Journal of Hydrology 519 (2014) 2202–2214

Contents lists available at ScienceDirect

Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Improvement and comparison of likelihood functions for modelcalibration and parameter uncertainty analysis within a Markov chainMonte Carlo scheme

http://dx.doi.org/10.1016/j.jhydrol.2014.10.0080022-1694/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +49 (0)30 838 70255; fax: +49 (0)30 838 70753.E-mail address: chengqinbo@gmail.com (Q.-B. Cheng).

Qin-Bo Cheng a,⇑, Xi Chen b, Chong-Yu Xu c,d, Christian Reinhardt-Imjela a, Achim Schulte a

a Freie Universität Berlin, Institute of Geographical Sciences, Malteserstraße 74-100, 12249 Berlin, Germanyb State Key Laboratory of Hydrology Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, Chinac Department of Geosciences, University of Oslo, P.O. Box 1047, 0316 Oslo, Norwayd Department of Earth Sciences, Uppsala University, Sweden

a r t i c l e i n f o s u m m a r y

Article history:Received 11 September 2013Received in revised form 25 September 2014Accepted 3 October 2014Available online 14 October 2014This manuscript was handled by AndrasBardossy, Editor-in-Chief, with theassistance of Niko Verhoest, Associate Editor

Keywords:Bayesian inferenceBox–Cox transformationNash–Sutcliffe Efficiency coefficientGeneralized Error DistributionSWAT-WB-VSA

In this study, the likelihood functions for uncertainty analysis of hydrological models are compared andimproved through the following steps: (1) the equivalent relationship between the Nash–Sutcliffe Effi-ciency coefficient (NSE) and the likelihood function with Gaussian independent and identically distrib-uted residuals is proved; (2) a new estimation method of the Box–Cox transformation (BC) parameteris developed to improve the effective elimination of the heteroscedasticity of model residuals; and (3)three likelihood functions—NSE, Generalized Error Distribution with BC (BC-GED) and Skew GeneralizedError Distribution with BC (BC-SGED)—are applied for SWAT-WB-VSA (Soil and Water Assessment Tool –Water Balance – Variable Source Area) model calibration in the Baocun watershed, Eastern China. Perfor-mances of calibrated models are compared using the observed river discharges and groundwater levels.The result shows that the minimum variance constraint can effectively estimate the BC parameter. Theform of the likelihood function significantly impacts on the calibrated parameters and the simulatedresults of high and low flow components. SWAT-WB-VSA with the NSE approach simulates flood well,but baseflow badly owing to the assumption of Gaussian error distribution, where the probability ofthe large error is low, but the small error around zero approximates equiprobability. By contrast,SWAT-WB-VSA with the BC-GED or BC-SGED approach mimics baseflow well, which is proved in thegroundwater level simulation. The assumption of skewness of the error distribution may be unnecessary,because all the results of the BC-SGED approach are nearly the same as those of the BC-GED approach.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Hydrologists have developed many models based on differenttheories and concepts, such as SWAT (Neitsch et al., 2005) basedon the principle of the hydrologic response unit (HRU), TOPMODEL(Beven and Kirkby, 1979) based on the topographic wetness index(TWI), TOPKAPI (Liu and Todini, 2002) based on the nonlinearreservoir theories, and HBV model (Wrede et al., 2013; Li et al.,2014) based on a modification of the bucket theory in that itassumes a statistical distribution of storage capacities in a basin.However, because of the hydrologic complexity and especiallythe hydrologic heterogeneity, these models cannot describe thenatural hydrologic processes entirely correctly, and their parame-ters can be interpreted only to the ‘‘effective parameters’’ which

represent the integrated behavior at the model element scale.Because it is difficult to determine the ‘‘effective parameters’’directly from field measurement, the model parameters shouldbe determined through calibration against the historical recorddata (Laloy et al., 2010). Owing to the lack of sufficient observationdata and the inter-dependence of model parameters, equifinality ofparameter sets must be expected instead of a single ‘optimal’parameter set in calibration against field data (Beven, 2001;Beven and Freer, 2001).

Additionally, errors in input data, model structure and mea-sured outcomes are all lumped into a single additive residual term,and then passed to the model parameters when calibrating thehydrological model (Yang et al., 2007a,b; McMillan and Clark,2009; Schoups and Vrugt, 2010). The parameter equifinality anderrors, individually or combined, result in parameter uncertainty.So, the calibration of model parameters is being developed toinclude estimation of the probability distribution of parameters

Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214 2203

that represents the knowledge about parameter values (Yang et al.,2007b), and the Bayesian approach (usually using Markov chainMonte Carlo scheme (MCMC)) is popularly proposed (Jin et al.,2010; Li and Xu, 2014). The Bayesian approach (or MCMC) triesto separate the observations (e.g. river discharges) into two parts:a deterministic component and a random component describingresiduals (Schoups and Vrugt, 2010). The deterministic componentis determined by the hydrologic model. The joint probability of therandom component, i.e. residuals/errors between observations andsimulations generated by hydrologic model with a particularparameter set, is estimated by a likelihood function. Byaugmenting the likelihood function with prior knowledge of modelparameters, the posterior distribution of model parameters isestimated.

There are many successful applications for calibration anduncertainty analysis of model parameters using the Bayesianapproach with the MCMC scheme. For example, McMillan andClark (2009) used a modified NSE (Nash–Sutcliffe Efficiency coeffi-cient (Nash and Sutcliffe, 1970)) as an informal likelihood functionto calibrate model parameters. However, the modified NSE failed toreveal the relationship between the NSE and the likelihood func-tion with statistical assumptions. Stedinger et al. (2008) indicatedthat the standard least squares (SLS), equivalent to maximizingNSE, is a kind of formal likelihood function under the assumptionthat the errors follow Gaussian distribution with zero mean anda constant variance. This theoretical derivation, however, is non-strict because of fixing the standard deviation of residuals/errors.

In recent years, some doubts have been expressed about the for-mal Bayesian approach. The two main reasons are summarized asfollows (Beven et al., 2012; Clark et al., 2012): First, the formalBayesian inference mistakenly treated all residuals as randomerrors; second, there is no generalized likelihood function thatcould be appropriate for all model structures. Beven et al. (2012)indicated that the model residuals include epistemic errors (suchas model structure and input errors) as well. The epistemic errorsresult in the correlative and heteroscedastic characteristics ofmodel residuals. In order to account for the errors’ correlationand heteroscedasticity, many researchers (Yang et al., 2007a,b;Schoups and Vrugt, 2010; Smith et al., 2010; Li et al., 2011) add a‘‘gray box’’ before calculation of the likelihood function in theMCMC scheme.

The first-order autoregressive (AR(1)) scheme and the Box–Coxtransformation method (BC) are widely used to remove errors’ cor-relation and heteroscedasticity, respectively (Vrugt et al., 2009a;Schoups and Vrugt, 2010; Smith et al., 2010; Li et al., 2011). TheBox–Cox transformation method needs to estimate transformationparameter (k). Most studies (Vrugt et al., 2009a; Engeland et al.,2010; Li et al., 2011) fixed the value of k, and some others (Yanget al., 2007a,b; Laloy et al., 2010) treat k as an inference parameter.Obviously, it is more effective to remove the errors’ heteroscedas-ticity when k varied as model predictions. Unfortunately, almost allthe inference results touch the boundary of k(0 6 k 6 1), such asthe result (k) of Yang et al. (2007b) approaches to zero, and k ofLaloy et al. (2010) approaches to one. The boundary value meansthe extreme situation, e.g. when k = 1, the BC is ineffective, i.e. notransformation of model residuals, and the BC becomes the logtransformation when k = 0, although it rarely occurs. Therefore, itis necessary to build a new efficient method to estimate the trans-formation parameter (k) in the MCMC scheme.

Another question is: which probability distribution is appropri-ate for the random errors? Gaussian distribution is widely used asthe probability distribution of the errors/residuals. However,recently some researchers have shown that there are many casesof non-Gaussian errors (Thiemann et al., 2001; Yang et al.,2007b; Schoups and Vrugt, 2010; Smith et al., 2010; Li et al.,2013). Some researchers proposed the Generalized Error

Distribution (GED) (Thiemann et al., 2001; McMillan and Clark,2009) and the Skew Generalized Error Distribution (SGED)(Schoups and Vrugt, 2010) that was developed from the GED.

The objective of this study is to assess the effect of differentlikelihood functions on the Bayesian inference in hydrologicalmodeling. The primary goal is achieved through the followingsteps. Firstly, we establish a relationship between the Nash–Sutcliffe Efficiency coefficient (NSE) and the likelihood function;then we introduce a constraint to estimate the Box–Cox transfor-mation (BC) parameter (k); finally we compare three likelihoodfunctions—NSE, Generalized Error Distribution with Box–Coxtransformation (BC-GED) and Skew Generalized Error Distributionwith Box–Cox transformation (BC-SGED) approaches—within theDiffeRential Evolution Adaptive Metropolis (DREAM) Markov ChainMonte Carlo (MCMC) scheme to discuss the effect of the form oflikelihood function.

2. Study area and hydrologic model

2.1. Study area

The Baocun watershed (86.7 km2) is a rural, mountainouswatershed located in the eastern Jiaodong Peninsula in China(Fig. 1). The elevation of the watershed ranges from 20 m at thewatershed outlet to about 220 m above mean sea level at thehead-watershed. The length of the watershed is 16.1 km, the aver-age width 5.4 km and the average slope 8.2‰. The climate of thewatershed belongs to the Western Pacific Ocean extratropicalmonsoonal region with 70% of the rain falling between June andSeptember (Fig. 2). The average annual precipitation is 805.6 mmwith average annual potential evapotranspiration loss of899.0 mm (measured by pan evaporation equipment termedE601). The average monthly temperature ranges from �0.8 �C inJanuary to 24.4 �C in August. Fig. 2 shows that the warmest monthscorrespond with the moistest months, and vice versa.

The geology of the watershed is mainly volcanic and metamor-phic rocks, and the dominant parent materials of the soil are gran-ite, diorite and gneiss. The dominant soil types are Luvisols,Regosols and Fluvisols covering about 94% of the area (FAO/IIASA/ISRIC/ISSCAS/JRC, 2009). The main land use is agriculture(terraced cropland), and the dominant crops are peanuts, cornand winter wheat. The agricultural lands are farmed three timesevery two years (termed crop rotation). The detailed schedules ofplanting crops are peanuts in May, winter wheat in October andcorn in June next year.

2.2. SWAT-WB-VSA model

The Soil and Water Assessment Tool (SWAT) is popularly usedfor water resource management all over the world (Gassmanet al., 2007). SWAT describes the spatial distribution of hydrologi-cal processes by dividing a watershed into multiple sub-basins,which are then further subdivided into hydrologic response units(HRUs) consisting of homogeneous land use, soil characteristicsand slope. HRU is the smallest element of SWAT. The Soil Conser-vation Service curve number procedure (CN) is widely used tosimulate the surface runoff generation in SWAT. However, the CNis an empirical method, which has some limitations in reflectingthat soil moisture affects the surface runoff generation (Hanet al., 2012). White et al. (2011) proposed a new model (termedSWAT-WB), which incorporated a physics-based rainfall-runoffapproach (i.e. the Water Balance (WB) method) into SWAT. In thisstudy, for reflecting the effect of topography on runoff, the VariableSource (runoff) Area (VSA) is incorporated into the SWAT-WBmodel. This new model is called SWAT-WB-VSA. In every HRU of

(a) (b)Fig. 1. Location and topography of the Jiaodong peninsula (a) and the Baocun watershed (b) as well as position of the gauging stations. Data base: 1:10,000 scale topographicmap.

-5

0

5

10

15

20

25

0

50

100

150

200

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Tem

pera

ture

(°C

)

Wat

er q

uant

ity (m

m)

PrecipitationPotential evaporationTemperature

Fig. 2. The average monthly precipitation, potential evaporation and temperature in Baocun watershed during 1953–2011.

2204 Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214

SWAT-WB-VSA, surface runoff occurs only when the precipitationamount is greater than the available soil moisture storage (s):

Q surf ¼Rday � s Rday > s0 others

�ð1Þ

where Qsurf is the daily overland flow (mm) and Rday is the dailyrainfall (mm).

The available soil moisture storage (s) is termed as thesaturation deficit in the soil profile:

s ¼ EDC � g� h ð2Þ

where EDC is the effective depth of the soil profile (unitless, rangingfrom 0 to 1), g is the total soil porosity of the soil profile (mm) and his the volumetric soil moisture for each day (mm).

In SWAT-WB-VSA, the soil topography index map is used toreplace the soil type map, which combines the soil type map with

the topographic index (TI, and TI = ln(A/tanb), where A is upslopecontributing area and tanb is slope) map. Each soil-topographicindex class has its own parameter of EDC (i.e. EDCi). SWAT-WB-VSA assumes the linear relationship between EDCi and TIi basedon the TOPMODEL concept (Beven and Kirkby, 1979; Chen et al.,2010):

EDCi ¼1� TIi

TI� 1� EDCð Þ TIi

TI� ð1� EDCÞ < 1

0 others

8<: ð3Þ

where TI is the catchment average TIi and EDC is the catchmentaverage EDCi.

The SWAT-WB-VSA model only improves the surface runoffgeneration approach without changing other parts in the SWATmodel, such as interflow, groundwater return flow and channelrouting. In SWAT, the kinematic storage model is used to estimatethe interflow, which linearly depends on the soil hydraulic

Table 1Description of model parameters and their ranges.

Categories Parametera Min Max Meaning

Evapotranspiration v__ESCO 0.01 1 Soil evaporation compensation factorv__EPCO 0.01 1 Plant uptake compensation factor

Surface water v__EDC 0 1 Effective depth of the soil profilev__OV_N 0.005 0.5 Roughness coefficient of sloping surfacev__SURLAG 0 24 Surface runoff lag coefficient

Soil water r__SOL_Z �0.9 2 Soil layer deepness (mm)r__SOL_BD �0.6 1 Moist bulk density (g cm�3)r__SOL_AWC �0.99 3 Available water capacity of the soil layerr__SOL_K �0.99 10 Saturated hydraulic conductivity (mm/d)

Ground water v__GW_DELAY 0 60 Groundwater delay time (days)v__ALPHA_BF 0 1 Baseflow recession constantv__GWQMN 0 1000 Threshold depth of water in the shallow aquifer required for return flow to occur (mm)v__RCHRG_DP 0 1 Deep aquifer percolation fractionv__REVAPMN 0 1000 Threshold depth of water in the shallow aquifer for revaporization (mm)v__GW_REVAP 0.02 0.2 Groundwater revaporization coefficient

Tributary/main channel v__CH_N1 0.005 0.15 Roughness coefficient of tributary channelsv__CH_N2 0.005 0.15 Roughness coefficient of main channels

a ‘‘v__’’ and ‘‘r__’’ refer to a replacement and a relative change to the initial parameter, respectively.

Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214 2205

conductivity (SOL_K, Table 1). The groundwater return flow is cal-culated via an Exponential-Decay model with the baseflow reces-sion constant (ALPHA_BF, Table 1). The Exponential-Decay modelwith the delay time constant (GW_DELAY, Table 1) is also usedto estimate the recharge from the soil layer to the shallow aquifer.The channel routing is calculated by the variable storage routingmethod with the Manning roughness constant (CH_N2, Table 1).Evaporations from soils and plants are calculated separately, whichdepend on soil depth and water content. The shallow aquiferevaporation (called ‘‘revap’’) is estimated by a linear function ofpotential evapotranspiration with the conversion coefficient(GW_REVAP, Table 1).

In this study, 17 key parameters will be identified by modelcalibration (Table 1). The so-called aggregated parameters areselected instead of the original parameters for simplifying calibra-tion parameters (Yang et al., 2007a). Aggregated parameters areformed by adding a modification term which includes two types:‘‘v__’’ and ‘‘r__’’, referring to a replacement and a relative changeto the initial parameter, respectively (the second column of Table 1).For example, the v__ESCO is the value of parameter ESCO, and ther__SOL_K is the relative change to the initial SOL_K:r__SOL_K = SOL_Know/SOL_Kinitial � 1, where SOL_Know and SOL_Kinitial are the active and the initial soil hydraulic conductivity(SOL_K), respectively.

2.3. Input data

The data of the digital elevation model (DEM) with 30 m resolu-tion are from resampling of the 1:10,000 scale topographic mapwith 2.5 m contour interval (Fig. 1b). The watershed is divided intoseven sub-basins according to the river system (Fig. 1b). Accordingto field investigations, the soil types in Boacun watershed can beclassified along with TI values because of the uniform geologicalcondition within a small mountain watershed. In this study, TI val-ues within the ranges of 3.2–7.5, 7.5–9.5 and 9.5–24.3 correspondto Regosols, Luvisols and Fluvisols (Fig. 3a), which account for 50%,30% and 20% of the total area that approximate the proportions ofsoil types in Baocun watershed according to the HarmonizedWorld Soil Database (FAO/IIASA/ISRIC/ISSCAS/JRC, 2009). The soilproperty data adopt field survey results (Cheng et al., 2011). InSWAT-WB-VSA, the soil type map of SWAT is replaced by the soiltopography index map to reflect the effect of topography on runoff.Because soil types depend on TI, soil-topographic indexes are

classified into ten units in terms of ten TI classes of equal areashown in Fig. 3b, where the value of TI in the legend is the averagedTI across each topographic index class.

The land use is only agriculture. However, the crops cyclicallyvary because of the crop rotation method. The average annual landuse map is charted to overcome the problem of land utilizationchange. The detailed steps are as follow: Firstly, extract the fixedland use from the 1:10,000 scale topographic map, such as waterbody, forest (mostly are apple orchard) and residential area; sec-ondly, randomly fill the rest area by peanut, winter wheat and cornwith the proportion of 0.25, 0.5 and 0.25 respectively according tothe growth time of crops. The average annual land use map isshown in Fig. 3c.

The meteorological data were collected at three nationalweather stations (Weihai, Chengshantou and Shidao, Fig. 1a). Theprecipitation data were observed at four rainfall gauges inside Bao-cun watershed (Fig. 1b). The potential evapotranspiration (PET)uses the data measured by E601 pan evaporation equipment atthe Baocun hydrometric station. The river discharge data at theBaocun hydrometric station are used as observed outcomes. Themodel running period is from 1990 to 2011, and the modelwarm-up period is from 1990 to 1992, which is used to avoid theeffects of the initial state of SWAT. The calibrated model is vali-dated using groundwater level data measured in the watershed(Fig. 1b) from the period of June 2007 to December 2011.

3. Methods

The likelihood function of a set of model parameter values (h)given observed outcomes (obs) is equal to the joint probability ofthe observed outcomes given the parameter values. The maximumlikelihood function is frequently used to estimate the parametersof statistical model.

3.1. Interpretation of the Nash–Sutcliffe Efficiency coefficient (NSE)from the likelihood function viewpoint

The errors/residuals (e) between the observed and simulatedoutcomes are treated as random variables:

ei ¼ obsi � simi ð4Þ

where obsi and simi are the observed and the simulated outcomes attime step i, respectively.

(a) (b) (c) Fig. 3. The spatial data in the Baocun watershed for SWAT-WB-VSA model: (a) soil type map; (b) soil-topographic index map; (c) average annual land use map.

2206 Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214

If the error ei is assumed to be independent and identicallydistributed (I.I.D.) according to Gaussian distribution with zeromean and a constant variance, the probability density function(PDF) of ei is:

p eijhð Þ ¼ 1ffiffiffiffiffiffiffi2pp

rexp � ei

2

2r2

� �ð5Þ

where r termed standard deviation is a parameter of the Gaussianprobability density function.

The logarithmic likelihood function of parameter set (h) can beexpressed as:

l hjobsð Þ ¼ lnYn

i¼1

p eijhð Þ !

¼ �n2

ln 2pð Þ � n2

ln r2� �

�Xn

i¼1

e2i

2r2 ð6Þ

where n is the length of time series of errors.When r2 ¼

Pn1e2

i =n that is the unbiased estimator of r, the like-lihood function (Eq. (6)) reaches the maximum value. So:

l hjobsð Þmax ¼ �n2

ln 2pð Þ � n2

ln r2� �� n

2

¼ �n2

ln 2peð Þ � n2

lnXn

1

e2i

n

!ð7Þ

where e is the base of the natural logarithm, �2.718.NSE (Nash and Sutcliffe, 1970) is:

NSE ¼ 1�Pn

1ðsimi � obsiÞ2Pn1 obsi � obs� 2 ¼ 1�

Pn1e2

iPn1 obsi � obs� 2 ð8Þ

where obs is the mean observed outcomes.From Eq. (8) we can obtain:

Xn

1

e2i

n¼ 1� NSEð Þn� 1

nr2

obs ð9Þ

where robs is the standard deviation of observed outcomes.By substituting Eq. (9) into Eq. (7), the logarithmic likelihood

function (Eq. (7)) is transformed to:

l hjobsð Þmax ¼ �n2

ln 2pen� 1

nr2

obs

� �� n

2ln 1� NSEð Þ ð10Þ

In Eq. (10), the � n2 ln 2pe n�1

n r2obs

� �is constant, so the likelihood

function is equivalent to NSE. In other words, the NSE is equivalentto a likelihood function under the assumption that the errorsbetween observed and simulated outcomes follow the Gaussianerror distribution with zero mean and a constant variance. In theNSE approach, the standard deviation of model residuals (r) is esti-mated by the unbiased equation that is also used by Vrugt et al.(2009a) and Laloy et al. (2010).

3.2. Formal likelihood function

3.2.1. Removal of the errors’ heteroscedasticityErrors between observed and simulated river discharges typi-

cally exhibit considerable heteroscedasticity, autocorrelation andnon-normality (Evin et al., 2013). These error characteristics (cor-relation, heteroscedasticity, etc.) need to be explicitly accountedfor before calculation of the likelihood function. In the Baocunwatershed, the floods present impulse form in daily river dischargechart shown in Fig. 4a because flood concentration time is only twohours, and rainfall events are short but very intense owing totyphoon storms. This implies that the autocorrelation of river dis-charges at daily step is relatively weak, or equivalently, the auto-correlation of errors between daily observation and simulationshould be weak for a good predictive performance. A graphic checkof the autocorrelation of errors confirms this assumption (resultsare not shown). Therefore, the errors’ autocorrelation will beignored in this study, which is adopted by many authors(Thiemann et al., 2001; Smith et al., 2010; Pianosi and Raso,2012). Evin et al. (2014) pointed out that the likelihood functionincorporating autocorrelation has disadvantages on the estimatedparameter uncertainty in some cases. However, the errors’heteroscedasticity resulted from larger rainfalls and streamflowsin Fig. 4a should be accounted for in the Baocun watershed (Evinet al., 2013).

Fig. 4. Comparison of the observed and the simulated river discharges of the NSE approach. The logarithmic vertical-axis (base 10) is used to emphasize the baseflow.

Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214 2207

The Box–Cox transformation (BC) method is widely used toremove heteroscedasticity of errors (Yang et al., 2007a,b; Smithet al., 2010; Li et al., 2011, 2013). One-parameter BC function(Box and Cox, 1964) is:

g yð Þ ¼yk � 1

kk – 0

lnðyÞ k ¼ 0

8<: ð11Þ

where k is the BC parameter, and y is simulated or observedoutcomes.

Because:

limk!0yk � 1

k¼ ln yð Þ ð12Þ

the BC can be simplified as:

g yð Þ ¼ yk � 1k

0 < k � 1 ð13Þ

The errors/residuals (e) between the observed and thesimulated outcomes after BC are expressed as:

ei ¼obsk

i � simki

k0 < k � 1 ð14Þ

k in Eq. (14) heavily affects the efficiency of the BC method.According to previous studies (Bates and Campbell, 2001; Yanget al., 2007a,b), the likelihood function should be changed forestimating k:

l hjobsð Þ ¼ l0 hjobsð Þ þ ðk� 1ÞXn

1

lnðobsiÞ ð15Þ

where l0 hjobsð Þ is the original likelihood function.Eq. (15) indicates that if k approaches zero, the likelihood

function reaches the maximum value (positive infinity) for thesmall observed outcomes (obsi), e.g. zero. In other words, the infer-ence result of k has always approached zero (Yang et al., 2007b), nomatter what the parameter sets are. So the constraint (Eq. (15)) of kis invalid, probably because the Eq. (15) incorrectly transplantedthe Jacobian determinant from the original BC (Eq. (11)) methodto the hydrological BC (Eq. (14)) method. The Jacobian determinantof the transformation from the model residual after BC (Eq. (14)) tothe raw model residual (Eq. (4)) may be non-existent. However, if

there is no constraint on the value of k (i.e. totally treating k as ahydrological model parameter), the inference result of k willalways approach one (i.e. no transformation of errors) when thereare many small observed outcomes (Laloy et al., 2010). It probablyresults from that the mode (i.e. the highest probability point) oferrors is zero, and the no-transformation of errors (closed to zero)contributes to the maximization of the likelihood function. There-fore, treating k as an additional inference parameter of the likeli-hood function cannot yield an effective BC parameter for removalof the heteroscedasticity of model residuals at the moment.

In fact, the hydrological Box–Cox transformation (BC) should betreated as an implicit model for removal of the heteroscedasticityof model residuals, in contrast to the explicit statistical model,e.g. the standard deviation of model residuals is modeled as a lin-ear function of simulated streamflow (Schoups and Vrugt, 2010) ora function of time (Pianosi and Raso, 2012). The parameters of theexplicit error-model can be directly inferred by the likelihood func-tion, because the explicit error-model has the Jacobian determi-nant of transformation (Schoups and Vrugt, 2010).

The implicit error model firstly uses a filter (e.g. BC method) totransform the raw model residuals (Eq. (4)) into the independentand identically distributed (I.I.D.) errors, and then estimates thelikelihood function of I.I.D. errors. Fixed transformation parameters(k) of BC by most studies obviously impaired the effectiveness ofBC (Bates and Campbell, 2001; Vrugt et al., 2009a; Engelandet al., 2010). Therefore, for improving the performance of BCthrough removal of the heteroscedasticity of model residuals, aconstraint is introduced for calculation of k to minimize the vari-ance of the time series of errors:

min variance eið Þð Þ ¼ 1n� 1

Xn

1

obski � simk

i

k� l

!2

min

ð16Þ

where l is the mean of errors after BC (Eq. (14)).The filter of BC with minimum variance constraint (Eq. (16)) in

the implicit error model is not isolated from the likelihood ofmodel residuals. The connection between the filter and likelihoodis implicit: the filter changes the transformation parameter toaffect the likelihood, and the likelihood affects the filter via hydro-logical model outcomes.

2208 Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214

3.2.2. Error distribution modelFor the non-Gaussian distribution, the Generalized Error Distri-

bution (GED) and the Skew Generalized Error Distribution (SGED)are selected for comparison.

The Generalized Error Distribution (GED) (Thiemann et al.,2001; McMillan and Clark, 2009) is expressed as:

p eijhð Þ ¼ xðbÞr

exp � cðbÞ ei � lr

b� �ð17Þ

where c bð Þ ¼ffiffiffiffiffiffiffiffiffiffiC½3=b�C½1=b�

q, x bð Þ ¼ bc bð Þ

2C½1=b�, b termed kurtosis is a parameter

of the probability density function of GED (b > 0), and C[x] is thegamma function evaluated at x.

GED is more flexible than Gaussian distribution: when b = 2,GED becomes the Gaussian distribution; when b = 1, GED is aLaplace distribution; and GED approaches a uniform distributionas b approaches infinity. However, GED is still a symmetric errordistribution.

Schoups and Vrugt (2010) used a more flexible error distribu-tion—Skew Generalized Error Distribution (SGED)—which is devel-oped from the GED:

p eijhð Þ ¼ 2xðbÞrn

rðnþ n�1Þexp � cðbÞn�sign lnþrn

ei�lrð Þ ln þ rn

ei � lr

� b� �ð18Þ

where ln = M1(n � n�1), rn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�M21

� n2 þ n�2� �

þ 2M21 � 1

r,

M1 ¼ C½2=b�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC½3=b�C½1=b�p , and n termed skewness is a parameter of SGED

(n > 0).SGED is positively (negatively) skewed for n > 1 (n < 1), and

when n = 1, SGED is symmetric and becomes the GED. The shapesof the three probability density functions (i.e. Gaussian distribu-tion, GED and SGED) are compared by Schoups and Vrugt (2010).

The statistical assumptions of three likelihood functions—NSE,Generalized Error Distribution with Box–Cox transformation(BC-GED) and Skew Generalized Error Distribution with Box–Coxtransformation (BC-SGED) approaches—are shown in Table 2. Thistable shows that NSE and BC-GED are the special cases of BC-SGED.However, the number of error model parameters of BC-SGED (k, l,r, b and n) is more than that of BC-GED (k, l, r and b), and morethan that of NSE (parameter-free).

3.3. Bayesian inference

The Bayesian theorem describes the relationship between theconditional distribution for model parameter set (h) givenobserved outcomes (obs) and the joint distribution of h and obs.The Bayesian theorem states:

p hjobsð Þ ¼ p hð Þf obsjhð ÞRp hð Þf obsjhð Þdh

/ p hð Þf obsjhð Þ ð19Þ

where p(h) is the set of prior distributions for parameter set (h),f(obs|h) is the likelihood function,

Rp hð Þf obsjhð Þdh is marginal

Table 2Comparison of the likelihood function characteristics among NSE, BC-GED and BC-SGED a

Likelihoodfunction

Statistical assumption Errors’ distributiona

NSE Independent and homoscedastic Gaussian with zero mean(=SGED(l = 0, b = 2, n = 1))

BC-GED Independent and homoscedastic afterBox–Cox transformation

GED (=SGED(n = 1)) after BoCox transformation

BC-SGED Independent and homoscedastic afterBox–Cox transformation

SGED after Box–Coxtransformation

a l is the mean of SGED; b is the kurtosis coefficient; n is the skewness coefficient.

likelihood (usually sets to unknown constant), and p(h|obs) is theposterior distribution for h given obs.

However, it is difficult to obtain the analytical or even thenumerical solution for the posterior parameter distribution (Eq.(19)) because of the unknown formulation of hydrologic modelsand too many model parameters (Yustres et al., 2012). The MarkovChain Monte Carlo (MCMC) scheme as a stochastic simulationapproach provides a simple and effective way around the compu-tational difficulties for the posterior distribution. The aim of MCMCscheme is to generate samples of the parameter set based on con-structing a Markov chain that has the posterior distribution as itsequilibrium distribution (Marshall et al., 2005). The DifferentialEvolution Adaptive Metropolis (DREAM) Markov Chain Monte Car-lo (MCMC) sampling is superior to other adaptive MCMC samplingapproaches in the presence of high-dimensionality and multimo-dality optimization problems, because DREAM scheme follows upon the SCEM-UA global optimization algorithm, runs multiple dif-ferent chains simultaneously for global exploration, and maintainsdetailed balance and ergodicity (Vrugt et al., 2009b). In this study,we select eight parallel chains and a total of 40,000 model evalua-tions/MCMC iterations for the DREAM algorithm parameterizationon R platform, and run them on the ‘‘Soroban’’ High-PerformanceComputing System at Freie Universität Berlin.

4. Results

4.1. NSE approach

The NSE approach first calculates the value of NSE for the modelparameter set (h), and then substitutes NSE in Eq. (10) to calculatethe value of the likelihood function at each MCMC iteration. Thebest (i.e. the maximum likelihood function) simulation resultsare shown in Fig. 4. In this figure (b), the logarithmic vertical-axisis used to emphasize the baseflow due to the considerable differ-ences between the high- and low-values. Fig. 4a shows that, withthe NSE approach, the hydrological model mimics the observedriver discharges well, which reproduces most major flood events.However, Fig. 4b shows that it cannot mimic the baseflow well.In Fig. 5, the range of y-axis approximates [l � 2r, l + 2r] (wherel is the mean of errors and r is the standard deviation of errors),and the dash line is the mean of errors. Figs. 5a and 6a respectivelyinspect the heteroscedasticity and probability distribution of thetime series of errors/residuals. Fig. 5a shows that the variance oferrors obviously increases with river discharges, suggesting heter-oscedasticity that violates the stationary assumption of errors inthe NSE approach. Fig. 6a shows that the error histogram is sub-stantially different from the assumed Gaussian distribution. Suchviolations (heteroscedasticity and non-Gaussian error distribution)demonstrate that SWAT-WB-VSA with the NSE approach cannotproduce the model residuals that fulfill the assumptions of theapproach in the Baocun watershed.

Optimization results of parameter sets are shown in Table 3. Inthis table, the ‘‘95% confidence’’ columns present the 95%

pproaches.

Advantages and disadvantages

Popular criterion; mimic flood well but baseflow badly; informallikelihood function because of violating its assumptions

x– Mimic baseflow well; formal likelihood function

Nearly the same as BC-GED, but over-parameterization

Fig. 5. Errors as a function of observed river discharges for heteroscedasticity diagnostics: (a) NSE approach; (b) BC-GED approach; (c) BC-SGED approach.

0

1

2

3

-4 -2 0 2 4

Den

sity

Errors (m3/s)

ErrorsDensityInferredGaussian

0

1

2

3

-2 0 2Errors (m3/s)

ErrorsDensityInferredGED

0

1

2

3

-2 0 2Errors (m3/s)

ErrorsDensityInferredSGED

(a) (b) (c)

Fig. 6. The empirical probability density of errors versus the assumed error distribution of the likelihood function: (a) NSE approach; (b) BC-GED approach; (c) BC-SGEDapproach.

Table 3Optimized parameters and 95% confidence interval of posterior parameter distributions for NSE, BC-GED and BC-SGED approaches.

Categories Parameter NSE BC-GED BC-SGED

95% Confidencea Optimal values 95% Confidence Optimal values 95% Confidence Optimal values

LB UB LB UB LB UB

Evapotranspiration v__ESCO 0.014 0.394 0.013 0.012 0.199 0.071 0.012 0.386 0.121v__EPCO 0.185 0.918 0.882 0.937 0.999 0.996 0.928 0.999 0.999

Surface water v__EDC 0.731 0.895 0.757 0.730 0.774 0.750 0.718 0.768 0.744v__OV_N 0.025 0.479 0.032 0.013 0.490 0.242 0.038 0.493 0.483v__SURLAG 4.209 23.648 19.613 0.238 1.568 0.954 0.375 1.525 1.289

Soil water r__SOL_Z 0.251 0.313 0.255 0.810 0.866 0.848 0.792 0.873 0.862r__SOL_BD 0.304 0.601 0.358 0.194 0.337 0.283 0.162 0.341 0.274r__SOL_AWC �0.150 0.203 �0.091 0.906 1.456 1.085 0.870 1.599 1.101r__SOL_K �0.769 �0.648 �0.725 �0.262 0.029 �0.046 �0.305 0.110 �0.031

Ground water v__GW_DELAY 0.009 0.268 0.019 0.478 1.933 0.656 0.474 3.765 2.193v__ALPHA_BF 0.963 0.999 0.999 0.481 0.988 0.549 0.529 0.999 1.000v__GWQMN 50.8 941.8 845.6 273.6 526.4 391.9 311.3 561.3 376.5v__RCHRG_DP 0.001 0.056 0.011 0.152 0.333 0.257 0.062 0.273 0.147v__REVAPMN 27.3 911.1 819.6 328.3 981.7 838.0 118.4 968.7 376.9v__GW_REVAP 0.174 0.200 0.199 0.026 0.194 0.103 0.021 0.200 0.198

Tributary/main channel v__CH_N1 0.009 0.145 0.005 0.010 0.147 0.136 0.015 0.148 0.111v__CH_N2 0.019 0.147 0.143 0.006 0.138 0.015 0.005 0.050 0.014

Box–Cox k (lambda) 0.432 0.445 0.440 0.432 0.445 0.439

Probability density function l (mean) �0.006 0.048 0.007 �0.054 0.045 �0.018r (sigma) 0.520 0.531 0.523 0.520 0.531 0.520b (beta) 0.662 0.692 0.672 0.656 0.687 0.667n (xi) 0.888 0.958 0.938

a LB and UB are the lower and the upper bounds, respectively.

Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214 2209

confidence interval of the posterior distribution for the parametersets; the ‘‘Optimal values’’ columns show the optimal parametervalues. The parameter is regarded as sensitive if its 95% confidenceinterval in Table 3 is significantly narrower than its initial range inTable 1. So the parameters related to soil evaporation (ESCO),effective soil depth (EDC), soil water storage and movement,and ground water movement are very sensitive, but Manningroughness coefficients (OV_N, CH_N1 and CH_N2), surface flow

lag (SURLAG) and groundwater storage (GWQMN and REVAPMN)parameters are insensitive in the NSE approach.

The kernel smoothing densities of the posterior parameter dis-tributions for the NSE approach are shown in Fig. 7. In this figurefor the NSE approach, values of ESCO gather to zero, meaning thatthe evaporation of soil layers is large; values of SOL_K are verysmall, resulting in less soil interflow; GW_DELAY approaches zeroand ALPHA_BF approaches one, indicating that the groundwater

0 0.2 0.4 0.60

5

10

v__ESCO

Ker

nel s

moo

thin

g de

nsity

NSE DensityBC-GED DensityBC-SGED Density

0 0.5 10

20

v__EPCO

NSE OptimizationBC-GED OptimizationBC-SGED Optimization

0.7 0.8 0.90

20

v__EDC0 0.2 0.40

2

v__OV_N

0 10 200

1

v__SURLAG0.2 0.4 0.6 0.80

20

r__SOL_Z0.2 0.4 0.6 0.8 1

0

5

10

r__SOL_BD0 0.5 1 1.5 2

0

2

4

r__SOL_AWC

-0.8 -0.6 -0.4 -0.2 0 0.20

5

10

r__SOL_K0 1 2 3 4

0

5

v__GW_DELAY0.4 0.6 0.8 1

0

25

50

v__ALPHA_BF0 500 10000

0.005

v__GWQMN

0 0.1 0.2 0.3 0.40

20

40

v__RCHRG_DP0 500 10000

0.005

v__REVAPMN0 0.05 0.1 0.15 0.2

0

50

v__GW_REVAP0 0.05 0.1 0.15

0

5

10

v__CH_N1

0 0.05 0.1 0.150

25

50

v__CH_N20.425 0.43 0.435 0.44 0.445050100

lambda-0.1 0 0.1 0.20

20

mean0.52 0.525 0.53 0.535

050100

sigma

0.62 0.64 0.66 0.68 0.70

25

50

beta0.86 0.9 0.94 0.980

10

20

xi

Fig. 7. The kernel smoothing densities of posterior parameter distributions and the optimized parameters for NSE, BC-GED and BC-SGED approaches.

Fig. 8. Comparison of the observed and the simulated river discharges of the BC-GED approach. The logarithmic vertical-axis is used to emphasize the baseflow.

2210 Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214

declines very fast, which leads to simulate baseflow badly (Fig. 4b);values of RCHRG_DP are close to zero, revealing that little ground-water percolates to deep aquifer; and values of GW_REVAP gatherto the right boundary (i.e. 0.2), showing strong revaporization/evaporation rates.

In summary, simulated results of the NSE approach show thatthe main way of groundwater loss is revaporization/evaporation,

and the main runoff component is the return flow of groundwaterwith rapid recession.

4.2. BC-GED approach

The BC-GED approach first uses the simulated river dischargesby the SWAT-WB-VSA model to calculate the value of Box–Cox

Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214 2211

(BC) transformation parameter (k) by the least squares methodbased on the minimum variance constraint (Eq. (16)), then calcu-lates errors after Box–Cox transformation (Eq. (14)), and finallyuses the Generalized Error Distribution (GED) model (Eq. (17)) tocalculate the logarithmic likelihood value (l(h|obs)) of errors ateach MCMC iteration.

The results of optimal simulation are shown in Fig. 8. Fig. 8ashows that the simulated results can reproduce most flood events,but in some cases the flood peaks are smaller than those ofobserved values. Surprisingly, Fig. 8b shows that the BC-GEDapproach mimics the baseflow well. Further, inspecting the statis-tical assumptions of BC-GED approach is shown in Figs. 5b and 6b.The scatter points in Fig. 5b almost fill the whole panel space,showing that after BC the variances of errors are nearly constantfor different river discharges. In other words, the time series oferrors after BC are nearly homoscedastic. Fig. 6b indicates thatthe inferred error distribution (GED) matches the empirical distri-bution of the residuals/errors well.

Optimal values and ranges of the parameter set and posteriorparameter distributions for the BC-GED approach are shown inTable 3 and Fig. 7, respectively. Table 3 indicates that most param-eters are sensitive in the BC-GED approach except the parameters ofshallow aquifer revaporization/evaporation (REVAPMN andGW_REVAP) and Manning roughness coefficients (OV_N, CH_N1and CH_N2). In Fig. 7 for the BC-GED approach, ESCO also gathersto zero and EPCO approaches one, indicating high evapotranspira-tion; CH_N2 is close to zero, meaning that the main channel storagecapacity is small; and the mean errors gather to zero, demonstrat-ing that the water balance is maintained in the BC-GED approach.

In summary, the simulation results of the BC-GED approachshow that the main path of water loss is through evapotranspira-tion, the main channel routing is quick and the bias betweenobserved and simulated river discharges after BC is small.

4.3. BC-SGED approach

The calculating procedure of the BC-SGED approach is similar tothe BC-GED approach, and the only difference being that the prob-ability distribution of errors in the BC-GED approach is replaced bythe SGED. The best simulation results of the BC-SGED approach areshown in Fig. 9. This figure shows that the simulated results of theBC-SGED approach are nearly the same to those of the BC-GEDapproach, which can mimic baseflow well. Similar to the BC-GED

Fig. 9. Comparison of the observed and the simulated river discharges of the BC-SG

approach, the errors produced by the BC-SGED approach also fulfillthe statistic assumptions of this approach well: errors after BC arenearly homoscedastic because the scatter points almost fill thewhole panel space (Fig. 5c), and the inferred SGED matches the his-togram of errors well (Fig. 6c).

The optimal parameters with the 95% confidence interval andthe kernel smoothing densities of posterior parameter distributionsfor BC-SGED approach are shown in Table 3 and Fig. 7, respectively.Table 3 shows that the sensitivity and the value of most parametersin the BC-SGED approach are nearly the same as in the BC-GEDapproach. Fig. 7 also shows the most posterior parameter distribu-tions of the BC-SGED approach are similar to those of the BC-GEDapproach. Figs. 6c and 7 and Table 3 all demonstrate that the modelerrors produced by the BC-SGED approach are symmetric, and theskewness (n/xi) of the BC-SGED approach is very close to one, i.e.no-skewness, that was assumed in the BC-GED approach.

4.4. Model validation using groundwater data

As shown in Fig. 1b, the location of the groundwater gauge liesin the flood plain and started working in June 2007. The boreholematerial shows that the soil can be separated into three layers:(i) 0.0–1.5 m, filled with loose soil, (ii) 1.5–3.5 m, silty and coarsesand, and (iii) below 3.5 m, coarse sand. The average groundwatertable depth is 2 m, and the groundwater table depth variesbetween 0.8 m and 3.5 m during the monitoring period. Thechange of groundwater level can reflect the change of soil waterresources because of the shallow water table depth and sand soillayers with high hydraulic conductivity, i.e. a little water remainsin the unsaturated soil layer. Comparison of the observed ground-water level and the simulated soil water volume among three like-lihood functions (i.e. NSE, BC-GED and BC-SGED approaches) isshown in Fig. 10. In this figure, R2 is the coefficient of determina-tion, which reflects the linear correlation between the observedgroundwater level and the simulated soil water volume. Sy is thespecific yield of unconfined aquifer and defines as:

Sy ¼DGWDh

� DSWDh

ð20Þ

where DGW and DSW are the change of groundwater (GW) and soilwater (SW) volumes per unit area, respectively, and Dh is thechange of groundwater level.

ED approach. The logarithmic vertical-axis is used to emphasize the baseflow.

Fig. 10. The observed groundwater level versus the simulated soil water volume for model validation.

2212 Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214

According to the studies by Fetter (1994), the experiential Sy ofsilt soil is between 0.03 and 0.19 with the mean of 0.18, of finesand is between 0.10 and 0.28 with the mean of 0.21, and the Sy

of unconfined aquifer around the groundwater gauge couldapproximate to 0.195. Fig. 10 shows that the values of R2 of theBC-GED and BC-SGED approaches are greater than that of theNSE approach, and their Sy are also much closer to the experientialvalue (Fetter, 1994). So, the SWAT-WB-VSA with the BC-GED or BC-SGED approach mimics the groundwater levels better than thatwith the NSE approach. However, there is little difference betweenthe BC-GED and BC-SGED approaches.

5. Discussion

5.1. Comparison of simulated results

Comparison of simulated river discharges corresponding to themaximum likelihood values among NSE, BC-GED and BC-SGEDapproaches shows that the SWAT-WB-VSA with NSE approachcan mimic flood events well, but simulate the baseflow badly. Bycontrast, the SWAT-WB-VSA with the BC-GED or BC-SGEDapproach mimics the flood peak worse but simulates the baseflowbetter (Table 2). The defect of the NSE approach is also indicated byKrause et al. (2005): NSE tends to put more weight on the highervalues than on the lower values because of the squaring of theerrors between the observed and simulated values. Similarly, fromthe likelihood viewpoint, it attributes the statistical assumptionthat errors/residuals follow Gaussian distribution with zero mean,which can be further explained by the relationship between theerror interval and its probability in Fig. 11 where the r is the stan-dard deviation of errors. There are high probabilities (68.3%) oferrors/residuals nearly uniformly distributed in the error interval[�r, r] instead of approaching to zero because of the evenly dis-tributed Gaussian error density in Fig. 6a and the linear trend forthe relationship between the error intervals and the probabilitiesin Fig. 11. It leads to a large relative error of the low-flow/baseflow

0 ±0.25σ ±0.5σ ±σ ±2σ ±3σ

y = 0.7047xR² = 0.9931

0

0.2

0.4

0.6

0.8

1

Prob

abili

ty

Error interval

GaussianGEDTrendline

Fig. 11. Probabilities of different error intervals in Gaussian distribution and GED.

as shown in Fig. 4b. The probability of error interval [�2r, 2r] is95.5%, and that of [�3r, 3r] is 99.7% in Fig. 11. In other words,the probability of the absolute error more than 3r is only 0.3%.So there are a few very large errors resulting in the small relativeerror of high-flow/flood (Fig. 4a).

In summary, the low probability of the large error and the equi-probability of the small error in the Gaussian error distributionassumed by NSE result in the small relative error of flood and thelarge relative error of baseflow, respectively.

By contrast, the error distribution of GED sharply concentrateson zero error (Fig. 6b), and the ratio between the probability andthe error interval quickly increases with the narrowing of the errorinterval (Fig. 11) in this study. Therefore, the optimization resultsof the likelihood function with GED guarantee that most errors/residuals approach zero. On the other hand, the Box–Cox transfor-mation of the BC-GED approach leads to amplification of the errorof low-flow/baseflow and mitigation of the error of high-flow/flood. So the BC-GED approach generates very small errors of lowflows (Fig. 8b) but relatively large errors of high flows (Fig 8a).For the BC-SGED approach, because the model-error distributionis similar to that of the BC-GED approach (Fig. 6), of which theskewness is insignificant and close to one, i.e. no-skewness (Table 3and Fig. 7), the BC-SGED approach also puts greater emphasis onlow-flow/baseflow. But the results of the BC-SGED approach easilyfall into the local optima because of over-parameterization (skew-ness coefficient) of the error model (Table 2).

The above analysis demonstrates that SWAT-WB-VSA cannotfully capture flow processes in the study area in terms of error dis-tribution of the three likelihood functions, possibly because of theextremely nonuniform spatio-temporal distribution of rainfall inthe Baocun watershed owing to the great intensity of typhoon rain-fall. Meanwhile, only using observations of the river dischargesmay be not enough to evaluate the three approaches (NSE, BC-GED and BC-SGED) because of too many parameters in the model.Comparatively, the formal likelihood (BC-GED and BC-SGED)approaches can mimic baseflow and groundwater level better(Fig. 10) because of the stronger linear relationship between theobserved groundwater level and the simulated soil water volume,and the more reasonable specific yield of unconfined aquifer. It isconfirmed that the formal likelihood approaches estimate the soillayer properties better than the NSE approach.

5.2. Posterior distribution of parameters

Most posterior parameter distributions are different betweenthe NSE and the formal likelihood (BC-GED and BC-SGED)approaches as shown in Fig. 7. But between the BC-GED and theBC-SGED approach, posterior parameter distributions are nearlythe same, except those of some parameters relating to the ground-water movement, probably because of the dependence and theequifinality of different parameters that lack physical meaning.For example, although the BC-GED and BC-SGED approaches allinferred the slow groundwater-recession processes (Figs. 8b and

Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214 2213

9b), they attributed the smallest baseflow recession coefficient(ALPHA_BF) for the BC-GED approach and the longest groundwaterdelay time (GW_DELAY) for the BC-SGED approach. By contrast,the NSE approach inferred an unreliable result (Fig. 4b), i.e. therapid recession of the groundwater with the largest ALPHA_BFand the shortest GW_DELAY.

For evapotranspiration processes, the plant transpiration isstrong (large value for EPCO in Table 3 and Fig. 7) for the formallikelihood (BC-GED and BC-SGED) approaches, which possiblyresults from good vegetation/crop cover in the Baocun watershed.For surface flow processes, the runoff capacity is nearly the samefor different topography profiles because of the large EDC for allapproaches, which corresponds to the terrace cultivation methodin the Baocun watershed because the terrace can weaken the effectof topography and improve infiltration capacity. For soil water, theresults of the formal likelihood approaches show larger soil thick-ness (SOL_Z), porosity (which is inversely proportional to the bulkdensity (SOL_BD)), available water capacity (SOL_AWC) andhydraulic conductivity (SOL_K) than those of the NSE approach.This means that the soil layer of the formal likelihood approachescan maintain more water and generate more lateral flow, and theparameter values of soil property for the formal likelihoodapproaches agree better with the field-investigated results, espe-cially SOL_K. For main channel routing, the channel storage capac-ity (CH_N2) of the NSE approach is much larger than that of theformal likelihood approaches, indicating that the formal likelihoodapproaches are more reasonable in neglecting the storage functionof the main channel because of only two hours concentration timeof flood in the Baocun watershed modeled by daily hydrologicmodel.

The mean errors of the NSE approach are larger than zero. Bycontrast, those of the formal likelihood approaches gather to zero.Therefore, the formal likelihood approaches can maintain waterbalance.

In conclusion, the posterior parameter distributions inferred bythe formal likelihood (BC-GED and BC-SGED) approaches are morereasonable than those inferred by the informal likelihood approach(NSE). For the two formal likelihood approaches, the results ofBayesian inferences including the best simulated river dischargesand soil water volumes, and the posterior parameter distributionsare all nearly the same, because the model residuals after BC (Eq.(14)) are all symmetric and the SGED degrades into the GED. Mean-while, it demonstrates the assumption of skewness of the errormodel may be unnecessary, because the ideal model-residualsshould be unbiased (i.e. the mean is zero) and most of them shouldbe zero (i.e. the mode (the highest probability point) is zero), whichwill result in the symmetry of the error distribution with zero-mean and zero-mode.

6. Conclusions

By strict derivation, the NSE is proved to be equivalent to a kindof likelihood function with the assumption that the errors follow aGaussian distribution with zero mean. However, these assump-tions cannot be satisfied totally, as is confirmed by the applicationof the NSE approach in the Baocun watershed, so the NSE is aninformal likelihood function. The minimum variance constraint isproved to be an effective method to estimate the Box–Cox transfor-mation (BC) parameter (k) for removal of the heteroscedasticity oferrors. The range of the inferred values of lambda (k) (i.e. [0.432,0.445] in Table 3) is very small, which means that the scheme ofBC with minimum variance constraint approximates the schemeof BC with fixed lambda. However, the scheme of BC with mini-mum variance constraint increases the flexibility and effectivenessof BC to remove the heteroscedasticity of model residuals.

Comparison of the observed and the simulated river dischargesindicates that the results of the NSE approach can simulate floodwell, but baseflow badly owing to the assumption of Gaussianerror distribution, where the probability of the large error is low,but the error around zero approximates equiprobability. By con-trast, the results of the BC-GED and BC-SGED approaches can sim-ulate baseflow well, because the GED or SGED can well guaranteethat the model errors approach zero. The posterior parameter dis-tributions inferred by the BC-GED and BC-SGED approaches aremore reasonable than those inferred by the NSE approach becausethe values of parameters fit in with the field investigation results,the characteristics (e.g. small and mountainous area) of the Baocunwatershed, and the daily steps of the SWAT-WB-VSA model. Themodel validation using groundwater level shows that the BC-GEDand BC-SGED approaches are also better than the NSE approach.All the results of the BC-SGED approach are nearly the same asthose of BC-GED approach, because the skewness coefficient ofthe BC-SGED approach is close to one, i.e. no-skewness, and theBC-SGED approach degenerates into the BC-GED approach. There-fore, the assumption of skewness of the error distribution may beunnecessary.

Acknowledgements

This research was supported by the National Natural ScienceFoundation of China (Nos. 51190091, 51079038), the InternationalScience & Technology Cooperation Program of China (No.2012DFG22140) and a governmental scholarship from the ChinaScholarship Council (CSC). The third author thanks the support ofthe 111 Project of Hohai University under Grant B08048, Ministryof Education and State Administration of Foreign Experts Affairs,PR China. The authors would like to thank Weihai Hydrology andWater Resource Survey Bureau for providing hydrologic data, theNational Meteorological Information Center, China MeteorologicalAdministration (CMA) for providing climate data, and the comput-ing facilities of Freie Universität Berlin (ZEDAT) for computer time.The authors also wish to thank Niko Verhoest and the anonymousreviewer for their constructive comments that led to considerableimprovements of the paper.

References

Bates, B.C., Campbell, E.P., 2001. A Markov Chain Monte Carlo scheme for parameterestimation and inference in conceptual rainfall-runoff modeling. Water Resour.Res. 37 (4), 937–947.

Beven, K., 2001. How far can we go in distributed hydrological modelling? Hydrol.Earth Syst. Sci. 5 (1), 1–12.

Beven, K., Freer, J., 2001. Equifinality, data assimilation, and uncertainty estimationin echanistic modelling of complex environmental systems using the GLUEmethodology. J. Hydrol. 249 (1–4), 11–29.

Beven, K., Kirkby, M., 1979. A physically based, variable contributing area model ofbasin hydrology. Hydrol. Sci. Bull. 24, 303–325.

Beven, K., Smith, P., Westerberg, I., Freer, J., 2012. Comment on: ‘‘Pursuing themethod of multiple working hypotheses for hydrological modeling’’ by P. Clarket al. Water Resour. Res. 48 (11), W11801. http://dx.doi.org/10.1029/2012WR012282.

Box, G.E.P., Cox, D.R., 1964. An analysis of transformations. J. Roy. Stat. Soc. B 26,211–252.

Chen, X., Cheng, Q.-B., Chen, Y.D., Smettem, K., Xu, C.-Y., 2010. Simulating theintegrated effects of topography and soil properties on runoff generation in hillyforested catchments, South China. Hydrol. Process. 24, 714–725. http://dx.doi.org/10.1002/hyp.7509.

Cheng, Q.-B., Chen, X., Chen, X.-H., Zhang, Z.-C., Ling, M.-H., 2011. Water infiltrationunderneath single-ring permeameters and hydraulic conductivitydetermination. J. Hydrol. 398, 135–143.

Clark, M.P., Kavetski, D., Fenicia, F., 2012. Reply to comment by K. Beven et al. on‘‘Pursuing the method of multiple working hypotheses for hydrologicalmodeling’’. Water Resour. Res. 48, W11802. http://dx.doi.org/10.1029/2012WR012547.

Engeland, K., Renard, B., Steinsland, I., Kolberg, S., 2010. Evaluation of statisticalmodels for forecast errors from the HBV model. J. Hydrol. 384 (1–2), 142–155.

2214 Q.-B. Cheng et al. / Journal of Hydrology 519 (2014) 2202–2214

Evin, G., Kavetski, D., Thyer, M., Kuczera, G., 2013. Pitfalls and improvements in thejoint inference of heteroscedasticity and autocorrelation in hydrological modelcalibration. Water Resour. Res. 49, 1–7. http://dx.doi.org/10.1002/wrcr.20284.

Evin, G., Thyer, M., Kavetski, D., McInerney, D., Kuczera, G., 2014. Comparison ofjoint versus postprocessor approaches for hydrological uncertainty estimationaccounting for error autocorrelation and heteroscedasticity. Water Resour. Res.50, 2350–2375. http://dx.doi.org/10.1002/2013WR014185.

FAO/IIASA/ISRIC/ISSCAS/JRC, 2009. Harmonized World Soil Database (version 1.1).FAO, Rome, Italy and IIASA, Laxenburg, Austria.

Fetter, C.W., 1994. Applied Hydrogeology, third ed. Prentice Hall Inc., Upper SaddleRiver, NJ.

Gassman, P.W., Reyes, M.R., Green, C.H., Arnold, J.G., 2007. The soil and waterassessment tool: historical development, applications, and future researchdirections. Am. Soc. Agric. Biol. Eng. 50 (4), 1211–1250.

Han, E., Merwade, V., Heathman, G.C., 2012. Implementation of surface soilmoisture data assimilation with watershed scale distributed hydrologicalmodel. J. Hydrol. 416, 98–117.

Jin, X.-L., Xu, C.-Y., Zhang, Q., Singh, V.P., 2010. Parameter and modeling uncertaintysimulated by GLUE and a formal Bayesian method for a conceptual hydrologicalmodel. J. Hydrol. 383, 147–155.

Krause, P., Boyle, D.P., Bäse, F., 2005. Comparison of different efficiency criteria forhydrological model assessment. Adv. Geosci. 5, 89–97.

Laloy, E., Fasbender, D., Bielders, C.L., 2010. Parameter optimization and uncertaintyanalysis for plot-scale continuous modeling of runoff using a formal Bayesianapproach. J. Hydrol. 380 (1–2), 82–93. http://dx.doi.org/10.1016/j.jhydrol.2009.10.025.

Li, L., Xu, C.-Y., 2014. The comparison of sensitivity analysis of hydrologicaluncertainty estimates by GLUE and Bayesian method under the impact ofprecipitation errors. Stoch. Env. Res. Risk Assess. 28 (3), 491–504.

Li, L., Xu, C.-Y., Xia, J., Engeland, K., Reggiani, P., 2011. Uncertainty estimates byBayesian method with likelihood of AR (1) & normal model and AR (1) & multi-normal model in different time-scales hydrological models. J. Hydrol. 406, 54–65.

Li, L., Xu, C.-Y., Engeland, K., 2013. Development and comparison in uncertaintyassessment based Bayesian modularization method in hydrological modeling. J.Hydrol. 486, 384–394.

Li, H., Beldring, S., Xu, C.-Y., 2014. Implementation and testing of routing algorithmsin the distributed HBV model for mountainous catchments. Hydrol. Res. 45 (3),322–333.

Liu, Z., Todini, E., 2002. Towards a comprehensive physically-based rainfall-runoffmodel. Hydrol. Earth Syst. Sci. 6 (5), 859–881.

Marshall, L., Nott, D., Sharma, A., 2005. Hydrological model selection: a Bayesianalternative. Water Resour. Res. 41, W10422. http://dx.doi.org/10.1029/2004WR003719.

McMillan, H., Clark, M., 2009. Rainfall-runoff model calibration using informallikelihood measures within a Markov chain Monte Carlo sampling scheme.Water Resour. Res. 45, W04418. http://dx.doi.org/10.1029/2008WR007288.

Nash, J.E., Sutcliffe, J.V., 1970. River flow forecasting through conceptual models 1: adiscussion of principles. J. Hydrol. 10 (3), 282–290.

Neitsch, S.L., Arnold, J.G., Kiniry, J.R., Williams, J.R., 2005. Soil and Water AssessmentTool Theoretical Documentation and User’s Manual, Version 2005. GSWRAgricultural Research Service & Texas Agricultural Experiment Station, TempleTexas.

Pianosi, F., Raso, L., 2012. Dynamic modeling of predictive uncertainty by regressionon absolute errors. Water Resour. Res. 48, W03516.

Schoups, G., Vrugt, J.A., 2010. A formal likelihood function for parameter andpredictive inference of hydrologic models with correlated, heteroscedastic, andnon-Gaussian errors. Water Resour. Res. 46, W10531.

Smith, T., Sharma, A., Marshall, L., Mehrotra, R., Sisson, S., 2010. Development of aformal likelihood function for improved Bayesian inference of ephemeralcatchments. Water Resour. Res. 46, W12551. http://dx.doi.org/10.1029/2010WR009514.

Stedinger, J.R., Vogel, R.M., Lee, S.U., Batchelder, R., 2008. Appraisal of thegeneralized likelihood uncertainty estimation (GLUE) method. Water Resour.Res. 44, W00B06. http://dx.doi.org/10.1029/2008WR006822.

Thiemann, M., Trosset, M., Gupta, H., Sorooshian, S., 2001. Bayesian recursiveparameter estimation for hydrologic models. Water Resour. Res. 37 (10), 2521–2535.

Vrugt, J.A., ter Braak, C.J.F., Gupta, H.V., Robinson, B.A., 2009a. Equifinality of formal(DREAM) and informal (GLUE) Bayesian approaches in hydrologic modeling?Stoch. Env. Res. Risk Assess. 23 (7), 1011–1026. http://dx.doi.org/10.1007/s00477-008-0274-y.

Vrugt, J.A., Ter Braak, C.J.F., Diks, C.G.H., Robinson, B.A., Hyman, J.M., Higdon, D.,2009b. Accelerating Markov Chain Monte Carlo simulation by differentialevolution with self-adaptive randomized subspace sampling. Int. J. Nonlin. Sci.Numer. Simul. 10 (3), 273–290.

White, E.D., Easton, Z.M., Fuka, D.R., Collick, A.S., Adgo, E., McCartney, M.,Awulachew, S.B., Selassie, Y.G., Steenhuis, T.S., 2011. Development andapplication of a physically based landscape water balance in the SWATmodel. Hydrol. Process. 25, 915–925.

Wrede, S., Seibert, J., Uhlenbrook, S., 2013. Distributed conceptual modelling in aSwedish lowland catchment: a multi-criteria model assessment. Hydrol. Res. 44(2), 318–333. http://dx.doi.org/10.2166/nh.2012.056.

Yang, J., Reichert, P., Abbaspour, K.C., Yang, H., 2007a. Hydrological modelling of theChaohe Basin in China: statistical model formulation and Bayesian inference. J.Hydrol. 340, 167–182. http://dx.doi.org/10.1016/j.jhydrol.2007.04.006.

Yang, J., Reichert, P., Abbaspour, K.C., 2007b. Bayesian uncertainty analysis indistributed hydrological modelling: a case study in the Thur River basin(Switzerland). Water Resour. Res. 43, W10401. http://dx.doi.org/10.1029/2006WR005497.

Yustres, Á., Asensio, L., Alonso, J., Navarro, V., 2012. A review of Markov Chain MonteCarlo and information theory tools for inverse problems in subsurface flow.Comput. Geosci. 16, 1–20. http://dx.doi.org/10.1007/s10596-011-9249-z.