## 2 Materials and Methods
### Supervised Classification Algorithms
#### 2.3.1 Support Vector Machine (SVM)  
The SVM is widely applied in mineral exploration, especially in processing remote sensing data [6; 26; 29; 56]. As defined by [PERSON] [57], SVM can solve a quadratic optimization problem by creating nonlinear decision boundaries in high-dimensional variable space [58]. According to basic SVM theory, many hyperplanes can split different classes for a nonlinearly separable dataset, including points from two classes. Therefore, only a subset of \"k\" training samples, known as support vectors, are used to identify a hyperplane that best divides classes (i.e., the decision boundary). The optimal decision boundary between different classes is defined as those with the maximum margin M = \(\frac{2}{\|w\|}\) (distance) between the support vectors. SVM identifies M in nonseparable linear instances while considering a cost parameter C, which provides a penalty for misclassifying support vectors. The objective function must be adjusted to include this penalty component for wide-margined decision boundaries with misclassified support vectors, as shown below. Minimize  
\[\frac{\|w\|}{2}+C\sum_{n=1}^{k}\varepsilon_{n} \tag{1}\]  
Subject to the hyperplane boundary conditions  
\[ \begin{array}{l}y_{n}(w.x+b)\geq 1-\varepsilon_{n}\\ \varepsilon_{n}>0\text{ and }n=0,1,2,3\dots\dots\text{.}k\end{array}  \tag{2}\]  
where \(\|w\|\) is the Euclidean norm of \(w\), the hyperplane orientation vector that controls the hyperplane's orientation; b is the hyperplane's offset from the origin, and \(\varepsilon_{n}\) is the positive slack variable indicating the error distance between nth misclassified support vector and its marginal hyperplane. SVM uses an implicit transformation \(\varphi(x)\) of input data \"\(x\)\" for cases where classes are not linearly differentiable [59]. \[Z(xi,xj)=\varphi(xi).\varphi(xj) \tag{3}\]  
where \(w\) is the Euclidean norm of \(w\), the hyperplane orientation vector that controls the hyperplane's orientation; b is the hyperplane's offset from the origin, and \(\varepsilon_{n}\) is the positive slack variable indicating the error distance between nth misclassified support vector and its marginal hyperplane. SVM uses an implicit transformation \(\varphi(x)\) of input data \"\(x\)\" for cases where classes are not linearly differentiable [59]. \[Z(xi,xj)=\varphi(xi).\varphi(xj) \tag{3}\]  
Equation (3) provides the inner product kernel \(Z(xi,xj)\) of pairwise positions in variable space compared to input variables. Kernel functions allow SVM to handle nonlinear relationships efficiently by creating linear hyperplanes that separate nonlinearly separable support vectors by projecting samples from the original d-dimensional space into potentially infinite-dimensional kernel space. In this situation, the decision function's form is expressed as:  
\[M:M(x)=sgn(w.Z(xi,xj)+b) \tag{4}\]  
The choice of kernel functions is critical for SVM training and classification accuracy. Different kernel types available for SVM are polynomial kernel (PL), radial basis function (RBF) kernel, and sigmoid kernel (SIG). The polynomial with degree 1 is the simplest kernel; hence, it learns patterns faster than other kernels. The two most essential parameters other than kernel types that affect the performance of SVM are the penalty parameter C and the gamma coefficient. If the penalty parameter limits the error level to be accepted in the training data, the gamma parameter controls the degree of nonlinearity of the SVM model. For example, a very high value of the cost parameter C results in a complex margin, which reduces the training error. In contrast, a small value of the cost parameter creates a large margin, resulting in a significant error in the training data [60]. Google Earth Engine's built-in SVM classifier was used for lithological mapping. The training data selected for training the ML algorithm were directly fed to SVM. The most critical parameters that affect the performance of SVM are the kernel type, cost parameter (C), and gamma coefficient [4].