The parameters of hydrological models are usually calibrated to achieve good
performance, owing to the highly non-linear problem of hydrology process
modelling. However, parameter
calibration efficiency has a direct relation with parameter range.
Furthermore, parameter range selection is affected by probability
distribution of parameter values, parameter sensitivity, and correlation. A
newly proposed method is employed to determine the optimal combination of
multi-parameter ranges for improving the calibration of hydrological models.
At first, the probability distribution was specified for each parameter of
the model based on genetic algorithm (GA) calibration. Then, several ranges
were selected for each parameter according to the corresponding probability
distribution, and subsequently the optimal range was determined by comparing
the model results calibrated with the different selected ranges. Next,
parameter correlation and sensibility were evaluated by quantifying two
indexes,

Hydrological process modelling is an important tool for research on water resource management, flood control and disaster mitigation, water conservancy project planning and design, hydrological response to climate change, and so on (Zanon et al., 2010; Papathanasiou et al., 2015). The initial hydrological model was a black-box model in 1932 (Sherman, 1932) and conceptual and physically based models were subsequently put forward in 1960s (Freeze and Harlan, 1969). The three kinds of hydrological models have been significantly improved in recent years, with their structures becoming more mature. Theoretically, the physically based model has a definite physical mechanism of the water cycle and all parameters can be measured in situ (Abbott et al., 1986; Huang et al., 2014). Conceptual models express hydrological processes in the form of some abstract models which come from some physical phenomenon and experience. For example, the interflow and the base flow are simplified as the flow from linear reservoirs (Caviedes-Voullième et al., 2012; Lü et al., 2013). As a result, some parameters of conceptual models need calibrating. In general, conceptual models have better performance in modelling the streamflow at the catchment outlet than physically based distributed models do, especially for catchments lacking sufficient data (Bao et al., 2010; Cullmann et al., 2011). Thus, many conceptual models such as HBV model, TOPMODEL, Tank model and Xinanjiang model are of strong vitality (Abebe et al., 2010; Vincendon et al., 2010; Hao et al., 2015; Xie et al., 2015). Additionally, the performance of physically based distributed models can be improved after calibration of some parameters (Chen et al., 2016). Therefore, all of the hydrological models should be calibrated before engineering applications.

There are two kinds of calibration methods for hydrological models, the trial–error method and auto-calibration method. The trial–error method depends on plenty of trials for reducing the error of the objective. However, it is difficult to obtain an exact optimal solution due to limited enumeration (Boyle et al., 2000). The auto-calibration method is based on stochastic or mathematical calculations and thus more widely applied in the non-linear parameter optimization. Compared with the trial–error method, it is more efficient and effective, avoiding the interference of anthropogenic factors (Madsen, 2000; Getirana, 2010). The initial automatic optimization methods, such as the Rosenbrock method (Rosenbrock, 1960) and the simplex method (Nelder and Mead, 1965), are classical and useful methods, but at the same time have a negative side of being bounded by initial value ranges of parameters. Therefore, it can only be regarded as local optimization algorithms (Gupta and Sorooshian, 1985). Different from classical methods above, the genetic algorithm (GA), which is designed with random search strategy, can avoid the problem of local search and thus is a global optimization algorithm in its essence (Wang, 1991, 1997; Sedki et al., 2009; Chandwani et al., 2015). After that, many global optimization algorithms have been proposed inheriting the random search strategy. The shuffled complex evolution (SCE-UA) method combines many advantages of the GA and simplex methods, having a powerful capability of calibrating the rainfall–runoff model (Duan et al., 1994; Zhang and Shi, 2011). The particle swarm optimization (PSO) based on random solution can directly obtain the identification parameters through the iterative search for an optimal solution (Kennedy, 1997; Zambrano-Bigiarini and Rojas, 2013). Although the auto-calibration method has been intensively employed to calibrate parameters in the field of hydrology, the most advanced algorithm inevitably falls into local solution because of the strong non-linear problem of a hydrological model and parameter correlation (Chu et al., 2010; Jiang et al., 2010, 2015).

In general, parameter variables follow some specific probability distributions within the given range after multiple independent calibrations (Viola et al., 2009; Jin et al., 2010; Li et al., 2010). Graziani et al. (2008) stated that the shape of a parameter probability distribution can be significantly affected by a parameter range. Touhami et al. (2013) studied the effect of different probability distributions (e.g. normal distribution and uniform distribution) of parameter values on parameter sensitivity, and found that the probability distribution can provide a clue for realizing parameter sensitivity. Although normal and uniform distributions are greatly studied in practice, other types of probability distributions were seldom investigated in previous research (Kucherenko et al., 2012; Esmaeili et al., 2014).

Most hydrological models contain many parameters of different sensitive characteristics and correlation patterns. Some researchers believe that the sensitive parameter should be calibrated, while the insensitive parameter can be set as a fixed value by experience (Beck, 1987; Cheng et al., 2006). Inappropriate parameter ranges or fixed values may result in the instability of calibrated results. Furthermore, the range setting of one parameter may influence the calibration of other related parameters (Song et al., 2015). The model parameter sensitivity analysis has been a growing concern in recent years. Parameter sensitivity varies with catchment characteristics, objective functions, and parameter ranges (van Griensven et al., 2006). Wang et al. (2013) noted the different parameter ranges could lead to changes in parameter sensitivity. Shin et al. (2013) reported that reducing or extending ranges might render insensitive parameters into sensitive ones or vice versa. Thus, parameter ranges and correlation should be taken into consideration when the calibration of multi-parameter models is performed.

Parameter ranges are generally given roughly due to lack of knowledge concerning physical settings of a local catchment (Song et al., 2013; Hao et al., 2015). The more deviation between an optimal range and a given range, the more uncertainty of the calibration result. The selection of appropriate parameter ranges is critical for calibrating the model efficiently. However, there have not been many documented studies on how to select the appropriate parameter range for improving the calibration of hydrological models. Furthermore, the calibration of multiple parameters is more complex due to parameter sensitivity and correlation. Hence, it is necessary to find a way to coordinate the range settings of all parameters.

Considering the effect of parameter ranges on calibration efficiency of hydrological models, an approach of parameter range selection (PRS) is put forward to improve the calibration of hydrological models with multiple parameters. At first, probability distribution of each parameter was analysed based on many independent calibrations by using a GA method. Then the optimal range of a single parameter was specified for calibration according to its probability distribution. Finally, parameter correlation and sensitivity were estimated to determine the optimal combination of multiple parameter ranges. The proposed method is expected to be helpful for an effective and efficient calibration of hydrological models with multiple parameters.

The Chaotianhe River catchment is located in the northeast of the Guangxi
Zhuang Autonomous Region in southwest China (Fig. 1). The Chaotianhe River is
the major tributary of the Lijiang River of a well-known
karst landscape. The total catchment area is 476.24

The data concerning daily precipitation, evaporation and streamflow were
collected from national gauging stations for the 5-year period of 1996–2000.
Four precipitation stations, one streamflow gauging station, and one
evaporation station are selected for the investigation. Areal precipitation
was calculated using data from the four precipitation stations by using a
Thiessen polygon method under GIS environment (Cai et al., 2014). The
streamflow gauging station is at the catchment outlet. Some
hydro-meteorological statistical data of the studied catchment are summarized
in Table 1. From 1996 to 2000, the maximum of daily streamflow was about
719

The method of PRS is designed for most of hydrological models. At present, there have been many hydrological models for hydrological process simulation. Considering the climate characteristics of the study area, the Xinanjiang model, which is suitable for humid regions, was chosen to serve as a hydrological model for the investigation. The Xinanjiang model mainly includes three evapotranspiration layers and three runoff components (i.e. surface-, subsurface runoff and groundwater) (Zhao, 1992). The surface runoff is routed by the Unit Hydrograph (UH) which is derived from the observed streamflow, and other runoff components are simplified as linear reservoirs (Ju et al., 2009). With regard to the Xinanjiang model, there are 10 parameters that should be calibrated. The definitions of the parameters are given in Table 2 (Lin et al., 2014; Hao et al., 2015). The proposed PRS method is introduced as follows, when a Xinanjiang model is taken as an example.

Metro-hydrological statistical data of the study area.

Location of the study area.

In theory, the parameter values calibrated by using a stochastic-based
auto-calibration method are not the same as each other but follow a specific
probability distribution under a reasonable convergence condition (Jiang et
al., 2015). The stochastic-based auto-calibration is used to calibrate the
model, and samples of calibrated parameter values are obtained in order to
analyse the probability distribution of parameter values. The sample size of
100 is adequate for estimating the probability distribution of calibrated
parameter values in the investigation, which is deduced from the results of
trial tests as shown in Fig. 2. It can be seen that both maximum and minimum

Parameters of Xinanjiang model.

Variation curves of maximum and minimum

A GA was selected as the auto-calibration method in the
investigation, because GA is a common and widely used global optimization
algorithm based on stochastic and evolutionary optimization. Many studies
show that evolutionary algorithms provide equal or better performance of a
model than other algorithms do (Cooper et al., 1997; Jha et al., 2006; Zhang
et al., 2009). The Nash–Sutcliffe efficiency (

The probability distributions of calibrated parameter values can be
estimated roughly by using box-plot charts, cumulative frequency curves, and
frequency histograms. The symmetry of the box-plot chart (including one box
and two whiskers) and the length ratio of the whisker to the box, the shape
of the cumulative frequency curve, and the frequency histogram are important
indicators for the identification of the distribution type. Based on these
indicators, three types of probability distributions are listed as follows:
(1) normal distributions, where the box and whiskers are approximately symmetrical
along the

Different probability distribution types of calibrated parameter
values.

A Kolmogorov–Smirnov test (K–S test) is geared to examine whether a data
set fits a reference probability distribution or not (Haktanir, 1991). In a
K–S test, for any variable

In order to improve

Selection of minimum and maximum range (MINR and MAXR) with a cumulative frequency of 50 %.

Variation curves of maximum and minimum

In general, there is more or less correlation between parameters for most
hydrological models. As far as a Xinanjiang model is concerned, parameters WM
and

If there is a negative influence between two parameters, the optimal range of
the parameter of higher sensitivity is used and the initial range of the
other parameter is kept for calibration generally to mitigate the negative
impact. It is due to the fact that sensitive parameters play more important
roles than insensitive parameters do in a multi-parameter calibration. In
order to assess the sensitivity of parameter range change to

Given the fact that there are more than two parameters in most hydrological
models, the accumulative influence and the coordination of range selection
were investigated in the study. The mean value of

The flow chart of multiple parameter range selections.

Range changes and K–S tests (

The flow chart of the PRS method is shown in Fig. 6. In
stage 1, a set of initial ranges of parameters are given for a hydrological
model and the probability distribution for each parameter analysed based on
the 100 independent parameters values calibrated by an auto-calibration
method. In stage 2, there are three range adjustment methods with response to
a probability distribution of parameter values: for a normal distribution,
the optimal range of a single parameter is obtained by reducing the initial
range; for an exponential distribution, the initial range of a single
parameter is extended to specify the optimal range, or the initial range is
reduced to seek the optimal range for calibration when the extension of the
parameter range is limited; for a uniform distribution, the initial range is
kept. In stage 3, the single-parameter range selection (S-PRS) is performed
on each parameter. Based on the indexes

A series of calibrated parameters values were obtained through 100
independent calibration runs by using a GA method. Trial tests were employed
to determine the optimal GA control parameters: crossover probability of 0.5,
mutation probability of 0.7 for the individual, mutation probability of 0.5
for each gene, population size of 21, maximum generation number of 500, and
maximum iteration number of 50. These parameters were kept constant for GA
calibrations in the investigation. The initial and calibrated ranges of
parameters are presented in Table 3. The ratio of the calibrated range length
to the initial one in Table 3 is less than 60 % for most parameters (i.e.
parameter CI, Kc, KI, SM,

The box-plot chart of normalized calibrated values for parameters of Xinanjiang model.

were employed to determine the probability
distributions of parameters and the corresponding results are listed in
Table 3. It is shown that only a normal distribution is accepted for
parameters CI and SM. Despite the fact that both normal and uniform
distributions are accepted for parameter KC, the probability distribution of
parameter KC is regarded as a normal distribution. It is because the

Results of range selection of parameter CI. Probability distribution
of parameter values for schema initial range

Since the probability distribution of a single parameter has a direct relation with the PRS, the range adjustment pattern of a single parameter was discussed on the basis of the parameter probability distribution in the investigation.

For a normal distribution, the range was reduced to find the optimal range.
Figure 8 shows the calibration results of parameter CI when the different
ranges are selected. The MINR (0.679–0.713) and the MAXR (0.623–0.694) were
picked out based on the cumulative frequency curve derived from calibrations
with the initial range (0–0.900). From the cumulative curves and the
histograms in Fig. 8a, b, and c, it is found that the probability distribution
of parameter CI values is converted from a normal distribution to a uniform
distribution when the initial range is reduced to the MINR, whereas the
probability distribution approximates an exponential one when the MAXR is
used. Figure 8d reveals the contribution of the PRS to

Results of range selection of parameter KI. Probability distribution
of parameter values for schema initial range

Results of range selection of parameter

The indexed

To an exponential distribution, both reduced ranges and extended ranges of
reasonable meaning were used to select the optimal range for parameter
calibration. Figure 9 shows the calibration results under three different
input ranges of parameter KI. Since the initial range of parameter KI cannot
be extended, the two reduced ranges (i.e. the MINR, 0.660–0.700, and the
MAXR, 0.522–0.660) were picked out according to the cumulative frequency
curve. From the cumulative curves and the histograms in Fig. 9a, b, and c, it
is found that the probability distribution of parameter KI values is similar
to a uniform distribution in the case of the MINR, whereas that is still
exponential in the case of the MAXR. The contributions of the three parameter
ranges to

Figure 10 shows the calibration results of parameter

The S-PRS method was employed to determine the optimal range for each
parameter. According to the optimal ranges and the corresponding initial
ranges, indexed

To coordinate with negatively related parameters, the index

Parameter range setting for different cases.

The symbol “I” represents the initial range of the parameter in Table 3, and “O” the optimal range of the parameter in Table 4.

In order to determine the optimal range combination of multiple parameters,
seven cases were investigated with different range combinations of parameters
(Table 5). Case 1 was defined as the initial case using all initial ranges.
Cases 2–4 were defined as the single parameter range selection
(S-PRS) cases. Cases 5–7 were set as the multiple parameter range
selection (M-PRS) cases. The box plots of

The box-plot chart of

The variation curves of maximum and minimum

Through a calibration run, a set of calibrated values of all parameters and
the corresponding

Considering that there is a relation between the selection of multi-parameter ranges and the calibration effect of a hydrological model, an approach to determine an optimal combination of ranges for the multi-parameter calibration was put forward by analysing parameter probability distribution, parameter sensitivity, and correlation between parameters. The newly proposed method was applied for the calibration of a Xinanjiang model for karst areas, and some findings are presented as follows.

In the Xinanjiang model, parameters CI, Kc, SM, and

The proposed PRS method improves the minimum and mean values of

The M-PRS method is superior to the
S-PRS one for calibrating hydrologic models with multiple
parameters. The

Please contact the corresponding author to access the data in this study.

The authors declare that they have no conflict of interest.

The investigation is supported by the Non-profit Industry Financial Program of Ministry of Water Resources of China (no. 201401057), NFSC (no. 91225301), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (no. 2013-1792). Edited by: G. Di Baldassarre Reviewed by: two anonymous referees