Scaling of entropy and multi-scaling of the time generalized qq-entropy in rainfall and streamflows

• Entropy of rainfall series exhibits a power law with aggregation time with exponent 0.5 before saturation.
• Scaling exponents for rainfall do not depend on record length.
• Entropy of streamflow series exhibits a power law with time with exponent marginally larger than zero.
• Scaling of entropy differs for rainfall and streamflows due to zeros in rainfall series.
• The time generalized qq-entropy exhibits multi-scaling for rainfall and streamflows.

We investigate the behavior of the Shannon entropy, S(T)S(T), and the time generalized qq-entropy, Sq(T)Sq(T), at increasing aggregation intervals, TT, using series of 15-min and hourly rainfall records in the tropical Andes of Colombia, spanning from 21 months to 40 years, as well as average daily streamflows in Colombia, the Amazon River basin, and USA, spanning from 34 to 69 years. Results for rainfall show that S(T)S(T)∼∼TβTβ with β=0.5β=0.5, valid up to a timescale 〈TMaxEnt〉=83〈TMaxEnt〉=83  h, and β=0β=0 for T>TMaxEntT>TMaxEnt. Scaling exponents (β=0.5β=0.5) are statistically independent of record length, although not so the values of TMaxEntTMaxEnt, owing to the greater amount of zeros in rainfall series during El Niño in Colombia. Maximum entropy is reached through a dynamic Generalized Pareto distribution, whose parameters are not statistically affected by record length. Entropy for daily streamflows behaves as S(T)∼TβS(T)∼Tβ with β≳0β≳0, consistently with the theoretical limit. The scaling behavior of entropy differs between rainfall and streamflows due to the presence of zeros in rainfall series, and by the rate at which they vanish upon temporal aggregation. The amount of zeros exhibit two different scaling regimes with TT, separated at 〈Tb〉=24〈Tb〉=24  h. Rainfall series get devoid of zeros at 〈Tnoz〉=164〈Tnoz〉=164  h, but later on for longer record lengths. A power law relates both timescales: TMaxEnt∼Tnoz0.62. Besides, Sq(T)Sq(T) consists of a continuous set of power laws with respect to TT, for increasing values of qq, whose scaling exponents vary nonlinearly with qq, thus implying multi-scaling of the time generalized qq-entropy, which in turn inherits all the conclusions from the Shannon entropy, since it is recovered at Sq=1(T)Sq=1(T).