Estimating CCN number concentrations using aerosol optical properties : 1 Role of particle number size distribution and parameterization 2 3

16 The concentration of cloud condensation nuclei (CCN) is an essential parameter affecting 17 aerosol-cloud interactions within warm clouds. Long-term CCN number concentration (NCCN) 18 data are scarce, there are a lot more data on aerosol optical properties (AOPs). It is therefore 19 valuable to derive parameterizations for estimating NCCN from AOP measurements. Such 20 parameterizations have been made earlier, in the present work a new one is presented. The 21 relationships between AOPs, NCCN and particle number size distributions were investigated 22 based on in-situ measurement data from six stations in very different environments around the 23 world. The parameterization derived here depends on the scattering Ångström exponent (SAE), 24 backscatter fraction (BSF) and total scattering coefficient (sp) of PM10 particles. The analysis 25 showed that the dependence of NCCN on supersaturation SS% is logarithmic: 26 NCCN  ((287 ± 45)SAE10ln(SS%/(0.093 ± 0.006))(BSF – BSFmin) + (5.2 ± 3.3))sp. 27 At the lowest supersaturations of each site (SS%  0.1) the average bias, defined as the ratio of the 28 AOP-derived and measured NCCN varied from ~0.7 to ~1.5 at most sites except at a Himalayan site 29 where bias was > 4. At SS% > 0.3 the average bias ranged from ~0.7 to ~1.3 at all sites. In other 30 Atmos. Chem. Phys. Discuss., https://doi.org/10.5194/acp-2019-149 Manuscript under review for journal Atmos. Chem. Phys. Discussion started: 28 February 2019 c © Author(s) 2019. CC BY 4.0 License.


Introduction
Aerosol-cloud interactions (ACI) are the most significant sources of uncertainty in estimating the radiative forcing of the Earth's climate system (e.g., Forster et al., 2007;Kerminen et al., 2012), which makes it more challenging to predict the future climate change (Schwartz et al., 2010).An essential parameter affecting ACI within warm clouds is cloud condensation nuclei (CCN) concentration, the number concentration of particles capable of initiating cloud droplet formation at a given supersaturation.Determining CCN concentrations and their temporal and spatial variations is one of the critical aspects to reduce such uncertainty.
CCN number concentrations (NCCN) have been measured at different locations worldwide (e.g., Twomey, 1959;Hudson,1993;Kulmala et al., 1993;Hämeri et al., 2001;Sihto et al., 2011;Pöhlker et al., 2016;Ma et al., 2014).However, the accessible data especially for long-term measurement is still limited in the past and nowadays due to the relatively higher cost of instrumentation and the complexity of long-term operating.As an alternative to direct measurement, NCCN can also be estimated from particle number size distributions and chemical composition using the Köhler equation.Several studies have investigated the relative importance of the chemical composition and particle number distributions (Dusek et al., 2006;Ervens et al., 2007;Hudson, 2007;Crosbie et al., 2015).For the best of our understanding, the particle number size distributions are more important in determining NCCN than aerosol chemical composition.This makes particle number size distribution measurements capable of Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2019-149Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 28 February 2019 c Author(s) 2019.CC BY 4.0 License.
serving as a supplementary of direct CCN measurements.
Considering the tremendous spatiotemporal heterogeneity of atmospheric aerosol, neither direct measurements of NCCN of the concentrations estimated from particle size distribution are adequate for climate research.In order to overcome the limitation of current measurements, many studies have attempted to estimate NCCN using aerosol optical properties (AOPs) (e.g., Ghan et al., 2006;Shinozuka et al., 2009;Andreae, 2009;Jefferson, 2010;Liu et al., 2014;Shinozuka et al., 2015;Tao et al., 2018).Most of these studies attempted to link NCCN with extensive AOPs, such as the aerosol extinction coefficient (σext), aerosol scattering coefficient (σsp) and aerosol optical depth (AOD).Both NCCN and sp are extensive properties that vary with a varying aerosol loading.The most straightforward approach to estimate CCN is to utilize the ratio between CCN and one of the extensive AOPs (e.g.AOD, σext, σsp).However, the ratio is not a constant.Previous studies have also pointed out that the relationship between NCCN and extensive AOPs are nonlinear.On one hand, Andreae (2009) reported that the relationship between AOD and CCN number concentration at the supersaturation of 0.4% (CCN0.4)can be written as AOD500=0.0027•(CCN0.4) 0.640 , which indicates AOT and CCN depend in a non-linear way on each other: for a larger AOD there are more CCN per-unit change in AOD.On the other hand, Shinozuka et al. (2015) indicated that the larger the extinction coefficient σext was, the fewer CCN were per unit change of σext.Some studies have also involved intensive aerosol optical properties, such as the scattering Ångström exponent (SAE), hemispheric backscattering fraction (BSF) and single-scattering albedo (SSA) to build up a bridge between the NCCN and AOPs.Jefferson (2010) used BSF and SSA to parameterize the coefficients C and k to present NCCN(SS%) = C×(SS%) k , where SS% is the supersaturation percent (Twomey, 1959).Liu and Li (2014) discussed how different aerosol properties affect the ratio of NCCN to σsp, i.e., RCCN/σsp based on in-situ and remotesensing data.Shinozuka et al. (2015) used SAE and aerosol extinction coefficient to estimate NCCN.Tao et al. (2018) used a novel method to derive the ratio RCCN/σsp which they named as ARsp, based on SAE and aerosol hygroscopicity using a humidified nephelometer.All the In this study, we will introduce a new approach to estimate NCCN, along with a brief discussion on how the ratio between NCCN and σsp is related to BSF.The AOPs needed in our estimation are σsp, BSF and SAE obtained using a 3-wavelength nephelometer, either the TSI 3563 or Ecotech Aurora 3000.The main goal of this study is to provide a parameterization for calculating NCCN using AOPs, and to probe the physical explanations behind this parameterization.The method will be applied to six different sites worldwide.

Sites and measurements
In-situ measurements of AOPs, PNSDs, and NCCN were conducted at SMEAR II in Finland, SORPES in China, and 4 ARM Climate Research Facility (ACRF) sites (Mather and Voyles, 2013).The locations and measurement periods are listed in Table 1.
The Station for Measuring Forest Ecosystem-Atmosphere Relations (SMEAR II) is located at the Hyytiälä Forestry Field Station (61°51' N, 24°17' E, 181 m above sea level) of University of Helsinki, 60 km north-east from the nearest city.The station represents boreal coniferous forest, which covers ~8 % of the Earth's surface.Total scattering coefficient (sp) and hemispheric backscattering coefficient (σbsp) of sub-1 μm and sub-10 μm particles are measured using a TSI-3563 3-wavelength integrating nephelometer at  = 450, 550, and 700 nm.The calibration, data processing, and calculation of AOPs followed the procedure described by Virkkula et al. (2011) and Luoma et al. (2018).NCCN was measured at supersaturations (SS%) of 0.1%, 0.2%, 0.3%, 0.5% and 1.0% using a DMT CCN-100 CCN counter, likewise in Schmale et al. (2017).A whole measurement cycle takes around 2 hours; data were interpolated to hourly time resolution to compare with other measurements.Particle number size distributions (PNSD) were measured with a custom-made Differential Mobility Particle Sizer (DMPS) system in size range 3-1000 nm (Aalto et al., 2001).A more detailed description of The Station for Observing Regional Processes of the Earth System (SORPES) is located at a suburb of Nanjing, a megacity in the Yangtze River Delta municipal aggregation (32°07'14'' N, 118°57'10'' E; ~40m a.s.l.).sp and bsp of total suspended particles (TSP) are measured with an Ecotech Aurora-3000 3-wavelength integrating nephelometer at  = 450, 525, and 635 nm as described by Shen et al. (2018).NCCN is measured using a CCN-200 dual column CCN counter at 5 supersaturations: 0.1%, 0.2%, 0.4%, 0.6% and 0.8%.The two columns make the same cycle simultaneously to cross-check with each other.Each cycle takes 30 minutes.PNSD in the size range of 6 -800 nm are measured with a DMPS built by University of Helsinki.
The US Atmospheric Radiation Measurement (ARM) Mobile Facility (AMF) measures atmospheric aerosol and radiation properties all over the world.The first AMF (AMF1) was deployed in 2005 with both a CCN counter and a nephelometer.Between 2011 and 2018, AMF1 is operated at four locations: Ganges Valley (PGH) in the Himalayas, Cape Cod, Massachusetts (PVC) in a coastal area of U.S., Manacapuru (MAO) inside the Amazonian rain forest, and Ascension Island (ASI) on the South Atlantic Ocean downwind from Africa.Three of them are accompanied by a scanning mobility particle sizer (SMPS; Kuang, 2016).The SMPS is also part of the Aerosol Observing System (AOS) running side by side with AMF1 since 2012.Both PNSD and AOPs are available simultaneously at PVC, MAO, and ASI.sp and bsp of sub-1 μm and sub-10 μm particles are measured at all AMF1 locations by integrating nephelometers (Uin, 2016a).The size range of the SMPS is around 11 -465 nm with slightly different ranges for different periods.NCCN is measured at different supersaturations, details are in Table 1.The supersaturations are typically calibrated before and after each campaign at an altitude similar to measurement site by instrument mentors according to CCN handbook (Uin, 2016b).Detailed information about each dataset and measurement site can be found on AOS handbook Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2019-149Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 28 February 2019 c Author(s) 2019.CC BY 4.0 License.(Jefferson, 2011) or ARM web site (http://www.arm.gov/) and references thereby.
Ganges Valley (PGH) is located in one of the largest and most rapidly developing sections of the Indian subcontinent.The aerosols in this region have complex sources, including coal and fuel combustion; biomass burning; automobile emissions; and dust.In monsoon seasons, dust dominates the aerosol mass due to transportation (Dumka et al., 2017;Gogoi et al., 2015).
PVC refers to the on-shore data set for the 'first column' of the Two-Column Aerosol Project (TCAP) on Cape Cod, Massachusetts, USA.This is a marine site but still significantly affected by anthropogenic emissions (Berg et al., 2016).
MAO refers to Manacapuru in Amazonas, Brazil.Manaus pollution plumes and biomass burning impact the background conditions alternately.During the period we selected for this study, no severe pollution episodes were observed.The σsp for PM10 never exceeded 250Mm -1 in this study.
Ascension Island (ASI) locates in the southeast Atlantic where westward transport of southern Africa biomass-burning aerosols emphases heavy aerosol loading.Air mass at this site usually a mixture with aged biomass-burning plume and sea-salt aerosol.The aerosol loading can be very low without plume, in this case, there is substantial uncertainty on the backscatter fraction.
The primary purpose of this study is to use as basic and readily accessible measurement data as possible to estimate NCCN.Aerosol optical properties are measured for different cutoff diameters, usually 1 μm, 2.5 μm, 10 μm or TSP.At several stations there are two sets of AOPs using two cutoff diameters.For this study we chose to use AOP data with the 10 μm cutoff or TSP that are more universally used than smaller cutoff diameters.

Data processing
Regardless of the time resolution of raw data, all the data in this study were adjusted into hourly averages before further analyses.Suspicious data within the whole dataset were removed according to the following criteria: 1) for the size distribution data, all the data with unexplainable spikes were removed manually; 2) for CCN measurements, insufficient water supply may cause underestimation of CCN, especially at lower supersaturation ratios (DMT, 2009).NCCN reading at lower SS% has a sudden drop a few hours before the similar sudden drop for higher SS% under such conditions, so data from such periods were removed; 3) if any obvious inconsistencies between the AOPs and PNSD or between the NCCN and PNSD were found on closure study, all the data in the same hour were removed.Special treatments were carried out for ASI dataset.There will inevitably be a considerable uncertainty in the backscattering fraction if zero point of either σsp or σbsp is inaccurate in very clean conditions.The measured sp was in agreement with that calculated from the PNSD with the Mie model.However, in the data bsp approaches 0.3 Mm -1 whenever sp approaches 0. Thus, we subtracted from back scattering coefficients a constant 0.3 Mm -1 and no longer used any data points with σsp < 2 Mm -1 for this site to assure the data quality.

Light scattering calculated from the particle number size distributions
Light scattering coefficients were calculated using the Mie code similar to Bohren and Huffman (1983) for SMEAR II.The refractive index was set to the average value of 1.517+0.019ireported for SMEAR II by Virkkula et al.(2011).The wavelength for Mie modeling was set to 550 nm, which is same as in the measurements.The whole size range of the DMPS or the SMPS, depending on the station, was used.The total scattering coefficient (σsp) and hemispheric backscattering coefficient (σbsp) represent the scattering phase function integrated over the scattering angles of 0-180° and 90-180°, respectively.The backscatter fraction (BSF) is the ratio between σbsp and σsp.

CCN number concentration calculated from the particle number size distribution
Under the assumption of fully internally mixed particles, the CCN number concentration calculated from the particle number size distributions (NCCN(PNSD)) is obtained by integrating the PNSD of particles larger than the critical dry particle diameter (Dm): At a given SS.Dm is a diameter above which all particles can act as CCN.For a selected dry diameter of a particle having given hygroscopicity is computed from the maximum of: The Critical Diameter Dm is the minimum dry diameter (Dd ) that ensure the -Köhler curve (Petters and Kreidenweis, 2007) to have one real solution: Here Dd is the dry diameter, σs/a is the surface tension of the solution/air interface, R is the universal gas constant, T is temperature, and D is the diameter of the droplet, ρw is the density of water, Mw is the molecular weight of water, and κ is the hygroscopicity parameter.S(D) in this particular case is set to the same supersaturation ratio as CCN being measured (e.g., 0.1%, 0.2%, 0.3%, 0.5% and 1.0% for SMEAR II ).
The accuracy of NCCN(PNSD) is affected by the treatment of κ.In this study, we are not trying to achieve an accurate value of κ but instead want to illustrate that even an arbitrary setting of κ can yield reasonable CCN concentrations.This approach is named as 'unknown chemical approach' in (Kammermann et al., 2010) and as 'Prediction of NCCN from the constant κ' in Meng et al., (2014).Both of them give a detailed discussion of how this approach performs.
Arbitrary κ is not performing as good as a proper κ when calculating NCCN , yet we believe that it is good enough to be an alternative to measuring CCN in the empirical estimation of this study.Wang et al. (2010) also claimed that NCCN(PNSD) may be successfully obtained by assuming an internal mixture and using bulk composition few hours after emissions.For SORPES, ASI and PVC, we simply set a global-average value of 0.27 for κ (Pringle et al., 2010;Kerminen et al., 2012).For the forest sites, SMEAR II and MAO, we set κ = 0.12, which is close to the value of κ for Aitken mode particles reported previously by studies at forest sites (Sihto et al., 2011;Hong et al., 2014).

Aerosol optical properties and CCN concentrations of simulated size distributions
For studying the relationships of particle size, NCCN and AOPs we generated unimodal particle number size distributions num(GMD,GSD) with varying the geometric mean diameter (GMD) and geometric standard deviation (GSD).For them we calculated the same AOPs with the Mie model as were obtained from the real measurements from the stations sp and bsp and from these the BSF at the wavelengths  = 450, 550 and 700 nm.NCCN was calculated simply by integrating number concentrations of particles larger than a critical diameter of 80 nm, 90 nm, 100 nm, and 110 nm.
One year average NCCN(CSK) at SMEAR II for SS=0.1% is 197/cm 3 , 53% higher than NCCN(mea) for the same period.Also, R 2 of the linear regression between NCCN(CSK) and NCCN(mea) is 0.78 at SS=1.0%, which means that Jefferson's method performs approximately as well for SMEAR II as at the other sites presented by Jefferson (2010).However, our motivation is to develop a method that needs no absorption data.

AOPs and CCN calculated from particle size distributions
Aerosol optical properties calculated from particle number size distributions matched well with the measured scattering coefficients in PM1.For sp larger than about 40 Mm -1 , the calculated values were slightly lower than the measured ones.The measured and calculated BSF also matched well with r 2 =0.93 for the data with sp >10.Another quality check of the CCN data is that the NCCN(PNSD) calculated from Eq.( 1) was consistent with the measured CCN number concentration NCCN(meas): for the linear regression r 2 was 0.80, 0.91 0.94 and 0.92 for SS=0.1%, 0.2%, 0.5% and 1.0%, respectively, and the corresponding slopes varied between 0.85 and ~1.2 depending on the value of SS%.The correlation between NCCN(PNSD) and NCCN(meas) was the weakest for the lowest set of supersaturation (0.1%), most probably because the measurement uncertainty is much higher at lower values of SS% compared with higher SS% for DMT CCN counter (Rose et al., 2008).

Relationships between AOPs and CCN
The correlation between NCCN and σsp was weak at SMEAR II, especially for higher supersaturations (Fig 1).In spite of this, when color-coded with respect to BSF, the relationship between NCCN and σsp becomes clear: the scatter plot points of NCCN grows almost linearly as a function of σsp for a narrow range of values of BSF.This indicates BSF can serve as a good proxy for describing the ratio between NCCN and σsp at SMEAR II.
Hereafter, we will use the term RCCN/σ = NCCN/σsp to describe the relationship between CCN concentration light scattering and similar to Liu and Li (2014).Note that this same ratio was defined as ARscat in Tao et al. (2018).RCCN/σ varies over a wide range of values, so a proper parameterization to describe it is of significance.At SMEAR II this approach yields R 2 of 0.70, 0.86, 0.75 and 0.55 for SS=0.1%, 0.2%, 0.5% and 1.0%, respectively, and the slopes (and intercepts) are 0.95(13), 0.92(28), 0.86(52) and 0.76(87), respectively.All slopes are slightly less than 1 and the intercept are slightly over 0. One explanation is that when both x and y have uncertainties, the least-squares method in the linear regression trend to underestimates the slope (Cantrell, 2008).NCCN(AOP1) overestimates (or underestimates) NCCN(meas) by 4.8%, 1.2%, -4.2% and -12.5% at the above specified supersaturations.For the overall dataset regardless of supersaturations, R 2 , slope(intercept) and difference between NCCN(AOP1) and NCCN(meas) are 0.73, 0.81(56) and -5.1% respectively.
R 2 between NCCN(AOP1) and NCCN(meas) is higher at lower supersaturations than at higher supersaturations in most of the scatter plots shown in Figures 3 and 4. The reasonable explanation is that the higher the supersaturation is the smaller are the particles that can act as CCN.The smaller are the particles the less do they contribute to both total scattering and backscattering and the higher is the relative uncertainty of both of them and thus also the uncertainty of NCCN(AOP1).
For a given station, if there are simultaneous data of N CCN (meas) and  sp for some reasonably long period, (6) can be adjusted.Instead of subtracting (0.097 ± 0.013) from BSF the minimum BSF = BSFmin in the data set will be used.Further, when BSF = BSFmin the factor a 1 (BSF -BSFmin) = 0 and N CCN (AOP2)  Rminsp where Rmin is the minimum RCCN/ in the data set.It follows that The derivation of ( 7) is shown in the supplement.In the data processing the 1 st percentiles of both BSF and RCCN/ are used as BSFmin and Rmin, respectively.Here the free parameters are a 1 , BSFmin and Rmin.The coefficient a 1 is positively correlated with SAE.The linear regressions of a 1 and the average and median scattering Ångström exponent of PM10 particles (SAE10) (Table 3) at the 6 sites in the analyzed periods yield a 1  (298 ± 51)SAE10 cm -3 /Mm -1 and a 1  (287 ± 45)SAE10 cm -3 /Mm -1 , respectively (Fig. 7).The uncertainties are large but, the main point is that the correlations show that a 1 and thus NCCN(AOP) is the higher the higher SAE10 is.Rmin was estimated by calculating the 1 st percentile of RCCN/ at each site at each SS%.The average and standard deviation of Rmin was 5.2 ± 3.3 cm -3 /Mm -1 .Consequently the parameterization becomes 2 10 min % ( ) (287 45)SAE ln ( ) (5.2 3.3) 0.093 0.006

Comparison of NCCN from the AOP parameterization and measurements
The parameterization in Eq ( 8) was applied to the data of the 6 stations and NCCN(AOP2) was At the site-specific lowest SS% the scatter plots of NCCN(AOP2) vs. NCCN(meas) of data from most stations get clustered along the 1:1 line, but for the Himalayan site PGH the parameterization yields significantly higher concentrations (Fig 8a).It was mentioned above that we applied also the Jefferson ( 2010) parameterization to SMEAR II data.At SS=0.1% the average NCCN(CSK) was 53% higher than NCCN(meas) and R 2 of the linear regression was 0.78 at SS=1.0%.The bias of our method at SS% = 0.1 was ~0.66 so it underestimated measurements by 34% and R 2  0.65 (Fig 8), lower than that of Jefferson (2010).At SS% > 0.3 the bias varied from 1.1 to 1.3.At the highest SS% the deviations from the 1:1 line are smaller also for PGH (Fig 8b).At PGH at the lowest SS% the bias is > 4 but decreases to ~1.1-1.2 at SS% = 0.4% and even closer to 1 at higher SS%.At SS% > 0.4% the AOP-derived NCCN is higher than the measured concentration at four sites with their bias varying between ~1.1 and ~1.3.For the US coastal site PVC the parameterization constantly underestimates the CCN concentrations by about 30%.For the Amazonian site MAO the bias is close to 1 at the lowest SS% but for the higher SS% it varies from 0.68 to 0.79.

Evaluation of the effect of particle size distribution to the parameterization
The linear relationships of the coefficients of Eq. ( 5) are so clear (Fig. 6) that there should be some common underlying reason.To study this we generated lognormal unimodal size distributions as explained in section 2.5.GMD was given logarithmically evenly-spaced values from 50 nm to 1600 nm and GSD was given two values: 1.5 representing a relatively narrow size distribution and 2.0 a wide size distribution.We then calculated AOPs, NCCN and RCCN/ for these size distributions.
The reasoning for the approach of estimating NCCN from sp and BSF can easily be explained by the similar variations of RCCN/ and BSF (Fig. 9).RCCN/ is the highest for the smallest particles, i.e. for GMD = 50 nm and it decreases with the growing GMD as also BSF.Note that the width of the size distribution has very strong effects on RCCN/: for the wide size distribution it is approximately an order of magnitude lower than for the narrow size distribution.
Note also that the rates of decrease of RCCN/ and BSF.We used this information for estimating particle sizes with a stepwise linear regression.An example is given by the linear regressions of RCCN/ vs. BSF calculated for 5 consecutive size distributions, first for those that have their GMDs from 50 nm to 100 nm and the second for those that have their GMDs from 100 nm to 200 nm (Fig. 10).Note that it is obvious that linear regressions are applicable for short intervals but do not well for the whole size range.The absolute values of the slopes and offsets are clearly lower for the larger particle size range.The particle size that is used for describing the size range of each regression we define here as the equivalent geometric mean diameter GMDe, the geometric mean of the range of the GMDs of the unimodal size distributions used for each regression.It will be shown below that GMDe is a mathematical concept helping in explaining the observed relationships, not an actual GMD of the particle size distribution at the sites.
For the wide size distributions the slopes and offsets of the regressions of RCCN/ vs. BSF decrease and increase, respectively, monotonically with an increasing GMDe in the whole size range studied here (Fig. 11).For the narrow size distribution the slope decreases to GMDe  300 nm and then increases which means there is no unambiguous relationship between them.
Note also that the ranges of the absolute values of the slopes and offsets of the wide and narrow size distributions are very different.However, they decrease and increase simultaneously.This is the link to the observations from the field stations.We plotted the offset.vs slope of the unimodal size distributions and those obtained from the linear regressions of the field data at Several observations can be made of Fig. 12. First, for the simulated wide size distributions the relationship of the offset and slope is unambiguous but not for the narrow size distributions at sizes GMDe > ~200 nm (Fig 12b).Secondly, the field data points obviously follow the lines of the simulations.This supports the approach for the interpretation of the relationships presented above (Fig. 6) for the coefficients in Eq. ( 5).Especially, note the similar ranges of b0 vs a0 in the coefficients this suggests that the coefficients of Eq. ( 5) depend on the GMD and GSD of the particle size distributions.
Most field data agree well with the b vs. a line of the unimodal wide size distribution with the lowest activation diameter of 80 nm.For instance, the PVC data point corresponding to the highest supersaturation has the highest slope (1970 cm -3 /Mm -3 , Table 2) and it is close to the above-mentioned line (Fig. 12a).The corresponding GMDe of the unimodal size distribution is also ~80 nm (Fig 12b).The SMEAR II high SS% offset vs. slope fits best with the corresponding lines of the narrow unimodal size distributions with all activation diameters and the corresponding GMDe  150 -180 nm.
At the lowest SS% the offset vs. slope points of all stations agree with the lines derived from the lines derived from the unimodal modes.This is interesting considering the high uncertainties involved in the regressions at the lowest SS% (Fig. 2).For ASI the slopes and offsets of the lowest and highest SS% are especially close to each other, closer than at any other station (Fig. 12a), and the corresponding GMDe  750 nm and 400 nm, respectively, when the GMDe vs. a relationship of any of the wide distributions is used (Fig. 12b).For PGH at the lowest SS% the slope is actually negative which is not obtained from the simulations at all so no GMDe cannot be given for it.

Aerosol size characteristics for all site
As it was shown above, particle size distributions affect the coefficients of the parameterization.It is therefore discussed here how the size distributions vary at the six sites of the study and whether they support the interpretations presented above.The size distributions are discussed using the particle number size distribution and the ratios of sp of PM1 and PM10 size ranges data from those stations where they are available.

Diurnal variation of particle number size distribution
Fig. 13a shows the averaged diurnal cycle of PNSD at the sites where either a DMPS or SMPS is available.New particle formation (NPF) events is a significant source of uncertainty in the prediction of NCCN (Kerminen et al., 2012;Ma et al., 2016).Complete NPF events start from a burst of sub 10 nm particles and continuous growing up to a few hundred nanometers.As a result, the size distribution varies significantly.NPF is one possible explanation of the poor NCCN-σsp correlation.
SMEAR II and SORPES are reported to have an appreciable frequency of NPF (Kulmala et al., 2004;Dal Maso et al., 2005;Sihto et al., 2006;Qi et al., 2015).Continuous growth in particle size at SORPES can usually last for several days after NPF (Shen et al. 2018).Similar growth patterns have also been observed in the Two-Column Aerosol Project (TCAP; http://campaign.arm.gov/tcap/; refers as PVC in this study) according to Kassianov et al. (2014).NPF is rarely observed in the Amazon forest where MAO is located.However, it does take place also at MAO as is shown in the diurnal cycle of PNSD.At ASI, there no evidence of NPF according to the PNSD diurnal cycle.
These observations of the NPF are compared with the bias and correlation coefficients of the parameterization discussed in section 5.1 (Fig. 8).The correlation coefficient of NCCN(AOP2) vs.
NCCN(meas) is the highest, R 2  0.8 at all SS% at ASI where no NPF takes place and clearly lower

Distribution of geometric mean diameter
Figure 13b presents the normalized distribution of the geometric mean diameter at SMEAR II, SORPES, PVC, MAO and ASI.It varies from 20 nm to 200 nm at all sites, with the most frequent GMD between ~70 nm and ~120 nm depending on the site.This shows clearly that the above-presented equivalent geometric mean diameter GMDe calculated assuming a unimodal size distribution is not a quantitative GMD of the size distribution, it is a mathematical concept that explains partially the relationships of RCCN/ and BSF.
The frequency distribution of GMD at SMEAR II is the widest among five sites with PNSD data available, followed by SORPES and PVC.At MAO the frequency distribution of GMD has two peaks in this study, different from that at ATTO in Amazonas (Schmale et al., 2018)).The lower peak is possibly due to the burst of sub-20 nm particles since they have little chance to grow to sizes that can serve as CCN.The second peak around 100 nm possibly represents the GMD without the burst of sub-20 nm particles and it is distinctly narrower than at SMEAR II, SORPES and PVC.
A comparison of the correlation coefficients of NCCN(AOP2) vs. NCCN(meas) (Fig. 8) and the widths of the GMD frequency distributions (Fig. 13b) does not show any clear relationships between them, other than that of ASI.The frequency distribution of GMD is the narrowest at ASI indicating that the average particle size does not change much throughout the whole period.This is in line with the low variation of the slope and offset of the RCCN vs BSF of ASI (Fig 12a).At ASI also the correlation coefficient of NCCN(AOP2) vs. NCCN(meas) is the highest, R 2  0.8 at all SS%.

Contribution of light scattering by sub-μm particles
There is one more measure related to particle size distribution, the ratio between σsp of sub-1 μm and sub-10 μm aerosol (sp(PM1)/sp(PM10)).At SMEAR II, the contribution of submicron particles usually varies within range 0.8~0.9 and it is the highest among all sites in this study.PVC has two that particles larger than 1 μm contribute a considerable fraction of light scattering.For SORPES sp(PM1)/sp(PM10) is not available.
Among those five sites, when sp(PM1)/sp(PM10) decreases, the correlation between BSF and RCCN/ decreases.At some sites (e.g., ASI) the BSF of PM10 is often be even larger than that of PM1 which is most probably an error in the measurements but it may also be due to non-spherical particles like sea salt and dust, which will blur the correlation between BSF and RCCN/.In such a case the increase of the amount of large particles leads to an increase of BSF and a decrease of RCCN/ which is opposite to the usual positive correlation between BSF and RCCN/ in this study.
Thus, the lower sp(PM1)/sp(PM10) may in principle result in a poor performance of our method.
However, a comparison of the correlation coefficients and the sp(PM1)/sp(PM10) frequency distributions of each site shows the opposite.At the highest SS% of each site the R 2 in a decreasing order is ASI, PGH, MAO, SORPES, SMEAR II, and PVC (Fig. 8d).The peaks of the frequency distribution of sp(PM1)/sp(PM10) are, in a growing order, ASI: 0.375, PGH: 0.625, MAO: 0.65, PVC: 0.825, SMEAR II: 0.875.Note that at SORPES there is only one size range measured.Of these only PVC and SMEAR II are not in the same order.On the other hand, the bias at the highest SS% has no clear relationship with sp(PM1)/sp(PM10): for MAO our parameterization underestimates the NCCN the most (bias  0.68) and for ASI it overestimates the most (bias  1.28).

Conclusions
The relationships between aerosol optical properties, CCN number concentrations (NCCN) and There are many previous parameterizations for doing just the same.As a starting point we used the parameterization presented by Jefferson (2010).That one needs also absorption measurements since it includes single-scattering albedo.We instead studied how the parameterization would look like if only total scattering and backscattering data were available.
The basic idea for the parameterization is that NCCN is proportional to σsp and a function of the backscatter fraction (BSF), as is also in the parameterization of Jefferson (2010).One clear difference is that our data analysis showed that the dependence on supersaturation is logarithmic, different from that of Jefferson (2010).Actually this result is qualitatively in line with the relationship between AOD and CCN reported by Andreae (2010).
The coefficients of the parameterization derived for the different sites showed that they appear to be linearly related to each other.A simulation with unimodal size distributions showed that the relationships are affected by the size and width of the size distribution and the activation diameter. YS.
-dependent parameterization for each measured supersaturation, NCCN(AOP1) The first step in the development of the parameterization was to calculate linear regressions of RCCN/σ vs BSF.RCCN/σ depends clearly on BSF (Fig. 2) as RCCN/σ = a BSF + b (3) At SMEAR II the correlation between BSF and RCCN/σ is strong when σsp > 10 Mm -1 .At σsp < 10 Mm -1 the uncertainty of the nephelometer is higher which may at least partly explain the lower correlation.For each dataset and individual supersaturation, a and b the slope and offset Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2019-149Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 28 February 2019 c Author(s) 2019.CC BY 4.0 License. of the linear regressions has a different value as presented in Table 2.The parameterization gives the formula for calculating NCCN(AOP), ie, NCCN calculated from measurements of AOPs: NCCN(AOP1) =(aSS% *BSF+ bSS%) • σsp (4) The subscript 1 for AOP1 indicates the first set of parameterization.Scatter plots of NCCN(AOP1) vs NCCN(meas) are presented for the supersaturations used at the SMEAR II CCN counter in Fig 3 and for the highest and the lowest SS% used at the other stations in Fig 4.
compared with the NCCN(meas) at the supersaturations used in the respective CCN counters.The results are presented as scatter plots of of NCCN(AOP2) vs. NCCN(meas) (Fig 8a and 8b), the bias of the parameterization calculated as NCCN(AOP2)/NCCN(meas) (Fig 8c) and the squared correlation coefficient R 2 of the linear regression of NCCN(AOP2) vs. NCCN(meas) (Fig 8d).
Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2019-149Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 28 February 2019 c Author(s) 2019.CC BY 4.0 License. the supersaturatios presented in Table 2 and below it the GMDe vs. the slopes of the regressions of the unimodal size distributions (Fig 12).In Fig. 12 also the effect of the choice of the activation diameters of 80 nm, 90 nm, and 110 nm is shown.
Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2019-149Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 28 February 2019 c Author(s) 2019.CC BY 4.0 License.peaks in the sp(PM1)/sp(PM10) distribution, the peak around 0.2 corresponds to air masses from the sea, with a very low scattering coefficient and NCCN.By ignoring the cleanest air masses (σsp<5 Mm -1 ), the fraction of sp(PM1)/sp(PM10) is usually around 0.8, which is just slightly lower than at SMEAR II.At PGH and MAO, the distribution of the ratio is wider, and the peak position is around 0.65.The overall contribution of sub-µm particle light scattering at PGH is moderate among the sites in this study.At ASI sp(PM1)/sp(PM10) is the lowest among all sites in this study, indicating particle number size distributions were investigated based on in-situ measurement data from six stations in very different environments around the world.The goal of the work was to find Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2019-149Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 28 February 2019 c Author(s) 2019.CC BY 4.0 License.a parametrization to obtain NCCN from sites where AOPs are measured but no CCN counter is available.

Figure 3 .
Figure 3.Comparison between NCCN(AOP1) and NCCN(meas) at SMEAR II.NCCN(AOP) was calculated by using the constants a and b in Table2for each supersaturation.

Figure 5 .
Figure 5.The coefficients a and b of each station (Table2) as a function of supersaturation.

Figure 11 .
Figure 11.Size distributions of the coefficients of the linear regressions of RCCN/( = 550 nm) vs backscatter fraction BSF ( = 550 nm) of narrow and wide size distributions.a) slopes of RCCN/ vs. BSF, b) offsets of RCCN/ vs. BSF.RCCN/ was calculated assuming particles larger than 90 nm get activated.The regressions were calculated for 5 consequtive size distributions.GMDe is the geometric mean of the range of the unimodal size distributions used for the regressions.
KL, HK and PA carried out measurements, data collection and maintenance of measurement data of SMEAR II in Finland.YS, XC, XQ, WN and XH carried out measurements, data collection and maintenance of measurement data of SORPES in China.MK and TP provided the funding for YS in Finland.MK provided funding for the measurements and research at SMEAR II in Finland.TP and VMK formulated the goals of the research and supervised it.Atmos.Chem.Phys.Discuss., https://doi.org/10.5194/acp-2019-149Manuscript under review for journal Atmos.Chem.Phys.Discussion started: 28 February 2019 c Author(s) 2019.CC BY 4.0 License.