Copyright © 2009 IAHS Press
Identification of an appropriate low flow forecast model
for the Meuse River
MEHMET C. DEMIREL & MARTIJN J. BOOIJ
Water Engineering and Management, Faculty of Engineering Technology, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands
m.c.demirel@utwente.nl
Abstract This study investigates the selection of an appropriate low flow forecast model for the Meuse River based on the comparison of output uncertainties of different models. For this purpose, three data driven models have been developed for the Meuse River: a multivariate ARMAX model, a linear regression model and an Artificial Neural Network (ANN) model. The uncertainty in these three models is assumed to be represented by the difference between observed and simulated discharge. The results show that the ANN low flow forecast model with one or two input variables(s) performed slightly better than the other statistical models when forecasting low flows for a lead time of seven days. The approach for the selection of an appropriate low flow forecast model adopted in this study can be used for other lead times and river basins as well.
Key words low flows; linear regression model; ANN; ARMAX; uncertainty; appropriate model; Meuse River
INTRODUCTION
Low flow is defined as a seasonal phenomenon and an integral phase of a flow system of any river (Smakhtin, 2001). However, in Northern European countries high flows are often studied and low flows are generally neglected (Kwadijk & Middelkoop, 1994; Parmet & Burgdorffer, 1995). For the Rhine and Meuse basins, a limited number of studies on low flows have been published in refereed literature (Middelkoop et al., 2001; De Wit et al., 2007; Rutten et al., 2008), and non-refereed literature (Passchier, 2004; Arends, 2005; De Bruijn & Passchier, 2006). This is probably because low flow is a slow process which usually occurs during the dry season, unlike high flow events’ fast and eye-catching processes. Low flow events in the Rhine River and Meuse River in dry summers such as in 1921, 1976 and 2003, indicate the importance of considering these events. Moreover, the number of days with low flows in Northern European rivers is expected to increase due to climate change (Middelkoop et al., 2001; De Wit et al., 2007; Te Linde et al., 2008).
Elaborative studies on low flows started in 1976 when a Task Committee on Low Flow was organized by the American Society of Civil Engineers (Riggs, 1980). This committee drew attention to the consequences of hydrological droughts and to the need for further studies using standard low flow indexes. There are several low flow indexes used by different institutes since there are many ways of defining flow conditions as “low flow”. One typical way is the lowest flow that has been ever measured in the river. Another is the use of the annual minimum 10-day flow. A more common definition, however, is the flow level being exceeded 95% of the year.
The flow processes are generally represented by different functions embedded into a model. A perfect model including every physical process in a basin may never exist without a certain degree of uncertainty. Therefore, uncertainty analysis in hydrology is necessary, for instance to express the reliability of forecasts. The number of studies applying a systematic quantification of uncertainties has increased rapidly and complementary discussions began to appear in the literature to create consensus in hydrological uncertainty assessment terminology (e.g. Montanari, 2007). Different uncertainty analysis techniques are present for different models (e.g. Monte Carlo simulations, Generalized Likelihood Uncertainty Estimation, etc.).
In this paper, the model output uncertainty is used for the identification of an appropriate low flow forecast model. There have been other studies on model appropriateness (e.g. Booij, 2003; Dong et al., 2005), but the identification of an appropriate low flow forecast model based on uncertainty in predicted low flows has not been done. The objective of this study is therefore to identify an appropriate low flow forecast model for a lead time of seven days by comparing output
uncertainties from three different data driven models with different combinations of input variables. The study area is the Meuse basin in Western Europe.
STUDY AREA AND DATA
The Meuse basin covers an area of approximately 33 000 km2, including parts of France, Luxembourg, Belgium, Germany, and The Netherlands. About 60% of the Meuse basin is used for agricultural purposes (including pastures) and 30% is forested. The average annual precipitation ranges from 1000–1200 mm in the Ardennes to 700–800 mm in the Dutch and Flemish lowlands. The maximum altitude is just below 700 m a.s.l. Snowmelt is not a major factor for the discharge regime of the Meuse. The average discharge at the outlet is approximately 350 m3 s-1, this
corresponds with an annual precipitation surplus of almost 400 mm. Precipitation is equally distributed over the year. The seasonal variation in the discharge is a reflection of the variation in evapotranspiration (Booij, 2005).
Daily discharge data at Chooz (upstream area 10 000 km2) and Monsin (upstream area
21 000 km2), basin-averaged precipitation data and basin-averaged potential evapotranspiration
data for the period 1968–1998 are used; 20 years for calibration and 10 years for validation of the results. Low flows are discharge values measured at Monsin station in the Meuse River of less than 100 m3 s-1 analogously to Booij et al. (2006).
METHODOLOGY
The four steps to identify the appropriate low flow forecast model are: 1. Assessment of appropriate temporal input resolutions.
2. Determination of model structures for three data driven models with four different combinations of input variables.
3. Quantification of the model output uncertainty.
4. Identification of an appropriate low flow forecast model.
Following these steps, it is intended to test three different data driven models with different combinations of input variables with appropriate temporal resolutions. Model types used in this study differ in two different dimensions regarding model complexity: the number of input variables and the mathematical description of the models.
Assessment of appropriate temporal input resolutions
The assessment of the appropriate temporal input resolution is based on the determination of cross-correlation coefficients between the input variables at different temporal resolutions and the output variable. The input variables are the discharge at Monsin Qm, the discharge at Chooz Qc, the basin averaged precipitation P and the basin averaged potential evapotranspiration PET. The output variable is Qm seven days ahead of the input variables. For each input variable, the temporal resolution resulting in the largest cross-correlation coefficient is chosen as appropriate temporal resolution in the subsequent modelling steps.
Determination of data driven model structures
Data driven models are built based on input-output relations. Physical processes are generally ignored and model structures are less complex than physically-based or conceptual models. Three different data driven models for 7-day ahead low flow forecasts are compared in this study: a linear regression (LR) model, a multivariate auto regressive moving average model with exogenous inputs (ARMAX) and an artificial neural network (ANN) model. Each model is tested with four different combinations of input variables by adding the input variables according to their cross-correlation with the output variable in a descending order of cross-correlation. The LR model has the simplest structure compared to the other two models as shown in equation (1):
(
t)
a bx( )
tQLR + 7 = + i i (1)
where QLR is the 7-day ahead predicted discharge, a is the intercept, bi are regression coefficients
and xi are the independent input variables. The objective function in a regression model usually
aims to minimize the total sum of squared errors using the ordinary least squares method.
In ARMAX modelling several steps are distinguished: the identification of the model structure, parameter estimation and a diagnostic check to validate the model before using it in forecasting problems. Autocorrelation functions and partial autocorrelation functions are good indicators for univariate AR and MA model orders. The shape of the lagged correlogram generally gives an idea for modellers how to choose the best model order. The Yule-Walker equations are used to estimate the ARMAX multivariate model parameters, see equation (2):
(
t)
c x t e( )
tQARMAX +7 = i i()+ (2)
where QARMAX is the 7-day ahead predicted discharge, ci are model parameters, xi are the
independent input variables and e(t) is the white noise. Details of ARMAX modelling can be found in core text books such as Ljung (1986).
The ANN model structure has been designed based on the literature, user experience and trial-error processes. A network with one hidden layer with different numbers of hidden nodes has been mostly preferred in hydrological predictions (Raman & Sunilkumar, 1995; Coulibaly et al., 2001; Khan & Coulibaly, 2005; Demirel et al., 2008). More than one hidden layer requires many more parameters to be estimated as many new weights and bias values are necessary in the newly built connections. Following Rumelhart et al. (1986) and Govindaraju & Rao (2000) the feed forward ANN model is selected to model daily low flows. The Levenberg-Marquardt algorithm being a fast converging optimisation algorithm for training ANNs, and more efficient than many other present algorithms, is used for optimisation. The hyperbolic tangent transfer function and the logistic sigmoid function were both tested in a trial-and-error process and the former gave better results. The objective function is the Root Mean Squared Error (RMSE) for low flows. The performance goal for this objective function is defined as 1‰ of the observed data variance which is equal to 0.0015. Different guidelines for the number of hidden nodes have been proposed. For example, Ochoa-Rivera (2008) suggests starting with one hidden node and adding new nodes until a significant improvement in performance is achieved. Eberhart & Dobbins (1990) found it useful to commence simulations with a number of hidden nodes equal to half of the number of input nodes. In this study, the number of hidden nodes in the one and two input ANN models is selected by using a trial-and-error procedure following Ochoa-Rivera (2008). For the three and four input ANN models, the number of hidden nodes is assumed to be equal to the number of inputs multiplied with the lead time, hence 21 and 28 hidden neurons are used, respectively, as the lead time is seven days.
Quantification of uncertainty and appropriate low flow model identification
There are many different definitions of uncertainty (e.g. Walker et al., 2003; Refsgaard et al., 2007) Here, uncertainty is assumed to consist of inaccuracy and imprecision following Van der Perk (1997). Inaccuracy is defined as the difference between a simulated value and an observation, while imprecision refers to the possible variation around the average simulated values and observed values. Model inaccuracy can be assessed by, e.g. the RMSE. Obviously, the model inaccuracy does not cover all uncertainties and therefore underestimates the total uncertainty. However, it is expected to give an indication of the trend in uncertainty as a function of model complexity. The appropriate model is selected according to this indicator. An appropriate low flow forecast model is a model that produces output with the smallest uncertainty in low flows. This is quantified by the RMSE of observed and predicted low flows for the validation period.
RESULTS AND DISCUSSION
Assessment of dominant low flow indicators
Figure 1 shows the cross-correlation coefficients between the output variable and input variables as a function of the temporal input resolution. Different temporal resolutions were found to be appropriate for each of the four inputs: seven days for discharge values at Monsin, four days for discharge values at Chooz, and 150 days for basin-averaged precipitation and evapotranspiration.
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 14 21 28 30 60 90 120 150 180 210 240 270 300
Fig. 1 Cross correlation coefficients between discharge at Monsin Qm (t+7) and different inputs with a
lag time of seven days as a function of temporal input resolution (days) for four different inputs: precipitation P(t), potential evapotranspiration PET(t), discharge at Chooz Qc(t) and discharge at
Monsin Qm (t).
Determination of data driven model structures
LR model The results of the LR models are presented in Table 1. These results are not promising for low flow predictions and the model order of one can be an important limitation here. The uncertainty (RMSE values) decreased with an increasing number of input variables for both the calibration and validation. The contribution of precipitation to the model is very low. The negative relation between potential evapotranspiration and the discharge at Monsin is apparent in the four input LR model. After the inclusion of the discharge at Chooz (Qc) as an input to the
model, other input variables have marginal effects on the results.
Table 1 Estimated parameter values for LR model and RMSE values in calibration and validation. Estimated parameters RMSE
Inputs a b1 b2 b3 b4 Calibration Validation Qm(t) 0.356 0.680 0.313 0.316 Qm(t) Qc(t) 0.360 0.141 0.460 0.307 0.292 Qm(t) Qc(t) P(t) 0.150 0.115 0.458 0.093 0.296 0.278 Qm(t) Qc(t) P(t) PET(t) 0.519 0.092 0.405 0.095 –0.189 0.246 0.211
Multivariate ARMAX model The results of the ARMAX models are presented in Table 2. The model with two inputs is the better performing model in the ARMAX group. Precipitation and potential evapotranspiration have very low parameter values showing that their contribution to those models is not significant and can be excluded. For that reason, the uncertainty did not decrease when including this meteorological information. In general, the uncertainty is lower than for the LR model.
Temporal input resolution (day)
Correl ati on c oeffi ci ent (-) Qm (t+7)- Qm (t) Qm (t+7)-PET(t) Qm (t+7)- Qc (t) Qm (t)-P(t)
Table 2 Estimated parameter values for ARMAX model and RMSE values in calibration and validation. Estimated parameters RMSE
Inputs c1 c2 c3 c4 Calibration Validation Qm(t) 0.804 0.083 0.094 Qm(t) Qc(t) 0.366 0.384 0.077 0.081 Qm(t) Qc(t) P(t) 0.218 0.359 0.079 0.127 0.116 Qm(t) Qc(t) P(t) PET(t) 0.208 0.336 0.117 –0.038 0.124 0.108
ANN model The results of the ANN models are presented in Table 3. The most promising models were again the one and two input models as for ARMAX. Adding meteorological inputs did not improve the results. This might be due to the large temporal resolution of these inputs. Our aim is to capture daily variations in low flows; however, this can be difficult when using a temporal resolutions of 150 days. The addition of potential evapotranspiration in the other two models had small but positive impacts on the results; however, for the ANN model this is not the case. The possible reason is the model structure which has a very large number of hidden nodes causing difficulty in training and also weakening the effectiveness of the learning cycles. Other forecasting studies also had difficulties in determining the number of hidden nodes (Tingsanchali & Gautam, 2000).
Table 3 ANN model architecture and test scheme and RMSE values in calibration and validation. RMSE
Inputs Outputs Training Test Network
structure Epochs Calibration Validation
Qm(t) Qm(t+7) 7671 3652 1-10-1 5 0.063 0.062
Qm(t) Qc(t) Qm(t+7) 7671 3652 2-20-1 5 0.063 0.061
Qm(t) Qc(t) P(t) Qm(t+7) 7671 3652 3-21-1 4 0.091 0.085
Qm(t) Qc(t) P(t) PET(t) Qm(t+7) 7671 3652 4-28-1 10 0.098 0.090
Quantification of uncertainty and appropriate low flow forecast model identification
Table 4 and Fig. 2 show the model output uncertainty (RMSE values) as a function of model type and number of inputs. There is a significant decrease in uncertainty with an increasing number of inputs in the LR models in both the calibration and validation, while the ARMAX and ANN models do not show this behaviour.
Table 4 RMSE values for three data driven models (LR model, ARMAX model and ANN model) with four combinations of input variables for calibration and validation.
LR ARMAX ANN
Inputs
Calibration Validation Calibration Validation Calibration Validation
Qm(t) 0.313 0.316 0.083 0.094 0.063 0.062
Qm(t) Qc(t) 0.307 0.292 0.077 0.081 0.063 0.061
Qm(t) Qc(t) P(t) 0.296 0.278 0.127 0.116 0.091 0.085
Qm(t) Qc(t) P(t) PET(t) 0.246 0.211 0.124 0.108 0.098 0.090 The hypothesis is that more inputs to create more predictive accuracy requires more investigation when designing particularly ANN and ARMAX model structures. The possibility of a smaller number of hidden nodes and more efficient training algorithms should be critically tested. Accordingly, the governing trial-and-error processes in ANN modelling should be avoided and more deterministic approaches should be adopted. The improvement in the model results should be observed by the change of training cycles (i.e. epochs) and the number of the hidden
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 1 2 3 4 Number of inputs RM SE LR ARMAX ANN 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0 1 2 3 4 Number of inputs RM S E LR ARMAX ANN
Fig. 2 RMSE values as a function of number of inputs for three data driven models (LR model, ARMAX model and ANN model) for calibration (left) and validation (right).
nodes together. A longer lead time, such as 14 days, should also be tested to understand the behaviour of the models for longer terms. Data averaging for large window sizes should be carefully applied for low flow predictions as it is deteriorating the daily oscillations in low flows while increasing the persistence and correlation coefficient. Hence, we recommend four day to seven day data averaging for predictions with one or two week lead times. The LR and ARMAX models show different behaviour in the validation period, for example the one and two input ARMAX models revealed a higher uncertainty in the validation period than in the calibration period, but the three and four input ARMAX models performed better in the validation period.
The one input and two input ANN models have the smallest uncertainty indicating that they are the most appropriate low flow forecast models in this study. The observed and predicted low flows for these two models are illustrated in Figs 3 and 4. The low flow predictions are in general below the threshold of 100 m3 s-1. The magnitude of the low flows is more successfully captured in
the two input ANN model than in the one input model. As shown in Fig. 4 there are very low observed discharges in some validation years, e.g. in 1991, 1992, 1996 and 1997. The two input ANN model only approximated these events in 1992 and 1996. The one input ANN model was successful only for the low flows between 50 and 100 m3 s-1 and not for values below
25 m3 s-1. When the observed low flow is more stationary, as it is in the first and second year of the
validation period, the ANN models predict the low flows better. The arbitrary changes in the discharge values are not always well captured due to the learning rate of the networks.
0 25 50 75 100 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 D isch a rg e (m 3 s -1 ) Observed Predicted
0 25 50 75 100 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 D is cha rg e ( m 3 s -1 ) Observed Predicted
Fig. 4 Observed and predicted low flows at Monsin for two input ANN model in validation period. CONCLUSIONS
Three data driven models with different combinations of inputs were compared based on modelled output uncertainty. The aim was to identify the most appropriate low flow prediction model for the Meuse River. RMSE is used as an output uncertainty indicator and the ANN models with one and two input variables are found to be most appropriate for predicting low flows, i.e. they have the smallest RMSE values in the validation period. However, the LR model represents the idea that more information should result in less uncertainty very well as additional inputs created smaller RMSE values. This behaviour was not observed for the other two more complicated models, i.e. the ARMAX and ANN models, which might be due to possible obstacles in these two models such as the network structure in the ANN model and the delay factor in the ARMAX model. It can be concluded that low flow characteristics and data scales are the key factors in capturing the daily variations in hydrological low flow events. Futhermore, model inaccuracy can be used as an indicator for model output uncertainty. Finally, the four step methodology introduced in this study can be applied to other river basins and lead times as well.
Acknowledgements We acknowledge the financial support of Dr Ir Cornelis Lely Stichting (CLS), Project no. 20957310.
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