## 3 Equation Fitting
### Fitting Results  
The fitting equation employs a piecewise multiple regression, using oxide and elemental concentrations from the 791 records as input parameters. The oxide content is reported as weight percentage (%), while the concentrations of Cr, Ti, and Mn are expressed in \(\mu\)g/g. The fitted equation includes five sub-equations sequentially as follows. (1) For samples with SiO\({}_{2}\) < 33, excluding one outlier, a linear regression analysis was derived with residual errors within a logarithmic deviation of \(\pm\)0.2 as  
\[\text{lgCr}=0.0634\text{Al}_{2}\text{O}_{3}+0.00030\text{Ti}+0.67,\text{n}=28,\text{r}=0.921,\] (3a) where Al\({}_{2}\)O\({}_{3}\) contents range from 0.66 to 4.81%, Ti contents from 185 to 1510 \[\mu\)g/g, and Cr from 4.7 to 26 \[\mu\)g/g, yielding calculated Cr background values between 5.9 and 27.1 \[\mu\)g/g (Figure 2a).  
(2) For the residual samples after step (1) with MgO > 3.9, the sub-equation was fitted, excluding three outliers, with the residual errors within a logarithmic deviation of \(\pm\)0.2 as  
\[\text{lgCr}=0.4818\text{lgMgOlgNa}_{2}\text{O}+0.8408\text{lgMgO}-1.128 \text{lgNa}_{2}\text{O}+1.89,\text{n}=51,\text{r}=0.980, \tag{3b}\]  
where MgO contents vary from 3.94 to 41.51%, Na\({}_{2}\)O contents from 0.32 to 4.97%, and Cr from 76 to 2500 \[\mu\)g/g. The calculated Cr background values range from 82.2 to 2648 \(\mu\)g/g (Figure 2b).  
(3) For the residual samples after step (2) with WIG \(\leq\) 65, the sub-equation was derived, excluding 15 outliers, with residual errors within a logarithmic deviation of \(\pm\)0.2 (notably, the calculated results of five outliers were also located in this error margin) as  
\[\text{lgCr}=0.8753\text{lg}\text{TFe}_{2}\text{O}_{3}-91.15\text{Al}_{2} \text{O}_{3}/\text{Ti}+1.47,\text{n}=327,\text{r}=0.904 \tag{3c}\]  
where TFe\({}_{2}\)O\({}_{3}\) contents range from 2.53 to 8.30%, Al\({}_{2}\)O\({}_{3}\) contents from 8.26 to 24.9%, Ti from 2380 to 9034 \(\mu\)g/g, and Cr from 11 to 128 \(\mu\)g/g, yielding calculated Cr background values between 12.5 and 128 \(\mu\)g/g (Figure 2c). (4) For the residual samples after step (3) with Na\({}_{2}\)O \(\leq\) 3, the sub-equation was fitted, excluding 11 outliers, with residual errors within a logarithmic deviation of \(\pm\)0.2 as  
\[\text{lgCr}=-150.2\text{Al}_{2}\text{O}_{3}/\text{Ti}+0.1928\text{lg}((\text{ CaO}+\text{MgO})/(\text{Na}_{2}\text{O}+\text{K}_{2}\text{O}))+0.398\text{lg} \text{TFe}_{2}\text{O}_{3}+1.99,\text{n}=218,\text{r}=0.946 \tag{3d}\]  
where Al\({}_{2}\)O\({}_{3}\) contents range from 1.35 to 13.92%, K\({}_{2}\)O contents from 0.2 to 5.59%, Na\({}_{2}\)O from 0.17 to 3.00%, CaO from 0.76 to 15.67%, MgO from 0.46 to 3.50%, TFe\({}_{2}\)O\({}_{3}\) from 0.92 to 7.01%, Ti from 575 to 5727 \(\mu\)g/g, and Cr from 4.5 to 96.9 \(\mu\)g/g. The calculated Cr background values range from 5.0 to 100 \(\mu\)g/g (Figure 2d). (5) For the remaining samples, the sub-equation was derived, excluding 17 outliers, with residual errors within a logarithmic deviation of \(\pm\)0.2 as  
\[ \begin{split}\text{lgCr}=-4.567\text{lgMnlg}(\text{Al}_{2} \text{O}_{3}/\text{MgO})\text{lg}(\text{Na}_{2}\text{O}/\text{TFe}_{2}\text{O}_{3}) +6.612\text{lgMnlg}(\text{Na}_{2}\text{O}/\text{TFe}_{2}\text{O}_{3})\\ +12.99\text{lg}(\text{Al}_{2}\text{O}_{3}/\text{MgO})\text{lg}(\text{ Na}_{2}\text{O}/\text{TFe}_{2}\text{O}_{3})-1.117\text{lg}(\text{Al}_{2}\text{O}_{3}/\text{MgO})\\ -19.71\text{lg}(\text{Na}_{2}\text{O}/\text{TFe}_{2}\text{O}_{3})+2.47, \text{n}=120,\text{r}=0.987\end{split}  \tag{3e}\]where Mn contents range from 225 to 1440 \(\upmu\)g/g, Al\({}_{2}\)O\({}_{3}\) contents from 12.3 to 22.5%, MgO from 0.13 to 3.84%, Na\({}_{2}\)O from 3.01 to 6.52%, TFe\({}_{2}\)O\({}_{3}\) from 1.17 to 9.57%, and Cr from 1.8 to 100 \(\upmu\)g/g, with calculated Cr background values ranging from 1.8 to 100 \(\upmu\)g/g (Figure 2e).