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A general model building procedure is developed for employing the Akaike information criterion (AIC) to select the most appropriate stochastic model to describe a specified geophysical time series. To demonstrate the effectiveness of the proposed approach to model construction, formulas for the AIC are given for many types of stochastic models and the method is successfully employed for determining the most suitable models to fit a wide variety of data sets which arise in hydrology and the natural sciences. The AIC is used to discriminate between short and long memory models when modeling annual hydrological time series. An improved constrained autoregressive model is designed for describing yearly sunspot numbers while both stationary and nonstationary linear stochastic models are entertained for modeling a tree ring series. Three types of models are considered for modeling seasonal time series and in a practical application it is shown how to select the most appropriate model to fit to a monthly river flow time series. The model building procedure is also used to select an intervention model that describes the effects of the Aswan Dam on the average annual flows of the Nile River.