## 3 Results and Discussion
### Calibration and Validation
The selection of the optimal preprocessing strategy appeared to be element-dependent. In general, the Whittaker baseline treatment followed by SNV offers a conservative and stable approach, while airPLS combined with SNV can be beneficial when dealing with more complex or noisy signals, as observed in the case of Si. The usefulness of soft data-driven models lies in their predictive ability; specifically, in the case of the PLS-VIP approach proposed here, the estimation of the concentration of an element in a sample using h-LIBS. In this context, a critical aspect is the interplay between expected concentration in the sample and the limit of quantification (LOQ). Ideally, the LOQ should be significantly lower than the concentration in the sample, allowing for precise compositional differentiation. On the contrary, when the LOQ exceeds the sample concentration values, the predictions are not sufficiently accurate for practical application. To provide a clearer understanding of the PLS-VIP model's applicability, Figure 6 illustrates the LOQ, expressed as a percentage of average and maximum concentration in standard samples, for all the elements studied herein. For Ca, K, Si, Al, and Na among the major elements, as well as for Ti and Sr among the minor elements and for Rb, Zn, and Y among the trace elements, the LOQ is substantially lower than both the mean and maximum concentrations in the standard samples. This indicates that the model can reliably work when the concentration in a sample is near the mean values of the standard samples (see Table 2). However, for Fe, Mg, and K, despite the model's good predictive capabilities, the LOQ remains relatively high, making the model  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Whittaker + SNV**} & \multicolumn{3}{c}{**airPLS + SNV**} \\ \hline
**Element** & **RMSE\({}_{\text{cal}}\)** & **R\({}^{2}_{\text{cal}}\)** & **RMSE\({}_{\text{val}}\)** & **R\({}^{2}_{\text{val}}\)** & **RMSE\({}_{\text{cal}}\)** & **R\({}^{2}_{\text{cal}}\)** & **RMSE\({}_{\text{val}}\)** & **R\({}^{2}_{\text{val}}\)** \\ \hline Na & 2270 & 0.91 & 2314 & 0.95 & 2301 & 0.90 & 2554 & 0.94 \\ Si & 14,261 & 0.89 & 29,5 practical only when the element concentration exceeds the LOQ. Similar considerations apply to Pb, Cu, Be, Mn, and P. However, in all cases, the LOQ values are well below the maximum element concentration. From all of the above, despite the relatively limited number of standard samples available in this study, the proposed approach demonstrated satisfactory performance for a significant number of elements, highlighting the effectiveness of PLS-VIP calibration for h-LIBS devices. This outcome is particularly significant, because the calibration of h-LIBS instruments is a difficult, though desirable, task due to the rapidity of the measurements. Moreover, the importance of these finding is further emphasized when one considers that similar calibration strategies reported in the literature are mainly applied to bench-top instruments, often utilizing extensive sets of standard samples [45; 46; 47; 48]. A direct comparison of the uncertainties in prediction determined in this work and those reported in the literature by [PERSON] et al. [47] is proposed in Table 5. The authors published a reference article for the elemental quantification by means of LIBS analysis and trained a PLS model on a multi-matrix dataset containing more than 2000 standard geochemical samples including igneous, sedimentary, and metamorphic rocks. The literature RMSE in prediction, originally reported as oxide percentages, were converted to element percentage to match the concentration scale used in this work. The uncertain values are highly comparable for all the major elements (i.e., those expressed in%). In particular, the value reported in this work for Si (1.5%) is slightly lower than literature value (1.7%), which is understandable because this work is focused on silica-based materials.