After the matrix correction, in fact most regression equations are better because all r and [R.sup.2] values are higher for both OLR and UWLR than without the correction (Table S4; Figure 1 for r only).
Before the matrix correction, the intercepts of the IntConc regression lines were closer to zero for the UWLR (range ~-0.013 to +0.011) than for the OLR (range ~-2.098 to +11.47) model (Table S4; Figure 2).
Finally, the uncertainties on both intercept and slope parameters were mostly lower for the UWLR than the OLR (Table S4).
For the comparison of two models OLR and UWLR before the matrix correction, the uncertainty for the UWLR were lower than the OLR for 7 elements (positive [delta][u.sub.b] and [delta][u.sub.m]), whereas for after the matrix correction, it was so for 8 elements (out of 10; Figure 3).
The intercept values were closer to zero (zero being the theoretically ideal intercept) for the UWLR regression (~-0.113 to +0.104; Table 1) as compared to the OLR (~-47.8 to +12.3; Table 1).
All intercepts for the UWLR model, without exception, were closer to zero as compared to the OLR model.
The drift-corrected net intensities and the corresponding uncertainties were processed from the first set of two regression equations (Int-Conc OLR and UWLR models; Table S4) to obtain provisional concentration and uncertainty values.
Firstly, although the mean concentration values determined by the OLR and UWLR models showed a general agreement, the 99% uncertainty values ([u.sub.99]; Table 3) were generally lower for the UWLR models, which clearly indicates that this model should be used routinely, instead of the conventional OLR model.
An online computer program JSpectrom_XRFCalcUnknown will be available at our server https://tlaloc.ier.unam.mx for use for unknown samples, which will guide other users to achieve the UWLR calibration outside of the instrumental software and its routine application to unknown samples.
The XRF spectrometer calibrated under both the OLR and UWLR models clearly showed that the UWLR provides more reliable results (lower uncertainty estimates) than the OLR model commonly practiced for most XRF instruments.
Caption: Figure 1: Linear correlation coefficient (r) values for the ordinary least-squares linear regression (OLR) and uncertainty-based weighted least-squares linear regression (UWLR) models for the XRF calibration of major elements (Si[O.sub.2] to [P.sub.2][O.sub.5]) in rocks and minerals.
Caption: Figure 2: Intercept (b) values for the ordinary least-squares linear regression (OLR) and uncertainty-based weighted least-squares linear regression (UWLR) models for the XRF calibration of major elements (Si[O.sub.2] to [P.sub.2][O.sub.5]) in rocks and minerals.