## Mid and Long Term Forecasting

If quantitative short-term forecasting is useful for flood forecasting, mid- and long-term forecasting plays a major role in the management of water resources. Agriculture and water supply, among other water uses, can significantly benefit from the availability of forecasts of rainfall amounts that can be expected over a time horizon of a month or a season. This issue is particularly relevant for complex systems that strongly depend on joint management of surface and groundwater resources. Forecasting at the mid- and long-term scales involves problems that are similar to the one already observed for smaller time scales. Nonstationarity, nonlinearity as well as the identification of the correct predictors guided the development of methods.

Whereas regression methods and, more recently, artificial neural networks have been extensively used for the purpose, a few other approaches can be found in the literature to forecast rainfall on mid- and long-term time scales. A truncated normal distribution is, for instance, the basis of the formulation of a nonstationary multisite model of rainfall that Sanso and Guenni (2000) show to capture the year-to-year variability and suggest to be suitable for short-term forecasting as well. Stone et al. (1996) and de Jager et al. (1998) used a simple probabilistic rainfall forecasting technique that is based on the identification of lag relationships between the values of the Southern Oscillation Index (SOI)—which can be considered as representative of the phase of the El Nino Southern Oscillation (ENSO) cycle—and future rainfall. Probability distributions for the subsequent 3 months are thus derived conditioned on the state of the SOI. Sharma (2000) introduced a nonparametric probabilistic model for forecasting rainfall with 3 to 24 months of lead times. Specifically, nonparametric kernel methods (e.g., Scott, 1992) for probability density function (PDF) estimation are used to express the conditional probability density function. Then, probabilistic forecasts are made by resampling from the rainfall probability density conditioned on the current value of the associated predictor set. An interesting feature of this approach is that the shape of the PDF is directly built from the data, and this leads to forecasts that are expected to resemble the characteristics of the sample and therefore reproduce the variability of observed rainfall.

Regression-based techniques have been extensively used for predicting seasonal rainfall. The increased availability of predictor variables, like ENSO, in near real time, by means of either observations or numerical weather prediction has increased the applicability of regression-based models and the Kalman filter (e.g., Liu et al., 1998). Makarau and Jury (1997) forecasted summer rainfall in Zimbabwe based on a set of climatic predictors by means of a multivariate linear regression model in a forward stepwise approach. Fairly simple models including up to five predictors produced jack-knife skill test correlations of about 80 to 85% for a lead time of 2 to 3 months. A similar approach was used by Jury (1998) to forecast seasonal rainfall and other climatic variables for the KwalaZulu-Natal region in southern Africa, also obtaining a forecast skill of about 76% for rainfall and about two thirds of the variance in the other cases. Francis and Renwick (1998) focused on predicting seasonal (either 1 month or one 3-month season) rainfall anomalies.

Similarly, Thapliyal (1997) carried out a comparison of forecast models based on the correlation between predictors and predictands and a dynamic stochastic transfer model to predict monsoon rainfall in India. The dynamic stochastic transfer model corresponds essentially to an ARIMA model structure, the orders of which are estimated against observations. It should be observed that a critical issue in using regression-based techniques is the stability of the selected predictor and the robustness of the model describing its temporal evolution. Finally, ARMA models, which have been extensively applied to forecast streamflows, have also been used to simulate rather than to forecast mid- and long-term rainfall. Another application of ARIMA models for forecasting monthly rainfall series with the purpose of providing an input for flow forecasting in the management of water resources systems can be found in Delleur and Kavvas (1978).

As in the case of short-term forecasts, artificial neural networks have been proposed for forecasting seasonal rainfall. ANNs has been found to be useful for forecasting the behavior of complex and highly dynamic systems such as the

monsoon rainfall. Simple deterministic neural networks show, however, a limited robustness in terms of forecast skill, so that more complex networks are often introduced. An example of a simple four-layer input, two hidden layers, and one neuron output network for forecasting Indian monsoon rainfall can be found in Sahai et al. (2000). On the other hand, Navone and Ceccatto (1994), proposed an interesting but complex application of ANNs, as a nonlinear method to correlate preseason predictors to rainfall data and as an algorithm for reconstructing the rainfall time series dynamics. Accordingly, they implemented a hierarchical neural network, which is sketched in Figure 3. The network trained to correlate predictors and the network trained to learn the time series dynamics are combined by connecting their output units to a new neuron, which is then used to issue the forecasts. The authors refer about an improved forecast skill due to the hierarchical approach, especially in forecasting large anomalies. The performance of ANNs can be reduced if the parameters used to train and forecast are correlated. Guhathakurta et al. (1999) used principal component analysis—as suggested by Hsieh and Tang (1998)—to transform the original variables into a new set of uncorrelated variables. These are then used to train and issue forecasts by means of a three-layer, five-input, three hidden nodes in one single hidden layer ANN. The output from such ANN and the output of a simple deterministic ANN using the untransformed parameter set were then used each as input to a simple two-layer ANN without any hidden layer, which produced rainfall forecasts. The final hybrid model increased the overall forecast skill from about 40 to 80%.

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