A Satellite Data-Driven Framework to Rapidly Quantify Air Basin-Scale NOx Emission and Its Application to the Po Valley during the COVID-19 Pandemic

The evolving nature of the COVID-19 pandemic necessitates timely estimates of the resultant perturbations to anthropogenic emissions. Here we present a novel framework based on the relationships between observed column abundance and wind speed to rapidly estimate air basin-scale NOx emission rate and apply it at the Po Valley in Italy using OMI and TROPOMI NO2 tropospheric column observations. The NOx chemical lifetime is retrieved together with the emission rate and found to be 15–20 h in winter and 5–6 h in summer. A statistical model is trained using the estimated emission rates before the 5 pandemic to predict the trajectory without COVID-19. Compared with this business-as-usual trajectory, the real 2020 emission rates show two distinctive drops in March (−41%) and November (−35%) that correspond to tightened COVID-19 control measures. The temporal variation of pandemic-induced NOx emission changes qualitatively agree with Google and Apple mobility indicators. The overall net NOx emission reduction in 2020 due to the COVID-19 pandemic is estimated to be 21%.

and the spatial gradient is significantly less than many other polluted regions. The relative biases between OMI and TROPOMI NO 2 TVCD are assessed by comparing strictly collocated level 2 retrievals and given in Appendix A. OMI NO 2 TVCD is generally higher than TROPOMI in the cold season with monthly OMI-TROPOMI normalized mean bias (NMB) up to over 30%, whereas the TROPOMI TVCD is generally higher in the warm season with monthly OMI-TROPOMI NMB down to −20%.

Study domain and NO x emission inventories
The Po Valley air basin is delineated according to the boundary between the flat terrain in northern Italy and mountain ranges in the north, west, and south as well as the Adriatic Sea coastline in the east, as shown in Figure 1. The air basin area is 6.6 × 10 4 km 2 . The west-east length scale is ∼ 500 km, and the south-north length scale is ∼ 300 km, both larger than the square root of basin area (257 km) due to the irregularity of the basin shape. We contrast our derived monthly, air basin-scale NO x emission  (Miyazaki et al., 2019(Miyazaki et al., , 2020a. The NO x emissions from the JPL chemical reanalysis (Figure 1a) are constrained by assimilating O 3 , NO 2 , CO, HNO 3 , and SO 2 from the OMI, GOME-2, SCIAMACHY, MLS, TES, and MOPITT satellite instruments (Miyazaki et al., 2020a) and are considered to have the highest accuracy in spite of its relatively 80 low spatial resolution. The rest three are bottom-up emission inventories, including the Community Emission Data System (CEDS, McDuffie et al., 2020), Emissions Database for Global Atmospheric Research version 4.3.2 (EDGAR, Crippa et al., 2018), and the Peking University NO x (PKUNOx, Huang et al., 2017) inventories. The CEDS inventory is spatially resolved

Wind fields
We use wind fields gridded at 0.25 • ×0.25 • spatial resolution and hourly temporal resolution from the ERA5 reanalysis meteorology (Hersbach et al., 2020). The relevant ERA5 fields are spatiotemporally interpolated at each individual OMI/TROPOMI 90 level 2 observation. Previous observational data-driven emission inference studies represented horizontal advection of NO 2 (or similar short-lived tracers like SO 2 and NH 3 ) by 10m wind above the surface , 100m above the surface (Goldberg et al., 2020), vertically averaged wind from surface to 500m Liu et al., 2016;Goldberg et al., 2019a), or vertically averaged wind from surface to 1000m (Fioletov et al., 2017;Dammers et al., 2019). with stronger winds when higher altitudes are involved. The surface-1000m wind speed is almost twice as strong as the 10m wind, whereas the two intermediate options, the 100m wind and surface-500m wind are similar with a difference of 13%. The wind directions among those four options show much larger discrepancy, but only the wind speeds will be used in this study. The slopes labeled in the plot is from orthogonal regression, and r is correlation coefficient. Unit is m s −1 .

Construction of column-wind speed relationships by physical oversampling
A key step to estimate NO x emission from the observed NO 2 TVCDs is to construct the column-wind speed relationship by averaging column amounts over a range of wind speed intervals. Physical oversampling (Sun et al., 2018) provides a flexible way to spatiotemporally average satellite data with proper weighting and slice the data under different environmental conditions (e.g., wind speed). The averaged NO 2 TVCD ( Ω ) given sets of filtering criteria with respect to space (s), time (t), and other level 2 parameters (p) can be calculated as: Here j is the index of each level 3 grid cell at 0.01 • resolution, and j ∈ s includes all grid cells satisfying the spatial aggregation criterion s (e.g., within the boundary of an air basin). Ω i is NO 2 TVCD retrieved at level 2 pixel i. i ∈ t, p keeps only level 2 pixels satisfying time filtering criteria (e.g., within a calendar month) and parameter filtering criteria (e.g., wind speed at the 110 level 2 pixel within a certain interval). w i,j is the weight of level 2 pixel i at level 3 grid cell j and depends on the spatial response of pixel i at grid cell j as well as the retrieval uncertainty at pixel i Sun et al., 2018).
The column-wind speed relationship for an air basin over a certain time interval is an array of averaged NO 2 TVCDs over different wind speed intervals (every 0.5 m s −1 in this study):

115
where W is the horizontal wind speed that is interpolated at level 2 pixels and representative of horizontal advection. The four wind speed options shown in Figure 2 are tested in this study. Figure 3 shows the column-wind speed relationships for OMI and TROPOMI over the Po Valley in December 2018-November 2020 grouped into four seasons. TROPOMI provides 2-3 times more coverage than OMI, as indicated by the dot sizes, but ∼50 times more valid level 2 pixels due to much finer spatial resolution, as labeled in the legends.

Conceptual model of column-wind speed relationships
The emission rate over an air basin Q can be linked to the basin-average column amount through a box model where boldface symbols indicate vectors. The averaged NO 2 TVCD Ω and dynamic lifetime τ d are both vectors resolved over a range of wind speeds W , φ = 1.32 is the NO x /NO 2 ratio representative of cloud-free mid-day conditions in polluted air 125 mass (Beirle et al., 2019), A is the air basin area, and τ c is the NO x chemical lifetime. The NO x chemical lifetime may vary with wind speed depending on complicated nonlinear chemistry (Valin et al., 2013), so the scalar τ c here should be considered as the average value over the wind speed range. We further simplify the dynamic lifetime dimensionally as the ratio between wind speed and the horizontal length scale of the air basin L:

130
This implicitly assumes that the horizontal wind efficiently ventilates pollution away from the air basin, which is invalid at low wind speed. We thus limit our analysis over moderate wind speeds as will be shown in Figure 3. Then the conceptual model of  column-wind speed relationship can be written as the ratio between near-surface wind speed and half edge length of the square box . Similar box models have been also used to infer area-integrated CH 4 emission rates from column observations (Buchwitz et al., 2017;Varon et al., 2018). The chemical lifetime of CH 4 is negligible, and the dynamic lifetime was constrained by CTM simulations in these studies. Considering that the four wind options described in Section 2.3 (10m, 100m, surface-500m, and surface-1000m) give End-to-end emission rate estimates are performed using those four wind speed options with a range of L values in Section 4.1.
We found that using 100m wind and L = 280 km for the Po Valley air basin (close to √ A = 257 km) gives emission rate estimates that are most consistent with the JPL chemical reanalysis, which is considered to contain the smallest bias due to high level of observational constraints. This is deemed as a calibration for the dynamic lifetime and is specific to the Po Valley 145 air basin.
The behaviour of Eq. 5 is shown in Figure 3 as gray lines with prescribed emission rate Q and chemical lifetime τ c values for each season. Equation 5 implies that the column abundance should monotonously decrease with wind speed and, for the same chemical lifetime, scales with emission rate. When the chemical lifetime gets shorter, the 1/τ c term becomes larger relative to the dynamic lifetime term W /L, and hence the column abundance becomes a weaker function of wind speed. This 150 is demonstrated by the fact that the NO 2 TVCDs decrease more rapidly with stronger wind in winter, indicating a longer NO x chemical lifetime. The overall higher levels of NO 2 TVCDs in winter result from the combined effects of longer chemical lifetime and stronger emissions (see Section 4.3 for the seasonality of emission rates derived from this study as well as other top-down and bottom-up inventories).
As shown in Figure 3, the observed column-wind speed relationship deviates from Eq. 5 at lower and upper limits of wind 155 speed. The simple parameterization of dynamic lifetime by L/W assumes that the ventilation of the air basin is driven by horizontal advection, which is not valid when the basin air mass is stagnant. This is supported by the flattening of column-wind speed relationships at low wind speeds. At high wind speed, the number of valid observations rapidly decreases, leading to excessive noise. Therefore, we restrict our analysis to a moderate wind speed range of 3-8 m s −1 , as indicated by the shaded areas in cyan color in Figure 3. As shown in Eq. 5, Ω and W are vectors with elements separated by wind speeds, so we may directly fit Eq. 5 to the observed column-wind speed relationships and simultaneously obtain emission rate Q and chemical lifetime τ c . However, the information of τ c mainly comes from the flatness of the observed column-wind speed relationship, and thus the fitted τ c is highly sensitive to observational noise. Because Q and τ c are strongly anti-correlated, the error in τ c is efficiently propagated 165 to the fitted Q. For example, the spikes in the observed OMI column-wind speed relationships in Figure 3c-d would result in unphysically low chemical lifetime and unrealistically high emission rate without proper regularization. To reliably retrieve Q for each calendar month throughout the OMI and TROPOMI record, we build a monthly climatology of τ c from aggregated observation data and use it as prior information in a Bayesian optimal estimation framework (Rodgers, 2000;Brasseur and Jacob, 2017). The steps are summarized below, followed by described description in this section. 3. The optimally estimated τ c climatology is used as prior constraint to retrieve emission rate Q and τ c for each calendar month, separately for OMI and TROPOMI.

Constructing and fitting climatological column-wind speed relationships
The column-wind speed relationship of each climatological month is averaged from 3-month windows in all available years.
For example, the climatological month June is averaged from May-July in 2005-2020 for OMI and 2018-2020 for TROPOMI.

180
Although each climatological column-wind speed relationship is averaged from a significant number of calendar months (48-51 for OMI and 7-9 for TROPOMI), unregularized nonlinear fitting of Q and τ c is still highly unstable. Figure 4 shows the independent fitting of the column-wind speed relationships for each climatological month for OMI (a-b) and TROPOMI (cd) as black symbols. The gray symbols show 100 bootstrap realizations for each climatological month, where the calendar months used for averaging are selected randomly with replacement in each realization. This bootstrapping is necessary for 185 realistic error estimation, as the fitting errors are substantially biased low due to strong anti-correlation of fitted parameters.
The uncertainties are very large in certain climatological months. Some climatological months (April and September for OMI and August-October for TROPOMI) are characterized by nonphysically high emission rate and low chemical lifetime, whereas others (January and February for OMI) are subject to spurious high chemical lifetime. Those originate from irregular features on the column-wind speed relationship (observable in Figure 3) and tend to be more significant when satellite coverage is low. 190 We additionally remove "outlier" calendar months that would significantly alter the fitted τ c and Q from the climatological column-wind speed relationship. These outlier months are often characterized by anomalously high NO 2 TVCDs over a few wind speed bins. For each calendar month, the corresponding climatological month is processed twice, with and without that calendar month included in the averaging. The differences of the fitted Q and τ c the climatologial month with and without a specific calendar month are displayed in Figure 5. The calendar month is excluded as an outlier if the absolute value of its 195 impact on the climatological Q is larger than 70 mol s −1 or the absolute value of its impact on the climatological τ c is larger than 1.5 h. The long record of OMI enables a second round of outlier removal, where the climatology is averaged from a single month (instead of 3-month window). It is impossible to do that for TROPOMI as one climatological month would only have 2-3 calendar month to average from. In this round, the max Q difference with/without including a calendar month is still 70 mol s −1 , but the max τ c difference is relaxed to 5 h. The excluded calendar months are highlighted by red dots in Figure 5.

200
More winter months are excluded due to lower coverage and consequently noisier column-wind speed relationship. 53% of winter calendar months in the OMI record are excluded, while the overall removal rate is 30%.
After identifying and excluding the outlier calendar months, the climatological column-wind speed relationships are finalized and the climatology of emission rates and chemical lifetimes are fitted again. The results are shown by Figure 6. The fitting quality is significantly improved, as indicated by the reduced variation of bootstrap realizations.   climatological months April and September are unrealistically shorter than the summer months (Figure 6b), which is inconsistent with the TROPOMI values ( Figure 6d) and corresponds to suspiciously high emission rates in those two climatological 210 months ( Figure 6a). To further improve the climatology estimates, we incorporate the a priori information that the climatology should vary smoothly over the year through a Bayesian optimal estimation. The regularization from the optimal estimation will effectively suppress noise in the observed column-wind speed relationship.
In this optimal estimation setup, the 12 climatological column-wind relationships are concatenated into a single observation vector, and the 12 climatological chemical lifetimes and emission rates are retrieved simultaneously as a 24-element state 215 vector. The fitted OMI-and TROPOMI-based τ c values with outlier calendar months removed (black symbols in Figure 6b and d) are averaged together and smoothed by a first-order Savizky-Golay filter with a 3-month window (Savitzky and Golay, 1964). This smoothed curve is used as the prior values of chemical lifetimes for both OMI and TROPOMI. The prior value for the emission rates is a constant 260 mol s −1 for all climatological months. The prior error standard deviation is loosely set at 150% for both Q and τ c , and a time correlation scale of 1.5 month is assumed within the lifetime terms and the emission    Figure 8b compares the temporally averaged emission rates. The optimal L value, characterized by the lowest RMSE and the matching of temporal mean emission rates to the JPL mean value, increases in the order of 10m, 100m, surface-500m, and surface-1000m wind, consistent with the overall magnitude of those four wind options.

260
As shown by Figure 2, those four wind speeds are well linearly correlated. Therefore, the optimal L value scales with the wind strengths and partially "absorb" the systematic differences between wind speed options. We choose 100m wind due to its low optimal RMSE and better representation of horizontal advection than the 10m wind. The basin length scale L is selected to be 280 km, similar to the square root of the air basin area (257 km). One should note that this analysis is specific to the Po Valley air basin and should be repeated before applying such framework to other source regions.

NO x chemical lifetimes
The optimally estimated climatological chemical lifetimes, which are already shown in Figure 7b and d are replotted in Figure 9 to emphasize the confidence intervals and the prior values that is common for OMI and TROPOMI. The TROPOMI-based chemical lifetime estimates are consistently lower than the OMI-based values, but the error bars overlap in climatological   TROPOMI one spans 2018-2021, this difference implies a weak yet notable long-term decrease of NO x chemical lifetime. This is likely due to the decrease of NO x emissions (see Figure 11) and consequently the shifting of chemical regimes away from NO x -saturated conditions (Martin et al., 2004). Shifting in summertime NO x chemical lifetime due to change of NO x abundance and chemical regimes has been identified in North American cities using OMI observations and an EMG-based approach ( The TROPOMI-based climatological chemical lifetimes are suspiciously low after September. As the NO x sinks are driven by ambient temperature and solar radiation, we do not expect lower chemical lifetimes in September-October than June-July.

280
This anomaly likely results from abnormal TROPOMI column-wind speed relationships characterized by high NO 2 TVCDs in a few wind speed bins. Although the individual monthly column-wind speed speed relationship from OMI is noisier than TROPOMI (Figure 3), the much longer OMI record (197 calendar months vs. 34 calendar months for TROPOMI) enables more effective removal of outlier months and retrieval of climatological chemical lifetimes. As such, we focus on the OMI-based chemical lifetime climatology for the following analysis. The NO x climatological chemical lifetimes are 5-6 h in summer 285 and 15-20 h in winter, generally consistent with CTMs studies that consider NO x sinks comprehensively (Mijling and Van Der A, 2012;Stavrakou et al., 2013;Silvern et al., 2019;Shah et al., 2020). The summertime NO x chemical lifetime is also close to or slightly higher than other observational data-driven estimates, mostly through fitting the downwind decay of NO 2 plumes (Valin et al., 2013;de Foy et al., 2015;Liu et al., 2016;Goldberg et al., 2019a;Laughner and Cohen, 2019). This is consistent with the modeling verification by de Foy et al. (2014), which found the NO x chemical lifetime derived from 290 EMG-based approach to be biased low compared to the true lifetimes in the model simulations.
The OMI-based climatological chemical lifetimes in Figure 9 are then used as the prior to derive chemical lifetimes in each calendar month, for both OMI and TROPOMI. The resultant monthly NO x chemical lifetimes are shown in Figure 10a. Note that the chemical lifetimes in Figure 10a are retrieved from column-wind speed relationships for each calendar month, whereas the chemical lifetimes in Figure 9 are retrieved from column-wind speed relationships for each climatological month. The 295 observational information content of τ c for each calendar month, as indicated by the degrees of freedom for signal (DOFS), is only ∼ 0.02 (Figure 10b). This reflects our trade-off between emission rates and chemical lifetimes by applying relatively strong prior regularization to τ c in each calendar month. It also implies that the chemical lifetimes for calendar months are dominated by prior influences from the climatological chemical lifetimes. While the climatological chemical lifetimes are also derived from observations, the lack of observational constraints for the lifetime in each individual calendar month makes them 300 closely resemble the corresponding climtological month values (i.e., the prior) and prevents us from further interpretation of these monthly lifetime values.
The information of retrieved emission rate Q that is gained from observations, indicated by the corresponding DOFS, is however high and close to unity (Figure 10c). This indicates that we can confidently retrieve emission rates from the monthly column-wind speed relationships. The decaying DOFS for OMI-based emission rates from 2004 to 2021 and the higher DOFS 305 from TROPOMI than OMI are consistent with the instrument performances. month chemical lifetimes retrieved from TROPOMI are similar to OMI (Figure 10a), and hence the differences in OMI-and TROPOMI-based emission rates directly result from differences in their NO 2 TVCDs. This is supported by Figure 3; when 320 wind speed is controlled, the OMI TVCDs are higher in cold months while the TROPOMI TVCDs are higher in warm months.

NO x emission rates
A main advantage of the proposed satellite data-driven framework is to timely quantify rapid emission perturbations. The Po Valley region experienced two major COVID-19 outbreaks, one in February-May and the second one starting from October and ongoing (Dong et al., 2020). Both triggered lockdown measures that are expected to reduce NO x emission. However, the quantitative measure of net emission reduction due to the lockdowns is complicated by the long-term decreasing trend and 325 intra-annual variability. For instance, the simple difference between 2020 and 2019 values includes both the pandemic-induced emission changes and the business-as-usual decrease. Leveraging the long and consistent OMI record, we train a statistical model to present the inter-and intra-annual variability using the OMI-based emission rates from January 2010 to December 2019 (yellow shaded region in Figure 12a):

330
where t is time measured in fractional years, resolved by month, c, a, and b are model parameters, and e is an error term. The order of polynomial n p = 3 and the number of harmonics n h = 5 are chosen through the Akaike Information Criterion (Akaike, 1974). The fitted model and 95% confidence intervals are estimated using ordinary least squares and displayed in Figure 12a (a) Statistical model fitting using OMI-based emission rate OMI-based emission rate Prediction 95% confidence interval 1 9 -0 1 1 9 -0 2 1 9 -0 3 1 9 -0 4 1 9 -0 5 1 9 -0 6 1 9 -0 7 1 9 -0 8 1 9 -0 9 1 9 -1 0 1 9 -1 1 1 9 -1 2 2 0 -0 1 2 0 -0 2 2 0 -0 3 2 0 -0 4 2 0 -0 5 2 0 -0 6 2 0 -0 7 2 0 -0 8 2 0 -0 9 2 0 -1 0 2 0 -1 1 2 0 -1 2 2 1 -0 1 2 1 -0 2 Year-month grows. The well-documented emission perturbation during the 2008-2009 financial crisis (Castellanos and Boersma, 2012) is evident through the discrepancy between model extrapolation and real emission rates (Figure 12a). Similarly, since this statistical model is trained using data before the pandemic, the prediction in 2020 and beyond serves as a business-as-usual baseline. Compared to just using a previous year or multi-year averaged climatology as reference (Goldberg et al., 2020;Liu et al., 2020;Bauwens et al., 2020), the model prediction incorporates both the long-term trend and seasonality and is less The Apple mobility is measured by Apple user activity levels in driving and transit modes over the entire Italy relative to the baseline on 13 January 2020. Both Google and Apple mobility indicators are in daily native resolution and averaged weekly to remove day-of-week effects. The result is shown in Figure 12c. The relative NO x emission changes and the mobility in-355 dicators consistently show the "W" shape with the two troughs corresponding to large outbreaks. The impact of the second outbreak was lower than the first one, which is also consistent between the mobility indicators and OMI-based net emission changes. Discrepancies are noted in April 2020 and January 2021, when the mobility indicators stayed low after major control measures, but the NO x emission recovered quicker. We speculate this to be the impact of industrial NO x emissions that are not well represented by the human mobility indicators.

Conclusions and discussion
We present a satellite-data driven framework to rapidly quantify NO x emission rates over an air basin, and demonstrate it in Po Valley, Italy. Monthly emission rates and chemical lifetimes of NO x are retrieved from observed column-wind speed relationships, where the NO x column abundance is represented by OMI and TROPOMI NO 2 TVCD observations, and the wind speed is obtained from ERA5 reanalysis. To regularize the retrieval, we derive a NO x chemical lifetime climatology 365 and use it as prior information. The NO x chemical lifetime is 5-6 h in summer and 15-20 h in winter. Our observation-based emission rate estimates are consistent with top-down and bottom-up inventories and can be quickly updated as the method only depends on satellite and reanalysis data. Leveraging the long and consistent OMI record, a statistical model is trained to predict the business-as-usual trajectory without the pandemic. Compared with this trajectory, the real 2020-2021 emission rates show two distinctive dips that correspond to tightened COVID-19 control measures and reduced human activities. The overall net 370 NO x emission reduction due to the COVID-19 pandemic is estimated to be 20% with maximum reduction in March, followed by November.
Only observations under modest wind (3-8 m s −1 ) are used, so there is an implicit assumption that NO x emissions under modest wind can represent all wind conditions. Since NO x sources in air basins are mostly anthropogenic, this assumption is deemed to be valid. In addition, the satellite observations are made in the early afternoon local time, so the retrieved emission 375 rates may not necessarily represent the diurnal mean emission rate. This is a common limitation of all observational datadriven approach, and we note that the overall emission rate level is anchored to the overall emission rate level of JPL chemical reanalysis, which is spatiotemporally complete, through the selection of basin length scale L. The uncertainties of the retrieved monthly emission rates may also originate from the systematic biases of NO 2 TVCD products, but the relative emission variations should be insensitive to the observational biases. Updated satellite products (e.g., the version 2 TROPOMI NO 2 product to be released in 2021) can be readily adopted. The NO x :NO 2 ratio over the air basin has been fixed at a value of 1.32 (Beirle et al., 2011;de Foy et al., 2015;Liu et al., 2016). Appendix A: Pixel-based comparison between OMI and TROPOMI NO 2 TVCDs The TROPOMI retrievals within ±1 hour from an OMI retrieval are averaged using the relative pixel overlapping area as weight, and only OMI pixels that are > 80% covered by such TROPOMI pixels are used for comparison. This ensures that OMI and TROPOMI sample essentially the same air mass, and the NO 2 differences reflect the inherent differences of those two 395 products. Figure A1 compares the strictly collocated OMI and TROPOMI retrievals from December 2019 to November 2020.
Since the TROPOMI value in each TROPOMI-OMI pair is weight-averaged by 10-90 TROPOMI pixels, its random error is significantly lower than the OMI value, and hence we use the slope in ordinary least square (OLS) regression to represent the OMI/TROPOMI ratio. Figure A2 compares both the OLS slope and the OMI-TROPOMI NMB. In general, OMI is higher than TROPOMI in 400 the cold season, indicated by slopes larger than unity and NMB larger than zero, whereas TROPOMI is higher in the warm season. The temporal variation of the OLS slope and NMB show only moderate correlation (correlation coefficient r = 0.54), indicating that the discrepancy between OMI and TROPOMI is more complicated than an zero-level offset or proportional scaling.
Appendix B: Details in optimal estimations of Q and τ c

405
This appendix section provides technical details on the optimal estimation using column-wind speed relationships over both climatlogical months (Section 3.3.2) and calendar months (Section 3.  (B1)

415
Here f (β) is the forward model by concatenating Eq. 5 for each month, β a is the prior vector, and S a is the prior error covariance matrix. In the optimal estimations applied in this study, the strength of prior regularization is controlled by a single factor λ. Lower λ value, or weaker regularization, leads to smaller residuals but larger deviation from the prior; higher λ value, or stronger regularization, leads to smaller deviation from the prior but larger residuals. We select λ values for OMI/TROPOMI and climatology/calendar months separately, by finding the maximum curvature point. The corresponding L-curve plots in the 420 Q/τ c optimal estimations using column-wind speed relationships averaged to climatological months and in each calendar month are shown in Figure B1 and Figure B2, respectively. The selected λ values are labeled in the plots.
The cost function J (Eq. B3) is minimized by a Gauss-Newton approach, where the state vector is updated in each iteration by the following rule: Here β i is the state vector estimation in iteration i, and β 0 = β a . K i = ∂f (β i )/∂β i is the Jacobian matrix at iteration i.
Since the forward model is concatenated from column-wind speed relationship for each month (Eq. 5), and the state vector is concatenated from Q and τ c for each month, the Jacobian can be constructed using analytical derivations of Eq. 5: ∂ Ω ∂τ c = Q φA with a threshold that scales with the number of state vector elements.
After an optimal solution is found, the DOFS for each state vector element (Q or τ c ) is the corresponding diagonal element of the averaging kernel matrix where K is the Jacobian matrix at the final iteration and I is an identity matrix with the same dimension as the state vector.
Author contributions. KS designed and implemented this study and wrote the paper. LBL helped calculating the top-down and bottomup inventory emission rates. SJ helped curating satellite data. LBL and SJ contributed to satellite data analysis. DL provided expertise on atmospheric transport and helped with scientific interpretation and discussion.