The primary methodology employed in this study involved measuring meteorological parameters and PM_2.5 concentrations on the rooftops of high-rise buildings. Due to the absence of meteorological or television towers of approximately 200 meters in height in Hanoi, the research team opted for tall building rooftops to conduct simultaneous measurements. This approach enabled the study to effectively capture PM_2.5 data at various elevations within urban areas.

In this study, sampling was conducted at two urban locations within the inner city of Hanoi, with each site having simultaneous measurements taken at five distinct points situated on the rooftops of buildings at varying elevations as shown in Figure 1 . Vertical distribution data for the examined parameters were obtained by sampling at the highest floors of buildings, with elevations ranging from 40 meters to over 300 meters ( Table 1 ). Wind speed, humidity, and temperature were directly measured at different heights using a microclimate measuring device. At the same time, PM_2.5 concentrations were sampled using the Instrumex dust sampler and analyzed with an AD-421D-32 analytical balance.

Five Instrumex samplers, designed to concurrently collect PM_2.5 and PM₁₀ particles in compliance with U.S. EPA reference standards, were employed simultaneously for sampling. Each sample was collected over 6 hours, with two samples taken daily between 8 AM and 8 PM.

There are multiple approaches for analyzing the relationship between PM_2.5 concentrations and meteorological variables, with Multivariable Linear Regression (MLR) and Multivariable Nonlinear Regression (MNLR) being the most commonly employed techniques ( Liu et al., 2023 ; Li et al., 2017 ; Hao et al., 2022 ). In this study, both MLR and MNLR models were utilized to assess the relationship between PM_2.5 concentrations and key variables, including elevation, PM₁₀ concentrations, and various meteorological conditions.

Multivariable Linear Regression (MLR), an extension of simple linear regression, is utilized to predict a response variable based on multiple explanatory variables. This study employed a MLR to assess the extent to which fine dust concentration can be estimated using meteorological factors. The regression function was accordingly adjusted to incorporate several predictor variables as follows:

$Y={\text{β}}_0\ast X_0+{\text{β}}_1\ast X_1+{\text{β}}_2\ast X_2+\cdots {\text{β}}_n\ast X_n+\text{ε}$ (1)

In this statistical model, 𝑌 is designated as the dependent variable, with 𝑋₀, X₁ and additional terms representing the explanatory variables, also referred to as independent regressors. The termnotes the stochastic error component. It is observed that as the quantity of predictor variables escalates, the corresponding constants 𝛽 exhibit a proportional increase.

Multivariable Nonlinear Regression (MNLR) is a sophisticated statistical technique that enables the modeling of relationships between a dependent variable and one or more independent variables when linear representations cannot adequately capture such relationships. In this study, the regression model is formulated as a nonlinear function of the parameters in conjunction with one or more independent variables, allowing for greater flexibility and accuracy in representing complex data patterns.

Informed by the comprehensive analysis of MNLR in the studied ( Yin et al., 2016 ), the model proposed in this study is delineated as follows:

$Y=a+\underset{i=0}{\overset{n}{{\displaystyle \sum }}}{b}_i\ast X_i+\underset{j<i}{{\displaystyle \sum }}{b}_{ij}\ast X_{ij}+\underset{i=0}{\overset{n}{{\displaystyle \sum }}}{b}_{ii}\ast X_{ii}$ (2)

where Y represents the dependent variable (PM_2.5 dust), a is the constant, and $b_i$, $b_{ij}$, $b_{ii}$ are the coefficients associated with the independent variables $X_i$, which include PM₁₀ dust, temperature, wind speed, rainfall, humidity, and other relevant factors.

It is crucial to note that the considerations for interpreting the results of the nonlinear regression model are analogous to those applicable in the context of multivariate linear regression models.

This study collected observational data during two distinct periods across two separate areas, with five concurrent samples obtained at each location. Table 2 describes the geographic coordinates and sampling heights for each sampling point. The collected data encompassed wind speed, humidity, temperature, and concentrations of PM₁₀ and PM_2.5. Additionally, surface data were gathered from the Environmental Protection Agency’s air quality monitoring station located at 105.799˚E, 21.015˚N.

Table 2. Information about PM_2.5 sampling locations.

The trends in PM_2.5 concentrations were analyzed for two investigation periods: from August 25, 2023, to September 10, 2023, and from October 4, 2023, to October 22, 2023. These periods fall within the autumn season in Northern Vietnam and align with the pre-harvest and post-harvest phases of the summer-fall rice crop in the surrounding areas of Hanoi. PM_2.5 concentrations during these periods ranged from 9 to 109 μg/m³. The average concentration for the first period was 22.33 μg/m³ in Area 1 and 31.07 μg/m³ in Area 2, while for the second period, it increased to 33.39 μg/m³ and 31.93 μg/m³ for Area 1 and Area 2, respectively. These findings indicate higher fine particulate matter concentrations during the second period, likely due to the impact of agricultural residue burning, consistent with previous studies in Hanoi ( Le et al., 2022 ; Vuong et al., 2023 ; Pham et al., 2024 ; Pham et al., 2021 ).

In this study, the majority of PM_2.5 concentrations were within the lower range of the daily limit of 50 μg/m³ specified by QCVN 05:2013/BTNMT. However, these values exceeded the recommended 15 μg/m³ threshold established by the World Health Organization (WHO).

Figure 2 shows the variations in meteorological factors, along with PM₁₀ and PM_2.5 concentrations by Height during two measurement periods in Area 1. In the first measurement, PM_2.5 concentrations were observed to increase at a height of 40 m before decreasing sharply with further increases in height. This trend is attributed to the significant rise in wind speed at higher heights. At 40 m, the PM_2.5 concentration was approximately 34.76 ± 2.38 μg/m³, which declined to 13.95 ± 1.7 μg/m³ at 336 m, representing a reduction by a factor of two.

A similar pattern was observed in Area 2 during the first measurement period, where PM_2.5 concentrations also peaked at 40 m and then gradually decreased with increasing height ( Figure 3 ). At 40 m, the concentration was approximately 33.92 ± 2.98 μg/m³, decreasing to 23.99 ± 4.53 μg/m³ at 272 m, a reduction of about 1.4 times. Differences in the ground-level measurement location and the PM_2.5 monitoring methods can explain the concentration increase at 40 m. This trend of decreasing PM_2.5 concentration with height aligns with findings from several other studies worldwide ( Liu et al., 2018 ; Chan et al., 2005 ).

PM₁₀ and PM_2.5 concentrations decreased with increasing height during the first period; however, the second period did not show significant variations in these values. Despite these differences, fine particulate matter concentrations in both periods consistently remained within the permissible environmental limits.

Similarly, Figure 3 presents the results for meteorological factors and fine dust concentrations by Height in Area 2. As observed in Area 1, discrepancies were noted between ground-level measurements and those at 40 m. Temperature, wind speed, and humidity varied at different heights, showing consistent trends across both periods. PM_2.5 and PM₁₀ concentrations in Area 2 were comparable to those in Area 1, with no significant differences in concentration levels.

Meteorological factors are vital in influencing the distribution of PM₁₀ and PM_2.5 concentrations. Lower wind speeds contribute to poor dust dispersion in the air, and activities such as traffic and construction at ground level lead to higher dust concentrations. At greater heights, the influence of topographical obstructions diminishes, resulting in a gradual decrease in dust concentrations.

The measurements were conducted during a transitional period affected by light cold air masses, which emit higher dust values in the second period compared to the first. This pattern aligns with Hanoi’s climatic characteristics, as cold air masses hinder the dispersion of air layers, causing dust particles to remain stagnant.

A comparison of PM_2.5 values with those from published studies shows variations across regions. For instance, in the research by Zauli-Sajani et al. (2018) , PM_2.5 concentrations at height during warm conditions were measured at 15 μg/m³, while cold conditions yielded 22 μg/m³. Similarly, the research by C.Y. Chan ( Chan et al., 2005 ) in Beijing, which measured air quality at heights ranging from 8 to 325 m, found an average PM_2.5 concentration of 65 μg/m³ and PM10 concentration of 150 μg/m³. These studies also demonstrate a similar trend of minimal concentration changes with height, consistent with the results observed in Hanoi.

MLR and MNLR analyses were employed to assess the influence of variables such as height, temperature, wind speed, humidity, and PM₁₀ concentration on PM_2.5 levels. The detailed results of the MLR model are presented in Table 3 , while the MNLR model results are shown in Table 4 . The MLR analysis reveals that the selected factors account for approximately 50% to 80% of the variance in PM_2.5 concentrations. In comparison, the MNLR model explains between 80% and 94% of the variance in PM_2.5 concentrations. These findings demonstrate a clear influence of meteorological variables and PM₁₀ concentrations on PM_2.5 levels. Wind speed emerged as the most significant factor affecting PM_2.5 concentrations, aligning with the findings of several previous studies ( Vuong et al., 2023 ; Luong et al., 2021 ).

Since PM_2.5 is a subset of PM₁₀, numerous researchers in previous studies have posited that PM_2.5 concentration has a linear relationship with PM₁₀ concentration ( Makkonen et al., 2023 ). As a result, they have employed Multivariable Linear Regression (MLR) to estimate PM_2.5 concentrations based solely on PM₁₀ levels. However, as observed in Table 3 and Table 4 , meteorological factors such as temperature, wind speed, and humidity also significantly influence PM_2.5 concentrations with a proportional relationship with PM₁₀. Given that meteorological conditions vary by season, their impact on PM concentrations fluctuates accordingly. Multivariable Nonlinear Regression (MNLR) was applied for each survey period. Table 5 presents a comparison between the MLR and MNLR models. The results indicate that through all periods, the adjusted R² values for each area and survey period using MNLR are consistently higher than those obtained with MLR. Except for the first period in Area 1, the MNLR model also yields substantially better Mean Absolute Percentage Error (MAPE) values compared to the MLR model. Thus, MNLR proves to be a more reliable approach for estimating PM_2.5 concentrations. Figure 4 shows a comparison between the estimated PM_2.5 concentrations from both models and the observed values at different heights, under corresponding meteorological conditions.