## Abstract

This paper presents detailed analysis about the effects of spectral discrimination on the retrieval errors for atmospheric aerosol optical properties in high-spectral-resolution lidar (HSRL). To the best of our knowledge, this is the first study that focuses on this topic comprehensively, and our goal is to provide some heuristic guidelines for the design of the spectral discrimination filter in HSRL. We first introduce a theoretical model for retrieval error evaluation of an HSRL instrument with a general three-channel configuration. The model only takes the error sources related to the spectral discrimination parameters into account, while other error sources not associated with these focused parameters are excluded on purpose. Monte Carlo (MC) simulations are performed to validate the correctness of the theoretical model. Results from both the model and MC simulations agree very well, and they illustrate one important, although not well realized, fact: a large molecular transmittance and a large spectral discrimination ratio (SDR, i.e., ratio of the molecular transmittance to the aerosol transmittance) are beneficial to promote the retrieval accuracy. More specifically, we find that a large SDR can reduce retrieval errors conspicuously for atmosphere at low altitudes, while its effect on the retrieval for high altitudes is very limited. A large molecular transmittance contributes to good retrieval accuracy everywhere, particularly at high altitudes, where the signal-to-noise ratio is small. Since the molecular transmittance and SDR are often trade-offs, we suggest considering a suitable SDR for higher molecular transmittance instead of using unnecessarily high SDR when designing the spectral discrimination filter. These conclusions are expected to be applicable to most of the HSRL instruments, which have similar configurations as the one discussed here.

© 2014 Optical Society of America

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### Equations (28)

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(1)
$${B}_{C}^{\perp}=({\beta}_{m}^{\perp}+{\beta}_{a}^{\perp})\mathrm{exp}(-2\tau ),\phantom{\rule{0ex}{0ex}}{B}_{C}^{\parallel}=({\beta}_{m}^{\parallel}+{\beta}_{a}^{\parallel})\mathrm{exp}(-2\tau ),\phantom{\rule{0ex}{0ex}}{B}_{M}^{\parallel}=({T}_{m}{\beta}_{m}^{\parallel}+{T}_{a}{\beta}_{a}^{\parallel})\mathrm{exp}(-2\tau ),$$
(2)
$$\beta ={\beta}_{m}\frac{(1+\delta )}{(1+{\delta}_{m})}\frac{({T}_{m}-{T}_{a})K}{1-{T}_{a}K},$$
(3)
$$\delta ={B}_{C}^{\perp}/{B}_{C}^{\parallel}.$$
(4)
$$K={B}_{C}^{\parallel}/{B}_{M}^{\parallel}.$$
(5)
$$\tau =-\frac{1}{2}\mathrm{ln}\left[\frac{(1-K{T}_{a})(1+{\delta}_{m}){B}_{M}^{\parallel}}{({T}_{m}-{T}_{a}){\beta}_{m}}\right].$$
(6)
$${\epsilon}_{t}^{2}={\left(\frac{{\sigma}_{\beta}}{\beta}\right)}^{2}={\left(\frac{\partial \beta}{\beta \partial \delta}\right)}^{2}{({\sigma}_{\delta})}^{2}+{\left(\frac{\partial \beta}{\beta \partial K}\right)}^{2}{({\sigma}_{K})}^{2}+{\left(\frac{\partial \beta}{\beta \partial {T}_{a}}\right)}^{2}{({\sigma}_{{T}_{a}})}^{2}+{\left(\frac{\partial \beta}{\beta \partial {T}_{m}}\right)}^{2}{({\sigma}_{{T}_{m}})}^{2},$$
(7)
$${\epsilon}_{1}^{2}=\frac{{\overline{\delta}}^{2}}{{(1+\overline{\delta})}^{2}}[\frac{1}{{\mathrm{SNR}}_{C\parallel}^{2}}+\frac{1}{{\mathrm{SNR}}_{C\perp}^{2}}],$$
(8)
$${\epsilon}_{2}^{2}=\frac{1}{{(1-\overline{{T}_{a}}\overline{K})}^{2}}[\frac{1}{{\mathrm{SNR}}_{C\parallel}^{2}}+\frac{1}{{\mathrm{SNR}}_{M\parallel}^{2}}],$$
(9)
$${R}^{\parallel}=\frac{{\beta}_{a}^{\parallel}+{\beta}_{m}^{\parallel}}{{\beta}_{m}^{\parallel}}.$$
(10)
$${R}^{\parallel}=\frac{({T}_{m}-{T}_{a})K}{1-{T}_{a}K}.$$
(11)
$${\epsilon}_{1}^{2}={\left[\frac{({R}^{\parallel}-1){\delta}_{a}+{\delta}_{m}}{{R}^{\parallel}+({R}^{\parallel}-1){\delta}_{a}+{\delta}_{m}}\right]}^{2}\frac{1}{{\mathrm{SNR}}_{C\parallel C\perp}^{2}},$$
(12)
$${\epsilon}_{2}^{2}={[1+\frac{{R}^{\parallel}}{\overline{{T}_{m}}/{\overline{{T}_{a}}}_{}-1}]}^{2}\frac{1}{{\mathrm{SNR}}_{C\parallel M\parallel}^{2}},$$
(13)
$${\epsilon}_{3}^{2}={\left[\frac{{R}^{\parallel}-1}{\overline{{T}_{m}}-\overline{{T}_{a}}}\right]}^{2}{({\sigma}_{{T}_{a}})}^{2}$$
(14)
$${\epsilon}_{4}^{2}=\frac{1}{{(\overline{{T}_{m}}-\overline{{T}_{a}})}^{2}}{({\sigma}_{{T}_{m}})}^{2}.$$
(15)
$${({\sigma}_{\tau})}^{2}=\frac{1}{4}[{(1+\frac{{R}^{\parallel}}{{T}_{m}/{T}_{a}-1})}^{2}\frac{1}{{\mathrm{SNR}}_{M\parallel}^{2}}\phantom{\rule{0ex}{0ex}}+{\left(\frac{{R}^{\parallel}}{{T}_{m}/{T}_{a}-1}\right)}^{2}\frac{1}{{\mathrm{SNR}}_{C\parallel}^{2}}+{\epsilon}_{3}^{2}+{\epsilon}_{4}^{2}].$$
(16)
$$\mathrm{SDR}={T}_{m}/{T}_{a}.$$
(17)
$${\epsilon}_{1}^{2}={\left(\frac{\partial \beta}{\beta \partial \delta}\right)}^{2}\xb7{\sigma}_{\delta}^{2}=\frac{1}{1+\overline{\delta}}\xb7{\sigma}_{\delta}^{2}.$$
(18)
$${\sigma}_{\delta}=E[{(\delta -\overline{\delta})}^{2}],$$
(19)
$$\delta -\overline{\delta}=\frac{1}{\overline{{B}_{C}^{\parallel}}}({B}_{C}^{\perp}-\overline{{B}_{C}^{\perp}})+\frac{-\overline{{B}_{C}^{\perp}}}{{\overline{{B}_{C}^{\parallel}}}^{2}}({B}_{C}^{\parallel}-\overline{{B}_{C}^{\parallel}}).$$
(20)
$${({\sigma}_{\delta})}^{2}=\frac{{({\sigma}_{{B}_{C}^{\perp}})}^{2}}{{\overline{{B}_{C}^{\parallel}}}^{2}}+\frac{{({\sigma}_{{B}_{C}^{\parallel}})}^{2}\xb7{\overline{{B}_{C}^{\perp}}}^{2}}{{\overline{{B}_{C}^{\parallel}}}^{4}}.$$
(21)
$${({\sigma}_{\delta})}^{2}={\overline{\delta}}^{2}[\frac{{({\sigma}_{{B}_{C}^{\perp}})}^{2}}{{\overline{{B}_{C}^{\perp}}}^{2}}+\frac{{({\sigma}_{{B}_{C}^{\parallel}})}^{2}}{{\overline{{B}_{C}^{\parallel}}}^{2}}]\phantom{\rule{0ex}{0ex}}={\overline{\delta}}^{2}[\frac{1}{{\mathrm{SNR}}_{C\perp}^{2}}+\frac{1}{{\mathrm{SNR}}_{C\parallel}^{2}}].$$
(22)
$$\delta =\frac{{\beta}_{a}^{\perp}+{\beta}_{m}^{\perp}}{{\beta}_{a}^{\parallel}+{\beta}_{m}^{\parallel}}=\frac{{\beta}_{a}^{\parallel}{\delta}_{a}+{\beta}_{m}^{\perp}{\delta}_{m}}{{\beta}_{a}^{\parallel}+{\beta}_{m}^{\parallel}}=\frac{({R}_{\parallel}-1){\delta}_{a}+{\delta}_{m}}{{R}^{\parallel}}.$$
(23)
$${\mathrm{SNR}}_{M\parallel}\propto \sqrt{{T}_{m}{\beta}_{m}^{\parallel}+{T}_{a}{\beta}_{a}^{\parallel}}.$$
(24)
$${\epsilon}_{2}^{2}\propto {(1+\frac{{R}^{\parallel}}{\mathrm{SDR}-1})}^{2}(\frac{1}{{\mathrm{SNR}}_{C\parallel}^{2}}+\frac{1}{{T}_{m}{\beta}_{m}^{\parallel}+{T}_{m}{\beta}_{a}^{\parallel}/\mathrm{SDR}}).$$
(25)
$$\frac{\partial {\epsilon}_{2}^{2}}{\partial {T}_{m}}\propto {(1+\frac{{R}^{\parallel}}{\mathrm{SDR}-1})}^{2}\frac{-({\beta}_{m}^{\parallel}+{\beta}_{a}^{\parallel}/\mathrm{SDR})}{{({T}_{m}{\beta}_{m}^{\parallel}+{T}_{m}{\beta}_{a}^{\parallel}/\mathrm{SDR})}^{2}}<0.$$
(26)
$$\frac{\partial {\epsilon}_{2}^{2}}{\partial \mathrm{SDR}}\propto {(1+\frac{{R}^{\parallel}}{\mathrm{SDR}-1})}^{2}\frac{{T}_{m}{\beta}_{a}^{\parallel}}{{\mathrm{SDR}}^{2}{({T}_{m}{\beta}_{m}^{\parallel}+{T}_{m}{\beta}_{a}^{\parallel}/\mathrm{SDR})}^{2}}\phantom{\rule{0ex}{0ex}}-(1+\frac{{R}^{\parallel}}{\mathrm{SDR}-1})\frac{2{R}^{\parallel}}{{(\mathrm{SDR}-1)}^{2}}[\frac{1}{{\mathrm{SNR}}_{C\parallel}^{2}}+\frac{1}{{T}_{m}{\beta}_{m}^{\parallel}+{T}_{m}{\beta}_{a}^{\parallel}/\mathrm{SDR}}].$$
(27)
$$A=\frac{({R}^{\parallel}+\mathrm{SDR}-1){T}_{m}{\beta}_{a}^{\parallel}}{{({T}_{m}{\beta}_{m}^{\parallel}\mathrm{SDR}+{T}_{m}{\beta}_{a}^{\parallel})}^{2}}-\frac{2{R}^{\parallel}\mathrm{SDR}}{(\mathrm{SDR}-1)({T}_{m}{\beta}_{m}^{\parallel}\mathrm{SDR}+{T}_{m}{\beta}_{a}^{\parallel})}\phantom{\rule{0ex}{0ex}}=\frac{1}{{T}_{m}{\beta}_{m}^{\parallel}\mathrm{SDR}+{T}_{m}{\beta}_{a}^{\parallel}}[\frac{({R}^{\parallel}-1+\mathrm{SDR}){\beta}_{a}^{\parallel}/{\beta}_{m}^{\parallel}}{\mathrm{SDR}+{\beta}_{a}^{\parallel}/{\beta}_{m}^{\parallel}}-\frac{2{R}^{\parallel}\mathrm{SDR}}{\mathrm{SDR}-1}]<0.$$
(28)
$$A=\frac{1}{{T}_{m}{\beta}_{m}^{\parallel}\mathrm{SDR}+{T}_{m}{\beta}_{a}^{\parallel}}[{R}^{\parallel}-1-\frac{2{R}^{\parallel}\mathrm{SDR}}{\mathrm{SDR}-1}]\phantom{\rule{0ex}{0ex}}<\frac{1}{{T}_{m}{\beta}_{m}^{\parallel}\mathrm{SDR}+{T}_{m}{\beta}_{a}^{\parallel}}[{R}^{\parallel}-1-2{R}^{\parallel}]<0.$$