A Maximal Overlap Discrete Wavelet Packet Transform Coupled with an LSTM Deep Learning Model for Improving Multilevel Groundwater Level Forecasts

A Maximal Overlap Discrete Wavelet Packet Transform Coupled with an LSTM Deep Learning Model for Improving Multilevel Groundwater Level Forecasts | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A Maximal Overlap Discrete Wavelet Packet Transform Coupled with an LSTM Deep Learning Model for Improving Multilevel Groundwater Level Forecasts Dilip Kumar Roy, Ahmed A. Hashem, Michele L. Reba, Deborah L. Leslie, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3464867/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Developing precise groundwater level (GWL) forecast models is essential for the optimal usage of limited groundwater resources and sustainable planning and management of water resources. In this study, an improved forecasting accuracy for up to three weeks ahead of GWLs in Bangladesh was achieved by a coupled Long Short Term Memory (LSTM) network-based deep learning algorithm and a Maximal Overlap Discrete Wavelet Packet Transform (MODWPT) data preprocessing. The coupled LSTM-MODWPT model performance was compared with the LSTM model. For both standalone LSTM and LSTM-MODWPT models, the Random Forest feature selection approach was employed to select the ideal inputs from the candidate GWL lags. In the LSTM-MODWPT model, input GWL time series were decomposed using MODWPT. The ‘Fejér-Korovkin’ mother wavelet with a filter length of 18 was used to obtain a collection of scaling coefficients and wavelets for every single input time series. Model performance was assessed using five performance indices: Root Mean Squared Error; Scatter Index; Maximum Absolute Error; Median Absolute Deviation; and a-20 index. The LSTM-MODWPT model outperformed standalone LSTM models for all time horizons in GWL forecasting. The percentage improvements in the forecasting accuracies were 36.28%, 32.97%, and 30.77%, respectively, for one-, two-, and three-weeks ahead forecasts at the observation well GT3330001. Accordingly, the coupled LSTM-MODWPT model could potentially be used to enhance multiscale GWL forecasts. This research demonstrates that the coupled LSTM-MODWPT model could generate more precise GWL forecasts at the Bangladesh study site, with potential applications in other geographic locations globally. Groundwater level forecasts ‧ Long short-term memory networks ‧ Wavelet packet transform ‧ Variable selection ‧ Random forest Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 1 Introduction Groundwater aquifers are regarded as the primary sources of the world's clean water supplies and play a crucial role in the stability of irrigated agriculture, domestic water requirements, and industrial water supplies in locations where high-quality surface water is insufficient (Gong et al., 2016 , 2018 ; Roy et al., 2021a ). Groundwater supplies are under increased pressure due to population growth, rising water demand, and the inevitable influences of climate change (Wada and Bierkens, 2014 ). Consequently, groundwater systems are undergoing an accelerated decline. Although human action, such as excessive pumping, is thought to be the significant determinant of groundwater level (GWL) decline, recent forecasts suggest that the situation will worsen even sooner than predicted due to climate change (Wada and Bierkens, 2014 ). Due to the over-extraction of limited groundwater reserves, groundwater resources will continue to be depleted, causing various environmental, operational, and economic problems (Banerjee et al., 2009 ). Groundwater is a significant source of water supply in Bangladesh, where ~ 80% of the population relies on groundwater supplies primarily for the basic water requirements (Hoque and Adhikary, 2020 ). Consequently, prudent management and sustainable use of the aquifer's limited groundwater reserves are imperative to secure steady groundwater supplies for future generations. Precise prediction and projection of impending GWL fluctuations may generate a practical groundwater management approach in Bangladesh and globally (Roy et al., 2021a , b ). Numerical simulation models of groundwater flow processes have historically been used in groundwater hydrology to forecast GWLs and better understand the system's underlying mechanisms (Doble et al., 2017 ; Masterson and Garabedian, 2007 ; Park and Parker, 2008 ). However, accurate predictions of GWLs using simulation models require a thorough knowledge of the aquifer's characteristics and skilled modelers with a thorough understanding of aquifer geometry and modeling strategies. As such, data-driven modeling techniques have been developed and deployed to reduce modeling complexities in the hydrological research domains (Fahimi et al., 2017 ; Govindaraju, 2000a , 2000b ; Maier et al., 2010 ; Sadler et al., 2018 ; Yang et al., 2017 ). The attributes of the underlying physical processes do not need to be explicitly defined to develop data-driven models. In Machine Learning (ML) based data-driven modeling techniques, a model's inputs and outputs are directly mapped or correlated using an iterative learning process (Solomatine and Ostfeld, 2008 ). In the realm of the prediction of GWL time series data, it has been discovered that a data-driven model developed using an Artificial Neural Network (ANN) performed as well as or had improved prediction compared to a numerical simulation model (Karandish and Šimůnek, 2016 ; Mohanty et al., 2013 ). As such, there has been increasing interest in data-driven modeling techniques as alternatives to complex numerical simulation models. Sufficient and accurate prediction of GWL over the short- to medium-term helps develop a groundwater management plan in regions where droughts initiated by climate change or over pumping are the primary driving forces (Feng et al., 2008 ; Guzman et al., 2017 ; Sahoo et al., 2017 ). Different data-driven modeling techniques are increasingly being used since, compared to conventional hydrogeological modeling, as fewer datum points are necessary with also easier execution (Zhang et al., 2018 ). Numerous techniques have recently been employed in the research area related to predictions of GWL fluctuations. These approaches ranged from standalone data-driven modeling to hybrid modeling (Adamowski and Chan, 2011 ; Daliakopoulos et al., 2005 ; Obergfell et al., 2019 ; Roshni et al., 2019 ; Sakizadeh et al. 2019 ). The standalone modeling approaches comprise ML-based modeling (Samani et al. 2022 ; Dong et al., 2018 ; Guzman et al., 2017 ; Mohanty et al., 2015 ; Sahoo et al., 2017 ), ANNs (Ghorbani et al., 2018 ; Lee et al., 2019 ; Dadhich et al. 2021 ), NARX neural networks (Guzman et al., 2017 ), ANFIS (Roy et al., 2021a ; Nadiri et al., 2019 ; Nourani and Mousavi, 2016 ; Raghavendra and Deka, 2015 ; Wen et al., 2015 ; Zare and Koch, 2018 ), Gaussian Process Regression (Raghavendra and Deka, 2015 ), Prophet modeling technique (Aguilera et al., 2019 ), Support Vector Machine (SVM) (Nadiri et al., 2019 ; Tang et al., 2019 ), and Discrete Space-State model (Roy et al., 2021b ). On the other hand, the hybrid approaches used to predict GWL fluctuations are: hybrid Wavelet Transform–ML approaches (Adamowski and Chan, 2011 ; Barzegar et al., 2017 ; Peng et al., 2017 ; Raghavendra and Deka, 2015 ), hybrid ML and Ensemble Empirical Mode Decomposition (Gong et al., 2018 ), Wavelet – ANFIS (Moosavi et al., 2013 ), and Nonlinear System Identification models coupled with Linear Polynomials (Makungo and Odiyo, 2017 ). Another domain of hybrid ML-based GWL prediction includes the usage of evolutionary algorithms to tune the model parameters. These evolutionary algorithm tuned ML models include the application of Whale Algorithm – ANN (Banadkooki et al., 2020 ), Particle Swarm Optimization (PSO) – ARIMA (Boubaker, 2017 ), hybrid Self Organizing Map- and Multi-objective Genetic Algorithm-SVM (Fang et al. 2019 ), and hybrid SVM-PSO (Wei et al., 2020 ; Mozaffari et al. 2022 ). A thorough analysis of ML-based methods for modeling GWLs is provided by Rajaee et al. ( 2019 ) and by Sarma and Singh ( 2022 ). It is evident that a variety of modeling techniques have been used to forecast changes in GWL with differing extents of prediction accuracy. It is also clear that recommending a specific prediction model for projecting GWL changes is technically challenging, and possibly unachievable. Consequently, improving the prediction accuracy of GWL variations still necessitates more sophisticated approaches. Deep Learning (DL) has recently been used as an advanced well-developed sub-area of ML-based approaches. A growing number of scientific fields have successfully used DL-based modeling (Plappert et al., 2018 ; Fang et al., 2019 ; Fan et al., 2019 ; Cummins et al., 2018 ). The recent application of DL has also been noted in the prediction of time series data (Tien Bui et al., 2020 ; Xu et al., 2019 ; Yang and Chen, 2019), forecasting of GWLs (Bowes et al., 2019 ; Supreetha et al., 2020 ) and in estimating future water quality variables in the short term (Barzegar et al., 2020 ). As such, several recent groundwater modeling studies (Guzman et al., 2017 ; Chang et al., 2016 ; Daliakopoulos et al., 2005 ) have concentrated on the effective usage of DL-based Recurrent Neural Networks (RNNs). Long-term dependence between variables, however, is difficult for standard RNN designs to capture (Bengio et al., 1994 ), primarily because of the presence of two issues: vanishing and exploding gradients. The exploding and vanishing gradient difficulties of RNNs can be overcome via Long Short-Term Memory (LSTM) networks, an advanced version of traditional RNN topologies. However, LSTMs have only lately been used to predict hydrological time series, despite their widespread applicability in numerous study fields (Hu et al., 2018 ; Liang et al., 2018 ; Tian et al., 2018 ; Zhang et al., 2018 ). Jeong et al. ( 2020 ) used LSTM-based models to predict GWLs utilizing real-world ‘faulty’ data with outliers and noise. For a seaside city in the USA prone to episodic inundations, it was discovered that an LSTM network's predictive power was superior to an RNN when it came to predicting hourly GWL values (Bowes et al., 2019 ). In order to achieve accuracy and resilience in irrigation flow forecasting, Mouatadid et al. ( 2019 ) paired a Maximal Overlap Discrete Wavelet Transform (MODWT) and an LSTM model. Zhang et al. ( 2018 ) suggested an LSTM-based model for forecasting water table elevation in agricultural areas and achieved a reliable prediction outcome by deploying a straightforward data preprocessing method for data standardization. Considering recent literature, LSTMs have the potential to be utilized in the research domain of hydrological time series projections. Therefore, this study represents the first endeavor of employing LSTM-based models coupled with a data preprocessing tool for forecasting multi-step forward GWLs at designated observation wells in the Gazipur Sadar Upazilla, Bangladesh. Wavelet decomposition and wavelet packet transform as preprocessing tools have successfully been utilized in numerous research domains to improve the forecasting capability of ML-based models. However, recent studies related to wavelet packet-based prediction modeling techniques are associated with issues of leveraging the “future data.” Therefore, these applications are not appropriate for application in real-life forecasting scenarios (Quilty et al., 2021; Quilty and Adamowski, 2018 ). On the other hand, a Maximal Overlap Discrete Wavelet Packet Transform (MODWPT) approach is better suited for real-world applications because it can overcome the “future data” issues. This study utilized MODWPT as a preprocessing tool to improve the forecasting capabilities of an LSTM model for multiple-step forward GWL forecasting. Therefore, this study's key motivation and focus are to assess the potential use of MODWPT as a preprocessing tool to improve the forecasting capability of LSTM for predicting multiple-step forward GWLs for the designated observation well locations. 2 Methods 2.1 Study Area and Input Data The study area is in the Gazipur Sadar Upazilla, with approximate surface area of 446.38 km 2 . It is situated between the longitudes of 90.33° and 92.50°E and the latitudes of 23.88° and 24.18°N. This area falls within the physiographic unit referred to as the Madhupur jungle tract, locally known as Bhawal Garh . The terrace topography in many parts of the Madhupur jungle tract ranges from flat areas to lower-rounded mountains and ridges separated by closely spaced shallow ‘bides’ (valleys) (Bangladesh Bureau of Statistics (BBS), 2013 ). Additionally, a few ‘ beels ’ (lakes) are located near the periphery of the study area. The soil conditions in the Bhawal Garh are varied and often complex. A wide range of soil variability exists, from red laterite soils at the extreme to almost undeveloped soil of raw Pleistocene clay (Bangladesh Bureau of Statistics (BBS), 2013 ), with numerous intermediate soil layers. Four major constituents of land formation were reported (Bangladesh Bureau of Statistics (BBS), 2013 ): clay (70.02%), sandy loam (13.84%), sand (6.92%), and pebble (4.61%). Wet season rainfall is one of the critical sources of water for groundwater replenishment and recharge, which is often constrained by the thick clay soil surface and low rainfall periods leading to decreasing natural recharge. Pumped groundwater serves as the primary water source for residential and agricultural water requirements. In this study, secondary GWL data were collected from the Processing and Flood Forecasting Circle (PFFC) of the Bangladesh Water Development Board (BWDB) (Zahid and Hossain, 2014 ). GWL data were collected at various observation well locations. Two observation wells, GT3330001 and GT3330002, were selected based on the criteria of the fewest missing GWL entries. A data quality assurance procedure is often deployed to warrant the qualities of the obtained GWL datasets, which boosts the trustworthiness of GWL forecasts using ML tools (Estévez et al. 2016 ). Although a thorough quality assurance procedure was not accomplished for the current dataset, the quality of the acquired GWL data was assessed systematically for its exactness and comprehensiveness using range/limit tests. The basis of range testing is the straightforward verification that any observation is contained within a given range (Estévez et al. 2016 ). Measurements outside of this range are correctly flagged as invalid, and only measurements falling within this threshold are accepted (Feng et al., 2004 ; Shafer et al., 2000 ). The data occurring inside the acceptable range were utilized to simulate future GWL changes in the selected observation wells by focusing on providing a multiple-step ahead GWL projection. The study domain and locations of observation wells are shown in Fig. 1 . The weekly GWL data collected from BWDB for GT3330001 ranged between 07 January 1980 and 17 September 2018 or 2019 data points. There were 29 missing values or 1.4% of the data, leaving 1983 data points for analysis. On the other hand, the weekly GWL data at GT3330002 ranged from 07 January 1980 to 26 December 2016 with 1929 data points. There were 18 missing values or 0.93% of the data leaving 1919 data points for analysis. Those omitted GWL data were replaced using the ‘moving median’ method of data imputing in which a moving median with a specified window length was used. Finally, the observation wells GT3330001 and GT3330002 had 2012 readings (from 07 January 1980 to 17 September 2018) and 1937 readings (from 07 January 1980 to 26 December 2016) of weekly GWL entries after the imputation of missing entries. The weekly GWL dataset’s descriptive statistics are presented in Table 1 . The mean of the GWL data ranges from 12.96 m (at GT3330001) to 13.26 m (at GT3330002), while the standard deviation values range from 3.39 m (at GT3330002) to 7.92 m (at GT3330001) (Table 1 ). The positively skewed values indicate that the statistical distribution of the data at both observation wells has a larger right tail than a left tail (Table 1 ). The GWL data at GT3330001 had negative kurtosis values and light-tailed distributions. On the other hand, the datasets at GT3330002 showed heavy-tailed distributions because of the positive kurtosis value. Table 1 Values of the statistical parameters computed on the GWL data (m) at the designated observation wells Obs. wells Min Max Mean Median STD Skewness Kurtosis GT3330001 0.10 25.57 12.96 9.83 7.92 0.50 -1.39 GT3330002 6.30 23.50 13.26 13.26 3.39 0.74 0.44 2.2 Modelling Approaches Brief descriptions of various approaches, including the Long Short Term Memory Network (LSTM), Maximal Overlap Discrete Wavelet Packet Transform (MODWPT), integrated LSTM-MODWPT, input variable selection using Random Forest (RF), and data partitioning, are provided in the following sections. 2.2.1 Long-Short Term Memory (LSTM) Networks An LSTM model network is a subtype of advanced Recurrent Neural Networks (RNNs) able to acquire longer-term reliance amongst sequence data time steps. Because LSTMs incorporate state dynamics and gating functions, they overcome the exploding and vanishing gradient issues of conventional RNNs, making them particularly effective for predicting sequence data (Hochreiter and Schmidhuber, 1997 ). The LSTM network's architecture is made up of multiple memory blocks connected by layers, and each of which has a substantial number of memory cells with recurrent connections. The forget, input, and output gates are examples of the three multiplicative parts (called gates) that make up an LSTM memory cell (Yuan et al., 2018 ). A sequential input layer is used to provide time series data into an LSTM model network, making up the majority of a primary LSTM network. There are four layers in an LSTM network used to solve a simple regression problem: the LSTM network commences with a sequential input layer preceded by an LSTM layer, and the network concludes with an entirely interconnected layer and a regression output layer. The straightforward and deeper LSTM architectures as well as the mechanism by which LSTMs perform forecasting tasks are shown in Figs. S1, S2, and S2 of the Supplemental Information (SI). An LSTM network architecture with three hidden layers was used. Model overfitting was prevented by assigning a dropout layer to each hidden layer. The number of hidden neurons and the associated dropout rates in the 1st, 2nd, and 3rd hidden layers were selected upon several iterations. The optimal numbers of hidden neurons were 100, 50, and 20 for the 1st, 2nd, and 3rd hidden layers, respectively, while the corresponding optimal dropout rates were 0.4, 0.3, and 0.4, respectively. The LSTM architecture with multiple hidden units was employed. The numbers of ‘hidden neurons’ were decided via several trials by varying the number of ‘hidden neurons’ in each iteration. The LSTM architecture parameters were obtained upon numerous trials, and the optimum sets of parameter values are presented in Table 2 . These optimum parameter values were used for developing the LSTM models to predict one, two, and three-week(s) ahead GWL forecasting at the two observation wells. Table 2 The best pairings of various training alternatives Training options Relevant option values Solver used to optimize 'adam' Maximum number of epochs 1000 Gradient threshold value 1 Initial rate of learning 0.001 Minimum size of the batch 150 Length of the sequence 1000 2.2.2 Maximal Overlap Discrete Wavelet Packet Transform (MODWPT) The formulation of wavelet decomposition using MODWPT may be expressed as (Walden and Contreras Cristan, 1998 ): $${\stackrel{\sim}{W}}_{j,n,t}^{P}=\sum _{l=0}^{L-1}{\stackrel{\sim}{f}}_{n,l}{\stackrel{\sim}{W}}_{j-1,\left[n/2\right],\left(t-{2}^{j-1}l\right) \text{m}\text{o}\text{d} N}^{P}$$ 1 where, $${\stackrel{\sim}{f}}_{n,l}=\left\{\begin{array}{c}{\stackrel{\sim}{g}}_{l,} if n mod 4=0 or 3\\ {\stackrel{\sim}{h}}_{l,} if n mod 4=1 or 2\end{array}\right.$$ 2 where, \({\stackrel{\sim}{W}}_{j,n,t}^{P}\) represents the coefficients of MODWPT at time \(t\) for the level of decomposition \(j\) and band \(n\) \((\text{w}\text{h}\text{e}\text{r}\text{e} n=\text{0,1}, \dots ,{2}^{j}-1)\) , which is technically corresponding to frequencies for an interval of \(\left[\frac{n}{{2}^{j+1}},\frac{n+1}{{2}^{j+1}} \right].\) The MODWPT is a wavelet decomposition algorithm that uses energy, and the output of the MODWPT is time-delayed signals relative to the input signals. The MODWPT partitions the energy over the entire wavelet packets at each level. The total energy of any input signal equals the sum of the energies over all the wavelet packets. On the other hand, the features in the MODWPT details line up closely with the features contained in the input signal. The details, like the energies, produce the original signal when summing the details in each sample for a given level. The MODWPT executes a discrete wavelet packet transform and returns a ‘sequency-ordered’ wavelet packet tree. Time series decomposition process via MODWPT using a sequence-ordered wavelet packet tree can be found in Fig. S4 of the SI. 2.2.3 Proposed Model Development (Integrated LSTM-MODWPT) The essential phases for developing a model are presented in the following sub-sections and presented in Fig. 3 . The first step in developing DL-based forecasting modeling is the selection of potential input variables. Since weekly GWLs at two observation wells (GT3330001 and GT3330002) were the only data used in this effort, the possible inputs were the time-lagged forms of the measured GWLs (every station), and their wavelet packet transformed components. A Partial Autocorrelation Function (PACF) was used to obtain the time-lagged information from the GWL time series and to select which lags to incorporate as the potential inputs. The PACF plots for the GWL data obtained from two selected observation wells are illustrated in Fig. 2. From Fig. 2, it is possible to deduce that the present and previous five time lags are essential for three-weeks ahead GWL forecasting \(\left({GWL}_{t+1},{GWL}_{t+2}, \text{a}\text{n}\text{d} {GWL}_{t+3}\right)\) . The present and previous time lags for the observation well GT3330001 were \({GWL}_{t},{GWL}_{t-1},{GWL}_{t-3},{GWL}_{t-4},{GWL}_{t-5}, \text{a}\text{n}\text{d} {GWL}_{t-6}\) . On the other hand, the time lags for the observation well GT3330002 were \({GWL}_{t},{GWL}_{t-1},{GWL}_{t-2},{GWL}_{t-3},{GWL}_{t-4}, \text{a}\text{n}\text{d} {GWL}_{t-5}\) . Consequently, the potential input variables for the non-wavelet decomposition-based LSTM model utilized six time-lagged variables as inputs. For the MODWPT-based LSTM model, the candidate input variables include the same inputs plus their MODWPT decomposed counterparts. Each of the time-lagged GWL time series was wavelet decomposed independently. The MODWPT produced many candidate input variables for the wavelet-based LSTM models at both observation wells. Therefore, the most significant input variables were selected using a RF-based modeling approach. The choice of wavelet filters and decomposition levels and eradication of the boundary-impacted wavelet coefficients is the primary consideration during the wavelet decomposition (Quilty and Adamowski, 2021 ). This study used the Fejér-Korovkin scaling filter with a filter length of 18 and a decomposition level of 3. This essential elimination of 188 boundary-impacted wavelet coefficients from the start of the sets of possible input and target variables was based on the boundary correction approach adopted by Quilty and Adamowski ( 2021 ). Similar principles were used in the LSTM models that were not wavelet-based. MATLAB (Mathworks, 2020a ) commands and functions were used to develop the proposed models using the observed groundwater level data at the two observation wells. 2.2.4 Input Variable Selection The RF approach was used to select the scaling coefficients and wavelet family that are most effective in producing precise predictions of the output variable. The RF approach's main task was to ensure that only variables necessary for projecting the output variable were chosen. Simultaneously, unnecessary and/or inappropriate variables were omitted. Ideally, accurate models should not be complicated or limited in their ability to predict. This study used a ‘bagged’ ensemble of 500 regression trees to develop the RF model. All input variables were assigned at each tree node to ensure that each regression tree utilized all input variables to attain better accuracy. The number of levels among the inputs varied using a standard CART algorithm for selecting split inputs at each node of the trees in a RF model that might yield less accurate estimates. A curvature or interaction test was performed to select split inputs to avoid this problem (MathWorks, 2020b ). Variable importance was estimated by performing permutation of the out-of-bag observations among the trees. Time-lagged input variables, as well as their wavelet packet coefficients, produced 102 input variables of [GWL at the present time and five times lagged GWLs (6) + 16 wavelet packet coefficients for Fejér-Korovkin scaling filter with a specified filter length (16) × 6]. Using all the input variables is associated with a computational burden. Therefore, only the most significant input variables were selected for the model development to reduce the computational burden and enhance computational efficiency. This study used the 20 most influential input variables determined by the RF modeling technique to develop LSTM-MODWPT models at the observation wells for the one-, two-, and three-step ahead GWL forecasts. Figure 4 presents the plots of variable importance. 2.2.5 Data Partitioning Due to time lagging and removal of the boundary-affected coefficients, the observed GWL data were reduced at each observation well. At the observation well GT3330001, a total of 1816 records remained (from 09 October 1983 to 17 September 2018) after removing 188 boundary-affected coefficients and eight records due to time lagging (three time lags forward + five time lags backward) from the entire GWL time series of 2012 readings (from 07 January 1980 to 17 September 2018). At GT3330002, a total of 1741 records remained (from 09 October 1983 to 26 December 2016) after removing 188 boundary-affected coefficients and eight records due to time lagging (three time lags forward + five time lags backward) from the entire GWL time series of 1937 readings (07 January 1980 to 26 December 2016). The remaining dataset was separated into two distinct sets of training and testing samples where 80% of the data records were allocated for training, and the remaining 20% was allotted for testing. For GT3330001, the remaining 1816 readings were split into 1453 records (from 09 October 1983 to 03 October 2011) and 363 records (from 04 October 2011 to 17 September 2018), respectively, for training and testing purposes. For GT3330002, the remaining 1741 readings at GT3330002 were divided into 1393 records (from 09 October 1983 to 26 April 2010) and 348 records (from 27 April 2010 to 26 December 2016), respectively, for training and testing purposes. 3 Statistical Indices for Performance Evaluation Five statistical parameters were utilized to evaluate the model's performance (Equations 3 – 7 ). Generally, the Root Mean Squared Error (RMSE) criterion measures the error of the model. The lower RMSE value indicates the higher prediction power of the model. However, the value of RMSE largely depends on the magnitude of the data; therefore, a lower RMSE value does not necessarily mean better prediction performance. The Scatter Index (SI) criterion was used to eliminate the dimensionality effect of the data to overcome this issue. Model performance assessment criteria based on the SI index values were: Excellent when SI is less than 0.1, good when SI is between 0.1 and 0.2, fair when SI is between 0.2 and 0.3, and poor when SI is greater than 0.3 (Li et al., 2013 ). The \({a}^{20}- index\) value ranges between 0 and 1, and for an ideal model, the \({a}^{20}- index\) value is 1 (Xu et al., 2019 ). Root Mean Squared Error (RMSE): $$RMSE=\frac{\sqrt{\frac{1}{n}\sum _{i=1}^{n}{\left({GWL}_{i}^{A}-{GWL}_{i}^{P}\right)}^{2}}}{\overline{{GWL}^{A}}}$$ 3 Scatter Index, SI (Li et al., 2013 ): $$SI=\frac{RMSE}{\overline{{GWL}^{A}}}$$ 4 Maximum Absolute Error (MAE): $$MAE=max\left[\left|{GWL}_{i}^{A}-{GWL}_{i}^{P}\right|\right]$$ 5 Median Absolute Deviation, MAD (Pham-Gia and Hung, 2001 ): $$MAD\left({GWL}^{A},{GWL}^{P}\right)=median\left(\left|{GWL}_{i=1}^{A}-{GWL}_{i=1}^{P}\right|,\left|{GWL}_{i=2}^{A}-{GWL}_{i=2}^{P}\right|,\cdots ,\left|{GWL}_{i=n}^{A}-{GWL}_{i=n}^{P}\right|\right)$$ 6 a 20 – index (Xu et al., 2019 ): $${a}^{20}- index=\frac{{k}^{20}}{n}$$ 7 where, \({GWL}_{i}^{A}\) and \({GWL}_{i}^{P}\) are the actual and predicted \(GWL\) values for the \({i}^{th}\) data points of the dataset, respectively; \(\overline{{GWL}^{A}}\) and \(\overline{{GWL}^{P}}\) are the mean values of the actual and predicted \(GWL\) , respectively; \(n\) is the number of entries in the GWL time series data; \({k}^{20}\) is the amount of data that is associated with a \({GWL}_{i}^{A}/{GWL}_{i}^{P}\) value varying from 0.80 to 1.20 (Xu et al., 2019 ). 4 Results and Discussion The findings of the non-wavelet and wavelet-based LSTM models for multi-step (i.e., one-, two-, and three-week(s)) ahead GWL forecasting were evaluated using several performance evaluation indices. Graphical approaches were also used to evaluate the performances of the developed models. Generally, the performances of all models for both the training and testing phases exhibited a good tradeoff, indicating a reasonably fair generalization capability of the developed models in multi-step ahead GWL forecasting. However, the comparison of model performances between the non-wavelet and wavelet-based LSTM was based on their testing phase performances. Both training and testing phase model performances are presented in the following sub-sections. 4.2 Performance of the Standalone LSTM Models Determining the optimal LSTM model architecture is crucial in DL-based forecasting approaches. In this study, different combinations of various numbers of hidden layers and neurons were evaluated to determine the optimal LSTM model structure in multi-step ahead GWL forecasting. The RMSE criterion was utilized to assess how well the developed models performed during the training and testing phases in various scenarios of the hidden layer and hidden neuron combinations. The amounts of RMSE on the training and test dataset for different numbers of neurons are given in Fig. 5 . For GT3330001, the minimum values of the absolute difference between the training and test RMSE were 0.09 m (hidden neurons: 180-150-80), 0.03 m (hidden neurons: 150-100-50), and 0.06 m (hidden neurons: 150-100-50) for one-, two-, and three-week(s) ahead forecasting, respectively (Fig. 5 ). On the other hand, at GT3330002, the RMSE values were 3.13 m (hidden neurons: 150-120-80-50), 3.22 m (hidden neurons: 140-120-60), and 3.14 m (hidden neurons: 100-80-50-20) for one-, two-, and three-week(s) ahead forecasting, respectively. Therefore, the LSTM models with these hidden neurons were selected as the best-performing models. The findings (RMSE, Scatter index, MAE, MAD, and a-20 index) of the standalone LSTM models for forecasting GWLs at one-, two-, and three-week(s) ahead are presented in Table 3 . The overall accuracy across the two observation wells for the standalone LSTM models showed a very good performance in terms of scatter index (Li et al., 2013 ), MAD, and a-20 index (Xu et al., 2019 ) criteria whereas the performances were reasonably good with reference to the RMSE and MAE criteria. It is noted that the performances of the standalone LSTM models at the observation well GT3330001 were in general superior than those at the observation well GT3330002. The plausible reason for this discrepancy in modeling performances may be the quality and quantity of the observed data as well as the number of missing values that were imputed. Nevertheless, the developed LSTM models produced acceptable results at both observation wells (Table 3 ). Table 3 One-, two-, and three-week(s) ahead forecasting performance of the developed standalone LSTM models on test dataset Performance evaluation indices RMSE, m Scatter index MAE, m MAD, m a-20 index GT3330001 One week ahead 0.827 0.034 12.480 0.179 0.997 Two weeks ahead 0.910 0.037 10.237 0.319 0.997 Three weeks ahead 0.975 0.040 10.754 0.405 0.997 GT3330002 One week ahead 3.466 0.188 7.561 1.200 0.589 Two weeks ahead 3.623 0.196 7.727 1.326 0.531 Three weeks ahead 3.493 0.189 7.393 1.327 0.573 Although the accuracy of LSTM forecasts decreased with increasing lead times (Rahman et al., 2020 ), the LSTM models developed at the observation well GT3330001 showed very good performance (RMSE = 0.975 m, Scatter index = 0.040, MAD = 0.405, m and a-20 index = 0.997) for three-weeks-ahead GWL forecasting. For the monitoring well GT3330002, differences in the forecasting performances were also not substantial with respect to the increased lead times. It is worth mentioning that several previous studies compared the forecasting accuracies of ML algorithms such as SVR, MLR, ANN, and RF (Rajaee et al., 2019 ) and found SVR to be the best performing model (Rahman et al., 2020 ). Therefore, it is practically impossible to provide a direct comparison of these findings with the mentioned previous studies in GWL forecasting. Moreover, the study locations are also different. In the same study location, different ML approaches also provided superior performance over others for different observation well locations (Rahman et al., 2020 ). However, it can be argued from the findings of this study that LSTM models can effectively be utilized to provide reasonable GWL-level forecasts. 4.3 Performance of the MODWPT Coupled LSTM Models The findings of the MODWPT coupled LSTM models for the two observation wells are presented in this section. The obtained best numbers of hidden layers and hidden neurons for the standalone LSTM models were used to develop MODWTP-based LSTM models (LSTM- MODWTP). The training performance of the generated LSTM- MODWTP models at the two observation well locations is illustrated in Fig. 6 . These results support that MODWPT as a preprocessing tool significantly improved the training and testing performance of the LSTM models (Fig. 6 ). The smallest differences between the training and test RMSE values ensures that the coupled LSTM-MODWPT models were trained adequately and no model overfitting was observed during the model training. In addition, this study adopted the best practices proposed in Quilty and Adamowski ( 2018 ) to correct the data against the boundary affected datasets. This approach allowed us to use the wavelet transforms correctly as a data preprocessing tool for GWL forecasting. This also ensured that MODWPT did not universally lead to enhanced performances of the LSTM- MODWPT models over the standalone LSTM models. After checking for model overfitting using RMSE criterion, the trained models were used to compute several other statistical performance evaluation indices on the test dataset. The statistical performance outcomes for the one-, two-, and three-week(s)-ahead forecasting performance of the developed LSTM- MODWTP model on the test dataset are summarized in Table 4 . Table 4 reveals that similar to the standalone LSTM models (results are presented in Table 3 ), the performances of the LSTM- MODWTP for multi-step forward forecasts at GT3330001 were, in general, better than the forecasting abilities of the LSTM- MODWTP developed at GT3330002. The plausible reasoning for this performance deviation may have resulted from the length and quality of data and the number of missing data that were imputed. However, the developed LSTM-MODWPT models provided acceptable results at both observation wells. Similar metrics for evaluating the performance were calculated to assess the robustness of the proposed LSTM- MODWTP models (presented in Table 4 ). It is evident from Table 4 that MODWPT improved the LSTM performance at both observation wells and for all lead times as evidenced by the higher values of a-20 index and lower values of RMSE, Scatter index, MAE, and MAD criteria. Although the forecasting performance slightly decreased with the increased forecasting timeframes, the forecasting performance for the last forecasting horizon (three weeks ahead) is reasonably good and within the acceptable limits for the particular statistical indices. The improvements in forecasting performance of the LSTM-MODWPT over the standalone LSTM models are quite satisfactory at both observation wells for all forecasting horizons. The percentage improvements in the forecasting accuracy based on RMSE criterion were 36.28% for one-week-ahead forecasting at GT3330001, 32.97% for two-weeks-ahead forecasting at GT3330001, 30.77% for three-weeks-ahead forecasting at GT3330001, 78.68% for one-week-ahead forecasting at GT3330002, 76.37% for two-weeks-ahead forecasting at GT3330002, and 74.92% for three-weeks-ahead forecasting at GT3330001. The percentage improvements in the forecasting accuracy based on Scatter index criterion were 29.41% (one-week-ahead forecasting at GT3330001), 27.03% (two-weeks-ahead forecasting at GT3330001), 25% (three-weeks-ahead forecasting at GT3330001), 47.87% (one-week-ahead forecasting at GT3330002), 58.67% (two-weeks-ahead forecasting at GT3330002), and 54.50% (three-weeks-ahead forecasting at GT3330001). The percentage improvements in the forecasting accuracy based on MAE criterion were 56.49% (one-week ahead forecasting at GT3330001), 51.96% (two-weeks-ahead forecasting at GT3330001), 56.52% (three-weeks-ahead forecasting at GT3330001), 35.29% (one-week-ahead forecasting at GT3330002), 46.62% (two-weeks-ahead forecasting at GT3330002), and 46.04% (three-weeks-ahead forecasting at GT3330001). The percentage improvements in the forecasting accuracy based on MAD criterion were 14.53% (one-week-ahead forecasting at GT3330001), 31.97% (two-weeks-ahead forecasting at GT3330001), 11.85% (three-weeks-ahead forecasting at GT3330001), 27.75% (one-week-ahead forecasting at GT3330002), 32.58% (two-weeks-ahead forecasting at GT3330002), and 31.80% (three-weeks-ahead forecasting at GT3330001). The percentage improvements in the forecasting accuracy based on a-20 index criterion were 0.20% (one-week-ahead forecasting at GT3330001), 0% (two-weeks-ahead forecasting at GT3330001), 0.10% (three-weeks-ahead forecasting at GT3330001), 48.73% (one-week-ahead forecasting at GT3330002), 66.29% (two-weeks-ahead forecasting at GT3330002), and 59.69% (three-weeks-ahead forecasting at GT3330001). The most remarkable finding in Table 4 is that MODWPT especially improved the forecasting performance of the standalone LSTM models at the observation well GT3330002, where the standalone LSTM models performed poorly in all forecasting horizons. Table 4 One-, two-, and three-week(s)-ahead forecasting performance of the developed LSTM- MODWPT model on test dataset Performance evaluation indices RMSE, m Scatter index MAE, m MAD, m a-20 index GT3330001 One week ahead 0.527 0.024 5.430 0.153 0.999 Two weeks ahead 0.610 0.027 4.918 0.217 0.997 Three weeks ahead 0.675 0.030 4.676 0.357 0.998 GT3330002 One week ahead 0.739 0.098 4.893 0.867 0.876 Two weeks ahead 0.856 0.081 4.125 0.894 0.883 Three weeks ahead 0.876 0.086 3.989 0.905 0.915 In summary, Table 4 reveals the superiority of the proposed LSTM-MODWPT over the standalone LSTM models (Table 3 ) for the selected observation wells for the three forecast periods. This superior performance was evidenced through the five statistical performance evaluation indices considered for evaluating the models' performances in this study. Even though this study showed the promises of LSTM-MODWPT models under the conditions studied, further assessments for such modelling aspects would be required for other geographic locations. The outcomes of this research would improve the overall performance accuracy, reduce modelling complexity, and ease parameter selections for groundwater modelling. This finding is important in the water resources management since the early forecasting of GWLs is crucial in decision making in the fields of irrigation scheduling and planning, land development and in many other research domains including environmental sciences. 5 Conclusions An efficient and sustainable groundwater management plan can be developed using accurate and reliable predictions of GWLs. This planning will aid the optimal abstraction recommendations and groundwater usage for agricultural, domestic, and industrial purposes. However, due to the nonlinear nature of GWLs as well as their multiscale and time-varying behavior, it is frequently difficult to provide accurate GWL forecasts. One of the most influential pre-requisites of developing ML-based GWL forecast models is the appropriate ML algorithm choice. LSTM models have shown promising performance in hydrological and other time series predictions. Equally important is the incorporation of a suitable data preprocessing approach, which is believed to improve forecasting performance of ML-based algorithms. To address these issues, this research created a reliable forecasting tool using coupled LSTM-MODWPT models for one-, two-, and three-week(s)-ahead GWL fluctuations. MODWPT was used to acquire multiscale information from the GWL time series, which were incorporated in developing LSTM models to improve LSTM’s forecasting capability. The suitable weekly lag times of GWLs and their wavelet packet transformed counterparts were deployed as input variables to the forecast models. In contrast, the outputs were the one, two, and three week(s) ahead GWLs. The selection of an ideal blend of input variables for the proposed models was implemented through a RF-based modeling approach. The proposed models' performance assessment was executed using various statistical performance assessment metrics by which LSTM-MODWPT models were benchmarked against their non-wavelet-based counterparts, e.g., the standalone LSTM models. Results of the present study indicated that the LSTM-MODWPT models outperformed the standalone LSTM models for all three future time horizons and at each observation well. Therefore, it can be concluded that LSTM-MODWPT models could predict multi-step-ahead GWL fluctuations quite accurately for the study area. The proposed modeling approach was promising for short-term GWL forecasts at the specified observation wells of a water scarce region in Bangladesh that can be extended to other geographical areas. Moreover, this promising modeling framework can be applied to the research areas of hydrology and water resources for medium-term and long-term forecasting. Declarations Acknowledgements The authors are grateful to Emily Bellis [1] for providing an initial review of this manuscript, which improved the original manuscript. Author Contribution All authors contributed to the study conception and design . Dilip Kumar Roy: Conceptualization, Methodology, Formal analysis, Software, Validation, Visualization, Writing - original draft. Ahmed A. Hashem: Supervision, Writing - review & editing. Michele L. Reba: Conceptualization, Supervision, Writing - review & editing. Deborah L. Leslie: Assisted in analyzing the results, reviewed and edited the manuscript. John Nowlin: Reviewed the manuscript and provided constructive suggestions. Statements & Declarations Compliance with ethical standards: There are no potential conflicts of interest. The research does not include human participants and/or animals. Consent to participants: The research does not include human participants and/or animals. Consent to publish: The authors give their consent to publish in the Water Resources Management journal if accepted for publication. Funding: The authors declare that no funds, grants, or other support were received during the preparation of this manuscript. Competing interests: The authors have no relevant financial or non-financial interests to disclose. Availability of data and material : Datasets and other materials are available with the authors, and may be accessible at any time upon request. Code availability: MATLAB codes are available with the first author and will be provided upon request. 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Geotech Geol Eng 37:1661–1670. https://doi.org/10.1007/s10706-018-0713-6 Tian Y, Xu Y-P, Yang Z, Wang G, Zhu Q (2018) Integration of a parsimonious hydrological model with recurrent neural networks for improved streamflow forecasting. Water 10:1655 Tien Bui D, Hoang N-D, Martínez-Álvarez F, Ngo P-TT, Hoa PV, Pham TD, Samui P, Costache R (2020) A novel deep learning neural network approach for predicting flash flood susceptibility: A case study at a high frequency tropical storm area. Sci Total Environ 701:134413. https://doi.org/https://doi.org/10.1016/j.scitotenv.2019.134413 Wada Y, Bierkens MFP (2014) Sustainability of global water use: past reconstruction and future projections. Environ Res Lett 9:104003. https://doi.org/10.1088/1748-9326/9/10/104003 Walden AT, Contreras Cristan A (1998) The phase-corrected undecimated discrete wavelet packet transform and its application to interpreting the timing of events. Proc. R. Soc. A Math. Phys. Eng. Sci. 454, 2243–2266. https://doi.org/10.1098/rspa.1998.0257 Wang H, Zhao W (2009) ARIMA model estimated by particle swarm optimization algorithm for consumer price index forecasting BT - artificial intelligence and computational intelligence, in: Deng, H., Wang, L., Wang, F.L., Lei, J. (Eds.), International Conference on Artificial Intelligence and Computational Intelligence. Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 48–58 Wei Z-L, Wang D-F, Sun H-Y, Yan X (2020) Comparison of a physical model and phenomenological model to forecast groundwater levels in a rainfall-induced deep-seated landslide. J Hydrol 586:124894. https://doi.org/https://doi.org/10.1016/j.jhydrol.2020.124894 Wen X, Feng Q, Yu H, Wu J, Si J, Chang Z, Xi H (2015) Wavelet and adaptive neuro-fuzzy inference system conjunction model for groundwater level predicting in a coastal aquifer. Neural Comput Appl 26:1203–1215. https://doi.org/10.1007/s00521-014-1794-7 Xu H, Zhou J, Asteris PG, Jahed Armaghani D, Tahir MM (2019) Supervised machine learning techniques to the prediction of tunnel boring machine penetration rate. Appl Sci. https://doi.org/10.3390/app9183715 Yang T, Asanjan AA, Welles E, Gao X, Sorooshian S, Liu X (2017) Developing reservoir monthly inflow forecasts using artificial intelligence and climate phenomenon information. Water Resour Res 53:2786–2812. https://doi.org/10.1002/2017WR020482 Yuan X, Chen C, Lei X, Yuan Y, Muhammad Adnan R (2018) Monthly runoff forecasting based on LSTM–ALO model. Stoch Environ Res Risk Assess 32:2199–2212. https://doi.org/10.1007/s00477-018-1560-y Zahid A, Hossain A (2014) Bangladesh Water Development Board: A bank of hydrological data essential for planning and design in water sector. In: 2nd International Conference on Advances in Civil Engineering 2014 (ICACE-2014), 26 – 28 December, 2014, CUET, Chittagong, Bangladesh Zare M, Koch M (2018) Groundwater level fluctuations simulation and prediction by ANFIS- and hybrid Wavelet-ANFIS/Fuzzy C-Means (FCM) clustering models: Application to the Miandarband plain. J Hydro-environment Res 18:63–76. https://doi.org/https://doi.org/10.1016/j.jher.2017.11.004 Zhang J, Zhu Y, Zhang X, Ye M, Yang J (2018) Developing a long short-term memory (LSTM) based model for predicting water table depth in agricultural areas. J Hydrol 561:918–929. https://doi.org/https://doi.org/10.1016/j.jhydrol.2018.04.065 Additional Declarations No competing interests reported. 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Hashem","email":"","orcid":"","institution":"Arkansas State University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Ahmed","middleName":"A.","lastName":"Hashem","suffix":""},{"id":241472219,"identity":"a40cfb66-ddb6-4eed-a5d6-8e874a0599d0","order_by":2,"name":"Michele L. Reba","email":"","orcid":"","institution":"United States Department of Agriculture","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Michele","middleName":"L.","lastName":"Reba","suffix":""},{"id":241472220,"identity":"310c56e5-e02d-4c16-bddd-bd7b5113ef24","order_by":3,"name":"Deborah L. Leslie","email":"","orcid":"","institution":"University of Memphis","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Deborah","middleName":"L.","lastName":"Leslie","suffix":""},{"id":241472221,"identity":"b926738b-fbf2-44ff-954d-a49b2dfabcc7","order_by":4,"name":"John Nowlin","email":"","orcid":"","institution":"Arkansas State University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"John","middleName":"","lastName":"Nowlin","suffix":""}],"badges":[],"createdAt":"2023-10-19 04:29:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3464867/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3464867/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":45168254,"identity":"5a6a1aeb-af10-4752-8c3e-140f86492f9e","added_by":"auto","created_at":"2023-10-24 17:23:00","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":173695,"visible":true,"origin":"","legend":"\u003cp\u003eStudy area with two observation well locations in the Gazipur Sadar Upazilla\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3464867/v1/99ba8ce7046cab9820d8f1df.jpg"},{"id":45168252,"identity":"85b6ea31-7eaa-4ce1-a6e9-f9aefd18085e","added_by":"auto","created_at":"2023-10-24 17:23:00","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":58046,"visible":true,"origin":"","legend":"\u003cp\u003ePACF plots of the weekly GWL timeseries for the observation wells (a) GT3330001, (b) GT3330002\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3464867/v1/88c5fc460bf86d69523bfdd5.png"},{"id":45168248,"identity":"71b8ea84-aa2b-44a8-ba46-2ab0ab27d347","added_by":"auto","created_at":"2023-10-24 17:23:00","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":29060,"visible":true,"origin":"","legend":"\u003cp\u003eFlow chart of the main model development steps\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3464867/v1/7d8f3e151cd5cf46fa113834.png"},{"id":45168251,"identity":"945d7a51-8715-4b87-9c15-acb70bfffd93","added_by":"auto","created_at":"2023-10-24 17:23:00","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":83268,"visible":true,"origin":"","legend":"\u003cp\u003eMost significant inputs variables determined by RF approach: left panes (a, b, and c) represent one-, two-, and three-step weekly lead times, respectively, for GT3330001, whereas the second panes (d, e, and f) represent one-, two-, and three-step weekly lead times, respectively, for GT3330002\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3464867/v1/61e27c9441012531eb654ed2.png"},{"id":45169098,"identity":"a8d115c3-51f4-4851-856e-c4b85e92b2a8","added_by":"auto","created_at":"2023-10-24 17:31:00","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":128158,"visible":true,"origin":"","legend":"\u003cp\u003eTraining and testing RMSE values of the standalone LSTM models for different combinations of hidden layers and hidden neurons for the observation wells (a) GT3330001 and (b) GT3330002\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3464867/v1/cc07b8d6a77cd567e0628c8d.png"},{"id":45168249,"identity":"47180c02-e23d-479a-939e-6806382f6731","added_by":"auto","created_at":"2023-10-24 17:23:00","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":65488,"visible":true,"origin":"","legend":"\u003cp\u003eModel training and testing phase errors for the developed LSTM-MODWTP for the two observation wells (a) GT3330001, (b) GT3330002\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3464867/v1/9e9d9f1a9e3920eceb91f395.png"},{"id":47728886,"identity":"3065e7bc-39ec-4644-97a4-0ecd590fa06b","added_by":"auto","created_at":"2023-12-06 16:37:39","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1011053,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3464867/v1/37f0723d-051f-4d16-8120-652fae4036df.pdf"},{"id":45168253,"identity":"dd739746-a6f2-475e-b76a-c12cf9889664","added_by":"auto","created_at":"2023-10-24 17:23:00","extension":"docx","order_by":12,"title":"","display":"","copyAsset":false,"role":"supplement","size":172701,"visible":true,"origin":"","legend":"","description":"","filename":"SupplementaryInformation.docx","url":"https://assets-eu.researchsquare.com/files/rs-3464867/v1/05a24de6528cee1109c42b11.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Maximal Overlap Discrete Wavelet Packet Transform Coupled with an LSTM Deep Learning Model for Improving Multilevel Groundwater Level Forecasts","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eGroundwater aquifers are regarded as the primary sources of the world's clean water supplies and play a crucial role in the stability of irrigated agriculture, domestic water requirements, and industrial water supplies in locations where high-quality surface water is insufficient (Gong et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2016\u003c/span\u003e, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Roy et al., \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2021a\u003c/span\u003e). Groundwater supplies are under increased pressure due to population growth, rising water demand, and the inevitable influences of climate change (Wada and Bierkens, \u003cspan citationid=\"CR78\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Consequently, groundwater systems are undergoing an accelerated decline. Although human action, such as excessive pumping, is thought to be the significant determinant of groundwater level (GWL) decline, recent forecasts suggest that the situation will worsen even sooner than predicted due to climate change (Wada and Bierkens, \u003cspan citationid=\"CR78\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). Due to the over-extraction of limited groundwater reserves, groundwater resources will continue to be depleted, causing various environmental, operational, and economic problems (Banerjee et al., \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). Groundwater is a significant source of water supply in Bangladesh, where ~\u0026thinsp;80% of the population relies on groundwater supplies primarily for the basic water requirements (Hoque and Adhikary, \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Consequently, prudent management and sustainable use of the aquifer's limited groundwater reserves are imperative to secure steady groundwater supplies for future generations. Precise prediction and projection of impending GWL fluctuations may generate a practical groundwater management approach in Bangladesh and globally (Roy et al., \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2021a\u003c/span\u003e, \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003eb\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eNumerical simulation models of groundwater flow processes have historically been used in groundwater hydrology to forecast GWLs and better understand the system's underlying mechanisms (Doble et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Masterson and Garabedian, \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Park and Parker, \u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). However, accurate predictions of GWLs using simulation models require a thorough knowledge of the aquifer's characteristics and skilled modelers with a thorough understanding of aquifer geometry and modeling strategies. As such, data-driven modeling techniques have been developed and deployed to reduce modeling complexities in the hydrological research domains (Fahimi et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Govindaraju, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2000a\u003c/span\u003e, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2000b\u003c/span\u003e; Maier et al., \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Sadler et al., \u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Yang et al., \u003cspan citationid=\"CR84\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). The attributes of the underlying physical processes do not need to be explicitly defined to develop data-driven models. In Machine Learning (ML) based data-driven modeling techniques, a model's inputs and outputs are directly mapped or correlated using an iterative learning process (Solomatine and Ostfeld, \u003cspan citationid=\"CR73\" class=\"CitationRef\"\u003e2008\u003c/span\u003e). In the realm of the prediction of GWL time series data, it has been discovered that a data-driven model developed using an Artificial Neural Network (ANN) performed as well as or had improved prediction compared to a numerical simulation model (Karandish and Šimůnek, \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Mohanty et al., \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). As such, there has been increasing interest in data-driven modeling techniques as alternatives to complex numerical simulation models. Sufficient and accurate prediction of GWL over the short- to medium-term helps develop a groundwater management plan in regions where droughts initiated by climate change or over pumping are the primary driving forces (Feng et al., \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Guzman et al., \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Sahoo et al., \u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Different data-driven modeling techniques are increasingly being used since, compared to conventional hydrogeological modeling, as fewer datum points are necessary with also easier execution (Zhang et al., \u003cspan citationid=\"CR88\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Numerous techniques have recently been employed in the research area related to predictions of GWL fluctuations. These approaches ranged from standalone data-driven modeling to hybrid modeling (Adamowski and Chan, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Daliakopoulos et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Obergfell et al., \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Roshni et al., \u003cspan citationid=\"CR63\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Sakizadeh et al. \u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e2019\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThe standalone modeling approaches comprise ML-based modeling (Samani et al. \u003cspan citationid=\"CR70\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Dong et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Guzman et al., \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Mohanty et al., \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Sahoo et al., \u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), ANNs (Ghorbani et al., \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Lee et al., \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Dadhich et al. \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2021\u003c/span\u003e), NARX neural networks (Guzman et al., \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), ANFIS (Roy et al., \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2021a\u003c/span\u003e; Nadiri et al., \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Nourani and Mousavi, \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Raghavendra and Deka, \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Wen et al., \u003cspan citationid=\"CR82\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Zare and Koch, \u003cspan citationid=\"CR87\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), Gaussian Process Regression (Raghavendra and Deka, \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2015\u003c/span\u003e), Prophet modeling technique (Aguilera et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2019\u003c/span\u003e), Support Vector Machine (SVM) (Nadiri et al., \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Tang et al., \u003cspan citationid=\"CR75\" class=\"CitationRef\"\u003e2019\u003c/span\u003e), and Discrete Space-State model (Roy et al., \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2021b\u003c/span\u003e). On the other hand, the hybrid approaches used to predict GWL fluctuations are: hybrid Wavelet Transform\u0026ndash;ML approaches (Adamowski and Chan, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Barzegar et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Peng et al., \u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Raghavendra and Deka, \u003cspan citationid=\"CR60\" class=\"CitationRef\"\u003e2015\u003c/span\u003e), hybrid ML and Ensemble Empirical Mode Decomposition (Gong et al., \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), Wavelet \u0026ndash; ANFIS (Moosavi et al., \u003cspan citationid=\"CR47\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), and Nonlinear System Identification models coupled with Linear Polynomials (Makungo and Odiyo, \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). Another domain of hybrid ML-based GWL prediction includes the usage of evolutionary algorithms to tune the model parameters. These evolutionary algorithm tuned ML models include the application of Whale Algorithm \u0026ndash; ANN (Banadkooki et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), Particle Swarm Optimization (PSO) \u0026ndash; ARIMA (Boubaker, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2017\u003c/span\u003e), hybrid Self Organizing Map- and Multi-objective Genetic Algorithm-SVM (Fang et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2019\u003c/span\u003e), and hybrid SVM-PSO (Wei et al., \u003cspan citationid=\"CR81\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Mozaffari et al. \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). A thorough analysis of ML-based methods for modeling GWLs is provided by Rajaee et al. (\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) and by Sarma and Singh (\u003cspan citationid=\"CR71\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). It is evident that a variety of modeling techniques have been used to forecast changes in GWL with differing extents of prediction accuracy. It is also clear that recommending a specific prediction model for projecting GWL changes is technically challenging, and possibly unachievable. Consequently, improving the prediction accuracy of GWL variations still necessitates more sophisticated approaches.\u003c/p\u003e \u003cp\u003eDeep Learning (DL) has recently been used as an advanced well-developed sub-area of ML-based approaches. A growing number of scientific fields have successfully used DL-based modeling (Plappert et al., \u003cspan citationid=\"CR57\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Fang et al., \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Fan et al., \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Cummins et al., \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). The recent application of DL has also been noted in the prediction of time series data (Tien Bui et al., \u003cspan citationid=\"CR77\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Xu et al., \u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Yang and Chen, 2019), forecasting of GWLs (Bowes et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Supreetha et al., \u003cspan citationid=\"CR74\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) and in estimating future water quality variables in the short term (Barzegar et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). As such, several recent groundwater modeling studies (Guzman et al., \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Chang et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Daliakopoulos et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2005\u003c/span\u003e) have concentrated on the effective usage of DL-based Recurrent Neural Networks (RNNs). Long-term dependence between variables, however, is difficult for standard RNN designs to capture (Bengio et al., \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1994\u003c/span\u003e), primarily because of the presence of two issues: vanishing and exploding gradients. The exploding and vanishing gradient difficulties of RNNs can be overcome via Long Short-Term Memory (LSTM) networks, an advanced version of traditional RNN topologies. However, LSTMs have only lately been used to predict hydrological time series, despite their widespread applicability in numerous study fields (Hu et al., \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Liang et al., \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Tian et al., \u003cspan citationid=\"CR76\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Zhang et al., \u003cspan citationid=\"CR88\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). Jeong et al. (\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) used LSTM-based models to predict GWLs utilizing real-world \u0026lsquo;faulty\u0026rsquo; data with outliers and noise. For a seaside city in the USA prone to episodic inundations, it was discovered that an LSTM network's predictive power was superior to an RNN when it came to predicting hourly GWL values (Bowes et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). In order to achieve accuracy and resilience in irrigation flow forecasting, Mouatadid et al. (\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) paired a Maximal Overlap Discrete Wavelet Transform (MODWT) and an LSTM model. Zhang et al. (\u003cspan citationid=\"CR88\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) suggested an LSTM-based model for forecasting water table elevation in agricultural areas and achieved a reliable prediction outcome by deploying a straightforward data preprocessing method for data standardization. Considering recent literature, LSTMs have the potential to be utilized in the research domain of hydrological time series projections. Therefore, this study represents the first endeavor of employing LSTM-based models coupled with a data preprocessing tool for forecasting multi-step forward GWLs at designated observation wells in the Gazipur Sadar Upazilla, Bangladesh.\u003c/p\u003e \u003cp\u003eWavelet decomposition and wavelet packet transform as preprocessing tools have successfully been utilized in numerous research domains to improve the forecasting capability of ML-based models. However, recent studies related to wavelet packet-based prediction modeling techniques are associated with issues of leveraging the \u0026ldquo;future data.\u0026rdquo; Therefore, these applications are not appropriate for application in real-life forecasting scenarios (Quilty et al., 2021; Quilty and Adamowski, \u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). On the other hand, a Maximal Overlap Discrete Wavelet Packet Transform (MODWPT) approach is better suited for real-world applications because it can overcome the \u0026ldquo;future data\u0026rdquo; issues. This study utilized MODWPT as a preprocessing tool to improve the forecasting capabilities of an LSTM model for multiple-step forward GWL forecasting. Therefore, this study's key motivation and focus are to assess the potential use of MODWPT as a preprocessing tool to improve the forecasting capability of LSTM for predicting multiple-step forward GWLs for the designated observation well locations.\u003c/p\u003e"},{"header":"2 Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Study Area and Input Data\u003c/h2\u003e \u003cp\u003eThe study area is in the Gazipur Sadar Upazilla, with approximate surface area of 446.38 km\u003csup\u003e2\u003c/sup\u003e. It is situated between the longitudes of 90.33\u0026deg; and 92.50\u0026deg;E and the latitudes of 23.88\u0026deg; and 24.18\u0026deg;N. This area falls within the physiographic unit referred to as the Madhupur jungle tract, locally known as \u003cem\u003eBhawal Garh\u003c/em\u003e. The terrace topography in many parts of the Madhupur jungle tract ranges from flat areas to lower-rounded mountains and ridges separated by closely spaced shallow \u003cem\u003e\u0026lsquo;bides\u0026rsquo;\u003c/em\u003e (valleys) (Bangladesh Bureau of Statistics (BBS), \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). Additionally, a few \u0026lsquo;\u003cem\u003ebeels\u003c/em\u003e\u0026rsquo; (lakes) are located near the periphery of the study area. The soil conditions in the \u003cem\u003eBhawal Garh\u003c/em\u003e are varied and often complex. A wide range of soil variability exists, from red laterite soils at the extreme to almost undeveloped soil of raw Pleistocene clay (Bangladesh Bureau of Statistics (BBS), \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), with numerous intermediate soil layers. Four major constituents of land formation were reported (Bangladesh Bureau of Statistics (BBS), \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2013\u003c/span\u003e): clay (70.02%), sandy loam (13.84%), sand (6.92%), and pebble (4.61%). Wet season rainfall is one of the critical sources of water for groundwater replenishment and recharge, which is often constrained by the thick clay soil surface and low rainfall periods leading to decreasing natural recharge. Pumped groundwater serves as the primary water source for residential and agricultural water requirements.\u003c/p\u003e \u003cp\u003eIn this study, secondary GWL data were collected from the Processing and Flood Forecasting Circle (PFFC) of the Bangladesh Water Development Board (BWDB) (Zahid and Hossain, \u003cspan citationid=\"CR86\" class=\"CitationRef\"\u003e2014\u003c/span\u003e). GWL data were collected at various observation well locations. Two observation wells, GT3330001 and GT3330002, were selected based on the criteria of the fewest missing GWL entries. A data quality assurance procedure is often deployed to warrant the qualities of the obtained GWL datasets, which boosts the trustworthiness of GWL forecasts using ML tools (Est\u0026eacute;vez et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). Although a thorough quality assurance procedure was not accomplished for the current dataset, the quality of the acquired GWL data was assessed systematically for its exactness and comprehensiveness using range/limit tests. The basis of range testing is the straightforward verification that any observation is contained within a given range (Est\u0026eacute;vez et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2016\u003c/span\u003e). Measurements outside of this range are correctly flagged as invalid, and only measurements falling within this threshold are accepted (Feng et al., \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Shafer et al., \u003cspan citationid=\"CR72\" class=\"CitationRef\"\u003e2000\u003c/span\u003e). The data occurring inside the acceptable range were utilized to simulate future GWL changes in the selected observation wells by focusing on providing a multiple-step ahead GWL projection. The study domain and locations of observation wells are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eThe weekly GWL data collected from BWDB for GT3330001 ranged between 07 January 1980 and 17 September 2018 or 2019 data points. There were 29 missing values or 1.4% of the data, leaving 1983 data points for analysis. On the other hand, the weekly GWL data at GT3330002 ranged from 07 January 1980 to 26 December 2016 with 1929 data points. There were 18 missing values or 0.93% of the data leaving 1919 data points for analysis. Those omitted GWL data were replaced using the \u0026lsquo;moving median\u0026rsquo; method of data imputing in which a moving median with a specified window length was used. Finally, the observation wells GT3330001 and GT3330002 had 2012 readings (from 07 January 1980 to 17 September 2018) and 1937 readings (from 07 January 1980 to 26 December 2016) of weekly GWL entries after the imputation of missing entries.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe weekly GWL dataset\u0026rsquo;s descriptive statistics are presented in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The mean of the GWL data ranges from 12.96 m (at GT3330001) to 13.26 m (at GT3330002), while the standard deviation values range from 3.39 m (at GT3330002) to 7.92 m (at GT3330001) (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The positively skewed values indicate that the statistical distribution of the data at both observation wells has a larger right tail than a left tail (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). The GWL data at GT3330001 had negative kurtosis values and light-tailed distributions. On the other hand, the datasets at GT3330002 showed heavy-tailed distributions because of the positive kurtosis value.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eValues of the statistical parameters computed on the GWL data (m) at the designated observation wells\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eObs. wells\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eMin\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eMax\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMean\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMedian\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003eSTD\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003eSkewness\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eKurtosis\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT3330001\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e25.57\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e12.96\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e9.83\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e7.92\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e-1.39\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGT3330002\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e6.30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e23.50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e13.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e13.26\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e3.39\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e0.74\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e0.44\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Modelling Approaches\u003c/h2\u003e \u003cp\u003eBrief descriptions of various approaches, including the Long Short Term Memory Network (LSTM), Maximal Overlap Discrete Wavelet Packet Transform (MODWPT), integrated LSTM-MODWPT, input variable selection using Random Forest (RF), and data partitioning, are provided in the following sections.\u003c/p\u003e \u003cdiv id=\"Sec5\" class=\"Section3\"\u003e \u003ch2\u003e2.2.1 Long-Short Term Memory (LSTM) Networks\u003c/h2\u003e \u003cp\u003eAn LSTM model network is a subtype of advanced Recurrent Neural Networks (RNNs) able to acquire longer-term reliance amongst sequence data time steps. Because LSTMs incorporate state dynamics and gating functions, they overcome the exploding and vanishing gradient issues of conventional RNNs, making them particularly effective for predicting sequence data (Hochreiter and Schmidhuber, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e1997\u003c/span\u003e). The LSTM network's architecture is made up of multiple memory blocks connected by layers, and each of which has a substantial number of memory cells with recurrent connections. The forget, input, and output gates are examples of the three multiplicative parts (called gates) that make up an LSTM memory cell (Yuan et al., \u003cspan citationid=\"CR85\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). A sequential input layer is used to provide time series data into an LSTM model network, making up the majority of a primary LSTM network. There are four layers in an LSTM network used to solve a simple regression problem: the LSTM network commences with a sequential input layer preceded by an LSTM layer, and the network concludes with an entirely interconnected layer and a regression output layer. The straightforward and deeper LSTM architectures as well as the mechanism by which LSTMs perform forecasting tasks are shown in Figs. S1, S2, and S2 of the Supplemental Information (SI).\u003c/p\u003e \u003cp\u003eAn LSTM network architecture with three hidden layers was used. Model overfitting was prevented by assigning a dropout layer to each hidden layer. The number of hidden neurons and the associated dropout rates in the 1st, 2nd, and 3rd hidden layers were selected upon several iterations. The optimal numbers of hidden neurons were 100, 50, and 20 for the 1st, 2nd, and 3rd hidden layers, respectively, while the corresponding optimal dropout rates were 0.4, 0.3, and 0.4, respectively.\u003c/p\u003e \u003cp\u003eThe LSTM architecture with multiple hidden units was employed. The numbers of \u0026lsquo;hidden neurons\u0026rsquo; were decided via several trials by varying the number of \u0026lsquo;hidden neurons\u0026rsquo; in each iteration. The LSTM architecture parameters were obtained upon numerous trials, and the optimum sets of parameter values are presented in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. These optimum parameter values were used for developing the LSTM models to predict one, two, and three-week(s) ahead GWL forecasting at the two observation wells.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe best pairings of various training alternatives\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTraining options\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRelevant option values\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSolver used to optimize\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e'adam'\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMaximum number of epochs\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGradient threshold value\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eInitial rate of learning\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.001\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eMinimum size of the batch\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e150\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eLength of the sequence\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1000\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2 Maximal Overlap Discrete Wavelet Packet Transform (MODWPT)\u003c/h2\u003e \u003cp\u003eThe formulation of wavelet decomposition using MODWPT may be expressed as (Walden and Contreras Cristan, \u003cspan citationid=\"CR79\" class=\"CitationRef\"\u003e1998\u003c/span\u003e):\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$${\\stackrel{\\sim}{W}}_{j,n,t}^{P}=\\sum _{l=0}^{L-1}{\\stackrel{\\sim}{f}}_{n,l}{\\stackrel{\\sim}{W}}_{j-1,\\left[n/2\\right],\\left(t-{2}^{j-1}l\\right) \\text{m}\\text{o}\\text{d} N}^{P}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere,\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${\\stackrel{\\sim}{f}}_{n,l}=\\left\\{\\begin{array}{c}{\\stackrel{\\sim}{g}}_{l,} if n mod 4=0 or 3\\\\ {\\stackrel{\\sim}{h}}_{l,} if n mod 4=1 or 2\\end{array}\\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\stackrel{\\sim}{W}}_{j,n,t}^{P}\\)\u003c/span\u003e\u003c/span\u003e represents the coefficients of MODWPT at time \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(t\\)\u003c/span\u003e\u003c/span\u003e for the level of decomposition\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(j\\)\u003c/span\u003e\u003c/span\u003e and band \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\((\\text{w}\\text{h}\\text{e}\\text{r}\\text{e} n=\\text{0,1}, \\dots ,{2}^{j}-1)\\)\u003c/span\u003e\u003c/span\u003e, which is technically corresponding to frequencies for an interval of\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left[\\frac{n}{{2}^{j+1}},\\frac{n+1}{{2}^{j+1}} \\right].\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003eThe MODWPT is a wavelet decomposition algorithm that uses energy, and the output of the MODWPT is time-delayed signals relative to the input signals. The MODWPT partitions the energy over the entire wavelet packets at each level. The total energy of any input signal equals the sum of the energies over all the wavelet packets. On the other hand, the features in the MODWPT details line up closely with the features contained in the input signal. The details, like the energies, produce the original signal when summing the details in each sample for a given level. The MODWPT executes a discrete wavelet packet transform and returns a \u0026lsquo;sequency-ordered\u0026rsquo; wavelet packet tree. Time series decomposition process via MODWPT using a sequence-ordered wavelet packet tree can be found in Fig. S4 of the SI.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e2.2.3 Proposed Model Development (Integrated LSTM-MODWPT)\u003c/h2\u003e \u003cp\u003eThe essential phases for developing a model are presented in the following sub-sections and presented in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The first step in developing DL-based forecasting modeling is the selection of potential input variables. Since weekly GWLs at two observation wells (GT3330001 and GT3330002) were the only data used in this effort, the possible inputs were the time-lagged forms of the measured GWLs (every station), and their wavelet packet transformed components. A Partial Autocorrelation Function (PACF) was used to obtain the time-lagged information from the GWL time series and to select which lags to incorporate as the potential inputs. The PACF plots for the GWL data obtained from two selected observation wells are illustrated in Fig.\u0026nbsp;2. From Fig.\u0026nbsp;2, it is possible to deduce that the present and previous five time lags are essential for three-weeks ahead GWL forecasting \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left({GWL}_{t+1},{GWL}_{t+2}, \\text{a}\\text{n}\\text{d} {GWL}_{t+3}\\right)\\)\u003c/span\u003e\u003c/span\u003e. The present and previous time lags for the observation well GT3330001 were \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({GWL}_{t},{GWL}_{t-1},{GWL}_{t-3},{GWL}_{t-4},{GWL}_{t-5}, \\text{a}\\text{n}\\text{d} {GWL}_{t-6}\\)\u003c/span\u003e\u003c/span\u003e. On the other hand, the time lags for the observation well GT3330002 were \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({GWL}_{t},{GWL}_{t-1},{GWL}_{t-2},{GWL}_{t-3},{GWL}_{t-4}, \\text{a}\\text{n}\\text{d} {GWL}_{t-5}\\)\u003c/span\u003e\u003c/span\u003e. Consequently, the potential input variables for the non-wavelet decomposition-based LSTM model utilized six time-lagged variables as inputs. For the MODWPT-based LSTM model, the candidate input variables include the same inputs plus their MODWPT decomposed counterparts. Each of the time-lagged GWL time series was wavelet decomposed independently. The MODWPT produced many candidate input variables for the wavelet-based LSTM models at both observation wells. Therefore, the most significant input variables were selected using a RF-based modeling approach.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe choice of wavelet filters and decomposition levels and eradication of the boundary-impacted wavelet coefficients is the primary consideration during the wavelet decomposition (Quilty and Adamowski, \u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). This study used the Fej\u0026eacute;r-Korovkin scaling filter with a filter length of 18 and a decomposition level of 3. This essential elimination of 188 boundary-impacted wavelet coefficients from the start of the sets of possible input and target variables was based on the boundary correction approach adopted by Quilty and Adamowski (\u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Similar principles were used in the LSTM models that were not wavelet-based.\u003c/p\u003e \u003cp\u003eMATLAB (Mathworks, \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020a\u003c/span\u003e) commands and functions were used to develop the proposed models using the observed groundwater level data at the two observation wells.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section3\"\u003e \u003ch2\u003e2.2.4 Input Variable Selection\u003c/h2\u003e \u003cp\u003eThe RF approach was used to select the scaling coefficients and wavelet family that are most effective in producing precise predictions of the output variable. The RF approach's main task was to ensure that only variables necessary for projecting the output variable were chosen. Simultaneously, unnecessary and/or inappropriate variables were omitted. Ideally, accurate models should not be complicated or limited in their ability to predict. This study used a \u0026lsquo;bagged\u0026rsquo; ensemble of 500 regression trees to develop the RF model. All input variables were assigned at each tree node to ensure that each regression tree utilized all input variables to attain better accuracy. The number of levels among the inputs varied using a standard CART algorithm for selecting split inputs at each node of the trees in a RF model that might yield less accurate estimates. A curvature or interaction test was performed to select split inputs to avoid this problem (MathWorks, \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2020b\u003c/span\u003e). Variable importance was estimated by performing permutation of the out-of-bag observations among the trees.\u003c/p\u003e \u003cp\u003eTime-lagged input variables, as well as their wavelet packet coefficients, produced 102 input variables of [GWL at the present time and five times lagged GWLs (6)\u0026thinsp;+\u0026thinsp;16 wavelet packet coefficients for Fej\u0026eacute;r-Korovkin scaling filter with a specified filter length (16) \u0026times; 6]. Using all the input variables is associated with a computational burden. Therefore, only the most significant input variables were selected for the model development to reduce the computational burden and enhance computational efficiency. This study used the 20 most influential input variables determined by the RF modeling technique to develop LSTM-MODWPT models at the observation wells for the one-, two-, and three-step ahead GWL forecasts. Figure\u0026nbsp;4 presents the plots of variable importance.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section3\"\u003e \u003ch2\u003e2.2.5 Data Partitioning\u003c/h2\u003e \u003cp\u003eDue to time lagging and removal of the boundary-affected coefficients, the observed GWL data were reduced at each observation well. At the observation well GT3330001, a total of 1816 records remained (from 09 October 1983 to 17 September 2018) after removing 188 boundary-affected coefficients and eight records due to time lagging (three time lags forward\u0026thinsp;+\u0026thinsp;five time lags backward) from the entire GWL time series of 2012 readings (from 07 January 1980 to 17 September 2018). At GT3330002, a total of 1741 records remained (from 09 October 1983 to 26 December 2016) after removing 188 boundary-affected coefficients and eight records due to time lagging (three time lags forward\u0026thinsp;+\u0026thinsp;five time lags backward) from the entire GWL time series of 1937 readings (07 January 1980 to 26 December 2016). The remaining dataset was separated into two distinct sets of training and testing samples where 80% of the data records were allocated for training, and the remaining 20% was allotted for testing.\u003c/p\u003e \u003cp\u003eFor GT3330001, the remaining 1816 readings were split into 1453 records (from 09 October 1983 to 03 October 2011) and 363 records (from 04 October 2011 to 17 September 2018), respectively, for training and testing purposes. For GT3330002, the remaining 1741 readings at GT3330002 were divided into 1393 records (from 09 October 1983 to 26 April 2010) and 348 records (from 27 April 2010 to 26 December 2016), respectively, for training and testing purposes.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"3 Statistical Indices for Performance Evaluation","content":"\u003cp\u003eFive statistical parameters were utilized to evaluate the model's performance (Equations \u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e). Generally, the Root Mean Squared Error (RMSE) criterion measures the error of the model. The lower RMSE value indicates the higher prediction power of the model. However, the value of RMSE largely depends on the magnitude of the data; therefore, a lower RMSE value does not necessarily mean better prediction performance. The Scatter Index (SI) criterion was used to eliminate the dimensionality effect of the data to overcome this issue. Model performance assessment criteria based on the SI index values were: Excellent when SI is less than 0.1, good when SI is between 0.1 and 0.2, fair when SI is between 0.2 and 0.3, and poor when SI is greater than 0.3 (Li et al., \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). The \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({a}^{20}- index\\)\u003c/span\u003e\u003c/span\u003e value ranges between 0 and 1, and for an ideal model, the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({a}^{20}- index\\)\u003c/span\u003e\u003c/span\u003e value is 1 (Xu et al., \u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e2019\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eRoot Mean Squared Error (RMSE):\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$RMSE=\\frac{\\sqrt{\\frac{1}{n}\\sum _{i=1}^{n}{\\left({GWL}_{i}^{A}-{GWL}_{i}^{P}\\right)}^{2}}}{\\overline{{GWL}^{A}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eScatter Index, SI (Li et al., \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2013\u003c/span\u003e):\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$SI=\\frac{RMSE}{\\overline{{GWL}^{A}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMaximum Absolute Error (MAE):\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$MAE=max\\left[\\left|{GWL}_{i}^{A}-{GWL}_{i}^{P}\\right|\\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eMedian Absolute Deviation, MAD (Pham-Gia and Hung, \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e2001\u003c/span\u003e):\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$MAD\\left({GWL}^{A},{GWL}^{P}\\right)=median\\left(\\left|{GWL}_{i=1}^{A}-{GWL}_{i=1}^{P}\\right|,\\left|{GWL}_{i=2}^{A}-{GWL}_{i=2}^{P}\\right|,\\cdots ,\\left|{GWL}_{i=n}^{A}-{GWL}_{i=n}^{P}\\right|\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ea\u003csup\u003e20\u003c/sup\u003e \u0026ndash; index (Xu et al., \u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e2019\u003c/span\u003e):\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$${a}^{20}- index=\\frac{{k}^{20}}{n}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({GWL}_{i}^{A}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({GWL}_{i}^{P}\\)\u003c/span\u003e\u003c/span\u003e are the actual and predicted \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(GWL\\)\u003c/span\u003e\u003c/span\u003e values for the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({i}^{th}\\)\u003c/span\u003e\u003c/span\u003e data points of the dataset, respectively; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\overline{{GWL}^{A}}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\overline{{GWL}^{P}}\\)\u003c/span\u003e\u003c/span\u003e are the mean values of the actual and predicted \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(GWL\\)\u003c/span\u003e\u003c/span\u003e, respectively; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(n\\)\u003c/span\u003e\u003c/span\u003e is the number of entries in the GWL time series data; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}^{20}\\)\u003c/span\u003e\u003c/span\u003e is the amount of data that is associated with a \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({GWL}_{i}^{A}/{GWL}_{i}^{P}\\)\u003c/span\u003e\u003c/span\u003e value varying from 0.80 to 1.20 (Xu et al., \u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e2019\u003c/span\u003e).\u003c/p\u003e"},{"header":"4 Results and Discussion","content":"\u003cp\u003eThe findings of the non-wavelet and wavelet-based LSTM models for multi-step (i.e., one-, two-, and three-week(s)) ahead GWL forecasting were evaluated using several performance evaluation indices. Graphical approaches were also used to evaluate the performances of the developed models. Generally, the performances of all models for both the training and testing phases exhibited a good tradeoff, indicating a reasonably fair generalization capability of the developed models in multi-step ahead GWL forecasting. However, the comparison of model performances between the non-wavelet and wavelet-based LSTM was based on their testing phase performances. Both training and testing phase model performances are presented in the following sub-sections.\u003c/p\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Performance of the Standalone LSTM Models\u003c/h2\u003e \u003cp\u003eDetermining the optimal LSTM model architecture is crucial in DL-based forecasting approaches. In this study, different combinations of various numbers of hidden layers and neurons were evaluated to determine the optimal LSTM model structure in multi-step ahead GWL forecasting. The RMSE criterion was utilized to assess how well the developed models performed during the training and testing phases in various scenarios of the hidden layer and hidden neuron combinations. The amounts of RMSE on the training and test dataset for different numbers of neurons are given in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFor GT3330001, the minimum values of the absolute difference between the training and test RMSE were 0.09 m (hidden neurons: 180-150-80), 0.03 m (hidden neurons: 150-100-50), and 0.06 m (hidden neurons: 150-100-50) for one-, two-, and three-week(s) ahead forecasting, respectively (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e5\u003c/span\u003e). On the other hand, at GT3330002, the RMSE values were 3.13 m (hidden neurons: 150-120-80-50), 3.22 m (hidden neurons: 140-120-60), and 3.14 m (hidden neurons: 100-80-50-20) for one-, two-, and three-week(s) ahead forecasting, respectively. Therefore, the LSTM models with these hidden neurons were selected as the best-performing models.\u003c/p\u003e \u003cp\u003eThe findings (RMSE, Scatter index, MAE, MAD, and a-20 index) of the standalone LSTM models for forecasting GWLs at one-, two-, and three-week(s) ahead are presented in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e. The overall accuracy across the two observation wells for the standalone LSTM models showed a very good performance in terms of scatter index (Li et al., \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2013\u003c/span\u003e), MAD, and a-20 index (Xu et al., \u003cspan citationid=\"CR83\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) criteria whereas the performances were reasonably good with reference to the RMSE and MAE criteria. It is noted that the performances of the standalone LSTM models at the observation well GT3330001 were in general superior than those at the observation well GT3330002. The plausible reason for this discrepancy in modeling performances may be the quality and quantity of the observed data as well as the number of missing values that were imputed. Nevertheless, the developed LSTM models produced acceptable results at both observation wells (Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eOne-, two-, and three-week(s) ahead forecasting performance of the developed standalone LSTM models on test dataset\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colspan=\"5\" nameend=\"c6\" namest=\"c2\"\u003e \u003cp\u003ePerformance evaluation indices\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRMSE, m\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eScatter index\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMAE, m\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMAD, m\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ea-20 index\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003e\u003cem\u003eGT3330001\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOne week ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.827\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.034\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e12.480\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.179\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTwo weeks ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.910\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.037\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10.237\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.319\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThree weeks ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.975\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.040\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e10.754\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e0.405\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cem\u003eGT3330002\u003c/em\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e\u0026nbsp;\u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOne week ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.466\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.188\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.561\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.200\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.589\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTwo weeks ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.623\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.196\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.727\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.326\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.531\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThree weeks ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3.493\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.189\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e7.393\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e1.327\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e0.573\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eAlthough the accuracy of LSTM forecasts decreased with increasing lead times (Rahman et al., \u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), the LSTM models developed at the observation well GT3330001 showed very good performance (RMSE\u0026thinsp;=\u0026thinsp;0.975 m, Scatter index\u0026thinsp;=\u0026thinsp;0.040, MAD\u0026thinsp;=\u0026thinsp;0.405, m and a-20 index\u0026thinsp;=\u0026thinsp;0.997) for three-weeks-ahead GWL forecasting. For the monitoring well GT3330002, differences in the forecasting performances were also not substantial with respect to the increased lead times. It is worth mentioning that several previous studies compared the forecasting accuracies of ML algorithms such as SVR, MLR, ANN, and RF (Rajaee et al., \u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) and found SVR to be the best performing model (Rahman et al., \u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Therefore, it is practically impossible to provide a direct comparison of these findings with the mentioned previous studies in GWL forecasting. Moreover, the study locations are also different. In the same study location, different ML approaches also provided superior performance over others for different observation well locations (Rahman et al., \u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). However, it can be argued from the findings of this study that LSTM models can effectively be utilized to provide reasonable GWL-level forecasts.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e4.3 Performance of the MODWPT Coupled LSTM Models\u003c/h2\u003e \u003cp\u003eThe findings of the MODWPT coupled LSTM models for the two observation wells are presented in this section. The obtained best numbers of hidden layers and hidden neurons for the standalone LSTM models were used to develop MODWTP-based LSTM models (LSTM- MODWTP). The training performance of the generated LSTM- MODWTP models at the two observation well locations is illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e6\u003c/span\u003e. These results support that MODWPT as a preprocessing tool significantly improved the training and testing performance of the LSTM models (Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e6\u003c/span\u003e). The smallest differences between the training and test RMSE values ensures that the coupled LSTM-MODWPT models were trained adequately and no model overfitting was observed during the model training. In addition, this study adopted the best practices proposed in Quilty and Adamowski (\u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e2018\u003c/span\u003e) to correct the data against the boundary affected datasets. This approach allowed us to use the wavelet transforms correctly as a data preprocessing tool for GWL forecasting. This also ensured that MODWPT did not universally lead to enhanced performances of the LSTM- MODWPT models over the standalone LSTM models. After checking for model overfitting using RMSE criterion, the trained models were used to compute several other statistical performance evaluation indices on the test dataset.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe statistical performance outcomes for the one-, two-, and three-week(s)-ahead forecasting performance of the developed LSTM- MODWTP model on the test dataset are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e reveals that similar to the standalone LSTM models (results are presented in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), the performances of the LSTM- MODWTP for multi-step forward forecasts at GT3330001 were, in general, better than the forecasting abilities of the LSTM- MODWTP developed at GT3330002. The plausible reasoning for this performance deviation may have resulted from the length and quality of data and the number of missing data that were imputed. However, the developed LSTM-MODWPT models provided acceptable results at both observation wells. Similar metrics for evaluating the performance were calculated to assess the robustness of the proposed LSTM- MODWTP models (presented in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e). It is evident from Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e that MODWPT improved the LSTM performance at both observation wells and for all lead times as evidenced by the higher values of a-20 index and lower values of RMSE, Scatter index, MAE, and MAD criteria. Although the forecasting performance slightly decreased with the increased forecasting timeframes, the forecasting performance for the last forecasting horizon (three weeks ahead) is reasonably good and within the acceptable limits for the particular statistical indices.\u003c/p\u003e \u003cp\u003eThe improvements in forecasting performance of the LSTM-MODWPT over the standalone LSTM models are quite satisfactory at both observation wells for all forecasting horizons. The percentage improvements in the forecasting accuracy based on RMSE criterion were 36.28% for one-week-ahead forecasting at GT3330001, 32.97% for two-weeks-ahead forecasting at GT3330001, 30.77% for three-weeks-ahead forecasting at GT3330001, 78.68% for one-week-ahead forecasting at GT3330002, 76.37% for two-weeks-ahead forecasting at GT3330002, and 74.92% for three-weeks-ahead forecasting at GT3330001. The percentage improvements in the forecasting accuracy based on Scatter index criterion were 29.41% (one-week-ahead forecasting at GT3330001), 27.03% (two-weeks-ahead forecasting at GT3330001), 25% (three-weeks-ahead forecasting at GT3330001), 47.87% (one-week-ahead forecasting at GT3330002), 58.67% (two-weeks-ahead forecasting at GT3330002), and 54.50% (three-weeks-ahead forecasting at GT3330001). The percentage improvements in the forecasting accuracy based on MAE criterion were 56.49% (one-week ahead forecasting at GT3330001), 51.96% (two-weeks-ahead forecasting at GT3330001), 56.52% (three-weeks-ahead forecasting at GT3330001), 35.29% (one-week-ahead forecasting at GT3330002), 46.62% (two-weeks-ahead forecasting at GT3330002), and 46.04% (three-weeks-ahead forecasting at GT3330001). The percentage improvements in the forecasting accuracy based on MAD criterion were 14.53% (one-week-ahead forecasting at GT3330001), 31.97% (two-weeks-ahead forecasting at GT3330001), 11.85% (three-weeks-ahead forecasting at GT3330001), 27.75% (one-week-ahead forecasting at GT3330002), 32.58% (two-weeks-ahead forecasting at GT3330002), and 31.80% (three-weeks-ahead forecasting at GT3330001). The percentage improvements in the forecasting accuracy based on a-20 index criterion were 0.20% (one-week-ahead forecasting at GT3330001), 0% (two-weeks-ahead forecasting at GT3330001), 0.10% (three-weeks-ahead forecasting at GT3330001), 48.73% (one-week-ahead forecasting at GT3330002), 66.29% (two-weeks-ahead forecasting at GT3330002), and 59.69% (three-weeks-ahead forecasting at GT3330001). The most remarkable finding in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e is that MODWPT especially improved the forecasting performance of the standalone LSTM models at the observation well GT3330002, where the standalone LSTM models performed poorly in all forecasting horizons.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eOne-, two-, and three-week(s)-ahead forecasting performance of the developed LSTM- MODWPT model on test dataset\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colspan=\"5\" nameend=\"c6\" namest=\"c2\"\u003e \u003cp\u003ePerformance evaluation indices\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRMSE, m\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eScatter index\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eMAE, m\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eMAD, m\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ea-20 index\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003e\u003cb\u003eGT3330001\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOne week ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.527\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.024\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5.430\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.153\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.999\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTwo weeks ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.610\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.027\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.918\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.217\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.997\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThree weeks ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.675\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.030\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.676\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.357\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.998\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"6\" nameend=\"c6\" namest=\"c1\"\u003e \u003cp\u003e\u003cb\u003eGT3330002\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOne week ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.739\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.098\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.893\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.867\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.876\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTwo weeks ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.856\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.081\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.125\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.894\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.883\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThree weeks ahead\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.876\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.086\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3.989\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.905\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e0.915\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn summary, Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e reveals the superiority of the proposed LSTM-MODWPT over the standalone LSTM models (Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) for the selected observation wells for the three forecast periods. This superior performance was evidenced through the five statistical performance evaluation indices considered for evaluating the models' performances in this study. Even though this study showed the promises of LSTM-MODWPT models under the conditions studied, further assessments for such modelling aspects would be required for other geographic locations. The outcomes of this research would improve the overall performance accuracy, reduce modelling complexity, and ease parameter selections for groundwater modelling. This finding is important in the water resources management since the early forecasting of GWLs is crucial in decision making in the fields of irrigation scheduling and planning, land development and in many other research domains including environmental sciences.\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Conclusions","content":"\u003cp\u003eAn efficient and sustainable groundwater management plan can be developed using accurate and reliable predictions of GWLs. This planning will aid the optimal abstraction recommendations and groundwater usage for agricultural, domestic, and industrial purposes. However, due to the nonlinear nature of GWLs as well as their multiscale and time-varying behavior, it is frequently difficult to provide accurate GWL forecasts. One of the most influential pre-requisites of developing ML-based GWL forecast models is the appropriate ML algorithm choice. LSTM models have shown promising performance in hydrological and other time series predictions. Equally important is the incorporation of a suitable data preprocessing approach, which is believed to improve forecasting performance of ML-based algorithms.\u003c/p\u003e \u003cp\u003eTo address these issues, this research created a reliable forecasting tool using coupled LSTM-MODWPT models for one-, two-, and three-week(s)-ahead GWL fluctuations. MODWPT was used to acquire multiscale information from the GWL time series, which were incorporated in developing LSTM models to improve LSTM\u0026rsquo;s forecasting capability. The suitable weekly lag times of GWLs and their wavelet packet transformed counterparts were deployed as input variables to the forecast models. In contrast, the outputs were the one, two, and three week(s) ahead GWLs. The selection of an ideal blend of input variables for the proposed models was implemented through a RF-based modeling approach. The proposed models' performance assessment was executed using various statistical performance assessment metrics by which LSTM-MODWPT models were benchmarked against their non-wavelet-based counterparts, e.g., the standalone LSTM models. Results of the present study indicated that the LSTM-MODWPT models outperformed the standalone LSTM models for all three future time horizons and at each observation well. Therefore, it can be concluded that LSTM-MODWPT models could predict multi-step-ahead GWL fluctuations quite accurately for the study area.\u003c/p\u003e \u003cp\u003eThe proposed modeling approach was promising for short-term GWL forecasts at the specified observation wells of a water scarce region in Bangladesh that can be extended to other geographical areas. Moreover, this promising modeling framework can be applied to the research areas of hydrology and water resources for medium-term and long-term forecasting.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgements\u0026nbsp;\u003c/strong\u003eThe authors are grateful to Emily Bellis\u003ca href=\"#_ftn1\" name=\"_ftnref1\" title=\"\"\u003e[1]\u003c/a\u003e for providing an initial review of this manuscript, which improved the original manuscript.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor Contribution\u0026nbsp;\u003c/strong\u003eAll authors contributed to the study conception and design\u003cstrong\u003e.\u0026nbsp;\u003c/strong\u003eDilip Kumar Roy: Conceptualization, Methodology, Formal analysis, Software, Validation, Visualization, Writing - original draft. Ahmed A. Hashem: Supervision, Writing - review \u0026amp; editing. Michele L. Reba: Conceptualization, Supervision, Writing - review \u0026amp; editing. Deborah L. \u0026nbsp;Leslie: Assisted in analyzing the results, reviewed and edited the manuscript. John Nowlin: Reviewed the manuscript and provided constructive suggestions.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eStatements \u0026amp; Declarations\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eCompliance with ethical standards:\u0026nbsp;\u003c/u\u003eThere are no potential conflicts of interest. The research does not include human participants and/or animals.\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eConsent to participants:\u003c/u\u003e The research does not include human participants and/or animals.\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eConsent to publish:\u003c/u\u003e The authors give their consent to publish in the Water Resources Management journal if accepted for publication.\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eFunding:\u003c/u\u003e The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eCompeting interests:\u003c/u\u003e The authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eAvailability of data and material\u003cstrong\u003e:\u003c/strong\u003e\u003c/u\u003eDatasets and other materials are available with the authors, and may be accessible at any time upon request.\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eCode availability:\u003c/u\u003e MATLAB codes are available with the first author\u0026nbsp;and will be provided upon request.\u003c/p\u003e\n\u003cdiv id=\"ftn1\"\u003e\n \u003cp\u003e\u003ca href=\"#_ftnref1\" name=\"_ftn1\" title=\"\"\u003e[1]\u003c/a\u003e Research Assistant Professor of Bioinformatics, Department of Computer Science, Arkansas State University, Jonesboro, AR 72467, United States\u003c/p\u003e\n\u003c/div\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAdamowski J, Chan HF (2011) A wavelet neural network conjunction model for groundwater level forecasting. 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J Hydrol 561:918\u0026ndash;929. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/https://doi.org/10.1016/j.jhydrol.2018.04.065\u003c/span\u003e\u003cspan address=\"10.1016/j.jhydrol.2018.04.065\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Groundwater level forecasts ‧ Long short-term memory networks ‧ Wavelet packet transform ‧ Variable selection ‧ Random forest","lastPublishedDoi":"10.21203/rs.3.rs-3464867/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3464867/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eDeveloping precise groundwater level (GWL) forecast models is essential for the optimal usage of limited groundwater resources and sustainable planning and management of water resources. In this study, an improved forecasting accuracy for up to three weeks ahead of GWLs in Bangladesh was achieved by a coupled Long Short Term Memory (LSTM) network-based deep learning algorithm and a Maximal Overlap Discrete Wavelet Packet Transform (MODWPT) data preprocessing. The coupled LSTM-MODWPT model performance was compared with the LSTM model. For both standalone LSTM and LSTM-MODWPT models, the Random Forest feature selection approach was employed to select the ideal inputs from the candidate GWL lags. In the LSTM-MODWPT model, input GWL time series were decomposed using MODWPT. The \u0026lsquo;Fej\u0026eacute;r-Korovkin\u0026rsquo; mother wavelet with a filter length of 18 was used to obtain a collection of scaling coefficients and wavelets for every single input time series. Model performance was assessed using five performance indices: Root Mean Squared Error; Scatter Index; Maximum Absolute Error; Median Absolute Deviation; and a-20 index. The LSTM-MODWPT model outperformed standalone LSTM models for all time horizons in GWL forecasting. The percentage improvements in the forecasting accuracies were 36.28%, 32.97%, and 30.77%, respectively, for one-, two-, and three-weeks ahead forecasts at the observation well GT3330001. Accordingly, the coupled LSTM-MODWPT model could potentially be used to enhance multiscale GWL forecasts. This research demonstrates that the coupled LSTM-MODWPT model could generate more precise GWL forecasts at the Bangladesh study site, with potential applications in other geographic locations globally.\u003c/p\u003e","manuscriptTitle":"A Maximal Overlap Discrete Wavelet Packet Transform Coupled with an LSTM Deep Learning Model for Improving Multilevel Groundwater Level Forecasts","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-10-24 17:22:55","doi":"10.21203/rs.3.rs-3464867/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"0a0d3ff3-3dfa-4021-a45c-a72168b0e491","owner":[],"postedDate":"October 24th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2023-12-06T16:29:27+00:00","versionOfRecord":[],"versionCreatedAt":"2023-10-24 17:22:55","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3464867","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3464867","identity":"rs-3464867","version":["v1"]},"buildId":"7rjqhiLT3MXkJMwkYKINL","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}