Knowable moments for high-order stochastic characterization and modelling of hydrological processes

Abstract

Classical moments, raw or central, express important theoretical properties of probability distributions but can hardly be estimated from typical hydrological samples for orders beyond two. L-moments are better estimated, but they all are of first order in terms of the process of interest; while they are effective in inferring the marginal distribution of stochastic processes, they cannot characterize even second-order dependence of processes (autocovariance, climacogram, power spectrum) and thus they cannot help in stochastic modelling. Picking from both categories, we introduce knowable (K-) moments, which combine advantages of both classical and L-moments, and enable reliable estimation from samples and effective description of high-order statistics, useful for marginal and joint distributions of stochastic processes. Further, we extend recent stochastic tools by introducing the K-climacogram and the K-climacospectrum, which enable characterization, in terms of univariate functions, of high-order properties of stochastic processes, as well as preservation thereof in simulations.