Orthogonal Bayesian calibration of XRF data for geological applications

X-ray fluorescence (XRF) calibration for quantitative analysis is typically performed using certified reference materials (CRMs), where the relationship between the concentration of an element and the net area of its peak in the spectrum is assumed, after properly applying corrections for possible matrix effects, to be linear. The calibration is often obtained by Ordinary Least Squares (OLS) by minimizing the vertical distance between regression line and data points taking into account the experimental uncertainty. However, uncertainties in CRM-certified concentrations are often neglected, potentially leading to underestimated errors in the calibration model. In this work, a Bayesian approach is employed to obtain calibration curves by minimizing the orthogonal distance, taking into account both experimental and certified uncertainties. Twenty-four geological, soil, and sediment CRMs were analyzed using an energy-dispersive XRF spectrometer, and the resulting spectra were employed for constructed calibration curves for eight oxides and seven elements. The model parameters were estimated via maximum a posteriori (MAP) optimization and the posterior was evaluated by a Gibbs sampling algorithm in an orthogonal parameter space. A principal components analysis (PCA) was implemented for this coordinate transformation to remove the linear correlation. Validation against independent samples shows that the Bayesian approach demonstrates that the ordinary least-squares regression calibration curve deviates from the orthogonal Bayesian MAP curve by up to one standard deviation, particularly for elements with higher reference material variability and lower concentrations. Mean absolute errors and root mean squared errors indicate improved predictive performance for major oxides, while trace elements exhibit variable agreement due to instrumental and sample-specific factors. The results demonstrate that orthogonal Bayesian XRF calibration can enhance both accuracy and uncertainty quantification in multi-elemental analyses. • Novel Bayesian calibration for XRF accounts for uncertainties in both net peak area and CRM-certified concentrations. • Significant discrepancies between Bayesian and OLS regressions are observed where CRM uncertainties are high. • MCMC sampling provides full posterior distributions for parameters, enabling credible intervals and LOD estimates.