## Abstract

Reliable long-term groundwater level (GWL) prediction is essential to assess the availability of resources and the risk to drinking water supply in changing climatic and socio-economic conditions, especially in areas with water deficits. The modern approach in this area involves the use of machine learning methods. However, the greatest challenge in these methods lies in the optimization of input selection. The presented research concerns the selection of the best combination of predictors using the Hellwig method. It served as a preprocessing technique before GWL prediction using support vector regression (SVR) and multilayer perceptron (MLP) for three wells in the Greater Poland Province, where the largest water deficits occur, in the period 1975–2014. The results of this method were compared with those of the regression method, general regression model. For the case study under investigation, the Hellwig method provided the best set of predictors consisted of GWL at lags of −1 and −2 months, precipitation from the current month, and delayed from −1 to −6 months, and past temperature at −1, −3, −4 and −6 months. Such input led to a model accuracy of 0.003–0.022 for a mean squared error and *r*^{2} of >0.8. The results obtained with SVR were slightly better than those with MLP. Moreover, every well required an individual set of predictors, and additional meteorological inputs improved the models’ performance.

## HIGHLIGHTS

The performance of the Hellwig method in the selection of input variables for groundwater level (GWL) forecasting using support vector regression and multilayer perceptron.

GWL dependence on meteorological and hydrological conditions within the past 6 months.

Additional meteorological inputs can improve the performance of forecasts.

Neighbouring wells require an individual set of predictors.

GWL modelling in areas threatened by water scarcity.

### Graphical Abstract

## INTRODUCTION

Differential access to water in the world causes temporary or permanent water deficits in various regions. This results from irregular water resources and the rainfall that supplies them, population increase, anthropogenic processes (like progressive urbanization), increase in the use of water for industry and agriculture, environmental pollution, climate change, etc. (Kubicz *et al.* 2018, 2021; Lochyński *et al.* 2019). Satisfying the water needs of various recipients in a sustainable way is, therefore, one of the major challenges of the modern world (Glorian *et al.* 2018). To meet this challenge, the potential of intelligent water resource management systems is currently being evaluated worldwide. The idea of these systems is to use complex IT techniques to model processes occurring within groundwater (and surface water). Rapid progress in the field of computer science enables the implementation of new algorithms for problem optimization in the area of water resource management. A very popular technique in this area is groundwater level (GWL) prediction.

In environmental engineering, GWL forecasting results in a specific (short-, medium-, and long-term) time horizon are important for the rational management of groundwater resources. Long-term predictions of the groundwater resources’ quantity and their trends’ analysis allow to assess the availability of resources in changing climatic (and socio-economic) conditions (Malakar *et al.* 2018), particularly in areas suffering from water scarcity (Ibrahim *et al.* 2015).

Until now, for this purpose, depending on the time horizon of forecast, machine learning methods such as genetic programming (GP) (Shiri & Kisi 2011; Sivapragasam *et al.* 2015), artificial neural networks (ANNs) (Tsanis *et al.* 2008; Jalalkamali *et al.* 2011; Chitsazan *et al.* 2015; Kumar & Bhattacharjya 2021), and support vector machines (SVMs) (Suryanarayana *et al.* 2014; Huang *et al.* 2017; Yadav *et al.* 2017; Rahman *et al.* 2020) were the most often used. The undoubted advantage of their use is the fact that detailed knowledge about the modelled phenomenon and information on the existence of dependencies between data, or whether such relationships exist at all, is not required. However, one of the greatest challenges in these methods lies in the optimal selection of input parameters for the prediction model (Chitsazan *et al.* 2015; Amaranto *et al.* 2018). Sivapragasam *et al.* (2015) indicated that the selection of predictors plays a crucial role in modelling efficiency, especially for limited data sets. Moreover, they stated that the study area needs a unique and individual approach in terms of GWL modelling because a model suitable for a particular site may not necessarily work well even for a neighbouring area. The authors use different data preprocessing techniques for this purpose. Wu *et al.* (2021) used the Boruta method to obtain the best input variable combinations for monthly GWL forecasting in the Zhangye Basin, northwest China. Sharafati *et al.* (2020) attempted the gamma test for monthly GWL prediction over the Rafsanjan aquifer in Iran. Rahman *et al.* (2020) used eXtreme Gradient Boosting and random forest to find predictors for GWL forecasts for Kumamoto City in southern Japan. Sahoo *et al.* (2017) employed an input variable selection (IVS) method based on singular spectrum analysis, mutual information, and genetic algorithms to model seasonal GWL changes in the High Plains aquifer system and the Mississippi River Valley alluvial aquifer, USA. However, less computational demanding methods for predictor selection in GWL forecasting based on correlations between input and output variables have been implemented. Zhao *et al.* (2016) tested the correlation between predictors and GWL before GWL modelling in the Shuping landslide in the Three Gorges Reservoir area, China. Iqbal *et al.* (2020) studied the correlation between meteorological parameters and target GWL for the area located between Ravi River and Sutlej River, Pakistan. Osman *et al.* (2021) used cross-correlation to choose the best input variables for predictive GWL models in Selangor, Malaysia. Kajewska-Szkudlarek & Łyczko (2021) assessed the Hellwig method for input selection and proved its effectiveness in filling the gaps in daily GWL time series on the basis of neighbouring wells. Obtained models with a root mean square error (RMSE) of 0.024–0.292 and *r*^{2} of 0.7–0.9 were considered as high quality. Moreover, they showed good performance for both high and low GWL values. In the present research, the Hellwig method was used to choose the best combination of input variables for monthly GWL forecasting; this method also fit in the correlation techniques for predictor selection. This method is otherwise known as the method of information capacity indicators (Hellwig 1968). Its high efficiency, confirmed by a comparison with other variable selection techniques (zero unitarization method, Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), etc.) (Omiotek *et al.* 2019; Wójcik-Leń *et al.* 2019; Szmidt *et al.* 2020), led the authors to apply it in GWL modelling. Authors who have used the Hellwig method in different applications have pointed out its several advantages like universality (Łopucki & Kiersztyn 2015), low complexity and easy implementation (Omiotek *et al.* 2019), wide availability (Omiotek *et al.* 2019), and low calculation effort (Kajewska-Szkudlarek & Łyczko 2021). Moreover, at the stage of predictor selection, it is based on the use of the variables that are strongly correlated with the target variable and poorly correlated with each other. (Łopucki & Kiersztyn 2015).

Previous studies indicate that the best GWL prediction models are obtained when the input variables are meteorological parameters describing weather conditions (or climatic conditions over longer periods) and conditions resulting from the hydrogeology of measurement sites. In addition, the authors consider the past levels of groundwater depth as the most significant variable, while the choice of other parameters depends on the location of the study area (Adamowski & Chan 2011; Jalalkamali *et al.* 2011; Shiri & Kisi 2011; Suryanarayana *et al.* 2014; Chitsazan *et al.* 2015; Sivapragasam *et al.* 2015). Among the most frequently considered input variables are: GWL, air temperature and humidity, rainfall, and evapotranspiration. The quality of predictive models is assessed based on the following measures: correlation coefficient *r* (0.95–0.98 in Chitsazan *et al.* 2015), squared correlation *r*^{2} (0.92–0.98 in Jalalkamali *et al.* 2011), mean squared error (MSE) (0.005–0.009 in Chitsazan *et al.* 2015), RMSE (0.131–0.276 in Jalalkamali *et al.* 2011), mean absolute error (MAE) (0.065–0.098 in Shiri & Kisi 2011), and mean absolute percentage error (0.157–0.423 in Jalalkamali *et al.* 2011).

Chitsazan *et al.* (2015) in their research used the feed-forward back propagation neural network to predict the monthly GWL of Aghili plain, Iran, with four different learning algorithms: descent with momentum, Levenberg Marquardt (LM), resilient back propagation (RP), and scaled conjugate gradient. Input data to the model included rainfall, evaporation, relative air humidity, air temperature, discharge of irrigation canals, recharge from the boundaries, and GWLs of wells in time *t*, *t* − 1, *t* − 2, and the water level value in *t* + 1. The climatological data covered 39 years of measurement. The authors adapted the clustering approach to the data, dividing them into three groups according to the hydrogeological characteristics of different parts of the plain surrounding each well. The measures that determined the quality of the prediction model at the individual stages of its creation were the MSE and correlation coefficient. The authors concluded that LM algorithm displayed best performance for all three hydrogeological groups.

Yadav *et al.* (2017) assessed the suitability of extreme learning machines (ELMs) and SVM for GWL prediction at two observation wells located in Canada. A monthly data set of 8 years from 2006 to 2014 consisting of both hydrological and meteorological parameters (rainfall, temperature, evapotranspiration, and GWL) was used for making a comparative study of the models. These variables were used in various combinations for performing univariate and multivariate analyses of the models. The prediction performance of the developed models was evaluated using the coefficient of determination, root mean square error, and Nash–Sutcliffe efficiency criterion. The study demonstrates that the proposed ELM model has better forecasting ability than the SVM model for monthly GWL forecasting.

Sivapragasam *et al.* (2015) used GP to model the spatial fluctuations of monthly GWL in the Arjuna Nadhi subbasin region, India, for a period of 10 years (2001–2010). Similar to Yadav *et al.* (2017), the authors proved that input parameter selection is essential for accurate prediction and that the parameters should be chosen by considering the individual conditions of research sites. They modelled the GWL in the output wells using information from neighbouring ones.

Jalalkamali *et al.* (2011) used ANN and neuro-fuzzy models for GWL prediction in Kerman plain, Iran. Their data set included 22 years of monthly records of air temperature, rainfall, and water levels in the output well as well as neighbouring wells. They looked for the best combinations among past GWL, temperature, and rainfall time series. However, they used only data from the current month and delayed by one month. Then, they evaluated the received models post factum on the basis of their performance.

Wunsch *et al.* (2018) used nonlinear autoregressive networks with exogenous input (NARX) to obtain GWL forecasts for several wells in southwest Germany. They determined the input and feedback delays by applying Seasonal and Trend decomposition using Loess on the temperature and rainfall time series and using auto- and cross-correlation functions on the remainders to choose significant time lags.

Currently, to improve the quality of forecasts (also the GWL), it is becoming increasingly common to create hybrids of predictive models combining several techniques or to use data preprocessing before implementing the models. It *increases the efﬁciency of the model by combining the unique features of the constituent models to capture different patterns in the data* (Suryanarayana *et al.* 2014). Adamowski & Chan (2011) hypothesize that introducing additional meteorological predictors, except for past hydrological time series, improves the prediction performance. Moreover, it is an attempt to answer the question if predictor selection may be universal for neighbouring wells or if every well should require an individual set of predictors.

The use of intelligent techniques for GWL forecasting is not a new phenomenon; however, the overwhelming majority of authors build predictive models based on relatively short time series of hydrological and meteorological parameters (usually less than 10 years), which are not sufficient to assess the response of groundwater to contemporary climate change.

Nash & Sutcliffe (1970) and Wheater *et al.* (1993) indicated that there is a need for including both simplicity and lack of duplication in the model structure as the most important principles of building models with optimized parameters. Driven by a motivation to use an IVS technique that has less computational burden than other methods in this area, the Hellwig method, which requires only a calculation of the correlation matrix between predictors and the output variable, was tested.

The purpose of the study was to assess how many and which lags in monthly temperature and precipitation as well as GWL time series will provide the most accurate GWL prediction for three wells located in the northern part of Greater Poland Province on the basis of data from the multiyear period 1975 to 2014. To find the optimal combination of hydrological and meteorological explanatory variables for GWL prediction modelling, the Hellwig method was used. To the best of the authors' knowledge, it is the first application of such methodology to GWL time-series forecasting approach apart from that of Kajewska-Szkudlarek & Łyczko (2021). Hydrological and basic meteorological time series (temperature and precipitation) were taken into account in the present research as the factors most influencing the GWL in areas with water scarcity. Another point examined in this study was to verify the hypothesis that introducing additional meteorological predictors, except for past hydrological time series, improves the prediction performance. Moreover, it is an attempt to answer the question if predictor selection may be universal for neighbouring wells or if every well should require an individual set of predictors.

The Greater Poland Province is the most vulnerable to water scarcity in the entire country. A great deal of local research is available on water problems in this province. Zieliński (2015) assessed the functioning of field farms from communes particularly threatened by agricultural drought and farms from other communes in that area. The author stated that farms suffering from drought were characterized by a greater labour input per 1 ha of used agriculture area (UAA), had smaller areas of UAA, and had worse economic conditions. Boczoń *et al.* (2016) studied the soil drought that occurred in 2015 in forests in almost the whole of Poland. The drought with the longest duration affected mostly Greater Poland, where a lack of water available to trees was mainly observed. This resulted in slow growth, a decrease in health status, and dying trees. Igliński & Skrzatek (2018) attempted to ask the question whether small hydropower plants could stop land *stepping*, while Nowak & Ptak (2018) focused on the important role of lakes as a component of small retention in that region. Waligórski *et al.* (2018) analysed reservoirs in the province in terms of their usefulness in water management. Kubicz & Bąk (2018) researched the effect of precipitation on the GWL and the impact of meteorological drought on the hydrogeological GWL in that area in the period 1981–2015.

Such a long database used in the present research (almost 40 years), describing GWL fluctuations and their predictions, provides information about groundwater resources’ response to contemporary climate change in areas threatened by water deficits. Moreover, the obtained results could support the process of groundwater resource management in environmental engineering from a long-term perspective.

The novelty in this research lies in the use of the Hellwig method for achieving the best combination of inputs in GWL modelling selection as well as defining GWL dependence on meteorological and hydrological conditions within the last 6 months. The main limitation is the low-resolution (monthly) data used, because sustainable groundwater management requires more specific (daily) results as well. In future research, the authors are planning to complement their studies with a short-term (higher-resolution) analysis of GWL variability on individual days and to attempt to answer the question about uncertainty in groundwater time-series modelling.

## MATERIALS AND METHODS

### Research area

GWL data were collected from three wells (I/170/4, II/401/1, II/430/1) located in the northern part of Greater Poland Province, where the largest water deficits occur (Bąk 2003; Kubicz & Bąk 2018) (Figure 1).

Monthly GWL time series in the first well covered the period 1975–2014, while in the other two sites, it was 1980–2014. These measuring points belong to the observation and research network of the Polish Geological Institute – National Research Institute (PIG–PIB). They are located in Borówiec (52°16′N, 17°02′E) – I/170/4, Ujście (53°04′N, 16°44E) – II/401/1, and Bęglewo (52°52′N, 16°11′E) – II/430/1. Summary information about the analysed wells is presented in Table 1.

Well number . | I/170/4 . | II/401/1 . | II/430/1 . |
---|---|---|---|

Site | Borówiec | Ujście | Bęglewo |

Hydrogeological region (Paczyński & Sadurski 2007) | Middle Vistula Region – Lowlands Subregion | Warta River Region – Lowlands Subregion | Warta River Region – Lowlands Subregion |

Coordinates (WGS 84) | 52°16′N, 17°02′E | 53°04′N, 16°44′E | 52°52′N, 16°11′E |

Type of research point | Piezometer | Drilled well | Drilled well |

Type of aquifer | Confined | Unconfined | Confined |

Stratigraphy | Quaternary | Quaternary | Quaternary |

Lithology | Sands and gravels | Sands | Sands |

Start of records | 1975 | 1980 | 1980 |

End of records | 2014 | 2014 | 2014 |

Well depth (m) | 50 | 30 | 27.5 |

Depth of aquifer top (m) | 28 | 13 | 23 |

Depth of aquifer bottom (m) | 46 | >30 | >27.5 |

WGW(1991–2015)^{a} (m) | 6.5 | 12.8 | 1.53 |

SGW (1991–2015)^{b} (m) | 7.58 | 13.7 | 2.93 |

NGW(1991–2015)^{c} (m) | 8.55 | 14.8 | 3.65 |

Well number . | I/170/4 . | II/401/1 . | II/430/1 . |
---|---|---|---|

Site | Borówiec | Ujście | Bęglewo |

Hydrogeological region (Paczyński & Sadurski 2007) | Middle Vistula Region – Lowlands Subregion | Warta River Region – Lowlands Subregion | Warta River Region – Lowlands Subregion |

Coordinates (WGS 84) | 52°16′N, 17°02′E | 53°04′N, 16°44′E | 52°52′N, 16°11′E |

Type of research point | Piezometer | Drilled well | Drilled well |

Type of aquifer | Confined | Unconfined | Confined |

Stratigraphy | Quaternary | Quaternary | Quaternary |

Lithology | Sands and gravels | Sands | Sands |

Start of records | 1975 | 1980 | 1980 |

End of records | 2014 | 2014 | 2014 |

Well depth (m) | 50 | 30 | 27.5 |

Depth of aquifer top (m) | 28 | 13 | 23 |

Depth of aquifer bottom (m) | 46 | >30 | >27.5 |

WGW(1991–2015)^{a} (m) | 6.5 | 12.8 | 1.53 |

SGW (1991–2015)^{b} (m) | 7.58 | 13.7 | 2.93 |

NGW(1991–2015)^{c} (m) | 8.55 | 14.8 | 3.65 |

^{a}Maximum GWL in a long-term period; the minimum value of the depth to water table.

^{b}Average GWL in a long-term period; the arithmetic mean of all measured values of the depth to water table.

^{c}Minimum GWL in a long-term period; the maximum value of the depth to the water table.

The source of the meteorological data was the IMGW-PIB. These data included precipitation and air temperature time series covering the same period. Precipitation time series were obtained from Trzcianka (53°02′N, 16°27′E) and Mosina (52°14′N, 16°50′E), while maximum temperatures were obtained from Szamotuły (52°36′N, 16°34′E) and Kórnik (52°14′N, 17°05′E) (Figure 1).

According to the Institute of Meteorology and Water Management – National Research Institute (IMGW-PIB) measurements, the mean annual air temperature in the province in the period 1991–2020 was about 9.0 °C (9.0 °C in spring, 19.0 °C in summer, 9.0 °C in autumn, and 0.0 °C in winter). In the case of precipitation, it was 550 mm (120 mm in spring, 200 mm in summer, 120 mm in autumn, and 110 mm in winter). The mean maximum temperature for the period 1991–2020 was 28.5 °C, while the minimum was −8.5 °C. The driest years when precipitation was lower than 400 mm were 1992, 2003, 2015, and 2018, and the wettest years with an annual rainfall of about 700 mm were 2010, 2016, and 2017 (klimat.imgw.pl; Szyga-Pluta 2018). On the basis of long-term meteorological data measured since the mid-19th century, the Greater Poland Province was already classified in the first half of the 20th century as the warmest region in Poland in terms of thermal conditions and as the driest in terms of precipitation with the lowest annual rainfall of 450–500 mm (Bąk 2003).

Before starting the analysis, the hydrological and meteorological data were standardized.

### Input variables’ selection

The literature indicates that past GWLs (Shiri & Kisi 2011; Suryanarayana *et al.* 2014; Chitsazan *et al.* 2015; Yadav *et al.* 2017), as well as rainfall and temperature time series, are the best predictors for GWL forecasting models (Jalalkamali *et al.* 2011). This is because these meteorological parameters are simple to measure and widely available and, thus, when chosen as predictors, they make the selected approach easily transferable (Wunsch *et al.* 2018). For achieving the optimal combination of explanatory variables’ selection, the Hellwig method, also known as the method of information capacity indicators (Hellwig 1968), was used. Although this method has not yet been used in GWL modelling (except for Kajewska-Szkudlarek & Łyczko 2021), its high efficiency is confirmed in the literature on the basis of a comparison with other variable selection techniques (Omiotek *et al.* 2019; Wójcik-Leń *et al.* 2019).

*m*is the number of input variables and

*S*is the number of combinations.

It is equalled as follows: 63 (GWL time series) and 127 (rainfall time series) as well as 127 (temperature time series), which is 317 for each well. Creating 317 models for each piezometer will involve a great deal of work and will be time-consuming, and, thus, the Hellwig method was used to create the rank of potential combinations which could serve as predictors to the forecasting models. The idea behind this method is to use input variables strongly correlated with the target variable and at the same time weakly dependent with each other. In addition, there is a numerical criterion, the so-called integral capacity of the combination of information carriers. The information carriers are all explanatory variables.

*j*is the variable number in the combination under consideration,

*r*is the correlation coefficient of the potential explanatory variable number

_{j}*j*with the explained variable (element of the vector of linear correlation coefficients between the explanatory variables and the explained variable

*R*

_{0}), and

*r*is the correlation coefficient between the

_{ij}*i*th and the

*j*th potential explanatory variables (elements of the correlation coefficients matrix between the potential explanatory variable

*R*).

The *H _{Q}* measures the magnitude of the information that variable

*Xj*gives about variable

*Y*in combination. When

*r*increases,

_{j}*H*also increases, whereas it decreases when the variable

_{Q}*X*is correlated more with the other explanatory variables.

_{j}*n*is the number of predictors,

*j*is the variable number in the combination under consideration, and

*h*is the individual capacity of the information carriers.

_{j}The integral capacity of the information carriers for the combination is the sum of the individual capacities of the information carriers, which belong to this combination. It is the criterion for choosing the appropriate combination of explanatory variables, and the chosen combination is the one with the highest *H _{Q}* (Hellwig 1968).

*Y*is the output variable (target),

*X*is the input variable (predictors),

*β*is the coefficient, unknown, necessary to estimate, and

*ε*is the random variable affected by the immeasurability of some factors, a lack of knowledge about them, or the inaccuracy of measurement.

The multiple linear general regression model (GRM) was used in this research for comparison with the results of the Hellwig method. The same hydrological and meteorological time series were used in variable selection as well as the regression modelling process. The input variable was the GWL in a given well, which was modelled on the basis of the above-mentioned predictors. The progressive and reverse stepwise and best-subset regression methods were used for IVS with which the best quality model was obtained.

Jalalkamali *et al.* (2011) also looked for the best combination between past GWL, temperature, and rainfall time series; however, they used only data from the current month and delayed by one month. In the present study, the authors hypothesized that such delay is not sufficient in the Greater Poland Province (in fact, the groundwater is relatively shallow; however, the analysed wells are characterized by a different type of aquifer), and, therefore, they proposed a 6-month lag during the creation of the input data sets. However, now the challenge was to find the optimal selection of precipitation and temperature data (monthly time series), which could serve as additional predictors apart from past GWL data.

### GWL modelling

The methodology consists in using support vector regression (SVR) to predict GWL fluctuations. This technique is considered accurate not only in GWL modelling but also in other environmental approaches. It refers to both a single model and a combination (hybrid) of several models and techniques (Shabib *et al.* 2018; Liu *et al.* 2019).

SVR, which is an element of the SVM method, was developed to solve problems related to regression. Both SVM and SVR allow to find a hyperplane that reflects the nature of the data in the best way, and they control whether it meets the binding requirements of correct element classification for SVM and computational error in a specific range for SVR. Its aim is to find a function with at most *ε* deviation from the obtained targets for training data that are as flat as possible (Elshorbagy *et al.* 2010). For SVM, the so-called *kernel trick* is used as a standard approach that allows to introduce nonlinearity to the model and, thus, to improve its ability to cope with data that are not separated in a linear way (Cortes & Vapnik 1995). The nonlinear nature of these data may be captured as a result of using radial basis function (RBF). For SVR, the values of the coordinates of the vector *w* and the value of the absolute term *b* are found by solving the following optimization problem:

*w*|| maintainingwhere

*||w||*is the length of vector

*w*,

*w*is the weight vector determining the hyperplane that separates points belonging to different classes,

*b*is the absolute term determined during optimization,

*x*is the set of data belonging to two classes defined by

_{i}*y*variables,

_{i}*y*is the actual value corresponding to

_{i}*x*input vector, and is the non-optimized method parameter that defines the acceptable level of error, i.e., of the difference between the predicted values and those that exist in the teaching data.

_{i}The characteristics of the created SVR models are presented in Table 2. In the present study, the GWL time series were predicted with the use of SVR supported by a RBF algorithm. The RBF was determined by the gamma parameter *γ* (width of the kernel function). The parameters of the SVR model that defined the width of the margin of trust and the maximum value of weight that may be determined for the given vector were *ε* and *C*. Optimization of the hyperparameters *C*, *ε*, and *γ* was conducted using the cross-validation method. First, the models were created using only GWL predictors (a), and then temperature and rainfall input variables were added to determine if the quality of the model changes after adding meteorological parameters (b).

SVR . | Parameters . | No. of support vectors . | The kernel . | ||
---|---|---|---|---|---|

C
. | ε
. | γ
. | |||

a) Only GWL predictors | |||||

I/170/4 | 10.0 | 0.100 | 1.000 | 117 | RBF |

II/401/1 | 10.0 | 0.100 | 0.500 | 93 | |

II/430/1 | 10.0 | 0.100 | 1.000 | 63 | |

b) All predictors | |||||

I/170/4 | 10.0 | 0.100 | 0.111 | 77 | RBF |

II/401/1 | 10.0 | 0.100 | 0.091 | 97 | |

II/430/1 | 10.0 | 0.100 | 0.091 | 48 |

SVR . | Parameters . | No. of support vectors . | The kernel . | ||
---|---|---|---|---|---|

C
. | ε
. | γ
. | |||

a) Only GWL predictors | |||||

I/170/4 | 10.0 | 0.100 | 1.000 | 117 | RBF |

II/401/1 | 10.0 | 0.100 | 0.500 | 93 | |

II/430/1 | 10.0 | 0.100 | 1.000 | 63 | |

b) All predictors | |||||

I/170/4 | 10.0 | 0.100 | 0.111 | 77 | RBF |

II/401/1 | 10.0 | 0.100 | 0.091 | 97 | |

II/430/1 | 10.0 | 0.100 | 0.091 | 48 |

To avoid the phenomenon of overfitting, the data set was divided into two subsets: training and testing, which were about 75 and 25%, respectively. The training set covered the period from 1975/1980 (depending on the well) to April 2006 and the testing set from May 2006 until the end of 2014. For I/170/4, the training subset consisted of 372 elements (the average GWL was 7.36, while *σ* = 0.523) and a testing of 104 (mean value 7.42 and standard deviation 0.306). In the case of II/401/1, the training set was of 310 values (the GWL average was 13.57 and *σ* = 0.347), whereas testing counted 104 (the mean GWL value was 13.72, *σ* was 0.332). For II/430/1, it was 307 values in learning (2.96 and 0.262, respectively) and 104 in testing (2.84 and 0.330, respectively) subsets.

To verify the results obtained, the SVR GWL models were compared with the most commonly used ANN type: multilayer perceptron (MLP). For MLP, the BFGS (Broyden–Fletcher–Goldfarb–Shannon) learning algorithm is the most frequently used, with a sum of squares (SOS) error function. For each well, from many created neural networks, the one with the best quality according to the SOS error was chosen to be compared with the SVR model. Different MLP models with differing architectures were made: (a) MLP 1-6-1, MLP 2-5-1, and MLP 1-6-1, where the numbers mean the number of neurons in input, hidden, and output layers; (b) MLP 9-7-1, MLP 11-12-1, and MLP 11-12-1, respectively, for I/170/4, II/401/1, and II/430/1 (Table 3). In ANN modelling, the same input variables (the best combination of predictors according to the Hellwig method presented in Table 4a) were used as in SVR models.

MLP . | No. of neurons . | No. of learning epochs . | Activation function of neurons . | |||
---|---|---|---|---|---|---|

Input layer . | Hidden layer . | Output layer . | Hidden layer . | Output layer . | ||

a) Only GWL predictors | ||||||

I/170/4 | 1 | 6 | 1 | 44 | tanh | Logistic |

II/401/1 | 2 | 5 | 1 | 57 | Exponential | tanh |

II/430/1 | 1 | 6 | 1 | 20 | tanh | Linear |

b) All predictors | ||||||

I/170/4 | 9 | 7 | 1 | 40 | tanh | Exponential |

II/401/1 | 11 | 12 | 1 | 91 | Logistic | Logistic |

II/430/1 | 11 | 12 | 1 | 13 | Exponential | Exponential |

MLP . | No. of neurons . | No. of learning epochs . | Activation function of neurons . | |||
---|---|---|---|---|---|---|

Input layer . | Hidden layer . | Output layer . | Hidden layer . | Output layer . | ||

a) Only GWL predictors | ||||||

I/170/4 | 1 | 6 | 1 | 44 | tanh | Logistic |

II/401/1 | 2 | 5 | 1 | 57 | Exponential | tanh |

II/430/1 | 1 | 6 | 1 | 20 | tanh | Linear |

b) All predictors | ||||||

I/170/4 | 9 | 7 | 1 | 40 | tanh | Exponential |

II/401/1 | 11 | 12 | 1 | 91 | Logistic | Logistic |

II/430/1 | 11 | 12 | 1 | 13 | Exponential | Exponential |

Activation function of neurons in the input layer was linear in all cases.

Well number . | GWL (m) . |
---|---|

(a) Hellwig method | |

I/170/4 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

II/401/1 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

II/430/1 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

P (mm) | |

I/170/4 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

II/401/1 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

II/430/1 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

T (^{o}C) | |

I/170/4 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

II/401/1 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

II/430/1 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

(b) General regression method | |

I/170/4 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

II/401/1 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

II/430/1 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

P (mm) | |

I/170/4 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

II/401/1 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

II/430/1 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

T (^{o}C) | |

I/170/4 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

II/401/1 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

II/430/1 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

Well number . | GWL (m) . |
---|---|

(a) Hellwig method | |

I/170/4 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

II/401/1 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

II/430/1 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

P (mm) | |

I/170/4 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

II/401/1 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

II/430/1 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

T (^{o}C) | |

I/170/4 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

II/401/1 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

II/430/1 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

(b) General regression method | |

I/170/4 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

II/401/1 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

II/430/1 | GWL-1, GWL-2, GWL-3, GWL-4, GWL-5, GWL-6 |

P (mm) | |

I/170/4 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

II/401/1 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

II/430/1 | P, P-1, P-2, P-3, P-4, P-5, P-6 |

T (^{o}C) | |

I/170/4 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

II/401/1 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

II/430/1 | T, T-1, T-2, T-3, T-4, T-5, T-6 |

Bold values denote the best combination of predictors obtained by the Hellwig method.

*N*is the number of instances (input–output) pairs used for learning,

*y*is the network prediction (network output), and

_{i}*t*is the observed value (output based on data) for the

_{i}*i*th instance.

The higher the difference between the actual and the predicted values, the larger the error and the more adjustment is required for the network (Dell Inc. 2016).

To eliminate the excessive adjustment phenomenon in SVR and MLP alike, the data were divided into learning and testing subsets.

### Models’ performance metrics

*r*

^{2}) for the actual and prognosed values, MSE as well as the MAE of the models according to the following formulas. Modelling was carried out using the Matlab and Statistica programs.where

*n*is the number of observations,

*x*is the predicted data, is the average value of predicted data,

_{i}*y*is the observed data, and is the average value of observed data.

_{i}## RESULTS AND DISCUSSION

### Input variables’ selection

To select the input variables for the SVR model, the Hellwig method was used to create the ranking of the predictors’ combinations. The best combinations for the three analysed wells are presented in Table 4a. Within each parameter (GWL, P, and T), the selected ones were marked in bold. The best combinations of input variables were GWL-1, P-1, P-2, P-3, P-4, P-5, and P-6 and T-1 and T-4 for I/170/4. In case the of II/401/1, it was a combination of GWL-1 and GWL-2, all P predictors, and T-3 and T-6. For II/430/1, it was a combination of GWL-1, all P input variables, and T-1, T-3, and T-6. The set of predictors created in this way served as an input to the SVR and MLP models. The validation of the above IVS method with general regression models (progressive, reverse, and the best-subset regression) provided different results (Table 4b). In that case, the role of precipitation in the model was significantly smaller and the temperature lags taken into account were unlike, especially for II/430/1, where temperature was not included in the predictors’ combination.

### GWL modelling

At the beginning, SVR modelling was performed only for predictors covering GWL time series (Table 5a). Then, temperature and rainfall time series were added (Table 5b).

. | . | I/170/4 . | II/401/1 . | II/430/1 . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Learning . | Testing . | Both . | Learning . | Testing . | Both . | Learning . | Testing . | Both . | ||

(a) For GWL predictors | ||||||||||

SVR | MSE | 0.018 | 0.022 | 0.019 | 0.017 | 0.013 | 0.016 | 0.006 | 0.005 | 0.006 |

MAE | 0.104 | 0.111 | 0.106 | 0.093 | 0.082 | 0.090 | 0.055 | 0.059 | 0.056 | |

r^{2} | 0.927 | 0.903 | 0.920 | 0.866 | 0.897 | 0.873 | 0.925 | 0.939 | 0.928 | |

MLP | ||||||||||

MSE | 0.019 | 0.016 | 0.019 | 0.014 | 0.022 | 0.016 | 0.006 | 0.005 | 0.006 | |

MAE | 0.107 | 0.097 | 0.105 | 0.085 | 0.101 | 0.089 | 0.057 | 0.051 | 0.056 | |

r^{2} | 0.918 | 0.933 | 0.925 | 0.885 | 0.838 | 0.861 | 0.923 | 0.942 | 0.933 | |

GRM | ||||||||||

MSE | 0.022 | 0.021 | 0.022 | 0.015 | 0.019 | 0.017 | 0.009 | 0.011 | 0.010 | |

MAE | 0.112 | 0.100 | 0.106 | 0.091 | 0.100 | 0.096 | 0.062 | 0.071 | 0.067 | |

r^{2} | 0.913 | 0.921 | 0.917 | 0.883 | 0.830 | 0.857 | 0.926 | 0.941 | 0.934 | |

(b) For GWL, rainfall, and temperature predictors | ||||||||||

SVR | MSE | 0.009 | 0.014 | 0.010 | 0.015 | 0.018 | 0.016 | 0.004 | 0.004 | 0.004 |

MAE | 0.071 | 0.088 | 0.076 | 0.088 | 0.093 | 0.089 | 0.044 | 0.047 | 0.045 | |

r^{2} | 0.964 | 0.937 | 0.958 | 0.877 | 0.864 | 0.871 | 0.958 | 0.960 | 0.958 | |

MLP | ||||||||||

MSE | 0.009 | 0.009 | 0.009 | 0.011 | 0.020 | 0.013 | 0.004 | 0.003 | 0.004 | |

MAE | 0.072 | 0.068 | 0.071 | 0.076 | 0.091 | 0.080 | 0.043 | 0.045 | 0.044 | |

r^{2} | 0.961 | 0.963 | 0.961 | 0.908 | 0.858 | 0.883 | 0.953 | 0.959 | 0.956 | |

GRM | ||||||||||

MSE | 0.006 | 0.010 | 0.008 | 0.014 | 0.018 | 0.016 | 0.002 | 0.006 | 0.004 | |

MAE | 0.082 | 0.086 | 0.084 | 0.091 | 0.093 | 0.092 | 0.050 | 0.056 | 0.053 | |

r^{2} | 0.964 | 0.966 | 0.965 | 0.860 | 0.880 | 0.870 | 0.953 | 0.959 | 0.956 |

. | . | I/170/4 . | II/401/1 . | II/430/1 . | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Learning . | Testing . | Both . | Learning . | Testing . | Both . | Learning . | Testing . | Both . | ||

(a) For GWL predictors | ||||||||||

SVR | MSE | 0.018 | 0.022 | 0.019 | 0.017 | 0.013 | 0.016 | 0.006 | 0.005 | 0.006 |

MAE | 0.104 | 0.111 | 0.106 | 0.093 | 0.082 | 0.090 | 0.055 | 0.059 | 0.056 | |

r^{2} | 0.927 | 0.903 | 0.920 | 0.866 | 0.897 | 0.873 | 0.925 | 0.939 | 0.928 | |

MLP | ||||||||||

MSE | 0.019 | 0.016 | 0.019 | 0.014 | 0.022 | 0.016 | 0.006 | 0.005 | 0.006 | |

MAE | 0.107 | 0.097 | 0.105 | 0.085 | 0.101 | 0.089 | 0.057 | 0.051 | 0.056 | |

r^{2} | 0.918 | 0.933 | 0.925 | 0.885 | 0.838 | 0.861 | 0.923 | 0.942 | 0.933 | |

GRM | ||||||||||

MSE | 0.022 | 0.021 | 0.022 | 0.015 | 0.019 | 0.017 | 0.009 | 0.011 | 0.010 | |

MAE | 0.112 | 0.100 | 0.106 | 0.091 | 0.100 | 0.096 | 0.062 | 0.071 | 0.067 | |

r^{2} | 0.913 | 0.921 | 0.917 | 0.883 | 0.830 | 0.857 | 0.926 | 0.941 | 0.934 | |

(b) For GWL, rainfall, and temperature predictors | ||||||||||

SVR | MSE | 0.009 | 0.014 | 0.010 | 0.015 | 0.018 | 0.016 | 0.004 | 0.004 | 0.004 |

MAE | 0.071 | 0.088 | 0.076 | 0.088 | 0.093 | 0.089 | 0.044 | 0.047 | 0.045 | |

r^{2} | 0.964 | 0.937 | 0.958 | 0.877 | 0.864 | 0.871 | 0.958 | 0.960 | 0.958 | |

MLP | ||||||||||

MSE | 0.009 | 0.009 | 0.009 | 0.011 | 0.020 | 0.013 | 0.004 | 0.003 | 0.004 | |

MAE | 0.072 | 0.068 | 0.071 | 0.076 | 0.091 | 0.080 | 0.043 | 0.045 | 0.044 | |

r^{2} | 0.961 | 0.963 | 0.961 | 0.908 | 0.858 | 0.883 | 0.953 | 0.959 | 0.956 | |

GRM | ||||||||||

MSE | 0.006 | 0.010 | 0.008 | 0.014 | 0.018 | 0.016 | 0.002 | 0.006 | 0.004 | |

MAE | 0.082 | 0.086 | 0.084 | 0.091 | 0.093 | 0.092 | 0.050 | 0.056 | 0.053 | |

r^{2} | 0.964 | 0.966 | 0.965 | 0.860 | 0.880 | 0.870 | 0.953 | 0.959 | 0.956 |

The relation between the observed and the predicted data indicated a strong dependence between them, which was confirmed by the high values of the squared correlation *r*^{2} (Table 5a and b, Figure 2). In the case of GWL predictors and SVR modelling, in the testing subset, the best model's fitting to real values (*r*^{2} = 0.939) was assessed for II/430/1, while the worst (*r*^{2} = 0.897) was assessed for II/401/1. Moreover, the lowest MSE value (0.005) was received for II/430/1 and the highest (0.022) for I/170/4. Compared with neural networks predictions (MLP), the best model's performance (*r*^{2} = 0.942, MSE = 0.005) was observed for II/430/1 and the worst (*r*^{2} = 0.838, MSE = 0.022) for II/401/1. The results obtained with SVR were slightly better than those obtained with MLP (Table 5a).

The received results indicate that adding temperature and precipitation time series as predictors slightly improves the models’ quality. In the testing subsets, for SVR forecasting, the highest (*r*^{2} = 0.960, MSE = 0.004) was in the case of II/430/1 and the lowest (*r*^{2} = 0.864, MSE = 0.018) was for II/401/1. The results obtained with the MLP models were relatively poor (Table 5b).

The prediction results obtained with the GRM models created on the basis of predictor combinations according to regression methods were very close to those obtained with SVR and MLP models (similar MSEs, MAEs and *r*^{2}s). All the above-mentioned techniques that provided GWL time-series forecasts demonstrated high efficiency.

Comparing the model's performance with the literature, it can be stated that the presented approach is reasonable and provides high-quality prediction results without using complicated techniques. MSE errors in the range of 0.003–0.022 correlate to those in Chitsazan *et al.* (2015) (0.004–0.052), Yadav *et al.* (2017) (0.0028–0.54), and Jalalkamali *et al.* (2011) (0.0004–0.076). The modelling results were better than those obtained by Sivapragasam *et al.* (2015), who created a GVL prediction model with *r*^{2} not higher than 0.82 (0.838–0.963) for the test subsets of ANN and SVR models in the current research. On the other hand, the presented performance (MSE error) is comparable to that of hybrid models combining prediction methods created by Suryanarayana *et al.* (2014): 0.039–0.285, Adamowski & Chan (2011): 0.002–0.306, as well as Sahoo *et al.* (2017): error in the test set is less than 2.0.

Figure 2 only shows the testing subsets of the SVR models created on the basis of GWL, rainfall, and temperature predictors. In I/170/4 and II/401/1 wells, the frequency distribution of SVR residues is similar in terms of negative-value domination, which means that the models overestimated the real GWL values. This research result converges with that of Yadav *et al.* (2017), Shiri & Kisi (2011), Suryanarayana *et al.* (2014), all of whom indicate a predominance of predicted values over the observed ones. In the case of I/170/4, the largest number of (46 and 30, respectively) intervals of differences between observed and predicted values were [−0.1; 0.0) and [−0.2; −0.1) (Figure 2(a)). For II/401/1, there were only about 13 positive differences (in cumulative classes from 0.0 to 0.4), while the largest were [−0.1; 0.0) and [−0.2; −0.1) where 33 and 37 events fell into (Figure 2(b)), respectively. In comparison with the above results, the histogram for II/430/1 showed that GWL residues were significantly lower and the most frequently (40) interval was [0.00; 0.05) (Figure 2(c)).

The course of the time series for all analysed wells indicated that the predictions clearly reflected reality in terms of GWL monthly fluctuations (Figure 3(a)–3(c)). These results confirmed the forgoing findings that the models obtained for I/170/4 and II/430/1 were more adjustable to the observed data than for II/401/1, but all received results indicated that SVR forecasts were characterized by high predictive ability.

## CONCLUSIONS

Long-term GWL forecasting supports optimization in water resource planning and management as well as assessing the risk to drinking water supply, particularly in areas with water scarcity. In this area, data-driven models using machine learning techniques are becoming increasingly popular and have great potential as they can be an alternative for multidimensional physics-based models. In the present study, SVR and MLP techniques, which are computational intelligence methods, were used to predict the monthly GWL time series in three wells located in the northern part of Greater Poland Province on the basis of the measurement period from 1975 to 2014. The main aim of the research was to find the optimal combination of predictors between hydrological and meteorological time series.

The obtained results indicated that two-stage approach to monthly GWL forecasting that combines the Hellwig method for input variables’ selection with machine learning (SVR and ANN) modelling is reasonable and improves the quality of prediction and minimizes error values. According to the Hellwig method, the most important predictors for the three wells are: GWL time series at lags −1 and −2, all precipitation from the current month, and delayed from 1 to 6 months, as well as −1, −3, −4, and −6 temperature lags. A comparison of the obtained results with general regression models (progressive, reverse, and the best-subset regression) provided different results. In this case, the most significant predictors for I/170/4 were GWL-1, P, P-1, P-3, P-4, T, T-2, and T-6 for II/401/1 GWL-1, GWL-2, GWL-5, P, and T-5, and GWL-1, P, P-1, and P-3 for II/430/1. Furthermore, with reference to the hypotheses stated in the Introduction, adding rainfall and temperature data while modelling provides better results (however marginally) than using only past GWL time series, whereas ANN (MLP) gives generally slightly worse forecasts than SVR. Additionally, each GWL measuring point needs an individual approach in terms of optimal input variables’ selection, which plays a crucial role in GWL time-series prediction.

High values of the model's quality measures like MSE errors in the range of 0.003–0.022 and *r*^{2} more than 0.8 were found to be acceptable in the field of regional hydrology in areas threatened by water deficits. Moreover, the created models showed good prediction ability for high and low GWL values. The GRM method using different set of predictors for each well provided predictive models’ accuracy very close to SVR and MLP based on input combinations according to the Hellwig method. However, the second approach required less computational effort (calculation correlation between inputs and target).

The main advantage of the proposed method is that it is simple and its implementation requires access only to basic hydrological and meteorological data. It may find application in groundwater management and planning, especially in actual conditions related to observed climate change and water deficit threats.

## ACKNOWLEDGEMENTS

The research was performed in, and supported by, the Institute of Environmental Engineering Wrocław University of Environmental and Life Sciences. This research did not receive any specific grants from funding agencies in the public, commercial, or not-for-profit sectors. The source of the hydrological data is Polish Geological Institute – National Research Institute (PIG–PIB). The source of the meteorological data is the Institute of Meteorology and Water Management – National Research Institute (PIB). Data from the Institute of Meteorology and Water Management – National Research Institute have been processed.

## CONFLICT OF INTEREST

On behalf of all authors, the corresponding author states that there is no conflict of interest.

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

## REFERENCES

*steppe. Ekologia i Technika*

*E3S Web of Conferences*44, 00127. doi:10.1051/e3sconf/20184400127

*Hydrological year*2018, Warsaw, Poland