A Physics-Informed Framework Linking Satellite AOD and Ambient Particulate Matter: A Pilot Study

Abstract

Recently, numerous studies have exploited satellite Aerosol Optical Depth (AOD) to estimate near-surface particulate matter (PM) concentrations, with the aim of overcoming the limited spatial and temporal coverage of ground-based air quality monitoring networks. Despite significant progress, the relationship between AOD and PM remains highly uncertain, mainly due to the inadequate representation of local aerosol microphysical properties and of hygroscopic growth effects. In particular, satellite AOD is retrieved at ambient relative humidity, whereas standard PM measurements are performed under dry conditions. This study proposes a physics-informed, semi-empirical approach that overcomes these limitations by directly relating satellite AOD to PM measured at ambient humidity. Co-located measurements, from a Light Optical Aerosol Counter (LOAC) in the urban area of Bologna (Po Valley, Italy) during 2023, are used. This study is designed as a pilot application to evaluate the physical consistency of the proposed framework under well-characterised observational conditions, including spatial co-location, temporal matching to satellite overpasses, and exclusion of precipitation and desert dust events. The LOAC provides particle number size distribution and particle-type classification, which are used to estimate key aerosol properties controlling the AOD–PM theoretical relationship, including the Effective Radius, Extinction Efficiency, and aerosol Mass Density. These quantities, together with Mixing Layer Height, are combined within a theoretical framework linking PM and AOD, allowing for the derivation of a physically based scaling coefficient without relying on empirical hygroscopic growth corrections. The results show that using ambient PM_2.5 alone already yields a moderate linear correlation with AOD normalized by Mixing Layer Height (Pearson’s R = 0.56) whereas no meaningful correlation is found when using standard dry PM_2.5. When aerosol microphysical properties derived from LOAC measurements are incorporated, the correlation substantially improves (R = 0.76), with regression slopes close to unity and reduced errors, independently of the season. These results demonstrate that explicitly accounting for aerosol size and optical properties enhances the physical consistency and robustness of satellite-based PM estimates. The proposed framework also provides a pathway to indirectly derive aerosol hygroscopic growth factors by coupling ambient PM estimates from satellite observations with conventional dry PM measurements. This opens new perspectives for characterizing aerosol–humidity interactions from space and for improving air quality monitoring in regions lacking of dense in situ networks.

1. Introduction

The health effects of aerosol pollution are well documented in the literature [1,2], with effects ranging from allergies to chronic respiratory diseases and even carcinogenic effects, as stated by the World Health Organization (WHO) in 2013 [3]. For this reason, the continuous monitoring of aerosol concentrations within urban areas and in background and remote locations is of primary importance for studies focused on exposure and risk assessment practices.

Aerosol monitoring is usually performed with conventional and standardized procedures that employ daily in situ measurements of particulate matter mass concentration (PM, in $\mathsf{\mu }$g/m³) at fixed stations. PM is defined according to particle size: PM₁, PM_2.5 and PM₁₀ refer to the PM of a particle whose diameter is lower, respectively, than 1, 2.5 and 10 $\mathsf{\mu }$m In the present work, only PM_2.5 is considered. According to Martin et al. [4], the European average is 3 to 4 PM monitors per million of inhabitants, while in other continents the station density is even more sparse. As a result, most surface areas are underrepresented in terms of the spatial and temporal characterization of PM. The Po Valley, which is the study area for the present work, has a denser monitoring network than the European average; nevertheless, spatial coverage remains insufficient to capture the full heterogeneity of PM concentrations at urban and local scales, supporting the need for satellite-based approaches.

In the few last decades, the scientific community has pursued the idea of using satellite observations to increase the time and space resolutions of the sparse ground-based network [5]. Satellite Aerosol Optical Depth (AOD) can be correlated to the ground-based PM by an approximate linear relationship if the aerosols are confined near the surface, such as in urban areas, where the aerosols are well-mixed below the Mixing Layer Height (MLH) [6,7]. For example, Wang and Christopher [6] found a linear correlation coefficient R equal to 0.7 (out of 1) between MODIS AOD and hourly PM_2.5; the correlation increased to values higher than 0.9 when lowering the time resolution to monthly averages. Koelemeijer et al. [7] considered several European cities and found a linear correlation coefficient of 0.6 between MODIS (Moderate-Resolution Imaging Spectroradiometer) AOD and hourly PM_2.5, over rural and background urban stations. Other studies obtained different results, depending on the location, time window and the type of linear model chosen to relate AOD and PM [8]. In general, multi-linear models perform worse when correlated variables are included without interaction terms, i.e., without product terms that capture the combined effect of two or more predictors. Multi-linear models that incorporate a variable selection algorithm, such as stepwise regression or regularization methods, to identify the most informative subset of predictors tend to perform better [9].

More complex methodologies employ data-driven models (statistical or machine learning) or chemistry transport models [10]. In the few last years, several studies trained ML models to estimate PM_2.5 from satellite AOD over Europe and world-wide (e.g., [11,12,13,14,15,16]). Each methodology has its strengths and weaknesses. For example, data-driven models require large datasets for the training phase, and identifying the most relevant predictor variables requires careful feature selection. On the other hand, the use of chemical transport models requires great computational power, and their accuracy depends on the quality of input data such as emission sources, land use, meteorology, etc. Lastly, semi-empirical methodologies employ measurements at selected locations to estimate the coefficients of the linear function that relates AOD and PM (e.g., [17]). This type of methodology presents low computational costs and is based on physical relationships, but the input data may be spatially and temporally sparse [18]. Nonetheless, a semi-empirical method can be implemented with some caution and tailored to the case study, depending on the available data. Ferrero et al. [19] considered factors such as the spatial representativeness of the ground-based PM with respect to the satellite footprint and the temporal difference between the satellite overpass time and the daily or hourly PM measurements, as well as the vertical characteristics of the atmosphere.

Table A1. Comparative table of previous studies, indicating the study region and year of the dataset used in the study; AOD and PM datasets; correlation models; and results.

Study	Region (Year)	Data	Model	Results
Wang and Christopher (2003) [6]	U.S.A. (2002)	MODIS AOD, Hourly PM2.5	Linear model	R = 0.7 (daily averages), R = 0.9 (monthly averages)
Koelemeijer et al. (2006) [7]	Europe (2003)	MODIS AOD, Daily and hourly PM2.5	Linear model with MLH correction	R = 0.6 (hourly averages)
Di Nicolantonio et al. (2009) [22]	Po Valley (2004, 2007, 2008)	MODIS AOD, Daily PM2.5	Linear model with MLH and RH corrections	R2 = 0.68 (MODIS Terra), R2 = 0.59 (MODIS Aqua), R2 = 0.72 (monthly averages)
Ferrero et al. (2019) [19]	Po Valley (2006)	MAIAC AOD, Daily PM1, PM2.5, PM10	(a) Linear model with MLH and RH corrections, (b) Multi-linear model with MLH correction (Dataset divided by season)	(a) R2 = 0.51, (b) R2 = 0.9 (spring), R2 = 0.7 (other seasons)
Arvani et al. (2016) [23]	Po Valley (2010–2012)	MAIAC AOD, Daily PM10	Linear model with MLH and RH correction (Dataset divided by season)	R2 = 0.54, R2 = 0.95 (AOD divided in bins)
Wang et al. (2010) [58]	China (2008)	MODIS AOD, Daily PM2.5 and PM10	Linear model with MLH and RH corrections	R2 = 0.47 (daily averages), R2 = 0.66 (monthly and regional averages)

summarizes some notable results from multiple previous studies.

In the revised Ambient Air Quality Directive from the European Commission [20], the annual limit for PM_2.5 has been lowered to 10 $\mathsf{\mu }$g/m³, while the 24 h average limit is 25 $\mathsf{\mu }$g/m³, to be exceeded less than 18 times during the year. The Po Valley, in Northern Italy, is an area known for frequent exceedance of air pollution limit values, including those for aerosol pollution. This is due to its high urbanization and industrialization density, and it is exacerbated especially in the winter period by its peculiar geography bounded by the Alps and the Apennine mountains limiting atmospheric circulation, as well as by weather patterns characterized by stability conditions that often trap and accumulate pollutants for long periods.

In recent decades, MODIS AOD has been used with empirical, statistical and machine learning methodologies to estimate the PM spatial variability in the area. Koelemeijer et al. [7] and Di Antonio et al. [21] managed to highlight the contrast between the high atmospheric pollution in the Po Valley compared to nearby areas. The study by Di Nicolantonio et al. [22] was one of the first to implement a semi-empirical methodology, and found a correlation of 0.68 and 0.59 between MODIS AOD and daily PM_2.5, respectively, using data from MODIS Terra and MODIS Aqua, reaching values of 0.7 when using monthly averages. More recent studies [19,23,24] exploited MAIAC (Multi-Angle Implementation of Atmospheric Correction) AOD product [25], retrieved by MODIS observations with a spatial resolution of 1 km × 1 km, to better characterize the spatial variation of PM.

Among recent studies in the area, Ferrero et al. [19] and Arvani et al. [23] considered the use of a hygroscopic growth function within their methodologies. Hygroscopicity refers to the property of some aerosols to grow in size according to their water uptake, changing their optical properties as well as their mass concentration. A corrective function is needed when relating the AOD, which is retrieved at ambient humidity, and the PM, which is measured in controlled environments with RH < 50%. The use of this function should improve the correlation between the two quantities. The main issue is the empirical nature of this function, whose coefficients depend on the local aerosol characteristics [26,27]. Ferrero et al. [19] found that hygroscopic corrections did not significantly improve the AOD–PM correlation for any of the size fractions considered (PM₁, PM_2.5 and PM₁₀).

In recent years, Optical Particle Counters (OPCs) have been employed as part of air quality monitoring networks because of their portability, higher temporal resolution and limited costs with respect to conventional reference instrumentation utilized for PM measurement [28]. OPCs detect the particle number (PN) concentration in specific size bins and retrieve the PM mass concentration through an internal algorithm. If the OPC is operated at ambient conditions, it estimates the ambient or “wet” PM, which is supposedly better correlated with AOD than the “dry” PM from standard measurements. In fact, the hygroscopic function is not needed when relating “wet” PM with AOD at ambient conditions.

Several previous studies have combined OPC data with satellite AOD. deSouza et al. [29] used OPC size distributions to constrain MISR component-specific AOD in Nairobi, deriving surface PM concentrations by scaling satellite retrievals with ground-based size distribution data. Kim et al. [30] incorporated OPC measurements alongside lidar and sunphotometer data to improve empirical MODIS AOD-PM_2.5 correlations at a single site in Korea, using aerosol profile information to screen thin cloud and dust cases. Fawole et al. [31] estimated PM_2.5 from satellite AOD using a Bayesian regression model with planetary boundary layer height and Relative Humidity as inputs, relying on reference ground-based PM measurements rather than OPC-derived ambient PM. However, none of these studies use OPC-derived aerosol microphysical properties (Effective Radius, Extinction Efficiency, and Mass Density) to estimate a theoretical scaling coefficient that directly links ambient PM_2.5,wet to satellite AOD, explicitly avoiding empirical hygroscopic corrections. The present study fills this gap.

The present study compares estimates of PM retrieved by a specific Optical Particle Counter, the LOAC (Light Optical Aerosol Counter), operated at ambient conditions, with MAIAC AOD for the year 2023 over the city of Bologna, in the Po Valley. The study also attempts to find a semi-empirical linear function to relate the “wet” PM_2.5 and the satellite AOD, with the use of auxiliary data. The use of a single, well-instrumented site allows for the minimization of spatial representativeness errors between satellite AOD and ground-based measurements, which are a major source of uncertainty in multi-site AOD–PM studies.

PN measurements from the LOAC are used as an estimate of the aerosol Particle Size Distribution (PSD), allowing for the computation of the Effective Radius of the aerosol mixture. The Effective Radius is an important variable in the correlation between the optical properties and the mass concentration of aerosols and strongly affects the radiative properties derived from satellite. In addition, the LOAC provides estimates of dominant aerosol type (such as salt, mineral, and carbonaceous aerosols) [32] which is used in the semi-empirical methodology together with Effective Radius and MLH.

The objective of this work is not to provide a spatially generalizable model, nor to develop an operational PM_dry estimation method, but to demonstrate that using ambient-humidity PM with co-located aerosol microphysical properties yields a physically consistent and direct relationship with satellite AOD, without relying on empirical hygroscopic corrections. The conversion from PM_wet to regulatory PM_dry using a hygroscopic growth factor is identified as a natural next step and left for future work. In contrast to previous multi-site statistical approaches, this study prioritizes the physical consistency of the AOD-PM_2.5 relationship over spatial coverage. The analysis is therefore conducted at a single, well-instrumented urban site (Bologna), enabling the consistent characterization of aerosols’ microphysical properties and their impact on the AOD-PM relationship. Given the limited spatial and temporal extent of the dataset, this work should be regarded as a pilot study aimed at assessing the robustness and interpretability of the proposed methodology. Extending the analysis to multiple sites will be essential to evaluate its transferability across different environments.

2. Materials and Methods

2.1. Study Area

Ground-based observations used in the present work were obtained during an intensive experimental field campaign, conducted in 2023 in the urban area of Bologna, a city located in the southern border of the Po Valley. The classification of the measurement site is urban.

The Po Valley is characterized by a distinct seasonality of pollutant concentrations, including particulate matter. During winter, the frequent stability conditions and low wind velocity tend to favour the trapping and accumulation of pollutants [33]. Particle accumulation gives rise to a concentration gradient at the MLH level, which correlates with a vertical gradient of Relative Humidity [34]. The concentration gradient is also linked to a vertical gradient of aerosol extinction. During winter, aerosol extinction tends to increase in the lowest levels of the atmosphere and concurrently to decrease at altitudes higher than 1 km; during spring and summer, the extinction is evenly distributed in the atmospheric column. As a result, during winter, more than 80% of AOD refers to particles below 1 km [35].

The relative composition of PM also varies during the year according to changes in aerosol emissions. In total, 75% of urban PM_2.5 in the Po Valley is composed of water-soluble inorganic compounds (nitrates and sulphates) and carbon [36]. Local emission sources in cities comprise different types of fuel due to traffic and residential heating, which leads to different fractions of black carbon and organic compounds during the year [37]. In the city of Bologna, at least 40% of PM_2.5 is due to traffic emissions, 24% is composed of road dust, and 8% is due to biomass burning; the levels of mineral dust, which are usually negligible, often increase in the summer during episodes of desert dust intrusion [38]. A change in the composition of the aerosol mixture affects the average optical properties retrieved from remote sensing observations, such as satellite AOD. For this reason, when comparing the aerosol mass concentration and the total extinction, it is important to obtain information about the chemical composition or the optical behaviour of the aerosol mixture.

2.2. Methodology

Figure 1 shows the methodological flowchart followed in the present work. Each block refers to one or more Sections presented below, from input data to final statistical verification.

A linear correlation between aerosol extinction at visible wavelengths and PM_2.5 can be established based on theoretical considerations (e.g., [7]):

$$k_{ext,aer}(z=0)=P{M}_{2.5}{\displaystyle \frac{3〈Q_{ext,aer}〉}{4{\rho }_{aer}{r}_{eff}}}$$

(1)

Equation (1) links the aerosol extinction coefficient $k_{ext,aer}$ at the ground level with the corresponding mass concentration $P{M}_{2.5}$. All the variables refer to ambient conditions. ${\rho }_{aer}$ is the aerosol average Mass Density and $r_{eff}$ is the Effective Radius, defined as follows:

$$r_{eff}={\displaystyle \frac{{\int }_0^{\infty }\pi r^{3}n\left(r\right)dr}{{\int }_0^{\infty }\pi r^{2}n\left(r\right)dr}}$$

(2)

where $n\left(r\right)$ is the aerosol PSD, and the average Extinction Efficiency is defined as follows:

$$〈Q_{ext,aer}〉={\displaystyle \frac{{\int }_0^{\infty }{Q}_{aer}\left(r\right)\pi r^{2}n\left(r\right)dr}{{\int }_0^{\infty }\pi r^{2}n\left(r\right)dr}}.$$

(3)

where $Q_{aer}\left(r\right)$ is the single-particle Extinction Efficiency and, for fixed wavelength and aerosol type, is a function of the particle radius.

The AOD is defined as the integral of the aerosol extinction coefficient along the atmospheric column. Assuming that all aerosols in the atmospheric column are well-mixed and placed below the MLH, AOD can be defined as

$$AOD={\int }_0^{TOA}{k}_{ext,aer}\left(z\right)dz\cong k_{ext,aer}\left(0\right)MLH$$

(4)

where TOA stands for Top of Atmosphere. The approximate formula relating $P{M}_{2.5,wet}$ (at ambient conditions) to AOD (at ambient conditions) is

$$P{M}_{2.5,wet}\cong {\displaystyle \frac{AOD}{MLH}}·{\displaystyle \frac{4{\rho }_{aer}{r}_{eff}}{3〈Q_{ext,aer}〉}}$$

(5)

The term $A={\displaystyle \frac{4{\rho }_{aer}{r}_{eff}}{3〈Q_{ext,aer}〉}}$ represents the theoretical linear coefficient between PM_2.5,wet and AOD scaled by MLH; that is, ${\displaystyle \frac{AOD}{MLH}}$. All the variables in A can be estimated from LOAC measurements, so they are consistent with the PM_2.5,wet measurements.

Equation (5) depends explicitly on MLH and implicitly on RH. Other meteorological variables such as wind speed, wind direction, solar radiation, and surface pressure do not appear in the theoretical derivation and are not required for the proposed scaling at a single urban site. Their inclusion would be relevant for spatial extrapolation or chemical transport modelling, which are beyond the scope of this study.

As mentioned previously, conventional ground-based PM measurements are performed under dry condtitions ($RH$ < 50%) to comply with regulatory standards. The relationship between PM at ambient and at dry conditions is defined as

$$P{M}_{2.5,dry}={\displaystyle \frac{P{M}_{2.5,wet}}{f\left(RH\right)}}$$

(6)

where $f\left(RH\right)$ is the hygroscopic growth function that depends on the local aerosol composition. As such, $f\left(RH\right)$ is usually derived empirically using co-located measurements of aerosol optical properties in ambient and dry conditions [27].

In the present study, the LOAC-derived PM_2.5,wet is compared to ${\displaystyle \frac{AOD}{MLH}}$, consistent with previous studies in the literature. Secondly, the LOAC PM_2.5,wet is compared with the PM_2.5,wet estimated using Equation (5). Since the relationship is approximated as linear, the comparisons are performed through a linear regression (least-squares fit) with the following function:

$$Y=a·X+b,$$

(7)

with $Y=P{M}_{2.5,wet}$ and $X={\displaystyle \frac{AOD}{MLH}}$ in the first case, and $X={\displaystyle \frac{4{\rho }_{aer}{r}_{eff}}{3〈Q_{ext,aer}〉}}{\displaystyle \frac{AOD}{MLH}}$ in the second case. The goodness of fit is evaluated using Pearson’s correlation coefficient R (95% confidence) and the linear regression performance is measured by the coefficient of determination $R^2$ and the Root-Mean Square Error (RMSE). By using a linear function, this methodology is able to describe the average linear correlation between PM and AOD, but does not account for possible non-linear and long-term effects.

Theoretically, the value of the slope a in the first case varies throughout the year according to the varying aerosol mixtures and the meteorological conditions that impact the location of interest. In general, the higher the MLH, for example during summer, the lower the linear correlation (e.g., [39]).

In the first case, the coefficient A is considered to be constant. In the second case, the value of A for each point is estimated, and the slope a is close to one if $X={\displaystyle \frac{4{\rho }_{aer}{r}_{eff}}{3〈Q_{ext,aer}〉}}{\displaystyle \frac{AOD}{MLH}}$ is a good estimate of PM_2.5,wet. In both cases, b is the offset between PM_2.5,wet and $AOD$, and, in theory, it does not vary seasonally.

Lastly, the variable importance assessment of $r_{eff}$ and the other LOAC-derived variables is performed by comparing the correlation of PM_2.5,wet with ${\displaystyle \frac{AOD}{MLH}}$, ${\displaystyle \frac{AOD}{MLH}}{r}_{eff}$, ${\displaystyle \frac{r_{eff}}{〈Q_{ext,aer}〉}}{\displaystyle \frac{AOD}{MLH}}$, and ${\displaystyle \frac{4{\rho }_{aer}{r}_{eff}}{3〈Q_{ext,aer}〉}}{\displaystyle \frac{AOD}{MLH}}$.

2.3. OPC Data

OPC measurements were obtained under ambient RH conditions by a LOAC [32] (manufactured by Meteomodem, France) mounted on the rooftop of the Department of Physics and Astronomy (DIFA) of the University of Bologna (lat: 44.499 N, lon: 11.354 E), at about 22 m above the ground (from now on named only DIFA). The surrounding area is a vegetated street inside the historical centre of Bologna. Possible local emission sources are trees and other vegetation (pollen and other biological particles), the low levels of vehicle traffic from Irnerio Street and the surrounding roads (road dust and combustion products), building heating systems during winter (biomass burning), and occasional desert dust events (dust particles). The LOAC dataset covers the months from February to September 2023.

The LOAC retrieves the particle number concentration (number of particles per cm³, #/cm³ or simply cm⁻³) in 19 size bins for particle diameters between 0.2 and 30 $\mathsf{\mu }$m (

Table 1. Definition of LOAC size bins, respective bin edges, and average diameters.

Size Bin	Bin Edges ($\mathsf{\mu }$m)	Bin Diameter ($\mathsf{\mu }$m)
1	0.2–0.3	0.25
2	0.3–0.4	0.35
3	0.4–0.5	0.45
4	0.5–0.6	0.55
5	0.6–0.7	0.65
6	0.70–0.85	0.775
7	0.85–1	0.925
8	1–2	1.5
9	2–3	2.5
10	3–5	4
11	5–7.5	6.25
12	7.5–10	8.75
13	10–12.5	11.25
14	12.5–15	13.75
15	15–17.5	16.25
16	17.5–20	18.75
17	20–22.5	21.25
18	22.5–25	23.75
19	25–30	27.5

) by the use of laser beams at a 650 nm wavelength. By using two scattering angles (12° and 60°), it also determines the degree of absorption by the detected particle, from transparent to highly absorbing. This estimate is used to classify the particles, divided in five size classes (

Table 2. Size classes of LOAC particle types and respective diameter edges (D).

Size Class	1	2	3	4	5	6
D ($\mathsf{\mu }$m)	0.2–0.5	0.5–0.9	0.9–5	5–12.5	12.5–20	20–40

), in four types: droplet, salt, mineral, and carbon. From the PN measurements and the particle type, PM₁, PM_2.5, and PM₁₀ ($\mathsf{\mu }$g/m³) are calculated with an internal algorithm which uses a suitable value of particle Mass Density (g/cm³) according to particle type. In general, all OPC instruments are highly affected by high Relative Humidity, cloudy conditions and precipitation events (e.g., [40]). As such, recent studies recommend to perform measurements with such sensors under low time resolutions (hourly and daily averages) and clear sky conditions [41].

The 1 min PN and PM_2.5,wet data are screened with a Hampel filter on a 10 min window [42] to remove outliers and then are averaged hourly. The Effective Radius $r_{eff}$ is also computed by discretizing Equation (2), which becomes

$$r_{eff}\cong {\displaystyle \frac{{\sum }_{i}P{N}_i{R}_i^3}{{\sum }_{i}P{N}_i{R}_i^2}}$$

(8)

where $P{N}_i$ is the particle number concentration in the size bin i and $R_i$ is half the nominal diameter of size bin i. This is because the particle number concentration in size bin i is the total value of $n\left(r\right)$ within the following size bin: $P{N}_i\cong {\int }_{D_i/2}^{D_{i+1}/2}n\left(r\right)dr$. The $r_{eff}$ is computed with the original filtered 1 min PN data and then averaged hourly.

The particle type dataset from LOAC is used together with PN data to compute the average aerosol Extinction Efficiency and Mass Density. This is achieved by computing the following:

• The Number Ratio (NR) of each particle type in each size bin;
• The average single-particle Extinction Efficiency of each particle type in each size bin.

Additionally, the proper Mass Density for each particle type is chosen. In Renard et al. 2016 [32], it is advised to use particle type information with a statistical approach. For each hour, the number of occurrences of each particle type x in each size bin i is divided by the total number of observations in each size bin in that hour (maximum of 60); the fraction is then converted into decimal percentage. This percentage defines the hourly Number Ratio $F_{i,x}$ in size bin i for the particle type x, where x can refer to salt (x = 1), mineral (x = 2) or carbon (x = 3).

In Renard et al. 2010 [43], the absorption of different particle types is defined in relation to different values of the imaginary part k of the Refractive Index (RI). The salt and mineral particle types refer to particles with low absorption at 650 nm, while the carbon particle type refers to particles with high absorption at the same wavelength. In the database OPAC (Optical Properties of Aerosols and Clouds) [44], the urban aerosols are defined as a mixture of soluble and insoluble aerosols, characterised by low absorption at 650 nm, and of black carbon (or soot) which has a high absorption at 650 nm. In this study, the assumption made is that each LOAC particle type can be related to each OPAC aerosol type:

• The salt LOAC type related to the “soluble” OPAC type (here considered at RH = 50%);
• The mineral LOAC type related to the “insoluble” OPAC type;
• The carbon LOAC type related to the “soot” OPAC type.

The respective single-particle Extinction Efficiency $Q_{aer}\left(r\right)$ values are taken from OPAC. This choice is made taking into account similar values of k between the LOAC and OPAC aerosol types. $Q_{i,x}$ is computed as the average value of $Q_{aer}\left(r\right)$ in the size bin i for type x. The average Extinction Efficiency for each LOAC observation is discretized from Equation (3) as

$$〈Q_{ext,aer}〉\cong {\displaystyle \frac{{\sum }_i\left(Q_{i,1}{F}_{i,1}P{N}_i+Q_{i,2}{F}_{i,2}P{N}_i+Q_{i,3}{F}_{i,3}P{N}_i\right)R_i^2}{{\sum }_{i}P{N}_i{R}_i^2}}$$

(9)

where $F_{i,x}$ denotes the previously defined Number Ratio in the size bin i for particle type x. Similarly, the average Mass Density ${\rho }_{aer}$ is computed as

$${\rho }_{aer}\cong {\displaystyle \frac{{\sum }_{i}P{N}_i(m_1{F}_{i,1}+m_2{F}_{i,2}+m_3{F}_{i,3})}{{\sum }_{i}P{N}_i}}$$

(10)

where $m_x$ are the nominal values of Mass Density for each LOAC particle type x (same value for all size bins). The single-particle Mass Density values from the LOAC algorithm are used in this computation, to be consistent with the PM calculation.

2.4. AOD

MAIAC AOD is retrieved at 550 nm from MODIS observations at high spatial resolution (1 km × 1 km) on a fixed grid. It has a better performance over bright surfaces compared to other algorithms and has been previously applied over urban areas [45]. In cases of anthropogenic aerosols, an AOD below 0.12 can be overestimated. AOD uncertainty indicated in the product files is related to surface brightness. A value below 0.05 allows the algorithm to perform a standard retrieval [45,46]. A 5% uncertainty value for the single retrieval values can be used. More information on the retrieval can be found at [25]. Intense pollution cases in urban areas within the Po Valley, especially in very high polluted cities such as Milan and Turin, can be sometimes interpreted as clouds and filtered out [23].

The co-location of AOD and ground-based data at DIFA site is performed on the fixed grid used in MAIAC retrievals. The cell containing the DIFA location is selected (Figure 2a). This grid cell refers to the innermost part of Bologna historical centre, characterised by medium-bright surfaces, traffic and low industrial emissions.

The AOD data is acquired for the whole period covered by the LOAC dataset (February–September 2023). AOD retrieval requires cloud-free conditions, limiting data availability. Of the 985 satellite overpasses during the study period, only 26.8% yielded successful retrievals. The retrieval success varied seasonally, ranging from 15–16% in April–May to 41% in August, reflecting regional cloud climatology and precipitation patterns.

Considering the dataset of co-located AOD retrievals, AOD uncertainty due to surface brightness is lower than 0.045, with a median of 0.016, and AOD values are lower than 0.45, with a median of 0.14. This means that the retrievals used in the following analysis are computed with the standard algorithm, and are likely not affected by overestimation.

2.5. Auxiliary Variables

The Mixing Layer Height is retrieved by a ceilometer Vaisala CL31 (manufactured by Vaisala, Finland) present at DIFA alongside the LOAC. The ceilometer continuously measures the vertical backscatter signal from the ground level to a maximum of 7.5 km and provides estimates of MLH with a temporal resolution of 16 s and a vertical resolution of 10 m. It also provides an estimate of Cloud Height if clouds are detected. The internal algorithm is based on the difference in backscatter signal between the mixing layer, where the signal is higher due to aerosols, and the free atmosphere, where the backscatter signal abruptly drops to very small values in absence of scattering layers (gradient method) [47]. The gradient method estimates the MLH as the height of the greatest negative gradient in the aerosol backscatter signal [48,49].

The ceilometer detects one to three possible MLH values; the highest one usually corresponds to residual layers or incorrectly detected clouds [50,51]. The gradient method requires a robust way to choose the correct MLH value. In the present study, MLH data obtained from radiosondes at the nearest IGRA (Integrated Global Radiosonde Archive) station [52,53] are used as reference. Radiosondes measure the vertical profiles of temperature, humidity, pressure and wind speed with high precision, and provide indirect estimates of MLH. The reference radiosonde measurements are taken at the IGRA station of San Pietro Capofiume (lat: 44.654, lon: 11.623), located 27.4 km northeast of Bologna (Figure 2b), and have a climatology of more than 30 years. The averages of IGRA MLH at 12:00 (local time) between 2021 and 2025 are considered as a reference. The IGRA MLH average for the day is chosen as a variable threshold to select the correct ceilometer MLH values in the same day. The ceilometer dataset, composed of all possible MLH values, is averaged on a 10 min window around the satellite overpass times. In general, the second and third MLH values are higher than the reference IGRA MLH average at the same time and day. The closest ceilometer MLH to the average radiosonde MLH in the same calendar day is chosen as MLH for the present study.

Auxiliary data include quality-controlled data from the local environmental agency (ARPAE) [54]: the 30 min Relative Humidity and hourly Precipitation Rate are measured from the closest meteorological station, Bologna Idrografico (lat: 44.499, lon: 11.346), located 620 m from DIFA; daily “dry” PM_2.5 from ARPAE Porta San Felice station (lat: 44.499, lon: 11.328), located 2120 m from DIFA. RH data is used in the comparison between PM_2.5 and AOD to verify that there is no dependence on humidity for both variables, while the “dry” PM_2.5 is used to compare the results of the linear regressions when using reference data and when using LOAC data. The stations are shown in Figure 2a alongside DIFA and the AOD cells from satellite, with the specific locations marked with blue asterisk symbols.

Lastly, data from AERONET (Aerosol Robotic Network) [55], a global network of ground-based sunphotometers that measure the AOD and related variables [56], is considered. The nearest AERONET stations (Figure 2b) are in the urban area of Modena (lat: 44.629, lon: 10.948), located 35.26 km northwest of Bologna, and in the rural area of Bologna at San Pietro Capofiume (lat: 44.650, lon: 11.620), located 26.94 km northeast of Bologna and 504 m northeast of the IGRA station. AERONET data provides useful information about aerosol columnar properties to complement the data analysis.

2.6. Uncertainties and Data Preprocessing

According to Lyapustin et al. [25], the relative uncertainty of AOD retrieved by the MAIAC algorithm is 0.05, so the uncertainty assigned to AOD is $\Delta AOD=0.05·AOD$. The MLH uncertainty is the maximum between the ceilometer measuring uncertainty of 10 m and the standard deviation of the 10 min average. The uncertainty assigned to $X={\displaystyle \frac{AOD}{MLH}}$ is calculated with the relative uncertainty rule, as follows:

$${\displaystyle \frac{\Delta X}{X}}=\sqrt{{0.05}^2+{\left({\displaystyle \frac{\Delta MLH}{MLH}}\right)}^2}.$$

(11)

The uncertainty assigned to $A{\displaystyle \frac{AOD}{MLH}}$ is computed in a similar way.

As the PM_2.5 and Effective Radius are averaged hourly from the 1 min LOAC data, and the standard deviation is assigned as uncertainty for both variables. The AOD retrieval time is set as the dataset index. The hourly dataset is composed of AOD, MLH, PM_2.5,wet, $r_{eff}$, $〈Q_{ext,aer}〉$, ${\rho }_{aer}$, A, and RH. All variables except AOD are temporally averaged around the time of satellite overpass.

To ensure consistency and avoid biases introduced by data gaps, only complete rows are retained for the linear regressions and the computation of statistical coefficients. This approach reduces the sample size, but preserves the physical consistency of co-located measurements, which is essential to the presented physics-informed approach. Imputation methods were not employed, as they would introduce artificial correlations between variables and undermine the direct physical relationship between AOD and ground-based PM that this study aims to characterise.

Noise and outliers in LOAC data are addressed using a Hampel filter applied to the LOAC data, as previously mentioned. To minimise the mismatch between the point measurements and the 1 × 1 km² area represented by the MAIAC grid cell, LOAC data is averaged hourly around the time of satellite overpass, while MLH data is averaged in a 10 min window around the time of satellite overpass, as stated above. The location of DIFA site is close to the border of the selected MAIAC grid cell, which could introduce biases related to different emission sources in the adjacent grid cells. However, as stated in Lyapustin et al. [25], AOD values are averaged in a 3 × 3 grid, and it is safe to assume that this spatial averaging accounts for possible mismatch due to the DIFA site location relative to the grid cell border.

A set of data selection criteria is designed to minimize confounding factors in the AOD-PM relationship. These include the following: (i) co-located measurements; (ii) the temporal matching of ground-based data to satellite overpass times; (iii) exclusion of precipitation events; (iv) the exclusion of desert dust intrusion events (Angstrom Exponent < 0.6 from AERONET); (v) uncertainty filtering (relative uncertainty < 20% to relate PM_2.5,wet and ${\displaystyle \frac{AOD}{MLH}}$; absolute uncertainty to relate PM_2.5,wet measured by the LOAC and estimated from AOD and auxiliary data); and (vi) outlier removal using a Hampel filter on LOAC data. These constraints ensure that the remaining data satisfy the theoretical assumptions of the framework (well-mixed boundary layer, clear sky conditions, typical urban aerosol conditions) and that the observed correlations reflect physical relationships rather than artifacts.

3. Results

3.1. Dataset Analysis

The measurement period considered in this work (February–September 2023) was characterized by a particular variability in the meteorological conditions impacting the study area. The month of May 2023 was affected by three exceptional rainfall events (2, 16, and 17 May), with cumulative daily rainfall values higher than 100 mm recorded at the Bologna Idrografico station. Other minor rainfall events occurred throughout the measurement period. Figure 3 shows daily averages of PM_2.5,wet from LOAC and RH from Bologna Idrografico for the whole period. Precipitation events with rainfall greater than 20 mm are highlighted in grey. As noted in [41], LOAC measurements are affected by the hygroscopic growth of particles under precipitation events and high RH conditions. This is apparent from the daily values of PM_2.5,wet recorded during the study period, which are all below 35 $\mathsf{\mu }$g/m³ except for observation records during precipitation events, in which PM_2.5,wet reaches values above 50 $\mathsf{\mu }$g/m³.

Desert dust intrusions were frequent during the months of June and July. The monthly PSD averages in Figure 4 show a corresponding increase in particles with a radius greater than 1 $\mathsf{\mu }$m, compared to particles with smaller radii during these months. The PSD is computed by dividing the PN by the width of each size bin. The Effective Radius, computed with Equation (8), shows values larger than 2 $\mathsf{\mu }$m during precipitation events on 2 and 16 May, and higher than 1.5 $\mathsf{\mu }$m between 19 and 22 June, and between 9 and 11 and 15 and 18 July (Figure 5). Data from AERONET near the stations of Modena and San Pietro Capofiume highlight that, during that time, the Ångström Exponent $\alpha$ was below 0.5 in June and below 0.6 in July. Low values of $\alpha$ are associated with desert dust transport which is frequent in Italy during summer, with dust plumes that also reach the Po Valley and Bologna [57]. Except for precipitation and dust transport events, the Effective Radius is constantly smaller than 0.6 $\mathsf{\mu }$m during the measurement period.

Figure 6 shows the monthly particle type occurrences, described by the Number Ratios as defined in Section 2. Only the first three size ranges are shown, consisting of particles with diameters between 0.2 and 5 $\mathsf{\mu }$m (as specified in Table 2). The salt and mineral types are present in the first two size ranges in varying proportions during the year. In the third size range (0.9–5 $\mathsf{\mu }$m), the mineral type is the most common, but also the carbon type is consistently present. Particles in the fourth size range (5–12.5 $\mathsf{\mu }$m) consist of water droplets or unidentified types. Particle types in the two largest size ranges (12.5–40 $\mathsf{\mu }$m) are mostly not identified, probably due to measured PN concentrations lower than 0.01 #/cm³.

3.2. Calculation of Aerosol Properties

Effective Radius. The smallest particle diameter detected by the LOAC is 0.2 $\mathsf{\mu }$m (Table 1), so that particles with radii less than 0.1 $\mathsf{\mu }$m are not detected. These particles are almost negligible in the computation of mass concentration, but have a significant contribution in terms of particle number concentration and in the extinction properties. Moreover, the Effective Radius should be calculated considering smaller particles, with radii starting, for example, from 0.05 $\mathsf{\mu }$m [44]. The same is valid for the computation of $〈Q_{ext,aer}〉$. Before computing the theoretical linear coefficient, the error made by neglecting particles smaller than 0.1 $\mathsf{\mu }$m in radius should be assessed.

A series of numerical simulations are performed using an original code written in MATLAB (Version 23.2). The aim is to compare a set of “theoretical PSDs” and “LOAC PSDs” with the same aerosol mixtures, and to compare their respective Effective Radii. The code calculates the urban aerosol PSD of as a superposition of three lognormal functions, representing the three aerosol types described in Section 2.3. The resulting PSD, from now on referred to as “theoretical PSD”, is the sum of the salt, mineral and carbon PSDs, each multiplied by the respective Number Ratio. The theoretical PSD is computed for particle radii between 0.05 $\mathsf{\mu }$m and 20 $\mathsf{\mu }$m. Using this PSD, the code computes the “theoretical Effective Radius” from Equation (2). As a second step, the code uses the theoretical PSD to compute the respective LOAC PN values in each size bin, therefore obtaining the approximated “LOAC PSD”, similar to those shown in Figure 4. From the LOAC PSD, the LOAC Effective Radius is computed by using Equation (8).

The comparison between theoretical and LOAC Effective Radii values shows that the LOAC $r_{eff}$ overestimates the theoretical value by a factor between 1.5 and 2, depending on the ratio between the fine and coarse particles used in the simulations. Therefore, the error made on the Effective Radius computed from LOAC PN data without corrections can be as high as 200%. To reduce this error, a simple correction algorithm is applied on the simulations, to quantify the improvement, and then on the actual LOAC data.

Firstly, considering the same set of PSD simulations, an additional size bin, referred to as PN₀, is defined as the PN of particles between 0.1 and 0.2 $\mathsf{\mu }$m. The value of PN₀ is extrapolated from the LOAC PSD, and specifically from the first four size bins. Secondly, the LOAC Effective Radius is computed again, considering the additional PN₀ size bin. The theoretical PSD is not used in this process, so the algorithm can be applied to real LOAC data. Using the same set of simulations from the previous analysis, the calculations show that the corrected Effective Radius ($r_{eff}^{\prime }$) overestimates the theoretical value by only a factor of 0.3 on average. This means that by applying the correction, the error made by computing the Effective Radius from LOAC data changes from approximately 100% to 30% of its theoretical value.

The last step is to apply the same correction to LOAC data. The corrected $r_{eff}^{\prime }$ values are lower than the original $r_{eff}$ values by a factor between 50% and 90%. The calculated hourly averages of LOAC Effective Radius (both original and corrected) have an uncertainty of 20% on average, rising above 50% during precipitation and desert dust events.

Extinction Efficiency and Mass Density. The average quantities $〈Q_{ext,aer}〉$ and ${\rho }_{aer}$ are calculated from the hourly Number Ratio, hourly PN, and single-particle $Q_{aer}\left(r\right)$ and $m_x$ respectively, applying Equations (9) and (10). Water particles are excluded from the calculation by setting to zero the Number Ratio of the droplet particle type, so the calculated values are not affected by precipitation events, unlike the Effective Radius; desert dust intrusions do not seem to affect them either.

Linear coefficient. Lastly, the coefficient A is computed using all the estimated aerosol properties. A is normalized considering the different units of measurement of PM_2.5 ($\mathsf{\mu }$g/m³), ${\displaystyle \frac{AOD}{MLH}}$ (m⁻³), $r_{eff}$ ($\mathsf{\mu }$m) and ${\rho }_{aer}$ (g/cm³). $〈Q_{ext,aer}〉$ is unitless. The variability of A is mostly affected by the variability of the Effective Radius.

3.3. Methodology Application

3.3.1. PM_2.5 vs. ${\displaystyle \frac{AOD}{MLH}}$

The linear regression performed uses $Y=P{M}_{2.5,wet}$ and $X={\displaystyle \frac{AOD}{MLH}}$. Data filtering selects PM_2.5,wet data with standard deviations less than $3 \mu g/m^3$ and relative uncertainty lower than 20%, AOD data between 0.01 and 0.35, to avoid errors in the retrieval algorithm, and ${\displaystyle \frac{AOD}{MLH}}$ with a relative uncertainty lower than 20%. The relative uncertainty threshold is set at 20% considering average LOAC data quality under stable conditions and ensuring that quantities represented by different physical units have comparable uncertainties. By using these rules, May and September data are filtered out, as well as several data points during June and July. The final dataset consists of a total of 56 observations available.

Pearson’s correlation coefficient R is 0.56 (out of 1), indicating the presence of a linear correlation between the two variables. The linear regression function has $RMSE=4.28$$\mathsf{\mu }$g/m³ and $R^2=0.31$. The linear function is $Y_1=7.26+0.28·X$. The residuals are normally distributed around 0, with a median of −0.06. The linear function is shown in Figure 7a with the selected data points. As expected, the relationship between the variables does not show any clear dependence on RH, suggesting that there is no need for a hygroscopic correction factor.

For comparison, the linear regression is repeated using daily “dry” PM_2.5 from the reference ARPAE station Porta San Felice (SF), in the same temporal window. As PM_2.5 data is obtained from reference instruments, the uncertainty is assumed to be always below 20%. The MLH is assumed not to change significantly between SF and DIFA; therefore, the ceilometer-derived MLH data is used. The selected AOD refers to the cell whose centre is closest to the ground-based station of Porta San Felice, where PM_2.5,dry is measured (blue box in Figure 2a). The AOD values assigned to DIFA and SF are retrieved from the same satellite swath, and as such refer to the same satellite overpass time. For comparison with daily $P{M}_{2.5,dry}$ data, daily estimates of ${\displaystyle \frac{AOD}{MLH}}$ are computed as follows. Firstly, the instantaneous ${\displaystyle \frac{AOD}{MLH}}$ values for each day are computed from the AOD values at satellite overpass time and the respective 10 min MLH averages. Secondly, the resulting ${\displaystyle \frac{AOD}{MLH}}$ values are averaged daily.

In this case, the regression is performed on 102 data points between February and September 2023. The value of R is 0.1558, indicating the absence of linear correlation. As is apparent from Figure 7b, “dry” PM_2.5 and the corrected AOD seem uncorrelated, and there is no correlation with the respective RH value (daily averaged). The linear function is $Y_1^{\prime }=10+0.15X_1^{\prime }$, with $R^2=0.0243$ and $RMSE=6.17$$\mathsf{\mu }$g/m³.

To assess whether the higher correlation of PM_2.5,wet with AOD is a physical result rather than an artifact of temporal resolution or filtering, two additional analyses are performed. First, daily averages of PM_2.5,wet and ${\displaystyle \frac{AOD}{MLH}}$ over DIFA are compared directly, yielding a dataset of $n_{Obs}=38$ observations. The linear regression of daily ${\displaystyle \frac{AOD}{MLH}}$ with daily PM_2.5,wet results in $R=0.51$, $R^2=0.26$, and $RMSE=4.33$$\mathsf{\mu }$g/m³. Despite the reduced temporal matching quality, the correlation with PM_2.5,wet remains substantially higher than the correlation with PM_2.5,dry, particularly in terms of the RMSE. Second, to address the difference in sample size between the two datasets, the daily PM_dry regression is restricted to the same $n=38$ days used in the daily PM_2.5,wet analysis. This yields $R=0.49$, $R^2=0.24$, and $RMSE=7.71$$\mathsf{\mu }$g/m³.

To verify that this result is not an artifact of the specific day selection, 50 random subsets of $n_{Obs}=38$ data points are drawn from the full PM_dry dataset ($n_{Obs}=102$). The resulting R values are distributed around $\overline{R}=0.39$ ($\sigma =0.12$), ${\overline{R}}^2=0.15$ ($\sigma =0.12$), and $\overline{RMSE}=5.5$$\mathsf{\mu }$g/m³ ($\sigma =1.1$$\mathsf{\mu }$g/m³). The fact that the 38-day subset selected via PM_2.5,wet filtering performs better than the average random subset ($R=0.49$ vs. $\overline{R}=0.39$) confirms that the day selection is not random: it preferentially retains days with low hourly PM variability, characterized by clear sky conditions, for which the AOD-PM relationship is physically better constrained. This information is accessible only through the use of hourly PM_2.5,wet data. These results confirm that the superior performance of PM_2.5,wet is a physical consequence of using ambient-humidity measurements, not an artifact of filtering or temporal resolution.

3.3.2. PM_2.5 vs. $A{\displaystyle \frac{AOD}{MLH}}$

The linear regressions between PM_2.5,wet measured by LOAC and PM_2.5,wet calculated using Equation (5) are performed by considering $X={\displaystyle \frac{4{\rho }_{aer}{r}_{eff}}{3〈Q_{ext,aer}〉}}{\displaystyle \frac{AOD}{MLH}}$. As the comparison refers to the same physical quantity, an absolute uncertainty filter should be applied. In the present work, the threshold of 2 $\mathsf{\mu }$g/m³ is considered, as a measure of the LOAC instrumental uncertainty. The filtered subset comprises 49 data points between February and July.

Pearson’s correlation coefficient is 0.76, indicating a greater linear agreement between the variables. The linear regression function has $RMSE=3.11$$\mathsf{\mu }$g/m³ and $R^2=0.58$, which indicates a slightly better linear fit compared to results from previous studies such as [22,58]. The linear function, presented in Figure 8a, is $Y=6.27+0.81·AX$.

The linear regression is repeated by substituting the original LOAC $r_{eff}$ with the corrected $r_{eff}^{\prime }$. In this case, the slope of the linear function reaches a value of $a=1.04$ with $R^2=0.54$, $RMSE=3.03$$\mathsf{\mu }$g/m³ and $R=0.73$ (shown in Figure 8b).

In both cases, the distribution of the residuals has a normal shape that is slightly negatively skewed, with an average close to 0 and a median of −0.2. The mean value of the residuals is also defined as Mean Bias. A Mean Bias close to zero indicates the absence of a systematic under- or overestimation in the model. The slight negative skewness simply indicates that the distribution of residuals has a slightly longer left tail, but their effects cancel out when considering the average tendencies of the model. For completeness, for both cases, the additional statistical parameters are the following:

• The Mean Absolute Error, defined as the average of the absolute value of residuals, is respectively $MAE=2.38$$\mathsf{\mu }$g/m³ and $MAE=2.34$$\mathsf{\mu }$g/m³;
• The relative $RMSE$, defined as the $RMSE$ divided by the average Y, is respectively $RMS{E}_{rel}=0.29$ and $RMS{E}_{rel}=0.28$.

To verify that the improvement from the previous correlation (R = 0.56) to the current one (R = 0.76) is mainly physical and not an artifact of the sample reduction (from $n_{Obs}$ = 56 to $n_{Obs}$ = 49), the first dataset ($X={\displaystyle \frac{AOD}{MLH}}$) is reduced to the same 49 data points used in the current Section. The correlation coefficient R between PM_2.5,wet and $X={\displaystyle \frac{AOD}{MLH}}$ is 0.58. Thus, there is an increase from 0.56 to 0.58 only due to the reduced sample size, while there is a larger improvement in linear correlation, from 0.58 to 0.76, due to the physical information carried by the aerosol properties contained in the scaling coefficient A.

Table 3. The results from all the linear regressions with the function $Y=aX+b$, where $Y=P{M}_{2.5,wet}$ and X is indicated in the first column. The regressions were performed on 49 data points apart from the first, which was performed on 56 data points. The coefficients a and b are listed with relative uncertainty based on 95% confidence bounds.

X	R	${\mathit{R}}^2$	RMSE ($\mathsf{\mu }$g/m3)	$\mathit{a}\mathbf{+}\mathbf{\%}$ Error	$\mathit{b}\mathbf{+}\mathbf{\%}$ Error
${\displaystyle \frac{AOD}{MLH}}$	0.56	0.31	4.28	0.28 + 40%	7.26 + 26%
${\displaystyle \frac{4{\rho }_{aer}{r}_{eff}}{3〈Q_{ext,aer}〉}}{\displaystyle \frac{AOD}{MLH}}$	0.76	0.58	3.11	0.81 ± 24%	6.27 ± 23%
${\displaystyle \frac{4{\rho }_{aer}{r}_{eff}^{\prime }}{3〈Q_{ext,aer}〉}}{\displaystyle \frac{AOD}{MLH}}$	0.73	0.54	3.03	1.04 ± 27%	6.38 ± 23%

summarizes the linear correlation results obtained in each case considered so far. All reported Pearson’s R values are statistically significant at the p < 0.001 level (two-tailed test), given the sample sizes ($n_{Obs}$ = 49–56). For $n_{Obs}$ = 0.49 ($n_{Obs}$ = 0.56), the minimum R required for p < 0.001 is 0.294 (0.279), well below the resulting R = 0.76 and 0.73 (R = 0.56). Thus, the observed correlations are extremely unlikely to arise by chance. Moreover, for $n_{Obs}$ = 102 the minimum R required for significance is 0.197, above the result of R = 0.16 for the case of daily PM_dry, consistent with the conclusion that no meaningful linear relationship between PM_dry and ${\displaystyle \frac{AOD}{MLH}}$ exists prior to selection of daily PM_dry data according to PM_wet data. In the case of the 38-day subsets, the minimum R for significance is 0.334, so the result of R = 0.49 is statistically significant.

The values of the offset b in all linear functions are compatible with each other within confidence bounds. This means that the offset depends on other factors, such as the difference between the point measurements (LOAC and ceilometer data) and the satellite AOD retrieval (with a footprint of 1 km²). Theoretically, the slope of the linear function between PM_2.5,wet and $X=A{\displaystyle \frac{AOD}{MLH}}$ should be close to one, as A represents the theoretical slope that relates PM_2.5 to AOD and should account for all the variability. Cross-validation was not performed due to the limited sample size (56 and 49 data points). This is a limitation of the dataset used in the present study. Future, multi-site studies with larger samples should include cross-validation.

3.3.3. Variable Importance Assessment

The variable importance of $r_{eff}$, $〈Q_{ext,aer}〉$ and ${\rho }_{aer}$ is assessed by adding one variable at a time to $X={\displaystyle \frac{AOD}{MLH}}$ and comparing Pearson’s coefficient values. Each time that one variable is added, only the data with relative uncertainty lower than 20% is considered.

The linear correlation coefficient between PM_2.5,wet and $X={\displaystyle \frac{AOD}{MLH}}$ is R = 0.5587 as previously stated. Multiplying X by $r_{eff}$, a value of R = 0.5992 is obtained between PM_2.5,wet and $X={\displaystyle \frac{AOD}{MLH}}{r}_{eff}$. Multiply by $〈Q_{ext,aer}〉$, and a value of R = 0.6082 is obtained. Lastly, multiplying by ${\displaystyle \frac{4}{3}}{\rho }_{aer}$, a value of R = 0.6103 is obtained between PM_2.5,wet and $X=A{\displaystyle \frac{AOD}{MLH}}$. The last value is lower than the one obtained in the previous analysis because in this case the considered data points have uncertainties greater than 2 $\mathsf{\mu }$g/m³ on X.

4. Discussion

Pearson’s correlation coefficient R = 0.56 shows that the “wet” hourly PM_2.5 is linearly correlated to AOD even without using information about aerosol properties, while MLH is always considered. For comparison, the correlation between AOD and “dry” PM_2.5 from reference ground-based measurements yields a coefficient R = 0.16, indicating the absence of a meaningful linear correlation. The lower value of R for “dry” PM_2.5 is partly attributable to its coarser temporal resolution compared to LOAC-derived “wet” PM_2.5, and underlines the need for reference data with better temporal resolution.

The analysed data show that higher values of both PM_2.5,wet and AOD are generally associated with higher values of RH, contrary to “dry” PM_2.5, which is measured at fixed RH values lower than 50%. The use of PM observations taken at ambient humidity allows for a direct correlation with satellite AOD without hygroscopic corrections, representing an element of novelty with respect to previous works on the same topic while reaching comparable results [7,19,22,23]. While a direct comparison of correlation coefficients with “dry” PM_2.5-based studies is complicated by differences in measurement protocols and RH treatment, the relative improvement observed here (R increasing from 0.56 to 0.76 when aerosol microphysical properties are incorporated) is consistent with the broader finding that accounting for aerosol physical properties enhances the AOD-PM relationship. The low value of $R=0.16$ obtained for “dry” PM_2.5 in this study should not be interpreted as a methodological flaw, but rather reflects the combination of factors that distinguish the present setup from previous works reporting $R\ge 0.6$: those studies used regional or multi-city datasets with large sample sizes and hourly PM measurements, while the presented “dry” PM_2.5 comparison relies on daily averages matched to a single urban site with a limited number of observations ($n_{Obs}=102$) and a higher-resolution AOD product (MAIAC 1 km vs. MODIS 10 km). Under these more stringent conditions, the temporal mismatch between daily “dry” PM_2.5 and instantaneous ${\displaystyle \frac{AOD}{MLH}}$ is the primary driver of the low correlation.

This interpretation is supported by the additional analysis presented in Section 3.3.1: when both datasets are reduced to the same $n=38$ daily pairs, “wet” PM_2.5 still outperforms “dry” PM_2.5 in terms of RMSE (4.33 vs. 7.71 $\mathsf{\mu }$g/m³), and the “dry” PM_2.5 results on those days are better than the average over random subsets of the same size ($\overline{R}=0.39$, $\sigma =0.12$), confirming that the day selection reflects physical constraints rather than a statistical artifact.

Furthermore, by comparing satellite-derived “wet” PM_2.5 with co-located reference “dry” PM_2.5 measurements, the local hygroscopic growth factor f(RH) can be inferred without assuming a fixed empirical parameterisation, potentially improving PM estimates in humid environments relative to approaches based on globally averaged RH corrections. A hygroscopic correction remains necessary to convert the “wet” PM_2.5 estimated from AOD into regulatory “dry” PM_2.5; this step is identified as future work.

The correlation between PM_2.5,wet and ${\displaystyle \frac{AOD}{MLH}}$ shows no significant dependence on the time of day within the sampling window (9 AM–2 PM). Analysis of Terra (morning) vs. Aqua (afternoon) overpasses revealed comparable scatter in both cases, despite the expected MLH evolution during this period. This suggests that normalisation by MLH adequately accounts for diurnal boundary layer variations within the satellite overpass timeframe.

A Pearson’s correlation coefficient of R = 0.76 is obtained when AOD is multiplied by the theoretical coefficient A, indicating a substantially greater linear agreement between the variables. The coefficient of determination $R^2=0.58$ is comparable to previous studies that used a linear fit for the correlation [22,58]. The linear relationship holds across the months represented in the dataset without requiring season-specific tuning, although this result cannot be extrapolated to the full annual cycle given that the dataset is weighted toward late winter and spring (February–May, approximately 75.5% of valid pairs); summer months (June–July) contribute only 24.5%, and August and September yielded no valid data points after filtering. The slope of the linear function is $a=0.81$, below the theoretical value of unity, reflecting the high relative uncertainty of A, which in turn derives from the high relative uncertainty of $r_{eff}$. By repeating the regression with the corrected $r_{eff}^{\prime }$, obtained by extrapolating particle number concentrations below the LOAC detection limit, the slope reaches $a=1.04$ while maintaining a comparable goodness of fit ($R^2=0.54$, R = 0.73).

The assessment of variable importance shows that the uncertainty assigned to $r_{eff}$ and X greatly impacts the linear correlation. By considering X data with relative uncertainties lower than 20%, Pearson’s coefficient increases by 5% between $X={\displaystyle \frac{AOD}{MLH}}$ and $X=A{\displaystyle \frac{AOD}{MLH}}$: in particular, adding $r_{eff}$ increases R by 4%, while adding $〈Q_{ext,aer}〉$ and ${\rho }_{aer}$ contributes a further 1%. For comparison, applying an absolute uncertainty threshold of 2 $\mathsf{\mu }$g m⁻³ on both Y and X increases R by a further 15%, for a total improvement of 20% from 0.56 to 0.76.

The dominant contribution of $r_{eff}$ is physically consistent with its large dynamic range at this site: as shown in Figure 5, $r_{eff}$ varies from below 0.6 $\mathsf{\mu }$m during typical conditions to above 1.5 $\mathsf{\mu }$m during desert dust intrusions, a range substantially larger than the seasonal variability of Extinction Efficiency and Mass Density. This confirms that particle size is the primary source of variability in the AOD-PM relationship at this urban site, and that even an observational estimate of $r_{eff}$ from an OPC instrument substantially improves the physical consistency of satellite-based PM_2.5 estimates.

5. Conclusions

PM_2.5 concentrations measured at ambient humidity by an LOAC optical particle counter were analysed alongside co-located satellite AOD retrieved at high spatial resolution. The use of ambient-humidity PM_2.5 in place of the standard dry measurement allowed a direct comparison with satellite AOD without relying on empirical hygroscopic correction functions, yielding a moderate linear correlation (Pearson’s $R=0.56$) when AOD was normalised by the Mixing Layer Height. No significant correlation was found when reference dry PM_2.5 measurements were used instead, highlighting the importance of temporal resolution and humidity consistency between the two quantities being compared.

The exploitation of aerosol microphysical and optical properties derived from LOAC measurements (Effective Radius, Extinction Efficiency, and Mass Density) enabled the derivation of a theoretical scaling coefficient linking PM_2.5,wet and AOD at ambient conditions. These properties directly control how aerosol mass concentration translates into column-integrated extinction, providing a physical basis for the observed improvement in correlation (Pearson’s $R=0.76$) upon inclusion of the physics-informed coefficient A. The regression slope approaches the theoretical value of unity ($a=0.81$ with the original $r_{eff}$, $a=1.04$ with the corrected $r_{eff}^{\prime }$), indicating that the theoretical relationship (Equation (5) accurately predicts both the magnitude and the variability of PM_2.5,wet from satellite AOD. The intercept remains consistent across all regression cases within confidence bounds, suggesting that the residual offset reflects the scale mismatch between point measurements and the satellite footprint rather than a systematic physical bias. The linear relationship holds across the analysed period, but the dataset coverage is insufficient to draw conclusions about autumn and winter conditions; multi-year observations are required for a robust seasonal characterization. Multi-year, continuous observations would be required to robustly characterize the seasonal and inter-annual variability in the AOD-PM relationship within this framework; the present results should therefore be regarded as a proof-of-concept rather than a definitive seasonal characterization.

The lower detection limit of the LOAC (0.2 $\mathsf{\mu }$m diameter) causes the overestimation of the Effective Radius, as the contribution of smaller particles is neglected. In the present work, this is addressed by extrapolating particle number concentrations to sizes below the detection limit using the first four size bins. The corrected Effective Radius, which is on average between 10% and 50% lower than the uncorrected value, improves the agreement between measured and satellite-estimated PM_2.5, bringing the regression slope closer to the theoretical value of one. This result highlights the central role of the Effective Radius in physically linking aerosol optical and mass properties, and points to the importance of accounting for the instrument detection limit when deriving aerosol microphysical properties from OPC data.

Although the analysed period does not cover a full annual cycle and the analysis is limited to a single site, this work was designed as a pilot study to evaluate the physical consistency of the proposed framework under well-controlled conditions. The use of a single, well-instrumented site allowed for the detailed and self-consistent characterization of aerosols’ microphysical properties, which is essential for testing the underlying assumptions of the methodology. Wind-driven transport, solar radiation, and atmospheric pressure were not explicitly accounted for; at a single urban site with co-located measurements, these factors are considered secondary relative to aerosol microphysical properties and MLH, but may become relevant in future multi-site applications. Future work will extend the analysis to multiple locations and longer time periods to assess the transferability and robustness of the approach across different aerosol regimes and environmental conditions.

For regulatory applications, ambient PM_2.5 estimated from satellite observations can be converted to dry PM_2.5 using co-located dry PM measurements and the relation $P{M}_{dry}=P{M}_{wet}/f\left(RH\right)$ (Equation (6)). The proposed framework therefore provides a basis for indirectly deriving local aerosol hygroscopic growth factors by coupling satellite-based ambient PM_2.5 estimates with conventional dry PM observations, opening new opportunities for characterising aerosol–humidity interactions from space and for improving air quality assessments in regions with limited ground-based monitoring.