Classifying Aerosol Particle Size Using Polynomial Coefficient of Aerosol Optical Depth–Wavelength Relationship ^†

Abstract

Aerosols are composed of suspended solid or liquid particles and interact with solar radiation through absorption, refraction, and scattering, influencing climate variability. The Ångström exponent (α) is commonly used to differentiate particle sizes, but its relationship with aerosol optical depth (AOD) and wavelength (λ) is non-linear. This relationship is modeled using higher-order polynomial expressions in this study based on the AOD data from the AErosol RObotic NETwork (AERONET). In the model, polynomial coefficients are used to effectively classify aerosol types, such as dust and biomass-burning aerosols, with a strong correlation among coefficients of the same order. Such a close correlation among the coefficients of the same polynomial order is attributed to a large variability. The coefficients of the same order exhibit a scaled relationship, where scaling factors are expressed as a function of wavelength.

1. Introduction

Aerosols consist of tiny solid particles or liquid droplets suspended in the atmosphere. They originate either from direct emissions—such as dust, sea salt, and carbon particles—or through chemical reactions, forming compounds including sulfate and nitrate salts, ammonium, and secondary organic aerosols. The size of aerosols varies largely, from 0.01 to 15 μm, and their composition is complex. These size differences influence the interaction of aerosols with solar radiation through absorption, refraction, and scattering, affecting atmospheric energy dynamics and contributing to climate variability [1,2]. Due to the diverse distribution and composition of aerosols across regions and seasons, understanding their spatial and temporal size variations is essential to reduce uncertainties in climate modeling and optical atmospheric assessments [3].

In remote sensing, the Ångström exponent (α) is used as an indicator of aerosol particle size. Low α values correspond to larger particles, such as dust and sea salt (particle size > 0.6 μm), while high values are indicative of smaller particles, typically from biomass burning (particle size < 0.6 μm) [4]. Ångström initially proposed a linear relationship between aerosol optical depth and the logarithmic wavelength of observed light, with the slope of the regression line defining the Ångström exponent. However, this logarithmic relationship deviates from strict linearity, instead of forming a curve with an as-yet undetermined nature [5].

This study aims to model this non-linear relationship using polynomial expressions of different orders. By identifying the coefficients of these polynomial models, the model can be used to classify indices and reveal underlying patterns in the relationships.

2. Methodology

2.1. Data Description

In response to the growing interest in aerosol characterization, global collaborations have emerged, with the AErosol RObotic NETwork (AERONET) serving as a prominent platform for remote sensing data. This extensive network provides accessible datasets on aerosol properties, including aerosol optical depth (AOD), represented as τ, and the wavelengths of detected light, indicated by λ. In this study, AOD data from five selected AERONET observation sites (

Table 1. Observation sites.

Location	Aerosol Types	Year	Month	Number of Data
Solar Village (24.907 N, 46.397 E)	Coarse	2011–2015	3–6	10,193
Dalanzadgad (43.577 N, 104.419 E)	Coarse	2011–2017	3–6	8458
Inner Mongolia (42.7 N, 116.0 E)	Coarse	2001	4	173
Mukdahan (16.607 N, 104.676 E)	Fine	2005–2009	3–4	4350
Chiang Mai Met Sta (18.771 N, 98.972 E)	Fine	2014–2018	3–4	22,924

) were examined. Each site provides AOD measurements at seven wavelengths. In this study, three wavelengths—440, 500, and 675 nm—were used and labeled as 1, 2, and 3, respectively.

Based on seasonal variations in aerosol properties elucidated in previous studies, the aerosol types in these regions were categorized into dust- and biomass-type aerosols. Dust-type aerosols, characterized by larger, coarser particles, are prevalent at three key observation sites: the Sun Village station on the Arabian Peninsula, the Dalanzadgad station in eastern Mongolia’s Gobi Desert, and the Inner Mongolian station near China’s Gobi Desert. During the pre-monsoon season (March to June), strong winds originating from arid areas facilitate the transport of dust aerosols, establishing the stations as major sources of dust aerosols across Asia [6]. Representative stations for biomass-burning aerosols, characterized by fine particles, include the Chiang Mai Metropolitan and Mukdahan stations in Thailand. A marked increase in aerosol optical depth (AOD) is observed from February to April, with a peak in March. This seasonal pattern corresponds with widespread biomass burning in Southeast Asia, driven by crop residue combustion and land clearing for crop rotation [7].

2.2. Polynomial Representation of Aerosol Extinction

Building on the spectral dependency of particle extinction, Ångström introduced the empirical Formula (1) to describe a power-law relationship between aerosol optical properties and wavelength [8]:

$${\tau }_i=\beta {{\lambda }_i}^{-\alpha },$$

(1)

where ${\tau }_i$ represents the aerosol optical depth (AOD) at the wavelength λ_i, α denotes the Ångström exponent, and β is the turbidity coefficient. Applying the logarithm to both terms of (1) yields the ensuing linear relationship.

$$-\alpha L_i+B=J_i,$$

(2)

where $J_i=\ln {\tau }_i$, $L_i=\ln {\lambda }_i$, and $B=\ln \beta$.

Similarly, the expression for second-order polynomial equations is articulated as (3).

$$J_i=a_2{L}_i^2+a_1{L}_{\mathrm{i}}+a_0.$$

(3)

where $a_n$ presents the coefficient of the nth order subject to estimation and analysis.

2.3. Analysis of Polynomial Coefficients

The following equations are formulated to explore the relationships among polynomial coefficients of various orders. By applying the logarithmic values of wavelengths, L_i and L_j, to (3), the resulting distinction between them eliminates the coefficient a₀ and is articulated as

$$-a_2\left(L_i+L_j\right)-a_1={\alpha }_{ij},$$

(4)

in which

$${\alpha }_{ij}=-{\displaystyle \frac{J_i-J_j}{L_i-L_j}}.$$

(5)

Employing a similar methodology while eliminating the coefficient a₁, the equation transforms into

$$-a_2\left(L_i·L_j\right)+a_0=B_{ij},$$

(6)

in which

$$B_{ij}={\displaystyle \frac{{L_{i}J}_j-{L_{i}J}_i}{L_i-L_j}}.$$

(7)

Formulas (5) and (7) show that determining the coefficients in (2) requires using two pairs of τ and λ. Substituting these pairs into (2) and averaging the results gives

$$-\left({\displaystyle \frac{L_i+L_j}{2}}\right){\alpha }_{ij}+B_{ij}={\displaystyle \frac{J_i+J_j}{2}}.$$

(8)

By applying three pairs of τ and λ to (4) and (6) and subsequently solving the simultaneous equations, the second-order polynomial coefficients (a₀, a₁, a₂) are derived. In addition, applying the same pairs to (4) and (6) and averaging the results yields (9) and (10), respectively.

$${\displaystyle \frac{2}{3}}\left(L_i+L_j+L_k\right)a_2+a_1=-{\displaystyle \frac{1}{3}}\left({\alpha }_{ij}+{\alpha }_{jk}+{\alpha }_{ki}\right),$$

(9)

$${\displaystyle \frac{1}{3}}\left(L_i{L}_j+L_j{L}_k+L_k{L}_i\right)a_2-a_0=-{\displaystyle \frac{1}{3}}\left(B_{ij}+B_{jk}+B_{ki}\right).$$

(10)

Formulas (8)–(10) are used to establish a connection between the polynomial coefficients of the nth order and those of the previous order. Specifically, the coefficients on the left side of (8) pertain to the first order, while those on the right-hand side correspond to the zero order. Analogous situations are noted in (9) and (10).

3. Result and Discussion

To evaluate the classification outcomes using first-order polynomial coefficients, the coefficients derived from (2) were plotted against the aerosol optical depth (AOD) values at a wavelength of 500 nm. Figure 1a,b illustrate the graphical representation of the coefficients α₁₃ and B₁₃, respectively, where the subscripts correspond to the wavelengths defined in Section 2.1. Both coefficients exhibit comparable classification patterns, with threshold values around 1 for α₁₃ and approximately 6 for B₁₃, effectively distinguishing between dust-type and biomass-type particles.

Notably, the B₁₃ values represent scaled versions of the α₁₃ values. To investigate this scaling effect, the first-order terms of (8) were analyzed. Figure 2 presents B₁₃ values plotted against the product of α₁₃ and the average logarithmic wavelength of L₁ and L₂. The results indicate a strong correlation with the 45-degree reference line, yielding a high R² value of 0.86. This finding suggests that the scaling factor is the average logarithmic wavelength, which serves as the coefficient of α₁₃ in (8).

A similar investigation was conducted for the second-order polynomial coefficients derived from (3). The coefficients a₂, a₁, and a₀ were plotted against the AOD values at a wavelength of 500 nm, as shown in Figure 3a–c. The patterns observed for these second-order coefficients aligned with those of the first-order coefficients with threshold values near zero for dust-type and biomass-type particles. Interestingly, the behavior of a₁ is an inverted reflection of a₂ and a₀ along the x-axis. Furthermore, the figures suggest that a₂, a₁, and a₀ are scaled versions of one another.

To examine scaling effects, the second-order terms from the left sides of Equations (9) and (10) were analyzed. In Figure 4a, a₁ is plotted against a₂, with the negative coefficient of a₂ in (9) serving as its scaling factor. Similarly, Figure 4b illustrates a₂ plotted against a₀, using the coefficient of a₂ in (10) as its scaling factor. Consistent with the first-order coefficients, these results exhibit a strong correlation with the 45-degree line, achieving exceptionally high R² values.

Equations (8)–(10) provide valuable information on the factors underlying the strong correlation among coefficients of the same order as classification indices. Viewing the terms on the left-hand side of (8)–(10) as x and y variables positions the terms on the right-hand side as intercepts in a linear algebraic framework. Figure 2 and Figure 4 indicate that the variations in intercepts are significantly smaller than those observed along the x and y axes.

To confirm this, the standard deviations for each term in (8)–(10), illustrated in Figure 5a–c, were compared. The standard deviation of first-order coefficients is about three times greater than that of zero-order coefficients, while second-order coefficients exhibit twenty times the variation in the first order. These pronounced differences in deviations enhance the linear correlation among coefficients within the same order.

4. Conclusions

The potential of polynomial coefficients, derived from the logarithmic relationship between aerosol optical depth (AOD) and wavelength (λ), was investigated as indices for classifying aerosol particle sizes. Utilizing AOD data from AERONET measurements, these polynomial coefficients effectively distinguish aerosol size distributions. A strong correlation is identified among coefficients of the same polynomial order, attributed to greater standard deviations within the respective order compared with lower orders. Additionally, the analysis results reveal a scaling relationship among indices of the same order, where the scaling factors are determined by functions of the wavelength.