Introduction

Brown carbon (BrC) aerosols, a subset of organic carbonaceous particulates exhibiting significant light absorption in the near-ultraviolet (300–400 nm) and visible light ranges1,2,3,4, can exacerbate global warming5,6,7. Absorption by BrC also diminishes atmospheric photolysis rates for key radicals8,9, influencing the secondary organic aerosol formation and air quality10,11. However, accurate quantification of BrC’s radiative effects remains hindered by insufficient characterization, particularly regarding the size-dependent properties.

Size distribution is one of the most important but poorly understood properties of BrC, and plays a crucial role in regulating radiative budgets by determining the aerosol optical depth (AOD), which measures the light extinction in the atmospheric column due to both scattering and absorption12,13. Understanding the accurate size distribution across an extensive size range can help reduce uncertainties in estimating AOD12. Furthermore, the size distribution of BrC determines their contribution to cloud droplet activation. Incorporating BrC into cloud droplets may facilitate cloud evaporation via light absorption, adding an additional uncertainty to the estimation of aerosol-cloud interactions14. However, our understanding of the size distribution remains limited due to the challenges in accurately measuring the size-resolved light absorption of BrC. The size-dependent BrC light absorption properties can be obtained by using cascade impactors15,16,17 or satellite remote sensing18,19, but they suffer from poor temporal (mostly 24 h) or size resolutions (mostly maximum 13 or 22 segments for the cascade impactor or satellite remote sensing, respectively). Additionally, satellite remote sensing has high uncertainty (30%–50%) when the BrC fraction is less than 10%, leading to a less precise description of BrC13,20.

Direct in-situ measurements of BrC light absorption as a function of size remain scarce due to limitations of current instrumentation in detecting BrC optical properties at high time and size resolutions. Some previous studies have provided valuable insights into the BrC size distributions. For example, Liu et al. 15 utilized cascade impactors to collect size-resolved aerosol samples for offline analysis, revealing that BrC chromophores predominantly exist in the accumulation mode with a mean aerodynamic diameter of 0.5 μm. Di Lorenzo et al. 16 collected size-resolved aerosols (10–18 μm) from aged wildfire plumes and found that BrC absorption was primarily in the fine-mode aerosols with molecular composition varying across different size fractions, showing distinct distribution patterns compared to the traditional biomass burning markers. Wu et al. 17. analyzed size-resolved particles in Xi’an, China, demonstrating that water-extracted BrC predominantly occurred in the fine mode (<2.1 μm), accounting for 81% and 65% of total BrC absorption in winter and summer, respectively. These studies provide crucial understanding of the size-dependent properties of BrC, however, they relied on offline filter-based methods with limited size resolution, primarily yielding mass-based size distributions rather than number-based distributions. Therefore, new techniques are needed to obtain higher time- and size-resolved measurements of BrC number concentration and light absorption.

In this study, we introduce a novel positive matrix factorization (PMF) approach constrained by light absorption and BrC marker fragment to derive the size distribution of the BrC number concentration and light absorption at high time and size resolutions. Our analytical framework primarily relies on optical measurements, using BrC absorption data from an aethalometer for PMF&ME-2 constrained analysis to derive the number concentration size distribution of BrC particles. As complementary validation, we applied the same analytical framework to the time series of 4-nitrocatechol (4NC) fragments (C6H5NO4+) measured by aerosol mass spectrometry. Through the application of this methodology, we obtain the number size distribution and the light absorption size distribution of sub-500 nm brown carbon aerosols.

Results

Determination of BrC size factor

In this study, we employed the following workflow (as shown in Fig. 1) to decompose and identify the size distribution of BrC using PMF and Multilinear Engine version 2 (ME-2): Scanning mobility particle sizer (SMPS) data was averaged hourly to create the original matrix X, with rows (j) representing the time series and columns (i) representing the particle size distribution. PMF was applied using ME-2 with time series constraints from absorption coefficient of brown carbon (babs,BrC) measured by an aethalometer and C6H5NO4+ from synchronous aerosol mass spectrometer (AMS) observations. babs,BrC represents changes in the BrC light absorption and was used to constrain particles related to BrC in the SMPS data. C6H5NO4+ has been confirmed in AMS studies as a characteristic fragment of nitrophenolic compounds such as 4-nitrocatechol (4NC)21. These compounds are recognized as important BrC components that contribute significantly to BrC light absorption in the near-UV and visible range22,23,24. We used this fragment to verify the reliability of the absorptivity constrained PMF results. PMF solutions with 2 to 8 factors were evaluated (Figs. S5 and S6) based on criteria explained in the methods section. Briefly, 6 factors were selected as the optimal solution based on the inflection point in Q/Qexp ratio (close to ideal value of 1), clear factor separation, and strongest correlations with the external tracers. The factor exhibiting the highest explained variance of external data and strongest correlation to the babs,BrC or C6H5NO4+ time series was identified as the candidate BrC size distribution factor. Using this constrained PMF approach, we identified a robust BrC size distribution factor as detailed in the following sections.

Fig. 1: Schematic of the workflow. SMPS number concentration data (original matrix X) is decomposed by PMF into factor number size distribution (F) and time series (G) matrices.
figure 1

Constraints from babs,BrC and C6H5NO4+ time series are applied to obtain the BrC number concentration size distribution factor.

Full size image

Figure 2 shows the 6 factors resolved by the constrained PMF analysis and their number concentration size distribution characteristics. The left side (Fig. 2a–f) presents the babs,BrC constrained results, while the right side shows the C6H5NO4+ constrained results. The patterns are similar between the two constraint methods.

Fig. 2: 6-factor PMF&ME2-resolved size distribution of BrC number concentration.
figure 2

Constraints applied: (a–f) aethalometer-derived babs,BrC and (g–l) AMS-derived C6H5NO4+ signal.

Full size image

Factors 1 and 2 exhibit size distributions similar to those reported for the traffic and mixed emissions in previous studies25,26. Further correlation analysis with chemical components revealed that Factor 1 correlates best with organic aerosol (R = 0.64; Fig. S7a, g). Factor 2 shows the highest correlation with nitrate (R = 0.71; Fig. S7b, h), while also maintaining a moderate correlation with organic aerosol (R = 0.60). Given that the light-absorbing components of organic aerosols were constrained to Factor 4 using the external time series, Factors 1 and 2 likely represent non-absorbing organic species and a mixture of non-absorbing organic species and nitrate (as evidenced by the improved correlation of Factor 2 with the sum of organic aerosol and nitrate, R increasing to 0.78, see Fig. S8), respectively. Factor 3 associates with sulfate (R = 0.70; Fig. S7c, i) and resembles the residential heating profiles26,27. Factor 4 represents the BrC factor and will be discussed in detail in the next section. Factor 5 represents sea spray (R = 0.74, Fig. S7d, k), with size distribution consistent with previous studies28,29. Factor 6 is the nucleation factor identified in previous studies25,27,30. The wind rose plot (Fig. S9f) shows that this factor reaches its highest concentration under low-speed (3–7 m/s) offshore winds (~135° direction), indicating the significant influence of land-sourced (mostly coastal seaweed beds during low tide) precursors on new particle formation at this coastal site. Under high-speed (>15 m/s) onshore wind conditions (180–210°), this factor also exhibits elevated concentrations, though slightly lower than those observed during offshore conditions.

The time series correlations (Fig. S7) and meteorological polar plots (Fig. S9) validate the factor determinations. Factor 5 derives from high offshore winds while Factors 1–4 originate onshore, confirming their gas- and particle-phase source differences. This supports the identification of Factor 4 as a robust BrC factor using the external constraints.

Number concentration size distribution of the BrC factor

Figure 3 shows the relative signal intensity and explained variance of factor 4 constrained by two external time series (Fig. 3a: babs,BrC and Fig. 3e: C6H5NO4+), as well as their corresponding number concentration size distribution (Fig. 3b, f), scatter plots (Fig. 3c, g), and time series results (Fig. 3d, h). Factor 4 accounts for over 99% of explained variance in babs,BrC and C6H5NO4+, confirming the resolved BrC factors. Additionally, the time series of the BrC factor is also highly correlated with the time series of babs,BrC (R = 0.93), and C6H5NO4+ (R = 0.99), providing further evidence for the determination of this factor.

Fig. 3: The BrC number concentration size distribution resolved by the PMF&ME-2 method.
figure 3

a, e are the results of the babs,BrC and C6H5NO4+ dominated factors among the 6 factors. b, f Are the corresponding number concentration size distribution; the right axis represents the explained variance of specific component of the factor. c, g Aare the numerical concentration of the BrC factor and the correlation between babs,BrC and C6H5NO4+, respectively, in addition to (d, h) their time series. Black vertical line in (b, f) represents the standard deviations of averaged iterative PMF&ME-2 result.

Full size image

The number concentration size distribution of the BrC factor shows a bimodal pattern, with peaks in both nucleation and accumulation modes. A higher peak appears in the nucleation mode at 20 nm, accounting for 62.6% (absorptivity constrained results, similar in the mass constrained results) of the total BrC particle number (from 20 nm to 30 nm). This size distribution correlates well with the nucleation factor (R = 0.62), suggesting BrC formation during new particle formation processes31. Previous chemical composition studies at MHD provide valuable insights into potential BrC sources. Lawler et al. 32 conducted thermal desorption chemical ionization mass spectrometry (TD-CIMS) to characterize the molecular composition of particles within the 15–85 nm diameter range at MHD. Their analysis revealed the presence of C7H7O2+ ions, indicative of benzoic acid. This aromatic compound exhibits significant absorption cross-sections in the near-UV spectral region33,34,35, suggesting its potential role as a primary chromophore contributing to the BrC optical signature observed in the nucleation and Aitken mode. Their investigation also identified the non-aromatic constituents, including nonanoic acid (C9H19O2+), formic acid (CH3O2+), and hexanoic acid (C6H13O2+), which predominantly absorb below 220 nm. These species thus likely contribute negligibly to near-UV BrC absorption. The second peak centers at 107 nm with a relative intensity weaker than the first peak, accounting for 31.0% (absorptivity constrained results, similar in the mass constrained results) of the total BrC number (from 77 nm to 143 nm), corresponding to the accumulation mode distribution. This finding aligns with previous studies that have reported BrC in the accumulation mode, often associated with secondary organic aerosol formation and aging processes15,36.

In addition, by comparing the results of the babs,BrC, and C6H5NO4+ constrained methods, we observed that the patterns of their number concentration size distribution are remarkably similar and their time series are also well correlated (R = 0.63, Fig. S10). The similarity between the BrC size distribution obtained from mass-constrained and light absorption-constrained PMF has led further confidence of robustness of this approach, especially considering that BrC and C6H5NO4+ were measured by different instruments based on different operational principles, indicating the validity of both constrained approaches.

Size-resolved light absorption of the BrC factor

We further calculated the size-resolved babs,BrC by converting the number concentration size distribution to mass size distribution and then multiplying it by the appropriate mass absorption efficiencies (MAE) of BrC at 370 nm. Recognizing that chemical composition likely varies across different particle sizes, we adopted a range-based approach to estimate the possible light absorption size distributions. Two boundary conditions were applied to estimate the possible light absorption size distributions: (1) assuming all particles have properties similar to primary emissions from biomass burning (MAE365 nm = 2.47 m2 g−1, representing an upper limit) and (2) assuming all particles have properties similar to clean marine environment (MAE365 nm = 0.98 m2 g−1, representing a lower limit)37. The results are shown in Fig. 4a, b, where the shaded areas represent the range between these upper and lower estimations, with the solid line indicating the average value.

Fig. 4: Size-resolved light absorption characteristics of BrC.
figure 4

a Absorption coefficient constrained by babs,BrC; (b) absorption coefficient constrained by C6H5NO4+; (c) direct solar absorption of BrC relative to BC (370–880 nm wavelength) in different size bins obtained by estimation. The shaded area represents the range of higher MAE results and lower MAE results.

Full size image

The size-resolved babs,BrC revealed a bimodal pattern of light absorption within the 20–484 nm range. Despite the presence of a large number of particles from 20 nm to 30 nm, their small mass results in negligible contribution to light absorption (0.006 ± 0.003 M m–1), accounting for only 0.03% of the total BrC light absorption in the 20–484 nm range. One of the peaks occurs at 137 nm with babs,BrC of 0.5 ± 0.2 M m–1 (babs,BrC constrained) and 0.6 ± 0.3 M m–1 (C6H5NO4+ constrained), representing around 3.2–3.3% of the total BrC particle number (from 127 nm to 147 nm), demonstrating light absorption by BrC in the accumulation mode, accounting for approximately 36–40% of the total BrC light absorption. The other distribution shows increasing values of babs,BrC starting from 211 nm (0.04 ± 0.02 M m–1 for babs,BrC constrained and 0.05 ± 0.02 M m–1 for C6H5NO4+ constrained) and continuing to rise rapidly until reaching its maximum value at the largest observable particle size of 484 nm (1.5 ± 0.6 M m–1 for babs,BrC constrained and 1.7 ± 0.9 M m−1 for C6H5NO4+ constrained), representing around 3.2–3.6% of the total BrC particle number (from 211 nm to 484 nm), accounting for approximately 57–70% of the total BrC light absorption. It is worth noting that while the specific magnitude of light absorption may vary within the shaded areas depending on the actual chemical composition at different particle sizes, our range-based approach using minimum and maximum MAE values provides reasonable boundaries for the likely absorption characteristics across the entire size spectrum.

Also, we calculated the size-resolved direct solar absorption of BrC relative to BC across the wavelength range of 370–880 nm wavelength, as shown in Fig. 4(c). The change in the ratio of the direct solar absorption of BrC relative to BC is not pronounced in the 20–484 nm size range, but a slight increase can still be observed in the size range above 100 nm. In the results with higher MAE, the ratio of the direct solar absorption of BrC relative to BC increased from 4.4% to 4.8%. In the results with lower MAE, the ratio increased from 1.7% to 1.8%. The value of MAE has a relatively large impact on the ratio of the direct solar absorption of BrC relative to BC, but the overall ratio (1.7%–4.8%) has a small impact on solar absorption.

Discussion

This study demonstrates a novel PMF-constrained methodology to derive size-dependent properties of BrC. The approach provides new insights into the number concentration and light absorptivity across a wide sub-500 nm size range. It should be noted that the size distribution of BrC was assumed to be identical throughout the analysed period, which may not always be the case. BrC is a collective term for a subset of light-absorbing organic substances. As such, its particle size distribution is expected to vary depending on the sources, chemical compositions, and atmospheric processes.

The BrC size distribution obtained in this study should be considered as a weighted average related to the atmospheric transport pattern and sources around the MHD research station during the measurement period. We consider this assumption is reasonable for several reasons. First, using a long-term average (~4.5 months of data) inherently smooths out short-term fluctuations, capturing the dominant size distribution patterns that persist throughout the study period. Second, our analysis of meteorological patterns during this period indicates relatively consistent source regions affecting our measurement site, suggesting some stability in the aerosol population characteristics.

In reality, factors such as changes in biomass burning, secondary organic aerosol formation, and marine organic emissions could alter the size distribution of BrC over time. We acknowledge that in environments with highly variable or multiple distinct BrC sources, this assumption would require careful reconsideration. For instance, locations with strong seasonal variations in biomass burning activities, or urban areas with complex mixtures of traffic, industrial, and residential combustion sources, might exhibit more pronounced temporal variations in the BrC size distributions.

Additionally, aging processes such as photochemical bleaching or browning of organic aerosols can alter BrC optical properties over time. Our approach inherently incorporates these effects as they manifest in the measured optical signals but does not explicitly separate fresh from aged BrC. The derived size distributions therefore represent the ambient BrC population in various stages of atmospheric processing rather than source-specific or age-specific distributions.

Further research is needed to characterize how the BrC size distributions fluctuate under different source influences and atmospheric aging processes. Future studies could address this limitation by implementing time-dependent constraints or by conducting source-specific analyses when appropriate source markers are available.

Despite limitations in the consistent size distribution assumption, the insights gained can help reduce uncertainties in estimating BrC radiative impacts for climate models. The consistency between our optically constrained and chemically constrained results provides confidence in the overall approach, while acknowledging the simplifications necessary in this initial application. While further instrumentation could validate the absorption contributions across fine sizes, this practical methodology opens up new possibilities to investigate relationships between size and optical characteristics of BrC in diverse environments.

Methods

Measurement site and instrumentation

The measurements were carried out at the Mace Head Atmospheric research station (MHD), which is located on the West coast of Ireland (www.macehead.org). The station is situated at the interface of land and ocean, where winds from a direction between 180 and 300° reach it without passing over any inhabited area, while the rest of the surrounding rural area is sparsely populated38. The data was collected from 10th Aug. to 31st Dec. in 2018. A scanning mobility particles sizer (SMPS; DMA 3080 and CPC 3010, TSI Incorporated, Shoreview, MN, USA) was used to measure aerosol number concentration size distribution ranging from 20 to 484 nm. In recent years, many studies have combined PMF with SMPS data to distinguish the size distribution of particles from different sources25,26,27,30,39,40,41,42. The BC and BrC concentration were measured by an aethalometer (AE33, Magee Scientific, Berkeley, CA, USA)43. The concentration of the C6H5NO4+ (Fig. S1), a fragment indicative of brown carbon species such as 4-nitrocatechol (4NC)21, was measured using a high-resolution time-of-flight aerosol mass spectrometer (HR-ToF-AMS, Aerodyne Research Inc., Billerica, MA, USA)44. Details and validation of the instrument can be found in our previous reports28,45. An overview of the data is presented in Figs. S2–S4. To facilitate consistent analysis and comparison across different measurements, all data were averaged to a common 1-h resolution.

PMF&ME-2

In this study, PMF and ME-2 were used to obtain the size-resolved number concentration and light absorption coefficient of BrC. Essentially, PMF is a bilinear receptor model capable of separating the input data matrix into distinct factor matrices and their corresponding time series of concentrations (weight matrices)46,47. It is described as follows:

$$X={GF}+E\,$$
(1)

where X represents the original matrix (i×j) of the number concentration (i) and size distribution (j) of aerosols measured by SMPS. F and G denote the factor matrix (p×j) of the size distribution (j) of aerosols under different factor numbers (p) and the weight matrix (i×p) of the time series of number concentration (i) for each factor, respectively. E indicates the residual between the reconstructed matrix, calculated by multiplying F and G, and the original matrix X. To reduce the residual, the weighted sum of squared residuals was introduced in the calculations. By using least squares to iteratively minimize the sum of squared residuals in the receptor model computation, the PMF results are optimized. It is described as follows:

$$Q=\mathop{\sum }\limits_{i=1}^{n}\mathop{\sum }\limits_{j=1}^{m}{\left(\frac{{e}_{{ij}}}{{\sigma }_{{ij}}}\right)}^{2}$$
(2)

where Q represents the sum of squared residuals after weighting, which is the ratio of the residual of the jth species (eij) in the ith hour in the matrix to its standard deviation (σij). PMF has been widely applied in the studies of particle size distribution25,27,39,40.

The multilinear engine method (ME-2) enables the incorporation of known external information to constrain PMF solutions48. In this study, we applied ME-2 constraining using the time series data of babs,BrC, and C6H5NO4+ in conjunction with SMPS as an external weight matrix to conduct ME-2 calculations on the results as follows:

$${g}_{i,{solution}}={g}_{i}\pm a{g}_{i}$$
(3)

where \({g}_{i,{solution}}\) represents the weight matrix G obtained by solving the ME-2 algorithm. \({g}_{i}\) denotes the introduced external weight matrix. This ensures the resolved the BrC factor corresponds to the external data, with the SMPS number concentration data entering each factor freely via unconstrained PMF.

In practice, we applied full constraints exclusively to the BrC factor by setting the a value to zero. This approach forced the BrC factor’s time series (G matrix) to match the external data exactly, while allowing all other factors to remain completely unconstrained. To account for potential variability, we ran the ME-2 algorithm 100 times at the a value of zero. Each run yielded a number concentration size distribution (F matrix) for the tightly constrained the BrC factor. The final reported BrC size distribution is the average of these 100 runs, providing a robust solution despite the complete constraints. This tailored application of ME-2 builds on previous work utilizing constrained PMF for aerosol source apportionment49,50. All calculations related to ME-2 were performed on SourceFinder (6.3en, Datalystica, Ltd, Park InnovAARE, Villigen, Switzerland)51 embedded in Igor Pro software (v6.37, WaveMetrics, Inc., Lake Oswego, OR, USA).

Error matrix

The uncertainty of size-resolved number concentration and external data was estimated by the method provided by the United States Environmental Protection Agency (USEPA) PMF 5.0 fundamentals and user guide as follows52:

$${Unc}=\frac{5}{6}\times {MDL}$$
(4)

where MDL represents the method detection limit of the instrument.

Factor number determination

The PMF model quality is evaluated using the ratio Q/Qexp. Q represents the sum of squared residuals weighted by uncertainty. Qexp corresponds to the expected Q value based on degrees of freedom in the data53,54.

Three criteria were sequentially followed to determine the optimal number of factors:

  1. I.

    An inflection point in the plot of Q/Qexp versus number of factors indicates the minimum number of factors needed to reproduce the data. Adding more factors beyond this point yields diminishing improvements in Q/Qexp55.

  2. II.

    A Q/Qexp value close to 1 suggests the model reproduces the true variability in the data based on the input error estimates56.

  3. III.

    The solution should minimize mixing or splitting of factors. Additionally, the BrC factor time series should correlate well with the external constraints.

The Q/Qexp values (Figs. S5d and S6d) exhibit an inflection point at 6 factors in both constrained analyses. From 5 to 7 factors, Q/Qexp ranges from 0.5–1.1 (babs,BrC constrained) and 0.6–2.4 (C6H5NO4+ constrained), consistent with the ideal value of 1.

Since the Q/Qexp ratio exhibited an inflection point at 6 factors, solutions with 5–7 factors were further evaluated. The particle size distribution of each factor and correlation of the BrC factor time series with external data were compared.

For 5 factors (Figs. S5a, S6a), no distinct BrC factor emerged. The sea spary distribution showed two peaks at 20–22 nm and 80–100 nm, indicating mixing with BrC. The external data correlation was lower (R = 0.39 for babs,BrC, R = 0.37 for C6H5NO4+) than for 6 or 7 factors.

With 7 factors (Figs. S5c, S6c), the results became unstable. The sea spray factor shape differed from literature reports29. For babs,BrC constrained results, factors 1 and 2 showed similar distributions, potentially splitting a single factor. However, the BrC factor time series (factor 5) was highly correlated (R = 0.99 for babs,BrC, R = 0.94 for C6H5NO4+).

Considering Q/Qexp values and factor characteristics, 6 factors were deemed optimal.

BrC light absorption coefficient

The light absorption Ångström Exponent (AAE) is used to characterize the wavelength dependence of aerosol optical absorption coefficient (babs(\({\rm{\lambda }}\))), and has been widely used in previous studies57,58,59,60. It describes as follows12:

$${b}_{{abs}}\left({\rm{\lambda }}\right) \sim {{\rm{\lambda }}}^{-{\rm{AAE}}}$$
(5)

At the Mace Head, impact of dust on the light absorption of aerosols is negligible, then babs(\({\rm{\lambda }}\)) can be divided into light absorption by BC (babs, BC(\({\rm{\lambda }}\))) and BrC (babs,BrC(\({\rm{\lambda }}\)))59. To determine the light absorption of BrC, it is necessary to obtain the light absorption of babs, BC at different wavelengths. Here, we assume that BC is the only absorbing substance at a wavelength of 880 nm61, then babs, BC at other wavelengths can be calculated as follows62:

$${b}_{{abs},{BC}}({\rm{\lambda }})={b}_{{abs}}(880)\times ({\frac{{\rm{\lambda }}}{880})}^{{-{\rm{AAE}}}_{{\rm{BC}}}}$$
(6)

where AAEBC represents the BC spectral dependence, and a value of 1.1 was utilized based on prior research59,63. Then, babs,BrC can be calculated as follows:

$${b}_{{abs},{BrC}}({\rm{\lambda }})={b}_{{abs}}({\rm{\lambda }})-{{\rm{b}}}_{{abs}}(880)\times ({\frac{{\rm{\lambda }}}{880})}^{{-{\rm{AAE}}}_{{\rm{BC}}}}$$
(7)

The maximum light absorption is observed at a wavelength of 370 nm in aethalometer. Consequently, in this study, light absorption of BrC at 370 nm (babs,BrC(370)) is used interchangeably with babs,BrC.

The direct solar absorption of BrC relative to BC was calculated using the method similar to Kirchstetter and Thatcher64 and Kirillova et al. 65, and described in detail in Huang et al. 66. It describes as follows:

$$f=\frac{\int {I}_{0}\left(\lambda \right)\left\{1-{e}^{{-b}_{{abs},{BrC},\lambda }{\rm{\cdot }}{h}_{{ABL}}}\right\}d\lambda }{\int {I}_{0}\left(\lambda \right)\left\{1-{e}^{{-b}_{{abs},{BC},\lambda }{\rm{\cdot }}{h}_{{ABL}}}\right\}d\lambda }$$
(8)

where \({I}_{0}\left(\lambda \right)\) is the clear sky Air Mass 1 Global Horizontal solar irradiance estimated by Levinson et al. 67\({b}_{{abs},{BrC},\lambda }\) and \({b}_{{abs},{BC},\lambda }\) for BrC and BC denote the light absorption coefficient derived from the analysis described above. \({h}_{{ABL}}\) represents to the boundary layer height (assuming 1000 m in this study).