Modeling the single and multiple scattering properties of soot-laden mineral dust aerosols

Guanglang Xu; Patrick G. Stegmann; Sarah D. Brooks; Ping Yang

doi:10.1364/OE.25.00A990

1. Introduction

Mineral dust aerosols play an important role in atmospheric radiation budget due to their significant contribution to the overall atmospheric aerosol loading [1–5] and its global distribution via long-range convective transport [6–8]. Specifically, the global emission of mineral dust is estimated to lie anywhere between 1000 and 3000 Tg yr⁻¹ [4]. Mineral dust aerosol particles influence the atmospheric radiative forcing both by interacting with the radiation field in the atmosphere and by providing condensation nuclei for water clouds. The the physical and chemical properties of mineral dust aerosol, essential to the evaluation of its radiative forcing effects, spatially and temporally vary [9,10]. On a global scale, dust properties and their tempo-spatial variations can only be estimated effectively from satellite observations.

Inferring key dust microphysical and optical properties, such as optical thickness, and particle size distributions of airborne dust from satellite observations requires a priori knowledge of the bulk single-scattering properties (such as the scattering phase function, and single-scattering albedo) of the individual mineral dust aerosol particles [11]. Subsequently these cloud properties are normally inferred by comparing the measured satellite radiance with pre-computed radiances (i.e. through best fitting) based on standard models [12]. To improve the accuracy of satellite retrievals for long-term climate study, there is a pressing need to develop realistic single-scattering properties of mineral dust aerosols.

One important factor influencing the single-scattering properties of dust is the particle nonsphericity [13–16]. In addition, the internal particle composition may also lead to substantial uncertainties in modeling the single-scattering properties of mineral dust aerosols. For instance, pronounced differences in single particle scattering by various compositions of dust were reported by Glen and Brooks [17]. Other studies show that small variations in the hematite content (Fe₂O₃) may largely alter the absorption of mineral dust particles [18,19]. These identified particle properties have been the primary causes of significant uncertainties in determining the single-scattering properties of pure mineral dust aerosols considered in previous studies.

As mineral dust aerosols are convectively transported across long distances [6–8], their optical properties are subject to variation because of aging and the mixing with other types of aerosol. Particularly, interactions between mineral dust and carbonaceous particles are commonly observed [14,15,20,21]. Among the various carbonaceous species, black carbon (BC) deserves special attention because of its increasing number of artificial sources, its long residence time in the atmosphere, and strong absorption in the visible band [22,23]. Mixing with black carbon may significantly alter both the scattering and absorption properties of mineral dust particles at the same time [24–27]. Thus, the impact of the degree of mixing on single-scattering property calculations needs to be further investigated.

The overarching goal of this study is to understand the effect of black carbon mixing on both single and multiple scattering properties of mineral dust aerosols. As a first step in this study, the single-scattering properties of soot, mineral dust, and mineral dust with soot attachments are calculated. The accuracy of the computed single-scattering properties is evaluated by comparison with both measurements and other numerical studies. Furthermore, we introduce a method to parameterize the resulting effective bulk scattering properties of the soot-laden dust aerosols. This method allows us to map a specific mixing state of aerosols to a set of effective single-scattering properties. Given the parameterized single-scattering properties, we numerically investigate the polarized reflectivity of an aerosol layer at various mixing states by solving the plane-parallel radiative transfer equation. Finally, the uncertainties caused by the mixing effect, on both forward and inverse modeling, are quantified.

This paper is divided into six sections. Section 2 introduces the conceptual framework on which the study is based. Section 3 describes the particle models used in the single-scattering calculations. Section 4 presents the computed single-scattering properties and their parameterization. Section 5 discusses the corresponding multiple scattering properties and the implications for retrieval. The conclusions are given in Section 6.

2. Conceptual framework

In the present model, we assume a situation where the solar irradiance $I_{0}$ impinges on the top of the atmosphere (TOA) along direction $Ω_{0}$ , and the radiance at TOA measured in direction $Ω$ , is defined as:

I_{T O A} (Ω) = μ_{0} R (Ω_{0}, Ω) I_{0}

where

μ_{0}

is the cosine of the solar zenith angle, and

R (Ω_{0}, Ω)

is a 4 by 4 matrix denoting the polarized reflectivity.

R (Ω_{0}, Ω)

is a solution to the radiative transfer equation (RTE),

\begin{array}{l} μ \frac{d}{d z} I (z, Ω) = (α_{p} (z) + σ_{p} (z)) I (z, Ω) \\ - \frac{ω_{p} (z)}{4 π} (α_{p} (z) + σ_{p} (z)) \int M_{p} (z, Ω, Ω^{'}) I (z, Ω^{'}) d Ω^{'} \end{array}

where

μ

is

\cos (θ)

,

θ

is the zenith angle of the radiance propagation direction,

z

is the vertical coordinate for the assumed plane-parallel geometry,

α_{p} (z)

and

σ_{p} (z)

are the absorption and scattering coefficients respectively of the particles in the medium,

ω_{p} (z)

is the single-scattering albedo, and

M_{p} (z, Ω, Ω^{'})

is the scattering phase matrix. The subscript ‘p’ indicates that the parameter is the bulk scattering properties of the particle ensemble. Two directions

Ω

and

Ω^{'}

form a plane called scattering plane, form an angle

Θ

called scattering angle. Assuming that each volume element contains an equal number of particles and mirror particles in random orientation, on a specific scattering plane, the scattering phase matrix has a block-diagonal form, given by

P_{p} (θ) = (\begin{matrix} \begin{matrix} \begin{matrix} P_{11} (θ) & P_{12} (θ) \end{matrix} & 0 & 0 \end{matrix} \\ \begin{matrix} \begin{matrix} P_{12} (θ) & P_{22} (θ) \end{matrix} & 0 & 0 \end{matrix} \\ \begin{matrix} 0 & 0 & P_{33} (θ) & P_{34} (θ) \end{matrix} \\ \begin{matrix} 0 & 0 & - P_{34} (θ) & P_{44} (θ) \end{matrix} \end{matrix})

where we omit the dependence on

z

.

The particle size distribution of dust particles $n (r)$ [28] is assumed to be a gamma distribution,

n (r) = \frac{d N (r)}{dr} = \frac{1}{Γ (\frac{1 - 2 ν_{e}}{ν_{e}})} {(ν_{e} r_{e})}^{\frac{2 ν_{e} - 1}{ν_{e}}} (r^{\frac{1 - 3 ν_{e}}{ν_{e}}}) (e^{- \frac{r}{v_{e} r_{e}}})

where

N

is the total number of particles,

r

is particle radius,

υ_{e}

is the effective variance (set to be 0.2), and

r_{e}

is the effective radius. This form of the size distribution ensures its normalization to unity and is convenient to use.

In addition to polarized reflectivity and scattering phase matrix, the following properties are also discussed in this study:

(1) The mass scattering and absorption coefficients (MSC, MAC): $M S C = \frac{C_{s c a} (r)}{ρ (4 / 3) π r^{3}},$ $M A C = \frac{C_{a b s} (r)}{ρ (4 / 3) π r^{3}} .$
where $ρ$ is the effective density of the particle.
(2) The single-scattering albedo: $ω = \frac{σ_{p}}{α_{p} + σ_{p}} .$
(3) The asymmetry factor: $g = \frac{1}{2} \int_{- 1}^{1} P_{11} (\cos (θ)) \cos (θ) d (\cos (θ)) .$
(4) The linear depolarization ratio $δ (θ) = \frac{P_{11} (θ) - P_{22} (θ)}{P_{11} (θ) + 2 P_{12} (θ) + P_{22} (θ)} \times 100% .$

3. Particle models

Freshly emitted black carbon tends to have a complex chain-like structure and becomes more compact as the particle ages [29,30]. Given such morphological complexity and variability, modeling the single-scattering properties of black carbon is not a trivial task. To circumvent this difficulty, it was commonly assumed in the past that soot particles could be sufficiently described by a spherical shape. Unfortunately, the spherical approximation may lead to large errors for both single-scattering calculations and radiative forcing estimates [31,32].

A more realistic model for soot particles is the aggregation of numerous spherical monomers [33]. However, this model introduces more degrees of freedom than using the spherical approach. In addition, the variable irregular shape of a mineral dust particle can lead to a large uncertainty in modeling its single scattering properties. All these uncertainties should be reasonably constrained before one can investigate the mixing effects on their bulk scattering properties.

The morphology of a soot particle can be characterized by [33]:

N_{m} = k_{o} {(\frac{R_{g}}{a})}^{D_{f}},

where N_m is the total number of monomers,

k_{o}

is a constant called the fractal pre-factor,

R_{g}

is the radius of gyration, a is the radius of each monomer, and

D_{f}

is the fractal dimension. The fractal dimension determines the compactness of the particle, and hence influences the efficiency of the particle to extinguish light. In this study, we focus on relatively aged soot particles and the fractal dimension is between 2 and 3. This scaling law provides a realistic description of the overall morphological structure of black carbon particles.

Table 1 lists the morphological parameters used in this study, where L is the major axis of the particle, defined as the maximum distance between any two monomers’ center at A and B, W is the minor axis of the particle, defined as the maximum distance between any two monomers in a plane perpendicular to and across the center of $\overset{⇀}{A B}$ , AR is the aspect ratio of L to W, r is the particle radius in terms of volume equivalent sphere. The particle aspect ratio is confined to the range of 1.3~1.8 [27]. Two fractal dimensions 2.5 and 2.2 are selected to represent particles with different degrees of compactness. Figure 1 illustrates the morphology of S6 from Table 1. To compute the single-scattering properties of these soot model particles, we employ the semi-analytical Multiple Sphere T-matrix method [34].

Table 1. The parameters of soot

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Fig. 1 Particle shape for case S6. A line connecting the red colored monomers is the major axis of the particle, while the green colored monomers denote the minor axis of the particle.

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To improve upon the commonly used spheroidal model with a smooth surface for mineral dust particles, we employ a fractal polyhedron with randomly tilted facets to model the single-scattering properties of pure mineral dust [35]. According to in situ measurements [36–38], the median particle aspect ratio of a mineral dust particle tends to be in the range of 1.4~1.9. In this study, 1.7 is assigned to the pure dust particle model. Figure 2 shows a particle shape with 24 facets used in the single-scattering calculation for a pure mineral dust particle.

Fig. 2 Particle shape for a pure mineral dust particle.

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The mixing of black carbon and mineral dust aerosol can be either external or semi-external [14, 39–41]. In this study, “semi-externally mixed” refers to cases in which soot is attached to the surface of the mineral dust. Due to the agglomeration of black carbon, the single-scattering properties of a semi-external mixture are generally different from their externally mixed counterparts. These differences may include a reduced single-scattering albedo, depolarization ratio, and may subsequently give rise to differences in the multiple scattering properties [42]. The semi-externally mixed model (aggregate model) has a morphology displayed in Fig. 3.

Fig. 3 The particle shape of a dust-soot aggregate

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The semi-externally mixed particle is modeled using simple spheres as a representation for the soot attached to the random Koch fractal. Using a fractal structure for the mineral dust particle and the attached soot at the same time would lead to a much poorer convergence in the single-scattering calculations, rendering such an approach impractical. Without providing appreciable differences in the results, the soot attachments are assumed to be comparatively small. Table 2 lists the parameters of the dust models used in this study, where VMR is the volume-mixing ratio (the ratio of the volume of soot to the volume of the host particle), N_s denotes the number of the attached soot particles, and $r$ denotes the radius of the whole particle. For the aggregate particle, one spherical soot globule has around 1.48% of the volume of the host particle. We apply the semi-analytical invariant imbedding T-matrix method [43,44] to compute the single-scattering properties for both pure dust and dust-soot aggregates.

Table 2. The parameters of dust

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In this study, single-scattering property calculations are performed at multiple wavelengths including 440 nm, 500 nm, 532 nm, 550 nm, 675 nm, 870 nm, 1020 nm, and 1064 nm. The refractive indices of black carbon and the mineral dust particle are those suggested by Chang and Charalampopoulos [45] and Wagner et al. [46], respectively.

4. Single scattering properties

4.1 Properties without angular dependence

The mass absorption coefficient (MAC) is important for evaluating the radiative forcing of black carbon because of its high value at visible wavelengths. Figure 4 shows the simulated MACs of the cases listed in Table 1 in comparison with the values from other simulation studies and in situ measurements at the specific wavelength of 550 nm. For the calculations, a density of 1.8 g/cm³ is assumed for soot particles. The range of MAC values is around 5.0 m²/g to 5.5 m²/g, which is in reasonable agreement with other studies. These results suggest that different cases of the model selected from Table 1 produce similar MAC values. For the wavelength range from 440 nm to 675 nm, the absorption and extinction Angstrom exponents are $1.15 \pm 0.11$ and $1.45 \pm 0.13,$ respectively.

Fig. 4 Simulated spectral mass absorption coefficients and extinction coefficients for the cases listed in Table 1.

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In the case of external mixing with soot, the single-scattering properties of different particles are additive because their light scattering events can be treated independently. On the other hand, when the two components are attached to each other, their combined optical properties are no longer a direct sum. The computation of the bulk scattering properties therefore requires the knowledge of the single-scattering properties of dust-soot aggregates. Figure 5 displays the spectral MSCs of pure dust and dust-soot aggregates with parameters listed in Table 2. The MSCs of both pure dust and the dust-soot aggregates exhibit a strong wavelength-dependency, especially for the cases with particle size of ~1.0 $μ m$ . As indicated by Fig. 5, the semi-external mixing by black carbon significantly alters the absorptivity of the dust aerosol. The single-scattering albedos and asymmetry factors of all cases of mineral dust and soot particles (Tables 1 and 2) can be summarized as follows: for soot particles, the asymmetry factor exhibits a strong wavelength dependence, ranging from ~0.1 to ~0.8, and their single-scattering albedo ranges from ~0.1 to ~0.4. For dust particles, the asymmetry factors of all cases are within the range of ~0.67 to ~0.8, and the semi-external mixing leads to a reduction of the albedo from ~0.9 to ~0.8. These results indicate a large variability of the bulk scattering properties of soot-laden mineral dust aerosols.

Fig. 5 Simulated spectral scattering and absorption mass coefficients of dust/dust-soot aggregation at wavelengths from 440 nm to 1064 nm.

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4.2 Properties with angular dependence

Accurately modeling the angular distribution of scattered intensity or polarization benefits the retrieval of important microphysical and optical properties, such as the effective particle size and optical depth of mineral dust aerosols on a large scale. The first-generation random Koch fractal can reproduce the laboratory-measured scattering phase matrix of feldspar sufficiently well [47], as shown in Fig. 6. The measured data are from the Amsterdam light scattering database [48], which provides an effective size of 1.0 $μ m$ , particle size distribution and refractive index of 1.50 + 0.001i for the simulations at wavelength 441.6 nm. The comparison demonstrates that the fractal polyhedral model captures the angular distribution pattern of intensity and polarization well. Spheroidal particle models may also be able to reproduce the measurement results, but multiple aspect ratios are often required to be mixed.

Fig. 6 Comparison between the simulated and laboratory-measured scattering phase matrix of mineral dust aerosols.

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The most characteristic effect of the mixing of black carbon with mineral dust is the enhancement of the absorption efficiency. External mixing generally leads to weaker absorption than semi-external mixing due to a lower degree of agglomeration of soot. Figure 7 shows the scattering phase matrix elements of the three possible soot-mineral dust configurations in a polluted dust layer. The properties of the pure soot are the average of cases listed in Table 1. Both the pure dust and the dust-soot aggregate have a particle size of 800 nm in radius, while the latter has 5.8% of the volume of attached soot. The result displays the changes induced by the attachment of soot on the scattering phase matrix. In particular, the degree of linear polarization –P₁₂/P₁₁ of an aggregate is slightly higher than pure dust, and P₂₂/P₁₁ gets closer to unity at the backward scattering angles, indicating that the aggregate behaves more similarly to a spherical particle than the pure dust. These changes seem to be modest at first glance, but the scattering cross-section has been drastically reduced.

Fig. 7 Scattering phase matrices of each possible component of a polluted dust aerosol. “d” denotes pure dust, “s” denotes pure soot, and “a” denotes dust-soot aggregate.

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4.3 Parameterization

Conventionally, the effective bulk properties of the mixed aerosol can be computed using the effective medium approximation. In the effective medium approximation, the number of aerosol components reduces to one effective component, and the effect of mixing can be characterized by a volume-mixing ratio. Alternatively, one could first compute the single-scattering properties of each possible component, and parameterize the bulk properties according to their number density weights. The preceding analysis suggests that semi-external mixing yields the single-scattering properties quite different from the external-mixing counterparts. This warrants the introduction of dust-soot agglomerates as a class distinct from pure dust and pure soot. Below we present a method to properly parameterize the bulk optical properties of the dust-soot mixed aerosol.

The mixing state can be characterized by the number density weight of each component, denoted as

S = {(w_{d}, w_{s}, w_{a})}^{T} .

The subscripts ‘d’, ‘s’, and ‘a’ denote the parameters associated with pure dust, soot, and their aggregates, respectively. The sum of the weights equals to the unity,

w_{d} + w_{s} + w_{a} = 1 .

A special case, the state of pure dust can be represented by

S_{o} = {(1, 0, 0)}^{T} .

The component of the dust-soot aggregate is characterized by so-called the aggregation ratio, i.e. the volume of the aggregated soot divided by the volume of the host dust particle,

β = \frac{V_{a g g r e g a t e d_s o o t}}{V_{h o s t_d u s t}}

where

β

should be considered as an effective parameter of all dust-soot aggregates. For instance, if it is a function of particle size, then

β = \frac{\int_{r_{m a x}}^{r_{m i n}} (\frac{4}{3} π r^{3}) \frac{β_{o} (r)}{β_{o} (r) + 1} \frac{d N_{a}}{d r} (r) d r}{\int_{r_{m a x}}^{r_{m i n}} (\frac{4}{3} π r^{3}) \frac{1}{β_{o} (r) + 1} \frac{d N_{a}}{d r} (r) d r}

where N_a (r) is the total number of dust-soot aggregates with particle size up to r,

β_{o} (r)

is the aggregation ratio of particle with size r, and

r_{m a x}

and

r_{m i n}

are the maximum and minimum size of the aggregates. The aggregation ratio describes how much soot is attached to a unit volume of dust for the aggregates. If the aggregation ratio goes to zero, the mixed aerosol reduces to simply two components, i.e. pure dust, and pure soot. The introduction of this parameter therefore becomes necessary when the semi-external mixing effects are to be evaluated.

Once the effective aggregation ratio $β$ is known, another two effective parameters of the bulk, namely the total volume mixing ratio $γ$ together with the degree of aggregation $η$ , can be used to characterize the mixing state,

γ = \frac{w_{s} V_{s} + w_{a} V_{d} β}{w_{a} V_{d} + w_{d} V_{d}},

η = \frac{w_{a} V_{d} β}{w_{s} V_{s} + w_{d} V_{d} β},

where

V_{s}

and

V_{d}

are the effective particle volumes of pure soot and clean dust before the mixing. The total volume mixing ratio

γ

is the volume ratio of soot to dust without specifying the ways of their mixing. The degree of adhesion

η

is the volumetric proportion of the attached soot to all mixed soot.

γ

can be viewed as an index of the total effect of the soot mixing, while

η

can be viewed as an index of the semi-external mixing effect.

Equations (6), (9) and (10) together lead to a parameterization of the mixing state,

S = F^{- 1} (γ, β, η) S_{0}

where

F

is

F (γ, β, η) = (\begin{matrix} 1 & 1 & 1 \\ γ V_{d} & - V_{s} & (γ - β) V_{d} \\ 0 & η V_{s} & (η - 1) β V_{d} \end{matrix})

The single-scattering properties of the mixture layer of aerosol can then be obtained by the following dot products,

σ_{p} = σ \cdot (F^{- 1} S_{0})

α_{p} = α \cdot (F^{- 1} S_{0})

σ_{p} P_{p} (θ) = P (θ) \cdot (F^{- 1} S_{0})

where the subscript ‘p’ denotes the parameterized bulk properties. The three vectors are called component vectors, formed by the effective scattering properties of each component,

σ = (σ_{d}, σ_{s}, σ_{a}),

α = (α_{d}, α_{s}, α_{a}),

P (θ) = (P_{d} (θ) σ_{d}, P_{s} (θ) σ_{s}, P_{a} (θ) σ_{a})

The above equations provide a set of single scattering properties labeled by three parameters, namely the total volume mixing ratio $γ$ , aggregation ratio $β$ , and the degree of adhesion $η$ . The aggregation ratio $β$ is assumed to be inherent to aerosol types.

Figures 8 and 9 show the parameterization sensitivity for the single-scattering albedo (as a scattering property without angular dependence), and the linear depolarization ratio (as a property with angular dependence) at 180 degree, respectively. The aggregation ratio is fixed at 5.8%. The effective size of the pure dust and aggregate is 1.0 $μ m$ , and the effective size of soot is around 0.1 $μ m$ . The figures clearly demonstrate how different ways of mixing, i.e. externally or semi-externally, change the bulk scattering properties of soot-laden dust aerosol.

Fig. 8 The parameterized single-scattering albedo of the bulk.

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Fig. 9 Parameterized linear depolarization ratio at 180 degree.

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5. Multiple scattering properties and analyses

5.1 Labeled polarized reflectivity

This section discusses the multiple scattering properties of the soot-contaminated dust aerosol by solving the radiative transfer equation with proper boundary conditions. The adding-doubling method [49] is applied for this purpose. In the multiple scattering calculations an idealized plane-parallel atmosphere with a single homogeneous aerosol layer and terminated by a Lambertian surface of albedo 0.1 is assumed. Using the parameterization introduced in previous section, we can label the RTE as follow,

\begin{array}{l} μ \frac{d}{d z} I (z, Ω) = (_{α_{p} (z) + σ_{p} (z)) γ, η} I (z, Ω) \\ - \frac{ω_{p} (z)}{4 π} (_{α_{p} (z) + σ_{p} (z)) γ, η} \int (M_{p} (z, Ω, Ω^{'}))_{γ, η} I (z, Ω^{'}) d Ω^{'} \end{array}

Consequently, the polarized reflectivity, as a solution to the RTE can be labeled as,

μ_{o} R_{i j} (Ω, Ω_{0}) = S o l [R T E_{γ, η}]

where “Sol” denotes the solving operation for the polarized reflectivity. We first compute the polarized reflectivity with five cases listed in Table 3. Figure 10 shows the polar plots of Stokes vector component or their ratios, I, -Q/I, U/I as a function of viewing direction in these five cases. The optical depth and effective particle size of dust are 0.60 and 0.80

μ m

respectively. The solar zenith angle is 45 degree. In each polar plot, the radius is the viewing zenith angle, the angle is the relative azimuthal angle, and the color represents the values of the component. To show the contour lines more clearly, the axes of the plots are displayed separately. The plot in the middle displays the pattern associated with pure dust, while the other surrounding plots show the pattern associated with the soot-laden dust. It is clear that the intensity patterns are fairly similar among these five cases, whereas the polarization patterns are quite distinguishable.

Table 3. The labels used for the RTE solutions

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Fig. 10 The polar plot of I, -Q/I, U/I as a function of view angle for labels listed in Table 3.

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5.2 Analysis

An analysis scheme is introduced to quantify the uncertainties of the multiple scattering properties due to the uncertainty on the aerosol mixing states. In accordance with observations [14,15,19,20], the amount of mixed soot, compared to mineral dust, is much smaller. Presumably, the interaction with black carbon plays a secondary role in determining the aerosols multiple scattering properties. Here the way of their mixing is treated as a source of uncertainty for modeling the multiple scattering properties. Under idealized conditions, the polarized reflectivity observed by a multiple-angle airborne satellite, can be viewed as a function of viewing direction, optical depth, particle size, and the mixing state:

μ_{o} R_{i j} = f (Ω, τ, r_{d}, γ, η)

where

τ

is the optical thickness of the aerosol,

r_{d}

is effective particle size of pure dust,

Ω = (μ, ϕ)

is the viewing direction in a spherical coordinate system,

R_{i j}

could be associated with any of the four components of the Stokes vector. It should be noted that the idealized condition is confined to the situation where the solar incidental direction is fixed, and the variability of soot particle size

r_{s}

and the aggregation ratio

β

are small. On the other hand, if we ignore the aerosol’s mixing effects, the reflectivity reduces to

μ_{o} R_{i j}^{o} = f_{o} (Ω, τ, r_{d}) = f (Ω, τ, r_{d}, 0, 0) .

The forward modeling deviation can be defined as the difference between the two functions integrated over an observational range of an instrument,

D (γ, η) = \frac{1}{Δ Ω'} \iint | \frac{f_{o} (Ω', τ_{o}, r_{d}^{o}) - f (Ω', τ_{o}, r_{d}^{o}, γ, η)}{f_{o} (Ω', τ_{o}, r_{d}^{o})} | d Ω' \times 100 % .

It should be noted that here the parameter pair

(τ, r_{d}) = (τ_{o}, r_{d}^{o})

is now constant, referred to as the standing point. Furthermore, in accordance with the instruments, the viewing direction is confined to

Ω' \in Ω_{\max},

where

Ω_{m a x}

is the maximum satellite observational range. For instance, the MISR-type instruments has a maximum observation range so that the scattering angle is confined to

82^{o}

to

148^{o}

, meaning that all viewing directions within this range constitute its maximum observational range. If the observational range is discretized into a finite number of directions with equal weightings, the above equation becomes

D (γ, η) = \frac{1}{N} \sum_{k = 1}^{N} | \frac{f_{o} (Ω^{k}, τ_{o}, r_{d}^{o}) - f (Ω^{k}, τ_{o}, r_{d}^{o}, γ, η)}{f_{o} (Ω^{k}, τ_{o}, r_{d}^{o})} | \times 100 % .

The value of this function can be computed when a standing point

(τ_{o}, r_{d}^{o})

and the satellite observational

Ω_{m a x}

range are specified. As we already mentioned, the functions

f_{o}

and

f

can be either associated with intensity or polarization. Therefore, the deviation function defined above could be either associated with intensity or the degree of polarization.

The deviation function at a specific standing point and observational range is associated with the forward modeling process, i.e. the theoretical observations by the satellite instruments. Given certain probability distribution of $(γ, η)$ , one can compute the average, minimum, and maximum deviations from the pure dusts on the polarized reflectivity. Figure 11 shows a contour plot of $D (γ, η)$ associated with the intensity component, and Fig. 12 shows the $D (γ, η)$ associated with the degree of linear polarization. From the two figures, it can be seen that the intensity deviation can go up to 60% when the total mixing ratio is around 3% to 5%, while the deviation on the linear polarization can go up to 150%. These results suggest that 1) the labeled multiple scattering properties have a similar pattern to those of the single-scattering properties; 2) compared to radiance intensity, much larger uncertainties are observed for the degree of linear polarization.

Fig. 11 Deviation function $D (γ, η)$ associated with the radiance intensity. The standing point for this computation is $(1.0, 0.8 u m)$ , meanwhile 7485 viewing directions with scattering angle ranging from $82^{o}$ to $148^{o}$ are used.

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Fig. 12 The deviation function $D (γ, η)$ associated with the degree of linear polarization. The parameters used are the same as used in Fig. 11.

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On other hand, it is also desirable to quantify the uncertainties in the retrieval process. Similar to the deviation function, we can introduce the following the so-called criterion function as

C (τ, r_{d}) = \frac{1}{Δ Ω'} \iint | \frac{f_{o} (Ω', τ, r_{d}) - f (Ω', \tilde{τ}, {\tilde{r}}_{d}, γ_{o}, η_{o})}{f_{o} (Ω', τ, r_{d})} | d Ω' \times 100 % .

Note that here two pairs of parameters

(\tilde{τ}, \tilde{r_{d}})

and

(γ_{o}, η_{o})

are constant. Now the two functions

f_{o}

and

f

can be interpreted in different ways.

f_{o} (Ω', τ, r_{d})

can be viewed as the quantities associated with the standard models in the retrieval process, whereas

f (Ω', \tilde{τ}, \tilde{r_{d}}, γ_{o}, η_{o})

can be viewed as the quantities associated with a quasi-realistic models. When

C (τ, r_{d})

becomes small enough, the model fits well with the observation, we then call it a retrieval event, and the fulfilled parameters

(τ, r_{d})

are called retrieved parameters. Since

f (Ω', \tilde{τ}, \tilde{r_{d}}, γ_{o}, η_{o})

is associated with the quasi-realistic model, the constant parameters

(\tilde{τ}, \tilde{r_{d}})

are referred to as goal parameters, and (

γ_{o}, η_{o}

) are referred to as confusion parameters. Because satellite observation is always limited to a limited number of viewing angles at a specific time and location, it is not able to reach its maximum observation range for a particular layer of mineral dust. Thus, the integration range for the criterion function must be a small subset of the maximum observational range, i.e.,

Ω' \in Ω \subset Ω_{\max} .

Again, if the integration range is discretized into a finite number of directions, then the criterion function becomes

C (τ, r_{d}) = \frac{1}{M} \sum_{k = 1}^{M} | \frac{f_{o} (Ω^{k}, τ, r_{d}) - f (Ω^{k}, \tilde{τ}, {\tilde{r}}_{d}, γ_{o}, η_{o})}{f_{o} (Ω^{k}, τ, r_{d})} | \times 100 % .

Accordingly the number of directions

M

is much smaller than

N

,

M < < N .

We then can introduce the following criteria using

C (τ, r_{d})

:

A). When $f_{o}$ or $f \equiv I$ , and $C_{A} (τ, r_{d}) \leq 1 %$ , namely the intensity criterion;
B). When $f_{o}$ or $f \equiv \sqrt{Q^{2} + U^{2}} / I$ , and $C_{B} (τ, r_{d}) \leq 1 %$ , namely, the polarization criterion;
C). $C_{C} (τ, r_{d}) = C_{A} (τ, r_{d}) * C_{B} (τ, r_{d}) \leq 1 %$ , namely, the intensity plus polarization criterion.

Given a set of goal parameters, under the “confusion” of the confusion parameters, it can be retrieved by applying the above three criteria. The retrieval quality can then be accessed. Figure 13 displays the fulfilled parameter range for the intensity, polarization and intensity plus polarization criterion. For all three criteria, the local minimum of the criterion function for two polluted cases (denoted by blue and red colors) can differ largely from the goal parameter. For example, for the goal parameters $(\tilde{τ}, \tilde{r_{d}}) = (0.6, 0.4 μm)$ , the retrieved parameter would be around (0.4, 0.7 $μ m$ ) using the intensity criterion and a confusion parameter $(γ_{o}, η_{o}) = (3 %, 100 %)$ . Such results suggest that by using a pure dust look-up table to infer the optical thickness and particle size of the polluted dust aerosol, the retrieved values might differ largely from the true values. However, noting that if more complex retrieval methods are used, the quality of the retrieval needs to be reevaluated.

Fig. 13 Simulated retrieval of particle size and optical thickness of the dust aerosols. The green color lines denote the fulfilled range associated with confusion parameters $(γ_{o}, η_{o})$ = (0,0), meaning the standard model perfectly matches the quasi-realistic model. The blue color lines denote the fulfilled range with confusion parameters $(γ_{o}, η_{o})$ = (3%, 0), while the red color is associated with the confusion parameters $(γ_{o}, η_{o})$ = (3%, 100%). Nine viewing directions are used in the computation.

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6. Conclusions

Accurately modeling the single and multiple scattering properties of mineral dust particles is a challenging task because of their irregular shapes, complex compositions and interactions with other aerosols. As an effort towards a more realistic modeling, more parameters should be taken into account in accordance with the observations. This study investigates the effect of soot mixing on the single-scattering characteristics as well as the uncertainties caused by this factor in the multiple scattering modeling.

In the single-scattering modeling, the parameters of the models are carefully selected, such that the applied models reasonably capture the characteristic scattering properties of the dust and soot particles. The introduced parameterization scheme quantitatively maps the mixing state to bulk single-scattering properties of aerosol.

In the multiple-scattering modeling, the polarized reflectivity is viewed as not only a function of optical depth and particle size, but also the mixing states of the aerosol under the assumed setups. The results of the deviation function and criterion function indicate that large uncertainty may be introduced by ignoring the state of mixing of soot on both forward and inverse modeling on the polarized reflectivity of mineral dust.

Funding

National Science Foundation (NSF) (ATMO-0803779, OCE-1459180); Endowment funds related to the David Bullock Harris Chair in Geosciences at the College of Geosciences, Texas A&M University.

Acknowledgments

This research was partly supported by the National Science Foundation (ATMO-0803779 and OCE-1459180) and by the endowment funds related to the David Bullock Harris Chair in Geosciences at the College of Geosciences, Texas A&M University. The present computations were carried out mainly at the Texas A&M University Supercomputing Facility.

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Case	N_m	L ( $μ m$ )	W ( $μ m$ )	AR	r ( $μ m$ )	D_f
S1	80	0.33	0.24	1.39	0.086	2.5
S2	120	0.40	0.24	1.66	0.099	2.5
S3	200	0.44	0.30	1.45	0.12	2.5
S4	280	0.53	0.33	1.60	0.13	2.5
S5	80	0.41	0.25	1.64	0.086	2.2
S6	120	0.53	0.30	1.76	0.099	2.2
S7	200	0.60	0.40	1.49	0.12	2.2
S8	280	0.78	0.47	1.67	0.13	2.2

Case	_$γ$	_$η$
E0	0%	0%
E1	2%	0%
E2	4.5%	0%
E3	2%	50%
E4	4.5%	50%

Case	N_m	L ( $μ m$ )	W ( $μ m$ )	AR	r ( $μ m$ )	D_f
S1	80	0.33	0.24	1.39	0.086	2.5
S2	120	0.40	0.24	1.66	0.099	2.5
S3	200	0.44	0.30	1.45	0.12	2.5
S4	280	0.53	0.33	1.60	0.13	2.5
S5	80	0.41	0.25	1.64	0.086	2.2
S6	120	0.53	0.30	1.76	0.099	2.2
S7	200	0.60	0.40	1.49	0.12	2.2
S8	280	0.78	0.47	1.67	0.13	2.2

Case	_$γ$	_$η$
E0	0%	0%
E1	2%	0%
E2	4.5%	0%
E3	2%	50%
E4	4.5%	50%

Modeling the single and multiple scattering properties of soot-laden mineral dust aerosols

Abstract

1. Introduction

2. Conceptual framework

3. Particle models

4. Single scattering properties

4.1 Properties without angular dependence

4.2 Properties with angular dependence

4.3 Parameterization

5. Multiple scattering properties and analyses

5.1 Labeled polarized reflectivity

5.2 Analysis

6. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (13)

Tables (3)

Equations (36)

Optics Express

Case	VMR	N_s	AR	r ( $μ m$ )
D1	0%	0	1.7	0.31
D2	0%	0	1.7	0.60
D3	0%	0	1.7	0.97
D4	2.9%	2	1.7	0.31
D5	2.9%	2	1.7	0.60
D6	2.9%	2	1.7	0.97
D7	5.8%	4	1.7	0.31
D8	5.8%	4	1.7	0.60
D9	5.8%	4	1.7	0.97