highlights
► Statistics and memory of tropical Andean rainfall for increasing aggregation times, T.
► Short (long) memory exists for small (large) lags, separated at 15–30 min and 24 h.
► Entropy behaves as a power law, S ∼ Tβ, with β = 0.5, before reaching a maximum value.
► Maximum entropy is reached through a dynamic Generalized Pareto distribution.
► Tsallis q-entropies behave as Sq (T) ∼ Tβ(q), with scaling exponents depending on q.