## Abstract

Single-scatter lidar signals carry information on the spatial atmospheric backscatter coefficient, attenuated by the path-integrated extinction. Assuming that the relationship between the backscatter and the extinction is known, the inverted extinction profile and the path-integrated extinction are uniquely related to the input boundary value. The integrated extinction over a certain range interval is a measure of the optical transmission along that path. In reverse, for a given transmission over the path of interest, the input boundary value is uniquely defined. An analytical expression is derived that describes the input boundary condition for the inversion of the single-scatter lidar equation in terms of the transmission losses over the path of interest. The proposed method is useful in situations in which independent transmission measurements are carried out or in situations in which targets such as multiple cloud layers or beam stops are available in the lidar path. Equations for both the forward and the backward integration method are presented. Compared with the widely accepted inversion schemes that are based on single-point reference extinction values, the proposed method is less sensitive to noise.

© 1996 Optical Society of America

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### Equations (15)

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(1)
$$P(R)\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}\frac{c}{2}\phantom{\rule{0.2em}{0ex}}E\phantom{\rule{0.2em}{0ex}}\frac{A}{{R}^{2}}\phantom{\rule{0.2em}{0ex}}\beta (R)exp\left[-2\phantom{\rule{0.2em}{0ex}}{\mathit{\int}}_{0}^{R}\sigma (r)\text{d}r\right]\phantom{\rule{0.2em}{0ex}},\phantom{\rule{0.4em}{0ex}}R\gg c\Delta t\phantom{\rule{0.2em}{0ex}},$$
(2)
$$\beta =C\phantom{\rule{0em}{0ex}}{\sigma}^{k}\phantom{\rule{0.2em}{0ex}},$$
(3)
$$S(R)=ln\phantom{\rule{0em}{0ex}}[\beta (R)]-2\phantom{\rule{0.2em}{0ex}}{\mathit{\int}}_{0}^{R}\phantom{\rule{0.2em}{0ex}}\sigma \phantom{\rule{0em}{0ex}}(r)\text{d}r\phantom{\rule{0.2em}{0ex}}.$$
(4)
$$\sigma (R)=\frac{exp\phantom{\rule{0em}{0ex}}[S\phantom{\rule{0.2em}{0ex}}(R)/k]}{\frac{exp[S({R}_{0}\phantom{\rule{0.2em}{0ex}})/k]}{\sigma ({R}_{0}\phantom{\rule{0.2em}{0ex}})}-\frac{2}{k}\phantom{\rule{0.2em}{0ex}}{\mathit{\int}}_{{R}_{0}}^{R}exp[S\phantom{\rule{0.2em}{0ex}}(r)/k]\text{d}r}\phantom{\rule{0.2em}{0ex}}.$$
(5)
$${T}_{{R}_{0}\phantom{\rule{0.2em}{0ex}}-{R}_{e}}=exp\phantom{\rule{0em}{0ex}}\left[-{\mathit{\int}}_{{R}_{0}}^{{R}_{e}}\sigma \phantom{\rule{0em}{0ex}}(r)\text{d}r\right]\phantom{\rule{0.2em}{0ex}}.$$
(6)
$$\Lambda =\phantom{\rule{0em}{0ex}}ln\left\{\frac{exp[S({R}_{0})/k]}{\sigma ({R}_{0})}-\frac{2}{k}{\mathit{\int}}_{{R}_{0}}^{R}exp[S(r)/k]\text{d}r\right\}\phantom{\rule{0.2em}{0ex}}.$$
(7)
$$\frac{\text{d}\phantom{\rule{0em}{0ex}}\Lambda}{\text{d}R}=-\frac{2}{k}\frac{exp[S(R)/k]}{\frac{exp[S({R}_{0})/k]}{\sigma ({R}_{0})}-\frac{2}{k}{\mathit{\int}}_{{R}_{0}}^{R}exp[S(r)/k]\text{d}r}\phantom{\rule{0.2em}{0ex}}.$$
(8)
$$\sigma \phantom{\rule{0em}{0ex}}(R)=-\frac{k}{2}\frac{\text{d}\phantom{\rule{0em}{0ex}}\Lambda}{\text{d}R}\phantom{\rule{0.2em}{0ex}}.$$
(9)
$$ln[{T}_{{R}_{0}\phantom{\rule{0.2em}{0ex}}-{R}_{e}}]=\frac{k}{2}{\mathit{\int}}_{{R}_{0}}^{{R}_{e}}\text{d}\phantom{\rule{0em}{0ex}}\Lambda \phantom{\rule{0.2em}{0ex}}.$$
(10)
$$ln\phantom{\rule{0em}{0ex}}[{T}_{{R}_{0}\phantom{\rule{0.2em}{0ex}}-{R}_{e}}]=\frac{k}{2}ln\left\{\frac{\frac{exp[S({R}_{0})/k]}{\sigma ({R}_{0})}-\frac{2}{k}{\mathit{\int}}_{{R}_{0}}^{{R}_{e}}exp[S(r)/k]\text{d}r}{\frac{exp[S({R}_{0})/k]}{\sigma ({R}_{0})}}\right\}\phantom{\rule{0.2em}{0ex}}.$$
(11)
$$\sigma \phantom{\rule{0em}{0ex}}({R}_{0})=\frac{k}{2}\frac{(1-{{T}_{{R}_{0}\phantom{\rule{0.2em}{0ex}}-{R}_{e}}}^{2/k})exp[S({R}_{0})/k]}{{\mathit{\int}}_{{R}_{0}}^{{R}_{e}}exp[S(r)/k]\text{d}r}\phantom{\rule{0.2em}{0ex}}.$$
(12)
$$\sigma \phantom{\rule{0em}{0ex}}(R)=\frac{k}{2}\frac{exp[S(R)/k]}{\frac{{\mathit{\int}}_{{R}_{0}}^{{R}_{e}}exp[S(r)/k]\text{d}r}{\left(1\phantom{\rule{0.2em}{0ex}}-{{T}_{{R}_{0}-{R}_{e}}}^{2/k}\right)}-{\mathit{\int}}_{{R}_{0}}^{R}exp[S(r)/k]\text{d}r}\phantom{\rule{0.2em}{0ex}}.$$
(13)
$${\mathit{\int}}_{{R}_{0}}^{R}exp[S(r)/k]\text{d}r$$
(14)
$${\mathit{\int}}_{{R}_{0}}^{{R}_{m}}exp[S(r)/k]\text{d}r-{\mathit{\int}}_{R}^{{R}_{m}}exp[S(r)/k]\text{d}r\phantom{\rule{0.2em}{0ex}},$$
(15)
$$\sigma \phantom{\rule{0em}{0ex}}(R)=\frac{k}{2}\frac{exp[S\phantom{\rule{0.2em}{0ex}}(R)/k]}{\frac{{\mathit{\int}}_{{R}_{0}}^{{R}_{m}}exp[S(r)/k]\text{d}r}{\left(\frac{1}{{{T}_{{R}_{0}-{R}_{e}}}^{2/k}}-1\right)}\mathit{+}{\mathit{\int}}_{R}^{{R}_{m}}exp[S(r)/k]\text{d}r}\phantom{\rule{0.2em}{0ex}}.$$