Fractionally differenced ARIMA models applied to hydrologic time series: Identification, estimation, and simulation

Abstract

Abstract. Since Hurst [1951] detected the presence of long-term persistence in hydrologic data, new estimation methods and long-memory models have been developed. The lack of flexibility in representing the combined effect of short and long memory has been the major limitation of stochastic models used to analyze hydrologic time series. In the present paper a fractionally differenced autoregressive integrated moving average (FARIMA) model is considered. In contrast to using traditional ARIMA models, this approach allows the modeling of both short-and long-term persistence in a time series. A framework for identification and estimation is presented. The data do not have to be Gaussian. The resulting model, which replicates the sample probability density of the data, can be used for the generation of long synthetic series. An application to the monthly and daily inflows of Lake Maggiore, Italy, is presented. Once a suitable model is chosen, this method gives a more accurate estimate of H. In this paper, the fractionally differenced autoregressive integrated moving average models (FARIMA, see section 3) are considered because they account for both the short-and long-memory components that are present in many hydrologic long-memory processes. An identification procedure consisting of several steps is introduced in order to determine the most suitable model. This method is then applied to daily flows to Lake Maggiore, Italy. For purposes of comparison we also apply it to monthly flows to Lake Maggiore and to monthly rainfall in Genoa, Italy. A stochastic simulation of the non-Gaussian observations is also performed.In the next section the determination of the Hurst exponent from observed hydrologic time series is discussed. Although heuristic graphical methods can detect the presence of long memory, they are not capable of estimating its intensity with a sufficient degree of accuracy. Therefore, in section 3 a stochastic modeling framework is introduced in order to combine long-memory effects with the traditional identification and estimation approach of autoregressive integrated moving average (ARIMA) models. The application of the model to daily (and monthly) flows is discussed in the section 4. An application to monthly rainfall is also presented in order to assess the flexibility of the proposed approach when modeling data not affected by persistence. Detecting Long Memory in Hydrologic Time SeriesThe relative simplicity of the heuristic estimation methods for H has made them popular as diagnostic tools. Their purpose is only to detect long memory and to provide a rough estimate of the value of the H exponent; they are not able to supply any additional information concerning the spectral density function of the data. Among the methods found in the literature, the best known is the R/S statistic, which was introduced by Hurst [1951]. Other methods have been proposed. 1035

Observed drought and wetness trends in Europe: an update

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