The SMEAR II measurement site at Hyytiälä is located at 61°51^′ N, 24°17^′ E, 181 m above sea level, and represents a boreal forest environment with some anthropogenic influence, particularly from the southern direction where many industrialized areas within Finland, Russia, and continental Europe are located (Patokoski et al., 2015; Riuttanen et al., 2013; Yttri et al., 2011; Tunved et al., 2006). The extent of the representativeness of SMEAR II for boreal forest environments varies seasonally and with air mass origin. The station is surrounded by mixed forest which covers 80 % of the land within a 5 km radius and 65 % within a 50 km radius (Williams et al., 2011). Primary local emission sources include a sawmill situated to the northeast and a pellet factory located around 6–7 km southeast of SMEAR II. Overall, the station can be considered a rural background site because the nearest major city, Tampere, is located about 60 km southeast of the measurement location. During the summer, local BVOC emissions (Hakola et al., 2012; Feijó Barreira et al., 2018), primarily those of monoterpenes, act as a major source of SOA at the station (Heikkinen et al., 2020; Heikkinen et al., 2021). New particle formation (NPF), which is an important process contributing to NCCN globally (e.g., Merikanto et al., 2009), is commonly observed at SMEAR II, especially in spring and fall (Nieminen et al., 2014). Sulfuric acid, bases and low-volatility BVOC oxidation products (e.g., Kulmala et al., 2014; Lehtipalo et al., 2018; Yan et al., 2018), have been identified as critical precursors for NPF at the site. During the winter, aerosol particles observed at the site are mainly from long-range transport (Riuttanen et al., 2013) and are frequently cloud-processed (Isokääntä et al., 2022). During this season, aerosol particles contain a larger inorganic component (about 36 % as compared to 23 % in summer, Heikkinen et al., 2020) increasing their hygroscopicity. However, during the winters, the increased contribution of black carbon (about 15 % as compared to 6 % in summer, Luoma et al., 2021), a hydrophobic aerosol component, decreases the overall hygroscopicity of the particles. SMEAR II is unique due to the comprehensive set of long-term measurements, crucial for answering questions related to aerosol-cloud interactions, which have been conducted for several years (Kulmala, 2018). Although facilities for measuring aerosol size distribution and CCN have existed for a long time (since 1996 and 1998, respectively), long-term composition measurements have become available more recently (Luoma et al., 2021; Heikkinen et al., 2020). This advancement has been due to the development and deployment of the ACSM and an aethalometer setup which provide near-real time data on the organics, sulfate, nitrate, ammonium, chloride and equivalent black carbon (eBC) in sub-micrometre aerosol particles (see also Sect. 2.1.4).

At SMEAR II, a Differential mobility Particle Sizer (DMPS) has been used for particle number size distribution (PNSD) measurements in a size range from 3 to 1000 nm since 1996 (Aalto et al., 2001). The DMPS data has the time resolution of 10 min. The data used in this study were accessed from SmartSMEAR database (https://smear.avaa.csc.fi/download, last access: 28 April 2022) for years 2016–2020 (see Fig. 2a). Medians of the size distribution data were taken over the start and end time periods of the respective co-located CCN measurements (see Sect. 2.1.3).

The twin-DMPS system consists of two Vienna-type Differential Mobility Analyzers (DMAs), each designed to classify aerosol particles into size bins across two distinct size ranges: 3–40 and 20–1000 nm. The sizing is based on the electrical mobility of the sampled and charged aerosol particles. Air is sampled at a height of 8 m above ground level with a common aerosol inlet. The common inlet line has a diameter of 100 mm and a flow velocity of 0.5 m s⁻¹. The sample flow for the instruments is taken from the centreline. The aerosol flow rates in the DMAs are 4 and 1 L min⁻¹, respectively. The sheath flows, with flow rates of 20 and 5 L min⁻¹, are dried to maintain RH of less than 40 %, while the aerosol flows are not dried. The particle concentration following each DMA is measured using Condensation Particle Counters (CPCs). For small particles (3–40 nm), a TSI 3025 CPC model was utilized (later changed to model TSI3776 after October 2016), while a TSI 3750 CPC is used for the detection of the larger particles in the size range 20–1000 nm.

As a first step toward the inverse closure (see also Sect. 2.2.3), we applied a Python implementation (Khadir, 2023) of the modal-fitting algorithm described by Hussein et al. (2005) to decompose the measured aerosol size distributions into two modes. The algorithm takes size distribution as input and returns the lognormal parameters (number concentration, geometric standard deviation, geometric mean diameter GMD) of different modes as output. While the algorithm would allow fitting up to four modes, bimodal fits (Aitken and accumulation mode, respectively; Supplement Fig. S1a) were selected to avoid overfitting (see also Liwendahl, 2021). The bimodal fits enabled us to reproduce the aerosol size distributions with a high correlation (Pearson correlation coefficient R= 0.99) between the observed total particle number concentration and that calculated from the fitted parameters (Fig. S1b).

The time series of observed NCCN were obtained using a CCN-100, a continuous-flow streamwise thermal-gradient CCN counter (CCNc), commercially provided by Droplet Measurement Technologies (Roberts and Nenes, 2005). The CCNc can be used in either monodisperse or polydisperse mode, where the former is utilized to determine size-segregated NCCN, as detailed in Paramonov et al. (2013). In contrast, the polydisperse mode, employed here, measures the overall NCCN at a given supersaturation.

The CCNc consists of a saturator unit and an Optical Particle Counter (OPC). The saturator is a vertically oriented flow tube, into which aerosol-laden sample air is introduced surrounded by a particle-free sheath air flow (1/10 flow ratio) under laminar flow conditions, forming a well-defined central flow path. The inner walls of the tube are wetted and subjected to a controlled temperature gradient. The sheath air flow is saturated with water vapor at the inlet temperature. A positive temperature gradient is maintained at the saturator column, inducing a quasi-constant supersaturation profile for a specific temperature difference. As the laminar flow progresses through the column, water vapor and heat diffuse from the moist walls toward the center. The effective supersaturation is influenced by factors such as flow rate, pressure, and temperature gradient. While moving through the tube, aerosol particles absorb water and grow and those particles with critical supersaturations lower than the centerline supersaturation are activated as cloud droplets. Droplets larger than 0.75 µm in diameter are detected by the OPC at the exit of the tube and those exceeding 1 µm are considered to be activated CCN. To measure at different supersaturations, the temperature gradient is increased in steps while the flow rate is held constant. Both polydisperse and monodisperse CCN concentrations were measured at each supersaturation setpoint (1.0 %, 0.5 %, 0.3 %, 0.2 %, 0.1 %). At each setpoint, the cycle includes 300 s polydisperse and 600 s monodisperse measurements, with additional stabilisation time after changing supersaturation, yielding a time resolution of about 2 h for this data set. Quantification and discussion of typical uncertainties related to the supersaturation and hence NCCN measured with this instrument are presented in e.g., Rose et al. (2008) and Topping et al. (2005). At SMEAR II, the air to the CCNc is sampled 8 meters above the ground level and features the same inlet as the DMPS (see Sect. 2.1.3.). The aerosol flow rate is 0.5 L min⁻¹, which is split into sheath flow of 0.45 L min⁻¹ and sample flow of 0.045 L min⁻¹. For quality assurance of the CCNc data, the CCNc calibration is conducted approximately twice a year using nebulised, dried, charge-equilibrated and size-segregated ammonium sulfate aerosol following procedure as per Rose et al. (2008).

Estimates of smallest activation dry diameter (Dact) were derived using the combination of the DMPS and the CCNc data by integrating the PNSDs from their maximum diameters to the diameter at which the integrated particle number was equal to the measured NCCN. Dact was then calculated by interpolating between the two adjacent size bins (Furutani et al., 2008). Essentially, variations in activation diameter reflect differences in the chemical composition of aerosol particles: the more hygroscopic the aerosol, the smaller the activation diameter.

An Aerosol Chemical Speciation Monitor (ACSM; Ng et al., 2011) was used at SMEAR II to measure the mass concentrations of non-refractory submicron particulate matter (NR-PM₁). The ACSM quantifies ions originating from non-refractory organic and inorganic species and reports them as mass concentrations of sulfate, nitrate, ammonium, and chloride ions, along with total organic aerosol mass. Briefly, the ACSM samples dried ambient air through a critical orifice (100 µm in diameter) with a flow rate of 1.4 cm³ s⁻¹ to an aerodynamic lens (Liu et al., 1995a, b), which focuses a submicron particle beam and directs it to the instrument vaporization and ionization chamber. The lens efficiently transmits particles with vacuum aerodynamic diameters (Dva) ranging from approximately 75 to 650 nm, yet it also passes through particles up to 1 µm in Dva with a less efficient transmission. These aerosol particles then undergo flash vaporization at 600 °C and are subsequently ionized using electron impact ionization (70 eV) and the mass spectrum is obtained with quadrupole mass spectrometry. While the vacuum system of the ACSM efficiently reduces the amount of air molecules entering the instrument detection unit, their distinction from the aerosol components is required. For this purpose, the ACSM contains a 3-way valve system to routinely measure the signals obtained from particle-free air, and this background is subtracted from the particle-laden sample. The detailed description of the ACSM measurements performed at SMEAR II since 2012 is provided in Heikkinen et al. (2020), which includes descriptions of the instrument ionization efficiency calibrations, collection efficiency corrections and data processing. The ACSM measurements were conducted < 100 m away from the DMPS, CCNc and aethalometer measurements in a separate container. A PM_2.5 cyclone was installed on the container roof, and the ∼ 3 m long inlet line had an additional make-up flow of 3 L m⁻¹. The air was dried to < 30 % RH with a Nafion dryer. The original time resolution of the ACSM data is ∼ 30 min.

We combined the ACSM measurements with measurements of eBC. The eBC concentration was determined based on PM light absorption measured by an aethalometer (Magee Scientific, models AE31 and AE33). For the period in question here (2016–2020), the instrument was changed in the middle as the old instrument broke down. AE31 operated until the end of 2017 and AE33 started measuring in the beginning of 2018. An aethalometer is a filter-based instrument and it measures aerosol light absorption at seven wavelengths (370, 470, 520, 590, 660, 880, and 950 nm). The aethalometer data were corrected for measurement artefacts caused by collecting the particles in a filter medium, the so-called loading effect and scattering caused by the filter material: AE33 applied the inbuilt dual-spot correction (Drinovec et al., 2015) with multiple scattering correction factor 1.39 whereas the AE31 data were corrected as suggested by Virkkula et al., 2007 with multiple scattering correction factor 3.14 (derived by Luoma et al., 2021, for SMEAR II data). The eBC concentration was derived from the absorption at 880 nm channel by using mass absorption cross-section of 7.77 g m⁻² for AE33 data (the default value suggested by the manufacturer) and 4.8 g m⁻² for AE31 data (derived from 6.6 g m⁻² at 637 nm used for multi-angle absorption photometer, which was used as a reference in Luoma et al., 2021). The head of the sampling line was located 4 m above the ground. The concentration of eBC was measured for PM₁₀. Sample air was dried with a Nafion dryer and data was marked as invalid if the relative humidity inside the instrument increased above 40 %. The aethalometer data was converted to STP conditions (273.15 K, 1013.25 hPa).

The published ACSM and eBC measurements data are averaged over 1 h intervals, but to align with the CCN measurements, the data set was further converted to the 2 h time grid by taking a median of the mass concentrations of each of the measured species over the time window of each CCN spectrum measurement. The time series (7 d running median) are shown in Fig. 2c. The data coverage is higher for the eBC data compared to the ACSM data, which has fewer observations during wintertime.

Figure 2 presents the overall data coverage along with the key aerosol properties observed (see Fig. S2 for the number of data points across different seasons). As mentioned earlier, SOA formation and NPF events lead to higher particle number concentrations during spring and summer. This is also reflected in the variability of CCN, particularly at higher supersaturations (see Fig. 2b), while lower seasonal variation is observed at lower supersaturations (SS = 0.1 %), where only larger particles (> 200 nm, see Fig. S3 and Table S1 in the Supplement) are activated. This suggests that most changes in aerosol particle number and chemical composition occur among smaller particles (Aitken and nucleation modes) between the winter and growing seasons (spring and summer). In terms of chemical composition, organics dominate the aerosol mass (see Fig. 2c), especially during the growing seasons, followed by sulfate and ammonium ions, with nitrate and black carbon contributing only minor fractions. However, given the significant seasonal variation in overall aerosol properties at the site, we present the results according to a seasonal classification. In this framework, March, April, and May represent spring; June, July, and August represent summer; September, October, and November correspond to autumn; and December, January, and February correspond to winter.

The classical Köhler theory (Köhler, 1936) utilizes information about the composition and size of aerosol particles. It estimates the critical supersaturation level SS_crit and wet particle diameter at which an aerosol particle becomes activated and grows through condensation to form a cloud droplet. The Köhler equation comprises two terms (see Eq. 1): one accounting for the influence of solutes (the soluble fraction of aerosol particles), which tends to reduce the equilibrium saturation ratio S (defined as 1 + SS), and the other known as the Kelvin term, which represents the increased surface tension over a spherical surface. In an aqueous solution, if P (Pa) is the partial vapor pressure of water and Ps (Pa) saturation vapor pressure of water over a pure flat liquid, the equilibrium saturation ratio S=P/Ps is represented as 1S=awexp⁡σρ4MwRTDp,wet where aw is the activity of water in the solution, ρ is the density of the solution (kg m⁻³), Mw is the molar mass of water (0.018 kg mol⁻¹), σ (N m⁻¹) is the surface tension of the solution, R is the universal gas constant (8.314 J mol⁻¹ K⁻¹), T is temperature (K), and Dp,wet is the diameter of the droplet (m). To facilitate the comparison to previous work, we use the modified version of Köhler theory (Eq. 1) described by Petters and Kreidenweis (2007) to calculate the activation dry diameter (related to the total amount of soluble mass) for a particular supersaturation SS (i.e., S-1, referred to as the κ-Köhler framework 2S=Dp,wet3-Dp,dry3Dp,wet3-Dp,dry31-κexp⁡σρ4MwRTDp,wet where Dp,dry is the dry diameter of the dry aerosol particle with a given composition described by a unitless hygroscopicity parameter κ. In our calculations, we have assumed that the density and surface tension of the solution are equivalent to those of water (1000 kg m⁻³ and 0.0728 N m⁻¹ respectively). Additionally, we have considered a constant ambient temperature (T) of 298.48 K for all seasons, corresponding to the median temperature inside the measurement hut.

Assuming internally mixed aerosol particles, the net hygroscopicity parameter κ for a mixture of n different chemical species is expressed as the linear combination of the individual species κi weighted by their respective volume fractions εi in the dry particle (Stokes and Robinson, 1966): 3κ=∑iεiκi.

The volume fractions εi of the individual components were calculated from the measured mass concentrations, mi, and their respective densities, ρi 4εi=miρi∑miρi

Assuming an internally mixed aerosol population is a key assumption made in this study. According to Paramonov et al. (2015), the aerosol in Hyytiälä indeed shows some seasonal and size-dependent mixing state characteristics. Specifically, they report that particles in the ∼ 75–300 nm range are internally mixed during late spring and early summer (May–July), with a very small CCN-inactive fraction (∼ 0.2 %). For the rest of the year, the aerosol becomes partially externally mixed, with the CCN-inactive fraction increasing to ∼ 6.6 %. However, within each size range – either below or above 100 nm – the κ distributions are relatively consistent, suggesting that particles are mostly internally mixed within those size classes.

In the forward closure, NCCN at supersaturations of 0.1 %, 0.2 %, 0.3 %, 0.5 %, and 1.0 % (corresponding to the supersaturations set in the CCNc, henceforth referred to as SS_CCNc) are predicted using observations of the aerosol number size distribution from the DMPS. As discussed above, two different assumptions about the hygroscopicity of the aerosol mixture were tested: (1) assuming constant hygroscopicity of 0.18 (κ0.18) (2) assuming mixture hygroscopicity (Eq. 4) using chemical composition information from the ACSM and aethalometer measurements (κbulk). κbulk therefore, does not depend on particle size, but is variable in time. For deriving κbulk the observed aerosol chemical composition was utilized, assuming that all sulfates are present as ammonium sulfate (NH4)2SO₄ (AS) and the observed nitrate was distributed between ammonium nitrate NH₄NO₃ (AN) and organic nitrate (ON), estimated using the method explained in Farmer et al. (2010) (see Supplement Sect. S1 and Fig. S4). For the calculation of the AS and AN mass concentration, only the measured sulfate and nitrate mass concentrations were used. The ammonium mass concentration required for yielding ion balance within the particles was calculated (see Fig. S5; Zhang et al., 2007). We acknowledge that the assumption that sulfate is present solely as AS can cause underestimations of aerosol hygroscopicity at SMEAR II (e.g., Riva et al., 2019). Finally, to retrieve the volume fractions of organics, AS, AN, ON and eBC from their estimated mass concentrations, the density information for each species is required. The chosen densities are shown in Table 1 along with the κi for each species.

The critical supersaturation SS_crit was then calculated for each of the size bins measured by the DMPS using κ-Köhler theory, assuming a uniform composition throughout the size distribution. Particles for which the calculated SS_crit was lower than the individual SS_CCNc were then considered as CCN corresponding to the respective SS_CCNc value. Linear interpolation was applied to estimate the exact activation diameter within a given size bin (see Lowe et al., 2016). The CCN spectra estimated by the forward closure were then compared to the observations made by the CCNc for the two different hygroscopicity assumptions i.e. κbulk and κ0.18.

In the inverse closure, our objective was to minimize the differences (e.g. through Normalized Root Mean Squared Error, NRMSE, see Sect. 2.2.4) between predicted and observed NCCN, while optimizing the size-dependent chemical composition and hygroscopicity parameter. More specifically, we assumed the size distribution to consist of internally mixed and log-normally spaced Aitken and accumulation modes.

The inverse closure and thus the optimization was performed implementing two different methods, namely the Nelder-Mead and the DREAM-MCMC (DiffeRential Evolution Adaptive Metropolis Markov Chain Monte Carlo) algorithms (see the descriptions of the Nelder–Mead and DREAM-MCMC algorithms). In both optimization methods, all AN and AS masses were combined and treated as inorganic mass for simplicity. The net ρ and κ of the inorganic fraction were derived using the corresponding observed mass fraction. While the κ for AN is slightly higher than that of AS (Table 1) and the density of AN is slightly lower of that of AS (Table 1), we consider this as a reasonable simplification given the low AN concentration at the site. Again, all ON is assumed to have the same κ and ρ as the organics (Table 1). Another key simplification is that eBC is assumed to have the same mass fraction in both modes.

The optimization procedures based on Nelder–Mead and DREAM-MCMC are illustrated in Fig. 3. In both approaches, the derivation of modal optimized hygroscopicity parameters (κoptAitken and κoptaccumulation from Nelder-Mead method and κMCMCAitken, κMCMCaccumulation from DREAM-MCMC, referred to as κopt and κMCMC for simplicity) begin with obtaining a bimodal fit of the aerosol number size distribution into Aitken and accumulation modes (see Sect. 2.1.2). Next, the fitted lognormal size distribution was binned onto the same diameter axes as the observational data, and the number of particles in each bin was scaled to match the particle number in measured size distribution (see a demonstration in Fig. S6 and Sect. S2). This way, the number contributions of the Aitken and accumulation modes to the observed aerosol size distribution could be estimated for each time point. Second, the masses of the Aitken and accumulation modes were estimated by approximating the density of both modes by the bulk density. The total masses of organics, inorganics and eBC to be distributed to the measured size distribution were then calculated using the mass fractions derived from the ACSM and aethalometer measurement. Finally, the Aitken vs. accumulation mode compositions, and hence κopt or κMCMC, were determined through optimization (see also Sect. S3 for details).

The Nelder–Mead simplex algorithm (Gao and Han, 2012) is suitable for both one-dimensional and multidimensional optimization problems and is relatively fast in our application. In our case, we need to optimize only one variable (the fraction of total organic mass in Aitken mode, morg,Ait) and the remaining masses can be derived from it through mass closure constraints – assuming PNSD to stay constant throughout each CCN measurement cycle. For each time step, the optimization begins with an initial simplex of three trial values of morg,Ait, and the NRMSE is evaluated at each point. The worst-performing value is reflected across the midpoint of the better two to explore whether a more accurate estimate can be found in the opposite direction. If this improves the fit, the algorithm attempts an expansion, pushing further in the same direction. If reflection does not improve the result, a contraction step is taken to move closer to the midpoint. If neither reflection nor contraction improves the outcome, the simplex undergoes shrinkage, tightening around the best-performing solution to focus the search locally. This process continues until the optimization converges, resulting in an estimate of morg,Ait that minimizes the NRMSE between modeled and observed CCN concentrations. Note that Nelder–Mead works well for simple, low-dimensional problems like optimizing just one parameter (e.g., morg,Ait), but it starts to struggle as the number of variables increases and have a tendency for converging to local minima.

In order to assess the importance of the variability of the bimodal size distribution parameters within each CCN cycle, we conducted a second inverse-closure experiment with the number concentration and mean diameter for both modes as additional optimization parameters (simultaneously with morg,Ait). Since optimizing both size distribution parameters and composition introduces a more complex and higher-dimensional parameter space, and we are interested parameter uncertainty, we use a Bayesian inference approach to estimate the parameter posterior distributions. Specifically, we chose the DiffeRential Evolution Adaptive Metropolis Markov Chain Monte Carlo (DREAM-MCMC) algorithm (Vrugt et al., 2009), which has been previously used for inverse CCN-closure studies in idealized cases (Partridge et al., 2012) and is available in the Python PINTS library (Clerx et al., 2019). DREAM-MCMC is an efficient MCMC method (Metropolis et al., 1953; Gelfand and Smith, 1990) that evaluates multiple Markov chains in parallel and automatically adapts its proposal strategy during sampling, making it particularly efficient for correlated, multi-modal problems such as aerosol-cloud microphysical interactions. To know more about MCMC and Bayesian inference, see Sect. S4.

We initialized the MCMC optimization with Cauchy priors for each parameter (see Sect. 5), centered on the median values of the fitted bimodal size distributions for each CCN cycle, specifically, the number concentration and GMD. For chemical composition we used the median of the ACSM observations during each CCN spectrum cycle. The scale value was the smaller of either 1 (resulting in a Student-t distribution) or the median absolute deviation (MAD) of the observations within the given CCN cycle. The priors were truncated to positive values only. We also constrained the total aerosol mass in each mode to remain within ±10 % of the total mass observed by the ACSM and aethalometer.

We used a heteroskedastic Gaussian likelihood function, which means that the highest likelihood is typically where the parameters provide the least squares fit to the CCN observations, analogous to minimizing the NRMSE described above. The likelihood is defined as 5Lθ|Y=∏i=1n12πsi2exp⁡-12si-2yi-ϕiθ2 where si is standard deviation of the measurement error, which we assume is 10 % of the CCN observations at each supersaturation value yi and ϕi is the model predictions of CCN spectra at each super-saturation given the calibration parameters θ (the log-normal parameters and mass fraction). We performed the optimization in a log-transformed parameter space, which improves sampler efficiency by normalizing scale differences between parameters. For each CCN observation window, we ran five chains with 40 000 iterations per chain, of which the first 15 000 were used as burn-in/adaptation. Up to two chains were discarded if they deviated strongly in central tendency after burn-in, and the last 20 000 steps of all accepted chains were then used to calculate posterior statistics. Convergence was assessed with the R^-statistic (Gelman and Rubin, 1992), using a relaxed threshold of R^<2.5 for all five parameters to retain a window in the analysis. The R^-statistic compares the variance within chains to the variance between chains; values close to 1 indicating well-mixed, converged chains. We used a relaxed threshold because the R^-statistic is quite conservative and because our problem has high correlation between parameters and the potential for multi-modality if there are multiple distinction aerosol populations within one window, which is penalized by the R^-statistic but realistic in this case. Overall, 19 % of windows were discarded due to high R^-statistic values. Even with the relaxed threshold, some windows were excluded where the MCMC identified reasonable parameter values and CCN spectra but the chains failed to mix well. Examples of the chain evolution and posterior parameter distributions are discussed in Sect. S5.

Unlike the Nelder–Mead optimization method, which uses the median of the size distribution during the CCN cycle period, the DREAM-MCMC setup requires the variability of the size distribution as input. To account for this, we calculate the median absolute deviation (MAD) of each lognormal parameter for every CCN cycle observation. The overall distribution of MAD values for the full 5-year dataset is presented in Sect. S6 and Fig. S10. MAD for individual CCN cycle period is calculated as follows:

Let Ic=[tcstart, tcend) be the time window for CCN cycle c; For a given lognormal parameter, k(among geometric mean diameter (GMD), geometric standard deviation (SD) and number concentration in each mode; so total 6 parameters), collect the samples inside this window as {xk(t):t∈Ic}={xk,1,xk,2,…,xk,nc}.

Median in the interval is mk(c): 6median{xk,1,xk,2,…,xk,nc}

MAD in interval c: 7median|xk,i-mk(c)|,whereivaries from 1 to nc

The Normalized Root Mean Square Error (NRMSE) between observed and predicted CCN concentrations was calculated as (see also Sect. S7 and Fig. S11): 8NRMSE=1n∑i=1nCCNpred,i-CCNobs,i2CCNobs‾ where CCN_pred,i is the predicted CCN concentration at supersaturation i, CCN_obs,i is the observed CCN concentration at supersaturation i,n is the number of data points (in this case five, as we have five different supersaturations) and CCNobs‾ is the mean of the observed CCN concentrations across all supersaturations.

To facilitate direct comparison with Schmale et al. (2018) we also calculated the Geometric Mean Bias (GMB) for each time point, defined as: 9GMB=exp⁡1n∑i=1nln⁡CCNpred,iCCNobs,i