## 4 Model Development and Performance Metrics  
The consideration of rainfall and runoff lag for model input combination is an important task in order to achieve better prediction performance. The best performance of ML models has been shown for an input combination in [PERSON] et al. (2018) and [PERSON] et al. (2010) was two days lag of discharge (Q\({}_{t-2}\), Q\({}_{t-1}\)), current day discharge (Q\({}_{t}\)), and current day precipitation (\(P_{t}\)) are used to forecast the next-day discharge (Q\({}_{t+1}\)) on daily R-R modeling. In this paper, auto-correlogram (ACF) and partial autocorrelation function (PACF) are used for selection of the optimal number of lags from the discharge series ([PERSON] et al., 2017). The optimal lags of the discharge series and a current day precipitation (\(P_{t}\)) are used to predict one-day ahead discharge value in the initial model development phase. The lag values of precipitation are already reflected in the lag discharge series. Figure 5a,c show ACF plot with 95% confidence bound for discharge series of Fal at Tregony and Teifi at Glanteifi, respectively, where both indicates (Figure 5a,c) that the correlation effect is weakened in subsequent lag values. In order to find a direct relationship (serial correlation) between number of lags with corresponding observation, PACF plots (Figure 5b,d) for Fal at Tregony and Teifi at Glanteifi, respectively) with 95% confidence are used in this study. PACF plots (Figure 5b,d) showed less correlation beyond some specific lag values or some specific lag values are highly correlated with corresponding observation. In this paper, the authors consider positive correlated lag values from PACF plots (Figure 5b,d). For the catchment Fal at Tregony (Figure 5b) 1, 2, 3, 4, 5, and 6 (6 preceding days) lags are considered for model input from the discharge series with current day precipitation value (here, 1-day lag). For the catchment of Teifi at Glanteifi (Figure 5d) 1, 3, 4, 5, and 6 (5 preceding days) lags are considered for model input from the discharge series with current day precipitation value (here, 1-day lag). The original rainfall and runoff data is decomposed by a DWT method. The decomposed rainfall and runoff data are given to model input and next day discharge data are the corresponding output. Therefore, next day value of discharge is predicted from antecedent information of rainfall and runoff (2\({}^{n}\)-day mode and a maximum level of approximation mode, where n = 0,1,... 9). The decomposition runoff series (one preceding lag) is selected from PACF plots where the previous day lag of runoff shows (Figure 5b,d) the highest positive correlation (greater than 0.8) with next day runoff value for both the catchments. Due to previous lags of precipitations already being reflected in runoff series, the previous one day behind precipitation is selected as decomposition precipitation series. In this paper, the most commonly used smooth mother wavelet, called Daubechies (Db), in the  
Figure 4: Process flow chart of the deep neural network (DNN). hydrometeorological study is selected for model deployment. The number of levels for decomposition is determined by Equation (19), which is based on the mother wavelet and sample data points [73]. \[D_{L}=\log_{2}[N/(2K-1)] \tag{19}\]  
where \(D_{L}\) is the maximum decomposition level, \(N\) is the number of sample data points, and \(K\) indicates the number of vanishing moments of a Db wavelet. A broader support of wavelet with higher vanishing moments are more appropriate for irregular and non-linear hydrological time series [74]. Therefore, an irregular wavelet Db7 is applied in this study where 7 is the vanishing moment. \(D_{L}\) is estimated from two same data series for Fal at Tregony and Teifi at Glanteifi (\(N\) = 5838). Where \(K\) = 7, then for both time series data the estimated value of \(D_{L}\) is found to be 9. It was showed that for \(K\geq\) 7, all approximations are similar [73].