## 2 Materials and Methods
### Methodology
#### 2.2.4 Data Post-Processing  
After obtaining the object-oriented anomaly score map, we need a binarization threshold defining whether a given image object is anomalous or not. In this article, the threshold value is empirically set as the 50% quantile (median) of the object-oriented anomaly score matrix, which is 0.089434 for the I-type elemental association, 0.091733 for the S-type association, and 0.12807 for all the ore-forming elements. In addition, as the anomaly score is an object-level summary statistics, so our post-processing operations will focus on the within-object statistical analysis as mentioned in the Introduction. From the geological perspective, differentiation-dominant processes are believed to be the pivotal cause of metallogenesis [30], and the fractal dimension (\(D\)) [15] is introduced here to describe the differentiation degree of each anomalous patch. As a within-object statistical parameter, \(D\) is used to quantify the power-law relationship between multi-elemental contents and their cumulative summation in a given image object. It can be computed and interpreted in the following way:  
\[N(\geq r)=Cr^{-D}\;(r>0) \tag{7}\]  
where \(r\) is the multi-elemental content value within an image object, and \(N(\geq r)\) denotes the summation of content values larger than or equal to a given \(r\); \(C>0\) is a proportionality coefficient, and the exponent \(D\) is known as the fractal dimension. Mathematically, a number of straight-line segments (namely scaleless intervals) can be derived from Formula (7) on the log-log paper. It aims to cluster a dataset into most similar groups in the same segment and most dissimilar groups in different segments [15]. For two adjacent straight lines with different slopes, \(D_{\mathrm{n}}\) & \(D_{\mathrm{n+1}}\), the inflection point (\(T\)) is routinely determined by RSS:  
\[\mathrm{RSS}=\sum_{i=1}^{\mathrm{i_{0}}}[\ln\mathrm{N}(r_{\mathrm{i}})+\mathrm{ D}_{1}\ln r_{\mathrm{i}}-\ln C_{1}]^{2}+\sum_{i=\mathrm{i_{0}}+1}^{\mathrm{N}}[ \ln\mathrm{N}(r_{\mathrm{i}})+\mathrm{D}_{2}\ln r_{\mathrm{i}}-\ln C_{2}]^{2 }\rightarrow\mathrm{Min} \tag{8}\]  
where RSS denotes the \"residual sum of squares\", and \(r_{i_{0}}\) is the dividing threshold (namely \(T_{\mathrm{n}}\), \(\mathrm{n}=1,2,\dots\)). In a similar fashion, the slopes of several scaleless segments, as well as the thresholds between them, can be quantitatively determined. As shown in Figure 7, in this step, they are customarily classified into two segments, \(D_{1}\) and \(D_{2}\). According to [PERSON] et al. [15], the dimension \(D\) can be used to measure the clustering degree of a dataset, which is a diagnostic spatial distribution pattern for ore-related geological objects. The larger the \(D\) value, the steeper is the line, the greater is the rate of change in elemental content, the lower is the degree of clustering, and vice versa. Note that \(D_{1}\) usually represents the low-content background information, and it does not change much over different image objects, so we use \(D_{2}\) as an indicator to quantify the multi-elemental clustering. Suppose there are \(L\) pixels in an image object, and each object has \(F\) data layers representing different geochemical variables, and then, all of the \(L\times F\) content values will be fed into the fractal algorithm aforementioned. As the selected elements are paragenetically associated with each other, and their content values are normalized to [0, 1] by subtracting the mean values, so if the metallogenesis or differentiation occurs, the content values of these elements within a given object will become more clustered, which corresponds to a smaller \(D_{2}\); otherwise, they remain scattered because of the weak feature of elemental association, which corresponds to a larger \(D_{2}\) value.