The degradation of air quality is a worldwide environmental problem: 91 % of the world's population have breathed polluted air in 2016 according to the World Health Organization (WHO), resulting in 4.2 million premature deaths every year (WHO, 2016). The recent study of Lelieveld et al. (2019) even suggests that the health impacts attributable to outdoor air pollution are substantially higher than previously assumed (with 790 000 premature deaths in the 28 countries of the European Union against the previously estimated 500 000; EEA, 2018). The main regulated primary (i.e., directly emitted in the atmosphere) anthropogenic air pollutants are carbon monoxide (CO), nitrogen oxides (NOx = NO + NO2), sulfur dioxide (SO2), ammonia (NH3), volatile organic compounds (VOCs) and primary particles. These primary air pollutants are precursors of secondary (i.e., produced in the atmosphere through chemical reactions) pollutants such as ozone (O3) and particulate matter (PM), which are also threatening to both human health and ecosystems. Monitoring concentrations and quantifying emissions are still challenging and limit our capability to forecast air quality to warn population and to assess (i) the exposure of population to air pollution and (ii) the efficiency of mitigation policies.

Bottom-up (BU) inventories are built in the framework of air quality policies such as The Convention on Long-Range Transboundary Air Pollution (LRTAP, http://www.unece.org, last access: March 2019) for air pollutants. Based on national annual inventories, research institutes compile gridded global or regional monthly inventories (mainly for the US, Europe and China) with a high spatial resolution (currently regional- or city-scale inventories are typically finer than 0.1∘×0.1∘). These inventories are constructed by combining available (economic) statistics data from different detailed activity sectors with the most appropriate emission factors (defined as the average emission rate of a given species for a given source or process, relative to the unit of activity in a given administrative area). It is important to note that the activity data (often statistical data) have an inherent uncertainty and that their reliability may vary between countries or regions. In addition, the emission factors bear large uncertainties in their quantification (Kuenen et al., 2014; EMEP/EEA, 2016; Kurokawa et al., 2013). Moreover, these inventories are often provided at the annual or monthly scale with typical temporal profiles to build the weekly, daily and hourly variability of the emissions. The combination of uncertain activity data, emission factors and emission timing can be a large source of uncertainties, if not errors, for forecasting or analyzing air quality (Menut et al., 2012). Finally, since updating the inventories and gathering the required data for a given year is costly in time, manpower and money, only a few institutes have offered estimates of the gaseous pollutants for each year since 2011 (i.e, European Monitoring and Evaluation Programme EMEP updated until the year 2017, MEIC updated until the year 2017 to our knowledge). Nevertheless, using knowledge from inventories and air quality modeling, emissions have been mitigated. For example, from 2010 to nowadays, emissions in various countries have been modified and/or regional trends have been reversed downwards (e.g., the decrease in NOx emissions over China since 2011; de Foy et al., 2016), leading to significant changes in the atmospheric composition. Consequently, the knowledge of precise and updated budgets, together with seasonal, monthly, weekly and daily variations of gaseous pollutants driven, amongst other processes, by the emissions are essential for understanding their role in the formation of tropospheric ozone and PMs at various temporal scales, for anticipating pollution peaks and for identifying the key drivers that could help mitigate these concentrations.

In this context, complementary methods have been developed for estimating emissions using atmospheric observations. They operate in synergy between a chemistry-transport model (CTM) which links the emissions to the atmospheric concentrations, atmospheric observations of the species of interest and statistical inversion techniques. A number of studies using inverse modeling were first carried out for long-lived species such as greenhouses gases (GHGs) (e.g., carbon dioxide CO2 or methane CH4) at the global or continental scales (Hein et al., 1997; Bousquet et al., 1999), using surface measurements. Later, following the development of monitoring station networks, the progress of computing power and the use of inversion techniques more appropriate to non-linear problems, these methods were applied to shorter-lived molecules such as CO. For these various applications (e.g., for CO2, CH4, CO), the quantification of sources was solved at the resolution of large regions (Pétron et al., 2002). Finally, the growing availability and reliability of observations since the early 2000s (in situ surface data, remote sensing data such as satellite data) and the improvement of the global CTMs, computational capacities and inversion techniques have increased the achievable resolution of global inversions, up to the global transport model grid cells, i.e., typically with a spatial resolution of several hundreds of square kilometers (Stavrakou and Müller, 2006; Pison et al., 2009; Fortems-Cheiney et al., 2011; Hooghiemstra et al., 2012; Yin et al., 2015; Miyazaki et al., 2017; Zheng et al., 2019).

Today, the scientific and societal issues require an up-to-date quantification of pollutant emissions at a higher spatial resolution than the global one, which will lead to a wide use of regional inverse systems. However, although they are suited to reactive species such as CO and NOx, and their very large spatial and temporal variability, they have hardly been used to quantify pollutant emissions. Some studies inferred NOx (Pison et al., 2007; Tang et al., 2013) and VOC emissions (Koohkan et al., 2013) from surface measurements. Konovalov et al. (2006, 2008, 2010), Mijling et al. (2012, 2013), van der A et al. (2008), Lin et al. (2012) and Ding et al. (2017) have also shown that satellite observations are a suitable source of information to constrain NOx emissions. These regional inversions using satellite observations were often based on Kalman filter (KF) schemes (Mijling et al., 2012, 2013; van der A et al., 2008; Lin et al., 2012; Ding et al., 2017).

Variational inversion systems allow for the solving of high-dimensional problems, typically solving for the fluxes at high spatial and temporal resolution, which can be critical to fully exploit satellite images. Here, we present the Bayesian variational atmospheric inversion system PYVAR-CHIMERE for the monitoring of anthropogenic emissions of reactive species at the regional scale. It is based on the Bayesian variational assimilation code PYVAR (Chevallier et al., 2005) and on the regional state-of-the-art CTM CHIMERE (Menut et al., 2013; Mailler et al., 2017). CHIMERE is dedicated to the study of regional atmospheric pollution events (e.g., Ciarelli et al., 2019; Menut et al., 2020), included in the operational ensemble of the Copernicus Atmosphere Monitoring Service (CAMS) regional services. The main strengths of PYVAR-CHIMERE come from the strengths of CHIMERE and from its high modularity for the definition of the control vector. CHIMERE is indeed an extremely flexible code, in particular for the definition of the chemical scheme.

The PYVAR-CHIMERE system takes advantage of the previous developments for the quantification of fluxes of long-lived GHG species such as CO2 (Broquet et al., 2011) and CH4 (Pison et al., 2018) at the regional to the local scales, but it now solves for reactive species such as CO and NOx. It has also a better level of robustness, clarity, portability and modularity than these previous systems. Variational techniques require the adjoint of the model to compute the sensitivity of simulated atmospheric concentrations to corrections of the fluxes. CHIMERE is one of the few CTMs for which the adjoint has been coded. Global models include GEOS-CHEM (Henze et al., 2007), IMAGES (Stavrakou and Müller, 2006), TM5 (Krol et al., 2008), GELKA (Belikov et al., 2016) and LMDz (Chevallier et al., 2005; Pison et al., 2009); limited-area models include CMAQ (Hakami et al., 2007), EURAD-IM (Elbern et al., 2007), RAMS/CTM-4DVAR (Yumimoto and Uno, 2006) and WRF-CO2 4D-Var (Zheng et al., 2018).

The principle of variational atmospheric inversion and the configuration of PYVAR-CHIMERE are described in Sects. 2 and 3, respectively. Details about the forward, tangent-linear (TL) and adjoint codes of CHIMERE are also given. Then, the potential of PYVAR-CHIMERE is illustrated in Sect. 4 with the optimization of European CO and NOx emissions, constrained by observations from the Measurement of Pollution in the Troposphere (MOPITT) and from the Ozone Monitoring Instrument (OMI) satellite instruments, respectively.

In what follows, we use the notations and equations used in the inverse modeling community (Rayner et al., 2019). The Bayesian variational atmospheric inversion method adjusts a set of control parameters, including parameters related to the emissions whose estimate is the primary target of the inversion.

The prior information about the parameters x to be optimized during the inversion process is given by the vector xb. The parameters to be optimized can be surface fluxes but may also include initial or boundary conditions for example, as explained in Sect. 3.4. The adjustments are applied to prior values, usually taken, for the emissions, from pre-existing BU inventories. The principle is to minimize, on the one hand, the departures from the prior estimates of the control parameters, which are weighted by the uncertainties in these estimates (called hereafter “prior uncertainties”), and, on the other hand, the differences between simulated and observed concentrations, which are weighted by all other sources of uncertainties explaining these differences (hereafter referred to collectively as “observation errors”). In statistical terms, the inversion searches for the most probable estimate of the control parameters given their prior estimates, observations, CTM and associated uncertainties. The solution, which will be called posterior estimate, is found by the iterative minimization of a cost function J (Talagrand, 1997), defined as follows: 1J(x)=x-xbTB-1x-xb+H(x)-yTR-1H(x)-y. H is the non-linear observation operator that projects the control vector x onto the observation space. In most of the variational atmospheric inversion cases (such as those described in Sect. 4), the observation operator includes the operations performed by the CTM in linking the emissions to the concentrations and any other transformation to compute the simulated equivalent of the observations such as an interpolation or an extraction and averaging of the simulated concentration fields (see Sect. 3.5). The observations in y could be surface measurements and/or remote sensing data such as satellite data. The prior uncertainties and the observation errors are assumed to be unbiased and to have a Gaussian distribution. Consequently, the prior uncertainties are characterized by their covariance matrix B and the observation errors are characterized by their covariance matrix R. By definition, the observation errors combine errors in both the data and the observation operator, in particular the following:

measurement errors and errors in the conversion of satellite measurement into concentration data,

For inversions with observation and control vectors with a high dimension, the minimum of J cannot be found analytically due to computational limitations. It can be reached iteratively with a descent algorithm. In this case, the iterative minimization of J is based on a gradient method. J is calculated with the forward observation operator (including the CTM), and its gradient relative to the control parameters x is provided by the adjoint of the observation operator (including the adjoint of the CTM). The gradient is defined as follows: 2∇J(x)=B-1x-xb+H*R-1H(x)-y, where H* is the adjoint of the observation operator.

The high non-linearity of the chemistry for reactive species makes it difficult to use its TL code to approximate the actual observation operator, and, more generally, it makes the inversion problem highly non-linear. Therefore, in PYVAR-CHIMERE, we use the M1QN3 limited-memory quasi-Newton minimization algorithm (Gilbert and Lemaréchal, 1989), which relies on the actual CHIMERE non-linear model to compute J at each iteration of the minimization. As with most quasi-Newton methods, it requires an initial regularization of x, the vector to be optimized, for better efficiency. We adopt the most generally used regularization, made by minimizing in the space defined by the following: 3χ=B12x-xb, instead of the control space defined by x. Although more advanced regularizations can be chosen, the minimization with χ is preferred because it simplifies the equation to solve. In the χ space, Eq. (2) can be re-written as follows: 4∇Jχ=χ+B12H*R-1H(x)-y.

The criterion for stopping the algorithm is based on a threshold set on the ratio between the final and initial gradient norms or on the maximum number of iterations to perform. As shown in Fig. 1, the minimization algorithm repeats the forward-adjoint cycle to get an estimate close to the optimal solution of the inversion problem for the control parameters. This approximation of the optimal estimate is found by satisfying the convergence criteria of the minimizer with a given reduction of the norm of the gradient of J. Nevertheless, due to the non-linearity of the problem, the minimization may reach a local minimum only, instead of the global minimum.

Finally, the calculation of the uncertainty in the estimate of emissions from the inversion, known as “posterior uncertainty”, is challenging in a variational inverse system (Rayner et al., 2019). Even though the posterior uncertainty can be explicitly written in various analytical forms, it requires the inversion of matrices that are too large to invert given the current computational resources in our variational approach. As a trade-off between computing resources and comprehensiveness, the analysis error may be evaluated by an approach based on a propagation of errors through sensitivity tests (e.g., as in Fortems-Cheiney et al., 2012). It can also be estimated through a Monte Carlo ensemble (Chevallier et al., 2007), implemented in PYVAR. Nevertheless, it should be noted that the cost of the Monte Carlo experiments used to derive these posterior uncertainties is huge.

The potential of the PYVAR-CHIMERE system to invert emissions of reactive species is illustrated with the inversion of CO and NOx anthropogenic emissions in Europe based on MOPITT CO data and OMI NO2 data, respectively. We have chosen to present an illustration of CO inversion over 7 d, the first week of March 2015. Considering the short lifetime of NOx of a few hours (Valin et al., 2013; Liu et al., 2016), we have chosen to present illustration of NOx inversion over 1 d, 19 February 2015. These particular periods have been chosen as they present a representative number of super-observations during winter, and as the emissions are high during that period. All the information required by the system to invert CO and NOx emissions is listed in Table 1.

This paper presents the Bayesian variational inverse system PYVAR-CHIMERE, which has been adapted to the inversion of reactive species such as CO and NOx, taking advantage of the previous developments for long-lived species such as CO2 (Broquet et al., 2011) and CH4 (Pison et al., 2018). We show the potential of PYVAR-CHIMERE, with inversions for CO and NOx illustrated over Europe. PYVAR-CHIMERE will now be used to infer CO and NOx emissions over long periods, e.g., first for a whole season or year and then for the recent decade 2005–2015 in the framework of the H2020 VERIFY project over Europe, and in the framework of the ANR PolEASIA project over China, to quantify their trend and their spatiotemporal variability. Nevertheless, as we have reported strong non-linear relationships between NOx emissions and satellite NO2 columns, a fully comprehensive scientific study is required, by analyzing the NOx lifetime through processes such as the NO2 + OH reactions and/or the reactive uptake of NO2 and N2O5 by aerosols (e.g., Lin et al., 2012; Stavrakou et al., 2013). Biogenic emissions will be also further studied to better understand the relationship between NOx emissions and NO2 spaceborne columns.

The PYVAR-CHIMERE system can handle any large number of both control parameters and observations. It will be able to cope with the dramatic increase in the number of data in the near future with, for example, the high-resolution imaging (pixel of 7×3.5 km2) of the new Sentinel-5P/TROPOMI program, launched in October 2017. These new space missions with high-resolution imaging have the ambition to monitor atmospheric chemical composition for the quantification of anthropogenic emissions. It will indeed entail using PYVAR-CHIMERE at higher spatiotemporal resolutions, but probably for smaller domains (i.e., over countries rather than over Europe) as a compromise between resolution and the computational cost. Moreover, a step forward in the joint assimilation of co-emitted pollutants will be possible with the PYVAR-CHIMERE system and the availability of TROPOMI co-localized images of CO and NO2. This should improve the consistency of the inversion results and can be used to inform inventory compilers and subsequently improve emission inventories. Moreover, this development will help in further understanding air quality problems and addressing air-quality-related emissions at the national to subnational scales.