JUAN B. VAL DÉS, PAOLO BURLANDO, AND JOSÉ D. SALAS

Over the past two decades, considerable research has been carried out in hydrology on developing mathematical tools and approaches for short- and long-term precipitation and streamflow forecasting. The forecasts may be concerned with flood warning, flood control, water quality control, navigation, energy production, and irrigation. Hydrologie forecasting signifies estimating the time of occurrence and the magnitude of a hydrological event before its actual occurrence (e.g., estimating daily streamflow with days or weeks in advance), i.e., an estimate of the future states of the hydrological phenomena is obtained in real-time. The adjective real-time is often used to reinforce the distinction between forecasting (the estimation of future hydrologie events based on the currently available data) and simulation, sometimes called long-term prediction (the estimation of equally likely scenarios of hydrologie events without necessarily conditioning on real-time data). In short, forecasting is generally used for operational and management purposes while simulation is used for design and planning purposes.

Forecasting of hydrological processes is an important tool for many water resources management and operational problems. For example, rainfall and stream-flow forecasting hours, days, weeks, and months in advance (depending on the particular case at hand) are important for many flood warning, evacuation, and mitigation plans and actions. The U.S. National Weather Service (NWS) routinely issues precipitation forecasts (throughout the year) for all the U.S. territories and flow forecasts at key control points of the stream network systems in the United

Handbook of Weather, Climate, and Water: Atmospheric Chemistry, Hydrology, and Societal Impacts, Edited by Thomas D. Potter and Bradley R. Colman. ISBN 0-471-21489-2 CD 2003 John Wiley & Sons, Inc.

States. Forecasting the number of hurricanes of certain strengths that may occur in the following year (Gray et al., 1994) and forecasting the path and the intensity of an ongoing hurricane, have been regular activities of the National Oceanic and Atmospheric Administration's (NOAA's) Hurricane Center. From the hydrologie and water resources perspectives, forecasting hurricanes has many implications, particularly as they relate to the occurrence of floods. In systems involving reservoirs, hurricane forecasts are useful for planning and implementing special operating rules to cope with impending floods. In small river systems that may be subject to flash floods, forecasting rainfall and streamflow a few hours in advance may be critical for implementing emergency actions such as alerting and warning the public. On the other hand, in large systems, such as the Mississippi River in the United States or the Paraná River in Argentina, flood occurrences may develop through several weeks and months. In these cases rainfall and flow forecasts are usually needed with lead times of weeks and months. Also in river systems where spring and summer runoff occurs from snowmelt, forecasts are usually needed weeks and months in advance for planning water supply and hydropower systems operations and for preparing for possible snowmelt floods. In such cases, determining the current amount of snow pack in the system and snow properties is of outmost importance. The development of reservoir operating rules and the real-time operation of reservoir systems may require hourly, daily, weekly, monthly, and yearly forecasts depending of the particular case at hand. Forecasts of rainfall, snowfall, snow pack, soil moisture, evaporation, streamflow, reservoir levels, river levels, and groundwater heads are generally needed in most cases of practical interest.

Forecasting of hydrologie processes has been developed using similar approaches as for simulation, although many models and techniques are unique either for simulation or forecasting. This chapter emphasizes forecasting based on stochastic and probabilistic techniques. Also, the emphasis will be on precipitation and streamflow processes, although many of the methods and models included herein are equally applicable for other hydroclimatic processes as well as évapotranspiration, soil moisture, surface and groundwater levels, and sea surface temperature.

In developing precipitation and streamflow forecasting models, one must be aware of the large uncertainty in the model parameters because of inadequate historical data of the relevant processes under consideration. Furthermore the model parameters may be expected to change slowly/rapidly with time, but the exact nature of the change is not predictable. In such cases, it is highly desirable to develop a model that has self-learning capabilities, so that it can adapt itself to the current situation (Brown and Hwang, 1997). For this purpose, filters have been formulated in the literature under the assumption that dynamic system parameters and input/measurement error statistics are known. This is not the case for precipitation and streamflow forecasting and additional estimation techniques are necessary. The sequential estimation procedure, known as the Kalman filter, is optimal under such conditions. However, if the actual values of system coefficients and covariances are different from those used in state estimation, then the filter is suboptimal: State estimates may contain more errors than is necessary and, in some cases, diverge from the neighborhood of the true values. State estimates could be improved by simultaneously estimating the uncertain parameters and the statistics. This additional information may be used to adapt the filter gains and model coefficients to the measurements. Adaptive filters may perform as well as optimal filters in the limit (Stengel, 1986).

Nonstationary characteristics are conventionally assumed to arise from the presence of one or more integrators in the stochastic part of the signal generation process. This applies in those cases where the model of the underlying time series data can only be characterized adequately by parameters, which vary over time in some significant manner. In all these situations the Kalman filter provides information on the possible nature of these parametric variations. Other statistical tools that are used for short- and long-term forecasting of precipitation and streamfiows include methods based on regression models, autoregressive integrated moving average (ARIMA) models, ARMAX models, transfer function noise (TFN) models, and models based in artificial neural networks (ANN). In the next section a brief description of the Kalman filter will be made because of the ample use of this technique in hydroclimatic forecasting and because many of the above models can be used in conjunction with the Kalman filter. Subsequent sections will include many of the referred models and techniques for precipitation and streamfiow forecasting.

2 ADAPTIVE PREDICTION: THE KALMAN FILTER

Since its introduction the Kalman filter has become a powerful tool in the fields of estimation and control theory (Kalman, 1960; Kalman and Bucy, 1961). As systems become more complex and noise becomes present in both input and output variables, it is then necessary to search for statistical solutions that can take advantage of past performance and adjust future forecasts accordingly. It is viewed as a complementary tool to the mathematical modeling of the rainfall-runoff process rather than a substitute because the knowledge of the underlying mechanism of the hydrologic process is essential for a successful implementation of the filter. The main purpose of this section is to present an introductory view of the Kalman filter rather than a thorough theoretical explanation of the statistical properties of the filter. For a successful application of the filter to real-time forecasting of hydroclimatic variables, the main hypothesis and limitations of the filter must be understood.

There are three different types of estimation problems depending on how the observations are used:

• Filtering The observations z = {zt z2,.. •, z,\ are used for filtering to obtain an estimate x,|, of the state of the system \t.

• Smoothing The observations z = {zl z2,..., z, zt+l} are used for smoothing to obtain an estimate x,|,+1 of the state of the system xt.

• Prediction The observations z = \zi z2, •.. , z,_i} are used in prediction to obtain an estimate xt\t~\ °f the state of the system x,.

For a detailed discussion on the topic the reader is referred to specialized books (e.g., Brown and Hwang, 1996). This reference also includes the software for some applications. Recursive algorithms are ideal for estimation of time-varying parameters. Modifications based on stochastic modeling of the parameter variations lead naturally to the development of the Kalman filter and the estimation of time-varying states in stochastic dynamic systems. Kalman considerably extended the state-estimation and filter theory of time-varying parameters or states so as to handle the analysis of nonstationary time series and provide a natural approach to the analysis of time series data that are assumed to be generated from stochastic state-space equations.

When modeling a system that evolves through time, specifically a stochastic process that is defined in discrete time, one would like to put the system in a state-space form or in the so-called state of the system vector x,. (Most linear models can be put into state-space form; nonlinear models can be linearized by using Taylor series expansion to reformulate them in state-space form.) If future values of the state of the system, xr+s, s = 1, 2,..., can be modeled using knowledge of x, (i.e., x, contains all the required information about the previous values î = 1, 2, . . .), wc obtain what is called a Markovian system. The best description of x, using x,_], x,_2,... can be modeled as

Equation (1) is called the state equation of the system, where <£(-) is the transition function, T( ) is the noise transition function, w( is the vector of system noises that describes the part of x, that is not explained by xs, s < t, and is assumed independent of xs and wv for s < t. When <£(•) and T( ) do not vary with time dependence, the system is referred to as stationary.

In most applications, the state of the system, x„ is not directly observed but rather measured in an observation vector z„ which is a function of x„ corrupted by measurement noise v,. This may be written as:

This equation is called the observation equation of the filter, and Eqs. (1) and (2) together constitute the heart of the Kalman filter; they may represent linear or nonlinear systems. The filtering problem is to estimate x, from the observations Z], ... ,zt, which are corrupted by measurement noises. If both the system and the observations are assumed linear, Eqs. (1) and (2) will have the following form:

The stochastic properties of the system and measurement noises have to be defined in order to apply the Kalman filter. Also the properties of the initial state

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