Enhanced Forecasting of Groundwater Level Incorporating an Exogenous Variable: Evaluating Conventional Multivariate Time Series and Artificial Neural Network Models

Abstract

Continuous and uncontrolled extraction of groundwater often creates tremendous pressure on groundwater levels (GWLs). As a part of sustainable planning and effective management of water resources, it is crucial to assess the existing and forecasted GWL conditions. In this study, an attempt was made to model and forecast GWL using artificial neural networks (ANNs) and multivariate time series models. Autoregressive integrated moving average (ARIMA) and ARIMA models incorporating exogenous variables (ARIMAX) were adopted as the time series models. GWL data from five monitoring wells from the study area of the Kushtia District in Bangladesh were used to demonstrate the modeling exercise. Rainfall (RF) was taken as the exogenous variable to explore whether its inclusion enhanced the performance of GWL forecasting using the developed models. It was evident from the results that the multivariate ARIMAX model (with the sum of squared errors, SSE, of 15.143) performed better than the univariate ARIMA model with an SSE of 16.585 for GWL forecasting. This demonstrates the fact that the multivariate time series models generated enhanced forecasting of GWL compared to the univariate time series models. When comparing the models, it was found that the ANN-based model outperformed the time series models with enhanced forecasting accuracy (SSE of 9.894). The results also exhibit a significant correlation coefficient (R) of 0.995 (model ANN 6-8-1) for the existing and predicted data. The current study conclusively proves the superiority of ANN over the time series models for the enhanced forecasting of GWL in the study area.

1. Introduction

Bangladesh is one of the most densely populated countries in the world. There is population growth and increasing urbanization in addition to climate change. For this reason, the stress on groundwater is growing rapidly. This earthly resource is essential, particularly for developing countries that rely on it for agricultural needs. Additionally, it plays a crucial role in the earth’s water cycle. Like many districts in Bangladesh, Kushtia relies heavily on agriculture, as a large irrigation project named the Ganges–Kobadak Project covers an area of 197,500 hectares and serves the region. Approximately 68% of the area is covered by irrigation [1]. According to the Statistical Yearbook of Bangladesh 2022 (by the Bangladesh Bureau of Statistics), irrigation in Kushtia District is carried out using various types of tube wells, including deep, shallow, and hand tube wells, as well as power pumps. Some authors [2] analyzed the importance of safe water sources and assessed conventional water sources, including groundwater. So, accurate and reliable forecasting of GWL is necessary. Furthermore, it is also essential for the sustainable management of water resources and to facilitate the conjunctive use of groundwater and surface water resources.

Degradation of GWL is common, and research [3] indicates that during the previous 25 years, the total volume of groundwater storage exhibited a declining trend throughout the Southeast Asian region. The increase in impervious surface caused by urbanization reduces infiltration, which has an impact on groundwater storage [4]. Groundwater is sometimes the only dependable freshwater supply in many developing nations since it is widely accessible, relatively inexpensive to collect, and usually of higher quality than surface water [5,6]. The dependence on groundwater is especially noticeable in areas with little or highly contaminated surface water [7]. The nation’s dense population, expanding agricultural needs, and quick industrialization are the main causes of this widespread reliance [8]. Lowering GWLs in agricultural areas might result in reduced crop yields because of limited water supplies, exacerbating food security problems [9,10]. Apart from the physical loss of groundwater, another major concern is its quality, especially in areas where contamination is common [11].

Time series analysis has recently become substantially more appealing as a statistical tool for developing forecast models, and its applicability has increased globally [12]. The authors’ studies found that RF is an essential component of groundwater recharge potential [13]. Furthermore, RF significantly influences groundwater recharge potential [14]. GWLs can be significantly impacted by the fluctuation of RF, which is a key source of groundwater recharge [15]. It is generally acknowledged that one of the most important ways to increase the precision of GWL predictions is to incorporate RF data into both ANN and ARIMA models [16]. Because of Bangladesh’s monsoon climate, the relationship between RF and GWL is significant [17]. For these reasons, RF is considered an endogenous variable in this study.

The goals of this study are to model and predict GWL fluctuations using ANN models and statistical time series ARIMA-based models, incorporating an exogenous variable to analyze its effects. It is widely accepted that ANN and ARIMA-based time series models are powerful tools for predicting various data. The performances of ARIMA-based ANN models were compared to determine which modeling strategy is better for predicting GWL fluctuations. In addition, assessing the ability of models to predict using various model evaluation criteria is also a primary focus of this study. One important issue in groundwater management is predicting GWL changes, which provides essential information about future groundwater availability and can help in developing policies for sustainable usage [18,19]. Furthermore, these models can be used to conduct additional research on groundwater issues.

2. The Study Area

The Kushtia District in the Khulna division of Bangladesh was selected for this study. The area covered by the district is 1608.8 km². It is positioned in southwest Bangladesh, encompassing the coordinates of 23°42′ to 24°12′ north and 88°42′ to 89°22′ east. Figure 1 shows the location of the study area. The area is characterized by several rivers, such as the Ganges, Mathabhanga, Kaligonga, and Kumar, which flow across the district. Additionally, the district has a flat, alluvial landscape, which is typical for the delta region. It has fertile soil that makes it favorable to agriculture and is highly dependent on irrigation. Groundwater is an almost worldwide source of superior freshwater [20]. The district also largely depends on it.

2.1. Aquifer Characteristics

The aquifers in the area consist of alluvial deposits, which include sand, silt, and clay. These formations typically exhibit varying permeability that influences groundwater movement. Shallow aquifers are usually found at lesser depths and are directly influenced by surface water and recharge primarily through RF and seasonal flooding. The authors found that the diffusivity (degrees of flow) varied from 181,143 m²/day to 256,788 m²/day, and the overall estimated parameter results of the aquifer system show that the area is hydrogeologically favorable for groundwater development provided the other conditions are fulfilled [21].

2.2. Data Description

For GWL modeling and prediction, in the current study, groundwater monitoring stations from five upazilas (i.e., Bheramara, Daulatpur, Kushtia Sadar, Kumarkhali, and Mirpur), along with an RF station of the district, were selected. The GWL data were collected on a weekly basis. Accordingly, 414 weekly GWL data points from 1999 to 2006 were sourced from the Bangladesh Water Development Board (BWDB). The RF data obtained here were exogenous input and collected for the same periods. The available GWL data were measured in m.PWD (meters above the public works datum (PWD)) used by BWDB. This datum is located 0.46 m below the mean sea level. Information on the selected GWL monitoring stations and RF stations, along with their locations, are given in

Table 1. GWL and RF station details.

SL No	Station ID	Station Type	Location of Station (Sub-District Name)	Latitude (Degree)	Longitude (Degree)
1	KG-1	GWL	Bheramara	24.09	88.96
2	KG-2	GWL	Daulatpur	23.98	88.83
3	KG-3	GWL	Kushtia Sadar	23.83	89.10
4	KG-4	GWL	Kumarkhali	23.84	89.20
5	KG-5	GWL	Mirpur	23.93	89.02
6	KR-1	RF	Mirpur	24.05	88.99

. The data are visualized for each currently available station in Figure 2 for the same period. The weekly data are provided in Appendix A.

3. Methodology

The primary objective of this study is to model and forecast GWL changes using ANN and statistical ARIMA-based time series models. Univariate and multivariate models were constructed, and the top-performing model was identified. In both cases, model development and evaluations were carried out. Afterward, GWL predictions were carried out based on the existing data and variables. Moreover, the impact of incorporating climate data was also observed.

3.1. Advantages of ANN Models

Time series modeling is considered one of the most robust statistical methods for studying the connection between climate variation, GWL, and forecasting [18]. The main advantage of using ANN models over conventional techniques is that they do not require underlying processes, which are complex in nature, to be explicitly defined in mathematical form [22]. In [23], researchers employed ANN models to forecast GWL changes for future periods. Results from [24] showed that forecasting time series is more accurate in ANN models than in ARIMA-based models. Accepting all complex parameters as input, ANN models generate patterns during model training, and then they use the same patterns to generate forecasts or predictions [25]. The strength of ANN models in prediction lies in their data-driven nature, their ability to detect previously unseen patterns, and their efficiency with large datasets [26].

3.2. Prediction Approaches of the Models

Several researchers have identified two main approaches in GWL prediction: data-driven models and numerical models. Researchers have also found that data-driven models are useful in assessing different aspects such as uncertainty, variability, and complexities in water resources and environmental problems [27]. In recent decades, artificial intelligence has been widely applied in studies related to water resources [27]. For example, ANN is used for daily weather forecasting [25], RF forecasting [26], and GWL prediction purposes [27,28]. However, this method has not been widely used until recently [28]. Due to the limited understanding of aquifer properties in Bangladesh, the applicability of conventional GWL prediction models is limited; for this reason, soft computing tools are good alternatives that provide higher efficacy [29]. The ANN model can discern connections in historical data that cannot readily be seen. In this way, it aids in prediction and forecasting. The authors of [30] worked on water quality forecasting. ARIMAX models have also been applied in forecasting electricity usage [31], grain production [32], domestic water consumption [33], and droughts [34]. “ARIMA models are well-known for their notable forecasting accuracy and flexibility in representing several different types of time series” [35].

3.3. Development Strategy of ANN Models

There are numerous techniques used to create and simulate a neural network [36]. In the current study, several model architectures were analyzed to find the most accurate prediction model via the MATLAB platform using the neural network toolbox. This toolbox offers proficiency in designing various neural network configurations with many applications [36]. Creating an ANN model requires defining its type, structure, variable processing, training algorithm, and stopping criteria [37].

3.3.1. Dataset Preparation

First, the datasets were collected and processed as inputs for the models. For univariate models, only GWL data were used. However, GWL and RF data were taken as inputs for multivariate models. The data were allocated as follows: 70% (i.e., 290 data points) for training, 15% (i.e., 62 data points) for validation, and 15% (i.e., 62 data points) for testing, which is typical. Before inputting the data into the ANN model, the data should be processed. In this study, weekly GWL and RF data were prepared on a one-week lag basis. This means that the data were sorted as follows: GWL_t, GWL_t−1, RF_t, and so on, depending on the number of inputs.

3.3.2. ANN Model Architecture and Model Training

The ANN model is inspired by the human brain and neurons are the basic units of its structure. It is similar to brain cells. Each neuron takes inputs, processes them, and produces an output. ANNs are organized into three layers. The input layer is where the network receives data. It contains neurons that represent input features. If there are n input features, this layer will have n neurons. Hidden layers are the layers between the input and output. Each hidden layer can have h_i neurons, where i indexes the hidden layer (e.g., h₁, h₂,…, h_k). Each neuron in a hidden layer transforms the input using weights and activation functions. Many researchers, e.g., in Ref. [27], have selected the hidden layer node count based on a trial-and-error procedure. Therefore, in the current study, the hidden layer size was specified by following the trial-and-error approach, based on the model’s performance. The output layer produced the final result or prediction of the network. The multilayer perceptron feedforward ANN model was applied. It is widely used in hydrological modeling [37]. In particular, a configuration with a single hidden layer is one of the most widely used ANNs for time series modeling and prediction [38]. The ANN 6-3-1 structure denotes that this model has six neurons in the input layer, three in the hidden layer, and one in the output layer.

Connections between neurons have weights, and each neuron has a bias. To compute the output, σ_j, of a neuron, the weighted inputs are summed up, a bias is added, and then an activation function is applied. The output can be expressed mathematically as Equation (1).

$${\mathsf{\sigma }}_{\mathrm{j}}=\mathrm{f}(\sum {\mathrm{w}}_{\mathrm{ij}}\times {\mathrm{x}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{j}})$$

(1)

where w_ij is the weight that connects input i to neuron j, x_i is the input value from the previous layer, and b_j is the bias term for neuron j.

The sigmoid activation function transforms the input (the weighted sum $\sum {\mathrm{w}}_{\mathrm{ij}}\times {\mathrm{x}}_{\mathrm{i}}+{\mathrm{b}}_{\mathrm{j}}$ passes through the activation function) into an output f(x). The function transforms the inputs into a range of (0, 1) so that the input value of the next layer is within a fixed range. In summary, it helps determine the output, σ_j, of a neuron. The function can be expressed mathematically using Equation (2):

$$\mathrm{f}\left(\mathrm{x}\right)={\displaystyle \frac{1}{1+{\mathrm{e}}^{-\mathrm{x}}}}$$

(2)

where x is the given input.

Model training involves adjusting the network’s weights and biases to minimize the difference between the predicted and actual outputs. This process allows the ANN to learn from the data. This study utilizes the commonly utilized Levenberg–Marquardt backpropagation technique for model training. The training was stopped after 1000 epochs of backpropagation, which is considered acceptable in this study. On the MATLAB platform (version R2018a), along with the prepared weekly GWL and RF data, the hidden layer size was also provided as input. The software performed model training and prediction.

3.4. ARIMA-Based Model Development

When predicting various extreme weather events such as heavy RF and droughts, ARIMAX models proved effective [39]. Apart from ANN models, this study adopted time series models for predicting GWL fluctuations: ARIMAX and ARIMA. One of the most frequently used time series models for analyzing and forecasting hydrologic data is ARIMA, which combines autoregressive (AR) and moving average (MA) components. For ARIMA-based models, the procedure involves specifying the model structure, estimating parameters, conducting residual diagnostics, and forecasting the data series. This study was carried out using the econometric modeler toolbox on the MATLAB platform. In the toolbox, the model parameters (p, d, and q) were provided along with the GWL data and RF as inputs. The toolbox performed the prediction using the model’s formula and provided the detailed results of different plots to evaluate performances. Explanations of the model’s parameters and formulas are provided in the following sections.

3.4.1. Mathematical Expression of the ARIMA Model

The ARIMA-based time series model contains three parts (p, d, and q), where p = the order of autoregression, d = the order of integration (differencing), and q = the order of moving average. The general mathematical form, or multiplicative equation, of an ARIMAX (p, d, q) model with one exogenous variable is given by Equation (3):

$$\mathsf{\varphi }\left(\mathrm{L}\right){(1-\mathrm{L})}^{\mathrm{d}}{\mathrm{Y}}_{\mathrm{t}}=\mathrm{c}+\mathsf{\beta }{\mathrm{x}}_{\mathrm{t}}+\mathsf{\theta }\left(\mathrm{L}\right){\mathsf{ϵ}}_{\mathrm{t}}$$

(3)

where c is the constant, ${\mathsf{ϵ}}_{\mathrm{t}}$ is the error term, L = lag operator, ϕ(L) = (1 − ϕ₁L − ϕ₂L² −… − ϕ_pL^p) is the AR polynomial, θ(L) = (1 + θ₁L + θ₂L² + … + θ_qL^q) is the MA polynomial, and β denotes the coefficient for the exogenous variable.

For example, this study considers a time series with AR(2), d(1), MA(1), and one exogenous variable. The model could be specified as Equation (4):

$${\mathrm{Y}}_{\mathrm{t}}=\mathrm{c}+{\mathsf{\varphi }}_1{\mathrm{Y}}_{\mathrm{t}-1}+{\mathsf{\varphi }}_2{\mathrm{Y}}_{\mathrm{t}-2}+{\mathsf{\beta }}_1{\mathrm{x}}_{\mathrm{t}-1}+{\mathsf{\theta }}_1{\mathsf{ϵ}}_{\mathrm{t}-1}+{\mathsf{ϵ}}_{\mathrm{t}}$$

(4)

where Y_t represents the prediction output, c is the constant term, ϕ₁ and ϕ₂ are the autoregressive coefficients for the lagged values of Y, specifically Y_t−1 and Y_t−2; β₁ is the coefficient for the lagged exogenous variable, X_t−1, θ₁ represents the coefficient associated with the lagged error term (ϵ_t−1) in the moving average component of the model, and ${\mathsf{ϵ}}_{\mathrm{t}}$ represents the error term at time t.

3.4.2. Model Identification

For the ARIMA-based models, the time series (GWL) was transformed into a stationary series by differencing (d = 1, 2, 3,…). From this step, the order of integration d was obtained and taken as 3 (as the data become stationary in this order). The values for p and q were selected based on ACF (autocorrelation function) and PACF (partial autocorrelation function) plots. With no significant correlation beyond lag 3, p and q were set to 3. The blue lines in the plots of the software outputs indicate the confidence intervals to assess whether the autocorrelations are statistically significant or not. If the autocorrelation value falls outside the significance lines, then the autocorrelation at that lag is statistically significant. After that, each possible model combination was analyzed and evaluated for accuracy to pick the best model. Once the model was identified, its parameters were estimated. The ACF and PACF plots are shown in Figure 3a,b.

3.5. Model Evaluation Criteria

In forecasting, accuracy is the primary concern, not the processing time. It has been observed that there are no models that can forecast with complete accuracy; however, errors can be reduced by using various techniques [25]. In this study, the mean squared error (MSE), root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), and the sum of squared errors (SSE) are used for ANNs performance evaluation. ANNs have proven to be an essential tool for accurate GWL modeling, along with various other AI methods [40]. Many authors used RMSE values to evaluate the performance of the ANN model [41]. Mathematically, NSE can be expressed using Equation (5). In this study, SSE is considered a crucial component of the model’s performance evaluation. Additional metrics are used, providing a more comprehensive assessment of the model’s performance. In order to analyze the accuracy of ARIMA-based models, the Akaike information criterion (AIC) and Bayesian information criterion (BIC) were also applied to evaluate the performance of the models.

$$\mathrm{NSE}=1-{\displaystyle \frac{{\sum }_{\mathrm{q}=1}^{\mathrm{n}}[{\mathrm{Y}}_{\mathrm{obs}}\left(\mathrm{q}\right)-{\mathrm{Y}}_{\mathrm{est}}\left(\mathrm{q}\right){]}^2}{{\sum }_{\mathrm{q}=1}^{\mathrm{n}}[{\mathrm{Y}}_{\mathrm{obs}}\left(\mathrm{q}\right)-{\overline{\mathrm{Y}}}_{\mathrm{est}}\left(\mathrm{q}\right){]}^2}}$$

(5)

where ${\mathrm{Y}}_{\mathrm{obs}}$ denotes the observed data, ${\mathrm{Y}}_{\mathrm{est}}$ denotes the estimated data, ${\stackrel{\mathrm{-}}{\mathrm{Y}}}_{\mathrm{est}}$ denotes the mean values of estimated data, and n denotes the number of observations.

4. Results and Discussion

In the current study, we analyzed ANN-based and ARIMA-based models to predict GWLs in the Kushtia District in Bangladesh, after developing and evaluating these models. In addition, an attempt was also made to test the prediction capability of the models. The results indicated that incorporating exogenous variables provided enhanced results. As further evidence, studies also indicate that the relationship between GWL changes and meteorological parameters, such as precipitation, is significant [42]. The study showed that ANN models yield more accurate predictions than ARIMA-based models.

4.1. Performance of ANN Models

The ANN 6-8-1 multivariate model with Station ID: KG-1 produced the lowest error (SSE 9.894) among the five stations. On the other hand, for univariate models, the best-performing model was found for the same station with an SSE value of 10.809; the model architecture was ANN 3-7-1. It is evident that the multivariate model performed more efficiently than the univariate model. The results of the best-performing models for each station are shown in

Table 2. Performances of the best-selected ANN (multivariate) models.

Station ID	Location	Model Architecture	Model Performance
			RMSE	NSE	SSE
KG-1	Bheramara	ANN 6-8-1	0.154	0.988	9.894
KG-2	Daulatpur	ANN 7-8-1	0.168	0.979	11.799
KG-3	Kushtia Sadar	ANN 10-4-1	0.231	0.965	26.910
KG-4	Kumarkhali	ANN 6-7-1	0.255	0.986	22.273
KG-5	Mirpur	ANN 8-9-1	0.180	0.984	13.434

and

Table 3. Performances of the best-selected ANN (univariate) models.

Station ID	Location	Model Architecture	Model Performance
			RMSE	NSE	SSE
KG-1	Bheramara	ANN 3-7-1	0.161	0.987	10.809
KG-2	Daulatpur	ANN 2-3-1	0.171	0.979	12.171
KG-3	Kushtia Sadar	ANN 4-9-1	0.258	0.957	26.802
KG-4	Kumarkhali	ANN 2-4-1	0.254	0.986	27.725
KG-5	Mirpur	ANN 5-10-1	0.181	0.984	13.595

.

4.2. Performance Overview of Different ANN Models in the Building Stages

Table 4. Performance overview of the selected ANN (multivariate) models for Station ID: KG-5.

Model	Training		Validation		Test
	MSE	NSE	MSE	NSE	MSE	NSE
ANN 8-2-1	0.030	0.984	0.066	0.964	0.046	0.978
ANN 8-3-1	0.032	0.982	0.061	0.967	0.045	0.978
ANN 8-4-1	0.046	0.975	0.069	0.963	0.076	0.963
ANN 8-5-1	0.036	0.980	0.074	0.960	0.037	0.982
ANN 8-6-1	0.031	0.983	0.078	0.958	0.042	0.980
ANN 8-7-1	0.030	0.983	0.201	0.892	0.040	0.981
ANN 8-8-1	0.035	0.981	0.079	0.958	0.042	0.980
ANN 8-9-1	0.032	0.983	0.083	0.955	0.032	0.984
ANN 8-10-1	0.027	0.985	0.071	0.962	0.039	0.981

presents thorough results for the KG-5 station for the training, validation, and testing stages of the models. The table illustrates the performances of these models at different stages of building ANN models. The performances were evaluated using MSE and NSE for all stages. Since ANN modeling has three phases, the performance of each stage is compared for greater insight and comprehensiveness. Here, two model evaluation criteria (MSE and NSE) were sufficiently used to provide an overall evaluation of the model’s performance.

4.3. Graphical Evaluation of the Selected Performances of the ANN Model

The scatterplot shown in Figure 4a depicts the ANN 8-9-1 model’s actual versus predicted data for station KG-5. This model’s output is illustrated to provide an overview of the iterative procedure involved in finding the best predictive model and its comparative performance. The correlation coefficient value (R = 0.993) indicates a highly significant positive linear relationship between the observed and predicted values. A graphical representation of the actual and predicted data plots for that station is shown in Figure 4b.

4.4. Performances of ARIMA-Based Models

The best performance of the ARIMAX models (SSE = 15.143) was observed for station KG-5 with the model architecture ARIMAX (3,0,2). However, the best performance of the univariate models was observed at the same station, with the model architecture ARIMA (2,0,1) and an SSE of 16.585. It is noticeable that multivariate models performed more efficiently than univariate models. Detailed results are provided in

Table 5. Best ARIMAX model results.

Station ID	Location	Model Architecture	Model Performance (SSE)
KG-1	Bheramara	ARIMAX (3,0,3)	15.361
KG-2	Daulatpur	ARIMAX (3,0,2)	18.721
KG-3	Kushtia Sadar	ARIMAX (1,0,3)	25.449
KG-4	Kumarkhali	ARIMAX (2,0,0)	63.680
KG-5	Mirpur	ARIMAX (3,0,2)	15.143

and

Table 6. Best ARIMA model results.

Station ID	Location	Model Architecture	Model Performance (SSE)
KG-1	Bheramara	ARIMA (2,0,1)	17.217
KG-2	Daulatpur	ARIMA (2,0,1)	26.880
KG-3	Kushtia Sadar	ARIMA (2,0,3)	28.207
KG-4	Kumarkhali	ARIMA (3,0,1)	64.582
KG-5	Mirpur	ARIMA (2,0,1)	16.585

.

4.5. Performance Overview of Different ARIMA Models

Several ARIMA-based models were built after the model’s identification; detailed results of the selected ARIMA (p, d, q) models for station KG-5 are presented in

Table 7. Performances of the selected ARIMA (p, d, q) models for Station ID: KG-5.

Model	SSE	MSE	RMSE	AIC	BIC
ARIMA (0,2,1)	20.688	0.050	0.224	−59.599	−47.521
ARIMA (1,2,2)	20.688	0.050	0.224	−55.600	−35.471
ARIMA (1,2,3)	20.680	0.050	0.223	−53.750	−29.594
ARIMA (2,0,0)	21.220	0.051	0.226	−47.077	−30.974
ARIMA (2,0,1)	16.585	0.040	0.200	−147.100	−126.971
ARIMA (2,0,2)	16.519	0.040	0.200	−146.764	−122.609
ARIMA (3,0,1)	16.537	0.040	0.200	−146.317	−122.162
ARIMA (3,2,1)	20.627	0.050	0.223	−54.820	−30.665
ARIMA (3,2,2)	20.563	0.050	0.223	−54.092	−25.911
ARIMA (3,2,3)	20.021	0.048	0.220	−63.151	−30.944

, along with AIC and BIC values. This table provides a comprehensive overview of the performances of different models and insights into how the procedure is incorporated to find an appropriate prediction model. It also illustrates model evaluation criteria with AIC and BIC values.

4.6. Evaluation of the Parameters of the ARIMA Models

Additionally, the model parameters and their statistics for ARIMA (2,0,1) at station KG-5 are provided in

Table 8. Model parameters for ARIMA (2,0,1) (Station ID: KG-5).

Parameters	Value	Standard Error	T Statistic	p Value
Constant	0.131	0.009	14.352	1.02 × 10−46
AR{1}	1.969	0.008	244.540	0
AR{2}	−0.984	0.007	−123.057	0
MA{1}	−0.931	0.020	−45.921	0
Variance	0.040	0.002	19.147	1.02 × 10−81

. The parameter values are the estimated coefficients for each parameter (e.g., AR coefficients); these parameters have a significant role in the predictive equations, and detailed values confirm statistical significance for the model.

4.7. Graphical Evaluation of the Selected ARIMA Model Performances

The ARIMA (2,0,1) model’s performance for Station ID: KG-5 is graphically presented in Figure 5a–e. It should be noted that 48 unique ARIMA models for each of the five GWL stations were tested (totaling 240 ARIMA models) to evaluate their fits. The plots illustrate a segment of an extensive iterative process in which the model is refined through the analysis of observed results. The blue lines in the ACF and PACF plots indicate the confidence intervals to assess whether the autocorrelations are statistically significant or not as described earlier. A linear alignment of points in the Q-Q plot along the reference line suggests that the residuals adhere to normality, and it indicates that the model adequately captures the underlying data patterns. The plot shows that there is little linear alignment of the points and the model building process should be repeated.

4.8. Forecasting of GWL

Finally, using the best-performing models (ANN 6-8-1 and ARIMAX 3,0,2), GWL data were predicted using existing data. It was observed that ANNs predicted GWL values, ranging from a maximum of 10.797 m to a minimum of 5.875 m. However, the highest and lowest values for the actual data were 12.610 m and 5.730 m, respectively. The main findings are summarized in

Table 9. Predicted highest, lowest, and average GWL values.

Station ID	Location	Model/Data	Highest (m.PWD)	Lowest (m.PWD)	Average (m.PWD)
KG-1	Bheramara	Existing (actual data)	12.610	5.730	8.148
KG-1	Bheramara	ANN 6-8-1 (multivariate)	10.797	5.875	7.742
KG-5	Mirpur	ARIMAX (3,0,2)	11.694	6.622	8.951

, along with the corresponding stations. The performances of ANN models can be attributed to the model’s architecture or training parameters. The ARIMAX results could be attributed to its incorporation of exogenous regressors, which might better account for factors influencing water levels. Future work should also consider incorporating a wider range of models and hybrid approaches to balance the strengths of each predictive technique. Moreover, additional validation against independent datasets will be crucial to confirm the robustness of the models and ensure their generalizability. In addition, a graphical representation of predicted GWL data is shown in Figure 6a for the ARIMAX (3,0,2) model and in Figure 6b for the ANN 6-8-1 model.

4.9. Relative Improvement in the Performances of Models and Comparison of the Best Models

A detailed analysis was conducted to validate the performances of the models and to provide an overall comparison between the ANN and ARIMA-based models. The evaluation was based on three primary performance metrics: performance enhancement, performance degradation, and reference model, against which other models are assessed.

4.9.1. Assessing the Efficiency of Models Relative to ANN

Compared to the ANN multivariate model with a 6-8-1 architecture, the results of other models indicate a notable drop; specifically, this model outperforms the best-performing ANN univariate, ARIMAX, and ARIMA models, with reductions of −8.459%, −34.658%, and −40.340%, respectively. In contrast to the ANN univariate model with a 3-7-1 architecture, the best-performing ANN multivariate model showed an enhancement of the performance with an increase of 9.241%, while ARIMAX and ARIMA experienced performance declines of −28.620% and −34.826%, respectively. The comparison indicates that, although the ANN univariate delivers enhanced predictive accuracy, it also exhibits substantial prediction deviations.

4.9.2. Assessing the Efficiency of Models Relative to the ARIMA-Based Model

Relative to the ARIMAX (3,0,2) configuration, both ANN multivariate and ANN univariate models showed significant performance improvements, with increases of 53.043% and 40.096%, respectively. However, the ARIMA model underperformed, demonstrating a performance decline of −8.694%. With the ARIMA (2,0,1) architecture, all models demonstrated a marked performance enhancement: the ANN multivariate model achieved a 67.617% improvement, the ANN univariate model improved by 53.436%, and ARIMAX showed a 9.522% enhancement. Comparative performance metrics of different forecasting models are presented in

Table 10. Evaluation metrics and comparative performance improvements of the predictive models.

			Performance Comparison of the Models				Remarks
Station ID	Results (SSE)	Model Architecture	ANN (Multivariate) (%)	ANN (Univariate) (%)	ARIMAX (%)	ARIMA (%)
KG-1	9.894	ANN 6-8-1 (multivariate)	0	−8.459	−34.658	−40.340	Reference model ANN (multivariate): Performance Enhancement: -; Performance Degradation: ANN (univariate), ARIMAX, ARIMA
KG-1	10.809	ANN 3-7-1 (univariate)	9.241	0	−28.620	−34.826	Reference model ANN (univariate): Performance Enhancement: ANN (multivariate); Performance Degradation: ARIMAX, ARIMA
KG-5	15.143	ARIMAX (3,0,2)	53.043	40.096	0	−8.694	Reference model ARIMAX: Performance Enhancement: ANN (multivariate), ANN (univariate); Performance Degradation: ARIMA
KG-1	16.585	ARIMA (2,0,1)	67.616	53.436	9.522	0	Reference model ARIMA: Performance Enhancement: ANN (multivariate), ANN (univariate), ARIMAX; Performance Degradation: -

.

4.9.3. Model Accuracy via Regression Plot

In order to provide a more comprehensive overview of the performance of the best models, ARIMAX (3,0,2) and ANN 6-8-1, the scatterplots are presented in Figure 7a,b. It is clear from the figures that the models achieved significant accuracy. Their corresponding correlation coefficient (R) values are also shown.

5. Conclusions

The purpose of this study was to assess and evaluate the efficacy of ANN-based and ARIMA-based models for predicting GWL in Kushtia, Bangladesh. Such modeling approaches, particularly those that incorporate exogenous factors, are not extensively used in this region. In order to achieve the best model fit, the ANN model was extensively tuned by testing a large number of model configurations, including hidden layer sizes and network architectures, on the MATLAB platform. To ensure that the ARIMA model accurately captured the underlying time series structure, ACF and PACF plots were used to determine the model orders.

The key findings of this study are as follows:

• The ANN model (6-8-1) outperformed the ARIMA-based best model (SSE 15.143) by 53.043% and achieved the highest predictive accuracy with an SSE 9.894.
• The multivariate ANN model showed 9.241% higher accuracy than the best univariate ANN model.
• The ARIMAX model improved prediction accuracy by 9.522% over the best ARIMA model.
• It is evident that models that included exogenous variables provided more reliable GWL predictions than univariate models.

Even though this work provides useful insights into GWL prediction, it is necessary to acknowledge several limitations. A primary constraint is the limited availability of GWL data. By including more monitoring wells, the extended dataset would significantly enhance the reliability of the model.

The current study is expected to be a useful tool for water resource managers and policymakers. Despite its limitations, this study provides significant insights into improving GWL forecasting, with practical implications for groundwater management, such as studying future GWL changes in regions with similar aquifer characteristics or hydrogeological conditions. Moreover, it lays the groundwork for future research. Overall, the findings of the current study could be helpful to improve groundwater monitoring, decision-making, and aquifer management strategies.