Spectrally dependent linear depolarization and lidar ratios for nonspherical smoke aerosols

Abstract

We use the numerically exact T-matrix method to model light scattering and absorption by aged smoke aerosols at lidar wavelengths ranging from 355 to 1064 nm assuming the aerosols to be smooth spheroids or Chebyshev particles. We show that the unique spectral dependence of the linear depolarization ratio (LDR) and extinction-to-backscatter ratio (or lidar ratio, LR) measured recently for stratospheric Canadian wildfire smoke can be reproduced by a range of model morphologies, a range of spectrally dependent particle refractive indices, and a range of particle sizes. For these particles, the imaginary part of the refractive index is always less than (or close to) 0.035, and the corresponding real part always falls in the range [1.35, 1.65]. The measured spectral LDRs and LRs could be produced by nearly-spherical oblate spheroids or Chebyshev particles whose shapes resemble those of oblate spheroids. Their volume-equivalent effective radii should be large enough (r_eff = 0.3 μm or greater) to produce the observed enhanced LDRs. Our study demonstrates the usefulness of triple-wavelength LDR measurements as providing additional size information for a more definitive characterization of the particle morphology and composition. Non-zero LDR values indicate the presence of nonspherical aerosols and are highly sensitive to particle shapes and sizes. On the other hand, the LR is a strong function of absorption and is very responsive to changes in the particle refractive index.

1. Introduction

Biomass burning is one of the largest contributors of both gaseous and particulate emissions to the atmosphere resulting in 34%–38% of total carbonaceous aerosol emissions [1]. Biomass burning aerosols (BBAs) are small particles that are released into the atmosphere from biomass fires. The two primary components of BBAs are organic and black carbon (OC and BC) aerosols with some inorganic species [2–4]. These aerosols affect the global radiation budget directly through scattering and absorbing the incoming solar radiation. They can also act as cloud condensation nuclei, increase cloud albedo and lifetime, and cause indirect radiative forcing [5–12]. Large uncertainties exist in the estimation of physical and optical properties of BBAs as well as their radiative forcing [13] and references therein). It is therefore essential to determine the global distribution of BBAs and their microphysical properties from remote sensing observations.

Observations with polarization lidars represent a powerful remote sensing technique for deriving aerosol microphysics actively from the ground, air, and space [14,15]. The extinction-to-backscatter ratio (or lidar ratio, LR) and the particle linear depolarization ratio (LDR) as measured by polarization-sensitive Raman lidars or high spectral resolution lidars (HSRLs) can be used for aerosol classification and for identification of the presence of different types of aerosols such as dust and smoke particles [16–19]. In addition, the particle LDR is an important aerosol characteristic that can be used for optical characterization of morphologically complex particulates [20–28]. A non-zero LDR indicates the presence of nonspherical aerosols and is rather sensitive to their size relative to the wavelength. LDRs observed for smoke aerosols at 532 nm typically are a few percent at most and are often discounted as negligibly close to zero and therefore hardly useful (e.g., [29,30]). However, as pointed out by Burton et al. [14] (and references therein), higher LDR values of 0.05–0.11 at 532 nm have sometimes been observed for layers of aged smoke. In their study, Burton et al. observed exceptionally high LDR values in a well-defined layer of wildfire smoke advected from the Pacific North-west of the Unites States to Boulder at an 8 km altitude using the NASA Langley airborne HSRL-2. The mean depolarization ratio had a peak of 20.3% at 355 nm, decreasing to 9.3% and 1.8% at 532 nm and 1064 nm, respectively. Burton et al. credited the high measured LDRs in the smoke case to the presence of coated soot aggregates. Later studies demonstrated that such spectral dependence of the measured LDR values can be reproduced by a range of model morphologies and a range of particle refractive indices (and hence different chemical compositions) [31–34]. Recently using the triple-wavelength polarization/Raman lidar called BERTHA, a similar spectral dependence of the LDR was observed by Haarig et al. [35] for the stratospheric (from 15 to 16 km) smoke layer at Leipzig, Germany, after sunset on 22 August 2017, with high LDRs of 22% at 355 nm and 18% at 532 nm and a comparably low value of 4% at 1064 nm. The smoke originated from a western Canadian wildfire. Haarig et al. argued that the high LDRs at 355 and 532 nm and the strong wavelength dependence of the depolarization ratios was probably caused by fine mode dry nonspherical soot particles following a specific size distribution. The findings by Haarig et al. were further corroborated by Hu et al. [36] who detected smoke layers that originated from the same Canadian wild fire using their triple-wavelength polarization lidar system at Lille, northern France, from 24 to 31 August 2017. The LDR was over 0.20 at 355 nm, 0.18–0.19 at 532 nm, and 0.04–0.05 at 1064 nm. Hu et al. concluded that the high LDR values and their spectral dependence were possibly caused by irregular-shaped aged smoke particles and/or by mixing with dust particles.

The LR is another quantity valuable for aerosol characterization. It strongly depends on the size, morphology, and chemical composition of aerosols [16]. The majority of aerosol lidars are elastic–backscatter ones, including the well-known Cloud–Aerosol Lidar and Pathfinder Satellite Observations (CALIPSO) lidar [25,37]. Such lidars rely on predefined LRs of key aerosol types to convert the retrieved backscatter coefficient profiles into extinction profiles [16]. So for elastic backscatter lidars, a priori knowledge of LRs is essential. Only Raman lidars [38] and HSRLs [39] are able to measure the LR directly and unambiguously. Müller et al. [16] were the first to summarize aerosol type dependent LR statistics solely based on Raman lidar measurements. According to the study, the mean values and one standard deviations of the particle LRs for forest fire smoke are 46 ± 13 and 53 ± 11 at 355 nm and 532 nm, respectively.

Although there are numerous LR observations at 355 nm and 532 nm available (e.g., [16,35] and references therein), only recently has it become possible to measure LRs at 1064 nm [40]. As a matter of fact, their triple-wavelength polarization/Raman lidar is the only lidar worldwide that permits the measurement of the particle extinction-to-backscatter ratio and the particle LDR at all three important lidar wavelengths of 355, 532, and 1064 nm [35].

In an attempt to interpret the LDRs and LRs at multiple wavelengths published by Haarig et al. [35], Gialitaki et al. [41] showed that near-spherical particle-shape assumption explains to a large extent such observed spectrally dependent LDR and LR values of the stratospheric smoke particles. The assumption of soot particles with a more complicated morphology used in the study however did not reproduce the observations. As an investigation study, the work by Gialitaki et al. [41] was limited in scope. For example, the range of refractive index (m = m_r + im_i) values was limited from 1.4 to 1.65 and from 0.005 to 0.04 for the real (m_r ) and imaginary (m_i) parts, respectively; the effective radius was set between 0.25 and 0.45 μm, and the spheroidal axial ratio varied from 0.7 to 1.2. It is well known that the chemical composition and optical properties of BBAs depend not only on the type of fire, but also on the environmental conditions in which combustion takes place. Moreover, after having been released into the atmosphere, the particles and vapors within biomass burning plumes undergo chemical and physical aging processes as they are transported downwind [42]. Naturally, large diversity exists in terms of the physical and chemical properties of BBAs, which can be well outside the ranges considered by Gialitaki et al. [41]. For example, in a recent study by Sarpong et al. [43], the real part of the retrieved refractive index for BBAs was found to be in the range [1.31, 1.56] and the imaginary part was in the range [0.045, 0.468] in the spectral interval 500–580 nm. The m_r and m_i values of BBAs at visible wavelengths had the ranges of 1.55–1.80 and 0.01–0.50, respectively, based on the Fire Laboratory at Missoula Experiments [44], and the corresponding geometric mean diameter ranged from 0.20 to 0.57 μm. In [41], the best fit between T-matrix simulations of light scattering and the measured LDR and LR values at 355, 532 and 1064 nm [35] were produced by nearly spherical spheroids with an aspect ratio of 1.11, refractive index of 1.42 + i0.02, and effective radius r_eff = 0.55 μm. The resulting LDR value at 1064 nm was 1.15% and hence deviated from the mean measured value of 4.3% considerably. In addition, the best-fit refractive index was fixed at 1.42 + i0.02 at all the wavelengths instead of being spectrally variable.

In this study, we use a different approach and allow the m values to vary spectrally while searching for good matches that can reproduce the LDRs and LRs observed by Haarig et al. [35] at all three lidar wavelengths. All in all, the understanding of the spectral changes of LDRs and LRs with respect to the aerosol particle morphology, refractive index, and size is far less complete. We believe that a more comprehensive study of LDRs and LRs of BBAs is valuable not only in terms of aerosol classification and characterization but is also helpful within the lidar community as the LR is a crucial parameter in the inversion of lidar signals [45].

2. Computations

As mentioned earlier, the two major components of BBAs are organic and black carbon aerosols. Freshly emitted soot particles typically form lacy fractal-like aggregates [46,47]. However after being released into the atmosphere, they undergo various aging processes which result in changes in their morphology and mixing state [47–50]. According to Ueda et al. [48,49], soot containing aerosols can be classified into seven types based on their dominant morphology: soot aggregates, spherical particles, coccoid particles, dome-like particles, clusters of spherical and coccoid bodies, crystalline coarse bodies, and re-crystallized droplet particles. Organic particles are abundant and quite often represent the most abundant particle type in biomass burning smoke [46]. Atmospheric tar balls, a member of the brown carbon (light absorbing organic carbon) family, have been widely described as being amorphous and spherical [51,52]. However, there are numerous studies that report observations of tar balls aggregated with other aerosols or forming fractal agglomerates [53,54]. According to China et al. [47], the relative abundance of tar balls can be 10 times greater than soot particles. Li et al. [46] showed the organic particles other than tar balls in the smoke sample did not have a spherical morphology but were mostly sub-rounded or irregularly shaped. It would be nearly impossible to account for the actual diverse morphologies of all kinds of biomass burning aerosols. Instead in this study we use simple rotationally symmetric spheroidal and Chebyshev particles to model light scattering and absorption by nonspherical smoke aerosols. One of our goals is to see if such simple models can reproduce the unique spectral dependence of LDRs and LRs observed for an aged smoke plume by Haarig et al. [35]. Their shapes are visualized in Fig. 1. In the polar coordinate system, the spheroidal shape is described by the equation

$$r(\theta ,\phi )=a{\left({\sin }^2\theta +\frac{a^2}{b^2}{\cos }^2\theta \right)}^{-1/2},$$ (1)

where θ is the polar angle, φ is the azimuth angle, b is the rotational (vertical) semi-axis and a is the horizontal semi-axis [55]. Chebyshev particles are obtained by continuously deforming a sphere by means of a Chebyshev polynomial of degree n [55,56]. Their shape is given by

$$r(\theta ,\phi )=r_0[1+\epsilon T_n(\cos \theta )],\text{ |}\epsilon \text{|<1,}$$ (2)

where r₀ is the radius of the unperturbed sphere, ε is the deformation parameter, and T_n( cos θ) is the Chebyshev polynomial of degree n . In this paper, only n = 2 is considered.

Fig. 1.

Oblate spheroid with an aspect ratio of 2 (a), prolate spheroid with an aspect ratio of 2 (b), and Chebyshev particles of degree 2 and deformation parameters 0.2 (c) and −0.2 (d).

We assume that the sizes of BBAs follow the standard power-law distribution [57]

$$n(r)=\{\begin{array}{cc}\frac{2r_1^2{r}_2^2}{r_2^2-r_1^2}{r}^{-3},& r_1\le r\le r_2,\\ 0,& \text{otherwise,}\end{array}$$ (3)

where r represents a volume-equivalent-radius. The effective radius r_eff and the effective variance v_eff of the size distribution are then defined according to

$$r_{\text{eff}}=\frac{1}{\langle G\rangle }{\int }_{r_1}^{r_2}drn(r)r\pi r^2$$ (4)

and

$$v_{\text{eff}}=\frac{1}{\langle G\rangle r_{\text{eff}}^2}{\int }_{r_1}^{r_2}drn\left(r\right){\left(r-r_{\text{eff}}\right)}^2\pi r^2,$$ (5)

respectively [55,57], where

$$\langle G\rangle ={\int }_{r_1}^{r_2}drn\left(r\right)\pi r^2.$$ (6)

We assume that spheroids and Chebyshev particles are randomly oriented [58] and form a statistically isotropic and mirror-symmetric ensemble, and apply the T-matrix method [55] to calculate their electromagnetic scattering and absorption properties. The elements of the normalized 4 × 4 Stokes scattering matrix have the following well-known block-diagonal structure [59]:

$$\tilde{F}(\Theta )=\left[\begin{array}{cccc}{a}_1(\Theta )& b_1(\Theta )& 0& 0\\ b_1(\Theta )& a_2(\Theta )& 0& 0\\ 0& 0& a_3(\Theta )& b_2(\Theta )\\ 0& 0& -b_2(\Theta )& a_4(\Theta )\end{array}\right],$$ (7)

where Θ ∊ [0°, 180°] is the scattering angle. The (1, 1) element of the scattering matrix, a₁ (Θ), is the conventional phase function; it is normalized according to

$$\frac{1}{2}{\int }_0^{\pi }\text{d}\Theta \sin \Theta a_1\left(\Theta \right)=1$$ (8)

and describes the angular distribution of the scattered intensity in the case of unpolarized incident light. The conventional LDR is defined by

$$\text{LDR}=\frac{a_1\left({180}^{\mathrm{°}}\right)-a_2\left({180}^{\mathrm{°}}\right)}{a_1\left({180}^{\mathrm{°}}\right)+a_2\left({180}^{\mathrm{°}}\right)},$$ (9)

While the LR is computed using the formula [59,60]

$$\text{LR}=\frac{4\pi }{\varpi a_1\left({180}^{\mathrm{°}}\right)},$$ (10)

where ϖ is the single-scattering albedo.

Table 1 lists the microphysical parameters of biomass burning aerosols that entered the T-matrix computations. Note that in this study we adopt the effective variance equal to 0.2, thereby representing a moderately wide distribution. Hansen and Travis [57] showed that scattering properties of most physically plausible size distributions of spherical particles depend primarily on only two characteristics of the distribution, the effective radius r_eff and the effective variance v_eff. This implies that if different size distributions of spherical particles have the same values of r_eff and v_eff then their scattering properties are quite similar. Mishchenko and Travis [61] extended this conclusion to randomly oriented spheroidal particles. The advantage of the power-law size distribution is that it helps minimize the CPU time in comparison with that required for the commonly used log-normal distribution having the same values of r_eff and v_eff. Note both LDRs and LRs change with effective variance. However the values are rather stable when v_eff ≥ 0.15. This is especially true of larger particles with effective radii greater than 0.2 μm. We believe that v_eff = 0.2 is a reasonable choice. The distribution is wide but not so wide as to make T-matrix computation prohibitive for larger and highly nonspherical particles.

Table 1.

Model parameters for T-matrix simulations

Real part of the refractive index	1.35, 1.4, 1.45, 1.5, 1.55, 1.6, 1.65, 1.7, 1.75. 1.8
Imaginary part of the refractive index	0, 0.005, 0.01, 0.015, 0.02, 0.025, 0.03, 0.035, 0.04, 0.045, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7
Size distribution	Standard power law
Effective radius (μm)	0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1
Effective variance	0.2
Shapes	Oblate spheroids with aspect ratios a/b = 1.05, 1.1, 1.15, 1.2, 1.4, 1.6, 1.8, 2.0 Prolate spheroids with aspect ratios b/a = 1.05, 1.1, 1.15, 1.2, 1.4, 1.6, 1.8, 2.0 Chebyshev particles of degree 2 and deformation parameters ε = −0.2, −0.16, −0.12, −0.08, −0.04, 0.04, 0.08, 0.12, 0.16, 0.2

In this study, the real and imaginary parts of the refractive index are varied in the ranges from 1.35 to 1.80 and from 0 to 0.7, respectively, thereby covering a wide range of refractive indices of BC, brown carbon, and BBAs (e.g., [43,44,62–70]. In [41], the particles that reproduced the observed LDRs and LRs had a refractive index fixed at a single value across the spectrum from 355 nm to 1064 nm. We, however, while searching for a qualified particle candidate which generates LDR and LR values that match the lidar measurements by Haarig et al. [35], allow the values of m vary as long as m_i (355 nm) ≥ m_i (532 nm) ≥ m_i(1064 nm) based on our analyses of the refractive indices for biomass burning aerosols as a function of wavelength. For example, Kirchstetter et al. [68] showed that OC extracted from biomass smoke samples had the imaginary part of the refractive index decreasing with wavelength, while BC had a relatively constant m_i across the spectral range [350 nm, 700 nm]. Bluvshtein et al. [69] derived a wavelength-dependent effective complex refractive index from 350 to 650 nm for freshly emitted and slightly aged BBAs and showed that both the real and imaginary parts of the refractive index decrease with increasing wavelengths. The study by Sumlin et al. [70] showed that the real part of the refractive index for brown carbon aerosols is constrained between 1.5 and 1.7 with no obvious functionality in wavelength, while the imaginary counterpart decreased from 0.014 to 0.003 when the wavelength increased from 375 nm to 532 nm. The combustion aerosol particles in Kim et al. [71] revealed stronger absorption at shorter wavelengths which rapidly decreased with increasing wavelength. Chang and Charalampopoulos [72] showed that the real part of the refractive index of flame soot continuously increased with increasing wavelength, while the m_i values as a function of wavelength exhibited an opposite trend, consistent with the results published by Ackerman and Toon [73]. So there is a rather general consensus that the imaginary part of the refractive index for BBAs decreases with increasing wavelength, while the spectral pattern for m_r is less conclusive.

Tables 2 and 3 list all the relevant physical properties of particles that reproduce the observed LDR and LR values for stratospheric smoke particles by Haarig et al. [35]. The results vividly show that the spectral dependence of the measured LRs and LDRs can be reproduced by a range of model morphologies, a range of particle refractive indices, and a range of particle sizes. If we keep the refractive index fixed across the spectrum like in [41], then there are only two valid candidates: oblate spheroids with a/b = 1.2, r_eff = 0.5 μm, m = 1.45 + i0.02 and oblate spheroids with the same aspect ratio and effective radius but with a slightly different refractive index m = 1.45 + i0.015. However, there are many more particles that successfully reproduce the LDRs and LRs observed at the three lidar wavelengths when we allow the refractive index to be spectrally variant with the only constraint of m_i(355 nm) ≥ m_i(532 nm) ≥ m_i(1064 nm), with m_r often varying by as little as 0.005. We notice that the imaginary part of the refractive index is always less than (or close to) 0.035, consistent with the finding by Bi et al. [74]. Indeed, to produce enhanced LDRs, the imaginary part of the refractive index should be small since the external reflection from spheroids produces no depolarization. The real part of the refractive indices always falls into the [1.35, 1.65] range and is predominantly between 1.4 and 1.55, which is typical of BBAs. The measured spectrally dependent LDRs and LRs are mostly reproduced by nearly-spherical oblate spheroids as well as by Chebyshev particles with negative deformation parameters whose shapes resemble those of oblate spheroids, while their volume-equivalent effective radius should be sufficiently large (r_eff = 0.3 μm or larger).

Table 2.

Physical properties of spheroids that reproduce the spectrally dependent LDR and LR values observed by Haarig et al. [35].

								LDR (%)			LR (sr)
								355 nm	532 nm	1064 nm	355 nm	532 nm	1064 nm
Observations								22.4 ± 1.5	18.4 ± 0.6	4.3 ± 0.7	40 ± 16	66 ± 12	92 ± 27
T-matrix simulations (spheroids)
355 nm		532 nm		1064 nm
mr	mi	mr	mi	mr	mi	aspect ratio	reff (μm)
1.55	0.035	1.5	0.035	1.65	0.035	p1.2a	0.35	21.5	17.9	3.7	49	76	73
1.55	0.03	1.5	0.03	1.65	0.03			21.8	18.0	3.6	41	67	68
1.55	0.035	1.5	0.035	1.65	0.03			21.5	17.9	3.6	49	76	68
1.55	0.035	1.5	0.03	1.65	0.03			21.5	18.0	3.6	49	67	68
1.35	0.01	1.35	0.01	1.4	0.01	o1.1b	0.7	23.5	17.9	4.2	49	72	106
1.35	0.01	1.35	0.01	1.4	0.005			23.5	17.9	4.1	49	72	91
1.35	0.01	1.35	0.01	1.4	0.			23.5	17.9	4.1	49	72	78
1.35	0.01	1.4	0.01	1.4	0.01			23.5	19.0	4.2	49	58	106
1.35	0.01	1.4	0.01	1.4	0.005			23.5	19.0	4.1	49	58	91
1.35	0.01	1.4	0.01	1.4	0.			23.5	19.0	4.1	49	58	78
1.4	0.015	1.35	0.01	1.4	0.01	o1.1	0.75	23.3	18.1	4.6	54	61	85
1.4	0.015	1.35	0.01	1.4	0.005			23.3	18.1	4.6	54	61	73
1.45	0.01	1.35	0.01	1.4	0.01			21.5	18.1	4.6	28	61	85
1.45	0.01	1.35	0.01	1.4	0.005			21.5	18.1	4.6	28	61	73
1.45	0.015	1.4	0.015	1.35	0.005	o1.1	0.8	23.3	18.7	4.8	45	64	104
1.45	0.015	1.4	0.015	1.35	0.			23.3	18.7	4.9	45	64	91
1.45	0.02	1.45	0.02	1.45	0.02	o1.2	0.5	21.0	18.8	3.8	50	66	104
1.45	0.02	1.45	0.02	1.5	0.02			21.0	18.8	4.7	50	66	82
1.45	0.02	1.45	0.02	1.45	0.015			21.0	18.8	3.8	50	66	94
1.45	0.02	1.45	0.02	1.5	0.015			21.0	18.8	4.6	50	66	73
1.45	0.02	1.45	0.02	1.45	0.01			21.0	18.8	3.8	50	66	84
1.45	0.02	1.45	0.02	1.5	0.01			21.0	18.8	4.6	50	66	65
1.45	0.02	1.45	0.02	1.45	0.005			21.0	18.8	3.7	50	66	75
1.45	0.02	1.45	0.02	1.45	0.			21.0	18.8	3.7	50	66	66
1.45	0.02	1.45	0.015	1.45	0.015			21.0	18.8	3.8	50	54	94
1.45	0.02	1.45	0.015	1.5	0.015			21.0	18.8	4.6	50	54	73
1.45	0.02	1.45	0.015	1.45	0.01			21.0	18.8	3.8	50	54	84
1.45	0.02	1.45	0.015	1.5	0.01			21.0	18.8	4.6	50	54	65
1.45	0.02	1.45	0.015	1.45	0.005			21.0	18.8	3.7	50	54	75
1.45	0.02	1.45	0.015	1.45	0.			21.0	18.8	3.7	50	54	66
1.45	0.015	1.45	0.015	1.45	0.015			21.2	18.8	3.8	39	54	94
1.45	0.015	1.45	0.015	1.5	0.015			21.2	18.8	4.6	39	54	73
1.45	0.015	1.45	0.015	1.45	0.01			21.2	18.8	3.8	39	54	84
1.45	0.015	1.45	0.015	1.5	0.01			21.2	18.8	4.6	39	54	65
1.45	0.015	1.45	0.015	1.45	0.005			21.2	18.8	3.7	39	54	75
1.45	0.015	1.45	0.015	1.45	0.			21.2	18.8	3.7	39	54	66
1.45	0.02	1.45	0.02	1.4	0.015	o1.2	0.55	22.4	18.1	4.4	49	57	118
1.45	0.02	1.45	0.02	1.4	0.01			22.4	18.1	4.3	49	57	105
1.45	0.02	1.45	0.02	1.4	0.005			22.4	18.1	4.3	49	57	93
1.45	0.02	1.45	0.02	1.4	0.			22.4	18.1	4.3	49	57	82
1.55	0.03	1.55	0.03	1.4	0.01	o1.4	0.45	21.4	18.7	4.1	50	59	111
1.55	0.03	1.55	0.03	1.4	0.005			21.4	18.7	4.2	50	59	101
1.55	0.03	1.55	0.03	1.4	0.			21.4	18.7	4.2	50	59	92

^ap1.2 means prolate spheroids with b/a = 1.2.

^bo1.1 means oblate spheroids with a/b = 1.1

Table 3.

As in Table 2, but for Chebyshev particles of degree 2.

								LDR (%)			LR (sr)
								355 nm	532 nm	1064 nm	355 nm	532 nm	1064 nm
Observations								22.4 ± 1.5	18.4 ± 0.6	4.3 ± 0.7	40 ± 16	66 ± 12	92 ± 27
T-matrix simulations (Chebyshev particles)
355 nm		532 nm		1064 nm
mr	mi	mr	mi	mr	mi	ε	reff (μm)
1.55	0.025	1.55	0.025	1.65	0.025	−0.2	0.3	22.9	18.5	4.1	53	72	78
1.55	0.025	1.55	0.025	1.65	0.02			22.9	18.5	4.1	53	72	74
1.55	0.025	1.55	0.025	1.65	0.015			22.9	18.5	4.1	53	72	70
1.55	0.025	1.55	0.025	1.65	0.01			22.9	18.5	4.2	53	72	66
1.55	0.025	1.55	0.02	1.65	0.02			22.9	18.4	4.1	53	64	74
1.55	0.025	1.55	0.02	1.65	0.015			22.9	18.4	4.1	53	64	70
1.55	0.025	1.55	0.02	1.65	0.01			22.9	18.4	4.1	53	64	66
1.55	0.025	1.55	0.015	1.65	0.015			22.9	18.2	4.1	53	57	70
1.55	0.025	1.55	0.015	1.65	0.01			22.9	18.2	4.2	53	57	66
1.55	0.02	1.55	0.02	1.65	0.02			23.0	18.4	4.1	45	64	74
1.55	0.02	1.55	0.02	1.65	0.015			23.0	18.4	4.1	45	64	70
1.55	0.02	1.55	0.02	1.65	0.01			23.0	18.4	4.2	45	64	66
1.55	0.02	1.55	0.015	1.65	0.015			23.0	18.2	4.1	45	57	70
1.55	0.02	1.55	0.015	1.65	0.01			23.0	18.2	4.2	45	57	66
1.55	0.015	1.55	0.015	1.65	0.015			23.1	18.2	4.1	38	57	70
1.55	0.015	1.55	0.015	1.65	0.01			23.1	18.2	4.2	38	57	66
1.55	0.03	1.55	0.03	1.35	0.	−0.16	0.5	22.5	18.0	4.0	50	54	108
1.5	0.025	1.5	0.025	1.4	0.015	−0.12	0.5	21.1	18.0	3.9	50	60	117
1.5	0.025	1.5	0.025	1.4	0.01			21.1	18.0	3.9	50	60	105
1.5	0.025	1.5	0.025	1.4	0.005			21.1	18.0	3.9	50	60	95
1.5	0.025	1.5	0.025	1.4	0.			21.1	18.0	3.8	50	60	85
1.45	0.015	1.4	0.015	1.4	0.015	0.04	0.75	23.2	18.1	4.7	35	60	97
1.45	0.015	1.4	0.015	1.4	0.01			23.2	18.1	4.8	35	60	84
1.45	0.015	1.4	0.015	1.4	0.005			23.2	18.1	4.9	35	60	72
1.45	0.015	1.4	0.015	1.35	0.			23.2	18.1	5.0	35	60	107

Fig. 2 shows the LDR and LR at 355 nm as functions of the particle refractive index and effective radius. The particle are oblate spheroids with an aspect ratio of 1.2. Obviously, when the particles are small, i.e., r_eff = 0.1 μm, the LDR values are predominantly smaller than 1%, with the highest value not even reaching 2.5%. As the particles grow, i.e., r_eff = 0.5 μm and 1 μm, the LDR values are significantly enhanced and exceed 30% when both m_r and m_i are at the lower end of their corresponding ranges. The LDR values rapidly decrease, for example from 31.7% to 8.8% when r_eff = 1 μm, as the imaginary part of the refractive index increases from 0.05 to 0.1. Generally speaking, the LDRs decrease with increasing m_r and m_i. The latter trait is consistent with the finding by Mishchenko et al. [59]. As explained by Bi et al. [74], when the real part of the complex refractive index increases, the portion of externally reflected light also increases thus preventing light from penetrating into the particle and then exiting in the direct backscattering direction (hence producing depolarization). Moreover, the imaginary part of the refractive index should be sufficiently small so that the high-order transmission has noticeable contribution to backscattering. For strongly absorptive particulates, the diffraction and external reflection dominate while producing no backscattering depolarization. The LR is inversely proportional to the single-scattering albedo, and is also affected by the particle size and shape. When absorption is strong, the LRs are rather high and are well beyond the lidar-measured values for aged smoke particles. This indicates that the stratospheric smoke aerosols observed by Haarig et al. [35] are not particularly absorptive.

Fig. 2.

LDR (in%) and LR at 355 nm as functions of the real and imaginary parts of the refractive index. The particles are oblate spheroids with an aspect ratio of 1.2.

Similarly to Fig. 2, Fig. 3 shows the LDR and LR at 355 nm as functions of the particle shape and effective radius. The refractive index is fixed at 1.45 + i0.02. For a better visualization, Fig. 4 parallels the left-hand column of Fig. 3 and displays the LDR as a function of particle nonsphericity for five different aerosol sizes. It is demonstrated once again that LDR values are close to zero for smaller particles. As the particle size grows, so does the LDR. As a matter of fact, the LDRs are already mostly over 20% when the particle effective radius reaches 0.2 μm. However the LDR does not grow with particle size monotonically, as evidenced by Fig. 4. We also notice that as the value of r_eff increases, the transition region from small to enhanced LDRs indicated by blue colors narrows (Fig. 3), which means that when the particle dimension is large, even a slight deviation from a perfect sphere can induce significant depolarization. This phenomenon was first discovered by Mishchenko and Hovenier [21] and Mishchenko and Sassen [75], and was further studied in detail by Bi et al. [74,76]. Interestingly, for particles of the same size, the transition from small to enhanced LDRs is more gradual for Chebyshev particles than for spheroids. The LR on the contrary is not a strong function of particle shape. The maximum LR values are observed for particles with an effective radius of 0.15 μm in this case (Chebyshev particles).

Fig. 3.

LDR (in%) and LR at 355 nm as functions of the axis ratio a/b (spheroids) or the deformation parameter (Chebyshev particles) and the particle effective radius. The refractive index is fixed at 1.45 + i0.02.

Fig. 4.

LDR as a function of particle nonsphericity for five different aerosol sizes. The refractive index is fixed at 1.45 + i0.02.

Figs. 5–7 demonstrate convincingly the power of triple-wavelength polarization lidars with the capacity to measure the particle extinction-to-backscatter ratio directly for the derivation of aerosol microphysical properties. Fig. 5 shows the LDR and LR as functions of the particle effective radius at the three lidar wavelengths of 355, 532, and 1064 nm. The refractive index is fixed at 1.45 + i0.02. The particle are spheroids, and different axis ratios are designated by different colors. The shaded regions illustrate the corresponding ranges of the lidar observations by Haarig et al. [35]. Clearly when the value of m is spectrally independent and fixed at 1.45 + i0.02, the LDRs and LRs produced by spheroids with an aspect ratio of 1.2 (shown by turquoise color) at r_eff = 0.5 μm are all within the ranges of the corresponding lidar measurements. We therefore mark them as qualified candidates that satisfactorily reproduce the lidar observations and list them in Table 2. The LDR tends to increase with particle effective size parameter, which makes its measurement at 355 nm particularly useful. Indeed, the LDRs at 1064 nm for particles with r_eff = 0.3 μm and smaller are nearly zero and too weak to be useful. However, the LDRs for the same aerosols at 355 nm can exceed 40% and they are very sensitive to the change of the degree of asphericity. It should be mentioned that the LDR does not increase with particle size indefinitely. As a matter of fact, based on their calculations of depolarization of lidar returns by small ice crystals, Mishchenko and Sassen [75] found that maximal LDR values for most shapes are observed at effective size parameters close to and sometime smaller than 10 (corresponding to r_eff about 0.85 μm at a wavelength of 532 nm). For most cases in this study, the LDRs have not reached their maximal values by r_eff = 1 μm. This may be because we use a different refractive index 1.45 + i0.02 instead of the value 1.308 + i1.328+10⁻⁶ adopted by Mishchenko and Sassen [75]. The relationship between the LRs and aerosol effective radii does not change markedly with particle shape for near spherical spheroids, i.e., those with axis ratios within the range of [1/1.4, 1.4], and the LRs are not particularly responsive to the change of particle size at shorter wavelengths as evidenced by the rather flat curves at r_eff ≥ 0.4 μm. However, when the asphericity is substantial, with aspect ratios near 2 for example, the LR values can exceed 200 for large particles.

Fig. 5.

LDR (%) and LR as functions of particle size at the three lidar wavelengths. The refractive index of the particles is 1.45 + i0.02. The particles are spheroids with varying axis ratios a/b. The shaded regions represent the corresponding ranges of lidar observations.

Fig. 7.

LDR (in%) and LR as functions of particle size at 355, 532, and 1064 nm. m_i is fixed at 0.02, while different colors designate different values of m_r. The particles are oblate spheroids with an aspect ratio of 1.2.

Similarly, Figs. 6 and 7 show size-dependent LDR and LR values as functions of the imaginary and real parts of the refractive index, respectively. The particles are oblate spheroids with an aspect ratio of 1.2. In Fig.6, the m_r value is fixed at 1.45, while the m_i value is set to be 0.02 in Fig.7. Fig. 6 is a strong example demonstrating that LR measurements combined with LDR observations lead to a better and more accurate particle characterization. Although strong absorption tends to suppress depolarization for larger particles in general, the LDRs are rather insensitive to the change of the m_i value as manifested by the bottom left panel in which the LDRs calculated for different values of m_i are completely indistinguishable until r_eff = 0.7 μm. The LR, on the other hand, increases with increasing m_i and is a sensitive function of particle absorption. At 355 nm, even for small particles with r_eff = 0.1 μm, there is a ~50 sr difference in the LR values between nonabsorbing (the black curve) and weakly absorbing (the purple curve) aerosols. The size dependent LDRs and LRs both decrease with increasing real part of the refractive index (Fig. 7). This is understandable, since when m_r increases, the probability increases for light to get externally reflected, which means that less light penetrates into the particle, leading to higher single-scattering albedos (and hence smaller LRs) and smaller LDRs (less refracted light exiting the particle). Overall in terms of distinguishing size dependent LRs among particles with different real parts of the refractive index, lidar observations at short wavelengths work well for smaller particles, while a longer wavelength is preferable for larger ones.

Fig. 6.

As in Fig. 5, but the real part of the refractive index is fixed at 1.45 while different values of the imaginary part are designated by different colors. The particles are oblate spheroids with an aspect ratio of 1.2.

3. Discussion and conclusions

In this study, backscattering LDRs and LRs by aged nonspherical smoke particles at three lidar wavelengths of 355, 532, and 1064 nm have been calculated using the numerically exact T-matrix method. We have used spheroid and Chebyshev models to account for particle nonsphericity. Our goal was to find aerosols that reproduce the observed spectral dependence of the LDRs and LRs by Haarig et al. [35] in the context of using lidar measurements for characterization of particle morphology and composition. If we use a fixed aerosol refractive index across the spectrum like in Gialitaki et al. [41] then only two cases would provide a good match with the lidar measurements: oblate spheroids with a/b = 1.2 and r_eff = 0.5 μm and with (slightly different) refractive indices m = 1.45 + i0.02 and 1.45 + i0.015, respectively. However, there are significantly more particles, as shown in Tables 2 and 3, that successfully reproduce the observed LDRs and LRs when allowing the refractive indices to be spectrally variable with the only constraint of m_i(355 nm) ≥ m_i(532 nm) ≥ m_i(1064 nm). Among these particles, the imaginary part of the refractive index is always less than (or close to) 0.035, and the corresponding real part always falls into the range [1.35, 1.65]. The measured spectrally dependent LDRs and LRs were likely observed for particles represented by nearly-spherical oblate spheroids and similarly shaped Chebyshev particles with negative deformation parameters. Their volume-equivalent effective radii should be large enough (r_eff = 0.3 μm or greater) to produce the enhanced LDRs.

Our study of the LDR and LR at multiple wavelengths as functions of the complex refractive index, effective radius, and shape demonstrates the advantage of using triple-wavelength lidar observations yielding the capability to measure the particle LR directly and thereby improve the characterization of particle morphology and composition. Non-zero LDR values always indicate the presence of nonspherical aerosols and can be highly sensitive to particle shapes and sizes. LDRs tend to increase with particle effective size parameter and decrease with increasing refraction and absorption. The LR is a strong function of absorption and is very responsive to changing refractive index. LRs increase significantly with increasing absorption and decrease with the real part of the refractive index. All in all, we believe that our study of LDRs and LRs of BBAs is valuable not only in terms of analyzing the specific observations by Haarig et al. [35] but will also find broader use by the lidar community.