## 3 Results and Discussion
### Calibration and Validation
Such a range corresponds to where the calibration is more reliable but indicates that the model provides less robust concentration estimation for values below 4\(\cdot\)10\({}^{4}\) ppm and above than 7\(\cdot\)10\({}^{4}\) ppm. The diagrams reporting predicted vs. measured concentration for all elements are given as Supporting Information together with the variables selected (i.e., emission wavelengths) through VIP analysis. In Table 3, the quality parameters for each individual PLS-VIP model, corresponding to a different element, are reported. Specifically, the table provides RMSE, bias, and R\({}^{2}\) for each step of the multivariate regression (i.e., calibration, cross-validation, validation), as well as the number of variables selected after the VIP analysis, the number of retained latent variables (LV), and the limit of detection (LOD). As for the calibration step, the bias is close to zero in all cases, and the correlation coefficients are generally high (>0.8, reported in bold), particularly for the major elements (Na: 0.91; Mg: 0.95, Al: 0.97; Si: 0.89; Ca: 0.99). However, some elements exhibit correlation coefficients in calibration lower than 0.7, including Be (R\({}^{2}\) = 0.67), Cu (R\({}^{2}\) = 0.53), and Ti (R\({}^{2}\) = 0.67), for which a standard sample (BCR2) was determined to be an outlier. For Y, the calibration phase resulted quite satisfactory (R\({}^{2}\)\({}_{\rm cal}\) = 0.95), although one standard sample (GB07103) was excluded. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**cal**} & \multicolumn{3}{c}{**CV**} & \multicolumn{3}{c}{**val**} \\ \hline \multirow{3}{*}{**Element**} & \multicolumn{3}{c}{**Number of Selected Variables**} & \multirow{3}{*}{**n. LV***} & \multirow{3}{*}{**RMSE\({}_{\rm cal}\)**} & \multirow{3}{*}{**Bias**} & \multirow{3}{*}{**R\({}^{2}_{\rm cal}\)**} & \multirow{3}{*}{**RMSE\({}_{\rm CV}\)**} & \multirow{3}{*}{**Bias**} & \multirow{3}{*}{**R\({}^{2}_{\rm CV}\)**} & \multirow{3}{*}{**RMSE\({}_{\rm val}\)**} & \multirow{3}{*}{**Bias**} & \multirow{3}{*}{**R\({}^{2}_  
In the validation step, the quality parameters indicate that the models obtained for Na, Mg, and Rb, are featured by the highest R\({}^{2}\)val values, being 0.95, 0.90, and 0.96, respectively. Overall, it can be inferred that among the major elements, the predictive capabilities of the models are highly satisfactory for both the calibration and validation steps for Na, Ca, Fe, K, Mg, and Si. However, the validation performance for Al could be improved by including additional samples that cover the entire concentration range (see Figure 5 and the low correlation coefficient in Table 3). For the minor elements (i.e., P, Ti, Mn, Sr, and Ba), the results are generally satisfactory in the calibration phase, whereas the validation phase shows less consistent performance. Notably, Ti and Sr achieved relatively high correlation coefficients (R\({}^{2}\)val = 0.63 and 0.60, respectively), and the performance for P (R\({}^{2}\)val = 0.47) can be considered moderately acceptable. In contrast, Mn exhibited a low correlation coefficient (R\({}^{2}\)val = 0.02), which can be attributed to the clustering of validation samples within a narrow region at low concentration (see Supporting Information), similar to what was observed for Al. For trace elements, the correlation coefficient is excellent for Rb (R\({}^{2}\)val = 0.96) and reasonably satisfactory for Be (R\({}^{2}\)val = 0.63), Cu (R\({}^{2}\)val = 0.63), and Y (R\({}^{2}\)val = 0.43). Conversely, poor correlation was found for Zn (R\({}^{2}\)val = 0.18), Mo (R\({}^{2}\)val = 0.05), and Pb (R\({}^{2}\)val = 0.02). As was observed in the case of Al above, low R\({}^{2}\)val values featured by the models obtained for these elements can be interpreted considering the tight concentration range in which the validation samples fall. In the case of Pb, validation samples comprised between 20 and 60 ppm, which is a much narrower range compared to that of the calibration samples (i.e., 1-300 ppm).