The propagation of uncertainty on interpolated scales, with examples from thermometry

The propagation of uncertainty on measurement scales that are based on polynomial interpolation is described. Reformulation of the interpolation in terms of Lagrange polynomials, which are orthogonal over the set of measured variables, provides the key mathematical simplification. Consequently, with many uncertainty calculations the correlation terms do not need to be carried, and the effects of the various sources of uncertainty are easily visualized. Indeed, once a user is familiar with Lagrange interpolation, both the interpolation equation and the uncertainty equation can often be written down by inspection without the need for any intermediate calculation. The method is applied to both the International Temperature Scale of 1990 (ITS-90) and multiwavelength radiation thermometry to highlight the advantages of the Lagrange approach and to illustrate some of the advantages and disadvantages of interpolated scales. Several means of estimating the additional uncertainty arising from interpolation error are also discussed.