## Abstract

Taking advantage of the broad spectrum of the Cabannes-Brillouin scatter from atmospheric molecules, the high spectral resolution lidar (HSRL) technique employs a narrow spectral filter to separate the aerosol and molecular scattering components in the lidar return signals and therefore can obtain the aerosol optical properties as well as the lidar ratio (i.e., the extinction-to-backscatter ratio) which is normally selected or modeled in traditional backscatter lidars. A polarized HSRL instrument, which employs an interferometric spectral filter, is under development at the Zhejiang University (ZJU), China. In this paper, the theoretical basis to retrieve the aerosol lidar ratio, depolarization ratio and extinction and backscatter coefficients, is presented. Error analyses and sensitivity studies have been carried out on the spectral transmittance characteristics of the spectral filter. The result shows that a filter that has as small aerosol transmittance (i.e., large aerosol rejection rate) and large molecular transmittance as possible is desirable. To achieve accurate retrieval, the transmittance of the spectral filter for molecular and aerosol scattering signals should be well characterized.

© 2013 OSA

## 1. Introduction

Quantitative measurements of atmospheric aerosol optical properties are required for studies of the Earth’s radiation budget and climate change [1]. Backscatter lidars are widely used to measure atmospheric aerosols. In the data processing of backscatter lidars, however, the extinction-to-backscatter ratio (or the lidar ratio) has to be assumed to be a constant and known in order to solve the ‘one-equation, two-unknown’ problem to retrieve aerosol optical properties. The retrieval accuracy of backscatter lidars is therefore limited by the accuracy of lidar ratio selection. In reality, however, the lidar ratio can vary in a large range for the atmospheric aerosols depending on their composition, size distribution, shapes and refractive index [2].

Taking advantage of the broad spectrum of the Cabannes-Brillouin scattering from atmospheric molecules, the high spectral resolution lidar (HSRL) technique employs a narrow spectral filter to reject the aerosol Mie scattering component in the lidar return signals. Therefore, an HSRL can directly measure the extinction and backscatter coefficient as well as the lidar ratio [3–5]. Numerous HSRL systems employing either atomic/molecular absorption filter [5–8] or Fabry-Perot interferometer (FPI) [3,9] have been built. An absorption filter can suppress the aerosol scattering by a factor of at least 10^{−5}. The retrieval process is simple as the residual Mie scattering in the output signal from the filter is ignorably small and hence no additional correction is needed [10]. Unlike the absorption filter, an interferometric filter produces two complementary outputs, and can only suppress the aerosol signal by a factor of about 10^{−1} to 10^{−3} [11–13]. Because there is a significant amount of residual Mie scattering left in the filtered signal, correction is employed to deal with the cross talk between the two outputs of the filter.

A polarized HSRL instrument, which employs an interferometric spectral filter, is under development at the Zhejiang University (ZJU), China. In this paper, the retrieval of the aerosol optical properties, such as extinction-to-backscatter ratio and aerosol depolarization ratio, is presented. Sensitivity of the aerosol retrieval to errors in characterizing the transmittance of the molecular and aerosol scattering signal of the spectral filter is also investigated.

The paper is constructed as follow. Section 2 presents the layout of the ZJU polarized HSRL system. Section 3 presents the mathematical basis to retrieve the aerosol optical properties. Error analyses of the aerosol retrieval are performed in Section 4, which is followed by the conclusions of this paper in Section 5.

## 2. ZJU polarized high spectral resolution lidar

#### 2.1 General description of interferometric HSRL

An HSRL takes advantage of the broad spectrum of the Cabannes-Brillouin scattering from atmospheric molecules and employs a narrow spectral filter to separate the molecular component from the aerosol component which has a much narrower spectrum in the lidar return signal. The spectral transmission function of an HSRL using an interferometric filter is schematically illustrated in Fig. 1, where Fig. 1(a) shows the spectral transmission of the HSRL interferometric filter and Fig. 1(b) shows the molecular and aerosol components in the output spectrum of the filter. In Fig. 1(a), the black dashed curve is the atmosphere backscatter signal while the pink solid and turquoise solid curves are the backscatter transmission and output signal from the spectral filter, respectively. The detected atmospheric backscatter signal is composed of a narrow spike from aerosols superimposed on a broad Gaussian-like distribution from molecules. The laser used in an HSRL system is quasi-monochromatic and has very narrow spectrum. When the laser is transmitted into the atmosphere, the scattered lights will be Doppler broadened by the random motion of the atmospheric aerosols or molecules. Because the Doppler broadening in the aerosol backscatter spectrum for a monochromatic source is 1-2 orders in magnitude smaller than the transmitter bandwidth, the aerosol backscatter spectral distribution is essentially the same as the laser spectral distribution. In contrast, the molecular signal is very broad and is about some GHz wide depending on the temperature and working wavelength. Both signals distribute in a Gaussian-like shape and center at the central frequency of the laser, as illustrated in Fig. 1(a). As is indicated, an interferometric filter produces two complementary outputs. Usually, the valley of one transmission curve is chosen to locate at the central frequency of the laser, thus suppressing the aerosol scattering. As is shown in Fig. 1(a), the output spectrum mainly contains the molecular scattering signal while the aerosol component is hardly to find.

Note that, the filter transmission function in Fig. 1(a) is idealized. Yet even that, because of the nature of interferometric filters, the aerosol scattering cannot be blocked totally and there is still some remaining in the transmitted signal. Figure 1(b) shows the molecular (blue) and aerosol (green) components of the output spectrum of the interferometric filter, respectively. We can find that, there is some aerosol scattering signal left in the output spectrum. In contrast to the absorption filters that can suppress the aerosol scattering by a factor of at least 10^{−5}, the interferometric filter can only suppress the aerosol signal by a factor of about 10^{−1} to 10^{−3}. As a result, correction should be employed in the retrieval process to deal with the cross talk between the aerosol and molecular components in the output spectrum.

#### 2.2 System layout

A schematic layout of the ZJU polarized HSRL system is shown in Fig. 2. The quasi-monochromatic laser is transmitted into the atmosphere and scattered by the molecules and aerosol particles in the atmosphere. The backscattered signals [2,14] from the molecules and aerosols are then collected by the telescope of the HSRL receiver. After some pre-processing optical systems, the backscatter signal is then split by a polarized beam splitter,_{$PBS$}, into two orthogonally polarized beams. The reflected beam is polarized perpendicularly to the polarization direction of transmitted laser beam, called the *combined perpendicular channel beam*, which is recorded by a photomultiplier tube (PMT), PMT1. The transmitted beam is polarized parallelly, and is further split by a beam splitter, _{$BS$}. Note that, $BS$ is not a 50%:50% beam splitter but one with very small reflectance and very large transmittance. The reflected part by _{$BS$} is recorded by PMT2 and called the *combined parallel channel beam*, while the transmitted part is then guided into the interferometric spectral filter. The filtered signal is collected by PMT3 and called the *molecular channel signal*.

Note that, the ZJU polarized HSRL is just a traditional polarized lidar added with an interferometric spectral filter. As mentioned earlier, traditional backscatter lidars encounter problems of one equation with two unknowns, which makes difficult in retrieving the aerosol optical properties. With the aid of the spectral filter, additional measurements can be made and therefore the ‘one-equation, two-unknown’ problem can be solved. In the next section, the retrieval process of the aerosol optical properties in the ZJU polarized HSRL will be described in detail.

## 3. Retrieval of aerosol optical properties

The signal in the *combined perpendicular channel* and *combined parallel channel* of the ZJU polarized HSRL shown in Fig. 2 can be presented as

*molecular channel*) is given by

In the above equations, ${C}_{\text{C}}^{\parallel}$, ${C}_{\text{C}}^{\perp}$ and${C}_{\text{M}}^{\parallel}$are the system constants for the combined parallel and perpendicular channels and the molecular channel, respectively. Subscriptions M and C represent the *molecular channel* and *combined channel*s, respectively, and superscriptions $\parallel $ and $\perp $indicate the parallel and perpendicular polarization, respectively; _{$r$} is the range of the scatter volume from the lidar; $\Psi $ is the transmitter-receiver geometric overlap function and approaches unity as the range from the lidar increases; ${\beta}_{m}$ and ${\beta}_{a}$ are the volume backscatter coefficients for the molecule and aerosols, respectively; ${T}_{m}$ and ${T}_{a}$ are the transmittance of the spectral filter for the molecular and aerosol scattering, respectively, and are determined by

Introducing attenuated backscatter $B=P{r}^{2}/(C\Psi )$, i.e., the normalized, range and geometric overlap-corrected lidar return signals, Eqs. (1a) – (1c) can be rewritten as

and*r*,

From the retrieval process above we can find that, in contrast to traditional backscatter lidars, the HSRL technique can obtain the optical properties of the aerosols without assumptions of their conditions. With the aid of the spectral filter, more measurements can be made. The aerosol backscatter coefficient,${\beta}_{a}$, the aerosol extinction coefficient,${\alpha}_{a}$, the aerosol depolarization ratio, ${\delta}_{a}$, and the lidar ratio (extinction-to-backscatter ratio),${S}_{a}$, can all be retrieved from the lidar return signals, the transmission constants of the filter, and the well-known Rayleigh scattering properties of the atmospheric gas composition [16].

## 4. Analysis and discussions

In the previous section, the mathematical basis to retrieve the aerosol scattering properties from the ZJU polarized HSRL have been presented. And we can conclude that the measurement accuracy of the HSRL is affected by a) the detected lidar return signals, b) the transmission characteristics of the filter, and c) the optical properties of the atmosphere molecules. Since factors (a) and (c) can be assured by choosing of high SNR (signal-to-noise ratio) detectors and more accurate atmosphere models [17], respectively, only factor (b) remains to be investigated. In this section we will perform error analysis and examine the sensitivity of the retrieval to the transmission constants of the filter.

From Eqs. (9) and (11),we can get the relationship between volume backscatter coefficient, $\beta $ and optical depth $\tau $ as

Then through the derivative of $\beta $ with respect to$\tau $, we haveIt is very interesting to find that the*absolute uncertainties*of optical depth are half the

*relative uncertainties*of backscatter coefficient.

In fact, this relation can also be got from the lidar equation from a mathematical view. From Eqs. 5(a) and 5(b) we can get the lidar equation

where, ${B}_{C}^{\perp}$ and ${B}_{C}^{\parallel}$ are the normalized, range and geometric overlap-corrected lidar return signals, $\beta $ is the volume backscatter coefficient of the atmosphere, and $\tau $ is the optical depth from lidar to the measured volume atmosphere. Eq. (16) describes the relation between the factors of the lidar setup, and the power received and the atmosphere examined. One can note that since the measured signal_{${B}_{C}^{\perp}$}and ${B}_{C}^{\parallel}$ can be regarded as constants, $\beta $ and $\mathrm{exp}\left(-2\tau \right)$ play parallel roles in attenuating the laser power. In more detail, the backscatter coefficient, $\beta $, describes the ability of the atmosphere to scatter light back into the direction from which it comes while the transmission term, $\mathrm{exp}\left(-2\tau \right)$, shows how much light gets lost on the way from the lidar to height $r$ and back. So we can conclude that these two terms have the same relative variations when subjecting to system parameter changes and that is

Comparing Eqs. (17) and (18) we can obtain the same conclusion with that from Eq. (16). Since the optical depth,$\tau $, is dependent of its value from lidar to the atmosphere at _{$r$} while the backscatter coefficient is not, the relative uncertainty of the optical depth can only be obtained after the measurement is made while the relative uncertainty of the backscatter coefficient can be obtained without dependence of the value at other distance.

We introduce the parallel aerosol scattering ratio ${R}_{}^{\parallel}=({\beta}_{a}^{\parallel}+{\beta}_{m}^{\parallel})/{\beta}_{m}^{\parallel}$, which is analogue to the aerosol scattering ratio $R=({\beta}_{a}^{}+{\beta}_{m}^{})/{\beta}_{m}^{}$ [18], then have

The relative uncertainty in the volume backscatter coefficient ($\beta ={\beta}_{a}+{\beta}_{m}$) due to the determination error in the spectral filter constants,$\Delta {T}_{a}$ and $\Delta {T}_{m}$, can be estimated usingFigure 3 shows the relative uncertainties in the backscatter coefficient estimated using Eq. (20). In Fig. 3(a) eight cases of combination of filter transmittances *T*_{a} = 0.01 and 0.05 and *T*_{m} = 0.3 and 0.7 and aerosol transmittance errors Δ*T*_{a} = 3% and 10% ($\Delta {T}_{m}$ = 0) are examined. The calculation results for ${T}_{a}$ = 0.05 and 0.01 are colored with blue and red, respectively, while the results for ${T}_{m}$ = 0.3 and 0.7 are marked with “+” and “□”, respectively. In general, _{${\eta}_{\beta}^{{T}_{a}}\text{\hspace{0.17em}}$}shows an increasing dependence on ${R}^{\parallel}$. For low aerosol loading conditions, there is little aerosol scattering in the detected return signals. In this case, the ${T}_{a}$ determination error does not cause significant impact on the measurement result. However, as the aerosol loading increases, $\Delta {T}_{a}$ increases its impacts on the retrieval accuracy. When ${T}_{m}$ = 0.3, ${T}_{a}$ = 0.01 and $\Delta {T}_{a}$ = 10%, the relative measurement uncertainty of the volume backscatter coefficient is smaller than 0.1% for ${R}^{\parallel}$ = 1.1 and larger than 30% for${R}^{\parallel}$ = 100.

From Fig. 3(a) we can also find that, a smaller ${T}_{a}$ and larger ${T}_{m}$ tend to produce better retrieval accuracy. A smaller ${T}_{a}$ value corresponds to a situation where there is a smaller fraction of the aerosol signal in the molecular channel that needs to be corrected and therefore the determination error in ${T}_{a}$ causes smaller error in the retrieval. Similarly, a larger ${T}_{m}$ value indicates the molecular channel is more dominated by the molecular signal and the determination error in ${T}_{a}$ will have smaller effect on the retrieval. Note that, the analysis present here is of general meaning and can be applied to all configurations of spectral filters.

It can also be seen in Fig. 3(a), as expected, that a better determination of ${T}_{a}$ can produce a better retrieval accuracy for the same ${T}_{a}$ and ${T}_{m}$values. However, it is worthy to note that the case of ${T}_{a}$ = 0.01 and $\Delta {T}_{a}$ = 10% produces better result than the case of ${T}_{a}$ = 0.05 and $\Delta {T}_{a}$ = 3%. This implies that a small *T*_{a} is desired.

The backscatter coefficient retrieval error resulted from the ${T}_{m}$ determination error is presented in Fig. 3(b). As in Fig. 3(a), we assume ${T}_{a}$ = 0.05 or 0.01, and ${T}_{m}$ = 0.3 or 0.7. Two error values of $\Delta {T}_{m}$ = 1% and 3% ($\Delta {T}_{a}$ = 0) are examined for each condition. Unlike the simulation results for _{${\eta}_{\beta}^{{T}_{a}}\text{\hspace{0.17em}}$}in Fig. 3(a), the retrieval error resulted by the uncertainty in ${T}_{m}$ remains the same for different aerosol loading conditions. Similar to the _{${\eta}_{\beta}^{{T}_{a}}\text{\hspace{0.17em}}$}simulation in Fig. 3(a), the spectral filter with smaller ${T}_{a}$ and larger ${T}_{m}$ is less sensitive to the transmittance determination error and can produce better retrieval. The resultant uncertainties range from 1% to 4%. Compared with $\Delta {T}_{a}$, $\Delta {T}_{m}$ has much smaller impact on the retrieval when the aerosol loading is large [11].

Overall, when the aerosol loading is very small, the impact from $\Delta {T}_{a}$ is very small and the volume backscatter retrieval is limited by _{$\Delta {T}_{m}$}. However, when the aerosol loading is large, the contribution from $\Delta {T}_{a}$ dominates the retrieval error which increases as increasing$\Delta {T}_{a}$.

Since the aerosol backscatter coefficient can be obtain by subtracting the known molecular backscatter coefficient from the total backscatter coefficient, the aerosol backscatter coefficient,${\beta}_{a}$, has the same measurement error as the total backscatter coefficient,$\beta $. As already stated, the absolute uncertainty of optical depth follows the behavior of the relative uncertainties of the backscatter coefficient. But note that, the plots of the relative uncertainties in Fig. 3 share percentage longitudinal axes while the plots for the absolute uncertainties of optical depth should range from 10^{−4} to 10^{0}. For other parameters, the volume extinction coefficient which is the derivative of the optical depth for instance, the accuracy can be estimated from the relative uncertainty of the volume backscatter coefficient or the absolute uncertainty of the optical depth. But we should note that, the trends of the measurement accuracy are similar for all the detected parameters: smaller ${T}_{a}$ and larger ${T}_{m}$ both with better determination errors tend to produce better measurement accuracy. The ${T}_{m}$ induced error plays more important role when the aerosol loading is very low while the impact of ${T}_{a}$ dominate the measurement accuracy when the aerosol loading gets higher.

## 5. Conclusions

A polarized HSRL that employs an interferometric spectral filter and is under development at the Zhejiang University was introduced in this paper. The mathematical basis for the retrieval of the aerosol optical properties such as extinction-to-backscatter ratio, and aerosol depolarization ratio, extinction and backscatter coefficients were presented. Error analyses and sensitivity studies have been carried out on the transmittance characteristics of the spectral filter. The result shows that a filter that has as small aerosol transmittance (i.e., large aerosol rejection rate) and large molecular transmittance as possible is desirable. To achieve accurate retrieval, the transmittance of the spectral filter for molecular and aerosol scattering signals should be well characterized.

## Acknowledgments

This work was partially supported by the National Defense Key Program of China (0205010803.18), the State Key Lab. of Modern Optical Instrumentation Innovation Program (MOI201208), the “985” III: First-Class Discipline Construction Program, and the Fundamental Research Funds for the Central Universities (2013QNA5006). The authors would like to express their great appreciation to the reviewers and the editor for the improvement of this paper.

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