Introduction

Black carbon (BC) is an important aerosol component and is primarily emitted from fossil fuel combustion and wood burning. BC has strong absorption of light from ultraviolet to near-infrared wavelengths1 and it contributes to global warming, following CO22,3,4. Adverse impacts of BC on regional air quality5,6 and human health7,8,9 have been widely reported. Considering its short atmospheric lifetime (4–12 days)10,11,12, reducing BC emissions would have synergistic benefits against its warming13, air pollution14, and adverse human health effects8,9. For reducing BC emissions, it is important to quantify the contributions of its various sources.

There are five methods to apportion the BC sources including receptor model, radiocarbon method, Marco-tracer, air quality modeling, and the Aethalometer method. The key strengths and weaknesses of these common BC source apportionment methods are summarized elsewhere15. Among these methods, the Aethalometer model has the advantages of high temporal resolution and is easy to be operated16. Therefore, it has been widely adopted in BC source apportionment studies16,17,18,19,20,21,22. This method uses the aerosol light absorption at two wavelengths to apportion the equivalent BC (eBC) into fossil fuel combustion-related eBC (eBCff) and wood burning-related eBC (eBCwb)16. The accuracy of the Aethalometer model results is mainly dependent on the absorption Ångström exponent (α) for fossil fuel combustion (αff) and wood burning (αwb). Previous studies always applied fixed α values in the Aethalometer model17,22,23,24,25,26,27. However, αff and αwb varied in the ranges of 0.9–1.128,29 and 0.9–3.530,31,32, respectively, determined by combustion efficiency33, mixing state of aerosol components33,34,35, aerosol size36, and chemical composition37. It is problematic to use fixed α values in the Aethalometer model. Previous studies also showed spatial heterogeneity in the optimal combination of αff and αwb. αff = 0.90 and αwb = 1.68 were recommended in Switzerland38. αff = 1.10 and αwb = 1.60 were adopted in Helsinki of Finland21. A combination of αff = 0.90 and αwb = 1.82 was reported in metropolitan Milan, Italy39. The optimal αff and αwb were reported in the ranges of 0.97–1.12 and 1.63–1.74, respectively, in the urban and rural areas of Spain40. Therefore, the determination of site-specific αff and αwb is essential prior to using the Aethalometer model.

To get the site-specific αff and αwb, previous studies used auxiliary measurements such as radiocarbon (14C)16,38,39,41 and levoglucosan (LG)20,21,40,42. 14C is an ideal tool to distinguish BC from the combustion of contemporary carbon and fossil fuel43. The constraint using 14C can precisely obtain the optimal αff and αwb38,39. The attribution using 14C still has some issues such as sample contamination, improvement in instrument analysis, and the separation of organic carbon (OC) and elemental carbon (EC)43. These issues inevitably introduce uncertainty to the source apportionment result. Constraint using the linear regression between LG and eBCwb can calculate the optimal αff, while the optimal αwb value is empirically chosen. The usage of LG to get the optimal α combination also has shortages. LG undergoes chemical degradation from the source to receptor44,45,46 and it has other sources such as coal combustion47,48, cooking emission49, and garbage burning50, etc. These sources would result in an overestimated contribution of wood burning to BC. Additionally, previous studies with 14C or LG to find the optimal α values were low in temporal resolution, which was determined by the sampling duration (i.e., >23 h)51. However, the BC emission sources showed obvious diurnal variations, which suggested that the previous studies with the 14C and LG constraints failed to describe the dynamic variations of optimal α values within a day. Therefore, a wood burning tracer with properties of chemical inertness, high-temporal resolution, and easy to be measured is needed to optimize the α values in the Aethalometer model38.

Potassium has been widely used as a tracer of wood smoke/biomass burning52,53,54. Compared to the sample collection and chemical analysis of 14C and LG, potassium has the advantage of high-temporal resolution (i.e., 1 h) in measurement55,56. Despite the exitance of additional sources of potassium (sea salt and soil dust), the potassium from wood burning \(\left({\rm{K}}_{\rm{wb}}^{+}\right)\) can be corrected19. The \({\rm{K}}_{\rm{wb}}^{+}\) can be further used to optimize the α values in the Aethalometer model. Currently, almost all the environmental monitoring supersites in China synchronously monitor the real-time potassium and BC with an online ion chromatography analyser and Aethalometer, respectively. Whether \({\rm{K}}_{\rm{wb}}^{+}\) can be adopted to improve the hourly BC source apportionment results by the Aethalometer model has not been reported.

In this study, the method to calculate the αff and αwb was first developed and then applied to calculate the optimal αff and αwb values at hourly resolution. The BC source apportionment results using the fixed and dynamic constraining α values were compared. Finally, the uncertainty of this method was estimated.

Results

General characteristics

Figure 1 shows the hourly variations and distribution characteristics of ambient EC, \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\), absorption coefficient at 880 nm (babs_880), and α370_950 during the observational period. EC, \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\), babs_880, and α370_950 varied in the ranges of 0.20–9.82 µg m−3, 0.03–3.25 µg m−3, 2.27–98.7 Mm−1, and 0.80– 1.76, respectively. The highest values mostly occurred in winter with the mean (± standard deviation) values of 2.22 ± 1.30 µg m−3, 1.22 ± 0.58 µg m−3, 27.6 ± 14.2 Mm−1, and 1.34 ± 0.11 correspondingly. The lowest mean values were found in summer, as 1.09 ± 0.50 µg m−3, 0.36 ± 0.22 µg m−3, 15.5 ± 7.81 Mm−1, and 1.16 ± 0.09, respectively, for EC, \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\), babs_880, and α370_950.

Fig. 1: Time series and distribution characteristics of key variables during the observational period.
figure 1

a Elemental carbon (EC). b Aerosol light absorption at 880 nm (babs_880). c Absorption Ångström exponent was calculated by power-law fit at 7 wavelengths from 370 to 950 nm (α370_950). d Potassium from wood burning (\({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\)). The colors in the figures represent different seasons (Spring: March, April, and May; Summer: June, July, and August; Autumn: September, October, and November; Winter: December, January, and February).

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Similar seasonal patterns of these variables were also reported elsewhere (Supplementary Table 1). For instance, higher ambient aerosol α was reported in winter compared to other seasons. A higher α value is indicative of wood burning and a lower α value implies fossil fuel combustion21,42. Aerosols from wood burning contain abundant light-absorbing organic compounds (known as brown carbon) such as humic-like substances and polycyclic aromatic hydrocarbons. These organic compounds can strongly enhance the light absorption at ultraviolet wavelengths compared to those in the near-infrared wavelengths, where BC dominates the absorption57,58,59. Aerosols from fossil fuel combustion, however, contain a higher fraction of BC than organic compounds16. As summarized in Supplementary Table 2, aerosol from wood burning generally has a higher α value compared to fossil fuel (coal and oil) combustion. It should be noted that if aerosol from fossil fuel combustion has the same or higher α value as wood burning, the Aethalometer model would overestimate the contribution of wood burning. Anyway, a higher α value in winter suggested a larger fraction of BC from wood burning.

Method of determining the optimal α combination

As shown in Fig. 2a, the Pearson correlation coefficients between \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\) and eBCwb did not vary with αwb increasing while it decreased with the increase of αff. It suggested that the relationship between eBCwb and \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\) was only determined by αff20,21,42. The slope also showed a reduction with the increase of αff. A higher slope was therefore found with a lower αwb value if αff was fixed at a certain value (Fig. 2b). The intercept decreased with the increase of αff and the approximate zero value was only possible when the αff value was 1.09 (Fig. 2c). Therefore, αff = 1.09 was chosen as its physical meaning that eBCwb and \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\) were only from wood burning and they held similar atmospheric removal rates20,21,42.

Fig. 2: Diagnostic parameters of linear regression equation as the function of different α combinations.
figure 2

a Pearson correlation coefficient (r). b Slope. c Intercept (int).

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To find the optimal αwb, previous studies determined it empirically according to the diurnal variations of eBCwb and eBCff calculated with the optimal αff value and different αwb values21,42,60,61. The method failed to determine the optimal αwb in this study (Supplementary Fig. 1). As explained mathematically, the fixed variables except for αwb in Eq. (5) can only result in the different values of eBCwb. Only changing the αwb in eBCwb calculation in different hours within a day cannot modify the diurnal variations of eBCwb. To determine the optimal αwb value, the statistical parameters (NMB, RMSE, and IOA) for the linear regression between eBCwb and \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\) were used. The optimal statistical results of NMB, RMSW, and IOA were obtained when αwb were 1.60, 1.79, and 1.77, respectively, at the optimal αff value (1.09) (Supplementary Fig. 2). To find the optimal αwb, the Taylor Diagram62 was further adopted to evaluate the model performances with different αwb values (Fig. 3). The diagram showed how r, standard deviation, and RMSE varied simultaneously, which can be represented through the Law of Cosines63. The combination of αff = 1.09 and αwb = 1.79 showed the lowest RMSE and the standard error, which was close to the observed standard deviation of \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\). Therefore, αwb = 1.79 was considered as the optimal value in this study.

Fig. 3: Taylor diagram plot of different models.
figure 3

The models were built using the optimal αff value (1.09) and different αwb values ranged from 1.60 to 1.79 with a 0.01 step.

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The optimal αff value here (1.09) was higher than the optimal αff reported in Milan (0.90)39 and London (0.96)20 but was comparable with that in Granada (1.10)42. Compared to the optimal αwb of 1.68 in Switzerland38 and 1.82 in Milan39, the optimal αwb value here (1.79) was between them. The optimal αwb value here was also within the reported values (1.63 ± 0.32) from smog chamber studies for fresh and aged wood burning emissions64, verifying that it was reasonable to use it in the Aethalometer model.

Constraining α combination at hourly resolution

Following the method developed above, the optimal αff and αwb were calculated at hourly resolution and they varied in the ranges of 1.02–1.19 and 1.71–1.90, respectively (Fig. 4a). To check the improvement of BC source apportionment results using the dynamic optimal α values, the mass concentrations of eBCff and eBCwb were calculated using both the fixed α values (αff = 1.0 and αwb = 2.0), optimal α values (αff = 1.09 and αwb = 1.79), and dynamic α values. As shown in Fig. 4b, eBCff calculated from the dynamic α combination significantly correlated with NO2 (r = 0.43, p = 0.04), while their positive correlations calculated from the fixed and optimal α values were not statistically significant (p > 0.05). Additionally, the fraction of eBCff calculated from the dynamic α values clearly showed the increase during the morning and evening traffic rush. eBCff fractions calculated from the fixed and optimal α values decreased (Supplementary Fig. 3), which was not in accordance with the actual situation of enhanced vehicle emissions during the rush hours. For eBCwb, the dynamic constraint also improved the reasonability of BC source apportionment results. In Fig. 4c, the Pearson correlation coefficient between eBCwb and \({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\) was 0.75 (p < 0.01) for the dynamic optimal α values, which was higher than those calculated from the fixed (r = 0.42, p = 0.04) and optimal (r = 0.35, p = 0.10) α combinations. The diurnal variations in physical-chemical properties of aerosol (chemical composition, particle size, and source emission strengths) can be characterized by aerosol α65. The increasing of optimal α combination from about 08:00 to 10:00 and from 16:00 to 20:00 in this study (Fig. 4a) suggesting that the dynamic α values reflected the relative abundance of traffic and wood burning emissions. Therefore, BC source apportionment using the dynamic optimal α values showed more reasonable results compared to those calculated with fixed α values.

Fig. 4: Diurnal variations of optimal absorption Ångström exponent and black carbon source apportionment results.
figure 4

a Diurnal variations of optimal absorption Ångström exponent for aerosol from fossil fuel (αff) and wood burning (αwb). b Diurnal variation of black carbon from fossil fuel (eBCff). c Diurnal variation of black carbon from wood burning (eBCwb). The red, orange, and green lines in panel b and c represent the source apportionment results using the dynamic optimal (values in panel a), fixed (αff = 1.0, αwb = 2.0), and optimal (αff = 1.0, αwb = 1.79) α combinations, respectively. The filled areas in panels b and c represent the 95% confidence intervals of mean values.

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BC source apportionment results and uncertainty estimation

The BC sources at an urban station of Wuhan, Central China were apportioned with the dynamic optimal α combinations. The temporal variations of BC sources are shown in Fig. 5 and Supplementary Fig. 4. eBCff contributed to 77.5 ± 18.9% of eBC with the highest contribution in summer (90.5 ± 9.81%) and lowest in winter (63.9 ± 17.6%). eBCwb accounted for 22.5 ± 18.9% of eBC. Contrary to eBCff, the lowest percentage of eBCwb was found in summer (9.52 ± 9.81%) and the highest percentage occurred in winter (36.1 ± 17.6%). On the annual scale, eBCff showed a positive correlation with ambient temperature (r = 0.11, p < 0.05) while eBCwb was negatively correlated with ambient temperature (r = −0.51, p < 0.01). eBCff showed a significant positive correlation with NO2 (r = 0.56, p < 0.01) during the entire year, indicating that vehicle emission could be an important source of BC in urban areas23,38,39. The high levels of eBCwb in winter and its negative correlation with ambient temperature suggested that more wood was consumed during winter for heating27. The diurnal variations of BC sources showed two peaks in the morning and evening due to the increased traffic emissions39,42. The minimum levels of eBC occurred in the afternoon, which was related to better dispersion conditions61,66. eBC was negatively correlated with the mixing layer height and wind speed with Pearson correlation coefficients of −0.38 (p = 0.06) and −0.12 (p = 0.57), respectively. Therefore, the higher mixing layer height and larger horizontal wind speed in the afternoon contributed to the reduction of eBC levels61,66,67.

Fig. 5: BC source apportionment results using the dynamic absorption Ångström exponents.
figure 5

a Hourly variations of black carbon sources during the observational period. b Boxplot of their relative contributions to BC mass concentration. c Diurnal variations of BC sources in different seasons. The filled areas in panel c represent the 95% confidence intervals of mean values.

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In this study, the uncertainties in the mass concentrations of eBCff and eBCwb were estimated by the propagation of errors41. It should be noted that the uncertainties of eBCff and eBCwb in this study were associated with the measurement uncertainties rather than the true uncertainties of fossil fuel and wood burning derived BC constrained by the radiocarbon method38. Several measurements including babs, EC, and α would introduce uncertainties to the BC source apportionment results. The absorption coefficients measured by AE31 had an uncertainty of 5%68. The uncertainty of OC/EC analyser was reported as 24%69. αff and αwb held the uncertainties of 4.63 and 3.32%, respectively, estimated from the dynamic optimal α values in this study. The uncertainty of MAC calculated from corrected babs and EC was estimated as 24.5% which produced an uncertainty of 25.0% for eBC using the ratio of babs to MAC. Finally, the uncertainties of eBCff and eBCwb were estimated as 28.6 and 56.2%, respectively. The average uncertainties estimated in this study were within the uncertainty ranges of eBCff (50%–96%) and eBCwb (4%– 50%) reported by Favez et al. (2010)17. Compared to Martinsson et al. (2017)41, the uncertainty of eBCff in this study was lower than it (41%) and the uncertainty of eBCwb was higher than that study (42%). Despite the high uncertainties, eBCff and eBCwb calculated with the optimal α combination in the hourly resolution were well correlated with NO2 and \({\rm{K}}_{\rm{wb}}^{+}\), respectively (Fig. 4). It suggested that the use of \({\rm{K}}_{\rm{wb}}^{+}\) to optimize the α combination in the Aethalometer model can accurately estimate the BC from fossil fuel combustion and wood burning.

Discussions

The Aethalometer model is originally developed to attribute BC into liquid fossil fuel combustion (vehicle emissions) and biomass/wood burning in western Europe16. In other places, where coal combustion is an important source of BC, the application of the Aethalometer model would overestimate the contribution of liquid fossil fuel combustion assuming BC is only from liquid fossil fuel combustion and wood burning. For instance, vehicle emissions, coal combustion, and biomass burning contributed 31, 45, and 24% to EC during winter in Xi’an China with 14C and stable isotope (13C)70. Therefore, BC sources attributed by the Aethalometer model should be adjusted to fossil fuel (liquid + solid) combustion and wood/biomass burning in China and other places with large coal consumption. For instance, a study conducted in Xiamen, China suggested that the average contributions of BC from fossil fuel combustion and biomass burning derived by the source-originated model were 67.4 and 32.6%, respectively. They were very close to the results (66.7% for eBCff and 33.3% for eBCwb) obtained by the Aethalometer model71. The intercomparison results suggested that it was reasonable to attribute BC into fossil fuel combustion and wood/biomass burning using the Aethalometer model in China.

The optimal αff obtained with \({\rm{K}}_{\rm{wb}}^{+}\) also implied the reasonability to attribute BC into fossil fuel combustion rather than liquid fossil fuel combustion. The α for solid fossil fuel combustion was generally higher than that for liquid fossil fuel combustion, although the α value was determined by combustion efficiency33 and fuel types (Supplementary Table 2). If fossil fuel combustion-related BC was only from liquid fossil fuel combustion, the optimal αff in this study should be within the ranges of reported α values for liquid fossil fuel combustion (i.e., as 0.91 ± 0.08 from Supplementary Table 2). However, the derived optimal αff values in this study (1.02–1.19) were higher than the reported α ranges for liquid fossil fuel, but lower than that from solid fossil fuel combustion. It suggested that the additional sources of BC like coal combustion. Therefore, the derived αff suggested that the eBCff resolved by the Aethalometer model contained the contributions from both the liquid and solid fossil fuel combustion-related BC. It also raised a question that how to further separate the fossil fuel combustion-related BC into liquid and solid fossil fuel using the Aethalometer model.

Although the dust and sea salt originated potassium was subtracted from the measured potassium (\({\rm{K}}_{\rm{wb}}^{+}\) accounting for 93.6 ± 6.68% of the measured K+), the usage of \({\rm{K}}_{\rm{wb}}^{+}\) as a wood burning tracer to optimize the α values had several issues. For instance, the highest Pearson correlation between \({\rm{K}}_{\rm{wb}}^{+}\) and eBCwb in this study (r2 = 0.66) was lower than that between LG and eBCwb (r2 = 0.91)21 assuming an αff of 0.80. Using the least squared method38 to solve the fitting between \({\rm{K}}_{\rm{wb}}^{+}\)/EC and babs_370/babs_880, the optimal αff and αwb values were calculated as −0.39 and 1.96, respectively in this study. The solved αwb seems plausible while the αff was much lower than the reported α for fossil fuel combustion as discussed above. Anyway, the dynamic constraint using hourly measured potassium improved the BC source apportionment results compared to the fixed α values. Further researches should focus on the use of radiocarbon or LG to get the site-specific optimal α values prior to BC source apportionment with the Aethalometer model in China. Furthermore, the methodology should be developed to apportion the BC into vehicle emissions, coal combustion, and biomass burning using the optical method with the help of 14C and 13C.

Methods

Observation data

Near real-time (1 h resolution) ambient light absorption coefficients (babs), water-soluble ions, OC, and EC from March 2018 to February 2019 were measured at a supersite of Wuhan, Central China (Supplementary Fig. 5). Ambient light absorption coefficients at seven wavelengths (370, 470, 520, 590, 660, 880, and 950 nm) were continuously measured using an Aethalometer (AE31, Magee Scientific, USA) equipped with a PM2.5 inlet. Simultaneous measurements of OC and EC in PM2.5 were conducted with a semicontinuous thermal-optical transmittance carbon analyser (Sunset RT-4, USA) using a simplified version of the NIOSH 5040 protocol72. Detailed information about OC/EC measurements can be found in Supplementary Table 3, Supplementary Fig. 6, and Supplementary Methods. Hourly water-soluble ions including \({\mathrm{NH}}_{4}^{+}\), Na+, Mg2+, K+, Ca2+, Cl−, \({\mathrm{NO}}_{3}^{-}\), and \({\mathrm{SO}}_{4}^{2-}\) were measured using an online ion chromatography analyser (MARGA-1S, Metrohm) equipped with a PM2.5 inlet (Supplementary Methods). The scatter plot between the total anions and cations during the entire period showed a high correlation with r2 of 0.95 and slope close to 1, indicating the good quality of the dataset (Supplementary Fig. 7). On-site meteorological parameters during the observational period were synchronously recorded (Supplementary Fig. 8).

AE 31 data correction

BC concentration measured by AE 31 is determined by the light absorption (babs) and the mass absorption cross section (MAC) and it is reported as eBC73. Due to the loading and multiple scattering effects, the babs of aerosol deposited on the filter was different from the ambient air. A method developed by Weingartner et al. (2003)74 was used to correct the babs (more details can be found in Supplementary Methods). The MAC is determined by the particle size, mixing state of aerosol components, and morphology1,28,75 and it shows spatial heterogeneity (Supplementary Methods). To determine the local MAC, the linear regression between the corrected babs and EC mass concentration was conducted (Supplementary Fig. 9) and it showed temporal variations (Supplementary Fig. 10). After the correction of babs and MAC, the eBC level was calculated as the ratio of babs to MAC. The α of ambient aerosol was calculated using a power-law fitting with the absorption coefficients at seven wavelengths1,22,27 and denoted as α370_950 in this study.

Wood burning derived potassium (\({{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ +\))

In this study, potassium from wood burning was corrected by the following equation, considering sodium and calcium as the tracers of sea salt and dust, respectively19:

$${{{\mathrm{K}}}}_{{{{\mathrm{wb}}}}}^ + = \left( {{{{\mathrm{K}}}}_{{{\mathrm{m}}}}^ + - 0.036 \times {{{\mathrm{Na}}}}_{{{\mathrm{m}}}}^ + - \left[ {{{{\mathrm{K}}}}^ + /{{{\mathrm{Ca}}}}^{2 + }} \right]_{{{{\mathrm{dust}}}}} \times {{{\mathrm{Ca}}}}_{{{{\mathrm{nss}}}}}^{2 + }} \right)/\left( {1 - 0.1 \times \left[ {{{{\mathrm{K}}}}^ + /{{{\mathrm{Ca}}}}^{2 + }} \right]_{{{{\mathrm{dust}}}}}} \right)$$
(1)

where \({\rm{K}}_{\rm{wb}}^{+}\) is the wood burning derived K+; K+m and Na+m represent the measured ambient mass concentrations of K+ and Na+, respectively; 0.036 is the standard ratio of K+/Na+ in sea salt76; [K+/Ca2+]dust is the ratio of potassium to calcium in local dust (0.053 in this study, see Supplementary Methods and Supplementary Fig. 11). \({\mathrm{Ca}}_{\mathrm{nss}}^{2+}\) is the non-sea salt Ca2+ concentration in the ambient air corrected by that from sea salt19. After the correction, the statistical robust correlation between \({\rm{K}}_{\rm{wb}}^{+}\) and EC showed that \({\rm{K}}_{\rm{wb}}^{+}\) was a good indicator of wood burning (Supplementary Methods).

Aethalometer model

With the light absorption and α values to apportion the eBC into fossil fuel combustion and wood burning, the Aethalometer model can be expressed as follows38:

$$b_{{{{\mathrm{abs}}}}}\left( {\lambda _1} \right)_{{{{\mathrm{ff}}}}}/b_{{{{\mathrm{abs}}}}}\left( {\lambda _2} \right)_{{{{\mathrm{ff}}}}} = \left( {\lambda _1/\lambda _2} \right)^{ - \alpha _{{{{\mathrm{ff}}}}}}$$
(2)
$$b_{{{{\mathrm{abs}}}}}\left( {\lambda _1} \right)_{{{{\mathrm{wb}}}}}/b_{{{{\mathrm{abs}}}}}\left( {\lambda _2} \right)_{{{{\mathrm{wb}}}}} = \left( {\lambda _1/\lambda _2} \right)^{ - \alpha _{{{{\mathrm{wb}}}}}}$$
(3)
$$b_{{{{\mathrm{abs}}}}}\left( \lambda \right) = b_{{{{\mathrm{abs}}}}}\left( \lambda \right)_{{{{\mathrm{ff}}}}} + b_{{{{\mathrm{abs}}}}}\left( \lambda \right)_{{{{\mathrm{wb}}}}}$$
(4)
$${{{\mathrm{eBC}}}}_{{{{\mathrm{wb}}}}} = \frac{{\frac{{b_{{{{\mathrm{abs}}}}}\left( {\lambda _1} \right) - b_{{{{\mathrm{abs}}}}}\left( {\lambda _2} \right) \times \left( {\lambda _1/\lambda _2} \right)^{ - \alpha _{{{{\mathrm{ff}}}}}}}}{{\left( {\lambda _1/\lambda _2} \right)^{ - \alpha _{{{{\mathrm{wb}}}}}} - \left( {\lambda _1/\lambda _2} \right)^{ - \alpha _{{{{\mathrm{ff}}}}}}}}}}{{b_{{{{\mathrm{abs}}}}}\left( {\lambda _2} \right)}} \times {{{\mathrm{eBC}}}}$$
(5)

where λ1 and λ2 are the wavelengths near-ultraviolet and near-infrared, respectively; αff and αwb are the absorption Ångström exponent for fossil fuel combustion and wood burning, respectively. Combined the Eq. (2)–(4), the mass concentration of eBCwb can be calculated with Eq. (5).

Sensitivity analysis

The simultaneous measurements of \({\rm{K}}_{\rm{wb}}^{+}\) and eBC provided the potential to conduct the sensitive analysis of α on the Aethalometer model result. It is expected that the intercept of the linear regression between eBCwb and \({\rm{K}}_{\rm{wb}}^{+}\) should be zero if eBCwb and \({\rm{K}}_{\rm{wb}}^{+}\) are only from the wood burning and they have the same atmospheric removal rates20,21,42. In practice, the intercept of the linear regression between eBCwb and \({\rm{K}}_{\rm{wb}}^{+}\) can approach zero by changing the αff and αwb in the calculation of eBCwb. The combination of αff and αwb which resulted in the zero or approaching zero of the intercept was considered as the optimal α values20,21,42. To calculate the eBCwb, the variation step was set as 0.01 for αff and αwb, which varied in the ranges of 0.80–1.30 and 1.60–2.20, respectively. The Pearson correlation coefficient, slope, intercept, normalized mean bias (NMB), root mean squared error (RMSE), and index of agreement (IOA) (Supplementary Methods) of the linear regression between eBwb and \({\rm{K}}_{\rm{wb}}^{+}\) were calculated to find the optimal α combination.