Journal of Hydrology 486 (2013) 384–394
Contents lists available at SciVerse ScienceDirect
Journal of Hydrology
journal homepage: www.elsevier.com/locate/jhydrol
Development and comparison in uncertainty assessment based Bayesian
modularization method in hydrological modeling
Lu Li a,⇑, Chong-Yu Xu a,b, Kolbjørn Engeland c
a
Department of Geosciences, University of Oslo, P.O. Box 1047, 0316 Oslo, Norway
Department of Earth Sciences, Uppsala University, Sweden
c
SINTEF Energy Research, 7465 Trondheim, Norway
b
a r t i c l e
i n f o
Article history:
Received 4 October 2012
Received in revised form 13 December 2012
Accepted 3 February 2013
Available online 13 February 2013
This manuscript was handled by Andras
Bardossy, Editor-in-Chief, with the
assistance of Ezio Todini, Associate Editor
Keywords:
High flows
Uncertainty assessment
Modularization Bayesian
Hydrological model
WASMOD
s u m m a r y
With respect to model calibration, parameter estimation and analysis of uncertainty sources, various
regression and probabilistic approaches are used in hydrological modeling. A family of Bayesian methods,
which incorporates different sources of information into a single analysis through Bayes’ theorem, is
widely used for uncertainty assessment. However, none of these approaches can well treat the impact
of high flows in hydrological modeling. This study proposes a Bayesian modularization uncertainty
assessment approach in which the highest streamflow observations are treated as suspect information
that should not influence the inference of the main bulk of the model parameters. This study includes
a comprehensive comparison and evaluation of uncertainty assessments by our new Bayesian modularization method and standard Bayesian methods using the Metropolis-Hastings (MH) algorithm with the
daily hydrological model WASMOD. Three likelihood functions were used in combination with standard
Bayesian method: the AR(1) plus Normal model independent of time (Model 1), the AR(1) plus Normal
model dependent on time (Model 2) and the AR(1) plus Multi-normal model (Model 3). The results reveal
that the Bayesian modularization method provides the most accurate streamflow estimates measured by
the Nash–Sutcliffe efficiency and provide the best in uncertainty estimates for low, medium and entire
flows compared to standard Bayesian methods. The study thus provides a new approach for reducing
the impact of high flows on the discharge uncertainty assessment of hydrological models via Bayesian
method.
Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
Conceptual hydrological models are widely used tools for calculating the runoff dynamics and the water balance at various scales
and regions (e.g. Xu et al., 1996, 2012; Widén-Nilsson et al., 2007,
2009; Kizza et al., in press; Zhang et al., 2012), for predicting
hydrological impact of climate change (e.g. Veijalainen and
Vehviläinen, 2008; Chun et al., 2010; Jung et al., 2012; Wetherald
and Manabe, 2002; Yang et al., 2012; Yuan et al., 2012), for
estimating impact of land use changes (e.g. Niehoff et al., 2002;
Petchprayoon et al., 2010; Rose and Peters, 2001; Wagener,
2007; Jiang et al., 2012), for simulating runoff in ungauged basins
(e.g. Vandewiele and Elias, 1995; Xu, 1999; Kokkonen et al.,
2003), and for flood forecasting (e.g. Kitanidis and Bras, 1980;
Krzysztofowicz, 1999, 2002; Jasper et al., 2002; Nester et al., in
press; Faulkner et al., 2012). Since the observations have unavoidable errors, computer models are not a perfect representation of
reality, and model parameters are usually not uniquely identifi⇑ Corresponding author. Tel.: +47 22856678; fax: +47 22854215.
E-mail address: lu.li@geo.uio.no (L. Li).
0022-1694/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jhydrol.2013.02.002
able, the model results are uncertain. Understanding and quantification of modeling uncertainty has become an important research
field in hydrology.
During almost three decades Bayesian method is widely used
for uncertainty assessment in various hydrological models and
catchments (e.g. Ajami et al., 2007; Engeland and Gottschalk,
2002; Engeland et al., 2005; Frost et al., 2007; Kuczera, 1983;
Kuczera et al., 2006; Li et al., 2010a; Marshall et al., 2007; Reggiani
and Weerts, 2008; Romanowicz et al., 1994; Thiemann et al.,
2001). The Bayesian method requires a specification of a likelihood
function. The likelihood function describes the probability density
of the observations when the model parameters are given. This
probability density can be derived as a function of the stochastic
properties of the simulation errors which are the difference between the observed and simulated streamflows. Many different
hypotheses of distribution laws for the errors have been proposed
(Johnson and Kotz, 1970; Venables and Ripley, 1997). A distribution that accounts for the temporal dependency among the errors
must be defined since simulation errors are correlated in time
(Liu et al., 2005). In order to derive the joint distribution or the
predictive conditional distribution from a treatable multivariate
L. Li et al. / Journal of Hydrology 486 (2013) 384–394
distribution, the common approach is to assume the errors are normally and identically distributed. Then a data transformation is often needed to satisfy the assumption (Jin et al., 2010; Maranzano
and Krzysztofowicz, 2004; Montanari and Brath, 2004; Todini,
2008). The discrete-time and continuous-time autoregressive error
models (Engeland et al., 2010; Krzysztofowicz, 2002; Mantovan
and Todini, 2006; Yang et al., 2007a) are commonly used.
Various regression and probabilistic approaches have been employed in hydrological applications for model calibration, parameter estimation, and analysis of uncertainty sources (Li et al., 2010b;
Liu et al., 2005; Looper et al., 2009; Wang et al., 2009; Yang et al.,
2008, 2007b). Some methods build a statistical model of the residuals to consider all uncertainty sources (Engeland and Gottschalk,
2002; Jin et al., 2010; Krzysztofowicz, 2002; Montanari and Grossi,
2008). Some others use parameter uncertainty to address all uncertainty sources, like the GLUE method (Beven and Freer, 2001).
While the rest analyze the input error and/or model structure error
jointly or separately (Chowdhury and Sharma, 2007; Kavetski
et al., 2006a, 2006b; Kuczera et al., 2006). However, the problem
with all the probabilistic approaches is that it is difficult to derive
error models whose assumptions are fulfilled by the data. None of
the above methods aim to improve the assessment of the uncertainty by considering different levels of flows, i.e. low, medium
and high flows. The success of the Bayesian method depends on
the realism in the formulation of the likelihood function. It is, however, difficult to find a proper likelihood function that is suitable
for all flows (low, medium and high). One important question that
has not yet been investigated is how high flows affect the uncertainty intervals of discharge and parameter estimates in the Bayesian method for hydrological modeling.
Modeling high streamflows is challenging due to quality of both
data and models. The quality of high streamflow observations depends on the quality of the rating curve, and will in many cases be
based on a few data points or even an extrapolation (Guerreroa
et al., 2012). In this case, high flows easily become outliers. In statistical analysis, the outlier’s problem is a complex one. Many studies of Bayesian analysis have attempted to treat the problems
regarding outliers, influence data, error diagnostics or suspect data,
i.e. assigning prior probabilities to different possible explanations
of the outliers; deletion or down weighting is taken as a result of
finding outliers (Barnett and Lewis, 1978; Hawkins, 1980; Cook
and Weisberg, 1982; Chaloner and Brant, 1988). Kuczera et al.
(2006) characterized model error by using storm-dependent
parameters and pointed out that there is a need to deal with outliers. McMillan et al. (2010) designed a method to quantify uncertainty in river discharge measurements caused by stage-discharge
rating curve uncertainty. In their study the discharge error distribution which is nonstationary in time, was parameterized. The results showed that flood flows’ uncertainty is high due to the
limited number of observations, up to ±23% of the median discharge and ignoring rating curve uncertainty could lead to significant underestimation of the uncertainty associated with the model
flow predictions, particularly during flood events. It also pointed
out those natural uncertainties, i.e. hydraulic geometry shifts during flood events, are more significant than measurement uncertainties. Furthermore for high streamflows, the results are
sensitive to the numerical solution of the hydrological model, i.e.
how are the differential equations integrated in time (Clark and
Kavetski, 2010; Kavetski and Clark, 2010, 2011). It is also difficult
to calibrate the model to high streamflows since small timing errors might result in large errors in streamflow values and timing.
Many non-Bayesian calibration strategies try to solve this problem.
Sorooshian and Dracup (1980) developed maximum likelihood criteria for various types of streamflow errors, including the heteroscedastic maximum likelihood estimator (HMLE), which deals
specifically the type of errors which is streamflow-value depen-
385
dent (i.e., large flows have higher errors). Other studies shows that
the parameters could be calibrated to fit different parts of the hydrograph using the multi-objective method or flow transformation
(e.g. Engeland et al., 2010; Fenicia et al., 2007; Madsen, 2000; Yapo
et al., 1998; Xu, 2001) or a fuzzy multi-objective method (Yu and
Yang, 2000). These papers show that model calibration is non-unique since no unique calibration will be optimal with respect to all
performance measures considered (Madsen et al., 2002) and model
parameters should be time-invariant in a given catchment (Fenicia
et al., 2008). One solution to this problem is to follow Hostache
et al. (2011) who calibrate two hydrological models, one for low
flow and one for high flows, and then derived a forecasting chain
based on these two models. To establish the two models they
use two objective functions, one enhancing the model error with
respect to low flow simulation, and the other enhancing model error with respect to high flows based on Fenicia et al. (2007). Coccia
and Todini (2010) modeled the predictive uncertainty of flood forecasting by using a joint truncated normal distribution, in order to
improve adaptation to low and high flows. The truncated normal
distributions (TNDs) for low flows and high flows were used to divide the entire normal domain into two sub-domains in their Model Conditional Processor (MCP) framework, which means that the
MCP can be applied assuming that the joint distribution in the normal space is not unique but can be divided into two TNDs.
The aim of this paper is to improve the uncertainty assessment
in hydrological modeling by using a Bayesian modularization
method. In this method reliable and suspect information used in
parameter inference are separated. In this study, the 2% highest
flows were considered as suspect data. This application of modularization method therefore aims to reduce the impact from suspected high flows and get more reliable uncertainty estimates of
discharge for different levels of flow magnitudes, especially for
low and medium flows. The primary aim was achieved by performing a comprehensive comparison and evaluation of uncertainty
estimates by the new Bayesian modularization method and standard Bayesian methods using the Metropolis-Hastings (MH) algorithm. The study is exemplified using a daily conceptual
hydrological model (WASMOD) (Li et al., 2011; Xu, 2002). In the
standard Bayesian methods, 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 Multinormal model. We also tested error models where the statistical
parameters were dependent and independent on time, respectively. This study can be considered as a continuation of the study
by Li et al. (2011).
2. Study area and data
2.1. Study area
The Jiuzhou catchment, a tributary of the Dongjiang River in
southern China, was used to perform the study (Fig. 1). The catchment has a drainage area of 385 km2 at Jiuzhou gauge station. It is
characterized by hills and plains. The catchment is located in a
Monsoon region in the tropical and sub-tropical humid climate
with a mean annual temperature of about 21 °C and only few days
of air temperature dropping below 0 °C in the mountainous areas.
The average annual precipitation for the period of 1978–1988 is
around 1802 mm, with 80% falling in April to September. The average annual runoff is 993.4 mm or half of the annual precipitation.
2.2. Data analysis
Daily Precipitation data of 1978–1982 were the main input to
the model. The Penman–Monteith formula was used to calculate
386
L. Li et al. / Journal of Hydrology 486 (2013) 384–394
Fig. 1. Main river and methodological stations and discharge station of Jiuzhou river basin.
the potential evaporation. All data have gone through quality control in earlier studies (Jin et al., 2010; Li et al., 2011). To make a
proper statistic model for the residuals of discharge, the Box–Cox
transformation method was used:
(
gðy; kÞ ¼
yk 1
k
k–0
ð1Þ
lnðyÞ k ¼ 0
where y is the variable in original space, g is the transformed variable and k is a transformation parameter. Criteria from Lilliefors
(1967) were used to verify the normality of the transformed variable. Different values of k were evaluated and the best value was
found to be 0.6 (Fig. 2). The residuals, defined as the difference between observed and simulated discharge values, are often heteroscedastic and autocorrelated at daily time step. The residuals nt of
M
the transformed observations gt(yt) and simulations gM
t ðyt Þ are
modeled as:
M
nt ¼ gt ðyt Þ gM
t yt
ð2Þ
Probability
where t represents the index of time, M indicates that the variable is
simulated by hydrological model. It was also necessary to verify
whether nt were stochastically independent. In our case the residuals were auto-correlated. We therefore fitted a AR(1) model. Fig. 4
shows that the variances of AR(1) innovations from daily residuals
have different values in different seasons. We therefore divided
them into two periods, i.e. a wet period from March to September
and a dry period from October to April.
3. Methodology
3.1. Standard Bayesian methods
Bayes’ theorem is expressed as follows:
pðujnÞ ¼ R
LðujnÞ f ðuÞ
LðujnÞ f ðuÞdu
ð3Þ
where the posterior density pðujnÞ of the model parameters
u = {h, x} conditioned on a sequence of observations n = {n1, . . . , nn}
is derived from the prior density f(u) and the likelihood function
LðujnÞ; n is the number of observations, and the model parameters
were specified to be u = {h, x} in which h represents the hydrological model parameters and x represents statistical parameters. The
0.999
0.99
0.95
0.75
0.25
0.05
0.01
0.001
-10
-5
0
5
10
Residual
Fig. 2. Normal probability plot of the residual after Box–Cox transformation k ¼ 0:6 for daily WASMOD from 1979 to 1988 at Jiuzhou station.
L. Li et al. / Journal of Hydrology 486 (2013) 384–394
integral in the denominator will in our application not be evaluated
since we will use a Metropolis–Hastings algorithm.
In view of the results of data analyzes mentioned above, we
tested three statistical models, whose likelihood functions LðujnÞ
are described in the following sub-sections (Li et al., 2011).
3.1.1. AR(1) plus Normal model independent of time (Model 1)
The likelihood function of the AR(1) plus Normal model (Yang
et al., 2007b) for the prediction errors were:
LðujnÞ ¼
1
r
Y
n
n
1 nt b aðnt1 bÞ
q 0 q
r
t¼1
r
r
ð4Þ
where 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.
3.1.2. The AR(1) plus Normal model dependent on time (Model 2)
The likelihood function for the daily WASMOD for which the
statistical parameters depend on the time periods is:
LðujnÞ ¼
1
r0
q
n0
r0
Y
n
1
n bt at ðnt1 bt1 Þ
q t
t¼1
rt
rt
ð5Þ
where xt{bt, at, rt} are the statistical parameters and which are labeled according to two periods: wet period, W (March to September) and dry period, D (October to February).
3.1.3. The AR(1) and Multi-normal model (Model 3)
The likelihood function (Mantovan and Todini, 2006) is built for
time steps ranging from 1 to t in order to predict t + 1:
xt fbt ; at ; rt g ¼
xW fbW ; aW ; rW g t 2 ðMar: to SepÞ
xD fbD ; aD ; rD g
others
P
exp 12 ðn bÞT 1 ðn bÞ
LðujnÞ ¼
P 1=2
ð2pÞn=2 ½detð Þ
ð6Þ
ð7Þ
P
where
is the covariance matrix that can be approximated with
(Basilevsky, 1983, p221; Mantovan and Todini, 2006):
2
1
6 r
6 1
6 2
6
2 6 r1
R ¼ rc 6
6 6 n2
4 r1
rn1
1
r1
1
r 21
r1
rn2
1
rn3
1
r1
1
rn4
1
r n3
1
r n4
1
1
r n2
1
r n3
1
r1
3
r n1
1
7
r n2
7
1
7
7
r n3
1
7
7
7
7
r1 5
387
on the results of uncertainty assessment. Liu et al. (2009) give a
methodological suggestion that is called modularization for dealing with this problem in computer models. The modularization
methods suggest keeping inference for different components or
modules in the model totally or partly separated so that suspect
information will be contained within one module and not influence
all parts of the model.
In hydrological modeling the streamflow observations might be
more or less suspect depending on the flow value. In this study we
attempted to use the modularization concept, and we assumed that
implausibly large flows are suspect information. This is partially
supported by Fig. 2 which shows that only the residuals between
3 and 3 fit the normal distribution line. We therefore, in a first step,
excluded implausibly large streamflow observations in the estimation of hydrological and statistic parameters estimation. In a second
step the statistical parameters were re-estimated using both reliable
and suspect information. We will describe this Bayesian modularization approach in more details in the following subsections.
3.2.1. Step one – inference using trusted information
The first step was to identify the trusted information and use
these data for inference in a separate module. High flows are probably more suspect compared to low and medium flows, e.g. extremely sensitive to the numerical solution of the hydrological model;
difficult to find a statistical model to address its property and in
some condition the extrapolation of statistics can result in significant errors for high flows. In this study, we classify the runoff into
three groups, i.e. high flows, medium flows and low flows, which
are accounting for 10%, 70% and 20% of total runoff, respectively.
We found that 71 points accounting for around 2% of total points
are out of normality line, in which 49 out-boundary spots are high
flows accounting for 69% of total points and none are low flows
(Fig. 3). So in this study, we assumed that all low and medium
flows are trusted. We divide the entire data set into two sample
groups: 2% highest flows was considered to be suspect sample
group and the rest points are belonging to the trusted group, i.e.
high flows (8%), all medium flows (70%) and all low flows (20%).
For the trusted sample, the autocorrelation of residuals has been
checked which is almost the same as before. It indicates that the
elimination of some high flows did not break the condition of
AR(1) model. Both hydrological and statistical parameters were
estimated within the trusted sample. In this likelihood function
we only considered those flows of g 6 ga:
ð8Þ
1
where
r2c ¼ v ar½ft t ¼ 1; . . . ; n
Pn
r1 ¼
t¼2 ðnt nÞðnt 1 Pn
2
t¼1 ðnt fÞ
nÞ
ð9Þ
ð10Þ
in which rc2 > 0, r 1 is the first-order autocorrelation coefficient, n is
the mean of residuals.
3.2. Bayesian modularization method
Bayesian analysis incorporates different sources of information
into a single analysis through Bayes’ theorem. This might be a
problem if part of information has flaws. Then the other sources
of information might be overly influenced, which impact directly
Fig. 3. Plot of residuals vs. observed runoffs of daily WASMOD at Jiuzhou station
(1979–1988): (1) red dash divided low flows and medium flows and purple dash
divided medium flows and high flows; (2) two black lines represents the normal
region of residuals (3, 3) according to Fig. 2. (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
LðujnÞ ¼
L. Li et al. / Journal of Hydrology 486 (2013) 384–394
1
r0
q
n0
r0
Y
m
1
n bt at ðnt1 bt1 Þ
q t
r
t¼1 t
rt
ð11Þ
Q g gM b a g gM b t
t ð t1
t1 Þ
t
t
1
t1
m
f ðuÞ
t¼1 rt q
rt
pðujgÞ ¼ R M Q M
gt gt bt at ðgt1 gM
bt1 Þ
g0 g0
1
1
t1
m
f ðuÞdu
t¼1 rt q
r0 q
r0
rt
1
r0 q
g0 gM
0
r0
ð12Þ
xt fbt ; at ; rt g ¼
Variances of residuals
388
1.2
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
Month
xW fbW ; aW ; rW g t 2 ðMarch to SeptemberÞ
xD fbD ; aD ; rD g
others
Fig. 4. The variances of daily residuals after Box–Cox transformation & AR(1) model
at Jiuzhou station (1979–1988).
ð13Þ
where m is the time dimension without suspected high flows, which
means that for all t = 1, . . . , m, we have gt 6 ga; ga represents threshold
value of high flows; g0 and gM
0 are the transformed observed variables
and the transformed simulated variables at the initial time. The same
as Model 2, Bayesian theorem updating of the prior distributions of
the statistical parameters given in Table 1 are obtained separately
for the two periods, i.e. wet period and dry period, for Bayesian modularization method. Other notations are as before.
In this case, the 2% highest flows that are considered as outliers,
were deleted in Bayesian analysis of step one. For some cases that
down weighing can be taken instead of deletion in Bayesian analysis if there is some knowledge or confidence about suspect data.
3.2.2. Step two – inclusion of suspect information
In the second step, the estimated values of the hydrological
parameters from step one were used to provide streamflow realization. The suspect data, i.e. 2% highest flows were used together
with the trusted data for inference of the statistical parameters,
from which the statistical parameters for high flows were estimated. In this step, the likelihood function of statistical parameters
depends on the wet and dry periods as well. The formulas are the
same as for Model 2 (Eqs. (6) and (7)).
The statistical parameters derived by Step one were used for uncertainty analysis of the trusted flows, i.e. low, medium and part of high
flows, while the statistical parameters from Step two were used for
the uncertainty analysis of suspect flows, i.e. 2% highest flows.
3.3. MCMC simulations
The Metropolis-Hastings (MH) algorithm (Hastings, 1970), a
Markov Chain Monte Carlo (MCMC) methodology, was used to
get the posterior distributions of the parameters (Chib and Greenberg, 1995; Engeland and Gottschalk, 2002; Kuczera and Parent,
1998; Li et al., 2010a). The prior densities of all parameters in
the daily WASMOD
pffiffiffi are shown in Table 1. Furthermore, the Scale
Reduction Score R (Gelman and Rubin, 1992) was used to check
whether the Monte Carlo chains converged.
3.4. Estimation and evaluation of uncertainty intervals
All the MH-samples of the hydrologic parameters were used to
calculate the uncertainty intervals for discharge due to parameter
uncertainty, whereas the total predictive distribution was calcu-
lated by adding the model residuals to each of the sample simulations (Engeland et al., 2005; Li et al., 2011).
We wanted the predictive distribution for the streamflow to
be narrow and reliable. An ideal uncertainty analysis technique
would lead to a 95% probability band that is as narrow as possible
while still being a correct estimate (i.e. on average 95% of the
observations are inside this interval). The sharpness was measured by the Average Relative Interval Length (ARIL) by Jin et al.
(2010), which should be as small as possible. The percentage of
observations bracketed by the Confidence Interval (PCI) (Li
et al., 2009) was used for reliability. The PCI counts the ratio of
the observed data within a range of intervals of p confident interval. PCI plotted as a function of p and should be close to the 45
degree diagonal line. Two indices that combine sharpness and
reliability were also used, i.e. the Continuous Rank Probability
Score (CRPS) (Hersbach, 2000) and the percentage of observations
bracketed by the Unit Confidence Interval (PUCI) (Li et al., 2011).
The CRPS is averaged over the whole time series. The minimal value (zero) of CRPS is only achieved when the empirical distribution is identical to the predicted distribution, that is, in the case
of a perfect deterministic forecast (Hersbach, 2000; Yang et al.,
2008). The PUCI was proposed by Li et al. (2011) for uncertainty
assessment of 95% confidence interval of discharge based on ARIL
and PCI.
PUCI ¼ ð1:0 AbsðPCI 0:95ÞÞ=ARIL
ð14Þ
The PUCI ranges from zero to infinity, and the upper boundary is
not clear (Li et al., 2011). The larger the PUCI the lower the uncertainty of 95% confidence interval of discharge is. Finally to be in
line with tradition in hydrology the maximum Nash–Sutcliffe value
(MNS) (Nash and Sutcliffe, 1970) is used for measuring efficiency.
The MNS measures the efficiency of the model predictions, which
should be as close to 1 as possible.
4. Results and discussion
The prior distributions specified in Table 1 were used. The
MH-algorithm was used to obtain 5 independent chains with
5000 iterations each giving totally 25000 samples. The acceptance
pffiffiffi
rate was between 20% and 30%. The Scale Reduction Scores R for
all the parameters have been limited between 0.9 and 1.1 to guarantee the convergence of the iterative procedure for a chain.
Table 1
The prior distributions of all parameters in daily WASMOD.
Parameters
a1
a2
a3
a4
a
b
r
Prior distributions
U[0, 1]
U[0, 1]
U[0, 1]
U[0, 1]
U[0, 1)
U[10, 10]
/r1
Note: a1 is evapotranspiration, a2 is slow flow parameter, a3 is fast flow parameter and a4 is routing parameter.
U[a, b] means the prior distribution of the parameter is uniform over the interval [a, b].
/ r1 Means the prior density of the parameter at value r is proportional to r1.
389
L. Li et al. / Journal of Hydrology 486 (2013) 384–394
0.3
0.3
0.3
0.3
0.2
0.1
0
0.95
0.2
0
0.1
1
0.2
0.1
0.1
0.975
Frequency
0.4
Frequency
0.4
Frequency
0.4
Frequency
(a) 0.4
0.2
0
0.3
0.2
0.1
0
0.05
0
0.3
0.1
0.3
0.3
0.3
0.3
0.2
0.1
0
0.95
0.2
0
0.1
1
0.2
0.1
0.1
0.975
Frequency
0.4
Frequency
0.4
Frequency
0.4
Frequency
(b) 0.4
0.2
0
0.3
0.1
0
0.05
0
0.3
0.1
0.3
0.3
0.3
0.1
0
0.95
0
0.1
1
0.2
0.2
0
0.3
0
0.05
0
0.3
0.1
0.3
0.3
0.3
0.1
0
0.8
1
0
0.2
0.1
0.1
0.9
Frequency
0.3
Frequency
0.4
Frequency
0.4
Frequency
0.4
0.2
0
a1
0.2
0.4
0
0.5
0.2
(d) 0.4
0.2
0.4
0.1
0.1
0.1
0.975
Frequency
0.3
Frequency
0.4
Frequency
0.4
Frequency
0.4
0.2
0.5
0.2
(c) 0.4
0.2
0.4
0.4
0.5
0.2
0.1
0
0.25
a2
0
0.5
a3
0
0.25
0.5
a4
Parameter
Fig. 5. Histograms of hydrological parameters of daily WASMOD derived by: (a) Model 1; (b) Model 2; (c) Model 3; (d) Bayesian modularization. (Note the scale difference in
the x-axis.)
4.1. Parameter estimates
Fig. 5 shows the marginal posterior parameter densities calculated by Bayesian modularization method and standard Bayesian
methods: the AR(1) plus Normal model independent of time (Model 1), the AR(1) plus Normal model dependent on time (Model 2)
and the AR(1) plus Multi-normal model (Model 3). The x-axis
scales of parameters densities of Bayesian modularization method
Table 2
The comparison of uncertainty measures for the 95% confidence interval.
Model 1
Model 2
Model 3
Bayesian Modularization
Methods
ARIL
PCI (%)
ARIL
PCI (%)
ARIL
PCI (%)
ARIL
PCI (%)
Bayesian _P
Bayesian _PM
0.451
2.489
28.2
96.4
0.28
1.963
12.1
94.8
0.433
2.362
26.6
96.4
0.767
1.150
44.1
92.3
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.
390
L. Li et al. / Journal of Hydrology 486 (2013) 384–394
Table 3
The ARIL, PCI, PUCI and MNS based on low, medium, high and all flows due to 95% confidence interval for 1978–1982 daily runoffs in Jiuzhou river
basin.
50
Index
Classify of flow
Model 1
Model 2
Model 3
Bayesian modularization
ARIL
Low flows
Medium flows
High flows
All flows
3.791
2.248
1.022
2.489
3.122
1.664
1.176
1.963
3.582
2.138
0.972
2.362
1.768
0.984
0.773
1.150
PCI
Low flows
Medium flows
High flows
All flows
0.989
0.987
0.769
0.964
1.000
0.954
0.795
0.948
0.989
0.992
0.744
0.964
0.977
0.933
0.744
0.923
PUCI
Low flows
Medium flows
High flows
All flows
0.254
0.428
0.802
0.396
0.304
0.599
0.718
0.508
0.268
0.448
0.816
0.417
0.550
0.999
1.026
0.846
CRPS
Low flows
Medium flows
High flows
All flows
0.196
0.367
2.106
0.512
0.150
0.338
2.245
0.496
0.200
0.363
2.224
0.523
0.125
0.311
2.756
0.527
MNS
All flows
0.805
0.823
0.827
0.834
(a)
95% confidence interval
Observed
Simulated
Q (mm)
40
30
20
10
0
1
51
101
151
201
251
301
351
Day of year 1982
50
(b)
95% confidence interval
Observed
Simulated
Q (mm)
40
30
4.2. Reliability
20
10
0
1
51
101
151
201
251
301
351
Day of year 1982
50
(c)
95% confidence interval
Observed
Simulated
Q (mm)
40
30
20
10
0
1
51
101
151
201
251
301
351
Day of year 1982
50
(d)
95% confidence interval
Observed
Simulated
40
Q (mm)
are different from that of other three models, which are much
wider. It shows that the posterior distributions derived by standard
Bayesian methods are narrower and sharper than those obtained
by Bayesian modularization method, which means less uncertainty
in parameters. Besides, the posterior distribution of parameter a1
in Bayesian modularization method is not symmetrical. All these
indicate that the uncertain range of hydrological parameters is larger derived by Bayesian modularization method than others. It is
probably because there are mainly low and medium flows to estimate hydrological parameters in step one of Bayesian modularization method. So there is less data to estimate the posterior
distribution of hydrological parameters in Bayesian modularization method which leads to a larger estimation variance.
30
20
10
0
1
51
101
151
201
251
301
351
Day of year 1982
Fig. 6. The 95% confidence intervals of daily discharge (1982) due to parameter
&model uncertainty (gray bands), simulated discharge at the maximum of the
posterior distribution (solid) and observed discharge (dots) for (a) Model 1; (b)
Model 2; (c) Model 3; (d) Bayesian modularization.
The PCI counts the ratio of the observed data within a range of
intervals p, which can be plotted as a function of p and should be
close to the 45 degree diagonal line, i.e. the 95% PCI should be as
close to 0.95 as possible. The difference of 95% confidence intervals
between all statistical models can be seen in Table 2. From the table, we can find that PCI of Bayesian modularization method is
92.3% which is close to 94.8% of Model 2, while Model 1 and Model
3 get the same result of 96.4%. Furthermore, PCI based on low,
medium, high and all flows for the 95% confidence interval are
shown in Table 3, which indicates (1) all models overestimated
low flows and underestimated high flows to some extent; (2) Model 1 and Model 3 over-estimated PCI for all flows and medium
flows, while which were slightly under-estimated by Bayesian
modularization method; and (3) most of the observed data has
been covered by the 95% confidence intervals for all flows, which
can also be seen more clearly from Fig. 7 which presents the plots
of change of the percentage of observations bracketed by the Confidence Interval (PCI) for different confidence interval values for
daily WASMOD. It indicates that (1) PCI estimated by Model 1,
Model 2 and Model 3 are more similar compared with the results
from Bayesian modularization method; (2) for low, medium and
all flows, Model 1, Model 2 and Model 3 obviously over-estimated
the intervals; (3) for high flows, all models under-estimated 95%
confidence interval; and (4) the PCI derived from Bayesian modularization method is most stable compared with those from other
three models for low, medium and all flows, which means Bayesian
modularization method performs the best in Reliability.
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L. Li et al. / Journal of Hydrology 486 (2013) 384–394
0.8
0.8
PCI of mid flow
(b) 1
PCI of low flow
(a) 1
0.6
model 1
0.4
model 2
model 3
0.2
0.6
model 1
0.4
model 2
model 3
0.2
model 4
model 4
0
0
0
0.2
0.4
0.6
0.8
Confidence Interval
1
(c) 1
0.2 0.4 0.6 0.8
Confidence Interval
1
(d) 1
model 1
model 2
model 3
model 4
0.6
0.8
PCI of all flow
0.8
PCI of high flow
0
0.4
0.6
model 1
0.4
0.2
0.2
0
0
model 2
model 3
model 4
0
0.2
0.4
0.6
0.8
Confidence Interval
1
0
0.2 0.4 0.6 0.8
Confidence Interval
1
Fig. 7. Proportion of observation inside confidence intervals (PCI) for Model 1, Model 2, Model 3 and Model 4 (Bayesian modularization) of daily runoffs in Jiuzhou river basin
for (a) low flows, (b) medium flow, (c) high flows and (d) all flows. High flows are 10% highest observed flows, low flows are 20% lowest flows and medium flows are the rest
flows.
(a)
(b) 2.5
4
3.5
ARIL of mid flow
ARIL of low flow
2
model 1
3
model 2
2.5
model 3
2
model 4
1.5
1
model 1
model 2
1.5
model 3
model 4
1
0.5
0.5
0
0
0
0.2
0.4
0.6
0.8
Confidence Interval
0
1
(c) 1.4
(d)
1.2
ARIL of all flow
ARIL of high flow
1
model 2
0.8
model 3
0.6
model 4
0.4
1
3
2.5
model 1
0.2 0.4 0.6 0.8
Confidence Interval
model 1
model 2
2
model 3
1.5
model 4
1
0.5
0.2
0
0
0
0.2
0.4
0.6
0.8
Confidence Interval
1
0
0.2 0.4 0.6 0.8
Confidence Interval
1
Fig. 8. Average Relative Interval Length (ARIL) for Model 1, Model 2, Model 3 and Model 4 (Bayesian modularization) of daily runoffs in Jiuzhou river basin for (a) low flows,
(b) medium flow, (c) high flows and (d) all flows. High flows are 10% highest observed flows, low flows are 20% lowest flows and medium flows are the rest flows.
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L. Li et al. / Journal of Hydrology 486 (2013) 384–394
4.3. Sharpness
The sharpness measured by ARIL should be as small as possible.
The 95% confidence intervals of discharge from 1982 estimated by
all statistical models due to parameter and model uncertainty for
daily WASMOD are plotted in Fig. 6 for illustrative purpose. It
shows that the confidence interval derived by Bayesian modularization method is much narrower than that from the standard
Bayesian methods, which indicates that the uncertainty of discharge estimated by this Bayesian modularization method is reduced. While comparing the three standard Bayesian methods in
Fig. 6a–c, it indicates that the confidence intervals in low flow periods derived by Model 2 are narrower than those derived by Model
3 and Model 1. Fig. 8 shows the plots of ARIL from Models 1–3 and
Bayesian modularization method for all flow classes. In this figure
we see that the confidence interval from the Bayesian modularization method is the sharpest for low, medium, high and all flows.
For high flows, ARIL estimated by Model 1 and Model 3 are almost
identical, which are slightly less sharp than that from Model 2. The
difference of 95% confidence intervals between all statistical models can be seen in Table 2. From the table, it is confirmed that ARIL
of 95% confidence interval due to parameter and model uncertainty
derived from Bayesian modularization method is the smallest with
a value of 1.150, followed by Model 2 of 1.963, and Model 1 and
Model 3 of 2.489 and 2.362, respectively.
4.4. Measures combing sharpness and reliability
From results of Table 2, it is not yet clear enough to identify
which statistical model is the best when considering both ARIL
and PCI as the indices for confidence interval evaluation. For instance Model 1 has the biggest PCI value while Bayesian modularization method has the smallest ARIL value for 95% confidence
interval. Therefore, two combined measures, PUCI and CRPS are calculated and their values are shown in Table 3 for 95% confidence
intervals of all flow classes. It can be clearly seen that Bayesian
modularization method has the largest PUCI values for low, medium, high and all flows. Besides, the CRPS results show that for
low and medium flows accounting for 90% of total flows, the
Bayesian modularization has the smallest value which indicates
better perform in the credibility intervals estimates, while Model
1 performs slightly better in CRPS for high flow.
4.5. Efficiency
The maximum Nash–Sutcliffe (MNS) values of Models 1–3 and
Bayesian modularization method are shown in Table 3, which are
0.805, 0.823, 0.827 and 0.834, respectively. Again the MNS from
Bayesian modularization method is the best. It indicates that
Bayesian modularization method has higher accuracy although it
has the widest interval of posterior distribution for parameters.
5. Conclusions
This study performed a comprehensive comparison and evaluation of uncertainty estimation between the proposed Bayesian
modularization method and standard Bayesian methods by applying the Metropolis-Hastings (MH) algorithm to the well-tested daily conceptual hydrological model (WASMOD). The parameters
uncertainty, reliability, sharpness and efficiency of uncertainty
intervals have been compared. From results of this study the following conclusions are drawn:
In terms of the PCI criterion, the uncertainty interval derived
from the Bayesian modularization method is most reliable.
Other three Bayesian models performed equally well.
In terms of the ARIL criterion, the uncertainty interval derived
by the Bayesian modularization method is the sharpest.
The maximum Nash–Sutcliffe efficiency value obtained by the
Bayesian modularization method is the highest.
According to the results of PUCI, Bayesian modularization
method outperforms other standard Bayesian methods over
the entire flow range.
The values CRPS show that the Bayesian modularization method
performs best in low and medium flows which is accounting for
90% of total flows.
The study show that the Bayesian modularization method performs better in terms of most criteria and flow classes than the
standard Bayesian methods as exemplified by the daily WASMOD
in the Jiuzhou river basin. However, it is worth of noting that the
Bayesian modularization method still under-estimated the uncertainty interval of high flows in this study. The reason is probably
that the normal distribution assumption for high flows is not good
enough. One of the major challenges is therefore to find an appropriate likelihood function for the error distribution for different
flow classes. Although this study has led to interesting findings, a
topic for future research is to assess the generality of the results
by testing the method in other areas using other models.
Acknowledgements
We thank two anonymous reviewers for their valuable manuscript comments. This paper is financially supported by China Ministry of Water Resources’ special funds for scientific research on
public causes (No. 200901042), 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), and the
111 Program of Introducing Talents of Hydrology and Water Resources Discipline to Hohai University. We would like to thank Prof
Yongqin David Chen of the Chinese University of Hong Kong for
providing the hydrological data.
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