Hydrological model parameters play an important role in the
ability of model prediction. In a stationary context, parameters of
hydrological models are treated as constants; however, model parameters may
vary with time under climate change and anthropogenic activities. The technique of
ensemble Kalman filter (EnKF) is proposed to identify the temporal variation
of parameters for a two-parameter monthly water balance model (TWBM) by
assimilating the runoff observations. Through a synthetic experiment, the
proposed method is evaluated with time-invariant (i.e., constant) parameters
and different types of parameter variations, including trend, abrupt change
and periodicity. Various levels of observation uncertainty are designed to
examine the performance of the EnKF. The results show that the EnKF can
successfully capture the temporal variations of the model parameters. The
application to the Wudinghe basin shows that the water storage capacity
(SC) of the TWBM model has an apparent increasing trend during the
period from 1958 to 2000. The identified temporal variation of SC
is explained by land use and land cover changes due to soil and water
conservation measures. In contrast, the application to the Tongtianhe basin
shows that the estimated SC has no significant variation
during the simulation period of 1982–2013, corresponding to the relatively
stationary catchment properties. The evapotranspiration parameter
(
Hydrological model parameters are critically important for accurate simulation of runoff. Parameters of conceptual hydrological models can be considered as a simplified representation of the physical characteristics in hydrologic processes. Therefore, parameter values are closely related to the catchment conditions, such as climate change, afforestation and urbanization (Peel and Blöschl, 2011). In hydrological modeling, parameters are usually assumed to be stationary; i.e., the calibrated parameters are constants during the calibration period, and have extrapolative ability outside the range of the observations used for parameter estimation (Merz et al., 2011). The estimated parameters usually depend on the calibration period since the calibration period may contain different climatic conditions and hydrological regimes compared to the simulation period (Merz et al., 2011; Zhang et al., 2011; Coron et al., 2012; Seiller et al., 2012; Westra et al., 2014; Patil and Stieglitz, 2015). The model parameters may change as a response to the variations in climatic conditions and catchment properties. For example, land use and land cover changes contribute to temporal changes of model parameters (Andréassian et al., 2003; Brown et al., 2005; Merz et al., 2011). Therefore, it is no longer appropriate to treat parameters as time invariant.
Time-variant hydrological model parameters have been reported in a few recent publications (Merz et al., 2011; Brigode et al., 2013; Jeremiah et al., 2013; Thirel et al., 2015; Westra et al., 2014; Patil and Stieglitz, 2015). For example, Ye et al. (1997) and Paik et al. (2005) mentioned the seasonal variations of hydrological model parameters. Merz et al. (2011) analyzed the temporal changes of model parameters, which were calibrated by using six consecutive 5-year periods between 1976 and 2006 for 273 catchments in Austria. Recently, Westra et al. (2014) proposed a strategy to cope with nonstationarity of hydrological model parameters, which were represented as a function of a time-varying covariate set before using an optimization algorithm for calibration. Previous studies provided two main methods to estimate the time-variant model parameters: (1) available historical records are divided into consecutive subsets, and parameters are calibrated separately for each subset using an optimization algorithm (Merz et al., 2011; Thirel et al., 2015); (2) a functional form of selected time-variant model parameters is constructed, and the parameters for the function are estimated using an optimization algorithm based on the entire historical record (Jeremiah et al., 2013; Westra et al., 2014).
The data assimilation (DA) actually provides another method to identify the potential temporal variations of model parameters by updating them in real time when observations are available (Liu and Gupta, 2007; Xie and Zhang, 2013). The DA method has been widely applied in hydrology for soil moisture estimation (Han et al., 2012; Kumar et al., 2012; Yan et al., 2015) and flood forecasting (Y. Li et al., 2013; Liu et al., 2012; Abaza et al., 2014). It has also been successfully used to estimate model parameters (Moradkhani et al., 2005; Kurtz et al., 2012; Montzka et al., 2013; Panzeri et al., 2013; Vrugt et al., 2013; Xie and Zhang, 2013; Shi et al., 2014; Xie et al., 2014). For example, Vrugt et al. (2013) proposed two Particle-DREAM (DiffeRential Evolution Adaptive Metropolis) methods, i.e., Particle-DREAM for time-variant and time-invariant parameters, to track the evolving target distribution of HyMOD parameters, while both results were approximately similar and statistically coherent since only 3 years of data were used. Xie and Zhang (2013) used a partitioned forecast-update scheme based on the ensemble Kalman filter (EnKF) to retrieve optimal parameters in a distributed hydrological model. Although the DA method has been used to estimate model parameters, these studies are focused on the estimation of constant parameters. Little attention has been paid to the identification of time-variant model parameters by using the DA method.
The aim of this study is to assess the capability of the EnKF to identify the temporal variations of the model parameters for a monthly water balance model. Thus, a synthetic experiment, including four scenarios with different parameter variations and one scenario with time-invariant parameters, is designed for parameter estimation at different uncertainty levels. Furthermore, two case studies are implemented to estimate the model parameter series and to interpret the parameter variations in response to the changes in catchment characteristics, i.e., land use and land cover. The remainder of this paper is organized as follows. Section 2 presents a brief review of the monthly water balance model and the EnKF method. Following the methodology, Sect. 3 describes the synthetic experiment and the application to two case studies. Results and discussion are presented in Sect. 4, followed by conclusions in Sect. 5.
The two-parameter monthly water balance model (TWBM), developed by Xiong and
Guo (1999), has been widely applied for monthly runoff simulation and
forecast (Guo et al., 2002, 2005; Xiong and Guo, 2012; S. Li et al., 2013;
Zhang et al., 2013; Xiong et al., 2014). The inputs of the model include
monthly areal precipitation and potential evapotranspiration. The actual
monthly evapotranspiration is calculated as follows:
The monthly runoff is dependent on the soil water content and is calculated
by the following equation:
States and parameters of the two-parameter monthly water balance model.
As a sequential data assimilation technique, EnKF is essentially the Monte Carlo implementation of the Kalman filter, producing an ensemble of state simulations for updating the state variables and their covariance matrices (Evensen, 1994; Burgers et al., 1998; Moradkhani et al., 2005; Shi et al., 2014). It is applicable to a variety of nonlinear problems (Evensen, 2003; Weerts and El Serafy, 2006) and has been widely applied to hydrological models (Abaza et al., 2014; DeChant and Moradkhani, 2014; Delijani et al., 2014; Samuel et al., 2014; Tamura et al., 2014; Xue and Zhang, 2014; Deng et al., 2015). Furthermore, the EnKF has been successfully used in time-invariant parameter estimations for hydrological models (Moradkhani et al., 2005; Wang et al., 2009; Xie and Zhang, 2010, 2013).
In this paper, the EnKF is applied to simultaneously estimate state
variables and parameters (Table 1) in the TWBM model. The augmented state
vector includes both states and model parameters (Wang et al., 2009), i.e.,
The observation ensemble member can be written as
Based on the available state and observation equations, the model parameters
and state are updated according to the following equation:
Since the parameters are limited within a range, the constrained EnKF (Wang
et al., 2009) is used in this study. The ensemble size, uncertainties in
input and output have significant impacts on the assimilation performance of
the EnKF, and they are specified following the previous studies (Moradkhani
et al., 2005; Wang et al., 2009; Xie and Zhang, 2010; Nie et al., 2011;
Lü et al., 2013; Samuel et al., 2014). The ensemble size is set to 1000
for the synthetic experiment and the two case studies. In the present study,
the uncertainties, including state variable and parameter errors
(
Two evaluation criteria, including the Nash–Sutcliffe efficiency
(NSE) (Nash and Sutcliffe, 1970) and the volume error (VE)
are used to evaluate the runoff assimilation results for the synthetic
experiment and the application to real catchments (Deng et al., 2015; Li et
al., 2015).
Different variations of model parameters in the synthetic experiment.
Proportional factors of the standard deviations for precipitation
(
The assimilated parameter results are evaluated using the following
criteria, including the Pearson correlation coefficient (
Location and mean monthly precipitation and runoff from 1956 to 2000 of the Wudinghe basin.
A synthetic experiment is designed to evaluate the capability of the
assimilation procedure to identify the temporal variation of model
parameters. Five scenarios of different parameter variations are developed,
as shown in Table 2. The model parameters in the first four scenarios are
time variant, and those in the last scenario are constant. Parameter
Time series of model parameters are synthetically generated, including
the time-variant parameters and the constant parameters. Model parameter
sets are produced using a sinusoidal function and/or a linear trend function
within the specified ranges shown in Table 1. The runoff observations for
each scenario are computed from the TWBM model taking monthly potential
evapotranspiration, monthly precipitation and the parameters as inputs. The initial ensembles of model parameters and state variables are
generated using uniform distributions within the specified ranges in Table 1. The ensemble size and the total number of assimilation time steps are
specified. After the initialization of parameters and state variables, the
hydrological model parameters and states are updated by assimilating the
runoff observations obtained in step (1). The additive errors for generating
the ensemble members of model parameters, state variables and runoff
observations are obtained from Gaussian distributions with zero mean and
specified variance.
To evaluate the effects of errors on identifying parameter variation,
different levels of observation uncertainty are considered in the
synthetic experiment, as detailed in Table 3. The uncertainties from the
observed precipitation and runoff are characterized by adding Gaussian
noises, where standard deviations are assumed to be proportional to the
magnitude of the true values, and the corresponding proportional factors are
denoted as
Location and mean monthly precipitation and runoff from 1980 to 2013 of the Tongtianhe basin.
The method is applied to the Wudinghe basin (Fig. 1), which is a sub-basin
of the Yellow River basin and located in the southern fringe of the Maowusu
Desert and the northern part of the Loess Plateau in China, where the climate is semiarid
climate. It has a drainage area of approximately 30 261 km
Comparison between estimated
Comparison between estimated SC and its true values for various parameter changes under different uncertainty levels. The gray areas represent the 95 % prediction uncertainty intervals.
The soil erosion is severe in the Wudinghe basin, owing to the highly erodible loess and sparse vegetation. Since the 1960s, the soil and water conservation measures have been undertaken. Several engineering measures, including tree and grass plantation, check dam and reservoir construction, and land terracing, were effectively implemented during several decades. The land use changes caused by the soil and water conservation measures had a significant effect on increasing water storage capacity (Xu, 2011).
The Tongtianhe basin (Fig. 3) is located in the southwestern Qinghai
Province,
China, with a continental climate. It belongs to the source area of the Yangtze
River basin with a drainage area of about 140 000 km
The data sets used in this study include monthly precipitation, potential
evapotranspiration and runoff in the Wudinghe basin (from 1956 to 2000) and
the Tongtianhe basin (from 1980 to 2013). The potential evapotranspiration is
estimated using the Penman–Monteith equation (Allen et al., 1998) based on
the meteorological data from the China Meteorological Data Sharing Service
System (
Estimations of time-invariant
Performance statistics for various changes of
Performance of runoff estimations for various parameter changes under different levels of uncertainty in the synthetic experiment.
The comparisons of the estimated and true model parameters under different
scenarios are presented in Figs. 3, 4 and 5. Tables 4 and 5 show the
evaluated statistics for the parameters and runoff estimations. The
assimilated parameter values are obtained from the ensemble mean at each
time step. The estimation of parameters
It should be noted that there are time lags between the assimilated and true
The results for the scenario of constant parameters are shown in Fig. 5, demonstrating that the estimated parameters can approach their true values after the initial 24 assimilation steps. The gray areas represent the 95 % prediction uncertainty intervals, which reduce quickly and approach a stable spread. The performance of the estimated parameters is correlated with the uncertainty level. Higher precipitation and runoff observation errors correspond to the greater RMSE values (Table 4) of estimated parameters and uncertainty ranges. The performance of runoff estimations for various parameter changes under different levels of uncertainty is shown in Table 5, suggesting that the EnKF perfectly matches the observations with NSEs higher than 0.95 and absolute VEs smaller than 0.02. The EnKF can successfully capture the temporal variations of the true parameters, although the uncertainty levels of the observations can affect its performance to a certain degree. The above results demonstrate that the EnKF is able to identify the temporal variation of the model parameters by updating the state variables and parameters based on the runoff observations.
Figure 6 shows the double mass curve between monthly runoff and precipitation for the Wudinghe and Tongtianhe basins, respectively. Figure 6a shows the linear relationship between cumulative runoff and precipitation pre- and post-1972 in the Wudinghe basin, which is similar to the result presented by Xu (2011) and Li et al. (2014). The results show two straight lines with different slopes for the relationships between precipitation and runoff, indicating that an abrupt change occurred in 1972; i.e., the runoff generation had been changed from this year due to the soil and water conservation measures. On the other hand, Fig. 6b demonstrates that a single linear relationship fits all the data for the Tongtianhe basin, suggesting a stable precipitation–runoff relationship during the 1982–2013 period.
Double mass curve between monthly runoff and precipitation for
Wudinghe basin within the period of 1958–2000
Estimated parameter values of
The estimated parameters and the associated 95 % prediction uncertainty
intervals are shown in Fig. 7. The time series of estimated SC
shows an apparent increasing trend, with two different trends for pre- and
post-turning points in Fig. 6a. The temporal variation of the water storage
capacity is correlated with the changes of land use and land cover. Both the
trends in Fig. 7c show an increase of SC because the
implementation of the large-scale engineering measures significantly
improved the water holding capacity of the Wudinghe basin, especially for
the reservoir and check dam construction. The trend slopes of the two
periods, one from 1956 to 1971 and the other from 1972 to 2000, are
different because the degree of implementing engineering measures varied
during the period of 1958–2000. Moreover, the increase of the water holding
capacity slowed down during the 1980s due to the sedimentation in reservoirs
and check dams after periods of operation (Wang and Fan, 2003). Figure 8a
shows the long-term time series of precipitation and potential evaporation
in the Wudinghe basin. The result shows that the runoff decreases significantly
while precipitation changes slightly and potential evaporation has no trend,
indicating that the actual evaporation increases significantly due to
impacts of human activities, i.e., soil and water conservation measures.
Figure 8b presents the runoff reduction caused by all the soil and water
conservation measures, i.e., land terracing, tree and grass plantation and check dam and reservoir construction. The runoff reduction positively
relates to the water holding capacity, namely the SC value. The
slope for the period of 1958–1971 is higher than that for the period of
1972–1996, suggesting that the SC in the former period has a higher
increasing trend. On the other hand, results of Tongtianhe basin show that
the estimated SC has no detectable trend with a small
For parameter
In summary, the above results demonstrate that the EnKF can identify the
temporal variation of model parameters well by updating both state variables
and parameters based on the runoff observations. The trends of parameter
SC can be explained by the changes of catchment characteristics (i.e.,
land use and land cover) in the Wudinghe basin. However, the estimated SC
for the Tongtianhe basin is approximately stable with a small standard
deviation because the basin is located in a water protection zone and has no
significant changes on water storage capacity caused by human activities.
The parameter
This study proposes an ensemble Kalman filter (EnKF) to identify the temporal variation of model parameters of the two-parameter monthly water balance model (TWBM) by assimilating runoff observations. A synthetic experiment, which contains four scenarios with different changes of model parameters and one scenario with constant parameters, is designed to examine the capability of the proposed approach. Furthermore, three different levels of observation uncertainty are taken to assess the performance of the EnKF. The main conclusions are as follows. For the time-variant parameters, the EnKF provides superior performance even though slight time lags exist for parameters with periodic variations. The true values of the constant parameters can be approached quickly after 24 time steps of the assimilation process. The temporal variations of the parameters can be successfully captured even under a high level of observation uncertainties, which would have an influence on the performance of the EnKF.
The EnKF method is applied to the Wudinghe basin in China, aiming to detect
the temporal variations of the model parameters and to provide an
explanation for the parameter variation from the perspective of
catchment characteristic changes. Meanwhile, a comparison is implemented to
investigate the variation of model parameters in the Tongtianhe basin, which
is barely affected by human activities. The parameter of water storage
capacity (SC) for the monthly water balance model shows a significant
increasing trend for the period of 1958–2000 in the Wudinghe basin. The soil
and water conservation measures, including land terracing, tree and grass
plantation and check dam and reservoir construction, were implemented
from 1958 to 2000, resulting in the increase of the water holding capacity
of the basin, which explains the increasing trend of SC. Moreover, the
magnitudes of the engineering measures in different time periods play an
important role in the degree of increasing trend for SC. In the
Tongtianhe basin, the parameter SC has no significant trend for the period
of 1982–2013, which is consistent with the relatively stationary catchment
characteristics. The evapotranspiration parameter (
The method proposed in this paper provides an effective tool for the time-variant model parameter identification. Future work will be focused on the influence of the correlations between/among model parameters and performance comparison of multiple data assimilation methods.
The meteorological data can be requested and obtained from the China
Meteorological Data Sharing Service System (
This study was supported by the Excellent Young Scientist Foundation of NSFC (51422907) and the Open Foundation of State Key Laboratory of Water Resources and Hydropower Engineering Science in Wuhan University (2015SWG01). The authors thank the China Meteorological Data Sharing Service System for providing part of the data used in this study. The authors would like to thank the editor and the anonymous reviewers for their comments that helped to improve the quality of the paper. Edited by: A. Guadagnini Reviewed by: two anonymous referees