The purge flows that protect the high-reflectivity mirrors in the CAPS PMssa shorten the effective optical path length of the cavity and may slightly dilute the instrument sample flow at the cavity inlet. Therefore, a correction factor must be applied to the measured extinction coefficients in order to account for these changes (Massoli et al., 2010; Onasch et al., 2015; Petzold et al., 2013). We refer to this correction factor as the geometry correction factor, α, which can be determined by external calibration, i.e., by comparing CAPS PMssa measurements against independently measured or calculated bext (e.g., Mie-calculated bext values for spherical, monodisperse test aerosols, Petzold et al., 2013, or measured bext values for non-absorbing test aerosols obtained with a reference nephelometer, Pfeifer et al., 2020).

Onasch et al. (2015) applied the Mie calculation approach to measurements of polystyrene latex (PSL) spheres of varying diameter to determine an α value of 0.73 for a CAPS PMssa unit operating at 630 nm. This is lower than the general value of 0.79 quoted by Onasch et al. (2015) for CAPS PMex monitors, which they note is expected due to small differences in the cavity geometries. The CAPS PMssa units used in this study (Table 2) participated in European Center for Aerosol Calibration (ECAC; http://www.actris-ecac.eu/, last access: 29 January 2021) workshops (CAPS630b in August 2016 and CAPS450, CAPS630a, and CAPS780 in January 2017) where their geometry correction factors were determined against reference instrumentation (CAPS PMex, nephelometer) using ammonium sulfate test aerosols. The units were determined to have α values of 0.78 (CAPS450), 0.71 (CAPS630a), 0.7 to 0.73 (CAPS630b), and 0.78 (CAPS780). Therefore, it appears that α is instrument-unit-dependent. The stability of α over time is still an open question. However, regular and frequent measurements of α in CAPS PMex monitors performed at the ECAC suggests that it does not drift by more than 3 % over the period of a year. By default, the CAPS PMssa firmware automatically applies an α factor of 0.73 to calculate bext (Fig. 2).

The scattering cross-calibration factor (β) is used to relate the PMT-measured scattering signal of the CAPS PMssa to an absolute aerosol scattering coefficient. The value of β can be determined by cross-calibrating the uncalibrated aerosol scattering coefficient bsca,uncalib. against bext measured by the extinction channel (Onasch et al., 2015). This approach is possible because the scattering and extinction coefficients are measured simultaneously for the same air sample, and bext measured using the CAPS method is effectively “calibration free” (apart from the geometry correction factor, as discussed in Sect. 2.2.1, as well as potential non-linearities at high baseline losses). Amongst other factors, β depends on the PMT detector response, which can vary over time. Therefore, regular cross-calibrations should be performed.

Non-absorbing test samples are required to perform the cross-calibration and to determine a value for β (i.e., purely scattering samples for which bext=bsca, or SSA = 1). In principle, the calibration can be performed with gases or aerosol particles. In practice, we performed all calibrations in the present study with particles because readily available calibration gases such as CO2 span a much smaller range in bsca than is achievable with aerosols of different concentrations, additional corrections are required to account for the changes in optical path length and dilution with the purge flows for different gases (see Sect. 2.2.1), and we have observed that the instrument can take a long time (∼ hours) to adjust and stabilize when filled with different gases (as expected due to the low flows and large filter areas in the purge flow setup).

When using the particle-based calibration method, non-absorbing aerosol particles with size parameters x in the Rayleigh light-scattering regime should be used to ensure well-defined scattered light truncation, since the scattering-phase function is independent of particle size in the Rayleigh regime. We term cross-calibration constants derived in this specific manner as βRayleigh. The size parameter x relates the aerosol particle diameter Dp to the wavelength of light λ through the expression πDp/λ. The Rayleigh regime is defined by the condition x≪1. In practice, there is a trade-off between selecting particle sizes that are small enough to lie within or near the Rayleigh regime limit but large enough to generate scattering and extinction signals with sufficiently high signal-to-noise ratios. This means particles with diameter less than approximately 150 nm should be used to determine βRayleigh in the 450 nm CAPS PMssa, while slightly larger particles (e.g., Dp∼200 nm) can be used with 630 or 780 nm CAPS PMssa instruments.

Formally, the Rayleigh-regime, particle-based cross-calibration approach can be expressed as

3βRayleigh=bsca,uncalib.non-abs,Rayleighbextnon-abs,Rayleigh=α⋅bsca,uncalib.non-abs,Rayleighbext,uncorr.non-abs,Rayleigh=αβRayleigh′, where bextnon-abs,Rayleigh and bsca,uncalib.non-abs,Rayleigh are the extinction and uncalibrated scattering coefficients, respectively, for a population of non-absorbing particles with size parameters in the Rayleigh regime. The right-hand side of Eq. (3) is obtained by substitution of the relationship bext=bext,uncorr./α into the left-hand side ratio. From this substitution, it can be seen that βRayleigh (and β, generally) is directly proportional to the geometry correction factor α, which is required to measure bext accurately as discussed in Sect. 2.2.1 (the remaining fraction of the cross-calibration constant is termed β′Rayleigh to distinguish it from βRayleigh). Thus, Eq. (3) demonstrates how the cross-calibration approach quantitatively links bsca to bext in the CAPS PMssa. Following application of βRayleigh, we refer to the calibrated aerosol scattering coefficient corrected for the truncation of Rayleigh scattered light as bsca,Rayleigh. This is to recognize the fact that the cross-calibration approach represented by Eq. (3) implicitly corrects for the truncation of light scattered from the calibration aerosol, which has been chosen specifically to have the well-defined phase function corresponding to Rayleigh light scattering.

Onasch et al. (2015) demonstrated that the linearity shown by Eq. (3) is valid up to extinction coefficients of ∼1000 Mm-1, which is higher than typical ambient aerosol extinction coefficients, excluding perhaps coefficients in heavily polluted urban environments. The precise limit of linearity should be examined for individual instrument units if it is relevant for a particular experiment. For very high aerosol loadings above the limit of linearity the CAPS PMssa cross-calibration approach can still be used. However, this requires the addition of empirically derived higher-order terms in bextnon-abs,Rayleigh to Eq. (3). In addition, the potential occurrence of multiple scattering effects needs to be considered at very high aerosol loadings (Wind and Szymanski, 2002).

The final quantitative correction factor that must be applied to the scattering coefficients measured with the CAPS PMssa is the truncation correction factor, γ. The truncation correction factor γ is applied to bsca,Rayleigh to compensate for the light scattered in near-forward and near-backward directions that is not measured by the instrument due to geometric restrictions. The truncation correction factor γ depends on both the instrument properties as well as the angular distribution of light scattered from the aerosol sample being measured (referred to in short as the ensemble scattering-phase function, Sp), which depends on the aerosol size distribution, morphology, mixing state, and composition (refractive indices). The existing methods for calculating the CAPS PMssa truncation correction factor γ either do not include the process of scattered light reflection from the inner surface of the glass sampling tube (Onasch et al., 2015) or are computationally expensive (Liu et al., 2018) and not well suited for calculating time-resolved truncation factors for large datasets (e.g., as required for the example Cabauw dataset in Sect. 6.2). Therefore, we present here a new truncation calculation framework that overcomes both of these limitations.

The new calculation framework is presented visually as a flowchart in Fig. 3. The full set of details and equations is given in Appendix A. Briefly, we define γ as the normalized ratio of the true integrated scattering coefficient, bsca,true, to the truncation-affected scattering coefficient that is actually accessible to measurement, bsca,meas. The true scattering coefficient bsca,true represents the coefficient that would be measured by an ideal integrating nephelometer capable of collecting light scattered in all possible directions. The ratio requires normalization by a factor kRayleigh to represent the fact that some scattered light truncation is already included implicitly in the cross-calibration constant, due to the way in which it is measured. For the recommended case of cross-calibration with Rayleigh scatterers according to Eq. (3), kRayleigh represents the truncation of the Rayleigh scattered light from the calibration aerosol. That is,

4γ=bsca,truebsca,meas⋅kRayleigh=bsca,truebsca,meas⋅bsca,measRayleighbsca,trueRayleigh.

Defined in this manner, γ equals 1 for aerosols in the Rayleigh regime. For aerosols containing larger particles or non-spherical particles that produce more forward-focused light scattering, γ is always greater than 1.

The equations for calculating the integrated scattering coefficients in Eq. (4) are detailed in Appendix A. These equations have been given in several previous publications (Anderson et al., 1996; Heintzenberg and Charlson, 1996; Moosmüller and Arnott, 2003; Müller et al., 2011b; Peñaloza, 1999). The novel aspect of our formulation is that we explicitly define a function representing the efficiency with which an integrating nephelometer is able to collect scattered light, η(θ,λ), which is a simple function varying between 0 and 1. Values of 0 indicate that a nephelometer collects no light of wavelength λ at some scattering angle θ, while values of 1 indicate that a nephelometer collects all the light scattered at angle θ. Considering η(θ,λ) explicitly has a number of advantages: (i) it allows transparent representation of an instrument's truncation angles (i.e., by setting η equal to 1 between two truncation angles and 0 beyond them), (ii) it allows for the simple and explicit introduction of additional physical processes into light-scattering calculations (e.g., reflection from the glass sampling tube can be considered by combining the Fresnel equation for reflection probability with η, as shown in Appendix A), (iii) it provides a clear and intuitive way to compare the abilities of different nephelometers to collect scattered light, and (iv) it emphasizes the modular nature of the truncation calculation.

One of the important characteristics of integrating sphere-type reciprocal nephelometers like the CAPS PMssa is that truncation is a function of position along the central axis of the optical cavity (which we denote as the z dimension, Fig. 1). This characteristic is represented by the small subplots in Fig. 3 that show light collection efficiency curves (termed ηspot in Appendix A) for six different z positions in the CAPS PMssa, including for two positions at 1 cm outside of the integrating sphere (i.e., z=-6 and 6 cm). Positions outside of the integrating sphere must be considered since it is possible for particles outside the sphere to scatter light into the sphere (e.g., Varma et al., 2003), even if only through a narrow range of scattering angles. We term the extra length that needs to be considered outside the sphere's boundaries as the l parameter. The geometrical limits for the l parameter are 0 (i.e., no extra path length considered) and 4.7 cm (the distance between the integrating sphere and the sample inlet and outlet ports to the optical cavity). Onasch et al. (2015) and Liu et al. (2018) both used l=1 cm in their calculations (i.e., they considered a z range from -6 to 6 cm). The ηspot subplots in Fig. 3 also demonstrate the effect of glass tube reflection: between a sphere's truncation angles, reflection decreases the probability of light collection from 1 to some value less than 1. Therefore, glass tube inner surface reflection serves to increase scattered light truncation. A single, integrated light collection efficiency function for the CAPS PMssa can be generated by integrating ηspot over all possible z positions (Eq. A13). CAPS PMssa integrated η functions are shown in Fig. 3 for the two cases of without and with glass reflection.

It is important to stress the implications of the modularity of truncation calculation. This modularity means that once the η(θ,λ) and angular sensitivity functions are known for a particular instrument, they can be combined with any measured or calculated ensemble scattering-phase function in order to calculate γ. In the present study, we used Mie theory and co-located particle size distribution measurements to efficiently calculate hourly resolved Sp functions and γ values for a month-long field campaign (Sect. 6.2; Fig. S12). This Mie calculation method assumes spherical, homogeneous particles. If one wished to consider more complex particle morphologies, a more sophisticated optical model could be used to calculate the scattering-phase functions Sp, or if co-located polar nephelometer measurements of the scattering-phase function were available (e.g., Espinosa et al., 2018), these could be input directly into the truncation calculation.

In this section we detail the experimental methods that we applied to investigate and characterize the ability of the CAPS PMssa to measure atmospheric aerosol absorption coefficients. A total of four different CAPS PMssa monitors were used in this study: one operating at 450 nm, two at 630 nm, and one at 780 nm. The four units are listed in Table 2 along with their relevant specifications.

A multi-angle absorption photometer (MAAP; Thermo Fisher Scientific, Waltham, MA, USA) was used during the Cabauw field campaign (Sect. 3.4.1) to measure absolute aerosol absorption coefficients at a wavelength of 637 nm (Petzold and Schönlinner, 2004). As discussed in the Introduction, the MAAP is a filter-based absorption photometer that incorporates additional measurements of back-scattered light and a two-stream radiative transfer scheme in order to constrain aerosol absorption coefficients more tightly than is possible with simple light attenuation measurements. The MAAP is a well-known and well-characterized instrument for measuring light absorption by atmospheric aerosols. The accuracy of MAAP absorption coefficients was investigated against laboratory reference EMS absorption measurements in the Reno Aerosol Optics Study (RAOS), and the two methods were found to agree within 7 % for a range of different black-carbon-containing aerosols (Petzold et al., 2005). Müller et al. (2011a) demonstrated that the unit-to-unit variability between six different MAAP instruments was less than 5 %. These authors also showed that the true operation wavelength of the instrument was 637 nm, not the nominal value of 670 nm. Assuming an absorption Ångström exponent of 1.02, a 5 % correction factor should be applied to the firmware output of the MAAP to account for this wavelength difference (Müller et al., 2011a). This correction factor was applied in the present study. A mass absorption cross-section value of 6.6 m2 g-1 was used to convert the equivalent BC mass concentrations reported in the firmware output of the MAAP to absorption coefficients (as specified by the manufacturer).

During the RAOS campaign (Petzold et al., 2005), MAAP absorption coefficients were observed to have no relationship with aerosol SSA. However, at extremely high SSA values the absorption coefficient measurements from the MAAP can be biased high. It is also important to consider that – to the best of our knowledge – no dedicated study has yet been performed to assess the precision and accuracy of MAAP measurements of samples containing a large fraction of super-micrometer particles. To quantitatively compare the MAAP and CAPS PMssa absorption coefficients during the Cabauw field campaign, both coefficients were adjusted to standard temperature (273.15 K) and pressure (1 atm). It should be stressed that in this comparison we do not consider the MAAP to be a true reference standard for measuring aerosol absorption coefficients. Rather, the value of the instrument for the present study lies in the fact that it displays very low instrument unit-to-unit variability, which means it can provide a common and stable reference point against which CAPS PMssa absorption measurements can be compared.

A single particle soot photometer (SP2; Droplet Measurement Technologies, Longmont, CO, USA) was used to measure black carbon mass concentrations at high time resolution from a mobile laboratory deployed during the Bologna field campaign (Sect. 3.4.2). The SP2 measures the mass of individual black carbon particles on a single-particle basis using the principle of laser-induced incandescence. The instrument has been described in detail previously (Schwarz et al., 2006; Stephens et al., 2003). Due to its very high sensitivity and responsiveness, its specific purpose in the present study was to provide a high time resolution reference time series of relative absorbing aerosol concentration. Its configuration during the present study is described by Pileci et al. (2020a).

Two different types of aerosol size classifiers were used to generate monodisperse test aerosols for the purposes of measuring cross-calibration constants and scattered light truncation: an aerodynamic aerosol classifier (AAC; Cambustion Ltd, Cambridge, UK) and a differential mobility analyzer (DMA; custom-built version of the same design as the TSI Model 3081 long-column DMA, TSI Inc. Shoreview, MN, USA). The correct operation and sizing of both types of classifiers was confirmed throughout all the experiments by measuring nebulized PSL particles of different diameters (i.e., by operating the classifiers in scanning mode with downstream concentration measurements performed by a condensation particle counter).

The AAC classifies particles based on their relaxation time under the action of a centrifugal force generated in the annular gap between two rotating coaxial cylinders (Johnson et al., 2018; Tavakoli and Olfert, 2013). The particle relaxation time is related to the aerodynamic-equivalent diameter in a straightforward manner. In the context of highly size-dependent optical measurements, the major advantage of such a classification method is that it does not depend on particle electrical charge (unlike the DMA), which means the AAC can produce truly monodisperse distributions of particles (i.e., of finite width but without the presence of additional size distribution modes due to multiply charged particles). This charge-independent classification approach also enables higher aerosol transmission efficiencies than is possible with the DMA, which improves the signal-to-noise ratio of any downstream optical measurements. An additional advantage of the AAC relative to the DMA is that it can classify particles over a wider diameter range, including particles with diameters of up to ∼5 µm. The AAC was operated in the present study with the sheath-to-aerosol flow ratio of around 10:1, which results in geometric standard deviations for the classified aerosols of around 1.14. The set point aerodynamic diameters were converted to volume-equivalent diameters using literature values of particle density and assuming the classified particles were spherical.

Measurements of ambient particle size distributions were required during the Cabauw field campaign (Sect. 3.4.1) as inputs for the truncation correction calculations. These measurements were obtained by a scanning mobility particle sizer (modified version of the TSI SMPS 3034; TSI Inc. Shoreview, MN, USA) covering the mobility diameter range from 10 to 470 nm and an aerodynamic particle sizer (TSI APS 3321; TSI Inc.) nominally covering the aerodynamic diameter range from 0.54 to 20 µm.

The hourly averaged SMPS and APS size distributions were merged to create total aerosol size distributions covering the diameter range from 0.0104 to 10 µm for use in the truncation calculations. This was achieved by first converting the measured diameters of the respective instruments to volume-equivalent diameters. The SMPS electrical mobility diameters were simply taken to represent volume-equivalent diameter (i.e., shape effects were neglected). The APS aerodynamic diameters were divided by the square root of particle effective density to translate them into volume-equivalent diameters. A constant effective density of 2 g cm-3 was assumed. The joined size distributions were then created by linearly interpolating the SMPS and shifted APS measurements onto a common diameter scale between 0.0104 and 10 µm. The APS measurements of particles with physical diameters of less than 0.6 µm were not used in this joining calculation since they are known to display counting efficiency problems (Pfeifer et al., 2016).

The Cabauw field campaign was conducted from 11 September to 20 October 2016 at the KNMI (Koninklijk Nederlands Meteorologisch Instituut) Cabauw Experimental Site for Atmospheric Research (the Netherlands; 51∘58′ N, 4∘55′ E; -0.7 m a.s.l.). The campaign was conducted in the framework of the ACTRIS project (WP11) and occurred simultaneously with the CINDI-2 MAX-DOAS intercomparison campaign (Kreher et al., 2020). The CAPS630b unit was the CAPS PMssa instrument deployed during this campaign to measure absorption coefficients at 630 nm. Absorption coefficients were also measured at 637 nm with a MAAP (Sect. 3.1.1). Particle size distributions were measured with an SMPS and APS (Sect. 3.1.3). All instruments were housed in a laboratory at the base of the KNMI-mast Cabauw behind identical inlets consisting of PM10 sampling hats protruding 4.5 m from the laboratory roof. The inlets contained large diameter Nafion driers, which kept relative humidity in the sampling lines below 50 %. All the data used in the present study were averaged over 1 h periods. This includes the joined SMPS and APS size distributions (Sect. 3.1.3), which used to calculate hourly resolved truncation correction factors using the model presented in Appendix A.

The Bologna field campaign was conducted from 5 to 31 July 2017. This campaign was also conducted within the framework of the ACTRIS project. The full campaign consisted of multiple stationary measurement sites that were centered around the city of Bologna in Italy's Po Valley. Additionally, a heavily instrumented mobile measurement van (the MOSQUITA; Bukowiecki et al., 2002; Weimer et al., 2009) travelled between the stationary sites to perform spatially resolved measurements of black carbon concentrations and properties. The results of these mobile measurements are presented by Pileci et al. (2020a). In the present study we use only 1 h of mobile measurements that were performed from the MOSQUITA while it was travelling on the heavily trafficked A1 highway between Bologna and Lodi on the morning of 25 July 2017. During this time period absorption coefficients were being measured with the CAPS780 and black carbon concentrations with an SP2 (Sect. 3.1.1).

The Payerne field campaign was conducted from 26 August 2019 to 14 January 2020 in Payerne, Switzerland, and involved the PMssa units CAPS450 and CAPS780. The goal of this campaign was to compare the hygroscopic properties of aerosols measured using remote sensing and in situ techniques. In the present study we only present the results of the CAPS PMssa cross-calibrations that were performed for the campaign: no ambient measurements are shown. In addition to the cross-calibrations that were performed at the Payerne field site with the CAPS450 and CAPS780 units, we also present the results of calibrations performed immediately before and after the campaign in the Aerosol Physics Laboratory of the Paul Scherrer Institute. In addition to the other two PMssa units, the CAPS630a was also included in these laboratory calibrations.

The scattering and extinction baseline measurements over the campaign are displayed in Fig. S8. These measurements were performed for 1 min every 10 min using the auto-baselining feature of the instrument. The average standard deviations of the extinction and scattering baseline measurements over all 1 min baseline periods were 0.35 and 0.66 Mm-1, respectively. We take these values to represent the precision-based uncertainty estimates in bext,baseline and bsca,baseline in Table 1. Both baseline time series indicate that the optical cavity was generally clean and suffered no major contamination events during the campaign. Figure S8 also contains time series of the baseline drift in each channel, which were calculated from the differences between two successive baseline measurements (i.e., over a period of 10 min, which means the calculated metric does not include possible variations over shorter timescales). On average, the extinction baseline drifted by 0.026 Mm-1 min-1 and the scattering baseline by 0.013 Mm-1 min-1. Individual values of up to 0.3 and 0.09 Mm-1 min-1 were observed in the extinction and scattering channels, respectively. To minimize the impacts of these drifts, both the extinction and scattering measurements were reprocessed using the method of linear interpolation between successive baseline measurements (Pfeifer et al., 2020). Figure S9 indicates that this reprocessing had a noticeable effect at a 1 s time resolution for bext values less than ∼20 Mm-1 and bsca values less than ∼5 Mm-1. However, these effects are averaged out when considering hourly averaged data.

The cross-calibration constant for the CAPS630b could be measured with high precision (∼2 %) and appeared to be very stable based on measurements at the beginning (0.82; Fig. S10) and at the end of the campaign (0.80; Fig. S11), which differed by only 2.5 %. This is comparable with the optimum performance we have observed for CAPS PMssa units with regards to cross-calibration (see Sect. 4).

In the absence of direct scattering-phase function measurements, hourly time-resolved truncation correction factors were calculated with the Mie-theory-based model presented in Appendix A. Joined size distributions measured by SMPS and APS were input into the model. To estimate the uncertainty in γ a sensitivity analysis was performed with respect to the model's input parameters. This analysis is presented in the Supplement as Sect. S1 and visualized in Fig. S12. Specifically, we investigated the sensitivity of γ to the ambient aerosol refractive index (real parts between 1.50 and 1.59 and imaginary parts between 0.00 and 0.01, based on the summary of measurements presented by Espinosa et al. (2019), the l parameter (varied from 0 to 4.7 cm based on the results presented in Sect. 5), and the accuracy of the coarse-mode size distribution measurements between diameters of 2.5 and 10 µm. The truncation correction was found to be most sensitive to the l parameter (∼4 % across the range of tested parameters), weakly sensitive to the real part of the refractive index and size distribution information between 2.5 and 10 µm, and barely sensitive at all to complex part of the refractive index (not shown). Overall, we estimate an uncertainty in γ of 6 % for the Cabauw campaign based on this analysis (difference between the minimum and maximum average values of all the simulated distributions shown in Fig. S12). However, it must be stressed that this estimate is limited by our Mie-theory-based calculations, which do not include potential effects due to morphologically complex particles such as freshly emitted fractal BC aggregates. That is, the estimate only covers the parametric uncertainty in our Mie-theory-based calculations.

The distributions of the time-resolved truncation correction factors calculated over the Cabauw campaign are clearly bi-modal (Fig. S12). This indicates that there were two limiting types of aerosols measured during the campaign: (i) fine-mode-dominated aerosol that only required a minor truncation correction; and (ii) aerosol with a substantial coarse-mode fraction that required a larger truncation correction. For further investigation, we extracted two subsets of data representing the fine-mode-dominated and coarse-mode-containing samples. This was done by selecting those aerosols whose coarse-mode number fractions (defined as NDp>1µm/Ntotal calculated from size distributions) are in the lower and upper quartiles of all the data, respectively. The median normalized size distributions for these two categories are plotted in Fig. S13. The figure clearly shows a coarse mode of particles with diameters between ∼0.6 and 5 µm that was present in the coarse-mode-containing samples but not the fine-mode-dominated aerosols. Despite the small number fractions, these coarse-mode particles can make substantial contributions to the scattering coefficients, and they produce a greater fraction of forward- and backward-scattered light, thereby affecting truncation disproportionately.

The distributions of the time-resolved γ values for the fine-mode-dominated and coarse-mode-containing groups of samples are displayed in Fig. S14 (for the same ranges of model inputs that were examined for the full Cabauw dataset in Fig. S12). As expected, the required truncation correction values are substantially smaller for the fine-mode-dominated group than the coarse-mode-containing samples. This follows from the normalized ensemble scattering-phase functions for each group, which are displayed in the right panel of Fig. S13. These functions were calculated from the median size distributions with Mie theory and an assumed refractive index of 1.59+0.01i (functions like these are required as inputs for the γ calculation, as shown in Fig. 3). It is seen that the phase function for the fine-mode-dominated category is less focused in the near-forward (scattering angles <25∘) and backward (scattering angles >125∘) scattering directions than that of the coarse-mode-containing category, which is why these samples are associated with lower γ values.

Figure S13 also compares the Mie calculated scattering-phase functions with measurements obtained by Espinosa et al. (2018) from a broad range of aircraft flights conducted throughout the USA. Three categories of measurements are shown: two categories of coarse-mode-containing aerosols (coarse categories 1 and 2) and one category of “fine” aerosols (which is comprised of measurements for aerosols classified as “urban”, “biomass burning” and “biogenic”, all of which were observed to have very similar scattering-phase functions). The calculated scattering-phase functions agree reasonably well with the corresponding measurements at near-forward scattering angles (comparing the fine-mode-dominated and “fine” categories; and the coarse-mode-containing and the two coarse categories). However, at near-backward scattering angles the Mie calculations predict proportionally more light scattering than is observed in the measurements. If it is assumed that the measurements by Espinosa et al. (2018) are reasonably representative of the average scattering-phase functions of these aerosol types also in European continental air masses, this comparison suggests that the Mie calculations would tend to slightly overestimate the truncation correction factors displayed in Figs. S12 and S14, and that the degree of overestimation would be greater for the coarse-mode-containing group than the fine-mode-dominated samples. Considering only the parametric uncertainty in the calculated γ values (i.e., that uncertainty related to the Mie truncation model inputs), we estimate values of 4 % and 9 % for the fine-mode-dominated and coarse-mode-containing groups, respectively, based on the minimum and maximum mean values for each of the groups shown in Fig. S14.

Considering all of the underlying CAPS PMssa uncertainties for the Cabauw dataset that are summarized in Table 1, it is clear that the largest individual source of uncertainty is related to the truncation correction that must be applied to bsca. To investigate the effect of this uncertainty on the ultimate derived babs, Fig. 8 compares these measurements against independent babs measurements obtained with a MAAP under three different truncation correction scenarios (corresponding to the three rows of the figure). It is worth recalling that the coefficients from both instruments were adjusted to standard temperature (273.15 K) and pressure (1 atm) for this quantitative comparison. The plot is further split into three columns, following the data grouping done in the previous subsection, so that the results for the fine-mode-dominated samples, the coarse-mode-containing samples, and the full dataset taken as a whole can be inspected separately. Each subplot contains an uncertainty envelope that represents the 95th percentile of hourly resolved theoretical uncertainty values calculated with the error model and inputs presented in Sect. 2.3 and Table 1. Uncertainties in γ of 4 %, 9 %, and 6 % were used for the fine-mode-dominated samples, coarse-mode-containing sample, and full dataset, respectively. The uncertainty envelopes are plotted around straight lines (shown as solid orange lines) representing the mean ratios of the CAPS to MAAP babs values displayed in each subplot. Mean ratios were chosen rather than standard linear fits to avoid biasing the plotted lines towards the highest measured values and because averaging to 1 h effectively removed the random noise at the lowest measured values (e.g., as can be seen in Fig. S9).

The first row of this figure (Fig. 8a, b, and c) displays the CAPS PMssa babs measurements that were processed with time-resolved truncation correction factors calculated with the Mie-theory-based model (Appendix A) with m set to 1.59+0.01i and l to 4.7 cm. This processing results in generally good agreement between the CAPS PMssa and MAAP babs measurements. Strong correlation is seen between the two independent measurements for all three subsets of the data (Fig. 8a, b, and c). However, on average, the absolute values of the two measurements are systematically offset by about 20 %. In particular, the mean ratios of CAPS babs to MAAP babs (i.e., as indicated by the solid orange line in each sub-figure) varies from 0.78 to 0.81. The precise reasons for these systematic offsets are not clear. One possibility is the geometry correction factor applied to the CAPS630b PMssa data (0.73) was inaccurate. However, we consider it unlikely that this factor alone could fully explain the observed discrepancy (it would need to be lower by ∼20 % – since babs is directly proportional to 1/α, Eq. (6) – and this is beyond the range of α values that have so far been measured for PMssa units; see Sect. 2.2.1). Another possibility is that the MAAP absorption coefficients were systematically overestimated (uncertainties of up to 7 % can be expected based on previous studies; see Sect. 3.1.1). Regardless of the precise reasons for the systematic offset between the measurements, it is important to note that the bias between the CAPS PMssa and MAAP measurements displays no, or only minor, dependence on SSA (which would be detectable as colour trends across the scatterplots). This indicates that it is likely that the CAPS PMssa bsca measurements have not been over-corrected for truncation, since such over-correction would affect the high SSA samples more than the low SSA samples. For these reasons, we believe that Fig. 8a, b, and c represent an example of reasonably well-estimated truncation correction. We note that this result was achieved with an l value of 4.7 cm, which is within the plausible range for this model input parameter based on the results presented in Sect. 5.

The second row of the figure (Fig. 8d, e, and f) displays measurements corrected with time-resolved γ values calculated with the Mie model with l=1 cm, which represents the best guess for this parameter based on the laboratory truncation curve measurements presented in Sect. 5. Under this truncation correction scenario, a sizeable number (17.3 %) of the hourly averaged coarse-mode-containing CAPS babs measurements shown in Fig. 8e are negative because bsca>bext, which is physically not possible. In addition, the bias between the CAPS PMssa and MAAP measurements displays an SSA dependence, most clearly seen in Fig. 8e and f. Together, these pieces of evidence indicate that the CAPS bsca measurements have been slightly overestimated, leading to absolute babs values that are biased low. Despite the poor agreement between the absolute CAPS PMssa and MAAP values for this truncation scenario, it is noteworthy that the two measurements still correlate well with each other.

The third and final row of this figure (Fig. 8g, h, and i) displays measurements that have been corrected for Rayleigh light scattering only. This means a constant γ value of 1 was applied to the bsca measurements. As shown in Fig. 2, this is the same as simply taking the bsca values output by the instrument firmware (i.e., bsca,Rayleigh), assuming that the firmware contained the correct α and β values. Here it is clear that the truncation correction has now been underestimated, leading to babs values that are greater than the corresponding y-axis values in the first row of the figure. Again, this is most clearly seen for the coarse-mode-containing group of samples (Fig. 8h). On average, the babs values for these samples are 1.86 times higher than the corresponding MAAP measurements.

To further examine the sensitivity of CAPS PMssa babs measurements to the scattering truncation effect, five additional truncation correction scenarios are displayed in the five rows of Fig. S15. These results generally support the results displayed in Fig. 8. Additionally, the following conclusions can be drawn: (i) for a given l value (4.7 cm in this case), variation of the ambient aerosol refractive index between 1.5+0.01i and 1.59+0.01i has only a minor effect on CAPS PMssa measured babs, which can be seen by comparing Figs. S15a–f and 8a–c, (ii) setting l to 0 cm in the model (the lower limit for l that we derived from Fig. 6) results in substantially overestimated truncation and bsca, which in turn leads to a substantial fraction (35.3 %) of negative babs values for the coarse-mode-containing group (Fig. S15h), and (iii) using constant γ values calculated from campaign-averaged size distributions (the averaged joined SMPS and APS particle size distribution, Fig. S15j–l; and the averaged SMPS distribution only Fig. S15m–o) generally leads to poorer results for the coarse-mode-containing samples compared to the corresponding results obtained with time-resolved truncation correction, but the results for the fine-mode-dominated samples are similar. This latter result is pertinent to field studies where time-resolved scattering-phase function information is not readily available (either directly or indirectly, e.g., in the form of measured size distributions), such that a user might be forced to use a constant truncation correction factor.

The Cabauw example demonstrates that the biggest hurdle that must be overcome when measuring atmospheric aerosol absorption coefficients with the CAPS PMssa is accurate accounting of the scattered light truncation effect. Unfortunately, there are many potential sources of errors in γ, as discussed in a general sense in Sect. 2.3. Even if the scattering-phase functions for the atmospheric aerosols being measured were known with high accuracy, γ would still carry substantial uncertainty related to the instrument geometry that must be considered in the truncation calculation, which we choose to represent through the glass sample tube extension length l. This uncertainty can lead to a broad range of different final babs outcomes, as we have shown by examining the sensitivity of babs to variation in the l parameter.

Although the uncertainty in γ cannot be totally avoided, its effect on babs can be substantially mitigated by restricting datasets to only those aerosol samples that do not display strongly forward-focused light scattering. For the Cabauw example, we successfully achieved this by separately analysing the fine-mode-dominated samples. The results for this group of samples were generally very consistent over the range of truncation correction scenarios we investigated. Considering the cases shown in Figs. 8 and S15, the correlation coefficients between CAPS PMssa and MAAP babs varied between 0.91 and 0.96, while the average ratios of the two measurements varied from 0.61 to 0.95 (for a sample size of 150). Although the precise reasons for the systematic offset between the CAPS PMssa and MAAP measurements are still unclear, the consistency of the results against the changes in γ is highly encouraging. This suggests that despite the remaining truncation uncertainties, the CAPS PMssa can still provide a reliable babs measurement for fine-mode-dominated atmospheric aerosols.

In contrast, the equivalent results for the coarse-mode-containing samples were more problematic. For this group and for the truncation correction scenarios shown in Figs. 8 and S15, the correlation coefficients between CAPS PMssa and MAAP babs varied between 0.88 and 0.95, while the average ratios of the two measurements varied widely from 0.03 and 1.86 (again for a sample size of 150). The truncation problems for coarse-mode-containing samples are 2-fold. Firstly, the scattering-phase functions for such samples are highly asymmetric, with enhanced forward- and backward-scattering, which means the corresponding γ values are large and more sensitive to small changes in particle size, shape, and/or composition (as demonstrated with respect to particle size, for example, by the steepness of the truncation curves displayed in Fig. 6 at particle diameters greater than ∼1 µm). This sensitivity is likely responsible for the large variability in the mean ratios of CAPS PMssa to MAAP babs that were obtained for the coarse-mode-containing samples across the tested truncation scenarios.

Adding to this is a second problem, which is that the Mie-theory-predicted phase functions are likely to be inaccurate for super-micrometer particles of complex morphology (e.g., mineral dust aerosols; Curtis et al., 2008). This problem could potentially be overcome by performing truncation calculations with scattering-phase functions that have been measured directly or calculated with consideration of complex particle morphologies (e.g., using more sophisticated optical models or potentially with scattering-phase functions parameterized according to the asymmetry parameter, as inspired by the truncation relationships presented by Liu et al., 2018). However, further work is required to determine how much such truncation correction methods could improve the reliability of CAPS PMssa measurements of aerosols with substantial coarse-mode number fractions.