Semi-distributed parameter optimization and …faculty.sites.uci.edu/jasper/files/2016/04/36.pdfLUC FEYEN1, MILAN KALAS1,2 & JASPER A. VRUGT3 1 European Commission – DG Joint Research

Hydrological Sciences–Journal–des Sciences Hydrologiques, 53(2) April 2008

Open for discussion until 1 October 2008 Copyright © 2008 IAHS Press

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Semi-distributed parameter optimization and uncertainty assessment for large-scale streamflow simulation using global optimization LUC FEYEN1, MILAN KALAS1,2 & JASPER A. VRUGT3

1 European Commission – DG Joint Research Centre, Institute for Environment and Sustainability, Italy [email protected]

2 Department of Land and Water Resources Management, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia

3 Center for Nonlinear Studies (CNLS), Los Alamos National Laboratory, Los Alamos, New Mexico 87545,USA Abstract In catchments characterized by spatially varying hydrological processes and responses, the optimal parameter values or regions of attraction in parameter space may differ with location-specific characteristics and dominating processes. This paper evaluates the value of semi-distributed calibration parameters for large-scale streamflow simulation using the spatially distributed LISFLOOD model. We employ the Shuffled Complex Evolution Metropolis (SCEM-UA) global optimization algorithm to infer the calibration parameters using daily discharge observations. The resulting posterior parameter distribution reflects the uncertainty about the model parameters and forms the basis for making probabilistic flow predictions. We assess the value of semi-distributing the calibration parameters by comparing three different calibration strategies. In the first calibration strategy uniform values over the entire area of interest are adopted for the unknown parameters, which are calibrated against discharge observations at the downstream outlet of the catchment. In the second calibration strategy the parameters are also uniformly distributed, but they are calibrated against observed discharges at the catchment outlet and at internal stations. In the third strategy a semi-distributed approach is adopted. Starting from upstream, parameters in each subcatchment are calibrated against the observed discharges at the outlet of the subcatchment. In order not to propagate upstream errors in the calibration process, observed discharges at upstream catchment outlets are used as inflow when calibrating downstream subcatchments. As an illustrative example, we demonstrate the methodology for a part of the Morava catchment, covering an area of approximately 10 000 km2. The calibration results reveal that the additional value of the internal discharge stations is limited when applying a lumped parameter approach. Moving from a lumped to a semi-distributed parameter approach: (i) improves the accuracy of the flow predictions, especially in the upstream subcatchments; and (ii) results in a more correct representation of flow prediction uncertainty. The results show the clear need to distribute the calibration parameters, especially in large catchments characterized by spatially varying hydrological processes and responses. Key words distributed modelling; parameter identification; streamflow simulation; uncertainty

Optimisation de paramètres semi-distribués et évaluation de l’incertitude pour la simulation de débits à grande échelle par l’utilisation d’une optimisation globale Résumé Dans les bassins caractérisés par des processus et des réponses hydrologiques variables dans l’espace, les valeurs optimales des paramètres ou régions d’attraction dans l’espace des paramètres peuvent différer en fonction de caractéristiques et de processus variables dans l’espace. Cet article évalue l’utilité des paramètres de calage semi-distribués pour la simulation de débit à grande échelle à partir du modèle spatialement distribué LISFLOOD. Nous utilisons l’algorithme d’optimisation globale connu sous le nom de Shuffled Complex Evolution Metropolis (SCEM-UA) pour inférer les paramètres de calage sur la base d’observations de débits journaliers. La distribution de paramètres postérieurs qui en résulte reflète l’incertitude sur les paramètres du modèle et fournit une base pour effectuer des prévisions probabilistes de débit. Nous évaluons la pertinence des paramètres de calage semi-distribués en comparant trois stratégies de calage. Avec la première, des valeurs uniformes sur l’entièreté de la zone d’intérêt sont adoptées pour les paramètres inconnus, lesquels sont calés par rapport aux débits observés à l’exutoire du bassin. Avec la deuxième stratégie de calage, les paramètres sont également distribués uniformément mais ils sont calés par rapport aux débits observés à l’intérieur du bassin ainsi qu’à l’exutoire du bassin. Dans la troisième stratégie de calage, une approche semi-distribuée est adoptée. En partant de l’amont, les paramètres de chaque sous-bassin sont calés à partir des débits observés aux exutoires de ces sous-bassins. Pour ne pas propager les erreurs d’amont dans le processus de calage, les débits observés aux exutoires des bassins amonts sont utilisés comme débits d’entrée lors du calage des sous-bassins aval. A titre d’exemple, nous présentons la méthodologie sur une partie du bassin de la Morava couvrant approximativement 10 000 km2. Les résultats du calage révèlent que la valeur ajoutée des stations intérieures dans le cas de la deuxième stratégie est limitée. Evoluer d’une stratégie globale à semi-distribuée: (i) améliore la précision des prévisions, en particulier dans les sous-bassins amont; et (ii) conduit à une réprésentation plus correcte de l’incertitude de prévision du débit. Les résultats montrent clairement la nécessité de distribuer les paramètres de calage, surtout pour les grands bassins caractérisés par des processus et des réponses hydrologiques variables dans l’espace. Mots clefs modélisation distribuée; identification de paramètres; simulation de débit de rivière; incertitude

Luc Feyen et al.

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INTRODUCTION

Spatially distributed hydrological models are increasingly applied to address a range of environ-mental and water resources problems. These types of models have the potential to evaluate the spatially varying effects of land use or climate changes on catchment behaviour, and to meet the request for more accurate predictions of streamflow at any point along the river system. To date, the capabilities of such models have not yet been fully exploited, primarily due to their computat-ional, distributed input and parameter estimation requirements. However, the rapid increase in computer power, the development of efficient computational methods, the availability of sophisticated Geographical Information Systems, and the increasing wealth of spatial data of different types facilitate the proper calibration of these models, and consequently their use in making reliable predictions. LISFLOOD (De Roo et al., 2000, 2001) is a spatially distributed, mixed conceptual-physically based hydrological model developed for flood forecasting and impact assessment studies at the European scale. The model simulates the spatial and temporal patterns of catchment responses in large river basins as a function of spatial information on topography, soils and land cover. The accuracy of the predictions depends on the ability of the model to capture the dominating hydrological processes that transfer precipitation into river runoff at the catchment scale, and on its ability to reproduce historical time series of observed river discharges. Owing to the general nature of LISFLOOD, its application to any given river basin requires that certain parameters of lumped conceptual functions be identified for the particular basin. Hence, a crucial step that contributes significantly to the accuracy of LISFLOOD predictions is the calibration of the model for all European catchments. The application of automatic parameter estimation techniques has received considerable attention over the last decades (e.g. Sorooshian & Dracup, 1980; Kuczera, 1983; Duan et al., 1992; Thyer et al., 1999; Vrugt et al., 2003). Several studies have reported difficulties in obtaining unique global parameter estimates because of the presence of multiple local optima, nonlinear interaction between model parameters, and the shape and roughness of the response surface defined by the selected objective function (Bates & Campbell, 2001). This has led to the develop-ment of global optimization techniques such as the Shuffled Complex Evolution (Duan et al., 1992) and simulated annealing (Sumner et al., 1997). Often, the optimal parameter estimates are used without taking due account of the uncertainty in the parameter estimates in subsequent predictions. However, without a realistic assessment of parameter uncertainty it is not possible to undertake with confidence such tasks as evaluating prediction/confidence limits on future hydrological responses, assessing the significance of deviations in split-sample tests, and assessing the value of regional relationships between model parameters and catchment characteristics (Kuczera & Parent, 1998). Several approaches have been developed to assess parameter uncertainty and its effect on subsequent predictions. These include the use of multi-normal approximations (Kuczera & Mroczkowski, 1998), simple uniform random sampling (URS) over the feasible parameter space (Uhlenbrook et al., 1999), parametric bootstrapping and Markov Chain Monte Carlo (MCMC) methods (Kuczera & Parent, 1998; Campbell et al., 1999; Bates & Campbell, 2001; Vrugt et al., 2003). Traditional statistical theory based on first-order approximations and multinormal distributions is typically unable to cope with the nonlinearity of complex hydrological models. The URS method allows easy exploration of the parameter space but is computationally inefficient. Unless a large number of random samples are drawn from the multidimensional parameter space, importance sampling can produce seriously misleading results. MCMC methods on the other hand generate samples from a Markov Chain that adapts to the stationary posterior parameter distribution, and can be applied to complex inference, search and optimization problems. The posterior parameter distribution quantifies the uncertainty about the model parameters after considering the observed catchment responses, and forms the basis for making probabilistic predictions.

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Other methods have been developed that focus on assessing global uncertainty in rainfall–runoff modelling, such as the generalised likelihood uncertainty (GLUE) method (Beven & Binley, 1992) and the meta-Gaussian approach (Krzysztofowicz & Kelly, 2000; Montanari & Brath, 2004). These methods estimate the aggregated model uncertainty without attempting to separate the individual effects of input, parameter and model uncertainty. We specifically focus on estimating confidence limits for the conceptual model parameters to quantify the effect of parameter uncertainty on river flow forecasting. In this work, we employ the Shuffled Complex Evolution Metropolis (SCEM-UA) algorithm (Vrugt et al., 2003) to infer the posterior parameter distributions for the LISFLOOD model. The SCEM-UA algorithm is a modified version of the original SCE-UA global optimization algorithm (Duan et al., 1992). The algorithm is Bayesian in nature and operates by merging the strengths of the Metropolis algorithm, controlled random search, competitive evolution, and complex shuffling to continuously update the proposal distribution and evolve the sampler to the posterior target distribution. The SCEM-UA algorithm has been successfully applied to calibrate the five-parameter conceptual rainfall–runoff model HYMOD for a 1944 km2 watershed (Vrugt et al., 2003) and to estimate vadose zone properties for a small watershed using the distributed fully coupled surface–vadose zone–groundwater model MODHMS (Vrugt et al., 2004). Feyen et al. (2007) used SCEM-UA to calibrate the LISFLOOD model for the Meuse catchment, assuming uniform parameters for the large catchment of area approximately 22 000 km2. In catchments characterized by spatially varying hydrological processes and responses, the optimal parameter values or regions of attraction in parameter space may differ with location-specific characteristics and dominating processes. Relatively few studies have investigated the value of the level of spatial detail of calibration parameters for large-scale streamflow simulation. Andersen et al. (2001) used different spatial levels of manual calibration and found that calibration against one station revealed significant shortcomings for some of the upstream tributaries. Further calibration against additional discharge stations improved the performance in several sub-catchments. Boyle et al. (2001) used an automated multi-criteria calibration framework to evaluate the benefit of spatially distributing the model input (precipitation), structural components (soil moisture and streamflow routing computations), and surface characteristics (parameters). Improvements in model performance were related to the spatial distribution of the model input and streamflow routing, whereas no improvement was associated with the distribution of surface characteristics (model parameters). Ajami et al. (2004) compared lumped, semi-lumped and semi-distributed versions of the SAC-SMA model to evaluate the effect of the spatial detail of forcing data (precipitation) and model parameters. They also observed improvements when increasing the resolution of the forcing data from the entire to sub-basins. Varying the parameters between sub-basins did not further improve the simulation results, neither at the outlet nor at an interior test point. The above studies have shown the importance of spatially distributing the forcing data, but showed little additional value of increasing the level of spatial detail of the model parameters. Because these studies all used deterministic calibration strategies, the relation between the level of spatial detail of the parameters and parameter uncertainty, and its consequent effect on the predictive capabilities of the model, was not considered. In the LISFLOOD model, the forcing data (e.g. precipitation, potential evaporation) and information on topography, soil properties, land cover and channel geometry, are fully distributed, i.e. variable at the model grid scale. Hence, we only focus on the spatial variability of the lumped conceptual parameters of the model, which lack the physical meaning to be derived from existing spatial data sets. The aim is to evaluate the value of semi-distributing the calibration parameters for: (a) a better spatial representation of the calibration parameters and their uncertainty, and (b) improved streamflow simulation at the catchment outlet and interior points along the river system, in terms of accuracy and spread of the prediction uncertainty. In the approach followed, the resolution of the discharge stations defines the spatial detail of the parameters.

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THE HYDROLOGICAL MODEL LISFLOOD

In this study we employ LISFLOOD (De Roo et al., 2000; 2001), a rainfall–runoff model that has been devised to predict floods and to assess the impact of land use and climate changes on river streamflow in large European river basins. To meet this objective the model: (a) is distributed, (b) has a physical basis, and (c) uses readily available inputs. The model is raster based and is embedded in a dynamic modelling GIS-environment (PCRaster), which facilitates the handling of large European spatial data sets such as the Corine Land Cover and the European Soils Database. A conceptual scheme of the model with its different components is presented in Fig. 1. The model is driven by meteorological input time series such as precipitation, temperature, wind speed, sunshine duration, cloud cover and actual vapour pressure. It is based on simplified storage concepts to represent the soil and groundwater zone. Processes simulated for each grid cell include snowmelt, soil freezing, interception of rainfall by vegetation, evaporation from the soil surface, infiltration into the soil, water uptake and transpiration by plants, surface runoff, redistribution of soil moisture within the soil profile, drainage of water to the groundwater system, preferential flow, groundwater flow and 1-D channel routing.

INFactESact

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Fig. 1 Schematic overview of the LISFLOOD model. P = precipitation; Int = interception; EWint = evaporation of intercepted water; Dint = leaf drainage; ESact = evaporation from soil surface; Tact = transpiration (water uptake by plant roots); INFact = infiltration; Qsr = surface runoff; Dus,ls = drainage from upper to lower soil zone; Dls,ugw = drainage from lower soil zone to upper groundwater zone; Dpref,gw = preferential flow to upper groundwater zone; Dugw,lgw = drainage from upper to lower groundwater zone; Qugw = outflow from upper groundwater zone; Qlgw = outflow from lower groundwater zone; Qloss = loss from lower groundwater zone. All variables in mm/d. Note that snowmelt is not included in the figure, even though it is simulated by the model.



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MODEL CALIBRATION WITH SHUFFLED COMPLEX EVOLUTION METROPOLIS (SCEM-UA) ALGORITHM

LISFLOOD can be cast as a nonlinear regression model according to:

ttt qq ε+= ),(L, θξ (1)

where qt is the discharge at time t, qt,L is the LISFLOOD predicted discharge for time t, θΘ nℜ⊂∈θ is the vector of unknown model parameters, ξΞ nℜ⊂∈ξ comprises the forcing

model inputs such as precipitation or ET0 rates, and εt is the modelling residual. In Bayesian inference, the unknown parameters θ are treated as random variables distributed according to a probability density function (pdf), which describes the uncertainty about the parameters. Prior to considering the observed discharges, information about the parameters is summarized by the prior pdf, denoted by p(θ). The posterior pdf p(θ|q) captures the beliefs about the parameters given the observed discharge measurements T

1 ),...,(qnqq=q . This distribution is

proportional to the product of the likelihood function and the prior pdf. Assuming that the residuals are mutually independent, identically and normally distributed, the likelihood function is given by (Box & Tiao, 1973):

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡= ∑

=

−q

qn

t

tnL1

222 exp)2(),(

εεε σ

υπσσ qθ (2)

where: )()( L, ttt qGqG −=υ (3)

is a transformation that allows us to handle non-normal, heteroscedastic and auto-correlated errors. Assuming a non-informative prior of the form 1),( −∝ εε σσθp for the residual error variance, the influence of σε can be integrated out, resulting in the following posterior density of θ:

qq

nn

ttυCp

21

1

21)(−

=

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡= ∑qθ (4)

where θd21

1

2∫ ∑

−

= ⎥⎥⎦

⎤

⎢⎢⎣

⎡=

qq

nn

ttυC is the normalising constant.

The posterior pdf is typically highly dimensional and complex, with strong nonlinear parameter interdependences. Hence, it is difficult, if not impossible, to summarize this distribution by direct calculation and it is necessary to resort to Monte Carlo methods to approximate the distribution. The SCEM-UA algorithm uses the Metropolis-Hastings (Metropolis et al., 1953) search strategy to generate a sequence of parameter sets {θ1, θ2,…, θn} that adapts to the target posterior distribution. It starts with an initial population of points (parameter sets) randomly distributed throughout the feasible parameter space defined by the prior parameter distributions. For each parameter set, the posterior density is computed using the Bayesian inference scheme presented in equations (2)–(4). The population is partitioned into q complexes, and in each complex k (k = 1, 2, …, q) a parallel sequence is launched from the point that exhibits the highest posterior density. A new candidate point in each sequence k is generated using a multivariate normal distribution either centred around the current draw of the sequence k, or the mean of the points in complex k, augmented with the covariance structure induced between the points in complex k. The Metropolis-annealing criterion is used to test whether the candidate point should be added to the current sequence. Subsequently the new candidate point randomly replaces an existing member of the complex. Finally, after a certain number of iterations new complexes are

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formed through a process of shuffling. This series of operations results in a robust MCMC sampler that conducts a robust and efficient search of the parameter space. Due to the computational demands of the LISFLOOD model and the large number of iterations typically needed to obtain a stable posterior parameter distribution, it was required to implement the SCEM-UA algorithm using parallel computing (Vrugt et al., 2006). RIVER BASIN, MODEL SETUP AND PARAMETERISATION

The Morava River is situated in Central Europe, with parts in Austria, the Czech Republic and the Slovak Republic, and it forms one of the most important tributaries of River Danube. Figure 2 shows the Morava catchment, including its river network, with an overlay of the topography and the location of meteorological and hydrological stations. The river rises in the forested Jeseniky Mountains. Further downstream it flows through wide valleys and plains. The main part of the Morava River has a typical lowland character with small slopes and alluvial flood plains on both sides. The Morava basin has a typical continental climate with annual precipitation of about 640 mm. In this study, we only consider the northeastern part of the Morava catchment upstream of Straznice, covering an area of approximately 10 000 km2. In the model, the area is discretized in 1 km × 1 km grid blocks. The calibration period spans three years, from 01/10/1997 to 30/09/2000. The first year is used as a warming-up period, hence only predicted discharges from 01/10/1998 to 30/09/2000 are used for the calibration.

Fig. 2 Morava catchment with river network, overlay of topography and locations of meteorological and hydrological stations. Note that only the part upstream of Straznice is considered.



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Table 1 Calibration parameters of the LISFLOOD model with upper and lower bounds of the prior uniform distributions.

Parameter Lower bound Upper bound Upper Zone Time Constant (UZTC, d mmα) 1 50 Lower Zone Time Constant (LZTC, d) 10 1500 Ground Water Percolation Value (GWPV, mm d-1 ) 0 1.5 Xinanjiang parameter b (Xb, -) 0.01 1 Power Preferential Bypass Flow (PPF, -) 1 15 Channel Manning Coefficient Multiplier (ChMM, -) 0.5 2.5 Snow factor (SnFact, -) 0.5 4.0 GwAlpha (Gwα, -) -0.5 2.0 Snowmelt coefficient (SnMC, mm °C-1 d-1) 1.0 6.0 To avoid problems of over-parameterisation and to reduce the dimensionality of the model calibration, input parameters and variables of LISFLOOD for the area are estimated a priori from available databases as much as possible. For example, soil physical properties are derived from the European Soil Geographical Database (King et al., 1994). The HYPRES database (Wösten et al., 1999) is used to estimate porosity, saturated hydraulic conductivity and moisture retention properties for each texture class. Vegetation and land use information are obtained from the Corine land cover database (EEA, 2000). Meteorological input data are derived from the observed variables at the meteorological input stations shown in Fig. 2. Temperature was interpolated using a modified universal kriging approach to correct for the altitude effect. Precipitation was interpolated using local anisotropy kriging. Digital elevation data were obtained from the Catchment Information System, which has a spatial resolution of 1 km (Hiederer & De Roo, 2003). Although LISFLOOD is essentially based on physics, some processes are only represented in a lumped conceptual way. As a result, some parameters lack physical basis and cannot be directly obtained from field data. In the current version of LISFLOOD, nine parameters need to be estimated by calibration against measured streamflow records. The calibration parameters are tabulated in Table 1, with the respective upper and lower bounds adopted for the prior uniform distributions. The Upper Zone Time Constant (UZTC) and Lower Zone Time Constant (LZTC) control the amount and timing of outflow from the simplified upper and lower groundwater reservoirs, respectively. The Groundwater Percolation Value (GWPV) controls the flow from the upper to the lower groundwater zone. The Xinanjiang parameter b (Xb) is an empirical shape parameter in the Xinanjiang model (Zhao & Lui, 1995) that is used to simulate infiltration. In controls the fraction of saturated area within a grid cell that is contributing to runoff, hence it is inversely related to infiltration. The Power Preferential Bypass Flow parameter (PPF) is an empirical shape parameter in the power function relating preferential flow with the relative saturation of the soil. The Channel Manning Coefficient Multiplier is a multiplier that is applied to the Manning’s roughness maps of the channel system. The Snow Factor is a multiplier that is applied to the rate of precipitation when precipitation falls as snow. Since snow is commonly underestimated in meteorological observation data, setting this multiplier to some value greater than one can counteract for this. GwAlpha defines the nonlinearity of the relation between the storage in and outflow from the upper groundwater zone to the channel. For GwAlpha equal to zero, the relation is linear. The Snowmelt Coefficient is a degree-day factor in the snowmelt model. CALIBRATION STRATEGIES

To evaluate the effect of semi-distributed parameters, three different calibration strategies are adopted, as depicted in Fig. 3. In the first, the calibration parameters are uniformly distributed over the area upstream of Straznice and are calibrated against observed discharge measurements at

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Calibration Strategy 1: Uniform parameters, calibration against discharge at the outlet of the catchment (Straznice).

Calibration Strategy 2: Uniform parameters, calibration against discharge at the outlet of the catchment (Straznice) and the internal discharge stations.

Calibration Strategy 3: Semi-distributed parameters, calibration against discharge at the outlet of the corresponding subcatchment.

Fig. 3 Schematic overview of the three calibration strategies adopted. Straznice (outlet of the area of interest). The parameters are also uniformly distributed in the second strategy, but they are calibrated against observed discharge measurements at Straznice and at the internal stations. In the third strategy a semi-distributed approach is adopted; starting from upstream, parameters in each subcatchment are calibrated against the observed discharges at the outlet of the subcatchment. In order not to propagate upstream errors in the calibration process, observed discharges at upstream catchment outlets are used as inflow when calibrating downstream subcatchments. RESULTS AND DISCUSSION

For all (sub-)catchments and calibration strategies the SCEM-UA algorithm was run with a population size s = 500 and q = 10 complexes, or 50 points in each complex, and with a maximum of 15 000 iterations. To avoid problems with heteroscedastic and non-Gaussian error distributions, a Box-Cox power transformation with λ = 0.2 was applied to the observed and simulated discharge time series. Convergence of the MCMC sampler to a stationary distribution was evaluated using the Scale Reduction score ( SR ) defined by Gelman & Rubin (1992). For all parameters, the parallel sequences converged to the target distribution in less than 7500 iterations. The remaining 7500 samples generated after convergence of SCEM-UA were used to represent the posterior parameter distributions and formed the basis for making predictions with the LISFLOOD model. Figure 4 presents box-whisker plots for the nine calibration parameters, indicating the minimum, maximum, median, lower and upper quartiles of the marginal posterior parameter distributions. The first seven entries along the x-axes correspond to the seven sub-catchments; the last two entries correspond to the uniform calibration strategies calibrated against the catchment outlet (entry 8: Qo) and against the catchment and internal outlets (entry 9: Qo + Qi). Note that the limits of the y-axes in Fig. 4 do not necessarily correspond to the ranges specified for the prior uniform parameter distributions (see Table 1). For all calibration strategies, SCEM-UA was able to identify a clear region of attraction in parameter space for all parameters. For some parameters, these regions differ considerably in location and spread between the different sub-catchments and calibration strategies. For the uniform approaches, parameter uncertainty was somewhat larger when calibrating against only the observations at Straznice (Strategy 1). This strategy uses the least information in the calibration process. When the observations at internal stations are included



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Fig. 4 Box-whisker plots for the nine calibration parameters, with indication of the minimum, maximum, median, lower and upper quartile of the posterior parameter distributions.

in the uniform calibration, less parameter combinations will be able to reproduce the calibration data. For the semi-distributed calibration strategy, a more correct identification of the ranges of attraction in parameter space compared to the uniform approaches does not necessarily imply that the ranges are smaller, which may have its effect on the subsequent predictions (as discussed later). These results indicate that uniform parameter values over the entire basin do not fully

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capture the spatially varying hydrological characteristics. This leads to locally non-optimal parameter values and a false interpretation of parameter uncertainty (e.g. UZTC and SnMC parameters for the sub-catchment of Raskov). In the semi-distributed approach, correlation among parameters was evaluated by calculating correlation coefficients using the 7500 parameter sets sampled from the posterior parameter distributions after convergence of SCEM-UA. The average and standard deviation of the correlation coefficients obtained for the seven sub-catchments revealed that only two parameters (UZTC and Gwα) showed a consistent pattern of strong positive correlation over the different sub-catchments, as indicated by a very high average correlation coefficient (0.94) and small standard deviation (0.03). These two parameters control the outflow from the upper groundwater zone. Higher values of the UZTC increase the residence time of water in the upper groundwater zone. This is offset by higher values of Gwα, which increases outflow from the upper groundwater zone. For all other parameter combinations, relatively low average correlation coefficients were obtained, indicating a lack of consistent correlation among the other parameters. Some parameters showed opposite signs of correlation, depending on the sub-catchment (e.g. Xb and SnFact), which shows that correlation among some parameters is location-specific. To ensure that the posterior parameter distributions under the different calibration strategies adequately describe parameter uncertainty, we performed diagnostic checks on the modelling residuals. Simulations of the hydrograph were obtained by running the LISFLOOD model for 7500 parameter combinations sampled from the respective posterior parameter distributions. For each simulation the transformed residuals were obtained as the difference between the Box-Cox transformed observed and simulated discharge series. Figure 5 presents normal probability plots of the mean transformed residuals (transformed residuals averaged over the 7500 simulations) at the catchment outlet, Straznice, for the uniform calibration strategy using discharge observations only at Straznice (Fig. 5(a)) and for the semi-distributed calibration strategy (Fig. 5(b)). The normality plots reveal that the residuals closely conform to a normal distribution under both calibration strategies. This was confirmed by the Lilliefors test at a 5% significance level. However, whereas for the semi-distributed approach the residuals are centred on zero, they are not for the uniform calibration strategy. The same results were observed at the other stations, with stronger deviations from a normal distribution centred on zero for the uniform calibration strategies. It is important to note that we did not account for auto-correlation in the residuals. Correlation of the residuals in time can be accounted for by fitting an autoregressive (AR), moving average (MA) or mixed (ARMA) error model to the residuals (see e.g. Sorooshian & Dracup, 1980; Kuczera, 1983; Bates & Campbell, 2001). Such implementations may impact the posterior

-3 -2 -1 0 1 2 3

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Fig. 5 Normal probability plots for the mean transformed residuals: (a) uniform calibration strategy using discharge observations at catchment outlet Straznice; (b) semi-distributed calibration strategy).

(a) (b)



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Raskov: distributed parameters - Q internal EF = 0.85bias = 0.23R2 = 0.85

Fig. 6 Hydrograph prediction uncertainty at Raskov. Observed discharges are represented by black dots; the dark shaded area denotes the prediction uncertainty that results from parameter uncertainty; and the light shaded area the additional prediction uncertainty that results from model and measurement uncertainty.

parameter pdfs. However, this is beyond the scope of the current paper and will be explored in future studies. For each calibration strategy, probabilistic predictions of the hydrograph were obtained from the discharge series simulated by the LISFLOOD model using the 7500 parameter combinations sampled from the posterior parameter distribution after convergence of SCEM-UA. Results at a selection of gauging stations are presented in Figs 6–8. Similar results were obtained for the other stations. The plates in these figures represent the observed discharges and the hydrograph prediction uncertainty. The black dots represent the observed discharges. The dark shaded area represents the uncertainty in the predictions due to uncertainty in the parameters, whereas the light shaded area represents the total uncertainty in the predictions (due to parameter, model and measurement uncertainty). The latter is computed as follows. The standard deviation of the error model is obtained from the RMSE between the Box-Cox transformed series of the observed discharges and of the simulated discharges using the most likely parameter set. For each of the 7500 simulations a constant error term equal to ±(1.96 × RMSE) is added to the transformed discharges at each time step. The obtained prediction uncertainty limits in the transformed space are then transformed back to the original output space, which explains the varying width of the total prediction uncertainty limits with time. The top plate in each of Figs 6–8 corresponds to the

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0

100

200

300q

(m3 /s

)

10/1/98 2/1/99 6/1/99 10/1/99 2/1/00 6/1/00time in days

Dluhonice: uniform parameters - Q at outlet (Straznice) EF = 0.69bias = 0.34R2 = 0.71

0

100

200

300

q (m

3 /s)

10/1/98 2/1/99 6/1/99 10/1/99 2/1/00 6/1/00time in days

Dluhonice: uniform parameters - Q outlet + internal EF = 0.69bias = 0.35R2 = 0.71

0

100

200

300

q (m

3 /s)

10/1/98 2/1/99 6/1/99 10/1/99 2/1/00 6/1/00time in days

Dluhonice: distributed parameters - Q internal EF = 0.84bias = 0.25R2 = 0.85

Fig. 7 Hydrograph prediction uncertainty at Dluhonice. See Fig. 6 for explanation.

first uniform calibration strategy. The middle plates represent the second uniform calibration strategy, and the bottom plates show results of the semi-distributed approach. To quantify a posteriori the model performance under the different calibration scenarios, values have been calculated for the following three statistical measures based on the simulation with the most likely parameter set (i.e. the mode of the posterior distribution derived by SCEM-UA):

∑

∑

=

=

−

−−= n

tt

n

ttt

QQ

QQ

1

2obs,obs

1

2,sim,obs

)(

)(1EF (5)

∑

∑

=

=

−= n

tt

n

ttt

Q

QQ

1,obs

1,sim,obs ||

bias (6)



305

0

200

400

600q

(m3 /s

)

10/1/98 2/1/99 6/1/99 10/1/99 2/1/00 6/1/00time in days

Straznice: uniform parameters - Q at outlet (Straznice) EF = 0.85bias = 0.26R2 = 0.87

0

200

400

600

q (m

3 /s)

10/1/98 2/1/99 6/1/99 10/1/99 2/1/00 6/1/00time in days

Straznice: uniform parameters - Q at outlet + internal EF = 0.84bias = 0.28R2 = 0.86

0

200

400

600

q (m

3 /s)

10/1/98 2/1/99 6/1/99 10/1/99 2/1/00 6/1/00time in days

Straznice: distributed parameters - Q internal EF = 0.90bias = 0.20R2 = 0.89

Fig. 8 Hydrograph prediction uncertainty at Straznice. See Fig. 6 for explanation.

2

5.0

1

2sim,sim

5.0

1

2obs,obs

1sim,simobs,obs

2

)()(

))((

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡−

−−=

∑∑

∑

==

=

n

tt

n

tt

n

ttt

QQQQ

QQQQR (7)

The coefficient of efficiency EF (Nash & Sutcliffe, 1970) is the ratio of the mean square error to the variance in the observed data, substracted from unity. It ranges from minus infinity to 1.0, with higher values indicating a better agreement. The coefficient of determination R2 describes the proportion of the total variance in the observed data that can be explained by the model. It ranges from 0.0 to 1.0, with higher values indicating better agreement. Because of the squared differences EF and R2 are overly sensitive to extreme values. The bias coefficient reflects the absolute simulation error relative to the observations. It varies between 0.0 and infinity, with lower values indicating a better agreement. The values calculated for the statistical measures are presented in the upper right corner of each plate in Figs 6–8. Two measures have been calculated to evaluate the prediction uncertainty. Table 2 presents a measure that quantifies the spread of the predictive uncertainty. It is obtained by scaling for each simulation time step (day) the range of the 95% prediction uncertainty bounds resulting from parameter uncertainty by the corresponding observed streamflow and averaging over the simulation

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Table 2 Average range of the 95% prediction uncertainty bounds resulting from parameter uncertainty as a fraction of daily observed streamflow.

Uniform parameters Qoutlet

Uniform parameters Qoutlet + Qinternal

Distributed parameters Qinternal

Raskov 0.31 0.27 0.24 Moravicany 0.30 0.27 0.25 Lostice 0.35 0.32 0.36 Olomouc 0.31 0.27 0.23 Dluhonice 0.36 0.31 0.39 Kromerir 0.31 0.28 0.35 Straznice 0.33 0.29 0.22 Table 3 Percentage of streamflow observations captured by the 95% prediction uncertainty bounds resulting from parameter uncertainty.

Uniform parameters Qoutlet

Uniform parameters Qoutlet + Qinternal

Distributed parameters Qinternal

Raskov 26 23 54 Moravicany 48 48 64 Lostice 25 28 63 Olomouc 47 49 60 Dluhonice 34 31 69 Kromerir 43 41 67 Straznice 34 29 51 period. For example, a value of 0.30 centred on a discharge of 100 m3/s corresponds to a 95% prediction uncertainty interval stemming from parameter uncertainty between 85 and 115 m3/s. It should be noted that the averaging over the simulation period is merely done for reasons of comparison. The predictive hydrographs in Figs 6–8 clearly show that the uncertainty bounds are variable throughout the simulation period. Table 3 presents the percentage of streamflow observations that is captured by the 95% prediction uncertainty bounds resulting from parameter uncertainty, which can be seen as a measure of the reliability of the ensemble of predictions. From the figures presented and statistical measures tabulated the following important findings should be noted. First, the calibration of uniform parameter values against discharges at the catchment outlet resulted in good reproductions of the hydrograph at the downstream stations (Olomouc, Kromeriz and Straznice), but inferior fits were obtained for more upstream subcatchments (Raskov, Lostice, Dluhonice and Moravicany). This is because at larger scales (i.e. a larger upstream area), interference of the inferiorly simulated discharge time series of different upstream areas can still result in a good reproduction of river discharges more downstream. Including observations from internal discharge stations in the objective function slightly improved the results for all subcatchments, expect for the most downstream one (Straznice). This can be attributed to the fact that the uniform parameter values also have to reproduce internal observations, and not only those at the outlet. The semi-distributed approach considerably improved the predictions, especially in the upstream catchments, because it allows accounting for location-specific characteristics of the sub-catchment. For example, the lumped approaches failed to reproduce the peaks at Dluhonice (Fig. 7). Downstream, improvements were less pronounced, largely because the uniform approaches yielded good results in these stations due to interference of the discharge time series of upstream areas. Nevertheless, the semi-distributed approach did considerably improve base flow predictions in the downstream areas (see Straznice, Fig. 8). A second important finding is the observed changes in prediction uncertainty between the different calibration strategies. For the uniform approaches, the predictive spread due to parameter uncertainty decreased when including internal stations during calibration. However, in most cases this resulted in a lower percentage of predictions captured by the 95% prediction bounds resulting



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from parameter uncertainty. For the semi-distributed strategy, the contribution of parameter uncertainty to the prediction uncertainty was in some cases (Lostice, Dluhonice, Kromerir) larger compared to the uniform approaches. This is because an improved (in terms of more correct) identification of the ranges of attraction in parameter space does not necessarily imply that the ranges are smaller (see Fig. 3). However, the prediction uncertainty stemming from parameter uncertainty obtained for the semi-distributed strategy can be seen as a more correct and reliable representation of the true effect of parameter uncertainty on the predictions. This is confirmed by the considerable increase in the percentage of predictions captured by the 95% uncertainty bounds stemming from parameter uncertainty when going from a uniform to a semi-distributed approach. The semi-distributed approach also resulted in a considerable reduction in the total prediction uncertainty with respect to the uniform approaches (see Figs 6–8). The improved representation of the spatially varying hydrological characteristics by the calibration parameters results in more accurate simulations and consequently a smaller variance of the error model (in terms of the RMSE calculated based on the simulation with the most likely parameter combination). Because the input to the model and the measurement error do not differ between the calibration strategies, the increase in total prediction uncertainty can be attributed to the combined effects of parameter and model uncertainty. It should be stressed that the increase in model uncertainty here reflects the change in parameterization, i.e. uniform instead of semi-distributed parameter values, which can be assumed to be a less correct representation of the unknown distribution of the hydrological properties. CONCLUSIONS In this work, we used SCEM-UA, an automatic Bayesian parameter inference algorithm based on Markov Chain Monte Carlo methods, to calibrate the conceptual parameters of the LISFLOOD model for a part of the Morava catchment (area of 10 000 km2) upstream of Straznice. Uniform and semi-distributed calibration strategies were applied to evaluate the value of spatially distributing the calibration parameters. In the semi-distributed approach, the spatial detail was defined by the resolution of the discharge stations. In all calibration strategies, SCEM-UA was able to efficiently and effectively explore the feasible parameter space and to converge to the target posterior parameter distributions after less than 7500 iterations. All the calibration parameters of the LISFLOOD model were well identifiable using 2 years of measured daily discharges. For some parameters, the regions of attraction in parameter space differed considerably in location and spread between the different sub-catchments and calibration strategies. Results indicated that lumped parameters do not fully capture the spatially varying hydrological characteristics in the area, which leads to locally non-optimal parameter values and a false interpretation of parameter uncertainty. For all strategies, probabilistic streamflow predictions were obtained using 7500 parameter combinations sampled from the respective posterior parameter distributions after convergence of the SCEM-UA algorithm. The improved representation of parameter uncertainty when calibrating sub-catchments individually yielded a more correct representation of the effect of parameter uncertainty on the predictions. The semi-distributed approach also considerably improved the accuracy of the predictions, especially in the upstream catchments, because it allows the para-meters to better account for location-specific characteristics of the sub-catchment. The more accurate simulations and consequently smaller variance of the error model resulted in less spread of the total prediction uncertainty. The increase in model uncertainty when going from a semi-distributed to a lumped approach can be attributed to the less correct representation of the spatially varying hydrological characteristics of the system. Acknowledgements The Czech Hydrometeorological Institute (CHMI) provided the hydrological and meteorological data for this study. We are grateful to the SARA centre of parallel computing at the University of Amsterdam for the computational support. The third author is supported by the LANL Director’s funded postdoctoral programme. We also wish to acknowledge Johan van der

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Knijff of the Land Management and Natural Hazards Unit at the Joint Research Centre for his input to this paper. We are grateful to two anonymous reviewers for their comments which helped us to improve this manuscript. REFERENCES Ajami, N. K., Gupta, H. V., Wagener, T. & Sorooshian, S. (2004) Calibration of a semi-distributed hydrological model for

streamflow estimation along a river system. J. Hydrol. 298, 112–135. Andersen, J., Refsgaard, J. C. & Jensen, K. H. (2001) Distributed hydrological modelling of the Senegal River Basin—model

construction and validation. J. Hydrol. 247, 200–214. Bates, B. C. & Campbell, E. P. (2001) A Markov Chain Monte Carlo scheme for parameter estimation and inference in

conceptual rainfall–runoff modeling. Water Resour. Res. 37(4), 937–947. Beven, K. J. & Binley, A. (1992) The future of distributed models: model calibration and uncertainty analysis. Hydrol.

Processes 6, 279–298. Box, G. E. P. & Tiao, G. C. (1973) Bayesian Inference in Statistical Analysis. Addison-Wesley-Longman, Reading, MA, USA. Boyle, D. P., Gupta, H. V., Sorooshian, S., Koren, V., Zhang, Z. & Smith, M. (2001) Toward improved streamflow forecast:

value of semidistributed modeling. Water Resour. Res. 37(11), 2749–2759. Campbell, E. P., Forx, D. R. & Bates, B. C. (1999) A Bayesian approach to parameter estimation and pooling in nonlinear flood

event models. Water Resour. Res. 35(1), 211–220. De Roo, A. P. J., Wesseling, C. G. & Van Deurzen, 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., Odijk, M., Schmuck, G., Koster, E. & Lucieer, A. (2001) Assessing the effects of land use changes on floods

in the Meuse and Oder catchment. Physics and Chemistry of the Earth (B), 26(7-8), 593–599. Duan, Q., Sorooshian, S. & Gupta, V. K. (1992) Effective and efficient global optimization for conceptual rainfall–runoff

models. Water Resour. Res. 28(4), 1015–1031. EEA (European Environment Agency) (2000) The European Topic Centre on Terrestrial Environment: Corine Land Cover

Raster Database 2000–100 m. Feyen, L., Vrugt, J. A., Ó Nualláin, B. van der Knijff, J. & De Roo, A. (2007) Parameter optimisation and uncertainty

assessment for large-scale streamflow simulation with the LISFLOOD model. J. Hydrol. 332, 276–289. Gelman, A. & Rubin, D. B. (1992) Inference from iterative simulation using multiple sequences. Statistical Sci. 7(4), 457–472. Hiederer, R. & De Roo, A. (2003) A European flow network and catchment data set. Report of the European Commission, Joint

Research Centre, EUR 20703 EN. King, D., Daroussin, J. & Tavernier, R. (1994) Development of a soil geographical database from the soil map of the European

Communities. Catena 21, 37–56. Krzysztofowicz, R. & Kelly, K. S. (2000) Hydrologic uncertainty processor for probabilistic river stage forecasting. Water

Resour. Res. 36(11), 3265–3277. Kuczera, G. (1983) Improved parameter inference in catchment models. 1. Evaluating parameter uncertainty. Water Resour.

Res. 19(5), 1151–1162. Kuczera, G., & Mroczkowski, M. (1998) Assessment of hydrologic parameter uncertainty and the worth of multi-response data.

Water Resour. Res. 34(6), 1481–1490. Kuczera, G. & Parent, E. (1998) Monte-Carlo assessment of parameter uncertainty in conceptual catchment models. J. Hydrol.

211(1-4), 69–85. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. (1953) Equations of state calculations by fast

computing machines. J. Chemical Physics 21, 1087–1091. Montanari, A. & Brath A. (2004) A stochastic approach for assessing the uncertainty of rainfall-runoff simulations. Water

Resour. Res. 40, W01106, doi:10/1029/2003WR002540. Nash, J. E., & Sutcliffe J. V. (1970) River flow forecasting through conceptual models. Part I, A discussion of principles. J.

Hydrol. 10 (3), 282–290. Sorooshian, S. & Dracup J. A. (1980) Stochastic parameter estimation procedures for hydrologic rainfall-runoff models:

correlated and heteroscedastic errors. Water Resour. Res. 16(2), 430–442. Sumner, N. R., Fleming, P. M. & Bates, B. C. (1997) Calibration of a modified SFB model for twenty-five Australian

catchments using simulated annealing. J. Hydrol. 197, 166–188. Thyer, M., Kuczera, G. & Bates, B. C. (1999) Probabilistic optimization for conceptual rainfall-runoff models: a comparison of

the shuffled complex evolution and simulated annealing algorithms. Water Resour. Res. 35(5), 767–773. Uhlenbrook, S., Seibert, J., Leibundgut, C. & Rodhe, A. (1999) Prediction uncertainty of conceptual rainfall–runoff models

caused by problems in identifying model parameters and structure. Hydrol. Sci. J. 44, 779–797. Vrugt, J. A., Gupta, H. V., Bouten, W. & Sorooshian, S. (2003) A Shuffled Complex Evolution Metropolis algorithm for

optimization and uncertainty assessment of hydrologic parameter estimation. Water Resour. Res. 39(8), 1201, doi:10.1029/2002WR001642.

Vrugt, J. A., Schoups, G. H., Hopmans, J. W., Young, C. H., Wallender, W., Harter, T. & Bouten, W. (2004) Inverse modeling of largescale spatially distributed vadose zone properties using global optimization, Water Resour. Res. 40(6), W06503, doi:10.1029/2003WR002706.

Vrugt, J. A., Ó Nualláin, B., Robinson, B. A., Bouten, W., Dekker, S. C. & Sloot, P. M. A. (2006) Application of parallel computing to stochastic parameter estimation in environmental models. Computers & Geosci. 32(8), 1139–1155.

Wösten, J. H. M., Lilly, A., Nemes, A. & Le Bas, C. (1999) Development and use of a database of hydraulic properties of European soils. Geoderma 90(3-4), 169–185.

Zhao, R. J. & Liu, X. R. (1995) The Xinanjiang model. In: Computer Models of Watershed Hydrology (ed. by V. P. Singh), 215–232. Water Resources Publications, Littleton, Colorado, USA.

Received 7 August 2006; accepted 13 November 2007

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Semi-distributed parameter optimization and …faculty.sites.uci.edu/jasper/files/2016/04/36.pdfLUC FEYEN1, MILAN KALAS1,2 & JASPER A. VRUGT3 1 European Commission – DG Joint Research