ME-2 (Paatero, 1999) is a powerful engine for solving the positive matrix factorization algorithm (PMF, Paatero and Tapper, 1994). Model configuration and post-analysis are performed by Source Finder (SoFi Pro 6.8, Datalystica Ltd., Villigen, Switzerland) within the Igor Pro software environment (Wavemetrics, Inc., Portland, OR, USA) as described in Canonaco et al. (2013). In its bilinear mode, PMF describes the measured data matrix X as a product of two matrices, G and F, and the residual matrix E. In element notation the equation is 1xij=∑k=1pgik⋅fkj+eij. In the measured matrix X the columns j are the m/zs, and each row i represents a single mass spectrum. p is defined as the number of factors of the selected model solution, i.e., the number of columns of G and the number of rows of F. Each column of the matrix G represents the time series of a factor, whereas each row of F represents the factor profile (i.e., mass spectrum); both are indexed by k. In an unconstrained PMF run in ME-2, the model is initialized with random entries in G and F (“seed”) and the quantity Q is minimized with respect to all model variables by means of the conjugate gradient algorithm (Paatero, 1999): 2Q=∑i=1n∑j=1meijσij2, where eij are the elements of the residual matrix E and σij represents the measurement uncertainty for the input point xij.

To compare Q values from various PMF runs with a different size and/or number of factors, Q is normally scaled by the remaining degrees of freedom (Qexp, which depends on the size of the input data and the number of chosen factors): 3Qexp=m⋅n-p(m+n). PMF is subject to rotational ambiguity, in which different combinations of G and F yield similar Q values. Some of these combinations may contain mixed factors and/or environmentally unreasonable descriptions of the data. Previous work has shown that constraining expected factor profiles using the a-value approach for AMS/ACSM data is an efficient method for isolating the set of environmentally interpretable PMF runs (Lanz et al., 2008; Canonaco et al., 2013; Crippa et al., 2014). The a value determines the extent to which the m/z in the mass spectral profile, also referred to as anchor (fkj), is allowed to vary during the model iteration according to 4fkj′=fkj±a⋅fkj. The index j stands for the actual variable (m/z) of the kth factor, and the a value is its scalar product. For example, an a value of 0.1 allows for a variability of ±10% during the iterative process. This implies that some variables might increase and some might decrease within this limit. Note that after renormalizing the solution, the extent to which the constrained values changed might be slightly outside the defined a-value range. For example, consider a case where the a value is set to 0.1 for all variables of a factor profile. The values of all variables but one could decrease by 10 %, while the value of this single variable might increase by 10 % during the iteration. After renormalizing the factor profile outside ME-2 by, e.g., the sum of the profile, the intensity of this single variable will exceed the boundaries set with the a values during the PMF iteration. Moreover, note that the a-value approach defines only the boundaries of a solution and does not imply any weighting within these boundaries.

The organic data and error matrices (Allan et al., 2003) are computed using the ACSM local tool version 1.5.3.2 (Aerodyne Research, Inc., Billerica, MA, USA) in Igor Pro. Weak (signal-to-noise ratio between 2 and 0.2) and bad (signal-to-noise ratio below 0.2) variables were downweighted according to the recommendations in Paatero and Hopke (2003). The m/z 16, 17, 18 and 28 variables that are replicates of the variability of m/z 44 were removed for the PMF calculation and recalculated a posteriori as a function of the m/z 44 contribution attributed to each factor profile (Elser et al., 2016). This approach is preferable to downweighting (Ulbrich et al., 2009), as it maintains a direct mathematical relationship between m/z 44 and its dependent variables, which can otherwise be distorted by dynamic weighting of outliers in the PMF robust mode.

PMF analysis is conducted on a subset of data defined by a small window that is moved in 1 d increments across the entire dataset and as such allows capturing of seasonal variations of the factor profiles. Note that rolling windows containing less than 10 % of real data are automatically skipped by the rolling algorithm. This avoids performance of PMF runs over large gaps due to, e.g., calibrations or instrument failures. The window size (swin) is a free parameter that requires optimization. The rolling window PMF analysis of Parworth et al. (2015) utilized a 2-week window, arguing that this length is representative of the average lifecycle of aerosols in the atmosphere. Even for (low-time-resolution) ACSM data, 2 weeks have been shown to provide enough temporal variability to distinguish sources with similar factor profiles such as HOA and COA (Fröhlich et al., 2015). In the present study, likewise a 14 d window is selected after additionally assessing the performance of 3, 7, 21, and 28 d windows.

The model performance in response to swin is assessed by monitoring the value of Q/Qexp (which decreases as the mathematical goodness of fit improves) and the number of non-modeled time points (nnon-modeled) as a percentage of the total number of measurements. nnon-modeled is defined as any ACSM time point for which the user-defined criteria (see Sect. 2.3.3 and 2.3.4) are not met for any PMF runs that include this measurement (note that for most points this will include PMF runs from multiple overlapping windows). Figure 1a shows Q/Qexp and nnon-modeled as a function of swin. The Q/Qexp values are minimized for a 7 d window and are approximately 15 % higher for the 3 and 14 d windows and 45 % higher for the 21 and 28 d windows. nnon-modeled shows a minimum for 14 d with a slight increase for larger windows and a steep increase for smaller swin.

A 14 d window has been chosen for the current dataset, as this avoids significant increases in Q/Qexp without inducing unacceptably high nnon-modeled. Moreover, because the 1 d step of the rolling window is smaller than the 14 d width, each time point is included in 14 different window runs (except for those within the first or last 14 d of the dataset). As discussed later, these repeats aid the uncertainty analysis.

The rolling strategy described above defines a new window after every window shift. Within this new window, a PMF run, referred to as repeat in the text, is generated via ME-2, which initializes new seeds, a values, and bootstrap resampling of the PMF input. The seed initializes all model entries in G and F, and unconstrained information therein is randomly initialized. Additionally, a priori information on the factors from the seasonal pre-tests is used to confine the solution space and thus to decrease the rotational ambiguity of the solution.

In the current study, constraints are applied only to profiles of the POA factors, namely traffic (HOA), cooking (COA) and biomass burning (BBOA). The HOA and COA profiles are taken from Crippa et al. (2013), while BBOA is the averaged mass spectrum reported by Ng et al. (2011a). These anchor profiles were also successfully used for the seasonal analysis of the Zurich–Kaserne data (Canonaco et al., 2013, 2015).

Every constrained factor profile applied in a PMF run requires a sensitivity analysis of the a value to identify the range of reasonable solutions (Canonaco et al., 2013; Crippa et al., 2014; Elser et al., 2016). Typically, variation of the a value of one or more constrained factor profile(s) allows exploration of a region in the solution space that includes environmentally reasonable solutions. In the present analysis, the goal is to consider all PMF runs (not just the best one) that are mathematically and environmentally reasonable. Recent studies have systematically investigated the entire solution space allowed by the a values, e.g., by conducting PMF runs covering every combination of a values over the range 0 to 1 with a step of 0.1 (Elser et al., 2016; Bozzetti et al., 2017; Daellenbach et al., 2017). However, this approach is not computationally practical for moving window analysis. For instance, given that three factors are constrained in this present study, the above a-value exploration strategy would require 113=1331 PMF runs for a-value exploration per window run. Also, each combination of a values would require a minimum of 100 PMF runs for bootstrap analysis (Norris et al., 2014). Furthermore, the seasonal pre-tests indicated that both four- and five-factor solutions should be assessed (corresponding to one or two OOA factors). In total, this would require 1331×100×2∼2.66×105 PMF runs per window. Moreover, the daily shift of the rolling window will initialize the window runs 351 times (1 year minus the swin), resulting in 1331×100×2×351∼9.35×107 PMF runs for a systematic analysis. This will require several months of computation even on modern PCs with multicore processors. To overcome these computational issues, two strategies were considered for reducing the number of runs required for a-value exploration. In both cases, a systematic exploration of the a-value space is replaced by randomly generated a values between zero and an upper limit (amax⁡). For the first strategy, the amax⁡ limit was fixed at one, and the number of repeats (xPMF) was adjusted until the same criteria described above for swin optimization were satisfactory. However, this approach was rejected, as executing the full set of PMF runs required computational times on the order of months (see Part A of the Supplement) and therefore was impractical on regular PCs.

The second strategy, which is used here, exploits the a priori information of the sources. If some factor profiles are known to be present and their source profiles are known to some extent, there is no need to explore regions in the solution space, for which these factor profiles may drastically depart from their realistic anchors.

Therefore, amax⁡ undergoes a systematic scan from zero upwards, with model performance assessed by Q/Qexp and nnon-modeled, as described above for the swin estimation. The current strategy counts as a local-minimum algorithm, as the full parameter space (swin, amax⁡, xPMF) is not fully investigated. Moreover, pre-tests based on literature data, i.e., a 14 d PMF window for swin (Parworth et al., 2015) and an upper a value of 0.3 amax⁡ (Crippa et al., 2014), represented the starting condition for the parameter optimization discussed in Fig. 1.

Figure 1b shows an almost flat Q/Qexp, while that of the nnon-modeled behaves as a quadratic function with a minimum at a=0.4. For a values below 0.4 the constrained fingerprints cannot optimally adapt to the current input. Given only 50 random a-value explorations out of 1331 (see above) of the entire a-value space for every PMF window, outcomes for higher amax⁡ may be purely stochastic and lead to a high degree of mixing and consequently rejection of the PMF runs (high nnon-modeled). a=0.4 represents the optimum amax⁡ and is set as a free parameter for the a-value exploration.

The random resampling of the PMF input uses the bootstrap approach for every repeat. A window comprising 14 d with at most 48 (number of scans per day) × 14 (d) = 672 time points will create resamples containing again 672 new time points, where some time points may occur multiple times and others may be absent. As above, Q/Qexp and the percentage of nnon-modeled are monitored as a function of the xPMF. Figure 1c reveals a constant Q/Qexp, whereas the number of nnon-modeled decreases and stabilizes from 50 repeats onwards. We conclude that 50 repeats per window are sufficiently high for the bootstrap strategy. Note that the final number of PMF runs per time point may be higher than xPMF due to the overlapping PMF runs resulting from the rolling strategy. The total number of PMF runs for this study equals 50 (xPMF) × 351 (number of days) × 2 (four- and five-factor) = 35 100 runs and required approximately 3 d on a modern multicore PC.

Manual inspection of all generated PMF runs is impractical and is replaced by an automated procedure based on pre-defined user criteria that (1) identifies and sorts unconstrained factors and (2) determines whether each PMF run should be accepted or discarded. Examples of user-defined criteria could include the factor correlation with an external tracer in terms of either the overall time series or diurnal pattern or characteristic temporal features, e.g., a prominent lunch peak for a cooking factor. Modeled PMF factors for which no factor criteria are satisfied, i.e., very poor score values due to factor mixing/swapping or sampling of transient sources not accounted for, typically yield nnon-modeled.

In addition to determining whether an individual PMF run should be accepted or rejected, the criteria are used to determine the identity of unconstrained factors. While the positions of constrained factors within the F and G matrices are pre-defined for constrained factors, the same is not true of unconstrained factors, and these must be correctly identified prior to further data analysis. Consequently, all possible combinations for sorting unconstrained factor positions are evaluated (factor identification) and their scores combined together. As criteria with various score ranges are potentially possible, e.g., correlation coefficient, lunch peak ratio, the explained variation (EV; see Eq. 5) of m/z 60 and variable fractions, these score values must be corrected before being added up. z-score transformation as a linear correction is applied, where at the end the score distribution of each criterion possesses a mean value of 0 and a standard deviation of 1. Finally, the z-score transformed combination with the highest values is chosen to represent the PMF result for a specific PMF run. This is essential in the case of the two unconstrained factors SV-OOA and LV-OOA in this study. Note that this requires criteria to be defined for a minimum of all factors but one (i.e., p-1 factors).

Considering the large amount of PMF runs by the rolling window algorithm, the main advantage of this criteria-based inspection is that the complexities of a factor profile and time series are reduced to single values (“scores”). Based on the score plots, potentially promising PMF runs can be further investigated and validated. This significantly improves the efficiency of PMF analysis by discarding PMF runs where the score for any criterion falls below the user-defined threshold (“bad PMF runs”). In contrast to conventional analyses, where a single PMF run often represents an optimal description of the dataset, the entire set of PMF runs classified as environmentally reasonable is used for the analysis and presentation. This provides a more comprehensive and robust representation of the dataset and supports uncertainty assessment.

To determine whether an individual PMF run is accepted or rejected, acceptance thresholds are defined for each of the selected criteria. These thresholds are free parameters and must be defined for each criterion separately. A threshold is inferred either from previous studies or from significance tests or similar statistical analyses (see the discussion for the HOA and COA thresholds in Sect. 2.3.4 for such an example).

The computational time required for criteria application subsequent averaging is typically on the order of minutes to hours with a modern multicore PC, depending on the number of accepted PMF runs. Thereafter, the results can be inspected in real time, allowing the user to efficiently investigate the set of PMF runs and, if needed, test various criteria.

In this study one criterion per factor was defined, although it is possible to apply multiple criteria to the same factor, as each criterion is assessed individually on an accept/reject basis.

Figure 2 shows the criterion scores calculated for each PMF run, with each plot representing an individual factor. The grey points show the score values for all PMF runs, the blue points denote PMF runs where criterion thresholds are satisfied, and the green points represent PMF runs where criterion thresholds for all criteria are simultaneously fulfilled. These green points are then used to compute the final PMF solution. The criteria and their corresponding thresholds applied for each criterion (blue points in Fig. 2) are also reported in Table 2 (first value).

In the current study, the thresholds for the criteria of HOA and COA were determined based on statistical analyses with the help of the results from conventional (no rolling technique) seasonal PMF from previous studies (Canonaco et al., 2013, 2015). The contributions of HOA and its tracer eBCtr were bootstrapped together and the correlation coefficient (RPearson) was evaluated each time, leading to a distribution for RPearson. Similarly, the time series of COA was bootstrapped and the lunch peak enhancement in COA evaluated each time (COA11+12+13h/COA9+10+14+15h), leading to a distribution for the lunch peak concentration. Finally, the 10th percentile value was chosen as the threshold score value. These seasonal thresholds are also visible as steps in the score plots (blue points in Fig. 2a and b, respectively) and are also reported in Table 2 (second value in brackets). For spring 2011, summer 2011 and winter 2012, however, the resulting thresholds for HOA either caused too many missing time points (RPearson=0.8) or had rather non-significant correlation coefficients (RPearson=0.2, with a p value of 0.4, n=24 as for the other seasons). Hence, these thresholds were systematically lowered for spring 2011 and increased for winter 2012 to achieve the highest possible correlation coefficient with maximal data coverage, i.e., the same nnon-modeled when considering all PMF runs for these periods in these criteria.

NOx is a typical tracer for HOA in urban areas. However, due to incomplete NOx measurement coverage in this campaign (especially during spring and fall), eBCtr is used as a traffic tracer and the RPearson correlation coefficient is computed between the diurnal cycle of eBCtr and the HOA factor.

As is frequently the case, no chemical tracers for COA were available in this study. Previous measurements in Zurich (Canonaco et al., 2013, 2015) have demonstrated a strong diurnal pattern for COA, with an increased concentration during lunchtime. As a proxy for COA, the lunchtime COA enhancement is monitored (Table 2).

The wood burning contribution to black carbon (eBCwb) as determined by the eBC source apportionment (eBC-SA) method of Sandradewi et al. (2008) was considered as a possible criterion for BBOA but then rejected. The eBC-SA analysis applies to air masses highly influenced by biomass burning and has been validated for winter data only. Uncertainties in eBCwb during warm seasons, when the biomass burning contribution is small, have been shown to be quite high (Harrison et al., 2013). Therefore, it was decided to use another metric for BBOA, exploiting the key spectral feature at m/z 60. For BBOA the explained variation (EV) (Paatero, 2010) for m/z 60 is monitored as follows: 5EVj,k=∑i=1ngik⋅fkj/σij∑i=1n∑h=1pgih⋅fhj+eij/σij. This threshold is chosen following the recommendation in Paatero (2010), where a variable modeled by its mean explains already ∼25 % of the variation. If the measured variability of a variable is explained by a specific factor, that factor must capture more than the mean value of the variable, and hence Paatero (2010) recommended 30 %–35 % as a minimum EV. However, using 30 % or 35 % as a threshold resulted in several weeks of non-modeled time points, in particular for spring and fall 2011. An a value of 25 % resulted in a reasonable compromise between EV and the number of non-modeled time points. Note that this approach requires the assumption that m/z 60 should be predominantly explained by BBOA, which is likely true when the fraction of the OA signal occurring at m/z 60 (f60) is relatively high. However, for measurements where f60 is low, m/z 60 is more likely to also have contributions from other sources. A rough guideline for utilizing this criterion is a threshold for biomass burning influence of f60=0.003 as identified by Cubison et al. (2011). In the current dataset, ∼85% of all measured time points exceeded this threshold. Every measured day was observed to comprise at least some time points (in winter, spring and fall almost all points but in summer mostly evening points) above this threshold, suggesting that the criterion is valid throughout the dataset.

Ng et al. (2010) described higher f43 and lower f44 for the mass spectrum of SV-OOA and vice versa for LV-OOA. Therefore, f43 and f44 are used as proxies for SV-OOA and LV-OOA or OOA, respectively. For LV-OOA (Fig. 2d, Table 2) all score values are allowed here, whereas for SV-OOA (Fig. 2e, Table 2) the PMF runs meeting the thresholds for the five-factor solutions are selected. This threshold corresponds to the point where nnon-modeled is minimal with respect to this criterion; i.e., considering all PMF runs in this criterion leads to the same nnon-modeled at the highest possible f43 for SV-OOA.

The criterion of SV-OOA is further used to differentiate between four- and five-factor solutions on the window runs. For the PMF windows where no five-factor solution with SV-OOA is selected, the set of four-factor solutions in the corresponding PMF window is automatically selected (green points at zero in Fig. 2e). Finally, the averaging procedure also controls and prevents that four- and five-factor solutions are simultaneously considered for the averaging of single time points by privileging five-factor solutions; i.e., any time point containing accepted PMF runs with both four- and five-factor solutions retains only the five-factor solution.

Figure 3a shows the time series of each factor for the entire dataset as a mean, averaged over all accepted PMF runs. The data from Fig. 3a are re-averaged to monthly and seasonal means and shown in Fig. 3b and c, respectively. For Fig. 3c, seasons are defined as follows: winter is December–February, spring is March–May, summer is June–August, and fall is September–November.

In winter, spring and fall the concentrations of primary organic aerosols (HOA, COA and BBOA) are approximately 40 % compared to the 60 % of the (secondary) oxygenated organic aerosols (SV-OOA, LV-OOA or OOA). In summer the primary fraction decreases to reach minimum values of 30 % compared to 70 % of OOA. The relative fractions of HOA and COA are rather constant, contributing on average between 0.4–0.7 µgm-3 (7.8 %–9.0 %) and 0.7–1.2 µgm-3 (12.2–15.7 %), respectively, throughout the year. BBOA shows a strong yearly cycle with the lowest mean concentrations in summer (0.6 µgm-3, 12.0 %), slightly higher mean concentrations during spring and fall (1.0 and 1.5 µgm-3, or 15.6 % and 18.6 %, respectively) and the highest mean concentrations during winter (1.9 µgm-3, 25.0 %). Only during summer, the bulk OOA is completely separated into SV-OOA and LV-OOA, with mean concentrations of 1.4 µgm-3 (26.5 %) and 2.2 µgm-3 (40.3 %), respectively.

For the remaining seasons the seasonal concentrations of SV-OOA, LV-OOA and OOA comprise 0.3–1.1 µgm-3 (3.4 %–15.9 %), 0.6–2.2 µgm-3 (7.7 %–33.7 %) and 0.9–3.1 µgm-3 (13.7 %–39.9 %), respectively.

The time series of the primary OA factors HOA, COA and to some extent BBOA are rather spiky (Fig. 3a), underlining a strong influence of local sources. The COA spikes that are present from May 2011 through the end of September 2011 are likely due to local barbecuing events during the evening, as also observed in an earlier study at this site (Lanz et al., 2007). The highest COA concentrations are observed in early July 2011, where the NR-PM1 mass concentrations reached 70 µgm-3, and correspond to three consecutive evenings/nights of a yearly Latin American dance and grill festival (Caliente). During this festival, the courtyard containing the measurement site was filled with food and grill stands, explaining the dominant contribution of COA. Throughout the summer and spring and less frequently in fall/winter SV-OOA was modeled in addition to LV-OOA. This warm period was characterized by high daily temperatures and induced on the one hand variability in the condensed OOA allowing for separation of SV-OOA and LV-OOA and on the other hand increased emissions of biogenic SV-OOA precursors (Canonaco et al., 2015).

Figure 4 summarizes the weekday (left) and weekend (right) daily cycles for the modeled factors. The daily cycle of HOA follows the averaged daily cycles of the estimated traffic of eBC (eBCtr) and of NOx. The same is true for the daily cycle of BBOA following that of the biomass burning of eBC (eBCwb). HOA, eBCtr and NOx exhibit a clear rush-hour peak on weekdays and none on the weekend. During the weekdays, a small lunch peak is visible for COA, underlying the meal activity during the working days and the presence of many restaurants in this area. There are no evident differences between the weekday and weekend daily cycles of LV-OOA, SV-OOA and OOA. LV-OOA and OOA show rather flat daily cycles, similarly to their inorganic aerosol tracers SO42- and NH4+, respectively. This is in line with their most-likely regional background, as already suggested earlier (Canonaco et al., 2015). Only the concentration of SV-OOA tends to decrease during the afternoon, suggesting its volatile nature, similarly to its inorganic aerosol tracer NO3-. The weekly cycle for HOA, COA, BBOA and the OOAs, including their tracers eBCtr, NOx, eBCwb, SO42-, NO3- and NH4+, respectively, are reported in Supplement B. Apart from OOA, the weekly cycles for HOA, BBOA, SV-OOA and LV-OOA are in good agreement with their tracers.

The analysis and further validation of the PMF runs using the criteria-based selection are performed on the PMF results of the rolling windows, and therefore correlations are performed over 14 d in this study. The performance of the rolling strategy can then be verified by the factor–tracer correlation, e.g., on average over the seasons (Table 3). Moreover, the same factor-to-tracer correlations are also evaluated for the seasonal pre-tests (PMF runs over the seasons with no rolling strategy) and are reported in brackets in Table 3.

NOx data are available only in winter and fall 2011. Both NOx and eBCtr are correlated with HOA over the full year and within individual seasons. The correlation values with NOx, are lower compared to those found in Canonaco et al. (2013). However, in Canonaco et al. (2013) the data covered mostly the two winters including some parts of spring and fall. For the latter two seasons NOx data were not properly validated and was consequently removed from further analysis (no NOx data are available for spring and summer). Moreover, in Canonaco et al. (2013) the model validation was strongly based on the first winter period, and when performing the correlation between HOA and NOx data for that period only, the correlations were similar also in the current study (not shown in the table).

BBOA shows substantial correlation with eBCwb in fall and winter, as also found in Canonaco et al. (2013), while the correlation is low in spring and very low in summer. These low correlations are expected, since the determination of eBCwb is highly uncertain when the eBCwb/eBCtraffic ratio is low. Wood burning source apportionment of eBC data, as already stressed above, is not suited under warm conditions with low biomass burning contributions. However, the correlation is good over the full year, as the problematic data yield eBCwb concentrations near zero anyway, and the correlation is thus driven by the data with high signal-to-noise ratios.

High correlations between LV-OOA and SO42- are seen over the year as well as for spring and fall, whereas they are lower in summer, as shown in Table 3, in contrast to Lanz et al. (2007) (RPearson=0.5 between LV-OOA and SO42- during a summer AMS campaign). The correlation between SV-OOA and NO3- is higher for winter 2011 and summer but lower in spring and fall. This is understandable, as the spring and fall represent the transition between modeling SV-OOA and LV-OOA (summer) compared to one OOA only (winter). The correlation between SV-OOA and LV-OOA for winter 2012 is not shown due to the low number of time points for which both OOAs were modeled. OOA correlates well with NH4+ throughout the year in accordance with summer and winter data reported previously (Lanz et al., 2007, 2008; Canonaco et al., 2013). In contrast to the OOAs, few differences are observed for BBOA, HOA, or COA between the two winters. This supports the conclusion that the different OOA behavior in these two winters reflects actual meteorological and chemical differences rather than mixing and/or splitting between the POA and SOA factors.

Importantly, the rolling results show generally higher correlations with the external tracers than do the conventional seasonal PMF runs (values in brackets in Table 3). This demonstrates that the rolling approach generally outperforms the conventional seasonal PMF analysis.