Eigen-system analysis of X-ray diffraction profile deconvolution methods explains ill-conditioning

Several deconvolution methods common in X-ray diffraction profile studies have been examined using an eigen-system analysis in which an error-bound function is used to represent the maximum difference between the solution and true specimen profiles. This approach quantifies the sources of misfitting and ill-conditioning that appear in the solution profile and expresses them as a function of the control parameter for a particular method. A simulation of an instrument-broadened profile overlaid with random noise and background signals was used to evaluate the error-bound function for the iterative and constrained deconvolution methods, and the properties of the error-bound function were related to the features of the solution profile for each method. This analysis quantifies the terms that contribute to the ill-conditioning of the solution profile. It shows that, even for optimum values of the control parameters, positivity is not preserved and spurious oscillations are present in the solution profile.