Propagation of uncertainty for non-linear calibration equations with an application in radiation thermometry

Abstract

Analytic expressions are derived for the sensitivity coefficients for the uncertainties in both the x and y calibration points for an arbitrary non-linear calibration equation, y = y(x). Formalisms for the propagation of uncertainty are given for both exact fitting and least-squares determinations of the adjustable parameters. These formulae are applied to the approximation of ITS-90 above the silver point by radiation thermometry, using the recently developed metal–carbon and metal-carbide–carbon eutectic fixed points. A modified form of the non-linear Sakuma–Hattori calibration equation is analysed. It is shown that for temperatures between the fixed points, the combined uncertainty can be kept below the uncertainty in the individual calibration points, and reduces as the number of degrees of freedom for least-squares fitting increases.