Most time series models in hydrology are used for river flow forecasting, for generation of synthetic data sequences or for the study of physical characteristics underlying the hydrological processes. The models are formulated as linear stochastic difference equations. Three phases are considered for the selection of a model based on a satisfactory representation of a given empirical time series: identification, estimation and validation. Several criteria have been proposed for the selection of the order of ARMA models. The Akaike information criterion (Ale) is popular among hydrologists, but the posterior probability criterion has the advantage of optimality and asymptotic consistency. There are numerous applications of AR or ARMA models to annual streamflow series which are stationary. Seasonal, monthly, weekly or daily streamflow series are cyclically stationary and generally exhibit periodicities in the mean and variance and possibly in the autocorrelation structure. Removal of the periodicity has been accomplished by fitting harmonic series or by subtracting the seasonal mean and dividing by the seasonal standard deviation, and a time series model is then fitted to the residual series. Alternatively, ARMA models with time-varying coefficients are also used. The multiplicative ARlMA model of Box and Jenkins is less frequent in hydrology because of the difficulty in the identification of the parameter structure. Multivariate models are used when river flows at different sites are considered. Parameter estimation in multivariate time series models can become cumbersome because of the dimensionality of the problem. Often the covariance matrix of the noise term is not known in advance and limited information estimates are used. Multivariate models have been used for annual and monthly series. Disaggregation models have been used to subdivide a yearly series into monthly or weekly series or to disaggregate a main river flow into tributary flows while maintaining certain space and time cross-correlations. The aggregation of monthly into yearly time series has been shown to improve the parameter estimation of the yearly series. Hydrologic time series occasionally exhibit changes in level due to natural or man-made causes such as forest fires, volcanic eruption, climatological change, urbanization etc. These situations can be treated making use of intervention analysis.
Breakdown coefficients and scaling properties of rain fields