Figure 1a shows the scatterplot between the observed H2SO4 concentrations and that derived by the proxy using the full Eq. (2). The correlation coefficient was 0.84 (1860 data points). The data were related to 3 h medians, as the monoterpene concentration was measured only every third hour. In Fig. 1b–d, the proxy is refitted after removing one of the source or sink terms (Eqs. 4–6), in order to evaluate the sensitivity of the proxy to each of the terms and to show the improvement of the proxy using the additional source and sink (Fig. 1a) in comparison to the simple proxy that was used by Petäjä et al. (2009) (Fig. 1d). Our results show that the integration of additional terms of H2SO4 formation (i.e., the stabilized Criegee Intermediates) and loss (atmospheric cluster formation) gives the new proxy the ability to accurately capture the diurnal variation of the H2SO4 concentration, demonstrating a clear improvement over the earlier physical proxy (Petäjä et al., 2009).

In Fig. 1b, the corresponding data are shown without the alkene term (Eq. 4). The correlation is substantially weaker (0.74) than with the full equation. Even more importantly, we cannot estimate the contribution of the alkene term to the sulfuric acid concentration (Fig. 2 – Fit 2) as the fit also results in an unphysical coefficient for cluster formation (Kürten et al., 2015) and the fit fails to capture the diurnal pattern during dark hours after 16:00 LT (Fig. 2 – Fit 2). When fitting the data without the cluster source term (Eq. 5), the correlation coefficient is high (Fig. 1c), yet the goodness of the fit is not as good as when the cluster source term is taken into account (Table S4 in the Supplement – Fig. S9). Furthermore, we derived an additional proxy equation using CS corrected for hygroscopic growth (Laakso et al., 2004) to be used when calculating a more robust proxy for Hyytiälä. The details, equation and results are shown in Figs. S10–S12 in the Supplement.

Note that we opted for deriving a bulk proxy (daytime and nighttime together) instead of two independent proxies, one for daytime and one for nighttime separately. Our results show that one bulk equation is able to explain the Hyytiälä sulfuric acid daytime and nighttime sources accurately. Additionally, separating the bulk equation into two distinct equations results in bias towards the pattern of one of the predictor variables. For instance, the k1 value during daytime follows the cycle of global radiation, while that of k2 follows the cycle of alkenes. Therefore, in order to accurately reflect the continuum of source and sink terms throughout the day, we decided on the bulk proxy. Additionally, one bulk equation was able to predict sulfuric acid concentrations during daytime and nighttime with high accuracy (slope of ∼ 1), as further discussed in Sect. 4.5.

The fit was able to reproduce the sulfuric acid concentration in such a clean environment without the cluster term (Fig. 2 – Fit 3), perhaps due to low concentrations of bases participating in clustering in Hyytiälä (Jen et al., 2014). Finally, the corresponding data without both the alkene source term and cluster formation source term (Eq. 6, Fig. 1d) shows a weaker correlation between the measured and modeled sulfuric acid concentration (0.70), but more importantly it deviates far from the 1:1 line during both daytime and nighttime (Fig. 2 – Fit 4). It is important to note here that when deriving the Petäjä proxy (Petäjä et al. 2009), the model relied on summer data between April and June 2007, which could explain the misfit with the current data from Hyytiälä that spans the whole year. See also Figs. S13 and S14 in the Supplement for scatterplots comparing the measured sulfuric acid concentrations of the training data set with Petäjä et al. (2009) and Mikkonen et al. (2011), respectively. In general, using all four terms in Eq. (2) shows improvement over all other combinations (Eqs. 4–6), not only in terms of correlation coefficients and accurate diurnal cycle between measured and calculated concentrations of sulfuric acid as shown in Figs. 1 and 2 but also by showing a better goodness of the fit as shown in Table S4 and Fig. S9 when using the AIC statistical method. The final equation for the boreal forest environment can be found in Table 1, Eq. (9).

Since there were no direct measurements of alkenes in Agia Marina, we had to exclude the formation of H2SO4 in the oxidation by sCI from the proxy, and therefore we derived only the daytime H2SO4 proxy concentration. The correlation between the measured and proxy concentration of H2SO4 was 0.88 (96 data points), which shows that the chosen predictors were able to largely explain the measured sulfuric acid concentration (Fig. 3). However, the slope deviates from the 1-to-1 line, which could be attributed to the additional formation mechanisms that we could not include with the current data. However, the addition of the cluster loss mechanism shows a noticeable improvement over the simple proxy in Fig. 3b (R=0.80). The cluster loss term starts to become more important in this rural environment in comparison to the boreal forest, which could be due to a higher concentration of stabilizing bases in Agia Marina compared with Hyytiälä. Although both fits of Eqs. (4) and (6) show similar diurnal patterns (Fig. 4, Fits 2 and 4), the loss term due to H2SO4 cluster formation improved the precision of the new proxy (Fig. 3). According to the statistical AIC method, the goodness of the fit improved from 70 to 33, with and without the clustering term, as shown in Fig. S9. Also, even without the alkene term, the newly derived coefficients improved the proxy in comparison to Petäjä et al. (2009) and Mikkonen et al. (2011), as shown in Figs. 4, S13 and S14. The final equation for the rural site can be found in Table 1, Eq. (10).

Next we try to understand the mechanisms of sulfuric acid formation and losses in an even more complex environment, i.e., urban Budapest (Figs. 5 and 6). Since there were no direct measurements of alkenes there or of its proxies, such as monoterpenes, or anthropogenic volatile organic compounds, we derived the sulfuric acid proxy excluding the formation due to stabilized Criegee intermediates pathway, as in Eq. (4). In comparison to the simple proxy (Fig. 5b; R=0.49; 263 data points), the correlation between the measured and proxy concentration of H2SO4 improved with the addition of the loss term due to cluster formation, R=0.59 (Fig. 5a). The correlation between measured and modeled values of sulfuric acid became weaker in Budapest in comparison to Hyytiälä and Agia Marina, which could be attributed to a more complex environment, and additional pathways of sulfuric acid formation and losses. Additionally, we observed a sudden SO2 concentration change in the middle of the campaign, possibly due to sudden change in local meteorology and air mass transport, which could also explain the weaker correlation (See Fig. S1). The loss term due to H2SO4 dimerization improved the precision of the new proxy in comparison to the simple model, as well as the Petäjä et al. (2009) or the Mikkonen et al. (2011) derivation, as shown in Figs. 6, S13 and S14. We think that the overestimation in the Petäjä proxy is because of its dependence on the SO2/CS ratio. The proxy is originally derived in Hyytiälä, and when we apply the same coefficients to Budapest it gives a higher estimated concentration when compared to the measured one since the SO2/CS ratio is smaller in Budapest (Fig. 9). Although the proxy developed by Mikkonen et al. (2011) has been shown to work in varying environments, it clearly overestimates the sulfuric acid concentration in Budapest for perhaps the same reasons (its dependence on the SO2/CS ratio). It is also visible from Figs. 5 and 6 that the addition of the dimerization term was capable of better capturing the lower H2SO4 concentrations in comparison to fitting the data without the dimerization term. In comparison to both Hyytiälä and Agia Marina, the coefficient associated with dimerization in Budapest is slightly higher, which can be attributed to the availability of a possibly facilitated clustering due to higher abundance of stabilizing bases such as amines and ammonia (discussed in Sect. 4.6). The final equation for the urban environment can be found in Table 1, Eq. (11).

In megacities, in our case Beijing, the sulfuric acid concentration is particularly high during nighttime, which confirms the need for determining the contribution of sources other than OH (radiation) to its formation. Our observations emphasize the contribution of the alkene pathway, as without considering this route we would not replicate morning hours correctly. During daytime, there is enhanced dimerization and cluster formation due to the abundance of stabilizing bases (Yao et al., 2018). We assessed the derivation of the proxy equation first using daytime data and nighttime data separately and found that such a separation results in an unphysical k3 value since clustering in Beijing happens mostly during daytime (Zhou et al., 2020). This obstacle was also observed when deriving a bulk equation. To overcome it, we set an upper limit for the k3 value at 7 × 10-9 obtained from the fitting of daytime data (GlobRad ≥ 50 Wm2). The reason for such an observation is that, in such a complex environment, sulfuric acid might originate from sources other than the ones we accounted for in our calculation, especially during nighttime, e.g., through the hydrolysis of SO3 formed from non-photochemical processes (Yao et al., 2020). The alkenes or volatile organic compounds during daytime are different from those during nighttime and might vary between seasons, which could be attributed to a different fleet composition during those times or the biogenic activity (Yang et al., 2019). However, the derived Eq. (12) (derived from spring data) is able to predict the daytime and nighttime sulfuric acid concentrations during summer and autumn (See more in Sect. 4.5).

In Fig. 7, we see an improvement of the new proxy (Eq. 2) in comparison to the simple proxy (Eq. 6) derived by Petäjä et al. (2009) as the former takes into the account the additional sources and sinks of H2SO4 that were not considered in previous works (see also Fig. S9). Introducing the alkene production term improved the accuracy of the H2SO4 concentration during both daytime and nighttime (Figs. 7 and 8), which supports our assumption that H2SO4 formation during nighttime is driven by stabilized Criegee intermediates. In Fig. 7b we show the proxy without the alkene term is unable to capture the nighttime concentrations. In Fig. 8, we see the importance of all sources and sinks predicted for sulfuric acid, as Fit 1 (Eq. 2) best predicts the measured sulfuric acid concentration. Additionally, according to the statistical AIC method, using the full equation has the least probability of inaccuracy and error in estimating the sulfuric acid concentration (Fig. S9). Moreover, it is clear that the addition of the cluster sink term in the megacity environment is required due to its large contribution as a sink for H2SO4, especially due to higher concentrations of stabilizing molecules; the cluster mode (sub-3 nm) particle concentration is the highest in Chinese megacities (Zhou et al., 2020). The final equation for the megacity can be found in Table 1, Eq. (12).

Each of the proxies of the boreal forest environment, the rural background and the megacity were tested for predictive power on independent data sets using extended data sets from the same location or using measurements from locations with similar characteristics. The sulfuric acid concentrations at each of these locations is modeled using the equation (with median k per source/sink term) relevant to the site and compared to the measured concentrations. The derivation of the sulfuric acid concentrations using 10 000 combinations of k values, as well as the error in the predictions, are shown in the Supplement. Note that the testing data sets are not subject to any bootstrap resampling or uncertainty additions but are instead used as is for testing the predictive power of the suggested proxy.

The variations of coefficients related to Eq. (3) can be used to get insights into the general chemical behavior under current atmospheric conditions, as well as into the mechanisms of sulfuric acid formation and losses in various environments. The contribution of different terms in different locations seem to vary significantly. The new loss term taking into account clustering starting from dimer formation needs to be taken into account in all the environments in daytime. On the other hand, without alkene term it is in practice impossible to get nighttime concentrations correct.

In Table 2, we have presented the fitted coefficients (Eq. 3) for all our sites, whereas the contributions of the different terms in the balance equation are given during daytime in Fig. 11 and Table 3. The contribution of the various source and sink terms to the change of H2SO4 concentrations are determined using Eq. (2). The median-derived k1, k2 and k3 values, together with the measured H2SO4, CS, trace gases and GlobRad per site, were used to calculate each of the terms. Source term 1 refers to k1 × GlobRad × [SO2], source term 2 refers to k2 × [O3] × [Alkene] × [SO2], sink term 3 refers to k3 × [H2SO4]2 and sink term 4 refers to CS × [H2SO4]. The contribution of each term is then calculated as the median or percentiles of the normalized term to the sum of all terms. The variability of the coefficients (Table 2), as well as the relative contributions of each term to the change in sulfuric acid concentration (Table 3), could give valuable information on the mechanisms resulting in sulfuric acid formation and losses. At steady state (Eq. 2), the sources and sinks are in balance with each other during both daytime and nighttime, but there were clear differences in the individual contributions. For instance, a variation in k1 could be due to variations in OH sources and sinks. Although in urban locations OH sinks are expected to be higher and therefore k1 to be lower, additional sources of OH are available in such locations, for example HONO (Zhang et al., 2019). The alkene and Criegee intermediates terms was found to be an important H2SO4 source (Figs. 1, 2, 7 and 8), as without it we are not able predict night or morning concentrations of H2SO4 properly. The alkene source term contributed up to almost 100 % of the H2SO4 sources during nighttime in Beijing and up to 90 % of the sources during nighttime in Hyytiälä (Fig. 12). The Criegee intermediates term showed its importance mostly when global radiation is low, not only in nighttime but also during winter (Fig. 12) in both Hyytiälä and Beijing. It is important to note here that Criegee intermediates vary between locations, and they also form in different yield percentages from different alkenes (Novelli et al., 2017; Sipilä et al., 2014). These stabilized Criegee intermediates also react differently under different environmental conditions.

The CS term had the highest contribution to the total sink in Hyytiälä. Its contribution decreased when moving towards more polluted environments (Fig. 11) to become regardless of the relatively high condensation sink in Megacities, smaller in Beijing than that of the cluster sink term (Laakso et al., 2006; Monkkonen et al., 2004, 2005; Yao et al., 2018). This observation might be attributed to decreased effectiveness of condensation sink in more polluted environments (Kulmala et al., 2017), but also to increased contribution of the clustering sink term in such environments where the concentration of stabilizing bases is highest, particularly in daytime (Yao et al., 2018; Yan et al., 2018). It should be noted that measurements of ammonia and similar bases are rare, so their exact contribution is difficult to estimate. The cluster term is found to contribute most during spring daytime in Hyytiälä (Fig. 12a and c), which is the time window during which clustering and thus new particle formation events happen (Dada et al., 2017, 2018). The same is observed for Beijing, where the clustering term contributed up to 70 % of the total sink terms during daytime (Fig. 12d), especially during summer when the CS is lowest (Deng et al., 2020).