Resolution unfolding with limits imposed by statistical experimental errors.

A typical form of the resolution equation is derived by considering the physical measurement of an energy dependent spectrum. It is shown that the information contained in a data set may be expressed by writing the spectrum as a linear combination of a set of resolution functions. Introduction of other functions to describe the spectrum involves extra physical information. An iterative conjugate gradient technique to obtain a spectrum consistent with the data is described. At each iteration the residual discrepancy between the currently predicted yield and the measured data is used to generate the form and magnitude of the next term to be added to the spectrum. Other unfolding techniques are described and analysed, some faster than the conjugate gradient technique in special cases, but restricted in usefulness by implicit assumptions about the resolution functions. The nature of residual errors is considered. The variations of independently measured data sets are discussed, and hence, the variations of the sequence of terms appearing in a consequent conjugate gradient analysis. An approximate measure is obtained for the expected variation of independently obtained spectra. Refinements are briefly considered which apply to a resolution function that is not known precisely or which make use of a requirement that the spectrum be positive throughout its range. It is concluded that a conjugate gradient technique is best if sufficient computer facilities are available, and that, of the less demanding techniques, the best is one that is essentially a more slowly convergent version of a conjugate gradient method