Solution of an ‘‘impossible’’ diffusion-inversion problem

Abstract

The recent conclusion of Craig and Thompson [Comp. Phys. 8, 648 (1994)] that the inversion of spatial diffusion data to estimate accurately an initial delta‐function temperature source distribution is an unstable, impossible problem, is demonstrated to be far too pessimistic. Instead of inverting such data with a method appropriate for continuous distributions as they did, inversion was carried out with one appropriate for a discrete distribution. The inversion involved simultaneous estimation of both line strengths and their positions. It yielded highly accurate estimates of the strength and position of the source, even with very large amounts of noise. Several inversions of noisy data are illustrated for the normalized time value used by Craig and Thompson and for very much longer times. Further, it is demonstrated that even for very noisy data it is possible to discriminate between the presence of one line or of several lines, and to conclude unambiguously whether a given source distribution is discrete or continuous. Finally, a closely related method appropriate for estimation of continuous source distributions was used to invert diffusion data calculated from a very narrow, continuous source distribution approximating a delta function. Even with noise present, such continuous source distributions can be reliably estimated, but the narrower the distribution the less well can it be distinguished from a single line, and the less important such discrimination becomes. © 1995 American Institute of Physics.