## 2 Material and Methods
### Geophysical and Remote Sensing Data
#### 2.4.3 Fuzzy Logic Modelling of Radiometric and ASTER Satellite Imagery  
Fuzzy logic modelling is a widely and successfully used technique in mining mapping and mainly for the development of mineral prospectivity maps [8; 66; 68; 69; 70]. Mathematically, it is a form of multi-valued logic based on the fuzzy set theory where the real values of the variables are included in the interval [0-1]; zero corresponds to non-membership, and 1 corresponds to full membership [71]. This method was first proposed in 1965 by [PERSON] and is defined as follows:  
\[\text{A}_{ij}=\left\{\begin{array}{ll}\left(\text{x}_{ij},\text{ }\mu_{\text{A}}\right)/\text{X}_{ij}\in\text{X}_{i}\end{array}\right\}, \text{ }0\text{ }\leq\text{ }\mu_{\text{A}}\text{ }\leq 1\]  
where \(\mu_{\text{A}}\) is called the degree of membership function of x in A and X corresponds to a set of layers X\({}_{i}\) (\(i\) = 1, 2, 3, \(\ldots\), n), and each layer to r classes defined as (\(j\) = 1, 2, 3, \(\ldots\), r).  
The degree of membership \(\upmu\) A plays in the interval [0-1], so: If \(0\leq\mu_{\text{A}}<0.5\): x\({}_{ij}\) is not conducive to mineralization; if \(\mu_{\text{A}}\) = 0.5: we cannot determine is x\({}_{ij}\) conducive to mineralization or not; if 0.5 < \(\mu_{\text{A}}\)\(\leq\) 1 it means that x\({}_{ij}\) is conducive to mineralization.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Subsystem** & **Band No.** & **Spectral Range** & **Radiometric** & **Absolute Accuracy** & **Spatial** & **Signal** \\  & **(\(\upmu\)m)** & **Resolution** & **Absolute Accuracy** & **Resolution (m)** & **Quantization (Bits)** \\ \hline \multirow{4}{*}{VNIR} & 1 & 0.52–0.60 & & & & \\ \cline{2-6}  & 2 & 0.63–0.69 & & NE\(\upalpha\)\(\leq\) 0.5\% & \(\leq\)\(\pm\)4\% & 15 & 8 \\ \cline{2-6}  & 3 & 0.78–0.86 & & & & \\ \cline{2-6}  & 3N & 0.78–0.86 & & & & \\ \hline \multirow{4}{*}{SWIR} & 4 & 1.60–1.70 & NE\(\upalpha\)\(\leq\) 0.5\% & & & \\ \cline{2-6}  & 5 & 2.145–2.185 & NE\(\upalpha\)\(\leq\) 1.3\% & & & \\ \cline{2-6}  & 6 & 2.185–2.225 & NE\(\upalpha\)\(\leq\) 1.3\% & & & \\ \cline{2-6}  & 7 & 2.235–2.285 & NE\(\upalpha\)\(\leq\) 1.3\% & & 30 & 8 \\ \cline{2-6}  & 8 & 2.295–2.365 & NE\(\upalpha\)\(\leq\) 1.0\% & & & \\ \cline{2-6}  & 9 & 2.360–2.430 & NE\(\upalpha\)\(\leq\) 1.3\% & & & \\ \hline \multirow{4}{*}{TIR} & 10 & 8.125–8.475 & & \(\leq\)3 K (200–240 K) & & \\ \cline{2-6}  & 11 & 8.475–8.825 & & \(\leq\)2 K (240–270 K) & & \\ \cline{1-1} \cline{2-6}  & 12 & 8.925–9.275 & NEAT \(\leq\) 0.3 K & \(\leq\)1 K (270–340 K) & 90 & 12 \\ \cline{1-1} \cline{2-6}  & 13 & 10.25–10.95 & & \(\leq\)2 K (340–370 K) & & \\ \cline{1-1} \cline{2-6}  & 14 & 10.95–11.65 & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Characteristics and performance of the ASTER sensor [67].  
In this study, we applied this analysis to produce mineral prospectivity maps following the methodological approach presented in the Figure 3. The first map combines the four layers corresponding to the band ratios extracted from the ASTER image (Table 3). Meanwhile, the second map combines the geophysical data corresponding to the K, K/eTh ratio and K/eU ratio layers. These layers were fuzzified individually using the linear membership function. Then the fuzzy gamma operator was applied to combine our thematic layers. The choice of this operator is based on the fact that it is a compromise between the fuzzy algebraic sum and its product [72]. In addition, it is possible to develop improvements on the input of the resulting maps [73]. After several trials, we took 0.72 as the value of parameter \(\gamma\). The fuzzification parameters used for the input data are represented in the Table 4.