The inversion matrix and error estimation in data inversion: Application to diffusion battery measurements

Abstract

Judicious selection of the measurement conditions and analysis methods to be used can make it less difficult to produce accurate data inversions in the presence of experimental error. The response (data) vector (b) of a multi-channel instrument, such as an optical particle counter or multi-stage impactor or diffusion battery, to an input distribution vector (x) can be modelled as a set of linear equations given by the vector-matrix equation b = Ax. For low resolution instruments, one of several methods of data inversion is usefully employed: simple inversion, least-squares inversion, various smoothing inversions and various non-linear approaches. One non-linear approach [Twomey, S. (1975) J. Comput. Phys. 18, 188-200.] we found to be sensitive to starting conditions and to show cycling during iteration, similar to equations leading to 'chaos' [Wu, J. J. et al. (1989) J. Aerosol Sci. 20, 477-482.]. Simple inversion, least-squares inversion and smoothing are alike in that they produce their solutions from x = Zb, where Z(i, j) are the elements of what one could call the 'inversion matrix', Z, a kind of transfer function. Z gives the sensitivity of the inferred values to changes (or errors) in the data values. A criterion for the best measurement instrument or measurement conditions could be the minimum largest absolute Z(i, j) or the mean absolute value or some other weighting. Propagation of error analysis indicates that another measure of Z(i, j) that would be useful would be its root mean square. The 'condition number' is another measure that has also been suggested [Cooper, D. W (1974) Ph.D. dissertation. Division of Engineering and Applied Science, Harvard University, Cambridge, MA, (1975) 68th Annual Meeting of the Air Pollution Control Assoc., Boston, MA; Yu, P.-Y. (1983) Ph.D. dissertation. Department of Chemical and Nuclear Engineering, College Park, MD; Farzanah, F. F. et al. (1984) Environ. Sci. Technol. 19, 121-126; Hirleman, E. D. (1987) 1st Intl. Conf. on Particle Sizing, Rouen, France.]. Some comparisons of these measures are made. The inversion matrix gives the clearest indication of the relationship between the data and the results of inversion. We recommend that proposed experimental conditions should be adjusted based on inversion matrix studies in order to lessen ill-conditioning and the reliance on various data analysis methods to cope with ill-conditioned systems. © 1990.