The reporting of the national air pollutants follows international guidelines that are available via the European Monitoring and Evaluation Programme (EMEP) Centre on Emission Inventories and Projections (https://www.ceip.at/reporting-instructions, last access: 1 November 2022). The reported inventory data for Germany, in the form of the detailed informative inventory report (IIR), describing the technical methodology may be found at https://iir.umweltbundesamt.de/2022/ (last access: 1 November 2022).

The data are arranged in time series per gas species, considering the different emission sources of NOx in sectors such as, for example, public power, industry, and traffic in a very detailed, disaggregated form at the national level. The bottom-up creation of these inventories is driven by statistical data provided by the German Federal Statistical Office (Destatis), and complex models use this data to compile the emissions for a specific gas (or aerosol) for a specific emission source in a specific sector and year. The uncertainties for each reported emission source depend on the availability of the data used for the emission calculation and may vary considerably. As an example, uncertainties in emissions from sectors, which are quite accurately described by statistical data and models such as emissions from large power plants, show much lower uncertainties than sectors that are governed by a great complexity such as the natural variability in the NOx from agricultural emissions in soils (e.g. uncertainties can be more than 300 % for agricultural soils; ).

In this study, both the gridded and non-gridded reported emission datasets are retrieved directly from the respective Convention on Long-range Transboundary Air Pollution (CLRTAP) inventories, which follow the Nomenclature for Reporting (NFR) standard. The gridded dataset is only available for 2019, while the non-gridded data are available for both 2019 and 2020. The 2021 data are a prognosis based on the trends observed between 2012–2019 for all emission classes under the assumption that the patterns in most emission sectors rebound after the 2020 COVID-19 lockdowns. An overview of the relative contributions of individual sectors to the total gridded emissions is shown by sector in Fig. .

All emissions except the MEMO items (MEMO items are additional reported emissions on non-standard emission such as volcanoes and forest fires) are selected from the CLRTAP inventories. Two natural sources are added to this set, namely non-agricultural soils and lightning. Globally, the lightning NO constitutes about 3 % of the total NOx emission budget . According to the guidebook , only 20 % of the lightning NO is formed in the lowest 1000 m of the atmosphere and the remaining 80 % at higher altitude (all inter-cloud lightning above 5 km height). A rough estimate for the lightning emissions can be made on the basis of the number of flashes per kilometre squared and the expected NOx emissions released per flash. A study by gives an average of about two flashes per kilometre squared throughout Germany, with fewer flashes in the central and northern parts. Assuming that on average the number of lightning flashes did not increase significantly in combination with a production of about 180 mol NO per flash and a German surface area of about 357 000 km2 gives us a German lightning NOx emission total of about 5 kt (NO2) NOx per year. This emission total is very minor and spread out over a large domain and is not expected to be a significant source of error when comparing the satellite-derived emission estimates with the emission inventory. From this point forward in this work, kt (NO2) NOx is written as kt NOx.

Due to widespread nitrogen pollution and deposition in Germany, it is complicated to make an estimate of purely non-anthropogenic and non-agricultural soil emissions. There are several studies that looked at soil NOx emissions for the European domain, which are mostly based on the anthropogenic emissions reported but with few that focus on purely natural emissions. gave an estimate of 3–90 kt NOx for forest emissions and 20 kt NOx for grassland soils. This estimate was more recently updated by and is available as the CAMS-GLOB-SOIL inventory , with a reported 2018 German emission total of about 160 kt NOx. Within the inventory, the emissions are split between fertilizer-induced, biome, deposition-related, and pulsed-soil emissions. There is always a danger of double counting such emissions, but the fertilizer-induced emissions of 100 kt NOx match closely to those included within the 2019 GNFR (Gridded Nomenclature for Reporting) data of approximately 110 kt NOx (classed under L_AgriOther sources). The remaining 60 kt of NOx per year is a combination of biome, deposition-related, and pulsed-soil emissions. The non-agricultural source emissions are quite uniformly distributed throughout Germany, peaking somewhat towards the northeastern part of the country. Note that stress that the derived soil emissions still have a large uncertainty range, mostly related to a lack of observations, missing data for some biomes, and the uncertainty in the input parameters such as soil temperatures. Annual variations are expected to be large, depending on variations in the soil temperatures. do not provide an upper and lower range of the emissions.

Additionally, we use the European Pollutant Release and Transfer Register (E-PRTR) for the emission locations and strengths of the largest industrial emission sources within Germany. The latest dataset (v18) can be accessed via https://www.eea.europa.eu/data-and-maps/data/industrial-reporting-under-the-industrial-6 (last access: 1 November 2022). Only sources with an emission strength above 0.25 kt NOx per year are selected for later comparison to the satellite-derived emissions. Note that the most recent data available are based on reported emissions of the year 2017, and thus we only use the data as a rough indication of source strength.

The TROPOMI instrument, on the Sentinel-5P satellite platform, was launched on 13 October 2017. The satellite instrument achieves almost full daily coverage of the globe through a sun-synchronous orbit with a local overpass at around 13:30 LST . TROPOMI has an unprecedented horizontal resolution of 3.5 × 5.5 km2 for the NO2 product. Details on the retrieval are described in the Copernicus user manuals (Algorithm Theoretical Basis Document, ATBD, https://sentinels.copernicus.eu/documents/247904/2476257/Sentinel-5P-TROPOMI-ATBD-NO2-data-products, last access: 1 November 2022), as well as in earlier publications such as . The TROPOMI NO2 operational product has three data streams: the near-real-time product available within 3 h (NRTI), the offline (OFFL) version that follows 1 d later and receives a more stringent quality control (now spanning 2019–2021), and a complete reprocessed version that is provided at more irregular intervals (RPRO or reprocessed, April 2018–November 2018). Over time, several improvements in the retrieval algorithm lead to processor updates and new product versions. Finally, independently from the operational steams, a reanalysis of the full dataset with the most up-to-date retrieval algorithm became available at the end of 2021, named the PAL product, which is currently available until the end of November 2021, connecting seamlessly to OFFL v2.3.1 from November 2021 to July 2022. The TROPOMI NO2 product went through several upgrades concerning its product versions over the years, with the most recent three upgrades from version v1.3.2 to version v1.4.0, then v2.2.0, then v2.3.1, and then v2.4.0 taking place, respectively, in November 2020, July 2021, November 2021, and July 2022. The most recent upgrade to version 2 involved a more major overhaul that greatly improved the overall quality of the retrieval .

The TROPOMI NO2 PAL product includes a reanalysis of the earlier data and provides a consistent version throughout (v2.3.1). This product is recommended to be used for any longer time series analysis and has been used in this study. We combine this product with 2 months of the newest OFFL data (v2.3.1) to complete the data series for 2021. The PAL product is available through the PAL data portal (https://data-portal.s5p-pal.com/, last access: 1 November 2022).

The quality of the TROPOMI NO2 PAL and OFFL products based on the v2.3.1 processor version is discussed by . Furthermore, the previous dataset versions 1.2.x and 1.3.x were relatively well validated . The TROPOMI NO2 data correlate well when compared to ground-based MAX-DOAS and PANDORA instruments but tend to show an underestimation of the tropospheric column. The median negative bias ranges from -15 % to -35 % in most clean to slightly polluted regions and up to -50 % over highly polluted regions for versions 1.2.x and 1.3.x. This bias is reduced in the PAL dataset , with reported improvements for the tropospheric columns from an average low bias of -32 % to -23 %. The range of the differences for individual sites are, however, quite wide with, for example, MAX-DOAS in De Bilt, the Netherlands, showing a range of around -75 % up to around +50 % (25th and 75th percentiles) with a median of around -20 %. The negative bias can be explained by the low spatial resolution of the a priori profiles, as well as the treatment of clouds and aerosols in the retrieval . As for the TROPOMI data quality criteria, the requirements recommended in the ATBD were used, which means observations with a cloud fraction below 0.03 were used, based on the cloud_fraction_crb_nitrogendioxide_window variable in the data files. Furthermore, observations with a quality value (qa_value) below 0.75 were filtered from the dataset. It is important to note that the MAX-DOAS and PANDORA instruments are not completely free of bias themselves; however, the ground-based instruments typically have much lower uncertainties than the TROPOMI NO2 product, as stated in .

Figure  shows yearly averages of the TROPOMI NO2 (PAL, v2.3.1) data. Here the reduced column densities that occur concurrent with the COVID-19 lockdown measures in 2020 is clearly visible. The industrialized Ruhr valley at the western border of Germany shows far reduced levels of NO2 if compared to 2019. The same is also observed in the industrial centres further to the south-southwest, which almost vanishes in 2020 and shows only a very slow recovery of emissions in 2021.

The methodology in this study makes use of the wind rotation approach, as explained in detail in , , and . The required wind data are taken from ECMWF's ERA5 dataset which was downloaded at a 0.25° × 0.25° resolution and 1 h temporal resolution. To match each of the satellite footprints, the meteorological fields are interpolated (spatially and temporally) to each of the observations. We assume that the majority of the NOx mass from local emissions is located in the lower boundary layer , and for the transport of NOx, an average of the wind fields of the first 100 hPa (around the first kilometre) is taken above the surface. These are approximately the levels between ∼1000–900 hPa for a typical sea level location, and for a location with a surface pressure of 800 hPa, winds between 800 and 700 hPa are averaged. The surface pressure at the location of the satellite observations is used to determine the 100 hPa layer.

Emissions were derived using the multi-source plume method (MSPM) , which was originally developed by and can be used for an assessment of emissions from both area and point sources. For a more detailed explanation, we refer readers to those publications. In short, the method relates observations and emission sources by creating a linear system of plume functions, which effectively establish a system of source and receptor relations in which the total tropospheric column density of each observation is described as a combination of the total column densities of all source plume functions. This is expressed as 1Ax=B, where A is the linear system of source–receptor relations, x is the emission sources, and B is the satellite-observed vertical column density (in our case the TROPOMI NO2 tropospheric columns). Several additional terms can be incorporated in x to account for regional product biases and for background concentrations. While the TROPOMI NO2 product does have local biases, the small number of validation stations hampers an accurate determination and correction for the product bias. To account for the bias, we apply a correction on an overpass to overpass basis, following , removing the lowest 5 % of the observed total column density within the larger domain. The short lifetime of NO2 ensures that further corrections to background concentrations are not needed.

Any plume function can be used to represent the relations in matrix A; here we use the exponentially modified Gaussian (EMG) plume function, which has been successfully applied in previous studies . Using this method, observations are rotated around a single point, the emission source, so that each is positioned in a similar upwind–downwind frame with respect to the wind direction. This enables us to describe the position of each observation as a point within a downwind plume. For more details on the plume rotation method see, Fig. S4 in . The EMG plume function describes the vertical column density (VCD) concentrations downwind of a source. The VCD at each position x,y near a source can be described by Eq. (), where a represents the emission enhancement, f(x,y) the crosswind diffusion (Eq. ), and g(y,s) (Eq. ) a convolution of the downwind advection and diffusion. Within all functions, x represents the crosswind position, y the downwind position, and s the wind speed. 2V(x,y,s)=a⋅f(x,y)⋅g(y,s)+B,3σ1=σ2-1.5y,y<0σ,y≥0,4f(x,y)=1σ12πexp⁡(-x22σ12),5λ1=λs,6g(y,s)=λ12exp⁡(λ1(λ1σ2+2y)2)erfc(λ1σ2+y2σ). Parameters σ and λ represent the plume spread and decay rate of NO2, with τ=1/λ being the decay time or lifetime. The parameters σ1 and λ1 shown in Eqs. () and () represent the adjusted form of a plume upwind of the source (σ1) and the decay rate divided by the wind speed (λ1). Each observation j can then be described by the sum of the enhancements of all sources i, forming Eq. (). 7ColumnNO2j(ψj,θj,sj)=∑iaif(xi,j,yi,j)g(yi,j,sj)+Bi,j The emission rate of each source i can then be calculated by dividing the emission enhancement ai by the decay rate τ; E=a/τ=aλ.

In this work, a grid with a resolution of 0.1° × 0.1° is used to describe the emission, covering the full domain of Germany, with a 2° padding added to the edges to reduce any edge effects . The resolution is chosen as a compromise between computational burden, the limitations of the instrument, the level of detail required, and the conditioning of the linear system in Eq. ().

The lifetime of NOx depends on both the chemical decay rate and loss to surfaces (dry deposition). Within our domain of interest, the chemical decay will be the dominant factor. Commonly used lifetimes in the literature are typically based on either modelled lifetimes or derived lifetimes from (satellite) observed plumes. Modelled lifetimes are commonly estimated via the availability of OH and production thereof (often including radiation) . Several studies have explored this route before and either estimate the availability of OH by some basic assumptions on production or by using modelled OH fields (with the drawback of a potential bias within the simulated concentrations). Either route is possible, and estimates for the effective lifetimes end up at around 2–5 h for spring and summertime values . Outer estimates for wintertime lifetimes are 12–24 h . Alternatively, lifetimes can be derived from tagging emitted molecules and tracking these within the model domain . The study reported that for a region representative of Germany (Benelux), approximately 50 % of the modelled satellite signal (Ozone Monitoring Instrument, OMI; ) result from NOx emissions in the 3 h prior to OMI overpass. Assuming a relatively constant source, this translates to a lifetime of about 4 h (at column level and assuming a basic mass balance). Several other studies report on effective lifetimes derived from fits to observed plumes from cities and large industrial areas. Using the EMG plume functions, the studies derived lifetimes between 2–5 h, based on the decay downwind of major sources worldwide , with a recent study by giving a value of 3.3 h representative of larger emissions within the US and Canada (2018–2022).

Following the modelled and observed lifetimes, we assume a mean lifetime of 4±1 h to account for local and seasonal variations. A potential point of concern remains with respect to how representative the lifetime is for the whole year. Most of the estimates are biased towards spring, summer, and autumn as there are typically more observations available within these months. To correct for the representativity bias, a seasonal variation factor (1.11) will be included (explained in next section); additionally, by choosing a value of 4.0 h, we remain on the high end of the lifetime estimates. The standard deviation of ±1 h ensures that common values within 3–5 h remain within the uncertainty range. Furthermore, also notes that while lifetime has a large impact on the emission estimates, relative changes do not have a major impact when comparing individual years to one another. They point out that 1 h deviations from the 3.3 h mean only changed the emission estimates between years by about 1 %.

The plume spread can be seen as a combination of the diffusion, satellite footprint size, and the spatial size of the sources . Taking into account the effective TROPOMI footprint, as well as the added diffusion, we use a value of 7 km for the plume spread (similar plume spreads are used in , and ). A dampening factor is added to the linear system in Eq. (), forming Eq. () to reduce oscillation effects within the solution. The resulting sparse linear system can be solved efficiently with the SciPy sparse.linalg.lsqr package . 8AγCx=B0 Satellite observations of short-lived species are only representative of emissions near the overpass time. A correction factor should be applied to the satellite-based estimated emissions to account for the diurnal variability. To account for this, we can use a basic box model to approximate the mass over time and apply a posterior correction. Assuming a mass m(t) and a constant lifetime (τ=1/lifetime) and the emission E at time t, the mass can be calculated with 9m(t)=m(t-1)e-τ+E(t). This equation is applied to the domain-wide emissions that are injected into the domain for a whole year, including a few days of spin-up, and averaged and normalized for a selection of expected lifetimes. For the temporal distribution, we use the average NOx emission profile for all NOx sources within the German domain, as used in the LOTOS-EUROS model . A lifetime of about 4 h and an overpass time of around 13:00 LST results in a correction factor of 1.24, meaning that the estimated emissions can be expected to be overestimated by around 24 %.

Depending on the source location and time of year, this value is expected to vary due to variations in the temporal emission profile. However, as actual measurements of diurnal cycles of NO2 emissions are rare and only exist for larger power plants, only the variability in the model emissions can be used to create a regional adjustment parameter. Surface concentration observations should in turn be used to analyse and optimize the modelled diurnal emission profiles for individual sectors. To calculate the viability of such a regional factor, the adjustment parameter was calculated for each cell. The standard deviation of the regional parameters is around ∼0.05. Therefore, to reduce complexity, the value of 1.24 is assumed for the entire domain. A similar parameter is derived to account for the seasonal variability in the emissions in combination with the variable availability of TROPOMI NO2 observations passing the data quality filters. The correction parameter is calculated as the weighted mean of the number of observations per month and the mean correction factor for each month. Using this approach, a value of 1.05 is found. Combined with the diurnal parameter, this gives a factor of approximately 1.30.

TROPOMI is only capable of observing NO2. Therefore, an additional correction is needed to account for the NO mass. The NOx : NO2 concentration ratio depends on the local chemistry that is influenced by ozone concentration, photolysis frequency of NO2 (solar-zenith-angle- and cloud-cover-dependent), and the rate constant of the NO + O3 reaction (temperature), with values commonly falling within the 1.2–1.5 range for polluted regions . In this study, we apply the 1.32±0.26 factor, as used by , and include the standard deviation of 0.26 (20 %) to further account for the variations in the uncertainty budget.

Based on the methods and choice of parameters described in the previous sections, a summary can be made of the total expected uncertainty in our method. An overview of the uncertainty parameters with a short summary of the chosen parameter values and impact on the emissions is given in Table .

One of the major parameters of uncertainty is the TROPOMI NO2 data product. As stated earlier, the current TROPOMI NO2 product overestimates concentrations in background/low-emission regions (± a few percent) while having a negative bias in source regions, with -35 % up to -50 % in extreme cases , which, according to , adds up to a potential mean bias of around -32 % to -23 %. Assuming -23 % for a larger industrialized region such as Germany, we end up with an underestimation of the emissions by a factor of -30 %. Locally, these values can decrease further (high-emission zones) or increase (low-emission zones; up to a positive percentage). The main cause for the bias can be found in the inaccuracies of the air mass factor (AMF) which come from uncertainties in the underlying modelled concentration fields and missing variations in the stratospheric NO2 concentrations . Local variations due to errors in the AMF cannot be corrected without the use of a chemistry transport model (CTM) and lead to an under- or overestimation of emissions in high source and background regions. A recent approach using the modelled CAMS Europe profiles shows that the large negative bias can be resolved with the help of higher-resolution a priori profiles. Beside this systematic uncertainty, the VCDs will also have a random uncertainty (of up to 30 %–50 % for individual observations). Due to the large number of observations used to constrain each source, the impact of those uncertainties is expected to be minor. Furthermore, there is the detection limit of the TROPOMI instrument, which limits the ability to detect smaller sources. The study by gives a limit of about 0.11 kg s-1, based on the divergence method. An emission source of 0.11 kg s-1 equals about 3.5 kt NOx per year. This is based on a peak fit which typically has a radius of 25 km, which roughly gives us a 2500 km2 area that, when divided the detection limit over the area, results in a detection limit of around 1.4 t km-2. To summarize, the total expected uncertainty in the emissions due to the TROPOMI product will add up to around -30 %.

The second major parameter with a large uncertainty is the choice of lifetime. An underestimation of the chemical losses could lead to an overall overestimation of the emissions, and conversely, an overestimation of the lifetime can lead to an underestimation of the emissions. A doubling of the lifetime roughly halves the emissions, which shows the importance of the parameter. Lifetimes, as stated, are location-dependent and to more accurately estimate them will require further detailed plume and chemistry (model) studies. Examples of recent studies give an indication of the typical ranges of the NOx (chemical) lifetime and give a range of 2–5 h, with the study by giving a value of 3–5 h representative of the Germany domain. The 4±1 h results lead to a -33 % to +20 % under-/overestimation of the emissions.

The NOx : NO2 ratio can also have local variations which affect the total emissions. At source level, the majority of NOx is emitted as NO, which can rapidly turn into NO2, after which an equilibrium is reached, the speed of which depends on the availability of O3. recently gave a modelled estimate of the ratio, which was very close to the factor 1.32 (±20 %) given in his original study, with values moving towards 1.0 for industrial areas just north of the Equator, while values tended towards higher ranges (1.6) for less industrialized and high-latitude regions.

Next up, there is the influence of the wind speed and direction for which we assume an uncertainty of up to 1 m s-1 in both the u and v wind field parameters, leading to the realistic situation of a higher uncertainty in direction at low wind speeds. The effect translates to an uncertainty of around 15 %–20 % for average conditions over Germany (based on the matched wind fields), which matches earlier uncertainty estimates by , .

Finally, the diurnal and seasonal variations show some variations of the order of a few percent (<5 %). Note that a fixed parameter was determined for the whole German domain, but locally the diurnal correction factor can be lower/larger for the more continuous/strongly varying emissions. For example, in the case of power plants, which run more continuously than road transport, this can result in a negative bias for the emissions.

Taken together, these error terms result in a Germany-averaged error range between -50 % and +35 % for the Gaussian plume method. The low error estimate corresponds to source regions where the low bias of the TROPOMI VCDs, effectively biasing the emissions low, are counteracted by the potentially high bias in the emissions of the NOx:NO2 ratio and effective lifetimes. Both values should be seen as conservative estimates which would occur in the unlikely case that the inaccuracy in the NOx:NO2 ratio, lifetime, AMF, and wind fields all nudge the estimate in the same direction for all locations in the domain of interest. In reality, not all errors point in a similar direction (like the product bias pointing in opposite directions for background and source regions).

A direct sector-based attribution of emissions is not feasible when using the satellite data only. Therefore, additional data need to be taken into account to attempt to estimate a potential sectoral attribution of the emission. We used the GNFR/CLRTAP sector outputs to create a spatial index filter for the emission data. The GNFR data are used as a basis and summed and regridded for all the NFR classes to match the 0.1° × 0.1° grid used in this study. A Gaussian filter (scipy.ndimage; ) is applied to the data with a sigma of one grid cell. The posteriori smoothing is only there to bridge the limitations of the method and instrument. The spatial limit to resolve two sources of a similar size depends on the effective lifetime, the pixel size, and meteorological factors such as typical diffusion. Of these, the pixel size and lifetime are dominant at the TROPOMI pixel limit (5.5×3.5 km2). The pixel size combined with diffusion gives us a typical plume width of around 7 km (e.g. σ2=σplume2+σpixel2+σsource2). This value varies, depending on typical size of a source, but most sources of NO2 are limited in size (except for large mines, very large cities, etc.). Based on , a plume width of 7 km combined with a lifetime of 4 h gives an effective resolvability limit of 15–20 km, which for 0.1° × 0.1° source cells (e.g. ∼10×10 km2) explains the choice for a sigma of one grid cell. More smoothing can produce better results but also reduces the observable details. The structural similarity index measure (SSIM) should be seen more as a metric to judge the comparability and not the accuracy of the emissions, as the inventory emissions are not perfect either. The resulting masks are divided by the total emissions of all sectors to derive each sector's fraction of all emissions (emission fraction) and are shown in the Appendix Fig.  for the non-smoothed version and Fig.  for the Gaussian smoothed version. For further sectoral emissions, analysis-only locations with an emission fraction above 50 % are selected, and the resulting mask is shown in Fig. .