The LUR models used in this work were both calibrated and validated by surface NO2 concentrations measured by in situ stations from the Hong Kong Air Quality Network (HK-AQN). These stations are maintained by the Hong Kong Environmental Protection Department . Between 2005 and 2015 11 monitoring stations measuring ambient pollutant concentrations were in operation (see Fig. ). These stations provide hourly measurements of CO, SO2, O3, NOx, NO2, and particulate matter. NO2 concentrations are measured through a combination of chemiluminescence and differential optical absorption spectroscopy DOAS; . These stations are placed on buildings, away from traffic junctions, and so are thought to be representative of ambient conditions. Of these stations, 10 are located in developed regions while one (Tap Mun) is located in the Sai Kung Country Park, and so can be considered a rural background station. Throughout the study period, the HKEPD have reported that the precision and accuracy of the NO2 measurements have been within the ±20 % control limit.

The number and spatial sampling of these in situ stations is smaller than those typically chosen for LUR modelling . However, it is not entirely without precedent, as used 14 in situ stations from the local regulatory monitoring network in their LUR model to predict NO2 concentrations over Jinan, China. Therefore, it may be possible to model an equivalent Chinese megacity using a similarly limited in situ network.

Between 2005 and 2015 there were three satellite instruments measuring tropospheric NO2 VCDs: OMI, GOME-2A, and SCIAMACHY. These instruments and their retrieval algorithms are briefly summarized in this section. All retrieval algorithms based on these instruments derive these VCDs by first retrieving a total slant column density (SCD) from the measured visible (400–500 nm) reflectance spectrum using the DOAS technique. The stratospheric component of the total column is then separated, either by empirical estimation based on unpolluted regions e.g. or by model assimilation e.g. . In addition to this, the column is also weighted by an air mass factor (AMF) calculated from a priori information to account for biases resulting from scene-specific features (e.g. viewing geometry, scene albedo, NO2 vertical profile).

Tropospheric NO2 VCDs from these instruments were previously verified over China and Japan between 2006 and 2011 using ground-based multi-axis DOAS (MAX-DOAS) measurements by . It was found that the biases between these instruments and the MAX-DOAS observations were small enough to be considered insignificant, suggesting that data from these instruments could be combined for use in air quality studies.

Because of their varying ground pixel sizes, all satellite data products used in this work were reprojected onto a 0.01∘ grid. To avoid biases from cloud contamination, only ground pixels where the reported cloud fraction was < 30 % were used from all instruments. For scanning instruments, only pixels observed during forward scans were used.

LUR models are typically fixed effect models, in which the concentration of a pollutant is expressed as the linear sum of variables approximating the influence of various emission sources and sinks. These variables are “fixed” in the sense that they are temporally invariant, and apply to the mean atmospheric state over the entire observation period. As a result, traditional LUR models are sensitive to unobserved heterogeneity arising from temporal variability in emissions or other ambient conditions. In this work, an additional variable is required to cover time-dependent effects (so-called “random” effects) in order to model daily NO2 concentrations. In practice, time-dependent effects are modelled in linear regression through the inclusion of a discrete “dummy” variable to describe a property of the data, such as the in situ station where particular measurement was made. These effects are considered to be “random”, as the magnitude and/or sign of the effect is not expected to be the same over all measurements. A model combining both fixed and random effects is therefore known as a “mixed-effects” model, in which the concentration is expressed as the sum of fixed variables along with other variables whose effects vary with time or other properties classified by the dummy variables. In this work, these models are fitted from the observation dataset using the lme4 R software package .

The mixed-effects LUR models considered in this work are similar to the one developed by . The daily ambient NO2 concentration at a location i on day j is assumed to be a linear function of the gridded daily satellite tropospheric NO2 VCD retrieved over the same location, Ωij: NO2,ij=α+uj+β1+vjΩij+∑mβmXijm+ϵijujvj∼N00,Σ.

This approach accounts for day-to-day variations in the surface NO2/Ω ratio, while also reducing the influence of days with insufficient in situ or satellite data.

In Eq. () α and uj are the fixed and random intercepts, respectively, while β1 and vj are the fixed and random slopes of Ωij, respectively. The βm are the fixed slopes of additional predictor variables Xijm at point i and day j. The error term of the model is represented by, ϵij(ujvj)∼N(0,σ2), while Σ represents the variance–covariance relationship for the day-specific random effects.

The main source of spatiotemporal information in the model is the NO2⋅Ω relationship derived from the in situ and satellite measurements, while the other parameters are used to give a local context for probable NOx emission sources and sinks. The fixed terms in Eq. () represent the spatial average of the NO2⋅Ω relationship, while the random terms model the day-specific variations. The day-specific relationship may be the consequence of daily variations in the NO2 vertical profile caused by changes in boundary layer height, emissions, or other influences. For this work the daily mean NO2 concentration from each of the stations shown in Fig.  was used as the dependent variable in Eq. (). These concentrations were log-transformed to ensure that the input dataset was normally distributed.

As this is a purely empirical model, the modelled surface NO2 concentration is primarily a function of the in situ and satellite data used to train it. Therefore, surface concentrations are only modelled for a particular day, j, if at least one in situ station has ≥75 % of the expected hourly measurements and a cloud-free satellite observation on that day.

Figure  shows the mean surface NO2 concentration predicted by all models between 2007 and 2012, which was a time period common to all of them. Because of differences in instrument spatial resolution and ground coverage, only 38 days in this time period were found to have cloud-free measurements by all three satellite instruments. As a compromise, Fig.  shows the mean of all data predicted by each model.

Over the Hong Kong SAR, all models show clear enhancements over the areas already noted in Fig. . The models also all predict a negative longitudinal gradient; concentrations predicted by the models over Lantau South Country Park (22.24∘ N, 113.93∘ E) were on average 2.6 times higher than those over Sai Kung Country Park (22.40∘ N, 114.35∘ E). This gradient may potentially be the result of in situ station coverage; the most eastern station (Tap Mun) is situated in the Sai Kung Country Park, while the most western station (Tung Chung) is within a residential area and nearby Hong Kong International Airport.

The distribution of elevated NO2 concentrations over the Hong Kong SAR does not significantly change between models, though the longitudinal gradient is more pronounced in some models than others. In models 2–8 the gradient is strong enough to result in mean surface NO2 concentrations predicted over Lantau South Country Park to be ∼40 µgm-3. These values seem unrealistic, as the MODIS and WorldPop datasets suggest that the region is mostly uninhabited and undeveloped compared to districts like Aberdeen and Yantian, which show similar concentrations.

Outside of Hong Kong, the distribution of the Bao'an and Shenzhen enhancements change considerably between models, depending on whether road networks or population density and urban area coverage were used. Because of a lack of available surface concentration data from mainland China, these regions cannot be validated in this work.

All models including satellite data were found to predict higher surface NO2 concentrations during the winter than in the summer, particularly over urban areas. This seasonal dependence may be caused by lower boundary layer height and longer NOx lifetime during winter, as well as increased emissions from residential heating. Figure  shows this seasonal gradient in the mean surface NO2 concentration during 2005–2015 predicted by models 1 and 2 over both seasons.

Both models in Fig.  are highly correlated in the summer (R2=0.97), as they are largely based on the same variables, though Model 2 does not feature a longitudinal gradient. However, in winter the models are much less correlated (R2=0.78), with Model 2 showing a much stronger longitudinal gradient than Model 1. As in Figure , this gradient leads to unphysically high concentrations being predicted over uninhabited regions such as Lantau South Country Park, making it unlikely that this is a realistic model of winter air quality over Hong Kong.

From Table  it is clear that the winter model had over 1000 fewer observations to use compared to the summer model. During winter there are fewer cloud-free observations, which would lead to the model overfitting the data available. Despite having fewer observations to use the winter model-adjusted R2 is higher than the summer model, suggesting that overfitting has occurred. The spatial footprint size of GOME-2A and SCIAMACHY are much larger than OMI, which would result in fewer cloud-free observations being available in the same time period, and so lead to the null results observed when seasonal models involving these datasets were attempted.

Based on the adjusted R2 values for each model shown in Table , it appears that models 6 and 8 are the best performing models, suggesting that using more than one satellite dataset improves model prediction. However, the adjusted R2 statistic may be artificially inflated by overfitting to the input data, and so may be overly optimistic descriptors of model performance. Ideally these models would be validated against additional measured concentrations from stations independent of the current dataset. However, in the absence of other stations measuring ambient NO2, the LUR models in this work were validated using cross-validation (CV), in which subsets of the data used to initially train the model are iteratively removed from the training process and used to compare against the model forecast.

LUR models are typically validated using two major CV approaches: leave-one-out cross-validation (LOOCV) and k-fold cross-validation. LOOCV involves data from a particular station being reserved from the model training process and used to validate the model, such that data from any one station are validated against a model trained using data from every other station. Conversely, k-fold cross-validation involves randomly partitioning the data into k equal-sized subsets (i.e. from all stations), and then using each subset to validate the model trained using the remaining k-1 subsets. Because of the limited number of stations available for this work, removing entire stations from the training dataset would remove significant information from the model training process, and so unfairly bias the validation results. The limitations of LOOCV compared to k-fold cross-validation when applied to LUR models based on limited in situ data have previously been discussed in and .

Because of the limited number of in situ stations available, this work used a 5-fold CV approach to validate the models, in which 80 % of the available data is used to calculate the coefficients and intercepts of each of the models shown in Table . These models are then used to estimate the surface concentrations of the remaining 20 % of the data. This process is repeated until every data point has been estimated by a model that has not been trained using it.

In this work the predictive performance of each model is determined through comparing the cross-validated model dataset against the original in situ measurements through linear regression. Agreement between the two datasets is quantified by calculating the adjusted R2, gradient, intercept (referred to henceforth as the model bias), and root mean square error (RMSE, µgm-3). Because the models developed in this work are purely statistical, the CV gradient and bias against the in situ data are considered to be the main measures of model accuracy in this work. The RMSE of a model was calculated as the square root of the mean of the squared errors. Table  shows the results of the cross-validation on each of the models considered in this work.

From considering the CV-adjusted R2 and RMSE, it is clear that all the models including satellite data perform better than Model 9, suggesting that there is some utility in incorporating satellite data in LUR models. Model 2 has the highest CV-adjusted R2 and lowest RMSE, suggesting that OMI data offered the best agreement with in situ measurements, so long as seasonal effects are accounted for. Sources of error reflected by the RMSE in models 1–8 may be from coarse spatial sampling by the satellite instrument, or retrieval algorithm errors in the satellite dataset.

Models using only one satellite dataset also perform better than those combining two or more datasets. A likely cause behind this difference is that the models using more than one satellite dataset had fewer cloud-free observations to use, because of complications arising from different spatial resolutions and orbital coverage. A lack of available data would have therefore resulted in these models overfitting the input data available.

For all models in this work the CV dataset can be grouped by station, which allows for side-by-side comparisons of model performance over all regions to be made. Figure  shows the CV-adjusted R2 and RMSE for each model over each station. It is clear from the CV that with the exception of Tap Mun, models 1–4 agree much better with the in situ data overall compared to models 5–9. Figure  also shows that models 1–4 also on average have much lower RMSEs over most stations excluding Tap Mun, which suggests that they offer a higher precision than models 5–9.

However, over Tap Mun almost all models (excluding Model 2) perform poorly, with lower adjusted R2 values and RMSE values that are higher than the mean of the other stations. This result suggests that the models all have poor spatial representivity over unpopulated areas, which is because such regions are largely unrepresented by the in situ stations.

Model 2 is somewhat of an outlier to this trend, as over Tap Mun the CV-adjusted R2 is 0.73, which is comparable to values retrieved over the other stations. Similarly, the CV RMSE retrieved over Tap Mun is also lower than the value retrieved by Model 1. This suggests that training a season-specific model may better account for variability between rural and urban areas.

The CV datasets produced in this work can also be grouped and validated by year to determine whether annual or decadal changes in NO2 are successfully predicted by models trained with all available data. The inclusion of satellite data as a predictor variable also raises the possibility of instrument degradation affecting model performance. Unlike in situ stations, satellite instruments can only be passively recalibrated over their lifetime, leading to a possible drift in retrieval precision that may progressively bias surface NO2 models e.g. .

The LUR models are affected by the number of observations available, which in turn are also dependent on instrument degradation. One example of this is the OMI row anomaly, which since 2007 has grown to affect half of the instrument orbital coverage. Over time, this would lead to fewer available observations, which may lead to biases in the LUR models. The degradation in available measurements, combined with the potential decrease in precision of the DOAS fit over time, may result in a decline in the annual CV-adjusted R2 and a corresponding rise in the CV RMSE because of the increased uncertainty in the model.

Table  shows the annual CV-adjusted R2 and RMSE of models 1 and 2 between 2005 and 2015. While no statistically significant trend is observed in the CV-adjusted R2 values for either model, both models show a statistically significant decline in RMSE over time (Model 1: -0.28 % yr-1; Model 2: -0.11 % yr-1), which suggests that coverage losses or instrument degradation are not significant influences on model accuracy or precision. Table  also shows that on average the adjusted R2 of Model 2 is ∼8.0 % higher than Model 1, while the RMSE is ∼23 % lower, suggesting that the better performance Model 2 showed in Table  compared to Model 1 was not the result of anomalously high correlation with in situ measurements over certain years.

For models 1–8, the inclusion of temperature and wind speed from ERA-Interim was not found to significantly improve the adjusted R2 compared to the other considered variables. One possible reason for this may be that the spatial resolution of the ERA-Interim is too coarse to capture the true variation in temperature and wind speed. Another possibility is that the satellite data implicitly contain information about ambient atmospheric conditions observed as part of the VCD measurement, so additional meteorological data may not be needed in the LUR model.

In order to determine whether meteorological data substantially improve the LUR model, Model 1 was trained again while forcing surface temperature and wind speed from ERA-Interim as predictor variables. The training process again selected the same variables shown in Table , with the addition of the total tertiary road length within 400 m. Wind speed and temperature were found to have a negative effect on surface concentration; the ERA-Interim temperature may represent the ambient actinic flux, while high wind speeds would increase mixing and therefore act to lower concentrations. Figure  shows the seasonal average surface NO2 concentration predicted by Model 1 with and without meteorological data for 2005–2015. The addition of meteorological data causes a ∼17 % mean increase in surface NO2 concentrations across the region, though no new emission sources are visible.

As with the other models, this model variant can be validated against the in situ measurement data using 5-fold CV and compared with the results in Table . When meteorological data were forced the CV-adjusted R2 was 0.806, compared with 0.775 before, suggesting that the inclusion improves the model agreement. Similarly, the model CV RMSE decreased to 12.0 µgm-3 (22.1 %) after including meteorological data. The CV gradient also decreased to 0.846, while the CV bias became 7.17 µgm-3. The decrease in gradient and increase in bias against in situ data suggests that the inclusion of ERA-Interim data does not adequately improve the LUR model accuracy, though the increase in CV-adjusted R2 and decrease in RMSE shows that it does improve the precision of the model.

For this work it is thought that the effect of meteorological data in the LUR model is limited by the spatial resolution of the satellite instruments, or the ERA-Interim dataset. Previous LUR models incorporating daily meteorological data e.g. have typically used measurements from weather stations either close to or at the sites where the NO2 concentrations have been measured, with the ambient temperature and wind field therefore interpolated from these fixed points. Because of the comparatively fewer number of NO2 stations available for this work, it was thought that a harmonized dataset like ERA-Interim would reduce the spatial uncertainty otherwise introduced by discrete weather stations. Future iterations of this work should investigate whether using in situ weather data would provide a better outcome.

An alternative technique to deriving surface NO2 concentrations from satellite measurements is to use a chemical transport model to estimate the vertical profile at the time of the satellite overpass . The profile can then be used to partition the tropospheric VCD into its surface and free-tropospheric components, thereby estimating a scaling factor that can be applied to the measured VCDs. This approach is advantageous in that it allows for surface NO2 concentrations to be mapped at a higher spatial resolution than many CTM grids.

For this work a similar approach to was used to infer surface NO2 concentrations from OMI data. Daily mean NO2 vertical profiles over Hong Kong were sampled from the MACC-II reanalysis dataset Monitoring Atmospheric Composition and Climate; for this purpose. For an OMI ground pixel O, the surface NO2 concentration SO is estimated from the OMI tropospheric VCD, ΩO, using the following relation: SO=νSGνΩG-ν-1ΩGF×ΩO.

Here, the terms ΩG and SG are the tropospheric VCD and the surface concentration derived from the MACC-II daily average profile, for which the surface is defined as the lowest layer of the profile (20 m). To obtain the tropospheric VCD the profile is integrated up to the tropopause height taken from the OMNO2 dataset. The modelled free-tropospheric NO2 column, ΩGF, is taken to be horizontally invariant over the MACC-II grid cell, in order to represent the longer NOx lifetime in the free troposphere. As the spatial resolution of the MACC-II dataset is much larger than the OMI nadir resolution (1.125∘×1.125∘), the S/Ω conversion factor is weighted by an additional term, ν, which is defined as the ratio of the local OMI tropospheric VCD to the mean OMI field over the MACC-II grid cell.

MACC-inferred surface concentrations were calculated for all cloud-free OMI pixels measured over Hong Kong between 2005 and 2012 and compared against the daily ambient NO2 concentrations recorded at the in situ stations. Figure  shows the mean surface NO2 concentration estimated using MACC-II and OMI data for winter and summer over Hong Kong. Compared to Fig. , it is clear that the MACC-inferred concentrations are much lower and capture much less spatial information than the LUR models, because of limitations caused by the OMI spatial resolution. Over both seasons, NO2 concentrations appear to peak north of the Hong Kong SAR, potentially caused by emissions from Shenzhen and Bao'an, or transported further north from the Pearl River Delta.

Because of this lack of spatial detail, the MACC-II concentrations correlate very poorly with the in situ data (R2=0.11, RMSE = 41.9 µgm-3), with a linear gradient of ∼0.58. This analysis was repeated with MACC-II profiles modelled at 14:00 local time (the closest available time to the daily OMI overpass), with similarly poor agreement. As well as this, previous comparisons of tropospheric NO2 VCDs inferred from MACC-II profiles with SCIAMACHY data over East Asia suggest that the dataset underestimates tropospheric NO2 by a factor of 2 in winter , which may also partially explain the lack of agreement with the in situ data. It is clear from this result that the mixed-effects LUR model offers better spatial resolution and predictive capability than the MACC-II reanalysis over Hong Kong.

The Model 1 dataset covers a decade of near-continuous measurements, from which it may be possible to determine whether NO2 concentrations have significantly changed after accounting for noise and seasonal variation. To determine whether a statistically significant trend can be observed from this dataset, surface concentrations modelled over Kowloon and Hong Kong Island (see Fig. ) were binned to monthly averages between 2005 and 2015. Following , a linear trend with a seasonal component was fitted to this time series. The surface concentration at month t (Y(t), where t=0 is January 2005), was modelled as a combination of a fixed intercept μ and linear trend ω: Y(t)=μ+ωt+1+ξ×∑j=14β1,jsin⁡2πjt12+β2,jcos⁡2πjt12+Nt.

The time series may be subject to variations in the seasonal component caused by changes in emissions and NOx lifetime. To reflect this, an additional term, ξ, is introduced to Eq. () to dampen or drive the seasonal oscillation over time. The term N(t) represents the noise component (i.e. the remaining signal in the time series that cannot be explained by the model)

Equation () is first solved using nonlinear regression to determine the values of μ, ω, and ξ that minimize N(t). The seasonal components have a negligible impact on the estimation of the other parameters in Eq. () , so these are subtracted from the time series. In addition to this, the autocorrelations are also accounted for using a linear matrix transformation. Finally, linear regression is applied to determine μ and ω .

In order to determine the linear trend error, it is assumed that the noise N(t) is autoregressive with lag 1 (AR(1)). Following the approach defined by , the linear trend is considered to be statistically significant only if the following condition is satisfied: PH0ω^>2σω^=erfω^σω^2>95%, where erf(x) is the Gauss error function.

The monthly average time series and the fitted model are shown in Fig. , along with an annual bottom-up NOx emission inventory estimated by the HKEPD . The linear trend was estimated to be: -0.0208 µgm-3 yr-1 (-0.430 % yr-1 relative to the average 2005 concentration). The seasonal dampening term ξ was estimated to be: -0.0287 µgm-3 yr-1. However, the trend was found to be statistically insignificant. This analysis was repeated on the raw OMI tropospheric VCDs observed over the region, which resulted in a statistically insignificant trend of -2.52 % yr-1. A similar result was found when analysing satellite data between 1996 and 2012 over Hong Kong by , who also found that the signs of μ and ξ were the same. Another investigation by using only SCIAMACHY data also found a statistically insignificant negative trend, as well as a statistically significant trend of -3.8 % yr-1 over Shenzhen.

A statistically insignificant negative trend was also estimated when this analysis was repeated using data predicted by Model 2 (-0.537 % yr-1), as well as the spatial mean concentration reported by the in situ stations in this region (-0.240 % yr-1). By contrast, the HKEPD inventory shows a statistically significant trend of -1.60 % yr-1. A possible reason behind this discrepancy could be influence from NOx emissions transported from mainland China which may obscure any decline in local emissions. The coarse OMI spatial resolution can also cause a smoothing of sub-pixel plumes over urban areas, and so the resulting retrieved column may be an underestimate of the true value , which would therefore result in a negative bias in the modelled surface concentrations.