## 2 Materials and Methods  
_Key formulas. Accurate accounting for the background signal._ We assume that to determine the analytical-line intensity it is necessary to sum the signals \(M\) of the CCD pixels. In the given case study, these are pixels located within the yellow sector in Figure 1. The \(N\) number of standard samples (SS) is used to construct the calibration for this element. We number them in ascending order of concentration, i.e., the lower standard is \(N1\), etc. Thus, the analytical-line intensity of a certain wavelength \(I\) of some element, when the standard sample \(i\) is used, transformed into the signal of the recording system is:  
\[I_{i}=\sum_{k=1}^{M}I_{ki}, \tag{1}\]  
where \(I_{ki}\) is the intensity in the pixel \(k\) of the analytical line of the element being determined in the analysis of the standard sample \(i\). We should note that the intensities are calculated considering the model background (the part of the intensity that is lower than the cutoff line is omitted in each pixel in Figure 1). The relative (or the absolute, according to the analytical task) impurity concentration in sample \(i\) is \(C_{i}\). The standard procedure for plotting the calibration is to use the data sets \(I_{i},C_{i},i=1,\ldots,N\) to plot the dependence  
\[C=F(I), \tag{2}\]  
where \(F(I)\) is usually a polynomial of no greater than degree 4. It should be noted that we previously discussed the dependence of the analytical-line intensity on the element concentration, which is justified from a physical point of view, although Equation (2) is an inverse function. The point is that, from a mathematical point of view, it is Equation (2) for which the process of determining the unknown concentration by the measured intensity is trivial, while for the inverse dependence \(I=F^{-1}(C)\), it is necessary to solve an algebraic equation. As indicated, the non-coincidence of the shape of the actual plasma background radiation (which is unknown in advance) with the cutoff-line shape results in distortions of the calibration curve at low concentrations \(C\); as such, the calibration dependence does not meet the requirement of zero intensity at zero concentration of the determined element. Therefore, we will proceed as follows. The intensity \(I_{ki}\) consists of the actual analytical signal \(I_{ak}\), determined by the impurity in the sample, and \(I_{fk}\), which is the difference in the pixel \(k\) between the intensity of the actual plasma background and the background that has the shape of the cutoff line (hereinafter we will refer to this difference as simply the background). The latter weakly depends on a standard sample number, i.e., on the content of the impurities determined within it. This dependence will be neglected. Thus, for the standard sample number \(i\), we have:  
\[I_{ki}=I_{aki}+I_{fk}. \tag{3}\]  
Summing Equation (3) by pixels, we have:  
\[I_{i}=I_{ai}+I_{F}, \tag{4}\]  
where \(I_{ai}=\sum_{k=1}^{M}I_{aki}\); \(I_{F}=\sum_{k=1}^{M}I_{fk}\) is the actual (not the measured) intensity of the analytical line of standard sample \(i\) and the background at this line location, therefore, is converted by the recording system into an electrical signal. To calculate value \(I_{F}\) we will compile dataset \(\Delta I_{i}=I_{i}-I_{1}=I_{ai}-I_{a1}\); \(\Delta C_{i}=C_{i}-C_{1}\); \(i=1,\ldots,N\). We should note that the dataset also has \(N\) values of \(\Delta I_{i}\) and \(\Delta C_{i}\), and there is no value \(I_{F}\).