Although the parameters of hydrological models are usually regarded as constant, temporal variations can occur in a changing environment. Thus, effectively estimating time-varying parameters becomes a significant challenge. Two methods, including split-sample calibration (SSC) and data assimilation, have been used to estimate time-varying parameters. However, SSC is unable to consider the parameter temporal continuity, while data assimilation assumes parameters vary at every time step. This study proposed a new method that combines (1) the basic concept of split-sample calibration, whereby parameters are assumed to be stable for one sub-period, and (2) the parameter continuity assumption; i.e. the differences between parameters in consecutive time steps are small. Dynamic programming is then used to determine the optimal parameter trajectory by considering two objective functions: maximization of simulation accuracy and maximization of parameter continuity. The efficiency of the proposed method is evaluated by two synthetic experiments, one with a simple 2-parameter monthly model and the second using a more complex 15-parameter daily model. The results show that the proposed method is superior to SSC alone and outperforms the ensemble Kalman filter if the proper sub-period length is used. An application to the Wuding River basin indicates that the soil water capacity parameter varies before and after 1972, which can be interpreted according to land use and land cover changes. A further application to the Xun River basin shows that parameters are generally stationary on an annual scale but exhibit significant changes over seasonal scales. These results demonstrate that the proposed method is an effective tool for identifying time-varying parameters in a changing environment.

Conceptual models describe the physical processes that occur in the real world by means of certain assumptions and empirically determined functions (Toth and Brath, 2007). In spite of their simplicity, conceptual models are effective in providing reliable runoff predictions for widespread applications (Refsgaard and Knudsen, 1996; Quoc Quan et al., 2018), such as real-time flood forecasting, climate change impact assessments (Stephens et al., 2019; Deng et al., 2019), and water resources management. Conceptual hydrological models typically have several inputs, a moderate number of parameters, state variables, and outputs. Among these, the parameters play an important role in accurate simulation and should be related to the catchment properties. However, parameter values often cannot be obtained by field measurements (Merz et al., 2011). An alternative approach is to calibrate parameters based on historical data.

Parameters are usually regarded as constants in time, because of the general idea that catchment conditions are temporally stable. Constant parameters become inaccurate in differential split-sample test (DSST) conditions (Klemes, 1986). For example, parameters calibrated based on data from a wet (or dry) period may fail to simulate runoff in a dry (or wet) period for the same catchment. Boderick et al. (2016) used DSST to assess the transferability of six conceptual models under contrasting climate conditions. They found that performance declines most when models are calibrated during wet periods but validated in dry ones. Fowler et al. (2016) pointed out that the parameter set obtained by mathematical optimization based on wet periods may not be robust when applied in dry periods. Additionally, the catchment properties can change over time, such as in the case of afforestation and deforestation (Siriwardena et al., 2006; Guzha et al., 2018). These changes need to be taken into account through model parameters (Bronstert, 2004; Hundecha and Bardossy, 2004). Hence, temporal variations in parameters should reflect the changing environment.

One challenge here is the methodology used to identify time-varying parameters. In the literature, three approaches have been discussed. The first is split-sample calibration (SSC), whereby available data are split into a moderate number of sub-periods and the parameters are calibrated individually for each period (Thirel et al., 2015). The second method is data assimilation (Pathiraja et al., 2018; Deng et al., 2016). This method assimilates observational data to enable errors, states, and parameters to be updated (Li et al., 2013), making it possible to identify time-varying parameters. The third approach is to construct a functional form or empirical equation according to the correlation between parameters and some climatic variates such as precipitation and potential evapotranspiration (Jeremiah et al., 2013; Westra et al., 2014; Deng et al., 2019). Note that this study focuses on methods to identify time-varying parameters rather than modelling them; hence, only comparisons between SSC and data assimilation are discussed.

SSC is the most commonly used method (Paik et al., 2005; Coron et al., 2012; Fowler et al., 2018; Xie et al., 2018). Merz et al. (2011) investigated the time stability of parameters by estimating six parameter sets based on six consecutive 5-year periods. Lan et al. (2018) clustered calibration data into 24 sub-annual periods to detect the seasonal hydrological dynamic behaviour. Despite broad application, it remains debatable whether a particular mathematical optimum gives the parameter value during one period. Many equivalent optima can exist simultaneously for one dataset when calibrating the model against observations (Poulin et al., 2011). Several studies addressed this question by adding more constraints to the objective function over the respective period. For example, Gharari et al. (2013) emphasized consistent performance in different climatic conditions, while Xie et al. (2018) modified SSC by selecting parameters with good simulation ability for both the current sub-period and the whole period. Some conceptual hydrological parameters reflect the catchment characteristics. When climate change and human activities occur, watershed characteristics, such as soil water storage capacity, are difficult to change dramatically in a very short time. Hence, parameter continuity, defined as differences between the parameters in consecutive time steps to be small, is required for hydrological modelling. However, few reports have considered the continuity of parameters in the SSC method.

This assumption of parameter continuity is the basic idea behind data assimilation methods. For example, the a priori parameters in ensemble Kalman filter (EnKF) methods are commonly derived from updated values from the previous time step (Xiong et al., 2019; Moradkhani et al., 2005). From this, a trade-off between simulation accuracy and parameter continuity is established, and parameters that enable greater continuity are more likely to be selected. Deng et al. (2016) validated the ability of the EnKF to identify changes in two-parameter monthly water balance (TMWB) model parameters. Pathiraja et al. (2016) proposed two-parameter evolution models for improving conventional dual EnKF and obtained superior results for diagnosing the non-stationarity in a system. EnKF and its variants are relatively advanced approaches for identifying time-varying parameters (Lu et al., 2013). However, for a hydrological model, the states may change over every time step, whereas the parameters may not, in particular for hourly timescales. This can be offset by SSC, which assumes that the parameters remain stable for a pre-determined period (such as decades, years, or months). Compared to EnKF, the simplicity of SSC is another advantage, as it has a less complex mechanism and reduced redundancy (Chen and Zhang, 2006).

The aim of this study is to present a new method for time-varying parameter estimation by combining the strengths of the basic concept of SSC and the continuity assumption of data assimilation, which is a useful tool for diagnosing the non-stationarity caused by a changing environment. Compared with data assimilation, the proposed split-sample calibration based on dynamic programming (SSC-DP) avoids overly frequent changes of parameters, such as hourly or daily variations. Compared with SSC, the distinctive element is that SSC-DP considers the parameters to be related over adjacent sub-periods and selects parameter sets with good performance for each period and small differences between adjacent time steps. In this study, three aspects of the proposed method are evaluated: (1) the performance of SSC-DP is compared with that of existing methods in terms of the estimation of time-varying parameters; (2) the applicability of SSC-DP to more complex hydrological models with a considerable number of parameters; (3) the ability of SSC-DP to provide additional insights on parameter variations and their correlations with the properties of real catchments. To investigate the above issues, the proposed method is compared with SSC and EnKF in two synthetic experiments (one with a two-parameter monthly model, the other with a 15-parameter daily model). SSC-DP is also applied to two real catchments for parameter estimation under different environmental conditions.

The remainder of this paper is organized as follows. Section 2 describes the proposed method, reference methods, and performance evaluation indices. Section 3 describes two synthetic experiments and two real catchment case studies for comparison among different time-varying parameter estimation methods. Sections 4 and 5 present the results and discussion, respectively, before the conclusions to this study are drawn in Sect. 6.

Flow chart of the methodologies.

In this section, a SSC-DP method is proposed to identify the time-varying parameters of hydrological models. The two hydrological models considered in this study are the TMWB and Xinanjiang models. Their concepts and differences are presented in Sect. 2.1. A sensitivity analysis is employed to focus efforts on parameters important to calibration and avoid prohibitive computational cost, as outlined in Sect. 2.2. Three time-varying parameter estimation methods (SSC, SSC-DP, and data assimilation) are presented in Sect. 2.3. The SSC and data assimilation are provided for comparisons with the SSC-DP. Finally, to evaluate the performance of the time-varying parameter estimation methods, six evaluation criteria are selected and formulated in Sect. 2.4. The flow chart of the methodologies is shown in Fig. 1.

Parameters of the TMWB model.

The TMWB model developed by Xiong and Guo (1999) is efficient for monthly runoff simulations and forecasts (Kim et al., 2016; Dai et al., 2018; Yang
et al., 2017; Guo et al., 2002). The model requires monthly precipitation and potential evapotranspiration as inputs. Its simplicity and efficiency of
performance mean that TMWB can easily be used to investigate the impacts of climate change (Deng et al., 2016; Luo et al., 2019). Its outputs include
monthly streamflow, actual evapotranspiration, and the soil moisture content index. The model has only two parameters (Table 1),

Flow chart of the Xinanjiang model.

Parameters of the Xinanjiang model.

The Xinanjiang model (Zhao, 1992) is widely used in China (Yin et al., 2018; Si et al., 2015; Li and Zhang, 2017). It takes precipitation and pan-evaporation data as inputs and estimates the actual evapotranspiration, soil moisture storage, surface runoff, interflow, and groundwater runoff from the watershed. The simulated streamflow is calculated by summing the routing results of the surface, interflow, and groundwater runoff (Sun et al., 2018). In this study, the surface runoff is routed by the instantaneous unit hydrograph (Lin et al., 2014), while the interflow and groundwater runoff are routed by the linear reservoir method (Jayawardena and Zhou, 2000). A schematic overview of the model is presented in Fig. 2. The meaning, range, and units of all the parameters in the Xinanjiang model are listed in Table 2.

There are two important differences between the TMWB and Xinanjiang models: (1) the TMWB model has two parameters, while the Xinanjiang model has 15 parameters; (2) TMWB is a monthly rainfall–runoff model, whereas the Xinanjiang model can run on hourly or daily step sizes.

Sensitivity analysis is used to identify which parameters significantly affect the performance of the Xinanjiang model and reduce the number of parameters to be calibrated. Numerous sensitivity analysis methods are available, such as the Morris method (Morris, 1991) and Sobol analysis (Sobol, 1993). The Morris method provides similar results to Sobol analysis with a reduced computational burden (Rebolho et al., 2018; Teweldebrhan et al., 2018; Yang et al., 2018).

The Morris method assumes that if parameters change by the same relative amount, the parameter that causes the larger elementary effect is the more
sensitive (King and Perera, 2013). The elementary effect is calculated as follows:

Each parameter is changed in turn and every parameter set produces an elementary effect. The parameter sensitivity is evaluated using the mean
value

Flow chart of SSC-DP.

SSC provides a simple way of diagnosing parameter non-stationarity under a changing environment (Merz et al., 2011). As illustrated in Fig. 3a, the method usually has two steps (Kim et al., 2015; Hughes, 2015): (1) available data are divided into several consecutive periods, which can be arbitrarily chosen as hours, days, months, seasons, or years; (2) parameters are calibrated separately for the respective period. This procedure gives better simulation performance than using constant parameters but leads to the estimated parameters fluctuating strongly over adjacent sub-periods, producing false temporal variants.

To overcome this problem, the SSC-DP method identifies time-varying parameters with consideration of temporal continuity. SSC-DP has five steps
(Fig. 3b):

As the decision-making process during the current sub-period is related to that of the previous sub-period, the parameter estimation over

Another approach for diagnosing variations in parameters is data assimilation, using methods such as the EnKF and ensemble Kalman smoother (EnKS). These are used here as reference methods. The EnKF has been widely applied to conceptual models, including TMWB (Deng et al., 2016). Li et al. (2013) noted that the EnKF struggles to handle the time-lag in routing processes. However, the routing component is vital to the Xinanjiang model. EnKS can efficiently determine the states of the Xinanjiang model (Meng et al., 2017), but the estimation of routing parameters deserves discussion. Most previous studies have used a fixed distribution of the routing hydrograph in data assimilation (Lu et al., 2013); i.e. the parameters are constant for routing processes. With respect to these issues, a modified EnKF (named SSC-EnKF) is established as a third data assimilation reference method in the synthetic experiment with the Xinanjiang model (described in Sect. 3.1).

The EnKF includes two main steps: model prediction and assimilation. The state vector is augmented with parameter variables so that time-varying
parameters can be estimated simultaneously with model states. For model prediction, the augmented vector is derived by adding noise on that from the
previous time step through the following equation:

In the assimilation process, the augmented vector is updated using the following equations if suitable observations are available:

The EnKS is based on the EnKF. Whereas the EnKF updates the model states and parameters at the current time step, the EnKS takes account of those
values over the past time steps. The main steps of the EnKS are identical to those of the EnKF, but the equation of the assimilation process is
formulated as follows:

A third data assimilation approach is constructed based on the SSC. Instead of assimilating one observed variable, it assimilates the observed
variables during a given period in one assimilation process. Assuming that the parameters are constant in the given period, the equation of the
assimilation process for the

This approach addresses the routing-lag issue by allowing parameters of the routing processes, such as the instantaneous unit hydrograph, to remain constant for each sub-period and to be time-varying over the whole period.

The streamflow simulations given by the proposed method are verified using the NSE, relative error (RE), and NSE on logarithm of streamflow
(

Different cases of synthetic experiments and real catchment case studies for comparison and evaluation.

Two synthetic experiments and two real catchment case studies were designed to assess the performance of SSC-DP. The experiments are described in Table 3.

The two synthetic experiments examine the ability of SSC-DP to identify the time-varying parameters of the TMWB and Xinanjiang hydrological models. The merit of synthetic experiments is that the parameters can be synthetically generated to be either constant or time-varying. Hence, it is convenient to compare the estimated values with the pre-determined parameters to evaluate different parameter estimation methods. Note that synthetic experiments have been successfully used in several time-varying parameter identification studies (Deng et al., 2016; Pathiraja et al., 2016; Xiong et al., 2019).

Synthetic data of monthly precipitation and potential evapotranspiration were collected from the 03451500 catchment of the Model Parameter Estimation Experiment (MOPEX) (Duan et al., 2006). The data cover 252 months. Runoff was derived by the TMWB model using synthetic precipitation, potential evapotranspiration, and the pre-determined parameters. Gaussian noise was added to the simulated runoff to represent uncertainties. The mean of the noise was set to zero, and the SD was assumed to be 3 % of the magnitude of the values (Deng et al., 2016).

True parameters of different scenarios in the synthetic experiment with the TMWB model.

Eight scenarios with different pre-determined parameters were investigated (Table 4). The first scenario considered constant parameters. Scenarios 2
and 3 considered month-by-month variations in TMWB model parameters; i.e. the parameters remain constant during each month but change from month to
month. Scenarios 4 and 5 considered parameters that change every 6 months. Scenarios 6–8 considered year-by-year variations. The changes in both

Results of the Morris method for the synthetic experiment with the Xinanjiang model. The sensitivity analysis is based on three different kinds of model responses:

Hourly precipitation and pan evaporation data were collected from the Baiyunshan Reservoir basin in China. The data cover a period of 18 000

True parameters of different scenarios in the synthetic experiment with the Xinanjiang model

Similar to the experiment with the TMWB model, the synthetic runoff was derived from the Xinanjiang model with added Gaussian noise. The mean of the noise was set to zero, and the SD was assumed to be 5 % of the magnitude of the values. As presented in Table 5, all 15 parameters were set to be constant in the first scenario. The pre-determined sensitive parameters were considered to vary with a certain trend and periodicity in scenarios 2 and 3, respectively. Scenario 4 considered a combined variation of trend and periodicity for the parameter KE, with the other free parameters set to vary linearly. The parameter variations in scenarios 2–4 were assumed to occur once a month.

Location of

The Wuding River basin (Fig. 5a) examined in the first case study is a large sub-basin of the Yellow River basin located on the Loess Plateau (Xu, 2011). The Wuding River has a drainage area of 30 261

Soil and water conservation measures, such as the construction of the check dams and afforestation, have been undertaken since the 1960s. The areas of two soil and water conservation measures are plotted in Fig. 5e, the data of which were collected from Zhang et al. (2002). The areas of tree planting have an increasing trend, but the slope gets much larger after 1972. It indicates that greater efforts have been made for afforestation since the turning point. Similarly, the areas of dammed lands also increase, but the rate gets slower after 1972. These two soil and water conservation measures had changed the underlying surface of the watershed and impacted the relationship between precipitation and runoff (Jiao et al., 2017; Gao et al., 2017).

The proposed method was also applied to the Xun River basin, China (Fig. 5b). Located between 108

It can be observed from Fig. 5d that no trend is found in annual precipitation, pan evaporation, and streamflow, suggesting that the relationship between precipitation and runoff of the Xun River basin is rarely affected by human activities during 1991–2001. However, strong seasonal patterns are exhibited in these three climatic and hydrological variables, suggesting that seasonal variations in hydrological parameters should be considered.

Comparison between the EnKF and SSC-DP methods for

When using SSC-DP, the first task is to define how the hydrological data series should be split into the

Comparison among different methods for

Figure 6a presents the runoff simulation performance for various scenarios. In scenario 1, the NSE values of the three SSC-DP methods are all higher
than that of EnKF. The results of

Figure 6b and c focuses on the ability of the four methods to identify time-varying parameters. It can be seen that the RMSE and MARE values of the 3-SSC-DP are larger than those of other methods in most cases. That is because the sub-period length that serves as a calibration period for MCMC is so short (i.e. 3 months) that the estimated parameters are associated with higher uncertainties.

Regarding the synthetic true parameters having constant values (scenario 1), 12-SSC-DP gives the best performance with the lowest RMSE and MARE and highest

When the synthetic true parameters vary linearly (scenarios 2, 4, and 6), 12-SSC-DP produces the best estimations in comparison with EnKF, 3-SSC-DP, and 6-SSC-DP. The performances of 6-SSC-DP and EnKF are similar.

When the synthetic true parameters vary sinusoidally from month to month, EnKF gives the best estimations in scenario 3. The poor performances of
6-SSC-DP and 12-SSC-DP can be explained by the sub-period length being much longer than the actual one. When the parameters vary periodically at
6-month intervals (scenario 5), 6-SSC-DP yields the best performance with the lowest RMSE and MARE and highest

The Xinanjiang model is more complex than TMWB, and so some sensitivity analysis is necessary. As stated in Sect. 3.1.2, the sensitive parameters are KE, CI, CG, KI, KG, and NK. The 18 000 hourly hydrological data points were divided into 25 sub-periods (monthly timescale) and 12 sub-periods (bimonthly timescale). It is considered that a monthly timescale helps diagnose seasonal variations, whereas a 2-month timescale provides data for longer calibration lengths.

Comparison among EnKF, SSC-EnKF, and EnKS in the synthetic experiment with the Xinanjiang model.

Three data assimilation methods (see Sect. 2.3.2 for details) were applied to the synthetic data: (1) EnKF, (2) EnKS, and (3) SSC-EnKF. The results in Fig. 8 indicate that EnKS is superior to EnKF, as previously observed (Li et al., 2013), although SSC-EnKF gives the best results. This is probably because SSC-EnKF is based on the assumption that the parameters remain constant during each sub-period.

The simulated streamflow and identification of time-varying parameters were compared across four methods: 1-SSC, SSC-EnKF, 1-SSC-DP, and 2-SSC-DP. The
simulation performance is summarized in Fig. 9a. For all scenarios, the NSE of 2-SSC-DP is the lowest, but it performs better for low flows. The
SSC-EnKF produces the highest RE in scenarios 2, 3, and 4, indicating the problem of simulating water balance. The SSC and 1-SSC-DP perform well for
all scenarios in terms of NSE, RE, and

Comparison among the SSC, SSC-EnKF, and SSC-DP methods in the synthetic experiment with the Xinanjiang model for

Figure 9b and c compares the time-varying parameter estimation performance among the four methods. In scenarios 1 and 2, 2-SSC-DP produces the lowest
RMSE, MARE, and

Comparison between estimated parameters and their true values for scenario 3 of the synthetic experiment with the Xinanjiang model.

When the synthetic true parameters vary sinusoidally from month to month (scenario 3), the estimated parameters are plotted in Fig. 10. It can be seen
that 1-SSC-DP successfully detects a seasonal signal in every parameter. The SSC-EnKF performs well for

Double mass curves between daily runoff and precipitation for

Simulation performance for streamflow in the Wuding River basin. The results of NSE and NSE

Figure 11a and b show the double mass curves between daily runoff and precipitation for the Wuding River basin. Similar to the work of Deng
et al. (2016), the two linear slopes (

The simulation results given by 12-SSC-DP were benchmarked against those from 12-SSC, data assimilation, and the conventional method in which all
Xinanjiang model parameters remain constant. The simulation performance is presented in Fig. 12. The values of the NSEs are relatively low, because
the streamflow in dry regions is difficult to simulate. It can be seen that the 12-SSC-DP gives the best simulation results among different methods
with the highest NSE and

Although the objective function of 12-SSC-DP considers the trade-off between simulation accuracy and parameter continuity, 12-SSC-DP gives a higher
NSE value. This may be because 12-SSC locates a local peak over one sub-period, resulting in unreasonable model states for the beginning of the next
sub-period, whereas 12-SSC-DP uses dynamic programming to explore more reasonable parameter values and model states. Figure 13 shows the
quantile–quantile plots, from which it can be seen that if the parameters are assumed to be constant, streamflow is highly underestimated. The
underestimation mainly derives from the deficiencies of the model structure. Methods 12-SSC and 12-SSC-DP reduce this underestimation by using
time-varying parameters. Additionally, 12-SSC-DP is slightly inferior to 12-SSC in terms of peak flows but is superior in terms of simulating
streamflow lower than 100

The estimated time-varying parameters estimated by 12-SSC-DP are plotted in Fig. 14. The results show that WM remains constant before and
after 1972, but WUM varies significantly over this period, indicating that the distribution of soil water capacity may change, i.e.
WUM decreases but WLM increases. A Person correlation analysis is applied to investigate the relationship between the areas of tree
planting and WUM as well as WLM. It is found that there is a significant negative correlation (Pearson correlation efficient

The simulated and observed streamflow using the conventional method, SSC-EnKF, SSC, and SSC-DP for the Wuding River basin.

Estimated sensitive parameters of the Xinanjiang model for the Wuding River basin. The blue and orange solid lines represent the estimated parameters pre- and post-1972, respectively.

Figure 11c and d show the double mass curves between runoff and precipitation for the Xun River basin. The linear slope of the curve is generally
stationary for the whole 10-year period shown in Fig. 11c, with a correlation coefficient of 99.6 %. In contrast, the linear slope for an
intra-annual timescale is non-stationary (Fig. 11d). Based on these results, it can be inferred that the relationship between precipitation and runoff
is stable from 1990–2000 but varies over the intra-annual timescale. Hence, sub-periods of 3 and 12 months were examined in the Xun
River basin using models 3-SSC-DP and 12-SSC-DP. From the Xun River basin data from 1991–2000, sensitivity analysis suggested that five parameters of
the Xinanjiang model are relatively sensitive, namely KE,

Simulation performance for streamflow in the Xun River basin. The results of NSE and NSE

The simulated and observed streamflow using the conventional method, SSC-EnKF, SSC, and SSC-DP for the Xun River basin.

Similar to the case study of the Wuding River basin, the simulation performance of 3-SSC-DP was benchmarked against that of 3-SSC, data assimilation,
and the conventional calibration method. Among the data assimilation methods described in Sect. 2.3.2, 3-SSC-EnKF gives the highest simulation
accuracy. The simulation performance is presented in Fig. 15. All methods performed well, with NSE values of 92.5 %, 93.0 %, 95.0 %, and
94.8 % for the conventional method, 3-SSC-EnKF, 3-SSC, and 3-SSC-DP, respectively. 3-SSC and 3-SSC-DP also perform well for

Estimated sensitive parameters of the Xinanjiang model for the Xun River basin over

The estimated parameters using 3-SSC-DP are presented in Fig. 17a. Some parameters vary significantly over an intra-annual timescale. Among them, the
parameter KE, representing the ratio of potential evapotranspiration to pan evaporation, exhibits the most distinct seasonal variations. A
fast Fourier transform was used to calculate the spectral power of the KE time series to explore its periodic characteristics. As can be
observed from Fig. 17b, 3-SSC-DP had the greatest spectral power, for a period of 4.0

As noted in the methodology section, the performance of the proposed method is influenced by several factors, such as the weights in the objective function and the choice of lengths. Some suggestions regarding the improvement of the proposed approach are now discussed in detail.

In the conventional method, a parameter set is identified as optimal for providing the best simulation over the calibration period. However, other
parameter sets with slightly worse (but still good) performance can also be candidates. Allowing for input data uncertainty and local optima, SSC-DP
identifies parameter sets that perform near-optimally and display fewer fluctuations over sub-periods. This can be adjusted by weights in the
objective function of the dynamic programming approach (see Eq.

Correlation efficiency results of SSC-DP using different weights of parameter continuity for synthetic experiments with

Figure 18a shows the

As mentioned by Gharari et al. (2013), there are different ways of determining the sub-period lengths. The sub-periods can be non-continuous
hydrological years (Seiller et al., 2012), months or seasons (Deng et al., 2018; Paik et al., 2005), and discharge or precipitation events (Singh and
Bardossy, 2012). This introduces a controversial issue whereby parameters are impacted by the length of the calibration period. Merz et al. (2009)
suggested that 3–5 years is an acceptable calibration period, whereas Singh and Bardossy (2012) indicated that a small number of events may be
sufficient for parameter identification. It is suggested that the determination of the sub-period length considers three factors:

However, many studies are based on the conventional assumption that the parameters of different sub-periods are independent. Hence, the sub-period lengths should be long enough to reduce the degree of uncertainty. In this study, the assumption of parameter continuity is introduced to give another constraint that considers correlations between parameters of adjacent sub-periods. It appears that the determination of sub-period lengths deserves further investigation.

This paper has described a time-varying parameter estimation approach based on dynamic programming. The proposed SSC-DP combines the basic concept of
SSC and the continuity assumption of data assimilation to estimate more continuous parameters while providing comparably good streamflow
simulations. Two synthetic experiments were designed to evaluate its applicability and efficiency for time-varying parameter
identification. Furthermore, two case studies were conducted to explore the advantages of SSC-DP in real catchments. From the results, the following
conclusions can be drawn:

The proposed method with a suitable length not only produces better simulation performance, but also ensures more accurate parameter estimates than SSC and EnKF in the synthetic experiment using the TMWB model with two parameters. The impact of sub-period lengths on the performance of SSC-DP is significant when the pre-determined parameters vary sinusoidally.

The proposed method can be used to deal with complex hydrological models involving a large number of parameters, demonstrated by the synthetic experiment using the Xinanjiang model with 15 parameters. A sensitivity analysis was performed to reduce the probable computational cost and improve the efficiency of identifying the time-varying parameters.

The proposed method has the potential to detect the relationship between the time-varying parameters and dynamic catchment characteristics. For example, SSC-DP produces the best simulation performance in the case study of the Wuding River basin and detects that parameters representing soil water capacity and impervious areas changed significantly after 1972, reflecting the soil and water conservation projects carried out from 1958–2000. Additionally, SSC-DP detects the strongest seasonal signal in the case study of Xun River basin, indicating the distinct impacts of seasonal climate variability.

This study has demonstrated that the proposed method is an effective approach for identifying time-varying parameters under changing environments. Further work is still needed, such as to determine an objective method for choosing the sub-period lengths.

The data and code that support the findings of this study are available from the corresponding author upon request.

All of the authors helped to develop the method, designed the experiments, analysed the results, and wrote the paper.

The authors declare that they have no conflict of interest.

This study was supported by the Innovation Team in Key Field of the Ministry of Science and Technology (2018RA4014), the National Natural Science Foundation of China (51861125102), and Natural Science Foundation of Hubei Province (2017CFA015). The authors would like to thank the editor and anonymous reviewers for their comments that helped improve the quality of the paper.

This research has been supported by the Innovation Team in Key Field of the Ministry of Science and Technology (grant no. 2018RA4014) and the National Natural Science Foundation of China (grant no. 51861125102).

This paper was edited by Dimitri Solomatine and reviewed by two anonymous referees.