Optimised Stacking Generalisation Methodology for Groundwater Level Prediction
JamelSeidu1✉Emailjseidu@umat.edu.gh
AnthonyEwusi1
JerrySamuelYawKuma1
YaoYevenyoZiggah2
1Department of Geological Engineering, Faculty of Mineral Resources TechnologyUniversity of Mines and Technology University of Mines and Technology (UMaT)TarkwaGhana
2Department of Geomatic Engineering, Faculty of Mineral Resources TechnologyUniversity of Mines and Technology (UMaT)TarkwaGhana
Jamel Seidua*, Anthony Ewusia, Jerry Samuel Yaw Kumaa, Yao Yevenyo Ziggahb
aDepartment of Geological Engineering, Faculty of Mineral Resources Technology, University of Mines and Technology University of Mines and Technology (UMaT), Tarkwa, Ghana;
bDepartment of Geomatic Engineering, Faculty of Mineral Resources Technology, University of Mines and Technology (UMaT), Tarkwa, Ghana; orcid: 0000-0002-9940-1845
*Corresponding author
Corresponding author’s email: jseidu@umat.edu.gh
Corresponding author’s ORCID: 0000-0003-4729-765X
Anthony Ewusi ORCID: 0000-0002-5367-6018
Abstract
In this current study, the stacking method, which is based on optimised artificial intelligence (AI), was employed. The methods used are the particle swarm optimization-artificial neural network (PSO-ANN), genetic algorithm-artificial neural network (GA-ANN) and self-adaptive differential evolutionary extreme learning machine (SaDE-ELM). In the first level classifier, rainfall, temperature and evaporation were used as the input parameters to predict groundwater level (GWL), which was the output parameter. The best predictor among the PSO-ANN, GA-ANN and SaDE-ELM was selected and used as the meta-learner. At this stage, the predicted GWL data from the optimal base-learners was used as input. Statistical analyses show that the proposed stacking method performed better than the standalone hybrid AI models, with average RMSE and R values of 0.2209 m and 0.78, respectively. The superiority of the stacking method was further revealed using the Taylor diagram to present the statistical comparison with the observations of all models used.
Keywords:
stacking
optimization
groundwater level prediction
self-adaptive differential evolutionary extreme learning machine
A
1 Introduction
A
Groundwater is a major resource for most countries, as people rely on it for domestic, agricultural and industrial uses. Unfortunately, groundwater resources in most countries have come under a lot of stress because aquifers are being overexploited due to factors such as population increase, urbanisation and industrialisation. Additionally, the pollution of groundwater has also affected the sustainability of the global supply of the resource (Mohanty et al., 2010). For most countries, sustainable use of groundwater has come under stress, which is a major concern because of the lack of strategies to effectively maintain aquifer yields (Mohanty et al., 2010; Kisi, Alizamir and Zounemat-Kermani, 2017). To ensure better management strategies for the sustainability of groundwater, there is a need for accurate and reliable prediction of groundwater level (GWL) changes. This is to say that the importance of GWL forecasting in any hydrogeological studies cannot be overemphasised. Given this, there have been several types of research works conducted to develop models to accurately predict groundwater level fluctuations. Models used to predict GWLs have evolved over the years, from the use of mathematical and conceptual models to the application of data-driven models, particularly artificial neural networks (ANNs). Although physically-based numerical models have been used to represent the physical conditions of hydrogeological regimes, their underlying limitations, such as the need for vast and accurate data requirements, make them unsuitable for prediction, especially in areas where the quantum of data needed is not available (Yang et al., 2009). Alternatively, conceptual models have also been used to understand the basic principles of groundwater systems. Yet, they have an underlining limitation due to their inability to consider the complexities associated with the groundwater systems (Chang et al., 2016).Data-driven artificial intelligence (AI) models have been found to capture different data patterns and can imitate complex groundwater systems at diverse spatio-temporal scales (Chang et al., 2016). In hydrological studies, AI has been successfully used for rainfall-runoff modelling, streamflow computations, water quality analysis, and groundwater level predictions (Srinivasulu and Jain, 2006; Wang et al., 2006; Palani, Liong and Tkalich, 2008; Mohanty et al., 2015). The application of AI techniques in groundwater level prediction has been reviewed by Rajaee et al. (2019), which consists of 67 papers spanning from 2001 to 2018. In their research, the AI methods widely employed included ANN, support vector machine (SVM), adaptive neuro-fuzzy inference system (ANFIS), genetic programming (GP), a hybrid of these methods, among others. A similar review was carried out by Tao et al. (2022), where AI methods applied spanning from 2008 to 2020 were considered. Methods such as ANN (Nair and Sindhu, 2016; Yadav et al. 2017), fuzzy logic and neuro-fuzzy (Fallah-Mehdipour et al. 2013; Bak et al, 2021), kernel-based (Sahoo et al. 2018; Yoon et al. 2016), deep learning (Zhang et al. 2018; Shin et al. 2020; Wunsch et al. 2022) and hybrid AI models (Adamowski and Chan 2011; Malekzadeh et al. 2019; Seidu et al. 2022a; Wei et al 2023). In Khan et al. (2023), emphasis was on the performance of different deep learning models that are used for groundwater studies. In addition, wavelet-bidirectional-long short-term memory was proposed in the study to simulate yearly GWL. It can be stated that over the last four years, three important studies have been undertaken that look at a review of AI approaches for GWL predictions (Rajaee et al. 2019; Tao et al. 2022; Khan et al. 2023), totalling 314 publications (value may be lower due to the possibility of overlapping papers).
Based on the reviews (Rajaee et al. 2019; Tao et al. 2022; Khan et al. 2023), it was evident that the independent use of the traditional AI methods for predictions has some limitations, such as slow learning speed, converging to a local minima and overfitting (He et al., 2019). Therefore, as a means of overcoming these setbacks, researchers have applied several hybrid techniques, some of which are optimised-ANN methods like particle swarm optimisation-ANN (PSO-ANN) (Kisi, Alizamir and Zounemat-Kermani, 2017; Balavalikar et al., 2018), and genetic algorithm-ANN (GA-ANN) (Dash et al., 2010; Pandey et al., 2020). Additionally, hybrid models like wavelet transform-ANN (WT-ANN) and empirical mode decomposition-ANN (EMD-ANN), which are based on a combination of signal decomposition and traditional ANN, have been applied to improve the performance of GWL predictive models (Adamowski and Chan, 2011; Yang et al., 2014; Gong et al., 2018). In the aforementioned studies, it has been established that the use of hybrid prediction models results in better model performance as opposed to the use of single traditional AI methods. For example, in PSO-ANN, the PSO is used to optimise the input weights and threshold of ANN, leading to improved prediction capability of the ANN when independently applied. Similarly, the WT as used in WT-ANN served as a data pre-processing technique to either denoise, compress or decompose the input data (Rajaee et al. 2019). The combined WT and ANN (WT-ANN) has also been found to outperform the use of a single ANN. These research works further affirmed that the hybrid models have the capacity to enhance the prediction capability of single AI methods.
One other effective hybrid methodology in literature that has gained wider attention in other areas of geosciences is the stacking generalisation proposed by Wolpert (1992). For example, authors such as Jia et al. (2021) investigated the capability of the stacking methodology for classification of rock types and for 3D geological modelling. In Anifowose et al. (2015), a stacked generalisation model was proposed for predicting the permeability and porosity of reservoir. Asante-Okyere et al. (2023) proposed a stacking model to predict total organic carbon using mineral composition from shale rocks. Information gathered from these studies demonstrates that the stacked approach produces superior prediction accuracy as compared to the standalone AI predictions. The reason is that the stacking method deals with the combination of multiple AI algorithms into a single framework called the base-level classifier (base-learners) and a meta-level classifier (meta-learner) (Sigletos et al., 2005). Here, the individual predictions from the base-learners serve as the input data to the meta-learner, leading to enhanced prediction accuracy.
Given the enumerated strengths, this study investigated for the first time the prediction strength of the stacking generalisation method of machine learning in groundwater level studies. In the proposed approach, three different optimised algorithms, namely: PSO-ANN, GA-ANN and self-adaptive differential evolutionary extreme learning machine (SaDE-ELM), are employed as the base learners, while the best-performing model amongst these three is subsequently selected as the meta-learner. These models bring complementary strengths, that is optimization, computational efficiency and adaptability to the complex task of groundwater level prediction. This approach provides a good foundation for developing a predictive model that is not only accurate but also able to generalize to diverse groundwater systems. This makes the study both innovative and practically relevant in advancing groundwater modelling research.
To this end, the main contributions of this study are as follows:
i.
To investigate the prediction capacity of the stacking generalisation method for groundwater level, which has not yet been discussed in the literature.
ii.
To establish that the optimised AI methods' performance in groundwater level prediction can be enhanced using the stacking generalisation methodology.
The rest of the paper is organised as follows: Information about the study area and data description is presented in section 1.1. The AI methods used are presented in section 2. In section 3, the results and discussion are presented. The conclusion is discussed in section 4.
1.1 Study Area and Data Description
This current research was conducted in the Tarkwa Mining Area, which is located in (Fig. 1). The Tarkwa area lies within the southwestern equatorial climatic zone, characterised by distinct seasonal patterns. These are controlled by two dominant wind systems: moisture-laden southwestern monsoon winds originating from the South Atlantic Ocean, and the dry, dust-bearing northeastern Harmattan winds that originate from the Sahara Desert's subtropical high-pressure zone. The area has two rainfall peak periods in a year, which is controlled by the times when the Inter-Tropical Convergence Zone (ITCZ) crosses over the Tarkwa area. The two wet seasons stretch from April to July (with a peak in June) and from September to November. Meteorological data recorded at the University of Mines and Technology (UMaT) station between 1990 and 2021 reveal significant interannual variability, with annual rainfall totals ranging from 1,381 mm (1998) to 2,174 mm (2021) (Fig. 2). The Tarkwa area's hydrogeological system comprises two aquifer types: shallow weathered aquifers that develop in the regolith above the fresh bedrock interface, and deeper fractured zone aquifers that occur in competent rock below the weathered zone. In some areas, the weathering depth rarely exceeds 20 m, and also because of the presence of very fine materials, such as clay and silt, aquifers tend have high porosity and storage but relatively low permeability.
Groundwater level data for this study consisted of monthly measurements obtained between 2013–2018 from 11 monitoring wells (a total of 351 datapoints), obtained from AngloGold Ashanti Iduapriem Limited's (AAIL) Environmental Department in Tarkwa, Ghana. The data on GWL is useful in planning for mine pit dewatering, designing facilities such as tailings storage and for general groundwater studies. Similarly, weather stations have been installed to collect meteorological data, which is used for planning purposes within the mine and its catchment communities. Therefore, meteorological data including rainfall, evaporation and temperature records for the corresponding period under consideration were also obtained as exogenous factors that influence groundwater level fluctuations.
Descriptive statistics for all eleven (11) boreholes used for the study are presented in Table 1, for both the training and testing datasets. The values indicate that BH7 has the least static water level measured from the surface. For GWL monitoring, the minimum, mean, and maximum values recorded in BH7 are 40.27 m, 41.08 m and 42.07 m, respectively, for the training data. Also, the minimum, mean, and maximum values for the testing data for BH7 were 40.75 m, 41.40 m and 41.67 m, respectively.
Additionally, the highest static water level was recorded in BH10 with the minimum, mean and maximum values recorded as 69.71 m, 70.18 m and 70.46 m, respectively, for the training data. Generally, the minimum, mean and maximum GWL values are in the ranges of 40.27 m – 69.71 m, 41.17 m – 70.16 m and 42.27 m – 69.71 m. The standard deviation (SD) and coefficient of variation (CV) for the training data are in the ranges of 0.25–0.55 and 0.06–0.30, respectively. The values obtained for the standard deviation in the training and testing data indicate that they are close to the mean, which suggests low variability in the groundwater level data.
ID | Training | Testing | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
Min | Mean | Max | SD | CV | Min | Mean | Max | SD | CV | |
BH1 | 45.08 | 45.72 | 46.28 | 0.32 | 0.10 | 45.92 | 46.02 | 46.18 | 0.10 | 0.01 |
BH2 | 43.72 | 44.74 | 45.57 | 0.45 | 0.20 | 44.19 | 44.84 | 45.14 | 0.34 | 0.12 |
BH3 | 43.30 | 44.03 | 44.65 | 0.39 | 0.15 | 43.30 | 44.05 | 45.32 | 0.59 | 0.35 |
BH4 | 58.56 | 59.71 | 60.36 | 0.55 | 0.30 | 59.30 | 59.58 | 60.24 | 0.32 | 0.10 |
BH5 | 47.06 | 47.74 | 48.26 | 0.40 | 0.16 | 47.38 | 47.73 | 48.16 | 0.26 | 0.07 |
BH6 | 46.71 | 47.49 | 48.11 | 0.42 | 0.18 | 47.06 | 47.53 | 47.86 | 0.33 | 0.11 |
BH7 | 40.27 | 41.08 | 42.07 | 0.46 | 0.21 | 40.75 | 41.40 | 41.67 | 0.34 | 0.11 |
BH8 | 44.44 | 45.17 | 45.76 | 0.35 | 0.12 | 45.10 | 45.26 | 45.68 | 0.22 | 0.05 |
BH9 | 44.65 | 45.25 | 45.83 | 0.32 | 0.11 | 44.65 | 45.26 | 45.89 | 0.43 | 0.18 |
BH10 | 69.71 | 70.18 | 70.46 | 0.25 | 0.06 | 69.92 | 70.11 | 70.86 | 0.37 | 0.14 |
BH11 | 46.78 | 47.55 | 48.15 | 0.42 | 0.18 | 47.37 | 47.70 | 47.96 | 0.22 | 0.05 |
| Min: minimum; Max: maximum; SD: standard deviation; CV: Coefficient of variation | ||||||||||
Fig. 1
Study Area showing Spatial Distribution of the Boreholes
Fig. 2
Annual Rainfall for the Tarkwa area between 1990 and 2021
2 Methods used
The general framework for this research is shown in Fig. 3. As earlier stated, the three methods used as base-learners are PSO-ANN, GA-ANN and SaDE-ELM, where the best performing model among them was selected as the meta-learner (level-1 classifier). Details of the methods that were employed for this study are presented in sections 2.1 to 2.4. In this research, the input parameters used were rainfall, temperature and evaporation, whereas GWL was the output parameter. These inputs were selected because there has been known in literature to be exogenous factors that affect water level (Moosavi et al 2014; Djurovic et al 2015; Seidu et al. 2022b). The data set was divided into training and testing using a partitioning percentage of 70 − 30. The partitioning percentages were selected based on recommendations from the research conducted in the study area by Seidu et al. (2022b).
Fig. 3
Framework for Stacking Method Used
2.1 PSO-ANN
The PSO-ANN is a hybrid of the particle swarm optimization (PSO) algorithm and artificial neural network (ANN). The PSO algorithm, as proposed by Eberhart and Kennedy, (1995), is a nature-inspired evolutionary algorithm that mimics the movement seen in fish schooling or bird flocking. In this hybrid method, the PSO is used to optimise the input weights of ANN such that the prediction accuracy is improved as compared to the application of a single ANN. In the PSO, a cost function that should be optimised is defined, where the particles created are treated as a point in the m-dimensional space of the problem (Ghosh, Mandal and Mondal, 2019; Shariati et al., 2019). Each particle keeps information of its best coordinate value attained, and this is termed the personal (local) best. Additionally, particles also look out for personal best coordinates from other particles, where a comparison is made to determine whether or not there is a better coordinate value. Ultimately, the particle that returns the best coordinate after all the comparisons have been made is called the global best (Ghosh, Mandal and Mondal, 2019). In the optimization process, the particle updates its velocity (v) and position (x) using equations 1 and 2. The pseudocode for the PSO is shown in Fig. 4.
1
2
where vi = {vi1, vi2, vi3,…, vin} is the velocity component, xi = {xi1, xi2, xi3,…, xin} is the position, i = 1, 2,. . .,n, n is the population size, w is the inertia weight, c1 and c2 are two positive constants, representing the cognitive and social components, respectively, and r1 and r2 are random numbers that is uniformly distributed (0, 1).
Fig. 4
Pseudocode for the PSO Algorithm
2.2 GA-ANN
A
The hybrid GA-ANN method is an intelligence system that can improve generalization as well as ease the ANN design process (Ahmad et al., 2010). The GA algorithm is a type of evolutionary algorithm, whose techniques are based on evolutionary biology such as inheritance, selection, crossover and mutation to find solutions to optimisation problems (Tahboub et al., 2016). The principle governing the GA dictates that an initial population of chromosomes should be generated. Afterwards, selection and crossover operators are used to generate a new and more effective population which will result in the optimal value among them (Inthachot, Boonjing and Intakosum, 2016). A simplified algorithm of the GA-ANN is presented as a flowchart in Fig. 5.Fig. 5
Flowchart of the GA-ANN
2.3 SaDE-ELM
The SaDE-ELM was proposed by Cao, Lin and Huang (2012) to overcome the limitations of manually choosing trial vector generation strategies and its control parameters that are associated with Differential Evolution (Storn and Price, 1997). The SaDE-ELM algorithm uses the self-adaptive differential evolution algorithm to optimise the network input weights and hidden node biases, and the extreme learning machine to derive the network output weights. With learning datasets, L hidden nodes and the activation function g(x), the SaDE-ELM can be formulated as (Cao, Lin and Huang, 2012; Malekzadeh et al., 2019; Yosefvand and Shabanlou, 2020):
i.
The optimization process begins by initializing a population of candidate solutions, represented as population vectors (NP) that involve hidden nodes as defined in Eq. (7)
7
where G is the number of generations, aj and bj (j = 1,2,…,L) are randomly generated, k = 1,2,…, NP.
ii.
For each population vector, compute the output weight matrix and the corresponding root mean square error (RMSE) metric with the following equations (Equations 8–11)
8
9
where H*k,G is Moore–Penrose generalized inverse (MPGI) of Hk,G expressed as:
10
11
The population vector with the best RMSE value is stored in RMSEθbest,1 and θbest,1. The results obtained as the output weights and RMSE value are optimised as differential evolution (DE) operators using mutation and crossover. The pseudocode of the SaDE-ELM algorithm is presented in Fig. 6.
Fig. 6
Pseudocode for the SaDE-ELM algorithm (after Seidu et al., 2022b)
2.4 Stacking Method Modelling
The stacking method consists of two levels, namely, level 0 (multiple base learners) and level 1 (meta learner). The meta-learner uses prediction results from the multiple base learners to maximise the generalisation ability of the model. In this study, the base learners used were PSO-ANN, GA-ANN and SaDE-ELM, out of which the best model was selected as the meta-learner. For each of these methods, rainfall, temperature and evaporation were used as input data and GWL as the output. The model with the best performance in the testing stage was selected as the meta-learner. The output from the base learners served as the new dataset for prediction. GWL values from the base learners were used as input data, which were fed to the meta-learner to train the model.
2.5 Performance Indicators
For this research, well-known statistical indicators, namely root mean square error (RMSE) and correlation coefficient (R), were used to assess the performance of the models. RMSE is used to measure the accuracy of prediction, such that the closer the RMSE value to zero, the better the prediction ability of the model. The correlation coefficient (R), which varies between 0 and 1, indicates the closeness of the observed values fitted to a regression line. The closer the value is to 1, the stronger the correlation that exists between the observed and predicted values.
12
13
where n is the data size, Fi is the predicted values, Oi is the observed values, 
and 
are the means of the observed and predicted values, respectively.
3 Results and Discussions
3.1 Optimum Models Developed
The AI models used in this study, GA-ANN, PSO-ANN, and SADE-ELM, were developed and run using MATLAB 2019a. To partition the data, we applied the widely recognised hold-out cross-validation technique, splitting the dataset into training and testing sets with a 70:30 ratio for all eleven (11) boreholes. This approach was chosen to minimise the risks of overfitting and underfitting, ensuring robust model performance.
For model evaluation, the Root Mean Square Error (RMSE) was used as the loss function. The model with the smallest RMSE difference between the training and testing datasets was identified as the most effective. To handle the high variability in the dataset, the data was normalized to a range of [-1,1] using MATLAB's mapminmax function. This normalization step ensured consistency and improved the models' ability to generalize.
The optimum model developed for this study is based on the SADE-ELM technique. For the SADE-ELM technique, the control parameters include crossover rate, mutation scaling factor, number of hidden neurons and position (Table 2). These parameters were carefully optimised to ensure the model's effectiveness and accuracy.
ID | CR | FP | NP | NHN | Position |
|---|---|---|---|---|---|
BH1 | 0.40 | 1.10 | 80 | 4 | 2064 |
BH2 | 0.50 | 2.00 | 100 | 5 | 3885 |
BH3 | 0.80 | 0.70 | 78 | 17 | 1357 |
BH4 | 1.00 | 1.20 | 65 | 9 | 2389 |
BH5 | 0.70 | 1.90 | 80 | 3 | 3723 |
BH6 | 0.90 | 1.20 | 78 | 17 | 2377 |
BH7 | 0.50 | 1.90 | 110 | 6 | 3686 |
BH8 | 0.30 | 1.00 | 50 | 9 | 1849 |
BH9 | 0.50 | 1.50 | 80 | 3 | 2883 |
BH10 | 0.90 | 1.00 | 80 | 8 | 1968 |
BH11 | 0.50 | 1.30 | 80 | 5 | 2485 |
CR crossover rate, F mutation scaling factor, NP Number of Population, NHN number of hidden neurons
3.2 Evaluation of Base Learners
The performance of the base learners was evaluated using the RMSE statistical index. Among the investigated methods, a model with the least RMSE values is regarded as the best model and most superior. For example, BH1 produced RMSE values of 0.36716, 0.38394 and 0.23077 representing GA-ANN, PSO-ANN and SaDE-ELM models, respectively (Table 3). Similar patterns were observed in the remaining 10 boreholes (BH2–BH11), where the SaDE-ELM produced the best model. Consequently, the SaDE-ELM in all boreholes produced the model with the least RMSE and was therefore selected as the meta-learner. Results in Table 2 have been presented graphically in Figure. 7 to give a visual impression of the information in the table.
ID | GA-ANN | PSO-ANN | SaDE-ELM |
|---|---|---|---|
BH1 | 0.36716 | 0.38394 | 0.23077 |
BH2 | 0.43117 | 0.47623 | 0.25196 |
BH3 | 0.43720 | 0.39313 | 0.28372 |
BH4 | 0.37380 | 0.46232 | 0.33297 |
BH5 | 0.45664 | 0.36561 | 0.27859 |
BH6 | 0.26916 | 0.28595 | 0.20709 |
BH7 | 0.39375 | 0.40341 | 0.36235 |
BH8 | 0.18330 | 0.26020 | 0.13667 |
BH9 | 0.48854 | 0.43174 | 0.37828 |
BH10 | 0.32743 | 0.37767 | 0.22622 |
BH11 | 0.31871 | 0.35240 | 0.19358 |
Fig. 7
Graphical representation of the performance of the Base learners
3.3 Performance of the Stacking Method
In this section, we evaluate the prediction efficacy of the stacking method for groundwater level prediction. The performance of the proposed stacking method is compared to efficient artificial intelligence techniques like PSO-ANN, GA-ANN and SaDE-ELM. Based on the results presented in Table 4, the RMSE and R statistical indices show that the stacking method has a better prediction capability as compared to the other investigated methods. For example, the RMSE value for the stacking method is 0.1949, which shows a significantly better model performance than the 0.3672, 0.3839 and 0.2308 values recorded for the GA-ANN, PSO-ANN and SaDE-ELM, respectively. Similarly, the R values recorded for the stacking method as compared to the other methods indicate that there is a strong relationship between the observed and predicted groundwater level values. In the stacking method, the R-value for BH1, for example, is 0.7263, indicating a better prediction capability of the model as compared to the R values recorded in the GA-ANN (-0.2768), PSO-ANN (0.5047) and SaDE-ELM (-0.8708). This scenario is observed in all the other boreholes and therefore indicates that the stacking method has the best prediction capability among all the models used. In furtherance, based on the RMSE values, the best model is the stacking method, followed by the SaDE-ELM. However, such consistency is not observed in the GA-ANN and PSO-ANN. For instance, in boreholes BH3, BH5 and BH9, PSO-ANN outperforms the GA-ANN based on their RMSE values. However, in all other boreholes, the RMSE values indicate that GA-ANN outperforms the PSO-ANN model. To visualise the statistical summary of how well observed and predicted groundwater level values match each other, Taylor diagrams for all the boreholes have been shown in Figs. 8 and 9. Taylor showed statistical information such as correlation, root-mean-square difference and the ratio of variances based on the prediction ability of the proposed stacking method.
Models | STACKING | GA-ANN | PSO-ANN | SaDE-ELM | ||||
|---|---|---|---|---|---|---|---|---|
ID | RMSE | R | RMSE | R | RMSE | R | RMSE | R |
BH1 | 0.1949 | 0.7263 | 0.3672 | -0.2768 | 0.3839 | 0.5047 | 0.2308 | -0.8708 |
BH2 | 0.2268 | 0.7423 | 0.4312 | -0.6027 | 0.4762 | -0.6656 | 0.2520 | 0.0175 |
BH3 | 0.2219 | 0.9286 | 0.4372 | 0.6880 | 0.3931 | 0.7398 | 0.2837 | 0.7680 |
BH4 | 0.2612 | 0.6884 | 0.3738 | 0.6787 | 0.4623 | 0.6593 | 0.3330 | 0.6243 |
BH5 | 0.2306 | 0.5794 | 0.4566 | -0.2513 | 0.3656 | -0.3632 | 0.2786 | -0.1426 |
BH6 | 0.1235 | 0.9236 | 0.2692 | 0.5744 | 0.2859 | 0.5339 | 0.2071 | 0.8535 |
BH7 | 0.3439 | 0.5850 | 0.3937 | 0.3115 | 0.4034 | 0.2761 | 0.3624 | 0.5731 |
BH8 | 0.1283 | 0.8539 | 0.1833 | 0.6989 | 0.2602 | -0.3260 | 0.1367 | 0.7200 |
BH9 | 0.3277 | 0.7570 | 0.4885 | 0.1676 | 0.4317 | 0.2992 | 0.3783 | 0.4781 |
BH10 | 0.1959 | 0.9557 | 0.3274 | -0.1198 | 0.3777 | -0.2985 | 0.2262 | 0.8913 |
BH11 | 0.1754 | 0.7918 | 0.3187 | -0.2685 | 0.3524 | -0.7412 | 0.1936 | 0.6555 |
The current study has demonstrated that the stacking method is more effective when used in groundwater level prediction. The strength of the proposed stacking method is based solely on the different stages of modelling, where at the initial stage, the objective was to select among three models the one with better prediction capability. The second and final stage uses the results from the initial stage as input and subsequently for prediction. The stages of prediction make the stacking method more refined as compared to the standalone methods. In furtherance, the stacking method has demonstrated that the prediction power of the model is enhanced. Even though we have established that rainfall, temperature and evaporation (which are exogenous factors that influence GWL) influence the groundwater level, because of the variability that exists within the individual parameters, the resultant groundwater level prediction values are affected. However, the stacking method has addressed this limitation by using GWLs from the base learner as the new input for prediction. This action has revealed that the ultimate prediction power of the model has been enhanced significantly.
3.4 Sensitivity Analysis
Sensitivity analysis was performed to determine the influence and relative importance of the input variable values on the output variable using the cosine amplitude method (Eq. 14). The sensitivity values for precipitation ranged from 0.79037 (BH8) to 0.86393 (BH11) (Table 5). This indicates moderate to high sensitivity, with an average score of 0.83542 across all boreholes. Sensitivity values for temperature were consistently high, ranging from 0.99782 (BH6 and BH7) to 0.99850 (BH4). The sensitivity scores for evaporation ranged from 0.97846 (BH5) to 0.98216 (BH6).
14
Among the three variables, temperature showed the highest and most consistent sensitivity across all boreholes, highlighting its dominant role in influencing groundwater levels. Precipitation exhibited some variability but still maintained a strong influence, likely due to its direct contribution to groundwater recharge.
ID | Precipitation (mm) | Temperature (°C) | Evaporation (mm) |
|---|---|---|---|
BH1 | 0.83846 | 0.99824 | 0.97923 |
BH2 | 0.84233 | 0.99847 | 0.98167 |
BH3 | 0.84496 | 0.99848 | 0.98174 |
BH4 | 0.85710 | 0.99850 | 0.98029 |
BH5 | 0.83898 | 0.99782 | 0.97846 |
BH6 | 0.84439 | 0.99817 | 0.98216 |
BH7 | 0.82738 | 0.99782 | 0.98022 |
BH8 | 0.79037 | 0.99828 | 0.97965 |
BH9 | 0.80733 | 0.99824 | 0.98138 |
BH10 | 0.83375 | 0.99805 | 0.98088 |
BH11 | 0.86393 | 0.99790 | 0.98176 |
4 Conclusions
Groundwater is an important natural resource that is used in many systems, including drinking water, agriculture, and environmental security. However, there is a threat to the water supply due to over-reliance on groundwater resources, which is most likely an environmental hazard. Groundwater level studies, such as the current work, are vital to ensure efficient resource utilization. In this study, we introduce the stacking method for groundwater level prediction, which is based solely on the application of optimised AI techniques as base learners and a meta learner. The stacking method leverages the strength of the individual base learners and therefore produces results that perform better as compared to the use of standalone AI techniques. A comparison of the stacking method to the GA-ANN, PSO-ANN and SaDE-ELM revealed that it has a greater prediction accuracy and has therefore been able to reliably predict groundwater levels. In the stacking method, the base learners used were all optimised AI techniques, and this gives uniqueness to the proposed stacking method. The successes associated with the stacking method can open a new dimension to the application of AI methods in groundwater level modelling. We proposed that for future studies, the stacking can be done by considering the traditional AI methods and the most recent deep learning approaches, to leverage their strengths and weaknesses for improved predictions.
Fig. 8
Taylor diagram displaying a statistical comparison with observations of four model estimates of groundwater level in (a) BH1 (b) BH2, (c) BH3, (d) BH4, (e) BH5 and (f) BH6
Fig. 9
Taylor diagram displaying a statistical comparison with observations of four model estimates of groundwater level in (a) BH7 (b) BH8, (c) BH9, (d) BH10, and (f) BH11
Acknowledgement
The authors would like to thank Anglogold Ashanti Iduapriem Limited (AAIL), Tarkwa for providing data for this research.
Author Contribution Statement
JS was in charge of data curation, conceptualization, writing of the draft and final manuscript. AE assisted in the data collection, conceptualization, prepared and edited the draft and final manuscript. JSYK was in charge of data validation, software scrutiny, conceptualization and edited the draft and final manuscript. YYZ provided software and modelling tools, part of conceptualization, prepared the methods and edited the final manuscript. All authors were involved in the manuscript review during several stages of its preparation.
Data Declaration
The data used from this research was obtaind from the archives of Anglogold Ashanti Iduapreim Limited (AAIL), Tarkwa. Statistics of the data is provided in the manuscript. The authors request that the full data be in the possession of them until a time where the journal would request for the data (especially for review purposes), because of confidentiality.
A
Funding
Declaration
The authors declare that there was no funding for this research.
Ethics approval and consent to participate
Not applicable
Ethics and Consent to Participate declarations
Not applicable
A
Author Contribution
JS was in charge of data curation, conceptualization, writing of the draft and final manuscript. AE assisted in the data collection, conceptualization, prepared and edited the draft and final manuscript. JSYK was in charge of data validation, software scrutiny, conceptualization and edited the draft and final manuscript. YYZ provided software and modelling tools, part of conceptualization, prepared the methods and edited the final manuscript. All authors were involved in the manuscript review during several stages of its preparation.
A
Data Availability
The data used from this research was obtaind from the archives of Anglogold Ashanti Iduapreim Limited (AAIL), Tarkwa. Statistics of the data is provided in the manuscript. The authors request that the full data be in the possession of them until a time where the journal would request for the data (especially for review purposes), because of confidentiality.
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