Analysis of parameter uncertainty in semi-distributed hydrological models using bootstrap method: A case study of SWAT model applied to Yingluoxia watershed in northwest China
Introduction
A typical hydrological model generally consists of a large number of equations describing the hydrological processes and a small number of parameters describing the watershed properties. In ideal cases, where intensive data collection is possible, model parameters are measured or estimated from watershed characteristics (e.g. Vertessy et al., 1993, Nandakumar and Mein, 1997). More commonly, however, the parameters are determined by calibrating the model against the observations due to the unknown spatial heterogeneity of parameter values and the high cost involved in those experiments or measurements (e.g. Abbott et al., 1986, Refsgaard et al., 1992, Nandakumar and Mein, 1997). Parameters obtained from calibration are affected by several factors such as correlations amongst parameters, sensitivity or insensitivity in parameters, spatial and temporal scales and statistical features of model residuals, and may lead to the so-called equifinality (non-unique parameter estimation and hence uncertainty in the estimated parameters) (Gupta and Sorooshian, 1983, Kuczera, 1988, Duan et al., 1992, Beven and Binley, 1992). Without a realistic assessment of parameter uncertainty, it is difficult to have confidence in tasks, such as evaluating prediction limits on future hydrological responses and assessing the value of regional relationships between model parameters and watershed characteristics (Kuczera and Parent, 1998, Vrugt and Bouten, 2002).
Much attention has been focused on the parameter uncertainty issues in hydrological modelling and their effects on model performance (e.g. Gupta and Sorooshian, 1983, Kuczera, 1983, Kuczera, 1988, Beven and Binley, 1992, Duan et al., 1992, Abbaspour et al., 1997, Vrugt and Bouten, 2002, Muleta and Nicklow, 2005, Schuol and Abbaspour, 2006, Yang et al., 2007a and Yang et al., 2007b). Several approaches for addressing such uncertainty have been proposed over the past several decades. The most traditional statistical method is first-order approximation providing approximate confidence intervals of parameter uncertainty with few computing burdens (Kuczera, 1988) and yielding reasonable results under the condition that the linearization of objective function near its minimum is valid in the parameter domain. One of the weaknesses of this classical method is that it does not account for correlations between the parameter estimates (Kuczera, 1988, Vrugt and Bouten, 2002). Another method for the calculation of parameter confidence intervals is the contour plots, including the uniform grid sampling, uniform random sampling and Sequential Uncertainty Fitting algorithm (SUFI) (Abbaspour et al., 1999, Abbaspour et al., 1997, Uhlenbrook et al., 1999). Such methods are robust but require massive computing resources for high dimensional parameter space, additionally, if the initial sampling of the parameter space is not dense enough, such sampling scheme probably cannot ensure a sufficient precision of the statistics inferred from the retained solutions (Bates and Campbell, 2001, Vrugt and Bouten, 2002). With the advances in computing technology, Monte Carlo-based methods are much more popular than traditional methods, with strengths in dealing with the nonlinearity and interdependency of parameters in complex hydrological models. Two Monte Carlo-based approaches are commonly seen in recent literatures: importance sampling and Markov Chain Monte Carlo (MCMC) simulation (e.g. Beven and Binley, 1992, Smith and Roberts, 1993, Kuczera and Parent, 1998, Bates and Campbell, 2001, Gallagher and Doherty, 2007). Importance sampling is a technique for randomly sampling from a probability distribution and was implemented in Generalised Likelihood Uncertainty Estimation (GLUE) by Beven and Binley (1992). The efficiency of this algorithm depends strongly on the choice of the importance distribution. If one or more importance weights dominate, the algorithm can produce unreliable inferences (Gelman et al., 1995). MCMC is one of the most important numerical technique for creating a sample from the posterior distribution, which has been widely used in hydrological modelling to quantify parameter uncertainties (e.g. Kuczera and Parent, 1998, Campbell et al., 1999, Makowski et al., 2002, Vrugt et al., 2003). Its underlying rationale is to set up a Markov chain to simulate the true posterior distribution by generating samples from a random walk. An obvious advantage of this method is that it does not require linearity assumptions in model or even that model outputs do not need to be differentiable with respect to parameter values (Gallagher and Doherty, 2007). Because of its robust performance, MCMC is often used to assess parameter uncertainties in combination with GLUE or Bayesian inference by estimating a probability density for model parameters conditioned on observations. Blasone et al. (2008) recently found that using a MCMC sampling scheme coupled with GLUE significantly improves the efficiency and effectiveness of the methodology of GLUE. Another widely used method for parameter estimation and uncertainty analysis is the Bayesian method which provide more information than single-point estimates (Bates and Campbell, 2001, Engeland et al., 2005, Gallagher and Doherty, 2007, Yang et al., 2007a). The posterior distribution can be obtained by applying Bayes’ theorem based on a prior distribution and observed data (Gelman et al., 1995). The Monte Carlo-based approaches are often used to generate a large-enough sample from the posterior distribution so that desired features of the posterior distribution may be summarized.
The motivation of this study comes from a desire to obtain the parameter distributions and the associated statistical features for hydrological models without strong programming effort and heavy computational burden. The bootstrap method, a simple nonparametric technique (Efron, 1979) is proposed in this paper as it is simple to describe and easy to implement. It involves little or no further programming. Many hydrological models, especially semi-distributed and distributed hydrological models, are quite complicated and require substantial programming effort before implementation. The bootstrap can be directly used via existing computer program and therefore can easily be used by anyone with little programming training. Its use in hydrological models is increasing (e.g., Tasker, 1987, Zucchini and Adamson, 1989, Abrahart, 2001, Srinivas and Srinivasan, 2005), with many applications in the estimation of parameter uncertainties for simple time series and its variation. For example, Cover and Unny (1986) used the bootstrap to estimate parameter uncertainty in autoregressive moving average (ARMA) models of streamflow; Tasker and Dunne (1997) used the bootstrap position analysis to assess the effects of parameter uncertainty on the periodic autoregressive moving average (PARMA) model. However, to the best of our knowledge, this method has not been used in parameter uncertainty analysis for semi- or highly-distributed hydrological models. In this study, the bootstrap method is used to quantify parameter uncertainties in a semi-distributed hydrological model, Soil and Water Assessment tool (SWAT) (Arnold et al., 1998), and then to evaluate the effects of parameter uncertainty on model simulation results. As a comparison, the widely used Bayesian method is also conducted to assess the parameter uncertainties. Not only the uncertainty ranges, but also the effects of parameter uncertainties on simulation results as well as the computational efficiency of both methods are investigated and compared. The case study is based on the Yingluoxia watershed located in the upper reaches of Heihe River basin, the second largest inland basin in northwest China.
The paper is organised as follows. An explanation of bootstrap method is given in the next section, followed by the description of study area and data, and then the selection of parameters for model calibration and validation. The results and discussion, including parameter estimation and uncertainty analysis, model simulation results and the effects of parameter uncertainty on model simulation uncertainties, are presented before the major conclusions.
Section snippets
Bootstrap method
The bootstrap method, first introduced and named by Efron (1979), is a resampling technique for estimating the properties, such as the variance, of an estimator or statistic. The idea behind bootstrap is that the sample values are the best guide to the underlying true distribution even when the information about the true distribution is lacking. The advantage of this method over analytical approximation in the classic method is its relative simplicity in implementation. It has been widely used
Study area description
Yingluoxia watershed, with an area of 10,009 km2, lies in the upper reaches of the Heihe River basin, a typical inland river basin covering approximately 130,000 km2 in the middle of the Hexi Corridor of Gansu Province in northwest China (with longitude between 97°37′–102°06′E and latitude between 37°44′–42°40′N) (Qi and Luo, 2006). The elevation in the watershed ranges from 1674 m above sea level (ASL) at the lower point to about 5120 m ASL at its headwaters. A half of the watershed is located
Model parameter selection
Based on analysis of parameter sensitivity using LH-OAT method (Latin Hypercube-one factor At a Time; Van Griensven et al., 2006) and taking previous studies into account (Wang et al., 2003, Huang and Zhang, 2004), nine key parameters are identified for model calibration and validation. Of those nine parameters (see Table 2), three are related to snow melting (TIMP, SMTMP, SMFMX), three to runoff and base flow (CN2, ALPHA_BF, SURLAG), two to soil types (SOL_AWC, SOL_K) and one to
Parameter uncertainty analysis
Fig. 3 shows the marginal frequency distributions of nine aggregate parameters from bootstrap method. The degree of uncertainty in each calibrated parameter is usually expressed by confidence interval. The 95% confidence interval (CI) is derived by ordering the samples and then identifying the 2.5% and 97.5% thresholds values. Table 3 shows the 95% CI for each parameter along with the results of one-sample Kolmogorov–Smirnov nomality test.
It is clearly seen from Fig. 3 and Table 3 that not all
Conclusions
Bootstrap method, as another candidate for uncertainty analysis, was employed in this study to assess the parameter uncertainty in SWAT model applied to Yingluoxia watershed. The parameter distributions were obtained and then the associated uncertainty ranges were estimated according to the related confidence intervals. Furthermore, contributions of parameter uncertainty on simulation uncertainty were also investigated. As a result, six of nine aggregate parameters were not normally
Acknowledgements
The first author thanks the financial support by the China Scholarship Council, as well as the support provided by CSIRO during her study in Australia. Thanks are also given to “Environmental and Ecological Science Data Center for West China (http://westdc.westgis.ac.cn/), National Science Foundation of China and Digital River Basin (http://heihe.westgis.ac.cn/)” for providing all the dataset. The authors also thank three anonymous referees for their comments that have led to improved quality
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