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

We investigate the behavior of the Shannon entropy, , and the time generalized -entropy, , at increasing aggregation intervals, , 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 with , valid up to a timescale h, and for . Scaling exponents ( ) are statistically independent of record length, although not so the values of , owing to the greater amount of zeros in rainfall series during El Nino 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 with , 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 , separated at h. Rainfall series get devoid of zeros at h, but later on for longer record lengths. A power law relates both timescales: . Besides, consists of a continuous set of power laws with respect to , for increasing values of , whose scaling exponents vary nonlinearly with , thus implying multi-scaling of the time generalized -entropy, which in turn inherits all the conclusions from the Shannon entropy, since it is recovered at .