Numerical Analysis of the Analytical Relationships between Angstrom Coefficients of Aerosols and Their Optical Properties for Four Types of Aerosols

In this paper, the authors numerically analyzed the analytical relationships between angstrom coefficients and optical properties of aerosols to the existing data extracted from OPAC at the spectral length of 0.25 μm to 2.5 μm at eight relative humidity for desert, urban, marine clean and continental clean aerosols. That is apart from their relationships with the wavelength that was determined, in this paper their relation with respect to aerosols’ type and RHs are determined. The properties extracted are scattering, absorption, and extinction coefficients and single scattering albedo. The results showed that the extinction and single scattering albedo are correct for all the aerosols but single scattering co-albedo is satisfied for only sahara and continental clean.


Introduction
The Angstrom exponent (AC) is a parameter that is being widely used in atmospheric sciences to analyze the optical properties of aerosol particles.Since the early publications of Angstrom [1,2] and his later publications [3,4], where this parameter was mainly applied to the description of the spectral behavior of the atmospheric extinction and transmission, respectively, it is now being applied to a variety of similar but slightly different optical properties, for instance to the atmospheric, optical depth, extinction coefficient, scattering or backscattering coefficients etc.It is very popular not only because of the simplicity of the equation, but because it enables extrapolation or interpolation of aerosols' optical properties, because it is connected to particle microphysics (related with the mean size of aerosols) as it describes, approximately for a certain radius range, and a spectral range, a power law (Junge) aerosol size distribution [5][6][7][8].It was refined by O'Neill and Royer [9] who derived bimodal size distribution radii using these parameters.
The Angstrom exponent being an indicator of the aerosol spectral behaviour of aerosols [10], has been adopted by a number of authors in the literature to characterize biomass burning aerosols [11,12], urban and desert dust aerosol [13] and maritime aerosols [14].In general, the aerosol optical depth (AOD) and AC parameters can be used to differentiate between coarse and fine particles [15].
Simple analytical relationships between extinction, scattering, and absorption coefficients and single scattering albedo (SSA) [16], and the corresponding relationships for ACs [17] exist.Such relationships are useful to compare ACs obtained from extinction, scattering, and absorption, including the ground truthing of remote sensing and satellite measurements.For example, aerosol extinction can be obtained from ground-based and satellite remote sensing at multiple wavelengths yielding extinction Angstrom coefficients (EACs).Simple analytical relationships between EACs, scattering Angstrom coefficients (SACs), and absorption Angstrom coefficients (AACs) will help attribute the EACs to the underlying physical phenomena, namely scattering and absorption, and analyzing closure between the different Angstrom coefficients.In addition, SSA is the key parameter that normally determines the sign and magnitude of aerosol radiativeforcing.SSA can be obtained at multiple wavelengths from in-situ measurements [7,8,18], ground-based remote sensing measurements [19,20], and potentially from satellite measurements [21,22].Relating the SSA Angstrom coefficient (SSAAC) to the underlying SAC, AAC, and EAC will help with data interpretation and closure and physical understanding.The SSAs are some of the most dominant input factors that determine the aerosols type in radiative transfer models [23][24][25][26] and depend on the microphysical properties of the aerosols and therefore their value can be used for the characterization of the aerosol type.SSA can be interpreted as the probability that light will be scattered, giving an extinction event or the ratio between the scattering coefficient and the extinction coefficient while the Single scattering co-albedo (SSCA) can be considered as the probability of absorption per extinction event or ratio between the absorption coefficient and the extinction coefficient.Therefore if SSAAC is less than 0 it indicates that SSA increases with wavelength, while if SSAAC is larger than 0, SSA decreases with wavelength.This shows that SSAAC can be used to determine the increase or decrease in the radiative forcing and while for single scattering co-albedo Angstrom coefficient (SSCAAC) is the reverse.From the various plots we observed some spectral intervals where SSA decreases with the wavelength as well as some spectral intervals where SSA increases with the wavelength.
In addition, ACs can be obtained from simple linear or nonlinear regression of data plotted on a log-log scale or more complicated non-linear fits of data that may also yield higher order terms which give additional information about the type of aerosols using the curvature [27].Relationships between different ACs that include the SSA (ω) have only been derived by Moosmuller and Chakrabarty, [17]as for single-and two-wavelength ACs, while for ACs obtained from linear or non-linear fits the mathematics gets much more complicated due to the difficulty of appropriately attributing the influence of the SSA at different wavelengths.However, in most cases, the single-wavelength equations still give a good approximation depending on the type of aerosols and relative humidity.
In aerosol optics, the ACs that are of most interest are scattering, absorption, and extinction coefficients and for the SSA (ω) and single scattering co-albedo (SSCA).The relationships between these ACs are analytically determined by Moosmuller and Chakrabarty, [17] as: As suggested by Moosmuller and Chakrabarty [17], in this paper we are going to apply these relationships to the existing data extracted from OPAC at the spectral length of 0.25 μm to 2.5 μm and eight RHs (0%, 50%, 70%, 80%, 90%, 95%, 98%, and 99%) for desert, urban, marine clean and continental clean to determine its accuracy and its dependence on the types of aerosols, the power of the polynomials and RHs (that is hygroscopic growth as a result of the change in RHs).

Methodology
The models extracted from OPAC are given in Table 1.
The spectral behavior of the aerosol's optical parameter (X, say), with the wavelength of light (λ) is expressed as inverse power law [3]: where: N i is the mass concentration of the component, water soluble components (WASO, consists of scattering aerosols, that are hygroscopic in nature, such as sulfates and nitrates present in anthropogenic pollution), water insoluble (INSO), soot (SOOT, not soluble in water and therefore the particles are assumed not to grow with increasing relative humidity), mineral nucleation mode (MINM), mineral accumulation mode(MIAM) , mineral coarse mode (MICM), Sea salt accumulation mode (SSAM) and Sea salt coarse mode (SSCM).Urban aerosol represents strong pollution in urban areas.Continental clean aerosol represents remote continental areas without or with very low anthropogenic influences.Desert aerosol is used to describe aerosol over all deserts of the world, and no distinction with respect to the local properties is made.It consists of the mineral aerosol components in a combination that is representative for average turbidity, together with a certain part of the water-soluble component.Maritime aerosol types contain sea salt particles and Maritime clean is given to represent undisturbed remote maritime conditions with no soot, but with a certain amount of water-soluble aero-sol, which is used to represent the non-sea salt sulfate.

 
where X(λ) can represent extinction, scattering, and absorption coefficients, single scattering albedo and single scattering co-albedo while β is the turbidity and α is the Angstrom exponent (AC) [9,29].The wavelength dependence of X(λ) can be characterized by the AC, which is a coefficient of the following regression: However the Angstrom exponent itself varies with wavelength, and a more precise empirical relationship between aerosol extinction and wavelength is obtained with a 2nd-order polynomial [13,[30][31][32][33][34][35][36][37][38] as: Here, the coefficient α 2 accounts for a "curvature" often observed in sunphotometry measurements.Eck et al. [12,13], Schuster et al., [27], O' Neill et al., [34] and Kaskaoutis et al., [39,40] reported the existence of negative curvatures for fine-mode aerosols and near zero or positive curvatures are characteristic of size distributions with a dominant coarse-mode or bimodal distributions with coarse-mode aerosols having a significant relative magnitude.Now differentiating Equation ( 5) with respect to lnλ we obtained Also differentiating Equation ( 6) with respect to lnλ we obtained Assuming that Equations ( 7) and ( 8) are evaluated at a wavelength, this implies we can substitute Equation ( 7) into (8) to obtain Equation ( 9) now shows the relationship between α and wavelength.
We now also proposed a cubic relation of the form to determine whether cubic relation can improve the accuracy of Equations ( 1)-(3).Also differentiating Equation (10) with respect to lnλ we obtained Assuming that Equations ( 7) and ( 11) are evaluated at a wavelength, this implies we can substitute Equation ( 7) into (11) to obtain In this paper we are going to determine the correlation of Equations ( 1)-(3) with Equations ( 7), ( 9) and ( 12) for all the four types of the aerosols with respect to wavelengths and RHs.In Equation (1) since it involves products, we determined the average, but in Equations ( 2) and (3), since they have linear relations, we compared the coefficients.
Table 2(a) shows good correlations at 90%, 95%, 98% and 99% RH, but bad correlations 0%, 50%, 70%, and 80% RH for linear.The increase in the power of the polynomials and RHs caused increase in the correlations.
Figure 1(b) shows that the plots can be approximated by the power law.
Table 2(b) shows very good correlations, and the correlations increase with the increase in the power of the polynomials and RHs.
Figure 1(c), spectral extinction coefficients decrease with wavelength and can be approximated with a powerlaw wavelength dependence and also a bi-modal type of particle size distributions [13].The increase of the coefficients with RH has occurred because of the increase in mode size as a result of the increase in RHs.The increase of the extinction with RH at the deliquescence point (90 to 99) is that the growth increase substantially, making the process strongly nonlinear with RH [41,42].
Table 2(c) shows good correlations between extinction and  using Equations ( 5), ( 6) and (10).The correlations increase with the increase in the power of the polynomials and RHs.
Observing Figures 1(a) and Table 2(a), it can be seen that at RHs 0% to 90% the scattering coefficients have not satisfied the power law, and within this range it can be observed that in Table 2(d), Equation (1) underestimated Equation (7) on the linear part but overestimated Equations ( 9) and ( 12) on the quadratic and cubic part respectively at all the RHs.The gaps decrease with the increase in RHs.
Figure 1(d) shows that the plots can be barely approximated by the power law.
Table 2(e) shows that the correlations decrease with the increase in RH, but increase with the increase in the power of the polynomials.
Comparing the coefficients at )

Table 2(b).
The results of the Angstrom coefficients of absorption coefficients using Equations ( 5), ( 6) and (10) for sahara model at the respective relative humidities using regression analysis with SPSS16.0.

Table 2(c).
The results of the Angstrom coefficients of extinction coefficients using Equations ( 5), ( 6) and (10) for sahara model at the respective relative humidities using regression analysis with SPSS16.0.

Table 2(e).
The results of the Angstrom coefficients of single scattering albedo using Equations ( 5), ( 6) and (10) for sahara model at the respective relative humidities using regression analysis with SPSS16.0.
RH Linear Equation ( 5) Quadratic Equation ( 6) Cubic Equation ( 10)  can be observed that they are approximately the same with some to three places of decimal while some to four places of decimals.Figure 1(e) shows it is almost the opposite of Figure 1(d), and the plots can be barely approximated by power law and it decreases with the increase in RH.
Table 2(g) shows that the correlations decrease with the increase in RH, but increase with the increase in the power of the polynomials.
Comparing the coefficients in Tables 2(g) and (h) it can be observed that they are approximately the same, some to one place of decimals while some to two places of decimals.
Figure 2(a) shows a steep but smooth decrease of the extinction coefficients with wavelengths and all the plots satisfy power law.
Table 3(a) shows very good correlations for all the polynomials, and the correlations increase with the increase in the powers of the polynomials.
Figure 2(b) shows a steep but smooth curves that decrease with the increase in wavelength, but shows little effect with the increase in RH.They all satisfy power law.
Table 3(b) shows very good correlations for all the polynomials, and the correlations increase with the increase in the powers of the polynomials.
Figures 2(c) and (a) are almost similar.Table 3(c) shows very good correlations for all the equations, but the correlation increases with the increase in the power of the polynomials.
From Table 3(d) it can be seen that they are approximately the same, with some to one place of decimal while some to two places of decimals.
Figure 2(d) shows that not all can satisfy power law.Table 3(e) shows that the correlations decrease with the increase in RH, but increase with the increase in the power of the polynomials.
Comparing the coefficients in Tables 3(e) and (f) it can be observed that they are approximately the same with some to two places of decimals while some to four places of decimals.Table 3(g) shows very good correlations between, and the correlation increases with the increase in the power of the polynomials.
Comparing Tables 3(g) and (h), the linear part shows that at 0%, 50% and 70% RH, they are the same to one place of decimal places.After that they are completely different.
Comparing Figures 3(a) and 2(a) it can be observed that they are similar.
Table 4(a) shows very good correlations for all the polynomials, and the correlations increase with the increase in the powers of the polynomials.
From Figure 3(b), the plots barely obey power law.Table 4(b) shows that the correlations decrease with the increase in RH, but increase with the increase in the power of the polynomials.
Comparing Figures 3(c) and 2(c) it can be observed that they are similar.
Table 4(c) shows very good correlations, and the correlations increases with the increase in the power of the polynomials.
From Table 4(d) it can be seen that the coefficients are approximately the same to one place of decimal and some to two places of decimals.
Figure 3(d) shows that power law is not obeyed.Table 4(e) shows that there are poor correlations in the linear part, but the correlation improves with the increase in the power of the polynomials.
Comparing the coefficients of Tables 4(e) and (f) it can be observed that they are all approximately the same within two places of decimals, while some to four places of decimals.Table 4(g) shows that there are poor correlations in the linear part, but the correlation improves with the increase in the power of the polynomials.

Table 2(g).
The results of the Angstrom coefficients of single scattering co-albedo using Equations ( 5), ( 6) and (10) for sahara model at the respective relative humidities using regression analysis with SPSS16.0.

Table 3(a)
. The results of the Angstrom coefficients of scattering coefficients using Equations ( 5), ( 6) and (10) for urban model at the respective relative humidities using regression analysis with SPSS16.0.

Table 3(c).
The results of the Angstrom coefficients of extinction coefficients using Equations ( 5), ( 6) and (10) for urban model at the respective relative humidities using regression analysis with SPSS16.0.

Table 3(e).
The results of the Angstrom coefficients of single scattering albedo using Equations ( 5), ( 6) and (10) for urban model at the respective relative humidities using regression analysis with SPSS16.0.

Table 3(g).
The results of the Angstrom coefficients of single scattering co-albedo using Equations ( 5), ( 6) and (10) for urban model at the respective relative humidities using regression analysis with SPSS16.0.

Table 4(a)
. The results of the Angstrom coefficients of scattering coefficients using Equations ( 5), ( 6) and ( 10) for continental clean model at the respective relative humidities using regression analysis with SPSS16.0.

Table 4(b)
. The results of the Angstrom coefficients of absorption coefficients using Equations ( 5), ( 6) and ( 10) for continental clean model at the respective relative humidities using regression analysis with SPSS16.0.

Table 4(c).
The results of the Angstrom coefficients of extinction coefficients using Equations ( 5), ( 6) and ( 10) for continental clean model at the respective relative humidities using regression analysis with SPSS16.0.

Table 4(g).
The results of the Angstrom coefficients of single scattering co-albedo using Equations ( 5), ( 6) and (10) for continental clean model at the respective relative humidities using regression analysis with SPSS16.0.
RH Linear Equation ( 5) Quadratic Equation ( 6) Cubic Equation ( 10)  can be observed that they are approximately the same within two places of decimals, with some to three places of decimals.Figure 4(a) shows that power law decreases with the increase in RHs.
From Table 5(a), it can be observe that the correlations decrease with the increase in RHs, but increases with the increase in the power of the polynomials.
From Figure 4(b) it can be observed that power law is not obeyed.
Table 5(b) shows poor correlation in the linear part, but good correlations at second and third order polynomials, and the correlations increase with the increase in order of the polynomials and RHs.
Comparing Figures 4(c) and (a), it can be observed that they are similar.
From Table 5(c), it can be seen that the correlations decrease with the increase in RHs, but increases with the increase in the power of the polynomials.
From Table 5(d), from the linear part it can be seen that Equation (1) underestimated Equation ( 7) at 0% to 70% RH, and overestimated it at 90% to 99% RH.At the quadratic part Equations ( 1) and ( 9) are equal within two places of decimals, but at RHs 95% to 99% Equation (1) underestimated Equation (9).At the cubic part, Equations (1) and ( 12) are the same within two places of decimals except at 98% and 99% where Equation (1) underestimated Equation (2).
Figure 4(d) shows that power law is not obeyed.
From Table 5(e) it can be seen from the linear part that there is a poor correlation between single scattering albedo and wavelength, though as the power of the polynomials increase the relation also improves.
Comparing Tables 5(e) and (f) it can be observed that in the linear part only the values at 80%, 95% and 99% agree to three places of decimals.This can be attributed to the poor correlations at Table 5(e).From the quadratic and cubic some coefficients agree to two places while some to three places of decimals.5), ( 6) and ( 10) for maritime clean model at the respective relative humidities using regression analysis with SPSS16.0.
RH Linear Quadratic Cubic (%) Equation ( 7) Equation ( 1) Equation ( 9) Equation ( 1) Equation ( 12) Equation ( 1)   Comparing Tables 5(g) and (h) it can be seen that from the linear part they are different, and this can be attributed to the poor correlations in 5(g).From the quadratic and cubic, the coefficients from Tables 5(g) and (h)

Conclusions
From all the tables and graphs obtained, it can be seen that it is only urban aerosols that scattering, absorption and extinction coefficients that satisfy power laws excellently at this spectral range and can be seen that it is the only aerosols that show very good relations between the estimated Equations ( 1)-( 3) and the linear Equation ( 5).Linear models are considered most important, because they are the values that are obtainable from remote sensing and ground truthing instruments.Another observation made is that Equations (1) and (2) are sufficient.This is because the information that can be obtained from equation is almost the same as that of Equation (3).
Additional important observation made from the various graphs is that, at the spectral range of 0.4 to 1.5 power laws are obeyed by all the aerosols.Therefore, since 99% of sun's radiation falls between 0.2 -5.6 μm; and 80% falls between 0.4 -1.5 μm (visible and near infrared) and the atmosphere is quite transparent to incoming solar radiation with the maximum radiation at 0.48 μm (visible) and in the study of the earth's surface, most of the remote sensing instruments are designed to operate within solar spectral window (0.4 -0.7 μm) and near infrared (0.7 -1.5 μm), where cloudless atmosphere will transmit sufficient radiation for detection, which also shows that these formulas can be useful in remote sensing.

Figure 2 (
e) is almost the inverse of Figure 2(d).

Figure 3 (
e), is the inverse of Figure3(d) and the power law is not obeyed.

Figure 4 (
e) in the inverse of Figure 4(d).

Table 5 (
g) shows poor correlations in the linear part.There are good correlations at quadratic and cubic polynomials, though poor correlation can be observed at 99% RH throughout.