AbstractsA large number of models are available for streamflow forecasting. In this paper we classify and compare nine types of models for short, medium and longterm flow forecasting, according to six criteria:
We first distinguish between empirical and conceptual models, the difference being that conceptual models correspond to simplified representations of the watershed, while empirical model only try to capture the structural relationships between inputs to the watershed and outputs, such as streamflow. Amongst empirical models, we distinguish between stochastic models, i.e. models based on the theory of probability, and nonstochastic models. Three types of stochastic models are presented: statistical regression models, BoxJenkins models, and the nonparametric knearest neighbor method. Statistical linear regression is only applicable for long term forecasting (monthly flows, for example), since it requires independent and identically distributed observations. It is a simple method of forecasting, and its hypotheses can be validated a posteriori if sufficient data are available. BoxJenkins models include linear autoregressive models (AR), linear moving average models (MA), linear autoregressive  moving average models (ARMA), periodic ARMA models (PARMA) and ARMA models with auxiliary inputs (ARMAX). They are more adapted for weekly or daily flow forecasting, since the yallow for the explicit modeling of time dependence. Efficient methods are available for designing the model and updating the parameters as more data become available. For both statistical linear regression and BoxJenkins models, the inputs must be uncorrelated and linearly related to the output. Furthermore, the process must be stationary. When it is suspected that the inputs are correlated or have a nonlinear effect on the output, the knearest neighbor method may be considered. This databased nonparametric approach simply consists in looking, among past observations of the process, for the k events which are most similar to the present situation. A forecast is then built from the flows which were observed for these k events. Obviously, this approach requires a large database and a stationary process. Furthermore, the time required to calibrate the model and compute the forecasts increases rapidly with the size of the database. A clear advantage of stochastic models is that forecast uncertainty may be quantified by constructing a confidence interval. Three types of nonstochastic empirical models are also discussed: artificial neural networks (ANN), fuzzy linear regression and multivariate adaptive regression splines (MARS). ANNs were originally designed as simple conceptual models of the brain. However, for forecasting purposes, these models can be thought of simply as a subset of non linear empirical models. In fact, the ANN model most commonly used in forecasting, a multilayer feedforward network, corresponds to a non linear autoregressive model (NAR). To capture the moving average components of a time series, it is necessary to use recurrent architectures. ANNs are difficult to design and calibrate, and the computation of forecasts is also complex. Fuzzy linear regression makes it possible to extract linear relationships from small data sets, with fewer hypotheses than statistical linear regression. It does not require the observations to be uncorrelated, nor does it ask for the error variance to be homogeneous. However, the model is very sensitive to outliers. Furthermore, a posteriori validation of the hypothesis of linearity is not possible for small data sets. MARS models are based on the hypothesis that time series are chaotic instead of stochastic. The main advantage of the method is its ability to model nonstationary processes. The approach is nonparametric, and therefore requires a large data set. Amongst conceptual models, we distinguish between physical models, hydraulic machines, and fuzzy rulebased systems. Most conceptual hydrologic models are hydraulic machines, in which the watershed is considered to behave like a network of reservoirs. Physical modeling of a watershed would imply using fundamental physical equations at a small scale, such as the law of conservation of mass. Given the complexity of a watershed, this can be done in practice only for water routing. Consequently, only short term flow forecasts can be obtained from a physical model, since the effects of precipitation, infiltration and evaporation must be negligible. Fuzzy rulebased systems make it possible to model the water cycle using fuzzy IFTHEN rules, such as IF it rains a lot in a short period of time, THEN there will be a large flow increase following the concentration time. Each fuzzy quantifier is modeled using a fuzzy number to take into account the uncertainty surrounding it. When sufficient data are available, the fuzzy quantifiers can be constructed from the data. In general, conceptual models require more effort to develop than empirical models. However, for exceptional events, conceptual models can often provide more realistic forecasts, since empirical models are not well suited for extrapolation. A fruitful approach is to combine conceptual and empirical models. One way of doing this, called extended streamflow prediction or ESP, is to combine a stochastic model for generating meteorological scenarios with a conceptual model of the watershed. Based on this review of flow forecasting models, we recommend for short term forecasting (hourly and daily flows) the use of the knearest neighbor method, BoxJenkins models, water routing models or hydraulic machines. For medium term forecasting (weekly flows, for example), we recommend the knearest neighbor method and BoxJenkins models, as well as fuzzyrule based and ESP models. For long term forecasting (monthly flows), we recommend statistical and fuzzy regression, BoxJenkins, MARS and ESP models. It is important to choose a type of model which is appropriate for the problem at hand and for which the information available is sufficient. Each type of model having its advantages, it can be more efficient to combine different approaches when forecasting streamflow. KeywordsReview, forecasting, flow, stochastic model, conceptual model, artificial neural network, fuzzy set theory. Corresponding author Bernard Bobée, Chaire en Hydrologie Statistique, INRSEau,
Terre & Environnement, 2800 rue Einstein, C.P.7500, SainteFoy, Québec,
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