Journal of Hydrology 406 (2011) 54–65
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Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Uncertainty estimates by Bayesian method with likelihood of AR (1) plus Normal
model and AR (1) plus Multi-Normal model in different time-scales
hydrological models
Lu Li a,⇑, Chong-Yu Xu a,b, Jun Xia c, Kolbjørn Engeland d, Paolo Reggiani e,f
a
Department of Geosciences, University of Oslo, P.O. Box 1047, Blindern, 0316 Oslo, Norway
Department of Earth Sciences, Uppsala University, Villavägen 16, SE-752 36 Uppsala, Sweden
Key Laboratory of Water Cycle & Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, CAS, Beijing 100101, China
d
SINTEF Energy Research, 7465 Trondheim, Norway
e
Section of Hydraulic Structures and Probabilistic Design, Delft University of Technology, P.O. Box 5048, 2600GA Delft, The Netherlands
f
Deltares, P.O. Box 177, 2600MH Delft, The Netherlands
b
c
a r t i c l e
i n f o
Article history:
Received 6 October 2010
Received in revised form 1 May 2011
Accepted 30 May 2011
Available online 16 June 2011
This manuscript was handled by
Dr. A. Bardossy, Editor-in-Chief, with the
assistance of Ezio Todini, Associate Editor
Keywords:
Uncertainty assessment
Hydrological models
Bayesian methods
Multi-Normal
Time scales
s u m m a r y
Bayesian revision is widely used in hydrological model uncertainty assessment. With respect to model
calibration, parameter estimation and analysis of uncertainty sources, various regression and probabilistic approaches have been used in different models calibrated for either daily or monthly time step. None
of these applications however includes a comparison of uncertainty analysis in hydrological models with
respect to the time periods, at which the models are operated. This study pursues a comprehensive intercomparison and evaluation of uncertainty assessments by Bayesian revision using the Metropolis Hasting
(MH) algorithm with the hydrological model WASMOD with daily and monthly time step. In the daily
step model three likelihood functions are used in combination with Bayesian revision: (i) the AR (1) plus
Normal time period independent model (Model 1), (ii) the AR (1) plus Multi-Normal model (Model 2),
and (iii) the AR (1) plus Normal time period dependent model (Model 3). In addition an index called
the percentage of observations bracketed by the Unit Confidence Interval (PUCI) was used for uncertainty
evaluation. The results reveal that it is more important to consider the autocorrelation in daily WASMOD
rather than monthly WASMOD. Firstly, the resulting goodness of fit of the daily model vs. observations as
measured by the Nash–Sutcliffe efficiency value is comparable with that calculated by the optimization
algorithm in monthly WASMOD. Secondly, the AR (1) model is not sufficiently adequate to estimate the
distribution of residuals in daily WASMOD since PUCI shows that Model 2 outperforms Model 1. Furthermore, the maximum Nash–Sutcliffe efficiency value of Model 2 is the largest. Thirdly, Model 3 performs
best over the entire flow range, while Model 2 outperforms Model 3 for high flows. This shows that additional statistical parameters reflect the statistical characters of the residuals more efficiently and accurately. Fourthly, by considering the difference in terms of application and computational efficiency it
becomes evident that Model 3 performs best for daily WASMOD. Model 2 on the other hand is superior
for daily time step WASMOD if the auto-correlation of parameters is considered.
Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction
The Bayesian method is widely used for uncertainty assessment
in hydrologic modelling. During the last two decades several applications to various models and catchments have been pursued in
the literature (Bates and Campbell, 2001; Duan et al., 2007;
Engeland and Gottschalk, 2002; Li et al., 2010b; Marshall et al.,
2007; Kavetski et al., 2006a,b; Kuczera et al., 2006; Kuczera and
Parent, 1998; Krzysztofowicz and Kelly, 2000; Liu et al., 2005;
⇑ Corresponding author.
E-mail address: lu.li@geo.uio.no (L. Li).
0022-1694/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.jhydrol.2011.05.052
Thiemann et al., 2001; Vrugt et al., 2003). The Bayesian method requires a specification of a likelihood function. For it to provide
unbiased and reliable results several conditions need to be satisfied. The likelihood function provides a probability distribution of
all simulation errors. It is therefore necessary to specify a correct
distribution for the simulation errors and a possible mutual correlation structure. The most common approach is to transform the
data for them to become normally and identically distributed (constant variance).
There are two commonly used transformation methods in
uncertainty assessment for hydrological models. One is the Box–
Cox method (Engeland et al., 2005; Jin et al., 2010; Wang et al.,
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
2009; Yang et al., 2007b) and the other one is the Normal Quantile
Transform (NQT) (Krzysztofowicz, 1997, 1999; Todini, 2008; Van
der Waerden, 1952, 1953a,b). Krzysztofowicz and Herr (2001) proposed the Normal Quantile Transform (NQT) method in his Hydrological Uncertainty Processor (HUP) to convert both observation
and model output into the Gaussian space with the aim to estimate
the predictive uncertainty. This method is one of the most commonly used in hydrological uncertainty analysis in deriving the
joint distribution and predictive conditional distribution from a
multivariate distribution (Krzysztofowicz and Kelly, 2000; Liu
et al., 2005; Maranzano and Krzysztofowicz, 2004; Montanari
and Brath, 2004; Reggiani and Weerts, 2008; Todini, 2008). Because hydrological processes have a ‘‘memory effect’’, the model
residuals are often inter-correlated. In this context the discretetime and continuous-time autoregressive error models have been
used universally nowadays (Yang et al., 2007b), amongst which,
the first order auto-regressive model (AR (1) model) is the most
widely used. The latter is employed in combination with the transformation method to make the residuals satisfy the statistical
assumptions of identical and independent distribution (Engeland
et al., 2010; Krzysztofowicz, 2002; Yang et al., 2007a). Mantovan
and Todini (2006) proposed a Multi-Normal AR (1) model as likelihood function by considering the autocorrelation of errors.
All uncertainty assessment methods account for different
sources of uncertainty. Some methods use a statistical model of
the residuals to address all the uncertainty sources (Engeland
and Gottschalk, 2002; Jin et al., 2010; Krzysztofowicz, 2002;
Montanari and Grossi, 2008). Others consider all uncertainty
sources lumped into parameter uncertainty such as the GLUE
method and the multi-objective method (Beven and Freer, 2001;
Engeland et al., 2006; Yapo et al., 1998). Others consider the input
error and/or model structure error jointly or separately
(Chowdhury and Sharma, 2007; Kavetski et al., 2006a, 2006b;
Kuczera et al., 2006). Recently, some criteria have been proposed
to evaluate the performance of uncertainty analysis methods: (1)
indices for goodness of fit at the level of the posterior distribution
(efficiency) based on the Nash–Sutcliffe (NS) coefficient or other
objective functions (Jin et al., 2010; Yang et al., 2008); (2) indices
used to compare the width of the model uncertainty intervals (resolution) mainly based on 95% confidence interval of discharge
(95CI), which include the width of 95CI, the average distance between the upper and the lower limits of 95% confidence intervals,
95PPU (d-factor), (Li et al., 2009; Xiong et al., 2009; Yang et al.,
2008); and (3) indices to compare the distribution of the uncertainty estimates (reliability) which include the percentage of
observations bracketed by the 95CI, the Continuous Rank Probability Score (CRPS). Engeland et al. (2010) proposed a hierarchy of all
the criteria which are reliability, resolution and efficiency.
According to Todini (2007, 2011) all these studies treats the
problem of ‘‘uncertainty’’, which can be distinguished into ‘‘Validation Uncertainty’’ and ‘‘Predictive Uncertainty’’. The ‘‘Validation
Uncertainty’’ is the probability that the simulated value (water level, discharge, water volume, etc.) will be less than or equal to a
prescribed value given prior knowledge, the historical information
and the observed value, which refers to running the model in historical mode based on observations. Numerous approaches for
quantifying the ‘‘Validation Uncertainty’’ have been proposed,
including the Generalized Likelihood Uncertainty Estimation
(GLUE) (Beven and Freer, 2001; Blasone et al., 2008a,b; Vogel
et al., 2008; Xiong and O’Connor, 2008) and formal Bayesian methods (Montanari and Brath, 2004; Vrugt et al., 2003; Engeland et al.,
2005; Yang et al., 2008; Jin et al., 2010; Li et al., 2010b). In which,
the uncertainty has been estimated by 95% confidence interval
(95CI) (Engeland et al., 2005; Li et al., 2009, 2010b) or 95% prediction uncertainty band (95PPU) (Abbaspouretal et al., 2007; Yang
et al., 2008) calculated at the 2.5% and 97.5% levels of the
55
cumulative distribution function of the model output variables.
Krzysztofowicz (1999, 2001) defines ‘‘Predictive Uncertainty’’ as
expressing the uncertainty on what may happen at a future stage
conditional on the forecast of model, which was considered as a
known value, based on the all available information, like input,
parameter and model structure. It refers to running the model in
forecast mode. After this, Todini (2007) redefined the predictive
uncertainty that it is based on the posterior density of parameters
and the conditional probability density of a predict and conditional
on the covariates and a set of parameters. According to this, three
approaches for the assessment of predictive uncertainty have been
motioned (Todini, 2011), which are Hydrological Uncertainty Processor (HUP) (Krzysztofowicz, 1999), the Bayesian Model Averaging (BMA) (Raftery et al., 2003, 2005) and the Model Conditional
Processsor (MCP) (Todini, 2008; Coccia and Todini, 2010). This definition of predictive uncertainty is restricted to hydrologic forecasting. It might, however, argued that the predictive uncertainty
might be defined for other applications that include predictions,
e.g. streamflow in an ungauged catchment (conditioned on streamflow in a nearby catchment), or streamflow for a period of missing
data (conditioned on observations outside this period). For this
reason the difference between the ‘‘Predictive Uncertainty’’ and
‘‘Validation Uncertainty’’ as defined by Todini might be more related to the problem addressed (forecasting/non-forecasting) and
not to a general Bayesian definition of predictive uncertainty. In
both types of uncertainty analyses, the Bayesian method requires
a specification of a likelihood function. In order to provide unbiased and reliable results several conditions need to be satisfied.
The likelihood function provides a probability distribution of all
simulation errors. It is therefore necessary to specify a correct distribution for the simulation errors and a possible mutual correlation structure. The most common approach is to transform the
data for them to become normally and identically distributed (constant variance).
Hydrological models at monthly time-step have been widely
utilized for long-range streamflow forecasting, for assessing the
long-term climatic change impact on water resources, and for regional water resources management (Xu and Vandewiele, 1995;
Chen et al., 2007). Whereas daily or shorter time step hydrological
models are required for real-time flood forecasting, for generation
of synthetic sequences of hydrologic data for facility design, for
studying the potential impacts of changes in land use or climate,
and for interdisciplinary researches. Therefore, both monthly and
daily hydrological modeling approaches are essential in water cycle and water resources researches. With respect to model calibration, parameter estimation, uncertainty sources analyses, various
regression and probabilistic approaches have been employed in
hydrological applications. Some include monthly water balance
hydrological models (Benke et al., 2008; Engeland et al., 2005; Jin
et al., 2010; Li et al., 2010b), others are daily conceptual hydrological models or physically-based distributed hydrological models
(Blasone et al., 2008a; Engeland and Gottschalk, 2002; Li et al.,
2010a; Liu et al., 2005; Looper et al., 2009; Wang et al., 2009; Yang
et al., 2008, 2007b). However, none of the above applications
seems to include comparisons of uncertainty analyses in timevariant hydrological models. It is common knowledge that the
hydrological response variables are dependent on the time scale
of model simulation. However, the question remains how model
time scales impact on the performance of the uncertainty analysis
methods and how models of similar structure operated with different time steps affect the uncertainty assessment via Bayesian
methods. The objective of this paper is to attempt to fill this gap.
This study performs a comprehensive comparison and evaluation of uncertainty estimates by Bayesian methods using the
Metropolis Hasting (MH) algorithm in two validated conceptual
hydrological models (WASMOD) with monthly and daily time
56
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
steps. It aims at explaining the differences between the Bayesian
method applied to hydrological models of similar concept and
structure but with different time steps. The Box–Cox transformation in this study was used to obtain variables which are closer
to a normal distribution. In daily WASMOD, two likelihood functions were used to remove the time dependence of residuals in
the Bayesian method: (i) the AR (1) plus Normal model and (ii)
the AR (1) plus Multi-Normal model. We also tested models, where
the statistical parameters were dependent and/or independent on
time, respectively. The Average Relative Interval Length (ARIL) defined by Jin et al. (2010), the percentage of observations bracketed
by the Confidence Interval (PCI), and a new index called the Percentage of observations bracketed by the Unit Confidence Interval
(PUCI) were used in this study for evaluation of model performance. Furthermore, the 95% confidence interval of daily discharge
in low flow, medium flow and high flow were selected for comparison. Finally, the paper tries to give some advice to users on application of Bayesian method in validation uncertainty in timevariant hydrological model applications.
2. Study area and hydrological model
2.1. Study area
The selected study area is the Jiuzhou basin (Fig. 1) a tributary
of the Dongjiang River in southern China. The river flows from the
north-east to the south-west estuary with a drainage area of
385 km2 (upstream area of Jiuzhou gauge station). The landscape
is characterized by hills and plains. The Jiuzhou basin is located
in a Monsoon region in a tropical and sub-tropical humid climate
with a mean annual temperature of about 21 °C and only occasional incidents of winter daily air temperature dropping below
0 °C in the mountainous areas of the upper catchment part. Precipitation is generated mainly from two types of storms: frontal type
and typhoon-type rainfalls. Eighty percent of the annual precipitation occurs during the wet season from April to September and the
mean annual precipitation for the period of 1978–1988 is
1802 mm. The mean annual runoff is 993.4 mm or 55% of the annual precipitation. Precipitation data from Heduobu, Tangwei and
Jiuzhou stations for the period of 1978–1988 are used for the study
as the main input data. The potential evapotranspiration was calculated through the Penman–Monteith formula using meteorological data of the Heyuan meteorological station. All data have gone
through quality control in earlier studies (Jin et al., 2010).
2.2. Wasmod
WASMOD is a well-tested simple conceptual water balance
model with multiple time scales (Xu, 2002). The input data to
the model include precipitation, potential evapotranspiration.
Model output data are fast flow, slow flow, actual evapotranspiration and soil moisture. The model can be set up with different spatial and temporal scales varying from global to catchment and from
daily to monthly (Gong et al., 2009; Jin et al., 2010; Widen-Nilsson
et al., 2009, 2007; Xu, 2002). For this study a monthly and a daily
version of the model at the catchment scale are used. The main
Fig. 1. Main river, meteorological stations and discharge station of Jiuzhou River Basin.
57
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
Table 1
The main equations in WASMOD.
Actual evapotranspiration
Slow flow
0 6 a1 6 1
0 6 a2 6 1
et ¼ minfðsmt1 =Dt þ pt Þð1 expða1 ept ÞÞ; ept g
st ¼ a2 ðsmt1 Þ2
ft ¼ a3 ðsmt1 Þðpt ept ð1 expðpt = maxðept ; 1ÞÞÞÞ
smt ¼ smt1 þ ðpt et dt Þ Dt
SC t ¼ SC t1 þ ft Dt
Q t ¼ a4 SC t
SC t ¼ SC t Q t Dt
dt ¼ st þ Q t
Fast flow
Water balance
Routing of fast flow
Total runoff
0 6 a3 6 1
0 6 a4 6 1
In which, t is t th month or day, pt is precipitation, ept is potential evaporation, smt is t th monthly or daily soil moisture, SCt is channel
storage at time step t, Dt is time step which is either one month for the monthly modal or one day for the daily model. The unit for all
the fluxes is either mm/month for the monthly model and mm/day for the daily model. The unit for the state variables soil moisture and
channel storage is mm.
transformation yields residuals closest to normal for values of
k = 0.6 in daily and k = 0.2 in monthly WASMOD.
It was also necessary to verify whether nt were stochastically
independent. In case of stochastic dependence, an AR (1) model
needs to be fitted to the residuals as follows:
model equations are shown in Table 1. Snowmelt is not considered
in the simulation because of the rare appearance of snow in the
Jiuzhou River basin in the South of China. There are three parameters in monthly WASMOD and four parameters in daily WASMOD,
with one additional routing parameter with respect to the former.
Furthermore, parameter a2 and a3 have multiplied 103 and 104
respectively, aiming to be scaled into [0, 1] in this paper.
nt b ¼ aðnt1 bÞ þ et
There is no need to use AR (1) model in monthly WASMOD,
since the monthly residuals were not significantly correlated.
On the contrary, daily residuals were more heteroscedastic and
autocorrelated. Using the Box–Cox transformation to reduce the
heteroscedasticity of residuals increased their degree of autocorrelation. We therefore used the AR (1) model to make the Box–Cox
transformed residuals independent. However, the AR (1) innovations were still auto-correlated. We therefore introduced AR (1)
plus Multi-Normal model to account for all autocorrelations. This
method has been used before in hydrological applications
(Mantovan and Todini, 2006; Romanowicz et al., 1994).
Fig. 2 shows that the variances of AR (1) innovations from daily
residuals have different values in different seasons. We therefore
tested two statistical models: (i) in Model 1 the statistic parameters were constant and (ii) in Model 3 the parameters were allowed
to depend on the season. Two simulation sub-periods were defined: a wet period from April to September and a dry period from
October to March.
3. Methods
3.1. Residual analysis
The variable nt represent the difference of the observed random
M
variable gt ðyt Þ and the simulated random variables gM
t ðyt Þ:
M
nt ¼ gt ðyt Þ gM
t ðyt Þ
ð1Þ
where t is an index for time, M indicates that the variable is simulated, y is the variable in original space and g is the transformed
variable.
In this study, Box–Cox transformation method is used, which
can be expressed as:
(
hðy; kÞ ¼
yk 1
k
k–0
ð2Þ
lnðyÞ k ¼ 0
ð3Þ
3.2. Statistical models for the prediction errors
where k is a transformation parameter. It is often fixed, e.g. at the
value of 0.3 in Yang et al. (2007) or 0.2 (Engeland et al., 2010).
The square-root transformation and log-transformation are special
cases of the Box–Cox transformation (Engeland and Gottschalk,
2002; Engeland et al., 2005; Jin et al., 2010; Langsrud et al., 1998;
Wang et al., 2009).
The Lilliefors test (Lilliefors, 1967) was used to verify the normality of the distribution of the residuals of the Box–Cox transformed variable with k fixed at different values for the monthly
and daily models. The test statistics is called LSTAT and it should
be as small as possible. Calculations indicated that the Box–Cox
3.2.1. Statistical model for monthly WASMOD
The assumption that there is neither bias nor autocorrelation in
monthly residuals results in the following likelihood function for
monthly WASMOD:
Lðx; hjnÞ ¼
T
Y
1
t¼1
n
qð t Þ
ð4Þ
r r
where h represents the hydrological model parameters, x represents the statistical parameters including in this case only the
6
Residual
4
2
0
-2
-4
-6
1
2
3
4
5
6
7
8
9
10
11
12
month
Fig. 2. Variances of daily runoff residuals in Jiuzhou basin (1979–1988) after Box–Cox transformation and AR (1).
58
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
standard deviation of the residuals r, n ¼ fn1 ; . . . ; nT g is the residuals, t is the time index, and q is the normal density operator.
xt fbt ; at ; rt g ¼
xW fbW ; aW ; rW g t 2 ðApriltoSeptemberÞ
xD fbD ; aD ; rD g
others
ð11Þ
3.2.2. Statistical models for daily WASMOD
3.3. Prior density
(a) The AR (1) plus Normal model in which statistical parameters independent on periods (Model 1)
The likelihood function for the AR (1) plus Normal model for the
prediction errors in daily WASMOD were:
Lðx; hjnÞ ¼
1
T
Y
n
1 nt b aðnt1 bÞ
qð 0 Þ qð
Þ
r r
t¼1
r
r
ð5Þ
where h represents the hydrological model parameters. x represents the statistical parameters including an autoregressive term
a, a bias term b and a standard deviation of the residuals r, n0 is
the initial residual value, t is the time index, and q is the normal
density operator.
(b) The AR (1) plus Multi-Normal model (Model 2)
The likelihood function is built for time steps ranging from 1 to t
(t = n) given the AR (1) plus Multi-Normal model.
Lðx; hjnÞ ¼
where
P
expð 12 ðn bÞT
1
P
r1
r21
. . . r n2
1
1
r1
. . . r n3
1
6 r
6 1
6 2
X
6 r1
¼ r2n 6
6 ...
6
6 n2
4 r1
r1
1
...
r n4
1
...
...
...
...
rn3
1
r n4
1
...
1
rn1
1
rn2
1
r n3
1
...
r1
rn1
1
3
7
rn2
7
1
7
n3 7
r1 7
... 7
7
7
r1 5
ð7Þ
1
where
r2n ¼ var½nt ð8Þ
n1
P
ðnt nÞðntþ1 nÞ
r1 ¼
t¼1
n
P
ð9Þ
ðnt nÞ
(c) The AR (1) plus Normal model in which statistical parameters depend on periods (Model 3)
The likelihood function for the daily WASMOD for which the
statistical parameters depend on the periods is:
1
qð
n0
r0 r0
ð12Þ
where f ðuÞ is the prior density, u ¼ fx; hg and n ¼ fn1 ; . . . ; nT g. The
likelihood L is given in Eqs. (4), (5), (6), and (10) above.
The Markov chain Monte Carlo method known as the Metropolis Hasting (MH) algorithm was used to sample from the posterior
distributions. The Markov chain is initiated from a random starting
value, whereby a new sample is derived from the proposal distribution. To avoid long ‘‘burn in’’ time, the Markov chain is initiated
in correspondence of an approximated maximum Nash–Sutcliffe
value calculated with the help of an optimization algorithm named
VA05A (Hopper, 1978). A new sample was suggested from the proposal distribution. The new sample was accepted according to an
acceptance probability. If not, the chain kept the old sample. The
necessary steps of the Metropolis Hasting method have been
explained in detail in the literature (Chib and Greenberg, 1995;
Engeland and Gottschalk, 2002; Kuczera and Parent, 1998).
Here the proposal distributions d were assumed as normal in
the Random walk M–H Chain:
dðxÞ N½xL ; Sx ð13Þ
dðhÞ N½hL ; Sh ð14Þ
2
t¼1
Lðx; hjnÞ ¼
LðujnÞ f ðuÞ
LðujnÞ f ðuÞdu
ð6Þ
is the covariance matrix that can be approximated with:
1
pðujnÞ ¼ R
3.4. Metropolis Hasting algorithm
ðn bÞÞ
P 1=2
n=2
ð2pÞ ½detð Þ
2
There is no information about the distribution of parameters.
Parameter r is an unknown model standard deviation and with
Jeffreys’ uninformative prior it is proportional to r1 (Bernardo
and Smith, 1994; Yang et al., 2007b). Besides, the prior probability
densities of other parameters are generally taken as non-informative multi-uniform distribution in hydrological applications (Engeland et al., 2005; Jin et al., 2010; Liu et al., 2005; Yang et al., 2008).
The prior densities and intervals of all parameters in the two
hydrological models are shown in Table 2. The posterior density
pðujnÞ is:
Þ
T
Y
1
t¼1
rt
n bt at ðnt1 bt1 Þ
qð t
Þ
rt
ð10Þ
where all statistical parameters (bt , at and rt ) are labelled according
to two periods: the wet period (April to September) and the dry period (rest months):
where x ¼ fa; b; rg are the statistical parameters, h are the hydrologic parameters, L index the current state of the chain, Sx and Sh
are the covariance matrixes of the proposal distributions. The normal distribution is not appropriate for r > 0. However, in this study,
the standard deviation of r was much smaller than the mean of r,
which means that the normal distribution density functions are
mostly inside the feasible region. Other statistical parameters and
model parameters in this paper satisfied the same assumption.
The algorithm works best if the proposal density matches the target
distribution which is generally unknown. The tuning of the variance
was performed by calculating the acceptance rate. The ideal
Table 2
The prior distributions of all parameters in two hydrological models.
a
b
Hydrological models
Monthly WASMOD
Parameter
Prior distributionsa
Hydrological models
Parameter
Prior distributionsa
a1
U[0, 1]
Daily WASMOD
a1
U[0, 1]
rb
a2
U[0, 1]
a3
U[0, 1]
/r1
a2
U[0, 1]
a3
U[0, 1]
a4
U[0, 1]
U½a; b means the prior distribution of the parameter is uniform over the interval ½a; b.
/ r1 means the prior density of the parameter at value r is proportional to r1 .
a
U[0, 1)
b
U[10, 10]
rb
/r1
59
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
A1
0.4
0.1
0.3
Frequency
0.2
0.2
0.1
0
0.7
0.8
0
0.1
0.9
A3
0.4
0.3
Frequency
0.3
Frequency
A2
0.4
0.2
0.1
0.2
0.3
0.4
0
0.1
0.2
0.3
0.4
Fig. 3. Histograms of parameters of monthly WASMOD derived by Metropolis Hasting method.
acceptance rate is approximately 40–50% for one parameter updating, decreasing to approximately 20–30% for multi-parameters
updating in a one block (Chib and Greenberg, 1995; Engeland and
pffiffiffi
Gottschalk, 2002). Furthermore, the Scale Reduction Score R
(Gelman and Rubin, 1992) was used to check whether the MC
chains converged.
3.5. Uncertainty confidence intervals estimation
The uncertainty intervals for streamflow due to parameter
uncertainty were calculated using all the MH-samples of the
hydrologic parameters. The samples were then sorted for every
day in order to get credibility intervals. The empirical distribution
P B ðy < yi Þ based on the rank i was used to calculate the probability
using the sorted sample yi
Table 3
The statistic characteristic of parameters estimated by Bayesian method for monthly
WASMOD.
Method
Parameters
Min
Max
Average
Variance
Skewness
Estimated
2.50% Quantile
5% Quantile
50% Quantile
95% Quantile
97.50% Quantile
MNS
a1
a2
a3
0.783
0.872
0.830
1.59E04
0.134
0.829
0.804
0.809
0.830
0.850
0.854
0.856
0.132
0.340
0.224
1.38E03
0.166
0.207
0.158
0.166
0.223
0.286
0.300
0.148
0.405
0.258
1.54E03
0.248
0.244
0.186
0.195
0.257
0.325
0.341
MNS: Maximum Nash–Sutcliffe.
PB ðy < yi Þ ¼ i=n
ð15Þ
where n is the number of sample simulations; PB ðy < yi Þ is the
quantile of discharge yi .
The 95% confidence intervals due to parameter uncertainty and
model uncertainty for discharge were calculated by adding the
model residuals in the form of a normally distributed random error
with zero mean and variance r2 to each of the sample simulations
that were available for each time step (Eq.(16)). For the daily model
the bias and autoregressive terms were included as well (Eq.(17)).
The inverse of the Box–Cox transformation was then applied.
gi;t ¼ gMi;t þ rnormð0; rÞi
ð16Þ
gi;t ¼ gMi;t b þ a ðgi;t1 gMi;t1 bÞ þ rnormð0; rÞi
ð17Þ
1
yi;t ¼ h ðgi;t Þ
ð18Þ
where i is the sample index, t is the time index, rnorm is the random
value extracted from a normal distribution with zero mean and var1
iance r2 , and h represents the inverse operation of the Box–Cox
transformation. Then the 5% percentile and 95% percentile of discharge due to parameter uncertainty and model uncertainty are derived by sorting new discharge values yi;t were sorted at each time
step (Eq. (18)). And Eq. (15) was used to obtain the empirical
distributions.
3.6. Evaluation of uncertainty interval estimates
Three indices were used to evaluate the resolution, reliability
and efficiency of the uncertainty interval estimates: (1) the Average Relative Interval Length (ARIL) by Jin et al. (2010), (2) the percentage of observations bracketed by the Confidence Interval (PCI)
Table 4
The statistic characteristic difference of parameters estimated by three methods for daily WASMOD.
Parameter
Min
Max
Mean
Variance
Skewness
5% Quantile
50% Quantile
95% Quantile
MNS
Model 1
a1
a2
a3
a4
0.967
0.112
0.033
0.34
0.993
0.177
0.062
0.443
0.981
0.143
0.045
0.393
1.83E05
1.26E04
1.84E05
2.18E04
0.164
0.074
0.42
0.022
0.974
0.125
0.038
0.368
0.981
0.143
0.044
0.393
0.988
0.162
0.052
0.416
0.805
Model 2
a1
a2
a3
a4
0.955
0.129
0.053
0.364
0.988
0.204
0.088
0.459
0.974
0.17
0.07
0.411
3.81E05
1.90E04
4.21E05
2.77E04
0.156
0.206
0.136
0.038
0.963
0.145
0.059
0.383
0.974
0.17
0.071
0.411
0.984
0.192
0.08
0.437
0.831
Model 3
a1
a2
a3
a4
0.959
0.159
0.057
0.329
0.983
0.217
0.088
0.462
0.972
0.184
0.073
0.394
1.72E05
1.18E04
2.55E05
3.91E04
0.186
0.715
0.393
0.186
0.965
0.17
0.064
0.363
0.972
0.182
0.074
0.394
0.979
0.206
0.081
0.429
0.821
MNS: Maximum Nash–Sutcliffe.
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
(Li et al., 2009) and (3) the maximum Nash–Sutcliffe value (MNS)
(Nash and Sutcliffe, 1970).
ARIL measures the resolution of the predictive distributions:
NQ in;p is the number of observations which are contained within the
p confidence interval. PCI was plotted as a function of p and should
be close to the 45° diagonal line.
The maximum Nash–Sutcliffe value (MNS) measures the efficiency of the model predictions.
ð19Þ
LimitUpper;t;p and LimitLower;t;p are the upper and lower boundary values of the p confidence interval, n is the number of time steps,
Robs;t is the observed discharge. ARIL should be as small as possible
and was plotted as a function of p.
The PCI measures the reliability of the predictive distributions
and it is based on the count, the number of the observed data within a range of simulated intervals, which is a function of p.
N
MNS ¼ maxfNSi g
a1
a2
a3
a1
a2
a3
1.000
0.003
0.029
0.003
1.000
0.725
0.029
0.725
1.000
T
P
NS ¼ 1 0.4
0.1
0
0.95
(b)
0.4
0
0.1
1
A1
0.4
0.2
0.1
(c)
0.4
0.975
A1
0.2
0.1
1
A2
0
0.3
0.1
A3
0.4
0.1
0
0.05
A3
0.4
0.3
0.5
A4
0.1
0.4
0.4
0.5
A4
0.3
0.2
0
0.2
0
0.3
0.1
0.1
0.2
0.4
0.3
0.3
0.2
0
0.1
0.05
0.2
0
0.3
0.1
0.975
0
0.4
0.3
Frequency
Frequency
0.2
0.2
0.1
0.3
0.1
0.4
0.2
0
0.3
A2
0.2
0
0.1
1
0.3
0
0.95
0.2
A4
0.3
0.1
0.3
Frequency
Frequency
0.3
0
0.95
0.2
0.1
0.975
0.4
0.3
Frequency
0.2
A3
0.4
0.3
Frequency
Frequency
0.3
A2
Frequency
A1
ð22Þ
ðRobs;t Robs Þ2
in which, i is the acceptable sample index, t is the time index, Robs;t is
the observed discharge, Rsim;t is the simulated discharge, and Robs is
the average value of Robs;t . MNS should be as close to 1 as possible.
However, the previous research (Li et al., 2010b) shows that it is
not adequate to judge the uncertainty results by only ARIL or PCI,
because they vary simultaneously. To facilitate a more direct
uncertainty assessment for 95% confidence interval of discharge,
Frequency
0.4
ðRobs;t Rsim;t Þ2
t¼1
T
P
t¼1
Frequency
(a)
ð21Þ
i¼1
Table 5
The Correlation between the parameters for monthly WASMOD estimated by
Bayesian method.
Correlation
ð20Þ
Frequency
1 X Limit Upper;t;p LimitLower;t;p
ARILp ¼
n
Robs;t
NQ in;p
n
PCIp ¼
Frequency
60
0.2
0.1
0
0.05
0.1
0
0.3
0.4
0.5
Fig. 4. Histograms of parameters of daily WASMOD derived by Metropolis Hasting method using: (a) Model 1; (b) Model 2 and (c) Model 3.
61
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
this paper proposes a new index based on ARIL and PCI, called the
Percentage of observations bracketed by the Unit Confidence Interval (PUCI):
PUCI ¼ ð1:0 AbsðPCI 0:95ÞÞ=ARIL
ð23Þ
From Eq. (23), we can see that the PUCI is ranges from zero to
infinity, and the upper boundary is not clear, which is a weak point
of the index. It’s only used in 95% confidence interval evaluation. In
fact the larger the PUCI the lower the uncertainty of 95% confidence
interval of discharge is.
4. Results and discussion
4.1. Parameter estimates and parameter uncertainty
Five parallel simulation chains are used in the MH-algorithm
with 5000 iterations each. The acceptance rate is ranging between
20% and 30%. The total number of samples
of each parameter is
pffiffiffi
25,000. The Scale Reduction Score R for all the parameters is
approximately 1.0, indicating convergence of the iterative procedure for a chain.
Fig. 3 shows the marginal posterior parameter densities calculated by Bayesian method for monthly WASMOD. Sharp and peaky
distributions show well identifiable parameters, while flat distributions indicate a larger parameter uncertainty. Fig. 3 shows that
all parameters have well-defined posterior distributions, from
which parameter estimates can be clearly inferred as modal values.
The marginal posterior distributions for all parameters are nearly
symmetric as indicated by the skewness coefficients in Table 3
and also visible in Fig. 3. Besides, it is evident that the interval of
parameter a1 is very narrow and parameter a3 has the most
skewed distribution.
Correlations between the parameters of monthly WASMOD
estimated from Bayesian samples are represented in Table 5. The
results show that the correlation coefficients range between
0.003 and 0.725. The correlation coefficients of all parameters
are quite small except between parameter a2 and a3 with a value
of 0.725. It shows that the slow flow and fast flow are highly correlated since both of them are related to soil moisture content.
For daily WASMOD, the marginal posterior parameter densities
calculated by Models 1–3 are shown in Fig. 4. The following can
be deduced from the Figure: (1) by comparing the posterior distributions of parameters derived by the three models, those estimated by Model 1 are slightly different; the variances and
Table 6
The Correlation between the parameters for daily WASMOD estimated by Bayesian method.
Model 1
a1
a2
a3
a4
Model 2
Model 3
a1
a2
a3
a4
a1
a2
a3
a4
a1
a2
a3
a4
1
0.468
0.617
0.283
0.468
1
0.189
0.259
0.617
0.189
1
0.327
0.283
0.259
0.327
1
1
0.338
0.380
0.272
0.338
1
0.259
0.148
0.380
0.259
1
0.231
0.272
0.148
0.231
1
1
0.341
0.488
0.363
0.341
1
0.352
0.305
0.488
0.352
1
0.282
0.363
0.305
0.282
1
Q(mm)
800
95% confidence interval
Observed
Simulated
600
400
200
0
1979-01 1980-01 1981-01 1982-01 1983-01 1984-01 1985-01 1986-01 1987-01 1988-01
Month (Y-M)
Fig. 5. The 95% confidence intervals of monthly discharge (1979/1–1988/12) due to parameter and model uncertainty (grey bands), simulated discharge at the maximum of
the posterior distribution (solid) and observed discharge (dots).
2
(a)
1
1.2
0.6
0.8
0.4
0.4
0
(b)
0.8
PCI
ARIL
1.6
0.2
0
0.2 0.4 0.6 0.8
Confidence Interval
1
0
0
0.2 0.4 0.6 0.8
Confidence Interval
1
Fig. 6. The Average Relative Interval Length (ARIL) and the percentage of observations bracketed by the Confidence Interval (PCI) due to different confidence interval value
(Probability) for 1979–1988 monthly runoff in Jiuzhou river basin.
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
effective parameter space obtained from Model 3 is more similar to
that from Model 1 than Model 2 and (2) the posterior distributions
obtained from Model 1 are somehow sharper and narrower than
those obtained from Model 2 and Model 3, indicating that a slightly
more optimal parameter set has been identified.
This is confirmed by the variances of parameter samples given
in Table 4, which reports posterior summary statistics for each
parameter estimated by three likelihood functions for daily WASMOD. It shows several interesting features: (1) the variance of
parameter samples estimated by Model 1 is smaller than those
from Model 2 and Model 3; (2) the interval length for parameter
a4 is nearly the same for the three methods, while for the other
three parameters, the samples by Model 2 and Model 3 have
slightly larger intervals than those by Model 1; (3) the marginal
posterior distributions for all parameters except for a3 of Model
1 and a2 of Model 3, are nearly symmetric as demonstrated by
the skewness coefficients and (4) the maximum Nash–Sutcliffe
(MNS) values of Model 1, Model 2 and Model 3 are 0.805, 0.831
and 0.821, respectively. Obviously, the MNS from Model 1 is the
smallest. It indicates that Model 1 has low accuracy although it
has the narrowest interval of posterior distribution for parameters.
It means that residuals autocorrelation in daily WASMOD is more
1
0.8
0.6
PCI
62
0.4
0.2
0
0.8 1.2
ARIL
1.6
2
important, which needs to be considered for uncertainty assessment purposes.
Table 6 shows the correlations between the parameters samples
estimated from the three Models. We see that compared to the
monthly WASMOD, the average correlation between the parameters has increased slightly, but is still in the same range. And
95% confidence interval
Observed
Simulated
30
Q(mm)
0.4
Fig. 8. The relation of ARIL and PCI due to different confidence interval value
(Probability) for 1979–1988 monthly runoff in Jiuzhou river basin.
(a)
40
0
20
10
0
1
51
101
151
201
Day of year 1982
251
(b)
40
351
95% confidence interval
Observed
Simulated
30
Q(mm)
301
20
10
0
1
51
101
151
201
Day of year 1982
251
(c)
40
351
95% confidence interval
Observed
Simulated
30
Q(mm)
301
20
10
0
1
51
101
151
201
Day of year 1982
251
301
351
Fig. 7. The 95% confidence intervals of daily discharge (1982) due to parameter and model uncertainty (grey bands), simulated discharge at the maximum of the posterior
distribution (solid) and observed discharge (dots) for (a) Model 1; (b) Model 2 and (c) Model 3.
63
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
Table 7
The comparison of uncertainty measures between two hydrological models for the 95% confidence interval.
Monthly WASMOD(79–88)
Daily WASMOD (78–82)
Model 1
Model 2
Model 3
Methods
ARIL
PCI (%)
ARIL
PCI (%)
ARIL
PCI (%)
ARIL
PCI (%)
Bayesian_P
Bayesian_PM
0.250
1.516
29.2
94.2
0.451
2.642
28.2
96.4
0.433
2.512
26.6
96.2
0.290
1.556
11.5
92.9
Bayesian_P represents the 95% confidence interval is only due to parameter uncertainty.
Bayesian_PM represents the 95% confidence interval is due to parameter and model uncertainty.
Table 8
The PUCI based on low, medium, high and all flows due to 95% confidence interval for
1978–1982 daily runoffs in Jiuzhou river basin.
Model 1
Model 2
Model 3
Low Flows
Medium Flows
High Flows
All Flows
0.255
0.263
0.610
0.428
0.442
0.607
0.805
0.813
0.675
0.373
0.393
0.629
now the largest correlation is between a1 and a3 and not a2 and a3
as in the monthly WASMOD.
4.2. Confidence interval of discharge due to parameters and model
uncertainty
For monthly WASMOD, the 95% confidence intervals of discharge from 1979 to 1988 due to parameter and model uncertainty
are plotted in Fig. 5. It shows that most of the observed data has
been covered by the 95% confidence intervals, which can also be
seen more clearly from Fig. 6 presenting the plots of change of Average Relative Interval Length (ARIL) and the percentage of observations bracketed by the Confidence Interval (PCI) due to different
confidence interval values for monthly WASMOD. It indicates that
both ARIL and PCI are increasing with the increase of confidence
interval. Furthermore, the relation of ARIL and PCI due to different
confidence interval value (Probability) was shown in Fig. 8.
Table 7 reports the uncertainty measures for different likelihood models for two different time-scale hydrological models for
the 95% confidence interval. Due to parameter uncertainty and
combined parameter and model uncertainty for monthly WASMOD, ARIL values derived by Bayesian method are 0.250 and
1.516, respectively, while the PCI values are 29.2% and 94.2%,
respectively. This indicates that the uncertainty in the simulated
discharges resulting from parameter uncertainty is less important
than that resulting from total uncertainty.
The 95% confidence intervals of daily discharge from 1982 estimated by three statistical models due to parameter and model
uncertainty for daily WASMOD are plotted in Fig. 7 for illustrative
purpose. Fig. 7a and b shows that there is no difference between
results obtained by Model 1 and Model 2 on the basis of visual
comparison, while the confidence intervals estimated by Model 3
are different in the two periods (Fig. 7c). It indicates that the confidence intervals in dry periods derived by Model 3 are narrower
than those derived from Model 2 and Model 1.
The difference of 95% confidence intervals between three statistical models can be seen more clearly in Table 7, which shows (1)
ARIL of 95% confidence interval due to parameter and model uncertainty derived from Model 2 is narrower than those from Model 1.
Furthermore, the PCI obtained by Model 2 is nearly the same as
that of Model 1; (2) ARIL of 95% confidence interval due to parameter and model uncertainty derived from Model 3 is the smallest
with a value of 1.556, or less than 60% of 2.512 and 2.642 from
Model 2 and Model 1, respectively. Meanwhile, the PCI of Model
3 is 92.9% which is quite close to 96.2% of Model 2 and 96.4% of
Model 1. The explanation lies in the fact that the statistical parameters of Model 3 depend on simulation periods which makes the
interval of dry period discharge to be much narrower and the result
thus more reliable and less uncertain and (3) ARIL of 95% confidence interval due to parameter uncertainty is much narrower
than those due to both of parameter uncertainty and model uncertainty for all statistical models in monthly and daily WASMOD.
Above results reveal that the parameter uncertainty is not the most
important key in uncertainty estimates for both monthly and daily
hydrological models.
However, from results of Table 7, it is difficult to identify which
likelihood function is the best for daily WASMOD when considering both ARIL and PCI as indices for confidence interval evaluation.
For instance Model 1 has the biggest PCI value while Model 3 has
the smallest ARIL value. In order to settle this problem and make
the comparison easier and clearer, a new index as defined in Eq
(23) is used for comparison and the results are shown in Table 8.
In which the values of PUCI based on low, medium and high flows
due to 95% confidence interval for 1978–1982 daily runoffs in the
Jiuzhou basin are compared for the three likelihood functions. It
can be clearly seen that (1) Model 3 has the largest PUCI of 0.629
and Model 1 has the smallest PUCI of 0.373 for all flows. Consequently Model 3 is the best for daily time steps, highlighting the
importance for the time-dependent statistical parameters; (2) the
PUCI of Model 2 is the largest based on the high flow, implying that
Model 2 is less uncertain and better in high flow estimate than the
other two models and (3) an inter-comparison of Model 1 with
Model 2 without time dependence shows in terms of PUCI values
that Model 2 is superior to Model 1 over the whole range of flows.
It reflects that Model 2 which uses more auto-correlation steps
over the time periods outperforms Model 1.
4.3. Comparison of application and computational efficiency
Table 9 shows the comparison of application and computational
efficiency of statistical models in WASMOD with different
time-scales. From the running time, we can see that Model 2 is
Table 9
Comparison of application and computational efficiency (Computer with Intel(R) Core(TM) 2 Duo CPU T8300 @ 2.40 GHz and 2.39 GHz, 1.96 GB of RAM).
Monthly WASMOD (1979–1988)
Difficulty of implement
Number of runs number of chains
Time
Very easy
5000 5
1 min
Daily WASMOD (1978–1982)
Model 1
Model 3
Model 2
Little complicated
5000 5
141.6 min
More complicated
5000 5
188.4 min
Very complicated
2000 5
23 days
64
L. Li et al. / Journal of Hydrology 406 (2011) 54–65
computationally more demanding than Model 1 and Model 3.
Implementation of model 1 is very easy compared to Model 2. This
is due to the fact that Model 2 considers the autocorrelation of
whole sample periods, which implies hundreds or thousands of
matrix calculations for the more complex likelihood function and
subsequent processing. For monthly WASMOD, the computational
efficiency is much higher than that of daily WASMOD. For daily
WASMOD, although it is more efficient for Model 2 to find the correct posterior distribution of parameters than for Model 1 and
Model 3, Model 2 is computationally more expensive than Models
1 and Model 3, which is certainly the main disadvantage of using
this model in the Bayesian method for uncertainty assessment.
5. Conclusions
This study performed a comprehensive comparison and evaluation of uncertainty estimation through the Bayesian method by
applying the Metropolis Hasting (MH) algorithm to two welltested conceptual hydrological models (WASMOD) operated with
monthly and daily time step. Several likelihood functions have
been constructed to supply the uncertainty estimates in monthly
and daily hydrological models. The Box–Cox transformation was
used on the observed and simulated flows in all likelihood functions. For monthly WASMOD a statistical Gaussian error model
was constructed for the residuals after the transformation. For daily WASMOD, the AR (1) plus Multi-Normal model was used after
the transformation (Model 2). Furthermore, Model 1 and Model 3
compared the results derived by two likelihood functions whose
statistical parameters dependent on time and independent on
time, respectively. A novel index called Percentage of observations
bracketed by the Unit Confidence Interval (PUCI) has been proposed
for uncertainty evaluation of discharges. The following conclusions
are drawn from the results of this study:
The autocorrelations of residuals in monthly WASMOD and daily WASMOD are different. In monthly WASMOD, it is not important
to consider the autocorrelation, whereas in daily WASMOD it is
necessary.
In monthly WASMOD, the posterior distributions of parameters
were sharp and peaky which is associated with well identifiable
model parameter sets. Besides, the goodness of model fit, as measured by the maximum Nash–Sutcliffe efficiency value, is comparable to the value calculated by the optimization algorithm.
The AR (1) model is not adequate enough to estimate the distribution of residuals in daily WASMOD. The values of PUCI show that
Model 2 outperforms Model 1, although the posterior distributions
of parameters derived by Model 1 are narrower than those obtained by the Model 2. Furthermore, the maximum Nash–Sutcliffe
efficiency value obtained by Model 2 is 0.831, which is much higher than 0.805 achieved by Model 1. We conclude that the estimated
parameters derived from Model 2 are more accurate than that of
Model 1.
For daily WASMOD, the values of PUCI show that Model 3 performs the best over the entire flow range, while Model 2 is the best
for high flows. Moreover, the MNS derived from Model 3 is 0.821,
which is slightly smaller than that of Model 2 which is in turn larger than that of Model 1. This is because Model 3 uses more statistical parameters which reflect more appropriately the statistical
properties of the residuals than Model 1. In fact Model 3 performs
superior on 95% confidence interval of discharge estimates than
Model 1.
Considering the difference in application and computational
efficiency, we can see that a simple Gaussian error model is suitable for monthly WASMOD since there is no auto-correlation in
transformed monthly residuals, while the value of PUCI shows that
Model 3 is the best for daily WASMOD. But if considering the
uncertainty of high flows and accuracy of estimated parameters,
Model 2 is superior compared to Model 1 and Model 3 for daily
WASMOD.
Despite that Model 2 outperforms in high flows for daily WASMOD, the PUCI of Model 2 is smaller than that of Model 3 for all
flows, which shows that Model 2 is also sub-optimal for daily
WASMOD. It is better for statistical models to split the simulation
period. Two periods (dry and wet period) for the statistic parameters are probably not enough, which means that additional time
sections need to be considered for Bayesian analyses in daily WASMOD. Furthermore, some low flows and high flows do not fit the
normal distribution after the Box–Cox transformation and AR (1)
modelling in daily WASMOD, which implies that further efforts
are required to improve the formulation of likelihood functions
used in hydrological applications since the assumption of normal
distribution of the residuals in daily WASMOD over the entire flow
range is inappropriate. Finally, one of the major challenges is to
find an appropriate likelihood function for the error distribution
when extreme flows are outside the normal distribution. In view
of the computational effort required by Model 2, another important issue is to find an improved way to account for error auto-correlation in daily hydrological models. We observe that although
the study has led to interesting findings, the generality of the results requires using additional hydrological models and study regions in future research.
Acknowledgments
This study was supported by Key Project of the Natural Science
Foundation of China (No. 40730632), the Norwegian Research
Council (RCN) for sponsoring project number 171783 (FRIMUF)
and the Environment and Development Programme (FRIMUF) of
the Research Council of Norway, project 190159/V10 (SoCoCA).
The authors would like to thank Mario L.V. Martina for his advice
and help in this study. We also appreciate the comments and suggestions from two reviewers and the associate editor, Prof. Ezio
Todini. Their comments greatly helped to improve the quality of
this paper.
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