2.1 Two-stream radiative transfer model

A 1-D two-stream radiative transfer framework for multiple scattering in absorbing media was developed in Bohren (1987) – widely known as the Kubelka–Munk theory (Kubelka, 1948) – and subsequently discussed in relation to aerosol-filter systems in several studies (Arnott et al., 2005; Clarke, 1982; Gorbunov et al., 2002; Petzold and Schönlinner, 2004). Solving a radiation balance for an aerosol-laden filter medium yields the following expressions for transmittance (T_l) and reflectance (R_l), respectively:

Here, ω_l, g_l, and τ_e, l denote the single scattering albedo (SSA), asymmetry parameter, and extinction optical depth, respectively, of the composite layer. The parameter K is defined as

Arnott et al. (2005) used the above model to derive the form for an approximate correction factor for the aethalometer. The aethalometer uses optically thick quartz fiber filters, which are strongly multiple scattering, transmitting only ∼10 % of light in the visible wavelengths. A mathematical consequence of strong multiple scattering is that the term Kτ_e, l is much greater than unity and Eqs. (1a) and (1b) can be replaced by simplified approximations. In contrast, the Teflon filters used in this study are optically thin and constitute a weak multiple scattering medium: they transmit 70 %–80 % of incident visible light. Therefore, the full equations for T_l and R_l were solved for the filter-particle system, using a range of plausible values of dimensionless aerosol optical properties: absorption optical depth (τ_a, s) and SSA. Two other required inputs could not be measured: the penetration depth of aerosols into the filter was assumed to be 10 % of the total filter thickness, and the asymmetry parameter of the aerosols was fixed at 0.6, based on the typical values reported for biomass burning emissions (Martins et al., 1998; Reid et al., 2005). A schematic representation of the two-layer system – the aerosol-laden layer with properties T_l and R_l and a clean filter layer with properties T_f and R_f – is shown in Fig. 1. Transmittance and reflectance (T_s and R_s, respectively) through the filter, when light is first incident on the aerosol-laden layer (or “sample-side”), are given by (Gorbunov et al., 2002)

If the light first passes through the clean filter layer, the model predicts that transmittance, T_c, is still given by Eq. (3a). However, filter substrates are not uniform over their depths and have visually distinguishable front and back surfaces. Therefore, measurements of T_s and T_c are expected to differ. For the model substrate, reflectance, R_c, is given by

Attenuation (ATN) due to the aerosol deposit is calculated by applying Beer–Lambert's law relating to the reduction in transmittance of an exposed filter (T_s) relative to a blank (T_b) (Bond et al., 1999; Campbell et al., 1995). For a non-absorbing filter substrate, all attenuation of incident light must be caused by aerosol light absorption. Therefore, ATN is a measure of τ_a, s.

Here, R_b is the reflectance of the blank. An alternate measure of filter-aerosol optical depth utilizes transmittance and reflectance of the clean face of the filter (Campbell et al., 1995; White et al., 2016). This reflectance measurement R_c can be assumed to be approximately equal to R_b. PTFE blanks are non-absorbing; therefore the numerator in Eq. (5) can be replaced by $T_{\mathrm{b}}=\mathrm{1}-R_{\mathrm{b}}\approx \mathrm{1}-R_{\mathrm{c}}$. It should be noted that for translucent Teflon filters, R_b and R_c cannot be assumed to be exactly equal (Campbell et al., 1995; Clarke, 1982). Therefore, we represent this measure of optical depth by a separate variable, OD_c, where the subscript denotes that the transmittance and reflectance values used correspond to the clean side of the filter:

Finally, we define an optical depth measure using sample-side transmittance and reflectance, which can be interpreted as a measure of transmission of the fraction of incident radiation that is not backscattered by the filter-aerosol system:

Values of ATN, OD_c, and OD_s for a range of τ_a, s values (0–1) are shown in Fig. 2 for two cases: (1) highly absorbing aerosols (SSA =0.3) and (2) highly scattering aerosols (SSA =0.95). We illustrate that OD_c is nearly equal to ATN for the absorbing aerosol case, but there are significant differences between the two optical measures when the filter is loaded with highly scattering aerosols. This is because a translucent substrate with a reflective coating on its back behaves like a mirror: reflectance of the substrate increases when such a coating is applied. We also find that ATN shows the largest variation with SSA for a given value of τ_a, s, while OD_s exhibits the smallest variation. This can be attributed to the changing relationships between R_s and filter loading for different SSA values (see Fig. S2 in the Supplement). For large SSA, R_s>R_b, and therefore, from Eqs. (5) and (7), OD_s< ATN. The converse is true for small SSA values. It should be noted that fixed blank optics – based on the mean of transmittance and reflectance measurements (measurement techniques are described in the following section) on 20 blank filters – were used to model ATN, OD_c, and OD_s. The purpose of this exercise was to illustrate the sensitivity of the above optical depth measures to SSA values of aerosols deposited on identical media.

Figure 1Two-layer model of a filter sample consisting of an aerosol-laden layer “l” and a clean layer “f”.

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Figure 2Modeled values of filter optical depth measures (OD_c, ATN, and OD_s) with increasing aerosol optical depth (τ_a, s) of deposited highly absorbing (SSA =0.3) or highly scattering (SSA =0.95) aerosols. Fixed blank optics were assumed.

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The variation in filter optical measures with SSA was quantified by calculating the means and standard deviations (SDs) of ATN, OD_c, and OD_s, over SSA values ranging 0.2–0.99 for each input value of τ_a, s. A total of 500 linearly spaced points along the SSA range were used. Blank filter properties were also varied within the model in accordance with the range observed over the 20 lab blanks. For every model sample, defined by a given SSA and τ_a, s combination, a model blank was generated assuming a normal distribution of blank transmittance values (mean =0.7, SD =0.02). Figure 3 shows the ratio of standard deviation to the corresponding average values of each filter optical depth measure. The relatively low standard deviation in OD_s (for most loading values) implies that this variable is a good candidate for estimating aerosol light absorption from filter optical measurements, for a wide range of aerosol types. The increase in standard deviation with increasing loading is mainly contributed by very high SSA points which are typically associated with lower absorption per unit mass; therefore very high mass loadings of such aerosols would be required to yield the upper range of the τ_a, s. For SSA <0.9, modeled attenuation values show little spread (<15 % variability around the mean) with changing SSA. A surface plot of OD_s for all model data points ($\mathrm{0.2}<\text{SSA}<\mathrm{0.99}$ and $\mathrm{0}<{\mathit{\tau }}_{\mathrm{a},\mathrm{s}}<\mathrm{1}$) is shown in Fig. S4.

Figure 3Ratio of standard deviations to means of modeled filter optical depth measures (OD_c, ATN and OD_s) averaged over 500 equally spaced SSA points ranging 0.2–0.99, corresponding to each value of aerosol optical depth (τ_a, s). Blank optics were randomly generated for each sample point from a normal distribution, with mean =0.7 and SD =0.02.

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2.2 Experiments

Diverse biomass fuels including wood and needles from pine, fir, and sage trees, grass, peat, and cattle dung were burned in a 21 m³ stainless steel combustion chamber located at Washington University (Sumlin et al., 2018, 2017). Flaming, smoldering, and mixed combustion phases were employed (see Sect. S1 in the Supplement) to generate a range of intrinsic aerosol properties: SSA values at 375, 405, and 532 nm ranged 0.4–0.99 and absorption Ångström exponents (AÅEs) for 375–532 nm ranged 1.2–6.8. A kerosene lamp was used to generate soot particles, with an SSA of ∼0.3 and AÅE within 0.70–1.1. A schematic of the experimental setup is shown in Fig. 4. Experimental conditions and intrinsic aerosol optical properties for each fuel–combustion phase combination are listed in Table 1. Approximately 10–50 g of a given type of woody biomass, grass, or dung was placed in a stainless steel pan and ignited using a flame. It was either allowed to continue flaming or brought to a smoldering phase by starving the flame with a lid. In the same type of pan, 5–15 g of peat was smoldered by using a ring heater to raise its temperature to 200 ^∘C. In one set of experiments, smoke from the chamber was directly sampled, while in another set, a hood placed over the pan was used for sampling the aerosols. The chamber exhaust was closed during the burns. The outlet from the hood or chamber was passed through a diffusion dryer and a semi-volatile organic compound denuder into a mixing volume, from which aerosols were continuously sampled by the four IPNs.

Figure 4Schematic representation of the experimental setup. Inlet to the semi-volatile organic compound denuder was taken from either the chamber sampling port or the hood. “IPNs” denotes the integrated photoacoustic-nephelometer spectrometers.

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Table 1Number of burns conducted and filter samples collected for each fuel type and combustion phase in this study. Intrinsic optical properties of emissions from each study condition are also given.

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During each burn, optical (absorption and scattering coefficients) signals were monitored using IPNs until a steady state was reached. During the steady state, particle samples were collected on 47 mm PTFE membrane (Pall) filters. The filter sampling flow rate was set to 5 L min⁻¹, and the sampling durations were between 2 and 20 min. For each filter sample, τ_a, s of the deposited aerosols was calculated from the absorption coefficients measured using the IPNs:

where b_abs, av is the average absorption coefficient (in Mm⁻¹) during the sampling duration t_s (in min), Q is the flow rate (in liters per minute or L min⁻¹) through the filter, and A_s is the filter sample area (in m²). Optical depth τ_a, s for the samples in this study ranged between 0.01 and 0.68. The uncertainty in these estimates was predominantly from the standard deviation in b_abs, av over the averaging interval and was within 10 % for all samples. Values of b_abs, av at 532 nm ranged from ∼300 Mm⁻¹ for smoldering samples to ∼20 000 Mm⁻¹ for flaming phase samples; the corresponding range at 375 nm was ∼3000–30 000 Mm⁻¹.

Sample-side transmittance (T_s) and reflectance (R_s) for the filter samples were measured using a PerkinElmer LAMBDA 35 UV–Vis spectrophotometer (described in Zhong and Jang, 2011). This instrument contains an integrating sphere and two sample holders. Transmittance was measured by placing the sample in the first holder ahead of the sphere, in the direction of the sample beam, while a white standard was placed in the second holder (behind the sphere). Reflectance was measured by keeping the first holder empty and placing the sample in the second holder. Both measurements were performed on the sample face of the filter: light was incident on the side that was exposed to the sample air. Each measurement was normalized to the baseline transmittance and reflectance value of the measurement system: between every 10 sample scans, transmittance and reflectance were measured with no sample placed in the first holder, and a white standard was placed in the second holder. Sample transmittance and reflectance values were then divided by the corresponding baseline. Only T_s and R_s were measured for all samples in this study as model results indicated that OD_s is better suited than OD_c for estimating τ_a, s (Fig. 3). To test the validity of this assumption, transmittance and reflectance were also measured on the clean side of the filter (T_c and R_c, respectively) for a subset of the samples (n=54). This subset corresponded to samples collected during 17 biomass burning experiments which yielded aerosols with SSA (375, 405, and 532 nm) ranging 0.54–0.99. For all samples, we found T_s>T_c.

From normalized T_s and R_s measurements, OD_s was calculated using Eq. (7). When this equation is applied to blank filters, it results in OD_s values between 0.01 and 0.03. A wavelength-dependent “blank optical depth” was subtracted from the sample data. Triplicate transmission and reflection measurements were used to estimate measurement uncertainty, which is attributable to random fluctuations in the measurements. Means and standard deviations of the OD_s values calculated from the replicate measurements yielded an uncertainty (ratio of standard deviation to mean) of 5 % in OD_s. Similarly, OD_c was calculated for the 54-sample subset.

A correction factor (C) that captures the net effect of multiple scattering and aerosol loading can be defined as