Hydrological modelling under climate change considering nonstationarity and seasonal effects

The impacts of climate change are of increasing interest to water resources managers (Compagnucci et al. 2001; Bates et al. 2008). The uncertainty of the impacts of climate change may arise from numerous factors such as Global Circulation Model (GCM), downscaling method, the structure and parameters of hydrological model and emission scenarios (Wilby & Harris 2006). Numerous researches argue that GCM structure is the largest source of uncertainty and hydrological model parameterisation is almost the last (Wilby & Harris 2006; Arnell 2011; Chen et al. 2011; Teng et al. 2012). However, some studies indicate that the rank of hydrological modelling may depend on the type of hydrological model used and the study catchment (Wilby 2005; Blöschl et al. 2007; Blöschl & Montanari 2010) which shows that hydrological modelling should not be disregarded as insignificant in climate change impact analyses.

Generally, ensembles of these sources are used to quantify and reduce the uncertainty of the impact of climate change (Minville et al. 2008; Chiew et al. 2009; Boyer et al. 2010; Vaze et al. 2010). One assumption of these studies is that the catchment does not change with time (i.e. stationary conditions) which means the model calibrated for the historical period is valid for the future period. The premise of this assumption is that if the model is calibrated for a long time period of observation data then these calibrated parameters can be assumed to be still effective for the future climate conditions (Arnell 1994). However, in reality, due to changes of climate and catchment conditions, this stationarity assumption may not be valid in the future. Therefore, the model should be reliably calibrated under current climate conditions in order to estimate the impact of climate change on future hydrological system. However, it is a challenge to make the hydrological model adaptive to the future climate and catchment conditions that are not observable at the present time.

The main sources of uncertainty in the hydrological modelling are model structural errors and parameterisation errors. The uncertainty of model structure errors is generally quantified by using several different models, and numerous methods are proposed regarding quantification of the uncertainty of parameterisation problem. The time varying parameters which arise from climate change and catchment change (such as land use/cover change) may be another source of uncertainty in climate change impact studies. Recently, there have been some studies about the stability of the model performances and the effect of parameter values (Xu 1999; Li et al. 2014; Yan & Zhang 2014; Patel & Rahman 2015). The reasons of time varying model parameters can be explained by several reasons (Merz et al. 2011). First, the hydrological model has structure errors and the calibrated parameters may change for different time periods in order to compensate these problems with the model structures (Wagener et al. 2003). Secondly, catchment characteristic change (Brown et al. 2005) such as land use and climate variations (Merz & Blöschl 2009) can also lead to the change of calibrated parameters. However, the correlation between parameters is complicated and may be related with catchment conditions (Wagener 2007) which make it hard to understand the reason of the parameter changes in time (Wagener et al. 2010).

The purpose of this study is to assess the validity of the assumption of hydrological stationarity and to improve the traditional time invariant model parameterisation for nonstationary hydrological system. Catchment change, such as land use/cover change, may be a source for the temporal change of the model parameters but it is not taken into account in this study due to difficulties in obtaining the data. We only consider the relationships between the trend of parameters and climate conditions (assuming that the climate change could be used as proxies for catchment changes such as vegetation change). Long observation data from 1961 to 2008 are used and seasonal calibration (in this study only the summer period is further explored because it is more sensitive to climate and land cover change than the other three seasons) has been done since there are significant seasonal rainfall patterns. The data are split into calibration and validation periods with the intention of using the validation period to represent the future unobserved situations. The performance of three different models, Static model, Nonstationary model and Stationary model, are compared with the calibrated model. The calibration has been conducted with the use of Nash–Sutcliffe efficiency (NSE) to minimise the difference between observed and simulated flow for the summer period and the optimised parameters have been tested in the validation period.

To explore modelling of the nonstationary hydrological system we propose two nonstationary parameterisation schemes. The idea is that the model parameters are changing with time and if we can find some parameter trends against time or some meaningful correlation between parameters and weather variables, this nonstationary model performance might be better in the future than the static model assuming that the catchment does not change with time (i.e. stationary conditions). The first model is adapting the parameters using the step forward and backward selection schemes and the second model is optimising only one parameter while the other parameters are set as the fixed which are described in the following sections.

The calibration has been done every consecutive 10-year period by moving the window one year from 1961 to 1990; hence we get 21 data points, each data representing every 10 years (i.e. the first and the last data points represent 1961–1970 and 1981–1990 respectively). Since some model parameters show trends over 30 years and some do not, the idea of a new calibration method is to constrain the model parameters one by one and calibrate the model step by step. We assumed the trend to be linear and the statistically significant level is set at 5% for detecting the trend. Here we propose the two new calibration methods, Forward Stepwise Method (FSM) and Backward Stepwise Method (BSM). FSM is to start the calibration by setting a parameter as a fixed value, whose p-value is the largest among the other parameters that do not show any trend (i.e. with the first 10-year value) and the rest parameters are set free during optimisation. We repeat this calibrating process step by step until all the rest of the parameters show trends in time. Next, when we encounter a situation where all the parameters show trends in mid process, we choose the parameter which has the lowest P-value and fix this linear regression equation followed by calibrating the rest of the free parameters for optimisation. As we have eight parameters, calibration has carried out seven times by incrementing one fixed (either constant or keeping the trend) parameter each step. Then finally we get all the parameters optimised, and with each step the parameters are optimised under previously constrained parameters.

The FSM process is as follows:

1. Calibrate the model in every 10-year window from 1961–1970 to 1981–1990.

2. Test each parameter if it has a trend over the calibration period.

3. Among the parameters which do not have trends, choose the parameter that has the least trend (i.e. the largest P-value).

4. Fix this parameter for all the calibration periods while the other parameters are set free and do the optimisation.

5. Go to step 2 and repeat the processes until all parameters are constrained.

6. During the process when the parameters which do not show any trends are all constrained and only the parameters that have trends are left, then choose the parameter that has the strongest trend (i.e. the smallest P-value) and fix that trend (i.e. linear regression equation) for the calibration period, then do the calibration.

7. Repeat the processes until all parameters are constrained.

For example, Table 2 represents calibration setting and optimisation results for the FSM for the summer period from 1961 to 1990. First, the Parameter l is set as a fixed value for all the calibration period in Step 1 since l in the calibration stage has the highest P-value among the other parameters. The other seven parameters are set free and optimisation has been done. Next, Steps 2–5 have been carried out in the same way. In Step 5, optimisation result shows that all three optimised parameters show significant trends along time, among which f shows the lowest P-value. Hence, in Step 6, the trend of f along time is fixed, i.e. f value of each calibration period is calculated from a linear regression equation. Then optimisation has been done for the rest two parameters, c and β_q. Step 7 has been done in the same way. Finally, for this nonstationary model, five parameters are set as fixed values and three parameters are set to have trends along time. This nonstationary model is compared with the other models described in the following sections.

The BSM process is the same as FSM except starting to constrain the parameters that show the strongest temporal behaviour (i.e. the smallest p-value) instead of constraining the parameter that does not show any trend along time. Table 3 represents the calibration setting and optimisation results for the BSM for the summer period from 1961 to 1990. For this nonstationary model, two parameters are set as a fixed value and six parameters are set to have trends along time.

An alternative parameterisation scheme for the nonstationary system is to select only one parameter for optimisation while the other parameters are set as fixed. In other words, the stationarity assumption is valid for all parameters except one. The reason why we adopt this method is due to the issue of equifinality in modelling complex environmental system. The concept of equifinality is that many different parameters of the model may reproduce the observed behaviour of the system which are acceptable (Beven & Freer 2001). In other words, similar model performances can be achieved with different sets of parameters. Therefore, in parameterisation of a hydrological model, detecting the change of parameters with time may be a difficult problem. This is in part due to the interdependency of parameters. Since the parameters may be linked with each other to a certain degree, setting all parameters free in the optimisation process may lead to difficulty in detecting the trend of nonstationary parameters. Hence, we constrain all the parameters except one to resolve this issue. The success of this model depends on whether the optimised one parameter shows time stability or strong correlation with climate variables, given the fixed rest parameters. This is because, in this condition, the model parameter can be stably predicted with changing time and changing climate in the future. For the fixed parameter values, the first 10-year period (1961–1970) calibrated parameters are applied. The one varying parameter is selected that has a strong relationship with climate variables or shows a statistically significant linear trend along time.

The poor performance of the nonstationary model in the validation period means that both the regression of parameters with time and weather variables is not reliable since the trend of parameters in the calibration period may not be in a monotonic linear relationship. In other words, the relationship that expresses the trend of parameter in the calibration period cannot capture the relationship between the parameter and time or parameter and climate variables in the validation period for this catchment and input data. For example, the temporal distribution of the temperature modulation of drying rate f shows quite a linear relationship during the calibration period (the data point from 1961 to 1981 in Figure 6); however, the trend tends to be opposite afterwards. Another issue is that changing multiple parameters may not make every parameter optimised since they are interdependent and may be unnecessarily correlated with each other (e.g. their effects could offset each other which result in equifinality).

We cannot assume that the trend of parameter τ_w in Figure 10 may appear in the far future as well. However, it can be justified to extrapolate the trend to the future since the observation data are quite long (1961–2008, 47 years) and the trend is stable.

In this study, the trend of the hydrological model parameters are found and extrapolated in order to adapt to the future unobserved situations by functionally relating them with time or climate variables. However, the obstacle of this approach is that different parameter sets can produce a similar model performance which is known as parameter equifinality and this may result in large uncertainty in prediction (Beven 1993; Niel et al. 2003; Wilby 2005; Minville et al. 2008). These diverse sets of possible parameters may lead to different results when they are applied to assess the impacts of climate change on flow (Uhlenbrook et al. 1999) and make it difficult to find the trend of parameters. This may be one likely reason for unsatisfactory performance of the stepwise calibrated models in the validation period. Another reason can be when the parameters are extrapolated just in time or in functionally related climate variables, the estimated parameter set may not be an optimised one since there are complex correlations among them. It is not straightforward to understand the temporal change of model parameters (Wagener et al. 2010). Therefore, we propose a calibrating scheme that optimises only one parameter which shows an apparent trend and has strong correlation with climate variables.

There are some possible research areas to be explored further. First, although the proposed methodology, which is adjusting one parameter against time and weather variable, works well for this catchment it does not assure the same result for different catchments and different climate conditions. Therefore, further research is needed to explore the proposed scheme in different catchments in order to find out whether the trend of model parameters and climate variables show consistent spatial correlations which may add credibility to our proposed method. Second, in this study, the catchment conditions are not considered. However, the parameter trends in time may be related to the changes of catchment characteristics and should be explored further. A possible approach to this issue could be based on the Normalized Difference Vegetation Index (NDVI). Third, for decision making regarding the impact of climate change on water resources, water allocation model should be used to see the difference of various adaptation options (e.g. how much larger the reservoir volume should be expanded?).

The main findings of this paper are as follows: The temporal trends of summer precipitation and runoff are apparently decreasing while the air temperature tends to increase. Some model parameters such as the reference drying rate τ_w, the slow flow recession time constant τ_s and the relative volume of slow flow to total flow v_s show clear trends in time when calibrated every 10-year period from 1961–1970 to 1999–2008 for this catchment. We have proposed two parameterisation schemes when conceptual hydrological model is used to assess the impact of climate change. The first method is to adapt the parameters using the step forward and backward selection schemes. However, in the validation period, both the forward and backward multiple parameter changing models do not show much improvement compared with the model which uses time invariant parameters (i.e. static model). One problem is that the regression with time is not reliable since the trend may not be in a monotonic linear relationship with time. The second issue is that changing multiple parameters makes the selection process complex which is time consuming and not effective in the validation period. As a result, a new scheme is explored: only one parameter is selected for adjustment while the other parameters are fixed and regressions of parameters are made against climate conditions and time. It has been found that such a new approach is effective and this nonstationary model works well both in the calibration and validation periods. Although the catchment is specific in southwest England and the data are for the summer period only, the methodology proposed in this study is general and applicable to other catchments. We hope this study will stimulate the hydrological community to explore a variety of sites so that valuable experiences and knowledge could be gained to improve our understanding of such a complex modelling issue in climate change impact assessment.

The first author is grateful for the financial support from the Government of South Korea for carrying out his PhD study at the University of Bristol. The second author was supported by a grant (15RDRP-B076564-02) from the Regional Development Research Program funded by the Ministry of Land, Infrastructure and Transport of Korean government. The authors acknowledge the UK Met Office for providing the data.