Introduction

Water is an essential component of life since it maintains life (Akintorinwa et al. 2020). According to Arsene et al. (2018) and Srinivasan et al. (2013), these valuable but finite natural resources are generally required for home and industrial applications; without them, production and sustainability are near to impossible. Molden (2007) and Al Sabahi et al. (2009) define groundwater as water enclosed in a geologic substance in the earth's subsurface known as an aquifer. These aquifer units might be weathered layers, voids, cracks, faults, and so forth. Groundwater is a major source of water in Nigeria for home use and agriculture. As a result, groundwater evaluation and acquisition are critical for obtaining portable drinking water, assisting in irrigation operations, and also in enterprises (Yousefi et al. 2018; Ouedraogo et al. 2016). According to Obaje (2009), groundwater occurrence in Nigeria's basement complex is relatively low and irregular, owing to the prevalence of abrupt discontinuities in features and faults responsible for groundwater accumulation, as well as low aquifer overburden (Olabode 2019; Anbazhagan et al. 2011; Chandra et al. 2008). Because of the nature of the crystalline basement rocks underlying the hard rock terrain, these discontinuities are frequently characterised by low porosity and low permeability of fluids percolation, which causes localisation of groundwater potential zones and scattered presence of pockets of groundwater accumulation present in the basement complex terrain, making groundwater occurrence generally low and irregular (Olabode 2019; Ajayi 2017; Akintorinwa 2014; Odeyemi et al. 1985). According to Ajayi (2017) and Mogaji et al. (2011), few crystalline rocks, such as schist, worn quartzite, gneiss, and so on, have exhibited great groundwater potential, and their aquifer units often serve as a conduit to the natural resource in huge quantities. Geophysical surveys are one of the most commonly used methodologies for evaluating groundwater potential zones since the conduit medium cannot be seen immediately on the earth's surface (Braga et al. 2006). Several researchers have done groundwater hydrology studies, particularly in hard rock terrain, with promising results, including Omosuyi (2010), Mogaji et al. (2014), Oladunjoye et al. (2019), and Akintorinwa et al. (2020).

In the realm of groundwater hydrology, the use of geographic information system (GIS) and remote sensing (RS) techniques has proven to be good in spatial analysis related hurdles (Mogaji et al. 2021; Rahmati and Meselle 2016; Rahmati et al. 2015; Manap et al. 2014; Nasiri et al. 2012). This is because satellite database archives provide convenient and rapid access to data, saving time and money in analysis (Zare et al. 2013). In prior study in this subject, the combined use of GIS, RS, and multicriteria decision analysis (MCDA) functioned as a superior decision support system in demarcating groundwater potential zones (Ajaykumar et al. 2020; Mogaji and Lim 2016; Al-Abadi 2015). MCDA approaches have been used to give weights to index parameters from various research (Arshad et al. 2020; Mogaji and Lim 2016; Adiat et al. 2012; Chowdhury et al. 2009). The Analytical Hierarchy Process is one of the most extensively utilised MCDAs (AHP). Based on fundamental mathematics and psychology, the AHP model is a systematic approach for organising and evaluating complicated decisions (Saaty 1987). It is a measuring theory based on pairwise comparisons that depends on expert opinion to establish priority scales. Yet, due to the model's subjectivity, the assessment may be inconsistent and prejudiced. Yet, it has been successfully applied in the subject of groundwater hydrology (Mogaji and Omobude 2017; Akinlalu et al. 2017; Adeyemo et al. 2017). Because of AHP's subjectivity and biases, academics have looked ahead and moved to research models that are devoid of human or expert input or viewpoints. They are also known as objective models or data-driven approach models. These models include frequency ratio (Arshad et al. 2020; Pourtaghi and Pourghasemi 2014; Elmahdy and Mohammed, 2014; Ozdemir 2011; Oh et al. 2011), artificial neural networks (Adiat et al. 2019; Lee et al. 2012; Corsini et al. 2009), weights of evidence (Chen et al. 2018; Pourtaghi and Pourghasemi 2014; Lee et al. 2012), maximum of entropy (Rahmati et al. 2015), preference ranking organisation method for enrichment evaluation (PROMETHEE-II) (Widianta et al. 2018; Roodposhti et al. 2012), and evidential belief functions (Pourghasemi and Beheshtirad 2015; Nampak et al. 2014; Mogaji et al. 2014). These data-driven techniques employ mathematical algorithms with known well/spring locations as dependent variables, as well as other known factors, to assess groundwater occurrence in a certain geo-spatial setting (McKay and Harris 2015). The weights of the parameters evaluated for groundwater zonation of the research region were calculated using the entropy technique of weightage calculation in this study.

According to Lin and Weng (2009) and Wang et al. (2020), the entropy technique is an object (non-expert) driven strategy that comprises computing the entropy and entropy weight to determine the weight values of specific indicators. Entropy is inversely related to entropy weight, therefore the lower the entropy, the higher the associated entropy weight. The quantity of meaningful information provided by the target to the decision-maker is lowered. Many scholars in several branches of science have employed entropy extensively in the literature (Al-Abadi et al. 2017; Al-Abadi and Shahid 2016; Li et al. 2012; Safari et al. 2012; Qi et al. 2010).

Brans (1982) created the PROMETHEE-II algorithm, which was further upon by Vincke and Brans (1985). PROMETHEE-II employs a full ranking approach on a limited set of possibilities, ranging from best to worst. This approach compares options pairwise along each identified criterion and takes into account the preference function of criteria over others, which tends to provide superior results (Abdullah et al. 2019; Nasiri et al. 2012; Roodposhti et al. 2012). This approach has been widely employed in a variety of disciplines of study, including banking, water resource management, investments, healthcare, tourism, the industrial sector, dynamic management, and so on (Widianta et al. 2018; Roodposhti et al. 2012). This approach has been widely employed in a variety of disciplines of study, including banking, water resource management, investments, healthcare, tourism, the industrial sector, dynamic management, and so on (Widianta et al. 2018; Roodposhti et al. 2012).

The PROMETHEE-II approach based on entropy weight was employed in this study to assess the groundwater potentiality index and rating of the study region. The creation of these combined MCDA techniques was prompted by the combination of the properties of both model algorithms. To begin, the PROMETHEE-II considers the preference functions of each criteria to enhance decision-making regarding the outcome. Furthermore, the entropy technique employs a data-driven strategy in determining the weights of criteria, making this task object-driven and devoid of human input in producing the final algorithm index. Additionally, in the realm of groundwater hydrology, the coupling of these model techniques with GIS and RS has not been used to groundwater potential mapping. Lastly, because of its robust libraries and high-speed computing capabilities, the model algorithms' procedures were calculated using the Python programming language (Butwall et al. 2019). These integrated methodologies were used to estimate the groundwater potentiality of the study region using geoelectrical and remote sensing-derived metrics such as lithology (L), hydraulic conductivity (K), lineament density (Ld), transmissivity (T), and transverse resistance (TR). The methodology was demonstrated in typical multifaceted geologic settings.

Methodology, study area, and data used

Methodology

As indicated in Fig. 1, the study technique was carried out in four stages, employing geology, remote sensing, and geophysical data. The initial phases required collecting, processing, and interpreting remote sensing, geology, and geophysical characteristics for this study's groundwater potential evaluation. Following that, thematic maps of the conditioning factors were created in ArcGIS, and uniformly spaced fishnet (observation) points were placed to extract pixel values at those points. Furthermore, the Python-based entropy method of weighting was used on the fishnet points, as well as the PROMETHEE-II algorithm, producing the PROMETHEE-Entropy-based groundwater potential index (GPI), which was synthesised in the GIS environment to produce the groundwater model map of the study area. Finally, the suggested model was validated using borehole data and a reflection coefficient map of the research region. Figure 1 depicts the process flowchart.

Fig. 1
figure 1

The adopted methodology flowchart for the study

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Description of the study area

The research area is in northern Ondo state, Nigeria, and includes the local government areas of Akoko south-west, Akoko south-east, Akoko north-west, Akoko north-east, and Ose, as indicated in Fig. 2. The region is designated by the Minna-Nigeria 31N datum of the UTM (Universal Traverse Mercatum) system and extends from longitude 5°30″E to 6°0″E and latitude 7°10″N to 7°45″N. It is surrounded to the west, north, and east by Ekiti, Kogi, and Benin states, respectively, and to the south by other sections of Ondo state. The overall area covered by the research area is approximately 1492.55 square kilometres. The terrain in the research region is relatively low to highly undulating, with surface elevations ranging from 162 to 1500 m above sea level. Ikare and Oke agbe, located in the northern half of the research region, have high elevations characteristic of dispersed batholiths, but communities to the south, such as Idoani, Oba, and Idosale, have low-lying rock exposures (Olabode 2019; Oyedotun and Obatoyinbo 2012).

Fig. 2
figure 2

Location maps showing: a map of Ondo State and Nigeria and b study area map displaying the data acquisition points and other features

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The environment is hot and humid, with rain-bearing south-west monsoon winds blowing in from the Sahara Desert. The rainy season lasts from April to October, with repeated maxima occurring from August to October, with annual rainfall ranging from 1500 to 2000 mm. The average annual temperature is roughly 23–26 °C (Duze and Ojo 1982), with a relative humidity of 75–95%. Forest-savannah vegetation is the natural vegetation.

The research area is underlain by the Precambrian basement complex of Southern Nigeria (Rahaman 1988). The geological units identified in the region include granite, migmatite gneiss, and quartzite. More than 70% of the area is underlain by migmatite gneiss. Figure 3 shows a geological map of the area.

Fig. 3
figure 3

Geologic map of the study area (Modified after Geological and mineral resources map of Ondo State) Nigeria, 2006)

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Data used

Lithology

The lithologic map of the research area was derived from Ondo State's regional geology and mineral resources map (NGSA 2006). After georeferencing the scanned copy of the regional map in an ArcGIS environment, the shape file of the research area was extracted from the state map to generate the lithological map of the study area shown in Fig. 3.

Remote sensing

The remote sensing techniques employed in this study include lineament extraction from LANDSAT 8 images of the subject region. In the year 2020, this image was retrieved from the United States Geological Survey (USGS) Earth Explorer with route 189 and row 055. Visual interpretation of both LANDSAT 8 photos was used to identify the lineaments (Fig. 3). To extract the area of interest (AOI), a 432 (RGB) false-colour composite band combination was employed, followed by principal component analysis (PCA), band rationing, directional filtering, and edge enhancement using PCI Geomatica 2012. These lineaments were identified, photographed, scanned, and placed on a lithologic map (Fig. 3).

Lineaments are typically examined using frequency or length against azimuth histograms (Zakir et al. 1999; Mostafa and Zakir 1996), rose diagrams (Karnieli et al. 1996), and/or lineament density maps (Zakir et al. 1999; Mostafa and Zakir 1996). (Zakir et al. 1999). The most popular way is to construct density maps of lineaments (Zakir et al. 1999). Equation (1) expresses the Ld definition mathematically:

$${\text{Ld}}=\sum \limits_{i=1}^{i=n}\frac{{L}_{i}}{A}\left({\text{km}}^{-1}\right),$$
(1)

where \(\sum Li\) = total length of all the lineaments (km) and A = area of the grid (km2).

The lineament density map is shown in Fig. 4 below.

Fig. 4
figure 4

Lineament density thematic layer for the study area

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Geophysical investigation

Data acquisition and interpretation

The geophysical data in the study region were collected using the Schlumberger array of electrical resistivity techniques. The array was utilised to collect 118 vertical electrical soundings (VES) data from 1 to 100 m utilising half-electrode spacing (AB/2). This approach takes into account vertical differences in the apparent resistivity of the ground, which were measured with a fixed centre of the array. The survey was carried out by progressively expanding or increasing the electrode spacing around a fixed centre of the array. Normally, electrodes are positioned in a straight line, with a pair of potential electrodes placed between two pairs of current electrodes. In this work, the Global Positioning System (GPS) was utilised to spatially identify VES sites for spatial analysis in a GIS setting. The apparent resistivity of the VES data is the product of the resistance and the matching geometric factor (G) of the electrode spacing for each spread length (AB/2). On a log–log graph sheet, these apparent resistivity values were plotted against the electrode spacing. The VES curves that were generated were displayed and divided into types. These classifications demonstrate the qualitative character of subsurface lithology. In addition, quantitative interpretations of the partial curve matching findings, which are the layer thickness and layer resistivity, were determined. The results were entered into the WinResistTM Software as model parameters (Vander-Velper 2004). The theoretical model curve, the primary geoelectric parameters (layer resistivity, layer thickness), and the depth to the top of each layer provide good insight into the aquifer's subsurface information, which is vital for groundwater potential research. Figure 5 and Table 1 show typical curves depending on underlying geology and a summary table representation of geoelectric characteristics.

Fig. 5
figure 5

Typical resistivity model curves obtained in the study area; a migmatite gneiss, b quartizite series and c granite gneiss rock unit

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Table 1 Summary of the interpreted geoelectric parameters
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The derived secondary geoelectric parameter

The primary geoelectric parameters, layer resistivity and layer thickness, were utilised to determine the secondary geoelectric parameters, which are important conditioning variables in delineating groundwater potential zones in the research region. The validation method takes into account hydraulic conductivity (K), transmissivity (T), transverse resistance (TR), and reflection coefficient (Rc). The primary geoelectric characteristics in Table 1 were reanalysed using Eqs. 2–5 to generate the aforementioned groundwater conditioning factors. Table 2 displays the values of the calculated K, T, TR, and Rc parameters.

Table 2 Summary of the derived secondary geoelectric parameters
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Hydraulic conductivity, K (m/day), is given by:

$$K \, = \, 0.0{\text{538e}}^{{ - 0.00{72}\rho }},$$
(2)

where ρ = aquifer resistivity

$${\text{Transmissivity}}, \, T \, \left( {{\text{m}}^{{2}} /{\text{day}}} \right) \, = {\text{ Kh}},$$
(3)

where, K = hydraulic conductivityh = aquifer thickness

$${\text{Transverse resistance}},{\text{ TR }}\left( {\Omega {\text{m}}} \right) \, = h_{n} *\rho_{n} = \, h_{1} *\rho_{1} + \, h_{2} *\rho_{2} + \, \ldots \ldots \ldots . \, + \, h_{n} *\rho_{n},$$
(4)

where n = number of layers overlying the aquifer unith = layer thickness, ρ = layer resistivity

$$\mathrm{Reflection \; coefficient},\mathrm{Rc }= \frac{{\rho }_{n}-{\rho }_{n-1}}{{\rho }_{n}+{\rho }_{n-1}},$$
(5)

where ρn is the layer resistivity of the nth layer and ρn-1 is the layer resistivity overlying the nth layer.

Selection of groundwater potentiality conditioning factors (GPCF) and production of their thematic layers in the GIS environment

The groundwater potential of the research was evaluated using five (5) factors: lithology, hydraulic conductivity, lineament density, transmissivity, and transverse resistance. Thematic maps of these factors were generated using ArcGIS 10.3's inverse distance weighting (IDW) approach, and data from Table 2 were utilised to develop the geo-electrically linked thematic layers displayed in Figs. 6, 7, and 8.

Fig. 6
figure 6

Hydraulic conductivity map of the study area

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Fig. 7
figure 7

Transmissivity map of the study area

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Fig. 8
figure 8

Transverse resistance map of the study area

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The groundwater potential conditioning factors (GPCF) thematic maps shown in Figs. 2, 4, 6, 7, and 8 were created to model the groundwater potential of the research region. These layers were used as input parameters for developing the suggested PROMETHEE-Entropy model method in Python. Table 3 was created for ease of computation by guaranteeing uniformly dispersed fishnet points (Fig. 9) over the study area. Akintorinwa et al. (2020), Olabode 2019, Alhassan et al. (2015), and Nasiri et al. (2012) have highlighted the hydrological relevance of the GPCF for the suggested model method (2012).

Table 3 Decision matrix for the groundwater potential conditioning factors
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Fig. 9
figure 9

Fishnet template map of the study area

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Models review

The entropy method

According to Lin and Weng (2009), the entropy method is an object-driven (non-expert) significance approach that involves determining the weight values of individual indicators by calculating the entropy and entropy weight (Qi et al. 2010). The method is based on the idea of discreet probability distribution where uncertainty is depicted with broad distribution (Zou et al. 2006) and since entropy is the measure of a system’s disorder nature, it can be used to extract useful information from a given data (Al-Abadi et al. 2016). The difference in the values of the evaluating objects on a criteria has a direct effect on the entropy, which is inversely proportional to the entropy weight. Many scholars in several branches of science have employed entropy extensively in the literature (Al-Abadi et al. 2017; Al-Abadi and Shahid 2016; Li et al. 2012; Safari et al. 2012; Qi et al. 2010).

Shannon proposed the notion of the index of entropy first (1948). The approach allows us to calculate the weights by assuming that the issue has m viable alternatives, Ai (i = 1,2,…,m), and n assessment criteria Cj (j = 1,2,…,n) (Aysegul and Esra 2017). The first step is to create the choice matrix (Eq. 6). The essence is that it compares the performance of several alternatives based on many parameters.

$$X = \left(Xij\right)_{ m\times n} =\left(\begin{array}{ccc}{X}_{11}& {X}_{\begin{array}{c}12\\ .\\ .\\ .\end{array}}\dots & {X}_{1n}\\ {X}_{2n}& {X}_{\begin{array}{c}2n\\ .\\ .\\ .\end{array}}\dots & {X}_{2n}\\ {X}_{m1}& {X}_{m2}\dots & {X}_{mn}\end{array}\right)$$
(6)

for i = 1,2…,m; j = 1,2,…,n, where Xim = feasible alternatives, Xjn = evaluation criteria, m = number of alternatives, n = number of criteria

After the preceding steps, the decision matrix is normalised by dividing each criterion value (Xij) by the total arithmetic column sum of the criteria. The normalised equation is written as Eq. (7):

$$r_{ij} = \frac{{x_{ij} }}{{\mathop \sum \nolimits_{i = 1}^{m} x_{ij} }}i{ } = { }1,2 \ldots ,m;{ }\; j{ } = { }1,2, \ldots ,n.$$
(7)

The entropy values of each criterion are then calculated. After the above stages, the entropy values (ej) (as in Eq. 8) are computed.

$$e_{j} = - h\mathop \sum \limits_{i = 1}^{m} r_{ij} \in r_{ij} {\text{ i }} = { }1,2 \ldots ,m; \; { }j{ } = { }1,2, \ldots ,n,$$
(8)

where \(h=\frac{1}{m}.\)

Then entropy weight (wj), Eq. 9, is calculated:

$${w}_{j}=\frac{1-{e}_{j}}{n-{\sum }_{i=1}^{m}{e}_{j}},$$
(9)
$$\mathrm{where} \sum \limits_{j=1}^{n}{w}_{j}=1.$$
(10)
Entropy Python class

The entropy weights for each groundwater potential conditioning factor (GPCF) studied in this study were computed using a Python programming class attribute. After embedding the Python-based entropy function into a class property, the appropriate libraries were imported, input variables were assigned, and the table was imported as a csv file. The normalised weights of the GPCFs computed using the Python-based entropy weighting approach are shown in Table 4. The entropy technique codes are presented in the Appendix.

Table 4 Calculated entropy values, degree of divergence and entropy weights for groundwater potential conditioning factors
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PROMETHEE-II

The PROMETHEE-II technique is a comprehensive outranking flow method created by Brans (1982) and later extended by Vincke and Brans (1985). This approach is one of the PROMETHEE methods created to help decision-makers rate a finite number of choices partially or entirely (Nasiri et al. 2012; Roodposhti et al. 2012). The preference function of each criterion versus other criteria is considered, resulting in a smooth and robust decision-making process when employing PROMETHEE-II (Nasiri et al. 2012).

PROMETHEE-II is based on a pairwise analysis of options for each criterion. The procedure consists of four phases, which are highlighted below.

Determination of deviations based on pairwise comparisons by computing the desired degree value for each pair of possible decisions and each criterion is stated in Eq. (11). The preference function is used in the way shown in Eq. (12):

$${d}_{j}\left(a,b\right)={g}_{j}\left(a\right)+{g}_{j}\left(b\right)$$
(11)
$${P}_{j}\left(a,b\right)={F}_{j}\left[{d}_{j}\left(a,b\right)\right] j = 1,\dots ,k,$$
(12)

where gj(a) and gj(b) show the performance (value of a criterion j) of alternatives a and b (for a decision a or decision b), respectively.dj(a,b) = difference of the performance (value of a criterion j for two decisions a and b).

Pj (a, b) is the preference of alternative a regarding alternative b on each criterion, as a function of dj (a, b).

Aggregation of all criteria's preference degrees for each pair of potential options, followed by computation of the global preference index for each of these possible decisions, is indicated in Eq. (13):

$$\forall a,b \in A,\;\pi \left( {a,b} \right) \, = \sum\limits_{j = 1}^{k} {P_{j} } \left( {a,b} \right)w_{j},$$
(13)

where π(a, b) is the preference of an over b (from zero to one) and wj is the weight associated with jth criterion.

Calculation of the outranking flows (positive and negative outranking flow) is shown in Eqs. 14 and 15.

$$\phi^{ \mp } (a) = \frac{1}{n - 1}\sum_{x \in A} {\Pi \left( {a,x} \right)}$$
(14)
$$\phi^{ - } (a) = \frac{1}{n - 1}\sum_{x \in A} {\Pi \left( {x,a} \right)},$$
(15)

where ϕ+(a) and ϕ− (a) are positive and negative outranking flows for each alternative, respectively.

Calculation of complete outranking flow index is shown in Eq. 6.

$$\phi \left( {\text{a}} \right) \, = \phi^{ + } \left( {\text{a}} \right) \, - \phi^{ - } \left( {\text{a}} \right),$$
(16)

where ϕ(a) is the net outranking flow for each alternative.

PROMETHEE-II Python class

In this study, the PROMETHEE-II steps were implemented utilising the class feature of the Python programming language. The constructor, which initialises the class object, and the net outranking flow method are included in the Python PROMETHEE-II class. The class feature was activated when the required libraries were imported. The following Python programmes were used to compute the PROMETHEE-net II's outranking flow.

Preparation of the proposed PROMETHEE-Entropy model for groundwater potential mapping

The object-oriented prowess of the PROMETHEE-Entropy algorithm was a key consideration in this study, which was accomplished by modelling five groundwater potential conditioning factors (GPCFs): lithology (L), hydraulic conductivity (K), lineament density (Ld), transmissivity (T), and transverse resistance (TR). The number of criteria (GPCF), number of alternatives (fishnet points), weights of each criterion, and types of criteria were among the input parameters utilised in constructing the Python-based PROMETHEE-Entropy algorithm. The algorithm procedures are divided into two phases. The first is the computation of the objective weights of each criterion using the entropy approach, which was accomplished using the previously stated Python-based entropy algorithms. These weights were then supplied into the PROMETHEE-II class along with other input parameters. The criteria were divided into two types: beneficial and non-beneficial criteria, denoted numerically by 1 and − 1, respectively. (Table 5). Beneficial criteria are those in which the maximum values are preferred, whereas non-beneficial criteria are those in which the minimum values are preferred. The net outranking flow index of this study was calculated using the combined Python-based entropy and PROMETHEE-II classes.

Table 5 Ratings, groundwater storage potential type-classification and entropy weight of the GPCF produced thematic layers
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Results and discussion

Discussion of groundwater potential conditioning factors

As shown in column 3 of Table 5, groundwater potential conditioning factors (GPCFs) have a significant importance to groundwater accumulation in the aquifer. The relevance is discussed further below.

Lithology

Lithology is a significant component that affects groundwater accumulation in an ecosystem, particularly in terms of quantity. There are three primary geological units in the research area: Migmatite gneiss, Granitic gneiss, and Quartzite Schist (Fig. 3). As compared to other rock groups in the study region, Quartzite Schist exhibits the greatest degree of fracture (Olabode 2019). The amount and potential of groundwater in the research region is directly related to the degree of weathering of the rocks in the study area. Olabode (2019) assigned the following values to each of the lithologic units in Table 3: magmatite gneiss (17), granite gneiss (29), and quartzite schist (57). For migmatite, granite, and quartzite, the area extents are 1096.91, 108.54, and 287.1 km2 correspondingly. The Migmatite gneiss rock unit covers over 75% of the study region. The majority of the towns in the north, north-central, and north-west are underlain by Migmatite gneiss rock units, whereas the centre and southern parts of the study region are underlain by Quartzite Schist rock units, and the south-east is underlain by Granite gneiss rock units.

Lineament density

Lineament is defined as observable geomorphic linear features typified by weak zones with characteristics of fissures/joints, fractures, and possibly weathered formation that can be attributed to geological structures, most notably fractures or lithologic contacts (Chowdhury et al. 2009 and O'Leary et al. 1976). The distribution of lineament (Fig. 4) and density of lineament (Fig. 5) demonstrate that over 95% of the research region is underlain by lineament at a very low to low density. As a result, the region may be classified as having low to medium groundwater potential. Nevertheless, the zones with significant groundwater potential in the studied region are attributable to secondary porosity and permeability, which are also characterised by high lineament density (Olabode 2019). According to the lineament density map (Fig. 5), the geographical distribution of the lineament density of the study region is separated into five zones (very low, low, medium, medium–high, and high), with area extents of 965.25, 484.05, 25.25, 9.05, and 8.95 km2 correspondingly.

Hydraulic conductivity

This refers to an aquifer's physical capacity to transfer water through pore spaces and cracks when subjected to a hydraulic gradient (Diminescu et al. 2019). According to Table 3, the hydraulic conductivity, K, of the research region varies from 0.5 to 12.5 m/day. The resulting thematic map (Fig. 7) depicts the regional variance of hydraulic conductivity in the research area. The area extend coverage of the categorised zones is 38.55, 387.25, 428.25, 324.25, and 268.25 km2 correspondingly. According to Okogbue and Omonona (2013), Lee et al. (2012), and Tizro et al. (2010), groundwater potentiality is directly related to an aquifer's K rating. The K parameter greatly influences the variability of groundwater potential from location to location. According to the areal extent computation, about 68% of the research area is underlain by medium to high hydraulic conductivity zones, whereas 32% is inhabited by relatively low and low hydraulic conductivity zones. It implies that the aquifer units in the area have a high groundwater potential in general. Unfortunately, the K parameter cannot alone confirm this assessment of the area's groundwater potential.

Transmissivity

The ability of water to pass through pores spaces of a unit width of the aquifer under a unit hydraulic gradient is referred to as aquifer transmissivity (Salem 1999). This means that aquifer units with high transmissivity have high fluid permeability across their pore spaces. The transmissivity of the research region ranges from 0.02 to 33.3 m3/day, as shown in Table 3. The resulting aquifer Transmissivity map (Fig. 8) depicts the geographical variance of the research area's transmissivity, which is divided into five zones: very low, low, medium, medium–high, and high. According to the characteristic shown in Fig. 8, the research region has typically poor transmissivity, with low groundwater potential zones covering almost 71% of the area. Many communities in Akoko's south-west, east, and north-west, including Ora Ojora, Ikeram, Yaya, Ishe, Isua, Igbe, Ikare-Akoko, and Oba, are within the medium to high transmissivity zones.

Traverse resistance

Figure 9 depicts the total transverse resistance of the research region, which varies spatially from 24 to 5414 m (Table 3). Figure 9 depicts the generated TR map, which is divided into five zones (high, medium–high, medium, low, and very low zones). According to the produced thematic map, Akoko north-east and north-west have primarily low transverse resistance, which might be suggestive of possibly low groundwater potential zones in the area, whereas towns like Ikarram, Ijora, and Efifa may be experiencing the reverse. This situation of low transverse resistance also applies to other places where towns such as Oba, Ikun, and Afo are expected to face a mixed destiny due to limited groundwater potential. This submission is consistent with the findings of Adiat et al. (2012), who found that high transverse resistance indicates high permeability of water through the aquifer's overlaying strata, implying that TR is directly proportional to an area's groundwater potential.

The PROMETHEE-Entropy index application results

The data-driven ability of the suggested PROMETHEE-Entropy algorithm to get an object-oriented output is the primary emphasis of this work, which was accomplished by following the flowchart methodology in Fig. 1. The initial step is to extract the fishnet point pixel values from each GPCF (Table 3). This table functioned as the decision matrix for the following phase of the process, which required determining the entropy weightage. Table 4 shows the weights of each GPCF after passing the matrix through the Python-based entropy weight class. Table 5 displays the ratings, classifications, criterion type, and potential of GPCFs values to groundwater potential. The details from Tables 4 and 5 were used as input into the PROMETHEE-II class to compute the PROMETHEE-Entropy index, and the implementation codes are presented in the Appendix.

Table 6 displays the resultant data frame from above, and higher values indicate fishnet sites with more groundwater potential, while lower index values indicate fishnet points with lower groundwater potential.

Table 6 Computed model results of PROMETHEE-Entropy class algorithms
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Groundwater potential PROMETHEE-Entropy model map of the study area

The PROMETHEE-Entropy method, which is written in Python, produced the net outranking flow (groundwater potential index), as shown in Table 6 column 6. The groundwater potential model map of the study area was generated using the computed net outranking flow. The map obtained using the index is shown in Fig. 11. Using the Natural Breaks Method, the generated map was divided into five (5) groundwater potential zones (Jenks 1967). The study area's geographic variation of the groundwater potential index is split into five zones: extremely low, low, medium, medium–high, and high. According to a further analysis of the generated GPM, the area extents of the very low, low, medium, medium–high, and high classes are 442.1, 629.1, 273.5, 122.5, and 25.35 km2, respectively (Table 7). The chart shows that the majority of the towns in the Akoko north-east and north-west lie within the very low to low groundwater potential zones, with the exception of a few towns in the north-eastern section of the study, which include Ikeram, Efifa, and Ora-Ojora. Furthermore, the majority of the remaining areas of the research fell mostly within the extremely low to low groundwater potential zones.

Table 7 Characteristics of the groundwater potentiality model (GPM) map based on PROMETHEE-Entropy model
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Validation of model

The PROMETHEE-Entropy GPM map was validated by two means: qualitative and quantitative. The resulting GPM map was qualitatively confirmed using accessible records of borewell yield data from the study area. The Ondo State Water and Sanitation Department provided these scanty borehole yield statistics (WATSAN). According to Mogaji (2016), the area actual yield descriptions of 0.2 l/s, > 0.2 l/s, 1 l/s, > 1 l/s, 1.5 l/s, and > 2 l/s are for the very low rate, low rate, medium rate, and high rate, respectively (Table 8). These were utilised to assess the prediction maps' percentage accuracy, which is based on the comment of coincide and not coincide between the predicted yield descriptions from the prediction map (Fig. 10) and the actual borehole yields descriptions. This analysis is as follows:

Table 8 The groundwater potentiality model map validation result based on in situ borehole yield data
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Fig. 10
figure 10

The groundwater potentiality model (GPM) map of the study area based on PROMETHEE-Entropy approach

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Total number of boreholes available = 7.

Number of boreholes where the expected and the actual yield classifications coincide = 5

Number of boreholes where the expected and the actual yield classifications do not coincide = 2

Success rate (accuracy) of the prediction = \(\frac{5}{7}\times 100=71\text{\%}\)

The quantitative validation, on the other hand, was carried out using the Reacting Operating Characteristics (ROC) curve technique, in which the confusion matrix (ROC) was applied to the actual data, i.e. the derived Reflection coefficient parameter, and the predicted data (Groundwater potential index, GPI) of the study area. The reflection coefficient (Rc), Fig. 11, parameter as a groundwater potentiality indicator is significant because it provides an in situ evaluation of the rate of fracture that an area bedrock has undergone. This Rc feature, according to Ariyo et al. (2011) and Alhassan et al. (2015), will frequently improve aquifer unit rechargeability through the fracture conduit medium and therefore enable groundwater accumulation. The AUC (Area under the curve) of the PROMETHEE-Entropy GPM index map was calculated using the ROC confusion matrix. The True Positive Rate (TPR) and False Positive Rate (FPR) were calculated for the determined Reflection coefficient parameter values of the study area, as shown in Table 9, by first marking the cut-off points on the GPM map index (Table 10) and binary (True (1) or False (0)) cut-off points, with values between 0.5 and 0.87 as True and 0.87 and 0.99 as False (Table 2). The predicted groundwater potential index (GPI) values were extracted from the GPM map at various cut-points and compared with the actual reflection coefficient (Rc) values using Table 9 criteria. The Rc values were compared to the appropriate GPI values to calculate the True Positive (TP), False Positive (FP), True Negative (TN), and False Negative (FN). Equations 17 and 18 were used to get the TPR (Sensitivity) and FPR (1-Specificity). The area under the curve (AUC) was computed after the ROC curve (Fig. 12) was displayed using the relevant Python libraries. The AUC value indicates the accuracy of the PROMETHEE-Entropy groundwater potential map.

Fig. 11
figure 11

Reflection coefficient map of the study area

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Table 9 Confusion Matrix application in generating the ROC curves
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Table 10 Groundwater Potential cut-off points
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Fig. 12
figure 12

ROC curve for the PROMETHEE-Entropy model map

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$$\mathrm{Sensitivity }=\mathrm{ TPR }=\frac{\mathrm{FP}}{\left(\mathrm{TN}+\mathrm{FP}\right)}$$
(17)
$$1-\mathrm{ Specificity }=\mathrm{ FPR }= \frac{\mathrm{TP}}{\left(\mathrm{FN}+\mathrm{TP}\right)}$$
(18)

The AUC value of the prediction rate for the PROMETHEE-Entropy-based model is 0.86 prediction accuracy based on the findings of this investigation, as shown in Fig. 12. According to Naghibi et al. (2014), the following AUC classes correspond to the prediction accuracy of each model map as reported: 0.9–1.0 (excellent), 0.8–0.9 (very good), 0.7–0.8 (good), 0.6–0.7 (average), and 0.5–0.6 (poor). Given the categorisation above, it is possible to infer that the PROMETHEE-Entropy-based model performs "very good."

Conclusion

The use of object-driven multicriteria decision analysis (MCDA) techniques on geophysical, geological, ancillary, and remote sensing parameters in establishing groundwater resource sustainability management in typical hard rock geologic settings in south-western Nigeria was investigated in this study. The object-oriented PROMETHEE-Entropy MCDA approach was applied to 114 fishnet (observation) points distributed over the study area's generated thematic maps. Thematic maps have been generated by using GIS on processed data from Schlumberger VES data, remote sensing, and auxiliary data. The Python programming language was utilised to compute the model algorithms employed in this study for ease of calculation. This is due to Python's flexibility and high-level computational skills as a result of the existence of vast libraries and other resources.

The groundwater potential conditioning factors (GPCFs) of the study area were weighted using the entropy approach. Lineament density, hydraulic conductivity, transverse resistance, transmissivity, and lithology make up the GPCFs studied. The result of the Python-based entropy weighting approach reveals that lineament density has the greatest impact on the groundwater potential of the study area, whereas hydraulic conductivity has the least weight and influence on groundwater storage potentiality. The PROMETHEE-II was then utilised to rank the alternatives using Python programming by calculating the net outranking flow of each alternative. The groundwater potential index (GPI) of the study area was computed using the net outranking flow. The PROMETHEE-Entropy model map of the study area was generated using the calculated GPI values in a GIS context. According to the groundwater potential model (GPM) map, 72% of the region has low groundwater potential, while the rest has medium to high groundwater potential. As a result, the aquifer units in the area are largely unproductive and have low groundwater potential.

The PROMETHEE-Entropy model map generated in Python was evaluated and compared using qualitative and quantitative validation methods. The qualitative validation was carried out using borehole pumping test data from the study area, and the PROMETHEE-Entropy success rate was 71%. The ROC curve was also used for quantitative validation, and the AUC for the PROMETHEE-Entropy method was 86%. The above validation findings indicate that the object-oriented PROMETHEE-Entropy-based groundwater potential model (GPM) is of good accuracy. The developed PROMETHEE-Entropy-based GPM tool may be used as part of a decision support system by water resource stakeholders to improve the development and management of groundwater resources in the investigated area and other places with similar geologic settings throughout the world.