Abstract:
The inverse problem of obtaining particle size distributions from observations
of the angular distribution of near forward scattered light is reexamined.
Asymptotic analysis of the forward problem reveals the information
content of the observations, and the sources of non-uniqueness and
instability in inverting them. A sampling criterion, such that the observations
uniquely specify the size distribution is derived, in terms of the
largest particle size, and an angle above which the intensity is indistinguishable
from an asymptote. The instability of inverting unevenly spaced
data is compared to that of super-resolving Fourier spectra. Resolution is
shown to be inversely proportional to the angular range of observations.
The problem is rephrased so that the size weighted number density is
sought from the intensity weighted by the scattering angle cubed. Algorithms
which impose positivity and bounds on particle size improve the
stability of inversions. The forward problem can be represented by an
over-determined matrix equation by choosing a large integration increment
in size dependent on the frequency content of the angular intensity, further
improving stability.
Experimental data obtained using a linear CCD array illustrates the theory, with standard polystyrene spheres as scatterers. The scattering
from single and tri-modal distributions is successfully inverted.
