the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Potential of European ^{14}CO_{2} observation network to estimate the fossil fuel CO_{2} emissions via atmospheric inversions
Yilong Wang
Grégoire Broquet
Philippe Ciais
Frédéric Chevallier
Felix Vogel
Lin Wu
Rong Wang
Shu Tao
Combining measurements of atmospheric CO_{2} and its radiocarbon (^{14}CO_{2}) fraction and transport modeling in atmospheric inversions offers a way to derive improved estimates of CO_{2} emitted from fossil fuel (FFCO_{2}). In this study, we solve for the monthly FFCO_{2} emission budgets at regional scale (i.e., the size of a mediumsized country in Europe) and investigate the performance of different observation networks and sampling strategies across Europe. The inversion system is built on the LMDZv4 global transport model at 3.75^{∘} × 2.5^{∘} resolution. We conduct Observing System Simulation Experiments (OSSEs) and use two types of diagnostics to assess the potential of the observation and inverse modeling frameworks. The first one relies on the theoretical computation of the uncertainty in the estimate of emissions from the inversion, known as “posterior uncertainty”, and on the uncertainty reduction compared to the uncertainty in the inventories of these emissions, which are used as a prior knowledge by the inversion (called “prior uncertainty”). The second one is based on comparisons of prior and posterior estimates of the emission to synthetic “true” emissions when these true emissions are used beforehand to generate the synthetic fossil fuel CO_{2} mixing ratio measurements that are assimilated in the inversion. With 17 stations currently measuring ^{14}CO_{2} across Europe using 2week integrated sampling, the uncertainty reduction for monthly FFCO_{2} emissions in a country where the network is rather dense like Germany, is larger than 30 %. With the 43 ^{14}CO_{2} measurement stations planned in Europe, the uncertainty reduction for monthly FFCO_{2} emissions is increased for the UK, France, Italy, eastern Europe and the Balkans, depending on the configuration of prior uncertainty. Further increasing the number of stations or the sampling frequency improves the uncertainty reduction (up to 40 to 70 %) in high emitting regions, but the performance of the inversion remains limited over lowemitting regions, even assuming a dense observation network covering the whole of Europe. This study also shows that both the theoretical uncertainty reduction (and resulting posterior uncertainty) from the inversion and the posterior estimate of emissions itself, for a given prior and “true” estimate of the emissions, are highly sensitive to the choice between two configurations of the prior uncertainty derived from the general estimate by inventory compilers or computations on existing inventories. In particular, when the configuration of the prior uncertainty statistics in the inversion system does not match the difference between these prior and true estimates, the posterior estimate of emissions deviates significantly from the truth. This highlights the difficulty of filtering the targeted signal in the model–data misfit for this specific inversion framework, the need to strongly rely on the prior uncertainty characterization for this and, consequently, the need for improved estimates of the uncertainties in current emission inventories for real applications with actual data. We apply the posterior uncertainty in annual emissions to the problem of detecting a trend of FFCO_{2}, showing that increasing the monitoring period (e.g., more than 20 years) is more efficient than reducing uncertainty in annual emissions by adding stations. The coarse spatial resolution of the atmospheric transport model used in this OSSE (typical of models used for global inversions of natural CO_{2} fluxes) leads to large representation errors (related to the inability of the transport model to capture the spatial variability of the actual fluxes and mixing ratios at subgrid scales), which is a key limitation of our OSSE setup to improve the accuracy of the monitoring of FFCO_{2} emissions in European regions. Using a highresolution transport model should improve the potential to retrieve FFCO_{2} emissions, and this needs to be investigated.
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CO_{2} emitted from fossil fuels (FFCO_{2}) is the major contributor to the increase of atmospheric CO_{2} (Ballantyne et al., 2015). Knowledge of FFCO_{2} emissions and their trends is essential to understand the drivers of their variations and assess the effectiveness of agreed upon emission reduction policies over time (Pacala et al., 2010). At national scale, FFCO_{2} emission inventories are derived based on energy and fuel use statistics, combustion efficiencies and emission factors. These inventories have low uncertainties in OECD countries and large uncertainties in developing countries due to uncertain energy data and fuelspecific emission factors (Liu et al., 2015; Ballantyne et al., 2015; Andres et al., 2014; Ciais et al., 2010). At subnational and intraannual scales, the uncertainties in the estimates of FFCO_{2} emissions are higher than at national and annual scale (Ciais et al., 2010; Wang et al., 2013) because subnational intraannual estimates require either the topdown disaggregation of national annual emissions relying on uncertain socioeconomic proxies (Wang et al., 2013; Pregger et al., 2007; Oda and Maksyutov, 2011; Andres et al., 2012), or a detailed knowledge of local activity data for a bottom upscaling of emissions (Gurney et al., 2009). The comparison of different emission maps of that kind also suggests large uncertainties due to, for example, treatment of administrative or land–water borders, the use of different proxies and different spatial resolutions of the maps (Andres et al., 2016). In consequence, national budgets obtained by aggregation of emission maps may have larger uncertainties than those based on national energy use and fuel accounting systems.
Atmospheric inversions exploit the observed variability in atmospheric mixing ratios of CO_{2} to quantify CO_{2} fluxes. Inversions have been applied for natural CO_{2} sources and sinks based on CO_{2} observations (Broquet et al., 2011; Chevallier et al., 2010; Peylin et al., 2013). Recent attempts to quantify FFCO_{2} emissions with inversions based on atmospheric CO_{2} measurements have stressed the importance of measuring mixing ratio gradients very close to the emitting source, such as a city (Staufer et al., 2016; Cambaliza et al., 2014; Lindenmaier et al., 2014) or a power plant (Turnbull et al., 2016). Away from the emitting source, the atmospheric signals of FFCO_{2} emissions mixes with those of natural fluxes, so that FFCO_{2} emissions can hardly be monitored by atmospheric CO_{2} measurements only (Shiga et al., 2014). Because of this, monitoring FFCO_{2} emissions at national scales, using continental networks of stations located outside the vicinity of the largest sources, is only possible when measuring an additional tracer specially sensitive to the signal of FFCO_{2} emissions (Miller and Michalak, 2017; Basu et al., 2016). Radiocarbon in CO_{2} is arguably the best tracer (Levin et al., 2003; Turnbull et al., 2006). Pacala et al. (2010) proposed that national fossil fuel emissions of the US be estimated with an inversion based on measurements of radiocarbon in CO_{2}. Assuming 10 000 atmospheric ^{14}CO_{2} observations at 84 sites per year and a transport model of 5^{∘} × 5^{∘} horizontal resolution, they suggested that the inversion could reduce the relative uncertainty in monthly emissions of the US from 100 % (prior) to less than 10 % (posterior). Ray et al. (2014) assumed virtual FFCO_{2} observations are sampled every 3 h from a network of 35 measurement towers, and their inversion at 1^{∘} × 1^{∘} resolution could reduce errors on 8 days countrylevel fossil fuel emissions from about 15 % (prior) down to 7 % (posterior). Basu et al. (2016) developed an inversion system at 1^{∘} × 1^{∘} resolution to account for the fact that ^{14}CO_{2} is not a perfectly accurate tracer of FFCO_{2} alone and that its mixing ratio is also affected by natural fluxes. They showed that given the coverage of ^{14}CO_{2} measurements available in 2010 over North America (969 measurements per year), the US national total fossil fuel emissions can be constrained with a relative precision of 1 % for the annual mean and less than 5 % for most months.
In all these pioneer studies, the actual spatial scale of the areas emitting FFCO_{2} is smaller than the grid sizes of the transport models (from 100 to 500 km). The misfits between the spatial scales controlled or modeled within the inversion system and those of actual emissions or those of the FFCO_{2} patterns in the atmosphere generate errors known as aggregation and representation errors (see Sect. 2.2.2), which strongly affect the inversion of FFCO_{2} emissions (Wang et al., 2017). Those errors were not formally accounted for in previous FFCO_{2} inversion studies.
In recent years, as part of the ICOS project, a rather dense network of standardized, longterm and highprecision atmospheric measurements of CO_{2} has been set up in Europe. Some of the ICOS sites also measure ^{14}CO_{2} and this type of measurement will be extended in the near term with the aim of determining gradients of FFCO_{2} mixing ratios across the European continent. The ICOS atmospheric network is expected to sample 2week integrated ^{14}CO_{2} at about 40 stations (1000 analyses per year; ICOS Stakeholder handbook 2013 at http://www.icosuk.org/ukicos/sites/ukicos/files/documents/Stakeholders Handbook 2013.pdf). In this context, network assessment studies are needed to understand how much this ^{14}CO_{2} network will improve the knowledge on FFCO_{2} emissions.
In this study, we study the potential of an atmospheric inversion system to quantify FFCO_{2} emissions at regional scales (i.e., the size of a mediumsized country in Europe like France or Germany) over the European continent based on continentalscale networks of atmospheric CO_{2} and ^{14}CO_{2} measurements. Special attention is paid to the representation and aggregation errors induced by the use of a coarse grid transport model. Wang et al. (2017) derived the statistics of these errors for the inversion system that we apply here, which is based on the Laboratoire de Météorologie Dynamique's LMDZv4 global transport model (Hourdin et al., 2006) and our study strongly relies on their results. They highlighted that both the representation and aggregation errors have large magnitudes and could thus strongly reduce the ability of the inversion to filter the information on the uncertainties in regional FFCO_{2} emissions. They also stressed the fact that the spatial scales of the correlations in the representation and aggregation errors are smaller than that of the projection in the atmospheric observation space of the typical uncertainties in the prior estimates of regional emissions (called “prior FFCO_{2} errors” hereafter). More precisely, with their modeling configuration they obtained values smaller than 200 km and larger than 700 km, respectively, for these spatial scales. Therefore, if the observation networks are dense enough to provide information at finer spatial scale (typically with distances from a given station to the closest ones being systematically smaller than 700 km), the impact of aggregation and representation errors on the inversion of the regional budgets of FFCO_{2} emissions could be small (Wang et al., 2017). In this study, we account for the aggregation and representation errors using their detailed and quantitative characterization and check whether using dense networks could overcome the limitations brought by coarseresolution transport models and by the uncertainties in the distribution of the emissions at high resolution when retrieving regional emission budgets. Using the error estimates from Wang et al. (2017) ensures that our inverse modeling system does not overestimate the potential of measurement networks that are dense compared to our coarse transport model resolution but whose distances between the sites are larger than the spatial scales of local atmospheric signals from the anthropogenic emissions.
Our inversion system solves for monthly FFCO_{2} emissions in different regions of Europe over a period of 1 year by assimilating synthetic observations of atmospheric gradients of FFCO_{2} mixing ratios obtained from colocated CO_{2} and ^{14}CO_{2} measurements at ICOSlike stations. The study primarily aims at providing a typical quantification of the inversion performances and at understanding qualitatively how the inversion behaves depending on the level of FFCO_{2} emissions, on the knowledge on these emissions and on the network density. Furthermore, we assume here that the uncertainties in the signals from ^{14}CO_{2} fluxes other than the FFCO_{2} emissions, such as that from terrestrial biosphere, oceans, nuclear power plants and cosmogenic production, should have a moderate impact on the order of magnitude of the inversion performances that are analyzed in this study. This leads us to ignore these uncertainties and consider that the only uncertainties in the FFCO_{2} mixing ratios data are related to the instrumental precision of CO_{2} and ^{14}CO_{2} measurements. In practice, in the frame of this study, which focuses on the propagation of uncertainties, this is mathematically equivalent to assuming that ^{14}CO_{2} is a perfect tracer of FFCO_{2}. However, this does not imply that the signal from natural fluxes and nuclear power plants could be ignored when processing real data.
Although the results are presented only over Europe, we use a global inversion system and the global transport model LMDZv4 to ensure that uncertainties in FFCO_{2} emitted over other regions of the globe are properly accounted for and to study their impact on the inversion of the FFCO_{2} emission in Europe. LMDZv4 has a 3.75^{∘} × 2.5^{∘} longitude × latitude horizontal resolution and 19 layers in the vertical between the surface and the top of the atmosphere. This spatial resolution is comparable to that of transport models used in stateoftheart global inversions (Peylin et al., 2013). We assess the potential of this inversion to improve the estimates of regional fossil fuel emissions based (1) on the statistics of the theoretical prior and posterior uncertainties provided by a Bayesian statistical framework and (2) on the statistics of the misfits between the prior and posterior estimates of emissions against the assumed “truth” generated by the choice of another emission inventory independent of the one used as prior (see Sect. 2.3). The second type of assessment is used to test the impact of error structures that can hardly be accounted for by the representation of the prior and model uncertainties in the theoretical framework of the atmospheric inversion.
The presentation of the results first focuses on regional FFCO_{2} emission budgets over 1 year. It also explores the monitoring of the decadal changes of FFCO_{2} emissions, compared to a baseline year, which is also of importance since it corresponds to climate mitigation targets set for the Kyoto Protocol and the Intended Nationally Determined Contribution. The trends of FFCO_{2} emissions over multiple years can be computed using simple regression of series of annual emissions estimates from inventories or atmospheric inversions. The relative uncertainties in decadal trends (e.g., the relative uncertainties in regression slopes) tend to be lower than that in the emission budget of a given year (Pacala et al., 2010), implying that changes can be monitored more accurately than annual budgets. Here, we provide a quantitative analysis of how accurate the trends of national annual FFCO_{2} emission can be monitored using measurements of FFCO_{2} mixing ratios.
The paper is organized as follows. Section 2 gives a full description of the inversion and framework of Observing System Simulation Experiments (OSSEs). Section 3 analyzes the statistics of the posterior uncertainties and misfits from inversions using different observation networks. Section 4 evaluates the potential of atmospheric inversion for the monitoring of decadal changes and discusses the relevance of using a coarseresolution transport model in the inversion system to quantify regional FFCO_{2} emissions. Conclusions are drawn in Sect. 5.
2.1 The configurations of the observation network
We consider three different observation networks, in which the number of the stations ranges from 17 to 233. The minimum network (NE17) includes 17 sites, based on existing European ICOS ^{14}CO_{2} stations in 2016. Using these sites and possible future additional ^{14}CO_{2} stations listed in the 2013 ICOS Stakeholder handbook (available at http://www.icosuk.org/ukicos/sites/ukicos/files/documents/Stakeholders Handbook 2013.pdf), we also consider an intermediate ^{14}CO_{2} network of 43 sites (NET43). The NET17 and NET43 networks have high densities in France, Germany, UK and Switzerland, but remain sparse in eastern Europe (Fig. 1). The corresponding site locations are given in Table S2 in the Supplement. We also test a very dense network of 233 sites (NET233), in which two sites are placed in each European land pixel of the LMDZv4 transport model (Fig. 1c). The NET233 network is denser than NET17 and NET43 in the high emitting regions, e.g., Germany, and also covers the region that is not well sampled by NET17 and NET43. However, the location of its 233 sites is not intended to be optimal since the emissions have a very heterogeneous spatial distribution. Their homogeneous spreads allow us to reduce the impact of representation and aggregation errors (Trampert and Snieder, 1996; Kaminski et al., 2001) and to assess the impact of having a dense network for all control regions.
The highaltitude station Jungfraujoch (JFJ) at 3450 m a.s.l. (meters above sea level) in Switzerland samples free tropospheric air over Europe, assumed to be representative of the “background” concentration. In all the three configurations of the observation network, JFJ is chosen as the reference station. In this study, we assimilate gradients of FFCO_{2} between other sites and JFJ in the inversion. Measurements at other sites than JFJ are all assumed to be made at 100 m a.g.l. (meters above the ground level), the typical height of ICOS tall towers (Kadygrov et al., 2015; Marquis and Tans, 2008).
Wang et al. (2017) have already made a detailed characterization of the distributions of representation errors at the sites considered here and characterized two types of stations based on the population density of the grid cells within which a station is located and on the locations of large point sources (e.g., large power plants). All the sites in different networks are thus categorized as “urban” or “rural” sites according to their results. In the NET233 network, the two sites in each land pixel of the transport model are assumed to be one urban and one rural, distanced by more than 200 km in order to combine data for the structures of representation errors that are different (i.e., which have a different view in terms of the scale of FFCO_{2} emissions). Any of the transport model pixels provides such locations since they have areas of nearly 10^{5} km^{2} (Wang et al., 2017).
2.2 Configuration of the inversion system
The assessment of the potential of different networks to constrain fossil fuel emissions is based on the inversion framework presented by Wang et al. (2017). In this section we summarize the main elements of this framework for which the details can be found in Wang et al. (2017).
2.2.1 Theoretical framework of the Bayesian inversion and diagnostics of the inversion performance in OSSEs
The inversion relies on a Bayesian statistical framework. The estimate of the fossil fuel emission budgets at monthly and regional scales over 1 year, called hereafter the control variables x, is corrected from a prior knowledge of these variables, x^{b} (from a gridded inventory covering the globe). This correction is based on (i) a set of gradients of FFCO_{2} mixing ratios between the different measurement sites and JFJ sampled during the afternoon (see Sect. 2.2.2) across Europe, called hereafter the “observations” y_{o}; (ii) the observation operator H linking y with x, based on the spatial and temporal distribution of the emissions within a control region and within a month, on a linear CO_{2} atmospheric transport model and on the sampling of the gradients between the corresponding sites; and (iii) and (iv) a modeling of the covariances B and R of the distributions of the uncertainties in the prior estimate and of the observation errors. The observation error is a combination of the measurement error, the errors from the model transport, representation errors and aggregation errors. In this study, we ignore the impact on the FFCO_{2} gradients from the transport model initial conditions that are not controlled by the inversion because it is assumed to be negligible (Wang et al., 2017). Assuming that the prior uncertainties and observation errors are uncorrelated with each other and have unbiased and Gaussian statistical distributions, the statistical distribution of the estimate of x, given x^{b} and y_{o}, is also unbiased and Gaussian, and its corresponding mean x^{a} and covariance matrix A are given by
where ^{T} and ^{−1} denote the transpose and inverse of a matrix, respectively.
Equation (1) shows that A depends on neither the value of the observations y_{o} nor the prior emission budgets x^{b} themselves, but rather on the prior and observation error covariance matrices, on the observation times and locations (through the definition of H corresponding to the y space) and on the observation operator. Equation (2) shows that the actual value of x^{a} also depends on the observations y_{o} and on the prior emission budgets x^{b}.
A common performance indicator is the theoretical uncertainty reduction (UR) for specific budgets of the fossil fuel emissions (at control or larger space and timescales), defined by
where σ^{a} and σ^{b} are the standard deviations of the posterior and prior uncertainties in the corresponding budget of emissions. Such an indicator can directly be derived from the modeling of B and from the theoretical computation of A by Eq. (1). Of note is that the scores of uncertainty and of UR given in this study will refer to the standard deviation of the theoretical uncertainty in a specific emission budget.
However, if the modeling of B and R does not match the actual statistics of the prior and observation uncertainties, or if the theoretical framework of the inversion (assuming that all sources of uncertainty have unbiased and Gaussian distributions, prior and observation errors are uncorrelated and the observation operator is linear) is not well satisfied, such a theoretical computation of UR may not reflect the actual performance of the inversion. Wang et al. (2017) derived the statistics of the different components of the observation errors for the same inversion framework as used here. Their statistics of the representation and aggregation errors were based on the comparison of transport model simulations made at high and low spatial resolutions. They highlighted the fact that the distribution of these errors depart from purely Gaussian distributions and that their covariances can hardly be characterized by the relatively simple models traditionally used in atmospheric inversion systems. In this study, we thus test the inversion system with OSSEs using synthetic truth and errors to build x^{b} and y_{o}, which better reflect the type of observation errors found by Wang et al. (2017). We use Eq. (2) to derive the estimates of x^{a} and we analyze the misfits between x^{b} and x^{a} against the synthetic true emission budgets x^{t}. This leads us to define an alternative indicator of the inversion performance, called misfit reduction (MR) hereafter. While this indicator does not provide an exhaustive statistical view of the uncertainty in the inverted emissions, it is used to evaluate the confidence in the more complete (with a full covariance estimate rather than just a realization of the distribution) but more theoretical computation of the posterior uncertainties and of the UR based on Eq. (1). We write the MR for specific budgets of the fossil fuel emissions (at control or larger scales of space and time) as follows:
where ε^{a} and ε^{b} are the posterior and prior misfits between the inverted and prior emission budgets against true values for the corresponding emission budgets. MR range from negative values (when the inversion deteriorates the precision of the estimation) to 1 (or “100 %”, when the inversion provides a perfect estimate of the emissions).
We focus on uncertainties and misfits at both monthly and annual scales. However, we can have only one practical realization for x^{b}, y_{o} and x^{a} following the protocol presented in Sect. 2.3. Therefore, the assessment of the performance of the inversion for a given region–month using the corresponding score of MR may be over or underestimated due to the lack of sampling of the prior and observation errors. Consequently, at monthly scale, in order to strengthen the evaluation of the theoretical uncertainties based on these single realizations of the prior and posterior misfits, we compare, for a given region, the quadratic mean of the 12 monthly misfits (called “monthly misfits” without mention of a specific month in Sect. 3) to the quadratic mean of the standard deviations of the 12 monthly uncertainties (called “monthly uncertainties” without mention of a specific month in Sect. 3), which characterizes the average monthly uncertainties over the year. This computation implicitly assumes that the 12 monthly misfits through a year follow the same statistical distribution and represent 12 independent realizations of this distribution. In such a situation, the comparison between the averages of the prior and posterior monthly misfits give a good indications of the error reduction that should not be highly skewed by sampling errors. In the result section, for a given region i, UR and MR scores derived at the “monthly” scale without mention to a specific month will correspond to the relative difference between the prior and posterior values of these average monthly uncertainties and misfits from a whole year of inversion:
At the annual scale, the diagnostics of UR will have to be compared to MR values for single realizations of the annual misfits. In addition, we discuss the scores of the relative uncertainty and misfit, defined as the ratios of the absolute uncertainties and misfits to the absolute prior emission budgets.
2.2.2 Practical setup
Control vector
The inversion system has a global coverage and controls monthly budgets of FFCO_{2} emissions for a set of regions during the year 2007. The map of these regions is given in Fig. 2a. The space discretization of regions is higher where emissions are the largest in Europe (area of interest, Fig. 2b) and also in the US and China. In other areas with lower emissions or where observational data to further constrain the prior emissions are lacking (Fig. 2a and Table S1), the size of the control regions is much larger and can reach that of a continent. The spatial resolution of the control vector (a region) in central and eastern Europe corresponds to the typical size of a mediumsized European country, but in western Europe apart from Spain, Portugal and Ireland, where emissions are the highest, the control variables correspond to subnational regions (e.g., southern and northern UK, southern and northern Italy, western and eastern Germany, western and eastern France; Fig. 2b). Monthly emissions over the ocean are included in the control vector, but the ocean is considered as one large region. In total, the world is divided into 54 land regions and 1 ocean region (Table S1). The inversion solves for the 12 monthly budgets of emissions for these regions, but not for the spatiotemporal distributions within each region and month. In our framework, choosing the year 2007 for the inversion only impacts the meteorological conditions and thus the atmospheric transport conditions. We assume that the atmospheric transport conditions in 2007 are representative of average conditions. We also ignore the impact of interannual variations of FFCO_{2} emissions, which is usually less than 4 % (Levin and Rödenbeck, 2008), and of their prior uncertainty (see below the configuration of the prior uncertainty matrix, which is a function of the emissions).
Time selection of data to be assimilated
Current atmospheric ^{14}CO_{2} samples in Europe are usually filled continuously over the course of 2 weeks (Vogel et al., 2013; Levin et al., 2013). However, stateoftheart inversion systems generally make use of data during the afternoon only due to limitations of transport models in simulating nighttime mixing ratios near the ground. Given the ability to have an intermittent filling of air samples for ^{14}C analysis (Turnbull et al., 2016; Levin et al., 2008), we thus define the observations to be selectively sampled only during the afternoon (12:00–18:00 local time). Since the cost of the ^{14}CO_{2} analysis of one sample is presently high, monitoring of ^{14}CO_{2} (and thus FFCO_{2}) during a whole year favors the choice of integrated samples at the weekly to 2week scale (Levin et al., 1980; Turnbull et al., 2009; Vogel et al., 2013). In this study, we first consider 2week integrated afternoon data. More precisely, we first consider 2week averages of afternoon FFCO_{2} gradients with respect to JFJ. In addition, we present tests with daily afternoon gradients, for which the corresponding sampling scheme would be more costly. Sampling FFCO_{2} observations at high temporal resolution should decrease the weight of the random errors on longer timescales, which should improve the potential of the inversions of monthly to annual emission budgets. While inversions are conducted with 2week samplings for the three networks, daily sampling is tested for NET43 only, which is sufficient to evaluate the usefulness of high frequency sampling.
Observation operator
The atmospheric FFCO_{2} mixing ratios are influenced by the 3D initial FFCO_{2} distribution and by surface emissions during the year. In this study, the inversion rescales all emissions during 1 year (here 2007) and we ignored initial conditions on 1 January which are rapidly transported out of Europe and do not cause subsequent FFCO_{2} gradients between European sites (Wang et al., 2017). The observation operator is restricted to a matrix H which consists of a chain of three suboperators, H =H_{samp}H_{transp}H_{distr}, where H_{distr} distributes regional monthly emission budgets into a gridded emission map at the resolution of the transport model, H_{transp} is the atmospheric transport model and H_{samp} samples the FFCO_{2} gradients with respect to JFJ corresponding to the observation vector from the transport model outputs (Wang et al., 2017).
We use the highresolution (0.1^{∘}) annual FFCO_{2} emission map from the PKUCO_{2} inventory in the year 2007 (Wang et al., 2013) to distribute the emissions in space within each region. PKUCO_{2} is an annual emission map with no temporal profile, so that the modeled temporal distribution in H_{distr} is flat between months. This implementation of H_{distr} is denoted ${\mathbf{H}}_{\mathrm{distr}}^{\mathrm{PKU}}$.
The offline version of the general circulation model of LMDZv4 forms H_{transp}. Atmospheric transport simulations was nudged to analyzed wind fields from the European Centre for MediumRange Weather Forecasts (ECMWF) Interim Reanalysis (ERAInterim; Dee et al., 2011) for the year 2007. We denote this implementation of H_{transp} by ${\mathbf{H}}_{\mathrm{transp}}^{\mathrm{LMDZ}}$.
The sampling of FFCO_{2} gradients relies on the extraction of individual simulated mixing ratio data at the measurement locations and chosen temporal sampling frequency, followed by the computation of differences (gradients) between time series of FFCO_{2} mixing ratios at each site and that at the JFJ reference site. The mixing ratio data for a given site are sampled at the chosen sampling height in the transport model grid cell containing this site. We recall that the sampling height is 100 m a.g.l., the first level of LMDZv4, except for JFJ being at 3450 m a.s.l., the sixth level. The resulting implementation of H_{samp} is denoted ${\mathbf{H}}_{\mathrm{samp}}^{\mathrm{coloc}}$.
In sum, the observation operator used in the practical configuration of the inversion system is defined by H^{prac}= ${\mathbf{H}}_{\mathrm{samp}}^{\mathrm{coloc}}$ ${\mathbf{H}}_{\mathrm{transp}}^{\mathrm{LMDZ}}$ ${\mathbf{H}}_{\mathrm{distr}}^{\mathrm{PKU}}$.
Prior error covariance matrix
Emission estimates from inventories are limited to annual and national scales and rarely provide systematic assessments of uncertainties. There are a limited number of datasets providing emission maps at higher spatial–temporal resolutions. Although there have been some efforts to compare such FFCO_{2} emission maps (Macknick et al., 2009; Ciais et al., 2010; Andres et al., 2012, 2016), the ability to characterize the uncertainties of an emission inventory is limited, especially for subnational and subannual scales. In this study, we use different streams of information to model the prior emission uncertainty covariance matrix B and we use two different configurations of this matrix in the inversions.
The first configuration of the B matrix, called here notional or B^{notion}, is related to the notional estimates of (1sigma) uncertainties for national emissions claimed by inventory compilers to range from 1 to 2.5 % for the USA (US EPA, 2015), 2 to 7 % for European countries (Andres et al., 2014; Ballantyne et al., 2015) and 7.5 to 10 % for China (Gregg et al., 2008; Liu et al., 2015). However, Ciais et al. (2010) found that the ratios between geographically distributed emission maps, even after correction for inconsistencies and aggregated at national scale, ranged from 0.86 to 1.5, which is larger than the uncertainties claimed by inventory compilers. In this study, the prior uncertainty covariance B^{notion} of monthly emissions is set up based on three constraints: (1) the relative uncertainty in annual emission equals 10 % for US and European national budgets, 15 % for China and 10 % for individual control regions outside US, Europe and China; (2) uncertainties in monthly emissions have a 2month exponentially decaying temporal autocorrelation; and (3) spatial correlations between uncertainties in monthly emissions across adjacent regions within the same country are fixed to −0.2, a negative value to account for the fact that subnational emissions are usually disaggregated from national inventories, so that a positive bias in part of a country must be compensated by a negative one in another. All other spatial correlations in B^{notion} are assumed to be null, and the overall correlation matrix in B^{notion} is derived from the Kronecker product of temporal and spatial correlation matrices (assuming that the correlation between two control variables is given by the product of the spatial and temporal correlations between the two corresponding control regions and the two corresponding time window, respectively). The full computation of B^{notion} is detailed in Appendix A. With this setting, prior uncertainties in monthly emissions can exceed 10 % and be as large as 30 % for some subnational control regions.
The second configuration of the B matrix, known as empirical or B^{empiric}, is based on the empirical derivation of the statistics of the differences between two spatially gridded emission maps (which will be used to define the prior and true estimate of emissions in the OSSEs; see Sect. 2.3). The two maps are PKUCO_{2} (Wang et al., 2013, http://inventory.pku.edu.cn/) and IEREDG (available at http://carbones.ier.unistuttgart.de/wms/index.html), both corresponding to the year 2007. The IEREDG map combined EDGAR annual map with countryspecific temporal profiles (monthly, daily and hourly) from IER. In general, the differences in annual emissions from the control regions in Europe between these two emission maps range from 3 to 20 %, except for the Balkans where they reach up to 44 %. We assume that there is no spatial correlation of the prior uncertainty between different control regions. For each control region of the globe, the statistics of the difference between the monthly emission budgets from the two maps are fitted by a covariance model that combines four different covariance matrices, with exponentially decaying temporal correlations at timescales of 1 month, 3 months and 6 months for the first three ones, respectively, and a full temporal correlation over the year for the fourth one (representing the annual bias on the prior emissions). The mathematical formulation for this computation and the full derivation of B^{empiric} is detailed in the Appendix B.
B^{empiric} is built using an error covariance model which cannot perfectly characterize the structure of the differences between the PKUCO_{2} and IEREDG budgets at the control resolution, which will be used to derive realistic x^{b} and x^{t}, respectively, and thus the “actual prior errors“ in the OSSEs with synthetic data (see Sect. 2.3). However, by construction, B^{empiric} better fits these errors in our OSSEs than the B^{notion} matrix in terms of both the standard deviation of the uncertainty at the 1 month–regional scale and the temporal correlations. The differences between the results of the inversions using either B^{empiric} or B^{notion} will be used to give an estimate of the range of the inversion skills as a function of different assumptions regarding the prior uncertainty in emission budgets.
Observation error covariance matrix
Wang et al. (2017) derived estimates of the observation errors in FFCO_{2} gradients across Europe when using the same inverse modeling framework as in this study. They analyzed four sources of observation errors (i.e., sources of misfits when comparing the modeled to the measured FFCO_{2} gradients other than the uncertainties in the estimates of the emission budgets at the 1month and regional scale), one related to the FFCO_{2} data and three to the observation operator:

The measurement error ε_{i} on FFCO_{2} gradients is simply assumed to be 1 ppm with no temporal and spatial correlations, which corresponds to the typical precision of the analysis of air samples by accelerator mass spectrometry (AMS) for ^{14}CO_{2} (2–3 ‰) (Hammer et al., 2016; Turnbull et al., 2014).

The representation error ε_{r} arises from the mismatch between the coarse resolution of modeled emissions and concentrations in the observation operator (here the transport model) and the spatial variability of the actual emissions and concentrations.

The transport error ε_{t} is due to discretized and simplified equations for modeling transport, using a given meteorological forcing in practice.

The aggregation error ε_{a} arises from the mismatch between the control resolution (budgets of regions in each month) and the resolution of the emission modeled in the observation operator (here the transport model). It reflects uncertainties in H_{distr}.
In this study, we use the estimates of the standard deviations and of the correlation functions for these different types of observation errors from Wang et al. (2017) to set up the R matrix. Wang et al. (2017) sampled representation and aggregation errors by using simulations with a mesoscale (with higher resolution than LMDZv4) regional transport model and by degrading the spatial and temporal resolution of the emission maps in the input of this model and in the output FFCO_{2}. Based on these samples, the standard deviation of ε_{r} was characterized by a function of season and on whether a station is “urban” or “rural” (see Sect. 2.1). For ε_{a}, the standard deviation for spring–summer and autumn–winter were derived. The standard deviation of the transport error at a given site is set up proportionally to the temporal standard deviation of the 1yearlong time series of the highfrequency variability of the detrended and deseasonalized simulated daily mean afternoon mixing ratios in the grid cell of the transport model, at which the sites are located. Such an estimation of transport error which relies on some results from Peylin et al. (2011) aims at representing the typical value for global transport models, not that of the specific transport model used in this study. The temporal autocorrelations in the representation and aggregation errors were characterized by Wang et al. (2017) using the sum of a longterm component and a shortterm component: r(Δt)=a $\times {e}^{\mathrm{\Delta}t/b}+(\mathrm{1}a)\times {e}^{\mathrm{\Delta}t/c}$, where Δt is the time lag (in days) and a, b and c are parameters optimized by regressions against the samples of the errors. Furthermore, we do not include temporal autocorrelations in the transport error for simulated daily to 2week mean afternoon FFCO_{2} gradients, since previous studies of the autocorrelations of the transport errors have not evidenced that they should be significant at daily scale (Lin and Gerbig, 2005; Lauvaux, 2009; Broquet et al., 2011). This choice follows the corresponding discussion by Wang et al. (2017) and implicitly ignores that transport model errors likely bear longterm components (often referred to as “biases”; Miller et al., 2015) even when being dominated by components on short timescales. The corresponding values of the standard deviation and the modeling of temporal autocorrelation of the observation errors for 2week/daily mean afternoon FFCO_{2} gradients are listed in Table S3 and Table S4.
A simpler account of the spatial correlations in the observation errors is derived from the diagnostics of Wang et al. (2017). We do not account for the spatial correlation in the representation error, as the scale of the spatial correlation according to Wang et al. (2017), i.e., 55–89 km, is much smaller than the size of the grid cells of the global transport model ${\mathbf{H}}_{\mathrm{transp}}^{\mathrm{LMDZ}}$ used for the inversion. When there are more than two sites located in the same grid cell of the transport model, we consider that the aggregation errors and the transport errors are fully correlated between these sites, according to the definition by Wang et al. (2017). We do not account for spatial correlations between aggregation errors for measurements made at sites in different grid cells, because the scale of the spatial correlation is 171 km and is smaller than the size of the grid cell, according to Wang et al. (2017). Finally, we do not account for spatial correlations between transport errors or measurements made at sites in different grid cells.
Assuming that all these sources of errors are independent from each other and have Gaussian and unbiased distributions, i.e., ε_{i} ∼ N(0,R_{i}), ε_{r} ∼ N(0,R_{r}), ε_{t} ∼ N(0,R_{t}) and ε_{a} ∼ N(0,R_{adistr}), R is given by the sum of the covariance matrices corresponding to each of them: $\mathbf{R}={\mathbf{R}}_{i}+{\mathbf{R}}_{\mathrm{r}}$ + R_{t}+R_{a}.
2.3 Configurations of the OSSEs
In this study, we consider two types of OSSEs corresponding to the two configurations of prior error covariance matrix B^{notion} and B^{empiric}. The first OSSEs use B^{notion} (called here INVN), while the second type of OSSEs uses B^{empiric} (called here INVE). As discussed in Sect. 2.2.1, in both types of OSSEs, the theoretical computation of the posterior uncertainty and UR is based on Eq. (1). These diagnostics would perfectly characterize the performance of the system if the prior uncertainty and the observation errors have Gaussian and unbiased distributions that are perfectly characterized by the setup of the prior uncertainty covariance matrix B and observation error R in the inversion system. In both types of OSSEs, these diagnostics are evaluated based on a practical application of Eq. (2) and on the analysis of posterior misfits and MR, with a synthetic truth (true emissions and true observation operator) and observations that are generated in a similar way as in Wang et al. (2017). Here, the “actual” prior and observation errors have a complex origin and structure which are not perfectly adapted to the unbiased and Gaussian assumptions and not perfectly reflected by the setup of the prior uncertainty covariance matrix B and observation error covariance matrix R in the inversion system, even in INVE where B=B^{empiric} and R are fitted to the “actual” prior and observation errors. Of note is that in INVN, B^{notion} has significant inconsistencies with the actual differences between x^{b} and x^{t}, so that, in this experiment, the analysis of the posterior misfits and MR will be used to evaluate the performance of the inversion when using a poor configuration of the prior uncertainty covariance matrix in the inversion system in addition to accounting for errors which hardly fit with the assumption that their distribution is Gaussian and unbiased. This corresponds to situations for which there is little knowledge about the uncertainties in the inventories used for inversions with real data. The analysis of misfits and MR in INVN is thus more pessimistic than that in INVE.
In the OSSEs, the synthetic prior estimate of the regional–monthly emissions x^{b} is built based on the emissions from PKUCO_{2} (x^{PKU} hereafter). The synthetic true emission budgets and synthetic observations are modeled using a realistic representation the “actual” emission budgets x^{t} and of the “actual” H_{distr} operator based on the relatively independent IEREDG inventory. The synthetic true regional–monthly emissions and the synthetic true H_{distr} operator are thus referred to as x^{IER−EDG} and ${\mathbf{H}}_{\mathrm{distr}}^{\mathrm{IER}\mathrm{EDG}}$ hereafter. The synthetic observations are generated using x^{IER−EDG} and the operator ${\mathbf{H}}^{\mathrm{OSSE}}={\mathbf{H}}_{\mathrm{samp}}^{\mathrm{coloc}}{\mathbf{H}}_{\mathrm{transp}}^{\mathrm{LMDZ}}{\mathbf{H}}_{\mathrm{distr}}^{\mathrm{IER}\mathrm{EDG}}$, which relies on the same ${\mathbf{H}}_{\mathrm{samp}}^{\mathrm{coloc}}$ and H_{transp} operators as the H^{prac} observation operator used in the inversion system. Consequently, the difference between H^{OSSE} and H^{prac} underlies aggregation errors only. Therefore, in order to account for the transport, representation and measurement errors, the data H^{OSSE}x^{IER−EDG} are perturbed following the statistics of the corresponding errors as detailed in Sect. 2.2.2.
The parameters of the two inversion configurations are summarized in Table 1 and Fig. 3. All the combinations of networks and data temporal sampling described in Sect. 2.1 and 2.2.2 are tested with the two configurations of OSSEs. The resulting eight OSSEs are listed in Table 2.
3.1 Assessment of the performance of inversions when using the NET17/NET43 and 2week integrated sampling
3.1.1 Analysis of the results at the regional and monthly scale
Figure 4 shows the URs of monthly emissions using the NET17 and NET43 networks and 2week sampling (N17W, E17W, N43W and E43W in Table 2). With NET17, INVN and INVE inversions show similar spatial patterns of UR scores. The largest UR occurs in the region of western Germany, being 34 % for inversion N17W and 38 % for E17W. The URs are also significant in eastern Germany for both inversions. This stems from the fact that several stations are located around and within these regions and that the emission in these regions are higher than those in other regions. Moderate UR values are found for Benelux (12 %) and eastern France (15 %) in inversion E17W and the UR values elsewhere are marginal. Going from NET17 to NET43 adds a significant increase (improvement) of the UR for southern UK (from 3 to 23 %), northern Italy (from 3 to 18 %) and eastern Europe (from 2 to 15 %) in INVN (Fig. 4e). The increase of UR in E43W, compared with the UR in E17W, mainly occurs in eastern France (from 16 to 33 %) and the Balkans (from 3 to 13 %). Because the added stations in NET43, compared to NET17, are mostly located outside Germany, the URs over western and eastern Germany are not significantly improved (Fig. 4e and g). Despite their different URs for specific regions, both types of inversions highlight the overall increase in the UR for western European regions by increasing the number of sites from NET17 to NET43.
The differences in the spatial patterns of UR between INVN and INVE inversions shown in Fig. 4 reveal the high sensitivity of UR to the configuration of the prior uncertainties. Figure 5a and b show the prior uncertainties associated with the two configurations of B^{notion} and B^{empiric}. The regions where these uncertainties and thus the potential for reducing these uncertainties from the inversion are the highest are very different between B^{notion} and B^{empiric}. For example, B^{empiric} defines a much larger uncertainty than B^{notion} over eastern France (43 % vs. 16 %) while the opposite is true for southern UK (4 % vs. 14 %). As a result, the UR of eastern France is 33 % in E43W and 8 % in N43W, and the UR of southern UK is 2 % in E43W and 23 % in N43W.
Complementing the uncertainty reduction, Fig. 5 shows the prior and posterior uncertainties and provides insight into the precision of the estimates of monthly FFCO_{2} emissions after inversion with NET17 and NET43 and 2week sampling. For example, using NET17, uncertainties in monthly FFCO_{2} emissions are reduced from 29 % (or 17 %) in the prior estimates to 17 % (or 9 %) in the posterior estimates for western Germany in INVN (or INVE). Using additional sites in NET43 reduces the uncertainties in monthly FFCO_{2} emissions in southern UK from 25 % in the prior estimates to 19 % in the posterior estimates in INVN and reduces the uncertainties in monthly FFCO_{2} emissions in eastern France from 44 % in the prior estimates to 29 % in the posterior estimates in INVE. Like the UR, posterior uncertainties and their spatial variations are different between INVN and INVE inversions and demonstrate a strong dependence on the choice of B=B^{notion} or B=B^{empiric}.
The scores of the MR and misfits of monthly emissions in both inversions using NET17 and NET43 and 2week sampling are shown in Fig. 4 (b, d, f, h) and Fig. 5 (b, d, f, h, j, l). In INVE, there are slight differences between posterior misfits and uncertainties and between MR and UR. For example, for E43W, the MR (21 %) for Iberian Peninsula is larger than the UR (5 %), while the MR (40 %) for western Germany is slightly smaller than the UR (47 %). Despite such differences, the spatial patterns of the MRs in Fig. 4 and posterior misfits in Fig. 5 are close to those of the URs and posterior uncertainties. In contrast, there are large differences between the statistics of posterior misfits and posterior uncertainties and between MRs and URs in INVN. In some regions, such as southern UK (MR = −0.9 in N17W and MR = −1.4 in N43W) and northern Italy (MR = −0.4 in N17W and MR = −1.5 in N43W), the MRs are negative and far below zero. This means that the posterior misfits are even larger than the prior misfits (comparing Fig. 5f and j with b), and thus a degradation of the emission estimates from the inversion is seen in these regions when assimilating FFCO_{2} data. This suggests that the theoretical computation of posterior uncertainty poorly characterizes the actual performance of the inversion in practice when the configuration of the prior uncertainty covariance matrix and the actual prior errors are not consistent.
Figure 6 shows the correlations in the prior and posterior uncertainties in monthly emissions from different regions, and their differences in inversions N43W and E43W. After assimilating the observations, the change of correlations mainly occurs among regions that have large URs. In both inversions, there are negative correlations between the posterior uncertainties in monthly emissions from some neighboring regions, in particular between western Germany and eastern Germany (from −0.27 to −0.18 depending on the months). The negative correlations between the posterior uncertainties in monthly emissions of different regions indicate that NET43 brings a strong constraint on the budgets over a large area but does not separate individual regions so well. At the same time, the temporal correlations in the posterior uncertainties between different months for a given region also change after the inversion. For example, in INVN, temporal correlations between posterior uncertainties in monthly emissions for a specific region are smaller than those between prior uncertainties for that region when the time lag is smaller than 3 months, while they are larger than the ones in prior uncertainties when the time lag exceeds 3 months (Fig. 6e). Because our setup of B^{notion} only considers an exponentially decaying temporal correlation with a correlation length of 2 months (Sect. 2.2.2), these longerterm correlations in monthly posterior uncertainties must hence be driven by the temporal correlations in observation error, which contains a longterm component (see Sect. 2.2.2). In contrast, in INVE, where B^{empiric} includes a component with annualscale temporal correlations, the temporal correlations between posterior uncertainties in the monthly emissions are smaller than those between prior uncertainties. The analysis of the correlations in the prior or posterior uncertainties from N17W and E17W leads to very similar conclusions, but is not shown here.
3.1.2 Analysis for annual emissions
We compare the performance of different inversions to constrain annual mean FFCO_{2} emissions. Corresponding UR and MR values are shown in Fig. 7. The patterns and values of UR for annual emissions are very similar to those at monthly scale (Fig. 4). High URs and MRs occur mostly in regions where the observation networks are dense and the emissions are high. For example, up to 47 % UR is achieved for annual emissions in western Germany when using network NET43 and 2week sampling. As a result, the posterior uncertainties of annual fossil fuel emissions, when using NET43 with 2week sampling, are 10 % (or 4 %) for southern UK, 8 % (or 8 %) for western Germany and 15 % (or 28 %) for eastern France in INVN (or INVE).
Both the spatial spread and the magnitude of the MR of annual emissions in INVE (Fig. 7d and h) are larger than those of the UR. The differences between MR and UR are much larger at annual than at monthly scale (when comparing Figs. 4 and 7). The cause of the discrepancy between UR and MR was presented in Sect. 2.2.1, and it may have a larger impact at the annual scale than at the monthly scale due to the evaluation of annual UR scores to annual MR values corresponding to single realizations of the misfits. In INVN, the spatial spread and the magnitude of the MR are still significantly different from those of the UR and the MRs for some regions are still negative and far below zero.
3.2 Impact of using daily measurements and using a dense observation network
Figure 8 shows the URs and MRs of monthly emissions from inversions using NET43 and daily sampling and from inversions using NET233 network and 2week sampling (N43D, E43D, N233W and E233W in Table 2). When using NET43 and daily sampling, the URs of monthly emissions are generally larger (improved) than when using 2week sampling for all regions. The differences between the UR values of monthly emissions with daily and with 2week sampling are larger (meaning more improvement with daily sampling) over the regions where the network is dense and the emissions are high. For instance, the URs of monthly emissions for western Germany are as high as 62 % (or 67 %) in INVN (or INVE). When using the much denser NET233 network but with a lower 2week sampling (Fig. 8d–f), we found that URs of monthly emissions in some regions that were poorly sampled by networks NET17 and NET43 are largely improved. For instance, the UR value in eastern Europe is 36 % in N233W (compared with 15 % in N43W) and is 73 % in the Balkans in E233W (compared with 13 % in E43W). In principle, large regions tend to encompass more sites and to be surrounded by more sites than small regions and thus may have more observations to improve their estimates of emissions. However, in both N233W and E233W, the URs for regions with a large area like northern Europe are still limited to below 5 %. Large URs are identified over the regions whose absolute uncertainties are high, revealing the important roles of the absolute prior uncertainties when using the coarseresolution transport model in the inversion of FFCO_{2} emissions over Europe. The scores of MR match relatively well those of UR only in E43D and E233W (INVE inversions) but not in N43D and N233W (INVN inversions) (comparing Fig. 8d versus Fig. 8c, and Fig. 8h versus Fig. 8g). Even though the temporal frequency or spatial coverage of the sampling of the FFCO_{2} mixing ratios is largely improved using NET43 and daily sampling, or NET233 and 2week sampling, the MRs are still negative and below zero for a large number of regions in Europe.
4.1 Implication for longterm trend detection of fossil fuel emissions
In the Copenhagen conference of parties, the European Union (EU) set up the goal to decrease its emissions (in CO_{2} equivalents) by 80–95 % below 1990 by 2050 (European commission, 2010). In 2015, the EU Intended Nationally Determined Contribution (INDC) submitted to the UNFCCC set a target of 40 % domestic greenhouse gas emissions reduction below 1990 levels by 2030. These targets translate into annual reductions compared to 1990 of roughly 1 % per year in the 2020s, 1.5 % in the decade from 2020 until 2030 and 2 % in the two decades until 2050 (European commission, 2010). Levin and Rödenbeck (2008) showed that, taking into account the interannual variations of the atmospheric transport, changes of 7–26 % between two consecutive 5year averages of FFCO_{2} emissions in southwestern Germany could be detected at the 95 % confidence level with monthly mean gradients of ^{14}CO_{2} observations between two stations (Schauinsland and Heidelberg) and the reference site JFJ. Such a detectability skill is clearly insufficient to support the “verification” of 1–2 % annual change of emissions per year (meaning 5–10 % changes between two consecutive 5year averages) corresponding to the EU targets. Here, we evaluate the skill to detect trends when using the much larger ^{14}CO_{2} networks and the atmospheric inversion framework detailed in this study.
The uncertainty in the trend of FFCO_{2} emissions calculated from the linear regression of a series of annual estimates is independent of this trend itself (see Appendix C). This allows us to extrapolate posterior uncertainties in annual emissions from this study to investigate the detectability of emissions trends. Assuming that the absolute values of the standard deviations of the uncertainties in annual emissions of different years (in Tg year^{−1}) are identical and that these uncertainties are fully independent, we calculate the uncertainty in relative trends for different time lengths as a function of the posterior uncertainty in annual emissions (Table 3). Here, the relative trend is defined as the ratio of the linear regression slope of emissions to the emission in the base year. Using NET17 or NET43 and 2week sampling, the posterior uncertainty in annual emissions of some wellsampled regions, e.g., Germany, is largely below 10 % (Sect. 3.1.2). In this case, given Table 3, the uncertainty in the relative trends over 20 years is in the range of 0.27 to 0.43 % yr^{−1}. However, the uncertainty in trend estimation over 10 years would be 1 % yr^{−1}. The EU target of 1–2 % annual reduction could thus be verified using NET17 or NET43 in these wellsampled regions over a period of 20 years but not over a period of 10 years. For other regions with sparser coverage of stations, either the posterior uncertainty in annual emissions are much larger than 10 % (e.g., in Ireland and Balkans in INVE) or the URs (or MRs) of annual emissions are marginal (meaning no improvement in the estimate of annual emissions from the inversion), so that the verification of the trend in these regions based on the inversion framework of our study is thus challenging.
Our assumption that the posterior uncertainties in annual emissions have the same amplitude from year to year should not strongly drive the results, so the results here give a good indication of the level of uncertainty in the trend detection for a typical level of uncertainty at the annual scale. However, changes of the transport from year to year or on decadal scales (Aulagnier et al., 2009; Ramonet et al., 2010) may change the level of the sensitivity of the observations to the emissions, i.e., the level of the atmospheric constraint of the inversions which leads to uncertainty reduction, and thus the level of posterior uncertainties on the same timescales. A more complex model accounting for varying levels of annual posterior uncertainties may thus be useful to refine the quantification of the uncertainty in the trends. Of note is that the level of uncertainties in the trends could be increased if the modeling framework accounts for the trends in the transport or in the sources of ^{14}CO_{2} other than the fossil fuel emissions. Such trends in the modeling errors may have to be considered for applications with real data.
4.2 Adequacy of largescale atmospheric inversion for the monitoring of fossil fuel emissions and potential improvements of the inversion skills
In this study, we showed that given the NET17 ^{14}CO_{2} measurement station network, the potential of our atmospheric inversion of fossil fuel emissions at large scale using a coarseresolution model is limited (Figs. 4 and 5). When using the denser NET43 network and 2week sampling and assimilating ∼ 1000 measurements per year, the potential of the inversion system is improved, yet mainly over high emitting regions. In particular, Sect. 3 indicates that the inversion can significantly reduce the uncertainties and misfits in the estimate of monthly emission budgets for large or high emitting regions, even though the observation operator used by the inversion assumes flat temporal profiles for the emissions while the true emissions have diurnal, weekly and seasonal temporal profiles. This confirms that the 2week mean afternoon ^{14}CO_{2} samplings integrate the atmospheric signal transported from both daytime and nighttime emissions across Europe, which can be filtered from the signal from local emissions to provide largescale information on the emissions.
We made sure (as compared to previous OSSEs published for the USA) to account for aggregation and representation errors, which is the reason why our inversions do not provide as impressive error reductions (uncertainty and misfit) as the misfit reduction of Ray et al. (2014) and Basu et al. (2016). However, we still did not account for all sources of uncertainty. Indeed, we assumed that atmospheric FFCO_{2} gradients can be derived from the ^{14}CO_{2} measurements with a precision of 1 ppm. This 1 ppm standard deviation approximately corresponds to the errors in the atmospheric measurements and ignores uncertainties in the conversion of ^{14}CO_{2} and CO_{2} measurements into FFCO_{2}. Uncertainties in various fluxes that influence atmospheric ^{14}CO_{2}, such as those from cosmogenic production, ocean, biosphere and nuclear facilities, bring errors to the conversion of ^{14}C measurements into FFCO_{2} (Lehman et al., 2013; Vogel et al., 2013). Over land regions, heterotrophic respiration is expected to be one of the main contributors to the largescale signals of atmospheric ^{14}CO_{2} (Turnbull et al., 2009). Over some areas of Europe, ^{14}C emissions from nuclear facilities may have even larger influences than plant and heterotrophic respiration (Graven and Gruber, 2011). The level of uncertainties in these fluxes and how much their influences on the FFCO_{2} gradients will introduce additional errors remains to be quantified. According to the simulations by Graven and Grubber (2011), Turnbull et al. (2009) and Miller et al. (2012), one can expect that the impact of signals from the uncertainties associated with the estimate of these fluxes, on the conversion of atmospheric ^{14}CO_{2} measurements to FFCO_{2}, is typically below 1 ppm, i.e., much smaller than the observation errors that have been accounted for in this study, thus justifying that we have ignored these fluxes. However, these signals may have complex spatial and temporal patterns, leading to significant impact on the quantification of the inversion performances. Uncertainties in the trends of these fluxes could also impact that in the fossil fuel trend detection. Therefore, in future studies, especially if working with real data, the impacts from uncertainties in the ^{14}CO_{2} fluxes other than the anthropogenic fossil fuel emissions need to be investigated and accounted for by modeling all these ^{14}CO_{2} fluxes, their atmospheric ^{14}CO_{2} signals and associated uncertainties.
In Sect. 3.3, we explored the concept of having more observations assimilated in the inversion system by increasing the sampling frequency and expanding the observational network. Wang et al. (2017) showed that because the representation error, aggregation error and the prior FFCO_{2} errors have very similar error structures in time, it is difficult to use daily sampling to filter uncertainties in the prior estimate of the emissions. However, we showed that when using NET43 and daily sampling, the UR of monthly emissions is still much larger than using 2week sampling. This stems from the fact that having daily sampling decreases the weight of the measurement errors at the 2week to annual scales, which are assumed not to have temporal autocorrelations. We also tested the concept of extending the observation network to a very dense configuration, NET233, with a wide coverage across Europe. It exhibits a significant increase in the UR of monthly emissions across Europe, especially over eastern Europe. Emissions in Northern Europe, however, remain poorly constrained. This illustrates the limitation of using a coarseresolution transport model to quantify fossil fuel emissions. Such a limitation is attributed to the following facts: (1) the observation errors in the inversions are larger than the prior FFCO_{2} error (typically 0.21 ppm for 2week mean afternoon FFCO_{2} gradients and 0.49 ppm for daily mean afternoon FFCO_{2} gradients; Wang et al., 2017) and (2) the observation errors bear complex temporal and spatial correlations which are close to the prior FFCO_{2} errors (Wang et al., 2017). Such a result illustrates the need for using a suitable observation error characterization (here based on the results from Wang et al., 2017) to prevent the stations having a full coverage of information on the emissions in the model framework shown here even when the observation network is as dense as NET233. A proper account for the observation errors and their temporal and spatial correlations avoid overestimating the potential of the atmospheric inversion in OSSEs when using a coarseresolution transport model.
This study provides understanding of the inversion behavior and sensitivity to network density, but the precise quantification of the performance of the inversion is largely dependent on the spatial resolution of the transport model. Wang et al. (2017) showed that the representation error contributes the most to the observation errors, followed by the transport and measurement errors. Following the definition of the observation errors in Wang et al. (2017) and in this study, both the representation and the transport error are highly dependent on the transport model resolution. Increasing the transport model resolution will reduce the representation errors and (potentially) reduce the transport error if topography effects and synoptic variations are better simulated by finerresolution models. We thus assume that using a regional mesoscale transport model with higher resolution than LMDZv4 (like for the regionalscale natural flux inversions in Kadygrov et al., 2015; Broquet et al., 2013; Gourdji et al., 2012; Lauvaux et al., 2008) should be the most efficient way to improve the results from atmospheric inversion of FFCO_{2} emissions at regional scale. A proper quantification of the change of representation and transport error as a function of spatial resolution and of the impact of this change on the performance of the inversion system would require a series of transport models and inversions at varying spatial resolution, which is out of the scope of this study but would be worth investigation in the future.
However, unlike such regional transport models, a global transport model can propagate uncertainties in emissions in other continents to Europe and thus allow one to account for them when estimating the European emissions. To quantify the impact of the uncertainties in emissions from other continents, we conducted additional inversions that only solve for emissions in European regions, ignoring those of other continents. The results show that fossil fuel emissions from other continents have negligible impacts on UR, MR and posterior emission budgets of European regions (the relative differences between these estimates being smaller than 1 %; not shown). This indicates that the inversion system mainly exploits the signals of the gradients between the European sites to constrain the European emissions, and the incoming FFCO_{2} over the European airshed from emissions outside the European continent results in very small FFCO_{2} gradients between JFJ and other stations in Europe. As a result, it highlights the possibility of using a mesoscale regional transport model and a regional inversion framework to derive monthly and nationalscale emission budgets from ^{14}CO_{2} networks in Europe. In such a framework, the uncertainties in the signals of fossil fuel emissions from remote emissions outside Europe could be neglected or coarsely accounted for by controlling the regional transport model boundary conditions. However, such a conclusion may need to be reevaluated when processing real data and accounting for uncertainties in other types of ^{14}CO_{2} fluxes, since, e.g., parts of the Atlantic ocean fluxes may have a significant signature on the European ^{14}CO_{2} gradients.
4.3 The need for good estimates of the uncertainties in the prior estimate of the emissions from inventories
The inconsistencies between the posterior misfits and the theoretical computation of posterior uncertainties and between the scores of MR and UR in INVN inversions indicate that the theoretical computation of posterior uncertainty is not sufficient to characterize the actual performance of the inversion, especially when the prior uncertainty covariance matrix does not capture the actual error statistics of the prior estimate of the emissions. Moreover, in INVN, there is a degradation of the emission estimates for many regions, characterized by negative and farbelowzero MRs in Sect. 3. This degradation occurs even when using daily measurements or the network NET233. A first explanation is that the signature of the errors in the prior emission estimates in the FFCO_{2} fields has a smaller amplitude than the observation errors and thus the ability to filter this information for a proper correction of the emissions strongly relies on the knowledge of the prior uncertainty covariance. If B misses the amplitude and the temporal and spatial correlations of the actual errors, the system can translate observation errors into corrections to the emissions. Furthermore, some of the region–months are poorly constrained by the observations (due to the meteorological conditions and/or to the observation network spatial distribution), and the corrections to such region–months are imposed by the extrapolation of the corrections to other region–months following the uncertainty structures characterized by B. If those structures do not represent the actual errors correctly, the system could apply corrections with a wrong sign or amplitude to the poorly observed region–months. A similar problem occurs when the network can constrain the sum of the budgets for several region–months but not the individual budgets of these region–months (due to being too coarse). If the structure of B is wrong, the repartition of the constraint from the observations between these different region–months can be erroneous. All these analyses reveal the difficulty of capturing the signatures of uncertainties in the prior emission estimate from the assimilated prior model–data misfits in our specific inverse modeling problem and thus to derive good corrections when the prior uncertainty covariance matrix is not configured properly.
In such a situation, only a precise configuration of the prior uncertainty covariance matrix can support the filtering of the prior errors. Consequently, even though both B^{empiric} and B^{notion} are derived from realistic assumptions on the uncertainties in the inventories and, to some different extent, from the analysis of inventory maps, the inconsistencies between these two matrices lead, in general, to positive MRs when using the former and negative ones when using the latter.
In real applications, having such a good fit between the configuration of the prior uncertainty covariance matrix in the inversion system as between B^{empiric} and the synthetic prior errors in our OSSEs could appear to be unlikely, especially since the difference between B^{empiric} and B^{notion} illustrates the range of assumptions we could have on the uncertainties in the existing inventories. Consequently, in order to improve the estimate of FFCO_{2} emissions, on the one hand, more detailed and systematic evaluations of the uncertainty in the FFCO_{2} emission inventories and of their potential temporal–spatial correlations (Andres et al., 2014, 2016) would be required. On the other hand, as mentioned in Sect. 4.2, using a regional mesoscale transport model with higher resolution would reduce the representation error and (potentially) the transport error, and thus the observation error. Such a model would be needed to decrease the ratio of the observation error to the prior FFCO_{2} error and thus increase the ability to filter the prior errors from the prior model–data misfits.
In this study, we present the application of a global atmospheric inversion method to quantify FFCO_{2} emissions over Europe at regional scale using three continental networks of ^{14}CO_{2} measurement sites. Its framework has been introduced by Wang et al. (2017). This method combines a prior emission estimate from an inventory with the information from atmospheric observations of FFCO_{2} gradients to provide improved emission estimates with reduced uncertainties. A set of inversions are performed to test the potential of such a global atmospheric inversion system and the relevance of the largescale inverse modeling (using coarseresolution transport model and controlling the emissions at regional scale) to monitor FFCO_{2} emissions. The results show that given the 17 ^{14}CO_{2} measurement stations that are available in 2016 and the typical 2week sampling frequency, the inversion reduces the uncertainties in monthly emission estimates for western Germany by 34 to 38 %, depending on the setup of the prior uncertainty. By using a plausible network containing 43 measurement stations which is planned for the future and using 2week sampling, one could expect higher URs of the emissions over the high emitters in Europe, e.g., eastern France (16 to 33 %) and southern UK (3 to 23 %). In addition, given the posterior uncertainty in the emissions that could be achieved in such an inversion system, the uncertainties in the regressed trends can be significantly reduced below 1 % yr^{−1} by monitoring the FFCO_{2} emissions for more than 20 years.
Increasing the number of observations assimilated in the inversion system by using daily sampling or a very dense observational network could potentially increase the UR over European regions. However, even though the inverse modeling framework used here can be assumed to be optimistic, e.g., regarding the assumption of the FFCO_{2} data precision (see Sect. 2.2.2), its potential to improve the estimate of FFCO_{2} emissions is often limited. The concept of using a coarseresolution transport model in a global inversion system to solve for fossil fuel emissions of the regions whose emissions are not as high as those of Germany and France is challenged by the fact that coarseresolution transport model can hardly filter the signature of the uncertainties in the emission budget from other signals and sources of errors within their coarse grid cells. Thus, regional highresolution transport models could thus be required for the monitoring of FFCO_{2}. At the same time, the posterior estimate of the emissions are much degraded when the configuration of prior uncertainty in the inversion system is improper, implying that systematic evaluations of the uncertainties and temporal and spatial correlations in FFCO_{2} emission inventories are also needed to improve the estimate of FFCO_{2} emissions when applying such an inversion system to actual data.
The inversion system is available upon request from Yilong Wang (yilong.wang@lsce.ipsl.fr).
The B^{notion} is a block diagonal matrix. The ith main diagonal block B_{i} represents the prior uncertainty covariance of the emissions for 12 months for a given region i. Assuming the relative error δ_{i} for ${\mathit{x}}_{i}^{\mathrm{b}}$ is the same for 12 months and ${\mathit{x}}_{i,m}^{\mathrm{b}}$ is the emission for region i and month m (m= 1 means January, m= 12 means December), the diagonal entries of the B^{notion} are
The assumed 2month temporal autocorrelation (Sect. 2.2.2), expressed by an exponential decaying function, leads to the nondiagonal entries in B_{i}. Accordingly, the covariance between the uncertainties in the emissions of 2 months (month m and n, for instance) is
If regions i and j are within the same country, the offdiagonal block B_{i,j} is built to account for the spatial correlation between these two regions. We assume that ${\mathit{\delta}}_{i}={\mathit{\delta}}_{j}={\mathit{\delta}}_{ij}$ and the spatial correlation between this two regions for a given month m is −0.2 to account for the fact that present emission estimates at such scales are generally disaggregated from national inventories, that is
We assume that the correlation between two control variables are given by the product of the spatial and temporal correlations between the two corresponding control regions and the 2 months, respectively. Lastly, the δ for each region is determined so that the prior annual emission uncertainty is satisfied, i.e., 10 % for US, 10 % for European countries, 15 % for China and 10 % for other large regions.
The B^{empiric} is also a block diagonal matrix. For a given region i and a specific month m, assuming the prior control parameter corresponding to PKUCO_{2} emission is ${\mathit{x}}_{i,m}^{\mathrm{b}}$, the “true” value of x, corresponding to IEREDG, is ${\mathit{x}}_{i,m}^{\mathrm{t}}$, so that the errors of the prior monthly emissions are
The longterm error component at annual scale ε_{ann} equals
The residues are
Then the 6month variation ε_{6 m} equals the standard deviation of the 6month mean residues:
Again, the residues become
In the same way, the 3month variation ε_{3 m} equals the standard deviation of the 3month mean residues:
The corresponding residues are
The 1month variation ε_{1 m} equals the standard deviation of these residues:
Using such a decomposition, the root mean square errors (RMSE) between the prior and the “true” values Δx_{i,j} satisfy the following equation:
Finally, for the diagonal entries of the B matrix corresponding to the monthly emissions of region i, they are equal to the RMSE_{i}; for the nondiagonal entries, the covariance between month j and month k for a given region is expressed as the sum of the products of the different variations multiplied by corresponding correlations (expressed by exponential decay functions) at different timescales:
Assuming the linear trend of the FFCO_{2} emissions in an nyear period is to be calculated, which satisfies the function
where y is the vector of annual emissions for the n years, $\stackrel{\mathrm{\u0303}}{\mathit{y}}$ is the predicted value by the regression and x is the vector of corresponding years, and the slope a is the linear trend we are going to calculate by linear regression. We rewrite Eq. (C1) as follows:
Thus the linear trend a and the interception b can be solved using linear algebra. With the notations used in Eq. (C2), the result of the linear regression is
The associated uncertainties in the regression parameters in vector p is thus given by the following covariance matrix:
where cov(.) is the covariance matrix for a set of variables.
Since X is a fixed matrix filled by the numbers of years and 1's, the uncertainties in the linear trend (first item in main diagonal of cov(p)) is independent of the annual emissions themselves but only depends on the uncertainties and associated correlations of annual emissions. As seen in Fig. C1, this error covariance of y should include two independent parts: (1) the uncertainties associated with the estimation of the emissions for each year in y and (2) the interannual variability (IAV) in the detrended y.
In this study, based on the time series of national annual emissions from IEREDG, we assume a 5 % IAV in the annual fossil fuel emissions for European countries. In general, this 5 % IAV is the upper limit of the typical values for European countries (Levin and Rödenbeck, 2007). Ballantyne et al. (2015) assumed that in the selfreported fossil fuel emission inventories, the emission error in 1 year could be highly correlated with the error from the previous year by an autoregressive coefficient of 0.95 due to potential errors that are not corrected retroactively after about 20 years. However, we do not conduct a multiyear inversion to get a typical estimate of the correlations in the posterior uncertainties in annual emissions and assume that there is no correlations between the posterior uncertainties in annual emissions. This assumption is fairly conservative, since Eq. (C4) implies that the larger (either positive or negative) the correlations between the estimation of fossil fuel emissions from different years, the smaller the uncertainties in the regressed trends.
The supplement related to this article is available online at: https://doi.org/10.5194/acp1842292018supplement.
The authors declare that they have no conflict of interest.
The authors acknowledge the support of the French Commissariat à
l'énergie atomique et aux énergies alternatives (CEA). This study is
cofunded by the European Commission under the EU Seventh Research Framework
Programme (grant agreement no. 283080, geocarbon project). Grégoire Broquet and Frédéric
Vogel acknowledge funding from the industrial chair BridGES (supported by the
Université de Versailles SaintQuentinenYvelines, the CEA, the Centre National
de la Recherche Scientifique, Thales Alenia Space and Veolia). We are also
grateful to Ingeborg Levin for the useful discussions on this topic. We also
would like to thank the partners of the ICOS infrastructure for details of
radiocarbon samplings and FFCO_{2} monitoring.
Edited by: Qiang Zhang
Reviewed by: Peter Rayner and one anonymous referee
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