## 6 Future Research Direction and Discussion
### Wavelet-Bi-LSTM (W-Bi-LSTM)  
As discussed above, wavelet transform (WT) is a data pre-processing tool to decompose time series in the time-frequency scale. WT is capable of decomposing the time series at various scales and into several sub-time series that give insights into the relationships between time-dependent features. To capture high-frequency information, short time intervals are used and, conversely, long-duration intervals analyze low-frequency information. Researchers report that wavelet-coupled ML models have often achieved higher prediction accuracy than regular ML models [87; 88]. WT is categorized into two types: continuous wavelet transform (CWT), and discrete wavelet transform (DWT). CWT is time-consuming and computationally expensive; therefore, DWT is mostly preferred in hydrological problems, particularly in groundwater level simulation. The mathematical equation of a discrete wavelet can be represented as [89]. \[g_{\left(i,j\right)}^{\left(t\right)}\ =\ \left(1/\sqrt{a_{0}}\right)g(\left(t-jb_{0}a_{0 }^{i}\right)/\left(a_{0}^{i}\right) \tag{1}\]  
In Equation (1), \(i\) and \(j\) represent the integral values, and \(a_{0}\), and \(b_{0}\) are the location parameter with specified fixed dilation steps and the most common values are 1 and 2, respectively. For details refer to ([PERSON] and [PERSON]) [90]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference** & \
\begin{tabular}{c} **Performance Evaluation** \\ **Metrices** \\ \end{tabular} & **Prediction** & **Target Prediction** & **Year** \\ \hline [PERSON] et al. [74] & NSE, RMSE & Daily & GWL & 2020 \\ \hline [PERSON] et al. [77] & NSE, MAE, RMSE, R & Monthly & GWL & 2020 \\ \hline [PERSON] et al. [96] & RMSE, \(R^{2}\) & Monthly & GWL & 2020 \\ \hline [PERSON] et al. [97] & RMSE, \(R^{2}\) & Monthly & GWL & 2020 \\ \hline [PERSON] et al. [98] & MAE & 3-month interval & Depth to GW & 2020 \\ \hline [PERSON] et al. [99] & MSE, \(R^{2}\) & Weekly & GWL & 2020 \\ \hline [PERSON] et al. [100] & \(R^{2}\), RMSE & Monthly & GWL & 2019 \\ \hline [PERSON] et al. [101] &
\begin{tabular}{c} Bias, R, VAF, RMSE, SI, MAE, \\ NSE, RMSRE, MAPE \\ \end{tabular} & Monthly & GWL & 2019 \\ \hline [PERSON] et al. [102] & \(R^{2}\), RMSE, NSE & Monthly & GWL & 2019 \\ \hline [PERSON] et al. [6] & \(R^{2}\), RMSE, NSE & Half-hourly & GWL & 2019 \\ \hline [PERSON] and [PERSON] [103] & MaxAE, MAE & Monthly & GWL & 2011 \\ \hline [PERSON] et al. [81] & \(R^{2}\), MAPE, RMSE & Monthly & GWL & 2011 \\ \hline [PERSON] et al. [83] & MSE & Daily & GWL & 2010 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
### Wavelet-Bi-LSTM (W-Bi-LSTM)  
Unlike the conventional LSTM [52], Bi-LSTM has a simultaneous two-way flow of prior information to better understand the contextual dependencies between the variables using forward hidden layers and backward hidden layers [91]. Bi-directional LSTM manages the flux of the input and output variables using several gates called memory cells, while classical recurrent neural networks (RNNs) use hidden layer nodes with nonlinear activation functions [92]. Figure 6 shows the graphical representation of Bi-LSTM. hf and hb are two memory cells in the Bi-LSTM network which manage the forward and backward computed values. \[hft=f(wf1 xt+wf2 ht-1) \tag{2}\]  
\[hbt=f(wb1 xt+wb2 ht+1) \tag{3}\]  
In the above equations, hf represents the forward layer LSTM output and \(hb\) is the backward layer LSTM output. The final output value of the hidden layer is computed by combining the results of forward and backward layers [93]. \[ot=g(w01*hf+w02*hb) \tag{4}\]  
In Equations (2)-(4), \(wi\) is the weight coefficient matrix that is repeatedly applied at each time step. I hypothesize that using wavelet transform (WT) with Bi-LSTM (W-Bi-LSTM), groundwater levels can be simulated yearly with higher prediction efficiency. To the best of my knowledge, no such model has been proposed, given that, in the literature, most of the studies focused on monthly, weekly, and daily GWL predictions.