## 3 Results and Discussion  
### Calibration and Validation  
Although the number of available standard samples (seventeen) was relatively limited, it was sufficient to calibrate the h-LIBS device. However, not every standard sample contained all elements of the periodic table at concentrations exceeding the declared limit of detection (LOD) [76; 77; 78]. Consequently, the calibration of the h-LIBS device was performed only for those nineteen elements that surpassed the LOD across all seventeen standard samples. Selected elements for the CRMs and GEO standard samples, together with their concentrations in parts per million (ppm), are listed in Table 2. A specific portion of this table--comprising the complete set of element concentrations measured in the standard samples--constitutes the previously mentioned Y matrix. Additionally, the table includes descriptive statistical parameters, namely, the mean value (\(m\)), standard deviation (\(s\)), range (r), minimum value (min. ), and maximum value (max.). The elements Na, Mg, Al, Si, K, Ca, and Fe can be regarded as major elements, since their mean concentration (\(m\)) is equal or greater than 10,000 ppm (i.e., 1%); P, Ti, Mn, Sr, and Ba can be regarded as minor elements, with an average concentration between 100 and 10,000 ppm, whereas Be, Cu, Zn, Rb, Y, Mo, and Pb are considered as trace elements, since their average concentration is equal to or less than 100 ppm. Spectral data acquired from the standard samples, allocated in the X matrix, were pre-processed prior to performing multivariate regression vs. the Y concentration matrix. As was previously stated, common issues for LIBS spectra analysis include baseline fluctuations and varying sensitivity among elements. To address these issues, the most commonly adopted approach involves baseline correction and spectral normalization. To this end, a pre-processing step based on the Whittaker filter, followed by a SNV, was carried out. The parameters of the Whittaker filter were chosen to effectively model the baseline while minimizing low-frequency oscillations. A regularization parameter of \(\lambda=100\) was selected to impose sufficient smoothness on the estimated baseline without excessively flattening relevant spectral features. The penalty factor \(p=0.001\) was chosen to allow a degree of sensitivity to subtle baseline variations, improving the separation of weak peaks from the background. This combination enhances the signal-to-noise ratio, particularly for low-intensity spectral lines, which is critical for accurate elemental identification in LIBS analysis. The parameters were initially based on values reported by [PERSON] et al. [58], and subsequently adjusted to optimize performance on the specific dataset used in this work. The impact of such pre-treatments on the LIBS spectra of standard samples, shown in different colors, is illustrated in Figure 4. In parallel, the airPLS algorithm [72] was also tested as an alternative method to the Whittaker filter for baseline correction, offering an adaptive, data-driven approach to suppress background noise while preserving peak integrity. Also, in this case, after baseline correction, the pre-treated spectra were normalized with a SNV transformation. Parameters for the airPLS algorithm were: \(\lambda=10^{5}\), order of the difference of penalties = 2, Weight of Exception Proportion (WEP) = 0.1, asymmetry \(p=0.5\), maximum number of iterations = 20. The impact of the use of two different baseline pre-treatments on the calibration and validation performance is discussed later. The X and Y matrices were split into calibration and validation subsets using the Kennard-Stone algorithm [73], which was chosen because it generates subsets with similar sample density, being a distance-based splitting method. This approach ensures that both subsets maintain a comparable range of element concentration, leading to a more reliable internal validation process.