Introduction

Mineral prospectivity mapping (MPM) aims to delineate target areas that are most likely to contain mineral deposits of a certain type in the region of interest. To achieve this goal, one of the challenges in MPM is to assign weights to the individual evidential layers that are used as indictors. A variety of methods are used to weight individual evidential layers and to integrate them into a single potential map. These methods can be categorized into three types (Bonham-Carter 1994; Carranza 2008; Yousefi and Carranza 2015c). (1) Knowledge-driven MPM methods qualitatively assess the relationship between each evidential layer and the presence of deposits of the type sought based on expert knowledge. Boolean logic (Bonham-Carter 1994), index overlay (Carranza et al. 1999), fuzzy logic (An et al. 1991; Knox-Robinson 2000), wildcat mapping (Carranza and Hale 2002; Carranza 2010), and outranking method (Abedi et al. 2013a) are examples of knowledge-driven MPM methods, which are properly used in frontier or less-explored areas (so-called greenfields) with no or very few mineral deposits of the desired type. (2) Data-driven MPM methods analyze and quantify spatial associations between each evidential layer and the locations of known deposits that share a common genesis. The ultimate aim of this process is to obtain a mineral potential map by combining quantified spatial associations. Data-driven MPM methods include weights of evidence (Liu et al. 2014; Zuo 2011), evidence belief functions (Carranza 2014; Carranza and Hale 2003; Liu et al. 2015), logistic regression (Agterberg and Bonham-Carter 1999; Carranza and Hale 2001), neural networks (Porwal et al. 2003a; Singer and Kouda 1996), etc. These methods are commonly applied in well-explored areas with sufficient known mineral deposits of the type sought. (3) There are other methods that assign weights to evidential layers using neither known mineral occurrences nor the judgments of experts (Yousefi and Carranza 2014, 2015a, b, c; Yousefi and Nykänen 2015).

The knowledge and data-driven MPM methods each have their weaknesses in application. In terms of data-driven methods, enough known mineral deposits are needed as “training points” to ensure well performance. For the knowledge-driven methods, the assignment of meaningful weights to each evidential layer is a highly subjective exercise that usually involves trial and error, even in cases where “real-expert” knowledge is available, and particularly when a number of different experts are involved. Nevertheless, the analytical hierarchy process (AHP) proposed by Saaty (1980) can resolve this difficulty in evaluating the relative importance of each evidential layer, aided by making pairwise comparisons (Carranza 2008). In addition, this method is straightforward for decision-makers (DMs) to use to structure a complex problem into a systematic hierarchy using the AHP technique. Despite of the above strengths, the AHP is criticized for expressing human judgment in crisp values in that DMs usually feel more confident to provide fuzzy judgments than crisp numbers (Dagdeviren 2008; Wang et al. 2008). As a result, fuzzy AHP and its extensions have been proposed to solve fuzzy justification problems (Laarhoven and Pedrycs 1983; Buckley 1985; Chang 1996; Xu 2000). Chang’s (1996) extent analysis for fuzzy AHP is applied in this study, which has proved to be a useful tool in mineral prospectivity mapping (Abedi et al. 2013b; Najafi et al. 2014; Pazand et al. 2014).

The purpose of this study is to highlight high potential zones of porphyry-Cu deposits in the Dananhu metallogenic belt, NW China, where only five porphyry-Cu deposits have been discovered. For this purpose, a fuzzy AHP method was applied to determine the weights of 12 evidential layers obtained from geological, geochemical, and geophysical data, based on the opinions of three DMs who are professionals in porphyry-Cu exploration. Prospectivity maps were then generated by combining the weighted evidential layers using fuzzy operators. Subsequently, prediction-area (P-A) plot (Yousefi and Carranza 2014, 2015a, b, c; Yousefi and Nykänen 2015) was then used for comparison and evaluation of the results to obtain the optimum result, and concentration-area (C-A) model (Cheng et al. 1994) was used to determine the thresholds of the optimum result for getting a ternary prospectivity map (Yousefi and Carranza 2015b, c; Yousefi and Nykänen 2015). Ultimately, the ability of the applied method was demonstrated in a spatial domain.

Study area

Geological setting

The study area is located in the southern margin of the Turfan-Hami (commonly abbreviated as TuHa) Basin in eastern Tianshan, which contains parts of the Dananhu metallogenic belt (Dong et al. 2010) and covers an area of about 15,800 km2 (Fig. 1). The region is characterized by extensive occurrences of Quaternary gravel and aeolian sand covering an area of more than 6500 km2, in which a number of massive sulfide and porphyry copper-zinc deposits were discovered. The region is chiefly composed of Devonian to Carboniferous volcanic and intrusive rocks with several genetically affiliated porphyry-Cu deposits of different sizes, including the Yandong, Tuwu, Linglong, and Chihu deposits (Zhang et al. 2006). The base of the belt is represented by basaltic to andesitic volcanic rocks, with locally overlying Lower Carboniferous carbonates and calcareous mudstones (Mao et al. 2005). The structures of the region are dominated by a series of NW- and NE-trending strike-slip faults, including the Kanggurtag fault and the Dacaotan fault. Permian and older strata have been regionally metamorphosed to lower greenschist and prehnite-pumpellyite facies (Han et al. 2006). As far as the geological setting, the Dananhu metallogenic belt is similar to the Gobi desert of southern Mongolia, in which a world-class Oyu Tolgoi Cu–Au–Mo porphyry deposit was discovered. Generally, the Dananhu metallogenic belt has prospective potential for porphyry-Cu deposits (Zhang et al. 2006).

Fig. 1
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Simplified geological map of the study area (modified from Bureau of Geology and Mineral Resources of Xinjiang)

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Porphyry-Cu mineralization model

Porphyry-Cu deposits are formed by magmatic-hydrothermal transport of metals along fractured conduits within porphyritic intrusive rocks. Commonly, once the magma solidifies, hydrothermal fluids are converted into porphyries and its surrounding host rocks (Lindsay et al. 2014; Yousefi and Carranza 2014).

In the Dananhu metallogenic belt, the heat sources for porphyry-Cu mineralization were Carboniferous to Permian intermediate-acid porphyritic intrusions, such as plagiogranite and dioritic porphyrite, which provided heat and metal-bearing magmatic-hydrothermal fluids for porphyry-Cu mineralization (Zhu 2003). The emplacement of deposits is predominantly controlled by proximity to NE-trending regional faults, which facilitate the channeling of magma and the circulation of hydrothermal fluids. In particular, the most important structural zone, the Kanggurtag and Dacaotan faults along which intense deformation, magmatic activity, and associated mineralization took place, can be regarded as a favorable indicator for the occurrence of porphyry-Cu deposits. Chen (2006) and Xiao (2013) indicate that the porphyry-Cu mineralization is mainly hosted within Carboniferous intermediate–basic volcanic rocks. In the formation of porphyry-Cu deposits, the process is commonly accompanied by hydrothermal alteration. The pattern of this alteration is characterized by sericite, silicified, pyritized, and propylitic alteration. Mineralogy and geochemistry show that the ore-bearing plagiogranite porphyries have an affinity with the early Carboniferous adakitic tonalitic rocks. It indicates that early Carboniferous tonalitic rocks play an important role for porphyry-Cu mineralization (Wang et al. 2015; Zhang et al. 2006). To summarize, the occurrence of heat sources, regional faults, favorable host rocks, and alteration are essential for the deposition of porphyry-Cu mineralization. The mineral assemblages of porphyry-Cu mineralization exhibit high concentrations of Ag, Au, Cu, Mo, Pb, and Zn, which are indicator elements for porphyry-Cu mineralization (Xiao 2013; Zhuang 2003; Zhuang et al. 2003 ). Additionally, geophysical anomalies in the airborne magnetic and Bouguer gravity data, represented by locally high magnetic and low gravity data, are symptomatic of porphyry-Cu mineralization (Zhu et al. 2003; Zhuang et al. 2003). The characteristics of porphyry-Cu mineralization discussed above are considered as a model that informs the choice of spatial data as evidential layers.

A typical Porphyry-Cu deposit

The Tuwu porphyry-Cu deposit, which reserves about 2.04 million tons of copper at an average grade of 0.67 % Cu, is one of the largest porphyry-Cu deposits in western China (Liu et al. 2003; Wang et al. 2001). The Tuwu deposit is hosted in the Carboniferous Qi’eshan Group, which can be divided into three sections. The lowest section, exposed north of the Kanggurtag fault, is composed of volcaniclastics and tuff with minor biolithite and glutenite. The middle section is represented by basalt, andesite, and dacite. The composition of volcanic rocks varies from calc-alkaline to alkaline. The upper section, distributed south of the Dacaotan fault, is mainly composed of sandstone and basalt with intercalated tuff and andesite (Wang et al. 2001). The ore-bearing plagiogranite porphyries yielded SHRIMP zircon U–Pb ages of 333 ± 2 Ma and Re–Os isochron ages of 323 ± 2.3 Ma (Liu et al. 2003; Rui et al. 2002a). Primary fluid inclusion indicates that mineralizing temperatures are of 150 to 280 °C (Rui et al. 2002b). The mineral assemblages are chiefly chalcopyrite, pyrite, chalcocite, molybdenite, quartz, and sericite. Wall-rock alteration is divided into five zones from the core to the margin in sequence: quartz core zone, biotite zone, phyllic zone, argillite zone, and propylitic zone (Wang et al. 2001).

Data and method

Data

In this study, the spatial dataset was derived from established multi-source geological spatial databases containing geological, geochemical, and geophysical data. Geological maps at a scale of 1:200,000 were collected from the Bureau of Geology and Mineral Resources of Xinjiang. The stream sediment geochemical data at a 1:200,000 scale were obtained from the National Geochemical Mapping Project of China (Xie et al. 1997). Geophysical datasets include Bouguer gravity data and airborne magnetic intensity data with a 2-km spatial resolution.

Fuzzy AHP method

This study presents the extent fuzzy AHP (Chang 1996), in which the weights of the nine-level fundamental scales of judgments are expressed via the triangular fuzzy numbers (TFNs) in order to represent the relative importance among the hierarchy criteria (Karimi et al. 2011).

The steps involved in applying the fuzzy AHP in MPM based on the paper published by Abedi et al. (2013b) are summarized as follows.

  1. Step 1.

    Construction of a hierarchy

In the first step, a complex decision problem is simplified into a hierarchy of interrelated decision elements (criteria, decision alternatives). A hierarchy has at least three levels: the first hierarchy is the goal; the middle hierarchy refers to multiple criteria that define alternatives; and the final hierarchy consists of decision alternatives (Albayrak and Erensal 2004). The hierarchical structure established to represent the interrelationships in MPM for this study is illustrated in Fig. 2.

Fig. 2
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Hierarchical structure of mineral prospectivity mapping

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  1. Step 2.

    Construct pairwise comparison matrix

A group of t decision-makers (DM p ) compares pairwise criteria and alternatives according to their relative importance with respect to a proposition, and uses the fundamental comparison scale of nine levels (Table 1) (Carranza 2008). Each DM will individually construct a pairwise comparison matrix (PCM), as shown in Eq. (1), for each criterion:

Table 1 The fundamental scale for pairwise comparisons (Saaty 1980)
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$$ D{M}_p=\left[\begin{array}{cccc}{a}_{11p}& {a}_{12p}& \cdots & {a}_{1mp}\\ {}{a}_{21p}& {a}_{22p}& \cdots & {a}_{2mp}\\ {}\vdots & \vdots & \vdots & \vdots \\ {}{a}_{m1p}& {a}_{m2p}& \cdots & {a}_{mmp}\end{array}\right]p=3,4,\dots, t, $$
(1)

where \( {a}_{ijp} \) is the quotient of weights of the alternatives, m is the number of alternatives for each criterion, and t is the number of DMs.

  1. Step 3.

    Check for consistency ratio (CR)

If the pairwise comparison matrix DM p  = (a ijp ) m × n satisfies a ijp  = a ikp  × a kjp for any i, j, k = 1, …, m, then DM p is considered to be perfectly consistent; otherwise, it is said to be inconsistent. The consistency index (CI) is:

$$ \mathrm{C}\mathrm{I}=\frac{\lambda_{\max } - n}{n-1} $$
(2)

where λ max is the maximum eigenvalue of DM p .

The final consistency ratio (CR) determines whether the evaluations are sufficiently consistent, and is calculated as the ratio of the CI and the random index (RI) (Table 2):

Table 2 Random index (RI) (Saaty 1980)
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$$ \mathrm{C}\mathrm{R}=\frac{\mathrm{CI}}{\mathrm{RI}} $$

If CR ≤ 0.1, the consistency of a pairwise comparison matrix is accepted; otherwise, the pairwise comparisons must be revised in step 2.

It should be noted that the consistency of the pairwise comparison judgments not only measures the consistency of decision makers but also evaluates the quality of the model (Albayrak and Erensal 2004; Pazand et al. 2014).

  1. Step 4.

    Construct fuzzy evaluation matrix

A fuzzy number M on R is a TFN if its membership function μ M (x) : R → [0, 1] is equal to

$$ {\mu}_M(x)=\left\{\begin{array}{c}\hfill \frac{x}{m-l}-\frac{l}{m-l}, \kern1.25em x\in \left[l,\ m\right],\hfill \\ {}\hfill \frac{x}{m-u}-\frac{u}{m-u},\kern1em x\in \left[m,\ u\right],\kern0.5em \hfill \\ {}\hfill 0,\kern5.75em \mathrm{otherwise},\kern0.5em \hfill \end{array}\right\} $$
(4)

where l < < m << u, l and u stand for the lower and upper values of the support of M, respectively, and m gives the modal value of the membership function μ M (x). The triangular fuzzy number can be denoted by (l, m, u). The support of M is the set of elements {x ∈ R|l < x < u} (Chang 1996).

First, a comprehensive PCM is constructed by integrating the grades of all DMs via Eq. (5). In this way, the PCM values of DMs are transformed into TFNs to make the fuzzy evaluation matrix:

$$ \begin{array}{l}{l}_{ij}= \min \left({a}_{ijp}\right),\kern0.5em {m}_{ij}=\kern0.5em {\sum}_{p\kern0.5em =\kern0.5em 1}^t\kern0.5em \frac{a_{ijp}}{t},\kern0.5em {u}_{ij}= \max \left({a}_{ijp}\right),\kern0.5em {M}_{ij}=\left({l}_{ij},\ {m}_{ij},\ {u}_{ij}\right)\\ {}p=1,\dots, t;\ i=1,\dots, m;\ j=1,\dots, m\end{array} $$
(5)

where min (a ijp ) and max (a ijp ) indicate minimum and maximum values of the PCMs prepared by DMs for each i and j, respectively.

Second, compute the value of the fuzzy synthetic extent with respect to the ith object of m alternatives for each criterion via Eq. (6):

$$ {S}_i=\sum_{j=1}^m{M}_{ij}\otimes \kern0.5em {\left[\sum_{k=1}^m\left(\sum_{j=1}^m{M}_{kj}\right)\right]}^{-1}i,j,k=1,\dots, m $$
(6)

where all the M ij are TFNs after construction of the fuzzy evaluation matrix. Considering two TFNs, M 1 = (l 1, m 1, u 1) and M 2 = (l 2, m 2, u 2), their operational laws are as follows:

$$ \begin{array}{l}\left({l}_1,{m}_1,{u}_1\right)\otimes \left({l}_2,{m}_2,{u}_2\right)\approx \left({l}_1{l}_2,{m}_1{m}_2,{u}_1{u}_2\right)\\ {}\end{array} $$
(7)
$$ {\left({l}_1,{m}_1,{u}_1\right)}^{-1}\approx \left(\frac{1}{u_1},\frac{1}{m_1},\frac{1}{l_1}\right) $$
(8)

Third, calculate the degree of possibility (V) of M 2 > > M 1 via Eq. (9):

$$ V\left({M}_2>>{M}_1\right)=\kern0.75em \left\{\begin{array}{c}\kern1em 1,\kern8em \mathrm{if}{m}_2>>{m}_1\kern1em \\ {}\kern1em 0,\kern8.4em \mathrm{if}{l}_1>>{u}_2\kern1em \\ {}\kern1em \frac{l_1-{u}_2}{\left({m}_2-{u}_2\right)-\left({m}_1-{l}_1\right)},\kern2em \mathrm{otherwise}\kern1em \end{array}\right. $$
(9)

To compare M 1 and M 2, it is necessary to consider both values of V(M 2 > > M 1) and V(M 1 > > M 2).

Finally, the degree of possibility (V) that a convex fuzzy number is greater than k convex fuzzy numbers M i (i = 1, 2, …, k) can be defined by the following equation:

$$ \begin{array}{l}V\left(M>>{M}_1,{M}_2,\dots, {M}_k\right)=V\left[\left(M>>{M}_1\right)\ and\left(M>>{M}_2\right)\ \mathrm{and}\dots \mathrm{and}\left(M>>{M}_k\right)\right]\\ {}= \min V\left(M>>{M}_1\right)\kern0.75em i=1,\ 2,\dots, k\end{array} $$
(10)

Assume that d(B i ) = min V(S i  > > S k ), k = 1, …, m, and k ≠ i. The weight vector is then given by

$$ W^{\prime }={\left(d\prime \left({B}_1\right),\dots, d\prime \left({B}_m\right)\right)}^T $$
(11)

Where B i (i = 1, …, m) has m elements.

  1. Step 5.

    Calculate normalized weights

Via normalization, the normalized weight vectors are

$$ W={\left(d\left({B}_1\right), \dots,\ \mathrm{d}\left({B}_m\right)\right)}^T $$
(12)

where W is a non-fuzzy number.

As pointed out by Wang et al. (2008), the weights determined by the extent analysis method do not represent the relative importance of decision criteria or alternatives and could not be used to give their priority, on condition that irrational zero weights are assigned to some useful decision criteria and alternatives.

  1. Step 6.

    Using fuzzy operators

There are five fuzzy operators that are useful for integrating the weighted evidential layers to generate the final potential map: fuzzy AND, fuzzy OR, fuzzy algebraic product, fuzzy algebraic sum, and fuzzy gamma (γ) (An et al. 1991; Bonham-Carter 1994).

Prediction-area plot

The P-A method is a simple prediction rate-occupied area plot, which can be used not only to compare and evaluate the ability of different prospectivity models in predicting mineral deposits, but also to assign weights to evidential layers (Yousefi and Carranza 2014, 2015a, c; Parsa et al. 2016).

In a P-A plot, there are two curves, the curve of the percentage (prediction rate) of known mineral occurrences corresponding to the classes of the prospectivity map and the curve of the percentage of occupied areas corresponding to the classes of the prospectivity map. When an intersection point of the two curves is at a higher place in the P-A plot, it portrays a small area containing large number of mineral deposits. Furthermore, it chooses objectively a better model to give priority for mineral exploration (Yousefi and Carranza 2014, 2015a, c).

Application of the fuzzy AHP to porphyry-Cu deposits

Criteria for MPM

Considering expert opinions, the geological setting, the model of porphyry-Cu mineralization, a typical porphyry-Cu deposit, and the available data in the study area, three main criteria, including geological data, stream sediment geochemical data, and geophysical data, were used as input evidential layers to provide the mainstays in prospecting for porphyry-Cu deposits. The three main criteria consisted of (1) host rock lithology, (2) intrusive rock lithology as heat sources, (3) the density of faults and the distance to faults as structure, (4) different alteration zones, (5) geochemical anomalies of indicator elements Au, Ag, Cu, Mo, Pb, and Zn, and (6) the intensity of airborne magnetic and Bouguer gravity data, which are the most significant alternatives for exploration and characterization of porphyry-Cu deposits.

Preparation of evidential layers

Data preprocessing

The selection of evidential layers requires extensive consideration of the characteristics of porphyry-Cu deposits and the favorable conditions for mineralization. As discussed above, host rock lithology, intrusive rock lithology, alteration types, and faults were extracted and compiled from geological maps to obtain evidential layers.

The different types of intrusive rocks were extracted as separate maps, and each map of intrusive rocks and the host rock map were buffered into three zones, each 1 km wide, up to a distance of 3 km. Similarly, maps were generated for the different types of alteration, and these were buffered into three zones, each 0.5 km wide, up to a distance of 1.5 km. The density and distance of faults were analyzed by ArcGIS software. The density map was divided into nine equal-sized zones in the range 0–1.53 length per unit area, and values greater than 1.53 comprised the tenth zone. In this way, six 1-km-interval zones around the faults formed the sections of the distance map, and the seventh zone comprised area at a distance greater than 6 km.

Stream sediment geochemical anomalies were analyzed using the singularity mapping technique (Cheng 2007; Zuo et al. 2009). In simple terms, the element content of Au, Ag, Cu, Mo, Pb, and Zn were used as six other evidential layers, which were discretized into three classes. The thresholds of evidential layers corresponded to the second, fifth, and tenth quantiles. In addition, magnetic and gravity intensity data were assigned to ten classes, providing the evidential layers for airborne magnetic and Bouguer gravity data.

Data encoding

After categorizing evidential maps, the classes of the evidential maps must be ranked according to expert opinions of geoscientists. Based on Table 3, the above multi-class evidential maps were coded with integer values from 1 to 10. It should be noted that maps of different intrusive rocks, maps of different alterations, the density map, and the distance map were merged to generate the evidential layers for heat source, alteration, and structure. To simplify the process and improve efficiency, the above evidential layers were converted into grid format with a pixel size of 1 km.

Table 3 Summary of classes and class scores with respect to evidence layers
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Weights of evidential layers

In this study, three DMs with expertise in porphyry-Cu mineralization were invited to compare the relative importance of hierarchical elements using the scale in Table 1. In this phase, pairwise comparison matrices were formed to construct fuzzy evaluation matrices in case that CR ≤0.1.

All consistency ratios derived from the PCMs are less than 0.1 (Table 4). Consequently, the results were considered reasonable. The values of PCMs are then transformed into TFNs to construct fuzzy evaluation matrices presented in Tables 5, 6, 7, and 8.

Table 4 Consistency ratio (CR) for pairwise comparison matrix
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Table 5 Fuzzy evaluation matrix with respect to criteria
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Table 6 Fuzzy evaluation matrix with respect to geological alternatives
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Table 7 Fuzzy evaluation matrix with respect to geochemical alternatives
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Table 8 Fuzzy evaluation matrix with respect to geophysical alternatives
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After establishing fuzzy evaluation matrices, the weights of 12 evidential layers for mineral exploration can be calculated by the fuzzy AHP method. Taking geological alternatives as an example, details of calculating weights from the fuzzy evaluation matrix (Table 6) are given below.

From Eq. (6), the value of the fuzzy synthetic was computed as follows:

\( \begin{array}{l}{S}_{\mathrm{Host}\ \mathrm{rock}}=\left(1.8095,\ 3.0698,\ 5.2\right)\otimes \left(\frac{1}{42.95},\frac{1}{29.1396},\frac{1}{17.6191}\right)=\left(0.0421,\ 0.1053,\ 0.2951\right)\\ {}{S}_{\mathrm{Heat}\ \mathrm{source}}=\left(12,\ 15.6667,\ 20\right)\otimes \left(\frac{1}{42.95},\frac{1}{29.1396},\frac{1}{17.6191}\right)=\left(0.2794,\ 0.5376,\ 1.1351\right)\\ {}{S}_{\mathrm{Structure}}=\left(1.9167,\ 4.9833,\ 9.5\right)\otimes \left(\frac{1}{42.95},\frac{1}{29.1396},\frac{1}{17.6191}\right)=\left(0.0446,\ 0.1710,\ 0.5392\right)\\ {}{S}_{\mathrm{Alteration}}=\left(1.8929,\ 5.4198,\ 8.25\right)\otimes \left(\frac{1}{42.95},\frac{1}{29.1396},\frac{1}{17.6191}\right)=\left(0.0441,\ 0.1860,\ 0.4682\right)\end{array} \) The degrees of possibility of these fuzzy values were then determined from Eq. (9) as follows:

$$ \begin{array}{l}V\left({S}_{\mathrm{Hostrock}}\ge {S}_{\mathrm{Heatsource}}\right)=0.035,\kern1.5em V\left({S}_{\mathrm{Heatsource}}\ge {S}_{\mathrm{Hostrock}}\right)=1\\ {}V\left({S}_{\mathrm{Hostrock}}\ge {S}_{\mathrm{Structure}}\right)=0.7922,\kern1.8em V\left({S}_{\mathrm{Heatsource}}\ge {S}_{\mathrm{Structure}}\right)=1\\ {}V\left({S}_{\mathrm{Hostrock}}\ge {S}_{\mathrm{Alteration}}\right)=0.7567,\kern1.4em V\left({S}_{\mathrm{Heatsource}}\ge {S}_{\mathrm{Alteration}}\right)=1\\ {}V\left({S}_{\mathrm{Structure}}\ge {S}_{\mathrm{Hostrock}}\right)=1,\kern4em V\left({S}_{\mathrm{Alteration}}\ge {S}_{\mathrm{Hostrock}}\right)=1\\ {}V\left({S}_{\mathrm{Structure}}\ge {S}_{\mathrm{Heatsource}}\right)=0.4148,\kern1.2em V\left({S}_{\mathrm{Alteration}}\ge {S}_{\mathrm{Heatsource}}\right)=0.3494\\ {}V\left({S}_{\mathrm{Structure}}\ge {S}_{\mathrm{Alteration}}\right)=0.9706,\kern1.5em V\left({S}_{\mathrm{Alteration}}\ge {S}_{\mathrm{Structure}}\right)=1\end{array} $$

Finally, the weights were assigned and normalized using Eqs. (10) and (12):

$$ \begin{array}{l}V^{\prime}\left(\mathrm{Host}\ \mathrm{rock}\right)= \min \left(0.035,\ 0.7922,\ 0.7567\right) = 0.035\\ {}V^{\prime}\left(\mathrm{Heat}\ \mathrm{source}\right)= \min \left(1,\ 1,\ 1\right)=1\\ {}V^{\prime}\left(\mathrm{Structure}\right)= \min \left(1,\ 0.4148,\ 0.9706\right) = 0.4148\\ {}V^{\prime}\left(\mathrm{Alteration}\right)= \min \left(1,\ 0.3494,\ 1\right) = 0.3494\end{array} $$

The weights vector was (0.035, 1, 0.4148, 0.3494) and the normalized weights vector was calculated as (0.0195, 0.5558, 0.2305, 0.1492). In the same way, all normalized weights were obtained from fuzzy evaluation matrices. Final normalized weights for each criterion and alternative are presented in Table 9. It is apparent that heat source, structure, alteration, and airborne magnetic data have higher values, so they play a significant role in prospecting for porphyry-Cu mineralization in the study area.

Table 9 Weights of criteria and alternatives
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Integration of evidential layers

The corresponding evidential layers must be multiplied by final weights before employing fuzzy operators (Fig. 3). The evidential layers were combined using γ values of 0.78, 0.83, 0.88, and 0.93 to generate prospectivity maps (Fig. 4). Subsequently, these maps are divided into ten natural break (Jenks) classes. Each prospectivity map was evaluated and compared using the P-A plot. Figure 5 and Table 10 show that the prospectivity map obtained using a γ value of 0.83 was appropriate because it quantitatively depicts a smaller area containing the same number of mineral deposits. For further evaluation of this prospectivity map, the C-A model was used to determine the thresholds to defuzzify the map. The ternary map (Fig. 6) was generated using two thresholds, shown in Fig. 7, which identified the appropriate areas for porphyry-Cu mineralization.

Fig. 3
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Weighted evidential layers: a Host rock, b Heat source, c Structure, d Alteration, e Magnetic anomaly, f Gravity anomaly, g Au anomaly, h Ag anomaly, i Cu anomaly, j Mo anomaly, k Pb anomaly, and l Zn anomaly

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Fig. 4
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Prospectivity maps obtained using γ values of a 0.78, b 0.83, c 0.88, and d 0.93

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Fig. 5
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Prospectivity maps obtained using γ values of a 0.78, b 0.83, c 0.88, and d 0.93

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Table 10 Extracted parameters from intersection point of P-A plots in Figure 5
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Fig. 6
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Ternary prospectivity map generated by defuzzification of the prospectivity map (Fig. 4b)

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Fig. 7
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Concentration-area (C-A) model for the prospectivity map (Fig. 4b)

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Results and discussion

As mentioned above, the AHP’s pairwise comparison is made in crisp values, relying on expert knowledge. Any incorrect opinions of the expert can convey into the assignment of weights. In this process, the vagueness and uncertainty introduced into pairwise matrix lead to the difficulty providing exact weights for evidential layers (Feizizadeh and Blaschke 2013; Feizizadeh et al. 2014). To overcome this problem, fuzzy AHP use TFNs to simulate DMs’ preference in pairwise comparison process. In the fuzzy AHP method, each choice of the relative importance of hierarchy criteria is expressed by a vector, which is better to simulate human judgment than crisp comparison. Therefore, the fuzzy AHP method can provide more realistic weights than other knowledge-driven MPM methods

One of the key procedures in the implementation of the fuzzy AHP modeling is the selection of fuzzy operators. Knox-Robinson (2000) pointed out that fuzzy γ operator is useful and realistic, which focuses on balancing the “decreasive” and “increasive” effects of fuzzy algebraic product and fuzzy algebraic sum operators. Using appropriate values of γ can control the propagation of extreme-value noise to the final prospectivity map (Porwal et al. 2003b). The ultimate aim for carefully tempering the value of γ is to select one that provides the “best” result. A number of γ were tried, values of 0.78, 0.83, 0.88, and 0.93 were selected to generate prospectivity maps. However, Fig. 4 shows that the prospectivity maps for different γ values are remarkably similar. For avoiding subjective judgment, the P-A method (Yousefi and Carranza 2014, 2015a, b, c; Yousefi and Nykänen 2015) is used to select the “best” result. Consequently, the result of applying a γ value of 0.83 is the best one in that it reduces the target area of the study area while predicting the same number of known deposits, and it is meaningful in mineral exploration.

In the ternary map, high favorability zones occupy only 7.16 % of the study area, and these zones are mostly in the south and southwest; the moderate favorability zones are mainly close to the high favorability zones, occupying 18.61 % of the study area. The spatial distribution of high favorability zones is confined to specific intrusive rocks with a composition of diorite and monzodiorite, which emphasizes a strong heat sources control of porphyry-Cu mineralization in the study area. This conforms to characterizations of spatial associations between geological features and porphyry-Cu deposits. However, it fails to predict the Chihu deposit, a small, low-grade porphyry-Cu deposit, in high favorability zone. The metallogenetic epoch and the metallotectonic setting of the Chihu deposit, discovered in 1986, are uncertain as its formation age is vigorously debated (Ji and Sun 2011; Wu et al. 2006). Therefore, it is difficult to predict such a deposit in high favorability zones based on regional scale data and criteria for MPM. The larger scale data and local scale criteria for porphyry-Cu mineralization are synthesized to predict the Chihu deposit and demarcate other prospects within the predicted potential targets for guiding follow-up exploration using the same method. Therefore, high favorability zones and the specific moderate favorability area containing the Chihu deposit should be prioritized in the exploration of porphyry-Cu deposits.

The study area with only five porphyry-Cu deposits restricted the validation of prospectivity model using the known mineral deposits. From the perception of spatial domain, the spatial distribution of high favorability zones is in conformity with the model of porphyry-Cu mineralization. It well illustrates the effectiveness of fuzzy AHP method for porphyry-Cu deposits in this study. In addition, field observations will be used to evaluate target areas in the future.

Conclusions

  1. 1.

    The application of the fuzzy AHP approach simulates human judgment for the relative importance of hierarchy criteria, as well as reducing the vagueness and uncertainty of crisp comparison via the triangular fuzzy numbers. Furthermore, more realistic weights for evidence layers are provided than those given by other knowledge-driven MPM methods.

  2. 2.

    A key element of the proposed approach is the use of different values of γ to obtain the most suitable potential map, which has an advantage over other knowledge-driven MPM methods in that it avoids subjective opinions when selecting the final potential map.

  3. 3.

    The P-A plot provides an objective method to weight the relative effectiveness in terms of reducing the exploration area while decreasing the exploration costs, when the prospectivity maps for different values of γ are remarkably similar.

  4. 4.

    The ternary map shows a strong spatial correlation between high favorability zones and specific intrusive rocks composed of diorite and monzodiorite, which is consistent with the model of porphyry-Cu mineralization, thus, the fuzzy AHP method for mapping prospectivity for porphyry-Cu deposits is valid in this study.

  5. 5.

    In this study, potential targets are delineated in the Dananhu metallogenic belt. For obtaining a more detailed result, with the larger scale data, the same method needs to further delineate other prospects within the predicted potential targets to guide follow-up exploration.

  6. 6.

    The fuzzy AHP method described in this paper provides a simple yet effective method for prioritizing potential targets in “greenfield” areas.