## Abstract

The feasibility of using a generalized stochastic inversion
methodology to estimate aerosol size distributions accurately by use of
spectral extinction, backscatter data, or both is examined. The
stochastic method used, inverse Monte Carlo (IMC), is verified with
both simulated and experimental data from aerosols composed of
spherical dielectrics with a known refractive index. Various levels
of noise are superimposed on the data such that the effect of noise on
the stability and results of inversion can be
determined. Computational results show that the application of the
IMC technique to inversion of spectral extinction or backscatter data
or both can produce good estimates of aerosol size
distributions. Specifically, for inversions for which both spectral
extinction and backscatter data are used, the IMC technique was
extremely accurate in determining particle size distributions well
outside the wavelength range. Also, the IMC inversion results
proved to be stable and accurate even when the data had significant
noise, with a signal-to-noise ratio of 3.

© 2000 Optical Society of America

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### Equations (16)

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(1)
$$c\left(\mathrm{\lambda}\right)={\int}_{0}^{\infty}\mathrm{\pi}{r}^{2}{Q}_{\mathrm{ext}}\left(2\mathrm{\pi}r/\mathrm{\lambda}\right)\left(\frac{\mathrm{d}n}{\mathrm{d}r}\right)\mathrm{d}r,$$
(2)
$$c\left(\mathrm{\lambda}\right)={N}_{0}{\int}_{0}^{\infty}{K}_{\mathrm{ext}}\left(r,\mathrm{\lambda}\right)g\left(r\right)\mathrm{d}r,$$
(3)
$$c\left(\mathrm{\lambda}\right)={N}_{0}\sum _{i=1}^{M}{\overline{K}}_{\mathrm{ext}}\left({r}_{i},\mathrm{\lambda}\right)f\left({r}_{i}\right),$$
(4)
$${\overline{K}}_{\mathrm{ext}}\left({r}_{i},\mathrm{\lambda}\right)=\frac{1}{\mathrm{\delta}r}{\int}_{{r}_{i}-\mathrm{\delta}r/2}^{{r}_{i}+\mathrm{\delta}r/2}{K}_{\mathrm{ext}}\left(r,\mathrm{\lambda}\right)\mathrm{d}r$$
(5)
$$b\left(\mathrm{\lambda}\right)={N}_{0}\sum _{i=1}^{M}{\overline{K}}_{\mathrm{back}}\left({r}_{i},\mathrm{\lambda}\right)f\left({r}_{i}\right),$$
(6)
$${\overline{K}}_{\mathrm{back}}\left({r}_{i},\mathrm{\lambda}\right)=\frac{1}{\mathrm{\delta}r}{\int}_{{r}_{i}-\mathrm{\delta}r/2}^{{r}_{i}+\mathrm{\delta}r/2}\mathrm{\pi}{r}^{2}{Q}_{\mathrm{back}}\left(2\mathrm{\pi}r/\mathrm{\lambda}\right)\mathrm{d}r$$
(7)
$${c}_{j}={N}_{0}\sum _{i=1}^{M}{\left({\overline{K}}_{\mathrm{ext}}\right)}_{\mathit{ji}}{f}_{i},{b}_{j}={N}_{0}\sum _{i=1}^{M}{\left({\overline{K}}_{\mathrm{back}}\right)}_{\mathit{ji}}{f}_{i}.$$
(8)
$${c}_{j}={N}_{0}{w}_{0}\sum _{i=1}^{M}{\left({\overline{K}}_{\mathrm{ext}}\right)}_{\mathit{ji}}\prime {f}_{i},{b}_{j}={N}_{0}{w}_{0}\sum _{i=1}^{M}{\left({\overline{K}}_{\mathrm{back}}\right)}_{\mathit{ji}}\prime {f}_{i},$$
(9)
$${w}_{0}={\int}_{0}^{\infty}w\left(r\right)g\left(r\right)\mathrm{d}r$$
(10)
$${\overline{K}}_{\mathrm{ext}}\left({r}_{i},\mathrm{\lambda}\right)\prime =\frac{1}{\mathrm{\delta}r}{\int}_{{r}_{i}-\mathrm{\delta}r/2}^{{r}_{i}+\mathrm{\delta}r/2}\frac{{K}_{\mathrm{ext}}\left(r,\mathrm{\lambda}\right)}{w\left(r\right)}\mathrm{d}r$$
(11)
$$w\left(r\right)=4/3\mathrm{\pi}{r}^{3}.$$
(12)
$$c_{j}{}^{0}=\frac{1}{n\mathrm{walk}}\sum _{\mathit{nw}=1}^{n\mathrm{walk}}{\left({\overline{K}}_{\mathrm{ext}}\right)}_{\mathit{jrn}\left(\mathit{nw}\right)}.$$
(13)
$$N_{0}{}^{2}=\frac{1}{J}\sum _{j=1}^{J}{\left(\frac{{c}_{j}}{c_{j}{}^{0}}\right)}^{2}.$$
(14)
$${\tilde{\mathrm{{\rm X}}}}^{2}=\frac{1}{J}\sum _{j=1}^{J}\left[\frac{{\left({c}_{j}-{N}_{0}c_{j}{}^{0}\right)}^{2}}{\mathrm{\sigma}_{j}{}^{2}}\right],$$
(15)
$$g\left(r\right)=\frac{1}{\sqrt{2\mathrm{\pi}}r{\mathrm{\sigma}}_{m}}exp\left\{-\frac{{\left[ln\left(r/{r}_{m}\right)\right]}^{2}}{2{\left({\mathrm{\sigma}}_{m}\right)}^{2}}\right\},$$
(16)
$${\overline{K}}_{\mathrm{ext}}\left({r}_{i},\mathrm{\lambda}\right)=\frac{1}{\mathrm{\delta}r}{\int}_{{r}_{i}-\mathrm{\delta}r/2}^{{r}_{i}+\mathrm{\delta}r/2}\left[\mathrm{\pi}{r}^{2}{Q}_{\mathrm{ext}}\left(r,\mathrm{\lambda}\right)-{\int}_{\stackrel{\u02c6}{\mathrm{\Omega}}}\left(\frac{\mathrm{d}{\mathrm{\sigma}}_{\mathrm{scat}}}{\mathrm{d}\mathrm{\Omega}}\right)\mathrm{d}\mathrm{\Omega}\right]\mathrm{d}r,$$