Parameter identification and calibration of the Xin’anjiang model using the surrogate modeling approach

Abstract

Practical experience has demonstrated that single objective functions, no matter how carefully chosen, prove to be inadequate in providing proper measurements for all of the characteristics of the observed data. One strategy to circumvent this problem is to define multiple fitting criteria that measure different aspects of system behavior, and to use multi-criteria optimization to identify non-dominated optimal solutions. Unfortunately, these analyses require running original simulation models thousands of times. As such, they demand prohibitively large computational budgets. As a result, surrogate models have been used in combination with a variety of multi-objective optimization algorithms to approximate the true Pareto-front within limited evaluations for the original model. In this study, multi-objective optimization based on surrogate modeling (multivariate adaptive regression splines, MARS) for a conceptual rainfall-runoff model (Xin’anjiang model, XAJ) was proposed. Taking the Yanduhe basin of Three Gorges in the upper stream of the Yangtze River in China as a case study, three evaluation criteria were selected to quantify the goodness-of-fit of observations against calculated values from the simulation model. The three criteria chosen were the Nash-Sutcliffe efficiency coefficient, the relative error of peak flow, and runoff volume (REPF and RERV). The efficacy of this method is demonstrated on the calibration of the XAJ model. Compared to the single objective optimization results, it was indicated that the multi-objective optimization method can infer the most probable parameter set. The results also demonstrate that the use of surrogate-modeling enables optimization that is much more efficient; and the total computational cost is reduced by about 92.5%, compared to optimization without using surrogate modeling. The results obtained with the proposed method support the feasibility of applying parameter optimization to computationally intensive simulation models, via reducing the number of simulation runs required in the numerical model considerably.

Keywords

Xin’anjiang model / parameter calibration / multi-objective optimization / surrogate modeling

Introduction

Computer models comprise mathematical relations, data, and a calculation core, in order to simulate the behavior of a real-world system (Song et al., 2012c, 2013). Modeling of the rainfall-runoff process, which is dominated by watershed geomorphic and rainfall characteristics (Lee et al., 2008) owing to the tremendous spatial and temporal variability of watershed characteristics and precipitation patterns, and the number of variables (Jiang et al., 2013), has been the subject of research among hydrologists and engineers for a very long time (Srinivasulu and Jain, 2006; Song et al., 2012b). There is no doubt that a watershed hydrologic model is one of the most important tools in flood simulation, operational forecasting, and the detailed study of watershed response (Song and Kong, 2010; Song et al., 2012d; Yao et al., 2012; Xu et al., 2013). In recent decades, a large number of rainfall-runoff models have been developed, representing the main physical processes involved in the transformation of rainfall into runoff (Jones et al., 2008; Lü et al., 2013). Among these models, conceptual hydrologic models, as the most simple and effective rainfall-runoff modeling tools, have been widely applied in the world, such as the Sacramento model, the HBV model (Lindström et al., 1997), and the Xin’anjiang (XAJ) model (Zhao, 1992). Such models are used for various applications, including hydrologic process studies (Bingeman et al., 2006; Li et al. 2009), runoff estimates (Xu, 1999), soil moisture variations (Liu et al., 2007; Yu et al., 2010), and statistical analyses of hydrological processes in ungauged basins (Akhtar et al., 2008).

Generally, conceptual hydrologic models use a number of interconnected mathematical functions to approximate the soil moisture accounting phase of the hydrologic cycle at a basin scale. These models typically have ten or more parameters, some model input, some interim state variables, and some model output (Vrugt et al., 2006). Unfortunately, these parameters cannot be obtained directly, or measured from basin characteristics. A set of model parameters are determined by gradually adjusting the model parameters to match data, which is also known as parameter calibration or optimization (McCabe et al., 2005; Yang et al., 2011a). For the parameter calibration of hydrological modeling, the quality of the final model parameters derived depends primarily on four major factors: 1) the conceptual basis and structure of the model, 2) the power and robustness of the optimization algorithm (Yang et al., 2009), 3) the quality and the amount of information contained in the data available for model calibration and validation, and 4) the evaluation criteria used in the optimization procedure (Gan and Biftu, 1996; Yang et al., 2011b). In addition, the effect of the number or dimension of parameters on calibration, and the problem of equifinality and uncertainty (Song et al., 2011, 2012a) should not be ignored. Therefore, sensitivity and uncertainty analysis, as well as optimization for hydrological model parameters, should be considered simultaneously in the application.

Traditional calibration of hydrologic models is normally done with a single objective function. However, practical experience reveals that single-objective functions are often subjective, and inadequate to properly measure all of the characteristics of a hydrologic system (Vrugt et al., 2003b; Guo et al., 2013a). Therefore, it stands to reason that the calibration of hydrologic models should be made with multiple objective functions, which reflect different aspects of the system’s characteristics. In contrast to one optimal solution provided by single-objective optimization, the result of multi-objective optimization will not be a single best solution, but consists of a set of Pareto solutions. The important part for multi-objective optimization is to seek these approximate Pareto solutions. Recently, various multi-objective evolutionary algorithms have been developed in hydrological modeling (e.g., Yapo et al., 1998; Reed et al., 2003; Vrugt et al., 2003b; Khu and Madsen, 2005; de Vos and Rientjes, 2008, Dumedah et al., 2010; Dumedah 2012; Song et al., 2012c; Guo et al., 2013a). Over the years, evolutionary algorithms have been recognized and established as core optimization techniques for handling complex, multi-objective optimization problems, for example, the non-dominated sorting genetic algorithm (NSGA-II), Pareto archived evolutionary strategy (PAES), strength Pareto evolutionary algorithm (SPEA2), multi-objective shuffled complex evolution metropolis algorithm (MOSCEM), multi-objective shuffled complex differential evolution algorithm (MOSCDE), etc.

Nevertheless, calibrating parameters and evaluating their uncertainty in these models requires running the simulation models thousands of times, which also requires prohibitively large computational budgets (Razavi et al., 2012; Song et al., 2012c, 2013; Zhan et al., 2013). For this reason, surrogate modeling, or the meta-model approach, a second level of abstraction of the real-world system, is concerned with developing and utilizing cheaper-to-run “surrogates” of the “original” simulation models. Surrogate modeling has become increasingly popular over the last decade within the water resources community; and this is consistent with the increasing utilization of meta-models in the scientific literature since 1990 (Razavi et al., 2012).

The main motivation of developing surrogate modeling strategies is to make better use of the available, typically limited, computational budget. Many iterative water resources modeling analyses potentially stand to benefit from surrogate modeling (e.g., Khu and Werner, 2003; Broad et al., 2005, 2010; Mugunthan and Shoemaker, 2006; Yan and Minsker, 2006, 2011; Zou et al., 2007; Bliznyuk et al., 2008; Zhang et al., 2009, etc.). An incomplete list of classic or popular iterative modeling analyses in water resources, which are candidates for efficiency enhancement with surrogate modeling, include deterministic model parameter optimization (calibration) studies with evolutionary algorithms (e.g., Wang, 1991; Duan et al., 1992), uncertainty-based or Bayesian model calibration studies (e.g., Beven and Freer, 2001; Kavetski et al., 2006; Mugunthan and Shoemaker, 2006; Bliznyuk et al., 2008), multi-objective optimization algorithms (e.g., Reed et al., 2003; Zou et al., 2007; Razavi et al., 2012; Song et al., 2012c), global sensitivity analysis methods (e.g., Song et al., 2012d, 2013; Zhan et al., 2013), and any traditional Monte Carlo-based reliability or uncertainty analysis (e.g., Skaggs and Barry, 1997). Global optimization algorithms based on surrogate models have been developed to circumvent the computational budget limitations associated with computationally intensive simulation models, such as EGO (Efficient Global Optimization) (Jones et al., 1998), GMSRBF (Global Metric Stochastic Radial Basis Function), MLSLRBF (Multi Level Single Linkage Radial Basis Function) (Regis and Shoemaker, 2007), and Gutmann’s method (Gutmann, 2001).

Generally, example surrogate modeling-enabled multi-objective optimization algorithms are formed by 1) fitting response surface surrogate models to only one computationally expensive objective (or constraint) when other objectives are fast to run (Behzadian et al., 2009), 2) aggregating multiple objectives into one response function to be approximated by a single response surface surrogate (di Pierro et al., 2009; Zhang et al., 2010), and 3) using multiple surrogate models for multiple objectives (Wagner et al., 2010). In our study, the multiple surrogate models for multiple objectives were used to calibrate a conceptual hydrological model. However, the use of multiple surrogate models for multiple objectives has the potential to increase problems with inaccuracy, and definitely increases modeling time (Razavi et al., 2012). It is commonly assumed that for utilizing multiple surrogates, the approximation errors or uncertainties of these multiple surrogates are independent, despite the fact that the objectives are typically conflicting (Wagner et al., 2010). As a result, the uncertainty of the multi-objective optimization results based on surrogate models is also discussed.

The primary objective of our work is: (ⅰ) to construct the surrogate models for a conceptually-based distributed XAJ model, (ⅱ) to compare single-objective optimization with multi-objective optimization, and (ⅲ) to efficiently accomplish multi-objective parameter optimization for a complex and time-consuming hydrological model based on surrogate modeling. The remainder of this paper is organized as follows: Section 2 contains a brief description of the study area, the XAJ model, the surrogate modeling approach, and the evaluation criteria. Single objective and multi-objective optimization results are described in Section 3; and subsequently, Section 4 gives some conclusions of the study.

Materials and methods

The study area

The Yanduhe catchment (Fig. 1) of the Three Gorges is an upper tributary of the Yangtze River, with a drainage area of 601 km². The area, which is mountainous, with elevations ranging from 138 m above sea level at the watershed outlet to 3,031m in the upstream site, is covered by dense and thick vegetation. The vegetation cover accounts for approximately 80% of this area. There are five rain stations and one hydrological station in the catchment, as shown in Fig. 1. The mean annual temperature of this catchment is 11.5°C, the mean annual precipitation is almost 1,650 mm, and the mean annual runoff is 1,240 mm. Due to the dominance of a monsoon climate, more than 60% of annual rainfall occurs during the flood season (May to September). The DEM data on a grid scale of 30 m×30 m was used to extract the drainage network, and to derive topographic attributes of the watershed. The data set is provided by the International Scientific Data Service Platform, the Computer Network Information Center, and the Chinese Academy of Sciences (http://datamirror.csdb.cn). The catchment can be divided into five sub-catchments, in order to capture the spatial variations of precipitation and the underlying surface, based on the rain stations and the natural channel drainage network. The watershed characteristics for each sub-catchment are shown in Table 1 (Kong and Li, 2006). In this study, fifteen flood events from 1981 to 1987 were selected for analysis. The first 10 flood events were used to calibrate; and the other 5 flood events to validate.

XAJ model

The XAJ model is a conceptual watershed model developed in the 1980s, which can be used for forecasting flood, runoff, and stream flow over a catchment (Qu et al., 2012). The XAJ model has been widely used in many catchments since it was proposed, and much subsequent experience supports its validity for humid and semi-humid regions. A detailed description of the XAJ model can see in the work of Zhao (1984, 1992) and Zhao and Liu (1995). The model structure, inputs, outputs, and state variables of the XAJ model are shown in Fig. 2. The ranges and physical meaning of the model parameters are summarized in Table 2 (Li et al., 2009; Lü et al., 2013).

Surrogate model and design of experiments

Surrogate modeling has been widely used for uncertainty analysis and parameter optimization (Song et al., 2012c, 2013). Surrogate modeling is a collection of statistical and mathematical techniques useful for developing, improving, and optimizing processes. In general, a surrogate model utilizes some form of simplified function to approximate the underlying system using points that have already been evaluated, and is considered to be a fast surrogate model compared to the exact evaluation model. The simplified function may be a) linked to the processes to be modeled, b) some simplified solutions to a complex problem, c) a mathematical model of the input and output causal relationship, and d) a surrogate generated due to inherency or proximity. In this work, the third type of simplified function was used to construct a surrogate model. Therefore, a response surface model (RSM) was selected to explore the statistical relationships between original model parameters and output response objectives. The main idea of RSM is to obtain an optimal response between parameters (input) and objective functions (output), via a sequence of experiments designed from the original simulation model.

Generally, the formulation of a surrogate model involves the following three steps: a) experimental design—a means to generate data points for evaluation; b) model choice—the selection of a surrogate function as an approximator; and c) model fitting—tuning of the surrogate function to match the data points. First, most surrogate modeling enabled optimizers developed in the literature start with the formal Design of Experiments (DoE), in order to generate an initial set of design sites that is uniformly distributed in the explanatory variable space. A well-distributed initial set helps the surrogate models better represent the underlying (original) function (Razavi, 2013). However, as a priori knowledge about the underlying response surface is usually unavailable, DoE tends to assume uniformity of the distribution in the variable spaces. There are a wide variety of DoE methods available in the literature, such as full factorial design, fractional factorial design, central composite design, Latin hypercube sampling, LP-t method, and so on. In our study, the LP-t method is selected based on the PSUADE (Problem Solving environmental for Uncertainty Analysis and Design Exploration) platform to construct the surrogate model (Song et al., 2011), which was developed by the Lawrence Livermore National Laboratory (https://computation.llnl.gov/casc/uncertainty_quantification/).

The number of samples or design sites is another problem for constructing the surrogate model. The optimal size of an initial set of design sites is highly dependent on the shape and complexity of the original response surface, as well as the computational budget available. A small sample may not yield accurate statistical data, while a large sample will consume unnecessary computing resources (Zhan et al., 2013). The term “optimal” here reflects the fact that, for a given original response function, increasing the number of initial design sites would enhance the accuracy of fit. However, after some point, this enhancement would be at the expense of unnecessarily increasing the computational budget having to be initially spent on DoE, while it could have been spent more effectively in the next steps (Razavi, 2013). According to our previous work (Song et al., 2011, 2012a, 2012c, 2013), 4,000 samples based on the LP-t method are suitable for surrogate modeling. The first 2,000 samples were used to construct the surrogate model; and the other 2,000 samples were selected to validate.

In this work, the multivariate adaptive regression splines (MARS) method (Friedman, 1991) was selected to construct a surrogate model of the XAJ model due to its advantages (Song et al., 2013, Zhan et al., 2013). MARS is a nonparametric regression procedure that makes no assumption about the underlying functional relationship between the dependent and independent variables. Instead, it constructs this relation from a set of coefficients and basis functions that are entirely “driven” from the regression data. In a sense, it is based on the “divide and conquer” strategy, which partitions the input space into regions, each with its own regression equation (Quirós et al., 2009). This makes it particularly suitable for problems with higher input dimensions, where the curse of dimensionality would likely create problems for other techniques<FootNote>

http://documentation.statsoft.com/STATISTICAHelp.aspx?path=Gxx/MARSplines/MARSplinesIntroductoryOverview (2013-08-26)

</FootNote>. The resulting regression surface is piecewise linear and continuous. Basis functions are defined in pairs, using a knot or value of a variable that defines an inflection point along the range of a predictor. The basis functions together with the model parameters are combined to produce the predictions given the inputs. The detailed information for MARS can see the work of Friedman (1991).

Multi-objective optimization problem

Pareto solutions

In a single-objective model calibration approach, model performance is expressed by a single-objective function that reflects a subjective choice, highlighting a specific aspect of the hydrograph (Vrugt et al., 2003a; de Vos and Rientjes, 2008). This objective function is then optimized to find what is regarded as the optimal model parameters. Multi-objective methods, on the other hand, reveal a set of Pareto solutions that represent the trade-off between the objectives involved. The benefit of this approach is that more information from the data is used in the evaluation of the model. If a model performs well on multiple objectives it implies performance consistency; and thus, the model is likely to be more reliable.

The considerations of multi-objective calibration imply the design of a calibration strategy that has the ability to incorporate several objective functions (Reichert et al., 2011) simultaneously. A strategy that can address this challenge is multi-objective optimization, and can be stated as follows:

(1)\min F\left(\theta \right)=\min \left\{F_{\mathrm{1}}\left(\theta \right),F_{\mathrm{2}}\left(\theta \right),\mathrm{...},F_m\left(\theta \right)\right\},\theta \in \mathrm{\Theta },

where q represents the model parameter set, F_i(q) is the i^th of m objective functions, and Q is feasible parameter space. The solution to this problem will, in general, no longer be a single “best” parameter set, but will consist of a set of Pareto solutions in the feasible parameter space Q corresponding to various trade-offs among the objectives (Vrugt et al., 2003b; Tang et al., 2006; Guo et al., 2013a).

In this work, the surrogate modeling based multi-objective optimization method was used to estimate and optimize the parameters of the XAJ model. The flowchart of the integration method involves the following steps as shown in Fig. 3.

Multi-objective evaluation criteria

The selection of the length and information content of the experimental data series used in the model evaluation is a critical step, since it has strong influence on the model’s performance. As a result, methods for identifying critical time periods in the observations, which contain most of the information for parameter identification, can provide useful guidance when selecting the evaluation data series (Singh and Bárdossy, 2012). For addressing model performance evaluation, the use of qualitative and quantitative criteria is recommended, because both approaches allow for capturing distinct aspects of model performance (e.g., Clark et al., 2011; Kavetski and Clark, 2011; McMillan et al., 2011; Biondi et al., 2012; Pushpalatha et al., 2012; Ritter and Muñoz-Carpena, 2013). Success in the use of computer models for simulating environmental variables and processes requires objective model calibration and verification procedures. Several methods for quantifying the goodness-of-fit of observations against model-calculated values have been proposed (Song et al., 2012a, b). In this work, a combination of four evaluation tools was used to calibrate the models’ parameter values and assess the performance of their simulations.

First, in order to evaluate model performance, a plot of observations against the calculated values illustrates the degree to which the points match the identity line (denoted as the 1︰1 line). This scatter plot allows for a visual inspection of model performance, such that the higher the agreement between calculated and observed values, the more the scatters tend to concentrate close to the 1︰1 line. Another interesting aspect of this graphical method is its ability to show 1) if model performance is homogeneous along the prediction range, or depends on the magnitude of the calculated values; 2) other unexpected relationships between the two data sets: non-linear or a linear correlation different to the identity line; 3) the presence of potential outliers; and 4) appreciable model bias (Ritter and Muñoz-Carpena, 2013).

Secondly, as a dimensionless goodness-of-fit indicator, the Nash-Sutcliffe efficiency (NSE) coefficient (Nash and Sutcliffe, 1970) was selected. NSE represents the complement to unity of the ratio between the mean square error of observed vs. predicted values and the variance of the observations. This coefficient takes values –∞≤NSE≤1. NSE=1 indicates a perfect fit, while NSE≤0 suggests that the mean of the observed values is a better predictor than the evaluated model itself (Ritter and Muñoz-Carpena, 2013). This is calculated as follows:

(2)NSE=\mathrm{1}-{\displaystyle \frac{{\displaystyle {\sum }_{i=\mathrm{1}}^N{\left(O_i-S_i\right)}^{\mathrm{2}}}}{{\displaystyle {\sum }_{i=\mathrm{1}}^N{\left(O_i-\overline{O}\right)}^{\mathrm{2}}}}},

where O_i and S_i represent the sample (of size N) containing the observations and the model estimates, respectively; and \overline{O} is the mean of the observed values.

Thirdly, for quantifying prediction error in terms of the units of the variable calculated by the model, relative error of peak flow (REPF), and runoff volume (RERV) were selected (Madsen, 2000; Song et al., 2012b). These indicators are frequently used and their definitions are given by

(3)REPF={\displaystyle \frac{O_{\mathrm{p}}-S_{\mathrm{p}}}{O_{\mathrm{p}}}}\times \mathrm{100}\mathrm{\%},

(4)RERV={\displaystyle \frac{V_{\mathrm{o}\mathrm{b}\mathrm{s}}-V_{\mathrm{s}\mathrm{i}\mathrm{m}}}{V_{\mathrm{o}\mathrm{b}\mathrm{s}}}}\times \mathrm{100}\mathrm{\%},

where O_p and S_p represent the observed peak flow and the model estimates, while V_obs and V_sim mean the observed and simulated total runoff volume, respectively. These indicators range from 0 to ∞, where the value 0 indicates a perfect fit.

Performance of the multi-objective calibration problem

Section 2.4.1 reveals that multi-objective optimization problems can be stated as a minimum or maximum optimal problem for many evaluation criteria. However, the above three objective functions (NSE, REPF, and RERV) could not construct a minimum or maximum problem directly since these functions are not all good for smaller and smaller, or vice versa. As a result, the multi-objective optimization problem can be stated as follows:

(5)\min F\left(\theta \right)=\min \left\{F_{\mathrm{1}}\left(\theta \right),F_{\mathrm{2}}\left(\theta \right),F_{\mathrm{3}}\left(\theta \right)\right\},\theta \in \mathrm{\Theta },

where F_{\mathrm{1}}\left(\theta \right)=\mathrm{1}-NSE,\theta \in \mathrm{\Theta }, F_{\mathrm{2}}\left(\theta \right)=\left|REPF\right|,\theta \in \mathrm{\Theta } and F_{\mathrm{3}}\left(\theta \right)=\left|RERV\right|,\theta \in \mathrm{\Theta }.

The solution of Eq. (5) will not, in general, be a single unique set of parameters but will consist of the so-called Pareto set of solutions, according to various trade-offs between the different objectives. A member of the Pareto set will be better than any other member with respect to some of the objectives, but because of the trade-off between the different objectives, it will not be better with respect to other objectives. When solving a multi-objective calibration problem, the problem is usually transformed into a single-objective optimization problem by defining a scalar that aggregates the various objective functions. For example, some previous studies turned the multi-objective optimization problem into a single-objective optimization problem by taking the linear weighted sum of the multiple objectives (Bennett et al., 2004; Huang et al., 2013). In practical applications, the entire Pareto set may be too expensive to calculate, and one is only interested in part of the Pareto optimal solutions. In this case, it is proposed that an aggregated objective function that puts equal weights on the different objectives is used. The aggregated objective function is the Euclidean distance (Madsen, 2000)

F_{agg}\left(\theta \right)={[{\left(F_{\mathrm{1}}\left(\theta \right)+A_{\mathrm{1}}\right)}^{\mathrm{2}}+{\left(F_{\mathrm{2}}\left(\theta \right)+A_{\mathrm{2}}\right)}^{\mathrm{2}}}^{}

(6){+{\left(F_{\mathrm{3}}\left(\theta \right)+A_{\mathrm{3}}\right)}^{\mathrm{2}}]}^{\mathrm{1}/\mathrm{2}},

where A_i are transformation constants assigned to the different objectives. A balance measure can be defined by assigning transformation constants in Eq. (6) such that all (F_i+A_i) have about the same distance to the origin. The minimum values of F_i (F_i,min) are estimated from this initial population, and each of the objective functions is transformed to having the same distance to the origin as the objective function with the largest minimum value of F_i, i.e.,

(7)\begin{array}{ll}{A}_i=\max \left\{F_{j,\min }\right\}-F_{i,\min }\hfill & i,j=\mathrm{1},\mathrm{2},\mathrm{3}\hfill \end{array}.

Results and discussion

Parameter identification and selection

As stated above, the curse of dimensionality is increasingly hard to prevent due to the development and application of distributed hydrological models. Their model structures are much more complex than that of empirical models, with many sub-processes considered. Wheater (2002) noted that “a simple model structure does not reflect the complexity of the rainfall-runoff response; and a complex model structure is not always supported by the available data. A balance between the complexity of the model and available information is crucial for successful model identification”. Model complexity can be reduced by an appropriate degree through identification statistics (Dunn et al., 2008) or sensitivity analysis (Fenicia et al., 2008; van Werkhoven et al., 2009), and by holding insensitive parameters constant or formally re-structuring the model (McIntyre and Al-Qurashi, 2009). Therefore, the first step is to select an appropriate number of parameters for optimization (e.g., sensitivity analysis to reduce the dimensionality of model parameters).

A sensitivity analysis of model parameters was achieved in former work (Song et al., 2013). According to the results of the sensitivity analysis, ten parameters (e.g., K, WM, SM, KI, KG, CS, CI, CG, KE, XE) of the XAJ model are important for hour-scale simulation processes (Song et al., 2013). Actually, some sensitive parameters have clear physical meanings, such as Muskingum method parameters KE and XE, which can be initially determined by hydraulic formulas. KE may be approximately the travel time over a sub-reach; and XE may be determined by the special length and the length of the sub-reach. However, these parameters should also be estimated through manual or automatic calibration due to lack of available data and information. In addition, WM is usually considered as non-sensitive for output response (Li et al., 2009), whereas it is also important for the objective of water balance from our previous work (Song et al., 2013). By the degeneration principles mentioned before, taking a fixed value (e.g., 0.7) for the sum of KI and KG, optimizes the two parameters by a two-dimensional joint search. That is, one of the two parameters can be determined once the other parameter is estimated. In summary, nine parameters were selected for analysis in this study. The other non-sensitive parameters can be given fixed values, in accordance with prior knowledge and practice, for example, UM=20 mm, LM=70 mm, B=0.3, IM=0.01, C=0.15, EX=1.5, L=1 hr.

Establishment and validation of the surrogate model

In the MARS method, the number of basis functions is 1,000 and the number of knots is 500. In this study, the two parameters K and SM were selected as an example to construct the response surface models with different objectives, as shown in Fig. 4. The x-axis is parameter K, the y-axis is the parameter SM, and the z-axis shows the objective functions. From the response surface relationship between two parameters and the three different objective functions in Fig. 4, we can see that there are different response surfaces and optimal sites of parameters for different objectives. In a sense, it also indicates that the model calibration for single-objective functions is often inadequate to measure all of the characteristics of the model.

Another vital and indispensable step is to validate and test the response surface models. Two approaches were applied to check the responses for goodness-of-fit by examining the interpolation error and prediction error. The first method is cross-validation ( k-fold cross-validation method) (McLachlan et al., 2004), the other is an experimental test. The statistical histograms of interpolation errors based on the k-fold cross-validation method (k=500) for all the three objectives are shown in Fig. 5. The results demonstrate that all of the responses give acceptable interpolation errors, i.e., the sample is adequate to represent the input-output relationship. In addition, Fig. 6 depicts scatter plots between the surrogate-models’ simulated objective function values and the corresponding original XAJ model generated values. Note that the objective function values for these 4,000 samples for validation and test correspond to the values from the surrogate modeling. The data points do not deviate greatly from the 1︰1 line for the four objective functions, which is evident from the high R² values. The plots exhibit a very intense scatter, implying that the surrogate models mimic the performance of the original XAJ model fairly well. Once the surrogate models are deemed satisfactory, parameter optimization can rely on these response surface models, which are inexpensive to evaluate.

Single-objective optimization results

This section intends to estimate the efficiency of parameter calibration with a single-objective function. When the response surface models are acceptable and reasonable as a surrogate, the ten parameters of the XAJ model are optimized by searching the optimization values around the response surface models. As mentioned before, the parameters are optimized by the surrogate modeling using NSE, REPF, or RERV as the single-objective function. A population size of 100,000 from the three response objectives is used to estimate the optimal solution sets with respect to the 10 parameters, respectively. Table 3 gives the parameter values based on a single-objective optimal scheme for the three objectives. To illustrate a comparison between the optimal simulated discharge and the observed discharge, Figure 7 provides scatter plots for the optimization results in calibration and validation. For the calibration process, single-objective parameter optimization qualifies relative to the aspect of corresponding objective functions. However, it is not simultaneously perfect for all three of the objective functions in the calibration and validation periods. In other words, the models show large discrepancies in the values of one criterion when the other one is used as the objective function or evaluation criterion. For example, the NSE value drops below 0.8 when using REPF as an objective, while they are largely acceptable (above 0.9) when using NSE as the objective function. Unfortunately, the REPF value in the validation based on the REPF optimal parameters is –59.5%, which is far below the limit value of acceptability for model prediction (±20%). Furthermore, the simulation in the validation based on the RERV optimal parameters is also very poor, with an REPF value of –85.65%.

The reasons include the selection of the objective function and the problem of split sample. In this work, the first 10 flood events were taken as a seriate flood event to calibrate model parameters, which may be a key reason for the poor performance of the flood peak simulation. The results, as shown in Fig. 7, also reveal that single-objective functions, no matter how carefully chosen, are often inadequate to properly measure all of the characteristics of the observed data. In the validation period, most of the criteria are degraded; but the calibration using NSE as the objective function appears to be more robust, since the degradation on NSE from the calibration to validation mode is less pronounced than when using REPF and RERV as objective functions. To conclude, it is shown that the criteria based on REPF or RERV cannot be used as a single-objective function, since it may produce large discrepancies in water balance simulations, or erroneous estimation of peak discharges. However, these two criteria are usually considered as hydrological metrics for model calibration and application, because these indicators are suitable for describing rainfall-runoff response at a basin scale. NSE, as a statistical metric, focuses on the hydrograph (i.e., errors and trends). Therefore, it is evident that all of these metrics should be considered in multi-objective optimization.

Multi-objective calibration and uncertainty assessment

The defining limit of acceptability for model predictions depends on the model applications (Beven, 2006). Although subjective, we proposed four model performance classes based on the three objectives for model calibration in this work, which are denoted as Unsatisfactory, Acceptable, Good, and Very good (as shown in Table 4). In accordance with the accuracy standard (Standard for Hydrologic Information and Hydrological Forecasting, GB/T 22482-2008) established by the Ministry of Water Resources of the People’s Republic of China, if the absolute value of RERV and REPF is less than 20%, the simulation result qualifies relative to peak flow and total runoff volume. In addition, the criteria for our goodness-of-fit evaluation was proposed according to various recommendations that can be found in the literature (Moriasi et al., 2007; Biondi et al., 2012; Pushpalatha et al., 2012; Ritter and Muñoz-Carpena, 2013).

Because any conceptual watershed model will, in general, be unable to match all of the different aspects of the watershed’s behavior observed in the measured hydrograph, all three transition objective functions (F₁, F₂, F₃) are used to calibrate the XAJ model using surrogate modeling. Generally, if the performance rating is “good” or “very good” for a single-objective function, the optimal scheme is considered a perfect performance associated with the corresponding objective function. Besides, when the classes for the three objective functions are good, i.e., they all meet the performance thresholds as shown in Table 4, the optimal scheme could be considered to be a good solution; and it can be defined as an efficient solution for analyzing the Pareto front. A number of samples and tests were carried out in order to estimate the Pareto front and to analyze the trade-off solutions. Because in a multi-objective problem, several objectives are to be considered simultaneously, the concept of inferiority-superiority in multi-objective theory (Yapo et al., 1998) was used to rank the population and to obtain the Pareto optimal solution.

In this work, considering the accuracy standard (GB/T 22482-2008) and the concept of Pareto solutions, the maximum 100,000 samples were based on surrogate modeling and were used to estimate the Pareto sets. The single-objective optimizations provide the tails of the Pareto front, and the optimization based on the balanced aggregated measure approximate the balanced central part of the Pareto front (Madsen, 2000). The estimated Pareto front for the calibration of the three-objective problem is shown in Fig. 8. In this case the compromise solution corresponds to a break point on the Pareto front; that is, moving along the front in either direction implies only a small decrease of one of the objective functions, at the expense of a pronounced increase of the other objective function. In addition, many studies have demonstrated that the Pareto front for a three-objective problem should be a two-dimensional surface (e.g., Goel et al., 2007; Cao et al., 2011; Trautmann et al., 2013). Previous work presented by Guo et al. (2013b) demonstrated that the Pareto front for three-objective functions based on MSLE, M4E, and MSDE for the Xin’anjiang model is approximately a curve, the same as in Fig. 8.

In fact, this Pareto front is not a one-dimensional curve because the front is not consequent. That is, it is a two-dimensional surface. The variation of the optimum model parameter sets along the Pareto front is shown in Fig. 9. It presents normalized parameter plots for each of the sensitive parameters of the XAJ model using the surrogate modeling based on the 100,000 trials. The parameter values are normalized with respect to the upper and lower limits given in Table 2, so that the feasible range of all parameters is between 0 and 1. The ten XAJ model parameters are list along the x axis, while the y axis corresponds to the parameter values scaled according to their prior uncertainty ranges to yield normalized ranges. A remarkably large variability for some sensitive parameters (e.g., CS, CI, and CG) is observed in the parameter values when moving along the Pareto front. The variation of parameter combinations along the Pareto front also implies a large variability on the simulated hydrograph, as shown in Fig. 10.

The traditional concept of model calibration based on a single objective is built on the hypothesis that a unique optimum set of parameter values exists. It should be clear from the analysis above that such a unique global solution does not exist. In a multi-objective context, there is a multitude of parameter combinations that are “equally good”. Therefore, a calibrated model should produce a range of simulated hydrographs corresponding to the “equally good” parameter sets along the Pareto front. The hydrograph uncertainty ranges associated with these good solutions for ten sensitive parameters of the XAJ model in the calibration and validation periods are displayed in Fig. 10. The observed discharges are indicated with solid areas; while the good solutions for the three objective functions are indicated with shaded areas.

However, another task is much more challenging, that of finding a unique optimal solution from the Pareto front for model application. This is because the computational complexity is increased by generating multiple solutions, and also because of the difficulty associated with maintaining a desirable distribution of the alternative solutions (Huang et al., 2013). To overcome those difficulties, Bennett et al. (2004) and Huang et al. (2013) turned the multi-objective optimization problem into a single-objective optimization problem by taking the linear weighted sum of the multiple objectives. In this work, the balanced aggregated function method (Madsen, 2000) was used to find a “best” unique solution from the Pareto front, as shown in Fig. 10. The results presented in Fig. 10 emphasize that the solution based on the balanced aggregated function is good for the goodness-of-fit evaluation criteria, as shown in Fig. 11.

Efficiency gains of surrogate-enabled analyses

The most important question in assessing a surrogate-enabled analysis is how efficient or effective it is compared with other efficient alternative tools without surrogate modeling; because the computational efficiency achieved is the main factor motivating the research and application of surrogate modeling. Surrogate-enabled analyses typically sacrifice accuracy for efficiency; as they utilize approximate models to more efficiently achieve the analysis objectives. Broad et al. (2005) pointed out that surrogate models should only be used where time constraints prohibit the possibility of optimizing a problem with a simulation model. To be clear, a benchmark alternative analysis should be available for at least the same number of times of original model simulations as utilized in the surrogate-enabled analysis (Song et al., 2012d; Razavi, 2013).

In an optimization context, the most tangible measure of efficiency gains over a benchmark alternative is computational saving. Generally, the computational saving time can be obtained from the difference value between the computational budget, or time based on an original model, and surrogate modeling with the same quality. Computational budgets may be quantified as the total CPU clock time (e.g., Broad et al., 2005; Behzadian et al., 2009; Kourakos and Mantoglou, 2009), or the number of original function evaluations (Mugunthan and Shoemaker, 2006; Zou et al., 2007, 2009).

In any surrogate-enabled analysis, the available computational budget or time is divided between three main parts: 1) budget or time required to run the original model; 2) budget or time required to develop, run, and update the surrogate model; and 3) budget or time the analyst needs to identify and create an appropriate surrogate-enabled analysis framework. Parts 2 and 3, which are referred to as “surrogate-modeling time” and “analyst time”, respectively, should consume a small portion of the available computational budget, leaving the majority for part 1 (Razavi, 2013). However, analyst time is hardly quantified by total CPU clock time. As a result, in the majority of the water resources related surrogate modeling studies, we recognize that the analyst time based on surrogate-model optimizers is equivalent to the analyst time using an optimizer without surrogate models. Therefore, in our work, part 1 and part 2 can be considered to validate the efficiency of the proposed surrogate-enabled analyses.

First, the computational time required to run the original model for constructing the surrogate model should be considered. In this work, it took approximately 28 hours for the 4,000 runs of the original model. That is, a single evaluation of the original model takes about 25 seconds on average to execute on a 3.2 GHz Intel Core i5-3470 processor with 4 GB of RAM, and running the Windows 7 operating system. Based on this, if the number of runs is 100,000, it may be about 694 hours (approximately equal to 29 days) on the same computer, which might be practically infeasible. In addition, as mentioned above, constructing and validating the surrogate models also incurs computational cost. In our work, the surrogate modeling time including the development time, validation time, and analysis time, was about 8 hours for one objective response (e.g., 20–30 minutes for constructing the model, 420–450 minutes for cross-validation of the surrogate model, and 10–20 minutes for analyzing time). That is to say, for the three objective responses, it takes approximate 24 hours for model optimization by surrogate modeling. The MARS model can be developed based on PSUADE software with the running environment: Centos 5.5, Inter Pentium IV CPU, 1.80 GHz, 256 MB RAM. In summary, the total time of runs to optimize by surrogate modeling is reduced from the original 694 hours to 52 hours. Furthermore, the results confirmed that surrogate modeling is an effective and efficient approach when the original model is relatively complex and the computational cost is very high (e.g., larger than several hours or days). It can be seen that there is a significant advantage to using surrogate models for hydrological process simulation, design, and calibration.

Conclusions

Parameter optimization is a necessary and vital process within the application of conceptual or distributed hydrological models (Song et al., 2012c). However, the practice of model calibration in hydrological models also reveals that single objective functions are often inadequate to properly measure all of the characteristics of the observed values. Consequently, single-objective calibration methods do not usually provide parameter estimates that are considered to be acceptable by practicing hydrologists. As a result, more and more multi-objective optimization methods have been considered to estimate parameters for hydrological models or environmental models (Guo et al., 2013a), such as MOSCEM, NSGA-II, etc. Unfortunately, these methods exhibit some disadvantages, such as being time-consuming, having enormous computation cost, or being relatively difficult to implement in practice.

In this study, the surrogate modeling method based on multiple objective functions was used to calibrate the XAJ model. Surrogate-enabled analyses usually sacrifice accuracy for efficiency, as they utilize approximate models to achieve analysis objectives more efficiently, which is the main factor motivating the research and application of surrogate modeling, especially for complex computer models. Additionally, some criteria cannot be used or recommended as a single-objective function to calibrate model parameters for hydrological models, such as REPF and RERV. This result is understandable, since criteria based on REPF do not meet several key requirements for a discharge simulation, e.g., bias in discharge values (except peak flow).

In summary, the multi-objective calibration of hydrological models is a complex and non-linear optimization problem. The example presented in this paper indicates that the Pareto solutions of multi-objective functions can provide valuable information in model calibration and application, compared with parameter optimization based on single-objective functions. Clearly, it is difficult and precarious to generalize beyond the results presented here; and more research is needed on this aspect, especially for complex hydrological or environmental models with a high-dimension parameter space.

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