A number of investigators have evaluated the benefits of streamflow forecast using the Kalman filter. For example, Georgakakos (1986) studied the performance of a hydrometeorological model for streamflow forecasting using 6-h data for Bird Creek, a 2344 km2 catchment in Oklahoma. A precipitation forecasting model was developed and coupled to a modified version of the U.S. National Weather Service rainfall-runoff model to produce the streamflow forecast. Variables such as soil moisture storages and streamflow were updated through a Kalman filter. Eight 2-month forecast periods were examined. The results showed that the nonupdating forecasting model produced forecasts where the time-to-peak discharges were very different from those observed. The forecasts using updating techniques showed significant improvements.
Other applications in streamflow forecasting include the work of Takasao and Shiiba (1984) and Takasao et al. (1989) who developed a simple nonlinear stream-flow forecasting model and applied to the Haze River, a 370-km2 basin in Japan. Their work shows model performance for a flood in September 1965 with and without updating. As expected, the forecast errors of the model without updating are larger. The deterministic model NAMS11/MIKE 11 developed by the Danish Hydraulic Institute (DHI) uses a state variable updating procedure based on the Kalman filter for their conceptual rainfall-runoff model NAM (Refsgaard, 1997). Ahsan and O'Connor (1994) have expressed that the full capabilities of the Kalman filtering are not completely utilized since the predictions are expected to match the observed flows, which are considered to be noise free and that the filter will be more fully utilized in the future when remote sensing becomes more predominant.
Awwad and Valdes (1992) proposed an adaptive evaluation/forecasting algorithm for hydrologic forecasting and presented two multisite hydrologic forecasting approaches suitable for real-time applications. Their model is based on past and present flow rates, with the upstream inflows treated as exogenous inputs to the models. They applied the model to the Fraser River, Canada. In their original application Awwad and Valdes (1992) did not use precipitation terms, and even though their models performed very well in the one- and two-steps-ahead forecast error deteriorated rapidly. The authors later extended the adaptive evaluation/ forecasting algorithm to include precipitation inputs, upstream inflows forecasted/evaluated with uncertainty, and deterministic reservoir releases in the stochastic models (Awwad et al., 1994). This approach was adopted because it has a relatively simple dynamic structure in a black-box form and is calibrated online as additional information becomes available. The inclusion of precipitation information considerably benefited the multiple-period forecasting ability of the stochastic models.
The general form of the ARMAX models used in Awwad et al. (1994) followed the well-known state-space form of the Kalman filter presented above where optimal forecasts and updates of the states were obtained using the Kalman filter. Two other filters in the form of the Kalman filter, referred to as the parameter space and the noise-space filters, are used in parallel with the state-space filter to update the model parameters and noise statistics online along with the states. This adaptive estimation technique using parallel filtering does not require preassigned values for the Kalman filter coefficients and noise statistics, which are usually unknown in real-world applications. Other applications in short-term forecasting include: use of an ARMAX model to do predictions on the Fraser River in Canada (Ngan and Russell, 1986), use of the Kalman filter to estimate the parameters of a PARMA model with application to the Saugeen River in Ontario, Canada (Jimenez et al., 1989), and use of the ARMAX model with Kalman filter for short-term flow forecasting of snow-melt runoff in the Rio Grande at the Del Norte station (Haltiner and Salas, 1988).
As stated before in section 3 ANNs have been widely used for a number of hydrologic problems including forecasting of precipitation and streamflow. A vast literature already exists on the subject. For example, the ASCE J. Hydrol. Engr. (vol. 5, no. 2, 2000) is a dedicated issue on the subject. It includes the articles "Artificial Neural Networks in Hydrology I: Preliminary Concepts" and "Artificial Neural Networks in Hydrology II: Hydrologic Application," co-authored by the ASCE Task Committee on Applications of Artificial Neural Networks in Hydrology. The second article includes a review of various applications of ANNs on short-term and long-term flow forecasting. In addition, the book Artificial Neural Networks in Hydrology (Govindaraju and Rao, 2000) includes some chapters specifically on flow forecasting (e.g., Gupta et al., 2000; Salas et al., 2000; Deo and Thirumalaiah, 2000). The work by Gupta et al. (2000) discusses in some detail the training of ANNs based on multilayer feedforward neural networks (MFNNs), which are most commonly used for streamflow forecasting. It also presents some results illustrating some applications. The study by Salas et al. (2000) discuss some very basic concepts underlying ANNs, gives a simple detailed example, and two applications for daily and monthly streamflow forecasting. Finally, the work by Deo and Thirumalaiah (2000) includes the application of ANNs for real-time forecasting of daily flows and daily river stages.
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