A novel calibration approach of MODIS AOD data to predict PM 2 . 5 concentrations

Epidemiological studies investigating the human health effects of PM2.5 are susceptible to exposure measurement errors, a form of bias in exposure estimates, since they rely on data from a limited number of PM 2.5 monitors within their study area. Satellite data can be used to expand spatial coverage, potentially enhancing our ability to estimate locationor subject-specific exposures to PM 2.5, but some have reported poor predictive power. A new methodology was developed to calibrate aerosol optical depth (AOD) data obtained from the Moderate Resolution Imaging Spectroradiometer (MODIS). Subsequently, this method was used to predict ground daily PM2.5 concentrations in the New England region. 2003 MODIS AOD data corresponding to the New England region were retrieved, and PM 2.5 concentrations measured at 26 US Environmental Protection Agency (EPA) PM2.5 monitoring sites were used to calibrate the AOD data. A mixed effects model which allows day-today variability in daily PM2.5-AOD relationships was used to predict location-specific PM2.5 levels. PM2.5 concentrations measured at the monitoring sites were compared to those predicted for the corresponding grid cells. Both cross-sectional and longitudinal comparisons between the observed and predicted concentrations suggested that the proposed new calibration approach renders MODIS AOD data a potentially useful predictor of PM2.5 concentrations. Furthermore, the estimated PM2.5 levels within the study domain were examined in relation to air pollution sources. Our approach made it possible to investigate the spatial patterns of PM 2.5 concentrations within the study domain. Correspondence to: H. J. Lee (hlee@hsph.harvard.edu)


Introduction
Atmospheric aerosols originate from natural and anthropogenic emission sources.
Particularly, anthropogenic aerosols are considered to have major human health implications, and numerous studies have reported associations between mortality and morbidity and particulate matter with aerodynamic diameter ≤2.5 µm (PM 2.5 ) (Bell et al., 2007;Dominici et al., 2006;Franklin et al., 2007;Gent et al., 2003Gent et al., , 2009;;Schwartz et al., 1996;Slama et al., 2007).The PM 2.5 health effect studies generally use PM 2.5 measurements from ground monitoring sites, but there are many regions with no ground PM 2.5 measurements available due to their sparse monitoring networks.This limits the ability of estimating human exposures to PM 2.5 , which is likely to cause less reliable health effect assessments.
Satellite remote sensing can be used to assess PM 2.5 air quality for areas where surface PM 2.5 monitors are not available (Di Nicolantonio et al., 2009;Engel-Cox et al., 2004;Gupta and Christopher, 2008;Gupta et al., 2006;Koelemeijer et al., 2006;Liu et al., 2004;Schaap et al., 2009;van Donkelaar et al., 2010).The most applicable satellite-retrieved product for estimating PM 2.5 concentrations is aerosol optical depth (AOD), which measures the light extinction by aerosol scattering and absorption in the atmospheric column.Since the AOD reflects the integrated amount of particles in the vertical column, it has been used as an input parameter in statistical models predicting PM 2.5 levels.In addition to AOD values, several studies have also included other predictor parameters such as local meteorology and land use information (e.g., population density).As reported by previous studies, these parameters influence the relationship between AOD and ground-level PM 2.5 concentrations, thus can be used as additional predictors (Liu et al., 2005(Liu et al., , 2007a(Liu et al., , b, c, 2009)).However, these models, developed by us and others, generally predict <60 % of the variability in daily PM 2.5 concentrations, as shown by a review paper, Hoff and Christopher (2009).Additional time-varying parameters influence the PM 2.5 -AOD relationship, including PM 2.5 vertical and diurnal concentration profiles, PM optical properties, and others.Therefore, it is reasonable to expect that the relationship between PM 2.5 and AOD varies by day.In this paper we introduce a new approach to calibrate Moderate Resolution Imaging Spectroradiometer (MODIS) AOD data taking into account the daily variability in the prediction of PM 2.5 ground concentrations.
Our method is unique because it establishes day-specific PM 2.5 -AOD relationships using a mixed effects model to fully exploit satellite data.To the best of our knowledge, no previous studies have suggested a statistical approach establishing the PM 2.5 -AOD relations on a daily basis.

Ground-level PM 2.5 data
Our study region includes the States of Massachusetts (MA), Connecticut (CT), and Rhode Island (RI) in the Northeastern US.To calibrate satellite data, daily PM 2.5 concentrations measured at 26 US Environmental Protection Agency (EPA) PM 2.5 monitoring sites were used (Fig. 1).24-h PM 2.5 samples were collected on Teflon filters and were analyzed gravimetrically (US EPA, 2011).For days with collocated PM 2.5 samplers at a monitoring site, we calculated the daily averages of the PM 2.5 concentrations.The replicate measurements, which included two identical samplers located next to each other, were used to determine the method precision.Samples were collected at 15 Connecticut sites and 11 Massachusetts sites during the period 1 January through 31 December 2003.Sampling frequency differed by site including collecting samples every day, every third day, and every sixth day.

AOD retrieval
MODIS aboard the National Aeronautics and Space Administration (NASA)'s Earth Observing System (EOS) satellites, Terra and Aqua, was used to retrieve AOD (Collection 5; Level 2 aerosol product) for the year 2003.The Terra and Aqua satellites were launched in December, 1999 andin May, 2002, respectively.These polar-orbiting satellites, operating at an altitude of approximately 700 km, provide aerosol data every one to two days.However, aerosol data are often missing due to clouds, high surface reflectance (e.g., snow-and ice-cover), and retrieval errors.The Terra and Aqua satellites cross the equator at about 10:30 a.m.(descending orbit) and 1:30 p.m. (ascending orbit) local sun times, respectively, with a scanning swath of 2330 km (cross-track) by 10 km (along-track at nadir).The MODIS has 36 different wavelength channels, and seven of them (between 0.47 and 2.12 µm) are used for the aerosol retrieval.In the Collection 5 retrieval algorithm, three different channels of 0.47, 0.66, and 2.12 µm are primarily employed for over-land aerosol retrievals, while many other channels are used for screening procedures (e.g., cloud and snow-and ice-cover).These three channels (0.47, 0.66, and 2.12 µm) are simultaneously inverted to finally report AOD values at the wavelength of 0.55 µm.The uncertainty of the MODIS AOD measurements is expected to be AOD = ±0.05±0.15×AODover land.Furthermore, the maximum AOD value is constrained to be 5.0, and negative AOD values down to −0.05 were retained in order to avoid bias that can occur when truncating or omitting low exposure values.More details about the retrieval of MODIS satellite aerosol data are reported in Remer et al. (2005) and Levy et al. (2007Levy et al. ( , 2010)).Following the nominal resolution of MODIS (10 km), we created 387 grid cells of 10×10 km 2 covering our study region in ArcGIS (Version 9.3; ESRI).The current study is part of a large investigation examining the impact of PM 2.5 on human birth weight in New England.As a result, our study focuses on the areas where study subjects reside.We excluded any grid cells along the coast that had sea surface of at least 2/3 of the cell area.Subsequent analyses were based on the selected set of 387 grid cells.
Since Terra and Aqua satellites retrieve AOD data at two different times each day, the average of these two measurements should be used to predict daily PM 2.5 levels (Kaufman et al., 2000).However, there are many days where only one of the two retrievals is available.To fully exploit the measurements of both satellites we primarily used Terra AOD data for our predictions, and for days with no Terra data, Aqua AOD measurement values were used to estimate the missing Terra values.This was accomplished by multiplying Aqua AOD measurements by an adjustment factor, which was necessary to account for diurnal variations (Green et al., 2009) and potential calibration differences in two satellite sensors.This factor was equal to the average Terra AOD/Aqua AOD ratio which was calculated for days where both Terra and Aqua data were available.Due to the limited satellite data in winter (January, February, and December), the data were grouped into two seasons [warm (15 April-14 October) and cold (15 October-14 April)].The estimated (Terra AOD/Aqua AOD) ratios for the warm and cold seasons were similar, 1.15 and 1.17, respectively.

Statistical model
Since time-varying parameters such as relative humidity, PM 2.5 vertical and diurnal concentration profiles, and PM 2.5 optical properties influence the PM 2.5 -AOD relationship, our statistical model allows for day-to-day variability in this relationship.Furthermore, we hypothesize that these timevarying parameters exhibit little spatial variability and consequently the PM 2.5 -AOD relationship varies minimally spatially on a given day over the spatial scale of our study.Therefore, a quantitative relationship between PM 2.5 concentrations measured at 26 PM 2.5 monitoring sites and AOD values in their corresponding grid cells can be determined on a daily basis.A simple approach would be to calculate such PM 2.5 -AOD slopes separately for each day in the study.However, this simplistic approach can yield highly variable slope estimates, since some days might have a small to moderate amount of monitoring data.An alternative approach that pools daily slope estimates but uses data from all days to stabilize the estimates is to use a mixed effects model with random intercepts and slopes (Fitzmaurice et al., 2004), shown by the following equations: where PM ij is the PM 2.5 concentration at a spatial site i on a day j ; AOD ij is the AOD value in the grid cell corresponding to site i on a day j ; α and u j are the fixed and random intercepts, respectively; β and v j are the fixed and random slopes, respectively; s i ∼ N(0, σ 2 s ) is the random intercept of site i; ε ij ∼ N(0, σ 2 ) is the error term at site i on a day j ; and is the variance-covariance matrix for the day-specific random effects.In the statistical model, the AOD fixed effect represents the average effect of AOD on PM 2.5 for all study days.The AOD random effects explain the daily variability in the PM 2.5 -AOD relationship.The site bias may arise since an AOD value in a 10×10 km 2 grid cell is an average optical depth in the given grid cell, while the PM 2.5 concentrations measured at a given site may not be representative of the whole grid cell.Specifically, the bias can indicate spatial sites presenting high PM 2.5 levels due to their locations near high traffic areas.To control for the site bias, we added a site term as a random effect into the mixed effects model.It should be noted that the random estimates for the site term were omitted when estimating grid-specific PM 2.5 concentrations from AOD values, since AOD values are unbiased representatives of the corresponding grid cells.Because a slope cannot be estimated from a single data point, we excluded all the pairs of measured PM 2.5 concentrations and their corresponding AOD values when there was only one pair on a given day before running the mixed effects model.This resulted in the exclusion of 29 days.Furthermore, the model prediction was examined using the root mean squared error (RMSE) between the measured and predicted PM 2.5 concentrations on each day.Four sample days with RMSE > 5 µg m −3 , which was approximately the 95th percentile of the RMSE values, were excluded from the analysis, since the daily PM 2.5 -AOD relationships were not considered reliable enough to calibrate AOD data.Finally, PM 2.5 estimates covering the whole study area were produced using the AOD calibration model described above.
To demonstrate whether the mixed effects model improved the ability of AOD to predict PM 2.5 concentrations we compared our model performance to that of a previously used model which assumes that the PM 2.5 -AOD relationship remains constant over time (Wang and Christopher, 2003).In the previous model, measured PM 2.5 concentrations were regressed on AOD values retrieved in the corresponding grid cells as a fixed effect, establishing a single linear PM 2.5 -AOD relation applied to all sampling days.It is noted that the comparison of those two models was based on identical sampling days.As measures of accuracy and precision of the two models, we used coefficient of determination (R 2 ) and precision (% Precision) between the measured and predicted PM 2.5 concentrations.The precision was estimated as the square root of the mean of the squared errors, and % Precision was calculated as follows: % Precision = 100 × (precision/measured mean PM 2.5 ) (2)

Model validation
To test this new approach we analyzed the 2003 MODIS data for MA, CT, and RI.We utilized a cross-validation (CV) method to examine whether the model is generalizable to any grid cell in the study domain.Toward this end the data of one site (test site) were separated from those of the other 25 sites (calibration sites).Subsequently, a model was developed using the data from the calibration sites.Finally, the model was used to predict PM 2.5 concentrations for the test site.This process was repeated until each of the 26 spatial sites was tested, and the measured PM 2.5 concentrations were compared to those predicted at each site.Furthermore, Pearson correlation coefficients were used to examine the relationship between the measured and predicted concentrations in each site.Since time-series studies examine longitudinal associations between exposures and health outcomes, high correlation coefficients would imply that satellite AOD data can be used to assess exposures for these health investigations.In addition, we examined the agreement between the measured and predicted site mean PM 2.5 concentration levels for each of the 26 sites, which was assessed by the correlation between the measured and predicted mean PM 2.5 concentrations.This comparison is important for determining whether model predictions are reliable for cross-sectional studies, which require accurate assessment of spatial patterns in exposure.

PM levels in the study region
For each of the analyzed days, a concentration map of 387 grid cells was obtained using the day-specific PM 2.5 -AOD relationships determined by the mixed effects model.Since the AOD retrieval rate varies by location, the number of PM 2.5 concentration predictions varied by grid cell.Therefore, a direct comparison among cell means would not be adequate for the investigation of the PM 2.5 spatial patterns within the study domain.To minimize the potential impact of varying predictions per grid cell we estimated the mean differences between the predicted grid cell and regional PM 2.5 concentrations for the days where grid cell predictions were available.Note that daily regional PM 2.5 concentrations were calculated by averaging the predicted PM 2.5 concentrations for each of the grid cells on a given day.Since the number of AOD retrievals varied by day, the number of available PM 2.5 concentrations used to estimate the daily regional average levels varied by day as well.To obtain reliable and representative regional PM 2.5 concentrations, we chose our daily regional PM 2.5 estimations only for days with 50 or more grid cell predictions on a given day.All the other days with less than 50 grid cell predictions were not considered.Finally, the grid cell-specific PM 2.5 mean differences between the grid cell and the regional PM 2.5 concentrations were estimated for the selected days (50 or more grid cell predictions) and presented using septiles, which split the distribution of the mean differences into seven equallysized bins, in ArcGIS.Positive mean differences, expressed in µg m −3 , indicate that on average levels at a given grid cell are higher relative to the regional PM 2.5 levels, while the opposite is true for negative values.

Descriptive statistics
The mean PM 2.5 concentrations measured at the 26 EPA PM 2.5 monitoring sites in 2003 are summarized in Table 1.The mean (SE) PM 2.5 concentrations ranged from 9.0 (0.7) µg m −3 in Haverhill, MA (Site ID: 25-009-5005) to 17.0 (0.5) µg m −3 in New Haven, CT (Site ID: 09-009-0018).The mean PM 2.5 concentration at the New Haven site was exceptionally high as compared to those monitored at other sites, possibly because the site was located on a ramp connecting to interstate I-95.Many of the monitoring sites showed similar mean PM 2.5 concentrations.However, it should be noted that the number of samples used to estimate these means varied by site due to differences in sampling frequencies among sites and missing data.The average number of PM 2.5 samples across the monitoring sites was 136 (SD = 78) with 73 (SD = 41) in the warm season and 63 (SD = 39) in the cold season.Furthermore, mean (SE) daily AOD values observed for the 387 grid cells varied from 0.08 (0.02) to 0.36 (0.04).On average 67 AOD values were retrieved per grid cell which corresponds to 18 % of the entire study period of 365 days.

PM 2.5 prediction
In the mixed effects model, 99 different daily PM 2.5 -AOD relations were generated in 2003.The fixed effects of intercept and slope (AOD) were statistically significant [α=11.9(SE = 0.93), p <0.0001; β = 4.4 (SE = 1.53), p = 0.0049], respectively.The random effects of intercept and slope (AOD) varied considerably by day, with standard deviations of the daily intercepts and slopes of 8.0 and 2.3, respectively.The estimated σ 2 of the error term was 3.70.This supports our hypothesis that parameters influencing the relationship between PM 2.5 and AOD vary daily but not spatially.Therefore, it is possible to perform daily calibrations using data from the multiple PM 2.5 monitoring sites in the study domain.It is noted that the daily intercepts and slopes were in-dependent of the number of PM 2.5 -AOD pairs on a given day.
In addition, the averages of the daily intercepts and slopes were found to be 12.7 (SD = 8.7) and 4.6 (SD = 2.5) in warm season and 10.1 (SD = 5.4) and 3.8 (SD = 1.3) in cold season, respectively.The random effect estimates of the site term for densely populated and high traffic areas were positive as presented in Table 2. Therefore, inclusion of the site term was necessary to adjust for the site bias in our model.As shown by Table 3 and Fig. 2a, the mixed effects model performed quite well.Table 3 presents the site-specific comparisons between the measured and predicted PM 2.5 concentrations in the mixed effects model, and the model prediction was reliable for most spatial sites (mean % Precision = 13.16%, Range = 7.38 to 25.45 %).Moreover, Fig. 2a depicts the results of the linear regression model which was used to compare the measured and predicted daily concentrations for all 26 monitoring sites [R 2 = 0.97, slope = 0.96 (SE = 0.01), and intercept = 0.44 (SE = 0.11)].In addition, Fig. 2b shows the results of the linear regression model used to compare the measured concentrations to those obtained from the CV procedure [R 2 = 0.92, slope = 0.92 (SE = 0.01), and intercept = 0.88 (SE = 0.18)].It is noted that the predicted PM 2.5 concentrations from the CV procedure were not adjusted for the site bias, due to the fact that this term would not be available for location-specific predictions in an epidemiological health effects study.The more pronounced difference between the measured and predicted concentrations in Fig. 2b as compared to Fig. 2a is likely to reflect the bias.
As it can be seen, both the model fit and CV test resulted in high R 2 , slopes close to 1, and intercepts close to 0, indicating a good agreement between the measured and predicted concentrations.
In the mixed effects model, the differences between measured and predicted PM 2.5 levels can be attributed to a combination of monitoring site-specific characteristics as well as PM 2.5 measurement and AOD retrieval errors.For instance, the monitoring site location may not be representative of a given 10×10 km 2 grid cell for an average optical depth retrieved value.For example, concentrations measured at the New Haven site (Site ID: 09-009-0018), which was located on a ramp to interstate I-95, were significantly higher than those observed at the other sites, including the site (Site ID: 09-009-0026) located nearby (0.7 km).Therefore, the difference (4.65 µg m −3 ) between the measured and predicted mean PM 2.5 concentrations before taking the site bias into account at the New Haven site can be explained by the fact that this site is not representative of the corresponding grid cell 10 × 10 km 2 area, and it indicates that the approach of controlling for the site bias in the mixed effects model is reasonable for the comparisons between the measured and predicted PM 2.5 concentrations.However, considering that AOD-derived PM 2.5 concentrations reflect the overall PM 2.5 levels in the grid cell, the unadjusted predicted PM 2.5 levels may be more representative of the average population exposures to PM 2.5 .AOD retrieval errors due to unscreened clouds could introduce positive bias.The current cloud screening algorithm in AOD retrievals (Collection 5) effectively masks clouds, but it is still possible to have AOD values affected by clouds, particularly for isolated and residual clouds (Levy et al., 2007).The comparison between MODIS AOD and the Aerosol Robotic Network (AERONET) AOD (Level 2.0; within ±30 min of Terra measurements) in Billerica could indicate days with positive bias potentially from isolated and residual clouds in the area (correlation r = 0.92; slope = 1.20; intercept = −0.002 in a linear regression model between the MODIS AOD and the AERONET AOD data) (Holben et al., 1998).Consequently, the AOD values overestimated by the clouds may cause positive bias in predicted PM 2.5 concentrations.In part, PM 2.5 measurement errors might cause positive or negative bias in measured PM 2.5 levels.
The ability of the mixed effects model and the linear regression model (Wang and Christopher, 2003) to predict PM 2.5 concentrations was compared.For each model the predicted concentrations were regressed on the observed ones for each site separately (Table 4 and Fig. 3).It should be noted that the CV method produces less biased estimates than those obtained from the model fit (shown in Tables 3  and 4).The two models were compared using results from CV analyses to avoid over-fitting thus to produce more robust results.Note that the predicted PM 2.5 concentrations in the mixed effects model were not adjusted for the site bias (Table 4 and Fig. 3).The mixed effects model explained 95 % of the variability in the measured PM 2.5 concentrations on average, ranged from 82 % in Boston, MA (Site ID: 25-025-0027) to 100 % in Bridgeport, CT (Site ID: 09-001-0010).On the other hand, in the linear regression model, the mean variability of the measured PM 2.5 explained by the predicted PM 2.5 was 51 %, ranging from 12 % in Boston, MA (Site ID: 25-025-0027) to 88 % in Stamford, CT (Site ID: 09-001-2124).While the regression model yielded modest and considerably varying predictability by site, our model demonstrated consistently high predictability for most of the sites.These findings suggest that predicting PM 2.5 within a domain requires the use of daily calibrations.This explains Table 4. Comparisons of CV R 2 and % CV Precision (µg m −3 for CV precision) between the measured and predicted PM 2.5 concentrations using mixed effects model and regression model (Wang and Christopher, 2003)   why previous investigations have not demonstrated that AOD can be a robust predictor of PM 2.5 (Paciorek and Liu, 2009;Paciorek et al., 2008).
The predictive ability of our model was also compared to that of the regression model in terms of percent precision (% Precision) (Table 4 and Fig. 3).Note that this comparison was performed using the CV results as well.Since the R 2 does not reflect systematic differences between the measured and predicted PM 2.5 levels, the measure of precision (% Precision) is necessary to better assess model performance.In the mixed effects model, the % CV precision ranged from 8.   .Spatial variability in PM 2.5 levels in the study region.PM 2.5 levels are expressed as differences between grid-specific predicted and regional PM 2.5 concentrations (µg m −3 ).
performance of the mixed effects model to predict surfacelevel PM 2.5 concentrations was superior as compared to that of the regression model.Collectively, these performance tests suggest that the mixed effects model can be used to produce concentration data sets reliable for both time-series and cross-sectional health effect studies.

Spatial variability in PM 2.5 levels
The spatial patterns of PM 2.5 levels within the study domain are shown in Fig. 5.To highlight the spatial patterns, we used the mean differences between grid-specific PM 2.5 and regional PM 2.5 levels for the days with 50 or more grid cell predictions, as mentioned above.Mean concentration differences varied from −0.36 to 0.87 µg m −3 (mean=0.01 µg m −3 , SD = 0.17 µg m −3 ), and were lognormally distributed, which led us to use septiles for char-acterizing the spatial variability of PM 2.5 levels in our study region.The relatively small difference between the lowest and highest values (1.23 µg m −3 ) compared to the one presented in Table 4 can be explained by the fact that the result of Fig. 5 represented average cell concentrations which were based on the large number of overlapping days, while the large variability in average PM 2.5 concentrations between sites in Table 4 was derived from the limited number of samples, used to calculate the means, which did not correspond to the same time period.As expected, highly populated areas such as Bridgeport, New Haven, Hartford, Boston, Springfield, and Providence exhibited higher PM 2.5 levels.Also, higher PM 2.5 levels were predicted along the major interstate highways (e.g., I-91/95) and areas with high point emission sources (e.g., power plants located in coastal cities) (US EPA, 2008).The concentration spatial patterns observed in eastern Massachusetts were similar to those found by our previous studies (Gryparis et al., 2007).Furthermore, the estimated PM 2.5 levels in western Massachusetts were generally lower, which is due to the lower population density and traffic density in the area.However, it must be noted that the reported PM 2.5 spatial patterns may not be representative of the entire year, since AOD values are less likely to be collected during the cold season due to more frequent cloud conditions during this period.The average number of the predicted PM 2.5 concentrations in each grid cell was 39 (SD = 6) days in warm season and 14 (SD = 5) days in cold season.

Conclusions
Satellite AOD data have been increasingly used for PM 2.5 air pollution studies.Remote sensing technologies have a great potential to expand current ground-level PM 2.5 monitoring networks.To date, the application of satellite data to health effect studies has been limited mostly due to the insufficient power of AOD to predict PM 2.5 and the high frequency of non-retrieval days.We have introduced an AOD calibration method which made it possible to determine the temporal and spatial patterns of PM 2.5 in a large study domain comprising the States of Massachusetts, Connecticut, and Rhode Island.An approach to PM 2.5 prediction for non-retrieval days will be presented in our forthcoming paper.Finally, it is anticipated that future satellite technologies will provide data with finer spatial and temporal resolutions and more accurate data retrievals.In addition, the advanced capability of discriminating by aerosol species in satellite technologies will further contribute to health effect studies investigating species-specific health implications.Since satellite data are readily available, PM 2.5 concentrations can be predicted in a cost-effective way.Considering the sparse ground-level PM 2.5 monitoring networks, our method will help to investigate the associations between subject-specific exposures to PM 2.5 and their health effects.

Fig. 2 .
Fig. 2. Mixed effects model performance assessed by 576 measured and predicted daily PM 2.5 concentrations (µg m −3 ) from: (A) Mixed effects model and (B) CV mixed effects model.All pairs of the measured and predicted PM 2.5 concentrations at the 26 monitoring sites (576 pairs in total) are plotted.The solid line represents the regression line, and the dashed line displays the 1:1 line.

Fig. 3 .
Fig. 3. Cross-validation correlation coefficients and % Precision between the measured and predicted PM 2.5 concentrations for the: (A) Mixed effects model and (B) Regression model(Wang and Christopher, 2003).

Fig. 4 .
Fig. 4. Cross-sectional comparisons between the measured and predicted site mean PM 2.5 concentrations (µg m −3 ) for the: (A) Mixed effects model and (B) Regression model (Both from CV analyses).The solid line represents the regression line, and the dashed line displays the 1:1 line.

Fig. 5
Fig.5.Spatial variability in PM 2.5 levels in the study region.PM 2.5 levels are expressed as differences between grid-specific predicted and regional PM 2.5 concentrations (µg m −3 ).
a N indicates the number of PM 2.5 samples.
a Bias represents the random effect estimates of the site term in the mixed effects model.

Table 3 .
Mixed effects model performance by site a .
a The measured and predicted PM 2.5 concentrations, bias, and precision are in the unit of µg m −3 .b N indicates the number of days with both measured and predicted PM 2.5 concentrations.c Bias is defined as (PM 2.5 predicted -PM 2.5 measured ).d Precision is estimated as the square root of the mean of the squared errors.e % Precision is defined as [100 × (precision/PM 2.5 measured )].
The measured and predicted PM 2.5 concentrations, bias, and precision are in the unit of µg m −3 .b N indicates the number of days with both measured and predicted PM 2.5 concentrations.
a .a c Bias is defined as (PM 2.5 predicted -PM 2.5 measured ).d Precision is estimated as the square root of the mean of the squared errors.e % Precision is defined as [100 × (precision/PM 2.5 measured )].