**1. Introduction**

The conceptual hydrological models are simplified representations of the complex rainfall-runoff processes in the real world. Hence, great emphases are placed on the calibration of parameters to improve the performance of hydrological models. The optimal constant parameter set for a catchment can be inferred using various approaches [1,2] under the assumption that both the climate and the catchment characteristics are stationary. However, the ability to produce reliable predictions with the constant parameter set has been challenged under changing climate and watershed conditions [3–8].

The transferability of the model parameters between periods with different climate and watershed conditions is essential for reliable application of the hydrological models. Plenty of previous studies have revealed the fact that model parameters are strongly dependent on the data period with which the hydrological models are calibrated [9–11]. In general, loss of robustness and variation in model performance can be witnessed when the model parameters are inappropriately transferred [12]. For example, a model calibrated under wet conditions is usually unlikely to perform well under dry

conditions and vice versa. In addition, the optimal parameter set may become no longer suitable once the watershed conditions change significantly as a result of human activities (e.g., the construction of dams, afforestation and deforestation). The impact of such changes on hydrologic processes such as streamflow has been well documented [13,14]. In such cases, hydrological models with time-invariant parameters may show a poor predictive performance due to their inability to explicitly represent the changes within the catchment.

One approach to make the model perform robustly under changing environments is to identify the optimal parameter set with time consistent performance [15–22]. Such approach is based on a series of calibration procedures on different sub-periods, and careful identification of the parameter sets with consistent performance for all these sub-periods. All parameter sets are independently evaluated in every sub-period, and only the parameter set that performs well in all cases are finally selected. Nevertheless, this is achieved at the cost of a possible decline in model performance for some individual sub-periods.

Another approach to improve the predictive performance of hydrological models under changing environments is to allow model parameters to evolve over time [23–33]. For example, Wallner and Haberlandt [25] linked the time-varying model parameters of a modified version of the HBV (Hydrologiska Byråns Vattenbalansavdelning) model with the climate indices. The results showed a significant improvement in model performance for streamflow simulation when using the time-varying parameters, especially for low flows. Pathiraja et al. [26] demonstrated the potential for the data assimilation method to detect the temporal variation in all parameters of the Probability Distributed Model and to improve model performance for streamflow simulation. Westra et al. [27] reported a significantly improved streamflow simulation performance of the GR4J model when the parameter represents the production storage capacity was made dependent on covariates describing seasonality, annual variability, and longer-term trends. Deng et al. [28] assumed the parameters of the two-parameter monthly water balance model to be time-varying as functions of catchment properties. Their case study in southern China suggests that the incorporation of time-varying parameters can enhance the capability of hydrological models to produce reliable predictions under changing environments. These studies suggest that the time-varying parameter may serve as an effective compensation for the possible deficiency of model structure, which failed to represent changes in hydrological dynamics within the watershed.

However, few recent studies focus on the temporal transferability of the time-varying parameters (i.e., robustness under extrapolation). Some previous studies [25,27,28] have tested the temporal transferability of the time-varying parameter from only one calibration period to only one validation period. They reported that the streamflow simulation using a model with the time-varying parameter is better than the original model during the validation procedure. However, these studies failed to fully account for the cases where time-varying parameter is transferred between periods with contrasting climate and watershed conditions (e.g., from driest period to wettest period), which remains an interesting topic that deserves further researches.

This study aims to investigate whether a hydrological model with time-varying parameter would achieve a better streamflow simulation performance and parameter transferability under changing environments than its original form. The GR4J model was chosen for its simplicity in model structure and parsimony in model parameters [34]. The relatively sensitive parameters of the GR4J model were treated as time-varying and assumed to be a function of several external covariates. Several sub-periods with different climate and watershed conditions were set up for a series of split-sample test for the GR4J model with constant parameter and time-varying parameter. The investigation was carried out for Weihe Basin and Tuojiang Basin in western China.

#### **2. Methodology**

#### *2.1. The Original GR4J Model and GR4J Model with Time-Varying Parameter*

The original GR4J model has four parameters, namely the production storage capacity *x*1, the groundwater exchange coefficient *x*2, the one day ahead maximum capacity of the routing storage *x*<sup>3</sup> and the time base of unit hydrograph *x*4, as presented in Table 1. More details about the GR4J model can be found in the work of Perrin et al. [34].


**Table 1.** Description of the GR4J model parameters.

A comprehensive sensitivity analysis of all four parameters is needed to determinate which parameter is relatively more sensitive and should be treated as time-varying. To this end, the Sobol' sensitivity analysis method [35] was applied in this study. Considering that this study focuses on changing environments where stationary rainfall-runoff relationship is usually absent, the Sobol' sensitivity indices were computed at the monthly timescale. This temporally discretized sensitivity analysis was expected to provide significant information about model behavior and capture the diversity of parameter sensitivity during different periods [36].

The GR4J model with time-varying parameters is referred to as GR4J-T model hereafter. The selected time-varying parameters are treated as functions of several external covariates related to the alternation of climate or watershed conditions. The potential covariates for time-varying parameters are denoted as *Vj*(*j* = 1, 2, ... , *M*) and their long-term mean values as *Vj*(*j* = 1, 2, ... , *M*), where *M* is the total number of the external covariates. When the parameter *xi*(*i* = 1, 2, 3, 4) of the GR4J model is considered to be time-varying, it is denoted as *xi*,*t*, where *t* indicates the time. The general relationship of *xi*,*<sup>t</sup>* with the corresponding external covariates is assumed to be,

$$\frac{\mathbf{x}\_{i,t} - \mathbf{x}\_{i,c}}{\mathbf{x}\_{i,c}} = \sum\_{j=1}^{M} \beta\_{i,j} f\_j \left( \frac{V\_{j,t} - \overline{V\_j}}{\overline{V\_j}} \right) \tag{1}$$

which can be rewritten as,

$$\mathbf{x}\_{i,t} = \mathbf{x}\_{i,c} + \sum\_{j=1}^{M} \lambda\_{i,j} f\_j \left( \frac{V\_{j,t}}{\overline{V\_j}} - 1 \right) \tag{2}$$

where *xi*,*<sup>c</sup>* stands for the constant value of parameter *xi* over the calibration period, while β*i*,*<sup>j</sup>* and λ*i*,*<sup>j</sup>* (λ*i*,*<sup>j</sup>* = β*i*,*<sup>j</sup>* · *xi*,*c*), *j* = 1, ... , *M*, are the corresponding regression parameters. The *fj*(·) is the link function between the time-varying parameter *xi*,*<sup>t</sup>* and the external covariate *Vj*, which can take multiple forms such as *f*(*z*) = *z* (linear), *f*(*z*) = *ez* (exponential), or *f*(*z*) = ln(*z* + 1) (logarithmic), so as to account for possible linear or non-linear relationship of the time-varying parameter with the external covariates.

Choosing suitable external covariates is a critical issue for the utilization of Equation (1) and (2), but it is usually limited by the availability of dataset. Considering that in practice the daily changes in any external covariates could only cause negligible changes in the time-varying parameters, so only impacts of the monthly changes in any physical covariates on the parameter were investigated in this study, i.e., the time-varying parameters were updated monthly in every simulation run. Inspired

by recent work [27,28], 8 external covariates related to variation of climate or watershed underlying conditions, including 1-month, 2-month and 3-month antecedent monthly precipitation and potential evapotranspiration (*P*1, *P*2, *P*3, *PET*1, *PET*<sup>2</sup> and *PET*3) and *NDVI* (Normalized Difference Vegetation Index) of current month (*NDVI*0), one-month shifted *NDVI* (*NDVI*1) (as shown in Table 2), were considered as candidate covariates for the time-varying parameter.


**Table 2.** Description of external covariates for time-varying parameters.
