How to catch scaling in geophysics : some problems in stochastic modelling and inference from time series

Abstract

During recent decades, there has been a growing interest in research activities on change in geophysics and its interaction with human society. The practical aim is to improve our capability to make predictions of geophysical processes to support sustainable societal development in a changing environment. Geophysical processes change irregularly on all time scales, and then this change is hardly predictable in deterministic terms and demands stochastic descriptions, or random. The term randomness is usually associated to stochastic processes whose samples are regarded as a sequence of independent and identically distributed random variables. This is a basic assumption of classical statistics, but there is ample practical evidence that this wish does not always become a reality. It has been observed empirically that correlations between distant samples decay to zero at a slower rate than one would expect from not only independent data but also data following classical ARMA- or Markov-type models. Indeed, many geophysical changes are closely related to the Hurst phenomenon, which has been detected in many long hydroclimatic time series and is stochastically equivalent to a simple scaling behaviour of process variability over time scale. As a result, longterm changes are much more frequent and intense than commonly perceived and, simultaneously, the future states are much more uncertain and unpredictable on long time horizons than implied by typical modelling practices. The purpose of this thesis is to describe how to infer and model statistical properties of natural processes exhibiting scaling behaviours. We explore their statistical consequences with respect to the implied dramatic increase of uncertainty, and propose a simple and parsimonious model that respects the Hurst phenomenon. In particular, we first we highlight the problems in inference from time series of geophysical processes, where scaling behaviours in state (sub-exponential distribution tails) and in time (strong time dependence) are involved. Then, we focus on rainfall downscaling in time, and propose a stationary model that respects the Hurst phenomenon. It is characterized by a simple cascade structure similar to that of the most popular multiplicative random cascade models, but we show that the latter simulate an unrealistic non-stationary process simply inherent to the model structure.