## Abstract

A new method is considered that can be used for inverting data obtained from a combined elastic–inelastic lidar or a high spectral resolution lidar operating in a one-directional mode,
or an elastic lidar operating in a multiangle mode. The particulate extinction coefficient is retrieved from the simultaneously measured profiles of the particulate backscatter coefficient and the particulate optical depth. The stepwise profile of the column-integrated lidar ratio is found that provides best matching of the initial (inverted) profile of the optical depth to that obtained by the inversion of the backscatter-coefficient profile. The retrieval of the extinction coefficient is made without using numerical differentiation. The method reduces the level of random noise in the retrieved extinction coefficient to the level of noise in the inverted backscatter coefficient. Examples of simulated and experimental data are presented.

© 2007 Optical Society of America

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### Equations (15)

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(1)
$${\kappa}_{p}\left(h\right)={S}_{p}\left(h\right){\beta}_{p}\left(h\right)\text{.}$$
(2)
$${\tau}_{p}\left({h}_{j},h\right)={\displaystyle {\int}_{{h}_{j}}^{h}{\kappa}_{p}\left(h\prime \right)\mathrm{d}h\prime ={\displaystyle {\int}_{{h}_{j}}^{h}{S}_{p}\left(h\prime \right)}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}\text{.}$$
(3)
$${\tau}_{p}\left({h}_{j},h\right)={S}_{p\text{,}j}\left({h}_{j},h\right){\displaystyle {\int}_{{h}_{j}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}\text{,}$$
(4)
$${S}_{p\text{,}j}\left({h}_{j},h\right)=\frac{{\displaystyle {\int}_{{h}_{j}}^{h}{S}_{p}\left(h\prime \right){\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}}{{\displaystyle {\int}_{{h}_{j}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}}\text{.}$$
(5)
$$\u3008{\tau}_{p}\left({h}_{j},h\right)\u3009={S}_{p\text{,}j}{\displaystyle {\int}_{{h}_{j}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}\text{.}$$
(6)
$$\Delta {\tau}_{p\text{,}j}\left(h\right)={S}_{p\text{,}j}{\displaystyle {\int}_{{h}_{i}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime -{S}_{p\text{,}j}\left({h}_{j},h\right){\displaystyle {\int}_{{h}_{i}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}}\text{.}$$
(7)
$$\left|\Delta {\tau}_{p\text{,}j}\left(h\right)\right|\le \Delta {\tau}_{p\text{,}\mathrm{max}}\left(h\right)$$
(8)
$${\kappa}_{p}\left(h\right)\approx {S}_{p\text{,}j}{\beta}_{p}\left(h\right)\text{.}$$
(9)
$$\Delta {\tau}_{p\text{,}\mathrm{max}}\left(h\right)=\mathrm{max}\left[\Delta {\tau}_{p}\left({h}_{\mathrm{min}}\right);\text{\hspace{0.17em}}\Delta {\tau}_{p}\left({h}_{\mathrm{min}}+\Delta {h}_{d}\right);\Delta {\tau}_{p}\left({h}_{\mathrm{min}}+2\Delta {h}_{d}\right);\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}};\text{\hspace{0.17em}}\Delta {\tau}_{p}\left(h\right)\right]\text{.}$$
(10)
$$\u3008{\tau}_{p}\left(0,h\right)\u3009={\tau}_{p\text{,}b}\left(0,{h}_{j}\right)+\u3008{\tau}_{p}\left({h}_{j},h\right)\u3009={\tau}_{p\text{,}b}\left(0,{h}_{j}\right)+\left[{S}_{p\text{,}j}{\displaystyle {\int}_{{h}_{j}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}\right]\text{,}$$
(11)
$$\xi \left({h}_{k},{h}_{\mathrm{max}}\right)={\displaystyle \sum _{{h}_{k}}^{{h}_{\mathrm{max}}}{\left[\u3008{\tau}_{p}\left(0,h\right)\u3009-{\tau}_{p\text{,}b}\left(0,h\right)\right]}^{2}}\text{.}$$
(12)
$$\delta {\kappa}_{p}\left(h\right)=\sqrt{{\left[\delta {\beta}_{p}\left(h\right)\right]}^{2}+{\left[\delta {S}_{p}\left(h\right)\right]}^{2}}\text{.}$$
(13)
$$\left|\Delta {S}_{p\text{,}j}\left(h\right)\right|=\left|{S}_{p\text{,}j}-{S}_{p\text{,}j}\left({h}_{j},h\right)\right|\le \frac{\Delta {\tau}_{p\text{,}\mathrm{max}}\left(h\right)}{{\displaystyle {\int}_{{h}_{j}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}}\text{.}$$
(14)
$${S}_{p}\left(h\right)={S}_{p}\left({h}_{1},h\right)+\frac{{\displaystyle {\int}_{{h}_{1}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}}{{\beta}_{p}\left(h\right)}\text{\hspace{0.17em}}\frac{d}{\mathrm{d}h}\left[{S}_{p}\left({h}_{1},h\right)\right]\text{,}$$
(15)
$${S}_{p}\left({h}_{1},h\right)=\frac{{\tau}_{p\text{,}b}\left({h}_{1},h\right)}{{\displaystyle {\int}_{{h}_{1}}^{h}{\beta}_{p}\left(h\prime \right)\mathrm{d}h\prime}}\text{.}$$