Evaluation of Uncertainties in Input Data and Parameters of a Hydrological Model Using a Bayesian Framework: A Case Study of a Snowmelt–Precipitation-Driven Watershed

J. L. Zhang Sino-Canada Resources and Environmental Research Academy, North China Electric Power University, Beijing, China

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Y. P. Li Environmental Systems Engineering Program, Faculty of Engineering and Applied Science, University of Regina, Regina, Saskatchewan, Canada, and School of Environment, Beijing Normal University, Beijing, China

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G. H. Huang Environmental Systems Engineering Program, Faculty of Engineering and Applied Science, University of Regina, Regina, Saskatchewan, Canada, and School of Environment, Beijing Normal University, Beijing, China

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C. X. Wang Sino-Canada Resources and Environmental Research Academy, North China Electric Power University, Beijing, China

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G. H. Cheng Institute for Energy, Environment and Sustainable Communities, University of Regina, Regina, Saskatchewan, Canada

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Abstract

In this study, a Bayesian framework is proposed for investigating uncertainties in input data (i.e., temperature and precipitation) and parameters in a distributed hydrological model as well as their effects on the runoff response in the Kaidu watershed (a snowmelt–precipitation-driven watershed). In the Bayesian framework, the Soil and Water Assessment Tool (SWAT) is used for providing the basic hydrologic protocols. The Delayed Rejection Adaptive Metropolis (DRAM) algorithm is employed for the inference of uncertainties in input data and model parameters with global and local adaptive strategies. The advanced Bayesian framework can help facilitate the exploration of variation of model parameters due to input data errors, as well as propagation from uncertainties in data and parameters to model outputs in both snow-melting and nonmelting periods. A series of calibration cases corresponding to data errors under different periods are examined. Results show that 1) input data errors can affect the distributions of model parameters as well as parameters’ correlation, implying that data errors could influence the related hydrologic processes as well as their relations; 2) considering input data errors could improve the hydrologic simulation ability for peak streamflows; 3) considering errors of temperature and precipitation data as well as uncertainties of model parameters can provide the best modeling simulation performance in the snow-melting period; and 4) accounting for uncertainties in precipitation data and model parameters can provide the best modeling performance during the nonmelting period. The findings will help enhance hydrological model’s capability for simulating/predicting water resources during different seasons for snowmelt–precipitation-driven watersheds.

Corresponding author address: Y. P. Li, School of Environment, Beijing Normal University, Beijing 100875, China. E-mail: yongping.li@iseis.org

Abstract

In this study, a Bayesian framework is proposed for investigating uncertainties in input data (i.e., temperature and precipitation) and parameters in a distributed hydrological model as well as their effects on the runoff response in the Kaidu watershed (a snowmelt–precipitation-driven watershed). In the Bayesian framework, the Soil and Water Assessment Tool (SWAT) is used for providing the basic hydrologic protocols. The Delayed Rejection Adaptive Metropolis (DRAM) algorithm is employed for the inference of uncertainties in input data and model parameters with global and local adaptive strategies. The advanced Bayesian framework can help facilitate the exploration of variation of model parameters due to input data errors, as well as propagation from uncertainties in data and parameters to model outputs in both snow-melting and nonmelting periods. A series of calibration cases corresponding to data errors under different periods are examined. Results show that 1) input data errors can affect the distributions of model parameters as well as parameters’ correlation, implying that data errors could influence the related hydrologic processes as well as their relations; 2) considering input data errors could improve the hydrologic simulation ability for peak streamflows; 3) considering errors of temperature and precipitation data as well as uncertainties of model parameters can provide the best modeling simulation performance in the snow-melting period; and 4) accounting for uncertainties in precipitation data and model parameters can provide the best modeling performance during the nonmelting period. The findings will help enhance hydrological model’s capability for simulating/predicting water resources during different seasons for snowmelt–precipitation-driven watersheds.

Corresponding author address: Y. P. Li, School of Environment, Beijing Normal University, Beijing 100875, China. E-mail: yongping.li@iseis.org

1. Introduction

Physical and conceptual hydrological models aggregate the process-oriented, physically based, and spatially distributed surface and/or subsurface properties into much simpler homogeneous storages. The estimation of model parameters involves significant uncertainties derived from spatiotemporal heterogeneity. Specifically, some parameters are affected by meteorological conditions that vary temporally and characteristics of underlying surface that change spatially. These parameters cannot be determined through direct observation in the field but can be estimated by calibration against the input–output records of the watershed response, which inevitably contains errors (Vrugt et al. 2008; Joseph and Guillaume 2013). Bayesian analysis techniques have been widely employed to incorporate prior information to produce a posterior distribution on which statistical inferences about the model parameters are based (Robertson and Wang 2012; Panday et al. 2014; Li et al. 2015; Martin-Fernandez et al. 2015). Among them, Markov chain Monte Carlo (MCMC) methods overcome the limitations due to computational aspects through drawing samples of parameter values from a Markov chain (Steinschneider et al. 2012; Chandra et al. 2015; Zhang et al. 2016).

The above sources limit themselves to a comparison of observed and predicted streamflows, as if uncertainty in the input–output representation of hydrological model is attributable primarily to parameter uncertainties. Because of 1) inadequacies in the representation of a set of point-scale gauges for the entire areal meteorological field and 2) both systematic and random measurement errors of gauges themselves, errors in input data can be significant in hydrologic sciences (McMillan et al. 2011; Sun and Bertrand-Krajewski 2013). Streamflow data also contain significant errors because of discharge gauging errors, extrapolation of rating curves, unsteady flow conditions, flow regime hysteresis, and temporal changes in the channel properties (Renard et al. 2011). Model structural uncertainty mainly derives from the simplifications of reality in models’ principles (Kavetski et al. 2002). Addressing these uncertainties through model parameter adjustment may generate a satisfactory match in calibration period; however, misleading relations between parameters and watershed properties would be obtained for future responses. In the quest for the systematic framework of treating input data errors, much progress has been made in simultaneous estimation of data error and parameter uncertainty (Huard and Mailhot 2008; Renard et al. 2011), stochastic analysis methods for description of rainfall (Clark and Slater 2006; AghaKouchak et al. 2010), comparison between different rainfall product resolutions (Yilmaz et al. 2005; Gourley et al. 2011), global sensitivity analysis for analyzing forcing’s effect (Raleigh et al. 2015), and Bayesian inference framework (Kavetski et al. 2006a,b; Kuczera et al. 2010). For example, Guan et al. (2005) introduced a geostatistical model to map mountain precipitation considering precipitation spatial covariance and orographic and atmospheric effects. Ajami et al. (2007) proposed an input data error model for treating rainfall error in the form of a Gaussian multiplier; these multipliers were estimated along with model parameters using an MCMC scheme.

Most efforts made in analyzing input data errors have been limited to evaluating the effects of precipitation errors. In fact, changes in surface temperature have important consequences for the hydrological cycle, particularly in cold and arid watersheds with temperature-driven physics (Barnett et al. 2005; Nourani et al. 2015). Specifically, temperature changes translate into increasing variability in the magnitude, timing, and duration of the spring thaw and autumn freezing (Clark et al. 2011). Hydrologic research efforts have been conducted in investigating temperature and precipitation uncertain variations (Tao et al. 2011; Fu et al. 2013) and their influences on watershed hydrology (Chen and Chen 2014; Mankin and Diffenbaugh 2015). In fact, parameters that describe physical hydrological schemes can vary with various meteorological conditions (i.e., temperature and precipitation in different seasons have varied influences on runoff; Elsner et al. 2014; Raleigh et al. 2015). An analysis of the effect of input data errors on parameter uncertainty as well as their propagation to hydrological processes in different seasons is desired.

Therefore, this study aims to propose a Bayesian framework for investigating uncertainties in input data (i.e., temperature and precipitation) and parameters in a distributed hydrological model as well as their effects on the runoff response in the Kaidu watershed with snowmelt–precipitation-driven physics. The MCMC method is employed for investigating the inference of uncertainties in input data and model parameters. The Bayesian framework will be used for facilitating the exploration of 1) variations of model parameters due to input data errors and 2) propagation from uncertainties in data and parameters to model outputs. The findings will help enhance a hydrological model’s capability for simulating/predicting water resources for snowmelt–precipitation-driven watersheds.

2. The study system

a. Study area

The Kaidu watershed is located in the hinterland of the Xinjiang Uygur Autonomous Region (China) at the central southern slope of the Tian Shan (Fig. 1). It is enclosed within latitudes 42°43′–43°21′N and longitudes 82°58′–86°05′E, with a catchment area of 18 827 km2 (Fu et al. 2013; Zhang et al. 2014). The river flows through the Yulduz (Youerdusi) basin and the Yanqi basin into Lake Bosten (Bositeng) with a length of 610 km (Wang et al. 2015). As a typical arid region in China, the Kaidu watershed is characterized by low rainfall, low temperature, and high evaporation. The watershed has an average annual precipitation of less than 500 mm. The average annual temperature is −4.3°C, and the extreme minimum temperature is −48.1°C. The headwater region has a long snowfall period from November to the following March, with annual snow cover reaching 139.3 days and an annual snow depth of 12 cm (Chen and Chen 2014). The streamflow of this watershed is mainly contributed by snowmelt runoff in ablation period and rainfall runoff during summer, leading to multiple peak flows annually. Temperature and precipitation are considered the most important meteorological factors that drive hydrological mechanism in the Kaidu watershed (Li et al. 2008; Ma et al. 2013).

Fig. 1.
Fig. 1.

Topographic and land-use characteristics of the Kaidu watershed with the gauge stations.

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

b. Geographic and hydroclimatic data

The main hydrological characteristics of the watershed have been acquired from vector format data of land cover and soil properties as well as gridded maps of elevation at 30 m × 30 m cell size. The digital elevation model has been resampled to 100 m resolution, from which the input topography map, river network, and river basin boundaries are obtained. A monitoring program is set up to better understand the spatiotemporal variability of snowmelt and meteorology across the watershed. Meteorological information is available at six locations along the drainage network, including two official meteorological stations and four extra meteorological stations. In addition, two snow stations are also established across the watershed to collect the snow-melting information. The daily runoff data for the Kaidu River (from 1957 to 2001) are collected from Dashankou hydrometric station, which is the last station before the Kaidu River reaches the plain oases from the mountainous area. Table 1 presents the detailed information of the gauge stations. The established snow stations are configured with SHM30 (Jenoptik Instruments) with a sensor for snow-depth measurement. The meteorological stations are configured with wireless Vantage Pro2 (Davis Instruments). The monitoring data (e.g., temperature, precipitation, wind, dewpoint, and heat index) are sampled and recorded every 30 min. Such frequency of monitoring can ensure sufficient storage space for the data and electric power for the equipment.

Table 1.

Description of the meteorological, snow, and hydrometric stations.

Table 1.

3. Methodology

a. Hydrological model

The hydrological processes from precipitation and snowpack to runoff can be simulated using the Soil and Water Assessment Tool (SWAT). The foundation behind the hydrologic simulation is soil water balance, in which the model tracks precipitation, simulated soil water content, surface runoff, evapotranspiration, percolation, and return flow on a daily basis (Ghobadi et al. 2015). Generally, the simulation of the hydrologic cycle is based on the following water balance equation (Neitsch et al. 2009):
e1
where SWt is final soil water content (mm H2O), SW0 is initial soil water content (mm H2O), t is time (days), Rday is the amount of precipitation on day i (mm H2O), Ea is the amount of evapotranspiration on day i (mm H2O), wseep is the amount of water entering the vadose zone from the soil profile on day i (mm H2O), Qgw is the amount of return flow on day i (mm H2O), and Qsurf is the amount of surface runoff on day i (mm H2O).
Surface runoff occurs when the rate of precipitation to the ground surface exceeds the rate of percolation and evapotranspiration, which is simulated with the following equations (Neitsch et al. 2009):
e2
e3
where S is retention parameter (mm H2O); CN represents the curve number that varies spatially because of changes in soils, land use, management, and slope and temporally because of changes in soil water content; and Ia is initial abstractions that include surface storage, interception, and infiltration prior to runoff (mm H2O).
Snowmelt is affected by the air and snowpack temperature, the melting rate, and the areal coverage of snow. The snowmelt in SWAT is calculated as a linear function of the difference between the average snowpack-maximum air temperature and the threshold temperature for snowmelt (Neitsch et al. 2009):
e4
e5
where SNOmlt denotes the amount of snowmelt (mm H2O); bmlt is the melt factor [mm H2O (°C day)−1]; snocov represents the fraction of the hydrologic response unit area covered by snow; Tsnow is the snowpack temperature (°C); Tmx is the maximum air temperature (°C); Tmlt is the threshold temperature above which snowmelt is allowed (°C); SNO denotes the water content of the snowpack (mm H2O); SNO100 is the threshold depth of snow at 100% coverage (mm H2O); and cov1 and cov2 are coefficients defining the shape of areal depletion curve, which are determined by depth of snow corresponding to 50% snow cover.

b. Input data error model

To account for the uncertainties of input data, the daily meteorological observations from the six meteorological stations along the drainage network can be regarded as independent, latent variables and included in parameter uncertainty analysis. This could make the dimensionality of the parameter estimation problem grow manifold and affect the significance of the estimated parameters. An input data error model is proposed in this study by introducing several variables for precipitation and temperature errors at each time step, which can be formulated as follows:
e6
e7
where t is an index for time; Rt and Tt denote the observed precipitation depth and temperature data, respectively; and and represent error terms for precipitation and temperature that are independent of observed data, respectively. In this study, true meteorological data are assumed to be corrupted at all time steps by the error terms from the identical Gaussian distribution with unknown constant mean and variance . The form of error distribution is widely used to characterize the input data errors (Huard and Mailhot 2006; Kavetski et al. 2006a,b; Ajami et al. 2007; Li and Xu 2014). Thus, instead of addressing every meteorological event, four new parameters are introduced, . The input data error model can facilitate a direct treatment of data uncertainty with computational feasibility. The two most obvious forms of error are multiplier (Li and Xu 2014) and additive (Huard and Mailhot 2006). The magnitude of rain gauge errors is highly dependent on the precipitation intensity (Ciach 2003). Therefore, the error term of precipitation data is introduced in the form of multipliers to maintain the heteroscedastic nature of the error (Ajami et al. 2007). The error term for temperature is assumed to be additive since there is no evident correlation between the data and error.

c. Bayesian inference

The Bayesian paradigm can help generate posterior PDFs that combine useful information about the uncertainty associated with the model parameters, measurement error of input, and a priori knowledge about the parameters. The Bayesian theorem is expressed as follows (Wellen et al. 2014):
e8
where , in which denotes the hydrological model parameters and represents statistical parameters. The Bayesian approach implements statistical inference through a quantitative update of prior beliefs after considering measurement. The variable denotes the posterior probability that expresses our updated beliefs on after measurements and forcing conditions are accounted for. The effect of the prior density tends to vanish when dealing with large samples of observations. The likelihood function is the key issue of the Bayesian inference, which considers simulation errors through addressing their probability distribution.
In this study, a Box–Cox transformation with parameter is used to transform the observed discharge yobs and simulated discharge y to deal with a violation of the homoscedasticity assumption (Laloy et al. 2010):
e9
e10
where and are the transformed observed and simulated data. The transformed residuals between model outputs and observed data can be expressed as
e11
Autocorrelation exists in residuals in terms of observational errors and inadequacies in mathematical model representation of the actual hydrological processes, as well as natural process autocorrelation (Schaefli et al. 2007). A first-order autoregressive time series model (AR1), which is the most widely used type of autoregressive moving-average time series model in the hydrological literature (Yang et al. 2007), can be employed to deal with, at least partially, time dependence of the residuals. Thus, AR1-corrected transformed residuals can be formulated as follows:
e12
e13
e14
where is the daily correlation and represents the daily innovations with zero mean and variance . Combining Box–Cox transformation and AR1 model, the likelihood function can be formulated as follows:
e15
The prior distributions of all parameters are shown in Table 2. The mean areal precipitation has an error of approximately 10% for a sampling interval of 24 h from a network with one gauge per 140 km2 (Seed and Austin 1990; Villarini and Krajewski 2008). Thus, the range (indicated by square brackets) of the daily precipitation error is assumed to be [0.9, 1.1]. The bias between the observed and true temperature data could be as large as 4°C because of sparse observation and topographic inequality in the mountainous watershed. The temperature error in this study is ranged from −4° to 4°C. The prior distributions are set to U[0.95, 1.05] for , U[0.02, 0.03] for , U[−0.2, 0.2] for , and U[1.65, 1.75] for , such that the daily data errors that are generated using the simulation process can vary under the assumption.
Table 2.

Prior distributions of the model parameters.

Table 2.

d. Bayesian framework for uncertainties in input data and model parameters

The flowchart of the proposed Bayesian framework is shown in Fig. 2. To account for uncertainties in input data and model parameters, the input data error model is combined with Bayesian inference through introducing its parameters to the optimization process. Then, the integrated likelihood function of all parameters can be formulated as follows:
e16
where and . In this study, four calibration cases related to data errors were conducted: 1) case 1 is referred to as the classical calibration approach, which corresponds to analysis of uncertainty associated with SWAT model parameters and AR1 model coefficient; 2) case 2 corresponds to parameter uncertainty analysis and temperature error estimation; 3) case 3 is related to parameter uncertainty and precipitation error estimation; and 4) case 4 includes all the previous cases, addressing uncertainties in input data (precipitation and temperature) and model parameters. Correspondingly, the joint posterior PDFs for the four calibration cases can then be formulated:
  1. posterior PDF for case 1,
    e17
  2. posterior PDF for case 2,
    e18
  3. posterior PDF for case 3,
    e19
  4. posterior PDF for case 4,
    e20
MCMC provides an efficient way to draw samples of parameter values from complex, high-dimensional statistical distributions (Metropolis et al. 1953; Hastings 1970). An adaptive MCMC method, namely, Delayed Rejection Adaptive Metropolis (DRAM), is employed to perform Bayesian inference. DRAM benefits from its ability to make use of two powerful ideas, adaptive Metropolis samplers (Haario et al. 1999, 2001) and delayed rejection (Mira 2001). The algorithm adaptively adjusts the proposal function using the chain generated so far and is responsible for global adaptation (Haario et al. 2006). The delayed rejection searches the parameter space efficiently by reducing the probability that the algorithm will remain at the current state and allows for partial local adaptation (Smith and Marshall 2008). The DRAM can improve the computational efficiency with the above global and local adaptive strategies (Zhang et al. 2013). The implementation of DRAM that combines adaptive Metropolis with a two-stage delayed rejection algorithm consists of the following steps (Laine 2008):
  • Step 1: Start from an initial value and initial first stage proposal covariance . Select the initial nonadaptation period n0 and scalings for the higher-stage proposal covariances .

  • Step 2: Delayed rejection loop. Until a new value is accepted, or the second stage has been done. The covariance for the second stage is computed as a scaled version of the proposal of the first stage, .

    1. Propose from a multivariate Gaussian distribution centered at the current value .

    2. Accept according to the acceptance probability. The second stage is accepted with the probability
      eq1
      where α denotes the acceptance probability, and q represents the proposal function.
  • Step 3: Set or , according to whether we accept the value or not.

  • Step 4: After an initial period of simulation n0, adapt the proposal of the first stage of delayed rejection based on all previous states:
    eq2
    where sd is a parameter that depends on the dimension d of the state space, ε > 0 is a constant that we may choose to be very small, and denotes the d-dimensional identity matrix.
  • Step 5: Iterate from step 2 onward until enough values have been generated.

Fig. 2.
Fig. 2.

Flowchart of the proposed Bayesian framework.

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

In this study, the Box–Cox transformation was employed with of 0.35 to the observed and simulated discharge series. Uncertainty analysis was performed using the DRAM algorithm in the calibration period with 2192 continuous daily records (1995–2000) and a validation period with 365 daily streamflow observations in 2001. Ten hydrological model parameters, one AR1 model coefficient, and four statistic variables for data error terms were considered. Five chains were run in parallel with random starting points. For each simulation, the first 50 000 iterations were discarded and the final 50 000 iterations were used to explore the posterior uncertainty of the parameters. For each simulation, user interaction was needed in scaling the proposal distribution if acceptance rates were not around 20%–45% in initial runs. The convergence of the algorithm was confirmed using the Gelman–Rubin criterion (Gelman and Rubin 1992), which involved both the between-chain variance and the within-chain variance and quantitatively diagnoses if each parameter converges to a stationary distribution. The values of the summary statistic are almost equal to one for all parameters, indicating that the chains have converged and the number of iterations is sufficient.

4. Results and discussion

a. Effects of input data errors to parameter uncertainty

Figure 3 illustrates the marginal frequency distributions of the SWAT model and statistic variables for data error terms for the four cases. The deviations from probability distributions under cases 2, 3, and 4 to those under case 1 were investigated with the Kullback–Leibler divergence DKL, which was often used in statistics as a measure of similarity between two density distributions (Kullback and Leibler 1951; Weijs et al. 2010). Under case 1, parameters exhibit different posterior distributions from their prior distributions in terms of distribution shape and parameter range. Under cases 2–4 (uncertainties in input data and model parameters are considered), the parameters’ posterior distributions would be quite different from those obtained under case 1. The DKL of CN2 for cases 2, 3, and 4 is 0.122, 1.365, and 1.636, respectively. The results indicate that the posterior distributions under cases 3 and 4 are different from that under case 1. This may be because parameter CN2 indicates overland flow potential, which is affected by antecedent soil water conditions and land-use pattern. Precipitation errors can influence the soil moisture; temperature errors have an impact on the land use through altering snowmelt. The data errors could thus influence the overland flow generation in the Kaidu watershed. Besides, under cases 2–4, parameter SNO50COV, which represents the relationship between snow water and snow cover, exhibits different values for the mode of the distribution from that under case 1, with the DKL being 1.095, 0.851, and 5.090, respectively. This may be because precipitation affects the snow accumulation in winter and temperature dominates the snowmelt in spring. Errors of input data could thus influence the snow water generation within the watershed.

Fig. 3.
Fig. 3.

Marginal frequency distributions for parameters of SWAT and input data error model.

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

Under case 4, the uncertainty of most model parameters would be reduced (decreased parameter ranges), particularly for parameters of ESCO and TIMP (see Table 2 for parameter descriptions). The results indicate that considering errors in temperature and precipitation data could lead to more accurate estimates for parameters. ESCO can be employed to adjust the depth distribution used to meet the soil evaporative demand and to potentially improve the actual evaporation process in arid areas of the Kaidu watershed. If temperature is higher, the melting of winter snow occurs earlier in spring, leading to available water amount for evapotranspiration at the time when potential evaporation is low. Such a shift in snowmelt timing then reduces soil moisture and increases the evaporation resistance when potential evaporation is higher later in the year. Besides, precipitation tends to alter the available water amount for evapotranspiration. Thus, input data errors (temperature and precipitation) would have an influence on the distribution of ESCO. Parameter TIMP indicates the lag effect on snowpack temperature from air temperature influenced by snowpack density, depth, and exposure. Variations in temperature and precipitation could impact snow accumulation, depth, and density directly, which would thus affect the distribution of parameter TIMP. In addition, from the results, there is no evident difference among parameter uncertainty ranges under cases 1–3. This implies that the reduced uncertainty can be explained by the effects of errors in temperature and precipitation data, rather than individual effect of error in precipitation or temperature. It is thus essential to consider multiple sources of input data errors for parameter estimation in hydrological modeling.

The results also indicate that there is no obvious variation in the possible range of HRU_SLP and CH_K2 under the four calibration cases (Fig. 3). The DKL of distribution of HRU_SLP for cases 2, 3, and 4 is 0.296, 0.280, and 0.226, respectively, and the DKL for distribution of CH_K2 is 0.179 (case 2), 0.301 (case 3), and 0.120 (case 4). The results indicate little effect of data errors on the two parameters. This may be because HRU_SLP denotes the average steepness, which is dominated by the topographic factors of each hydrologic response unit. And CH_K2 describes the relationship between streams and the groundwater system; it is affected by the hydraulic conductivity of main channel alluvium. The results imply that input data errors would have little influence on model parameters that describe properties of the underlying surface.

The posterior distributions for the AR1 model coefficient and statistic variables for data error terms are also depicted in Fig. 3, which are quite different from their prior distributions. The mean of error terms for precipitation is larger than one for cases 3 and 4, indicating that the precipitation is underestimated by the observation data. The mean of error terms for temperature are different from zero. The result indicates that errors may exist in the temperature data. This may also indicate that the parameter is used to compensate model structural uncertainties. For example, base flow would be generated only from the shallow aquifer. Besides, the groundwater outflows would not be captured by the model.

Figure 4 depicts the joint posterior probability between CN2 and SNO50COV under cases 1 and 4. From the results, a week correlation between the two parameters can be identified under case 1. This may be because that larger value of SNO50COV indicates more snow water content in winter, which would increase the soil water content in snowmelt and thus the overland flow potential of land surface represented by CN2. Under case 4 (errors of precipitation and temperature data are considered), the correlation between the two parameters would become more evident. This may be because the variations in precipitation and temperature could have an influence on snow accumulation, snowmelt, and runoff generation. The results imply that the data errors could affect correlations between parameters and relations between the related hydrological processes.

Fig. 4.
Fig. 4.

Joint posterior probability between CN2 and SNO50COV: (a) case 1 and (b) case 4.

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

b. Propagation of uncertainties in input data and model parameters

Figure 5 shows the simulation uncertainty ranges from the propagation of estimated model parameters (under case 1). The observed discharges are separately indicated with red hollow circles. From the results, the uncertainty ranges at a quantile level of 0.05 (dotted lines) and a quantile level of 0.45 (yellow) seem to be unable to capture many of the observed discharges. This implies that attributing all the uncertainty sources to parameter uncertainty would lead to an inaccurate and biased modeling of the hydrological processes. In fact, observed streamflow data may contain errors due to discharge gauging errors. This study focuses on propagation from uncertainties in input data and parameters to model outputs; the output errors are not considered.

Fig. 5.
Fig. 5.

Streamflow prediction uncertainty ranges at quantile levels of 0.05 and 0.45 (case 1).

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

The estimated uncertainty bounds for the hydrograph associated with each of the three cases accounting for errors are illustrated in Figs. 68, respectively. From the results, the simulation intervals are wider than those obtained under case 1, leading to a much better coverage of the observed discharge when input data errors are explicitly considered. This is particularly obvious for the peak streamflows during the summer of 2000 and the spring of 2001, while classical calibration approach underestimates the observed storms (Fig. 5). This implies that considering uncertainties in temperature data could improve the modeling ability for peak streamflow in spring and considering precipitation errors could provide a better coverage of peak streamflow during summer. Figure 9 illustrates the normal probability of the mean transformed residuals, that is, transformed residuals averaged over all the iterations, under case 4. Similar results are obtained for the other cases. The results reveal that the residuals closely conform to a normal distribution. This is further confirmed by a Lilliefors test at a 5% significance level. Figure 10 depicts the autocorrelation function of the transformed residuals. From the results, the autocorrelations are very small except for the first-order coefficient. The results imply that there is no serious violation of the assumptions of independence and of distribution shape.

Fig. 6.
Fig. 6.

Streamflow prediction uncertainty ranges at quantile levels of 0.05 and 0.45 (case 2).

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

Fig. 7.
Fig. 7.

Streamflow prediction uncertainty ranges at quantile levels of 0.05 and 0.45 (case 3).

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

Fig. 8.
Fig. 8.

Streamflow prediction uncertainty ranges at quantile levels of 0.05 and 0.45 (case 4).

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

Fig. 9.
Fig. 9.

Normal probability plot for the mean transformed residuals (case 4).

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

Fig. 10.
Fig. 10.

Autocorrelation function plot of the mean transformed residuals (case 4).

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

The streamflow recessions were overestimated in 1999 and 2000, which can be inspected from the prediction uncertainty bounds for the hydrograph. Such errors in hydrological simulation (with SWAT) in similar watersheds can be found in Chen (2012) and Luo et al. (2012). Structural uncertainties in hydrological simulation with the SWAT model may account for this issue. Because of the steep slopes of the river basins in the Tian Shan, the quick recession of surface runoff, and the sluggish and stable baseflow processes, the streamflow process in the Kaidu watershed is featured with a quick percolation of rainfall and snowmelt waters during the summertime to an underground storage. The overestimation of recession process may be attributed to the improper simulation of base flow in autumn, which is associated with water discharged from groundwater storage. The limitation in modeling base flow with SWAT lies in the assumption that water entering the deep aquifer is not considered in the future water budget calculations and that base flow is generated only from the shallow aquifer.

The modeled snowmelt and soil water content under case 4 was further checked using limited snow coverage and soil moisture data. The hydrological processes were simulated based on the parameter set that maximizes the likelihood (under case 4). The snow coverage rate for the Kaidu watershed was collected from March to June 2000. The modeled snow coverage was calculated based on simulated snowmelt volume with Eq. (4). The data of soil depth and bulk density as well as monthly soil moisture in Shuidianzhan station was collected. The soil moisture was calculated based on the modeled soil water content and the data of soil depth and bulk density. Table 3 provides the simulated and observed snow cover and soil water content. The results indicate a low level of error, with the relative deviation being 12.56% (snow coverage) and 11.15% (soil moisture), respectively. This implies that, through combining the input data error model with Bayesian inference, the simulation for the related hydrological processes and the water cycle can be improved. However, the coarse and limited monthly verification cannot guarantee the validity of daily hydrologic simulation. Multivariable observation and multiobjective calibration technique are desired to recognize the detailed hydrological processes.

Table 3.

Observed and simulated snow cover and soil moisture in 2000.

Table 3.

c. Simulation performance under different periods

The entire simulation horizon was divided into snow-melting (March–May) and nonmelting (from June to the following February) periods. Hydrological modeling ability under different periods was investigated with the indicators of model sharpness , reliability bias , and interval skill score , as presented in the appendix. The results of the indicators for the three periods (i.e., snow melting, nonmelting, and entire simulation horizon) are illustrated in Fig. 11.

Fig. 11.
Fig. 11.

Model performance indicators at different quantile levels: (a) sharpness in entire simulation horizon, (b) sharpness in snow-melting period, (c) sharpness in nonmelting period, (d) reliability bias in entire simulation horizon, (e) reliability bias in snow-melting period, (f) reliability bias in nonmelting period, (g) interval skill score in entire simulation horizon, (h) interval skill score in snow-melting period, and (i) interval skill score in nonmelting period.

Citation: Journal of Hydrometeorology 17, 8; 10.1175/JHM-D-15-0236.1

Model sharpness is measured using the average bound spread, which describes the difference between the corresponding lower and upper quantile simulations. From the results depicted in Figs. 11a–c, under cases 2, 3, and 4 (model parameter uncertainty and data errors are considered), the hydrological simulations would have a bigger average bound spread (i.e., larger uncertainty bound) than that under case 1 at most quantile levels. In the snow-melting period (Fig. 11b), the average bound spread under case 4 would be the largest. It is further noticed that under case 2, the difference of the bound spread from that under case 1 would be largest in this period, with an average deviation being 14.639 m3 s−1, indicating that temperature errors would have an influence on streamflow in the snow-melting period. This may be because temperature dominates magnitude, timing, and duration of snowmelt and peak streamflows in the ablation period. In the nonmelting period (Fig. 11c), the hydrological modeling under case 3 can provide the largest bound spread, with the average being 46.873 m3 s−1. This indicates that precipitation errors would affect runoff response during the nonmelting period. This is mainly because variations in precipitation tend to alter the volume of runoff, and particularly the peak flows in summer for the watershed.

In Figs. 11d–f, the reliability bias for the three periods is plotted. Variable b indicates the percentage of time that the observations are within the confidence bounds. From the results, hydrological modeling under case 4 would be most reliable in the snow-melting period, with the average reliability bias being 0.175. Besides, the reliability bias under case 3 would be smallest in the nonmelting and entire periods. The results imply that considering uncertainties in input data (temperature and precipitation) and model parameters could provide a better coverage of the observed discharges. The results also indicate that the reliability under case 1 (only parameter uncertainty is considered) is particularly poor, with the average deviation from the nominal coverage being 0.371 in the entire horizon.

Interval skill score combines the information in terms of the sharpness and the reliability to evaluate the simulation performance directly. From the results (Fig. 11g), hydrological simulation under case 4 performs best in the entire simulation horizon, with a minimum average skill score being 216.431. When conditioning on the nonmelting period (Fig. 11i), the modeling simulation performance under case 3 would be the best with an average score of 229.834. In the snow-melting period (Fig. 11h), hydrological simulation under case 4 performs best at lower quantile levels (1 − β = 0.45–0.95); the average score is 165.130. The results indicate that considering uncertainties of temperature and precipitation data as well as model parameters could enhance a model’s capability for simulating water resources.

5. Conclusions

In the study, a Bayesian framework is proposed for investigating uncertainties in input data (i.e., temperature and precipitation) and parameters in SWAT model as well as their effects on the runoff response. The Bayesian framework can help facilitate the exploration of variation of model parameters due to input data errors, as well as propagation from uncertainties in data and parameters to model outputs in the snow-melting and nonmelting periods.

The presented framework has been applied to the Kaidu watershed with snowmelt–precipitation-driven physics. A series of calibration cases corresponding to data errors under different periods are examined. Several findings can be revealed as follows:

  1. Input data errors can affect the ranges and distributions of model parameters, such as CN2, SNO50COV, ESCO, and TIMP, implying that the data errors could influence the related hydrologic processes (i.e., overland flow generation, snowmelt, and evapotranspiration).

  2. Input data errors have an impact on correlations between CN2 and SNO50COV, implying that data errors could influence the relations between hydrological processes.

  3. There is no obvious variation in the possible range of HRU_SLP and CH_K2 under the four calibration cases; input data errors have little impact on model parameters that describe properties of underlying surface (e.g., topographic factors and main channel alluvium).

  4. Hydrological modeling under cases 2 and 3 could improve the simulation ability for peak streamflows in spring and summer, respectively.

  5. Hydrological simulation under case 4 has the largest average bound spread and least reliability bias in the snow-melting period; during nonmelting and entire simulation horizon, case 3 can provide a largest bound spread and best coverage of discharge data.

  6. In the snow-melting and entire simulation horizon, hydrological modeling under case 4 performs best; simulation under case 3 can provide the best modeling simulation performance in the nonmelting period.

The findings have significant implications in enhancing a hydrologic model’s capability for simulating/predicting water resources during different seasons for snowmelt–precipitation-driven watersheds.

Nevertheless, there are also potential extensions of this study. First, land use in the Kaidu watershed has significantly changed over the past few decades, which may result in variations of the characteristics of underlying surface and the specific mechanisms of runoff generation. The dynamic natures of model parameter uncertainties and the associated propagations to watershed hydrology are desired to be explored. Second, from the results, there are still observations failed to be captured; this indicates that although progress has been made in investigating errors in input data and model parameters as well as their effects on the runoff response, a further complete characterization of model uncertainties is desired to be achieved through accounting for other uncertainty sources, such as output error and model structural uncertainty.

Acknowledgments

This research was supported by the National Key Research Development Program of China (2016YFA0601502 and 2016YFC0502803), and the National Natural Science Foundation (51379075, 51225904, and 51520105013). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions.

APPENDIX

Probabilistic Performance Measures

To measure model ability under each case to generate useful confidence and to fit observed values at the Dashankou hydrometric station (Fig. 1), three goodness-of-fit performance measures are calculated in terms of width of the derived confidence bounds, referred to as the sharpness, and the percentage time that the observation data are within the confidence bounds, referred to as the reliability, as well as the interval skill score that combines the evaluation of the sharpness and the reliability.

The sharpness is an accuracy measure defined as the average width of the confidence bounds at any given quantile level β:
ea1
where and represent the simulated lower and upper bound at any given time step and quantile level during the entire simulation horizon. For a given simulation interval, a binary indicator variable is introduced corresponding to hits and misses of observations in each step:
ea2
where yk denotes the observed value at k day. The ideal coverage of the simulation bound is defined as the nominal coverage 1 − β. The reliability bias is calculated with the discrepancy between nominal and observed coverage of the confidence bounds (i.e., empirical coverage). A perfect fit is defined as = 0, when the empirical coverage is equal to the nominal coverage (Barnett et al. 2005):
ea3
Interval skill score combines performance measures discussed above to evaluate the modeling performance of the different models directly. The score at time instant k is calculated for the interval simulation as follows:
ea4
The skill score describes the distance between the simulated interval and the observation at each considered quantile. An increased score results in a reduced fit of simulation bounds (Thordarson et al. 2012).

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  • AghaKouchak, A., Bardossy A. , and Habib E. , 2010: Copula-based uncertainty modelling: Application to multisensor precipitation estimates. Hydrol. Processes, 24, 21112124, doi:10.1002/hyp.7632.

    • Search Google Scholar
    • Export Citation
  • Ajami, N. K., Duan Q. Y. , and Sorooshian S. , 2007: An integrated hydrologic Bayesian multimodel combination framework: Confronting input, parameter, and model structural uncertainty in hydrologic prediction. Water Resour. Res., 43, W01403, doi:10.1029/2005WR004745.

    • Search Google Scholar
    • Export Citation
  • Barnett, T. P., Adam J. C. , and Lettenmaier D. P. , 2005: Potential impacts of a warming climate on water availability in snow-dominated regions. Nature, 438, 303309, doi:10.1038/nature04141.

    • Search Google Scholar
    • Export Citation
  • Chandra, R., Saha U. , and Mujumdar P. P. , 2015: Model and parameter uncertainty in IDF relationships under climate change. Adv. Water Resour., 79, 127139, doi:10.1016/j.advwatres.2015.02.011.

    • Search Google Scholar
    • Export Citation
  • Chen, X., 2012: Hydrological Model of Inland River Basin in Arid Land. China Environmental Science Press, 119 pp.

  • Chen, Z. S., and Chen Y. N. , 2014: Effects of climate fluctuations on runoff in the headwater region of the Kaidu River in northwestern China. Front. Earth Sci., 8, 309318, doi:10.1007/s11707-014-0406-2.

    • Search Google Scholar
    • Export Citation
  • Ciach, G. J., 2003: Local random errors in tipping-bucket rain gauge measurements. J. Atmos. Oceanic Technol., 20, 752759, doi:10.1175/1520-0426(2003)20<752:LREITB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Clark, M. P., and Slater A. G. , 2006: Probabilistic quantitative precipitation estimation in complex terrain. J. Hydrometeor., 7, 322, doi:10.1175/JHM474.1.

    • Search Google Scholar
    • Export Citation
  • Clark, M. P., and Coauthors, 2011: Representing spatial variability of snow water equivalent in hydrologic and land-surface models: A review. Water Resour. Res., 47, W07539, doi:10.1029/2011WR010745.

    • Search Google Scholar
    • Export Citation
  • Elsner, M. M., Gangopadhyay S. , Pruitt T. , Brekke L. D. , Mizukami N. , and Clark M. P. , 2014: How does the choice of distributed meteorological data affect hydrologic model calibration and streamflow simulations? J. Hydrometeor., 15, 13841403, doi:10.1175/JHM-D-13-083.1.

    • Search Google Scholar
    • Export Citation
  • Fu, A. H., Chen Y. N. , Li W. H. , Li B. F. , Yang Y. H. , and Zhang S. H. , 2013: Spatial and temporal patterns of climate variations in the Kaidu River basin of Xinjiang, northwest China. Quat. Int., 311, 117122, doi:10.1016/j.quaint.2013.08.041.

    • Search Google Scholar
    • Export Citation
  • Gelman, A., and Rubin D. B. , 1992: Inference from iterative simulation using multiple sequences. Stat. Sci., 7, 457472, doi:10.1214/ss/1177011136.

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