Emissions from combustion of fossil fuels are the primary driver of increasing atmospheric CO_{2} (Ballantyne et al., 2015). Improved knowledge of FFCO_{2} emissions and their trends is necessary to understand the drivers of their variations, as well as to measure the effectiveness of mitigation actions (Pacala et al., 2010). Accurate estimates of emissions for the baselines years and the years after help verifying agreed-upon emission reduction targets. Implicitly, this requires that the uncertainties in the estimates of the emissions are much smaller than the amount of emissions to be reduced over a certain period of time.
Currently, fossil fuel CO_{2} emissions are established by inventories mainly at the scale of countries, based on energy or fuel use statistics. In these inventories, sectorial data concerning each activity that produces emissions are multiplied by combustion efficiencies and emission factors. Such inventories thus have uncertainties related to imperfect data of energy or fuel use statistics, combustion efficiencies and emission factors (Macknick, 2009; Andres et al., 2012; Liu et al., 2015). Emission inventories are self-reported by countries using non-comparable methodologies and different data-sets (Ciais et al., 2010), although the IPCC has published guidelines of good practice for emission reporting (IPCC, 2006). It is estimated that national annual FFCO_{2} emissions have two-sigma uncertainties ranging from 5% in OECD countries (Marland, 2008), 15–20% for China (Gregg et al., 2008) to 50% or more for less-developed countries (Andres et al., 2014). Global FFCO_{2} emission maps (e.g. EDGAR, http://edgar.jrc.ec.europa.eu (Olivier et al., 2005); PKU-CO2, (Wang et al., 2013); CDIAC, (Andres et al., 1996); ODIAC, (Oda and Maksyutov, 2011)) are compiled based on these national inventories and on the disaggregation of national (regional) emissions, or by bottom-up modelling of emissions based on local to regional activity data (Gurney et al., 2009). These products are available at a relatively high spatial resolution, typically down to 0.1°, but often without considering detailed spatial variations in emission processes. Also, different downscaling assumptions result in disagreements between emission maps (Oda and Maksyutov, 2011; Wang et al., 2013). These products usually provide annual values without temporal profiles associated with emissions at the intra annual scale. Thus, these emission maps often have larger uncertainties at sub-national and monthly scale (Ciais et al., 2010; Gregg et al., 2008).
An appealing method to independently assess FFCO_{2} emissions is to use an atmospheric inversion approach (Ray et al., 2014). The atmospheric inversion approach consists in adjusting the estimates of emissions to minimize the distance between modelled and observed mixing ratios, yielding an optimized posterior estimate. It uses a statistical method, which relies on statistics of the uncertainty in the prior estimate of the emissions and of the other sources of model-measurement misfits (transport errors, measurement errors, model-measurement mismatch due to different spatial representativeness, etc., which are grouped under the generic term observation errors). Atmospheric inversions have been used so far for estimating natural CO_{2} fluxes, with most studies being at the scale of large regions (Bousquet et al., 2000; Gurney et al., 2002), and few studies at the scale of small regions (Lauvaux et al., 2008; Broquet et al., 2011). These inversions have mainly used ground-based in situ atmospheric measurements while exploiting satellite measurements is presently challenging (Chevallier and O’Dell, 2013).
A first strategy to sample the atmosphere with in situ stations for the inversion of FFCO_{2} emissions would be to place stations very close to the largest fossil fuel CO_{2} sources (cities, power plants, etc.). This allows the detection of a clear signature of FFCO_{2} emissions in the measured CO_{2} gradients (Bréon et al., 2015). Very high-resolution inversion systems are required to exploit such data (Brioude et al., 2012; McKain et al., 2012; Newman et al., 2013; Bréon et al., 2015). A limitation of this sampling strategy is that it would necessitate dense networks and very high-resolution inversions around every large CO_{2} emitting area, while smaller sources will not be captured.
The second strategy is to sample the atmosphere away from local FFCO_{2} sources to monitor an atmospheric signal integrating their signature at the sub-continental scale. With this strategy, one may expect inversions to solve for fossil fuel emissions at the scale of sub-continental regions (e.g. middle-sized countries in EU, groups of States in the US, provinces in China) using a network of stations distributed across a large sub-continental domain (Pacala et al., 2010). This sampling strategy could benefit from the existing infrastructure of in situ networks already set-up for the monitoring of natural fluxes (e.g. the European Integrated Carbon Observing System, ICOS, https://www.icos-ri.eu/; NOAA-ESRL, http://www.esrl.noaa.gov/research/themes/carbon/).
A difficulty for inversions to solve for FFCO_{2} emissions based on atmospheric observations on a continental scale network is to separate the signal from fossil fuels from that of natural (biogenic and oceanic) fluxes in the atmospheric measurements. The effect of natural fluxes on atmospheric CO_{2} gradients is generally much larger than that of fossil fuel emissions at the sort of sites like ICOS or NOAA-ESRL, especially during the growing season (Shiga et al., 2014), and at least comparable to that of fossil fuel emissions during non-growing season (Levin and Karstens, 2008), if the stations are not immediately close to anthropogenic sources. A filtering of the FFCO_{2} signature based on knowledge on the spatial distribution and temporal profiles of FFCO_{2} emissions is presently challenging because of uncertainties in the spatial and temporal distribution of emissions and because large-scale transport models can hardly account for the potential of this information, which is concentrated at relatively high resolution. Shiga et al. (2014) analysed real measurements to study the potential of the surface observation networks to monitor anthropogenic emissions, and in particular to separate the signals of fossil fuel emissions from those of natural fluxes, in North America. However, in practice, there has not been any attempt at conducting inversions of the emissions at large-scale using real CO_{2} measurements alone from existing continental networks.
To circumvent the problem of separating natural fluxes and fossil fuel emissions in the atmospheric signals, it is possible to use proxies of the CO_{2} mole fraction from fossil fuel emissions in large-scale inversions. Several proxies have been proposed for FFCO_{2} (Gamnitzer et al., 2006; Rivier et al., 2006), but none of them is as close to a pure fossil fuel CO_{2} tracer as radiocarbon in CO_{2}. Measurements of radiocarbon in CO_{2} together with measurements of total CO_{2} can be used to separate FFCO_{2} (Levin et al., 2003) based on the principle that fossil fuel-emitted CO_{2} comes from geological deposits, and is radiocarbon-free. In this context, our study gives insights on the potential of the inversion of fossil fuel emissions in Europe based on hypothetical networks of collocated measurements of radiocarbon in CO_{2} and total CO_{2} measurements.
Note that radiocarbon in CO_{2} is only a proxy of FFCO_{2} and that its atmospheric gradients are also partly influenced by the transport of fluxes from stratosphere, ocean, biosphere and nuclear facilities as well as by that of fossil fuel emissions (Randerson et al., 2002; Naegler and Levin, 2006; Graven and Gruber, 2011). Within industrialized continents, radiocarbon gradients are however dominated by the signal of FFCO_{2} emissions (Graven and Gruber, 2011; Levin et al., 2011). In our studies, we postulate that atmospheric radiocarbon-CO_{2} observations are exact measurements of the FFCO_{2} component in atmospheric CO_{2}. We also postulate that numerous measurements of radiocarbon-CO_{2} could be made at many sites of a continental atmospheric network. In practice, radiocarbon is expensive to measure (e.g. can only be performed in discrete air samples, not in situ), so that the implementation costs of dense radiocarbon sampling networks could be a limitation as well.
Nevertheless, these two assumptions are not a limitation to the scope of this study focusing on evaluating whether the signal of FFCO_{2} gradients between continental sites that are not in the vicinity of high emission areas are large enough compared to modelling errors and radiocarbon measurement errors, and whether these gradients are representative enough of the emissions averaged at sub-national scales so that the use of a coarse-grid transport model remains valid for constraining sub-national FFCO_{2} emissions.
The recent OSSE study of Ray et al. (2014) demonstrated that using a network of 35 towers sampling atmospheric FFCO_{2} mixing ratios every 3 h across the U.S. with an uncertainty arbitrarily set to 0.1 ppm (which is very optimistic given the current precision of radiocarbon-CO_{2} measurements), an atmospheric inversion at 1° × 1° resolution could reduce errors on eight-day-averaged country-level fossil-fuel emissions by a factor of two. In the context of the US Inter-academy report on emission verification, Pacala et al. (2010) presented another OSSE experiment suggesting that, based on a hypothetical massive set of 10,000 atmospheric ^{14}CO_{2} measurements in one year and a perfect transport model of 5° horizontal resolution, an atmospheric inversion could reduce the uncertainty of the monthly mean fossil-fuel flux in the US from 100% to less than 10%. Moreover, Basu et al. (2016) developed a dual-tracer inversion framework assimilating both CO_{2} and ^{14}CO_{2}. They showed that given the actual coverage of ^{14}CO_{2} measurements available in 2010 over US, the dual-tracer inversion can recover the US national annual total FFCO_{2} emission to better than 1%.
In this study, we attempt at analysing in detail the weight of potentially critical limitations when targeting large-scale budgets of fossil fuel emissions based on atmospheric inversion. In particular, we characterize how much such an approach relies on the knowledge of the spatial distribution of the emissions at high resolution, while continental scale observation networks and inversion systems can hardly solve for it. When dealing with a large-scale inversion system, one also needs to carefully account for the fact that the grid size of transport models (typically 100–300 km for global models, down to 5–10 km for regional models; Law et al., 2008) is larger than the scale of emissions, which have very fine scale patterns. This ensemble of misfits between the scales solved for or modelled within the inversion system and that of actual emissions and patterns in the mixing ratios generates so-called aggregation and representation errors in the inversion (Gerbig et al., 2003; Lin et al., 2006) to which this paper gives a special attention. Actually, it will be shown in our study that the aggregation errors still characterize the impact of uncertainties in the distribution of the emissions on the potential for monitoring the emission at large scale when solving for them at the transport model resolution.
This study specifically addresses the derivation and analysis of the statistics of representation and aggregation errors in comparison to the typical FFCO_{2} signals at measurement sites. This work focuses on the derivation and analysis of these errors for an atmospheric inversion framework dedicated to the inference of national-scale monthly emissions over European countries using continental scale networks of measurement stations. This inversion framework uses a global atmospheric transport model and global maps of the emissions with spatial and temporal distributions within countries and one month. Having a global configuration ensures that uncertainties in fossil fuel CO_{2} emitted over other regions of the globe outside a target continent are properly accounted for. This study assumes that daily to monthly mean FFCO_{2} gradients can be estimated between numerous sites and a ‘reference’ site sampling the free tropospheric air over a continent by ^{14}CO_{2} measurements, with a precision of 1 ppm due to the typical measurement errors and to uncertainties in the conversion of ^{14}CO_{2} and CO_{2} measurements into FFCO_{2} (Levin et al., 2003).
The detailed objectives of this paper are:
Due to the link between the representation and aggregation errors with the configuration of the inversion, Section 2 first describes the large-scale fossil fuel emission inversion framework, and then develops the derivation for each term of the observation errors mentioned above. Special attention is given to the representation and aggregation errors when using a coarse-grid transport model and optimizing emissions at the scale of sub-continental regions. The section also discusses the significance of these two errors and the actual dependence of the underlying sources of uncertainties to the specific configuration of the inverse modelling framework. Practical ways to estimate these two errors are given in Section 3. Results for representation errors, aggregation errors and the errors due to the prior uncertainties in the simulation of observations in Europe are discussed in Sections 4 and 5. In Section 6, we compare these errors with the measurement error, model transport error, and typical signals of FFCO_{2}. We also discuss the effects of the spatial (temporal) resolution of the modelling (respectively observation) framework for the atmospheric inversion of FFCO_{2} emissions. Conclusions are drawn in Section 7.
The inversion framework considered here follows the Bayesian linear update (Enting et al., 1993; Tarantola, 2005) of a prior statistical knowledge p(x^{t}|x^{b}) on the actual value x^{t} for a set of control variables x (among which some variables underlie the target quantities i.e. budgets of FFCO_{2} emissions at large scale), where x^{b} is a prior estimate of these variables. The update relies on some observations y^{o} (here FFCO_{2} atmospheric measurements), on an affine observation operator x$\mapsto $Hx + y_{fixed} (including the global coarse-grid transport model and the distribution of the emissions at high resolution, and the signature of the influence of sources of FFCO_{2} that is not solved for by the inversion) linking the control space x to the observation space y and on statistics p(y^{o} − Hx^{t} − y_{fixed}|x^{t}) of the sources of observation errors (i.e. errors that are not due to the uncertainties in the estimate of x^{t} in the comparison between Hx^{t} + y_{fixed} and the observations y^{o}). It also follows the traditional assumption that the statistics of the prior and observation uncertainties are unbiased, Gaussian and independent of each other (Tarantola, 2005) so that p(x^{t}− x^{b}|x^{b}) ~ N(0, B) and p(y^{o} − Hx^{t} − y_{fixed}|x^{t}) ~ N(0, R) where B and R are the prior error and observation error covariance matrices, and so that the posterior statistical estimate of x^{t} from the optimal update given x^{b} and y^{o}, is a Gaussian distribution that can be written p(x^{t}|x^{b}, y^{o}) ~ N(x^{a}, A), where
We focus on the characterization of several critical terms of the observation error p(y^{o} − Hx^{t} − y_{fixed}|x^{t}), on the relevance of the assumption that the observation error can be represented by a Gaussian and unbiased distribution N(0, R), and on the derivation of a relevant R matrix for our configuration of a large-scale fossil-fuel emission inversion. The observation error plays a critical role in the estimate of the posterior uncertainty characterized by its covariance matrix A. If its projection back to the flux space (i.e. the term H^{T}R^{−1}H in Equation (1)) is far larger than the uncertainty in the fluxes that the inversion is expected to solve for (the B matrix), the assimilation of atmospheric observations will bring little and/or highly uncertain information about the fluxes and the potential of the inversion will be low. The observation error p(y^{o} − Hx^{t} − y_{fixed}|x^{t}) will be compared to an estimate of the projection of the prior uncertainty in the observation space p(H(x^{t} − x^{b})|x^{b}) to give insights on this (indicating whether the signature of the prior uncertainty should be easy to filter in the prior model–data misfits y^{o} − Hx^{b} − y_{fixed}) even though the full computation of Equation (1) is required to define whether the assimilation of atmospheric observation strongly decreases the uncertainty in the flux estimates.
The nature of the observation error strongly depends on the nature of the x and y space, and on the accuracy and precision of the observation operator x $\mapsto $ Hx + y_{fixed}. In the following we first present the practical configuration of these elements given our practical inversion framework. Then, we propose a theoretical decomposition of the observation errors with an emphasis on the terms that should be critical for our practical inversion framework and with a specific care at defining the representation and aggregation errors.
In practice the simulations, inversions and analysis are conducted for a 1-year period arbitrarily chosen to be a typical year 2007. This choice has consequences regarding the meteorological conditions and the level of emissions that are taken into account in our modelling framework but we expect that the conclusions from the analysis should not be strongly sensitive to this choice.
We divide the globe, according to administrative boundaries, into a set of emitting regions whose monthly mean fossil fuel emission budgets are solved for during a whole year (Fig. 1a). The corresponding space discretization is higher in continents that have the largest emission densities (Europe, US and China, Fig. 1b–d). The spatial resolution in Europe is in agreement with the typical size of European countries. It is finer in western Europe where emissions can be high in specific regions such as northern Italy, southern England, eastern and western Germany. In the US and China, the spatial discretization is also increased in the most populated and industrialized areas (i.e. the east and west coasts in the US, and the south-eastern coast in China). In a given emitting region, the inversion solves for the budget of FFCO_{2} emissions (in Mg C/hour) for each of the 12 months during one year, but does not solve for the space and time distribution within each region or at sub-monthly intervals.
The observation vector consists in FFCO_{2} gradients between sites of hypothetical ground-based European networks of atmospheric total CO_{2} and ^{14}CO_{2} measurements (that are used together to compute FFCO_{2}) throughout one year, at typical heights for continuous measurement sites. More precisely, we consider gradients between simultaneous FFCO_{2} observations at any site of these networks and a reference site sampling the free tropospheric air over Europe, as is traditionally done when analyzing ^{14}CO_{2} measurements (Levin et al., 2008). The actual sampling heights for the measurement sites are generally below 300 magl (Kadygrov et al., 2015). In this study, we first (in Section 4) consider a standard sampling height above the ground at all the sites except the reference site in each configuration of the observation vector: 100 magl. The choice of this standard height simplifies practical considerations for the analysis in this study but does not have the same physical impact at all sites due to the variations in the PBL and orography, and of the modelling skills depending on the locations and time. The sensitivity to other sampling heights will be discussed in Section 5.1. We select the High Alpine Research Station Jungfraujoch (JFJ, located at 3450 masl in Switzerland) as the reference site for European stations. Continuous measurement of total CO_{2} has been made for years in Europe (within the CarboEurope-IP, GHG-Europe and ICOS programs) and US (within the NOAA-ESRL framework) at tens of sites. A given radiocarbon measurement can be applied to a sample with any temporal integration time from 1 h to 1 month since air samples could be filled at constant rates over long periods. However, the cost of the ^{14}CO_{2} analysis of one sample is presently high so that monitoring of ^{14}CO_{2} during a whole year favours the choice of integrated samples at the daily to monthly scale (Levin, 1980; Turnbull et al., 2009; Vogel et al., 2013). In this study, we only consider a single sampling frequency at all sites in each configuration of the observation vector. The two-week mean sampling is considered as a standard sampling strategy in Section 4, while the sensitivity to other sampling strategies will be discussed in Section 5.2. We have also accounted for the technical ability to have an intermittent filling (Levin et al., 2008; Turnbull et al., 2016). Indeed, state-of-art inversion systems generally make use of data during afternoon only due to limitations in modelling the vertical mixing during other periods of the day. We thus assume that mean afternoon FFCO_{2} observations are sampled during 12:00–18:00 local time at the sites.
The locations of the stations where ^{14}CO_{2} measurements are made are assumed to be inland and distant from urban areas and other large sources, and aim to monitor the signature of the emissions at sub-continental scale. However, some sites will necessarily be closer to emitting areas (such as cities and power plants) than others, with consequences regarding the representativeness and amplitude of the measured FFCO_{2} signal. We thus define two types of sites, both corresponding to land model grid cells: ‘urban’ and ‘rural’ sites, based on a threshold on the population density (ORNL, 2015) within the grid cells where the stations are located. This threshold is country-dependent and matches the World Bank urbanization data (available at http://data.worldbank.org/indicator/SP.URB.TOTL.IN.ZS?page=1&order=wbapi_data_value_2011%20wbapi_data_value%20wbapi_data_value-first&sort=asc) for each country as done by Wang et al. (2013). In this study, the class ‘urban’ sites also include grid cells where large point sources exist, e.g. power plants (based on the CARMAv2 database, Ummel, 2012) and cement manufactures (Wang et al., 2013). The name ‘urban’ is used for convenience. One more straightforward approach would have been to select highest emitting grid cells according to EDG-IER with a threshold approximately consistent with the one on the population as discussed above. But we prefer to keep a level of independency to this inventory and to its uncertainties by taking an independent proxy of the high emitting locations.
The observations are only influenced by the initial condition and the emissions during the year. As indicated above, the emissions are solved for by the inversion. Through diffusion by atmospheric transport, the spatial gradients of FFCO_{2} from a pulse of emissions at a given time appear to become negligible (with an amplitude smaller than 0.1 ppm) within about 2 weeks, so that the influence of the global FFCO_{2} distribution on 1 January 2007 (i.e. the initial condition of the inversion experiments in our studies) is quite negligible for our simulations of gradients of FFCO_{2} in Europe in 2007, even for the results in January 2007 (not shown here). In our modelling of framework and corresponding simulations, initial conditions for the FFCO_{2} field in the atmosphere by the 1st day of the inversion year are thus ignored.
Consequently, the observation operator considered in this study is linear and does not bear an affine term y_{fixed} reflecting, in the observation gradients, the influence of a source or sink of FFCO_{2} that is not rescaled by our control vector. Therefore, it can be denoted as H. We decompose it into:
In this formulation, H is a chain of three operators denoting the distribution of emissions within each region-month corresponding to the control variables (H_{distr}), the atmospheric transport (H_{transp}), and the sampling of atmospheric gradients corresponding to the observation vector from the transport model outputs (H_{samp}), respectively. The spatial and temporal (sub-monthly) distribution operator x → f = H_{distr}x distributes the emission budgets for each region and month x into gridded emissions f at the spatial and temporal resolution expected as input of the atmospheric transport model. The atmospheric transport operator f → c = H_{transp}f simulates the FFCO_{2} field c using an atmospheric transport model with prescribed emissions f. The sampling operator c → y = H_{samp}c applies the atmospheric sampling procedure described above.
Each column of H represents the signature (the so-called response function) in the observation space of a unitary increment of the budget of the emissions in a given control region-month. Fig. 2 gives the frame of the observation operator and its link to control and observation vectors.
For the observation operator used in practice, we use a coarse-grid transport model and emission inventories which catch the typical spatial and temporal large-scale variations in the FFCO_{2} emissions and concentrations and thus ensure the realism of the typical estimates of uncertainties in our study. The corresponding products bear the typical precision/accuracy of the products that are used by state-of-the-art inversion systems when assimilating real data to quantify CO_{2} natural fluxes at large scale.
We use the PKU-CO_{2}-2007 global emission inventory for 2007 (Wang et al., 2013) to model, by the H_{distr} operator, the spatial distribution of emissions within the regions of control. PKU-CO_{2}-2007 is a high-resolution (0.1°) annual emission map based on the disaggregation of national emission budgets using sub-national statistics. Regarding the sub-monthly temporal distribution of emissions within each month, we assume a flat temporal profile, as in many large-scale natural flux inversion systems (Peylin et al., 2013). We denote by ${\mathbf{H}}_{\text{distr}}^{\text{PKU}}$ the practical implementation of the distribution operator H_{distr}.
An off-line version of the atmospheric general circulation model of Laboratoire de Météorologie Dynamique (LMDZ) (version 4) (Hourdin et al., 2006) is used as our atmospheric transport operator. The corresponding LMDZ simulation was nudged to the reanalysed wind fields from the European Centre for Medium-Range Weather Forecasts (ECMWF) Interim Reanalysis (ERA-Interim, (Berrisford et al., 2009)). LMDZ has participated to a series of intercomparison exercises for the simulation of CO_{2} concentrations (Law et al., 2008) and is able to reproduce most of the daily variations of the large-scale transport of FFCO_{2} (Peylin et al., 2011). The model configuration used here has a horizontal resolution of 3.75° × 2.5° (longitude × latitude) and 19 hybrid sigma-pressure layers to discretize the vertical profile between the surface and the top of the atmosphere. We denote by ${\mathbf{H}}_{\text{transp}}^{\text{LMDZ}}$ the resulting practical implementation of H_{transp}.
In the observation operator, the practical simulation of FFCO_{2} gradients corresponding to the observation vector relies on the simple extraction of individual concentration data at the measurement locations and then on the computation of differences between these concentration at different sites. We extract a concentration for a given location by taking the value in the transport model grid cell within which the site locates rather than interpolating values from several transport model grid cells. Usually, the height of the first level of LMDZ is about 150 magl. All the observations being assumed at 100 magl, they are all extracted from the first level of this version of LMDZ, except that of the reference site, Jungfraujoch (JFJ). JFJ is located at 3450 m above sea level (masl) but close to the ground level, at the top of a mountain. Since the LMDZ model poorly solves the topography in mountain areas, its ground level in the grid cell corresponding to JFJ is located far lower than this height. In order to ensure that the modelled concentrations are representative of the free tropospheric air, JFJ observations are extracted from the sixth level of LMDZ, which is usually located between 2700 and 3800 masl. 1-day to 1-month mean afternoon FFCO_{2} data are sampled in time by H_{samp} (depending on the corresponding single observation frequency, see Section 2.1.2). We denote by ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}$ the resulting practical implementation of H_{samp}.
To sum up, the observation operator that will be used in practice for inversions in the following can be written ${\mathbf{H}}^{\text{prac}}={\mathbf{H}}_{\text{samp}}^{\text{coloc}}{\mathbf{H}}_{\text{transp}}^{\text{LMDZ}}{\mathbf{H}}_{\text{distr}}^{\text{PKU}}$.
In this section, we are interested in decomposing the observation error p(y^{o} − Hx^{t}|x^{t}) for a typical H in order to isolate some critical sources of errors in practice. The observation operator H = H_{samp}H_{transp}H_{distr} maps low-resolution budgets of the emissions into a coarse spatial grid. But each term of this operator is likely not perfectly represented in the following ways: (1) the products for the distribution of emissions within countries such as the one used to build ${\mathbf{H}}_{\text{distr}}^{\text{PKU}}$ are necessarily imperfect; (2) the-state-of-the-art transport model such as the one used in ${\mathbf{H}}_{\text{transp}}^{\text{LMDZ}}$ are necessarily imperfect; (3) the spatial representativeness of the measurements close to the ground can be low with coarse-resolution transport models and it can be difficult to represent the measurements in the vertical grid of the coarse-resolution models (Broquet et al., 2011; Pillai et al., 2011) which impacts the precision/accuracy of practical models for H_{samp}H_{transp}. These add to the high measurement errors that have to be accounted for when monitoring FFCO_{2}.
Focusing on these sources of errors, the term y^{o} − Hx^{t} can be decomposed as follows:
where H_{transpHR} is a theoretical operator corresponding to the linear transport from emissions f_{HR} to y, f_{HR} and this transport being represented using the ‘infinitely high’ resolution (i.e. continuously instead of using a discrete form) needed for catching all the patterns in the emissions and concentrations; Superscripts t denotes the true value of the emissions or observation operators at their corresponding space and time resolution (a ‘true observation operator’ meaning here a perfect operator without any model error). Even if this decomposition primarily aims at giving a physical characterization of each resulting term, it is also made such that these different terms can be assumed to be independent (see the justification for Equation (6) below).
We define the different terms of the observation error y^{o} – Hx^{t} based on this decomposition:
The total observation error ε_{o} defined by p(y^{o} − Hx^{t} − y_{fixed}|x^{t}) can be expressed as:
Several of these terms are proportional to the value of x^{t} while x^{t} can take any value in the statistical framework of our inversion problem. This prevents, theoretically, from computing a fixed covariance R of the observation error assuming that this error can be represented by a distribution N(0, R). The configuration of such an error in the inversion systems generally ignore such a dependence of the model errors (transport, representation and aggregation errors) on the possible values for the actual fluxes which is a strong limitation for the application of the traditional data assimilation framework to flux inversion problems. In practice, we will derive R based on assumptions regarding the typical value for x^{t} in our inversion cases.
The errors on the different components of the observation operator relates to strongly different underlying input datasets and different types of model (see Section 2.1.3) and are thus considered to be independent. Assuming that they are all Gaussian and unbiased, one can write that ε_{i} ~ N(0, R_{i}), ε_{r} ~ N(0, R_{r}), ε_{t} ~ N(0, R_{t}), ε_{a} ~N(0, R_{a}), and compute R as the sum of the covariances of the different errors:
Of note is that our formulations of the representation error and of the aggregation error are similar to the derivations of representation error by Gerbig et al. (2003) and of aggregation error by Engelen (2002), respectively. However, our formulation of the aggregation error slightly differs from that of Kaminski et al. (2001) and Bocquet et al. (2011). We use a sort of ‘bottom-up’ approach to derive it, starting from the decomposition of the observation errors once having defined it as the sum of all sources of model data misfits other than the prior uncertainties and that are independent from these prior uncertainties. Kaminski et al. (2001) and Bocquet et al. (2011) rather followed what we consider as a ‘top-down’ approach to derive this aggregation error. Indeed, their introduction of the covariance of the aggregation error in the observation error covariance matrix ensures that the computation of the statistics for p(x^{t}|y^{o}, x^{b}) is the same regardless of the control resolution. Due to the use of the usual assumption of the atmospheric inversion that the observation error is independent of the prior uncertainty, our ‘bottom-up’ method ignores potential correlations between the aggregation errors and the prior uncertainties, which is not the case of the ‘top-down’ approaches. Therefore, the formulations of the covariance of the aggregation error in Kaminski et al. (2001) and Bocquet et al. (2011) include a component related to this correlation, which is ignored in our formulation. As discussed in Appendix A2, we have nevertheless computed the corresponding component and concluded that its weight is relatively small and negligible for our study. The mathematical details and a discussion regarding the potential correlations between the aggregation errors and the prior uncertainties are given in Appendix A2.
In theory, results for regions and months targeted by the inversion do not vary with the resolution of the control vector if the aggregation error ε_{a} is perfectly accounted for by R in the inversion configuration (see the demonstration in the Appendix based on the notations given above in Section 2.2). This assumes that the uncertainty in the emissions at the scale of interest is independent of the uncertainty at higher resolution (which is approximately verified with our modelling set-up, see the Appendix). In other words, an inversion at coarse resolution that accounts for aggregation errors ε_{a} should give the same results for monthly fluxes over large regions as the same inversion applied to solve for hourly fluxes at the highest resolution (transport model grid). This is due to the equivalence between accounting for the uncertainties of fluxes within regions/month through their projection in the observation error or through their assigned prior uncertainty (given the assumptions underlying the inversion framework). In this sense, even though they are formally a function of the control vector, the aggregation errors at a scale larger than the transport model resolution are not specific to a given inverse modelling set-up. Considering that the choice of the control resolution reflects a targeted resolution for the fluxes, aggregation errors rather reflect the impact for the monitoring of the fluxes at this targeted resolution of the uncertainties in the distribution of the fluxes at higher spatial or temporal resolutions. Increasing the control resolution would thus not, in theory, help solving for fluxes at the targeted resolution.
On the opposite, representation error is strongly linked to a specific inversion configuration. Increasing the resolution of the transport model used for the inversion necessarily decreases them without a full compensation of this decrease by the rise of prior uncertainties. The transport errors should also depend on the transport modelling configuration. For example, synoptic patterns and the influence of the surface topography on the transport are better simulated at higher resolution. However, different transport models are also based on different parameterizations and computational approach, etc., which makes the quantification and evaluation of the transport errors as a function of the model complicated and efforts have rather focused on the derivation of typical transport errors based on the spread of different transport models (Law et al., 2008; Peylin et al., 2011).
Finally, the errors in the measurements in our study should be fully independent of the inverse modelling framework. The 1 ppm measurement error for FFCO_{2} gradients between sites corresponds to typical values based on the analysis of air samples by accelerator mass spectrometry (AMS) for ^{14}CO_{2} (2–3‰, (Vogel et al., 2010; Turnbull et al., 2014)) and by typical analyzers for continuous CO_{2} samples (Chen et al., 2010; Turnbull et al., 2011). Apart from these errors, various fluxes that influence the atmospheric ^{14}CO_{2}, such as those from cosmogenic production, ocean, biosphere and nuclear facilities, make the direct conversion into FFCO_{2} gradients bear complex uncertainties whose typical values may exceed 1 ppm for some locations and periods of times (Hsueh et al., 2007; Bozhinova et al., 2013; Vogel et al., 2013). These additional sources of uncertainties are not included in this study. In addition, we assume that all ^{14}CO_{2} and CO_{2} samples will be analysed in the same laboratory such as the present ICOS Central Radiocarbon Laboratory, or at least if the samples are measured by different instruments and laboratories, they will follow the official target of compatibility made by the World Meteorological Organization (WMO) for ^{14}CO_{2} measurements (GGMT-2013). The consequence is that there should not be significant biases associated with instrumental errors impacting the gradients between sites analysed in this study.
In the inversion system, we use ${\mathbf{H}}^{\text{prac}}={\mathbf{H}}_{\text{samp}}^{\text{coloc}}{\mathbf{H}}_{\text{transp}}^{\text{LMDZ}}{\mathbf{H}}_{\text{distr}}^{\text{PKU}}$ as the observation operator. But here, we use a relatively independent representation of the ‘actual’ and higher resolution operators involved in the theoretical formulation of the observation errors in Section 2.2 in order to derive an estimate of these errors. These actual and higher resolution operators should bear patterns of the emissions, transport and concentration variability which should be realistic enough so that this estimate of the observation errors can provide a realistic characterization of the representation and aggregation errors when using real measurements.
A European configuration of the meso-scale transport model CHIMERE (Schmidt et al., 2001) run with a 0.5° horizontal resolution, with 25 hybrid sigma-pressure vertical layers from the surface to the pressure altitude of 450 hPa, and with hourly concentration outputs (to be aggregated into one-day to one-month mean afternoon data) is used to simulate ${\mathbf{H}}_{\text{transpHR}}^{\text{t}}$ and ${\mathbf{H}}_{\text{transp}}^{\text{t}}$. However, the LMDZ model is still used to model the practical H_{transp} when calculating the aggregation error. The CHIMERE simulations are initialized at 50 ppm at 1 January 2007.
We model ${\mathbf{H}}_{\text{transpHR}}^{\text{t}}$ by feeding CHIMERE with 0.5° resolution maps of the emissions and using the one-day to one-month mean afternoon 0.5°-resolution and 25 vertical-level concentration fields to extract simulated gradients of FFCO_{2}. Since the resolution of this CHIMERE configuration is not infinitely high, in practice, a sampling operator is still needed to model ${\mathbf{H}}_{\text{transpHR}}^{\text{t}}$. The vertical resolution of CHIMERE being about 35–45 m for the first three levels, the 100 magl observations involved in the computation of the FFCO_{2} gradients at high resolution are extracted in the third level of this model (and in the 0.5° horizontal model grid cell containing the horizontal position of the stations, using a sampling option similar to that used in ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}$). The FFCO_{2} concentrations at the reference site are extracted from the 23rd vertical level of CHIMERE corresponding to the altitude of 3450 masl in the 0.5° grid cell where the reference site located. This transport (and sampling) configuration that is used to model ${\mathbf{H}}_{\text{transpHR}}^{\text{t}}$ is denoted ${\mathbf{H}}_{\text{transpHR}}^{\text{CHIM}}$.
By degrading the horizontal and temporal resolution of the emissions in input of the CHIMERE model and by averaging (horizontally and vertically) the mole fractions in output of the CHIMERE simulations we model ${\mathbf{H}}_{\text{transp}}^{\text{t}}$. The spatial aggregation of the CHIMERE outputs consists first in a vertical aggregation, and then on a horizontal aggregation. The horizontal aggregation does not fully correspond to an aggregation within the LMDZ grid cells (i.e. to the interpolation of the 0.5° resolution fields for CHIMERE into the 3.75° × 2.5° resolution grid of LMDZ). For simplicity, the 0.5° CHIMERE grid cells are rather aggregated from blocks of 6 × 6 grid cells to yield coarse grid at the 3° resolution which is close to that of the LMDZ grid. ${\mathbf{H}}_{\text{transp}}^{\text{CHIM}}$ denotes the configuration where CHIMERE is fed with emissions maps aggregated at 3° resolution (close to that of the LMDZ model) and over 3-h time windows, and where CHIMERE one-day to one-month mean afternoon output concentrations are, again, aggregated at 3° resolution. For the modelling of H_{samp} in the computation of the error due to aggregation at the transport model resolution and in the computation of the representation error we apply an operator which follows the principle of ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}$ (and which we will thus also denote ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}$) i.e. FFCO_{2} observations are extracted in the first aggregated vertical levels for all the sites but the reference site, which is extracted in the sixth aggregated vertical level, and in the co-located aggregated 3° horizontal grid cells of the ${\mathbf{H}}_{\text{transp}}^{\text{CHIM}}$ outputs. When calculating the error due to the aggregation at region-month scale, we use ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}$ to model H_{samp} and apply it to ${\mathbf{H}}_{\text{transp}}^{\text{LMDZ}}$.
Associating CHIMERE at 0.5° resolution with ${\mathbf{H}}_{\text{transpHR}}^{\text{t}}$ (and consequently 0.5° resolution maps of the emissions with ${\mathbf{f}}_{\text{HR}}^{\text{t}}$) assumes that the main variations (i.e. those which have the largest impact for data at 100 magl) of emissions or concentrations within 3° resolution grid cells occur at scales larger than 0.5°. Furthermore, simulating ${\mathbf{H}}_{\text{transpHR}}^{\text{t}}$ and ${\mathbf{H}}_{\text{transp}}^{\text{t}}$ with CHIMERE which is a regional model (over Europe) assumes that the aggregation and representation errors in Europe due to the coarse representation of these emissions or of their signature outside Europe is negligible.
${\mathbf{H}}_{\text{distr}}^{\text{t}}$ (with outputs at 3° and 3-h resolution), the distribution of ${\mathbf{f}}_{\text{HR}}^{\text{t}}$ at 0.5° and 1-h resolution and x^{t} are modelled using the 0.1° × 0.1° EDGARv4.2 2007 emission map (http://edgar.jrc.ec.europa.eu) convoluted with temporal profiles (at 1-h resolution) from IER (available at http://carbones.ier.uni-stuttgart.de/wms/index.html). We denote this emission inventory EDG-IER afterwards. Aggregating this inventory at 1-h/0.5° resolution or at the scale of the inversion control region-month provides respectively ${\mathbf{f}}_{\text{HR}}^{\text{EDG-IER}}$ and x^{EDG-IER} that are used to model ${\mathbf{f}}_{\text{HR}}^{\text{t}}$ and x^{t}. Aggregating this inventory at 3-h/3° resolution (when computing representation errors at the coarse transport resolution using CHIMERE) or 3-h/3.75° × 2.5° (when computing aggregation errors using LMDZ) and then rescaling it homogeneously within each region/month of control for the inversions to get unitary budget of emissions provides ${\mathbf{H}}_{\text{distr}}^{\text{EDG-IER}}$ (using the same notation for the operator when the output ‘emission’ space is at 3° or 3.75° × 2.5° resolution) which is used to model ${\mathbf{H}}_{\text{distr}}^{\text{t}}$.
With these practical choices for modelling, the operators involved in the different types of observation errors defined in Section 2.2, the representation error writes:
and the aggregation error writes:
We only have one practical realization for each of these terms and thus of the corresponding errors, therefore, in order to derive their standard deviation and to investigate whether they bear potential temporal or spatial correlation, we make the strong assumption that the errors at different time and locations have relatively similar statistical distributions. However, this assumption of spatial and temporal homogeneity will be applied for adequate subset of observation time, locations and type, which will require a categorization of the observations.
Based on this assumption, we analyse the typical statistics of the representation and aggregation errors by using distributions of occurrences of these errors for different subsets (categories) of observations. Since observation sites of continental networks could locate in any grid cell, all the spatial grid cells and all one-day to one-month afternoon time windows are used and categorized among different subsets for this analysis. The different spatial and temporal categories will be defined based on the analysis of the spatial and temporal variations of the errors. The potential temporal auto-correlations and spatial correlations within/across categories are also analysed.
Using a similar approach, transport errors could have been evaluated using ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}{\mathbf{H}}_{\text{transp}}^{\text{CHIM}}{\mathbf{H}}_{\text{distr}}^{\text{EDG-IER}}{\mathbf{x}}^{\text{EDG-IER}}$ and ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}{\mathbf{H}}_{\text{transp}}^{\text{LMDZ}}{\mathbf{H}}_{\text{distr}}^{\text{EDG-IER}}{\mathbf{x}}^{\text{EDG-IER}}$ . However, the spatial resolution and orography of ${\mathbf{H}}_{\text{transp}}^{\text{CHIM}}$ and ${\mathbf{H}}_{\text{transp}}^{\text{LMDZ}}$ are not exactly the same and this could have artificially increased the transport error with representation error. Therefore, we make a simpler estimation of the transport errors for simulated FFCO_{2} gradients based on that of transport errors for simulated FFCO_{2} at individual sites.
For this estimation, we make several assumptions. First, we assume that there is no temporal auto-correlation of the transport error in simulated daily mean afternoon concentrations between different days at a given location. This assumption, however, may be violated if there are significant biases in the simulation of the meteorological conditions, i.e. the planetary boundary layer height and the vertical mixing strength (Miller et al., 2015; Basu et al., 2016). Nevertheless, the estimate of the structure of the temporal auto-correlations of the transport error is challenging (Lin and Gerbig, 2005; Lauvaux et al., 2009; Miller et al., 2015) and there is no clear evidence that such autocorrelations is significant at daily scale (Lin and Gerbig, 2005; Lauvaux et al., 2009; Broquet et al., 2011). In this study, we follow the assumption made by the majority of existing inversion studies (Peters et al., 2007; Chevallier et al., 2010; Niwa et al., 2012; Peylin et al., 2013), in which the temporal auto-correlations in the transport error are usually ignored. Second, we assume that the standard deviation of the transport error in simulated daily mean afternoon concentrations is constant in time at a given location. Finally, we assume that the ratio between this standard deviation and the temporal standard deviations of the 1-year long time series of the high-frequency variability of the detrended and deseasonalized simulated daily mean afternoon concentrations in the corresponding grid cell of the transport model is constant in space (i.e. that this ratio is identical for all grid cells of the transport model). The high-frequency variability is calculated by the method of Thoning et al. (1989). The underlying assumption is that the transport models should be less reliable at sites where the concentrations have a larger variability (Peylin et al., 2005; Geels et al., 2007).
These assumptions allow us to use the station of Saclay (SAC) near Paris for deriving a generalized estimate of the ratio between the transport errors and the simulated FFCO_{2} temporal variability. According to Peylin et al. (2011), the annual average of the standard deviations between simulated hourly mean FFCO_{2} concentrations at this site from a set of state-of-the-art transport models is 2.34 ppm. We use this value to define the standard deviation of the transport errors associated with the simulated daily afternoon mean concentrations. The standard deviation of the one-year long-time series of the daily afternoon mean concentrations simulated within one year with our practical implementation of the simulation of 3-hourly concentrations ${\mathbf{H}}_{\text{transp}}^{\text{LMDZ}}$${\mathbf{H}}_{\text{distr}}^{\text{EDG-IER}}$x^{EDG-IER} at SAC is 1.74 ppm. So the ratio between the standard deviation of the transport error for daily afternoon mean concentrations and the standard deviation of simulated time series for the daily afternoon mean concentrations within one year for any site is assumed to be 2.34/1.74 = 1.34. For any potential sites (in any grid cells in LMDZ), we thus multiply this ratio by the standard deviation of the simulated daily afternoon mean concentrations within one year to get an estimate of the transport error statistics for the daily afternoon mean FFCO_{2}. Transport errors for daily to monthly mean afternoon FFCO_{2} are then derived based on the value obtained for daily afternoon mean FFCO_{2} and on the above mentioned assumption that there is no temporal auto-correlation of the transport errors between afternoon mean concentrations in different days. For example, our estimate of the transport error for two-week mean afternoon concentrations (mean of 14 days) is equal to$1.74\times 1.34/\sqrt{14}=0.62$ ppm at SAC site.
Following this estimation, the transport error in the two-week mean afternoon FFCO_{2} concentration at JFJ site is 0.21 ppm. The transport error in the one-day to one-month mean afternoon FFCO_{2} gradients between any site and JFJ is calculated assuming no spatial correlation of the transport errors between sites, i.e. as √(ε${}_{\text{t,i}}^{2}$+ε${}_{\text{t,JFJ}}^{2}$) where ε_{t,i} is the transport error for concentrations at site i and ε_{t,JFJ} is the transport error in concentrations at site JFJ at the corresponding one-day to one-month scale. As a result, the transport errors in the two-week mean afternoon FFCO_{2} gradients from 100 magl sites to the JFJ reference site range from 0.42 to 1.07 ppm. The transport errors in the one-day mean afternoon FFCO_{2} gradients range from 1.58 to 3.99 ppm (and from 0.29 to 0.74 ppm in the case of errors on 1-month mean afternoon FFCO_{2} gradients, respectively).
As indicated in Section 2, we also want to compare the observation errors to the projection of the prior uncertainty in the observation space p(H(x^{t}− x^{b})|x^{b}) denoted Hε_{b} (and called ‘prior FFCO_{2} errors’ hereafter, ε_{b} corresponding to the prior uncertainties). Following the same approach as for the estimation of the representation and aggregation errors, and setting x^{b}, as in the companion inversion studies, with emission budgets from PKU-CO_{2}-2007 (hereafter x^{PKU}), we derive estimates of Hε_{b} based on statistics on H^{prac}(x^{EDG-IER} − x^{PKU}).
This Section characterizes the representation, aggregation and prior FFCO_{2} errors, derived from the method described in Section 3. This characterization consists in providing their typical values (estimates of their standard deviations), investigating whether they bear temporal or spatial correlations while such correlations of the observation errors are traditionally ignored by atmospheric inversions (Rödenbeck et al., 2003; Chevallier et al., 2005; Peylin et al., 2013), and in investigating the validity of the assumptions that these observation errors have Gaussian and unbiased distributions. Section 4 focuses on the errors for a standard sampling strategy i.e. two-week mean afternoon sampling at 100 magl. Section 5 will explore the sensitivity of the results to the temporal sampling strategy (from one-day to one-month mean afternoon sampling) and give insights on the errors that would have been obtained if considering measurements sites with a different measurement height.
Finally (in Section 6), we compare the typical values of the representation, aggregation and prior FFCO_{2} errors to the model transport errors (derived in Section 3), to the measurements errors (given in the introduction), and to the typical signal of FFCO_{2} modelled at the sites considered in this study.
The root mean square (RMS) of the representation, aggregation and prior FFCO_{2} errors for the one-year long time-series of two-week mean FFCO_{2} gradients at each of the 0.5° to 3.5° × 2.75° horizontal grid cells (depending on the error and thus on the scale at which it can be computed) are given in Fig. 3b–d. In general, the RMS of the representation error for land grid cells ranges between 0.4 and 4.0 ppm across Europe, while the RMS of the aggregation and prior FFCO_{2} errors are much smaller.
From Fig. 3b, the representation error at 100 magl shows higher values in the grid-cells classified as urbanized and large cities such as London, Paris, industrialized areas in Germany, etc. More generally, the spatial distribution of representation error shows a good consistency with the mask of the urban grid cells defined based on the population density and large point sources (Fig. 3a, see its definition in Section 2.1.2), indicating higher representation error in urban grid cells. We conclude that different statistics of the representation error need to be derived for the ‘urban’ 0.5° resolution land grid cells of the mask in Fig. 3a in one hand, and ‘rural’ 0.5° land grid cells of this mask on the other hand.
On the opposite, the aggregation error (Fig. 3c) being sampled at the coarse (~3° horizontally) grid resolution while urban area have generally smaller horizontal scales, their sub-sampling and thus the derivation of statistics for ‘urban’ and ‘rural’ grid cells (defined in Section 2.1.2) does not seem to be adapted.
The prior FFCO_{2} errors (Fig. 3d) have a magnitude similar to the aggregation error. Their spatial distribution is strongly linked with that of the differences between the emission budgets for the control regions/months from the two inventories EDG-IER and PKU-CO_{2}-2007. As a consequence, it is not systematically consistent with that of the most urbanized areas in Europe. As an example, mean prior FFCO_{2} errors reach 0.5 ppm in the Balkans but do not exceed 0.3 ppm in Northern Italy or in England. As for the aggregation error, the derivation of statistics of the prior FFCO_{2} error for ‘urban’ and ‘rural’ grid cells is not adapted.
Assuming that the statistics of the observation errors are independent of the location within the spatial categories defined above, we analyse the temporal variations of these statistics through that of the spatial RMS (over all corresponding 0.5° to 3.5° × 2.75° horizontal land grid cells) of the urban and rural representation errors, of the aggregation errors and of the prior FFCO_{2} errors for the two-week mean afternoon FFCO_{2} gradients to JFJ. The corresponding time series are given in Fig. 4.
There is a clear seasonal variation in the urban and rural representation errors (Fig. 4a). In spring and summer, when the vertical mixing of the lower atmosphere is stronger, the representation error drops to about 0.7 ppm for urban areas and 0.4 ppm for rural areas, while in winter, it can peak at about 2.0 ppm over urban grid cells and 0.8 ppm over rural grid cells, which is about twice the values in summer. Levene’s test shows that the variances of the representation errors are distinct between the four seasons, except when comparing values for urban representation errors in spring vs. summer (p < 0.05 between rural values of all other pairs of seasons, or between urban values of any pair of season). Therefore, different statistics of the representation errors need to be derived for the different seasons i.e. spring (March to May), summer (June to August.), autumn (September to November) and winter (December, January and February). The prior FFCO_{2} errors also show significant differences between the different seasons (p < 0.05 between all pairs of seasons). So the same seasonal categorization will also apply to them.
By contrast, there is only a small seasonal variation in the aggregation error (Fig. 4b). ε_{a} has lower values in spring and summer than in autumn and winter (Levene’s test, p < 0.05). Consequently, we use different statistics for ε_{a} in spring/summer and in autumn/winter.
The distributions of most of the different categories of representation, aggregation and prior FFCO_{2} errors defined by the Sections 4.1 and 4.2 are shown in Fig. 5. These categories used to build these distributions have a high number of values (at least 715 samples within one category, this minimum number applying to Hε_{b} in winter) so that the statistics from these distributions should be robust.
Two theoretical distributions are superimposed to each of the practical sampling of the errors in Fig. 5: a Gaussian distribution whose mean and standard deviation correspond to that of the practical sampling, and a Cauchy distribution whose location and scale parameters correspond to that of the practical sampling. For all categories of representation, aggregation and prior FFCO_{2} errors, the Gaussian distribution is a poor approximation of the practical distribution, whereas the Cauchy distribution, with a relatively narrow peak, generally fits better with the practical distribution.
All these distributions, which are based on all sites within a category, have near-zero means, which supports the assumption that observation errors are unbiased. Of note is that the potential consistency of the error in time at a given site or between neighbouring sites is not reflected in this distribution but is characterized through the analysis of the temporal and spatial correlations of these errors (see below). The standard deviations of these distributions are indicated in Table 1. The representation errors are much larger than the aggregation errors, and reach as high as 1.68 ppm for ‘urban’ grid cells in winter (Fig. 5c). The aggregation errors (Fig. 5e and f) are about one order of magnitude smaller than representation errors. The prior FFCO_{2} errors (Fig. 5g and h) are slightly larger than the aggregation errors but are still lower than the representation error by a factor of 3 to 6.
The temporal correlations of the different spatial categories of representation, aggregation and prior FFCO_{2} errors for two-week mean afternoon gradients are illustrated in Fig. 6a using the temporal auto-correlations for errors of all the occurrences of the gradients from all the potential 100 magl sites to the JFJ reference site. The autocorrelation for a given time lag is derived from the ensemble across all times and sites of all couples of errors which both apply to the same site and to two times separated by the given time-lag. Initial estimates accounting for the temporal categories (not shown) indicated that the temporal auto-correlations for different seasons are quite close to each other, so the temporal categorization is ignored here.
There are strong temporal auto-correlations in all types of errors for two-week mean afternoon gradients. The temporal auto-correlations of the representation errors, of the aggregation errors and of the prior FFCO_{2} errors are above 0.4 even when the time lag exceeds 3 months. The estimates of the autocorrelations of the errors computed separately for each potential site used in the gradient FFCO_{2} computations (not shown), are nearly null for time lag larger than 1 month for most of the potential sites. This indicates that the errors combine a sort of long term error component that is specific to each site (acting as a bias it does not show up in the site correlation), and a short-term error component whose typical correlation timescale is smaller than 1 month. A sum of two exponentially decaying functions r(Δt) = a × e^{−Δt/b }+ (1 − a) × e^{−Δt/c} is thus fitted to the estimate of the temporal auto-correlations (when using a sampling of the errors across all the times and sites) of each type of error, where Δt is the time lag (in days) and a, b, c are the parameters that are optimized by the regressions. The short timescale of correlation b arising from these regressions ranges from 9.3 days for the rural representation error to 16.6 days for the urban representation error. These values are close to the sampling integration time of 2 weeks. The long timescale of correlation c is larger than 1 year except for the rural representation error and prior FFCO_{2} errors. The relative weight of the short term component of the errors (a) for the two-week mean afternoon FFCO_{2} gradients is systematically below 40%. It is more important for the representation errors than for the aggregation and prior FFCO_{2} errors.
The spatial correlations within the different categories of error on two-week mean afternoon gradients or between the urban and rural representation errors on two-week mean afternoon gradients are shown in Fig. 6b. Their estimates for a given distance are based on the ensemble across all times and sites of all couples of errors which both apply to the same time and to two sites separated by the given distance (using intervals for this distance of ±20 km for the representation errors at the 0.5 horizontal resolution, and of ±150 km for the aggregation errors and the prior FFCO_{2} errors at the 3.75° × 2.5°resolution). Again, initial estimates accounting for the temporal categories (not shown) indicated that the spatial correlations for the different seasons are quite close, so the temporal categorization is also ignored here.
The spatial correlations of the representation and aggregation errors drop very fast with increasing distance which is not the case for the prior FFCO_{2} errors. The urban and rural representation errors even have negative spatial correlations when the distance is within the range of 100–300 km. This is driven by the fact that the representation and aggregation errors when using the average concentration and emissions (respectively) within a given area (a grid cell or a region) are necessarily balanced and have thus opposite signs over areas smaller than that of this area. An exponentially decaying function r(Δd) = e^{−Δd/a} is fitted to these estimates of spatial correlations, where Δd is the distance (in kilometers) and where a is the parameter that the regressions derive. The e-folding correlation lengths a are 75 and 89 km for the urban and rural representation errors respectively. The spatial correlations between urban and rural representation error has a similar e-folding correlation length of 55 km. In general, even though those correlations between rural and urban representation errors are smaller than that within a given category of representation error, they are very close to them, and the spatial correlations of the representation errors are weakly impacted by the categories which we have defined for these errors. The e-folding correlation length a is 171 km for the aggregation error. All of the correlation lengths derived for the representation and aggregation errors are thus smaller than the length of the LMDZ transport model grid cells. The spatial correlations of the representation and aggregation errors are thus negligible at this transport model resolution. However, the correlation length scale of the prior FFCO_{2} errors is approximately 700 km, which is larger than transport model resolution.
All the results above are derived for two-week mean afternoon gradients between sites at 100 magl and JFJ. Here, we investigate the variations of the different categories of representation, aggregation and prior FFCO_{2} errors for two-week mean afternoon gradients as a function of the sampling heights for all sites whose difference to JFJ correspond to these gradients, from near ground (20 magl) to top of planetary boundary layer (1000 magl, roughly). The height of the measurements at the JFJ reference site is not modified hereafter. The ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}$ and ${\mathbf{H}}_{\text{transpHR}}^{\text{CHIM}}$ operators (their selection of the LMDZ and CHIMERE vertical levels corresponding to the measurement locations; see Sections 2.1.3 and 3) are adapted for such a derivation of the vertical profiles of the errors. The sampling heights tested are 20, 50, 100, 200, 300, 500 and 1000 magl. These heights correspond to seven different vertical levels in CHIMERE (the bottom sampling height corresponds to the 1st CHIMERE level while the top sampling height corresponds to the 12nd to 15th CHIMERE level depending on the horizontal grid cells). The representation errors being computed at the spatial resolution of CHIMERE, we thus obtain different value of the different categories of representation errors per sampling height. However, due to its coarse vertical discretization, these sampling heights correspond to only the first five levels of the LMDZ model. This explains why only five values of the different type of categories of aggregation and prior FFCO_{2} errors (that are computed at the spatial resolution of LMDZ) are derived for these seven sampling heights.
Fig. 7 shows the corresponding vertical variations of the standard deviations of all the occurrences of each category of representation, aggregation and prior FFCO_{2} errors for two-week mean afternoon gradients. All categories of the errors decrease significantly with increasing sampling height. However, the different categories of errors have different vertical profiles. The standard deviation of the representation error for gradients between 300 magl sites in ‘urban’ grid cells and JFJ is equal to ~70% (in summer) or to ~50% (in winter) of the values for gradients between 20 magl sites in urban grid cells and JFJ. This seasonal variation in the decrease with height is likely due to the higher emissions but shallower depth of the vertical mixing in fall–winter than in spring–summer. The representation errors of FFCO_{2} gradients between rural grid cells and JFJ have relatively smaller vertical variations. This is likely due to the fact that the atmospheric signature of the emitting urban grid cells and thus the corresponding representation errors have been highly diffused in the vertical during the transport from such urban grid cells to the rural grid cells. The standard deviations of the aggregation errors for gradients between 300 magl sites in ‘urban’ grid cells and JFJ are equal to about 75% of those for gradients between 20 magl sites and JFJ. Of note is also that the aggregation errors drop significantly as the heights exceed 400 magl unlike the representation errors. The vertical distribution of the prior FFCO_{2} errors is similar to that of the aggregation errors. As a consequence, from the surface to 100–300 magl, the ratio of the prior FFCO_{2} errors over the sum of all observation errors increases. However, mainly due to the fact that measurement errors do not decrease with altitude, this ratio decreases with increasing heights above 300 magl.
All the results above about the representation and aggregation errors are derived for 2-week mean FFCO_{2} afternoon gradients. Here, we investigate the representation and aggregation errors for one-day, one-week, two-week and one-month mean afternoon gradients between 100 magl sites and the JFJ reference site. The ${\mathbf{H}}_{\text{samp}}^{\text{coloc}}$ and ${\mathbf{H}}_{\text{transp}}^{\text{CHIM}}$ operators (their temporal averaging of the afternoon FFCO_{2} gradients; see Sections 2.1.3 and 3) are adapted accordingly.
The standard deviations of all the occurrences of FFCO_{2} gradients within each category of the representation, aggregation and prior FFCO_{2} errors, and of the gradients generated using our practical simulation of the actual gradients ${\mathbf{H}}_{\text{transpHR}}^{\text{t}}{\mathbf{f}}_{\mathbf{\text{HR}}}^{\mathbf{\text{t}}}$ are shown in Fig. 8 as a function of the sampling integration time. All the errors and simulated gradients decrease significantly from one-day mean afternoon samplings to two-week mean afternoon samplings, while the decrease of the values from two-week to one-month mean afternoon samplings is relatively small. As analysed earlier when studying the temporal auto-correlations of the errors, this highlights the fact that these errors combine a long-term component specific to each site and a short term component. Fig. 8e and f show that this also applies to the simulation of the FFCO_{2} gradients. As for the analysis of the temporal autocorrelations of the errors, a sum of two exponentially decaying functions ε(l) = ε(1)×[a × e^{−(l−1)/b }+ (1 − a) × e^{−(l−1)/c}] is thus fitted to the values of the errors and simulated gradients as functions of the sampling integration time, where l is the integration time (in days) of the mean afternoon sampling, ε(1) is the standard deviations of the errors (or simulated gradients) with one-day sampling, and where a, b and c are the parameters that are optimized by the regressions. The values obtained for the b range between 2 and 5 days, and those of c often exceed 1 year (Table 3), reflecting, as when fitting the temporal auto correlations, the synoptic timescales and a long-term site specific error respectively. While for the representation and aggregation errors, a (the weight of the short term component, Table 3) ranges between 24 and 48%, it is lower for the prior FFCO_{2} errors (17 to 22% depending on the season) and for the simulated gradients (9 to 21% depending on the season).
Further analysis of the results when using different sampling integration time for the observations indicates that the spatial correlations of the errors do not evolve significantly as a function of this integration time. However, Fig. 9 shows that the temporal correlations of the representation errors associated with one-day mean afternoon gradients decrease a lot with increasing timelag until the timelag reaches 1 week, which could not be characterized when analyzing errors for two-week mean gradients in Section 4.3. Fitting the temporal correlations of the errors for one-day mean samplings with the sum of two exponentially decaying functions (as when analyzing the temporal correlations of the errors for two-week mean gradients, Table 2) indicate that the timescale of correlation for the short term components of the representation and aggregation errors is about 1 day and that for the prior FFCO_{2} error is 2.1 days. On the other hand, the timescale of the long-term component of the errors for one-day mean gradients still exceed 1 year except for the prior FFCO_{2} error, as when analyzing errors for two-week mean gradients. However, the relative weight of the short-term component (a) are one to two times higher than that for two-week mean afternoon gradients when analyzing the 1-day mean gradients.
Of note is the fact that the aggregation error for one-day gradients due to the coarse resolution of the control vector has a component (in addition to the short term and long term components already characterized) that has a weekly cycle which is shown by the cycle of the temporal autocorrelations of the error at this frequency, and which reflects the quite artificial differences between the flat temporal profiles of the emissions in the PKU-CO_{2}-2007 inventory and the hourly variations of the emissions in the EDG-IER inventory. The existence, for all types of errors, of a third component at the daily scale in addition to a short term component at the synoptic scale and to the long term component, which could not be detected by the analysis of the autocorrelations at the two-week mean scale, could explain the differences between the results obtained when analysing the results at the one-day vs. two-week scale.
In Section 2.2, we justified our estimation of the representation, aggregation and prior FFCO_{2} errors based on the assumption that the distributions of these errors are Gaussian and unbiased (as required for the application of the atmospheric inversion framework; (Lorenc, 1986)). Fig. 5 shows that the means of the representation, aggregation and prior FFCO_{2} errors are much smaller than the standard deviations of these errors, indicating that the assumption that their distribution is unbiased is relevant. A Cauchy distribution generally shows a better fit with the practical sampling of the errors than the Gaussian distribution, but the Gaussian distribution is still a good approximation of the former. However, the dependence of the long-term component of the errors to the sites used for the computation of the gradients could be interpreted as a sort of local bias in the representation and aggregation errors. While such a behaviour of the prior errors is generally well anticipated by inversion systems assuming long temporal autocorrelations of the prior uncertainties, these representation and aggregation errors can hardly be considered as a random noise for the observations as is done traditionally. However, accounting for both the short term and the long term error temporal correlations in the configuration of the inversion systems is feasible if using the type of regressions used in this study, and, if done, should enable a good characterization of these errors. And these analyses strengthen the posterior justification for our practical derivation of the representation and aggregation errors.
The analysis of the temporal autocorrelations of the errors for two-week to one-day mean errors and of the decrease in the standard deviation of the errors as a function of the temporal sampling integration time reveals three rather than two dominating component of the representation, aggregation and prior FFCO_{2} errors. The first component, at the daily scale, may be driven by the day to day variations of the emissions which are high compared to the long-term variations in these emissions and which are highly uncertain in present inventories. This component can be highlighted when analyzing errors for one-day mean gradients only. The second component, at the synoptic scale, is likely related to the transport of the error from the emission areas to the sites through synoptic events which are a critical component of the transport in Europe (Parazoo et al., 2008; García et al., 2010). The third component is related to the slowly varying (at the ‘long-term’ seasonal to inter-annual scales), continuous and direct influence of the emission in the vicinity (in the same transport model grid cell) of the different sites, which thus significantly varies from site to site and depending whether it applies to urban or rural sites. The urban sites receive a strong signature of the emissions in their vicinity while the rural sites are affected by these emissions in their corresponding grid cells through a more indirect process of aggregation, and with a weaker signal. This explains why the relative weight of the synoptic component compared to the long-term one is generally smaller for gradients between urban sites and JFJ than for gradients between rural sites and JFJ.
The analysis of these different components reveals that the daily components dominate in the errors and simulated signal. This component is cancelled when sampling the observations at the one-week to the one-month scale, in which case the long-term component dominates the errors and simulated signal. The differences between the numbers (Table 2) characterizing the temporal scales of these components and their relative weight depending on the specific analysis lead in this study can be explained by the uncertainties associated with these statistical analyses, by the difficulty to make a regression with a sum of three exponentially decaying functions when analyzing the errors and signal for one-day mean gradients (which could, in principle, help reconcile it with the results when analyzing the errors and the signal for two-week mean gradients), and by the difference between the analysis of the temporal correlations of errors for a given temporal sampling of the observations and that of the amplitude of the error for different temporal sampling of the observations. However, their general consistency gives insights into the typical correlation length scales to be used for accounting for such correlations when conducting atmospheric inversions, which will be critical given the amplitude of these correlations.
The analysis of the standard deviation and of the temporal scales of autocorrelations of the errors indicate that the representation error and the transport errors are the largest observation errors for 100 magl afternoon observations for any temporal scale of sampling when using the modelling framework of our study. They are larger than the measurement error (about 1 ppm) which is the third dominant type of observation errors in our modelling framework. The aggregation errors have a relatively small standard deviation compared to these errors (see their smaller standard deviation in Table 1 and their shorter temporal scale of auto-correlation in Table 2).
In total, the weight of the observation errors can reach up to 50% of the typical amplitude of the simulated 1-week to 2-week mean afternoon FFCO_{2} gradients, either for urban or rural sites, or for any season and temporal sampling. It can even reach up to 90% of this signal for one-day mean afternoon FFCO_{2} gradients. This questions the precision of the signature of the large-scale budget of FFCO_{2} emissions that could be filtered from the observations. Furthermore, the actual signal that the inversion aims at filtering from the assimilated prior-model data misfits in order to correct for the prior knowledge on the emissions at the control/one-month scale is that of the prior FFCO_{2} errors whose amplitude is generally smaller than that of the representation, transport and measurements errors. From our analysis, the signals at a given site from uncertainties in the distribution of the local emissions/concentrations (characterized by the representation error) exceed those of the uncertainties in the emissions at the regional scale (characterized by the prior FFOC_{2} error) and the signal from the uncertainties in the distribution of the emissions at the sub-regional scale (characterized by the aggregation error) at any temporal scale.
The temporal autocorrelations of the prior FFCO_{2} error have a structure that is similar to that of the representation errors. Its long-term component has a shorter temporal scale than that of the representation error but this can hardly be viewed as a basis for a practical separation of the prior FFCO_{2} and representation errors given that both these temporal scales exceed 300 days. The temporal length scale of the short term components for the prior FFCO_{2} and representation errors are different. However, the combination of representation, aggregation, measurement and transport errors which all have their own temporal scale of correlations will likely make it difficult to exploit the structures of the short term variations to filter the prior FFCO_{2} errors.
The most promising result for the potential filtering of the prior FFCO_{2} errors lie in the analysis of the spatial correlations of the errors. The prior FFCO_{2} errors are connected to uncertainties in emissions at large spatial scales which are not necessarily compensated between neighbouring control regions, while representation and aggregation errors should be compensated at the resolutions of the transport model to the control vector (and thus cancelled through atmospheric mixing faster than the prior FFCO_{2} errors). At the same time, transport errors and measurement errors should not be correlated in space. In consequence, the spatial correlations of the prior FFCO_{2} errors are larger than that of the observation errors. This could be exploited by the inversion to filter it if it can rely on a spatially dense network of measurement sites.
An analysis of the correlations between different types of errors and the transport conditions (e.g. wind direction and speed) could bring additional insights on the capability for isolating the prior FFCO_{2} errors from the observations. However, such an analysis could hardly be based on the framework of this study for which the different types of error (in particular the representation and prior FFCO_{2} errors) are derived using different modelling frameworks. A full assessment of this capability requires atmospheric inversions using the characterization of the errors from this study. However, this study already give insights into the challenges underlying the monitoring the emissions at large scale.
Of note is that the quite simple analysis of the sensitivity of the ratio between the prior FFCO_{2} errors and the observation errors supports the monitoring of FFCO_{2} at heights ranging between 100 and 300 magl when targeting the large-scale budgets of emissions. It is also the traditional heights for ICOS network (Kadygrov et al., 2015).
The results from this study do not strongly encourage to sample FFCO_{2} observations at high temporal resolution when targeting the regional/one-month scale budgets of emissions with atmospheric inversion built on a coarse-grid transport model. We have shown the relative weight of the short term components of the observation errors are larger than that of the prior FFCO_{2} errors and, as said previously, making it difficult to exploit the difference between the temporal correlation of these short term components to filter the prior FFCO_{2} errors. However, again, atmospheric inversion experiments would be required to check how much the correlations of the errors with the transport could be exploited, even at high temporal resolution. Still, having one-day mean sampling would dramatically decrease the weight of the measurement errors on longer time scales that can be considered as a random noise on each individual measurement of FFCO_{2}.
As explained in Section 2.3 and in the Appendix, in theory, solving for the emissions at high resolution or solving for emissions at large scale but perfectly accounting for the aggregation error would yield similar estimates of the emissions at large scale. In practice, as illustrated by Fig. 6, the complex structure of the correlations in the aggregation error may prevent such a perfect account for this error in the inversion set-up. Therefore, in principle, it is better to solve for the emissions at the highest resolution as possible.
On the other hand, the representation and transport errors are also highly dependent on the transport model resolution. Since the representation errors are the largest component of the observation error for our modelling framework, increasing the transport model resolution should be viewed as the most critical mean for improving (if needed) the results from atmospheric inversion for the large-scale monitoring of the emissions. Cancelling the representation errors would dramatically increase the ratio between the prior FFCO_{2} errors and the observation errors. However, this would require using a regional inverse modelling framework focusing on a specific area (such as Europe, the US or China) since the horizontal resolution of the atmospheric transport model used for global inversion hardly exceed 3° resolution.
This paper analyses the critical sources of errors that influence the estimate of FFCO_{2} emissions at sub-continental/monthly scale from atmospheric inversion based on continental networks of daily to monthly mean afternoon atmospheric FFCO_{2} observations. We provide a theoretical derivation of the representation and aggregation errors affecting daily to monthly mean afternoon FFCO_{2} gradients between possible measurement sites and a background station. This theoretical derivation is adapted to the practical estimation of these errors in Europe for our specific inverse modelling framework that is based on a global coarse-resolution transport model. Our analysis focuses on the derivation of the standard deviations, temporal and spatial correlations of the representation and aggregation errors, the standard deviation of the transport model and measurement errors, along with the standard deviation of the atmospheric signature of the prior uncertainty in the regional/1-month budgets of the emissions. These statistical parameters will be primarily used to set up a realistic configuration of the observation errors in inversion experiments described in future papers. In particular, the modelling of the spatial and temporal correlations using one or two exponentially decaying functions and the optimized parameters from the regressions in this study will be used to model the observation error correlations in these experiments.
Our comparison between these statistical parameters for the different types of errors also aims at assessing the ability to filter the signature of the prior uncertainty in the large-scale budgets of the emissions from the total observation errors when assimilating one-day to one-month mean afternoon FFCO_{2} gradients (which underlies the potential for estimating the emissions at large scale). It highlights that the representation, transport and measurement errors dominate the observations errors, while the weight of aggregation error is relatively small. In total, observation errors can reach up to 50% (90%) of the typical 2-week (1-day) mean FFCO_{2} gradients, and are larger than the signature of the prior uncertainty in the large-scale budgets of the emissions. Moderating the representation and transport errors by using a regional transport models at higher resolution could thus be a requirement for the monitoring of FFCO_{2}, even when targeting their large-scale budgets. The analysis also highlight the fact the critical weight of the temporal correlation of the representation and aggregation errors, and in particular that they have a long-term component, make it difficult to separate these errors from the signature of the prior uncertainty in the emissions at large scale when assimilating the one-day to one-month mean afternoon FFCO_{2} gradients. Filtering of the signature of the prior uncertainties could potentially rely on its spatial correlations scales which are significantly longer than that of the observation errors. This would require a network dense enough to capture the spatial coherence of this signature (at scales shorter than ~700 km), which could represent a larger number of sites than that of the present ICOS network. Finally, from this study, we do not recommend sampling FFCO_{2} data at high (one-day to one-week) rather than low (two-week to one-month) temporal resolution for the global atmospheric inversion based on coarse-grid transport model, since it demonstrates the difficulties associated with filtering the signal from prior uncertainties at such temporal scales.
More generally, while the statistics of representation and aggregation errors derived in this study primarily relate to the specific atmospheric inversion framework we use in this study, this study brings insights regarding these errors for a wide range of atmospheric applications, and more specifically to that dedicated to the inversion of the FFCO_{2} emissions. The practical derivation of their statistics can be easily generalized based on our theoretical framework and used for other studies. The structure and typical amplitude of the representation error derived for the transport of uncertainties in the fossil fuel emissions in Europe at ~3° resolution with LMDZ should be similar for other transport models with similar spatial resolution. The aggregation errors are shown to characterize the weight of uncertainties in the spatial and temporal resolution of the emissions even when solving for them at the transport model resolution. And the general conclusions raised above regarding the potential for filtering the signature of the uncertainty in the large-scale budget of the emissions using ‘remote’ measurement stations should be a general outreach of this study. While the correlations of the observation errors are generally ignored in atmospheric inversion, this study demonstrates how critical it should be to account for them. By conducting inversion experiments accounting for these correlations, the companion papers of this study should now indicate how much the potential for filtering the signature of the uncertainty in the large-scale budget of the emissions based on its specific spatial structure can be effectively exploited by atmospheric inversion to solve for the regional/monthly scale budgets of the emissions.
This work was supported by the European Commission under the EU Seventh Research Framework Programme [grant agreement number 283080], geocarbon project).
No potential conflict of interest was reported by the authors.
The authors acknowledge the support of the French Commissariat à l’énergie atomique et aux énergies alternatives (CEA). This study is co-funded by the European Commission under the EU Seventh Research Framework Programme (grant agreement No. 283080, geocarbon project). G. Broquet and F. Vogel acknowledge funding from the industrial chair BridGES (supported by the Université de Versailles Saint-Quentin-en-Yvelines, the Commissariat à l’Energie Atomique et aux Energies Renouvelables, 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.
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A1 Further decomposition of the representation error
In Equation (4), we derived the representation error as ${\mathbf{H}}_{\text{transpHR}}^{\text{t}}{\mathbf{f}}_{\mathbf{\text{HR}}}^{\mathbf{\text{t}}}$ – H_{samp}${\mathbf{H}}_{\text{transp}}^{\text{t}}{\mathbf{H}}_{\text{distr}}^{\text{t}}$x^{t}. As indicated in Section 2.2, it can be further decomposed to (1) errors arises from the modelling of concentrations at coarse resolution in the observation operator while it is compared to local measurements; (2) errors due to representing (averaging) the emissions at the coarse resolution of the transport model (into f^{t} = ${\mathbf{H}}_{\mathbf{\text{distr}}}^{\mathbf{\text{t}}}{\mathbf{x}}^{\mathbf{\text{t}}}$) and thus missing the true very-high resolution patterns in the emissions:
or in the other way round, accounting for the variability in the fluxes first:
where H_{transpHR → LR} is a theoretical operator corresponding to the transport from emissions f_{HR} but whose output concentrations are projected into the coarse resolution space of the outputs of H_{transp}, while H_{transpLR → HR} is a theoretical operator corresponding to the transport from emissions at low resolution but whose output concentrations are at high resolution.
Following the notations as in Section 3, we can derive a practical estimate corresponding to Equation (A1-1):
and
or the practical estimate corresponding to Equation (A1-2):
and
We have made those two practical computations for these two decomposition (not shown here). The dominant component is dependent on the order of the decomposition (A1-1 vs. A1-2), the first term being larger than the second term for each decomposition. Therefore, there is no robust characterization of each sub-component of the representation error, even though they correspond to different physical processes. We thus avoid further analysis of these decomposition of the representation error.
A2 Equivalence of the results for large region-months from an inversion at transport model resolution and from an inversion at large region-months resolution
In this section, we demonstrate that using the same atmospheric transport model but solving for emissions at the transport model resolution (called hereafter ‘pixel level’) or for large regions/time windows (accounting for aggregation error) should lead to the same results for large regions/time windows. The relationship between the control vectors at pixel level and at large region level is x_{PIX} = H_{distr}x. We define the aggregation operator $\mathbf{\Gamma}$ which aggregates the emissions at pixel level into large regions and time windows. The product between $\mathbf{\Gamma}$ and H_{distr} yields the identity matrix ($\mathbf{\Gamma}$H_{distr} = I) and $\mathbf{\Gamma}$x_{PIX} = x. The observation operator when using a pixel level control vector writes H_{PIX} = H_{samp}H_{transp} and H = H_{PIX}H_{distr}.
Assuming that errors have a Gaussian and unbiased distribution, at the pixel level, the prior uncertainty covariance is denoted B_{PIX}. The pixel level uncertainty ε_{PIX} is composed by an uncertainty at a resolution higher than that of the large region ε_{SUBR} ($\mathbf{\Gamma}$ε_{SUBR} = 0) whose covariance is denoted B_{SUBR} and verifies $\mathbf{\Gamma}$B_{SUBR} = 0 (see below), and by the projection H_{distr}ε of the uncertainty at the level of the large region ε, whose covariance can be written H_{distr}B${\mathbf{H}}_{\text{distr}}^{\text{T}}$. These 2 components are not necessarily independent, so that B_{PIX} = B_{SUBR} + H_{distr}B${\mathbf{H}}_{\text{distr}}^{\text{T}}$ + 2B_{cov} where B_{cov} is the matrix of covariance between H_{distr}ε and ε_{SUBR} which verifies $\mathbf{\Gamma}$B_{cov} = 0 (see below). Our bottom-up definition of the aggregation error is based on the assumption that B_{cov} = 0 and corresponds to R_{a} = H_{PIX}B_{SUBR}${\mathbf{H}}_{\text{PIX}}^{\text{T}}$ (i.e. on the projection of ε_{SUBR}). On the other hand, the top down approach of (Kaminski et al. (2001)) fully account for B_{cov} by defining R_{a} = H_{PIX}(B_{PIX – } H_{distr}B${\mathbf{H}}_{\text{distr}}^{\text{T}}$) ${\mathbf{H}}_{\text{PIX}}^{\text{T}}$ i.e. R_{a} = H_{PIX}(B_{SUBR} + 2 B_{cov})${\mathbf{H}}_{\text{PIX}}^{\text{T}}$. A common formulation of the covariance of the aggregation error is thus R_{a} = H_{PIX}(B_{SUBR} + 2 B_{cov})${\mathbf{H}}_{\text{PIX}}^{\text{T}}$ with the assumption that B_{cov} = 0 in our bottom-up framework (see the discussion regarding this assumption at end of this section).
The observation error covariance when using a pixel level control vector is denoted R_{PIX}. Applying the alternative formulation of Equation (2) (ignoring y_{fixed} which is null in our framework):
at the pixel level and at the large region levels isolating aggregation error yields:
Aggregating the results from Equation (A2-2) to large regions, we get
which simplifies into Equation (A2-3) and thus it means that the inverted emissions at large scale are the same when using the two types of control vectors.
Regarding the demonstration that $\mathbf{\Gamma}$B_{SUBR} = 0 and $\mathbf{\Gamma}$B_{cov} = 0: detailing B_{SUBR} (and B_{cov}) as the covariance of errors ε_{SUBR} (between H_{distr}ε and ε_{SUBR}) with $\mathbf{\Gamma}$ε_{SUBR} = 0, we get B_{SUBR} = E[ε_{SUBR}•${\mathit{\epsilon}}_{\text{SUBR}}^{\text{T}}$] and B_{cov} = E[ε_{SUBR}•(H_{distr}ε)^{T}]so that $\mathbf{\Gamma}$B_{SUBR} = $\mathbf{\Gamma}$E[ε_{SUBR}•${\mathit{\epsilon}}_{\text{SUBR}}^{\text{T}}$] = E[$\mathbf{\Gamma}$ε_{SUBR}•${\mathit{\epsilon}}_{\text{SUBR}}^{\text{T}}$] = 0 and $\mathbf{\Gamma}$B_{cov} = $\mathbf{\Gamma}$E[ε_{SUBR}•(H_{distr}ε)^{T}] = E[$\mathbf{\Gamma}$ε_{SUBR}•(H_{distr}ε)^{T}] = 0.
The assumption that B_{cov} = 0 i.e. of the independence between the prior uncertainty and the uncertainty in the distribution of the emissions in the control regions ε_{SUBR} (and consequently between the prior uncertainty and the aggregation error) is likely strong.
The typical inventories used to produce both the prior estimate of the emissions and H_{distr} are built on a mix of ‘top-down’ computations relying on a disaggregation of large-scale budgets of the emissions from statistics on the fossil fuel consumption, and on ‘bottom-up’ computations relying on emissions factors and local activity data. In the first type of computations, the data used for the disaggregation can be highly independent of the large-scale statistics which would qualitatively support the assumption that B_{cov} = 0. However, the statistics on the consumption of different fossil fuels can be used to derive the separation of the emissions between different types of anthropogenic activities (e.g. liquid fuel can be mainly related to traffic, gas and coal mainly to power generation …) and thus to derive information on the spatial and temporal distribution of the emissions. Through such a process, uncertainties at large scale can thus be downscaled into uncertainties at higher resolution. Furthermore, errors in the different emission factors used for the bottom-up computations generate uncertainties that are highly correlated between the high and low resolution.
The assumption that the prior uncertainty and the aggregation error are independent (e.g. B_{cov} = 0) is thus unlikely, and we have tried to analyse the relative weight of the term 2H_{PIX}B_{cov}${\mathbf{H}}_{\text{PIX}}^{\text{T}}$ that we have implicitly ignored in the derivation of the aggregation error R_{a} = H_{PIX}B_{SUBR}${\mathbf{H}}_{\text{PIX}}^{\text{T}}$ instead of R_{a} = H_{PIX}(B_{SUBR} + 2 B_{cov})${\mathbf{H}}_{\text{PIX}}^{\text{T}}$. Following a similar method and using the same notation as in Section 3, we can derive an estimate of the diagonal of H_{PIX}(B_{SUBR} + 2 B_{cov})${\mathbf{H}}_{\text{PIX}}^{\text{T}}$ = H_{PIX}E[ε_{SUBR}.(ε_{SUBR} + 2H_{distr}ε)^{T}]${\mathbf{H}}_{\text{PIX}}^{\text{T}}$ from statistics on the diagonal elements of $\left[{\mathbf{H}}_{{\text{samp}}^{\text{coloc}}}{\mathbf{H}}_{{\text{transp}}^{\text{LMDZ}}}\left({\mathbf{H}}_{{\text{distr}}^{\text{EDG-IER}}}-{\mathbf{H}}_{{\text{distr}}^{\text{PKU}}}\right){\mathbf{x}}^{\text{EDG-IER}}\right]\xb7$${\left[{\mathbf{H}}_{{\text{samp}}^{\text{coloc}}}{\mathbf{H}}_{{\text{transp}}^{\text{LMDZ}}}\left(\left({\mathbf{H}}_{{\text{distr}}^{\text{EDG-IER}}}-{\mathbf{H}}_{{\text{distr}}^{\text{PKU}}}\right){\mathbf{x}}^{\text{EDG-IER}}+2{\mathbf{H}}_{{\text{distr}}^{\text{PKU}}}\left({\mathbf{x}}^{\text{EDG-IER}}-{\mathbf{x}}^{\text{PKU}}\right)\right)\right]}^{\text{T}}$. Following the categorization discussed in Section 4.2, the difference between the corresponding estimate of the aggregation error and the one we have described in this study is about −0.03 ppm for both spring/summer and fall/winter. This would decrease the aggregation error by less than 20%. Considering the comparison with other sources of observation error (see Table 1), especially the representation error and the measurement error, such a modification would not impact the conclusions of this study.