## 4 Test  
### Sample Data  
A total of 138 fresh rock and soil samples from diverse geological settings, representing commonly observed geological materials in nature, were selected to test the fitting results of Equation (3) (Table 3). Within the 138 samples, there are 88 rock samples, including 20 andesitic specimens from the Bazman and Taftan volcanoes, located in the Makran subduction zone of southern Iran [33]; 58 calc-alkaline andesitic specimens from the Rishiri volcano in the southern Kuril Islands, Japan [34]; and 10 carbonatites from the Wushi region of the Tarim Basin in northwestern China [35]. These samples represent geological materials on different lith \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Eq.** & **Conditions** & **Equation** & \(\mathbf{x}\) & \(\mathbf{y}\) & \(\mathbf{z}\) & **n** & **r** \\
**No.** & & & & & & & \\ \hline (3a) & SiO\({}_{2}\) \(<\) 33 & lgCr = 0.0634\(x+0.000308y\) & Al\({}_{2}\)O\({}_{3}\) & Ti & / & 28 & 0.921 \\ (3b) & MgO \(>\) 3.9 & lgCr = 0.4818x\(y+0.8408x\) & lgMgO & lgNa\({}_{2}\)O & / & 51 & 0.980 \\ (3c) & WIG \(\leq\) 65 & lgCr = 0.8753\(x-91.15y-1.47\) & lgTe\({}_{2}\)O\({}_{3}\) & Al\({}_{2}\)O\({}_{3}\)/Ti & / & 327 & 0.904 \\  & & & & & & & \\  & & & & & & & \\ (3d) & Na\({}_{2}\)O \(\leq\) 3 & lgCr = -150.2x + 0.1928\(y\) & Al\({}_{2}\)O\({}_{3}\)/Ti & MgO/(Na\({}_{2}\)O & lgTe\({}_{2}\)O\({}_{3}\) & 218 & 0.946 \\  & & & & & & + K\({}_{2}\)O))) & & \\ (3e) & Others & lgCr = -4.567 xyz + 6.612 xz + 12.99 yz - 1.117y - 19.71z + 2.47 & lgMn & lg(Al\({}_{2}\)O\({}_{3}\)/MgO) & lg(Na\({}_{2}\)O/Te\({}_{2}\)O\({}_{3}\)) & 120 & 0.987 \\ \hline \end{tabular} Notes: Units of major oxides are weight percent (%) and others are \(\mu\)g/g. \(x\), \(y\), and \(z\) are used here for a brief writing of the equation.  
\end{table} 
Table 2: Equations derived from piecewise multiple regression on 791 records. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**References** & **Types** & **Counts** & **Cr (min)** & **Cr (max)** & **Sub-Equation** & **Fitting** \\  & & & & & **Used** & **Accuracy** \\ \hline [PERSON] et al. [33] & Andesites & 20 & 38 & 256 & (3b)–(3e) & 95\% \\ [PERSON] et al. [34] & Calc-alkaline andesites & 58 & 35.8 & 100.6 & (3b), (3e) & 100\% \\ [PERSON] et al. [35] & Carbonatites & 10 & 6.3 & 19.2 & (3a) & 90\% \\ [PERSON] et al. [36] & Soils & 50 & 18.6 & 80.9 & (3c), (3d) & 100\% \\ Total statistic & / & 138 & 6.3 & 256 & 5 sub-equations & 98.6\% \\ \hline \end{tabular} Notes: Unit of Cr concentration is \(\mu\)g/g. A logarithmic residual of \(\pm\)0.2 was used to calculate the fitting accuracy.  
\end{table} 
Table 3: Information of 138 samples to test the fitting results of Equation (3). Regarding the 20 andnesitic samples, they are from two volcanoes. The Bazman volcano samples were primarily formed from low to medium potassium basic to acidic magmas, generated through typical arc magmatism. In contrast, the Taftan samples, which display medium to high potassium characteristics, are attributed to an enriched mantle source, with a higher contribution of subducted sediments and the influence of a thicker continental crust. Cr concentrations in the Bazman and Taftan samples range from 38 to 256 \(\upmu\)g/g [33]. The 58 calc-alkaline andnesitic samples from the Rishiri volcano were mainly collected in the main structural body of the volcano. The volcano originated from the mixing of three distinct magmatic sources: a primitive mafic end-member, an evolved mafic end-member, and a felsic end-member. Cr concentrations in these samples range from 35.8 to 100.6 \(\upmu\)g/g [34]. The 10 carbonate samples from Wushi, Tarim Basin, are primarily composed of calcite (90-95%), with minor components of biotite (3-4%) and apatite (2-3%). These samples exhibit a reddish color and crystalline texture, with Cr concentrations ranging from 6.3 to 19.2 \(\upmu\)g/g. Their mineralogical and chemical characteristics classify them as igneous carbonates [35].