## 4 Methodology
### Spectrum-Area (S-A) Model  
The scale-invariant properties of various geological processes, such as erosion, mineralization, the magnetic field of the Earth's crust, earthquake distribution, and volcanic eruptions, often exhibit \"self-similarity\" or \"self-affinity\". These properties can be measured in both the frequency and spatial domains [50]. In the frequency domain, power spectra can be used to represent such properties [51]. Spectral energy density functions illustrate the distribution of power spectra in the frequency domain, whereas for certain complex convolution operations in the spatial domain, such as correlation analysis, filtering, and transformation, dealing with fields in the frequency domain can significantly simplify the process [5].  
Signal processing and time series analysis commonly employ Fourier and inverse Fourier transformations [51; 52]. The power spectrum-area (S-A) model is a fractal filtering technique based on Fourier spectral analysis that is used to separate anomalies from background values [51]. Scaling properties in the spatial domain are related to the spatial geometry of patterns, histogram distributions of values, and changes in shape corresponding to changes in value. This is employed in the multifractal inverse distance weighted (MIDW) interpolation method. A fractal filter, derived from the S-A model [51], defines components with similar scaling properties based on the power-law characteristics of a power spectrum in the frequency domain [53]. This filter not only identifies anomalies from the background but also extracts significant patterns from the original map. Due to the complex intrinsic structure of geochemical datasets and anisotropy, the filter has an irregular shape [54]. Detailed equations for S-A multifractal filtering are shown in Equations (11) and (12) [54].  
\[F(w_{x},w_{y})=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\text{cos} (w_{x}x+w_{y}y)dxdy-i\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\text {sin}(w_{x}x+w_{y}y)dxdy \tag{11}\]  
\[f(x,y)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}F(w_{x},w_ {y})\text{cos}(w_{x}x+w_{y}y)dxdy-\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{- \infty}^{\infty}F(w_{x},w_{y})\text{sin}(w_{x}x+w_{y}y)dxdy \tag{12}\]  
In Equation (11) and Equation (12), \(F(w_{x},w_{y})\) represents the signal in the frequency domain, \(f(x,y)\) represents the geochemical map in the space domain, \(w_{x}\) and \(w_{y}\) are the \"wave numbers\" of the x and y axes respectively, \(\hat{r}^{2}=-1\).  
From a generalized scale invariance (GSI) viewpoint, the S-A fractal model [55; 56] is based on the power-law relationships between areas of sets of data consisting of wave numbers with spectral energy density above \(S[A(>)S]\)] on the 2D frequency domain, and gives  
\[A(\geq S)\propto S^{-2d/\beta} \tag{13}\]  
where \(\beta\) is the anisotropic scaling exponent, and \(D\) is the GSI generator parameter representing the degree of overall contraction [57]. For a 2D linear case, \(d=1\); thus  
\[A(\geq S)\propto S^{-2/\beta} \tag{14}\]  
The expression \(A(S\geq s)\) denotes the area where the spectral density (\(S=\|\mathrm{F}(w_{x},w_{y})\|\)) exceeds a certain value \(s\). \(\beta\) represents the fractal dimension, and \(\infty\) stands for proportional.  
\[G(w_{x},w_{y})=\begin{cases}1,s_{1}\leq S(w_{x},w_{y})\leq s_{2}\\ 0,\hskip 28.452756 pt\mathrm{other}\end{cases} \tag{15}\]  
\[FF(w_{x},w_{y})=F(w_{x},w_{y})G(w_{x},w_{y}) \tag{16}\]  
Noise exists where spectral density is less than \(s_{1}\); background occurs when spectral density is greater than \(s_{2}\); and anomalies exist where spectral density is between \(s_{1}\) and \(s_{2}\).  
These power-law relations, including those in space domains [58] and Eigen domains [59] have been demonstrated to be useful tools for identifying geochemical anomalies for mineral resource exploration [5; 60; 61] and to determine geochemical baselines for environmental assessments [1] according to distinctive generalized self-similarity.