# Uncertainty and multiple objective calibration in regional water balance modelling: case study in 320 Austrian catchments

**ABSTRACT** We examine the value of additional information in multiple objective calibration in terms of model performance and parameter uncertainty. We calibrate and validate a semi-distributed conceptual catchment model for two 11-year periods in 320 Austrian catchments and test three approaches of parameter calibration: (a) traditional single objective calibration (SINGLE) on daily runoff; (b) multiple objective calibration (MULTI) using daily runoff and snow cover data; (c) multiple objective calibration (APRIORI) that incorporates an a priori expert guess about the parameter distribution as additional information to runoff and snow cover data. Results indicate that the MULTI approach performs slightly poorer than the SINGLE approach in terms of runoff simulations, but significantly better in terms of snow cover simulations. The APRIORI approach is essentially as good as the SINGLE approach in terms of runoff simulations but is slightly poorer than the MULTI approach in terms of snow cover simulations. An analysis of the parameter uncertainty indicates that the MULTI approach significantly decreases the uncertainty of the model parameters related to snow processes but does not decrease the uncertainty of other model parameters as compared to the SINGLE case. The APRIORI approach tends to decrease the uncertainty of all model parameters as compared to the SINGLE case. Copyright © 2006 John Wiley & Sons, Ltd.

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**ABSTRACT:**Large uncertainties in streamflow projections derived from downscaled climate projections of precipitation and temperature can render such simulations of limited value for decision making in the context of water resources management. New approaches are being sought to provide decision makers with robust information in the face of such large uncertainties. We present an alternative approach that starts with the stakeholder's definition of vulnerable ranges for relevant hydrologic indicators. Then, the modeled system is analyzed to assess under what conditions these thresholds are exceeded. The space of possible climates and land use combinations for a watershed is explored to isolate sub-spaces that lead to vulnerability, while considering model parameter uncertainty in the analysis. We implement this concept using classification and regression trees (CART) that separate the input space of climate and land use change into those combinations that lead to vulnerability and those that do not. We test our method in a Pennsylvania watershed for nine ecological and water resources related streamflow indicators for which an increase in temperature between 3°C to 6 °C and change in precipitation between -17% and 19% is projected. Our approach provides several new insights, for example we show that even small decreases in precipitation (~5%) combined with temperature increases greater than 2.5ºC can push the mean annual runoff into a slightly vulnerable regime. Using this impact and stakeholder driven strategy, we explore the decision-relevant space more fully and provide information to the decision maker even if climate change projections are ambiguous.Water Resources Research. 04/2014; - SourceAvailable from: Alberto Montanari[Show abstract] [Hide abstract]

**ABSTRACT:**A "Holy Grail" of hydrology is to understand catchment processes well enough that models can provide detailed simulations across a variety of hydrologic settings at multiple spatio-temporal scales, and under changing environmental conditions. Clearly, this cannot be achieved only through intensive place-based investigation at a small number of heavily instrumented catchments, or by regionalization methods that do not fully exploit our understanding of hydrology. Here, we discuss the need to actively promote and pursue the use of a "large catchment sample" approach to modeling the rainfall-runoff process, thereby balancing depth with breadth. We examine the history of such investigations, discuss the benefits (improved process understanding resulting in robustness of prediction at ungaged locations and under change), examine some practical challenges to implementation and, finally, provide perspectives on issues that need to be taken into account as we move forward. Ultimately, our objective is to provoke further discussion and participation, and to promote a potentially important theme for the upcoming IAHS Scientific Decade entitled "Panta Rhei".Hydrology and Earth System Sciences Discussions 07/2013; 10(7):9147-9189. · 3.59 Impact Factor - SourceAvailable from: Hongkai Gao[Show abstract] [Hide abstract]

**ABSTRACT:**Conceptual environmental systems models, such as rainfall runoff models, generally rely on calibration for parameter identification. Increasing complexity of this type of model for better representation of hydrological process heterogeneity typically makes parameter identification more difficult. Although various, potentially valuable, strategies for better parameter identification were developed in the past, strategies to impose general conceptual understanding regarding how a catchment works into the process of parameterizing a conceptual model has still not been fully explored. In this study we assess the effect of imposing semi-quantitative, relational expert knowledge into the model development and parameter selection, efficiently exploiting the complexity of a semi-distributed model formulation. Making use of a topography driven rainfall-runoff modeling (FLEX-TOPO) approach, a catchment was delineated into three functional units, i.e. wetland, hillslope and plateau. Ranging from simplicity to complexity, three model set-ups, FLEXA, FLEXB and FLEXC have been developed based on these functional units. While FLEXA is a lumped representation of the study catchment, the semi-distributed formulations FLEXB and FLEXC introduce increasingly more complexity by distinguishing 2 and 3 functional units, respectively. In spite of increased complexity, FLEXB and FLEXC allow modelers to compare parameters as well as states and fluxes of their different functional units to each other. Based on these comparisons, expert knowledge based, semi-quantitative relational constraints have been imposed on three models structures. More complexity of models allows more imposed constraints. It was shown that a constrained but uncalibrated semi-distributed model, FLEXC, can predict runoff with similar performance than a calibrated lumped model, FLEXA. In addition, when constrained and calibrated, the semi-distributed model FLEXC exhibits not only higher performance but also reduced uncertainty for prediction, compared to the calibrated, lumped FLEXA model.Hydrology and Earth System Sciences Discussions 12/2013; 10(12):14801-14855. · 3.59 Impact Factor

Page 1

HYDROLOGICAL PROCESSES

Hydrol. Process. 21, 435–446 (2007)

Published online 2 November 2006 in Wiley InterScience

(www.interscience.wiley.com) DOI: 10.1002/hyp.6253

Uncertainty and multiple objective calibration in regional

water balance modelling: case study in 320 Austrian

catchments

J. Parajka,*,†R. Merz and G. Bl¨ oschl

Institute for Hydraulic and Water Resources Engineering, Vienna University of Technology, Karlsplatz 13/222, A-1040 Vienna, Austria

Abstract:

We examine the value of additional information in multiple objective calibration in terms of model performance and parameter

uncertainty. We calibrate and validate a semi-distributed conceptual catchment model for two 11-year periods in 320 Austrian

catchments and test three approaches of parameter calibration: (a) traditional single objective calibration (SINGLE) on daily

runoff; (b) multiple objective calibration (MULTI) using daily runoff and snow cover data; (c) multiple objective calibration

(APRIORI) that incorporates an a priori expert guess about the parameter distribution as additional information to runoff and

snow cover data. Results indicate that the MULTI approach performs slightly poorer than the SINGLE approach in terms of

runoff simulations, but significantly better in terms of snow cover simulations. The APRIORI approach is essentially as good

as the SINGLE approach in terms of runoff simulations but is slightly poorer than the MULTI approach in terms of snow

cover simulations. An analysis of the parameter uncertainty indicates that the MULTI approach significantly decreases the

uncertainty of the model parameters related to snow processes but does not decrease the uncertainty of other model parameters

as compared to the SINGLE case. The APRIORI approach tends to decrease the uncertainty of all model parameters as

compared to the SINGLE case. Copyright 2006 John Wiley & Sons, Ltd.

KEY WORDS

multiple objective calibration; parameter uncertainty; water balance modelling

Received 20 May 2005; Accepted 4 November 2005

INTRODUCTION

Knowledge of the regional variability of water balance

components is important for solving a range of problems

in water resources management and planning. Simula-

tions of the water balance components are, however,

fraught with a range of problems, including uncertainty

in inputs, model parameters and model structure. These

problems are particularly acute in Alpine regions, where

data are sparse and the spatial variability of the hydro-

logical environment is enormous.

Although modellers have always been aware of model

parameter uncertainty it is only in the last few decades

that explicit efforts have been made towards assessing

this uncertainty. Because of multiple optima, non-linear

interactions between model parameters and data errors,

it may be difficult, if not impossible, to identify a

unique parameter set from runoff data. Methods for

assessing parameter uncertainty in hydrologic models

include the generalized likelihood uncertainty estimation

methodology (e.g. Beven and Binley, 1992; Beven and

Freer, 2001), multi-normal approximations to parameter

uncertainty (e.g. Kuczera and Mroczkowski, 1998), and

*Correspondence to: J. Parajka, Institute for Hydraulic and Water

Resources Engineering, Vienna University of Technology, Karlsplatz

13/222, A-1040 Vienna, Austria. E-mail: parajka@hydro.tuwien.ac.at

†On leave from the Institute of Hydrology, Slovak Academy of Sciences,

Bratislava, Slovakia.

Markov Chain Monte Carlo methods (e.g. Kuczera and

Parent, 1998).

Although these methods are useful in identifying the

degree of confidence one can attribute to a calibrated

parameter set they do not reduce their uncertainty.

For reducing parameter uncertainty the general line of

thought has been that information additional to runoff

data needs to be used to constrain the parameters over

what can be achieved from calibrations to runoff alone.

Various types of catchment response data can be used

depending on the application at hand. Seibert (2000)

and Madsen (2003) used runoff data and groundwater

level data jointly to calibrate model parameters. They

found that the groundwater data reduced the uncertainty

of the parameters representing groundwater dynamics

significantly. Beldring (2002) found that the inclusion of

groundwater levels reduced the uncertainty of most of

the model parameters. Other types of data that can be

used in constraining the uncertainty of model parameters

are snow cover and soil moisture data. The value of

snow cover data in distributed hydrologic simulations has

been demonstrated by Bl¨ oschl et al. (1991) and others.

Grayson and co-workers (Grayson and Bl¨ oschl, 2000;

Grayson et al., 2002) summarize numerous examples of

using snow and soil moisture data in addition to runoff

and suggest that these response data are particularly

useful if available as spatial patterns. Isotopes and

geochemical characteristics, such as stream chloride,

have also been used for identifying model parameters

Copyright 2006 John Wiley & Sons, Ltd.

Page 2

436

J. PARAJKA, R. MERZ AND G. BL¨OSCHL

(e.g. Holko and Lepist¨ o, 1997; Mroczkowski et al.,

1997).

As an alternative to measurements, the use of ‘soft

data’ or qualitative information from field surveys has

been suggested to constrain model parameters. ‘Soft’

information is widely used in practical applications of

catchment models where parameters are selected based

on all sources of information available to the analyst (e.g.

Bl¨ oschl, 2005). A recent contribution of using this type

of qualitative expert knowledge has been provided by

Seibert and McDonnell (2002).

Formal methods of incorporating information in addi-

tion to runoff in the calibration process are usually

referred to as multiple objective calibration. The various

objectives (related to runoff, groundwater levels, snow

cover, etc.) can be combined in various ways. The most

straightforward combination is by a weighted sum, as

in the weighted least-squares method used for calibrat-

ing groundwater models (e.g. Peck et al., 1988: 50). An

alternative is the use of fuzzy logics, where the compo-

nents of the objective function are combined based on

membership functions that indicate the relative degree of

satisfaction of each fuzzy objective (Seibert, 1997; Franks

et al., 1998; Yu and Yang, 2000; Cheng et al., 2002). A

third method is based on the concept of Pareto optimality.

A parameter set is considered Pareto optimal if there is no

other parameter set that performs at least as well on every

objective and strictly better on at least one objective. That

is, a Pareto-optimal solution cannot be improved upon

without hurting at least one of the objectives (Miettinen,

1999). Madsen (2003) proposed a Pareto-based approach

that emulates the ability of manual expert calibration of

using a number of complementary ways in evaluating

model performance. The method provided generally bet-

ter simulations of runoff compared with manual expert

calibration but virtually similar performance for ground-

water level simulations. Vrugt et al. (2003b) proposed an

optimization technique termed the multi-objective shuf-

fled complex evolution Metropolis (MOSCEM-UA) algo-

rithm that provides an estimate of the Pareto solution

space within a single optimization run. They found that

the MOSCEM-UA algorithm generates a fairly uniform

approximation of the entire set of Pareto parameter com-

binations for their problem. Although various methods

exist for making use of multiple data sources, the actual

merits of additional information have, to our knowledge,

never been identified for a large number of catchments

in the context of regional water balance modelling.

The aim of this paper, therefore, is to assess the value

of snow data in addition to runoff data as well as the value

of expert judgement in multiple objective calibration

of a catchment model. Specifically, we address two

science questions: (a) How will the multiple objective

calibration change the model performance over single

objective calibration? (b) To what extent does multiple

objective calibration reduce model parameter uncertainty

over single objective calibration? One would expect that

the use of additional snow data will improve the snow

cover simulations and decrease the model performance

with respect to runoff. The aim of this paper is to assess

the extent of such change. We use daily hydrologic data

from 320 catchments over a period of 22 years, which

will likely allow us to draw more generic inferences

than has been possible in most previous studies that used

smaller data sets.

DATA

This study was carried out in Austria using data from the

period 1976–1997. Austria is flat or undulating in the east

and north, and Alpine in the west and south. Elevations

range from 115 to 3797 m a.s.l. Mean annual precipita-

tion is less than 400 mm in the east and almost 3000 mm

in the west. Land use is mainly agricultural in the low-

lands and forest in the medium elevation ranges. Alpine

vegetation and rocks prevail in the highest catchments.

The dataset used in this study includes measurements of

daily precipitation and snow depths at 1091 stations and

daily air temperature at 212 climatic stations. To cali-

brate and verify a catchment model, daily runoff data

from 320 gauged catchments were used with areas rang-

ing from 10 to 9770 km2and a median of 196 km2.

Of these, 97 catchments range in area between 10 and

100 km2, 106 catchments between 100 and 300 km2, 64

catchments between 300 and 1000 km2and 55 catch-

ments have areas of more than 1000 km2. In preliminary

analyses we carefully screened the runoff data for errors

and removed all stations with significant anthropogenic

effects. We also removed stations where we were not able

to close the long-term water balance. The spatial distri-

bution of the climate stations and the boundaries of the

gauged catchments are shown in Figure 1.

The inputs to the water balance model were prepared

in two steps. First, the daily values of precipitation,

snow depth and air temperature were spatially interpo-

lated by methods that use elevation as auxiliary infor-

mation. External drift kriging was used for precipitation

and snow depths, and the least-squares trend predic-

tion method was used for air temperatures (Pebesma,

2001). The spatial distribution of potential evapotran-

spiration was estimated by a modified Blaney–Criddle

method (Schr¨ odter, 1985; Parajka et al., 2003) using

daily air temperature and potential sunshine duration cal-

culated by the Solei-32 model (M´ esz´ aroˇ s et al., 2002;

http://www.ih.savba.sk/software/solei/) that incorporates

shading by surrounding terrain. In a second step, a digital

elevation model with a 1 km ð 1 km grid resolution was

used for deriving 200 m elevation zones in each catch-

ment. Time-series of daily precipitation, air temperature,

potential evaporation and snow depth were then extracted

for each of the elevation zones to be used in the water

balance simulations.

METHODS

The model used in this paper is a semi-distributed

conceptual rainfall-runoff model, following the structure

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 21, 435–446 (2007)

DOI: 10.1002/hyp

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REGIONAL WATER BALANCE MODELLING

437

Figure 1. Map of Austria, including boundaries of gauged catchments and stations with precipitation and snow depth measurements

of the HBV model (Bergstr¨ om, 1976; Lindstr¨ om et al.,

1997). The model runs on a daily time step and consists

of a snow routine, a soil moisture routine and a flow

routing routine (Merz and Bl¨ oschl, 2004). A flow chart of

the model is presented in Merz (2002). The snow routine

represents snow accumulation and melt by a simple

degree-day concept, using a degree-day factor DDF and

a melt temperature parameter TM. The catch deficit of

precipitation gauges during snowfall is corrected by a

snow correction factor SCF. A threshold temperature

interval TR? TSis used to distinguish between rainfall,

snowfall and a mix of rain and snow. The soil moisture

routine represents runoff generation and changes in the

soil moisture state of the catchment and involves three

parameters: the maximum soil moisture storage FC, a

parameter representing the soil moisture state above

which evaporation is at its potential rate, termed the

limit for potential evaporation LP, and a parameter in

the non-linear function relating runoff generation to the

soil moisture state, termed the non-linearity parameter

ˇ. Runoff routing on the hillslopes is represented by an

upper and a lower soil reservoir. Excess rainfall enters the

upper zone reservoir and leaves this reservoir through

three paths: outflow from the reservoir based on a fast

storage coefficient K1; percolation to the lower zone with

a constant percolation rate CP; and, if a threshold of the

storage state LSUZ is exceeded, through an additional

outlet based on a very fast storage coefficient K0. Water

leaves the lower zone based on a slow storage coefficient

K2. The outflow from both reservoirs is then routed by a

triangular transfer function representing runoff routing in

the streams, where CRis a free parameter. More details

on the model are given in Appendix A.

The model is run for all 320 gauged catchments in

Austria. Inputs (precipitation, air temperature and poten-

tial evapotranspiration) are allowed to vary with elevation

within a catchment, so the soil moisture accounting and

snow accounting is performed independently in each ele-

vation zone of the 200 m altitudinal range. However, the

same model parameters are assumed to apply to all eleva-

tion zones of a catchment. These parameters (14 in total)

are estimated by model calibration.

We tested three cases of model calibration in this

paper. The first case conforms to the most widely used

procedure in hydrology, where the model parameters

are adjusted in a way that runoff simulations closely

match measured runoff. This case is termed SINGLE in

this study. The runoff objective function ZQfollows the

relation proposed by Lindstr¨ om (1997), which combines

the Nash–Sutcliffe efficiency ME and the relative volume

error VE and is defined as

ZQD ?1 ? ME? C wVE

?1?

where

ME D 1 ?

n

?

?

iD1

?Qobs,i? Qsim,i?2

n

iD1

?Qobs,i? Qobs?2

?2?

VE D

n

?

iD1

Qsim,i?

n

?

iD1

Qobs,i

n

?

iD1

Qobs,i

?3?

Qsim,i is the simulated runoff on day i,Qobs,i is the

observed runoff, Qobs is the average of the observed

runoff over the calibration (or verification) period of

n days, and the weight w D 0Ð1 was found in test

simulations to give the most plausible results.

The second case, referred to as MULTI in this study,

uses snow data in addition to runoff. In this study

we did not directly compare the observed snow depths

with simulated snow water equivalent, because the snow

density measurements necessary for such a comparison

were not available. Observed snow cover, therefore, was

estimated from daily grid maps constructed from the

observed snow depth data. If the catchment zone average

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 21, 435–446 (2007)

DOI: 10.1002/hyp

Page 4

438

J. PARAJKA, R. MERZ AND G. BL¨OSCHL

of snow depth was greater than 0Ð5 mm then the zone was

considered as snow covered, otherwise it was considered

as snow free. Simulated snow cover was derived from

the snow water equivalent simulated by the model: a zone

was considered snow covered if the water equivalent was

greater than 0Ð1 mm; otherwise, it was considered snow

free. The thresholds of 0Ð5 and 0Ð1 mm were set a little

above zero to avoid snow cover overestimation that may

result from the spatial interpolation. For local snowfall

events the interpolation of snow depth measurements into

neighbouring regions, where no snow was observed, may

lead to small values of catchment (or elevation zone)

snow depth average that, however, should be considered

as no snow. Snow simulations on a particular day were

considered to be poor if the absolute difference between

simulated and observed snow cover was greater than

50% of the catchment area. The 50% threshold was

determined in sensitivity analysis (not shown here) taking

into account different areal arrangements of elevation

zones in different catchments, where the sensitivity was

assessed on the basis of model performance. The snow

objective function ZSwas then defined as the ratio of the

number of days with poor snow cover simulation npsto

the total number of days in the simulation period n:

ZSDnps

n

?4?

The third calibration arrangement, termed APRIORI,

accounts not only for the runoff and snow cover objec-

tives, but also incorporates an a priori expert judgement

about the expected distribution of each model parame-

ter. In calibration procedures, the parameter values are

usually bounded between two limits (Duan et al., 1992)

and otherwise no a priori assumptions are made about

the parameters. This implies that the a priori distribution

of the parameters is a uniform distribution. We believe

that it is possible to make a more informed guess about

the shape of the a priori distribution and introduced a

penalty function ZP:

ZPD

k

?

jD1

fmax,j? fj

?pj? pl,j

fmax,j

?

pu,j? pl,j

?

?5?

fmax,jD fj

?pmax,j? pl,j

pu,j? pl,j

?6?

where pjis the model parameter j to be calibrated, pl

and puare the lower and upper bounds of the parameter

space respectively, pmaxis the parameter value at which

the a priori distribution is at a maximum and k is the

number of parameters to be calibrated. The probability

density function of the Beta distribution is

f?xjU,V? D

1

B?u,v?xu?1?1 ? x?v?1

for 0 < x < 1,u > 0,v > 0

?7?

with

B?u,v? D

?1

0

xu?1?1 ? x?v?1dx D?u??v?

?u C v?

?8?

We assumed values of u,v,pl,puand pmaxfor each

parameter j based on our own assessment of the hydro-

logic characteristics of the study region and on literature

values (Bergstr¨ om, 1992; Seibert, 1997). The order of

magnitude of the pmaxwas consistent with values found

by modelling studies in the region (Merz and Bl¨ oschl,

2004). In the absence of more detailed information we

have chosen the same values of u,v,pl,pu and pmax

(Table I) for all catchments. If more detailed informa-

tion was available (e.g. from catchment attributes or from

field studies), then the limits and parameters of the Beta

distributions for each model parameter could be assigned

differently from catchment to catchment. The resulting

Beta distribution functions are shown in Figure 2.

We calibrated the rainfall-runoff model for 320 catch-

ments using two automatic calibration methods: the

MOSCEM-UA (Vrugt et al., 2003a) and the SCE-UA

Table I. A priori distribution of parameter values. u and v are parameters of the Beta function (Equation (7)), pl and pu are the

lower and upper bounds of the parameter space and pmaxis the parameter value at which the Beta distribution is at a maximum

(Equation (5))

Parameter Model part

u

v

pl

pu

pmax

SCF

DDF (mm°C?1day?1)

TR(°C)

TS(°C)

TM(°C)

LP/FC

FC (mm)

ˇ

K0(day)

K1(day)

K2(day)

CP(mm day?1)

CR?day2mm?1?

LSUZ(mm)

Snow

Snow

Snow

Snow

Snow

Soil

Soil

Soil

Runoff

Runoff

Runoff

Runoff

Runoff

Runoff

1Ð2

2Ð0

2Ð0

2Ð0

3Ð0

4Ð0

1Ð1

1Ð1

2Ð0

2Ð0

1Ð05

2Ð0

1Ð05

3Ð0

4Ð0

4Ð0

4Ð0

4Ð0

3Ð0

1Ð2

1Ð5

1Ð5

4Ð0

4Ð0

1Ð05

4Ð0

1Ð05

3Ð0

1Ð0

0Ð0

1Ð0

1Ð5

5Ð0

3Ð0

1Ð0

3Ð0

1Ð0

20

2Ð0

30

180

8Ð0

50

100

1Ð03

1Ð25

1Ð5

?2Ð0

0Ð5

0Ð94

100

3Ð4

0Ð5

9Ð0

105

2Ð0

25

50

?3Ð0

?2Ð0

0Ð0

0Ð0

0Ð0

0Ð0

2Ð0

30

0Ð0

0Ð0

1Ð0

600

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 21, 435–446 (2007)

DOI: 10.1002/hyp

Page 5

REGIONAL WATER BALANCE MODELLING

439

00.1 0.20.3 0.40.50.60.70.80.91

(1.2,4)

(4,1.2)

(2,4)

(3,3)

(1.05,1.05)

(1.1,1.5)

Figure 2. Shapes of the Beta functions used for defining the a priori

distributions of the model parameters. Number in parentheses are u and

v (see Table I)

(Duan et al., 1992) methods. The MOSCEM-UA algo-

rithm we applied to the SINGLE and MULTI calibration

cases. As a result of the calibration, the MOSCEM-UA

provides a discrete set of possible parameter combina-

tions (a Pareto set) that represent tradeoffs between opti-

mal ways of constraining the model to be consistent with

observed daily runoff and snow cover data. From the

Pareto set solutions we selected two parameter combi-

nations. The single criterion end point in the Pareto set

with respect to the minimum ZQrepresents the parameter

combination used for the SINGLE calibration case. The

second parameter combination represents a compromise

solution between the runoff and snow cover objective

functions and is used for the MULTI calibration case.

We selected this parameter set in a way that yielded the

minimum value of

ZMULTID 0Ð8ZQC 0Ð2

ZS

ZS,MAX

?9?

The weights (0Ð8 and 0Ð2) were determined by test

simulations and gave a relative importance of 80% to ZQ

and 20% to ZSon average over the 320 catchments.

An example of the Pareto solution space for the

Wienerbruck catchment is presented in Figure 3. The

triangle in Figure 3 represents the parameter set selected

for the SINGLE case and the big black circle represents

the parameter set selected for the MULTI case.

For three objective functions, the MOSCEM-UA

method can be numerically quite taxing as it evaluates

the entire three-dimensional Pareto solution space. For

the APRIORI case, therefore, we used the SCE-UA cal-

ibration algorithm instead, which is numerically more

efficient, and minimized one compound objective func-

tion ZC:

ZCD w1ZQC w2ZSCC w3ZP

where the part representing the snow objective function

ZSCis defined as the ratio of number of days with poor

?10?

0.20.40.6

ZQ [-]

0.81.0

4

6

8

10

12

14

ZS [% of days]

single objective

multiple objective

multiple-selection

Figure 3. Pareto solutions in the two-dimensional objective function

space obtained by the MOSCEM-UA method; Wienerbruck catchment

is shown as an example. The small black dots correspond to 1500 Pareto

solutions, the filled triangle represents the minimum runoff objective

function (SINGLE) and the large filled circle shows the compromise

solution (MULTI) between the runoff and snow objective functions, ZQ

and ZS

snow cover simulation nps to the number of days with

observed snow cover nosexpressed as

ZSCD ZS

n

nos

?11?

The weights in Equation (10) were assigned in test

simulations as w1D 0Ð7, w2D 0Ð2 and w3D 0Ð1. These

test simulations consisted of sensitivity analyses that

showed that the model results were only moderately

sensitive to the choice of weights. The selection of

weights is arbitrary and always depends on subjective

user requirements and expectations. In this paper we

estimated the weights so that, on average, the runoff

ZQ, snow ZSCand a priori penalty ZPcontribute to the

final compound objective function ZC with 40%, 40%

and 20% respectively.

The evaluation of the calibration and verification effi-

ciencies of the SINGLE, MULTI and APRIORI opti-

mization approaches, along with the comparison of their

parameter uncertainties, were performed in two steps. In

a first analysis, we split the entire period of observations

(1976–1997) into two 11-year periods: from 1 January

1976 to 31 December 1986 and from 1 January 1987

to 31 December 1997. Warm-up periods from January

to October were used in both cases. For the efficiency

estimation and comparison between different calibration

procedures, we performed a split sample test in the ter-

minology of Klemeˇ s (1986). We used the 11-year periods

in turn for calibration and validation, and compared the

model performances from both arrangements. In a sec-

ond analysis, we judged the parameter uncertainty for the

SINGLE, MULTI and APRIORI methods by comparing

the parameters calibrated for the 1976–1986 period with

those calibrated for the 1987–1997 period. The compari-

son of parameter values obtained in two different periods

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 21, 435–446 (2007)

DOI: 10.1002/hyp

Page 6

440

J. PARAJKA, R. MERZ AND G. BL¨OSCHL

gives the total amount of uncertainty of these parame-

ters, including uncertainty due to input data and model

structure.

CALIBRATION AND VERIFICATION

EFFICIENCIES

The performance of the single- and multiple-objective

parameter optimization approaches is presented in terms

of their efficiency to simulate runoff ME (Equation (2)),

runoff volume errors VE (Equation (3)) and snow cover

errors ZS(Equation (4). For a favourable model perfor-

mance, the ME runoff efficiencies should be large, the

VE volume errors should be close to zero with a small

scatter and the ZSsnow cover errors should be small.

Table II and Figures 4 and 5 (left panels) show the ME

model efficiencies of the SINGLE, MULTI and APRI-

ORI optimization approaches. The statistical evaluation

Table II. Model efficiency ME of runoff according to Nash–

Sutcliffe for the calibration and verification periods using the

SINGLE, MULTI and APRIORI calibration approaches

MEa

SINGLEMULTIAPRIORI

Calibration 1976–1986

Calibration 1987–1997

Verification 1987–1997

Verification 1976–1986

0.74/0.17

0.75/0.12

0.70/0.13

0.68/0.20

0.72/0.18

0.74/0.12

0.70/0.14

0.64/0.20

0.74/0.17

0.75/0.12

0.71/0.13

0.68/0.19

aFirst value: median of ME efficiency over the 320 catchments. Second

value: difference of the 75% and 25% quantiles of model efficiencies, i.e.

a measure of scatter. High model performances are associated with large

medians and a small scatter

in Table II includes the median and, as a measure of scat-

ter, the differences of the 75th and 25th percentiles over

all 320 catchments. The SINGLE optimization approach

yields median model efficiencies of ME D 0Ð74 and 0Ð75

for the two periods when they are used for calibration.

Incorporating snow cover data in the MULTI approach

slightly decreases the median efficiencies to ME D 0Ð72

and 0Ð74. However, the incorporation of both snow cover

data and an expert judgement about the expected parame-

ter values in the APRIORI optimization scheme yields the

same ME performance as the SINGLE approach. This is

because, in the APRIORI objective function, less weight

is given to snow than in the MULTI case. This result

suggests that the use of a priori information on parame-

ters does not necessarily decrease model efficiency. This

is a typical result, which, of course, depends on the

weights chosen in the objective function. For the veri-

fication cases, there is a slight decrease in runoff model

efficiency for all methods compared with the calibration

case, but the relative performance of the methods remains

similar. It is interesting that in the APRIORI case some of

the lowest model efficiencies improve over the SINGLE

case (Figure 5, bottom left panel), whereas the opposite

seems to be true of the MULTI case (Figure 5, top left

panel).

The runoff volume errors VE are shown in Table III

and Figures 4 and 5 (top centre and bottom centre pan-

els). The median VE values around 0% for the calibration

periods indicate that the calibration is essentially unbi-

ased for all calibration procedures. In the verification

periods, the median biases are around š5%, which is

likely related to data issues. The scatter is around 3–4%

in the calibration periods and increases to around 11%

0.0 0.20.4

SINGLE

0.60.81.0

0.00.20.4

SINGLE

0.6 0.81.0

0.0

0.2

0.4

0.6

0.8

1.0

MULTI

-40-200 2040

-40 -2002040

SINGLE

-40

-20

0

20

40

-40

-20

0

20

40

0 1020 30

SINGLE

0102030

SINGLE

0

10

20

30

ME [-] VE [%]ZS [% of days]

0.0

0.2

0.4

0.6

0.8

1.0

APRIORI

SINGLE

0

10

20

30

ME [-]VE [%]ZS [% of days]

Figure 4. Comparison of the model efficiency of daily runoff ME, runoff volume error VE and snow over error ZSestimated by the single objective

(SINGLE) and the multiple objective (MULTI and APRIORI) calibration approaches. Each point in a panel relates to one out of 320 catchments for

the calibration period 1976–1986

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ME [-]VE [%]

0

10

20

30

0

10

20

30

ZS [% of days]

APRIORI

ME [-]VE [%]ZS [% of days]

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

MULTI

0.00.2 0.4

SINGLE

0.60.81.0

0.00.20.4

SINGLE

0.6 0.8 1.0

-40

-20

0

20

40

-40

-20

0

20

40

-40-2002040

SINGLE

-40

-2002040

SINGLE

0 102030

SINGLE

0102030

SINGLE

Figure 5. Comparison of the model efficiency of daily runoff ME, runoff volume error VE and snow cover error ZSestimated by the single objective

(SINGLE) and the multiple objective (MULTI and APRIORI) calibration approaches. Each point in a panel relates to one out of 320 catchments for

the verification period 1976–1986

Table III. Volume errors VE of runoff for the calibration and

verification periods using the SINGLE, MULTI and APRIORI

calibration approaches

VEa(%)

SINGLEMULTIAPRIORI

Calibration 1976–1986

Calibration 1987–1997

Verification 1987–1997

Verification 1976–1986

?0.4/3.3

0.1/3.6

5.7/11.5

?5.5/10.7

?0.8/4.2

?0.3/4.2

4.9/11.5

?6.5/10.5

?0.1/1.9

0.3/2.8

6.3/12.0

?5.0/10.1

aFirst value: median of VE over the 320 catchments. Second value:

difference of the 75% and 25% quantiles of VE, i.e. a measure of scatter.

High model performances are associated with medians close to zero and

a small scatter.

in the verification periods. The patterns of the VE verifi-

cation efficiencies (Figure 5) indicate that, for a number

of catchments, the water balance is not closed properly

for the data. This is the case for all three optimization

schemes.

The snow cover model performances ZSare presented

in Table IV and Figures 4 and 5 (right panels). The

poorest snow cover simulations are obtained by the

SINGLE optimization approach. The median errors over

the 320 catchments are around 7% for the calibration

periods and the scatter is more than 5%. The MULTI

and APRIORI approaches significantly improve the snow

simulations, with medians of less than 5% and around

6% respectively. More importantly, the scatter for the

MULTI and APRIORI approaches is only 3% and 3–4%

respectively. For the verification periods, the MULTI

and APRIORI methods, again, outperform the SINGLE

Table IV. Snow cover simulation errors ZSfor the calibration and

verification periods using the SINGLE, MULTI and APRIORI

calibration approaches

ZS(percentage of days)

SINGLEMULTIAPRIORI

Calibration 1976–1986

Calibration 1987–1997

Verification 1987–1997

Verification 1976–1986

7.7/5.5

6.2/5.2

7.5/6.1

6.6/4.4

5.0/2.9

4.7/2.9

5.7/3.4

4.8/2.7

6.3/3.7

5.9/3.2

6.8/4.0

5.9/3.5

aFirst value: median over the 320 catchments of the percentage of days

with poor snow cover simulations. Second value: difference of the 75%

and 25% quantiles, i.e. a measure of scatter. High model performances

are associated with small medians and a small scatter.

method. The median snow errors are around 5% and 6%

respectively in the MULTI and APRIORI cases, with

scatters of around 3% and 4% respectively, and the

SINGLE case gives median errors of around 7% with a

scatter of around 6%. It is clear that the use of snow data

in calibration improves the snow simulations significantly

in both the calibration and verification periods, but the

interesting issue is the extent of such an improvement.

The results show that the snow data have reduced

significantly not only the median of the snow model

performance, but also the regional differences represented

by the percentile difference.

PARAMETER UNCERTAINTY

We evaluated the uncertainty of the model parame-

ters obtained from the SINGLE, MULTI and APRIORI

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J. PARAJKA, R. MERZ AND G. BL¨OSCHL

Table V. Parameter uncertainty assessed by the coefficient of

determination r2as a measure of how similar are the model

parameters calibrated for two independent 11-year periods

(1976–1986 and 1987–1997)

ParameterModel part

r2

SINGLE MULTI APRIORI

SCF

DDF

TR

TS

TM

LP/FC

FC

ˇ

K0

K1

K2

CP

CR

LSUZ

Snow

Snow

Snow

Snow

Snow

Soil

Soil

Soil

Runoff

Runoff

Runoff

Runoff

Runoff

Runoff

0Ð50

0Ð26

0Ð15

0Ð23

0Ð27

0Ð29

0Ð40

0Ð44

0Ð26

0Ð49

0Ð48

0Ð49

?0Ð01

0Ð14

0Ð40

0Ð53

0Ð22

0Ð41

0Ð45

0Ð18

0Ð31

0Ð26

0Ð25

0Ð44

0Ð31

0Ð41

0Ð00

0Ð18

0Ð56

0Ð36

0Ð19

0Ð25

0Ð34

0Ð25

0Ð54

0Ð55

0Ð54

0Ð75

0Ð43

0Ð53

0Ð01

0Ð56

optimization approaches by a comparison of their values

calibrated for the 1976–1986 period with those calibrated

for the 1987–1997 period. As measures of the uncertainty

we used the coefficient of determination r2and the root-

mean-square deviation normalized to the parameter range

(RMSD) of each parameter for the two periods. Reliably

estimated parameters with small uncertainties are those

where r2is large, RMSD is small and they should cluster

around the 1:1 line in scatter plots.

The coefficients of determination r2are presented in

Table V. The most uncertain parameters in all three

optimization approaches are the routing parameter CR

and the threshold temperature for liquid precipitation

TR. Parameters with smaller uncertainties are the snow

correction factor SCF, the fast storage coefficient K1and

the percolation rate CP in the SINGLE approach, the

degree-day factor DDF and the threshold temperature

for snowmelt TM in MULTI, and practically all soil

and runoff parameters (except the routing parameter CR

and the soil LP/FC ratio) estimated by the APRIORI

approach. It is now interesting to examine how the

uncertainty changes when moving from the SINGLE

to the MULTI approach. Strikingly, the coefficients of

determination increase significantly for the degree-day

factor DDF (from 0Ð26 to 0Ð53), for the lower and upper

threshold temperatures of the precipitation state TRand

TS(from 0Ð15 to 0Ð22 and from 0Ð23 to 0Ð41 respectively)

and for the threshold temperature of melt TM(from 0Ð27

to 0Ð45). These are all parameters related to the snow

module of the model. In contrast, the other parameters

(related to the soil and runoff components) tend to get

slightly more uncertain or do not change much in terms

of their uncertainty. Clearly, if snow data are used in

calibration it is mainly the snow component of the model

that can be expected to improve and, interestingly, only

the snow component. The APRIORI case, in contrast,

tends to reduce the uncertainty of all model parameters

compared with the SINGLE case. However, the snow

Table VI. Parameter uncertainty assessed by the root-mean-

square deviation normalized by the parameter range (RMSD) as

a measure of scatter of the model parameters calibrated for two

independent 11-year periods (1976–1986 and 1987–1997)

ParameterModel partRMSD

SINGLEMULTI APRIORI

SCF

DDF

TR

TS

TM

LP/FC

FC

ˇ

K0

K1

K2

CP

CR

LSUZ

Snow

Snow

Snow

Snow

Snow

Soil

Soil

Soil

Runoff

Runoff

Runoff

Runoff

Runoff

Runoff

0Ð32

0Ð27

0Ð42

0Ð41

0Ð30

0Ð42

0Ð35

0Ð38

0Ð41

0Ð29

0Ð34

0Ð27

0Ð40

0Ð39

0Ð34

0Ð24

0Ð39

0Ð37

0Ð26

0Ð43

0Ð35

0Ð45

0Ð44

0Ð30

0Ð40

0Ð28

0Ð39

0Ð41

0Ð22

0Ð11

0Ð06

0Ð09

0Ð13

0Ð10

0Ð16

0Ð22

0Ð06

0Ð08

0Ð26

0Ð10

0Ð16

0Ð10

model parameters are more uncertain than in the MULTI

case. This is because, in the APRIORI objective function,

less weight is given to snow than in the MULTI case.

As an alternative measure of parameter uncertainty,

Table VI shows the root-mean-square deviation of the

parameters in the two periods normalized by the parame-

ter range (RMSD). The change in uncertainty when mov-

ing from the SINGLE to the MULTI approach is similar

to that identified by the coefficients of determination. The

RMSD values decrease for the snow parameters and tend

to increase or remain similar for the other parameters.

However, the RMSD values decrease significantly for

all parameters when moving to the APRIORI approach.

This indicates that the APRIORI approach constrains the

parameters much more drastically than the other meth-

ods do.

Examples of the visual appearance of the differences

between the calibrated parameters for the two periods are

plotted in Figure 6 for the degree-day factor DDF, the

maximum soil moisture storage FC and the fast storage

coefficient K1. The left panels relate to the SINGLE

case, the middle panels to the MULTI case and the right

panels to the APRIORI case. The scatter for the degree-

day factor DDF decreases when moving from SINGLE

to MULTI, but the scatter of the other parameters does

not change much. In contrast, the APRIORI case shows

significantly less scatter for all parameters. It should be

noted that the reduction in parameter variability in the

APRIORI case compared with the SINGLE case does

not come at the cost of decreased model performance,

as the runoff model performances for the two cases are

similar.

DISCUSSION AND CONCLUSIONS

In this study we have assessed the model performance

and the parameter uncertainty of two multiple objective

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 21, 435–446 (2007)

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DDF

APRIORIMULTI SINGLE

FC

K1

DDF

FC

1987-97

0

10

20

30

1976-86

0

10

20

30

1976-86

K1

0

1987-97

0

1

2

3

4

5

1976-86

0

1

2

3

4

5

1976-86

0

1

2

3

4

5

1976-86

DDF

0200400600

1987-97

0200400600

1987-97

0200400600

1987-97

0

200

400

600

1976-86

0

200

400

600

1976-86

0

200

400

600

1976-86

FC

0102030

0102030

1987-97

0102030

1987-97

0

10

20

30

1976-86

K1

12345

0

1987-97

123450

1987-97

12345

Figure 6. Selected model parameters estimated by the SINGLE (left), the MULTI (centre) and the APRIORI (right) calibration approaches. Horizontal

axes show the parameters calibrated for the period 1987–1997; vertical axes show the parameters calibrated for the period 1976–1986. The top

panels show the degree-day factor DDF, the middle panels the maximum soil moisture storage FC, and the bottom panels show the fast storage

coefficient K1

calibration approaches and compared their effectiveness

with a single objective calibration procedure that involved

only daily runoff data in the calibration. The verifi-

cation approach used for the validation of the three

calibration procedures corresponds to the undisturbed-

catchment multi-response split-sample and independent-

sample strategies proposed by Mroczkowski et al. (1997).

The first multiple objective approach (termed MULTI)

uses snow cover data in addition to runoff. The results

indicate that the use of snow cover data significantly

improves the model performance in terms of simulat-

ing snow cover, but slightly decreases the performance

in terms of simulating runoff. This is true of both the

calibration and the verification periods. It is interesting

that this finding applies to a large number of catchments

in diverse hydroclimatic regions of Austria. For a much

smaller number of catchments (only two) and different

processes (groundwater instead of snow), Seibert (2000)

concluded that, when calibrating a model against two

objectives, ‘the values of the objective functions were

about 5 per cent below their values from the single-

criterion calibration for both criteria’. Similar results have

been reported by Madsen (2003), who showed that the

calibration based only on groundwater levels provided

poor simulation of catchment runoff. However, a minor

relaxation of the performance of the groundwater level

simulations led to a significant improvement in runoff

simulation. Along similar lines, Seibert and McDonnell

(2002) reported that the decrease in model performance

was mainly caused by a conflict between the criteria used

in parameter optimization for their case of incorporating

soft data in multiple objective calibration.

We also used expert judgement to constrain the param-

eter distribution in addition to using runoff and snow

cover data (termed APRIORI case). For the calibration

periods, the runoff model efficiencies of this case were

similar to single objective calibration on runoff only and

for the verification periods the efficiencies were even

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 21, 435–446 (2007)

DOI: 10.1002/hyp

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J. PARAJKA, R. MERZ AND G. BL¨OSCHL

slightly better. This suggests that it is useful to make

a priori assumptions on the distribution of parameters

even if these are approximate estimates.

In the APRIORI case, the snow cover model efficiency

was somewhat lower than in the MULTI case where

runoff and snow cover data were used; this is because

less weight is given to snow in the objective function.

However, the MULTI case yielded significantly better

snow simulations than the SINGLE case where no snow

data were used. It appears that similar values of the runoff

objective function do not necessarily imply a similar

hydrological response of the catchment. As pointed out

by Gupta et al. (2003), ‘regardless of the objective

function, the response surface contains numerous very

similar solutions (in terms of objective function value) at

widely differing locations in the parameter space’. The

reduction of parameter uncertainty, hence, is clearly of

paramount importance.

We used a method for assessing parameter uncertainty

that is based on a split sample comparison of model

parameters for two periods following the suggestion of

Merz and Bl¨ oschl (2004). The advantage of the method

over, say, Monte Carlo methods is that it accounts for

a range of uncertainties, including data uncertainties and

non-stationarity, but it may only be used in a meaningful

way if a large set of catchments is available for testing, as

is the case here. Quantification of parameter uncertainty

by the coefficient of determination and the root-mean-

square deviation showed that the use of additional snow

cover data significantly reduced the parameter uncertainty

for the parameters representing the snow cover dynamics,

but tended to increase the parameter uncertainty for other

model parameters. This is in line with Seibert (2000),

who, for the case of using additional groundwater level

data, noted that ‘the parameter uncertainty decreases for

five parameters of the response routine, which is the

part of the model representing groundwater dynamics’.

Our results show that incorporating an expert guess

about the parameter distribution, additionally to runoff

and snow cover data, reduced the scatter significantly

between parameters optimized for different periods. It

is important that this finding applies to an ensemble

of 320 catchments. For individual catchments, expert

knowledge may or may not reduce parameter uncertainty.

An individual example where this has been the case was

reported by Seibert and Mc Donnell (2002), who found

a reduction of the parameter uncertainty when soft data

were added to the multi-criteria model calibration.

The use of expert knowledge about the distribution of

model parameters in the optimization procedure appears a

promising avenue for further reducing parameter uncer-

tainty. In this study we have used the same parameter

distribution for all catchments, but it may be of advan-

tage to use different distributions for different catch-

ments, depending on catchment characteristics for exam-

ple. Alternatively, neighbouring catchments could be

used, as is the case in regional calibration (e.g. Szolgay

et al., 2003). We believe that constraining the parame-

ter distribution in a regional calibration procedure will

improve the effectiveness of multiple objective calibra-

tion and provide even more robust regional patterns of

model parameters.

ACKNOWLEDGEMENTS

We would like to thank the Marie Curie Fellowship of

the European Community programme HUMAN POTEN-

TIAL under contract number HPMF-CT-2002-01872,

and the Austrian Academy of Sciences, project H¨O,

for financial support. We would also like to thank the

Austrian Hydrographic Service (HZB) for providing the

hydrographic data. We are grateful also to the three

anonymous reviewers for their thoughtful comments and

suggestions.

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algorithmfor optimizationand

APPENDIX A

Snow model

The snow routine represents snow accumulation and

melt by a simple degree-day concept. Mean daily precip-

itation P in an elevation zone is partitioned into rain PR

and snow PS based on the mean daily air temperature

TA:

PRD P

PRD PTA? TS

TR? TS

PRD 0

PSD P ? PR

where TS and TR are the lower and upper threshold

temperatures respectively. Melt starts at air temperatures

above a threshold TM:

if TA½ TR

if TS< TA< TR

if TA< TS

?A.1?

M D ?TA? TM?DDF

where M is the amount of melt water per time step,

DDF is the degree-day factor and SWE is the snow water

equivalent. The catch deficit of the precipitation gauges

during snowfall is corrected by a snow correction factor

SCF. Changes in the snow water equivalent from days

i ? 1 to i are accounted for by

SWEiD SWEi?1C ?SCFPS? M?t

where t is the time step of 1 day.

if TA> TMand SWE > 0

?A.2?

?A.3?

Soil moisture accounting

The soil moisture routine represents runoff generation

and changes in the soil moisture state of the catchment:

SSM,iD SSM,1?iC PRC M ? EA

where SSM is the soil moisture of a top soil layer

controlling runoff generation and actual evaporation EA.

The contribution SUZof rain and snowmelt to runoff is

calculated by an explicit scheme as a function of the

soil moisture of the top layer SSM using a non-linear

relationship with two free parameters, FC and ˇ:

?SSM

FC is the maximum soil moisture storage. The parameter

ˇ controls the characteristics of runoff generation and is a

non-linearity parameter. If the top soil layer is saturated,

i.e. SSMD FC, then all rainfall and snowmelt contributes

to runoff. The actual evaporation EAis calculated from

potential evaporation EPby a piecewise linear function

of the soil moisture of the top layer:

EAD EPSSM

LP

EAD EP

where LP is a parameter termed the limit for potential

evaporation EP.

?A.4?

SUZD

FC

?ˇ

?PRC M??A.5?

if SSM< LP

if SSM½ LP

?A.6?

Response and transfer functions

The response function represents runoff routing on the

hillslopes and consists of two reservoirs, representing two

soil zones. The storage states of the upper and lower

zones are SUZ and SLZ respectively. SUZ enters the

upper zone reservoir and leaves this reservoir through

three paths: outflow from the reservoir with a fast storage

coefficient of K1, percolation to the lower zone with a

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 21, 435–446 (2007)

DOI: 10.1002/hyp

Page 12

446

J. PARAJKA, R. MERZ AND G. BL¨OSCHL

constant percolation rate CP, and, if a threshold LSUZof

the storage state is exceeded, through an additional outlet

with a storage coefficient of K0. Water leaves the lower

zone with a slow storage coefficient of K2. The outflow

from both reservoirs QG is then routed by a triangular

transfer function, which represents the runoff routing in

the streams:

BQD BMAX? CRQG

BQD 1

if ?BMAX? CRQG? ½ 1

otherwise

?A.7?

where BQis the base of the transfer (triangular) function,

BMAX is the maximum base at low flows and CR is a

free scaling parameter. The BMAXmodel parameter was

set to a constant value of BMAXD 10 days on the basis of

sensitivity analyses, whereas the remaining 14 parameters

were found by the calibration.

Copyright 2006 John Wiley & Sons, Ltd.

Hydrol. Process. 21, 435–446 (2007)

DOI: 10.1002/hyp

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