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Abstract
Multiple scattering is always present in LiDAR measurements. It is one of the major causes of LiDAR signal depolarization when detecting backscattering from water clouds. For a given probing wavelength, the LiDAR signal is a function of the aerosol size distribution, cloud range, and optical depth, and of the LiDAR field of view (FoV). We present a relatively simple polarimetric multiple scattering model. It uses Poisson statistics to determine the photons’ scattering order distribution at a given optical depth and takes into account the aerosol’s properties as well as the characteristics of the LiDAR. The results are compared with Monte Carlo simulations performed on two types of cumulus clouds and on a moderate water fog. Good agreement is demonstrated for the total LiDAR signal and the depolarization parameter for a FoV of 1 mrad and a large FoV of 12 mrad.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION AND CONCEPT
For a given probing wavelength, the LiDAR signal is a function of the aerosol size distribution, of the cloud range and optical depth, and of the LiDAR field of view (FoV). Multiple scattering (MS) is always present in LiDAR measurements, even for very small optical depths [1,2]. The higher the optical depth, the higher the MS contribution to the signal will be. It is one of the causes of LiDAR signal depolarization when probing water clouds. MS effects can be reduced by restricting the LiDAR FoV to a value close to the laser beam divergence. Often considered a nuisance, its exploitation can lead to insights into cloud micro-physics such as droplet size estimation and better estimation of extinction coefficients [3,4]. In addition to Monte Carlo (MC) simulations, several models have been developed to estimate the MS contribution to the LiDAR signal [5–9]. A good review of the different models prior to 2005 can be found in [10]. Since then, important research papers have been published, leading to better exploitation of MS, especially from space LiDAR data, in particular the Hu equation [11], the work by Hogan et al. [12,13] and the physical model of Sato et al. [14,15]. In parallel, work has been done to better understand the physics of MS and depolarization [16–21]. A good review is done in [21].
All these research advances rely on mathematical modeling of the MS effects. Given the nature of the problem, modeling MS becomes quickly complicated. In this paper, we present a relatively simple model based on Poisson statistics [22]. Poisson statistics provide the probability that an event will occur a repeated number of times knowing the probability of a first occurrence. Photons entering an aerosol have a probability of passing through it without scattering described by an exponential decay $N = {N_0}\;\exp(- \tau)$, where ${N_0}$ is the initial number of photons and $\tau$ is the optical depth of the aerosol (also called optical thickness or attenuation length). Once a photon is scattered the first time, its probability to be scattered a second time still depends on $\tau$, but the location of the second scattering depends on the first scattering location. This problem can be solved by convolution of the original exponential decay by itself [23]. The convolution process can be used to find the population of any scattering order at any time. After propagation through an optical depth of $\tau$, it is found that the photon population in each of the scattering orders is simply dictated by a Poisson distribution. Poisson statistics are currently used to model scattering in biomedical and nuclear physics [24,25]. In this paper, we will apply it to estimate the MS contribution in LiDAR signals. The model will be referred to as the multiple scattering Poisson LiDAR (MSP LiDAR) model. It takes into account the droplet effective size, the LiDAR FoV and uses the equivalent-medium theorem to simplify propagation modeling. The contribution of the different scattering orders to the MS LiDAR signal and to the depolarization of the signal is compared to MC simulations.
The paper is divided as follows: in Section 2, the phase function and water cloud models are presented; in Section 3, the MSP LiDAR1 model is presented; in Section 4, the equivalent-medium theorem is presented, and its application to the present model is discussed; in Section 5, the model is applied to two types of cumulus clouds (C1 and C2), and the results are compared with MC simulations. Sections 6 and 7 are the discussion and the conclusion.
2. PHASE FUNCTION AND WATER CLOUD MODELS
The MSP LiDAR model is built on a few important premises:
- (1) The exact scattering phase functions of the different water cloud droplet distributions are obtained using Mie theory.
- (2) We assume that the forward scattering phase functions can be modeled using the summation of two Gaussian curves: one for the forward scattering diffraction peak and the second for the geometrical scattering of a round lens.
- (3) We assume that we can model the backscattering depolarization parameter as a function of the forward scattering diffraction peak.
- (4) We assume that we can use the convolution of the forward scattering phase function to model higher forward scattering orders (FSO).
- (5) We assume that small forward scatterings do not depolarize the laser beam.
- (6) We assume that the laser beam is a perfect match to the optical axis and is not affected by diffraction. The higher scattering orders generated during propagation are assumed to originate from the optical axis.
In addition to these premises, we assume that the conditions set by Eloranta [8] hold.
- (7) The multiply scattered photons returning to the receiver encountered only one backscattering event.
- (8) The extra path length produced by the different scatterings is negligible, meaning that the multiply scattered return is not delayed compared to the single scattering return.
- (9) The transverse dimension of the receiver’s FOV in the cloud is much less than the optical mean free path of photons in the cloud. This is usually the case for ground-based LiDAR.
The water clouds C1, C2, and moderate water fob (MWF) [26] used in the present study are assumed to be characterized by a gamma size distribution. The gamma distribution is
where $n({{r}})$ is the number of droplets of radius $r$, $a$ and $b$ are coefficients, whose values are provided in Table 1, and ${{\Gamma}}$ is the gamma function. The effective radius, ${r_{e \cdot}}$, of each cloud type is obtained by calculating ${\langle}{{{r}}^3} \rangle / \langle {{{r}}^2} \rangle $. The effective radius of the MWF, C1, and C2 clouds is 3, 5.99, and 11.92 µm, respectively. Their extinction profiles will be defined in Section 5.
Table 1. Values of the Gamma Distribution Parameters $a$, $b$ and the Calculated Effective Radius, ${r_{e \cdot}}$, for the C1-Type and C2-Type Water Cloud and for the Moderate Water Fog
In the following, the depolarization parameter will be used to quantify the depolarization. The LiDAR laser source is polarized either linearly (L) or circularly (C). Figure 1 shows the forward scattering phase functions and the depolarization parameters, ${D_p}$, calculated with Mie theory for a wavelength of 1064 nm. The ${D_p}$ parameter [27] is defined as the ratio of the perpendicular polarization component ($\bot$) of the phase function over the sum of the perpendicular and parallel ($\parallel$) components:
Fig. 1. Forward scattering peaks (full lines) and depolarization parameter ${D_p}$ (dashed lines) as a function of scattering up to 30° for ${{\rm{C}}_1},\;{{\rm{C}}_2}$, and MWF.
In Fig. 1, the strong forward scattering peak caused by diffraction and its droplet size dependence is clearly evident. However, forward scattering coming from diffraction does not significantly depolarize the incident polarized light. Figure 1 shows that the depolarization parameter has a value smaller than 2% for forward scattering angles up to 30 deg. This justifies our fifth premise.
Figures 2 and 3 show, respectively, the backscattering phase functions and the ${D_p}$ parameter functions as functions of the backscattering angle. The backscattered light is strongly depolarized for backscattering angles off 180°, but polarization is preserved at exactly 180°.
Fig. 2. Variation of the backscattering phase function, ${p_0}({{\beta _b}})$, as a function of the backscattering angle, ${\beta _b}$, for ${\rm{C1}},\;{\rm{C2}}$, and MWF.
Fig. 3. Variation of the depolarization parameter, ${D_p}$, as a function of the backscattering angle, ${\beta _b}$, for ${{\rm{C}}_1},\;{{\rm{C}}_2}$, and MWF.
Modeling of the phase function will be used and compared with the results of a MC simulator. The MC uses the phase function calculated by Mie theory. The MSP LiDAR models the forward Mie phase function and the backscattering depolarization parameter as a function of the width of the diffraction peak, ${\beta _d}$. The forward scattering phase function is written as the sum of two Gaussian curves [8,28]:
3. MODELING MULTIPLE SCATTERING LiDAR SIGNAL USING THE POISSON DISTRIBUTION
The probability for a photon to be in the $k$th FSO is expressed by a Poisson distribution:
where $\gamma$ corresponds to the optical depth. We are going to use the equivalent-medium theorem to determine the probability that all photons, backscattered after propagating through an optical depth $\gamma$, return to the receiver. The calculation considers forward scattering events followed by a single backscattering toward the detector.Multiply scattered LiDAR returns are dominated by two types of events. Events of type A consist of photons that have undergone small angle forward scatterings and a single backscattering toward the receiver, and events of type B consist of a single backscattering followed by forward scatterings toward the receiver. The equivalent-medium theorem [6,10] stipulates that events of type B are equivalent to events of type A. Because type A events are significantly simpler to calculate than type B events, calculation is done on events of type A using twice the attenuation. The resulting modified Poisson distribution, identified as ${\rm LiPoisson}({\gamma ,k}),$ is
The single scattering LiDAR equation corresponds to $k = {{0}}$. The higher scattering orders are obtained for $k = {{1}},{{2}},{{3}}{\ldots}$.
Figure 4 provides an example of ${\rm LiPoisson}({\gamma ,k})$ distribution for a cloud with constant extinction equal to ${0.0267}\;{{\rm{m}}^{- 1}}$ ranging from 500 to 650 m for a maximum optical depth equal to 4; hence, $\alpha (R) = 4/150\;{\rm m^{- 1}}$.
Fig. 4. ${\rm LiPoisson}({\gamma ,k})$ FSO as a function of penetration depth for ${{k}}$ ranging from 0 to 10. The optical depth $\gamma = ({Rc - 500})\;\alpha (R)$, with $\alpha (R) = 4/150\;{\rm m^{- 1}}$ is represented with the black dashed line.
When the LiDAR FoV is large enough to contain the forward scattered photons, applying the Poisson distribution leads directly to the relative contribution of the scattering orders obtained by Eloranta [8].
${\rm LiPoisson}({\gamma ,k})$ contains no information on the nature of the aerosol cloud or on the characteristics of the LiDAR. In order to obtain the LiDAR signal for the different scattering orders, the ${\rm LiPoisson}({\gamma ,k})$ distribution needs to be multiplied by the backscattering energy fraction (${{\rm BEF}_k}$), a function that will take into account:
- (a) the aerosol properties for a given probing wavelength as defined by scattering phase function; and
- (b) the geometry of the LiDAR (mainly, $\theta$ the FoV) and the cloud characteristics (range and optical depth).
The LiDAR signal and its perpendicular polarization components for scattering order $k$ are written as the product of ${\rm LiPoisson}({\gamma ,k})$, ${{\rm BEF}_k}$, and ${P^{*}}({{R_c}})$. This last parameter is the single scattering LiDAR signal. ${{\rm BEF}_k}$ and ${P^{*}}({{R_c}})$ will be developed later in Section 3.
The LiDAR return for the FSO $k$ is
Similarly, the perpendicular component of the LiDAR signal is
For single scattering occurring at exactly 180°, photon polarization remains the same. There is, thus, no contribution from the single scattering LiDAR equation in the perpendicular polarization:
Finally, the measured depolarization parameter is the ratio of the perpendicular signal over the total signal:
4. BACKSCATTERING ENERGY FRACTION FUNCTIONS
In the absence of MS considerations, the LiDAR signal return is given by
Figure 5 illustrates three forward scatterings followed by a backscattering toward the collection optics and the detector. Forward scatterings are represented by dashed colored lines illustrating the dispersion of the light according to their respective scattering phase functions ${p_0},{p_1},{p_2}$. The aerosol original phase function is ${{{p}}_0}$, and ${{{p}}_1}$ and ${{{p}}_2}$ are obtained by convolutions of ${p_0}$ (details below). Here, the angular dispersion after each forward scattering is simplified and modeled as originating from the optical axis (sixth premise) with a convoluted scattering phase function depending on the number of forward scatterings (fourth premise).
Fig. 5. Illustration of three forward scatterings occurring at ${{\rm{R}}_1},\;{{\rm{R}}_2},\;{{\rm{R}}_3}$ followed by a backscattering at ${{\rm{R}}_c}$. The angular scatterings are represented by dashed color cones and depend on the scattering phase functions ${{\rm{p}}_0},\;{{\rm{p}}_1}$, and ${{\rm{p}}_2}$.
The backscattered energy fraction depends on the system FoV, the range, and the aerosol phase function and can be noted ${{\rm BEF}_k}({\theta ,R,p({r,\lambda})})$. Its value ranges from 0 to 1. A value of 1 means that all the backscattered light is collected by the LiDAR. The calculation of ${{\rm BEF}_k}({\theta ,R,p({r,\lambda})})$ is done independently for the different FSOs, $k$. It bears close resemblance to the calculation of the second-order scattering (one forward scattering followed by a backscattering) with three major differences: first, the higher-order phase functions are obtained from convolutions of the aerosol scattering phase function; second, we use the ratio of the extinction over the optical depth, $\alpha (R)/\gamma ({Rc})$ for normalization; and third, a normalized backscattering phase function, $p_0^ + ({{\beta _b}})$, defined in Appendix B, is used.
Starting with $k = {{1}}$, ${{\rm BEF}_k}$ is
From Fig. 5, the relation between the scattering angle ${{\beta}}$ and the LiDAR FoV ${{\theta}}$ is
The scattering phase function of order $k - 1$ is noted ${p_{k - 1}}(\beta)$. It is obtained by calculating the convolution of the original phase function, ${p_0}(\beta)$, with the previously calculated phase function of FSO $k - 1$:
For the first FSO, the phase function is ${p_0}(\beta)$, the original phase function of the aerosols. For second, third, and fourth FSOs, we have
Fig. 6. Initial phase function ${{\rm{p}}_0}$ and phase functions of higher scattering orders obtained by convolution for the ${\rm{C2}}$-type water cloud.
The contribution to the LiDAR signal from FSO $k$ is
For the sake of clarity, the first three scattering orders are written explicitly below. The FSO zero corresponds to the single scattering LiDAR equation:
The first FSO is
The second FSO is
The MS perpendicular component, $S({\theta ,{R_C}})$, needs to be evaluated to estimate the depolarization parameter. It is calculated similarly to the total LiDAR signal:
Note that the normalized backscattering phase function, $p_0^ + ({{\beta _b}})$, is averaged as discussed in Appendix B and taken out of the integration.
5. MSP LiDAR MODEL PREDICTIONS AND COMPARISON WITH MC SIMULATIONS
In this section, we compare the MSP LiDAR model with MC simulations for full FoV of 1 and 12 mrad. The two water cloud types listed in Table 1 will be studied. The MC and MSP model results are normalized to the maximum value of the single scattering LiDAR signal, ${P_1}({{R_C}})$. Doing so, the reported LiDAR signals are unitless.
Polarimetric MC has been used successfully for at least 50 years [31]. Over the last 15 years, we have developed, validated, and used the Undique imaging MC simulator [32]. The simulator can reproduce a LiDAR system including the source, the propagation range, the target, and the receiver with a high level of fidelity. In Undique, photons are defined by an intensity, a 3D location in space, an orientation, and a polarization state. The polarization state is defined using Stokes parameters, and it is altered during propagation via Mueller matrices (see [33,34] for a detailed discussion on polarization alteration during propagation through aerosols). Photons are launched individually, and they interact with the various components of the simulation environment. Once a photon reaches the aperture of the receiver, it is focused on the detector using the thin lens equation, and its total time of flight, polarization state, and order of interaction are recorded. To increase the number of events registered by Undique, every interaction occurring within the FoV generates a result using the well-known semi-analytical method [35]. Undique was used and validated in various conditions [29,32–34,36,37]. Every MC simulation presented in this paper was made using 20 billion initial photons. Combined with the semi-analytical variance reduction method, it is enough to produce clean curves for low-order events that dominate the simulated systems. As we move to higher-order events, the number of events gets scarce, and the variance increases. No noise is introduced in the curves. The fluctuations seen on the curves are simply variance caused by under sampling.
A. Case 1–Constant Extinction C2 Cloud
In this first scenario, presented in more detail, the cloud extends from 500 m to 650 m and has a constant extinction. The cloud has an optical depth of 4, with an effective radius, $r_e$ of 11.92 µm. The ${\rm LiPoisson}({\gamma ,k})$ function is shown in Fig. 4.
Figure 7 shows the backscattered energy fraction, ${{\rm BEF}_k}({\theta ,R,p({r,\lambda})})$, as a function of range for a full FoV of 12 mrad; the forward scattering angle, $\beta$, has been limited to 15°. The ${{\rm BEF}_k}({\theta ,R,p({r,\lambda})})$ for the scattering order 2 shows the highest value and appears at the top in blue. The values of ${{\rm BEF}_k}$ decrease with the scattering order. Figure 8 compares the signals obtained at 12 mrad with the MC simulations for the different FSOs. The MC data show some statistical fluctuation, and model data are usually slightly lower than the MC data; the difference increases with the scattering order. The agreement is good for scattering orders up to 5.
Fig. 7. Constant extinction ${\rm{C2}}$ cloud: backscattered energy fraction, ${{\rm BEF}_k}({\theta ,R,p({r,\lambda})})$, as a function of range for a full FoV of 12 mrad for $k$ ranging from 1 to 7; the values of ${{\rm BEF}_k}$ decrease with the scattering order.
Fig. 8. Constant extinction ${\rm{C2}}$ cloud: comparison between MC simulation (colored lines) and MSP model (black lines) for FSOs 0 to 7 for full FoV of 12 mrad.
Figures 9 and 10 compare the LiDAR signals and D parameters obtained with the MSP model and the MC simulations for 1 mrad and 12 mrad FoVs. At 650 m, the LiDAR signals for the 1 mrad and 12 mrad are 1 order of magnitude apart. The agreement between the different models is excellent; all the 1 mrad curves superimpose—same thing for the 12 mrad. In Fig. 10, the depolarization parameter curves do not superimpose perfectly, but the agreement between the different models is good for both FoVs.
Fig. 9. Constant extinction ${\rm{C2}}$ cloud: LiDAR signals comparison between MC simulation and MSP model calculations for FoV of 1 and 12 mrad.
Fig. 10. Constant extinction ${\rm{C2}}$ cloud: comparison of D parameter values between MC simulation and MSP model calculations for FoVs of 1 and 12 mrad.
B. Case 2–Triangular Extinction C1 Cloud
In this scenario, the cloud extends from 500 m to 700 m. The extinction, $\alpha (R)$, increases linearly from 0 to ${0.04}\;{{\rm{m}}^{- 1}}$ over the first 100 m and then decreases linearly back to zero from 600 m to 700 m for a maximal optical depth, $\gamma$, equal to 4. Figure 11 shows ${\rm LiPoisson}({\gamma ,k})$, the MS distribution, as a function of range. The dashed black line is the variation of the optical depth also as a function of range.
Fig. 11. Triangular extinction ${\rm{C1}}$ cloud: ${\rm LiPoisson}({\gamma ,k})$ as a function of penetration depth for FSO $k$ ranging from 0 to 7. The optical depth, $\gamma$, is represented with the black dashed line.
Figure 12 shows the MSP signals calculated at 12 mrad with the MC for FSOs up to 7. The agreement between the MSP LiDAR model is good for scattering orders 2 to 5 and degrades quickly for higher scattering orders. Figures 13 and 14 compare the LiDAR signals and D parameters, respectively, obtained with the MSP model and the MC simulations for two FoVs. The agreement is excellent for the LiDAR signals and is considered good for the depolarization parameters.
Fig. 12. Triangular extinction C1 cloud: comparison between MC simulations (colored lines) and MSP model (black lines) for FSOs 0 to 7 for full FoV of 12 mrad.
Fig. 13. Triangular extinction ${\rm{C1}}$ cloud: LiDAR signals comparison between MC simulations and MSP model for FoV of 1 and 12 mrad.
Fig. 14. Triangular extinction ${\rm{C1}}$ cloud: comparison of D parameter values between MC simulation and MSP model calculations for FoV of 1 and 12 mrad.
Fig. 15. Two cloud layers of type ${\rm{C1}}$: ${\rm LiPoisson}({\gamma ,k})$ scattering order distribution as a function of penetration depth for FSOs $k$ ranging from 0 to 7. The optical depth $\gamma$ is represented with the black dashed line.
C. Case 3–Two Cloud Layers of Type C1
In this last scenario, the model is tested on two successive clouds. The first cloud extends from 500 m to 600 m, and the second extends from 650 m to 750 m. The extinction $\alpha (R)$ is constant and equal to ${17.08}\;{\rm{E}} \text{-} 03\;{{\rm{m}}^{- 1}}$ and ${{{r}}_e} = {5.99}\;{{\unicode{x00B5}{\rm m}}}$. The maximal optical depth, $\gamma$, is 3.4.
Figure 15 shows ${\rm LiPoisson}({\gamma ,k})$, the MS order distribution. The dashed black line is the variation of the optical depth as a function of range.
Fig. 16. Two cloud layers of type ${\rm{C1}}$: LiDAR signals comparison between MC simulation and MSP model calculations for FSOs 0 to 7 for full FoV of 12 mrad.
Fig. 17. Two cloud layers of type ${\rm{C1}}$: LiDAR signals comparison between MC simulation and MSP model calculations for FoV of 1 and 12 mrad.
Figure 16 compares the signal obtained with a 12 mrad FoV to the MC simulations for FSOs up to 7. The agreement is good for scattering orders up to 5. Figures 17 and 18, respectively, compare the LiDAR signals and ${{D}}$ parameters obtained with the MSP model and the MC simulations for two FoVs. The agreement between the different models is excellent for the LiDAR signals and is considered good for the depolarization parameters.
Fig. 18. Two cloud layers of type C1: comparison of ${{D}}$ parameter values between MC simulation and MSP model calculations for FoV of 1 and 12 mrad.
Note that the MC simulations register events in the range 600 to 650 m in the absence of clouds. What is recorded is delayed MS from the first cloud layer, and it explains the higher ${{D}}$ parameter.
6. DISCUSSION
The MSP LiDAR signals model compares very well with the MC simulations for optical depth close to 4. For the depolarization parameter values, the agreement is very good up to an optical depth of 2 and moderately good for larger values.
The models can be applied to MWF even though the forward scattering diffraction peak is not as predominant as for C1 and C2 water clouds. In the following example, the MWF has an effective radius ${{{r}}_e}$ of 3 µm and extends from 250 m to 700 m. The extinction $\alpha (R)$ is constant and equal to ${9.15}\;{\rm{E}}\text{-} 03\;{{\rm{m}}^{- 1}}$ for a maximal optical depth $\gamma$ of 4.18. Figure 19 compares the LiDAR signals and D parameters obtained with the MSP model and the MC simulations for two FoVs.
Fig. 19. MWF case: comparison of the LiDAR signals and ${{D}}$ parameters obtained with the MSP model and with MC simulations for MWF-type cloud for FoV of 1 and 12 mrad.
The MSP model models the forward Mie phase functions and the backscattering depolarization parameter as mathematical functions parametrized by the width of the diffraction peak, ${\beta _{{d}}}$. It has been verified that the direct use of the phase function and depolarization parameter obtained from Mie theory in the MSP model led to very similar results.
The MSP model presented here uses 1D convolutions of the forward scattering phase function. The use of 2D convolutions has been investigated. For the 1 mrad FoV, MS is highly dominated by second-order scatterings, and the results are practically identical. For the larger FoVs, 2D convolutions lead to a slight under estimation of the MS signal compared to the 1D convolution and MC simulations. A working hypothesis is that the 1D convolution compensates better the fact that the angular dispersion after each forward scattering is simplified and modeled as originating from the optical axis (premise number 6). Doing so, the actual broadening of the laser beam with scattering order is not fully taken into account. Additional research is required in that direction.
7. CONCLUSION
MS LiDAR models based on Poisson statistics are used to determine the distribution of the number of scatterings for a given optical depth. Combined with the equivalent-medium theorem and the specificities of the LiDAR and water clouds, good agreement has been demonstrated with MC simulations on C1-type and C2-type clouds for small and large FoVs for optical depth close to 4.
The MC uses the phase functions as calculated with Mie theory. The MSP model models the forward Mie phase functions and the backscattering depolarization parameter as mathematical functions parametrized by the width of the diffraction peak, ${{{\beta}}_{{d}}}$. The excellent agreement (for the case studied) between the two suggests that the MSP LiDAR model could be used to model and retrieve the effective droplet size and extinction coefficient of water clouds.
It is expected that the model provides good results for moderate extinctions and relatively short ranges (1 km or smaller). For large extinctions or ranges, MS becomes quickly important, and the rightfulness of the model will need to be validated for large FoVs.
In its present form, the model does not take into account the temporal broadening of the LiDAR signal by the MS and is not appropriate for LiDAR measurement from space and for large optical depth.
APPENDIX A: MODELIZATION OF THE DEPOLARIZATION PARAMETER NEAR THE BACKSCATTERING ANGLE
The detailed development of the model of $D({\beta ,{r_e}})$ is in [29,30]. We give here a summary.
Figure 3 of the present paper shows that the maximum value of the $D({\beta ,{r_e}})$ is practically independent of the forward scattering diffraction peak, ${\beta _d}$. However, the position of the maximum varies linearly with ${\beta _d}$ as ${\beta _{\rm{Max}}}({{\beta _d}}) = {{{179.67}}^\circ}{{- 0.92}}{\beta _d}$.
The depolarization parameter for backscattering angles from 160° to 180° is related to the forward scattering characteristic angle peak. The mathematical representation is as follows:
If, $\beta \ge {\beta _{\rm{Max}}}$,
If, $\beta \lt {\beta _{\rm{Max}}}$,
${D_{\rm{Max}}}({{\beta _d}})$ is practically independent of ${\beta _d}$ with a mean value equal to 0.75. The fit parameters ${w_1}$ and ${w_2}$ are also practically independent of ${\beta _d}$ with mean values equal to 0.93 and 1.37 respectively. Note that Eq. (A2) corrects the typo error of [29].
APPENDIX B: NORMALIZED BACKSCATTERING PHASE FUNCTION
By using the equivalence-medium theorem, calculations are done considering many forward scatterings and a single backscattering prior to a direct detection. These are known as type A events. Events of type B consist of a single backscattering occurring at 180° followed by forward scatterings before reaching the detector at a given FoV. By symmetry, there is an equivalent event of type A reaching the detector at the same angle. A type A event has first been forward scattered at various angles before having a backscattering that returns the photon directly on the detector without additional forward scatterings. Although equivalent in terms of scattering angles and number, process B will return a little more energy because the phase function is maximal at 180°.
The backscattering phase functions have been modified to take into account the fact that the magnitude of the backscattering occurring at 180° has a higher value than the backscattering occurring at other angles. The backscattered phase function is set equal to $p_0^ + ({{\beta _b}})\cdot {p_o}({{\beta _b} = \pi})$, where $p_0^ + ({{\beta _b}})$ is calculated as follows:
Doing so, ${p_0}({{\beta _b} = \pi})$ values are unchanged, and the backscattering values at other angles are decreased by a small amount.
For the C2, C1, and MWF water clouds, the average values obtained over the backscattering angle 165° to 180° are, respectively, equal to 0.67, 0.77, and 0.82, and when averaged over the backscattering angle 150° to 180°, they are 0.64, 0.70, and 0.74. This is in very good agreement with the value of 0.7 used in [3,8] for C1-type clouds.
Acknowledgment
We wish to thank Luc R. Bissonnette for careful revision of this paper and the anonymous reviewers for wise suggestions. This work was supported by the Defence Research and Development Canada Program.
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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