Inverse modeling of NOx emissions over eastern China : Uncertainties due to chemical 1 nonlinearity 2

8 Satellite observations of nitrogen dioxide (NO2) have often been used to derive nitrogen oxides 9 (NOx=NO+NO2) emissions. A widely used inversion method was developed by Martin et al. 10 (2003). Refinements of this method were subsequently developed. In the context of this inversion 11 method, we show that the local derivative (of a first-order Taylor expansion) is more appropriate 12 than the “bulk ratio” (ratio of emission to column) used in the original formulation for polluted 13 regions. Using the bulk ratio can lead to biases in regions of high NOx emissions such as eastern 14 China due to chemical nonlinearity. Inverse modeling using the local derivative method is applied 15 to both GOME-2 and OMI satellite measurements to estimate anthropogenic NOx emissions over 16 eastern China. Compared with the traditional method using bulk ratio, the local derivative method 17 produces more consistent NOx emission estimates between the inversion results using GOME-2 18 and OMI measurements. The results also show significant changes in the spatial distribution of 19 NOx emissions especially over high emission regions of eastern China. We further discuss a 20 potential pitfall of using the difference of two satellite measurements to derive NOx emissions. Our 21 analysis suggests that chemical nonlinearity needs to be accounted for and that a careful bias 22 Atmos. Meas. Tech. Discuss., doi:10.5194/amt-2016-140, 2016 Manuscript under review for journal Atmos. Meas. Tech. Published: 2 May 2016 c © Author(s) 2016. CC-BY 3.0 License.

analysis is required in order to use the satellite differential method in inverse modeling of NOx emissions.

Introduction
Nitrogen oxides (NOx=NO+NO2) play an important role in tropospheric photochemistry of ozone and secondary aerosol formation.There are normally emitted from both anthropogenic (e.g., fossil fuel combustion) and natural sources (e.g., lightning, soil, and wild fire).The rapid economic growth in East Asia has led to a significant increase of energy consumptions and thereby anthropogenic NOx emissions during last two decades (Ghude et al., 2009;Gu et al., 2013;Ma et al., 2006;Mijling et al., 2013;Richter et al., 2005;Stavrakou et al., 2008;van der A et al., 2006;Zhang et al., 2007).
The traditional bottom-up emission inventory rely on detailed information about sources and emission factors and therefore can have large uncertainties especially in countries such as China where the emission information is incomplete (Streets et al., 2003).In recent years, satellite measurements of nitrogen dioxide (NO2) from multiple instruments, including Global Ozone Monitoring Experiment (GOME), Scanning Imaging Absorption Spectrometer for Atmospheric Chartography (SCIAMACHY), Ozone Monitoring Instrument (OMI) and GOME-2, have been widely used to derive top-down estimation of NOx emissions as an alternative to the bottom-up emission inventory (Gu et al., 2014;Jaegle et al., 2005;Lamsal et al., 2011;Lin et al., 2010;Stavrakou et al., 2008;Zhang et al., 2012).
Inverse modeling of NOx emissions, incorporating model simulations and satellite observations, can help reduce uncertainties in bottom-up inventories especially in China (Lin et al., 2012;Mijling and van der A, 2012;Zhao and Wang, 2009).Martin et al. (2003) developed the first monthly inversion method by scaling the bottom-up emission inventory with top-down constraints from GOME NO2 column measurements.Zhao and Wang (2009) improved the method by carrying out the emission inversion iteratively on a daily basis.Gu et al. (2014) further refined this method by including column NO2 retrieval in inverse modeling, which removes the biases introduced by inconsistency between NO2 profiles in the retrieval and inverse modeling.Lin et al. (2010) developed a method of inverse modeling using column NO2 difference between GOME-2 and OMI.
In the emission inversion problem, the a posteriori emission is estimated by modifying the a priori emission with a term proportional to the difference between the simulated and observed column density, which can be expressed as: where E is the a posteriori emission, Ea is the a priori emission used in model simulation, Ωs is satellite observed column, Ωm is model simulated column and α is the correction rate.In the original formulation by Martin et al. (2003) and all subsequent studies using this or some variants of this method (Jaegle et al., 2005;Lamsal et al., 2011), α is calculated as the ratio between the a priori emission and simulated column (referred to as the bulk ratio hereafter): Note that this formulation has an implicit assumption that the emission is linearly proportional to the column density.However, previous studies of the emission and column trends suggested that the nonlinearity between NOx emission and tropospheric NO2 column is non-trivial (Gu et al., 2013;Lamsal et al., 2011;Lu and Streets, 2012;Stavrakou et al., 2008).This nonlinear relationship is mainly due to the nonlinear photochemical feedbacks between NOx and OH (the increase in NOx promotes OH production and reduces its lifetime in the low-emission condition, but suppresses OH production and increases its lifetime in the high-emission condition).Figure 1 shows tropospheric NO2 column densities as a function of surface emissions at GOME-2 and OMI overpass time over eastern China simulated in the Regional chEmical and trAnsport Model (REAM) (model details are given in section 2.2).Clearly tropospheric NO2 column is not linearly proportional to NOx emission, which was assumed in Eq. ( 2).The difference in the nonlinearity in the ratio of emission to column NO2 between GOME-2 and OMI overpass time reflects in part the difference in the relative importance between transport and chemistry at different time of a day.
In this study, we examine if considering the nonlinear ratio between NOx emissions and NO2 columns can improve the inverse modeling results and reduce the discrepancies in emission estimates using different satellite observations.Applying Taylor expansion to Eq. ( 1), we define a local derivative in place of the bulk ratio (Eq.( 2)).We implement the local derivative of emission to column NO2 ratio in the REAM model and examine its effects on the inverse modeling estimates of surface anthropogenic NOx emissions with GOME-2 and OMI measurements over China in August 2007.The new inversion results are compared to those using the bulk ratio method and the satellite differential method (Lin et al., 2010) and the implications for NOx inverse modeling are discussed.

Satellite data
We use the measurements from GOME-2 and OMI instruments in this study.Both instruments are nadir-viewing spectrometers (Boersma et al., 2004;2011;2007).The OMI instrument was launched onboard the Aura satellite in July 2004, and it has a spatial resolution of 24 × 13 km 2 at nadir (Levelt et al., 2006).The GOME-2 instrument was launched onboard the MetOp satellite in The error in the retrieval of NO2 tropospheric VCD is determined by those in total slant column density (SCD), stratospheric SCD, and tropospheric air mass factor (AMF) estimation.In this study, we use total and stratospheric SCD errors from KNMI DOMINO2 and TM4NO2A products, and compute the uncertainty of tropospheric AMF estimation following the KNMI algorithm.The details of error analysis were described by Boersma et al. (2004;2011;2007) and Hains et al. (2010).In general, the uncertainties of total and stratospheric SCD estimations are small (<0.7×10 15 molec.cm -2 ) relative to high tropospheric VCDs over eastern China (Zhao and Wang, 2009).The uncertainty in tropospheric AMF comes from surface albedo, cloud fraction, cloud pressure, and profile shape.The uncertainty from a priori profile can lead to ~10% error in tropospheric VCD retrievals.The total uncertainty of an individual retrieval is up to 50% over highly polluted eastern China for both OMI and GOME-2.

Inverse modeling method
As discussed in the introduction, the bulk ratio (α) used in the traditional method is based on the assumption of a linear relationship between NOx emission and tropospheric column NO2, which is only accurate under a low emission condition.As shown in Figure 1, using the bulk ratio for inverse modeling overlooks the nonlinear chemical feedback of NOx lifetime at different satellite overpass time, which not only affects the accuracy of inverse modeling results but also leads to inconsistency between emission inversions using different satellite measurements.To account for the nonlinearity, we apply Taylor expansion to Eq. ( 1) and obtain a local derivative ratio (α*) to replace Eq. ( 2): where ΔEa is the change of the a priori emission and ΔΩm is the change of model simulated column.(1) to estimate inversed NOx emission over eastern China in August 2007 with either GOME-2 or OMI observations.The results from the local derivative method are compared with those using the bulk ratio method in later sections.
The uncertainties of the a posteriori emissions come from those in a priori and top-down emission estimates.Uncertainties in top-down emission estimates are derived from those in tropospheric NO2 VCD retrievals and model simulations.The retrieval uncertainty is discussed in section 2.1.The uncertainty of model simulation is estimated at 30% and that of the bottom-up inventory is ~60% over China (Zhao and Wang, 2009).The overall uncertainty of the a posteriori emission is typically in the range of 20-40% over polluted eastern China.

Comparisons between α and α*
Figure 2 compares the relative difference between local derivative ratio (α*) and bulk ratio (α) values over eastern China for August 2007, at GOME-2 and OMI overpass time, respectively.For both satellites, α* is higher (>20%) than α over most low-emission rural regions but lower (-20~-60%) than α over most high-emission regions including eastern coastal areas and Sichuan Province.For emission estimates, the inversion biases over high emission regions tend to be more important than rural regions.The high bias of α relative to α* implies that top-down NOx emission estimates tend to over-correct for a given difference between observed and simulated column NO2.
We note here that the inversion bias can be either positive or negative depending on the column difference.

NOx emission inversion consistency between OMI and GOME-2
Another way to look at the effects of correction biases using the bulk ratio relative to local derivative method is to compare the inversion estimated NOx emissions using OMI and GOME-2 measurements.Gu et al. (2014) showed that inversion results using standard DOMINO products of GOME-2 tend to be higher than OMI due possibly to a bias in the TM4 NO2 profiles used in GOME-2 retrievals.Coupling this tendency of a high retrieval bias of GOME-2 with the different sensitivities discussed in the previous paragraph implies that the bulk ratio formulation would tends to overcorrect and estimate higher NOx emissions using GOME-2 than OMI measurements in polluted regions.
The bias expectation is confirmed in Figure 3.While the inversion results using DOMINO GOME-2 and OMI products show in general higher NOx emissions in the former, the difference between the inversions using two satellites is smaller using the local derivative than bulk ratio method, particularly over high emission regions.For example, we compare the emission estimates between the bulk ratio and local derivative methods in three largest megacities in China (e.g. Beijing, Shanghai and Guangzhou).Using the bulk ratio method, the relative difference of inversion emission estimates of using two satellites products relative to OMI results are 54.2%, 70.5% and 55.6% for the three megacities, respectively.The local derivative method reduces the corresponding relative difference down to 9.0%, 5.3% and 13.5%, respectively (Figure 3 and Table 1).These results demonstrate that a significant fraction of the discrepancy in inversed NOx emissions between different instruments can be attributed to neglecting the chemical nonlinearity in the traditional bulk ratio method and that improvement can be achieved with the local derivative method that we proposed in this study.The remaining discrepancy between inversion emission estimates using GOME-2 and OMI observations is likely due to NO2 profiles used in the retrieval and possible systematic biases between the two satellite instruments (Gu et al., 2014).With column NO2 measurements from two satellites, Lin et al. (2010) developed a method to utilize information from two different satellites (referred to as the satellite differential method in this study) to improve the emission estimates.The formulation of the method is as follows,

Comparison of inversion results between using the bulk ratio method and the satellite differential approach
where Ej is the a posteriori emission, Ea is the a priori emission, ΩOMI is OMI observed NO2 column, ΩGOME-2 is GOME-2 observed NO2 column, Ωm1330 is model simulated NO2 column at OMI overpass time, Ωm0930 is model simulated NO2 column at GOME-2 overpass time, t is the time gap between two satellite overpass time, and τ is the lifetime of NOx.The derivation of this method is analogous to the bulk ratio method (Lin et al., 2010) and is not directly comparable to the local derivative method.We therefore compare the inversion results using the satellite differential method with those using the bulk ratio method.
In Table 1, the inversion emission estimates using the satellite differential method are compared with those using the bulk ratio method over eastern China in August 2007.As discussed in the previous section, the bulk ratio method leads to consistently high emission estimates using GOME-2 products compared to using OMI over high emission ratios, e.g., 50-70% in the three megacities.
The estimates using the satellite differential method are even lower than the bulk ratio inversion estimates using OMI products.The reason is that in our analysis ΩGOME-2/Ωm0930 is consistently larger than ΩOMI/Ωm1330 over eastern China (Figure 4).The mathematical analysis in Appendix A shows that consequently a posteriori emission estimates using the satellite differential method are consistently lower than that using the bulk ratio method with either GOME-2 or OMI products.
When the differences between the two satellite products are not well characterized, we find that the local derivative method is more robust relative to the bulk ratio and satellite differential methods (Table 1).

Conclusions
We show in this study that the nonlinearity of NOx chemistry implies that the local derivative (of a first-order Taylor expansion) is better suited in the inversion formulation developed by Martin et al. (2003) than bulk ratio, in agreement with Vinken et al. (2014a;2014b) and Castellanos et al. (2014).In this study, single grid cell based perturbation sensitivity calculation was used instead of previous domain-wide perturbations such that upwind emission changes do not affect local derivative estimates.The latter effects can be more appropriately accounted for using the iterative method by Zhao and Wang (2009) and Gu et al. (2014).In the context of the inversion formulation by Martin et al. (2003), we compared the bulk-ratio and local derivative methods in inversing modeling of anthropogenic NOx emissions over eastern China for August 2007.At the observation time of OMI and GOME-2, the local derivative ratios (α*) are smaller (-20~-60%) than the bulk ratios (α) over most high-emission regions, but are higher (>20%) than bulk ratios (α) over most low-emission rural regions.Over high emission regions, the inversion emission estimates using the local derivative method produces more consistent results between OMI and GOME-2 products than the bulk ratio method.In our work, the observed to simulated tropospheric column NO2 are consistently higher for GOME-2 than OMI over high emission regions of eastern China, leading to a consistent low bias in a posteriori emission estimates by the satellite differential method relative to those by the other methods.
Computationally, the local derivative ratio is more complex to compute than bulk ratio.The iterative method in the previous work by Zhao and Wang (2009) and Gu et al. (2014) can largely mitigate the biases introduced by the bulk ratio method, although the inversion convergence time will likely be reduced when using local derivative than bulk ratios.For certain applications such as deriving the timeline of emission reduction during the Beijing Olympics (Yang et al., 2011), reducing the inversion convergence time is critically important.Further studies are needed to quantify the improvements.
We obtain, Given a priori emission estimation of Eq. ( 2) and (4), Eq. (A2) implies that where   is the inversion emission estimate by the satellite differential method,  , is the inversion emission estimate by the bulk ratio method using OMI measurements, and  ,−2 is the inversion emission estimate by the bulk ratio method using GOME-2 measurements.
Atmos.Meas.Tech.Discuss., doi:10.5194/amt-2016-140,2016   Manuscript under review for journal Atmos.Meas.Tech.Published: 2 May 2016 c Author(s) 2016.CC-BY 3.0 License.The Taylor expansion formulation of Eqs.(1) and (3) is equivalent to the previous works byVinken et al. (2014a;2014b) andCastellanos et al. (2014), who accounted for the nonlinear chemistry effect by adding the ratio of relative emission to relative column NO2 changes, first introduced byLamsal et al. (2011), to the formulation of Eqs.(1) and (2).For inversion of ship emissions,Vinken et al. (2014b) computed averaged nonlinear factors for selected regions with perturbations proportional to model-observation column difference.They also added a correction of profile change in their formulation.For polluted regions, the effect of profile change is small(Zhang et al., 2016) and is not included in this study.In the previous studies, model sensitivities are computed with domain-wide emission perturbations.This approach introduces uncertainties since the column change in a given grid cell is affected by emission changes in both that grid cell and upwind grid cells; the effects of upwind grid cells are a complex function of emissions, chemistry, and transport.As inGu et al. (2013), we carried out single-cell based emission perturbation in the 3-D REAM model to compute the value of α*.We first archived the 3-D influxes of all model tracers at each time step in the standard simulation.In the perturbation simulation (15% of anthropogenic NOx emissions), the influxes of all model tracers for all grid cells were replaced with the archived values at each time step.Consequently the emission perturbation only affects NOx chemistry, out flux, and concentration in the same grid cell.Using the perturbation and standard simulation results, we computed the value of α* and applied it to Eq.

Figure 1 .
Figure 1.Simulated NO2 column density as a function of surface NOx emission at GOME-2 (black) and OMI (yellow) overpass time over eastern China for August, 2007.The dots are grid average data from REAM simulations binned by an emission interval of 1x10 14 molec cm -2 and the solid lines are from least-square polynomial regression results.

Figure 2 .
Figure 2. Relative difference between monthly mean local derivative ratio α* and bulk ratio α, defined as 1α/α*, for GOME-2 (left) and OMI (middle) in REAM simulations.The right panel shows NOx emission distribution estimated from GOME-2 observations by using the local derivative method over eastern China for August 2007.

Figure 3 .
Figure 3. Relative difference of inversion emission estimates for NOx with GOME-2 relative to OMI products using the local derivative and the bulk ratio methods: regional distributions over eastern China, respectively, for August 2007 (left), and the differences in three megacities (right).