Parameter identification and global sensitivity analysis of Xin’anjiang model using meta-modeling approach

Xiao-meng SONG*, Fan-zhe KONG, Che-sheng ZHAN, Ji-wei HAN, Xin-hua ZHANG

1. Hydrology and Water Resources Department, Nanjing Hydraulic Research Institute, Nanjing 210029, P. R. China

2. School of Resource and Earth Science, China University of Mining and Technology, Xuzhou 221116, P. R. China

3. Key Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, P. R. China

4. State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, P. R. China

Parameter identification and global sensitivity analysis of Xin’anjiang model using meta-modeling approach

Xiao-meng SONG*1,2, Fan-zhe KONG2, Che-sheng ZHAN3,4, Ji-wei HAN2, Xin-hua ZHANG4

1. Hydrology and Water Resources Department, Nanjing Hydraulic Research Institute, Nanjing 210029, P. R. China

2. School of Resource and Earth Science, China University of Mining and Technology, Xuzhou 221116, P. R. China

3. Key Laboratory of Water Cycle and Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing 100101, P. R. China

4. State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, P. R. China

Parameter identification, model calibration, and uncertainty quantification are important steps in the model-building process, and are necessary for obtaining credible results and valuable information. Sensitivity analysis of hydrological model is a key step in model uncertainty quantification, which can identify the dominant parameters, reduce the model calibration uncertainty, and enhance the model optimization efficiency. There are, however, some shortcomings in classical approaches, including the long duration of time and high computation cost required to quantitatively assess the sensitivity of a multiple-parameter hydrological model. For this reason, a two-step statistical evaluation framework using global techniques is presented. It is based on (1) a screening method (Morris) for qualitative ranking of parameters, and (2) a variance-based method integrated with a meta-model for quantitative sensitivity analysis, i.e., the Sobol method integrated with the response surface model (RSMSobol). First, the Morris screening method was used to qualitatively identify the parameters’ sensitivity, and then ten parameters were selected to quantify the sensitivity indices. Subsequently, the RSMSobol method was used to quantify the sensitivity, i.e., the first-order and total sensitivity indices based on the response surface model (RSM) were calculated. The RSMSobol method can not only quantify the sensitivity, but also reduce the computational cost, with good accuracy compared to the classical approaches. This approach will be effective and reliable in the global sensitivity analysis of a complex large-scale distributed hydrological model.

Xin’anjiang model; global sensitivity analysis; parameter identification; meta-modeling approach; response surface model

1 Introduction

Computer simulation models (e.g., hydrological models or environmental models)comprise mathematical relations, data, and a calculation core to simulate or explore the behavior of a real-world system (Song et al. 2012c). The development, evaluation, and application of these models involve numerous choices and simplifications (Warmink et al. 2010). They are built in the presence of various types of uncertainties, e.g., parameter input variability, model algorithms or structure, model calibration data, scale, and model boundary conditions (Song et al. 2011b, 2011c). In general, hydrological models based on complex computer codes are highly non-linear, contain thresholds, and often have significant parameter interactions (Tang et al. 2007a). These computer models calculate several output values depending on a large number of input parameters and physical variables. In a broad sense, all sources of uncertainty that can affect the variability of the model output have been referred to as input factors. To provide guidance for a better understanding of this kind of modeling and in order to reduce the response uncertainties most effectively, sensitivity analysis (SA) of the input importance on the response variability can be useful (Marrel et al. 2009). Its role is to determine the strength of the relationship between a given uncertain input factor and the model outputs (Saltelli et al. 2004).

In past studies, SA has been categorized in multiple ways (Frey and Patil 2002), and Song et al. (2012b, 2012d) adopted the way that SA was divided into two broad categories: local SA and global SA. In the case of local SA, the sensitivity of a model output to a given input factor has been traditionally expressed mathematically as the derivative of the model output with respect to the input variation, sometimes normalized either by the central values where the derivative is calculated or by the standard deviations of the input and output values (Haan et al. 1995). Local one-at-a-time (OAT) sensitivity indices are only efficient if all factors in a model produce linear output responses, or if some types of averages can be used over the parameter space. However, in general, the model output responses to changes in the input factors are non-linear. Therefore, an alternative global SA approach, in which the entire parameter space of the model is explored simultaneously for all input factors, is needed. Currently, many global SA techniques have been applied to hydrological models (Kong et al. 2011; Zhan et al. 2013), involving qualitative global SA, such as the multiple regression model, Latin Hypercube one-at-a-time (LH-OAT) method (van Griensven et al. 2006), and Morris one-at-a-time method (Campolongo et al. 2007); and quantitative global SA, e.g., the variance-based Sobol method (Tang et al. 2007a, 2007b), Fourier amplitude sensitivity test (FAST) method (Xu and Gertner 2011), and extended FAST method (Xu and Gertner 2007; Ren et al. 2010). The advantage of the global approach over the local OAT method is that it provides the ranking of parameter importance and provides information not only about the direct (first-order) effect of the individual factors on the output, but also about their interaction (higher-order) effects (Mu?oz-Carpena et al. 2007).

However, these classical approaches directly estimate the parameter variances characterizing the sensitivity indices; they are conceived as black box methods and do not try to use information present in the samples. Though quantitatively dependable and robust intackling the specified SA settings, they have a noteworthy computational cost, requiring thousands upon thousands of model evaluations, which render impractical their applications to expensive computational models (Makler-Pick et al. 2011; Kong et al. 2011; Song et al. 2012b, 2012d; Zhan et al. 2013). To overcome the problem of long calculation time in sensitivity analysis, approaches based on nonparametric estimation tools have been proposed by Doksum and Samarov (1995). These nonparametric methods allow us to significantly reduce the number of function evaluations needed to accurately estimate sensitivity indices. Another solution that we want to focus on in this paper is to replace the complex computer code by a mathematical approximation, called a response surface model or a meta-model. The response surface model consists in constructing a function from a few experiments that simulate the behavior of the real phenomenon in the domain of influencing parameters. These methods have been generalized to develop surrogates for costly computer codes (Kleijnen and Sargent 2000), such as polynomial regression, state-dependent parameter modeling (Ratto et al. 2007), and some learning statistical models (Sathyanarayanamurthy and Chinnam 2009). However, these approaches have not yet been widely applied to complex hydrological models, or there have been few studies about it.

The objective of this paper is to demonstrate the potential of the meta-modeling approach, briefly presented in Section 2, for global sensitivity analysis of a hydrological model. Also, the Morris screening method and variance-based method are described and applied to this work. A case study of the Xin’anjiang model in the Yanduhe catchment (the upper tributary of the Yangtze River), with available data, model parameters, and evaluated criterions, is introduced in Section 3. The results and discussion of a sensitivity analysis are provided in Section 4. Subsequently, Section 5 presents the conclusions of our study.

2 Materials and methods

2.1 Morris screening method

Morris (1991) proposed an effective sensitivity screening measure to identify the important factors in models with many factors. The method is based on computing, for each input, a number of incremental ratios, called elementary effects, which are then averaged to assess the overall importance of a given input factor. Elementary effects are calculated by varying one parameter at a time across a discrete number of levels (p) in the space of input factors y (Zhan et al. 2013). The elementary effect is calculated from

The number of elementary effects (R) is obtained for each input factor. Based on this number of elementary effects calculated for each input factor, two sensitivity measures have been proposed by Morris (1991): (1) the mean of the elementary effects,μ, which estimates the overall effect of the parameter on a given output; and (2) the standard deviation of the effects,σ, which estimates the higher-order characteristics of the parameter (such as curvatures and interactions):

Campolongo et al. (2007) noticed the weakness of the original measureμin the method of Morris (1991) and proposed a modification of the original method in terms of the definition of this measure. They suggested considering the mean of distribution of the absolute values of the elementary effects,μ*, for the evaluation of a parameter’s importance in order to avoid the cancellation of the effects of opposing signs. The measureμi*, which is shown in Eq. (4), is a proxy of the variance-based total index, is acceptable and convenient (Campolongo et al. 2007), and can be used to rank the parameters according to their overall effects on model outputs.

2.2 Meta-model

The response surface model (RSM), which is also known as the meta-model or surrogate model, is a collection of statistical and mathematical techniques useful for developing, improving, and optimizing processes (Meyers and Montgomery 2002; Song et al. 2012c). The Problem Solving environment for Uncertainty Analysis and Design Exploration (PSUADE) software provides a number of response surface methods from parametric regression methods to nonparametric methods (e.g., the multivariate adaptive regression splines (MARS) method, support vector machine (SVM), and Gaussian process model) (Song et al. 2011b, 2011c). There are two steps to create a response surface. The first step is to construct a set of space-filling sample points to capture the dominant behaviors in the parameter space. These sample points and their corresponding outputs are then used to train a function approximator. The choice of response surface methods for a given simulation model depends on the knowledge about the simulation model itself. If no such knowledge is available about the mapping, the nonparametric models may be appropriate. We therefore selected the MARS method to create the meta-model of the simulation model.

The MARS method is essentially a combination of spline regression, stepwise model fitting, and recursive partitioning (Storlie et al. 2009). It uses a nonparametric modelingapproach that requires mild assumptions about the form of the relationship between the predictor and dependent variables (Quirós et al. 2009).

The general model equation representing the relationship between the predictor variable and the target variable is given as

where the summation is overMnonconstant terms in the model, andf(x) is predicted as a function of the predictor variablex={x1,x2,…,xk} and their interactions: the function consists of an intercept parameter (a0) and the weighted (byam) sum of one or more basis functionsBm(x), which is described as follows:

Kmis the number of truncated linear functions multiplied in themth basis function,xv(k,m)is the input variable corresponding to thekth truncated linear function in themth basis function,tkmis the knot value corresponding to the variablexv(k,m), andSkmis the selected sign: +1 or ?1.

The MARS algorithm for estimating the model functionf(x) consists of two algorithms: the forward stepwise algorithm and the backward stepwise algorithm (see Friedman (1991) for a detailed description).

2.3 Variance-based method

The RSMSobol method based on a response surface model (taking MARS as an example in this study) and the Sobol method was proposed to calculate the first-order, second-order, and total sensitivity indices for a complex hydrological model (Song et al. 2012d; Zhan et al. 2013).

The analysis method of the main effect (i.e., the first-order sensitivity index) proposed by McKay (1993) is a variance-based method. The variance decomposition based on theith conditional input is

whereV(Y) andE(Y) are the mean and variance of an outputY, respectively, andXiis theith input. The first term on the right-hand side is the variance of the conditional expectation ofY, conditioned onXi. It is also denoted asVCE(Xi), and it measures the variability in the conditional expected value ofYas the inputXitakes on different values. The second term is an error or a residual term, which represents the variability inYnot accounted for by the input subsetXi. The correlation ratioη2can be defined as

The total sensitivity indexSTifor theith input is defined as

whereMis the number of the input factors,Vijis the variance for inputsiandjtogether, and so on.STican be estimated by

where the subscript ～imeans all the inputs except inputi.

3 Case study

3.1 Xin’anjiang model

The Xin’anjiang hydrological model is a conceptual watershed model developed in the 1980s (Zhao and Wang 1988; Zhao 1992; Zhao and Liu 1995). Its main feature is the concept of runoff formation on the repletion of storage, which means that runoff is not produced until the soil moisture content of the aeration zone reaches the field capacity, and thereafter runoff equals the rainfall excess without further loss. This hypothesis was first proposed in China in the 1960s for daily rainfall-runoff and rainstorm flood forecasting, and much subsequent experience supports its validity for humid and semi-humid regions (Mohammed et al. 2010). In this study, we divided a catchment into a set of sub-catchments to capture the spatial variations of precipitation and the underlying surface (Shi et al. 2011).

3.2 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. There are five rain stations, Banqiao (at 110.15°E and 31.40°N), Xiagu (at 110.23°E and 31.37°N), Duizi (at 110.27°E and 31.30°N), Songziyuan (at 110.40°E and 31.33°N), and Yanduhe (at 110.30°E and 31.20°N)), and one hydrological station, Yanduhe (at 110.30°E and 31.20°N)) in the catchment, as shown in Fig. 1. The mean annual temperature of this catchment is 11℃ to 12℃, the mean annual precipitation is almost 1 650 mm, and the mean annual runoff is 1 240 mm. The catchment can be divided into five sub-catchments based on the rain stations and natural channel drainage network, using Arcview GIS 3.2 with the HEC-GeoHMS module (Song et al. 2012a).

3.3 Model performance and objective function criteria

In this study, the Xin’anjiang model was selected to validate this approach, and the parameters with their ranges are listed in Table 1 (Zhao and Wang 1988; Zhao 1992; Chenget al. 2006; Li et al. 2009; Song et al. 2012a). The time series of 30 flood events (Fig. 2) were used to analyze the sensitivity of model parameters. In general terms, the objective of model calibration can be stated as follows: selection of model parameters so that the model simulates the hydrological behavior of the catchment as closely as possible (Madsen 2000). In comparing the model simulation results with the observed data, criteria must first be identified, and then some statistical goodness-of-fit approaches must be employed to evaluate the model (Song et al. 2011a). Therefore, in order to make the results of parameter sensitivity analysis more persuasive, in this study, four objective functions, the Nash-Sutcliffe efficiency (NS), total water balance error coefficient (TE), water balance error coefficient for low flow events (DE) , and water balance error coefficient for peak flow events (GE), were used to evaluate the model performance. Their formulas are described as follows:

Table 1Parameters and their ranges in Xin’anjiang model

Fig. 2Discharge hydrograph for 30 flood events in Yanduhe catchment

4 Results and discussion

4.1 Qualitative screening

The Morris screening method was used to identify qualitatively important parameters for the Xin’anjiang model in this study. According to the Morris method, the required number of simulations (N) to perform the analysis isN=R(k+1). Previous studies have demonstrated that usingp= 4 andR= 10 produces satisfactory results (Campolongo et al. 1999; Saltelli et al. 2000). For example, in a case ofk= 15, only 160 model simulations are required for the Morris method, while variance-based methods would require approximately 10 000 or more simulations. In this work, we have the knowledge that the output varies more quickly with aparticular input, so it is more sensible to use a largerpfor this input; that is, the number of levelspis 10 and the number of simulationsNis 640.

Fig. 3Morris parameter screening for different objective functions

Fig. 4Scatter plots of modified mean value and standard deviation for different responses

The Morris method is qualitative in nature, and its sensitivity measures should not be used to quantify input factors’ effects on uncertainties of model outputs and to distinguish the nonlinearity of a factor from the interaction with other factors (Yang 2011). Rather, they provide qualitative assessment of parameters’ importance in the form of parameter ranking. Furthermore, this method cannot account for the spatial uncertainty of model inputs because it requires that all input factors are scalar values, and uses an analytical relationship between the model input and output for calculating sensitivity measures. Therefore, the results of this study indicate which factors are of potential importance. A subset of six to eight factors could be used for the more accurate and quantitative SA analysis (as in Mu?oz-Carpena et al. (2007)). Ten parameters (K,WM,SM,KI,KG,CS,CI,CG,KE, andXE) were selected to analyze their sensitivity indices. The reduction of the parameter input set from 15 original parameters to 10 identified as important by the screening method may result in the reduction of the number of simulations from approximately 15 000 to 10 000.

4.2 Response surface models

In this study, the response surface model was constructed and integrated with the variance-based method for subsequent sensitivity analyses using the PSUADE software. To construct a reliable and accurate response surface model, we need a suitable sampling design method and interpolation function. The quasi-random sequences (also called LP-τ sequences) (Sobol’ 1976) were utilized, as they produce samples with better space-filling properties and provide the best convergence properties (Elsawwaf et al. 2010). According to Sobol’ (1998),the sequences can be computed in a fast way using just one sub-route. In addition, as stated above, the MARS model was used as a meta-model to construct the response surface model.

Moreover, an efficient validation method was selected to check the response model. Thek-fold cross-validation method and L1-error approach were used. In thek-fold cross-validation, the original sample was randomly partitioned intoksubsamples. Of theksubsamples, a single subsample was retained as the validation data for testing the model, and the remainingk?1 subsamples were used as training data. Once the samples and response surface model were deemed satisfactory, subsequent analysis can rely on this response surface model. In this study, we divided the sample into 100 groups, held out one group at a time, and computed the prediction error statistics (Table 2). From Table 2, we can see that the maximum relative errors are 0.012 9 forNSresponse, 0.048 3 forTE, 0.028 3 forGE, and 0.020 6 forDE. The relative root mean square (RMS) error values are 0.003 81, 0.009 97, 0.004 99, and 0.005 16 for the four responses, respectively. The L1-norm (L1n) relative errors are 0.002 99, 0.007 37, 0.003 76, and 0.003 97, respectively. The relative error histograms and probability density functions for the four responses are shown in Fig. 5 and Fig. 6, respectively.

Table 2Characteristics of interpolation error for different response surface models

Fig. 5Histograms of prediction errors for responses

Fig. 6Probability density functions for different responses

The results show that all the responses give acceptable interpolation errors. We then selectedSMandWMas examples to construct the response surface with respect to the output objective functions. The response surface plots for the four responses are shown in Fig. 7.

Fig. 7Response surface plots for different functions

4.3 Quantitative sensitivity analysis

Using the meta-models, which are inexpensive and tractable, the first-order sensitivity indices for ten parameters of the Xin’anjiang model were computed by sampling 100 000 times from the response surface model. The running time, due to the response surface model, was 20 minutes for this process, while it would have been 12 days if we had used the classical method with 100 000 runs for sensitivity analysis. In this work, the first-order and total sensitivity indices were used to analyze the parameters’ sensitivity, as shown in Table 3 and Table 4. The results from Table 3 indicate that the parametersSMandWMare the most sensitive, with slight differences for the four responses. Apart from that, the Muskingum routing model parametersKEandXE, and the parameterCSare relatively less important for theNSresponse, while the impacts of the parameterCGon theTE,GE, andDEresponses are significant. Also, for the objective functionDE, the parameterCSshould not be neglected in the model simulation. We can see that the sum of the first-order sensitivity indices for all the responses is not equal to 1, i.e., there are some interactions in the parameters. Therefore, the total sensitivity index was used to evaluate the model parameter sensitivity based on different response surface models. The results of total sensitivity analysis from Table 4 correspond with those of the first-order sensitivity indices. The results are identical with the real physical processes. As mentioned above, the model combines the runoff generation process and flow routing process through the soil moisture content (WM). In addition, the FAST method has also been used to compare the results from the proposed method, as shown in Table 3. The FAST method was calculated with 3 281 runs of the Xin’anjiang model, and the samples were also generated using the PSUADE software. The results of the FAST method are consistent with those of the RSMSobol method. Therefore, the RSMSobol method is an effective tool for quantitative sensitivity analysis of complex models.

Table 3First-order sensitivity indices for different responses

Table 4Total sensitivity indices for different responses of interest

We can see that the first-order sensitivity index ofWMis 0.825 for the objective functionTE; that is,WMis the most important factor for the total water balance with the initial soil moisture. It is the sum ofWUMin the upper layer,WLMin the lower layer, andWDMin the deeper layer.WDMis therefore completely dependent on the other three and need not be considered for sensitivity analysis and optimization. Usually,WMis a measure of aridity, which varies from 80 mm in South China to 170 mm or more in North China. The flood event model operation is generally sensitive toWM, provided its value is large enough to ensure that the computed areal mean soil moisture contentWdoes not become negative. The areal mean free water storage capacity of the surface layerSMplays an important role in the distribution of runoff into the interflow and groundwater flow and represents the maximum possible deficit of free water storage. It controls the shape of the discharge hydrograph and runoff division into interflow and baseflow, i.e., it affects the model simulation performance for the objective functionsNS,GE, andDE. Usually, it is approximately 10 mm or less for thin soils, increasing to 50 mm or more for thick and porous surface soils. In addition, the ratio of potential evapotranspiration to pan evaporationKis also sensitive to the total water balance errorTE. Obviously, it controls the water balance and has a direct effect on the production of surface runoff. Increasing the value ofKresulted in higher evapotranspiration and this diminished the tension water storage and hence produced less surface runoff.

5 Conclusions

In this study, we proposed a new approach to conduct a global sensitivity analysis for distributed hydrological models using a statistical emulator (response surface model), which was used to analyze the sensitivity of Xin’anjiang model parameters. The following conclusions can be obtained:

(1) A two-step statistical evaluation framework using global techniques is presented based on (a) the Morris screening method for qualitative ranking of parameters, and (b) a variance-based method integrated with a meta-model for quantitative sensitivity analysis, i.e., the RSMSobol method. The computational cost of this method is largely reduced since the analysis employs a screening stage using a relatively fast method (the Morris method) to identify a subset of sensitive parameters that is subsequently used as input to the more intricate and computationally intensive quantitative sensitivity analysis method (the RSMSobol method).

(2) The Morris screening results indicated that only a small fraction of the model input parameters have an appreciable influence on the model outputs, and we qualitatively sifted ten relatively important parameters for quantitative global sensitivity analysis (GSA). The outcomes of the RSMSobol method provide a quantitative measure of the sensitivity of the output variables to the different parameters.

(3) The proposed efficient method can achieve the sensitivity assessment of a complex modeling system, improve the model efficiency, alleviate the high computational cost of uncertainty quantification for the complex modeling system, and lay a solid foundation for subsequent model calibration and parameter optimization.

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(Edited by Yun-li YU)

This work was supported by the National Natural Science Foundation of China (Grant No. 41271003) and the National Basic Research Program of China (Grants No. 2010CB428403 and 2010CB951103).

*Corresponding author (e-mail: wenqingsxm@126.com; xmsong@nhri.cn)

Received Oct. 11, 2011; accepted Jan. 17, 2012