ORACLES was a 3-year NASA-funded airborne field campaign to investigate the influence of BB emissions from southern Africa on regional and global climate (Redemann et al., 2021). We investigated aerosol optical properties from straight and level runs during seven research flights (RFs) from the ORACLES 2018 campaign (Fig. 1): RF05_1, RF05_2, RF05_3, RF06_1, RF06_2, RF10, and RF11. The specifications for each flight can be found in Table 1 in Dang et al. (2021). These flights were chosen as they provided investigations of aerosol properties in the marine boundary layer (MBL) and free troposphere (FT), from relatively aged aerosols to highly aged aerosols, and are comprehensive in online and offline measurements with few missing data. RF05_1 and RF05_2 were within the MBL at an altitude of ∼940 and 380 m, respectively; other flights were in the aerosol layer in the FT at an altitude equal to or greater than 2 km. The aerosol/plume age, modelled with a 2-week forecast using the Weather Research and Aerosol Aware Microphysics (WRF-AAM) model (Thompson and Eidhammer, 2014), was estimated to be ∼11 d for RF05_1 and RF05_2 and ∼7 d for RF05_3, RF06_1, RF06_2, RF10, and RF11 (Fig. S1 in the Supplement).

The mass and number concentrations of the refractory BC particles and the single-particle mass and mixing state were determined by the Single-Particle Soot Photometer (SP2; Droplet Measurement Technologies). Coating thicknesses of refractory-black-carbon-containing particles were derived from scattering measurements using the leading-edge-only (LEO) method of Gao et al. (2007). The SP2 detects BC particles with core diameters from 80–650 nm; a log-normal function was fit to BC number size distribution to account for BC particles with core diameters outside this range (Schwarz et al., 2006). The mass equivalent diameter of BC was calculated as (6m/ρBCπ)1/3, where m and ρBC represent the mass and density of BC, respectively, with ρBC assumed to be 1.8 g cm-3 (Bond and Bergstrom, 2006). The uncertainty of the measured BC mass concentration and coating thicknesses is estimated to be 20 % and 22 %, respectively (Laborde et al., 2012). The mass ratio (MR) of non-BC substance to BC for BC-containing particles was determined by MR=∑i(Dp,i3-Dc,i3)×ρBC-free∑iDc,i3∗ρBC, where Dp and Dc represent the diameter of coated BC particle and BC core, respectively; the i denotes the ith particle in the investigated time window. The determination of ρBC-free will be discussed in Sect. 2.2.

A particle soot absorption photometer (PSAP; Radiance Research) was used to determine 1 Hz aerosol absorption coefficients at wavelengths 470, 530, and 660 nm, which were corrected with the wavelength-averaged Virkkula correction (Virkkula, 2010) and smoothed to 10 s to reduce noise. The uncertainty of absorption coefficients is estimated to be 20 % (Fischer et al., 2010).

A high-resolution time-of-flight aerosol mass spectrometer (HR-ToF-AMS, Aerodyne Research) measured mass concentrations of sulfate (SO42-), nitrate (NO3-), ammonium (NH4+), chloride (Cl-), and OA from particles with vacuum aerodynamic diameters between 50 and 500 nm.

We determine single-particle mixing state and elemental composition from offline analysis of particles sampled on Paella TEM grids. Around 50 particles per filter were analysed. A JEOL™ JEM-2010F FEG-TEM with a ThermoNoran™ EDX detector was used. TEM was performed at 200 KeV accelerating voltage, with a take-off angle of 15.9∘ for X-ray emission from the sample. The elemental weight percentage per particle was determined using the NSS software (Thermo Electron Corporation) with the Cliff–Lorimer absorbance correction method. Detailed descriptions can be found in Dang et al. (2021).

All in situ instruments were mounted in the NASA P-3 aircraft and operated under dry conditions. Data for BC mass concentration less than 0.1 µg m-3 or absorption coefficients at 660 nm less than 1.5 Mm-1 were not included. Measurements were averaged to 10 s and adjusted to STP values at 273.15 K and 1013 hPa.

Most of the particles were found to be nearly spherical from the images obtained by TEM, with >70% of the particles having aspect ratios (the largest and smallest characteristic sizes of arbitrarily shaped particle) smaller than 1.5. Therefore, we applied Mie theory to determine aerosol optical properties. Models such as T-matrix or the discrete dipole approximation model were not used as they would require a large number of free parameters (Scarnato et al., 2013; He et al., 2016). We investigated the sensitivity of our closure simulations to four different models for the BC-containing particles – the ideal core–shell (CS) model and three homogeneous mixing models: (1) the volume mixing (VM) model, (2) the Maxwell–Garnett (MG) model, and (3) the Bruggeman (BG) model. The number size distribution of BC-containing particles was obtained from the BC core size distribution and the BC 2-D size and mixing state (i.e. coating thickness) distribution from SP2. The open-source PyMieScatt code was used in this study (Sumlin et al., 2018). Detailed descriptions and inputs of the four models can be found in Sect. S1 in the Supplement. We assumed that the non-BC components were homogeneously mixed and calculated the refractive index of the mixture, mBC-free, with the VM rule. The mass concentrations of SO42-, NO3-, NH4+, and Cl- measured by AMS were converted to those of inorganic salts to facilitate calculation, using the simplified ion-pairing scheme proposed by Gysel et al. (2007), modified as follows: 1nNH4Cl=nCl-2nNH4NO3=min(nNO3-nNH4+-nCl-)3nNH4HSO4=max(0,min(2nSO42--nNH4++nNO3-+nCl-,nNH4+-nNO3--nCl-))4n(NH4)2SO4=max(0,min(nSO42-,nNH4+-nNO3--nCl--nSO42-))5nKNO3=max(0,min(nNO3-,nNH4+-nCl--nNO3--2nSO42-))6nK2SO4=max(0,min(nSO42-,nSO42--nNH4++nNO3-+nCl-)), where n represents the number of moles. As potassium salts are the most frequently detected salts from TEM-EDX analysis (Dang et al., 2021), the residual anions were assumed to be combined with potassium, i.e. KNO3 and K2SO4; however, a small fraction of NO3- and SO42- detected by the AMS might be organics and thus lack an accompanying cation. Densities of the various salts and OA used to convert the mass concentrations to volume concentrations were taken from Kuang et al. (2020; Table 1) and Liu et al. (2015). The mBC-free is calculated as 7mBC-free=∑i=saltsmi×Vi+mOA×VOA/∑i=saltsVi+VOA, where mi and Vi represent the refractive indices and volumes of aforementioned salts; mOA and VOA are those for OA. The refractive indices of salts are taken from Fierz-Schmidhauser et al. (2010) and Cotterell et al. (2017). The real part of the refractive index of OA varies from 1.35 to 1.7 (Lack and Cappa, 2010; Liu et al., 2013; Saleh et al., 2014; Moise et al., 2015); 1.55 is used in this study. OA is assumed to be non-absorbing in the calculation. The average and standard deviation of calculated mBC-free is 1.52±0.015, consistent with the values used in BrC studies in literature (Saleh et al., 2013; Liu et al., 2021, 2015). The particle effective refractive indices for the VM, MG, and BG models can then be obtained using the corresponding mixing rules (Sect. S1) with mBC-free and mBC. The determination of the refractive index of BC will be discussed in the following section.

The numerical value selected for mBC has a large impact on the absorption attributed to BC (Fig. 3) and hence on the estimated absorption by BrC, but as noted above, there is no consensus on the best value of mBC. Additionally, the lack of measurements of possible absorbers other than BC and BrC such as magnetite makes it difficult to correctly estimate the BrC absorption, so approaches that take these components into consideration are required. Here, we introduce an effective refractive index of BC, meBC=1.95+ikeBC, to represent the refractive index of BC and any other absorbing components at 660 nm in this study. The meBC would be the same as mBC if the absorption at 660 nm is solely contributed by BC; however, if other absorbing components are present, the imaginary part (keBC) of meBC would be greater than that (kBC) of mBC, the amount of which depends on the absorptivity and relative amount of these absorbers. The real part ofmeBC is held constant at 1.95, the upper bound of the values commonly used for BC (Bond and Bergstrom, 2006; Saleh et al., 2013; Liu et al., 2015; Kahnert and Kanngießer, 2020). It only has a minor influence on absorption calculations (Liu et al., 2021): a sensitivity test of the calculated absorption to the real part ofmeBC when the latter is varied from 1.75 to 2.26 shows that, at the extreme case of RF10 with the thickest coating in this study, the absorption increased less than 5 % at 660 nm.

The retrieval of keBC is illustrated in Fig. 2. OA is assumed to be non-absorbing; thus, the imaginary refractive index kOA is taken as zero in this calculation. Values of keBC were determined by minimizing the squares, χ2, of the relative differences between the measured and calculated absorption coefficients at 660 nm for each model: 8χ2keBC=σabs,660cal-σabs,660meaσabs,660mea2, where σabs,660cal and σabs,660mea are the calculated and measured absorption coefficients at 660 nm, respectively. The value of keBC is searched from 0.3 to 2.6, over a greater range than the values of kBC recorded in literature (Chylek et al., 2019; Kahnert and Kanngießer, 2020) and with a span of 0.01. The keBC values with χ2 greater than 0.1 are excluded.

The absorption of BrC at the long visible, i.e. 660 nm, is expected to be very small or negligible considering the aged nature of aerosols in our study (Sedlacek III et al., 2018; Adler et al., 2019; Li et al., 2019). Therefore, we assume that the contribution of BrC to keBC can be neglected and use it to differentiate the BrC absorption from that of other substances, i.e. BC and any other absorbers. This assumption of nil contribution of BrC to keBC yields a lower bound of BrC absorption at 530 and 470 nm. An evaluation of the influence of this assumption to the BrC absorption at shorter wavelengths is discussed in Sect. S2. As illustrated in Fig. 2, meBC was held constant over the range of investigated wavelengths, i.e. 470–660 nm. This is supported by mBC that is usually considered to be very weakly dependent on, or independent of, wavelength in the visible spectrum (e.g. Chang and Charalampopoulos, 1990; Moteki et al., 2010; Saleh et al., 2014). As discussed later in Sect. 3.1, other absorbers contributing to meBC, if any, are hypothesized to be magnetite, whose refractive index is also mostly invariant between wavelengths 470 and 660 nm (Amaury et al., unpublished data, http://www.astro.uni-jena.de/Laboratory/OCDB/mgfeoxides.html, last access: 8 July 2022; Ackerman and Toon, 1981; X. L. Zhang et al., 2015). Thus, the assumption of constant meBC between 470 and 660 nm is reasonable in our study, although it may lead to overestimations of BrC absorption for highly aged particles, as discussed further in Sect. 3.2.

The absorption of BrC at wavelength λ (σabs,BrC,λ) is calculated as the difference between measured absorption coefficient (σabs,λmea) and that calculated with meBC (σabs,λcal): 9σabs,BrC,λ=σabs,λmea-σabs,λcal.

The fractional contribution of BrC to the total absorption at wavelength λ, RBrC,λ, is defined as 10RBrC,λ=σabs,BrC,λ/σabs,λmea and is assumed to be zero at 660 nm in our study. The mass absorption cross section (MAC) of OA at wavelength λ, MACOA,λ, can be determined from σabs,BrC,λ and the mass concentration of OA, MOA: 11MACOA,λ=σabs,BrC,λ/MOA. Similarly, the mass absorption coefficient of BC (MACBC) is defined as MACBC,λ=σabs,λmea/MBC, where σabs,λmea and MBC represent the measured absorption coefficient at wavelength λ and the mass concentration of BC, respectively.

In this section, we present and discuss the results of keBC derived from the aforementioned four models for each flight. Uncertainties of keBC retrieved from various models have been determined by Monte Carlo uncertainty analysis in Sect. S3. The values of keBC for each flight determined by the various models are shown in Fig. 4. No meBC for RF05_3 was obtained with the CS model because the criterion of χ2 smaller than 0.1 was not met, as the absorption coefficient calculated with the CS model using the largest possible mBC (i.e. 2.26+1.26i) was only half of the measurement for RF05_3 (Fig. S2). Variations of keBC show similar patterns among different models (Fig. 4). The values derived from the CS and VM models are the highest and lowest, respectively, and the values derived from the MG and BG models are between the other two models and very close to each other, in agreement with the absorption results calculated with various models in previous studies (Lesins et al., 2002; Fierce et al., 2017a; Taylor et al., 2020). The values of keBC for RF05_1, RF05_2, RF06_1, and RF06_2 from the CS model have greatly exceeded the largest value of kBC of 1.26 (grey shaded region in Fig. 4), determined by Moteki et al. (2010) for soot particles from an urban source (Tokyo). Little confidence has been placed in the value of keBC from the CS model, except for RF10 for two main reasons. First, the MR for all samples is less than 2, and application of the CS model can be inappropriate for particles with such low MR values (Liu et al., 2017). Furthermore, the TEM analysis found few core–shell structures (e.g. example of RF05_1 in Fig. 5) among investigated samples, with most BC-containing particles composed of condensed BC nodules internally mixed with salts/organics (e.g. example of RF05_3 in Fig. 5) or aggregates attached to non-BC components (e.g. example of RF05_2 in Fig. 5). Representative TEM images of different types of BC-containing particles for each flight are shown in Fig. 5. The use of the CS model for RF10 was supported by its high MR value of 7.4, which has exceeded the lower bound of MR for the use of CS model proposed by Liu et al. (2017), although a fair number of BC particles homogeneously mixed with salts and OA were detected as well (Fig. 5). In addition, results of the BrC contribution to the total absorption suggest that homogeneous models might be inappropriate for RF10, as discussed in detail in Sect. 3.3.

As discussed in Sect. 2.3, keBC should be the same as kBC if the absorption at 660 nm arises only from BC; higher keBC value indicates larger absorptivity of absorbers at 660 nm. The values of keBC for RF05_1, RF05_2, RF05_3, RF06_1, and RF06_2, for which iron (Fe) was detected from the TEM-EDX (as shown in the bar chart in Fig. 5), are mostly larger than unity and greater than those for RF10 and RF11, for which no iron was detected. Therefore, we suspect higher keBC values may be related to iron oxides (FeOx) that absorb at 660 nm. We have examined all possible compounds of detected elements from EDX and found that magnetite is the only compound that is not rare in the atmosphere and also absorbs strongly at 660 nm. Magnetite shows a strong and uniform absorptivity over the visible spectrum, with the reported imaginary refractive index ranging from 0.58 to 1.0 (Amaury et al., unpublished data, http://www.astro.uni-jena.de/Laboratory/OCDB/mgfeoxides.html, last access: 8 July 2022; Ackerman and Toon, 1981; X. L. Zhang et al., 2015), and therefore can contribute to the high values of keBC at 660 nm. Studies show that magnetite can be transformed from Fe(III) at high temperatures, such as goethite and hematite (Till et al., 2015; Ito et al., 2018), which happen to be the two most abundant forms of FeOx in African dust (Formenti et al., 2014). Therefore, we hypothesize that a part of the magnetite might be converted from Fe(III) during biomass burning. Additionally, magnetite can be emitted from anthropogenic activities, such as steel manufacturing, oil combustion, and engines and brakes of motor vehicles (Machemer, 2004; Liati et al., 2015; Moteki et al., 2017; Kurisu et al., 2019). The significance of anthropogenic magnetite in radiative forcing has been investigated and highlighted in recent studies (Moteki et al., 2017; Ito et al., 2018; Lamb et al., 2021). Although there is no investigation on magnetite in sub-Saharan Africa yet, the industrial and motor vehicle emissions in Africa are likely to contribute magnetite. Furthermore, the pyrometallurgical process is a widely used extraction method in copper mining, a major industry in the central African Copperbelt (https://www.pyrometallurgy.co.za/PyroSA/, last access: 8 July 2022, Vítková et al., 2010; Sikamo et al., 2016; Shengo et al., 2019), of which iron is a common unwanted slag element (Meter et al., 1999). Therefore, we suspect it might also contribute magnetite, considering the high temperature of the pyrometallurgical process.

RF05_3, collected at the uppermost aerosol layer (Fig. 4), is an exception among all flights, since the values of the absorption coefficient at 660 nm calculated with all four models are considerably smaller than the measured ones (Fig. S2). The MG, BG, and VM models yield median keBC values greater than 1.6; no values met the criterion of χ2<0.1 for the CS model. The MACBC of RF05_3 is the highest among all flights, with values (mean ± standard deviation) of 20.0±0.8, 17.8±0.8, and 14.3±0.7 m2 g-1 at 470, 530, and 660 nm, respectively. The absorption enhancement, EAbs, defined as the ratio of MACBC to the value for uncoated BC reported by Bond and Bergstrom (2006), is 2.3±0.1 for all three wavelengths. To the best of our knowledge, except for modelling or laboratory studies of thickly coated particles (Bond et al., 2006; Jacobson, 2012; Peng et al., 2016; Che et al., 2022a), such high values of EAbs, particularly at long visible wavelengths, are rarely reported in field measurements (Cui et al., 2016). Taylor et al. (2020) presented relatively high MACBC values of 20±4, 15±3, and 12±2 m2 g-1 at 405, 514, and 655 nm, respectively and an EAbs of 1.85±0.45 for CLARIFY 2017 (Cloud-Aerosol-Radiation Interaction and Forcing 2017 measurement campaign; Haywood et al., 2021), which are generally comparable to or smaller than those for RF05_3. However, particles in CLARIFY 2017 are universally thickly coated, with median MR values of 8–12, and therefore would be expected to have large values of Eabs, as opposed to RF05_3, which had a median MR (Fig. 4) of 1.6. We calculated absorption coefficients for RF05_3 at 660 nm using the VM model, which usually yields a higher estimation of the measurements (Taylor et al., 2020) and a mBC of 1.95+0.79i, an upper bound of the commonly used BC refractive index (Liu et al., 2021). Uncertainties related to calculated absorption coefficients can be found in Sect. S3. The calculated absorption, which is very likely an overestimation of the measurement, is only able to explain ∼57% of the measured absorption coefficient (Fig. S2), implying that the remaining amount of absorption which cannot be attributed to BC might result from other absorbers. This unexplained portion of absorption in RF05_3 is at least twice as high as that in other flights (e.g. Figs. 3 and S2). Back-trajectories show that air parcels in RF05_3 passed by the Copperbelt in Zambia (Fig. 1), where the pyrometallurgical process in copper mining might contribute large amounts of magnetite. In addition, we noticed titanium (Ti) on the particles from RF05_3. Although titanium oxides are not absorbing, several forms of Ti have been reported to strongly absorb at visible wavelengths (Pflüger and Fink, 1997; Palm et al., 2018). Therefore, we suspect this unexplained absorption in RF05_3 may also be related to absorbing titanium compounds.

Values of MACOA,470 calculated from Eq. (11) using the different models are shown in Fig. 6. The uncertainty of MACOA,470 has been determined using Monte Carlo uncertainty analysis in Sect. S3. The values of MACOA,470 from the three homogeneous models were fairly close to each other, as expected from previous studies (Lesins et al., 2002), while MACOA,470 from the CS model was much greater, highlighting the importance of model treatment. From Sect. 3.1, results of RF10 from the CS model and the results of other flights calculated with homogeneous models are more plausible, with the values (mean ± standard deviation) of MACOA,470 for RF05_3, RF06_1, RF06_2, RF10, and RF11 ranging from 0.30±0.27 to 0.68±0.08 m2 g-1. This result at 470 nm is generally comparable to the value of 0.31±0.09 m2 g-1 at 405 nm for highly aged aerosols sampled downwind of ORACLES in CLARIFY 2017 campaign (Taylor et al., 2020).

RF05_1 and RF05_2, whose particles are as least as aged as those in CLARIFY, showed unexpectedly high MACOA,470 values of 1.84±0.64 and 2.38±1.89 m2 g-1, respectively. Although the values of MOA for these samples are low (Fig. 6), measurements from all instruments are within the detection limit, which would argue that these values are valid. X. Wang et al. (2018) used airborne measurements to constrain their global model and found the best MACOA to represent the measurements was 1.33 m2 g-1 for freshly emitted BB OA at 365 nm. Lin et al. (2017) investigated relatively fresh BB aerosols subject to atmospheric processing during a night-long BB event in an urban environment and reported a MAC of 0.9 m2 g-1 for water-extractable BrC at 470 nm under the peak BB episode. Our values of MACOA,470 for RF05_1 and RF05_2 are much higher than those for much fresher BB aerosols, which seems unrealistic, as BrC bleaching is expected to occur during transport (Hems et al., 2021; Che et al., 2021). One possible explanation is that secondary BrC formation occurred, perhaps through aqueous-phase chemistry during transport (Hems et al., 2021), as RF05_1 and RF05_2 were sampled in the MBL with higher RHs. Saleh et al. (2013) reported that secondary BrC can be more absorbing than primary BrC at short visible wavelengths; however, to the best of our knowledge, such high MAC like those in RF05_1 and RF05_2 for secondary BrC have not been documented (Kasthuriarachchi et al., 2020).

We assume a constant meBC over the investigated spectrum, i.e. 470–660 nm, while the meBC would be underestimated at shorter wavelengths if there are some components with strong absorption at short visible but not at long visible wavelengths and therefore lead to an overestimation of MACOA.470. Hematite, whose imaginary refractive index ranges from 0 to 1.0 at 470 nm and 0 to ∼0.2 at 670 nm (X. L. Zhang et al., 2015; Go et al., 2022), is the second most abundant FeOx in western African dust (Formenti et al., 2014). Particles in RF05_1 and RF05_2 experienced over 11 d of transport and thus may have more opportunities to mix with hematite and therefore lead to an overestimation of MACOA. While lacking measurements to determine the compound, it is difficult to verify our speculation. A modified SP2 has been reported by Yoshida et al. (2016) that can discriminate black-coloured magnetite and red-coloured hematite; thus, we recommend adding such measurement capabilities in future BB investigations over Africa and the South-East Atlantic region.

The contribution of BrC to the total absorption at 470 nm, RBrC,470, which was calculated using the optical closure method from Eq. (10), is shown in Fig. 7. The uncertainties using various models can be found in Sect. S3. Similar to the results for MACOA,470, the MG, BG, and VM models yield fairly close results for RBrC,470, while those from the CS model are higher, consistent with previous studies (Lesins et al., 2002; Taylor et al., 2020). It is noted that the values of RBrC,470 from MG, BG, and VM models at 470 nm for RF10 are the lowest among all flights, with an average of 3.6±2.6%. Little confidence is placed on this value because particles with similar age in RF05_3, RF06_1, RF06_2, and RF11 all showed a much higher value of RBrC,470. In addition, Athanasios Nenes et al. (personal communication, 2021) reported the contribution of water-soluble organic carbon (WSOC) to the total absorption at 365 nm of 20 % (Fig. 7) for particles measured at a similar location in the plume 10 min later than RF10, with no distinct changes in either the meteorological conditions or aerosol properties. Therefore, we expect the AAE of BrC at 365/470 wavelength pair, AAE365/470,BrC, derived from σabs,365,BrCmea and σabs,470,BrCcal (absorption coefficients of BrC at 365 and 470 nm) to be within the range of ∼2–11 in literature (Laskin et al., 2015). Since σabs,365,BrCmea is unknown, we calculated σabs,365,extrapolate by extrapolating the measured σabs,470mea to 365 nm with the measured AAE470/530 and approximated the AAE365/470,BrC with the 20 % of σabs,365,extrapolate and the σabs,470,BrCcal from homogeneous models. This method yields an underestimation of AAE365/470,BrC, while its value can still reach 18, much higher than the upper limit of the AAE range reported in literature (Laskin et al., 2015), suggesting that the use of homogeneous models for RF10 may not be appropriate. As discussed in Sect. 3.1, the values of RBrC,470 for RF10 from the CS model and RF05_1, RF05_2, RF05_3, RF06_1, RF06_2, and RF11 from homogeneous models are more plausible. Particles of RF05_1 and RF05_2 may contain hematite, in which case their RBrC,470 may be overestimated (Sect. 3.2). Generally, in our study, the RBrC,470 ranges from ∼8 %–22 % at 470 nm (grey zone in Fig. 7), slightly lower than the contribution of 13 %–26 % at 500 nm for relatively fresh OA in the wood smoke from Kirchstetter and Thatcher (2012) and generally comparable to or slightly higher than the result of ∼11% at 405 nm for highly aged aerosols from CLARIFY 2017, which were sampled downwind of ORACLES, implying the bleaching of BrC during transport (Hems et al., 2021; Che et al., 2022b).

We calculated the contribution of BrC to the total absorption using the AAE attribution method (RBrC,470,AAE) and compared it with that from our optical closure method (RBrC,470). In the AAE attribution method, the absorption coefficient of BC at the investigated wavelength (470 nm in our case) is determined by extrapolating the absorption coefficient at a longer wavelength (530 nm in our case) with the AAE of BC. For example, in this study, the average AAE at 530/660 wavelength pair (AAE530/660) for all flights is 0.93±0.16, which is just below unity and within the range expected for BC (Liu et al., 2018; Taylor et al., 2020). Thus, we assume that the absorption coefficients at 530 nm (σabs,530mea) and AAE530/660 have no significant contribution from BrC and that the AAE for BC is independent of wavelength. The contribution of BrC to the total absorption, RBrC,470,AAE, can then be calculated as below.

12RBrC,470,AAE=1-σabs,470cal,AAEσabs,470mea13σabs,470cal,AAE=σabs,530mea530470AAE530/66014AAE530/660=-ln⁡(σabs,530mea/σabs,660mea)ln⁡(530/660) σabs,470mea, σabs,530mea, and σabs,660mea represent the measured absorption coefficients at 470, 530, and 660 nm, respectively. σabs,470cal,AAE is the absorption coefficient calculated from the AAE attribution method at 470 nm. Actually, this assumption is not always true as the AAE varies with both the wavelength and core size and coating thickness of BC particles (Lack and Cappa, 2010; Wang et al., 2016). The results of RBrC,470,AAE are shown in Fig. 7, with averages of all flights below 5 %, roughly a factor of 3 smaller than the RBrC,470 from the optical closure approach.

The AAE attribution method is widely used for its simplicity, while there is one caveat that need to be noted when applying this method. The absorption coefficients and AAE, in this case, σabs,530mea, σabs,660mea, and AAE530/660, should be for BC only, without contributions from other absorbers, such as BrC (Lack and Cappa, 2010). However, it is usually very difficult to completely exclude the impact of these absorbers in the calculation. The homogeneous models are usually reported to overestimate BC absorption (H. Zhang et al., 2015; Fierce et al., 2017b; Taylor et al., 2020) and hence underestimate BrC absorption, while the values of the contribution of BrC determined by the AAE attribution method are even lower than those from the homogeneous methods, which we suspect might be partly attributed to the impact of absorbers besides BC. Therefore, we evaluated the absorption attribution at detected wavelengths by taking RF06_1 as an example. The contribution of BrC derived from the AAE attribution method is 2 % for RF06_1, approximately a factor of 5 smaller than that from the optical closure method with the BG model, 11 % (Figs. 7 and 8). As illustrated in Fig. 8, σabs,530mea includes the contribution of BrC, which accounts for 4 % of the total absorption, and that of absorbers beyond BC and BrC, which accounts for 21 %. Note this attribution is based on the BC absorption coefficient calculated from the BG model with mBC=1.95+0.79i, which yields an upper bound of BC absorption (Liu et al., 2021). All calculated parameters are subject to uncertainties as discussed in Sect. S3. If we remove the 4 % BrC from σabs,530mea in Eq. (13), the BrC contribution to absorption at 470 nm would increase to 6 %, 2 times larger than 2 %, indicating a substantial impact of BrC on the result of the AAE attribution method, even though the BrC absorption may only exist as a small portion. Thus, we recommend that application of any optical property-based attribution method to use absorption coefficients at the longest possible wavelength to minimize the influence of BrC and, in the meanwhile, to account for potential contributions from other absorbing materials.