Analysis and Modeling of Precipitation Intermittency by Compound Markov-DARMA Models

Abstract

Intermittency in precipitation is analyzed by a new class of stochastic models consisting of the product of two binary stochastic models. The ingoing models in these compound models are confined to a one-parameter white noise model and either a first-order or a second-order autoregressive model, which is described by two or three parameters, respectively. The two resulting second-order compound models have three and four parameters, respectively. Expressions of these parameters are derived, as well as equations with which to assess the parameters in the ingoing models based on the parameters in the compound models. The compound models are shown to differ considerably from the usual Markov models and discrete autoregressive moving average (DARMA) models. Daily binary precipitation time series from Copenhagen, Denmark, and Alice Springs, Australia, are analyzed by depicting the Markovian transition probabilities in probability trees with time lags of up to 5 days, resulting in 32 branches. Each tree is a kind of signature for the time series considered, but only a few of the branches, typically 3–4, are significant. Furthermore, the significant parts of the trees are rather skewed due to the intermittency of the precipitation process. This skewness makes the application of usual Markov and DARMA models less effective, whereas the new compound models better explain the observations. Based on one of the two new models, explicit but approximate expressions for the cumulative distribution (or the return period) of the durations of wet and dry spells are derived and compared to the observations from Copenhagen.