Spectrum prediction in X-ray fluorescence analysis using Bayesian estimation

Reducing the measurement time is important in X-ray fluorescence (XRF) analysis. Micro-XRF and confocal-micro XRF analyses have been used to obtain elemental distributions. Because these techniques are performed in the scanning mode, reducing the measurement time per unit measurement point shortens the time needed for determining elemental distributions. Therefore, we applied the Bayesian theorem to XRF analysis to quickly and accurately estimate the XRF intensity. In the Bayesian formula, the posterior distribution was determined by the likelihood function and prior distribution. Because the obtained posterior function was a probability distribution, the expected value was used as the optimal value. We considered that determining the optimal likelihood function and prior distribution would make it possible to quickly estimate the XRF spectrum in a long measurement time. In this study, the Poisson distribution and the sum of two exponential functions were employed as the likelihood function and prior distribution, respectively. To estimate the XRF spectrum using the Bayesian formula, a standard glass sample containing several metal elements and metal plate were analyzed using a laboratory-made micro-XRF instrument. The micro-XRF measurements were performed at measurement times of 1, 3, 5, 7, 10, 20, 30, 60, 100, 180, and 3600 s, and then the net intensities of the elements in the glass and metal samples obtained with and without the Bayesian estimation were compared. A Cu Kα net intensity for a constantan sample that was close to that obtained in 3600 s could be obtained with the measurement times of 1 and 7 s with and without the Bayesian estimation, respectively. Thus, the measurement time for an accurate XRF net intensity measurement was reduced by >85%.