PMF is a well-established model based on the work by Paatero and Tapper (1994). This receptor model is useful for solving functional mixing models when the source number and source profiles are unknown. It fundamentally works on an assumption of mass conservation so that a mass balance analysis can be used to identify and apportion sources of the detected species in the atmosphere. The most important feature that distinguishes PMF from other receptor modeling (e.g., principal component analysis) is that it applies a least-squares algorithm that accounts for data uncertainties. It also constrains the solutions to the non-negative subspace so that they are environmentally reasonable. Due to these advantages, this model is widely used for source apportionment analysis. The PMF analysis in this work uses the IGOR based analyzing interface SoFi (solution finder, version 5.2) and ME-2 as described in Canonaco et al. (2013).

Using PMF on mass spectral data, the mass balance can be described as X=TS⋅MS+E. Matrix X is an m×n matrix, representing m measurements (in time) of n masses. The sizes of the factor matrices TS and MS are m×p and p×n, respectively, where p is the number of factors. In practice, the matrix TS is the time series of the p factors representing the source strength, and matrix MS contains the mass spectra of the p factors. Matrix E is the residual unexplained by the p factors. It should be noted that the value of p is not pre-fixed, and determination of the value will be based on the interpretability of the solutions.

The PMF runs to seek the minimum Q, the sum of squared residual weighted by the inverse of their respective measurement uncertainty, which can be described as Q=∑i=1m∑j=1n(EijSij)2. Here Sij is the estimated measurement uncertainty of mass j at time i, and Eij is the corresponding model residual. In this work, the uncertainty was estimated from laboratory data, which will be discussed in Sect. 2.3.3. Data points where Eij≫Sij have a large influence on the model iteration, and this needs to be reduced or removed by the model. A robust mode is applied to eliminate the strong outliers determined by α, meaning that any data points yielding Eij/Sij>α will be reduced to this threshold: ifEijSij>α,EijSij=α, where the value of α is a free parameter can be determined by the user, and a value of 4 was suggested by Paatero (1997).

Ideally, the modeled Q value should eventually approach to the expected Q values (Qexp), which is equal to the degree of freedom of the model solution. For mass spectra data, it roughly equals to the size of the matrix: Qexp≈(n×m).

The nitrate-ion-based (NO3-) CI-APi-TOF selectively measures HOMs with a ∼ 4000 Th/Th resolving power. In principle, this resolution allows us to fit peaks and in some cases resolve peaks with different composition at the same unit mass. However, the quality of the peak fitting strongly depends on mass calibration of the spectrum and the smoothness of the peaks. We found that the mass calibration may shift by 5 ppm by using data with 5 min integration time, and some HOM peaks are not smooth enough due to the weak signals. Fitting the peaks beforehand in such circumstances may introduce extra and nonuniform uncertainties that are difficult to estimate. Therefore, the data matrix used in this work is in unit-mass resolution, and peak fitting was performed afterwards to identify the elemental formula of peaks. Some examples of peak fitting are provided in Fig. S10 in the Supplement. The mass range of 201–650 Th was selected for PMF analysis, which covers most of the detectable HOMs. We continuously collected data from 4 April to 7 May 2012, with very few missing time points due to instrumental issues. The input data matrix consists of counts-per-second (cps) values, averaged from raw data using 5 min time resolution, resulting in a total of 9084 mass spectra. Thus, the final data matrix is in the size of 9084 (samples) × 450 (variables).

Due to the abovementioned model principle, the estimation of error matrix (Sij) is crucial. Suggested by Polissar et al. (1998), the error matrix in this work was estimated as Sij=σij+σnoise. There are two terms contributing to the total measurement uncertainty: σij is the analytical uncertainty from counting statistics, and σnoise is the standard deviation of instrument noise, also representing the instrument detection limit.

We estimate both σij and σnoise from laboratory data. The schematic of the corresponding experimental setup is provided in the Supplement (Fig. S1). Briefly, we generated stable signals with a temperature-controlled permeation source. A 100 mL min-1 (milliliter per minute) N2 gas served as carrier gas flowing through the source, which was then diluted by a 10 L min-1 (liter per minute) N2 flow before entering the CI inlet. The experiments were run under the following conditions:

Two different chemicals were used, i.e., perfluorobutanoic acid (CF3(CF2)2COOH) and perfluorononanoic acid (CF3(CF2)7COOH), respectively.

The σijwas estimated based on the assumption that the counting statistics follow the Poisson distribution (Allan et al., 2003): σij=aIts. I is the signal strength (ions s-1) of the ion, ts is the integration time in seconds, and a is an empirically determined factor incorporating any unaccounted contributions to the uncertainty, for example arising from shifting of mass calibration, or baseline correction when averaging data. It should be noted that the factor a is different from the factor α defined by Allan et al. (2003), which accounts for variability in the size of pulses generated when single ions impact the detector. In this work, our data acquisition card, a time-to-digital converter (TDC), only counted single ions crossing a threshold, and thus the pulse variability did not influence the error estimate, as all signal pulses were large enough to cross the threshold. It should be also noted that most lately manufactured CI-APi-TOFs use an analog-to-digital converter (ADC) as the data acquisition card, the same as AMS does, and in this case the empirical factor a would incorporate the factor α.

With the stable signals during these experiments, the analytical uncertainty was fitted to the signal strength based on Eq. (7). Detailed information and discussion are provided in the Supplement Sect. S1. Briefly, the results suggest that the analytical uncertainty is independent of mass-to-charge and instrument tuning; the a value was fitted as 1.28. In other words, the inclusion of the a parameter increases our uncertainty estimate by 28 %. For 300 s integration time, the overall error was estimated as Sij=0.074|Xij|+0.035. We also proposed a different statistical method based on ambient data (see Supplement Sect. S2). A comparison of different uncertainty estimation schemes is shown in Fig. 1, where the red curve denotes the revised error estimate in this work, the blue one is the customary estimate for AMS data, and the black one is the error estimated from ambient data with a different estimation scheme with its uncertainty shown as the gray area. Within the fitting uncertainty, all three estimates agree well.

In this work, two more steps were employed to further modify the error estimation:

For variables below 3σnoise, we fixed the signal as σnoise and the corresponding uncertainty as 6σnoise. A similar approach was suggested by Polissar et al. (1998), but this practice is criticized by the developer of the PMF model, P. Paatero (Paatero, 2016). The effect of this “data censoring” proved negligible in our work and is discussed in more detail in the Supplement (Sect. 3).

The nighttime type-1 factor is the largest contributor to nighttime HOM concentration. It exhibits elevated intensity during 20:00–04:00 and is less intense (about five times lower) in the daytime. The major peaks in this factor are identified as C10H14-16O6-13 and C19-20H28-32O10-18. As shown in Fig. 8, the profile of this factor is very similar to the reference spectrum from previous laboratory studies reported by Ehn et al. (2014), where only ozone and α-pinene were mixed. It should be noted that, in the atmosphere, there is always a mixture of monoterpenes likely contributing to these signals, in contrast to a single monoterpene precursor was used in the chamber experiments. Also humidity and temperature were typically different and changing constantly, and all these together can explain the minor difference in individual peak intensities; for example, 372 Th (C10H14O11) and 389 Th (C10H15O12) are higher in the reference spectrum than in the factor profile. The coefficient of uncentered correlations between the factor profile and the reference spectrum was calculated to be 0.91, confirming the high similarity between them. Thus, the source of this factor is very likely the ozonolysis of monoterpenes.

The diurnal variation of the nighttime type-2 factor has a similar pattern to that of the nighttime type-1 factor. Its intensity is about 30 % of nighttime type-1 factor during the nighttime and almost decreases to 0 during the day (Fig. 6). To our knowledge, no reference spectrum that matches the profile of this factor (shown in Fig. 5) has been reported. However, a set of masses can represent a new fingerprint. Figure 9a shows these fingerprint peaks in the dimer range, which are categorized and marked in different colors. In general, the vast majority of compounds contain nitrogen and we divide dimer peaks in this factor into six groups according to their elemental composition, i.e., C20H31O7-15NO3, C20H32O4-11(NO3)2, C19H29O6-13NO3, C19H31O8-11NO3, C18H29O8-11NO3, and other non-nitrogen-containing dimers. As dimers are closed-shell molecules, assumed to be formed through the reaction between two peroxy radicals (RO2) (Rissanen et al., 2014), the nitrogen atom(s) in the dimer molecule must come from its parent RO2 radical, suggesting NO3-initiated oxidation. Note that the possibility of NOx involvement can be ruled out, because when NOx reacts with RO2 it either ends up with an organonitrate HOM monomer or forms an alkoxy radical (RO) so that the nitrogen atom will not retain in the molecule. The fractions of different groups are shown in Fig. 9b. About 61 % of HOM dimers in this factor contain one nitrogen atom, suggesting that the major dimer formation process involves reaction between two RO2 radicals initiated by NO3 and O3, respectively. Also, about 22 % of these dimers contain two nitrogen atoms, meaning that both reacting RO2 radicals are NO3-initiated. The schematic illustrations given below show two examples of dimer formation containing one nitrogen atom (C20H31NO13, 555 Th including NO3-) and two nitrogen atoms (C10H32N2O12, 554 Th including NO3-), respectively. O3+C10H16⟶-OH⚫C10H15O2⚫⟶H-shift+O2…⟶H-shift+O2C10H15O8⚫⟶H-shift+O2C10H15O10⚫

NO3+C10H16→C10H16NO3⚫⟶H-shift+O2…⟶H-shift+O2C10H16O4NO3⚫C10H15O8⚫+C10H16O4NO3⚫→C20H31O10NO3+O2C10H16O4NO3⚫+C10H16O4NO3⚫→C20H32O6(NO3)2+O2 As NO3 is involved in the formation of more than 80 % dimer molecules, the nighttime type-2 factor is likely representing monoterpene oxidation by NO3.

As mentioned above, the nighttime factors are interpreted as representing nighttime oxidation of monoterpene initiated by the two major nocturnal atmospheric oxidants – O3 and NO3, respectively. Their nighttime patterns are similar, exhibiting an increase at 20:00 and a decrease at 04:00 in the next morning (Fig. 6). However, as the O3 concentration is relatively stable throughout day while NO3 is much lower in the daytime, the O3-initiated factor has finite level during the daytime while the NO3-initiated factor goes almost to 0. In general, the O3-initiated factor is a larger contributor than the NO3-initiated factor, suggesting that O3 is a more important nighttime oxidant for HOM formation at this measurement location. However, as shown in Fig. 10a, during the period (from 9 to 12 April) when polluted air masses containing high NOx were transported to this area, the NO3-initiated oxidation was significantly enhanced and became dominant. Since the NO3 chemistry could be one important pathway of forming HOMs, future laboratory study of this reaction channel is required.

The interpretation of daytime factors is more difficult, likely reflecting more complex daytime photochemistry. Nevertheless, certain conclusions can be drawn from spectral characteristics and temporal behavior of the three daytime factors.

As shown in Fig. 6, this factor concentration starts to increase in the early morning (around 04:00), concurrent with the increase of NO and the decrease of the two nighttime factors. The very similar temporal behavior of this factor and NO (Fig. 10b) indicates that NO reaction is likely plays an important role in this factor. The two highest peaks in this new factor are 355 Th (C10H15O6NO3) and 387 Th (C10H15O8NO3), which are likely formed through the reaction between the two most abundant (O3-initiated) RO2 radicals and NO, as shown below: C10H15O8⚫+NO→C10H15O6NO3,C10H15O10⚫+NO→C10H15O8NO3. We hereby interpret this factor as products from RO2 + NO reaction, which is also consistent with the observation that no dimer HOMs are present because NO is the dominating RO2 terminator in this pathway.

The daytime type-2 factor is one of the main daytime HOM contributors. The major peak in this factor is found at 339 Th (C10H15O5NO3), the single highest organonitrate molecule observed at this site and the representative of daytime HOMs previous reported by Kulmala et al. (2013). Another major peak in this factor is 224 Th (C5H6O6), possibly a fragment of monoterpene oxidation as observed in laboratory experiments (e.g., Tröstl et al., 2016). Besides these major peaks, this factor contains many other HOM monomer peaks.

This factor rises at around 05:00 and reaches a maximum between 11:00 and 15:00 (Fig. 6). Figure 10c shows that the time series of this factor and sulfuric acid are very similar. In cloudy days (UVB < 0.2 W m-2), the intensity of this factor is near 0. Note that this factor tracks sulfuric acid better than solar radiation. For example, the solar radiation was similar on 7 and 8 April, whereas the factor's intensity was much lower on 8 April, similar to the variation of sulfuric acid. Due to this reason, we interpret this factor as daytime oxidation of monoterpene controlled by OH, though NO must also be involved because the single highest peak is an organonitrate. Also, note that the participation of O3 cannot be entirely excluded.

The daytime type-3 factor shows maximum intensity in the afternoon around 14:00 (Fig. 6). Fingerprint peaks in this factor are organonitrate HOMs with smaller molecule weight, such as 213 Th (C3H5O3NO3), 241 Th (C4H5O4NO3), 255 Th (C5H7O4NO3), 269 Th (C6H9O4NO3), and 281 Th (C7H9O4NO3). Indicated by the smaller carbon number in the molecules, these light HOMs could come from anthropogenic VOCs (i.e., benzene and toluene). However, this possibility seems unlikely since the intensity of this factor does not show any significant increase during the period with transported pollution (9–12 April) when presumably benzene and toluene concentration were elevated. Another possibility is that these compounds are fragments from the oxidation of larger VOCs (e.g., monoterpene), and the presence of some HOM monomer peaks in this factor seems to support this assumption. This factor shows a good correlation with UVB (see Fig. 10d, and Table 2), indicating the HOM formation pathway represented by this factor is probably OH-initiated. Though the fingerprint peaks in this factor are organonitrates, the temporal variation of this factor shows no dependence on NO concentration. Instead, it exhibits a similar pattern with temperature, as shown in Fig. 10d. One possible explanation is that these small HOM molecules are relatively more volatile, so that their aerosol–gas partitioning is strongly affected by temperature – higher temperature leads to less condensation and high gas-phase concentration.

According to the mathematical diagnostics discussed in Sect. 4.2, the third factor is important for the model to account for a significant fraction of the variability in the ambient data. The only prominent peak in this factor is nitrophenol (C6H5NO3, 201 Th), a tracer for biomass burning suggested by previous studies (e.g., Mohr et al., 2013). The temporal behavior of this factor is similar to SO2, both showing a significant enhancement during the period of 9–12 April, when the measurement site was influenced by polluted air masses coming from eastern Europe (see Fig. S11). We therefore suggest that this factor is a signature of transported pollution from biomass burning from continental areas.