Hydrological Sciences–Journal–des Sciences Hydrologiques, 51(1) February 2006
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Parameter conditioning and prediction uncertainties of the LISFLOOD-WB distributed hydrological model XINGGUO MO1, FLORIAN PAPPENBERGER2, KEITH BEVEN2, SUXIA LIU3, AD DE ROO4 & ZHONGHUI LIN1 1 Key Laboratory of Ecological Network Observation and Modelling, Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China
[email protected] 2 Institute of Environmental and Natural Sciences, Lancaster University, Lancaster LA1 4YQ, UK 3 Key Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographical Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, China 4 Joint Research Centre, Ispra site,Via E. Fermi 1, I-21020 Ispra (VA), Italy
Abstract Distributed hydrological models are considered to be a promising tool for predicting the impacts of global change on the hydrological processes at the basin scale. However, distributed models typically require values of many parameters to be specified or calibrated, which exacerbates model prediction uncertainty. This study uses the generalized likelihood uncertainty estimation (GLUE) technique to analyse the parameter sensitivities of a distributed hydrological model, LISFLOOD-WB. Discharge time series and event volume data of the Luo River at upstream and downstream sites, Lingkou and Lushi, are used to analyse parameter uncertainty. Eight key parameters in the model are selected for conditioning and sampled using the Monte Carlo method under assumed prior distributions. The results show that maximum efficiency of model performance is lower and the number of behavioural parameter sets giving acceptable performance is fewer in the Lingkou sub-basin than in the Lushi sub-basin with the same criteria of acceptability. For both sub-basins the distribution shape parameter B in the fast runoff generation scheme is the most sensitive in predicting both discharge time series and event volume at the outlet. It is also shown that the value of parameter B at which the highest efficiency is derived is shifted from a high value for Lushi to a low value for Lingkou, consistent with past experience of model calibration that the larger the basin, the larger the B value is. The channel Manning coefficient Nc shows some sensitivity in the prediction of discharge time series, but less in the prediction of event volumes. The other key parameters show little sensitivity and good simulations are found across the full range of parameter values sampled. The uncertainty bounds of predicted discharges at the Lushi sub-basin are broad in the peak and narrow in the recession. The normalized difference between the upper and lower uncertainty bounds for both discharge and evapotranspiration are broad in summer and narrow in winter and that of recharge is the opposite. Key words distributed hydrological model; GLUE; parameter calibration; uncertainty
Conditionnement de paramétrage et incertitudes de prévision du modèle hydrologique distribué LISFLOOD-WB Résumé Les modèles hydrologiques distribués sont considérés comme un outil prometteur pour la prévision des impacts du changement global sur les processus hydrologiques à l’échelle du bassin versant. Cependant, les modèles distribués nécessitent généralement la spécification ou le calage des valeurs de nombreux paramètres, ce qui aggrave l’incertitude de prévision de la modélisation. Cette étude utilise la technique GLUE (generalized likelihood uncertainty estimation) pour analyser les sensibilités de paramétrage du modèle hydrologique distribué LISFLOOD-WB. Des séries temporelles de débit et des données de volume événementiel de la Rivière Luo, au niveau des sites amont et aval de Lingkou et Lushi, sont utilisées pour analyser l’incertitude de paramétrage. Huit paramètres clefs du modèle sont sélectionnés pour conditionner et utiliser après échantillonnage la méthode de Monte Carlo, sous conditions de distributions supposées a priori. Les résultats montrent que l’efficacité maximale des performances du modèle et que le nombre de jeux de paramètres d’état conduisant à des performances acceptables sont plus faibles dans le sous-bassin de Lingkou que dans celui de Lushi, avec le même critère d’acceptabilité. Pour les deux sous-bassins, le paramètre de forme de la distribution B du module de génération de l’écoulement rapide est le plus sensible pour la prévision des séries temporelles de débit ainsi que des volumes événementiels à l’exutoire. Il apparaît également que la valeur du paramètre B conduisant à la plus forte efficacité est grande pour Lushi mais petite pour Lingkou, ce qui est cohérent avec l’expérience passée de calage de modèle selon laquelle la valeur de B augmente avec la taille du bassin versant. Le coefficient de Manning en cours d’eau Nc présente une certaine sensibilité pour la prévision des séries de débit, mais moindre pour la prévision des volumes événementiels. Les autres paramètres clefs présentent une faible sensibilité, et de bonnes simulations sont obtenues à travers la gamme complète des valeurs de paramètres échantillonnées. Les bornes d’incertitude des débits
Open for discussion until 1 August 2006
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prévus pour le sous-bassin de Lushi sont larges pour le pic et étroites pour la récession. La différence normalisée entre les bornes d’incertitude supérieures et inférieures est grande en été et petite en hiver pour le débit et l’évapotranspiration, et inversement pour la recharge. Mots clefs modèle hydrologique distribué; GLUE; calage de paramètre; incertitude
INTRODUCTION Estimation of the temporal and spatial distribution of water resources and prediction of effects of climate and vegetation cover changes on hydrological processes are research topics of serious concern in many parts of the world. For example, there is a high demand for predictions of how the dramatic land use/cover changes might affect the discharge and evaporation in the middle part of the Yellow River catchment (Wang & Takahashi, 1999) in order to understand reduced discharges during dry periods. The hydrological cycles are closely linked with patterns of vegetation, soil, topography and climate conditions at the basin scale (Wigmosta et al., 1994). Distributed physical process-based hydrological models are designed to predict the spatial patterns of evapotranspiration and runoff generation and the impacts of land use/climatic changes, due to taking proper account of spatial variability of both inputs and basin characteristics. However, the prediction is still quite uncertain. The uncertainty may be associated with process parameterizations and parameter estimation, scale effects and the nonlinear response of hydrological processes to environmental forcing, as well as errors in the specification of input data and boundary conditions (Beven, 2002). For example, the basic physical equations generally depend on patch- or laboratory-derived relations used at the grid scale, which is usually larger than the scale of heterogeneity in vegetation and soil properties in large basin simulations (Famiglietti & Wood, 1994; Spear, 1997; Beven, 2001). Furthermore, although the argument is often used that the parameters of distributed models might be easier to estimate since they are more likely to reflect the local physical characteristics of a basin, it is clear that as a result of heterogeneity and scale effects it is necessary to estimate effective values of the parameters (Beven, 2000, 2001) for distributed modelling. As a consequence, the parameters in these equations may only be calibrated as effective values that will, to some extent, compensate for such sub-grid heterogeneities and any errors in the input data and model structure. However, since there are no adequate data available to constrain the values of these parameters spatially and temporally within the model framework, the calibration of effective parameter values often results in many parameter sets (called “behavioural” models by Spear & Hornberger, 1980), scattered throughout the parameter space, producing acceptable predictions on the basis of one or more performance measures (Franks & Beven, 1999; Beven & Freer, 2001). This is the so-called model equifinality problem. It is expected that the equifinality problem will be exacerbated as more distributed process representations and parameters are embedded into the model if no additional information is available to condition the parameter values (Beven, 2002, 2005; Schulz & Beven, 2003). One of the effective ways to assess the prediction uncertainty associated with multiple behavioural parameterizations is using the generalized likelihood uncertainty estimation (GLUE) technique (Beven & Binley, 1992; Beven & Freer, 2001). So far, there have been a number of rainfall–runoff model applications using GLUE to assess model performance (Freer & Beven, 1996; Nandakumar & Mein, 1997; Lamb et al., 1998; Franks & Beven, 1999; Cameron et al., 2000; Yu et al., 2001, Blazkova et al., 2002). However, few are based on distributed physical-based models (though see Copyright 2006 IAHS Press
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Christiaens & Feyen, 2001). As a way to reduce the equifinality and prediction uncertainty, multi-site and multi-variable calibration and conditioning should be performed for distributed models, where observational data are available to do so (Refsgaard, 1997; Beldring, 2002). In GLUE, Monte Carlo sampling is utilized to analyse the total sensitivity of model outputs to parameter variations and, as a consequence, a large number of simulations are necessary to characterize the parameter space adequately for any reasonably complex models. For each model realization the simulation residuals and likelihood weights are calculated with respect to some observations, such as discharge, to estimate the model performance efficiency and prediction uncertainty. One of the analysis tools is to draw the scatter plot, which is the projection of a high dimension parameter response surface onto a single axis. However, it is not easy to reveal any complex interactions between parameters that result in high efficiency of model performance, as well as the sensitivity of the model predictions to an individual parameter, unless some specific structure is evident. It is the interaction between the individual parameter values that leads to the model performance being behavioural or non-behavioural in the sense of providing acceptably good fits to any available observations. Therefore, it is helpful to reveal some sensitivity of parameters to the model performance with the generalized sensitivity analysis (GSA) of Spear & Hornberger (1980). Generalized sensitivity analysis evaluates the sensitivity of individual parameters by dividing their values into several sets and comparing the distributions of these sets that give rise to either behavioural or non-behavioural model performance. Similarity between the distributions of parameter sets suggests that the results are insensitive to a particular parameter, whereas marked dissimilarity between the distributions suggests the model predictions are sensitive to that parameter. In this study, the hydrological response of the Luo River, a tributary of Yellow River, China, above the Lushi gauge site, is simulated with the LISFLOOD-WB model. Discharges at the Lingkou and Lushi gauges are used for parameter conditioning and uncertainty estimation of the LISFLOOD-WB within the GLUE methodology. After briefly describing the LISFLOOD-WB model in the next section with special focus on those parts related to the parameters addressed in this paper, the GLUE methodology is briefly introduced. Next, the preparatory work, including the basin description, the GIS and remotely sensed data preparation for surface parameters, referenced parameter values and parameter selection and range for GLUE analysis are described. In the model results section, the comparisons of the distribution of behavioural parameter sets with GLUE among different data sets are given and the sensitivities of model predictions to individual parameters are analysed. Then the uncertainties of water balance components via uncertainty bound analysis are analysed. In the Discussion section, some key aspects related to the uncertainty analysis, such as the choice of the likelihood measure and reasons for low model performance are discussed. Finally, the conclusions are given. A BRIEF INTRODUCTION TO THE LISFLOOD-WB MODEL The original LISFLOOD-WB (De Roo et al., 2000, 2003) distributed hydrological model includes representations of rainfall interception, snowmelt, transpiration and evaporation, infiltration, percolation and capillary rise, groundwater flow, overland Copyright 2006 IAHS Press
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runoff and streamflow processes. The LISFLOOD-WB model used in this paper is a modified version in which a two-source canopy energy balance scheme is used to distinguish the canopy evapotranspiration and soil evaporation. The fast runoff/ infiltration scheme has been improved by using a grid-scale distribution of soil moisture storage. The introduction to the model in this section deals only with the parts related to the parameters used for uncertainty analysis. For more complete details about the model, please refer to Mo et al. (2005). In the model, water vapour fluxes to the atmosphere from canopy, intercepted water (Kergoat, 1998) and soil are described by the forms of the Penman-Monteith equation, with a canopy resistance calculated as a function of leaf area index (LAI), minimum leaf stomatal resistance, rsmin, and other environmental factors, such as global radiation, air water vapour deficit, effective surface temperature and soil moisture, θ2 in the root zone, respectively, in the form used by Jarvis (1976; see also Stewart, 1988; Noilhan & Planton, 1989). Net radiation is assumed to be a function of surface emissivity, ε and surface albedo, αref, that varies with canopy type. A significant difference from past forms of LISFLOOD is the introduction of a sub-grid variability in fast runoff generation. This has been implemented as a modification of the Xinanjiang runoff generation concept (Zhao, 1992), described in Mo et al. (2005). It treats storage deficit as a spatial distribution function, in which the fraction of saturated/unsaturated areas is determined by both the relative storage (maximum storage, Smax minus the storage before the rainfall event, S0) and rainfall amount, P (Fig. 1). The S0 is expressed as the product of initial soil moisture, θ0 and the effective depth of upper soil layer that affects infiltration, Leff. The scheme is controlled by a shape parameter, B. Overland runoff is calculated as the difference between precipitation and the amount of water infiltrating into the saturated portion, as. The dynamics of soil moisture are described with a two-layer scheme, in which the first layer acts as a significant source of soil evaporation, the second as the source of root uptake for canopy transpiration. The moisture fluxes between the two layers with the depth, Li (i = 1, 2) and recharge to the upper groundwater zone are calculated using Darcy’s law using effective hydraulic conductivities in each layer (Sellers et al., 1986). The related hydraulic conductivity and water potential are estimated by the relationships of Van Genuchten (1980) with parameters of saturated conductivity Ksat, air entry water potential ψsat in the unsaturated zone, shape parameters αvg and nvg, saturated soil moisture θs and residual soil moisture θr. Smax
R P
S = ( S max − S 0 )(1 − (1 − a ) 1 / B )
S0 a
as
1-a
Fig. 1 Diagram of the fast runoff generation scheme. Copyright 2006 IAHS Press
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Distributed lateral fluxes between grid squares are calculated using a Darcian saturated flux taking account of transmissivities in the upper (Tguz) and lower (Tglz) zone and head differences between grid squares in each of two groundwater zones. The distributed flow routing scheme computes the flow rate and water level simultaneously to approximate the actual unsteady inhomogeneous nature of flow propagation in a channel based on the continuity equation (Chow et al., 1988). A power-law function relating cross-sectional area and discharge is assumed, of the form: A = α mQβ
(1)
Assuming the flow is uniform, αm can be expressed as: β
æ NPw 2 / 3 ö ÷ αm = ç (2) ç S ÷ b è ø where N is an effective Manning channel roughness coefficient; Pw is the wetted perimeter; Sb is the bottom slope; and β is a parameter The kinematic wave formulation is used for both channel and overland flow routing with respective roughness coefficients Nc and Nof. THE GLUE METHODOLOGY In GLUE, the behavioural sets of effective parameter values are evaluated by random sampling within a Bayesian Monte Carlo framework. Prior knowledge of the parameter distribution can be reflected in a prior likelihood measure associated with each parameter set realization. Usually, the parameters are sampled uniformly within the potential parameter space, since it is difficult to specify prior distributions for scale-dependent effective parameter values and their interactions. A large number of model simulations should be carried out to explore the potential parameter space thoroughly. After applying the chosen criteria for model acceptability, the behavioural simulations are weighted by an appropriate likelihood weight for estimation of posterior parameter distributions and prediction uncertainty bounds. The results of the GLUE procedure depend on the choice of likelihood measures. This includes two aspects, namely the choice of the measure characterizing the behaviour and the choice of any criteria to reject the simulation. A traditional statistical likelihood measure based on a structural model of the errors can be selected if it is appropriate, but it is usually difficult to define a consistent error structure for hydrograph simulation (Freer & Beven, 1996; Freer et al., 2002; Pappenberger et al. 2005). The only requirements for a likelihood measure within GLUE are that it should increase monotonically with increasingly good simulations and should be zero for models that are rejected as non-behavioural. In this study, the Nash-Sutcliffe efficiency is employed as the likelihood measure (Freer & Beven, 1996). It is defined as the proportion of the variance of the observations explained by the model such that: 2 L(Y |θ i ) ∝ 1 − σ i2 / σ obs
2 σ i2 < σ obs
(3)
where L(Y|θi) is the likelihood measure for the ith model simulation for parameter vector θi on a set of observations Y; σi2 is the associated error variance for the ith 2 model; and σ obs is the observed variance for the period under consideration. Copyright 2006 IAHS Press
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Posterior distributions of the predicted variables over all behavioural parameter sets are derived directly from the likelihood weights for behavioural models by forming the cumulative distribution after rescaling the behavioural model likelihood values to a cumulative sum of unity so that: ∧
n
∧
L ( Z t < Z ) = å L ( θ i | Z t ,i < Z )
(4)
i =1
where n is the number of behavioural models, and Zˆ t ,i is the value of variable Z at time t simulated by parameter set θi. IMPLEMENTATION OF LISFLOOD-WB IN THE LUO RIVER BASIN Brief description of the basin In this study, LISGLOOD-WB is used to model the Luo River down to the gauging station at Lushi (4423 km2), a mid-basin tributary of the Yellow River in central China. Data are also available from the upstream gauging station at Lingkou (2665 km2). Over the Lushi sub-basin (henceforth simply Lushi basin), the altitudes range from 550 m in the east to 2500 m a.s.l. in the west (Fig. 2). Approximately 36% and 22% of the basin are covered by grass and upslope farmland, respectively. The remaining land cover consists of coniferous forest (16%), broadleaf forest (14%), dwarf shrub (9%) and irrigated farmland (2%). These ratios are derived from the land-use map of Wu (1990). The basin is mountainous, of which approximately 40% is with slopes more than 35°, 30% between 25 and 35°, 20% between 5 and 25°, and 10% less than 5° (Guo &
Fig. 2 Location of the Lushi basin (4423 km2) with contours, channel, hydrological stations and raingauges; within which is nested the Lingkou sub-basin (2665 km2). The basin is enclosed by the thick black line in the main map and shaded by the Yellow River in the inset map of China. Copyright 2006 IAHS Press
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Zheng, 1995). It has complex landscapes, rich in channel development and deep incisions. There are small flood plains along the river up to 1000 m wide. In the basin, the soil is mainly Chernozem, with medium loam texture. The vegetation types are mainly Betula albo-sinensis, Quercus liaotungensis, Populus davidiana, Pinus tabulaeformis, Platycladus orientalis, Sabina chinensis, Tilia dictyoneura, Paulownia fortunei and Sorbus hupehensis. The average annual precipitation, potential evaporation and river discharge over the Lushi basin are 703 mm, 960 mm and 21 m3 s-1, respectively, for the 1981–1997 period. Daily precipitation data recorded at 10 rainfall gauges, daily streamflow data at the Lushi and Lingkou sites, and meteorological data at the Lushi climate station are used for the study. The precipitation and meteorological variables are corrected for elevation effects with a 1-km2 digital elevation model (DEM) and spatially interpolated by an inverse square distance method. GIS and remotely sensed data for surface parameters The topography of the Lushi basin is described with a 30″ × 30″ DEM (US Geological Survey database). Based on the DEM, the channel network, slope and aspect of each 1-km2 grid element are calculated. Monthly maximum composite 1-km resolution normalized difference of vegetation index (NDVI) data obtained from NOAA-AVHRR for 1995 and 1996 are employed to estimate LAI. As in Sellers et al. (1996), LAI is retrieved from NDVI directly according to the vegetation types. The daily evolution of LAI is then interpolated linearly between the dates of the available images. All the spatial data are projected into a Lambert azimuthally equal area projection with 1-km2 grid size. Reference parameter values The LISFLOOD-WB model, as a physically-distributed hydrological model, has many parameters. Considering the computational burden, it is impossible to put all the parameters together for uncertainty analysis within GLUE. Therefore, some parameters—the reference parameters in Table 1—were assigned as spatially homogeneous and were not used for uncertainty analysis. The parameter values selected for the vegetation types were collected from references (Sellers et al., 1986; Chow et al., 1988; Kergoat, 1998; Habets et al., 1999; Wang & Takahashi, 1999; Mo et al., 2004). A total of eight key parameters (minimum stomatal resistance rsmin, land surface Manning roughness coefficient Nof, channel Manning coefficient Nc, shape parameter for overland runoff generation B, saturated conductivities in first and second unsaturated soil layers Ksat1 and Ksat2, upper and lower groundwater transmissivities Tguz and Tglz) were chosen to analyse the model parameter ranges and prediction uncertainty with GLUE methodology. These eight parameters are closely related to canopy, soil and channel hydrological processes in the basin. As the model is distributed, even for eight parameters, it is still necessary to think about the computation burden of making runs with many parameter realizations in such a medium-sized basin. It is therefore necessary to reduce the dimensionality of the parameter space before applying the GLUE procedure. Here this has been achieved by: (a) applying multipliers to the spatial patterns of the rsmin and Nof parameters; and Copyright 2006 IAHS Press
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Table 1 Reference values for the parameters assumed to be spatially homogeneous.
Parameter ε Smax Leff β L1 L2 αvg nvg θs θr
Description Surface emissivity Maximum storage capacity Effective depth of upper soil layer that affects infiltration Power parameter in the relationship between cross-sectional area and discharge Depth of upper soil layer Depth of lower soil layer Soil moisture characteristic parameter Soil moisture characteristic parameter Soil water content at saturation Residual soil water content
Units
Value 0.97 for vegetation canopy 250 600
mm mm
0.6 cm cm cm3 cm-3 cm3 cm-3
5 195 0.035 1.25 0.4 0.08
Table 2 Reference values for the parameters varied with each vegetation type. rsmin (s m-1)
Nof αref Arid farmland 4 100 0.2 0.035 Broadleaf forest 5 130 0.16 0.1 Coniferous forest 4 150 0.14 0.07 Dwarf shrub 4 120 0.16 0.05 Grass 4 110 0.2 0.04 Irrigated farmland 4 100 0.18 0.035 Note: αref represents the value for full canopy cover. It varies with LAI in a similar way to that used by Gao (1995). LAImax
Table 3 Sampled ranges of multipliers for model parameters assumed to be spatially variable.
Parameter rsmin Nof
Description Minimum stomatal resistance Overland flow Manning roughness coefficient
Multiplier range 0.4–2 0.1–3
Table 4 Sampled ranges of model parameters assumed to be spatially homogeneous.
Parameter B Ksat1 Ksat2 Nc Tguz Tglz
Description Fast runoff generation scheme First soil layer saturated hydraulic conductivity Second soil layer saturated hydraulic conductivity Channel Manning roughness coefficient Upper zone transmissivity of groundwater Lower zone transmissivity of groundwater
cm day-1 cm day-1 m day-1 m day-1
Range 0.1–10 0.1–840 0.1–840 0.01–1 0.005–40 0.0008–4
(b) assuming that the parameters of B, Nc, and the conductivities and transmissivities of the soil and groundwater layers (Ksat1, Ksat2, Tguz and Tglz) can be treated as homogeneous in the basin. The parameters of type (a) vary by vegetation type across the basin (Table 2). The reference values of rsmin and Nof are assigned, and then sampled in the Monte Carlo runs by applying a randomly chosen multiplier, whose ranges are given in Table 3. Parameters of type (b) are sampled uniformly across linear ranges as shown in Table 4. Copyright 2006 IAHS Press
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Model performance and data sets for GLUE analysis To analyse the model prediction uncertainty with the GLUE methodology, LISFLOOD-WB was run with a large number of parameter set realizations generated by Monte Carlo sampling from uniform prior distributions for the parameters varied as described above. A total of 14 400 sets of parameter samplings were run. Because it takes four hours to run the model on a Pentium 4 PC for the complete 17-year period (1981–1997) of available data for a single parameter set, it is infeasible to run a large number of realizations for the long-term simulations. Related to computation limits, one of the key questions is how many runs of the model can be made. For a standard hydrological simulation, Klemeš (1986) argued that three to five years of daily data should be used for calibration, while Sorooshian & Gupta (1995) suggested that time series of at least 500–1000 data points with large hydrological variability are necessary to activate all the operational modes of a model, resulting in reliable parameter estimates. As an alternative, the period of 1995–1996 was selected for GLUE analysis in this study: 1995 was a drought year with precipitation of only 480 mm (averaged over the 10 raingauges), whereas 1996 was a wet year with average basin precipitation of 766 mm. The model was run at 1-km spatial resolution, and time steps of 1 h for energy balance calculations and 6 h for river routing. The first two months of 1995 were taken as a spin-up period to reduce the sensitivity of the model prediction to assumptions about the initial conditions in the basin. The rest was used for the GLUE analysis. By calculating the contribution to the sum of efficiency values over all the behavioural simulations, the likelihood weight of each parameter set is derived. Four simulated data sets are used for model uncertainty analysis under GLUE as follows: – Dataset 1 is based on a two-year period of daily discharge data at the Lushi outlet (1995–1996). This is for testing the model’s ability to simulate the daily discharge. – Dataset 2 is based on streamflow volume data of 92 rainfall events from 1995 to 1996 at the Lushi outlet. The event periods are chosen as starting from the date having recorded rainfall to the day just before the next day with rainfall. This is to test the ability to simulate the runoff volume for individual events in the basin. – Datasets 3 and 4 are based on only a one-year period of daily discharge at the Lushi and Lingkou outlets, respectively, for 1996. They are supposed to compare the model’s ability to simulate the daily discharge in both upstream and downstream outlets. RESULTS Parameter sensitivities for Dataset 1 For Dataset 1, all models with Nash-Sutcliffe efficiency values (equation (3)) greater than 0.5 for the two-year period were accepted as behavioural, which resulted in 55% of the total simulations being rejected as non-behavioural. The highest efficiency is about 0.66. Generally speaking, the efficiency is low compared to operational hydrological forecasting. This is expanded on in the Discussion section. Scatter plots of the likelihood measure for the eight sampled parameters are shown in Fig. 3 for Dataset 1. Note that these scatter plots are projections of the surface of the Copyright 2006 IAHS Press
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Fig. 3 Scatter plots of the behavioural likelihood measure for daily discharge prediction (Dataset 1) for the Lushi basin from 1995 to 1996 relative to parameters: (a) B; (b) Ksat1; (c) Ksat2; (d) rsmin multiplier; (e) Nof multiplier; (f) Nc; (g) Tguz; and (h) Tglz.
likelihood measure within an eight-dimensional parameter space onto a single parameter axis. It can be seen that the better fits to the observed daily discharge are found across a wide range of the parameter space, even for the most sensitive parameters. Interaction among the parameter values in producing behavioural simulations means that the individual parameter values have little meaning unless related to the other parameter values required for the model. This reinforces the concept of equifinality of model parameter sets in producing behavioural simulations. Copyright 2006 IAHS Press
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In the Lushi basin, the most important rainfall events usually happen in the summer monsoon period with high intensities, and the annual evapotranspirative demand of the atmosphere is about 960 mm—much higher than the water amount available for evapotranspiration from annual precipitation of 703 mm. It commonly occurs that soil moisture in the root zone is lower than the field capacity and the percolation rate to the groundwater is a small portion of the rainfall amount. As a consequence, the groundwater discharge is less important, whereas the surface or near-surface runoff becomes the dominant component of channel discharge during and immediately after rainfall. The groundwater transmissivities and soil-saturated conductivities are not sensitive to discharge prediction in the Lushi basin. It is shown in the scatter plots that soil saturated conductivities (Ksat1, Ksat2) and groundwater transmissivities (Tguz and Tglz) are quite flat-topped across the ranges of parameter spaces. The discharge predictions are clearly sensitive to variations in parameter B. It is shown in Fig. 3 that there is a peak of the likelihood measure around 5.5 for B. The minimum stomatal resistance, rsmin, is the key parameter for canopy transpiration. The variation of this parameter will affect the soil moisture depletion through root extraction. Generally, as the stomatal resistance increases, the mean soil moisture content will increase and the infiltration amount will increase, resulting in a reduction of canopy transpiration. As a consequence, the discharge will increase as this resistance increases. It may be seen in the scatter plot that the minimum stomatal resistance is visually sensitive to the discharge likelihood, having a trend of likelihood increasing towards the upper limit of the range of values sampled. The cause of the trend may be due to the specified range of values of rsmin being below the “optimum” effective value, or the LAI (which is inversely proportional to rsmin in the calculation of canopy resistance) being underestimated from the NDVI images in this environment requiring direct compensation by an effective value of rsmin. Because of the lack of detailed prior information on Manning roughness coefficients for the basin channels, the same value is used for all the channels in the basin; therefore, it must be considered as an effective value. It is shown from the scatter plot that the channel Manning roughness coefficient, Nc, is also one of the more critical parameters to the model performance. The highest likelihood values obtained are around 0.5. As the Manning coefficient can affect the peak time and smooth the amplitude of the hydrograph by diminishing the stream velocity, in a basin of this size it significantly influences the apparent agreement between the simulated and observed daily discharge, if only by causing a peak to appear in one day or another. The reference pattern of values for the overland Manning roughness coefficient, Nof, has been based on the specified land cover types. It is found that the overland Manning roughness coefficient multiplier is not very sensitive to the likelihood measure within the range 0.1–3, with only a slight increase in the upper part of the sampled parameter range. Although there exists compensating interaction between the overland roughness and channel roughness, the most important reason is that the flow distance from a grid to the nearby channel only covers, at most, several grid lengths, which is much shorter than the duration of streamflow to the basin outlet. Consequently, at this scale of basin, the predicted timing of the storm hydrograph is expected to be more sensitive to routing in the channel (e.g. Beven & Wood, 1993).
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Fig. 4 Scatter plots of the behavioural likelihood measure for cumulative event discharge prediction (Dataset 2) for the Lushi basin from 1995 to 1996 relative to parameters (a) B; (b) Ksat1; (c) Ksat2; (d) rsmin multiplier; (e) Nof multiplier; (f) Nc; (g) Tguz; and (h) Tglz.
Parameter sensitivities for Dataset 2 To test that the model can simulate the runoff volumes for individual events in the basin, the scatter plots of likelihood based on individual hydrograph periods are analysed. As in the case of Dataset 1, there are only three parameters, i.e. B, rsmin and Nc, showing sensitivity to the output discharge under this condition (Fig. 4). It can be seen that parameter B is still the most sensitive parameter to model performance. The B scatter plot had a rainbow-like peaked band of dots, which means that the best model performances Copyright 2006 IAHS Press
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(efficiencies of the order of 0.6–0.8 over the 92 event volumes) are available for parameter sets having B values from 4 to 7. The scatter plots show that a higher efficiency is again obtained as rsmin increases in the specified range, as for the discharge time series data. The efficiency slightly increases as Nc increases. However, it is much less sensitive in predicting the event discharge than for the daily discharge in Dataset 1, since it can have a greater effect on the timing of the runoff than on the cumulative volume. Parameter sensitivities for Dataset 3 The model efficiency increases from 0.66 for Dataset 1, to 0.78 for Dataset 3. As Dataset 1 includes the dry year 1995, it is not unusual that model performance is lower because of the characteristics of the Nash-Sutcliffe efficiency, which favours fitting discharge peaks in periods of high flow variability. A more detailed analysis of this is given in the Discussion section. By comparing the scatter plot of efficiencies at Lushi outlet in one year of data (1996—Dataset 3; Fig 5(a) and (b)) with that for two years of data (1995–1996— Dataset 1; Fig. 3(a) and (f)), it is found that the peak likelihood weights of parameter B are shifted from 5 to 6 and that of Manning roughness coefficient Nc from 0.5 to 0.6. It is also found that the previous tendency of parameter rsmin for higher efficiency at higher values is not obvious for Dataset 3 (not shown).
Fig. 5 Scatter plots of likelihood for daily discharge in 1996 (Datasets 3 and 4) relative to parameters (a) B at Lushi; (b) Nc at Lushi; (c) B at Lingkou; and (d) Nc at Lingkou.
Parameter sensitivities for Dataset 4 Comparing the scatter plots based on time series of discharge in 1996 measured at the Lingkou site in Dataset 4 and the Lushi site in Dataset 3, it is found that only parameters B and channel Manning roughness Nc are sensitive to model performance at Copyright 2006 IAHS Press
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Lingkou (Fig. 5(c) and (d)), similar to the Lushi site. The Pearson correlation coefficient (r2) of the Nash-Sutcliffe coefficients between the Lushi and Lingkou time series is 0.92. This explains why the two scatter plots are similar to each other, identifying that the model performances are highly correlated in the two sites. The distinct difference between scatter plots of the two data sets is that the maximum efficiency of model performance is lower, and the number of behavioural parameter sets is fewer for Lingkou (Fig. 5(c) and (d)) than for the Lushi basin (Fig. 5(a) and (b)) with the same rejection criteria. It is also shown that the value of parameter B at which the highest efficiency is derived is shifted from 7 (Fig. 5(a)) to 5.8 (Fig. 5(c)). Detailed reasons for the lower model efficiency in the upstream sub-basin compared to the downstream basin are given in the Discussion section. Sensitivity to individual parameters Here the GSA technique has been extended by considering a subdivision of the behavioural parameter values (likelihood value > 0.5 for both years) into five sets based on Dataset 1. The five sets represent equal numbers of behavioural simulations (1034 in each set), with set 1 representing the lowest likelihood values above the rejection threshold, and set 5 the highest. In this way, one can identify parameter sensitivity in more detail over an a priori range or area of the parameter space. The results shown in Fig. 6 reinforce the conclusion that parameters B, Nc and rsmin are the most sensitive parameters. For B and Nc, the frequency density distributions of the five parameter sets are quite different and the highest likelihood is available along the parameter range of 4–6 for B and 0.5–0.7 for Nc, which is narrower than the other four sets, showing that B and Nc are very sensitive parameters. It is noted that the fifth set of rsmin markedly deviates from the other four sets, showing that the high efficiency is located in the high part of the parameter range. The parameters Nof and Ksat1 are less sensitive. The distributions of the other three parameters show little change as likelihood increases, which can be considered as insensitivity to the model performance. For the sensitive parameters, the good model fits still span a wide range of the parameter space. Prediction bounds After the non-behavioural models have been rejected and assigned a weighting of zero, the likelihood measures for the behavioural models are rescaled to a cumulative sum of unity. Rescaled likelihood weights are then applied to their respective predicted discharges at each time step (equation (4)), from which the selected discharge quantiles of upper and lower bounds can be calculated to represent the model prediction uncertainty bounds. Figure 7 shows the uncertainty bounds of 5% and 95% prediction quantiles for discharge, evapotranspiration and recharge based on Dataset 3. The greatest uncertainty of discharge is for the peak discharge of storm events. In some events, the measured discharge falls outside the uncertainty bounds, suggesting that errors of model structure and input data cannot in this case be completely overcome by the calibration process. The GLUE methodology does not allow such errors to be compensated for by use of an additive statistical error, such that model deficiencies are revealed more explicitly Copyright 2006 IAHS Press
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Fig. 6 Cumulative distributions of behavioural parameter values for daily discharge prediction for Lushi in 1995–1996 (Dataset 1) for parameters (a) B; (b) Ksat1; (c) Ksat2; (d) rsmin multiplier; (e) Nof multiplier; (f) Nc; (g) Tguz; and (h) Tglz. Sets 1 to 5 represent equal numbers of behavioural simulations, ranked by likelihood, where Set 1 has the lowest likelihood values and Set 5 the highest.
(Beven, 2005). In a large basin, water cycle processes are complex and the effect of human activities on them are difficult to assess and predict. More analysis of model errors is presented in the Discussion section. The uncertainty of evapotranspiration, integrated over the whole basin cannot be compared directly with observations. The distance between the upper and lower bounds is usually less than 1 mm day-1. Uncertainty of recharge is much less variable in time and the lower bound is close to zero. An increasing trend is apparent in summer when the rainy season begins. The variation of the upper bound of recharge is in the range of 0.06–0.12 mm day-1. Copyright 2006 IAHS Press
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Fig. 7 Uncertainty bounds (Bnd) of 95% and 5% quantiles for: (a) discharge; (b) evapotranspiration; and (c) recharge at Lushi in 1996 (Dataset 3).
For better comparison of the difference between the upper bound and lower bound for discharge and evapotranspiration (and recharge), which have different dimensions, this difference is normalized for each variable as follows:
∆x ′ =
∆x − ∆x σ ∆x
(5)
where ∆x is the difference between the upper bound (UB) and lower bound (LB) for discharge, evapotranspiration, or recharge; ∆x is the mean of ∆x; and σ∆x is the standard deviation of ∆x. The normalized difference between UB and LB, ∆x′, can be negative, positive, or zero. By comparing the normalized difference between UB and LB of discharge, evapotranspiration and recharge (Fig. 8), one can see that the differences for both evapotranspiration and discharge are broad in summer and narrow in winter, while the opposite is found for recharge. The variation of the uncertainty of evapotranspiration is opposite to the variation of the uncertainty of discharge. For 95% prediction bounds, the peak of discharge always corresponds to the bottom of evaporation and vice versa, indicating that during high-flow periods evapotranspiration is less sensitive than discharge and during the dry period evapotranspiration is more sensitive than discharge. In winter, while uncertainty bounds of discharge and Copyright 2006 IAHS Press
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Fig. 8 Normalized uncertainty bounds for discharge, evapotranspiration and recharge at Lushi in 1996 (Dataset 3).
evapotranspiration are low, the recharge becomes most sensitive. The evapotranspiration rate is dominated by both atmospheric demand and soil moisture storage in the root layer. The range of predicted values of evapotranspiration is much less than the total rainfall amount in the basin, suggesting that the prediction of root zone moisture is less affected by the different parameter sets, which results in narrower uncertainty bounds for evapotranspiration than for discharge in the rainy season. DISCUSSION The results of the GLUE analysis depend on the choice of likelihood measure used to evaluate the sample of models and the choice of acceptability criteria. It has been argued by previous authors that the utility of the Nash-Sutcliffe efficiency as a performance measure may be limited by bias in its evaluation (Garrick et al., 1978; Ma et al., 1998; Weglarczyk, 1998; Sauquet & Leblois, 2001), especially because σi2 is not 2 2 necessarily smaller than σ obs , when σ obs is small. When behavioural models under the GLUE framework were selected with Nash-Sutcliffe efficiency >0.5, for example, those models having a good match between observed and modelled high flow (with much higher observation variance) were more likely to be chosen; while those having a good match between observed and modelled low flow (even under the same model variance but with lower observation variance) had less chances to be chosen (with lower Nash-Sutcliffe efficiency). However, it has been suggested that the NashSutcliffe efficiency is an adequate measure of performance when the coefficient of variation for the observed data set is large (Ye et al., 1998). In the present study, the coefficient of variation for the observed data set is as high as 3.04 for the Lushi basin and 2.31 for the nested Lingkou sub-basin. The characteristics of the hydrographs in the study basin follow the monsoon pattern, with peaks occurring in summer and low flow in other seasons. In this case, it is found from the flow duration curve (Fig. 9) that, for the Lingkou sub-basin 3.5% of the total flow points belong to high flow, 8.5% to medium flow and 88% to low flow. For the Lushi basin, the corresponding values are 4.3, 10.4 and 85%. Therefore, even though the Nash-Sutcliffe efficiency favours a good match between observed and Copyright 2006 IAHS Press
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Daily discharge ( m 3 s -1)
Lushi 100
10
Lingkou
1 0.1
1 10 Exceedence probability
100
Fig. 9 Streamflow duration curve in 1996 at the Lushi and Lingkou sites, showing the higher discharges at low flows at Linkou.
modelled high flows, it still reflects the model performance for low flows, which make up a large number of data points in this case. In general, the Nash-Sutcliffe efficiency does not achieve very high values in this study. It is about 0.66 for the full two-year data series and 0.59 and 0.78 for the 1995 and 1996 discharge series, respectively. Nash-Sutcliffe efficiency values greater than 0.5 do not imply that the model simulations match observed flows well. According to the concept of GLUE, models that do not fall within the prior limits of acceptability should be rejected (see Beven, 2005). This allows the possibility of many feasible models satisfying the limits of acceptability and being accepted as behavioural. However, it also allows the possibility that none of the models will satisfy the limits of acceptability. This was the case for the distributed SHE model study in Parkin et al. (1996); for the application of TOPMODEL in Freer et al. (2002); and for an algal dynamics study in Van Straten & Keesman (1991). For the present study, there is also no model performance with Nash-Sutcliffe values greater than 0.8, which in China is considered as the minimum requirement for satisfying operational hydrological forecasting standards. Therefore, the limits of acceptability had to be lowered to 0.5 to obtain behavioural realizations of the model. This indicates that there are some difficulties for the present model structure and process description, to predict the daily discharge time series using the derived spatial input data series. This is, in part, a result of the use of limited spatial rainfall input and daily observed discharge data. In 1996, the coefficient of variation of areal precipitation is 0.08 for the Lingkou sub-basin over four raingauges and 0.13 for the Lushi basin over 10 raingauges. Variability in the daily values can be much higher. This is one of the sources of error causing the low model performance. The efficiency measure can also be affected by peak timing errors using a daily time step, especially in the smaller sub-basin, if the observed peak is in one day and the predicted peak either in the preceding or following day, depending on the actual space/time distribution of the rainfall. There is also an effect due to error in the observed discharges. By checking the original data records, the relative error of discharge rating curve is 4.4–5.1% (equivalent to the order of 25 m3 s-1 for the highest peak in 1996). In addition, by checking the observed discharge data more carefully, it is shown that, since 1980, the discharge data at the Lushi outlet from October to April are all smaller than those at the Lingkou outlet further upstream. This implies that some water volume Copyright 2006 IAHS Press
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is being lost. This is thought to be primarily caused by a diversion from the channel at Luobei, only 2 km upstream from the Lushi gauge. Fortunately, while this may have had a small effect on the model efficiencies, it occurred only in low-flow periods (Fig. 9). CONCLUSIONS Within the GLUE methodology, the equifinality and uncertainty of the distributed LISFLOOD-WB model have been analysed, based on the application to four data sets including daily discharge and event flow volume data at Lushi and Lingkou from 1995 to 1996. In such a distributed model, the parameters should be related to the characteristics of each of the 4423 grid elements in the basin. For example, the vegetation characteristic parameters, such as leaf stomatal resistance and aerodynamic roughness length, etc., are expected to be temporally dependent on the vegetation cover types, while the effective soil parameters are determined by soil physics and texture as well as local topography, vegetation and other factors. The spatial patterns of the effective parameter values will thus be very complex. Therefore, it is impracticable to sample the entire individual grid element or hydrological functional unit parameters within a Monte Carlo simulation strategy. Instead, only the key parameters are chosen for analysis and the basin-scale multipliers are used for some parameters to maintain reference patterns of soil and vegetation parameters. In applying the modified LISFLOOD-WB model to these data sets, obvious equifinality has been shown with a large number of parameter sets that result in behavioural models. Of all the eight key parameters, the highest sensitivity is seen for the fast runoff generation parameter B. Limited apparent sensitivity is observed for channel Manning coefficient Nc and minimum stomatal resistance rsmin. Apparently, the model has little or no sensitivity for the remainder of the parameters, for which good simulations are found across the full range of parameter values sampled, depending on the values of the other parameters in the set giving a behavioural simulation. The results of both the scatter plots and GSA indicate that the highest likelihood is available along the parameter range of 2.3–6.8 for B and 0.3–0.95 for Nc. The striking feature of the results is that the effects of parameter interactions within the model structure mean that the behavioural simulations are widely spread across the chosen ranges of most of the parameters. This clearly illustrates that it is the whole parameter set that is important in achieving a behavioural simulation. This is unlikely to be seen with single optimized model runs or with a limited sensitivity analysis around some optimal parameter set. Some interesting results are obtained based on the comparison between the upstream and downstream data, between time series and flow volume data, and between data in different years. Maximum efficiency of model performance is shown to be lower, and the number of behavioural parameter sets fewer at the smaller, upstream Lingkou sub-basin than those at the larger, downstream Lushi basin with the same criteria (Nash-Sutcliffe efficiency ≥ 0.5). It is also shown that the value of parameter B at which the highest efficiency is derived is shifted from a high value of 7 for Lushi to a low value of 5.8 for Lingkou. This is an example of the consistency between the GLUE uncertainty results and the general calibration experience in China Copyright 2006 IAHS Press
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that the larger the basin, the larger the value of B. The Nc scatter plots show much less sensitivity to the event discharge than to the daily discharge, corroborating the fact that Nc can have a greater effect on the timing of the runoff than on the cumulative volume. The daily data used also give rise to potentially lower efficiencies than event volume data, due to timing errors in the simulation associated with the actual timing of rainfalls in a particular day. The relatively low maximum values of efficiency found, particularly for the drought year of 1995, are not unusual in this type of environment where model performance can be constrained by the limitations of the input data as well as model structural errors. The uncertainty of the discharge and evapotranspiration predictions is broad in the peaks and narrow in the recessions; that of recharge is the opposite. Acknowledgements The first author was supported by a Royal Society Research Fellowship. Thanks to National Natural Sciences Foundation of China (90211007) and Innovation Knowledge projects of Chinese Academy of Sciences (KZCX2310 and SZV37800) for financially aiding the work. The LISFLOOD model is developed by the European Commission’s Joint Research Centre. The GLUE methodology is being developed under a UK Natural Environment Research Council Grant (no. NER/L/S/2001/00658). The authors also wish to thank the two anonymous reviewers and the Editor for pertinent points raised. REFERENCES Beldring, S. (2002) Multi-criteria validation of a precipitation-runoff model. J. Hydrol. 157, 189–211. Beven, K. J. (2000) Uniqueness of place and process representations in hydrological modelling. Hydrol. Earth System Sci. 4(2), 203–213. Beven, K. J. (2001) Dalton Medal Lecture: How far can we go in distributed hydrological modelling? Hydrol. Earth System Sci. 5(1), 1–12. Beven, K. J. (2002) Towards an alternative blueprint for a physically-based digitally simulated hydrologic response modelling system. Hydrol. Processes 16(2), 189–206. Beven, K. J. (2005) A Manifesto for the equifinality thesis. J. Hydrol. (in press). Beven, K. J. & Binley, A. (1992) The future of distributed models: model calibration and uncertainty prediction. Hydrol. Processes 6, 279–298. Beven, K. & Freer, J. (2001) Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. J. Hydrol. 249, 11–29. Beven, K. J. & Wood, E. F. (1993) Flow routing and the hydrological response of channel networks. In: Channel Networks (ed. by K. J. Beven & M. J. Kirkby), 99–128. Wiley, Chichester, UK. Blazkova, S., Beven, K., Tacheci, P. & Kulasova, A. (2002) Testing the distributed water table predictions of TOPMODEL (allowing for uncertainty in model calibration): the death of TOPMODEL? Water Resour. Res. 38(11), 1257–1257. Cameron, D., Beven, K. & Naden, P. (2000) Flood frequency estimation under climate change (with uncertainty). Hydrol. Earth System Sci. 4(3), 393–405. Chow, V. T., Maidment, D. R. & Mays, L. W. (1988) Applied Hydrology. McGraw-Hill, New York, USA. Christiaens, K. & Feyen, J. (2001) Analysis of uncertainties associated with different methods to determine soil hydraulic properties and their propagation in the distributed hydrological MIKE SHE model. J. Hydrol. 246(1/4), 63–81. De Roo, A. P. J., Wesseling, C. G. & Van Deursen, W. P. A. (2000) Physically based river basin modelling within a GIS: the LISFLOOD model. Hydrol. Processes 14, 1981–1992. De Roo, A. P. J. & 20 others (2003) Development of a European flood forecasting system. Int. J. River Basin Manage. 1(1), 49–59. Famiglietti, J. S. & Wood, E. F. (1994) Multiscale modelling of spatially variable water and energy balance processes. Water Resour. Res. 30, 3061–3078. Franks, S. W. & Beven, K. J. (1999) Conditioning a multiple patch SVAT model using uncertain time-space estimates of latent heat fluxes as inferred from remotely-sensed data. Water Resour. Res. 35(9), 2751–2761. Freer, J. & Beven, K. J. (1996) Bayesian estimation of uncertainty in runoff prediction and the value of data: An application of the GLUE approach. Water Resour. Res. 32(7), 2161–2173.
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(1980) A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898. van Straten, G. & Keesman, K. J. (1991) Uncertainty propagation and speculation in projective forecasts of environmental change—a lake-eutrophication example. J. Forecasting 10(1/2), 163–190. Wang Q. & Takahashi, H. (1999) A land surface water deficit model for an arid and semiarid region: impact of desertification on the water deficit status in the Loess Plateau, China. J. Climate 12, 244–257. Weglarczyk, S. (1998) The interdependence and applicability of some statistical quality measures for hydrological models. J. Hydrol. 206, 98–103. Wigmosta, M. S., Vail, L. W. & Lettenmaier, D. P. (1994) A distributed hydrology—vegetation model for complex terrain. Water Resour. Res. 30, 1665–1679. Wu, C. (1990) The 1:1 000 000 Land Use Map of China. The Science Press, Beijing, China. Ye, W., Jakeman, A. J & Young, P. C. (1998) Identification of improved rainfall–runoff models for an ephemeral lowingyield Australian catchment. Environ. Model. Software 13, 59–74. Yu, P. S., Yang, T.-C. & Chen, S.-J. (2001) Comparison of uncertainty analysis methods for a distributed rainfall–runoff model. J. Hydrol. 244(1/2), 43–59. Zhao, R. J. (1992) The Xinanjiang model applied in China. J. Hydrol. 135, 371–381. Received 9 March 2004; accepted 17 October 2005
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