## 3 Results
### Linear Relationships in Compost Samples  
The multiple linear equations obtained for the target elements in compost-type samples are:  
\[[\text{As(pred)}]=-0.407+0.0374[\text{Si(m)}]+0.0233[\text{Zr(s)}], \tag{1}\]  
\[[\text{As(pred)}]=-0.245+0.0381[\text{Si(m)}]+0.0157[\text{Zr(s)}]+0.0102[ \text{Pb(s)}], \tag{2}\]  
\[[\text{Cd(pred)}]=-0.0298+0.00162[\text{Zn(s)}]+0.00170[\text{Zr(s)}], \tag{3}\]  
\[[\text{Cd(pred)}]=-0.0222+0.00166[\text{Zn(s)}]+0.00180[\text{Zr(s)}]-0.0000654 [\text{Mn(s)}], \tag{4}\]  
\[[\text{Cr(pred)}]=-3.13+2.925[\text{Fe(m)}]+1.102[\text{K(s)}]-1.206[\text{S( s)}], \tag{5}\]  
\[[\text{Hg(pred)}]=-73.3+1.171[\text{Zn(s)}], \tag{6}\]  
\[[\text{Hg(pred)}]=29.2+1.213[\text{Zn(s)}]-0.171[\text{Mn(s)}]-3.323[\text{S( s)}], \tag{7}\]  
\[[\text{Ni(pred)}]=-42.7+2.259[\text{Fe(m)}]+0.442[\text{Bal(m)}], \tag{8}\]  
\[[\text{Se(pred)}]=444+2.877[\text{Zn(s)}]-4.628[\text{Zr(s)}], \tag{9}\]  
Their statistics are shown in Table 3. The correlation coefficients of aqua-regia-soluble elemental concentrations, X(ae), and the pXRF readings are shown as supplementary material (Table S1). The Si(m) reading was used for the first linear adjustment factor of As(ae) (Equations (1) and (2)); the Zn(s) was used for modelling Cd(ae) (Equations (3) and (4)), Hg(ae) (Equations (6) and (7)), and Se(ae) (Equation (9)); and the Fe(m) was used for Cr(ae) (Equation (5)) and Ni(ae) (Equation (8)). These predictors showed the highest correlation coefficient (Table S1) and valid readings were obtained for all compost samples (\(N=14\)). In the case of As, after the first linear adjustment, the residuals were successively correlated with the Zr(s) and Pb(s) readings, which were then introduced by obtaining the Equations (1) and (2). The best fit (r = 0.980, Table 3) was obtained with Equation (2), and hence it was used for the prediction  
\[[\text{As(pred)}]=-0.245+0.0381[\text{Si(m)}]+0.0157[\text{Zr(s)}]+0.0102[ \text{Pb(s)}], \tag{2}\]0 Canonical discriminant functions for the three types of organic amendments using pXRF readings as input variables (missing values corresponding to readings below LOD are replaced with a 0). of As, but only in \(n=13\) out of the 14 compost samples due to the lack of a Pb(s) datum in one of the samples. For the remaining sample, Equation (1) was used. It should be noted that the additional terms that are introduced into the equations did not necessarily correlate initially with the target element; in the case of As, Pb(s) did not show an initial correlation with As(ae) (Table S1), although its coefficient in Equation (2) was statistically significant (\(p=0.029\), Table 3). In the case of Cd, good fittings were obtained with Zn(s) and Zr(s) (Equation (3), used for \(n=1\) sample, r = 0.989) and Mn(s) (Equation (4) for \(n=13\) samples, r = 0.996). In the case of Cr, the best fitting was obtained with Fe(m), K(s), and S(s) (Equation (5), r = 0.975). Despite the fact that the Cr(s) readings could be obtained in 10 compost samples, the correlation with Cr(ae) of these readings (r = 0.764) was of lesser magnitude than the fitting obtained from Equation (5) (r = 0.975). In the case of Hg, a mediocre fitting (r = 0.949) was obtained with Zn(s) (Equation (6)), but it was improved (r = 0.995) by including Mn(s) and S(s) (Equation (7)). Ni was modelled with Fe(m) and Bal(m) (Equation (8), r = 0.977)). Selenium was modelled with Zn(s) and Zr(s) (Equation (9), r = 0.981). The elements Cd and Se were modeled with the same predictors Zn(s) and Zr(s) in Equations (3) and (9), which agreed with the significant correlation observed between Cd(ae) and Se(ae) in the compost samples (r = 0.918, \(n=14\)). All the terms included in the fitting equations were statistically significant (\(p\)-values \(<\) 0.05). In general, the equations for compost can be considered quantitatively acceptable for calibration (r \(>\) 0.98) [28].