Multiple parameters play a dominant role in groundwater recharge, and these are used to estimate the groundwater potential. The number of thematic layers that can be used also depends on the availability of data for the study region. As per the report by the Central Ground Water Board, the chief source of groundwater recharge in the Himalayan region is the glaciers, and in the foothill region, rainfall plays the dominant role contributing a copious amount. In Northeast India, groundwater occurrence can be witnessed in weathered residuum and fractures/ joints in consolidated formations in Assam and Meghalaya plateau, semi-consolidated porous formations of tertiary rocks, and unconsolidated formations of Quaternary age in the Brahmaputra & Barak valleys (Gupta 2014). Ahmed and Sajjad (2017) analyzed the groundwater potential zone in the lower Barapani watershed adjacent to the study area and found that gentle slopes, flood plain, monsoon rainfall, and the presence of wetlands are certain key causes for high groundwater potential. Based on the literature survey, expert opinion, and in consultation with local hydro-geologists the following parameters and their relative importance are considered for the estimation of groundwater potential in the study area: lithology, geomorphology, slope, land use land cover, rainfall, soil, lineament density, and drainage density. Lithology maps are prepared using data from the Bhukosh platform of the Geological Survey of India. For geomorphology maps, ISRO/ Bhuvan portal is used as a reference. Soil maps are prepared using datasets from the National Bureau of Soil Survey and Land Use Planning, Government of India. Slope and drainage density is prepared using the Digital Elevation Model (DEM) of the Shuttle Radar Topography Mission (SRTM). Lineament density and LULC maps are prepared using Landsat 8 satellite imagery. Rainfall data is collected from the GIOVANNI application of NASA. The groundwater yield map is prepared using data from Central Groundwater Board (CGWB). Static water table data is collected by field survey. Remote sensing and GIS are mainly used in preparing different layers. Arc GIS 10.7, ERDAS Imagine 10, and ENVI 5.6 are the prime GIS software used for the rectification, digitization, resampling, and processing of various data. Weight assignment to the parameters and their corresponding classes is carried out using Fuzzy Analytical Hierarchy Process (FAHP) based on their relative importance to groundwater recharge. A fuzzy set is characterized by a membership function that assigns to each object a grade of membership ranging between zero and one. Mathematically, if \(X\) is a universal set, a fuzzy set \(\stackrel{ˇ}{A}\) in\(X\) is characterized by its membership function denoted by\({\mu }_{\stackrel{ˇ}{A}}\), so that \({\mu }_{\stackrel{ˇ}{A}} : X \to \left[\text{0,1}\right]\)(Bhargava 2013). Though there are several different forms of membership functions, triangular and trapezoidal membership functions are extensively used due to their simplicity and efficiency in real-time computation (Azam et al. 2020). In this study, the triangular fuzzy membership function is used for computational ease. Before the implementation of the FAHP method, the Analytical Hierarchy Process (AHP) developed by Saaty is used to check the consistency of the decisions by the decision maker. It uses a pairwise comparison matrix calculating the weights for each criterion involved (Chakraborty and Joshi 2014). Though the nine-point AHP scale by Saaty gives information about the dominance of each parameter above the other, the fixed value discrete scale judgment cannot simplify the information unpredictability creating a weak spot in decision making However, fuzzy set theory with its interval judgment (Table 1) deal with such impreciseness (Kaganski et al. 2018). The matrix consistency is assessed using the Consistency Index (CI) formula: \(CI= \frac{{\lambda }_{\text{max}- }n}{n-1}\) where \({{\lambda }}_{\text{m}\text{a}\text{x}}\) is the principal eigenvalue of the pairwise comparison matrix, and n is the number of parameters used in the analysis. The Consistency Ratio (CR) is determined by dividing Consistency Index (CI) by Ratio Index (RI). For consistency of the weight, the CR value should be less than 0.1. If CR is higher than 0.1, re-evaluation within the matrix is needed (Tošovic´-Stevanovic' et al. 2020). The CR for this particular study is found to be 0.03, which is in the acceptable range.
Table 1
Linguistic scale for FAHP pair-wise comparison (Ayhan 2013)
Saaty Scale | Definition | Fuzzy Triangular Scale |
1 | Equally important | (1,1,1) |
3 | Weakly important | (2,3,4) |
5 | Fairly important | (4,5,6) |
7 | Strongly important | (6,7,8) |
9 | Absolutely important | (9,9,9) |
2, 4, 6, 8 | The intermittent values between two adjacent scales | (1,2,3), (3,4,5), (5,6,7), (7,8,9) |
Buckley’s column geometric mean method is used to get the final normalized weight. The process uses centre of area method as the de-fuzzification technique. The following mathematical equations are used in this study (Bayer and Karamasa 2018).
The membership function for triangular fuzzy numbers is defined by the triplet \(\left(l, m, u\right)\) indicating the smallest possible value, the most promising value, and the largest possible value. The membership function is defined by the following equation:
$${\mu }_{\stackrel{\sim}{M}}\left(x\right)= \left\{\begin{array}{c}\frac{\left(x-l\right)}{\left(m-l\right)}, l\le x\le m,\\ \frac{\left(u-x\right)}{\left(u-m\right)}, m\le x\le u\\ 0, otherwise\end{array}\right.$$
A triangular fuzzy number \(\stackrel{\sim}{M}\) is shown in the Fig. 2
Fuzzy pair-wise matrices can be represented as
$$\stackrel{\sim}{B}= {\left({\stackrel{\sim}{b}}_{ij}\right)}_{n\times k}\left[\begin{array}{cccc}\left(\text{1,1},1\right)& \left({l}_{12},{m}_{12},{u}_{12}\right)& \dots & \left({l}_{1k}, {m}_{1k}, {u}_{1k}\right)\\ \left({l}_{21},{m}_{21},{u}_{21}\right)& \left(\text{1,1},1\right)& \cdots & \left({l}_{2k}, {m}_{2k}, {u}_{2k}\right)\\ ⋮& ⋮& \cdots & ⋮\\ \left({l}_{n1},{m}_{n1},{u}_{n1}\right)& \left({l}_{n2},{m}_{n2},{u}_{n2}\right)& \cdots & \left(\text{1,1},1\right)\end{array}\right]$$
Where\({\stackrel{\sim}{b}}_{ij= }\left({l}_{ij}, {m}_{ij}, {u}_{ij}\right), {\stackrel{\sim}{b}}_{ij}^{-1}=(\frac{1}{{u}_{ji}},\frac{1}{{m}_{ji}},\frac{1}{{l}_{ji}})\)
$$\text{F}\text{o}\text{r} i=\text{1,2},3,\dots .n;j=\text{1,2},3,\dots .,k \text{e}\text{l}\text{e}\text{m}\text{e}\text{n}\text{t}\text{s} \text{a}\text{n}\text{d} i \ne j$$
New pair-wise comparison matrix averaging preferences for all decisions:
$$\stackrel{\sim}{B}= \left[\begin{array}{cccc}\stackrel{\sim}{{b}_{11}}& \stackrel{\sim}{{b}_{12}}& \cdots & \stackrel{\sim}{{b}_{1n}}\\ \stackrel{\sim}{{b}_{21}}& \stackrel{\sim}{{b}_{22}}& \cdots & \stackrel{\sim}{{b}_{2n}}\\ \cdots & \cdots & \cdots & \cdots \\ \stackrel{\sim}{{b}_{m1}}& \stackrel{\sim}{{b}_{m2}}& \cdots & \stackrel{\sim}{{b}_{mn}}\end{array}\right]$$
Geometric mean of each criterion is calculated by:
$$\stackrel{\sim}{{h}_{i }}= {\left[{\prod }_{j=1}^{n}\stackrel{\sim}{{b}_{ij}}\right]}^{1/n} \text{W}\text{h}\text{e}\text{r}\text{e}, i=\text{1,2},\dots .,m$$
Fuzzy weights of each criterion are obtained by:
$$\stackrel{\sim}{{\text{w}}_{\text{i}}}= \stackrel{\sim}{{\text{h}}_{\text{i}}} \otimes {\left(\stackrel{\sim}{{\text{h}}_{\text{i}}} \oplus \stackrel{\sim}{{\text{h}}_{2}}\oplus \dots . \oplus \stackrel{\sim}{{\text{h}}_{\text{n}}}\right)}^{-1}= \left({\text{l}}_{\text{i}}, {\text{m}}_{\text{i}}, {\text{u}}_{\text{i}}\right)$$
Centre of area de-fuzzification technique to transform fuzzy weights into crisp ones:
$${k}_{i}= \frac{{l}_{i}+ {m}_{i}+ {u}_{i}}{3}$$
Crisp weights to normalized final weight:
$${z}_{i}=\frac{{k}_{i}}{{\sum }_{i=1}^{m}{k}_{i}}$$
Finally, a weighted linear combination method (Das and Pal 2019; Mallick et al. 2019) is used in the GIS environment to delineate potential groundwater zones.
$$GWPZ=\left({RF}_{w}\times {RF}_{wc}\right)+ \left({LT}_{w}\times {LT}_{wc}\right)+\left({GM}_{w}\times {GM}_{wc}\right)+\left({LD}_{w}\times {LD}_{wc}\right)+\left({DD}_{w}\times {DD}_{wc}\right)+\left({S}_{w}\times {S}_{wc}\right)+\left({SL}_{w}\times {SL}_{wc}\right)+\left({LU}_{w}\times {LU}_{wc}\right)$$
Where, GWPZ = Groundwater Potential Zone, RF = Rainfall, LT = Lithology, GM = Geomorphology, LD = Lineament Density, DD = Drainage Density, S = Soil, SL = Slope, LU = Land Use Land Cover, \(w\) = Normalized weight of the parameters and \(wc\) = Normalized weight of the corresponding classes of the parameters