Maria Margarita Ntona

Konstantinos Chalikakis
email: konstantinos.chalikakis@univ-avignon.fr

Gianluigi Busico
email: gianluigi.busico@unicampania.it

Micòl Mastrocicco
email: micol.mastrocicco@unicampania.it

Kyriaki Kalaitzidou
email: kalaitzidou@cheng.auth.gr

Nerantzis Kazakis
email: nkazakis@upatras.gr

Maria Margarita Nton

Konstantinos Chalikak

Application of Judgmental Sampling Approach for the Monitoring of Groundwater Quality and Quantity Evolution in Mediterranean Catchments

Keywords: multivariate statistical analysis, groundwater pollution, evolution process, pearson correlation, groundwater level

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Groundwater quality is defined by physical, chemical, and biological characteristics, which are a complex function of many natural and anthropogenic influences on the surface and subsurface environment [START_REF] Chandnani | A comprehensive analysis of contaminated groundwater: Special emphasis on nature-ecosystem and socio-economic impacts[END_REF]. The chemical composition of groundwater is affected by recharge and discharge conditions, where the mineral contents found in groundwater samples are closely related to the dissolution processes of predominant materials in the soils [START_REF] Modibo Sidibé | Assessing Groundwater Mineralization Process, Quality, and Isotopic Recharge Origin in the Sahel Region in Africa[END_REF]. External factors, for instance, socioeconomic parameters due to agricultural activities, industrial structure, and urbanization, affect the groundwater sources qualitatively and quantitatively [START_REF] Luetkemeier | Anthropogenic Pressures on Groundwater[END_REF]. Thus, the frequency, quality, and quantity of field measurements are of utmost importance as they affect the performance of deduction [START_REF] Chaudhary | Qualitative and Quantitative Analysis of Water[END_REF]. For this purpose, each sampling campaign has to be organized based on the research target, specific hydrogeological conditions of the area, and the general limitations of the research approach.

In recent years, different techniques have been performed to investigate groundwater quality worldwide. Stable isotope analysis has been applied to identify nitrate pollution [START_REF] Zhang | Tracing nitrate pollution sources and transformations in the over-exploited groundwater region of north China using stable isotopes[END_REF] and strontium contamination [START_REF] He | Identifying strontium sources of flowback fluid and groundwater pollution using 87 Sr/ 86 Sr and geochemical model in Sulige gasfield, China[END_REF], while Voltaggio et al. [START_REF] Voltaggio | Assessment of groundwater pollution from ash ponds using stable and unstable isotopes around the Koradi and Khaperkheda thermal power plants (Maharashtra, India)[END_REF] applied stable and unstable isotopes for the assessment of groundwater pollution from ash ponds. Another approach that has seen significant development in recent years is hydrological modeling, which can represent the transport and transformation processes of pollutants in the environment [START_REF] Zhao | Groundwater pollution risk assessment based on groundwater vulnerability and pollution load on an isolated island[END_REF]. Groundwater quality indices (GQIs) have proven to be a straightforward and effective methodology for assessing the overall groundwater quality being performed in numerous studies in different regions of the world [START_REF] Giao | Groundwater quality assessment for drinking purposes: A case study in the Mekong Delta[END_REF]. Artificial intelligence (AI) and machine learning for groundwater quality prediction and simulation also proved to produce accurate results [START_REF] Nordin | Groundwater quality forecasting modelling using artificial intelligence: A review[END_REF][START_REF] Haggerty | Application of machine learning in groundwater quality modelling-A comprehensive review[END_REF]. The reliability of these methods for the determination of the hydrochemical process in different and complex aquifer systems is not certain since they all have advantages and drawbacks that can limit their application. Thus, the first step to identifying water quality variables responsible for spatial and temporal changes and the processes controlling groundwater quality should be the statistical analysis. Multivariate analytical statistical methods were used to characterize the hydrochemistry of surface [START_REF] Abuzaid | Combinations of multivariate statistical analysis and analytical hierarchical process for indexing surface water quality under arid conditions[END_REF] and groundwater systems [START_REF] Busico | Multivariate statistical analysis to characterize/discriminate between anthropogenic and geogenic trace elements occurrence in the Campania Plain, Southern Italy[END_REF][START_REF] Liu | Geochemical characterization of shallow groundwater using multivariate statistical analysis and geochemical modeling in an irrigated region along the upper Yellow River, Northwestern China[END_REF][START_REF] Parisi | Analyzing spatial and temporal evolution of groundwater salinization through Multivariate Statistical Analysis and Hydrogeochemical Facies Evolution-Diagram[END_REF]. A combination of methods, like electrical resistivity tomography (ERT) monitoring and physicochemical water analysis techniques to investigate groundwater vulnerability [START_REF] Park | Electrical resistivity tomography (ERT) monitoring of leachate seepage from an animal carcass waste (ACW) facility, South Korea[END_REF], as well as Bayesian isotope mixing model (SIAR) and multivariate statistical analysis (MSA) [START_REF] Matiatos | Nitrate source identification in groundwater of multiple land-use areas by combining isotopes and multivariate statistical analysis: A case study of Asopos basin (Central Greece)[END_REF][START_REF] Meghdadi | Quantification of spatial and seasonal variations in the proportional contribution of nitrate sources using a multi-isotope approach and Bayesian isotope mixing model[END_REF], have been applied for the enrichment of the results.

A multi-statistical analysis applied as a low-cost process and without complex input parameters can provide reliable results [START_REF] Banda | Application of Multivariate Statistical Analysis in the Development of a Surrogate Water Quality Index (WQI) for South African Watersheds[END_REF]. The application of various multivariate statistical techniques, including cluster analysis (CA), factor analysis (FA), principal component analysis (PCA), and/or a combination thereof [START_REF] Davis | Identifying natural and anthropogenic sources of metals in urban and rural soils using GIS-based data, PCA, and spatial interpolation[END_REF][START_REF] Machiwal | Identifying sources of groundwater contamination in a hard-rock aquifer system using multivariate statistical analyses and GIS-based geostatistical modeling techniques[END_REF][START_REF] Meng | Apportionment and evolution of pollution sources in a typical riverside groundwater resource area using PCA-APCS-MLR model[END_REF], as well as the incorporation of AI models [START_REF] Tizro | Estimation of total dissolved solids in Zayandehrood River using intelligent models and PCA[END_REF], has received significant attention in the study of water quality, especially in simplifying complex datasets. Therefore, the application of factors/clusters can identify the aspects that affect the water quality and offer a valuable tool for the reliable management of water resources [START_REF] Xiao | Investigating sources, driving forces and potential health risks of nitrate and fluoride in groundwater of a typical alluvial fan plain[END_REF]. Nevertheless, direct comparisons of localized studies are often hindered by inconsistencies in datasets and methodologies.

This study aims to investigate the main factors that affect groundwater quality in different semi-arid basins of the Mediterranean region under a multi-statistical analysis of the judgmental sampling approach. The selection of this method is dependent on the necessity for (i) the absence of pauses in data collection, (ii) quick and easy field measurements, and (iii) low-cost techniques. In this regard, groundwater samples were collected from three Mediterranean areas in Greece and Italy to analyze the characteristics and genesis of groundwater hydrochemical evolution. Based on the continuity of the sampling-detected results, Pearson correlation and PCA/FA were used to distinguish the contribution of different natural mineralization and human pollutants to the groundwater quality in the three study areas. Among these, groundwater level measurements were also implemented in all the study areas to determine the recharge and discharge conditions compared to previous years' accessible data.

Taking into consideration the drawbacks and limitations of the multi-statistical analysis, we decided to take the challenge and compare these three regions due to their different and interesting characteristics under the aspect of a non-probability sampling technique. In this term, a non-random sampling technique was practiced for the investigation of spatial and temporal groundwater quality status in the study areas. The validity and reliability of the results throughout the entire study were achieved based on the results of previous works. The current research can provide a pathway for water resources management and early warning under a low-cost and time-effective approach in Mediterranean regions.

Materials and Methods

Study Areas

The study areas are located in the Mediterranean region (Figure 1a-c) and are affected by different combinations of anthropogenic factors, geological formations, and meteorological conditions, which have already been analyzed in previous works [START_REF] Busico | Simulating future groundwater recharge in coastal and inland catchments[END_REF][START_REF] Ntona | Coupling SWAT and DPSIR models for groundwater management in Mediterranean catchments[END_REF]. The Eastern Thermaikos Gulf (Greece) includes the coastal area of Epanomi-Sozopoli and the Anthemountas basin and covers an extent of 932.7 km 2 .

Water 2023, 15, x FOR PEER REVIEW 3 of 19 reliability of the results throughout the entire study were achieved based on the results of previous works. The current research can provide a pathway for water resources management and early warning under a low-cost and time-effective approach in Mediterranean regions.

Materials and Methods

Study Areas

The study areas are located in the Mediterranean region (Figure 1a-c) and are affected by different combinations of anthropogenic factors, geological formations, and meteorological conditions, which have already been analyzed in previous works [START_REF] Busico | Simulating future groundwater recharge in coastal and inland catchments[END_REF][START_REF] Ntona | Coupling SWAT and DPSIR models for groundwater management in Mediterranean catchments[END_REF]. The Eastern Thermaikos Gulf (Greece) includes the coastal area of Epanomi-Sozopoli and the Anthemountas basin and covers an extent of 932.7 km 2 . The decision to combine the Anthemountas basin with the coastal part as a unified research area was driven by two main factors: (i) the hydrogeological connection of these systems and (ii) the great amount of previous data. The Mouriki basin (Greece) is in Macedonia Province and covers an area of 100 km 2 . The Volturno basin (Italy) is located in the Campania region and covers an area of 5613 km 2 , while the extent of the study area considered is around 1661 km 2 . The Upper Volturno-Calore basin is characterized mainly as strongly asymmetric with steep slopes. This study focused on the outlet of the Upper Volturno-Calore basin to assess the potential interaction between the river and the groundwater in the alluvial plain.

Sample Collection and Groundwater Level Measurements

Groundwater samples were collected during the wet and dry periods of 2021-2022 from the Greek regions and the dry period of 2021 from the Italian basin (Figure 1). The sampling points are shown in the maps below and were chosen to focus on the potential The decision to combine the Anthemountas basin with the coastal part as a unified research area was driven by two main factors: (i) the hydrogeological connection of these systems and (ii) the great amount of previous data. The Mouriki basin (Greece) is in Macedonia Province and covers an area of 100 km 2 . The Volturno basin (Italy) is located in the Campania region and covers an area of 5613 km 2 , while the extent of the study area considered is around 1661 km 2 . The Upper Volturno-Calore basin is characterized mainly as strongly asymmetric with steep slopes. This study focused on the outlet of the Upper Volturno-Calore basin to assess the potential interaction between the river and the groundwater in the alluvial plain.

Sample Collection and Groundwater Level Measurements

Groundwater samples were collected during the wet and dry periods of 2021-2022 from the Greek regions and the dry period of 2021 from the Italian basin (Figure 1). The sampling points are shown in the maps below and were chosen to focus on the potential influence of agricultural activities and surface water on groundwater quality. Forty-seven groundwater samples were collected and analyzed. Seventeen samples were from boreholes in TG, eight in the Mouriki basin, and twenty-two from the Upper Volturno basin.

After consideration of the local hydrochemical properties and hydrogeological conditions, 21 groundwater quality parameters were selected, including pH, electrical conductivity (EC), calcium (Ca), magnesium (Mg), potassium (K), sodium (Na), bicarbonate (HCO 3 ), chloride (Cl), sulfates (SO 4 ), nitrates (NO 3 ), ammonia (NH 4 ), strontium (Sr), phosphate (PO 4 ), iron (Fe), manganese (Mn), arsenic (As), chromium (Cr), zinc (Zn), boron (B), molybdenum (Mo), and silicon dioxide (SiO 2 ). The Chemical Balance Error (CBE) was applied for the internal consistency test. Under ideal conditions, cations and anions should exactly balance; thus, in satisfactory analysis, cations and anions should not differ by more than 5% [START_REF] Appelo | Groundwater and Pollution[END_REF]. Following the calculation of CBE, all the samples had relative errors within ±5%.

The design of groundwater level monitoring networks is essential for groundwater modeling and management. In most cases, the lack of monitoring is due to financial constraints [START_REF] Retike | Rescue of groundwater level time series: How to visually identify and treat errors[END_REF]. Issues in the field, such as transportation and work under different climate conditions, finding boreholes in and out of operation, and contact with farmers, make field research difficult. To avoid the high cost of monitoring and missing time series, the selection of specific boreholes that are easily accessible is suggested. For this purpose, boreholes that are already monitored in previous published works were selected. In the current work, the comparison of GWL measurements during the last decades is provided. Measurements of groundwater depth were applied in 68 and 20 boreholes in the TG and Mouriki basins, respectively. In the Upper Volturno basin, the selection of 15 boreholes was possible based on previous data to achieve the optimum spatial variation in GWL during the years.

In all sites, available data were considered from research projects and scientific publications. Specifically, a detailed time series (starting from 2011) of GWL was evaluated for the aquifers of TG. On the contrary, in the Mouriki basin, a comparison was obtained for the periods between 2004-2007 and 2021-2022, while in the Upper Volturno region, the comparison was obtained for the periods 2006 and 2016. All sites have representative sampling points for monitoring of water quality and quantity from national authorities in the framework of EU regulation for water sustainability. However, the number of sampling points per study area is limited and inappropriate for spatial analysis at the aquifer scale. For instance, in the Mouriki basin, there is one measurement point (http://nmwn.ypeka.gr/ accessed on 10 July 2023).

Systematic Judgmental Sampling

The determination of groundwater quality and quantity monitoring networks is crucial to conducting accurate research at a low cost and time [START_REF] Leach | Entropy based groundwater monitoring network design considering spatial distribution of annual recharge[END_REF][START_REF] Azadi | Optimal design of groundwater monitoring networks using gamma test theory[END_REF]. Vast sampling points and/or generic random sampling are high-cost methods and rely less on prior specific knowledge. On the other hand, in judgmental sampling, the hydrogeologists have to analyze general and detailed information about the area and select the optimum sampling campaign, minimizing the limits and biases that affect the reliability of the findings [START_REF] Ayvaz | Identification of the optimum groundwater quality monitoring network using a genetic algorithm based optimization approach[END_REF]. The main positive and negative effects of this method are provided in Figure 2. The weight of each effect may vary between regions, while each sampling campaign should be organized based on the results of the previous ones. By employing this cost-effective and time-efficient method, the researcher can gain useful insights by selecting the most representative points considering various site-specific factors and research queries (Table 1). The combination of multiple analysis techniques can improve the representativeness of the samples and provide accurate and reliable research outcomes. The main drawbacks are the unknown errors that can be detected after the chemical analysis and the scarce availability of previous hydrogeological analyses in less well-known areas.

In the current research, the sampling points are selected based on their relevance to the research question and their ability to provide meaningful information [START_REF] Ayhan | Non-probability Sampling Survey Methods[END_REF]. These points are selected with precision to ensure that they accurately represent the groundwater regime of the basin being studied. The main queries of Table 1 for the study areas are answered below, while the monitoring points were selected according to the characteristics of the study areas (Figure 3). In the current research, the sampling points are selected based on their relevance to the research question and their ability to provide meaningful information [START_REF] Ayhan | Non-probability Sampling Survey Methods[END_REF]. These points are selected with precision to ensure that they accurately represent the groundwater regime of the basin being studied. The main queries of Table 1 for the study areas are answered below, while the monitoring points were selected according to the characteristics of the study areas (Figure 3).

-

The principal purpose of the current work is the determination of groundwater qualitative and quantitative status of the Mediterranean regions to provide a fundamental river basin protocol for management. The selection of the research period was performed to protect the groundwater systems from the expected droughts according to the future climate scenarios and current human activities.

-

The pressures in the study areas, as well as the drawbacks of previously published works, are provided in the previous work of Ntona et al. [START_REF] Ntona | Coupling SWAT and DPSIR models for groundwater management in Mediterranean catchments[END_REF]. -Monitoring points were selected according to (i) the extent of the basin for the optimum distribution of the points, (ii) land use, (iii) geology, (iii) the easy access to boreholes to avoid gaps in time scale, and (iv) budget. The selection of the data is based on its use in groundwater modeling in the next steps of the research.

-

For the investigation of the variations during the year, a minimum of two sampling campaigns during the wet and dry period is suggested. Thus, the minimum period for field data collection is at least one hydrological year. The principal purpose of the current work is the determination of groundwater qualitative and quantitative status of the Mediterranean regions to provide a fundamental river basin protocol for management. The selection of the research period was performed to protect the groundwater systems from the expected droughts according to the future climate scenarios and current human activities. -

The pressures in the study areas, as well as the drawbacks of previously published works, are provided in the previous work of Ntona et al. [START_REF] Ntona | Coupling SWAT and DPSIR models for groundwater management in Mediterranean catchments[END_REF]. -Monitoring points were selected according to (i) the extent of the basin for the optimum distribution of the points, (ii) land use, (iii) geology, (iii) the easy access to boreholes to avoid gaps in time scale, and (iv) budget. The selection of the data is based on its use in groundwater modeling in the next steps of the research. -For the investigation of the variations during the year, a minimum of two sampling campaigns during the wet and dry period is suggested. Thus, the minimum period for field data collection is at least one hydrological year.

Statistical Analysis

Pearson correlation and principal component analysis (PCA)/factor analysis (FA) were used to find inter-relationships among the chemical parameters of the groundwater samples in the three basins. The PCA/FA was performed using R statistical software (version 4.2.1; R Foundation for Statistical Computing, Vienna, Austria).

In recent years, PCA has been widely used in hydrogeological studies to address a variety of environmental problems, including a comprehensive assessment of temporal and spatial changes in groundwater quality [START_REF] Jiang | Principal component analysis and hierarchical cluster analyses of arsenic groundwater geochemistry in the Hetao basin, Inner Mongolia[END_REF][START_REF] Zhang | Groundwater pollution source identification and apportionment using PMF and PCA-APCA-MLR receptor models in a typical mixed land-use area in Southwestern China[END_REF]. PCA is a multivariate statistical analysis method that depends on the orthogonal transformation to convert a set of observations of possibly correlated variables [START_REF] Sun | Integrated decision-making model for groundwater potential evaluation in mining areas using the cusp catastrophe model and principal component analysis[END_REF]. This method is applied to transform the original variables into new, uncorrelated variables (axes), which are linear compositions of the primary variables. FA is similar to PCA in the calculation principle. This internal analysis is the most widely used for its ability to directly obtain the common component number in a complex sample [START_REF] Audet | Performance indicators in multi objective optimization[END_REF]. FA uses a lower number of unobserved variables to explain more complex relationships in the observed variables. In this study, PCA/FA was performed on the normalized variables for 2021 and 2022.

The Kaiser-Meyer-Olkin measure (KMO) test was applied to assess the suitability of the data for PCA/FA. According to the Kaiser [START_REF] Kaiser | An index of factorial simplicity[END_REF] standard, the principle of eigenvalues being >1 was used to determine the number of principal components. Furthermore, values between 0.5 and 0.7 are mediocre, values from 0.7 to 0.8 are good, values between 0.8 and 0.9 are great, and values above 0.9 are superb. Values more than 0.5 signify a good justification for running factor analysis.

Statistical Analysis

Pearson correlation and principal component analysis (PCA)/factor analysis (FA) were used to find inter-relationships among the chemical parameters of the groundwater samples in the three basins. The PCA/FA was performed using R statistical software (version 4.2.1; R Foundation for Statistical Computing, Vienna, Austria).

In recent years, PCA has been widely used in hydrogeological studies to address a variety of environmental problems, including a comprehensive assessment of temporal and spatial changes in groundwater quality [START_REF] Jiang | Principal component analysis and hierarchical cluster analyses of arsenic groundwater geochemistry in the Hetao basin, Inner Mongolia[END_REF][START_REF] Zhang | Groundwater pollution source identification and apportionment using PMF and PCA-APCA-MLR receptor models in a typical mixed land-use area in Southwestern China[END_REF]. PCA is a multivariate statistical analysis method that depends on the orthogonal transformation to convert a set of observations of possibly correlated variables [START_REF] Sun | Integrated decision-making model for groundwater potential evaluation in mining areas using the cusp catastrophe model and principal component analysis[END_REF]. This method is applied to transform the original variables into new, uncorrelated variables (axes), which are linear compositions of the primary variables. FA is similar to PCA in the calculation principle. This internal analysis is the most widely used for its ability to directly obtain the common component number in a complex sample [START_REF] Audet | Performance indicators in multi objective optimization[END_REF]. FA uses a lower number of unobserved variables to explain more complex relationships in the observed variables. In this study, PCA/FA was performed on the normalized variables for 2021 and 2022.

The Kaiser-Meyer-Olkin measure (KMO) test was applied to assess the suitability of the data for PCA/FA. According to the Kaiser [START_REF] Kaiser | An index of factorial simplicity[END_REF] standard, the principle of eigenvalues being >1 was used to determine the number of principal components. Furthermore, values between 0.5 and 0.7 are mediocre, values from 0.7 to 0.8 are good, values between 0.8 and 0.9 are great, and values above 0.9 are superb. Values more than 0.5 signify a good justification for running factor analysis.

During the statistical analysis, samples characterized by over-high concentrations of a parameter were separated from the data base as they had a negative impact on the varifactor's separation. Thus, two sampling points characterized by salinization with extremely high values of EC and Cl, around 8806-10,522 µS/cm and 2703-3195 mg/L, respectively, were eliminated before PCA/FA calculation.

Water 2023, 15, 4018 7 of 19

Results

Basic Statistical Analysis

Basic statistical analysis was practiced as a first step to investigate the hydrochemical conditions of the study areas. The basic statistics of the groundwater quality parameters are summarized in Table S1. The geochemical evolution of groundwater can be recognized by plotting the concentrations of major cations and anions in a Piper diagram [START_REF] Piper | A graphical procedure in the geochemical interpretation of water analysis[END_REF]. On average, the cation concentrations of groundwater in Eastern Thermaikos Gulf (TG) were in the order Ca 2+ > Mg 2+ > Na + > K + , and those of the anions were in the order HCO 3 -> Cl -> SO 4

2-> NO 3 -. In the other two areas, the order is almost similar: Ca 2+ > Mg 2+ > Na + for cations and HCO 3 -> SO 4 2-for anions (Figure 4). No significant variations are noted during the wet and dry period of 2021-2022 in the Mouriki basin and TG.

During the statistical analysis, samples characterized by over-high concentrations of a parameter were separated from the data base as they had a negative impact on the varifactor s separation. Thus, two sampling points characterized by salinization with extremely high values of EC and Cl, around 8806-10,522 µS/cm and 2703-3195 mg/L, respectively, were eliminated before PCA/FA calculation.

Results

Basic Statistical Analysis

Basic statistical analysis was practiced as a first step to investigate the hydrochemical conditions of the study areas. The basic statistics of the groundwater quality parameters are summarized in Table S1. The geochemical evolution of groundwater can be recognized by plotting the concentrations of major cations and anions in a Piper diagram [START_REF] Piper | A graphical procedure in the geochemical interpretation of water analysis[END_REF]. On average, the cation concentrations of groundwater in Eastern Thermaikos Gulf (TG) were in the order Ca 2+ > Mg 2+ > Na + > K + , and those of the anions were in the order HCO3 -> Cl - > SO4 2-> NO3 -. In the other two areas, the order is almost similar: Ca 2+ > Mg 2+ > Na + for cations and HCO3 -> SO4 2-for anions (Figure 4). No significant variations are noted during the wet and dry period of 2021-2022 in the Mouriki basin and TG. According to the chemical analysis, pH ranges from 5.9 to 7.9 in the Italian region, while in Greek regions, it varies between 6.8 and 8.3. Higher concentrations of Ca appeared in the Mouriki and Upper Volturno basins, with values ranging from 30 mg/L to 97 mg/L and from 21 mg/L to 288 mg/L, respectively. The predominant cation in all the areas is HCO3, fluctuating from 57% to 83% of the samples concentration. The concentrations of Na and Cl in TG are up to 1365 mg/L and 3195 mg/L, respectively, highlighting the seawater intrusion phenomenon in the coastal part of the area (Figure 5a).

The concentration of SO4 in the Mouriki basin ranges from 9 mg/L to 35 mg/L, with an average value of 28 mg/L. In the same basin, the concentration of Fe ranges between 25 µg/L and 218 µg/L, with an average value of 117 µg/L, while Zn fluctuates from 5 µg/L to 550 µg/L (Figure 5b). Both chemical elements are related to the fertilization of apple trees, which cover a large extent of the basin. High concentrations of As occur mainly due to the According to the chemical analysis, pH ranges from 5.9 to 7.9 in the Italian region, while in Greek regions, it varies between 6.8 and 8.3. Higher concentrations of Ca appeared in the Mouriki and Upper Volturno basins, with values ranging from 30 mg/L to 97 mg/L and from 21 mg/L to 288 mg/L, respectively. The predominant cation in all the areas is HCO 3, fluctuating from 57% to 83% of the samples' concentration. The concentrations of Na and Cl in TG are up to 1365 mg/L and 3195 mg/L, respectively, highlighting the seawater intrusion phenomenon in the coastal part of the area (Figure 5a).

The concentration of SO 4 in the Mouriki basin ranges from 9 mg/L to 35 mg/L, with an average value of 28 mg/L. In the same basin, the concentration of Fe ranges between 25 µg/L and 218 µg/L, with an average value of 117 µg/L, while Zn fluctuates from 5 µg/L to 550 µg/L (Figure 5b). Both chemical elements are related to the fertilization of apple trees, which cover a large extent of the basin. High concentrations of As occur mainly due to the geological formations of TG [START_REF] Kouras | Distribution of arsenic in groundwater in the area of Chalkidiki, Northern Greece[END_REF][START_REF] Kazakis | The origin of uranium in groundwater of the eastern Halkidiki region, northern Greece[END_REF]. Moreover, variations between the wet and dry periods appeared in the concentrations of Fe and As in TG, as well as in Zn and Ni in the Mouriki basin. The concentration of NO 3 -in the Upper Volturno basin fluctuated from 0 to 128 mg/L (Figure 5c), with an average value of 27 mg/L.

Pearson Correlation Analysis

The Pearson correlation coefficient is the most common method to measure the linear relationship between pairs of variables and then identify the associations between parameters. The correlation matrix based on Pearson's correlation coefficient was applied to

Water 2023, 15, 4018 9 of 19 display the relationships between 10 main variables (Figure 6). A positive correlation is presented for the values close to +1, values approximately -1 indicate a strong negative correlation, and equal to zero designate no correlation. In the Mouriki basin, the negative weighting of K with all other components was observed, apart from pH, with a positive but weak correlation (r = 0.19), possibly due to the dissolution of clay minerals. A high positive correlation between Ca and SO 4 (r = 0.95) indicates the presence of gypsum and the thermal power plant of Amyntaio close to the NE part of the basin [START_REF] Tzampoglou | Hydrogeological Hazards in Open Pit Coal Mines-Investigating Triggering Mechanisms by Validating the European Ground Motion Service Product with Ground Truth Data[END_REF]. The presence of carbonates can explain the strong correlation found for Ca, Mg, and HCO 3 varying from 0.81 to 0.89.

play the relationships between 10 main variables (Figure 6). A positive correlation is pre-sented for the values close to +1, values approximately -1 indicate a strong negative correlation, and equal to zero designate no correlation. In the Mouriki basin, the negative weighting of K with all other components was observed, apart from pH, with a positive but weak correlation (r = 0.19), possibly due to the dissolution of clay minerals. A high positive correlation between Ca and SO4 (r = 0.95) indicates the presence of gypsum and the thermal power plant of Amyntaio close to the NE part of the basin [START_REF] Tzampoglou | Hydrogeological Hazards in Open Pit Coal Mines-Investigating Triggering Mechanisms by Validating the European Ground Motion Service Product with Ground Truth Data[END_REF]. The presence of carbonates can explain the strong correlation found for Ca, Mg, and HCO3 varying from 0.81 to 0.89.

In the Eastern Thermaikos Gulf, the pH has a negative correlation with all variables that can be explained by the pollutants released from industrial activities, which increases the concentrations of the rest of the ions. The high positive correlations between EC and the predominant ions (Na, K, Mg, Cl, SO4), with correlation coefficients higher than 0.9, indicate that the progressively increasing EC values may be ascribed to these ions released by the aquifer material as well as by seawater intrusion due to the high correlation between Na, Cl, and EC. Another strong correlation appeared between NO3, Na, Cl, and SO4 (r = 0.89 in all the cases), indicating the extensive use of N fertilizers in the area. In the Eastern Thermaikos Gulf, the pH has a negative correlation with all variables that can be explained by the pollutants released from industrial activities, which increases the concentrations of the rest of the ions. The high positive correlations between EC and the predominant ions (Na, K, Mg, Cl, SO 4 ), with correlation coefficients higher than 0.9, indicate that the progressively increasing EC values may be ascribed to these ions released by the aquifer material as well as by seawater intrusion due to the high correlation between Na, Cl, and EC. Another strong correlation appeared between NO 3 , Na, Cl, and SO 4 (r = 0.89 in all the cases), indicating the extensive use of N fertilizers in the area.

In the Upper Volturno basin, the dissolution of carbonate rocks is highlighted by the high correlation of HCO 3 with Ca (r = 0.98), Mg (r = 0.83), and EC (r = 0.92). A negative correlation of NO 3 with Na and Cl and a weak positive correlation with SO 4 (r = 0.09) indicates the low N loading from fertilizers. In addition, potassium sulfate (K 2 SO 4 ) and

Water 2023, 15, 4018 10 of 19 potassium chloride (KCl) fertilizers are absent according to the negative correlation between these components, with a range of values from -0.32 to -0.35. The correlation between Na and Cl (r = 0.76) can be attributed to some non-natural processes, such as leachates derived from the disposal of industrial waste septic tanks in rural areas and the use of manure.

In the third step of the proposed approach, ionic ratios were applied and plotted to confirm the current results. Figure 7a,b shows that the main hydrochemical composition of the sampling points in TG is evaporite rocks, while the samples from the Mouriki basin ranged between the silicate rock and evaporite rock, occurring in the silicate rock area. The samples from the Upper Volturno basin correspond mainly to carbonate rocks. The Na/Cl molar ratio (Figure 7c) was applied to recognize the evaporation state of the groundwater samples. Most groundwater samples plot in the freshwater evaporation line. Samples below the line correspond to Cl dissolution conditions (mainly halite dissolution). The results of the (Ca + Mg)/(HCO 3 + SO 4 ) ratio greater than unity further support the contribution of silicate weathering toward a higher concentration of Na instead of halite dissolution (Figure 7d). Additionally, climate conditions can affect the amount of evapotranspiration in a basin. The current results agree with those of Ntona et al. [START_REF] Ntona | Coupling SWAT and DPSIR models for groundwater management in Mediterranean catchments[END_REF], where the rate of evapotranspiration in the Anthemountas basin is higher than in the other two basins. Since the Na/Cl ratio is commonly greater than 1, the excess of Na over Cl suggests additional contributions of Na from other sources. The (Ca + Mg)/Na scatter plot (Figure 7e) supports the ion exchange process in all the areas except four samples in TG. A strong correlation between NO 3 , SO 4 , and Na is observed in TG due to the excessive use of fertilizers in contrast to the Mouriki basin (Figure 7f). In the Upper Volturno basin, the dissolution of carbonate rocks is highlighted by the high correlation of HCO3 with Ca (r = 0.98), Mg (r = 0.83), and EC (r = 0.92). A negative correlation of NO3 with Na and Cl and a weak positive correlation with SO4 (r = 0.09) indicates the low N loading from fertilizers. In addition, potassium sulfate (K2SO4) and potassium chloride (KCl) fertilizers are absent according to the negative correlation between these components, with a range of values from -0.32 to -0.35. The correlation between Na and Cl (r = 0.76) can be attributed to some non-natural processes, such as leachates derived from the disposal of industrial waste septic tanks in rural areas and the use of manure.

In the third step of the proposed approach, ionic ratios were applied and plotted to confirm the current results. Figure 7a,b shows that the main hydrochemical composition of the sampling points in TG is evaporite rocks, while the samples from the Mouriki basin ranged between the silicate rock and evaporite rock, occurring in the silicate rock area. The samples from the Upper Volturno basin correspond mainly to carbonate rocks. The Na/Cl molar ratio (Figure 7c) was applied to recognize the evaporation state of the groundwater samples. Most groundwater samples plot in the freshwater evaporation line. Samples below the line correspond to Cl dissolution conditions (mainly halite dissolution). The results of the (Ca + Mg)/(HCO3 + SO4) ratio greater than unity further support the contribution of silicate weathering toward a higher concentration of Na instead of halite dissolution (Figure 7d). Additionally, climate conditions can affect the amount of evapotranspiration in a basin. The current results agree with those of Ntona et al. [START_REF] Ntona | Coupling SWAT and DPSIR models for groundwater management in Mediterranean catchments[END_REF], where the rate of evapotranspiration in the Anthemountas basin is higher than in the other two basins. Since the Na/Cl ratio is commonly greater than 1, the excess of Na over Cl suggests additional contributions of Na from other sources. The (Ca + Mg)/Na scatter plot (Figure 7e) supports the ion exchange process in all the areas except four samples in TG. A strong correlation between NO3, SO4, and Na is observed in TG due to the excessive use of fertilizers in contrast to the Mouriki basin (Figure 7f).

Multivariate Statistical Analysis (MSA)

Multivariate statistical techniques are a tool for analyzing and interpreting datasets related to water quality and its temporal and spatial variations [START_REF] Li | A workflow for spatio-seasonal hydro-chemical analysis using multivariate statistical techniques[END_REF]. The multivariate statistical analysis (MSA) was employed to investigate the primary factors influencing the quality of groundwater in three Mediterranean regions. PCA/FA was used to identify potential pollution sources and their spatial distribution in the dry and wet periods of 2021-2022 and to evaluate the contribution of individual potential pollution sources. FA, with PCA as the extraction procedure, was applied to the datasets of all the parameters. Then, parameter commonalities were used to limit the number of variables that are significant contributors to the final factor model [START_REF] Okiongbo | Evaluation of major factors influencing the geochemistry of groundwater using graphical and multivariate statistical methods in Yenagoa city, southern Nigeria[END_REF]. The varifactors (VFs) were extracted, including the variables EC, pH, NO 3 , NH 4 , Cl, SO 4 , Na, K, Mg, and Ca. The samples with very high EC and Cl concentrations from TG had a negative impact on the varifactors separation in the first run of PCA/FA; therefore, they were eliminated. The results of this process are individually presented for each area for both sampling periods and one period for the Italian region and are also compared with previously published research.

Data Structure Determination Using PCA and FA PCA/FA analysis aims at data standardization and compares the composition patterns between groundwater samples to determine the important factors that affect each sample. The determined initial VFs in all the study areas are presented in Tables S2-S7, with their eigenvalues and cumulative % of the variance for the 2021-2022 periods. Based on the Kaiser Rule, in the samples of the Mouriki basin, the first two principal components were obtained with eigenvalue > 1 in the datasets of 2021 and 2022. This summarized 71.25% and 16.80% of the total variance in the 2021 groundwater quality datasets, respectively (Table S2), while for the groundwater samples of 2022, the variance of the VFs is 61.24% and 28.51%, respectively (Table S3). On the other hand, the samples from TG provide three VFs of 45.77%, 26.86%, and 15.15% of the total variance for the period 2021 (Table S4). The same number of VFs was obtained for samples in 2022, where an increase is observed in the first VF (63.66%), as shown in Table S5. Three VFs were obtained also in the Italian region of 49.07%, 20.56%, and 15.62% of the total variance for the period 2021 (Table S6). The variance factor rotation loading matrix of the selected groundwater quality parameters is presented in Table 2, where factor loadings > 0.75, 0.50-0.75, and 0.30-0.50 are considered to be strong, moderate, and weak, respectively. The results of the KMO test are provided in Table S7. In the framework of MSA, the PCA/FA method was also applied using the data from all study areas. The results indicated that there might be a statistically significant interrelationship between variables (chemical parameters), which rendered the PCA/FA valid. The KMO value was 0.69, and the statistical analysis provided only two VFs, underestimating the contribution of the variables. Thus, in this case, the combination of limited data from different areas cannot be successfully performed in the investigation for the determination of the hydrogeological complexities in the basins/areas. In addition, multi-statistical analysis with all 21 variables was avoided as the VFs could not be separated. Therefore, in the cases of limited available data, a multi-statistical analysis of a maximum of ten variables is suggested.

Identification of Varifactors

In the Mouriki basin for the dry period of 2021, VF1 was influencing the groundwater quality, accounting for 71.25% of the total variance. Its eigenvalue was 7.13, and it had strong and positive loadings on Na, Ca, Mg, K, SO 4 , HCO 3 , and EC. It showed moderately positive loading on NO 3 . VF2, accounting for 16.80% of the total variance, had strong and positive loadings on Cl, NO 3 , and Na and strong negative loadings on pH. In 2022, VF1 accounted for 61.24% of the total variance, its eigenvalue was 6.12, and it had strong and positive loadings on Cl, SO 4 , NO 3 , Na, and EC, while strong negative loadings on pH. VF2 for the period 2022, accounting for 28.51% of the total variance with an eigenvalue of 2.85, provided strong positive loading on HCO3, Ca, Mg, and EC. In Eastern Thermaikos Gulf, VF1 accounted for 74.85% and showed strong positive loadings on all the variables except pH and HCO 3 . VF2 accounted for 18.70% with strong negative and positive loadings on pH and HCO 3 , respectively. The same results with few alterations appeared for the samples of 2022, where strong negative loadings were noticed only on pH. For the Upper Volturno basin, water samples were collected only during the dry period of 2021. VF1 accounted for 49.07% of the total variance with an eigenvalue of 4.9, and it had strong and positive loadings on HCO 3 , Ca, Mg, and EC. VF2 had strong positive loadings on Cl and Na, medium positive on SO 4 . VF3 had strong positive loadings on pH, medium on SO 4 , and weak on NO 3 .

Groundwater Level Measurements

Long-period monitoring data of GWL measurements and rainfall data are provided in Figure 6 for TG. Wet and dry periods are separated to provide a clear view of the variations in each season over time. Two boreholes in the coastal (11 m above sea level) and inland (85 m above sea level) areas of TG were chosen to demonstrate the variations in the piezometric head from 2011 to 2022. The seasonal variation in the piezometric head in the coastal borehole is approximately 4 m during the wet and dry periods of the monitoring years (Figure 8a). On the other hand, minor GWL fluctuations are noted in the inland borehole (Figure 8b), where the over-pumping limits the increase in GWL. In addition, a significant decline in GWL is observed over time due to the over-pumping in the coastal/agricultural areas. In both cases, the variation in rainfall during the years affects the groundwater piezometry. In particular, the high values of 754 mm during 2014 contributed to the increase in the piezometric head.

In the Mouriki basin, the GWL data were used to provide the variations in the groundwater depth during the years 2004 and 2022 (Figure 9). In TG, the variation in the piezometric head between the wet periods of 2011 and 2022 is provided in Figure 10, considering previously published data. Figure 11 shows the GWL variation during the years 2006 and 2016 in the study area of the Upper Volturno basin. Groundwater depth in the Mouriki basin varies from 0 m to 19.7 m during the wet periods of the monitoring years. Figure 8 depicts the negative variation in piezometric heads over 18 years in the Mouriki basin based on the available GWL data.

The same trend is observed in TG over 11 years (Figure 8), where groundwater depth varies from 0.4 m to 125.3 m based on the measurements in the spring period. In the coastal area, the increase in the piezometric head is probably caused by the reduction in pumping due to the high salinity of water. The opposite condition of negative piezometric curves appears in the areas affected by agricultural activities, highlighting the results of the overexploitation during the years. Groundwater depth in the Upper Volturno basin varies from 25.5 m to 149 m during the monitoring periods. The variation in GWL in porous aquifers varies between -4.8 m and 8.1 m (Figure 9). The same trend is observed in TG over 11 years (Figure 8), where groundwater depth varies from 0.4 m to 125.3 m based on the measurements in the spring period. In the coasta area, the increase in the piezometric head is probably caused by the reduction in pumping due to the high salinity of water. The opposite condition of negative piezometric curve The same trend is observed in TG over 11 years (Figure 8), where groundwater depth varies from 0.4 m to 125.3 m based on the measurements in the spring period. In the coastal area, the increase in the piezometric head is probably caused by the reduction in pumping due to the high salinity of water. The opposite condition of negative piezometric curves

Discussion

Adopting new approaches, namely critical sampling, was tested within this study. This concept follows the current trend of global interest in research techniques to control the dynamics of groundwater pollution [START_REF] Zhu | Spatio-temporal variations of nitrate pollution of groundwater in the intensive agricultural region: Hotspots and driving forces[END_REF]. The assessment of groundwater quality and quantity variations is essential for groundwater management and protection [START_REF] Taghavi | Groundwater vulnerability assessment: A review including new statistical and hybrid methods[END_REF]; however, the creation of a large database and the application of different methodological approaches make the analysis costly [START_REF] Guo | Response of groundwater quality to river-aquifer interactions during managed aquifer recharge: A reactive transport modeling analysis[END_REF]. MSA of physicochemical parameters is suggested in the current work as a low-cost technique to achieve a more accurate evaluation of pollution sources based on judgmental sampling. This method was practiced in three different areas in the Mediterranean region to test it in different hydrological and hydrogeolog-

Discussion

Adopting new approaches, namely critical sampling, was tested within this study. This concept follows the current trend of global interest in research techniques to control the dynamics of groundwater pollution [START_REF] Zhu | Spatio-temporal variations of nitrate pollution of groundwater in the intensive agricultural region: Hotspots and driving forces[END_REF]. The assessment of groundwater quality and quantity variations is essential for groundwater management and protection [START_REF] Taghavi | Groundwater vulnerability assessment: A review including new statistical and hybrid methods[END_REF]; however, the creation of a large database and the application of different methodological approaches make the analysis costly [START_REF] Guo | Response of groundwater quality to river-aquifer interactions during managed aquifer recharge: A reactive transport modeling analysis[END_REF]. MSA of physicochemical parameters is suggested in the current work as a low-cost technique to achieve a more accurate evaluation of pollution sources based on judgmental sampling. This method was practiced in three different areas in the Mediterranean region to test it in different hydrological and hydrogeological environments. Pearson correlation, PCA, and FA were conducted to analyze the

Discussion

Adopting new approaches, namely critical sampling, was tested within this study. This concept follows the current trend of global interest in research techniques to control the dynamics of groundwater pollution [START_REF] Zhu | Spatio-temporal variations of nitrate pollution of groundwater in the intensive agricultural region: Hotspots and driving forces[END_REF]. The assessment of groundwater quality and quantity variations is essential for groundwater management and protection [START_REF] Taghavi | Groundwater vulnerability assessment: A review including new statistical and hybrid methods[END_REF]; however, the creation of a large database and the application of different methodological approaches make the analysis costly [START_REF] Guo | Response of groundwater quality to river-aquifer interactions during managed aquifer recharge: A reactive transport modeling analysis[END_REF]. MSA of physicochemical parameters is suggested in the current work as a low-cost technique to achieve a more accurate evaluation of pollution sources based on judgmental sampling. This method was practiced in three different areas in the Mediterranean region to test it in different hydrological and hydrogeological environments.

Pearson correlation, PCA, and FA were conducted to analyze the relationship between the variables and evaluate the factors (natural and anthropogenic) affecting hydrochemical components. These methods have been widely used for such analysis around the world to explore the relationship among water quality variables and optimize the monitoring network and basin management [START_REF] Haji Gholizadeh | Water quality assessment and apportionment of pollution sources using APCS-MLR and PMF receptor modeling techniques in three major rivers of South Florida[END_REF][START_REF] Abdelaziz | Groundwater quality index based on PCA: Wadi El-Natrun, Egypt[END_REF][START_REF] Chen | Assessment of Groundwater Quality Using APCS-MLR Model: A Case Study in the Pilot Promoter Region of Yangtze River Delta Integration Demonstration Zone, China[END_REF].

MSA requires a larger number of samples to obtain higher reliability [START_REF] Thurston | A quantitative assessment of source contributions to inhalable particulate matter pollution in metropolitan Boston[END_REF]. Nevertheless, the results of the current work show that in cases where the hydrogeological characteristics and pressures of the study areas are well known, fewer and more specific groundwater samples could be collected for the investigation of the temporal groundwater variability. More specifically, this research approach highlights the importance of comparison of fullfield hydrogeological studies.

According to the results, most of the samples are affected by a combination of factors. For instance, cultivable lands in TG have a distance of a few meters from the coastline; consequently, both seawater intrusion due to the over-pumping of groundwater and the unregulated use of fertilizers affect the water quality. As expected, the results of the multistatistical analysis showed that the geological formations and livestock activities are the main influences in the area. The selection of samples (even in low distribution compared to the whole extent of the basin) illustrated the factors' degree of influence. Thus, selecting the most representative sampling points plays a vital role in the optimal determination of groundwater quality status.

Positive and negative GWL variations are noted in all the study areas, while the most significant variations are observed in the TG. In the last years, coastal urban and agricultural zones have been the main study areas for climate-related ocean and sea level changes, with significant impacts on groundwater resources. The causes of GW rise in the coastal part of TG are not easy to identify with the current results. In a similar case, the determination of the extent of sea level and groundwater rise interaction was investigated by De Biase et al. [START_REF] De Biase | The Impact of Predicted Climate Change on Groundwater Resources in a Mediterranean Archipelago: A Modelling Study of the Maltese Islands[END_REF] with numerical modeling.

According to Li et al. [START_REF] Li | Quantitative decision making for a groundwater monitoring and subsurface contamination early warning network[END_REF], the establishment of a monitoring network is of utmost importance to investigate spatiotemporal variations in groundwater systems. This point is confirmed by the current work, where the pressures in the study areas are validated by Ntona et al. [START_REF] Ntona | Coupling SWAT and DPSIR models for groundwater management in Mediterranean catchments[END_REF]. The critical sampling analysis provides a lower-cost approach with the same results as other more expensive approaches due to a great deal of samples. The results of the applied methodology are in agreement with previous studies [START_REF] Kazakis | Multivariate statistical analysis for the assessment of groundwater quality under different hydrogeological regimes[END_REF][START_REF] Patrikaki | Vulnerability map: A useful tool for groundwater protection: An example from Mouriki basin, North Greece[END_REF], where a larger amount of data was collected to depict the GW status of the areas. A reduction of around 80% in the sampling points was chosen in the TG and Mouriki basin. Nevertheless, even more detailed sampling methods are crucial in the first research steps in unexplored areas. The framework of judgmental sampling was conducted based on the characteristics of semi-arid areas of the Mediterranean region.

Based on our knowledge, the random and capture-recapture sampling methods have not been applied for groundwater monitoring. Patently, the random method is characterized by a high percentage of errors. On the other hand, the capture-recapture method can be applied in the field but increases the difficulty of the fieldwork due to the frequency of the sampling collection and different sampling points in time. As concern the limitations of this methodology, it should be noted that the effectiveness of this application can be prosperous and with low risks in cases where the study areas have already been investigated. More details about the judgmental method can be found in Hannover [START_REF] Hannover | Evaluation of Performance: A Judgmental Approach[END_REF]. Moreover, the application of PCA/FA in different study areas with the same analysis approach does not provide satisfactory results due to the limited points. Along these, a multi-statistical analysis of a maximum of ten variables is suggested to avoid misinterpretations. The spatial distribution of the water variables has limitations due to the long distance between the monitoring points, so this approach is not recommended in cases where the spatial extent of pollution is investigated. Combined qualitative and quantitative groundwater measurements can eliminate the errors of the critical sampling approach by jointly analyzing the hydrological and hydrogeological characteristics of the study area. Thus, the methodology with the optimum results is applied according to the data availability and critical thought of the researchers. In the last decades, efforts have been made to create and enrich open databases for widely available data and avoid repeated chemical analyses by different agencies. In this way, the possible hazards of the current work could be minimized in future approaches.

We strongly believe that the current results can play a vital role in the decision making of stakeholders for groundwater management [START_REF] Karakatsanis | Optimization of dam operation and interaction with groundwater. An overview focusing on Greece[END_REF] and pollution control. This research approach proves that valuable information about the variations in the main parameters of groundwater can also be obtained by judgmental sampling approaches. This research will support future studies and practices. In future works, forecasting methods can be performed to explore predictive and normative scenarios. For instance, environmental scanning could generate the background information from which to forecast or develop scenarios. Overall, this standard information can be used in a primary monitoring stage, while in future steps, more detailed chemical analysis can be performed, including BOD, COD, microbiological, and isotope analysis, depending on the needs of each research. In addition, to specify the causes of positive GWL variations in the coastal part of TG, future research is suggested for the investigation of the sea level and groundwater rise interaction based on numerical modeling, including the MODFLOW model.

Conclusions

In the current work, an MSA approach was adopted based on a non-probability sampling technique to identify the groundwater hydrochemical evolution in three Mediterranean regions. The selection of the optimal groundwater monitoring technique in a study area is related to the natural characteristics of groundwater, the pressures, the available scientific data, and the financial budget. The judgmental sampling approach was applied for the conceptualization of the hydrogeological regime, concluding the following:

(1) The judgmental sampling approach can contribute to a cost and time-effective groundwater monitoring plan of a basin. (2) The application of the judgemental sampling approach provides reliable results regarding groundwater evolution. This information is crucial for the optimal application of water resource management and control techniques. (3) The implementation of a judgemental sampling approach should be performed periodically to minimize bias. (4) The methodological approach applied within this study is flexible and can be modified according to the specific characteristics of the site.

The results were verified by using PCA/FA and Pearson correlation analysis. The selected groundwater samples highlight the quality of groundwater in three study areas, the results of which are summarized below:

(1) No significant variations in the physicochemical parameters of groundwater samples appeared between the two sampling periods. The data from groundwater level measurements collected in the field and from previous works highlight the following results:

(1) A significant decline in GWL is observed over time in Eastern Thermaikos Gulf due to the over-pumping in the coastal/agricultural areas.

Figure 1 .

 1 Figure 1. Location of the three study areas.

Figure 1 .

 1 Figure 1. Location of the three study areas.

Figure 2 .

 2 Figure 2. Positive and negative effects of judgmental sampling.

Figure 2 .

 2 Figure 2. Positive and negative effects of judgmental sampling.

Figure 3 .

 3 Figure 3. Parameters for selection and judgmental groundwater sampling in the study areas.

Figure 3 .

 3 Figure 3. Parameters for selection and judgmental groundwater sampling in the study areas.

Figure 4 .

 4 Figure 4. Groundwater hydrochemical facies for the wet and dry periods in the three study areas.

Figure 4 .

 4 Figure 4. Groundwater hydrochemical facies for the wet and dry periods in the three study areas.

  15, x FOR PEER REVIEW 8 of 19 geological formations of TG[START_REF] Kouras | Distribution of arsenic in groundwater in the area of Chalkidiki, Northern Greece[END_REF][START_REF] Kazakis | The origin of uranium in groundwater of the eastern Halkidiki region, northern Greece[END_REF]. Moreover, variations between the wet and dry periods appeared in the concentrations of Fe and As in TG, as well as in Zn and Ni in the Mouriki basin. The concentration of NO3 -in the Upper Volturno basin fluctuated from 0 to 128 mg/L (Figure5c), with an average value of 27 mg/L.

Figure 5 .

 5 Figure 5. Distribution of Cl in the Eastern Thermaikos Gulf (a), Zn in the Mouriki basin (b), and NO3 in the Upper Volturno basin (c).

Figure 5 .

 5 Figure 5. Distribution of Cl in the Eastern Thermaikos Gulf (a), Zn in the Mouriki basin (b), and NO 3 in the Upper Volturno basin (c).

Figure 6 .

 6 Figure 6. Correlation matrix of chemical parameters in groundwater samples of (a) Mouriki basin, (b) East Thermaikos Gulf, and (c) Upper Volturno basin.

Figure 6 .

 6 Figure 6. Correlation matrix of chemical parameters in groundwater samples of (a) Mouriki basin, (b) East Thermaikos Gulf, and (c) Upper Volturno basin.

Water 2023 ,

 2023 15, x FOR PEER REVIEW 10 of 19

Figure 7 .

 7 Figure 7. Ionic ratio plots relative to groundwater samples from M = Mouriki basin, TG= Eastern Thermaikos Gulf, and V = Upper Volturno basin: (a) (Mg/Na)/(Ca/Na), (b) (HCO3/Na)/(Ca/Na), (c) Na/Cl, (d) (Ca + Mg)/(HCO3 + SO4), (e) (Ca + Mg)/Na, and (f) NO3/SO4 with Na.

Figure 7 .

 7 Figure 7. Ionic ratio plots relative to groundwater samples from M = Mouriki basin, TG = Eastern Thermaikos Gulf, and V = Upper Volturno basin: (a) (Mg/Na)/(Ca/Na), (b) (HCO 3 /Na)/(Ca/Na), (c) Na/Cl, (d) (Ca + Mg)/(HCO 3 + SO 4 ), (e) (Ca + Mg)/Na, and (f) NO 3 /SO 4 with Na.

Water 2023 , 1 Figure 8 .

 202318 Figure 8. Rain and piezometric head variation in monitoring boreholes of (a) coastal and (b) inland areas of Eastern Thermaikos Gulf during 2011-2022.

Figure 9 .

 9 Figure 9. Variation in the piezometric head in the Mouriki basin between May 2004 and 2022.

Figure 8 . 19 Figure 8 .

 8198 Figure 8. Rain and piezometric head variation in monitoring boreholes of (a) coastal and (b) inland areas of Eastern Thermaikos Gulf during 2011-2022.

Figure 9 .

 9 Figure 9. Variation in the piezometric head in the Mouriki basin between May 2004 and 2022.

Figure 9 .

 9 Figure 9. Variation in the piezometric head in the Mouriki basin between May 2004 and 2022.

Figure 10 .

 10 Figure 10. Variation in the piezometric head in Eastern Thermaikos Gulf between May 2011 and 2022.

Figure 11 .

 11 Figure 11. Variation in the piezometric head in the Upper Volturno basin between May 2006 and 2016.

Figure 10 . 19 Figure 10 .

 101910 Figure 10. Variation in the piezometric head in Eastern Thermaikos Gulf between May 2011 and 2022.

Figure 11 .

 11 Figure 11. Variation in the piezometric head in the Upper Volturno basin between May 2006 and 2016.

Figure 11 .

 11 Figure 11. Variation in the piezometric head in the Upper Volturno basin between May 2006 and 2016.

( 2 )

 2 Variations between the wet and dry periods appeared in the concentrations of Fe and As in Eastern Thermaikos Gulf as well as in Zn and Ni in the Mouriki basin.(3) NO 3 pollution occurs in all the areas except the Mouriki basin, where mainly Zn fertilizers are applied. (4) Salinization dominates as a pollution process in the coastal aquifer of Eastern Thermaikos Gulf. (5) Groundwater quality decline has been observed in Eastern Thermaikos Gulf and Mouriki basin over the years due to overexploitation for agricultural activities.

Table 1 .

 1 Parameters for judgmental groundwater sampling.

	Parameter	Description
	Definition of the target
		-	Groundwater level
	Aquifer types	-	Porosity
		-	Aquifers complexity
	Hydro morphological zones	Impacts of hydro morphological pressures (e.g., soil erosion)
	Land Use	--	Land use footprint Spatial and temporal variation
		-	Available and missing data
	Previous data	-	Methodologies and Results evaluation
		-	Unanswered questions
	Point representativeness	Comparison of sampling points
	Study area	--	Determination of the extent of the basin/area Morphology
	Cost	Funds availability
	Time	--	Research timetable Repetition of sampling

-

What is the main purpose of the research?

-Why now?

Table 1 .

 1 Parameters for judgmental groundwater sampling.

		-	Groundwater level
	Aquifer types	-	Porosity
		-	Aquifers complexity
	Hydro morphological zones	Impacts of hydro morphological pressures (e.g., soil erosion)
	Land Use	--	Land use footprint Spatial and temporal variation
		-	Available and missing data
	Previous data	-	Methodologies and Results evaluation
		-	Unanswered questions
	Point representativeness	Comparison of sampling points
	Study area	--	Determination of the extent of the basin/area Morphology
	Cost	Funds availability
		-	Research timetable
	Time	-	Repetition of sampling
		-	Researchers' availability
	Field issues	--	Sampling point accessibility Number and reliability of sampling points
	-

Parameter Description

Definition of the target -What is the main purpose of the research? -Why now?

Table 2 .

 2 Loadings of the selected groundwater quality parameters on varimax rotated factors in 2021 and 2022 sampling periods in the study areas.

					Mouriki basin			Thermaikos Gulf			Volturno basin
				2021	2022		2021		2022			2021
				Component	Component	Component	Component		Component
	Parameters	1	2	1	2	1	2	1	2	1	2	3
	pH		-0.040	-0.962	-0.882	0.268	-0.156	-0.910	-0.107	-0.990	-0.006	-0.093	0.867
	EC		0.945	0.323	0.652	0.751	0.996	0.071	0.970	0.187	0.937	0.239	-0.006
	HCO 3	-	0.846	0.148	-0.276	0.925	-0.132	0.915	0.907	0.186	0.965	0.103	-0.202
	Cl -		0.286	0.870	0.947	0.290	0.998	-0.018	0.978	0.109	0.448	0.848	-0.064
	SO 4	-	0.947	0.219	0.961	0.066	0.986	-0.029	0.976	0.082	0.063	0.581	0.696
	SO 4	-	0.685	0.657	0.961	0.066	0.915	-0.154	0.973	0.044	-0.493	-0.145	0.491
	Na +		0.375	0.767	0.826	0.393	0.988	0.044	0.954	0.155	-0.135	0.961	-0.033
	K +		-0.804	-0.409	-0.458	-0.593	0.966	0.162	0.883	0.323	-0.865	0.007	-0.373
	Ca 2+		0.982	0.149	-0.022	0.987	0.869	0.426	0.874	0.398	0.951	0.161	-0.114
	Mg 2+	0.910	0.319	0.583	0.808	0.973	-0.035	0.892	0.041	0.890	-0.158	-0.075
	Eigenvalue	7.13	1.68	6.12	2.85	7.49	1.87	8.21	1.03	4.9	2.06	1.56

Acknowledgments: We acknowledge the financial support by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "Second Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers" (Project Number: 00138, Title: Groundwater Depletion. Are Eco-Friendly Energy Recharge Dams a Solution?). The scholarship and research activities of Maria Margarita Ntona were part of the Environment, Design, and Innovation Ph.D. Program funded by the V:ALERE 2020 Program (VAnviteLli pEr la RicErca) of the University of Campania "Luigi Vanvitelli".

Funding: This research project was supported financially by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the "Second Call for H.F.R.I. Research Projects to support Post-Doctoral Researchers" (Project Number: 00138, Title: Groundwater Depletion. Are Eco-Friendly Energy Recharge Dams a Solution?). The scholarship and research activities of Maria Margarita Ntona were part of the Environment, Design, and Innovation Ph.D. Program funded by the V:ALERE 2020 Program (VAnviteLli pEr la RicErca) of the University of Campania "Luigi Vanvitelli".

Data Availability Statement: Data are contained within the article and supplementary materials.

(2) Negative and positive variations in GWL during the last decades appeared in all the areas. (3) Maximum recovery of GWL was noted in the Mouriki basin for the period 2014-2022.

Supplementary Materials:

The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/w15224018/s1, Table S1: Basic statistics of groundwater quality parameters in the study areas; Table S2

Conflicts of Interest:

The authors declare no conflict of interest.


Pavlos Antoniou
email: pavlos.antoniou@kcl.ac.uk

Costas S Iliopoulos

Řivoj Melichar

Pierre Peterlongo
email: pierre.peterlongo@univ-mlv.fr

Finding Common Motifs with Gaps Using Finite Automata ⋆

We present an algorithm that uses finite automata to find the common motifs with gaps occurring in all strings belonging to a finite set S = {S1,S2,...,Sr}. In order to find these common motifs we must first identify the factors that exist in each string. Therefore the algorithm begins by constructing a factor automaton for each string Si. To find the common factors of all the strings, the algorithm needs to gather all the factors from the strings together in one data structure and this is achieved by computing an automaton that accepts the union of the above-mentioned automata. Using this automaton we are able to create a new factor alphabet. Based on this factor alphabet a finite automaton is created for each string Si that accepts sequences of all non overlapping factors residing in each string. The intersection of the latter automata produces the finite automaton which accepts all the common subsequences with gaps over the factor alphabet that are present in all the strings of the set S = {S1,S2,...,Sr}. These common subsequences are the common motifs of the strings.

Introduction

The problem of finding common motifs in a set of strings has long been an area of interest in the academic community. Given a set of strings, the problem of finding common motifs in that set is the problem of finding similar substrings that lie in all of these strings. In some particular applications, like in biology, this requirement is more flexible in the sense that motifs do not have to be identical but have to share a certain degree of similarity. This degree is quantified using metrics such as Hamming and Levenshtein distances or by allowing don't care symbols to occur in the motifs. Don't care symbols are occurrences in the string that can match any symbol of the alphabet. In this paper, we are interested in finding common motifs in the strings that have don't care symbols concentrated in distinct parts of contiguous positions in the strings, i.e. common motifs with gaps.

This problem has engrossed biologists because of its applications in that area. It can be applied in understanding the fundamental process of gene expression [START_REF] Henry | Finding motifs with insufficient number of strong binding sites[END_REF]. Gene expression consists of two parts, transcription and translation. During transcription an mRNA molecule is created by copying a gene from the DNA and during translation the mRNA molecule is decoded to produce a protein. In order though for the transcription process to begin, one or more proteins, called transcription factors, have to bind to some specific regions of the gene called binding sites. These binding sites share common patterns which are the common motifs of the genes. If these common motifs are identified and extracted from the genes, they will give the opportunity to biologists to match these binding sites to their corresponding transcription factors in order to be able to fully understand the way gene expression works [START_REF] Henry | Finding motifs with insufficient number of strong binding sites[END_REF].

A classical approach to finding these motifs was by using artificial intelligence techniques [START_REF] Michael | An artificial intelligence approach to motif discovery in protein sequences: Application to steroid dehydrogenases[END_REF] but these methods are inexact methods that used machine learning to discover the motifs by training the machines to recognize them. Recently, microarray technology has been used particularly in this application of the problem but this technology is inexact, it is based on probabilities and is limited by weak signal sequences [START_REF] Henry | Finding motifs with insufficient number of strong binding sites[END_REF].

In text algorithm applications, finding common motifs with gaps has been mainly handled using suffix trees [START_REF] Charras | Exact string matching algorithms[END_REF][START_REF] Crawford | String matching techniques for musical similarity and melodic recognition[END_REF][START_REF] Crochemore | Text algorithms.O x f o r d U n i v e r s i t[END_REF][START_REF] Iliopoulos | A first approach to finding common motifs with gaps[END_REF] which provided exact results. In this paper we propose an algorithm using automata to index common gapped motifs. We believe that the use of automaton permit the indexation of bigger strings and allows more open definitions.

Section 2 formally introduces the general problem. Section 3 presents an algorithm in order to solve the question of finding common motifs with gaps. Moreover Section 4 presents a complete example following step by step the proposed algorithm. Eventually in Section 5 there is an analysis of the complexity of the proposed solution.

Definition of the Problem

Given a set of strings S = {S 1 ,S 2 ,...,S r } and p, q,1≤ p ≤ q ≤ min(|S j | : j ∈ 1,r ). The problem of finding common motifs with gaps consists in finding words B 1 ,B 2 ,...,B m such that:

1. m>1. 2. p ≤|B i |≤q for i ∈ 1,m . 3. B 1 • di,1 B 2 • di,2 ...• di,m-1 B m
occur in S i for all i ∈ 1,r , m>1a n dt h e size of the gap d i,j varies in each motif (Fig. 1), where • denotes don't care symbol matching any symbol of alphabet and • j denotes concatenation of j don't care symbols.

. . .

B 1 S 2 : S r : S 1 : B 1 B 2 d2,m-1 B 2 B m-1 B 2 B m-1 B m B m-1 B m B m B 1 dr,m-1 d1,m-1 d1,1 d2,1 dr,1
Fig. 1. An example of a motif with gaps that occurs in every string, where by di,j we mean a size of a gap

Algorithm

The algorithm takes as input a set of strings S = {S 1 ,S 2 ,...,S r } and two constants p, q, which will be the lower and upper bound respectively of the length each motif can have, and returns the common motifs with gaps found in all those strings. The algorithm begins with computing the set of all factors F of length between the constants p and q that appear in all strings belonging to the set S = {S 1 ,S 2 ,...,S r }.

In order to find all these factors in F that appear in all the strings, we begin by creating a factor automaton MF i for each string S i ∈ S. Each factor automaton MF i accepts all the factors of the particular string S i ∈ S. Then, the algorithm joins all the MF i automata together in one automaton. The resulting union automaton accepts the union of the languages accepted by each of the r automata. This automaton can either be deterministic or non-deterministic. If it is deterministic, then the algorithm finishes because this is a sign that there are no common symbols and therefore no common motifs are present in the strings from this set S. On the other hand, if the resulting union automaton is nondeterministic, the algorithm proceeds with transforming this non-deterministic automaton into a deterministic one.

From this deterministic union automaton we identify all factors having length between the two constants p and q that are repeating in all strings from the set S. These factors are subsequently used to create a repetition table RT,w h i c h is used to create a new factor alphabet containing only the symbols relevant to the factors extracted in the previous steps.

Based on the repetition table, we find the longest common subsequence over the factor alphabet of all the strings of S. To achieve this aim, we first create a finite automaton MS i for each string S i ∈ S accepting sequences of nonoverlapping factors using the factor alphabet as input alphabet. Then, we create the automaton MS by taking the intersection of these automata MS i .Theresulting automaton will accept the intersection of the languages accepted by each of the factor automata i.e. it accepts all sequences of factors occurring in all strings from the set S which are the common motifs of the strings with gaps.

The algorithm: Input: Set of strings S = {S 1 ,S 2 ,...,S r },p ,q . Output: Sequence of words B 1 ,B 2 ,...,B m occurring in all strings in S. Method:

1. (a) For each string S i ∈ S construct a factor automaton M iε by creating automaton M i , accepting string S i (i.e. L(M i )={S i }), then adding ε-transitions leading from the initial state to all states of M i and making all states final. of M DF we preserve a set of states of M F q ′ consists of. The set is called d-subset. 2. Find all states of M DF representing factors of length between p and q and having at least one state from each automaton M i in its d-subset. Construct a repetition table RT (the shortest path from the initial state to the state spells the repeated factor while members of d-subset identify locations). 4A n E x a m p l e

As an example let's consider a set of strings S = {aabccddab, babbcdacd}. We will find common motifs in this set of strings bounded from parameters p =2,q =3. First (step 1a of the algorithm) we construct finite automata M 1ε and M 2ε for both strings from S.S e eF i g .2 1 .

In the next step (step 1b of the algorithm) we construct automaton M ε accepting the union of languages L(M 1ε )a n dL(M 2ε ). See Fig. 3.

1 All states in the automata presented in this paper are final states.

ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε Fig. 3

. Transition diagram of finite automaton Mε from the example

According to step 1c of the algorithm, we construct automaton M F by replacing the ε-transitions by non ε-transitions. The transition diagram of the resulting automaton is given in Fig. 4.

In this example, automaton M F is nondeterministic. This means that there is a possibility that a common motif exists in set S.A c c o r d i n gt os t e p1 eo ft h e algorithm 3, we must construct its deterministic equivalent M D . Its transition diagram is given in Fig. 5.

Table 1 is the repetition table RT of common factors created as described in step 2 of the algorithm. The factor alphabet (step 3 of the algorithm) is FA = {ab, bc, cd, da}. Subsequently we will construct, according to step 4 of the algorithm, finite automata MS 1 and MS 2 accepting all non-overlapping sequences of factors of the both strings. Their transition diagrams are depicted in Fig. 6.

The last step (step 5 of the algorithm involves the construction of an automaton which accepts all sequences of factors occurring in both strings of S.T r a nsition diagrams of finite automata MS According to the step 5c of the algorithm, we need to construct a finite automaton accepting the language that corresponds to the intersection of the languages accepted by the two automata. The transition diagrams of the finite automaton accepting the intersection of the languages accepted by automaton MS N 1 and MS N 2 is depicted in Fig 9. Finally, from the above computation we may conclude that in this particular set S = {aabccddab, babbcdacd} and using parameters p =2,q = 3 the following common motifs occur:

2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 8 1 7 2 6 1 6 2 9 2 1 1 2 1 8 1 2 2 7 2 3 1 9 1 3 2
Factor d-subset Repetitions ab 3 1 9 1 3 2 (3 1 ,F), (9 1 ,G), (3 2 ,F) bc 4 1 5 2 (4 1 ,F), (5 2 ,F) cd 6 1 6 2 9 2 (6 1 ,F), (6 2 ,F), (9 2 ,G) da 8 1 7 2 (8 1 ,F), (7 2 ,F) cd 8 1 0 2 ab 6 2 9 1 cd 5 2
m 1 = {ab, cd}, m 2 = {ab, da}, m 3 = {bc, cd}, m 4 = {bc, da}.

Time and Space Complexity of the Algorithm

We shall discuss the time and space complexity for each step of the algorithm. As described in Section 3 the algorithm requires five steps.

In Step 1 we fist construct r finite automata, one for each string S i of the set of strings S. The time and space needed to construct each automaton depends on the length of each particular string S i which is the language to be accepted by the automaton. Therefore this process requires linear time and space [START_REF] Crochemore | Automata for matching patterns[END_REF] with respect to the length of the strings.

Subsequently we wish to construct the automaton which accepts the union of these languages. If we assume that the value n is the cumulated size of the input sequences i.e. n = r i=1 |S i |, then this step requires O(n) space and time for all the strings to be included in the automaton. Next, we wish to transform this union automaton into a deterministic factor automaton also called Direct Acyclic Word Graph (DAWG). In most cases the union automaton is non-deterministic. In order to create the deterministic factor automaton we need to transform the non-deterministic union automaton to a deterministic automaton which will be the factor automaton M F . Generally, the construction of a deterministic automaton from a non-deterministic requires exponential time and space. In the case of factor automata though the maximum number of states of the resulting deterministic automaton is 2|n|-1 and the maximum number of transitions is 3|n|-4 [START_REF] Crochemore | Direct construction of compact directed acyclic word graphs[END_REF]. Thus, this step is bounded overall by linear time and space.

In Step 2 we extract from M F the factors belonging to all the strings and which have length between the values p and q, and we add them to the repetition table RT . In order to find all these factors we need to reach all the states at depth q. Let δ = qp + 1 the length of the last interval. As we are looking for all the factors between p and q there are at most n×δ such factors. Thus the complexity of this step is O(n × δ).

In Steps 3 and 4 we construct the new factor alphabet and for each string S i in S we construct a finite automaton MS i accepting all non-overlapping factors from the factor alphabet. Each automaton requires O(n)t i m et obec on st ru c t e d and O(n) space. Thus, overall this step can be completed in linear time and requires linear space.

In Step 5 we construct the finite automaton MS accepting all common subsequences of the strings accepted by MS i . This is achieved by creating the finite automaton MS that accepts the intersection of the languages accepted by automata MS i taken from Step 4. The process of intersecting automata requires quadratic time and is usually done by cartesian product. In [START_REF] Holub | Approximate string matching using factor automata[END_REF], Holub and Melichar present an algorithm for the intersection of factor automata which does not employ cartesian product but uses state marking. Using this algorithm we avoid the creation of all inaccessible states during the automaton construction.

Although the resulting automaton from the algorithm in [START_REF] Holub | Approximate string matching using factor automata[END_REF] contains no inaccessible states as it would have if we had used cartesian product to construct it, nevertheless the time and space complexities of this step are still quadratic relative to the input. For the case of only two factor automata to be intersected, for example L 1 and L 2 having lengths n and m respectively then the state complexity of L 1 ∩ L 2 is O(nm). When we transfer this into a problem with many automata the complexity will become O(n k )-polynomial with n size of texts and exponential with k the number of automata. This is a familiar situation relating to the problem of finding the longest common subsequences of many (> 3) strings using Dynamic Programming which is an NP-complete problem so no better exact algorithm is destined to appear [START_REF] Skiena | The algorithm design manual[END_REF].

Overall, looking over all the steps we can see that the algorithm's time and space complexity is exponential due to the last step that requires the intersection of many finite automata.

6C o n c l u s i o n

We have presented a complete automaton based algorithm to solve the problem of identifying and indexing the common motifs with gaps in a set of strings. The algorithm takes advantage of the fact that one can find common motifs of a set of strings by intersecting their corresponding factor automata which were created by the common factors residing in the strings. Other solutions of the problem require some limit of gaps (fixed gap, bounded gap, bounded sum of gaps). The presented algorithm allows any gaps while keeping the same time and space complexity. Moreover it offers a sound application of finite automata on the problem of finding common motifs with gaps in a set of strings.

  (b) Construct automaton M ε , L(M ε )= r i=1 L(M iε ). (c) By eliminating ε-transitions in M ε we get M F . (d) If M F is deterministic, then strings in S have no common symbol and thus they cannot have a common motif. Set m = 0 and exit the algorithm. (e) Using determinisation of M F we construct M DF while for each state q ′

3 .

 3 Take all factors represented by states in the previous step and create a new "factor alphabet" FA . 4. For each string S i in S construct a finite automaton MS i accepting sequences of all non-overlapping factors from FA . 5. Construct automaton MS accepting all common subsequences of sequences accepted by automata MS i for i ∈ 1,r using the following approach: (a) Add ε-transitions parallel to each transition in each finite automaton MS i ,i ∈ 1,r . The resulting automata will beMS ε i . (b) By eliminating ε-transitions in MS ε i we get MS N i for each i ∈ 1,r . (c) Construct the automaton MS, L(MS )= r i=1 L(MS N i ).(d) Finite automaton MS is accepting all sequences B 1 ,B 2 ,...,B m which are sequences of factors occurring in all strings from set S.

Fig. 2 .

 2 Fig. 2. Transition diagrams of finite automata M1ε and M2ε for the set of strings S = {aabccddab, babbcdacd} from the example

ε 1 and MS ε 2

 2 are shown in Fig. 7 (step 5a). Transition diagrams of finite automata MS N 1 and MS N 2 are shown in Fig. 8 (step 5b of the algorithm).

Fig. 4 .6 1 7 1 6 2 9 2 b 3 1 9 1 1 2 3 2 4 2

 422 Fig. 4. Transition diagram of finite automaton MF from the example

4 1 5 2 4 1 5 1 5 2 8 2 Fig. 5 .

 225 Fig. 5. Transition diagram of finite deterministic factor automaton MD from the example

Fig. 6 .Fig. 7 .Fig. 8 .1 9 2 0 1 0 2 6 1 6 2 9 2 8 1 7 2 3 1 9 1 3 2 4 1 Fig. 9 .

 6782222219 Fig. 6. Transition diagrams of finite automata MS1 and MS2 accepting sequences of non-overlapping factors of both strings from the example

Table 1 .

 1 Repetition

table RT of common factors from the example (F -first occurrence, G -repetition with a gap)

⋆ This research has been partially supported by the Ministry of Education, Youth and Sports under research program MSM 6840770014 and the Czech Science Foundation as project No. 201/06/1039.


Nathan Trouvain
email: nathan.trouvain@inria.frbiographie

Deepayan Das

Xavier Hinaut
email: xavier.hinaut@inria.fr

Xavier Hinaut est chargé de

Nathan Trouvain est doctorant dans l'équipe Mnemosyne également, sous la direction de Xavier Hinaut, depuis octobre 2022. Sa thèse porte sur la modélisation de l'acquisition du langage oral par des mécanismes d'imitation et d'action-perception, à l'aide notamment de techniques de Reservoir Computing. Il est le principal développeur de la bibliothèque reservoirpy, qu'il a conçu alors qu'il était ingénieur contractuel auprès de l'Inria, entre 2020 et 2022.

Catégorie de la proposition

Sprint (jeudi/vendredi)

Titre de la proposition

Amélioration de ReservoirPy, un outil simple de Reservoir Computing

Description de votre proposition

Cette description sera visible sur le programme.

Le Reservoir Computing (RC) regroupe un ensemble de méthodes d'apprentissage artificiel (machine learning) pour les réseaux de neurones artificiels récurrents. Ici, pas de machine learning complexe, seulement des méthodes mathématiques et statistiques simples; mises en oeuvres correctement, elles permettent par exemple de concevoir des outils de reconnaissance automatique de son, de prédiction de mouvement, ou d'interprétation du langage.

reservoirpy est un outil Python conçu pour faire du RC rapidement et simplement. Il est développé pour s'intégrer le plus possible à l'écosystème scientifique Python, en utilisant uniquement numpy et scipy, ses outils de base, et en se fondant à l'univers de scikit-learn, la librairie de machine learning Python standard.

Au cours de ce sprint, nous -Xavier Hinaut, chargé de recherche, Nathan Trouvain, doctorant, et Deepayan Das, ingénieur, tous trois dans l'équipe Mnemosyne de l'Inria et principaux développeurs de reservoirpy -vous proposons de prendre en main cet outil pour tenter de l'améliorer. Ce sprint sera l'occasion pour les participants de pratiquer leur Python au sein d'un projet logiciel scientifique tout en (re)découvrant et (re)pratiquant un peu de machine learning en Python.

Vous aurez l'occasion de participer au projet à plusieurs niveaux. Certains sont très accessibles, même à des débutants en Python ou en machine learning ! 1. Refactoriser, documenter, expliquer

• Amélioration de la documentation (Sphinx, ReStructuredText); Quelques connaissances en machine learning peuvent être un plus, mais rien de nécessaire pour la plupart des développements proposés. Ce sprint pourra justement être l'occasion de s'initier un peu à ce domaine des sciences, ou d'élargir ses horizons en découvrant le Reservoir Computing.

Nous serons ravis de vous guider dans la découverte de notre domaine, et sans aucun doute nous apprendrons également beaucoup de vous !

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