A Novel Apportionment Method Utilizing Particle Mass Size Distribution across Multiple Particle Size Ranges

Abstract

Many cities in China are facing the dual challenge of PM_2.5 and PM₁₀ pollution. There is an urgent need to develop a cost-effective method that can apportion both with high-time resolution. A novel and practical apportionment method is presented in this study. It combines the measurement of particle mass size distribution (PMSD) with an optical particle counter (OPC) and the algorithm of normalized non-negative matrix factorization (N-NMF). Applied in the city center of Baoding, Hebei, this method separates four distinct pollution factors. Their sizes (ordered from the smallest to largest) range from 0.16 μm to 0.6 μm, 0.16 μm to 1.0 μm, 0.5 μm to 17.0 μm, and 2.0 μm to 20.0 μm, respectively. They correspondingly contribute to PM_2.5 (PM₁₀) with portions of 26% (17%), 37% (26%), 33% (41%), and 4% (16%), respectively, on average. The smaller three factors are identified as combustion, secondary, and industrial aerosols because of their high correlation with carbonaceous aerosols, nitrate aerosols, and trace elements of Fe/Mn/Ca in PM_2.5, respectively. The largest-sized factor is linked to dust aerosols. The primary origin regions, oxidation degrees, and formation mechanisms of each source are further discussed. This provides a scientific basis for the comprehensive management of PM_2.5 and PM₁₀ pollution.

1. Introduction

Particulate matter (PM) pollution poses a global health risk, causing millions of premature deaths annually due to cardiovascular and respiratory diseases and potential cognitive decline [1]. Effective management of air quality requires not only monitoring pollution levels of PM but also understanding the PM sources and their relative contributions [2]. Traditionally, this has been accomplished using expensive, research-grade instruments [3,4]. However, the high costs and logistical challenges associated with these instruments often limit the spatial and temporal resolution of collected data.

Optical particle counters (OPCs) have emerged as a cost-effective tool for monitoring particle size distributions [5,6,7]. Recent advancements have improved the accuracy and reliability of OPCs [6], especially for measuring particle mass concentrations [7,8], making them a viable tool for routine air quality monitoring and exposure assessment. These OPCs enable the collection of data at finer spatial and temporal scales, providing valuable insights into pollution levels and sources across diverse environments [9]. Despite these advancements, there remains a significant gap in understanding the underlying factors that control pollution levels of PM, such as the identification and quantification of pollution sources using OPC data [10].

Recently, several studies have utilized the non-negative matrix factorization (NMF) algorithm to analyze particulate pollution sources by examining data matrices composed of both particle number concentrations across various size bins and key gaseous pollutants (CO, NO, NO₂, and O₃). These studies, conducted in cities such as Delhi, India [5], Atlanta, Georgia, the USA [11], and Kinshasa, the Democratic Republic of the Congo [12], have compared the sharing of particles of different sizes among various resolved factors and correlated these with particulate factors derived from Aerosol Chemical Speciation Monitor (ACSM) measurements. Such approaches have provided substantial insights into the identification and explanation of pollution sources such as CO-dominated (combustion-related) aerosols and secondary aerosols. However, these resolved factors are often characterized by intensity in arbitrary units, which cannot quantify their contributions to PM₁ or PM_2.5. To address this limitation, Bousiotis et al. [13] applied a two-step positive matrix factorization (PMF) method on particle number size distribution (PNSD) measurements using OPCs (first PMF step), incorporating auxiliary datasets of PM₁, PM_2.5, and PM₁₀ and other criteria pollutants (second PMF step), to better estimate the relative contributions of pollution sources to PM [14].

In our study, we fully leverage the direct measurement of PMSD using an OPC to test the performance of the N-NMF algorithm on the PMSD data. The measurements were conducted in Baoding, Hebei, a typical industrial city in North China, during the summer. The PMSD is measured with a finer size bin resolution (68 channels ranging from 0.15 to 20 μm), allowing for detailed size profiles of the resolved factors. These profiles can be integrated to determine the PM concentrations at different sizes (PM₁, PM_2.5, PM₁₀, and PM₂₀). Additionally, the resolved factors are identified by correlating them with the criteria gaseous pollutants and chemical compositions of PM_2.5, providing a comprehensive understanding of the pollution sources. The primary origin regions, oxidation degrees, and formation mechanisms of each source are further discussed. This study provides a scientific basis for using OPC-based PMSD measurements for particle source apportionment and opens vast prospects for its application.

2. Experiment

2.1. Site Description

Continuous measurements were conducted from 10 June 2023 to 10 July 2023 at an urban site in Baoding, Hebei, China. The city, with a population of 11.2 million, is a prefecture-level city founded on its great manufacturing industries within the Beijing-Tianjin-Hebei Economic Region. Geophysically, Baoding is located on the northern side of the North China Plain, approximately 20 km away from the Taihang Mountains (1000 to 2000 m a.m.s.l. on average). Baoding has consistently experienced poor air quality due to its intensive industrial activities and unfavorable meteorological and geographical conditions [15]. Dust, secondary aerosol formation, coal burning, and biomass combustion are considered the largest contributors to PM in Baoding [16].

The sampling site was located on the rooftop of a four-story building at the Baoding Environmental Protection Bureau in Hebei Province, China (38.89° N, 115.47° E) (Figure 1). The rooftop is approximately 18 m above ground level. The site is in the central urban area, surrounded by busy roads and adjacent to government, commercial, and residential areas.

2.2. Instrument and Data Description

The PMSD (in units of μg m⁻³) was measured using an OPC-based particle mass sizer (CCLJP-100, Sailhero Co., Ltd., Shijiazhuang, China) with a time resolution of 5 min. This mass sizer measures PM concentrations within the size range of 0.16 µm to 20 µm across 68 channels.

An OPC is an optical-based particle counter with simultaneous output of the number and mass concentration of particles in 68 size bins in the range of 0.16 μm–20 μm. Aerosols containing particles of various sizes are sampled representatively within the PM range (<approximately 20 μm) and are dried to RH below 60% to remove moisture, ensuring accurate measurements. The particle size number distribution is first obtained. Then, the mass concentration distribution is determined using size-dependent conversion functions that account for the optical properties (refractive index) and particle density.

Here are some key features of the newly developed OPCs. The zero-count rate is below 0.015 cm⁻³; the counting efficiency is below ±20%; and the size measurement error is within 15%. The mass concentrations of PM_2.5 and PM₁₀ were calculated by summing the data of all corresponding channels. For the OPCs in this study, 2.5 μm was selected as the cut point for PM_2.5, summing all channels below 2.5 μm, and similarly, 10 μm for PM₁₀. During our measurement, the fitting slope for PM_2.5 measured using the OPC and regulatory-grade instruments is 1.08 ± 0.007 (R = 0.91), while the fitting slope for PM₁₀ is 1.00 ± 0.004 (R = 0.96). The absolute measurement error for PM_2.5/PM₁₀ with their 24 h average concentrations below 100 μg m⁻³ is ±15 μg m⁻³. The relative measurement error for PM₁₀ with the 24 h average concentration above 100 μg m⁻³ is ±15%. The instrument parallelism is within 10% for both PM_2.5 and PM₁₀.

In addition to the strong agreement with regulatory-grade (RG) instruments, which ensures the reliability of our analysis, we also calculate the concentrations of PM₁ (submicron particles) and PM₂₀ by aggregating the corresponding channels.

Three RG instruments were deployed at the same site to measure the chemical composition of PM_2.5. The concentrations of organic and elemental carbon (OC/EC), water-soluble ions, and trace metal elements in PM_2.5 were measured using a semi-continuous thermo-optical carbon analyzer (Sunset Laboratory, Portland, OR, USA), an In situ Gas and Aerosol Composition (IGAC, Model 63GA, Machine Shop, Fortelice International Co., Ltd., New Taipei City, Taiwan), and a Multi-Metals Monitor System with dispersive X-ray fluorescence analysis (HMAM-2000A, Sailhero Co., Ltd., Shijiazhuang, China), respectively.

Data for criteria pollutants, including PM_2.5, PM₁₀, CO, SO₂, NO₂, and O₃, were obtained from real-time monitoring with RG instruments at this site. Meteorological parameters, including wind speed (Ws), wind direction (Wd), surface pressure (Press), relative humidity (RH), and ambient temperature (Temp), were obtained through a weather station at the site.

2.3. Normalized Non-Negative Matrix Factorization (N-NMF)

NMF is a powerful algorithm for decomposing a non-negative matrix V into the product of two smaller non-negative matrices, W and H, such that V ≈ W × H. In this context, W represents the basis matrix, which captures the characteristic patterns of different factors; H is the coefficient matrix, which indicates the weight of each sample on these factors. Since its introduction by Lee and Seung in 1999 [17], NMF has become widely used in various fields due to its ability to reveal hidden structures in data through matrix decomposition. The non-negativity constraint ensures that both the basis and the coefficient matrices can be further decomposed if needed, providing an intuitive and interpretable model. From a matrix space perspective, the significance of NMF lies in identifying a new set of bases within the original space and projecting the original data onto these bases. The original non-negative matrix V represents the original data in the original space. The decomposed matrices W and H correspond to the new bases and the projected values on these bases, respectively.

To address the potential impact of magnitude differences between variables on the decomposition process, input data is often normalized before performing NMF. This specific approach, referred to as normalized NMF, involves dividing each variable by its mean value prior to decomposition [11,12]. After matrix decomposition, it is crucial to inverse-normalize the H matrix to obtain meaningful quantitative factor concentrations. This step ensures that the weight matrix accurately reflects the relative contributions of each factor across the samples [18].

We applied N-NMF to a PMSD (68 channels) matrix containing 744 sets of hourly averages. The model outputs average size profiles and time series for each factor. The intensity time series captures the sum of particle mass concentrations in the 0.16–20 μm range. Particles smaller than 0.16 μm are disregarded due to their negligible concentrations. Thus, this series reflects each factor’s contribution to PM concentrations below 20 μm.

2.4. Calculation of PM Concentration over Various Size Ranges

The concentrations of particles over various size ranges are calculated by summing the mass concentrations of the size bins within the size profiles of each resolved factor [19,20]. The calculation formula is as follows:

$$P{M}_{n-k}={\displaystyle {\displaystyle \sum }_{bin\_n}^{bin\_k}}\left(\begin{array}{c}k\\ n\end{array}\right)d{M}_{bin\_i}$$

where PM_n–k represents the mass concentration of particles over a size range of n–k, with n and k being the lower and upper size limits, respectively. dM_{bin_i} represents the mass concentration of the i^th size bin. In our study, the concentrations of PM₁, PM_2.5, PM₁₀, and PM₂₀, as well as the concentrations of PM_1–2.5, PM_2.5–10, and PM_10–20 for each resolved factor, have been determined using the source profiles obtained from N-NMF.

3. Results and Discussions

3.1. General Characteristics of PMSD

During the observation period, the wind speed was generally lower than 2 m s⁻¹, except on some days when it reached 4 m s⁻¹ at noon (Figure 2a). The average concentrations of PM_2.5 and PM₁₀ measured using the OPC were 24.6 μg m⁻³ and 44.7 μg m⁻³, respectively (Figure 2c). The Pearson correlation contour graph between PM concentrations over various size ranges generally shows the strongest correlations between adjacent size bins (Figure 2d). Regions with high Pearson’s R values (>0.9) are mainly concentrated in four particle size ranges, i.e., 0.16–0.3 μm, 0.3–1.0 μm, 1.0–10 μm, and 10–20 μm, suggesting four potential aerosols sources. Meanwhile, compared to our earlier study on PNSD in the submicron range [21], the wider coverage of high Pearson’s R regions in this study indicates that the aerosol sources encompass larger particle size ranges.

3.2. N-NMF Performance

Using the N-NMF algorithm, a solution with four factors is determined to be optimal, based on the “NMF rank survey” [16] and hierarchical clustering [22], as shown in Figure 3. Ideally, receptor models distinctly separate each source, but real-world results often show mixed sources. Therefore, the goal is to minimize mixed sources while identifying as many sources as possible for detailed air quality management. The NMF algorithm supports running multiple factors simultaneously and uses hierarchical clustering to evaluate the number of factors based on stability [23]. The resulting heat map helps assess the degree of separation among the factors. This section provides a detailed explanation of the parameters from the “NMF survey rank” (Figure 3a) and hierarchical clustering (Figure 3b) to clarify our approach in determining the optimal number of factors.

The cophenetic correlation coefficient [22], derived from the consensus matrix, assesses factor stability by measuring clustering consistency. Higher values indicate greater stability. For the factor numbers 2 and 4, the coefficients are 0.998 and 0.987, respectively. Since the two factors do not provide sufficient source apportionment, we initially chose a four-factor solution. Meanwhile, the dispersion coefficient [24] also evaluates NMF clustering reproducibility based on the consensus matrix, with the highest reproducibility observed for two- and four-factor solutions.

The evaluation metrics include explained variance (evar) and residuals. Evar measures how well the model explains data variability and should be maximized while other parameters are satisfactory. Conversely, residuals represent unexplained data segments, guiding model improvements by minimizing discrepancies between predicted and actual values. The residual sum of squares (rss) encapsulates these discrepancies, indicating model fidelity. However, a higher evar or lower residuals/rss with a larger number of factors is not always better, as it can sacrifice clustering consistency and factor stability. For solutions from 2 to 10 factors, evar increases, and residuals/rss decreases consistently, but the rate of change slows between four- and five-factor solutions.

Silhouette analysis [25] graphically represents clustering quality by measuring how well an object matches its cluster compared to others. The silhouette value, ranging from −1 to +1, indicates clustering effectiveness, with higher values showing better-defined clusters. For solutions from 2 to 10 factors, the silhouette parameter generally decreases, with the four-factor solution being the second highest after the two-factor solution. Furthermore, sparseness, as conceptualized by Hoyer [26], quantifies how concentrated the significant components are within a vector. In the context of NMF, these vectors are the columns of the factor matrices. Sparseness ranges from 0 to 1, where 1 indicates a very compact vector with few non-zero components. A higher sparseness value means that the vector is dominated by a few significant components, highlighting the importance of specific clusters. In this study, the four-factor resolution shows the almost highest sparseness parameter, suggesting a compact and meaningful clustering solution.

Finally, Figure 3b shows the hierarchical clustering of the original matrix, illustrating the degree of separation between subcategories [22]. A higher separation indicates a clearer distinction between factors and fewer mixed sources. Among 3, 4, and 5 factors, the four-factor solution shows the clearest separation. Thus, based on the “NMF survey rank” results, we determined that the four-factor solution is most appropriate. The final results confirm that this solution is both reasonable and practically significant.

3.2.1. Source Identification

The size profiles of the resolved factors, with an optimal number of 4, are presented in Figure 4a. Their particle size profiles exhibit clear distinctions, with N1–N4 (ordered from smallest to largest sizes) dominating (>50%) different particle size ranges, which are 0.16–0.3 μm, 0.3–1.0 μm, 1.0–8.0 μm, and >8.0 μm, respectively (Figure 4b). As illustrated in Figure 4c,d, the fractions at different size ranges (commonly used ranges such as PM₁, PM_2.5, PM₁₀, and PM₂₀) can be calculated from their source profiles (see Section 2.4). For PM₁, N1 and N2 are the primary contributors, accounting for 34% and 49%, respectively. As the particle size increases to below 2.5 μm (PM_2.5), the contribution of N3 rises to 33%, while the shares of N1 and N2 decrease to 16% and 37%, respectively. For larger particles, i.e., PM₁₀ and PM₂₀, the combined contribution of N3 and N4 exceeds 50%. This is because N3 and N4 are the predominant contributors to particles larger than 2.5 μm, accounting for more than 90% of the mass in this size range (Figure 4e).

To further investigate the source of these resolved factors, we compared their time series with criteria pollutants and PM_2.5 chemical compositions (Figure 5a). The correlations between each resolved factor are generally low (Figure 6). We found strong correlations with one or more substances, and these high-correlation tracers helped us identify the sources.

Combustion aerosols. N1 has the smallest size peak at 0.2 μm (Figure 4a) and shows a strong correlation (R = 0.77) with carbonaceous aerosols (total carbon, TC) in PM_2.5 (Figure 5, Figure 6 and Figure 7a). TC is made up of EC and OC. EC comes solely from combustion. OC includes primary organic carbon (POC), often co-emitted with EC, and secondary organic carbon (SOC), which forms from the oxidation processes of anthropogenic and biogenic volatile organic compounds (VOCs) [27,28]. Recent studies indicate that SOC in summer in this region is largely linked to combustion sources [29,30]. Hence, N1 is identified as combustion aerosols. Meanwhile, N1 is highly correlated with selenium (Se) (Pearson’s R = 0.66) and potassium (K) (Pearson’s R = 0.67), which are key markers for coal combustion [31] and biomass burning [32,33], respectively. This supports our identification of N1 as combustion aerosols. We refrain from extensive interpretations due to the limited availability of detailed data.

As illustrated in Figure 8a, the lowest percentile values of N1 generally coincide with low EC concentrations (<1 µg m⁻³). At higher EC concentrations (>1 µg m⁻³), the highest percentile values of N1 occur with higher OC concentrations (i.e., higher OC/EC ratios). OC/EC ratios varied between 3.0–7.4 (10th–90th percentiles), which matched with the values obtained in earlier studies [34]. A high OC/EC ratio typically indicates a greater degree of incomplete combustion (such as solid fuel combustion) or the formation of SOC. In June, which is neither the harvest season nor the residential heating period, ample solar irradiation and higher temperatures favor SOC formation. Therefore, the high OC/EC ratio is more likely due to SOC formation rather than fresh emissions, suggesting that N1 includes some aged components. N1 is not significantly correlated with SO₂ (Figure 6), a marker for fresh coal combustion emissions [21]. SO₂ mostly fluctuates within a narrow range of 2 ppb to 6 ppb. As illustrated in Figure 8b, Higher sulfate concentrations indicate a higher degree of oxidation and coincide with higher N1 values, which also suggests that N1 has undergone some oxidation.

During June 11–16, N1 shows an accumulating trend from midnight, peaking around 7:00 a.m.–9:00 a.m. (Figure 5a). However, over the entire observation period, N1, among all resolved factors, has the weakest diurnal variation on average (Figure 5b). Thus, morning rush hour commuting can only partly explain the observed peak of N1. Overall, N1 is identified as combustion aerosols related to coal combustion, with a certain degree of oxidation. Since the site is located in an urban area without nearby combustion sources using fossil fuels, the aerosols likely originate from suburban transmission. This transport provides aging time, contributing to the observed oxidation.

Secondary aerosols. N2 is characterized by a bimodal profile peaking at 0.2 and 0.5 μm (Figure 4a). N2 exhibits strong correlations with NO₃⁻ and NH₄⁺ with Pearson’s R values of 0.89 and 0.83, respectively (Figure 5 and Figure 7b). N2 is then identified as a secondary aerosol because ammonium nitrate is the main secondary inorganic component in PM_2.5 [35]. The N2 concentration increases at night, rising after sunset at 6 p.m. and peaking at 9 a.m. This phenomenon aligns with the volatilization-condensation process of nitrates in high summer temperatures. N2 predominantly originates from the north (Figure 9a). The distinct source region on this CPF does not align with the broad sources of secondary aerosols identified in previous studies [21]. We believe this is the result of a coupled process of nitrate formation and meteorological conditions. First, the night-time conditions at the observation site are conducive to the heterogeneous hydrolysis of N₂O₅, leading to nocturnal nitrate formation [36]. High concentrations of O₃ (61.2 ± 24.7 ppb) and NO₂ (7.1 ± 6.1 ppb) favor the gas-phase formation of N₂O₅ [37], while elevated levels of hygroscopic components (sulfates, nitrates, and chlorides with the night-time average concentrations of 4.6, 3.9 and 0.6 μg m⁻³) support this process. Nitric acid produced during the day under active photochemical conditions can condense onto particles after sunset with descending temperature [38]. The unique geographical features of Baoding result in obvious diurnal variations in wind speed and direction (Figure 9b). Starting at noon, the wind shifts from north to south, with wind speed gradually increasing. After sunset (around 6 p.m.), the wind speed decreases but remains southerly. By 5 a.m. the next day, the wind shifts to the north again, with a slight increase in wind speed around 9 a.m. The wind pattern changes cause pollutants (including gaseous nitric acid and precursors of the N₂O₅ reactions) originating from the city to accumulate at the base of the mountains. As night-time temperatures drop, nitrate concentrations increase. Ultimately, morning northerly winds blow these pollutants back to the observation site, creating a hotspot to the north on the CPF polar plot (Figure 9a). As previously discussed, N1 may also include secondary inorganic aerosols like sulfates, while N2 has a larger particle size mode (0.5 μm), likely related to the volatilization-condensation process of nitrates. Ambient particles around 0.5 μm, which significantly contribute to particle surface area, can easily capture gaseous nitric acid or N₂O₅ during condensation, forming particulate nitrate [39].

Industrial aerosols. N3 shows high correlations with trace metals such as manganese (Mn) (Pearson’ R = 0.75), iron (Fe) (Pearson’ R = 0.90) and calcium (Ca) (R = 0.82) in PM_2.5 (Figure 5 and Figure 7c). Mn and Fe mainly originate from industrial sources, particularly ferrous metal smelting and processing [40]. Iron is a well-established indicator of industrial aerosols because it is predominantly emitted from industrial sources when there are no dust events. Zhu, et al. [41] found that iron in aerosol particles is significantly contributed by industrial emissions, especially from metal smelting activities. The larger size profile of N3 (Figure 4a) matches with that of industrial aerosols reported from a size-resolved source apportionment study in China [42]. Hence, N3 is characterized as industrial aerosols. N3 shows a similar diurnal trend with N2 (Figure 4b). Excluding the specific episode from the southwest during 5–7 July (see more discussion in Section 3.2.2, the CPF hotspot is also located to the north of the city (Figure 10). We infer that N3 may also contain contributions of nitrate in the coarse mode. Although we only measured nitrates in PM_2.5, many studies have found that nitrates can enrich in the coarse mode [43], especially when abundant alkaline elements (such as Ca) are present [44]. N3 is also correlated with NO₂ with Pearson’s R of 0.66 (Figure 6). In China, NO_x originates from the industry, power, and transport sectors [45]. We cannot entirely exclude the impact of traffic dust on N3, especially given its high correlation with Ca. The size-resolved source apportionment study [41] has also shown that suspended dust mainly distributes in the coarse mode. However, due to the site’s central urban location, high-emission diesel vehicles are prohibited, and its elevated position minimizes local influences, resulting in no evident rush hour peaks (Figure 4b). It is more likely that the site is affected by combined emissions from suburban industrial areas and dense traffic networks after being transported.

Dust aerosols. N4′s size profile is the largest (Figure 4), identified as dust aerosols. It is the only factor not highly correlated with any criteria pollutants or PM_2.5 chemical compositions (Figure 6). With a primary particle size greater than 2.5 μm, this suggests the chemical compositions of PM_2.5–10 significantly differ from those of PM_2.5. N4 shows a relatively high correlation with PM_2.5–10 concentrations (R = 0.59, Figure 7d) and matches wind speed variations (Figure 2a). As illustrated in Figure 11, CPF analysis shows that high-speed southerly winds during the day (6:00 a.m.–6:00 p.m.) are the main source of N4, while night-time peaks (6:00 p.m.–6:00 a.m.) from the east indicate notable dust sources. These peaks do not occur under calm winds, indicating significant dust sources to the east of the site.

3.2.2. Pollution Episodes

Three pollution episodes (PE) were observed during the observation: PE1 (15–18 June), PE2 (1–3 July), and PE3 (5–8 July). Time series of nitrate, sulfate, and Fe concentrations in PM_2.5, along with N1–N3 concentrations, are marked in Figure 12 for these episodes. Backward trajectories are also plotted in Figure 13 to help analyze the meteorological conditions leading to these pollution events.

As illustrated in Figure 12, during PE1 and PE2, sulfate increased more significantly than N1, resulting in a weaker correlation between sulfate and N1 compared to TC (Figure 6), while N2 increased more significantly than nitrate. The sulfates in N1 mainly originate from local stationary combustion sources around Baoding. Under stagnant meteorological conditions across the North China Plain (Figure 13), sulfates from other cities (potentially indicating higher oxidation levels) influenced the site. These regional sulfates were ultimately incorporated into N2 rather than N1.

During PE3 (Figure 12), N3 (industrial aerosols) shows a significant accumulation, with Fe, Mn, and Ca following the same trend (not presented). This occurred as stagnant conditions transitioned to favorable dispersion (Figure 13), blowing high concentrations of pollutants accumulated in front of the Taihang Mountains toward the site. In contrast, N1 and N4, representing local pollution sources, did not exceed typical levels during PE3.

4. Conclusions

In this study, we apply the N-NMF algorithm to hourly PMSD data, achieving a successful four-factor resolution. These factors are objectively supported by model output parameters (NMF rank survey) and subsequently analyzed for correlation with time series data of criteria pollutants and PM_2.5 chemical compositions for identification. The four factors, ordered by increasing particle size profiles, correspond to combustion aerosols (N1), secondary aerosols (N2), industrial aerosols (N3), and dust aerosols (N4). Combustion aerosols primarily originate from stationary sources (coal-combustion related) around the city and urban commuting activities in morning rush hours, undergoing some oxidation before reaching the site. Secondary aerosols, mainly nitrates, are facilitated by the regular diurnal changes in wind speed and direction, with the northern suburbs of Baoding (at the foothills of the mountains) promoting nitrate formation, which then impacts the urban area in the early morning. Industrial aerosols, characterized by Fe and Mn from ferrous metal elements, are significant contributors to PM_2.5, influenced by both local and regional pollution sources. Dust aerosols are mainly influenced by wind speed, with notable night-time sources to the east of the site. We determined their contributions to PM₁, PM_2.5, PM₁₀, and PM₂₀, providing actionable insights for targeted regulatory measures. A recent study has shown that fine particle concentrations can be related to coarse particle concentrations [46].

Notably, PMSD-based source apportionment offers a new perspective on PM sources, which differs from PM_2.5 full composition analysis [47]. Combustion aerosols include sulfates, likely secondary but considered “primary sources” in Baoding. The unique geographical and meteorological conditions in Baoding facilitate the formation and dissolution processes of secondary aerosols with ammonium nitrate as the main component. Industrial aerosols, whether local or regional, show no significant size difference. The night-time dust aerosol source to the east of the site can also be identified from the PM_2.5–10 time series, with PMSD-based source apportionment indicating that this dust source is unassociated with other pollution sources.

We also recognize the limitations of the current study, such as the lack of support from size-resolved PM compositions. This is challenging due to insufficient equipment for simultaneous online measurements of PM_2.5 and PM₁₀ (using different cut-off heads). It is also difficult to obtain high-time-resolution data using offline methods due to the constraints of sampling volumes. Our study emphasizes that much valuable information is hidden in particle size distributions, warranting further research. This approach not only provides management tools but also deepens our understanding of PM pollution formation mechanisms. This methodology allows for cost-effective simultaneous source apportionment of PM_2.5 and PM₁₀. In cities with superstations measuring full PM_2.5 chemical compositions, this approach offers higher spatial resolution for source apportionment (including PM₁₀) by deploying more OPCs. Using higher time-resolution PMSD inputs (e.g., 5 min) can yield more detailed temporal resolution results, which can be used for future work. It is also suitable for underdeveloped areas that are unavailable for online PM_2.5 composition measurements, providing preliminary source apportionment results [12]. Although currently reliant on criteria pollutants and PM_2.5 compositions for identification, future applications across more sites and seasons could achieve more independent identification, potentially relying solely on LCS-measured criteria pollutants [11].