1. Introduction

The contribution of multiple scattering in lidar returns has been investigated for a long time. Up to date, it is generally recognized that the contribution on lidar returns due to the multiple scattering depends on the optical depth and optical properties (scattering phase function) of the scattering medium, the initial diameter and divergence angle of laser pulse, the distance to the volume sounded, the viewing angle of the receiving system, and the radiation wavelength [1]. It is recognized as well that the primary effect of the multiple scattering is to make the extinction coefficient of the medium appear to be less than it really is [2]. The prominent study on multiple scattering was performed by Platt, who introduced a multiple scattering parameter to describe the reduction in optical depth in the lidar equation [3]. In particular, different parameters which characterized different influences of multiple scattering on the elastic and Raman lidar returns were firstly introduced by Ulla Wangdinger [4]. To date, Monte Carlo techniques have been applied in a wide variety of ways to study the multiple scattering influences on the lidar returns [5-8]. A description of the experimental evidence of multiple scattering was presented in the work by Bruscaglioni et al. (1998) [6]. The MUSCLE (MUltiple SCattering Lidar Experiments) group, which took into account multiple scattering for a typical ground-based cloud-sensing scenario, started from the transport equation for radiation in a dense medium and developed different methods to account for multiple scattering by atmospheric particles [9-11]. And for the space-borne lidar returns, multiple scattering effects in the lidar retrievals were studied by Winker [12]. In addition to the multiple scattering parameters, the multiple-to-single scattering ratio which is the ratio between multiple scattering lidar equation and single scattering lidar equation was introduced by Bissonnette at the method of multiple scattering based lidar retrieval [13, 14]. However, most of the lidar inverse methods, such as Klett [15], Fernald [16], iterative algorithms [17, 18], the CESC technique [19] and Kunz’s Bipath method [20] etc., used for the retrieving of aerosol optical parameters, are all originally developed in the context of single scattering, and the multiple scattering effect is neglected during the process of lidar inversion method.

In this paper a modified CESC method and a Monte Carlo-based simulation are presented. The Monte Carlo-based simulation, in which multiple scattering parameters for both ground-based and space-borne lidar returns were retrieved, is used to test the feasibility of the modified CESC technique. Compared to the CESC method, the modified one is not only suitable for the single scattering signals, but also for signals influenced by the multiple scattering. In Section 2 the experimental evidence of different multiple scattering effects on CALIPSO (Cloud Aerosol Lidar and Infrared Pathfinder Satellite Observations) and Napoli (40.838°N, 14.183°E, 118 m above sea level, Italy) lidar returns are shown, respectively. Section 3 describes the theory of the modified CESC method used to retrieve correct value of backscattering and extinction coefficients from the combining ground-based and space-borne elastic lidar returns. In Section 4 the application of the modified method to a Monte Carlo-based simulation is discussed. The findings are briefly summarized in Section 5.

2. Experimental evidence of multiple scattering

We adopted the CALIPSO Level 1 products and the Napoli lidar signals together to analyze the difference of multiple scattering effects on both ground-based and space-borne lidar returns. The Napoli lidar station is part of EARLINET (the European Aerosol Research LIdar NETwork) and is based on a Nd:YAG laser source which produces simultaneous pulses at 1064nm, 532nm and 355nm at a pulse repetition rate of 20 Hz [21]. The CALIPSO lidar, which was launched on 28 April 2006, transmits its laser pulses in the atmosphere in opposite direction compared to that of Napoli lidar. Due to the counter-looking lidar returns, the comparison method described clearly in [22] is used to construct the downward attenuated backscatter β_G from the results of lidar measurements carried out at the Napoli lidar station.

To study the effects of multiple scattering, as the definition of accumulated single scattering fraction ,A_S [23], the ratio R_acc is defined as:

where β_S is the total attenuated backscatter of CALIPSO level 1 products. β_G is the constructed downward attenuated backscatter from the results of the ground-based lidar measurements. z ₀ is the near-range boundary of the medium being measured. The accumulated depolarization ratio, δ_acc(z), is retrieved from the CALIPSO lidar measurements at 532nm by the following formula:

where β _S,⊥ (z) and β _S,∥ (z) represent, respectively, the components of the range-corrected total attenuated backscatter signal, β_S, polarized parallel and perpendicular to the polarization plane of the laser transmitter. The relationship between lidar multiple scattering and depolarization for water clouds were introduced previously by Hu et al [23]. To compare with the ratio R_acc, the accumulated single scattering fraction, A_S for the water cloud, is given by [23]:

An integration of signals was performed in order to increase the signal to noise ratio for the vertical profiles of β_S and β_G. The choice of the number of CALIPSO profiles to integrate has been done by taking into account the closeness of the CALIPSO ground track to the Napoli lidar station. β_S was obtained by averaging 326 profiles (~ 16 seconds) corresponding to footprint path the order of the distance between CALIPSO ground track and the position of Napoli lidar, ~ 60 km [19]. Signals from ground-based lidar are acquired with a time resolution of 1 min. To increase the signal to noise ratio, we integrate ten profiles, centered at the overpass time. Figure 1 shows statistical results from CALIPSO and Napoli lidar measurements from the year of 2006 to 2008. In Fig. 1 the red circles describe the relationship between the ratio R_acc and the accumulated depolarization ratio, δ_acc(z), where the ratio R_acc and δ_acc(z) are obtained from Eqs. (1) and (2), respectively. The black squares illustrated in Fig. 1 represent the accumulated single scattering fraction, A_s, for the water cloud which is calculated by Eq. (3). The fit line F1 in Fig. 1 is the continuous curve retrieved by Eq. (3). From Fig. 1(a) it can be deduced that the ratio R_acc in the planetary boundary layer (PBL) is smaller than 1, which means the backscattered signal of Napoli lidar is smaller than that of CALIPSO lidar. At low altitude, the ground-based lidar returns are usually considered as the single backscattered signals due to the short distance between the laser source and scattering volume and narrow field of view. Thus the ratio R_acc has the same meaning as the accumulated single scattering fraction at this altitude range. As shown in Fig. 1(a), there are hardly any cases with value of R_acc similar to the accumulated single scattering fraction. This is probably due to the fact that A_s calculated from Eq. (3) is suitable for water clouds or aerosols with spherical particles. It is well known that the atmosphere within the PBL is easily influenced by human activities, such as industry pollutions, biomass burnings, which are usually considered to be the key reasons that make the scattering in the PBL more complex.

 

Fig. 1. Relation between R_acc and δ_acc for three different altitude ranges (a) from 0 to 2km, (b) from 2 to 5km, (c) from 8 to 15km. The black square represents values for water clouds, F1 is the continuous curve obtained by Eq. (3) and the red circle stands for the actual cases from the year 2006 to 2008.

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Figure 1(b) characterizes the relationship between the ratio R_acc and accumulated depolarization ratio at the range from 2km to 5km. At the same value of accumulated depolarization ratio, the cases in which the ratio R_acc is smaller than the accumulated single scattering fraction indicate that the shape of aerosol particles are not all spherical and the space-borne lidar returns were more influenced by multiple scattering than the ones received by the ground-based lidar. This is greatly due to the longer distance between the laser source and scattering volume for the CALIPSO lidar than that for the Napoli lidar. The relationship between the ratio R_acc and accumulated depolarization ratio at the range from 8km to 15km are illustrated in Fig. 1(c). However, compared with the Fig. 1(b), Fig. 1(c) shows that most of the measured ratios R_acc are greater than 1, meaning that at this altitude multiple scattering also influenced the ground-based lidar returns.

The number of cases with the ratio R_acc close to 1 gradually increases from the Fig. 1(a) to Fig. 1(c), which means that the signals detected by the ground-based lidar are gradually comparable with the ones received by the space-borne lidar. Due to this, the ground-based lidar returns are also recognized to be influenced by the multiple scattering in the same way as the space-borne lidar ones. As a result, from Fig. 1 it can be concluded that the effects of multiple scattering on both the ground-based and space-borne lidar returns should be considered differently along the laser path due to the different influences of multiple scattering on the two counter-looking lidar returns. This is mainly because of the different distance between laser source and scattering volume for the ground-based and space-borne lidar measurements, different aerosols detected by the two lidars, the optical depth and the optical properties of aerosols. In the next section, the question of how to obtain optical parameters of aerosols by considering the influences of multiple scattering is briefly described.

3. The modified CESC method

The effects of multiple scattering were not considered in the CESC technique firstly introduced by Wang et al [19]. Based on the CESC technique and Bipath method, a modified CESC method is proposed here which is also suitable for the cases when the lidar returns were influenced by the multiple scattering. As the same with the CESC technique, the two counter-looking lidars, a ground-based lidar and a space-borne lidar, detect the same atmosphere volume within the measurement time. This is also the precondition for the modified CESC algorithm.

We indicate RCS_g (Range Corrected Signal) and RCS_S the range corrected total elastic backscatter signals at the same wavelength as available from ground-based and space-borne lidar, respectively. In the case that multiple scattering influences the lidar returns, the two signals can be written as:

In Eqs. (4) and (5) z is the altitude (z=0 is the sea level. z=z_s is the space-borne lidar calibration altitude). R₀ and A₀ include all instrumental constants of the two lidars. β is the total volume backscattering coefficient. α_p(z) and α_m(z) denote volume extinction coefficients for particle and molecule, respectively. F_G and F_S stand for the multiple scattering parameters for the ground-based and space-borne lidar returns. For the sake of simplicity, the overlap function of the ground-based lidar is set to be unity and the multiple scattering parameters F_G and F_S are both considered to be independent on laser path within the aerosol layer.

The ratio of the Eqs. (5) and (4) can be expressed as:

here $k\text{'}=\genfrac{}{}{0.1ex}{}{A_0}{R_0}\exp \left(-2{\int }_0^{z_s}{\alpha }_p\left(z\text{'}\right)+{\alpha }_m\left(z\text{'}\right))dz\text{'}\right)$ is independent of altitude z. From Eq. (6) the extinction coefficient of the particle at altitude z can be obtained as:

where α_p^eff(z) is the effective extinction coefficient calculated by the CESC technique in which multiple scattering is not considered. If we do not consider the multiple scattering influences on the lidar signals, namely, F_G and F_S are equal to zero, the extinction formula of Eq. (7) is the same as obtained by the CESC technique [19].

Then the product of Eqs.(4) and (5) can be written as:

where k = R 0 A 0 exp(-2 ∫zs 0 (αp(z′)+αm(z′))dz′) is independent of altitude z. Considering an altitude z *, at which the total backscattering coefficient β(z *) is known, then the constant k can be determined. If the altitude z * is chosen within aerosol free region the total backscattering coefficient can be determined from a pure molecular atmospheric model or from radio sounding, β(z *)=βm(z *) and the total backscattering coefficient at altitude z is retrieved as:

here β^eff is the effective total backscattering coefficient obtained from CESC technique similar to the effective extinction coefficient α_p^eff(z). From Eq. (9), if there is no multiple scattering, that is, F_G and F_S equal zero, or the multiple scattering parameter F_G for the ground-based lidar measurement equals F_S for the space-borne lidar measurement, the total backscattering coefficient from Eq. (9) is the same as that in [19]. From Eqs. (7) and (9), it is easy to find that the extinction and backscattering coefficients calculated by CESC technique are correct only when the multiple scattering influences can be neglected, namely, F_G and F_S are equal to zero, otherwise they are only effective values. It can be easily understood from Eqs. (7) and (9) that the multiple scattering effects lead to an extinction coefficient less than it really is, and to a backscattering coefficient which depends on the difference between the multiple scattering functions F_G and F_S and thus can be larger or smaller than it really is, or even equal to the real one.

The multiple scattering parameters F_G and F_S, which are needed to calculate the extinction and backscattering coefficients can be obtained as in the following form [4, 7]:

here, the index i refers to either ground-based (G) or space-borne (S) lidar signals. $\genfrac{}{}{0.1ex}{}{P_i^{\left(\mathit{tot}\right)}\left(z\right)}{P_i^{\left(1\right)}\left(z\right)}$ is the ratio between the total received power and that contributing to the single scattering. α_p^eff(z) is the so-called effective extinction coefficient retrieved from the total backscattered lidar returns by the CESC technique as described above.

4. Monte Carlo-based multiple scattering parameters retrieving

In this work the Monte Carlo code previously developed by Bruscaglioni et al. [9] was used to simulate two lidar signals of typical cirrus cloud (c10527r01, which is one of cirrus phase functions from the datasets of Cirrus Optical Properties Enhanced version (COPE) provided by Dr. M. Hess, and the lidar ratio of this cirrus cloud is 22.7sr [24]). The phase function represents randomly oriented hexagonal ice crystals with maximum tilt angle of 1° [25, 26]. The parameters used in simulation are shown in table 1, where the values of field of view and laser divergence are those of Napoli and CALIPSO lidar system.

 

Fig. 2. The simulated RCS_g and RCS_S of three layers system-cirrus cloud (9-9.5km) and pure molecule Rayleigh scattering (7-9km, 9.5-10.5km), where ‘1’ and ‘t’ correspond to single and total scattering signal

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To simplify, we employed pure Rayleigh scatter below and above the cirrus and vacuum below 7km. The simulated extinction coefficients of cirrus and pure molecule below and above the cirrus are 5×10^-4m^-1, 5.5×10^-6m^-1 and 4×10^-6m^-1, respectively. Three different scatter (phase function) layers have been considered because the inversions of the volume backscattering coefficient need pure Rayleigh scattering as reference. The previous work [9] shows that five orders of scattering are enough for the geometric and scattering condition like this. The simulated single scattering and total scattering (from 1 to 5 orders) RCSg which stands for signals detected by the Napoli lidar and RCS_s standing for the CALIPSO lidar signals, are shown in Fig. 2.

Obviously, the single scattering signal follows Eqs. (4) and (5) when the multiple scattering parameters F_i are zero. As shown in Fig. 3, the cirrus extinction and backscattering coefficients obtained from the single scattering signal depicted as red circles are equal to the inputted values the black square ones, i.e., α_p=5×10^-4m^-1 and lidar ratio S_p=22.7sr. The total scattering signal deviates from single scattering nearly linearly increasing inside the cirrus cloud both for the ground-based and space-borne lidar as shown in Fig. 2. Above and below the cirrus cloud, where only rare pure Rayleigh scattering occurs, the difference between single and total scattering is little and nearly constant. The cirrus effective extinction coefficient and effective lidar ratio, which correspond to the total scattering signal, are α_peff=1.96×10^-4m^-1 and S_p^eff=8.71sr.

Table 1. Monte Carlo parameters used in the simulation

View Table

With the correct phase function, effective extinction coefficient, the total and single scattering signals simulated by Monte Carlo code, the multiple scattering parameters F_G and F_S can be calculated by the Eq. (10). The detailed information about how to choose the correct phase function is clearly described in [7]. With the multiple scattering parameters F_G and F_S, corrected extinction and backscattering coefficients can be obtained from Eqs. (7) and (9). The corrected extinction values were used in the Monte Carlo code to simulate the single and total scattering signals of RCS_g and RCS_S, thus the multiple scattering parameters F_G and F_S for the next step can be recalculated by the Eq. (10). Another new corrected extinction and backscattering coefficients can be retrieved again. Here several iteration steps are needed until the new extinction profile is compatible with the previous one within the set error margin. The multiple scattering parameters from the first to the fourth iteration step retrieved from the simulated signals both for the ground-based and space-borne lidar returns are displayed in Fig. 4. The supposed values, the extinction and backscattering coefficients calculated from the single scattering lidar returns, the effective values calculated from the total scattering lidar retuns and corrected extinction and backscattering coefficients at each iteration step are shown in Fig. 3.

From Fig. 4, it can be seen that the multiple scattering parameter for the ground-based lidar returns is different from that of space-borne lidar returns, and F_G is a little smaller than F_S approximately due to the longer distance between the laser source and the scatter volume for the space-borne lidar compared with the ground-based lidar. The effective values of extinction and backscattering coefficients illustrated in Fig. 3 as green triangles are both calculated by the CESC technique. From the simulation results associated with Fig. 3, the error of the effective extinction coefficient is as large as 70% and the error of the effective backscattering coefficient is also nearly 10%. Using the modified CESC method mentioned above, 4 iteration steps are required to correct the effective values of backscattering and extinction coefficients. As described in Fig. 3, the values of extinction and backscattering coefficients gradually approach the supposed values from the first to the fourth iteration step and the results of the fourth iteration are nearly the same as the values we supposed at first.

 

Fig. 3. The supposed, single, effective and corrected profiles of the extinction and backscattering coefficients, which are represented by ‘Supposed’, ‘Single’, ‘Effect’ and numbers, respectively. The number corresponds to the iteration step

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Fig. 4. Multiple scattering parameters F_G and F_S for the ground-based and space-borne lidar returns at the four iteration steps. The number corresponds to the iteration step

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5. Conclusion

Based on the lidar measurements carried out at the CALIPSO and Napoli lidar stations from the year 2006 to 2008, the effects of multiple scattering differ in the ground-based and space-borne lidar returns along the laser path. In the PBL, the ratio R_acc is smaller than the accumulated single scattering fraction of water cloud. This is primarily due to the fact that the non-spherical particles exist within PBL. At high altitude range, most of the ratio R_acc are higher than 1 because the lidar returns of ground-based lidar measurement were also influenced by the multiple scattering.

To analyze the multiple scattering influences, we introduce F_G and F_S to represent the multiple scattering parameters for the ground-based and space-borne lidar returns in CESC technique. The modified CESC algorithm proposed in this paper is not only suitable for the single scattering lidar returns but also for the lidar returns influenced by the multiple scattering. The algorithm has been applied to Monte Carlo-based simulation. From the simulation results, it can be concluded that the multiple scattering parameter for the ground-based lidar returns is different from the one for the space-borne lidar returns. Compared with the CESC technique, the modified one is more suitable to the actual lidar measurements. This is because that the multiple scattering effects should be considered on the space-borne lidar returns due to the long distance between the laser source and the scatter volumes. For the ground-based lidar returns, the effect of multiple scattering is also prominent and significant when the particle is non-spherical or the distance to the volume sounded is far away.

However, this method depends on the Monte Carlo code which is used to simulate the single and total attenuated backscatter signals. The two signals were used to calculate the multiple scattering parameters for correcting the effective values of extinction and backscattering coefficients. If we have suitable phase functions to characterize the practical aerosol or cloud layer, the multiple scattering parameters for the actual lidar measurements can be obtained by the Monte Carlo code, and then the extinction and backscattering coefficients can be retrieved more accurately by the modified CESC algorithm.