## Abstract

Unlike other errors in the lidar equation solution for the two-component atmosphere, the error of the measured aerosol extinction coefficient caused by inaccuracies in the assumed aerosol backscatter-to-extinction ratios significantly depends on the aerosol spatial inhomogeneity. In a slightly nonhomogeneous atmosphere, an incorrect value in the assumed aerosol backscatter-to-extinction ratio does not significantly corrupt the measurement result, whereas in an atmosphere with a large monotonic change of the aerosol extinction [e.g., in the lower troposphere], the incorrect value yields a large distortion of the retrieved extinction-coefficient profile. In the latter case, even the far-end solution can produce a large error in the retrieved extinction coefficient. The analytical formulas for the determination of the range errors, obtained for the Klett and the optical-depth solutions, show that these errors significantly depend on the method of the boundary-condition determination. Distortions of the retrieved aerosol extinction profiles are, in general, larger if the assumed aerosol backscatter-to-extinction ratio is underestimated in relation to the real value.

© 1995 Optical Society of America

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### Equations (17)

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(1)
$$S(r)=Cu(r)\text{exp}\left[-2{\int}_{{r}_{0}}^{r}u({r}^{\prime}){\text{d}r}^{\prime}\right],$$
(2)
$$u(r)={\mathrm{\sigma}}_{m}(r)[a(r)+R(r)],$$
(3)
$$R(r)=\frac{{\mathrm{\sigma}}_{a}(r)}{{\mathrm{\sigma}}_{m}(r)},$$
(4)
$$a(r)=\frac{{P}_{\mathrm{\pi},m}}{{P}_{\mathrm{\pi},a}(r)};$$
(5)
$$u(r)=\frac{0.5S(r)}{{I}_{b}-{\int}_{{r}_{b}}^{r}S({r}^{\prime}){\text{d}r}^{\prime}},$$
(6)
$${I}_{b}=\frac{0.5S({r}_{b})}{u({r}_{b})}.$$
(7)
$${I}_{b}=\frac{{\int}_{{r}_{0}}^{{r}_{m}}S({r}^{\prime}){\text{d}r}^{\prime}}{1-{T}^{2}({r}_{0},{r}_{m})\text{exp}\left\{-2{\int}_{{r}_{0}}^{{r}_{m}}{\mathrm{\sigma}}_{m}({r}^{\prime})[a({r}^{\prime})-1]{\text{d}r}^{\prime}\right\}}.$$
(8)
$$\u3008Y(r)\u3009={C}_{1}{a}_{\text{as}}(r)\text{exp}\left\{-2{\int}_{{r}_{0}}^{r}[{a}_{\text{as}}({r}^{\prime})-1]{\mathrm{\sigma}}_{m}({r}^{\prime}){\text{d}r}^{\prime}\right\}.$$
(9)
$$\u3008S(r)\u3009={C}_{2}D(r){u}_{\text{est}}(r)\text{exp}\left[-2{\int}_{{r}_{0}}^{r}{u}_{\text{est}}({r}^{\prime}){\text{d}r}^{\prime}\right].$$
(10)
$${u}_{\text{est}}(r)={\mathrm{\sigma}}_{m}(r)[{a}_{\text{as}}(r)+R(r)].$$
(11)
$$D(r)=\frac{1+\frac{R(r)}{a(r)}}{1+\frac{R(r)}{a(r)}\frac{{[{P}_{\mathrm{\pi},a}(r)]}_{\text{as}}}{{P}_{\mathrm{\pi},a}(r)}}.$$
(12)
$$\frac{\u3008u(r)\u3009}{{u}_{\text{est}}(r)}=\frac{D(r){{V}_{c}}^{2}({r}_{b},r)}{D({r}_{b})-2{\int}_{{r}_{b}}^{r}D({r}^{\prime}){u}_{\text{est}}({r}^{\prime}){{V}_{c}}^{2}({r}_{b},{r}^{\prime}){\text{d}r}^{\prime}},$$
(13)
$${{V}_{c}}^{2}({r}_{b},r)=\text{exp}\left\{-2{\int}_{{r}_{b}}^{r}{\mathrm{\sigma}}_{m}({r}^{\prime})[{a}_{\text{as}}({r}^{\prime})+R({r}^{\prime})]{\text{d}r}^{\prime}\right\}.$$
(14)
$$\mathrm{\delta}{\mathrm{\sigma}}_{a}(r)=\left[1+\frac{{a}_{\text{as}}(r)}{R(r)}\right]\hspace{0.17em}\left[\frac{\u3008u(r)\u3009}{{u}_{\text{est}}(r)}-1\right].$$
(15)
$$\frac{\u3008u(r)\u3009}{{u}_{\text{est}}(r)}=\frac{D(r){{V}_{c}}^{2}({r}_{0},r)}{\frac{2}{1-{{V}_{c}}^{2}({r}_{0},{r}_{m})}{\int}_{{r}_{0}}^{{r}_{m}}D({r}^{\prime}){u}_{\text{est}}({r}^{\prime}){{V}_{c}}^{2}({r}_{0},{r}^{\prime}){\text{d}r}^{\prime}-2{\int}_{{r}_{0}}^{r}D({r}^{\prime}){u}_{\text{est}}({r}^{\prime}){{V}_{c}}^{2}({r}_{0},{r}^{\prime}){\text{d}r}^{\prime}},$$
(16)
$$\frac{\mathrm{\Delta}a(r)}{a(r)}=\frac{-\mathrm{\Delta}{P}_{\mathrm{\pi},a}(r)}{{P}_{\mathrm{\pi},a}(r)+\mathrm{\Delta}{P}_{\mathrm{\pi},a}(r)}.$$
(17)
$$D(r)\approx \frac{{P}_{\mathrm{\pi},a}(r)}{{[{P}_{\mathrm{\pi},a}(r)]}_{\text{as}}},$$