Differential Probability Functions for Investigating Long-term Changes in Local and Regional Air Pollution Sources

Conditional probability functions are commonly used for source identification purposes in air pollution studies. CBPF (conditional bivariate probability function) categorizes the probability of high concentrations being observed at a location by wind direction/speed and investigate the directionality of local sources. PSCF (potential source contribution function), a trajectory-ensemble method, identifies the source regions most likely to be associated with high measured concentrations. However, these techniques do not allow the direct identification of areas where changes in emissions have occurred. This study presents an extension of conditional probability methods in which the differences between conditional probability values for temporally different sets of data can be used to explore changes in emissions from source locations. The differential CBPF and differential PSCF were tested using a long-term series of air quality data (12 years; 2005/2016) collected in Rochester, NY. The probability functions were computed for each of 4 periods that represent known changes in emissions. Correlation analyses were also performed on the results to find pollutants undergoing similar changes in local and regional sources. The differential probability functions permitted the identification of major changes in local and regional emission location. In Rochester, changes in local air pollution were related to the shutdown of a large coal power plant (SO2) and to the abatement measures applied to road and off-road traffic (primary pollutants). The concurrent effects of these changes in local emissions were also linked to reduced concentrations of nucleation mode particles. Changes in regional source areas were related to the decreases in secondary inorganic aerosol and organic carbon. The differential probabilities for sulfate, nitrate, and organic aerosol were consistent with differences in the available National Emission Inventory annual emission values. Changes in the source areas of black carbon and PM2.5 mass concentrations were highly correlated.


INTRODUCTION
Air pollution is decreasing in many developed countries (Colette et al., 2011;Guerreiro et al., 2014;Ahmed et al., 2015;Masiol et al., 2017a), including the United States (Parrish et al., 2011;Pouliot et al., 2015;Duncan et al., 2016;Nopmongcol et al., 2016;Emami et al., 2018;Masiol et al., 2018;Squizzato et al., 2018).Downward trends reflect the implementation of legislation and regulations (Gerard and Lave, 2005;Parrish et al., 2011), the application of increasingly stringent emissions standards, improved abatement technologies, changes in fuel sulfur content (Klimont et al., 2013;Kheirbek et al., 2014), road (Dallmann and Harley, 2010;Russell et al., 2012;U.S. EPA, 2016) and off-road (Eyring et al., 2010;IMO, 2013;Masiol and Harrison, 2014;U.S. EPA, 2016;Zetterdahl et al., 2016) fuels, economic drivers (Tong et al., 2016), and energy policy.Over the past decades, those measures took place at different scales across the U.S., from local, city, and metropolitan area to state, regional, and even continental scales.Recently, Emami et al. (2018) and Masiol et al. (2018) have reported the changes in air pollution concentrations measured in Rochester, NY, a moderate sized city typical of the northeastern United States.A major research challenge is to determine if the trends represent positive/negative feedbacks of specific mitigation measures or the synergy of implementation of multiple policies at different scales rather than the consequence of changes in human habits, a reflection of recent economic conditions, and/or the direct/indirect effects of climate change.
This study presents an extension of conditional probability methods helpful in the investigation of long-term series of air quality data.The differences between conditional probability values for temporally different data sets were used to explore changes in local and regional air pollution sources.CBPF (conditional bivariate probability function) categorizes the probability of high concentrations (> threshold criterion) being observed at a location by wind direction/speed and aims to investigate the directionality of local sources.PSCF (potential source contribution function), a trajectoryensemble method, allows the identification of the source regions most likely to be associated with high measured concentrations (> threshold criterion).Its utility has been previously examined by Cheng and Lin (2001) and Begum et al. (2005).The differential CBPF and differential PSCF were tested using a long-term series of air quality data (12 years; 2005/2016) collected in Rochester, NY, a typical medium-sized metropolitan area in northeastern United States.Negative differential probabilities highlight areas where emissions are decreased.Conversely, positive differential probabilities point out areas where emissions have increased.Probability functions were computed over 4 multiple year periods that represent known changes in emissions.

Data Sources, Handling, and Consistency
Air quality data used in this study were collected at the NYS Department of Environmental Conservation reference site for Rochester (Fig. S1).The site is representative of citywide air quality, but also lies ~300 m from the intersection of two major highways (I-490 and I-590) with an average traffic of ~230,000 vehicles day -1 .
Hourly CO, NO, total reactive nitrogen (NO y ), SO 2 , O 3 , and PM 2.5 concentrations were routinely measured in accordance with federally mandated methods.NO 2 was estimated as NO y -NO.Equivalent black carbon (BC) (Petzold et al., 2013) and Delta-C (DC, difference between absorbance at 370 and 880 nm used to estimate biomass burning PM) (Sandradewi et al., 2008;Wang et al., 2011a), were measured using aethalometers with PM 2.5 cut-off cyclones (Table S1).Twenty-four hour integrated PM 2.5 species data were retrieved from the U.S. EPA chemical speciation network, including elemental (EC) and organic (OC) carbon, nitrate, sulfate, ammonium, and K + (Solomon et al., 2014).Data were processed to return a consistent dataset over the 2005-2016 period.Details are reported in supplementary materials.Since secondary material needs to be estimated to account for local and regional sources (Wang et al., 2012a;Masiol et al., 2017b), OC was also split between primary (OC pri ) and secondary (OC sec ) using the EC tracer method (Turpin and Huntzicker, 1995;Lim and Turpin, 2002;Cabada et al., 2004).Details are provided in the supplementary materials.

Meteorological Data
Wind data from the Greater Rochester International Airport (KROC) were retrieved from the NOAA NCDC repository (https://www.ncdc.noaa.gov/data-access).Backtrajectories were calculated using the NOAA/ARL Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT_4) model (Stein et al., 2015;Rolph et al., 2017), using the NCEP/NCAR Reanalysis data (Kalnay et al., 1996).HYSPLIT was run backward in time for 120 h using the vertical mixing model with a starting height of 500 m a.g.l.(Cheng et al., 1993) and 1-h intervals (24 trajectories per day).

Overview of Probability Functions
Conditional probability functions (CPF, CBPF, and PSCF) are widely used to locate the potential local and external sources affecting a site.CPF assesses the probability of wind directions associated with specific threshold criteria (usually the 75 th -90 th percentiles).CPF was further extended to the bivariate case to produce a conditional bivariate probability function (CBPF) plot using wind speed as a third variable plotted on the radial axis (Uria-Tellaetxe and Carslaw, 2014) where m ∆θ,∆v and n ∆θ,∆v are respectively the occurrence of observations exceeding the threshold C and the total number of data in the wind sector ∆θ and wind speed interval ∆v.
For distant sources, PSCF identifies the potential source regions located within a grid cell at latitude i and longitude j with probabilities of exceeding a threshold criterion: where n i,j represents the number of times the trajectory endpoints fell into cell i,j in the domain grid, and m i,j is the number of times the observed concentration exceeds the threshold C.
In this study, differential CBPF and PSCF functions are defined as the difference between the probabilities estimated in each of the two different periods τ 1 and τ 2 , with ∆τ = τ 2τ 2 : A single criterion C (75 th percentile) was calculated over the whole period (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016), and was then applied to single periods, i.e., Cτ 1 = Cτ 2 .The criterion, C, values used for each variable (and relative concentrations) are listed in Table S2.Wind speed/direction data and back-trajectories have 1-h time resolution and are matched with the concentration of hourly-measured variables (gases, PM 2.5 , BC, and Delta-C).However, PM compositional data are available for integrated 24-hour samples.Each value is matched with the 24 wind values or for the 24 trajectories that were calculated for each day (Kim and Hopke, 2004).
PSCF values may be affected by grid cells containing only a few endpoints that may be overestimated.Multiple weighting functions, W i,j , have been proposed to avoid this issue (Polissar et al., 2001;Begum et al., 2005;Kim and Hopke, 2006).The weighting function used here was: where p c is the probability corrected for the function W(n i,j ), p is the probability value from the uncorrected function, and N̅ is the average number of endpoints over the grid cells with at least one endpoint.Since single PSCFs are weighted, ∆PSCF i,j,∆τ was not weighted further.However, the PSCF values were computed only for cells having with more than 100 endpoints.The PSCF analyses were made for grid cells of 1° latitude by 1° longitude.
In a similar way, CBPF probability is less reliable for small numbers of observations (n ∆θ,∆v ).A weighting function was applied: The CBPF analyses were made using a sector size ∆θ of 10° and wind speed increments, ∆v, of 1 m s -1 .The major issue in the computation of CBPF values is the choice of the best wind speed/wind direction bins (the choice of ∆θ,∆v intervals is arbitrary).The "original" CBPF approach proposed in Uria-Tellaetxe and Carlsaw (2014) applied smoothing to the CBPF values to avoid individual points having excessive impact on the surface prediction.The rationale for smoothing is presented in Carlsaw (2015).In this study, smoothing was applied to the simple CBPFs.However, due to the nature of ΔCBPFs (negative to positive), smoothed surfaces did not properly depict the calculated results.Therefore, smoothing was not applied to ΔCBPFs.In addition, ΔCBPFs were computed only if there were at least 4 observations available for both periods.

RESULTS AND DISCUSSION
The two periods (τ 1 and τ 2 ) to be compared in the differential probability functions should be chosen based on known changes in local and regional emissions.However, while the selection of relatively short periods may result in a high temporal resolution, a short time interval also decreases the number of data values and lowers the reliability of ΔCBPF and ΔPSCF values.
From 2005 to 2016, regional and local emissions have changed significantly (Emami et al., 2018).Vehicle emissions have been significantly reduced (Mariq, 2007;Bishop and Stedman, 2008;Bishop et al., 2012;McDonald et al., 2013;May et al., 2014) with improved emissions controls and cleaner fuels.In 2007, new heavy-duty diesel trucks were required to have particulate filters.NO x controls were added in 2010.However, credit flexibilities have allowed the sale of engines with NO x emissions greater than the 2010 limit (0.2 g bhp-hr -1 ) through model year 2014.The light-duty vehicle (< 6,000 lbs gross weight) Tier 2 emissions program (extended to vehicles up to 10,000 lbs) required an average sulfur standard of 30 ppm (from 120 ppm) with a sulfur cap of 80 ppm (from 300 ppm) phasedin from 2004 to 2009.On-road diesel fuel sulfur content dropped from < 500 ppm to ultralow sulfur diesel fuel (ULSD; < 15 ppm) in 2006.ULSD fuels were subsequently required for non-road vehicles by 2010 and for locomotives and marine vessels by 2014.Additionally, all distilled oil sold in NYS for any purpose (including building heating) were required to be ULS after July 1, 2012.The implemented regulations and associated technological improvements resulted in a sharp decrease of NO x (-52%), SO 2 (-85%), and CO (-45%) in NYS between 2005 and 2016.These reductions were mostly associated with the decrease in the emissions from highway and off-highway vehicles and fuel combustion for electric power generation (Emami et al., 2018;Masiol et al., 2018;Squizzato et al., 2018).
Local changes also occurred in Rochester, including the 2008 shutdown of a 260 MW coal-fired power plant (~15 km NNW of sampling site, noted as RG & E in Fig. S1), a major source of SO 2 and ultrafine particles (Kasumba et al., 2009;Wang et al., 2011b, b), and decreased emissions from a coal-fired cogeneration plant (~8.5 km NW).The monitoring site is adjacent to a mainline railroad and several interstate highways so the changes in on-road and non-road fuel sulfur content between 2006 and 2014 were expected to be significant.
There were also substantial changes in major upwind sources such as electric generating stations driven both by regulatory policies and economics.These changes are discussed in detail by Emami et al. (2018) and Squizzato et al. (2018).Although several major efforts by the U.S. Environmental Protection Agency (Clean Air Interstate Rule and Cross-State Air Pollution Rule) were voided by the courts, many utilities implemented controls.There were also reductions because of legal actions such as the consent decree between American Electric Power and the U.S. EPA.Ontario moved to eliminate all fossil fuel combustion for electricity generation that was completed in 2014.Thus, significant reductions in sulfate and nitrate have been observed in Rochester (Emami et al., 2018) and across NYS (Squizzato et al., 2018).

CBPFs
Individual CBPF plots calculated for the each 3-year period are provided in Figs.S3-S8.The plot surfaces were smoothed (Carslaw, 2015); wind speed/wind direction bins with less than 4 observations are omitted (grey surfaces).For most pollutants, it is possible to see changes in the conditional probabilities from one 3-year period to the next when all of the plots use the same criterion value across the entire period.The highest conditional probabilities for most variables were found for low/moderate westerly wind regimes with probability peaking toward the SW (PM 2.5 , BC, PNC 50-100 , PNC 100-500 , CO, NO, NO 2 , NO y , OC pri , OC sec , sulfate, ammonium).Most pollutants are emitted by vehicular traffic (both exhaust and resuspension) and the CBPFs point toward the I-490 and I-590 highways suggesting that local road emissions dominated other local sources.However, a railway is also adjacent to the south and west of the site.Therefore, the contribution of dieselpowered rail engine emissions must be considered.
Biomass burning (BB) accounts for up to 30% of Rochester's winter PM 2.5 mass (Wang et al., 2011a).BB tracers (DC and K + ) (Andreae and Merlet, 2001;Koppmann et al., 2005) showed CBPFs similar to the other pollutants, but also high probabilities toward southeasterly suburban areas (Figs. S3 and S8,respectively).BB from heating, cooking (wood-fired pizza), or recreational purposes is the likely local source.The probabilities for PNC 11-50 (Fig. S4) are similar to the other pollutants.However, high probabilities extend into the NW quadrant, i.e., toward the two power plants (Fig. S1).The effect of local emissions is evident for SO 2 (Fig. S6), showing two potential source locations, i.e., traffic (SW) and power plants (NW) in the first two periods, then only power plants.This result well depicts the drop of traffic emissions due to the application of stringent rules for fuel sulfur content.Ozone decreases with wind speed (lower probability under calm wind regimes, Fig. S6) suggesting a regional rather than local origin give the low probability values.Titration with locally emitted NO for low wind speeds may reduce the local values by depleting ozone on a timescale of minutes.
Nitrate exhibits increased probabilities for the SW and ESE sectors (Fig. S7).Particulate nitrate originates from gaseous nitric acid and ammonia.Rapid oxidation of NO 2 emitted by vehicles on the major adjacent roadways and by the railroad engines would be one contributor to nitrate.Cars as well as residences to the southwest would provide the ammonia.However, ammonia transported from several large sewage treatment plants along the Lake Ontario shore (north and northeast of the site) may also result in increased formation of particulate nitrate.

Differential CBPFs
The results of the ΔCBPFs between the first and last periods (2005/2007  where the probability of being a local source decreased.This region around the origin in these plots presents results for samples with low wind speeds and thus, poorly defined wind directions.Thus, the probabilities are very similar across the multiple periods and their differences are typically zero or very small.The probability changes were moderately low for most species, and thus, no smoothing was performed on the plot surfaces. Particulate variables prevailingly linked to primary road emissions (PM 2.5 , BC, PNC > 50 nm, OC pri ) show moderate (|0.3| < differential probability < |0.5|) to high (> |0.5|) ΔCBPF max decreases from the southern sectors, reflecting changes in highway and railway emissions.The incremental ΔCBPFs (Figs.S9-S10) for these pollutants suggest the effects of mitigation strategies for mobile emissions in North America, showing the largest declines in the first two periods and, then, lower changes during ΔCBPF3.These changes reflect reduction in the sulfur content of on-road and nonroad diesel fuels and the increasing impact of new heavy-duty diesel vehicles with catalytic regenerative traps that were required after July 1, 2007.Secondary inorganic aerosol (SIA) species exhibited a similar pattern reflecting reduced emissions from road and rail transport.However, nitrate showed increasing probabilities during ΔCBPF3.The formation of ammonium sulfate is favored with respect to ammonium nitrate (Seinfeld and Pandis, 2016), but the strong decline in sulfate concentrations in the last decade (Emami et al., 2018) has made more ammonium available to be neutralized by nitrate and, thus, has likely driven the increases seen for ΔCBPF3.
OC sec exhibited higher ΔCBPF max values toward the southern sectors mostly resulting from the strong incremental effect during ΔCBPF3 (Fig. S13).BB tracers (DC, K + ) showed low and noisy ΔCBPF max with an overall drop of probabilities in ΔCBPF2 and an increase in ΔCBPF3.The increase of OC sec in recent years may be in part due to the reductions in NO x and SO x emissions that would permit more oxidation of organic vapors.The rates of reaction of the inorganic species with OH are substantially higher than with organic compounds and thus, their decreased concentrations could lead to greater SOA formation.BB emissions provide both SOA precursors as well as directly emitted oxidized primary carbon during the smoldering phase.
CO, another tracer of mobile road emissions, experienced high decreases for all wind directions mostly during ΔCBPF2 (Fig. S11), while no changes were found in ΔCBPF3.The absence of directionality may indicate changes in local CO emissions from diffuse sources, such as domestic heating.However, the ΔCBPF max and ΔCBPF2 are probably biased by the change of monitors near the end of 2010 (lower limit of detection and higher sensitivity; Table S1).
SO 2 exhibited moderate to high decreases in the southeastern and both western quadrants, indicating an overall decline in concentrations for the major local sources (Fig. 1).However, larger incremental changes were observed in the first two periods (Fig. S12).The highest negative differential probabilities for ΔCBPF1 were found toward NW reflecting the shutdown of the coal-fired power plant in 2008.Probability decreases toward SW were dominant Ozone showed generally low but generally positive ΔCBPF max without any clear directionality.Winter ozone has been increasing in Rochester with the decreases in NO x (Emami et al., 2018).Period-to-period incremental changes (Fig. S12) show increases when wind speeds were greater than 5 m s -1 suggesting transport of regional air masses to Rochester contributing to local ozone.Nitrogen oxides were only measured beginning at the end of 2010 so there is generally insufficient data to observe clear patterns, except for NO that would only be affected by local sources and showed moderate decreases in all directions.

PSCFs
Figs. S15-S17 show single PSCFs.For completeness, PSCFs are reported for all the variables except for the particle number concentrations of particles with diameters below 100 nm.However, the overall consistency of PSCFs for highly reactive species (nitrogen oxides, SO 2 ) is likely biased by the known relationships between air parcel trajectories and local wind regimes.Generally, most variables exhibited higher probabilities for air mass pathways over the Ohio River Valley (PM 2.5 , BC, PNC 100 -500 , CO, OC pri , OC sec , and sulfate) and moderate probabilities extending northwesterly (central part of Canada).Nitrate presents similar patterns, but higher probability toward the northwest suggesting the role of colder temperatures that accompany northwestern flow into the area.Consequently, ammonium has a mixed behavior relative to nitrate and sulfate.The two BB tracers show different patterns.DC shows higher probability toward the northwest (similar to nitrate), while K + is stronger from the southeastern U.S., particularly the Gulf and southeast coasts.It is unclear why this difference in spatial patterns occurred.
Apart from combustion processes, CO is also generated by photochemical oxidation of methane and non-methane hydrocarbons.Although CO has a 30-90 day lifetime in the troposphere (Seinfeld and Pandis, 2016), PSCFs only reveal potential source areas extended over all the continent in 2005/2007 and no patterns afterward; thus, local emissions likely represent its dominant source in Rochester.

Differential PSCFs
The results of the ΔPSCFs between the first and last period (ΔPSCF max ) are presented in Figs.3-4 ).Generally, most variables (PM 2.5 , BC, CO, SO 2 , OC pri , nitrate, sulfate, and ammonium) showed moderate to large decreases in ΔPSCF max spanning the eastern continental U.S. and reflecting the implementation of national scale mitigation measures.However, incremental ΔPSCFs showed different temporal sequences of changes.Sulfate exhibited almost constant negative differential probabilities during the entire study period, while PM 2.5 , BC, and OC pri experienced larger declines for ΔPSCF1 and ΔPSCF3.Nitrate showed a similar ΔPSCF1, but increasing differential probabilities were recorded over the Gulf Coast and central U.S. regions during ΔPSCF2 possibly reflecting the increased electricity production following the end of the 2008 recession.At the same time, sulfate did not exhibit an increase in probabilities likely due to the shift from coal to cheaper natural gas for power generation, reducing SO 2 emissions.Between 2008 and 2009, coal used for power generation started to decline, and natural gas increased both at the state and national level because of changes in the operating costs driven by the relative costs of these fuels (Squizzato et al., 2018).The mirror image of this pattern was observed for ΔPSCF3 with net differential probabilities almost zero between 2008/2010 and 2014/2016.Consequently, patterns for ammonium were more similar to those of sulfate than nitrate.This result is expected since at low ambient ammonia concentrations, acidic sulfate neutralization is highly favored over ammonium nitrate formation.
Secondary OC experienced a general increase for ΔPSCF max driven largely by ΔPSCF3 (Fig. S20).The increased probabilities extended across North America, probably linked to decreasing SO 2 and NO x emissions combined with increasing biogenic volatile organic species with the generally increasing summer temperatures.There are increased OC sec probability areas in eastern coastal, southern, and midwestern areas of the U.S. and in the boreal forest areas of Canada east of Hudson Bay where wildfires are common (Begum et al., 2005).
DC and ozone maps show very low values but areas of increasing ΔPSCF max .The increase in DC suggests additional wood combustion including increasing rates of wildfires in areas to the west of the calculation domain.Increases in ozone primarily reflect higher winter values resulting from NO x emissions reductions across the region that are reflected in lower NO x concentrations (Emami et al., 2018).
The ΔPSCFs for SO 2 reflect the substantial reductions in emissions over this period.Decreases in PSCF values were observed only during ΔPSCF1 and ΔPSCF2 when there were shifts in fuel from coal to natural gas because of the change in their relative costs, changes in transport and heating fuel sulfur content and decreased economic activity from 2007 to 2009 and the related need for electricity.
To further examine the utility of the differential PSCF (DPSCF) plots, county-by-county emissions data for NH 3 , NO x , SO 2 , anthropogenic VOCs, and primary PM 2.5 have been retrieved from the U.S. EPA's National Emissions Inventory (NEI) for 2005, 2008, 2011, and 2014(U.S. EPA, 2018)).These values were normalized by the county area to provide an annual emission rate per unit area and  (2008-2005, 2011-2008, 2014-2011) were calculated as well as the overall difference  for each species.These differential emissions plots are provided in Figs.S26-S30.The U.S. EPA has changed methodology for developing the NEI over time and thus, there is an unknown level of uncertainty in what differences those changes have made relative to the changes in actual emissions.Substantial reductions in SO 2 and NO x emissions have occurred in the eastern U.S. and generally agree well with the areas in the PSCF with negative differential probabilities.There were both increases and decreases in the ammonia emissions between 2005 and 2014, but the DPSCF map for ammonium only shows strong negative probabilities for the overall period.Since the ammonium is only present in PM 2.5 when associated with either sulfate or nitrate, its DPSCF pattern is driven the counter ion to which it is bound.
The PM 2.5 map (Fig. 3) shows mostly negative probabilities whereas there were net increases in the central Midwest in primary PM 2.5 emissions between 2005 and 2014.However, since PM 2.5 is composed mostly of secondary ammonium sulfate, secondary ammonium nitrate, and secondary organic aerosol (SOA), the decreases in these secondary species substantially exceeds the increases in primary PM emissions and the estimated negative probabilities are then what would be expected based on the observed ambient concentration changes (Emami et al., 2018) and the secondary species DPSCF results.
For anthropogenic VOCs, there are both increases and decreases in the estimated emissions over the study period.The increases in more recent years likely reflects the increased economic activity as the economy rebounded from the 2008 recession.The secondary organic aerosol DPSCF map (Fig. 4) shows increased probabilities in these southern areas.However, we cannot relate secondary organic aerosol only to anthropogenic VOCs.Anthropogenic VOCs typically represent about 25% of the VOC emissions with the remainder being biogenic.The OC sec probabilities reflect the increase in oxidant concentrations as reflected by the small upward ozone trend in Rochester reported by Emami et al. (2018).Increased concentrations of OC sec were observed in 2014-2016 compared to 2010-2013 across NYS (Zhang et al., 2018).Thus, the decreases in SO 2 and NO x emissions may have reduced the sink for oxidants like hydroxyl radical and increased the formation of secondary organic aerosol.

Correlations among the Differential Probabilities
A correlation analysis was performed to qualitatively investigate the relationships among each set of differential probabilities.For the ΔCBPF max results: (i) the differential probabilities of all variables in each wind sector ∆θ and wind speed interval ∆v were merged; (ii) only the ∆θ, ∆v pairs with differential probabilities computed in τ 1 and τ 4 were retained; and (iii) a Pearson correlation matrix was computed.Similarly, the ΔPSCF max of all variables in each i,j cell were merged; only cells with probabilities in τ 1 and τ 4 were retained, and correlations were computed.The results were presented as correlograms (Fig. S31).The variables were ordered by applying an agglomerative hierarchical cluster analysis using complete linkage method to group similar variables together using 1-correlation as distance (Murtagh and Legendre, 2014).High correlations identify pairs of variables affected by similar changes of local or regional sources.
There were moderate (0.4 < r < 0.6) to strong (r > 0.6) ΔCBPF max correlations among CO, PM 2.5 , BC, and PNC in the two coarser size ranges, reflecting the negative differential probabilities over the SW-SE sectors (Figs.1-2), i.e., the drop of local mobile emission sources.The lack of correlation between SO 2 and this latter group of pollutants suggests local traffic is not the primary cause of the decrease in SO 2 concentrations, i.e., industrial emissions likely dominated the SO 2 concentrations in Rochester.Potassium was moderately correlated to PNC > 50 nm (BB emissions peak around 100 nm) (Petterson et al., 2011;Chandrasekaran et al., 2013), but also to OC sec and OC pri .Thus, changes in local BB emissions may be partially responsible for changes in differential probabilities for primary and secondary OC.Ammonium was strongly correlated with nitrate and sulfate indicating the local SIA production.However, neither sulfate nor nitrate were correlated with other variables indicating that changes in probabilities for SIA species are not linked to local primary emissions.
Results show generally high correlations for ΔPSCF max .A large group of variables linked to mobile sources and SIA exhibited moderate to strong inter-correlations.This result depicts the implementation of mitigation measures over North America.Ozone and OC sec were not correlated with any other variable.Potassium was moderately correlated with coarser PNC, OC pri , ammonium, and nitrate suggesting the influence of wood combustion on primary particles and ammonia concentrations (Hegg et al., 1988).

CONCLUSIONS
Differential probability functions are useful tools for investigating the changes in emission rates or locations of air pollutants.This study was performed over 12 years of air quality data (2005/2016) collected in Rochester, NY.The application of the differential probability functions allowed the identification of the major changes in local and regional emission scenarios that have affected the air quality.The major changes in air pollution due to local changes in emissions were related to (i) the shutdown of a large coal power plant (resulting in a reduction of SO 2 ) and (ii) the abatement measures applied to road and off-road traffic (resulting in reduced concentrations of primary pollutants).The concurrent effects of these changes in local emissions also produced a decrease in the number concentration of particles in the nucleation mode (< 50 nm).Changes in regional emission scenarios mostly drove the decreases in secondary aerosol and black carbon concentrations observed in Rochester.