Nitrogen dioxide (NO2) is an important atmospheric trace gas whose adverse health impacts have been extensively studied. Controlled human exposure experiments have shown associations between short-term exposure to very high levels of NO2 and airway inflammation (Blomberg et al., 1999), increased bronchial reactivity (Folinsbee, 1992), increased susceptibility to respiratory virus infections (Goings et al., 1989), etc. Chronic exposure to lower doses of NO2 (e.g. those currently observed in Europe and North America) has also been linked to lower lung function and deficits in lung function growth among children (Gauderman et al., 2000; Peters et al., 1999), chronic respiratory symptoms (Zemp et al., 1999) and increased cardiopulmonary mortality (Hoek et al., 2002) among adults in epidemiological studies. A key challenge for these epidemiological studies is to separate out the health effects due to NO2 exposure from those due to exposure to other pollutants, whose concentrations are often highly correlated with those of NO2.

NO2 belongs to the highly reactive group of nitrogen oxides (NOx), whose emissions occur primarily in the form of nitric oxide (NO) with a small proportion of NO2 (i.e. NOx = NO + NO2). NO is quickly oxidised by ozone (O3) to NO2, which, in daylight hours, rapidly photolyses to reform NO and (via O(3P)) O3. Thus, during daytime, NO, NO2 and O3 reach a photostationary state, typically on the timescale of a few minutes (Leighton, 1961). The presence of volatile organic compounds (VOCs) perturbs this null cycle by producing hydroperoxyl radicals (HO2) and organic peroxy radicals (RO2) which oxidise NO without consuming O3, leading to faster NO-to-NO2 conversion and net O3 production. This results in a non-linear response of O3 concentrations to reductions in the emissions of NOx and VOCs (Seinfeld and Pandis, 2016). It is therefore crucial to have accurate emission estimates for developing effective and synergetic control strategies for these interdependent pollutants (Cohan et al., 2005).

NOx can be produced from both anthropogenic and natural/biogenic sources such as fossil fuel combustion, biomass burning, soil microbial processes and lightning (Lee et al., 1997). Global total anthropogenic NOx emissions flattened around 2008, as reductions in Europe and North America were offset by increases in Asia (Hoesly et al., 2018). China, in particular, witnessed a rapid rise in anthropogenic NOx emissions until 2011–2012 (with the exceptions of a few regions where the emissions peaked earlier), which resulted from economic growth along with an absence of regulations (van der A et al., 2017; Liu et al., 2016; Zheng et al., 2018). Emission reduction targets were first announced in the 12th Five-Year Plan (2011–2015) (People's Republic of China, 2011), followed by the Action Plan on Prevention and Control of Air Pollution (2013–2017) (State Council of the People's Republic of China, 2013) and the Three-Year Action Plan for Winning the Blue Sky Defence Battle (2018–2020) (State Council of the People's Republic of China, 2018). The main measures implemented included the installation of selective catalytic reduction equipment in power-generating and industrial facilities and the implementation of stricter vehicle emission standards combined with accelerated fleet turnover (Liu et al., 2020). Decreases in anthropogenic sources are accompanied by an increased importance of soil NOx emissions, which are largely driven by nitrogen fertiliser application and can reach up to 20 % of the anthropogenic emissions in the crop-growing season in some regions with high agricultural activities (Lu et al., 2021). These emissions are relatively poorly quantified and currently unabated (State Council of the People's Republic of China, 2018).

Numerous studies have quantified China's NOx emissions and evaluated the short- or long-term trends in emissions. Some have used a bottom-up method that combines specific emission factors (i.e. mass of a pollutant emitted per unit fuel consumption or industrial production) with the corresponding activity rates (i.e. fuel consumption or industrial production), thus providing sector- or process-resolved emission estimates (Liu et al., 2016; Zhang et al., 2009; Zhao et al., 2013; Zheng et al., 2018). However, the underlying data are mostly not immediately available, resulting in an inevitable time lag between the occurrence of emissions and the establishment of an inventory (Janssens-Maenhout et al., 2015). Moreover, they can introduce potentially large and poorly quantified uncertainties into the emission estimates (Hong et al., 2017; Zhao et al., 2011), which can be further propagated through modelled pollutant concentrations into disease or mortality burden (Crippa et al., 2019) and economic loss estimates (Solazzo et al., 2018). Other studies have inferred top-down estimates of emissions using satellite observations (Ding et al., 2020; Lin et al., 2010; Qu et al., 2017; Zhang et al., 2012). This method requires tropospheric column densities of NO2, which are retrieved by transforming slant column densities to vertical column densities, removing the stratospheric contribution and correcting for the effects of albedo, cloud and aerosols (Leue et al., 2001). Earlier studies used a mass balance approach that assumes a linear relationship between emission rates and column densities (Martin et al., 2003) or between the normalised differences in the two quantities (Lamsal et al., 2011). The linear coefficient was determined from a chemical transport model (CTM) using an a priori emissions inventory. The linear relationship in one grid cell is assumed to be unaffected by atmospheric transport and chemistry in neighbouring grid cells (Mijling and Van Der A, 2012; Streets et al., 2013). For pollutants of longer lifetimes or at finer model resolutions, however, it is important to account for non-local sensitivities of pollutant concentrations to emissions. Advanced data assimilation techniques such as Kalman filter (Napelenok et al., 2008), ensemble Kalman filter (Miyazaki et al., 2012) and four-dimensional variational assimilation (Kurokawa et al., 2009) have been increasingly adopted to combine satellite observations and CTM simulations with prior emission estimates to derive a posteriori emission estimates. These inverse methods provide more timely emission estimates of high spatial and temporal coverage (based on the nature of satellite observations). Nonetheless, the derived emission estimates are not resolved by source sector. They are also subject to uncertainties propagated from the satellite retrievals and the model simulations. For instance, Archer-Nicholls et al. (2021) showed large differences in the NO2 column density simulated by two chemical mechanisms with different treatment of non-methane volatile organic compounds (NMVOCs), which are integrated into the same model with identical NOx emissions. When used in inverse modelling, these modelled NO2 quantities would result in different a posteriori NOx emissions.

This study introduces a novel approach that provides timely updates of a priori emission estimates by source sector using readily available in situ air quality observations. Using this approach, a priori NOx emissions in a bottom-up inventory compiled for Beijing for the year 2013 are updated for 2016. Uncertainties associated with emission trends between 2013 and 2016 were sampled by a perturbed emissions ensemble (PEE), which was constructed on the basis of an expert elicitation. The PEE was then input to an atmospheric dispersion model to generate an ensemble of air quality simulations. By comparing the simulated surface concentrations of NO, NO2 and O3 with observations from a dense monitoring network, the initially estimated uncertainties could be reduced, and a posteriori emissions could be derived. The sensitivity of the results to the chemical mechanisms in the model was also evaluated.

We first evaluated the performance of the adjusted PEE simulations in modelling NO2 concentrations at the long-term monitoring sites using mean square error (MSE). It is a compact indicator of model performance whose merit is demonstrated in the following. The MSE is calculated as 1MSE=1n∑i=1nmodi-obsi2, where obsi is the observed value for a given averaging period i (e.g. an hour, a day, a month), modi is the corresponding simulation output and n represents the number of the available observations.

At each site, MSE is calculated for the hourly NO2 concentration output by each adjusted PEE simulation and the base run. Figure 4a shows a distinct trend of increasing MSEs with growing annual total NOx emissions, such that at most sites, the base run with input emissions at the upper end of the scale (see Fig. 2a) is outperformed by most of the PEE simulations. Although a single MSE does not differentiate between over- and underestimation, this clear positive association between MSEs and NOx emissions suggests that NOx emissions are positively biased, both in the base emissions and in most members of the PEE. If NOx emissions were negatively biased in a considerable subset of the ensemble members, the MSEs would first decrease as emissions increased, until the absolute bias in the emissions reached a minimum. It is also evident that the base run is generally associated with larger errors at urban and traffic monitoring sites compared to other sites. Moreover, the increase in MSEs with increasing emissions is more rapid at these locations, resulting in a wider range of errors associated with the ensemble of simulations. This is an indication that the base emissions are larger in magnitude in the central areas (where these sites are situated; see Fig. 1) than in the periphery and are overestimated to a larger extent. Though spatially uniform scaling factors were applied within the study area, regions with higher base emissions would show larger variations in the perturbed emissions and thus model errors. This highlights a potential issue associated with spatially uniform perturbations to spatially non-uniform emissions.

Due to the important role of O3 in converting NO into NO2, the adjusted PEE simulations' performance in modelling the O3 concentrations was also evaluated. In other words, this was to ensure that the underlying chemical mechanisms were correctly modelled and that simulations with lower NOx emissions showed better agreement with NO2 observations for the right reasons. We calculated the MSEs in maximum daily 8 h mean (MDA8) O3 concentrations

The adjusted PEE simulations' performance in modelling the hourly O3 concentrations was also evaluated. The median MSEs in hourly O3 concentrations of all simulations are dominated by the mMSE, and their association with the input NOx emissions is substantially weaker than the association between median MSEs in MDA8 O3 concentrations and NOx emissions.

The MSE can be mathematically decomposed into the sum of three terms (Solazzo and Galmarini, 2016): 2MSE=mod‾-obs‾2+σmod-rσobs2+σobs2×1-r2, where σmod and σobs represent the standard deviation of the modelled and observed values, respectively, and r is the Pearson's correlation coefficient between model outputs and observations. The first term in Eq. (2) represents the bias component of model error, and it is largely introduced by external forcings, for example, input emissions and boundary conditions. The second term is the variance error which is associated with the processes resolved in a model. A trade-off between bias and variance, in other words, accuracy and precision, is often inevitable in complex models (Sun and Archibald, 2021). The last term, by definition, represents the proportion of the observed variance unexplained by the model. It summarises all non-systematic errors, including the noise and inherent variability (e.g. due to turbulence closure) in the observations as well as errors arising from the linearisation of non-linear processes and is referred to as the minimum achievable MSE (mMSE). The MSE is thus a well-rounded metric suitable for operational model evaluation, and its decomposition provides indications of possible sources of model error (Solazzo and Galmarini, 2016).

The values of MSEs shown in Fig. 4 were decomposed according to Eq. (2), and the term with the largest contribution to the MSE associated with each simulation at each site is shown in Fig. 5. There are striking differences in the attribution of error for hourly NO2 and MDA8 O3. Variance errors have the largest share in the MSEs in hourly NO2 concentrations associated with most of the adjusted PEE simulations at over half of the reference sites. At another 1/4 of the sites, all of which are urban or traffic monitoring sites, bias accounts for most of the errors in the bulk of the simulations. This is another indication of a higher degree of overestimation in the central areas in the base emissions and subsequently in many members of the adjusted PEE. For most sites (regardless of the site type), simulations using the lowest annual total NOx emissions are associated with MSEs that are mostly made up by the mMSE.

At the urban and traffic monitoring sites where the MSEs in hourly NO2 associated with the ensemble simulations are mainly made up of the bias error, the bias also happens to be the largest term in most MSEs in MDA8 O3 (Fig. 5b). At most other sites (including all suburban, clean and regional background sites), however, the MSEs in MDA8 O3 are dominated by the mMSE, irrespective of the input NOx emissions. This explains the weaker association between the total MSEs in MDA8 O3 and the emissions revealed in Fig. 4b, as the mMSE is much less dependent on model inputs. A further breakdown of the mMSEs (Fig. S4) reveals that variances in the observations of MDA8 O3 are substantially higher than those in the observed hourly NO2 concentrations. Despite considerably better correlation between model outputs and the observations, these large variances in the observations result in mMSEs in MDA8 O3 concentrations that are only moderately smaller than those associated with hourly NO2 concentrations. Meanwhile, the MSEs in O3 are considerably lower than those in NO2; the largest share of the mMSEs in the former is thus explained. Because of this dependence on the observations and the weaker connection to external drivers, the mMSE is often considered the least concerning component of model error (Solazzo and Galmarini, 2016).

As the distributions of the 33 MSEs (i.e. one for each long-term monitoring site) associated with individual PEE simulations are mostly non-Gaussian, we used the median MSE to represent a simulation's average performance for a certain pollutant across all sites within the modelling domain. A breakdown of the median MSEs (Fig. 6) is consistent with the findings described above: with more accurate (lower) input NOx emissions, a simulation's average performance for hourly NO2 can be improved substantially to a point that the remaining model error consists mostly of the non-systematic mMSE. The average performance for MDA8 O3 is less strongly associated with NOx emissions, as it is dominated by the mMSE in the majority of the PEE simulations. These associations between median MSEs and input emissions are also tested using simple linear regression (Fig. S5). Though both regression models are statistically significant (p value < 0.001), more variability in the modelled hourly NO2 is explained, compared to that in the modelled MDA8 O3. The scatter of the points around the regression line reveals the variations in model performance with varying mix of source sectors, given similar strengths of total emissions (note that these are represented on an ordinal scale in Figs. 4–6). This also demonstrates the importance of using network observations as constraints. With spatially uniform perturbations, it is likely that several different combinations of emission parameter values result in similar concentrations at a particular location. The risk of constraining the parameter values to just one of the possible combinations is reduced, when observations that sample a wide range of local environments are used.

On account of the analysis above, we only used observations of NO2 to constrain NOx emissions, which is also in line with numerous top-down emission optimisation studies using satellite observations of column NO2 (Lamsal et al., 2011; Martin et al., 2003; Napelenok et al., 2008; Qu et al., 2017). Figure 7 shows the average performance for hourly NO2 of individual PEE simulations against the value set for each emission parameter in Table 1. Figure 7f reveals a strong positive correlation between the median MSE in hourly NO2 of a simulation and the input transport sector NOx emissions. Simulations with a median MSE within the first quartile are forced with transport emissions 6 %–65 % of those in the base emissions. This range continues to reduce (from both ends) with improving simulation performance, such that when the median MSE falls below the 10th percentile, the corresponding traffic emissions are only 7 %–43 % of those in the base emissions. The range determined by the top-performing 5 % of the simulations (i.e. with a median MSE within the fifth percentile) remains the same, while it can be further constrained one-sidedly to 7 %–18 % if only the top 1 % were considered. However, as the top 1 % of a 140-member ensemble contains (maximally) two simulations, the difference in whose average performance is marginal (≤ 1.5 %), this range was considered not robust.

In addition to the total magnitude, the night-time fraction of the transport sector NOx emissions could also be constrained (Fig. 7g). Instead of hourly NO2, performance of the adjusted PEE simulations was evaluated against the observed annual mean diurnal variations of NO2 at each long-term monitoring site. The median of all 33 MSEs in the diurnal profiles at individual sites modelled by a simulation was also used to represent its average performance in modelling the different diurnal profiles observed within the study area (see Fig. S3). Though not as evident as the case of total transport emissions, the range of the night-time fraction also becomes narrower with improving model skill. Amongst simulations with a median MSE in diurnal NO2 profiles within the 15th percentile, 11 %–29 % of the transport emissions occur at night (in contrast to the 9 % in the base emissions). In the top 5 % of the simulations, this fraction varies between 15 % and 25 %.

Emission parameters for other source sectors could not be constrained with strong confidence (Fig. 7a–e), as the ranges of parameter values only start to noticeably differ from the full uncertainty range when the median MSE in hourly NO2 falls within the 10th percentile or even below. The residential sector is the smallest source sector of NOx in the base emissions (see Fig. 2a). A source apportionment of the base run reveals that its contribution to the annual mean NOx concentrations at individual long-term monitoring sites varies between 3 % and 8 % (Fig. S6). It can thus be expected that even a 150 % increase, i.e. its upper bound of uncertainty (see Table 1, “Adjusted PEE” column), is not sufficient to cause substantial changes in the simulation performance for NOx, based on which the emissions can be constrained.

Interestingly, Fig. S6 also reveals that at most sites, the contribution of NOx base emissions from the power sector to the modelled annual mean NO2 concentrations is even smaller than that from the residential sector, though the emissions are over 2 times higher (see Fig. 2a). This is attributable to the nature of the power sector, which is characterised by a few large point sources emitting at elevated levels. These sources undergo greater dispersion before reaching the surface-based monitoring sites (compared to ground-level emissions); thus their impact at most sites is small. As an example, the annual mean NOx concentration simulated at the site NZG is very similar to that modelled at the site GC, yet less than 3 % of this concentration stems from NOx emissions from the power sector, as opposed to the 11 % share at GC (Fig. S6). This can be explained by Fig. S7 in which the concentration resulted from power emissions is further apportioned to each grid cell. It is evident that the concentration at GC is predominantly contributed by the grid cell directly to its west. In fact, this grid cell contains the highest power emissions of NOx within the study area. In comparison, the site NZG is located at some distance from other grid cells with relatively large power sources. The contributions of these grid cells to the NOx concentrations at NZG are smaller. The differences are about 1 order of magnitude, as the contributions are log-transformed in the figure.

The fact that emissions released at higher levels are not well sampled by the existing surface-based monitoring sites also applies for those of NOx from industrial sources (emitted at up to 152 m). In addition, emissions from both source sectors were separated into two parameters and perturbed simultaneously (i.e. in an uncorrelated matter) for varying vertical distributions. The uncertainty range of the sum of the two parameters, i.e. the total power or industrial emissions, was thus smaller than the individual uncertainty ranges and may not be sufficiently large to capture the actual emissions.

The short-term, independent SNAQ measurements of NO2 were also used to constrain the emission parameters following the same approach (Fig. S8). Similarly, adjusted PEE simulations showing better performance for hourly NO2 concentrations and mean diurnal variations of NO2 concentrations (over the measurement period) at SNAQ sites are associated with lower transport sector total emissions, but a higher percentage of these emissions occur at night. In the top-performing 5 % of the simulations, total traffic emissions are 7 %–43 % of those in the base emissions, while the night-time fraction varies between 15 % and 26 %. The fact that the uncertainty ranges constrained by the short-term SNAQ measurements are consistent with those using long-term reference measurement demonstrates the robustness of both this approach and the findings. This also supports the use of low-cost sensors for this particular application, as they are more affordable for deployment in a dense network.

According to the base emissions, total NOx emissions from the transport sector were 2.1 × 105 Mg within the study area which extends over most of Beijing and parts of Hebei Province in 2013, 9 % of which occurred during 00:00–05:00 LT. Based on the top 5 % of the adjusted PEE simulations for modelling NO2 hourly concentrations and diurnal profiles, we found that transport NOx emissions were likely to have decreased to 1.5–9 × 104 Mg (i.e. a 57 %–93 % reduction) in 2016, and the night-time fraction was between 15 % and 25 %.

An exact comparison of these results with findings of previous studies is not possible, as the emissions were investigated over different spatial and/or temporal scales. However, it is possible to compare the relative changes in emissions. Biggart et al. (2020) found that total NOx emissions rates from the same a priori emissions inventory (and all source sectors) were 1.8 times higher than those from an optimised emissions inventory (with which ADMS-Urban simulated NOx and NO2 concentrations that were in much better agreement with the corresponding observations), though their investigation was focused on a small domain in urban Beijing and the duration of the APHH-Beijing winter campaign only. They also found that the modelled mean diurnal variations in NO2 concentrations at most sites could be substantially improved when the night-time fraction of NOx emissions was increased by 25 % and 50 %. Nonetheless, as mentioned in Sect. 2.2, this was defined as NOx emitted between 23:00 and 06:00 LT from all source sectors.

Squires et al. (2020) compared NOx emission rates in the same a priori emissions with flux measurements made from a tower in central Beijing (where SNAQ39 was also deployed; see Fig. 1) during the APHH-Beijing winter campaign. The study area was also smaller than that in this work, as the flux footprint (i.e. the upwind source area of the measured fluxes) was on average within 2 km (maximal 7 km) of the tower. Compared to the measured fluxes, the emissions rates were found to be overestimated by a mean factor of 9.9. They further considered emissions only from the transport and residential sectors (as no industrial or power sources were identified within the average footprint) and reduced these by 30 %, yet these were still on average 3.3 times higher than the fluxes. They also found much smoother diurnal variations in the NOx fluxes compared to those in the emission rates, indicating that the night-time fraction was underestimated in the latter.

In the standard MEIC v1.3 (from which the a priori emissions were downscaled and re-gridded), annual NOx emissions from the transport sector were estimated to be 1.05 × 105 Mg in Beijing and 6.59 × 105 Mg in Hebei Province in 2013. In 2016, these figures decreased to 8.87 × 104 and 5.68 × 105 Mg, respectively, corresponding to reductions by 15.5 % and 13.8 %, which are substantially lower than the reductions reported in this work. A slightly larger reduction of 20 % (from 1.44 × 105 Mg in 2013 to 1.15 × 105 Mg in 2016) was estimated for vehicle (including on- and off-road vehicles) sources of NOx in Beijing by Cheng et al. (2019), who used a bottom-up emissions inventory for Beijing which was compiled from the county level and associated with finer spatial resolutions than MEIC established from the province level. They also concluded that vehicle emission control measures contributed the most to the total reductions in NOx emissions over this period. According to the China Vehicle Environmental Management Annual Reports, NOx emissions from on-road vehicles were 7 × 104 Mg in Beijing and 5.2 × 105 Mg in Hebei Province in 2013. These fell by 14 % and 10 %, respectively, to 6 × 104 and 4.7 × 105 Mg in 2016, despite an increase in vehicle ownership (Ministry of Environmental Protection of the People's Republic of China, 2014, 2017).

The downward trends in traffic sources of NOx between 2013 and 2016 found by this study, and the studies and reports mentioned above contrast with the results in Xue et al. (2020). Based on yet another bottom-up emissions inventory for Beijing, they showed that mobile (i.e. on-road and off-road vehicles) sources of NOx increased from 9.6 × 104 to 1.1 × 105 Mg over the same period. Though they found a 20 % reduction in the total anthropogenic NOx emissions from 2013 to 2016, this was primarily attributed to optimised energy structure in the industry, power and residential sectors. In this study, we found that amongst the top-performing 5 % of the adjusted PEE simulations, the reduction in total NOx emissions from the base emissions varies between 59 % and 74 % (due to reductions from transport sources only, with other sources fixed). These conflicting findings again highlight the presence of inherent uncertainties in emissions inventories, the impact of which on the results of this work is discussed in the following, along with the impact of other sources of uncertainty.

Two types of uncertainties may be associated with the base emissions. Inherent uncertainties due to underlying emission factors and activity rates were estimated to be ±31 % for NOx emissions in MEIC (Cheng et al., 2019), which are smaller than the uncertainty ranges in Table 1, suggesting that the trends in (real-world) emissions from 2013 to 2016 represent a larger source of uncertainty when using the 2013 base emissions for 2016 simulations. Additional uncertainties may have been introduced when the standard MEIC v1.3 was downscaled and re-gridded to the a priori emissions inventory used in this work. As an example, Zheng et al. (2017) compared MEIC with another emissions inventory with a much larger share of point sources (which were allocated directly to grid cells) and different sets of spatial proxies to allocate non-point sources. They found that NOx emission fluxes from the most populated grid cells in Hebei Province were overestimated by 46 %–140 % in MEIC, mainly driven by spatial proxies that over-allocated industrial emissions to urban areas. Such uncertainties may have contributed to the spatial inhomogeneities of the biases in the base emissions revealed in Fig. 4 and, to a certain extent, propagated into the derived emission estimates, as spatially uniform perturbations were applied when constructing the adjusted PEE. Hence, biases in the spatial distribution of emissions may also be present in the a posteriori estimates, despite improvement in terms of the total magnitude. In comparison, the propagation of inherent uncertainties in the base emissions is of less concern. Though most emission parameters were defined relative to the corresponding values in the base emissions for an efficient perturbation, their uncertainty ranges were ultimately constrained solely by the observations.

Uncertainties in the observational constraints are also twofold. While those due to measurement errors are most likely small, as demonstrated by the consistency in the results derived using two independent sets of observations, the underrepresentation of the existing observations of power and industrial sources prohibited an update of emission strengths from these sources.

Another source of uncertainty is the input lateral boundary conditions which include meteorology and background pollution levels. The impacts of uncertainties in the input meteorological observations (i.e. due to measurement errors) are minimal, as reported in Yuan et al. (2021) using simulations forced with perturbed meteorological data. The effect of uncertainties associated with different background concentrations of NOx and O3 on the constrained NOx emissions is discussed next in the context of uncertainties in the underlying chemical mechanisms in ADMS-Urban.

The chemical partition of NOx emissions into NO2 concentrations by the model represents a potentially important source of uncertainty. Many studies that infer NOx emissions from satellite observations of the tropospheric NO2 column assumed an accurate representation of NOx chemistry in the CTM used. However, Valin et al. (2011) demonstrated the presence of biases in the modelled NO2 column that are dependent on the horizontal resolution of the model, which has implications on the inference of NOx emissions when matching the modelled column to satellite observations. These biases result from an inaccurate representation of NO2 lifetime (and thus concentration) at coarse resolution, which is determined primarily by OH concentration in daytime, which, in turn, has a non-linear dependence on NOx concentration.

While NO2 rapidly interconverts with NO in the presence of sunlight via NO2 photolysis and reactions of NO with O3 and peroxy radicals, it is also oxidised by hydroxyl radicals (OH) to nitric acid (HNO3), which can then be removed from the atmosphere via wet/dry deposition. At night, NO oxidation continues, but it cannot be converted from NO2 as no photolysis takes place. With very low concentrations of OH, NO2 is mainly oxidised by O3 to form the nitrate radical (NO3), which further reacts with NO2 and reaches an equilibrium with dinitrogen pentoxide (N2O5). NO2 also hydrolyses on aerosol surfaces to form nitrous acid (HONO). NO3, N2O5 and HONO are described as night-time reservoirs of NOx, as they can regenerate NO or NO2 after sunrise (Seinfeld and Pandis, 2016). These reservoir species are absent in the semi-empirical chemical mechanism in ADMS-Urban described in Sect. 2.3.

The SNAQ measurements which included NO allowed for an evaluation of the NOx photolytic chemistry in ADMS-Urban. Han et al. (2011) demonstrated the presence of a strong linear relationship between NOx and NO during night-time and between NOx and NO2 during the day in observations from Tianjin, another megacity not far from Beijing. Following this approach, simple linear regression models were fitted to the SNAQ-measured hourly averaged mixing ratios of NO and NO2, as a function of the corresponding mixing ratios of NOx. In addition, a linear function was fitted between log-transformed mixing ratios of O3 and (non-logarithmic) mixing ratios of NOx due to a negative correlation between the two (Fig. S9a). Additional linear functions were also fitted to daytime or night-time data only. Data from the individual SNAQ sites were not differentiated because of the short duration of operation at each site. Consistent with findings by Han et al. (2011), a good linear correlation can be found between the daytime mixing ratios of NO2 and NOx, while there is a strong correlation between night-time mixing ratios of NO and NOx. The lowest coefficients of determination are calculated for O3 as a function of NOx, which is consistent with the finding of Figs. 4 and 5 that O3 concentrations are less strongly dependent on NOx emissions (Fig. S9b).

We then fitted linear models to the corresponding output (i.e. for the same locations and time frame) from the top-performing 5 % of the adjusted PEE simulations (see Sect. 3). These are compared with models fitted to the SNAQ measurements with respect to the slope, as it indicates the number of changes in NO, NO2 and (log-transformed) O3 as NOx increases/decreases, which is directly related to the input NOx emissions. As can been seen from Fig. 8a, the modelled slopes of the linear models between NO2 and NOx are higher, whereas modelled slopes for NO as a function of NOx are lower compared to those observed, irrespective of the time of day. The discrepancies between modelled and observed slopes for O3 as a function of NOx are relatively small. This suggests that with similar concentrations of NOx as observed, the top-performing adjusted PEE simulations tend to overestimate NO2 while underestimating NO. In other words, lower NOx emissions may be needed for the model to simulate NO2 concentrations that are consistent with the observations. This suggests that the constrained emission estimates of NOx are indeed sensitive to the NOx photolytic chemistry in ADMS-Urban and, in this case, may be low-biased.

There are several possible explanations for the model's tendency to partition more (less) NOx into NO2 (NO). As described in Sect. 2.3, the reaction with O3 (Reaction R4) and reactions with the HO2 or RO2 (Reaction R2) are the two pathways of NO oxidation to form NO2 in ADMS-Urban. Organic radicals are produced via Reaction (R1) in which ROC is defined as the proportion of TVOC that is reactive and calculated by multiplying the TVOC concentrations with a reactivity coefficient. In ADMS-Urban version 4.2 (used in this study), a coefficient of 0.1 is adopted. It is set to 0.05 in the most up-to-date version 5 (Cambridge Environmental Research Consults Limited, 2017, 2020). As a secondary pollutant, modelled O3 concentrations depend on the input background levels, for which there is no widely accepted definition. To investigate the sensitivity of the model output to these two NO oxidation pathways, a series of sensitivity simulations were performed. All simulations were input with the same NOx (and VOC) emissions as R97, the adjusted PEE simulation with the best performance in simulating hourly NO2 concentrations at the long-term monitoring sites. Again, simple linear regression models were fitted between hourly mixing ratios of NO, NO2, (log-transformed) O3 and NOx output by these simulations.

Figure 8b shows that different definitions of background NO2 and/or O3 (see Table S3) indeed have an impact on the modelled NO-to-NO2 conversion. For example, in the sensitivity simulation S5 input with the lowest background levels of NO2 and O3, the modelled slopes for NO2 as a function of NOx are considerably lower than the corresponding slopes modelled by R97 and thus closer to the observed slopes. Also, the slopes between NO and NOx associated with outputs from S5 are higher than those found in outputs from R97 and agree better with the observed slopes. However, it is important to underscore that this does not suggest that the 10th percentile concentration is most representative of the background concentrations of NO2 and O3. It simply highlights the impact of the input background concentrations of reactive pollutants on the model outputs of relevant species (and thus on the emission estimates inferred on the basis of these model outputs), which can be comparable to the impact of varying the input NOx emissions (amongst the top-performing 5 % of the adjusted PEE simulations) shown in Fig. 8a. This calls for further research into appropriate definitions for background levels of NOx and O3 within a vast and heterogeneous urban area like the modelling domain in this study. Also, it is worth noting that the modelled chemistry is also influenced by the input background levels of NO. However, the specific sensitivities were not investigated, as the upwind concentrations extracted from the CAMS reanalysis dataset are the only set of NO observations available long-term.

The effect of organic radicals on the partition of NOx between NO2 and NO is shown in Fig. 8c by varying the concentrations of ROC. As explained above, ROC concentrations are controlled by both the TVOC concentrations (that result from the input emissions and background levels of VOC) and a reactivity coefficient which was set to 0.1 in R97 (as with other adjusted PEE simulations). With fixed TVOC concentrations, using a coefficient of 0.2 doubles the ROC available to produce HO2 and RO2, leading to an even more pronounced overestimation of NO2, accompanied by an underestimation of NO. In contrast, halving the ROC concentrations by using a coefficient of 0.05 partitions less of the NOx emitted into NO2. This highlights that the emissions and background concentrations of VOC (which are not evaluated in this study due to a lack of observations) also have an impact on the modelled NOx photolytic chemistry and thus the a posteriori emission estimates of NOx. It is also worth noting that biogenic VOCs are likely underestimated in the current simulations, as these are only represented by one of the eight species (i.e. isoprene) output by the CAMS reanalysis product used to approximate the background levels of TVOC and are not represented at all in the base emissions (which include anthropogenic sources only). Despite having low concentrations in the study area (compared to anthropogenic VOCs) (Mo et al., 2018), they are associated with high radical production and thus O3 creation potentials. Unlike background concentrations of NOx and O3, however, the effect of VOCs on the modelled NOx–O3 chemistry is restricted to daylight hours, as they only produce radicals in the presence of solar radiation in the model.

Another factor that affects the modelled NOx photolytic chemistry is the f-NO2 in the input emissions (see Sect. 3.1). Although not investigated with sensitivity experiments, it can be expected that a higher f-NO2 would result in higher NO2 concentrations and lower NO concentrations simulated by the model, thus further increasing the discrepancies between the modelled and observed slopes of linear functions, whilst a lower f-NO2 would have the opposite effect.

We developed a novel approach to update a priori emission estimates using ground-based network measurements as constraints and an ensemble of forward simulations which are input with a PEE. Using this approach, we were able to update the transport sector NOx emissions in Beijing from a 2013 emissions inventory for the year 2016. The updated emissions are substantially lower with a higher proportion occurring at night-time and are broadly consistent with findings of several previous studies. It would be possible to also update emissions from other (non-negligible) source sectors, provided that appropriate measurements were available.

As with existing emission optimisation techniques, this approach is sensitive to the chemical mechanisms in the underlying model, the uncertainties of which can be propagated into uncertainties in the emission estimates. Nonetheless, this approach has several unique advantages. Compared to inverse modelling techniques, the construction of a PEE and the forward simulations is rapidly executable. Even when the Gaussian dispersion model used in this study is replaced with a CTM for more explicit representations of chemistry, the efficiency can be maintained via parallel computing. Also, surface-based measurements of ambient concentrations are used as constraints, which are readily available and closer to the sources of emissions than the satellite-based measurements. This proximity to emission sources may be particularly important for capturing the high temporal and spatial variability of highly reactive species. For example, Qu et al. (2021) found that in comparison to surface concentrations, the NO2 column showed a muted response to the step decrease in NOx emissions in the United States during the COVID-19 crisis. Most importantly, this approach allows for an update of emissions by source sector, which is more relevant for policy interventions than total emissions, as they directly reflect the (in)effectiveness of the corresponding pollution control measures. Hence, we believe that this approach, particularly combined with low-cost sensors, has great potential in providing timely updates of emissions in regions undergoing rapid changes, where emissions inventories may be biased or outdated as soon as they have been compiled, and computing facilities may be limited.