The Aerosol Robotic Network (AERONET) is a federation of ground-based sun–sky radiometer networks, providing worldwide remote sensing aerosol data for more than 25 years (Holben et al., 1998). The current revision of the dataset is Version 3 (Giles et al., 2017). Due to its high quality, the data from AERONET have been regarded as theoretical true values to evaluate satellite-based products in related studies (Chen et al., 2020; Gao et al., 2016; Wang et al., 2019). AOD, FMF, and volume size distribution products with Level 2.0 (quality-assured) are applied to calculate the true values of the physical parameters and then to implement our modeling purpose (not involved in PM2.5 calculations). A total of nine AERONET sites corresponding to four typical aerosol types participate in the training. Table 1 shows the specific information.

MCD19A2, the Moderate Resolution Imaging Spectroradiometer (MODIS) Collection 6 Level 2 gridded (L2G) land AOD product (Lyapustin and Wang, 2015), is selected in this study. It is derived from the Multi-Angle Implementation of Atmospheric Correction (MAIAC) algorithm, which can improve the accuracy of cloud detection and aerosol retrieval (Lyapustin et al., 2011). Moreover, this new advanced algorithm jointly combines MODIS Terra and Aqua into a single sensor (Lyapustin et al., 2014). The product is produced daily with a 1 km resolution, including aerosol parameters such as 470 and 550 nm AOD, quality assurance (QA), and uncertainty factors.

The processing of MCD19A2 data (in Hierarchical Data Format, HDF) is mainly divided into five steps: AOD–QA band extraction, best-quality AOD selection, Terra–Aqua data synthesis, missing information reconstruction, and mosaic. Finally, the daily AOD distribution in GeoTiff format is obtained.

The original global land FMF products have poor data integrity and low accuracy. To enhance their reliability, Yan et al. (2022) have released a satellite-based dataset called Phy-DL FMF, which integrates physical and deep learning methods. Specifically, it selects the FMF data obtained by a physical method (i.e., lookup-table-based spectral deconvolution algorithm, LUT-SDA) as the optimization target (Yan et al., 2017). Then it combines the Phy-based FMF into a deep learning model along with multiple auxiliary data such as satellite observations for the final Phy-DL results. Note that the process is trained with AERONET data as the ground truth. The product has a spatial resolution of 1∘ and covers 2001 to 2020 (daily scale). In the comparison experiment against the ground FMF, Phy-DL FMF shows a higher accuracy (R=0.78, RMSE =0.100) than MODIS FMF (R=0.37, RMSE =0.282) (Yan et al., 2022).

The meteorological data are obtained from the ERA5 dataset, including the values of planetary boundary layer height (PBLH) and relative humidity (RH). As the fifth-generation reanalysis product released by the European Center for Medium-Range Weather Forecasts (ECMWF), ERA5 provides atmospheric data at 0.25∘ every hour based on the data assimilation principle (Hersbach et al., 2018). It should be noted that RH is not archived directly in ERA5 and thus should be calculated by 2 m temperature T and dew point temperature Td (refer to https://confluence.ecmwf.int/display/CKB/ERA-Interim:+documentation#ERAInterim:documentation-Computationofnear-surfacehumidityandsnowcover, last access: 20 July 2023): 1RH=100×es(Td)es(T), where es(t) represents the saturation vapor pressure related to a temperature t in degrees Celsius (Simmons et al., 1999) of 2es(t)=6.112×exp⁡17.67×tt+243.5.

The North China (NC) region is chosen as the main experimental validation area for the final PM2.5 calculations. The near-surface hourly PM2.5 values are obtained from the China National Environmental Monitoring Centre (CNEMC). Nowadays, over 1600 ground-based monitors are working continuously and a total of 232 stations (in 2017) participate in this work. Figure 1 displays the site distributions of the NC region.

Therefore, to overcome the above disadvantages, an optimized method called RF-PMRS is proposed. Figure 2b shows the process of our method, while optimizations for FMF and VEf are described separately below.

FMF dataset selection.

We introduce the Phy-DL FMF dataset into the PMRS method to improve the accuracy of size-cutting results. In terms of performance, it exhibits higher accuracy and wider space–time coverage than satellite products (Yan, 2021). See the Data section for details.

The simulation model of VEf is trained based on the data in Table 1. Specifically, the 10-fold CV result is used to determine the optimal combination of parameters for the model (see Appendix A3 for the adjustment of the model parameters). Considering that the completeness of the training data will optimize the generalization performance of the model, the experiment fine-tunes the model based on all the original datasets (the training period of Table 1) under the optimal parameters, and then the final RF model is constructed. This is also the most common method for ML model construction. Next, the IV experiment provides independent time validation of the final model.

Table 3 shows the CV and IV results to, respectively, demonstrate the internal and external accuracy of the final RF model. It can be seen that RF can capture the complex relationship between VEf and related variables well. R is as high as 0.974 (0.975), RMSE and MAE are both small, and RPE is around 30 %, which suggests the desired estimation accuracy. Overall, the CV results represent the great performance of the RF model for extracting information, i.e., the relationship of multisource data to VEf. In the meantime, the statistical results in the CV and IV experiments are similar, indicating that the RF model has no obvious overfitting phenomenon.

The purpose of RF-PMRS is to construct an optimal model from the obtained point matching data pairs and generalize it to the space–time continuous surface data for VEf derivation. In the subsequent experiments in Sect. 4.2 and 4.3, the VEf values are obtained by introducing the Phy-DL FMF dataset (surface data) to the final RF model. At the same time, the Phy-DL FMF data are also applied to the PM2.5 calculation process (FMF variable in Eq. 7) for a wide range of PM2.5 concentration.

Then, the experiment compares PM2.5 results of PMRS and RF-PMRS at the Beijing (BJ) and Beijing-CAMS (BC) AERONET sites in 2017. Here, RF-PMRS simulates VEf based on RF and replaces the polynomial of the PMRS method. Note that the results of the two sites are compared with their respective nearest ground PM2.5 stations (distances of 3.64 and 3.91 km, respectively, in line with the representative range of ground stations in previous studies; Shi et al., 2018). Figure 4 displays the time series of PM2.5 values for different models at two sites. The blue line fits the red line better than the gray one, confirming that the PM2.5 results of RF-PMRS are closer to the true values. Within the range of the black circles at positions 1 and 2, the variation in RF-PMRS results has better consistency with the ground truth, while the PMRS results show dislocation and excessive growth. The overall performance of the RF-PMRS estimations can signify the effectiveness of our proposed method framework. As observed in the red boxes at positions 3 and 4, both models have a certain degree of deviation, which is found to be consistent with the time regularity of the AOD high values. Meanwhile, Fig. B1 (in Appendix B) plots the bias time series between PMRS and RF-PMRS and in situ values. As can be seen, the bias of the optimization method (RF-PMRS) is stably distributed around zero, which greatly reduces the numerical uncertainty. It is worth noting that our method has mitigated the apparent overestimation of the original model (PMRS) well in the case of above-normal aerosol loadings. Furthermore, the average PM2.5 values from ground stations, PMRS, and RF-PMRS are compared. As for the two sites, the RF-PMRS results are satisfactory. As depicted in Fig. 5, the RF-PMRS and station mean values are close, with a difference of 4.82 (BJ) and 2.73 µg m-3 (BC), suggesting a good estimation. Nevertheless, the PMRS results have deviations greater than 40 µg m-3, and overestimation exists at both sites. It can be inferred that, in our proposed method, the optimization of VEf can greatly improve the PM2.5 estimation accuracy.

To visually compare the optimization effect, Fig. 6 plots the PM2.5 bias distribution patterns for two methods. From the boxplot, the average PM2.5 bias of RF-PMRS is close to zero (less than 5 µg m-3), which is greatly lower than that of PMRS. Moreover, PMRS PM2.5 has a larger deviation range, which manifests in two aspects. One is the maximum bias; specifically, it has exceeded 100 µg m-3 at the BC site. The other is the overall distribution of the data bias; the BJ site ones are mostly distributed below 0, indicating an obvious overestimation. As for RF-PMRS, the above circumstances are not obviously reflected in it. In addition, as can be seen from the indicators, the RMSE and MAE of RF-PMRS PM2.5 decrease by about half in comparison with PMRS. The experiment has confirmed that the RF-PMRS PM2.5 values have a strong linear relationship with the ground truth at both sites, with R around 0.8 (0.82 at BJ and 0.78 at BC). Such a large optimization effect is attributed to the VEf expression replacement to the fitted RF model.

Then, we estimate PM2.5 based on PMRS and RF-PMRS within North China in 2017 (Fig. 1 exhibits the distribution pattern of the validation stations). Table 4 shows the accuracy statistics. It can be seen that RF-PMRS greatly reduces the bias (about 44.87 %), with an MB of about 2.31 µg m-3. Similar to the results at the sites, the RF-PMRS method can derive PM2.5 concentration with practically no overestimation (underestimation). Although there is not much difference in the R values of the two models (R of RF-PMRS is only improved by 0.01), RMSE and MAE decrease by about 39.96 and 18.86 µg m-3, respectively. As a result, the optimized method deserves to be considered excellent.

Meanwhile, the PM2.5 scatterplots are presented below. As depicted in Fig. 7, there are sufficient estimated samples (28 305) in the NC region, which guarantees the credibility of our validation results. In general, the RF-PMRS PM2.5 values are distributed around the 1:1 reference line evenly, with a slightly higher R of 0.70 compared to that of the original method. The slope of the linear fitting relationship reaches 0.82, which indicates that the proposed method greatly reduces the overestimation of PMRS with a linear slope of 1.46. Although the overall performance of the RF-PMRS estimations maintains an excellent level, defects do remain. To be specific, in areas with high PM2.5 concentration (especially greater than 150 µg m-3), RF-PMRS results exist with a slight underestimation. It may be caused by the relatively small number of high-value PM2.5 points (only 1319 out of 28 305), which makes it difficult to adequately reflect the fitting effect of the method.

As for RF-PMRS, the deviation is reduced to a large extent, so the maps of the probability density function based on the bias of PMRS and RF-PMRS are further drawn. Figure 8 visualizes the probability densities within different bias ranges. In terms of distribution characteristics, the overall bias of RF-PMRS from the value of 0 (solid black line) is small. About the curve shape, it is high and narrow, manifesting in the fact that the bias has a lower standard deviation (SD) and is more prone to appear around the mean. However, PMRS shows a more discrete distribution pattern, and there are many outliers outside the range of greater than 600 µg m-3. Simultaneously, as can be concluded from the three boxes, within the bias range of ±20 and ±40 µg m-3, the data numbers of RF-PMRS results increase by 8.32 % and 12.81 %, respectively. Outside the range of ±100 µg m-3, the number decreases by 9.10 %. Therefore, as far as the accuracy is concerned, RF-PMRS results have lower bias and better stability.

In addition to the above general performance comparison in Sect. 4.3, Fig. 9 presents the annual average RMSE spatial distribution of PMRS and RF-PMRS PM2.5 at NC stations. The two methods show a large deviation in the middle and southeast, and the RMSE map of PMRS has more red points. However, RF-PMRS can weaken this phenomenon very well since its RMSE representative colors are generally light. In particular, the proportion of dark-red sites (RMSE greater than 60 µg m-3) decreases from 65.44 % (PMRS) to 4.15 % (RF-PMRS). In the areas where the ground stations are clustered, the deviation also reduces significantly.

In a word, the above analysis demonstrates that compared with the simple quadratic polynomial relationship (Eq. 9), the established RF model in RF-PMRS can more accurately capture the relationship between VEf and multiple variables, thereby improving the PM2.5 estimation accuracy.

To confirm the superiority of the Phy-DL FMF data adopted in our method framework, the experiment takes the BJ and BC sites as examples (in 2017) and then compares the PM2.5 accuracy and the number of effective days calculated by PMRS based on different FMF values. Table 5 presents the overall day-level results. Here, “DOY” means the day of the year and “valid” means that all variables related to the PM2.5 calculation are valid. As can be seen, after the FMF replacement, the number of valid DOYs grows (an increase of 113 d), which illustrates that the number of effective PM2.5 concentrations has gone up by about 5 times. Moreover, the accuracy has been significantly enhanced, with R having increased by about 0.30 and RMSE and MAE having decreased by 26.14 % and 16.47 %, accordingly. On the whole, Phy-DL FMF contributes to the improvement in PMRS results, signifying the first step in optimization of the proposed RF-PMRS method is effective.

Different machine learning models are suitable for diverse research data, and decision tree (DT) models can better fit experiments with fewer variables, such as this study. For comparison, except for RF, the extremely randomized tree (ERT) (Geurts et al., 2006) and gradient boosting decision tree (GBDT) (Friedman, 2001) models have also been established. The results of training VEf based on the above three DT models are presented in Tables 6 and 7. By contrast, RF performs best in CV and IV experiments, as indicated by the multiple accuracy indicators. Although the ERT and GBDT models are comparable to RF in some indicators, there exists a certain degree of overfitting in the above two models, which is manifested in the fact that their IV results are clearly worse than their respective CV ones. Thus, the RF model is applied to our study.

Additionally, the feature importance of RF is calculated to evaluate the contribution of model predictors to VEf simulation. Figure B2 (in Appendix B) shows the results by normalization (taking 100 as the total). Without a doubt, FMF accounts for the largest proportion, about 76.4 %, which is consistent with the analysis when selecting the VEf-related variables (see Sect. 3.2). The contribution of spatiotemporal variables is about 1/3 of FMF, which indirectly affirms the credibility of RF feature learning. Also, it provides a basis for further uncertainty optimization of VEf and PM2.5 accuracy.