Introduction

Geo-electrical resistivity technology, which deploys vertical electrical sounding (VES) and electrical resistivity tomography (ERT), has been employed by many authors in shallow/intermediate depth investigation for purposes such as hydrogeology, environment, archeology, engineering and mining surveys (Bandani 2011; Ibanga and George 2016; Mohammed et al. 2021; Ekanem et al. 2022; Ikpe et al. 2022). The direct current inverse resistivity problem is usually ill-posed with respect to incompleteness and uncertainties in datasets. In an attempt to make the solution well-posed, regularization schemes must be incorporated in the inverse problem to find a unique and stable solution (Binley et al. 1995; Udosen and George 2018a). Although regularization plays a substantial role in the problem of inverse theory, there is a vast ambiguity in choosing the regularization schemes (Ghanati et al. 2021). In view of the attendant ambiguity in inverse model and the imagined effect on its applications, mostly when it is used alone in the area with no ground truthing information, integrated methodologies such as direct (ISM) technique are needed. Variability of resistivity in the subsurface is often viewed to take a nonlinear dimension during the numerically iterative techniques of least squares inversion or during the process of finding the resistivity model with goodness of fits in the entire sequence of quadrupole measurements of apparent resistivities (Tripp et al. 1984). In minimizing the ambiguity in measurements, real resistivities are often considered when referring to the unknown reality. Computed resistivities were assumed when referring to resistivities resulting from inversion and apparent resistivities are construed when both real and computed resistivities unfold the ones resulting from direct voltage measurements of assumed homogeneous media. In electrical resistivity measurement technology, vertical electrical sounding (VES) and electrical resistivity tomography (ERT) are ill-posed/ill-constrained or undetermined inverse problems in as much as their solutions may be unstable or non-unique. Based on this realization, it is often ideal to couple the interpretation of VES and ERT data with depth dependent geological equivalence in areas where these geological data are available. Besides, the geological data of the subsurface are localized and the deductions may not be a generalized solution for areas that are remote from the sources of geological information. Therefore, comparison, calibration, validation and correlation of methodologies in phenomenological interpretation of non-unique geo-resistivity data are imperative for results that are believed to be geologically consistent in small and large scale geophysical and geological measurements. As a good practice, applying two or more complementary methods of interpretation in many a time can reduce the high degree non-uniqueness. Despite verifying the efficacy of the inverse slope method (ISM) for the interpretation of electrical sounding data in order to determine the primary geo-electric indices, the main aim of this research is to compare the direct/forward modelling with the ill-posed iterative inversion (inverse modelling), which relies on the mathematically theoretical curves to generate the primary geo-electrical indices. In order to compare the interpretation of geo-electric data using the LSIT with the inverse gradient method or technique (ISM/IGT), ground truthing information were required as real data to see the degree of convergence/divergence of independent interpretations of VES data using ISM and LSIT. ISM has unique quality of generating earth resistivity indices without requiring theoretical models like the conventional inverse methods of curve matching do (Ghanati et al. 2021). Bouadou et al. (2019) opined that findings from field survey and laboratory experiment showcased that the inverse gradient method has the potential to be employed in VES data analysis acquired with any electrode array. This unique quality is not accessible with all the other conventional analytical procedures that are currently in use.

Description of the site of study, geology and hydrogeological conditions

Study location

The study area is a sedimentary environment, located in a medium-sized newly designed residential estate, known and referred to as Shelter Afrique in Akwa Ibom State, southern Nigeria. The referred study area was spatially geo-referenced and the coordinates transformed from degree to meters. This was done by finding the respective latitude and longitude difference from their respective common reference points and multiplying them by 111,139 m. The resulting coordinates in meters were used for kriging. The survey area is positioned in the mid-western part of Akwa Ibom State in southern Nigeria. It massively cuts across Ibesikpo County in the south; with a small portion of the area located in Uyo, the headquarters of Akwa Ibom state (Fig. 1). The inhabiting landmass, which is landlocked outspreads between latitudes 4.958° and 4.9917° N and longitudes 7.9417° and 7.9750° E, with an area that is about 31 \({\text{km}}^{2}\). The area is a table land that is currently witnessing new dwellers and new state-of-the-art structures. It is also characterized by good road network and buildings that are well-planned to give room for long VES traverses. The climate is semi-temperate with scrubs. The study area witnesses dry and wet seasons. The wet season begins in earnest from April to September and dry season starts from October to March. The temperature ranges from 26 to 32° C while its annual rainfall range is from 2.0 and 2.5 m. The entire catchment is drained by the tributaries of Enyiong Creek, which is the main perennial reservoir of surface water in Itu Local Government Area (Ibuot et al. 2013; Ekanem et al. 2019; Uwa et al. 2o19; George 2021).

Fig. 1
figure 1

a Schematic map of Nigeria showing the geographic location of Akwa Ibom State in Southern Nigeria b map of Akwa Ibom State showing Atlantic Ocean and the geographical settings of the study area, c geographic and model domain map showing geology, VES and ERT points and Borehole locations

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Geology and hydrogeological settings

The Youngest Continental Plain sand/Benin Formation of the Niger Delta of southern Nigeria is where aquifer systems in the study area are located and subsurface water is drilled at various depths from this formation. The Benin Formation is characteristically noted by many researchers to be inundated with intercalations of pockets of arenites and slight/petty argillites (Peters 1982, 1989; Obianwu et al. 2011; George et al. 2014; Akpan et al. 2013; George et al. 2017).The Agbada and Akata Formation, respectively, lie below the Benin Formation in chronological depths of burial (Peters 1982, 1989). The study area explicitly, fits grossly into the Benin Formation and somewhat into the Beach Ridge Complex and Alluvium of Quaternary Period as Fig. 1 depicts. The constituents of Benin Formation preponderantly range from fine to gravelly sands (Fig. 2) and by composition; the light grey complementary petty argillites are comparatively insignificant and occasionally strewn. Morphological disparities caused by deposition, sedimentation and erosion are occasionally observed on the shallowest planes after downpour (Thomas et al. 2020). The deposits of sedimentary constituents are regularly observed in the dipping altitude and they are appealing to the gravitational attraction. The grayish muddy sand grains have a characteristic texture that varies from finer to coarsening textures and they are sequentially intercalating in nature according to Short and Stauble (1965). Myriad geological dynamics regulate the economic availability of groundwater and the depth of burial of aquifer in the survey area. According to Tizro (2012), some of these dynamics include—the stratigraphical disturbances and geological sequence of hydrogeological units. The hydraulic networks between seasonal drawdown and subsurface water level or groundwater conduit topography regulate the bathymetry of water wells in the study area (Ekanem et al. 2022).

Fig. 2
figure 2

Sampled correlations of VES 2, 7, 9 and 10 curves with their adjoining lithological log in the study area

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Material and methods

VES and ERT measurements

The geo-electrical resistivity technology, which ensures electrical sounding or electrical drilling, is soil conductivity—sensitive tool that maintains current and potential electrodes along a straight path at the equivalent relative spacing around a fixed central position and it is frequently deployed in the hydrogeological and solid mineral explorations (Akpan et al. 2018; Ibuot et al. 2019; Ekanem et al. 2021). Geo-sounding resistivity technicality deployed in this work involves a one-dimensional vertical electrical sounding (VES) and two-dimensional electrical resistivity tomography (ERT).These techniques were performed adjacent to water wells through the deployment of IGIS signal enhancement resistivity meter (SSP-MP-ATS) and its fittings in the area surveyed. Thirteen (13) and ten (10) closely spaced VES and ERT points, adjacent to water boreholes were, respectively, sounded (see Fig. 1). The procedure used in VES data acquisition was Schlumberger configuration with maximum current electrode separations (AB) of 400 m. Again, the ERT, leading to 2-Ddata acquisition used the Wenner electrode configuration with spread length of 105 m, taking through 5 m separations (Thomas et al. 2020). All the precautionary measures opined by Zohdy et al. (1974), Zohdy 1989; Evans and George 2007; Akpan et al. (2013) were observed for quality assurance. The apparent resistance of the earth for Schlumberger and Wenner techniques was determined as \(R_{{{\text{as}}}} {\text{ and }}R_{{{\text{aw}}}}\), respectively, for the VES and ERT technique. For VES and ERT methods, the apparent resistivities \(\rho_{{{\text{as}}}} {\text{ and }}\rho_{{{\text{aw}}}}\) were computed using the equations given below:

$$\rho_{{{\text{as}}}} = \pi \cdot \left[ {\frac{{\left( {{{{\text{AB}}} \mathord{\left/ {\vphantom {{{\text{AB}}} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)^{2} - \left( {{{{\text{MN}}} \mathord{\left/ {\vphantom {{{\text{MN}}} 2}} \right. \kern-\nulldelimiterspace} 2}} \right)^{2} }}{{{\text{MN}}}}} \right] \cdot R_{{{\text{as}}}}$$
(1)
$$\rho_{{{\text{aw}}}} = 2\pi aR_{{{\text{aw}}}},$$
(2)

where \({\text{AB}},{\text{MN}}\) and \(a\) are, respectively, electrical current electrode spacing, separation of potential electrodes and Wenner electrode spacings. The term on the right hand side is obtained by multiplying the \(R_{{{\text{as}}}}\) and \(R_{{{\text{aw}}}}\) by the geometric factor for Schlumberger and Wenner electrode configurations, respectively, in order to obtain the apparent resistivities \(\rho_{{{\text{as}}}}\) and \(\rho_{{{\text{aw}}}}\) for both the Schlumberger and Wenner spacing configurations in Eqs. 1 and 2, respectively. On a logarithmic scale, apparent resistivities were plotted manually against half of the current electrode spacings. Smoothening was performed on the data by removing the spurious signatures (outliers) that did not follow the prevailing curve trend entrenched by the geology of the layers that current passed through. The smoothed, VES apparent resistivities were interpreted further in two ways. The computer modeling least squares inversion technique (LSIT), which, according to Oladapo et al. (2004) and Obiora et al. (2015); George (2020) used the data to generate the final curves by comparing the field curves to the existing theoretical curves (Fig. 2). This was possible using a RESIST code developed by Vander Velpen and Sporry (1993). The field data were reduced to their geologically consistent equivalent models using computer-modeling techniques (Zohdy et al. 1974). Following a couple of iterations, a reasonably acceptable disparity observed between the field and theoretical data were obtained through absolute root-mean-square (RMS) error, which was generically found to be less than 10% (Fig. 2). For ERT, involving a two-dimensional (2D) inversion, RES2DINV software program was used and the same iterations in VES was applied until a reasonable RMSE indicating a good fit between the theoretical and field data were achieved (Loke and Barker 1996; Loke and Dalhin 2002; Loke et al. 2003). The inverse gradient method (ISM) (Kouassi et al. 2017; Bouadou et al. 2019), which applied direct/forward technique to predict the primary geo-electric indices in Fig. 3. In case 1, the curves were electronically deciphered quantitatively through a 1-D least squares computer-aided software program, known as WINRESIST (Vander Velpen and Sporry 1993) with the constraints of nearby mechanically logged borehole indices. The software program supplied details of the interpreted curve by delineating the primary geo-electric indices such as depth, layer resistivity, thickness and the root-mean-squares error (RMSE) (generally < 10%). The RMSE describes the degree of goodness of fit between the theoretical curve and the curve from the field data (see Fig. 2). Also, by inversion of apparent resistivities computed from Eq. 2, using RES2DINV VER 3.59 Geotomo software code, theorized by Loke and Barker (1996), Loke and Dalhin (2002) and Loke et al. (2003), the ERT images (Fig. 4) were modelled after preparing the separation and the apparent resistivity values in tandem with the RES2DINV VER. 3.59 Geotomo software format. The software package generated a resistivity model of the shallow subsurface based on iterative smoothness-constrained least squares, replicated on the resulting electrical resistivity tomography (Fig. 4). The inverse gradient method (ISM), theorized by Sankarnaryan and Ramanujachary (1967) for the analyses and interpretations of Wenner array compliant geo-sounding data or the Schlumberger method is used to generically compare, validate and calibrate VES and ERT data mostly when there is no borehole information, by applying the field equation to directly obtain the resistivities and thicknesses of the subsurface layers from the field datasets (Asfahani 2016; Bouadou et al. 2019). The method in Wenner array plots the inverse of resistance against the constant electrode separation (a) such that the number of linear segments identified on the graph corresponds to the number of layers the injected current penetrated. The inverse of the slope of each segment of the graph gives the true resistivity of the layer while the respective intersections of the segments give the depth of each of the layers. In Schlumberger array, ISM was deployed in VES data interpretation by plotting the ratio of half of current electrode separation to apparent resistivity \(\left( {\frac{{{\text{AB}}}}{2}/\rho_{a} } \right)\) against half of current Bouadou et al. (2019). Electrode separations \(\left( {\tfrac{{{\text{AB}}}}{2}} \right)\) as developed by Asfahani (2016), Kouassi et al. (2017) and each sounding station generates a graph with linear segments corresponding to the number of layers as showcased by Fig. 5. The ISM results in Fig. 3 show that the last segment on all the interpreted VES is characterized by a low or near negative slopes, which indicate the presence of a very high resistivity layer.

Fig. 3
figure 3

Samples of VES interpretation by IGM at VES points 2 and 4 showing the linear segments and their equations leading to the determination of different primary geo-electric indices (resistivity, thickness and depths)

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Fig. 4
figure 4

Representatives of ERTs 1, 2, 5 and 6 with their adjoining lithological log in the study area

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Fig. 5
figure 5

Sketch showing a plot of ratio of AB/2 to apparent resistivity against AB/2 for ISM in Schlumberger array

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The values of inverse of the slopes of the various segments were equaled to the true resistivity of the corresponding strata while the thicknesses were deduced from the intersections of the segments. The true resistivity of the layers was evaluated by Eq. (3):

$$\rho_{{\text{n}}} = \frac{1}{{\left| {m_{i} } \right|}},$$
(3)

where \(m_{i}\) is the gradient of the \(i{\text{th}}\) linear segment and \(i = 1,2,3...,{\text{n}}\) and \({\text{n}}\) is the deepest layer number assessed by the current at the maximum current electrode separation. The depth (d) to the different boundaries was obtainable generally by the use of intercept-slope relation in Eq. (4):

$$d_{i} = \frac{{2\left( {c_{i + 1} - c_{i} } \right)}}{{3\left( {m_{i} - m_{i + 1} } \right)}},$$
(4)

where m and c are the corresponding slopes and intercepts of the linear segments while \(i\) is the number of segments \(\left( {i = 1,2,3,...} \right)\), which corresponds to the number of strata. The thickness of each layer (h) is defined as the difference between the depths of two consecutive strata and at the point of intersection, \(y_{1} = y_{2}\) according to Fig. 5. As opined by Sanjiv (2010) and Bouadou et al. (2019), the intersections of the projected segments on the \(x\)-axis multiplied by a factor 2/3 give the true depth of the interfaces. Practically according to Bouadou et al. (2019), in the first segment, the value of the first thickness of the layer is equivalent to the value of the first intersection of the projected segments on the \(x\)-axis. Each line segment represents a layer and the intersections of the line segments, multiplied by a factor of 2/3, correspond to the depths of the particular layers. The ISM was identified with ease of interpretation of surface resistivity data as standard curves, partial and total curve matching were not required but the results were often characterized by reasonably good results (Bouadou et al. 2019). With the combination of the inverse modelling technique (LSIT) and direct modelling technique within the vicinity of mechanically drilled borehole data, the calibration of direct and inverse VES interpretation techniques were assessed and the primary geo-electric indices determined independently from the ISM and LSIT techniques as given in Table 1.

Table 1 Summary of VES interpretation from ISM and LSIT at various VES stations
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Results

Results

Employing the techniques of LSIT and ISM independently in VES interpretation, the qualitative and quantitative results in Table 1 show similar trend. Qualitatively, the curve type in both ISM and LSIT at VESs 1, 2, 8, 9, 10 and 12 is H with 46.1% composition. At VESs 3, 4, 5 and 13, the curve types are, respectively, AK and HK at 15.4% composition in each pair while curve type AH (VES 4), KQ (VES 6) and A (VES 11), respectively, has 7.7% each. Quantitatively, Table 1 shows the summary of primary geo-electric indices for the thirteen VES points, which show a strongly similar trend for the two techniques independently used in VES interpretations. The similarity in trend is demonstrated in the uniformity of curve types in the two techniques. The ERT results (Fig. 4) also correlate fairly with the logged boreholes and VES results. The marginal disparities between geo-electric results and geology are in agreement with the conclusion by Tomitope and Adeniyi (2016) that geo-electric sections are not totally in tandem with geologic sections. As the major preoccupation of this research, the most probable correlation with the nearby borehole is of paramount concern in comparing, calibrating or validating the results interpreted independently between ISM and LSIT. The results in Table 1 are used in generating the correlations of the earth’s consistent resistivities for the defined layers in ISM and LSIT that current passed through during geo-electric sounding as revealed in Fig. 6. The primary geo-electric indices (layer resistivities, thickness and depth) measured at various coordinate locations of the study area are apt as the basis of comparison and possible calibrations of ISM and LSIT. The independent joint interpretation of layered resistivities, depths and thickness from ERT, ISM and LSIT have also resulted in some noticeable pattern recognition images (Figs. 4, 6, 7, 8 and 9) and chats in Figs. 10, 11, and 12. These aimed at making comparison, calibration and possible validation of interpretation of the ill-posed problem of geo-electrical resistivity data standardized.

Fig. 6
figure 6

Graph showing IGM-LSIT plot for a layer one, b layer two and c layer three

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Fig. 7
figure 7

Comparison of resistivity image map of least-squares inversion technique (LSIT) and inverse slope technique (IGT) for layer one

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Fig. 8
figure 8

Comparison of resistivity image map of least-squares inversion technique (LSIT) and inverse gradient technique (IGT) for layer two

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Fig. 9
figure 9

Comparison of resistivity image map of least-squares inversion technique (LSIT) and inverse gradient technique (IGT) for layer three

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Fig. 10
figure 10

Delineated depth correlation of IGM and LSIT with approximate layer one depth from lithological data at different VES locations

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Fig. 11
figure 11

Delineated depth correlation of IGM and LSIT with approximate layer two depth from lithological data at different VES locations

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Fig.12
figure 12

Delineated depth correlation of IGM and LSIT with approximate layer three depth from lithological data at different VES locations

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Nash–Sutcliffe optimization criterion model efficiency coefficient (NSOCMEC)

The variational depths were estimated from the interpretation of electric sounding data by the ISM and LSIT. These estimated depths were regarded as calculated/model depths \(\left( {M_{{{\text{D}}i}} } \right)\) while the depth deduced from lithological logs were regarded as observed depths \(\left( {O_{{{\text{D}}i}} } \right)\). The mean depth of the observed was estimated as \(\left( {M_{{{\text{DO}}i}} } \right)\). The results of the calculated/modelled depth of geologic units penetrated by injected currents can be compared with the depths from drilling logs in order to verify the degree of reliability of ISM and LSIT using Nash–Sutcliffe optimization criterion model efficiency coefficient (NSOCMEC), given as one minus the ratio of error variance of a model values to the variance of the observed values and is mathematically expressed in percentage as:

$${\text{NSOCMEC}} = \left[ {1 - \frac{{\sum\nolimits_{i = 1}^{n} {\left( {O_{{{\text{D}}i}} - M_{{{\text{D}}i}} } \right)^{2} } }}{{\sum\nolimits_{i = 1}^{n} {\left( {O_{{{\text{D}}i}} - M_{{{\text{DO}}i}} } \right)^{2} } }}} \right] \times 100.$$
(5)

According to Yao et al. (2007), the optimization by NSOCMEC is judged according to the percentage ranges of values given below:

  • Nash ≥ 90%: the model is excellent;

  • 80% < Nash < 90%: the model is very satisfactory;

  • 60% < Nash < 80%: the model is satisfactory;

  • Nash < 60%: the model is bad.

Since the model uses depth as the criterion for optimization between the modelled and the calculated values, it is considered efficient when the calculated/modelled depths are close to the observed depth, i.e., to say when the value of the NSOCMEC is close to 100%.

Moreover, to assess the potential effect of extreme values caused by sup-optimal results caused by datasets with large outliers, re-scaling of NSOCMEC is necessary in order to obtain the normalized NSOCMEC (NNSOCMEC), which is given by Eq. (6):

$${\text{NNSOCMEC}} = \frac{1}{{2 - {\text{NSOCMEC}}}}.$$
(6)

Based on the realism, Eq. 6 indicates that at NSOCMEC = 1, NNSOCMEC corresponds 1. At NSOCMEC = 0, NNSOCMEC corresponds to 0.5 and at \({\text{NSOCMEC}} = - \infty\), NNSOCMEC corresponds to zero. The apt re-scaling of the NSOCMEC in Eq. 6 allows for quick and easier understanding and utilization of the NSOCMEC measurement of parameter estimation schemes used in model calibration.

Discussion of results

The quantitative (resistivity, thickness and depth) and qualitative (exact number of curves) results that are consistent when constrained with mechanical boreholes have been identified from the inverse (LSIT) and direct (ISM) techniques applied to interpret the VES data known to associate with ill-posed problems and non-unique interpretative results (Simms and Morgan 1992; Friedel 2003). For efficient delineation of subsurface resources, hybrid methods in phenomenological interpretation of non-unique geophysical techniques are apt for resourceful and result-oriented exploration that leads to a corresponding efficient exploitation of subsurface resources. The combination of ISM, which has the capacity to determine the different geo-electric layers while typifying their resistivities and true thicknesses, with LSIT through independent interpretation to compare and validate the interpretative results was aided by the borehole data. The quality of the curve types in the LSIT based on the resistivity trend in Table 1 is 100% in tandem with the trend in the ISM. The entrenched results show a trend that is marginally variable between the ISM and LSIT. Practically, the independent measurements of geo-electric indices from ISM and LSIT as shown in Table 2 show good match in the determined values of resistivities, depths and thicknesses for the layers that current passed through. For instance, the graphs in Fig. 6a–c indicate the respective ISM-LSIT plots of resistivities for layers 1–3 penetrated by current. The coefficients of determination for layers 1–3, respectively, give 73, 80.0 and 81% ISM-LSIT resistivity correlations for the regression lines generated from the plots (see Eqs. 7, 8, 9):

$${\text{ISM}} = 0.8001{\text{LSIT}} + 152.53$$
(7)
$${\text{ISM}} = 0.8073{\text{LSIT}} + 271.18$$
(8)
$${\text{ISM}} = 0.5858{\text{LSIT}} + 472.15.$$
(9)
Table 2 Ranges and averages of primary geo-electric indices determined from ISM and LSIT
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The slight variations in the correlation coefficients are in sync with the fact that fictitiousness of the subsurface medium decreases with depth due subsurface inhomogeneity (Zohdy et al. 1974). Generally, electrical resistivity measurements (VES and ERT) are often sensitive to electrode-spacing inaccuracies, potential errors as well as subsurface inhomogeneity (Mohammed et al. 2021). Interpreting resistivity data using inverse models (LSIT), which involve comparing the field data to theoretical data, will always have ambiguity except the interpretation is constrained with borehole data. In order to assess the level of goodness of fit, ISM technique was used to interpret the same data. Since the data were acquired closed to borehole, the two were compared to the borehole indices and the one with close fitness with the borehole data is regarded as the one that is less sensitive to the electrode-spacing errors, potential errors as well as subsurface inhomogeneity (Udosen and George 2018b; Mohammed et al. 2021). In terms of resistivities, Figs. 6, 7, and 8 showcase the resistivity pattern recognition for layers 1–3. Again, the uniformity or correlation of resistivity image gets better between LSIT and ISM with depth. However, the coarsening correlation in Fig. 6 (topmost layer) compared to the deeper layers is due to surface electrical effect caused by spurious effect of inhomogeneity in the interpretation of VES data using inverse and direct models. As resistivity inversion by numerical algorithms involves the minimization of error between the observed apparent resistivity in order to obtain a good fit between the field data and the assumed theoretical model, interpreting LSIT without a borehole may be misleading as the technique is an ill-posed problem (Ghanati et al. 2021). Therefore, this pretext advances the reason while there must be variations between LSIT and ISM or ISM noticed as either significant or marginal variability in correlating the interpretive results from ISM and LSIT in Figs. 6, 7, and 8 and in Table 2. To assess the goodness of fit between the two candidates of VES interpretation, correlation were made between their depth and thickness results with nearby boreholes and the results show that ISM is more perfectly correlated with the borehole data than LSIT especially for layers 2 and three where the subsurface layer is stable and not fictitious (see Figs. 10, 11, and 12).As observed on Fig. 10, VES 4, 6 and 12 did not have the real logged data and this is why only two bars representing depths interpreted from ISM and LSIT techniques are, respectively, shown. However, in the rest of the VES points, additional one bar representing approximate depth from lithological logged borehole is added. The heights of the three bars clustered at each VES point correlate the closeness of depths obtained from ISM and LSIT to the nearby approximate depth of borehole. In Figs. 11 and 12, the same diagrams are provided for layers 2 and 3. Based on these diagrams, the real drilling data are used as the reference point for validation/calibration of the methods (ISM and LSIT) employed. In Fig. 10, there are some spurious divergences of VES depths in layer one between ISM and LSIT while some convergences of depths between the two methods with respect to the nearby boreholes are also noticed. The same idea is also conveyed in Figs. 11 and 12 for layers 2 and 3. Although layer one (topmost) is often fictitious, the basis for spurious divergence of depth due to high degree of heterogeneity (Mohamed et al. 2014; George et al. 2015), layers two and three show extremely better harmonious convergence between ISM and borehole than LSIT, the traditional digital computerized method. As observed, the correlation amongst the primary geo-electric indices obtained from ISM and LSIT and even the drilling appears to be enhanced with depth (see Figs. 7, 8, 9, 10, 11, and 12). The observed lithological logs (Table 1) from boreholes are quite in-line with the trend of resistivity values obtained near the borehole (Mohamed et al. 2014; Bouadou et al. 2019). According to the results in Table 2, the layer one resistivity values ranged from 67.6 to 2000.0 \(\Omega {\text{m}}\) and 89.6–1464.2 \(\Omega {\text{m}}\) with average values 2000.0 \({\Omega m}\) and 682.3 \(\Omega {\text{m}}\), respectively, for ISM and LSIT. In layer two, resistivity ranged 384.6–3333.3 \(\Omega {\text{m}}\) and 308.3–2591.8 \(\Omega {\text{m}}\) with average 1467.0 \(\Omega {\text{m}}\) and 1481.3 \(\Omega {\text{m}}\) for ISM and LSIT, respectively. Similarly, layer three interpreted by ISM and LSIT showed resistivity ranges of 454.4–2500.0 \(\Omega {\text{m}}\) and 234.8–2301.8 \(\Omega {\text{m}}\) with corresponding averages of 1221.7 \(\Omega {\text{m}}\) and 1279.5 \(\Omega {\text{m}}\). In the VES identified with the fourth layer, resistivity spread between 256.0–1666.7 \(\Omega {\text{m}}\) and 542.2–1194.6 \(\Omega {\text{m}}\) with corresponding averages of 836.3 \(\Omega {\text{m}}\) and 880.8 \(\Omega {\text{m}}\).The interpreted datasets showed corresponding ranges for layers 1–3 in ISM as 0.4–17.8 m, 5.6–67.6 m and 61.2–110.0 m depths penetrated and 1.2–11.8 m, 6.5–76.0 m and 57.3–80.7 m, respectively, for layers 1–3 in the depth penetrated in LSIT. The ISM-LSIT depth averages for layers 1–3 are 5.4 m, 43.6 m an 87.9 m and 4.8 m, 46.3 m and 67.4 m, respectively. Similarly, the interpreted datasets showed corresponding thickness ranges for layers 1–3 in ISM as 0.4–17.8 m, 1.4–63.8 m and 55.6–92.7 m and thickness in LSIT as 1.2–11.8 m, 6.5–76.0 m and 57.3–80.7 m, respectively. In terms of averages, the ISM and LSIT showed thickness values of 5.6 m, 38.3 m and 66.2 m and 4.8 m, 41.5 m and 49.7 m, respectively. These ranges and mean values of interpreted primary geo-electric indices highlighted above indicate on the average, marginally enhanced depths and thicknesses from layer one to maximally enhanced depths in ISM (direct model) when compared to the conventional and digitally computerized method (LSIT) (see Table 2 and Figs. 10, 11, and 12). The analysis of the depth of penetration showed that the LSIT has correlation with borehole depth in the range of 56.3–88.6% (average: 70.3%) while ISM has average correlation of 79.0% with range of 68.0–87.7% in layers 1–3 clearly delineated in this study. On the basis of the results of this analysis, ISM is marginally more compliant with the drilling data than the conventional and digitally computerized method (LSIT). This according to (Ghanati et al. 2021) may be due to the ambiguity in choosing the regularization schemes, in solving the inverse problem associated with the least squares inversion technique in the traditional digitally computerized method, which often requires a fit between the field data and the theoretical datasets. Since geo-electrical resistivity technology is used in hydrogeology, environment, archeology, engineering and mining surveys (Panda et al. 2017; Mohammed et al. 2021) to explore and exploit the geo-resources, techniques leading to efficient prediction of depths of investigation, lateral and transverse spreads of resources sought for, reproducibility, versatility and precision/sensitivity as well simplicity in usage are the characteristics to be considered, which are all found in ISM.

Based on the modelled datasets obtained through ISM and LSIT modelling and the observed data from nearby borehole logs in the study area, model optimization were performed using depth parameter estimation schemes in model calibration. Using the expression in Eq. 6, the models obtained using ISM have 94%, 100% and 99% as the Nash–Sutcliffe optimization criterion model efficiency coefficients for layers 1–3, respectively. This shows excellent model according to Yao et al. (2007). The depth parameter estimation schemes in model calibration optimization criterion for LSIT according Eq. 6 also gives 99%, 100% and 96% as the percentage efficiency coefficient. This also implies excellent model as opined by Yao et al. (2007). The results of optimization as reflected by model error variance in both the scaled and re-scaled calibrations in Eqs. 5 and 6 show uniform optimization coefficients, reflecting excellent models of geologic units inferred using ISM and LSIT with respect to the observed geology. However, a novel observation indicates that the ISM is marginally better than LSIT based on the optimization indices realized in both the regression analyses and the Nash–Sutcliffe optimization criterion model.

Conclusion

This work was undertaken to assess the efficiency of ISM and LSIT in-line with ground truthing indices. Geo-electrical resistivity technology employs these exploratory tools (ISM and LSIT) for explorability of groundwater and related subsurface resources related to archeology, engineering, mining and environment. The appraisal between the results obtained from inverse slope method (ISM) and the conventional digitally computerized least-squares inversion technique (LSIT) of geo-electrical resistivity data interpretation indicate that the geo-electrical models derived from ISM is practically more correlated to the borehole indices than the LSIT, which assumes regularization schemes or models for inverting the resistivity of the earth. The results show that the first layer resistivity values ranged from 67.6–2000.0 \(\Omega {\text{m}}\) and 89.6–1464.2 \(\Omega {\text{m}}\) with average values 2000.0 \(\Omega {\text{m}}\) and 682.3 \(\Omega {\text{m}}\), respectively, for ISM and LSIT. The second layer resistivity ranged 384.6–3333.3 \(\Omega {\text{m}}\) and 308.3–2591.8 \(\Omega {\text{m}}\) with average 1467.0 \(\Omega {\text{m}}\) and 1481.3 \(\Omega {\text{m}}\) for ISM and LSIT, respectively. Correspondingly, layer three interpreted by ISM and LSIT showed resistivity ranges of 454.4–2500.0 \(\Omega {\text{m}}\) and 234.8–2301.8 \(\Omega {\text{m}}\) with averages of 1221.7 \(\Omega {\text{m}}\) and 1279.5 \(\Omega {\text{m}}\) while the fourth layer is identified with resistivity spread from 256.0–1666.7 \(\Omega {\text{m}}\) to 542.2–1194.6 \(\Omega {\text{m}}\) with corresponding averages of 836.3 \(\Omega {\text{m}}\) and 880.8 \(\Omega {\text{m}}\). Layer resistivity analysis indicates marginal correlation in layer one to maximal correlation of resistivities in layers two and three (Figs. 7, 8, and 9). Additionally, the curve types obtained from LSIT were 100% in agreement with the values of resistivity obtained for different layers in ISM. The depth of investigation showed that the LSIT showcased a correlation with borehole depth in the range of 56.3–88.6% (average: 70.3%) while ISM has average correlation of 79.0% with range of 68.0–87.7% in layers one to three distinctively delineated in this study. Based on the depth and thickness analyses displayed on Table 2 and Figs. 10, 11, and 12, ISM is more compliant with the drilling data than the conventional and digitally computerized method (LSIT). As conventionally known, LSIT is ill-posed problem that gives room for myriads of solutions based on interpreter’s subjectivity to the interpretation of data acquired. ISM is advantageous as its quantitative results (the determination of the exact number of layers of the subsoil) and qualitative results (true resistivities, depths and thicknesses of the different layers) are linear and simpler than the logarithmic curve-matching of LSIT. A unit layer identified by curve-matching method due to problem of equivalence and suppression can be robustly resolved into two or more layers by ISM in some case studies as opined by Asfahani (2016). Incorporation of LSIT with ISM, which is more correlated with the available lithological ground truths than the currently used conventional method (LSIT) as revealed in this work, can reduce ambiguity and the non-uniqueness in VES interpretation mostly in areas where there is ground truthing information.