1. Introduction

2. Materials and Methods

2.3. XAJ Model and Parameter Description

In this study, we used the Xinanjiang model, a conceptual hydrological model, designed by the Flood Forecast Research Institute of the East Chinese Technical University of Water Resources [33]. This model is utilized for humid and semiarid areas of China to estimate runoff generation within a basin [33,34]. The physical meaning of the model is robust. The simulated XAJ model utilized in this research comprises fifteen parameters as referred in Table 3 [33,35,36]. The complexity of the connections and interactions among the numerous parameters required in calibration can be minimized by the level of parameter estimation.

Here, the estimation of data adjustment parameter, C_ep, is detailed through the aridity index method[37] while the calculation of the recession constant parameter, c_g, employs the time constant, T.

$\varphi -0.34ex0$ denote the pre-optimized parameter vector as in Table 2 .

Let the function specify the XAJ model calibration, X and let the input datasets comprise the vector I^n,m where n is the length of datasets and m is the number of subsets. Let the simulated runoff by the XAJ model calibration for I^n,m be specified by the following equation:

$$Q_{cal}^{n,m}=X(I^{n,m},C_{ep},c_g\mid {\varphi }_0)$$

(1)

where C_ep represents the adjustment parameter (the most sensitive parameter at the annual scale in the XAJ model), c_g is the recession constant (sensitive at the annual scale when the adjustment parameters are kept constant), ${\varphi }_0$ represents the application of 13 pre-optimized parameter values, a subset of all 15 parameters in the XAJ model. These parameters are fixed in this study. As these parameter values are not inferred through the calibration process, ${\varphi }_0$ will be excluded from the following equation for clarity.

$$Q_{cal}^{n,m}=X(I^{n,m},C_{ep},c_g)$$

(2)

Table 2. Descriptive statistics of studied basins

Table 2. Descriptive statistics of studied basins

MOPEX ID	Mean precipitation (mm/year)	Median precipitation (mm/year)	Minimum precipitation (mm/year)	Maximum precipitation (mm/year)	Standard Deviation
903504000	1890	2051.98	1427.09	4424.72	571.450
902387500	1480	1481.05	1046.53	1930.67	228.240
902472000	1492	1550.62	1135.21	4615.13	674.880
903443000	2156	2064.90	1349.55	6646.70	1018.10
911532500	2687	2718.04	1644.96	7172.92	1176.72

2.8. Application of Data Adjustment Parameter and Recession Constant in Model Calibration

The simulated recession constant, c_g,ref, is initially estimated using the linear polynomial regression analysis. The values of c_g,ref were selected based on the c_g values with the best model performance in the longest datasets.

$$NSE(C_{ep}^{28,1},c_{g,ref})\ge NSE(C_{ep}^{28,1},c_g)$$

(11)

where $C_{ep}^{28,1}$ refers to the estimated adjustment parameter value using a 28-year data length.

Table 3. Description of the parameters in the Xinanjiang model.

Table 3. Description of the parameters in the Xinanjiang model.

Parameter	Physical meaning	Range	Pre-optimized values, ϕ0
			MOPEX ID
			903504000	902387500	902472000	903443000	911532500
Group I
Cp	Ratio of measured precipitation to actual precipitation	0.8–1.2	1	1	1	1	1
Cep	Ratio of potential evaporation to pan evaporation	0.8–1.2	0.7908	1.25	1.2806	0.9865	0.7184
Group II
SM	Areal mean free water capacity of the surface soil layer (mm)	1–50	40	30	50	40	30
EX	Areal mean of the free water capacity of the surface soil layer (mm)	0.5–2.5	1.2	0.5	0.5	1.2	0.5
KI	Outflow coefficients of the free water storage to interflow	0–0.7; KI + KG = 0.7	0.1	0.3	0.55	0.1	0.3
KG	Outflow coefficients of the free water storage to groundwater	0–0.7; KI + KG = 0.7	0.6	0.4	0.15	0.6	0.4
cs	Recession constant of the lower interflow storage	0.5–0.9	0.6	0.85	0.75	0.6	0.4
ci	Recession constant for the lower interflow storage	0.5–0.9	0.9	0.75	0.8	0.9	0.75
cg	Recession constant of the groundwater storage	0.9835–0.998	0.98	0.987	0.983	0.98	0.983
Group III
b	Exponent of the tension water capacity curve	0.1–0.3	0.3	0.15	0.15	0.3	0.15
imp	Ratio of the impervious to the total area of the basin	0–0.005	0.02	0.01	0.01	0.02	0.01
WUM	Water capacity in the upper soil layer (mm)	5–20	5–20	20	20	20	20
WLM	Water capacity in the lower soil layer (mm)	60–90	60–90	80	80	80	80
WDM	Water capacity in the deeper soil layer (mm)	10–100	10–100	60	160	160	160
C	Coefficient of deep evapotranspiration	0.1–0.3	0.1–0.3	0.15	0.15	0.15	0.15

Note: ϕ₀ indicates the pre-optimized parameter values.

To highlight the sensitivity impact of the recession constant in shorter datasets, the data adjustment parameter values for subsets are essential to optimize the recession constant values for subsets. To compare the impacts of parameter sensitivity in shorter data length, we first identify the estimated C_ep values for subsets based on three conditions: maximum, minimum and median annual model results, NSE. Let $C_{ep,best}^{n,m}$ , $C_{ep,worst}^{n,m}$ and $C_{ep,median}^{n,m}$ be the maximum, minimum and median $C_{ep}^{n,m}$ values in subsets (I^n,m), when running the model using c_g,ref.

• The maximum, minimum and median annual NSE, can be specified using $C_{ep}^{n,m}$ and c_g,ref in subsets using by the following equations,

$$max\left(NSE(C_{ep}^{n,m},c_{g,ref})\right)=NSE(C_{ep,best}^{n,m},c_{g,ref})$$

(12)

$$min\left(NSE(C_{ep}^{n,m},c_{g,ref})\right)=NSE(C_{ep,worst}^{n,m},c_{g,ref})$$

(13)

$$median\left(NSE(C_{ep}^{n,m},c_{g,ref})\right)=NSE(C_{ep,median}^{n,m},c_{g,ref})$$

(14)

For the comparison of the sensitivity influence of recession constants over the parameter estimation in shorter data length, we apply the maximum, minimum and median calibrated data adjustment parameter values ($C_{ep,best}^{n,m}$, $C_{ep,worst}^{n,m}$ and $C_{ep,median}^{n,m}$) in model calibration to receive the maximum, minimum and median recession constant values ($c_{g,best}^{n,m}$, $c_{g,worst}^{n,m}$ and $c_{g,median}^{n,m}$) in subsets. Finally, the maximum, minimum and median annual NSE results using $c_{g,best}^{n,m}$, $c_{g,worst}^{n,m}$ and $c_{g,median}^{n,m}$ are estimated by comparing NSE results with c_g,ref using the following equations.

$$NSE(C_{ep,best}^{n,m},c_{g,best}^{n,m})\ge NSE(C_{ep,best}^{n,m},c_{g,ref})$$

(15)

$$NSE(C_{ep,worst}^{n,m},c_{g,worst}^{n,m})\ge NSE(C_{ep,worst}^{n,m},c_{g,ref})$$

(16)

$$NSE(C_{ep,median}^{n,m},c_{g,median}^{n,m})\ge NSE(C_{ep,median}^{n,m},c_{g,ref})$$

(17)

where $c_{g,best}^{n,m}$ , $c_{g,worst}^{n,m}$ and $c_{g,median}^{n,m}$ are calculated based on the maximum, minimum and median model outputs while running with $C_{ep,best}^{n,m}$ , $C_{ep,worst}^{n,m}$ and $C_{ep,median}^{n,m}$ with c_g,ref.

3. Results and Discussions

3.1. Reference c_g for 28-Year Datasets

The pre-optimized recession constant values, ${\varphi }_0$, in five MOPEX studied basins were initially computed based on the time constant, T, as in Equation (7), to achieve in the interval of [0,1].These c_g values were applied in the XAJ model calibration alongside with the remaining pre-optimized parameters,${\varphi }_0$.Relying on the annual Nash results, the estimation of simulated c_g for each basin can be assessed by using the polynomila regression analysis.

To be able to attain better calibration, model users typically employ the longest accessible data series.This study considered the simulated recession constant as the reference, as it was estimated using 28-year of datasets. However, the real issue is the information included in the datasets and how effectively that information is extracted, not the length of the dataset itself [23,49]. Therefore, estimating the C_ep values ($C_{ep}^{n,m}$) for shorter datasets is essential to analyze the impact of the recession constant in subsets. Figure 2 reveals the best connection values, c_g,ref, comparing the annual Nash, NSE, and the pre-optimized c_g values.

Figure 2. Estimation of reference c_g (c_g,ref) from annual Nash NSE results using 28-year of datasets in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 2. Estimation of reference c_g (c_g,ref) from annual Nash NSE results using 28-year of datasets in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 3. Estimation of Cep for subsets in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 3. Estimation of Cep for subsets in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

3.3. Comparative Evaluation of Annual Nash Results, NSE, using $c_{g,best}^{n,m}$ and $C_{ep,best}^{n,m}$ in Subsets

After the model calibration using the reference c_g, the C_ep ($C_{ep,best}^{n,m}$) resulted from the highest annual Nash results were selected for each subsets as shown in Table 4.

Firstly,it is observed that as the data length increases, the NSE values derived from both c_g,ref and $c_{g,best}^{n,m}$ exhibit a declining trend. As indicated in Table, this implies that longer datasets might struggle to align with the model’s best performance. In the contrary, the 6-year subsets display the most significant upward trends in NSE results, attributed to shorter datasets being more compatible with $c_{g,best}^{n,m}$. Beyond this trend pattern, a comparison was conducted between NSE values in subsets when using c_g,ref and $c_{g,best}^{n,m}$ as shown in Figure 3.

Figure 4. Estimation of $c_{g,best}^{n,m}$ for subsets by using linear polynomial regression analysis in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 4. Estimation of $c_{g,best}^{n,m}$ for subsets by using linear polynomial regression analysis in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

According to Figure , the NSE curves in each subset showed a relative increase while running the model with $c_{g,best}^{n,m}$. However, no significant difference between the curves of NSE resulted from the model calibration using both c_g,ref and $c_{g,best}^{n,m}$ . In other words, the results suggest little space for improving the model performance while running the model with $c_{g,best}^{n,m}$. Therefore, the impact of the sensitivity of $c_{g,best}^{n,m}$ over C_ep in parameter estimation can be considered limited. Consequently, the potential impact of the sensitivity of $c_{g,best}^{n,m}$ can also be limited to the performance of the model as well as the minimum data length estimation. On the other hand, the recent estimation of the minimum data length remains consistent while considering the sensitivity of parameter estimation with c_g considering best parameter optimization scenario in subsets.

Figure 5. Comparison of Nash values using c_g,ref and $c_{g,best}^{n,m}$ for subsets in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 5. Comparison of Nash values using c_g,ref and $c_{g,best}^{n,m}$ for subsets in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 6. Estimation of $c_{g,median}^{n,m}$ for subsets by using linear polynomial regression analysis in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 6. Estimation of $c_{g,median}^{n,m}$ for subsets by using linear polynomial regression analysis in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 7. Comparison of Nash reulsts using c_g,ref and $c_{g,median}^{n,m}$ for subsets in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 7. Comparison of Nash reulsts using c_g,ref and $c_{g,median}^{n,m}$ for subsets in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Table 4. Detail Description of $c_{g,best}^{n,m}$ and $c_{g,best}^{n,m}$ values in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Table 4. Detail Description of $c_{g,best}^{n,m}$ and $c_{g,best}^{n,m}$ values in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

	MOPEX ID
Subsets	9903504000		9902387500		9902472000		9903443000		9911532500
	Cep,best	$c_{g,best}^{n,m}$	Cep,best	cg,best	Cep,best	cg,best	Cep,best	cg,best	Cep,best	cg,best
6	0.8189	0.979725	1.3559	0.98548	1.3108	0.98496	1.1154	0.98386	0.8562	0.98106
7	0.7970	0.979725	1.3500	0.98644	1.2181	0.91861	1.0891	0.98411	0.8353	0.98122
8	0.7810	0.978669	1.3524	0.98728	1.2805	0.98333	1.0831	0.98318	0.9835	0.98197
9	0.7937	0.978669	1.2974	0.98791	1.3074	0.98300	1.037	0.98359	0.9933	0.98253
10	0.7883	0.979033	1.3009	0.98791	1.3065	0.98333	1.0529	0.98460	0.9504	0.98293
11	0.8095	0.980980	1.2574	0.98597	1.2712	0.98411	1.0226	0.98260	0.9298	0.98293
12	0.8043	0.978669	1.2611	0.98644	1.2699	0.98365	1.0405	0.98411	0.9108	0.98212
13	0.7763	0.978669	1.2684	0.98687	1.2688	0.98300	1.0123	0.98411	0.8969	0.98226
14	0.7737	0.978669	1.2394	0.98728	1.2874	0.98282	0.991	0.98245	0.8732	0.97894
15	0.7666	0.978292	1.2292	0.98687	1.2843	0.98333	0.972	0.98060	0.8548	0.97791
16	0.7783	0.979725	1.2559	0.98728	1.2673	0.98365	0.96	0.98131	0.8634	0.97875
17	0.7955	0.979385	1.2197	0.98741	1.2666	0.98317	0.9566	0.98165	0.846	0.97952
18	0.7864	0.980054	1.2253	0.98741	1.2847	0.98349	0.9615	0.98131	0.8394	0.97970
19	0.7810	0.980054	1.2428	0.98754	1.2462	0.98191	0.9463	0.98149	0.8321	0.97875
20	0.7959	0.980054	1.2398	0.98766	1.2638	0.98229	0.9451	0.97982	0.8267	0.97894
21	0.7880	0.980681	1.2324	0.98741	1.3095	0.98454	0.932	0.98022	0.8095	0.97875
22	0.7959	0.980980	1.2268	0.98754	1.3090	0.98454	0.9377	0.98041	0.8475	0.97769
23	0.8087	0.981270	1.2249	0.98766	1.2804	0.98468	0.941	0.98060	0.8403	0.97701
24	0.8115	0.981270	1.2335	0.98791	1.2863	0.98482	0.9436	0.98096	0.8083	0.97629
25	0.8059	0.980681	1.2315	0.98802	1.3000	0.98454	0.9405	0.98096	0.7995	0.97552
26	0.7897	0.980054	1.2492	0.98802	1.2984	0.98426	0.9702	0.98198	0.8064	0.97629
27	0.7931	0.980054	1.2606	0.98791	1.2888	0.98396	0.9505	0.98096	0.7972	0.97701

3.4. Comparative Evaluation of Median $c_g^{n,m}$ ($c_{g,median}^{n,m}$) Results with Annual Nash (NSE) in Subsets

In this section, the values of C_ep ($C_{ep,median}^{n,m}$) were selected based on the median values of model output (NSE) after the model calibration with c_g,ref for 28-year of datasets, as shown in Table 5. After the $C_{ep,median}^{n,m}$ selection with the model outputs using c_g,ref, the calibration of the model was performed in order to achieve the relationship between NSE with $c_{g,median}^{n,m}$ results. In Figure 6, the $c_{g,median}^{n,m}$ can be assumed by the model output (NSE) using the linear polynomial regression analysis. Here, while taking the c_g values in the median range, the trend of the NSE values in all subsets for all studied basins has slightly increased as the data length increased.

While considering c_g values within the median parameter optimization performance scenario, NSE values show a slight upward trend across all subsets as data length increases. Furthermore, NSE values when utilizing $c_{g,median}^{n,m}$ are slightly higher than those with c_g,ref. Despite this minor difference, it is reasonable to conclude that model outcomes using median values did not indicate any limitations over the model performance while calibrating during the data scarcity as indicated in Figure 7.

Figure 8. Estimation of $c_{g,worst}^{n,m}$ for subsets by using linear polynomial regression analysis in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Figure 8. Estimation of $c_{g,worst}^{n,m}$ for subsets by using linear polynomial regression analysis in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

3.5. Comparative Assessment of Annual Nash results, NSE, Using $c_{g,worst}^{n,m}$ and $C_{ep,best}^{n,m}$ in Subsets

According to [50], to estimate the most sensitive parameter, it can be considered both the best and worse condition for the sensitivity analysis for the parameter optimization. However, to expect good model performance, the worst condition for the parameter optimization is essential to take into consideration.This approach achieving strong model performance under challenging conditions is imperative.A significant difference is evident when comparing NSE values obtained from the two categories (best and median) with those from $c_{g,worst}^{n,m}$ and c_g,ref. This result showed the significant impact over the model performance by utilizing NSE values with $c_{g,worst}^{n,m}$. Unlike the previously mentioned categories, the influence of c_g on the estimation of C_ep becomes pronounced in the worst-case scenario (Figure 4). Additionally, the sensitivity of c_g further magnifies its impact on model performance under these conditions. The insight provided by Figure 9 clearly illustrates the variations in NSE values among the three categories across all studied basins.

Table 6. Detail Description of $C_{ep,worst}^{n,m}$ and $c_{g,worst}^{n,m}$ values in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

Table 6. Detail Description of $C_{ep,worst}^{n,m}$ and $c_{g,worst}^{n,m}$ values in MOPEX ID (a) 903504000 (b)902387500 (c) 902472000 (d) 903443000 (e) 911532500.

	MOPEX ID
Subsets	903504000		902387500		902472000		903443000		911532500
	Cep,worst	cg,worst	Cep,worst	cg,worst	Cep,worst	cg,worst	Cep,worst	cg,worst	Cep,worst	cg,worst
6	0.5869	0.98155	1.1664	0.98673	1.3523	0.98702	0.9408	0.97941	0.9584	0.98318
7	0.6114	0.98235	1.1331	0.98548	1.2522	0.98411	0.9662	0.98022	0.9644	0.98413
8	0.6705	0.98235	1.1894	0.98548	1.2689	0.98381	0.9467	0.98096	0.9351	0.98413
9	0.6877	0.98182	1.2826	0.98728	1.2907	0.98426	0.952	0.98060	0.8556	0.97952
10	0.6992	0.98098	1.2653	0.98825	1.3156	0.98522	0.9688	0.98096	0.8752	0.98058
11	0.6954	0.98182	1.2574	0.98597	1.2822	0.98547	0.9863	0.98114	0.8559	0.98058
12	0.6940	0.98155	1.1682	0.98687	1.311	0.98496	0.9907	0.98131	0.8166	0.97678
13	0.7099	0.98209	1.1957	0.98715	1.3273	0.98559	0.9997	0.98165	0.8366	0.97791
14	0.7289	0.98235	1.1779	0.98741	1.3046	0.98534	1.0104	0.98165	0.7986	0.97653
15	0.7337	0.98209	1.2303	0.98754	1.3123	0.98702	0.9983	0.98214	0.815	0.97701
16	0.7322	0.98260	1.2374	0.98754	1.3199	0.98692	0.9905	0.98182	0.8059	0.97653
17	0.7379	0.97750	1.2769	0.98754	1.3247	0.98672	0.9973	0.98198	0.8224	0.97747
18	0.7377	0.98182	1.2919	0.98741	1.3103	0.98692	1.0057	0.98198	0.8428	0.97724
19	0.7517	0.97972	1.2381	0.98715	1.3158	0.98712	1.014	0.98317	0.8347	0.97653
20	0.7450	0.98037	1.2190	0.98741	1.2928	0.98454	0.9987	0.98275	0.8278	0.97701
21	0.7474	0.98155	1.2240	0.98754	1.2918	0.98454	0.9687	0.98165	0.805	0.97678
22	0.7543	0.98005	1.2330	0.98754	1.265	0.98349	0.9558	0.98198	0.7971	0.97578
23	0.7510	0.98037	1.2326	0.98766	1.2587	0.98265	0.9592	0.98182	0.8002	0.97604
24	0.7526	0.97938	1.2303	0.98766	1.2779	0.98349	0.9655	0.98198	0.8067	0.97653
25	0.7591	0.98037	1.2255	0.98754	1.2717	0.98282	0.9467	0.98131	0.8231	0.97791
26	0.7672	0.98037	1.2239	0.98766	1.2755	0.98426	0.9531	0.98149	0.8193	0.97813
27	0.7797	0.98005	1.2413	0.98766	1.2699	0.98349	0.9608	0.98149	0.801	0.97791

4. Conclusion

References

1. Devia, G.K.; Ganasri, B.P.; Dwarakish, G.S. A review on hydrological models. Aquatic procedia 2015, 4, 1001–1007. [Google Scholar] [CrossRef]
2. Peel, M.C.; Blöschl, G. Hydrological modelling in a changing world. Progress in Physical Geography 2011, 35, 249–261. [Google Scholar] [CrossRef]
3. Birkholz, S.; Muro, M.; Jeffrey, P.; Smith, H.M. Rethinking the relationship between flood risk perception and flood management. Science of the total environment 2014, 478, 12–20. [Google Scholar] [CrossRef] [PubMed]
4. Adikari, Y.; Yoshitani, J. Global trends in water-related disasters: an insight for policymakers. World Water Assessment Programme Side Publication Series, Insights. The United Nations, UNESCO. International Centre for Water Hazard and Risk Management (ICHARM), 2009; 1–24. [Google Scholar]
5. Modarres, R.; Ouarda, T.B. Modeling rainfall–runoff relationship using multivariate GARCH model. Journal of Hydrology 2013, 499, 1–18. [Google Scholar] [CrossRef]
6. Loucks, D.P.; Van Beek, E. Water resource systems planning and management: An introduction to methods, models, and applications; Springer, 2017.
7. Refsgaard, J.C.; Knudsen, J. Operational validation and intercomparison of different types of hydrological models. Water resources research 1996, 32, 2189–2202. [Google Scholar] [CrossRef]
8. Ramos, M.H.; Mathevet, T.; Thielen, J.; Pappenberger, F. Communicating uncertainty in hydro-meteorological forecasts: mission impossible? Meteorological Applications 2010, 17, 223–235. [Google Scholar] [CrossRef]
9. Raje, D.; Krishnan, R. Bayesian parameter uncertainty modeling in a macroscale hydrologic model and its impact on Indian river basin hydrology under climate change. Water Resources Research 2012, 48. [Google Scholar] [CrossRef]
10. Zhao, F.; Wu, Y.; Qiu, L.; Sun, Y.; Sun, L.; Li, Q.; Niu, J.; Wang, G. Parameter uncertainty analysis of the SWAT model in a mountain-loess transitional watershed on the Chinese Loess Plateau. Water 2018, 10, 690. [Google Scholar] [CrossRef]
11. Wu, Y.; Liu, S.; Huang, Z.; Yan, W. Parameter optimization, sensitivity, and uncertainty analysis of an ecosystem model at a forest flux tower site in the United States. Journal of Advances in Modeling Earth Systems 2014, 6, 405–419. [Google Scholar] [CrossRef]
12. Bárdossy, A.; Singh, S. Robust estimation of hydrological model parameters. Hydrology and earth system sciences 2008, 12, 1273–1283. [Google Scholar] [CrossRef]
13. Renard, B.; Kavetski, D.; Kuczera, G.; Thyer, M.; Franks, S.W. Understanding predictive uncertainty in hydrologic modeling: The challenge of identifying input and structural errors. Water Resources Research 2010, 46. [Google Scholar] [CrossRef]
14. Muleta, M.K.; Nicklow, J.W. Sensitivity and uncertainty analysis coupled with automatic calibration for a distributed watershed model. Journal of hydrology 2005, 306, 127–145. [Google Scholar] [CrossRef]
15. Jeremiah, E.; Sisson, S.A.; Sharma, A.; Marshall, L. Efficient hydrological model parameter optimization with Sequential Monte Carlo sampling. Environmental Modelling & Software 2012, 38, 283–295. [Google Scholar]
16. Celeux, G.; Hurn, M.; Robert, C.P. Computational and inferential difficulties with mixture posterior distributions. Journal of the American Statistical Association 2000, 95, 957–970. [Google Scholar] [CrossRef]
17. Arnold, J.G.; Youssef, M.A.; Yen, H.; White, M.J.; Sheshukov, A.Y.; Sadeghi, A.M.; Moriasi, D.N.; Steiner, J.L.; Amatya, D.M.; Skaggs, R.W.; et al. Hydrological processes and model representation: impact of soft data on calibration. Transactions of the ASABE 2015, 58, 1637–1660. [Google Scholar]
18. Bates, B.C.; Campbell, E.P. A Markov chain Monte Carlo scheme for parameter estimation and inference in conceptual rainfall-runoff modeling. Water resources research 2001, 37, 937–947. [Google Scholar] [CrossRef]
19. Vrugt, J.A.; Ter Braak, C.J.; Clark, M.P.; Hyman, J.M.; Robinson, B.A. Treatment of input uncertainty in hydrologic modeling: Doing hydrology backward with Markov chain Monte Carlo simulation. Water Resources Research 2008, 44. [Google Scholar] [CrossRef]
20. Gottschalk, F.; Sun, T.; Nowack, B. Environmental concentrations of engineered nanomaterials: review of modeling and analytical studies. Environmental pollution 2013, 181, 287–300. [Google Scholar] [CrossRef] [PubMed]
21. Kuczera, G.; Parent, E. Monte Carlo assessment of parameter uncertainty in conceptual catchment models: the Metropolis algorithm. Journal of hydrology 1998, 211, 69–85. [Google Scholar] [CrossRef]
22. Micevski, T.; Kuczera, G. Combining site and regional flood information using a Bayesian Monte Carlo approach. Water Resources Research 2009, 45. [Google Scholar] [CrossRef]
23. Sorooshian, S.; Gupta, V.K. Automatic calibration of conceptual rainfall-runoff models: The question of parameter observability and uniqueness. Water resources research 1983, 19, 260–268. [Google Scholar] [CrossRef]
24. Thyer, M.; Renard, B.; Kavetski, D.; Kuczera, G.; Franks, S.W.; Srikanthan, S. Critical evaluation of parameter consistency and predictive uncertainty in hydrological modeling: A case study using Bayesian total error analysis. Water Resources Research 2009, 45. [Google Scholar] [CrossRef]
25. Wagener, T.; McIntyre, N.; Lees, M.; Wheater, H.; Gupta, H. Towards reduced uncertainty in conceptual rainfall-runoff modelling: Dynamic identifiability analysis. Hydrological processes 2003, 17, 455–476. [Google Scholar] [CrossRef]
26. Bergström, S. Principles and confidence in hydrological modelling. Hydrology Research 1991, 22, 123–136. [Google Scholar] [CrossRef]
27. Beven, K.J. Rainfall-runoff modelling: the primer; John Wiley & Sons, 2011.
28. Lu, M. Recent and future studies of the Xinanjiang Model. J. Hydraul. Eng 2021, 52, 432–441. [Google Scholar]
29. Singh, V.P.; et al. Computer models of watershed hydrology; Vol. 1130, Water resources publications Highlands Ranch, CO, 1995.
30. Gan, Y.; Duan, Q.; Gong, W.; Tong, C.; Sun, Y.; Chu, W.; Ye, A.; Miao, C.; Di, Z. A comprehensive evaluation of various sensitivity analysis methods: A case study with a hydrological model. Environmental modelling & software 2014, 51, 269–285. [Google Scholar]
31. Zin, T.T.; Lu, M. Influence of Data Length on the Determination of Data Adjustment Parameters in Conceptual Hydrological Modeling: A Case Study Using the Xinanjiang Model. Water 2022, 14, 3012. [Google Scholar] [CrossRef]
32. Schaake, J.; Cong, S.; Duan, Q. US MOPEX data set. Technical report, Lawrence Livermore National Lab.(LLNL), Livermore, CA (United States), 2006.
33. Ren-Jun, Z. The Xinanjiang model applied in China. Journal of hydrology 1992, 135, 371–381. [Google Scholar] [CrossRef]
34. Lu, M.; Li, X. Time scale dependent sensitivities of the XinAnJiang model parameters. Hydrological Research Letters 2014, 8, 51–56. [Google Scholar] [CrossRef]
35. Hapuarachchi, H.; Li, Z.; Wang, S. Application of SCE-UA method for calibrating the Xinanjiang watershed model. Journal of Lake Sciences 2001, 13, 304–314. [Google Scholar]
36. Zhao, R.; Liu, X. The Xinanjiang Model, Computer Models of Watershed Hydrology; Singh, VP, Ed, 1995.
37. Li, X.; Lu, M. Application of aridity index in estimation of data adjustment parameters in the Xinanjiang model. 土木学会論文集B1 (水工) 2014, 70, I_163–I_168. [Google Scholar] [CrossRef] [PubMed]
38. Garrick, M.; Cunnane, C.; Nash, J. A criterion of efficiency for rainfall-runoff models. Journal of Hydrology 1978, 36, 375–381. [Google Scholar] [CrossRef]
39. Gupta, H.V.; Kling, H.; Yilmaz, K.K.; Martinez, G.F. Decomposition of the mean squared error and NSE performance criteria: Implications for improving hydrological modelling. Journal of hydrology 2009, 377, 80–91. [Google Scholar] [CrossRef]
40. Gupta, H.V.; Kling, H. On typical range, sensitivity, and normalization of Mean Squared Error and Nash-Sutcliffe Efficiency type metrics. Water Resources Research 2011, 47. [Google Scholar] [CrossRef]
41. Houghton-Carr, H. Assessment criteria for simple conceptual daily rainfall-runoff models. Hydrological Sciences Journal 1999, 44, 237–261. [Google Scholar] [CrossRef]
42. McCuen, R.H.; Knight, Z.; Cutter, A.G. Evaluation of the Nash–Sutcliffe efficiency index. Journal of hydrologic engineering 2006, 11, 597–602. [Google Scholar] [CrossRef]
43. Schaefli, B.; Gupta, H.V. Do Nash values have value? Hydrological processes 2007, 21, 2075–2080. [Google Scholar] [CrossRef]
44. Nash, J.E.; Sutcliffe, J.V. River flow forecasting through conceptual models part I—A discussion of principles. Journal of hydrology 1970, 10, 282–290. [Google Scholar] [CrossRef]
45. Kutner, M.H. Applied linear statistical models. (No Title), 2005. [Google Scholar]
46. Burden, R.L.; Faires, J.D.; Burden, A.M. Numerical analysis; Cengage learning, 2015.
47. Ostertagová, E. Modelling using polynomial regression. Procedia Engineering 2012, 48, 500–506. [Google Scholar] [CrossRef]
48. McCuen, R.H. The role of sensitivity analysis in hydrologic modeling. Journal of hydrology 1973, 18, 37–53. [Google Scholar] [CrossRef]
49. Li, C.z.; Wang, H.; Liu, J.; Yan, D.h.; Yu, F.l.; Zhang, L. Effect of calibration data series length on performance and optimal parameters of hydrological model. Water Science and Engineering 2010, 3, 378–393. [Google Scholar]
50. Li, X.; Lu, M. Multi-step optimization of parameters in the Xinanjiang model taking into account their time scale dependency. Journal of Japan Society of Civil Engineers, Ser. B1 (Hydraulic Engineering) 2012, 68, I_145–I_150. [Google Scholar] [CrossRef] [PubMed]