The estimate of uncertainties related to the different ICOS networks is based on an ensemble of inversions with the variational inversion system of Broquet et al. (2011), assimilating synthetic hourly averages of the atmospheric CO2 data from these networks (during the afternoon or during nighttime only, depending on the type of sites that are considered; see Sect. 2.2.2). A regional atmospheric transport model (see its description below) is used to estimate the relationship between the CO2 fluxes and the CO2 mixing ratios. The inversion system solves for 6 h mean NEE on each grid point of the 0.5∘ × 0.5∘ resolution grid used for the transport modeling. It also solves for 6 h mean ocean fluxes at 0.5∘ spatial resolution in order to account for errors from air–sea fluxes when mapping fluxes into hourly mean mixing ratios. However, analyzing the uncertainty reduction for ocean fluxes is beyond the scope of this paper.

Peylin et al. (2011) indicate that uncertainties in anthropogenic fluxes yield errors when simulating CO2 mixing ratios at ICOS stations that are smaller than atmospheric model errors. Furthermore, the relative uncertainty in anthropogenic emissions is smaller than that in NEE, while on short timescales, the anthropogenic signal is generally smaller than the signature of the NEE at sites that are not very close (typically distances less than 40 km) to strong anthropogenic sources such as cities (see the analysis for the Trainou ICOS station near Orléans, France; Bréon et al., 2015). Relying on such indications, we assume that the errors due to uncertainties in anthropogenic emissions are negligible compared to errors from NEE and atmospheric model errors. This is a reasonable assumption as long as most ICOS stations are relatively far from large urban areas, which should be the case because the ICOS atmospheric station specification document (https://icos-atc.lsce.ipsl.fr/?q=doc_public) recommends that the measurement sites be located at more than 40 km from the strong anthropogenic sources (such as the cities). Zhang et al. (2015) yielded conclusions from their transport experiments at 1∘ resolution, which contradict this assumption and this clearly raises an open debate. However, the evaluation of the inversion configuration from Broquet et al. (2013) supports the use of this assumption for our study.

In order to simulate the full amount of CO2 in the atmosphere, the inversion uses a fixed estimate of the fossil fuel emissions (see below) without attempting to correct it or account for uncertainties in these fluxes. The inversion also uses a fixed estimate of the CO2 boundary conditions at the lateral and top boundaries of the regional modeling domain without attempting to correct it or account for uncertainties in these conditions. This follows the protocol from Broquet et al. (2011), which assumed that the error from the boundary conditions for the European domain is mainly bias and which corrects for such a bias in a preliminary step that is independent to the subsequent application of the inversion. Such an assumption is supported by the evaluation of the inversion configuration by Broquet et al. (2013). The relatively weak impact of uncertainties in the boundary conditions in Europe (while studies in other regions, such as that of Göckede et al., 2010, indicate a high influence of such uncertainties) can be explained by the fact that the spatial scale of the incoming CO2 patterns at the ICOS sites from remote sources and sinks outside the European domain boundaries is relatively large compared to the typical distances between the ICOS sites, due to atmospheric diffusion (especially under west wind conditions, when the air comes from the Atlantic ocean). In principle, the inversion mainly exploits the smaller-scale signal of the gradients between the sites to constrain the NEE, and it is thus weakly influenced by the large-scale signature of the uncertainty in the boundary conditions. In this section we only summarize the main elements of the inversion system, starting with the theoretical framework, while the detailed description can be found in Broquet et al. (2011).

We define the control vector x of the atmospheric inversion as the 6 h and 0.5∘ × 0.5∘ mean NEE and ocean fluxes. The atmospheric inversion seeks the mean xa and covariance matrix A of the normal distribution N(xa, A) of the knowledge on x based on (i) the atmospheric transport model, (ii) the prior knowledge xb of x, (iii) the hourly mean atmospheric measurements y, (iv and v) the covariances B and R of the distributions of the prior uncertainty and of the observation error assuming that these uncertainties are normal and unbiased (i.e., equal to N(0, B) and N(0, R), respectively) and (vi) a Bayesian relationship between these distributions. The observation error is the combination of all sources of misfit between the atmospheric transport model and the concentration measurements other than the prior uncertainty, in particular the measurement errors, the model transport, aggregation and representation errors, and the errors from the model inputs that are not controlled by the inversion.

With this theoretical framework, xa is the minimum of the quadratic cost function J(x) (Rodgers, 2000): J(x)=12(x-xb)TB-1x-xb+12(H(x)-y)TR-1H(x)-y, where T denotes the transpose, and where H is the affine observation operator, which maps the 6 h (00:00–06:00, 06:00–12:00, 12:00–18:00 and 18:00–24:00; UTC time is used hereafter) and 0.5∘ × 0.5∘ mean NEE and ocean CO2 fluxes x to the observational space based on the linear CO2 atmospheric transport model with fixed open-boundary conditions, and with fixed estimates of the anthropogenic fluxes and natural fluxes at resolutions higher than 6 h and 0.5∘. The operator H: x→H(x) can be rewritten H: x→Hx+yfixed, where yfixed is the signature, through atmospheric transport, of the fluxes (in particular the anthropogenic emissions) and boundary conditions not controlled by the inversion. The operator H is the combination of two linear operators: the first operator distributing 6 h mean natural fluxes at the 1 h resolution, and the second operator simulating the atmospheric transport from the 1 h resolution fluxes at 0.5∘ resolution.

The inversion system derives an estimate of xa by performing an iterative minimization of J(x) with the M1QN3 algorithm of Gilbert and Lemaréchal (1989). The gradient of J is derived using the adjoint operator of H thanks to the availability of the adjoint version of the CHIMERE code. The covariance of the posterior uncertainty in inverted NEE A, of main interest for this study, is given by the formula A=(B-1+HTR-1H)-1. This equation demonstrates the point raised in the introduction for justifying the OSSE framework; A does not depend on the observations or on the prior flux values themselves but only on their error covariance matrices, on the observation network density and station location, and on the atmospheric transport operator. This allows assessing the performance of any observation system, whether existing or not. Of note is also that this calculation does not depend on yfixed, i.e., on the boundary conditions or on the anthropogenic fluxes in the domain; therefore, such components can be ignored for the estimate of A.

In this framework, a common performance indicator is the theoretical uncertainty reduction for specific budgets of the NEE estimates (averaged over specified periods of time and over specified spatial domains), defined by γ=1-σaσb, where σa and σb are the standard deviations of the posterior and prior uncertainties in the corresponding integrals in time and space (over the given periods of time and spatial domains) of the 6 h and 0.5∘ resolution NEE field. If the observations perfectly constrain the inversion of a given budget of NEE, then γ=1. If the observations do not bring any information to reduce the error from the prior, γ=0. By definition, γ is a quantity relative to the uncertainty in the prior fluxes, which depends on the type of prior information on NEE that is expected to be used (estimates from a biosphere model in our case, see Sect. 2.2.2). Of note is that the scores of uncertainty and of uncertainty reduction given in this study refer to the standard deviation of the uncertainty in a specific budget of NEE, and that, hereafter, the term standard deviation is generally omitted.

Due to the size of the observation and control vectors in this study, we could not afford the analytical computation of Eq. (2) based on the full computation of the H matrix (using a very large number of transport simulations, as done in Hungershoefer et al. (2010). Instead we use the Monte Carlo approach of Chevallier et al. (2007) to compute A. In this approach, an ensemble of posterior fluxes xai is derived from an ensemble of inversions using the synthetic prior flux xbi and data yi, whose random errors (xbi-xtrue for xbi and yi-Hxtrue for yi) with respect to a known truth (xtrue, whose value does not influence the results analyzed here, and which is thus ignored hereafter) sample the distributions N(0, B) and N(0, R). A is obtained as the statistics of the posterior errors xai-xtrue. The practical size of the ensemble is described below and its determination follows the discussion by Broquet et al. (2011). The convergence of the estimate of the inverted NEE for each inversion and the convergence of the statistics of the ensemble are necessary to ensure that the A matrix computed with this method corresponds to the actual covariance of the posterior uncertainty given by Eq. (2). These convergences cannot be perfect with a limited number of iterations for the minimization algorithm and a limited number of inversion experiments in the Monte Carlo ensemble imposed by computational limitations. Therefore, the estimate of A can depend on parameters other than H, B and R in practice, e.g., the number of iterations and of inversion experiments. However, it has been checked (see Sect. 2.2.2) that the convergence is sufficient, so that this dependence should not be significant for the quantities of interest.

In this study, the operator H is based on the, CHIMERE mesoscale atmospheric transport model (Schmidt et al., 2001) forced with European Centre for Medium-Range Weather Forecasts (ECMWF) winds. We use a configuration with a 0.5∘ × 0.5∘ horizontal grid and with 25 σ-coordinate vertical levels starting from the surface and with a ceiling at ∼ 500 hPa (such a ceiling being usual for regional transport modeling when focusing on mole fractions close to the ground, e.g., Marécal et al., 2015). The spatial extent of the corresponding domain is described below. CHIMERE is an off-line transport model. Hourly mass fluxes are provided by the ECMWF analyzes. The relatively high vertical and horizontal resolutions of CHIMERE allow for a good vertical discretization of the planetary boundary layer (PBL; the first 14 levels are below 1500 m) along with a good representation of the orography and dynamics to match high frequency observations better than with a global configuration, whose typical horizontal resolution is ∼ 3∘ (Peylin et al., 2013).

In this study, we use the European domain shown in Fig. 1a, which covers most of the European Union and some of eastern Europe, with a land surface area of 6.8 × 106 km2. Its southwest corner is at 35∘ N and 15∘ W, and its northeast corner is at 70∘ N and 35∘ E. Two temporal windows are considered, from 30 June to 20 July 2007 and from 2 to 22 December 2007 (of almost 3 weeks each). The choice of these periods of 3 weeks is a tradeoff between widening the scope of the study and computational burden. The Monte Carlo-based flux uncertainty reduction calculations require large computing resources, while we test three different network configurations for two different months, and for different setups of the error covariance matrices. Indeed, 3-week experiments allow for retrieving information about uncertainties at the 2-week scale without being biased by edge effects, i.e., they allow accounting for the impact of uncertainties from the days before the 14 targeted days and for the impact of the assimilation of measurements during the days after these 14 targeted days. The advection of CO2 throughout Europe can last more than 3 days, but atmospheric diffusion ensures that the signature at ICOS sites of the NEE during a 6 h window is generally negligible after 3 days of transport (not shown). Thus, the windows 3–17 July and 5–19 December were chosen for analysis. We consider the results for July and December to be representative for the summer and winter seasons (using the name of the seasons for the Northern Hemisphere hereafter), allowing an analysis of seasonal variations of the flux uncertainty reduction. Choosing year 2007 for the period of the inversion only impacts the meteorological conditions (i.e., the impact on the prior uncertainty, whose local standard deviations are scaled using data from this specific year, as detailed below in this section, is negligible) and thus the atmospheric transport conditions in the OSSEs. We assume that these conditions are not impacted by a strong inter-annual anomaly in 2007 so that they can be expected to be representative of average conditions for summer and winter. Hereafter, the mention of the year 2007 is thus often ignored and we assume that we retrieve typical estimates for July and December.

The setup of the error covariance matrix B follows the methodology of Chevallier et al. (2007). It is chosen to represent the typical uncertainty in estimates from the biosphere models (for NEE) and from climatologies (for ocean fluxes) used by traditional atmospheric inversion systems. The statistics have been derived for estimates from the Organising Carbon and Hydrology In Dynamic Ecosystems (ORCHIDEE) vegetation model (Krinner et al., 2005) and the ocean climatology from Takahashi et al. (2009). The uncertainties in NEE are assumed to be autocorrelated in space and in time and are modeled using isotropic and exponentially decreasing functions with correlation lengths that do not depend on the time or location. A Kronecker product of the matrices of temporal and spatial correlations is applied to define the correlations between uncertainties for different locations and time windows. The e-folding spatial and temporal correlation lengths are set according to the estimation of Chevallier et al. (2012) based on comparison of the NEE derived by the ORCHIDEE model and eddy-covariance flux tower data, for our specific prior flux spatial and temporal resolution, i.e., 30 days in time and 250 km in space over land. NEE uncertainties for different 6 h windows of the day are not correlated, i.e., the temporal correlations only apply to a given 6 h window of consecutive days. The standard deviations of the prior uncertainties in B are set proportional to the heterotrophic respiration fluxes from the ORCHIDEE model (the corresponding proportional coefficient between the heterotrophic respiration and the prior uncertainty at the daily and 0.5∘ scale is approximately 2). We apply time-dependent scaling factors to these fluxes so that the NEE uncertainties have lower values during the night than during the day, and during winter than during summer, providing typical values for grid scale and daily errors of ∼ 2.5 g C m-2 day-1 in summer (maximum value 3.4 g C m-2 day-1) and of ∼ 2 g C m-2 day-1 in winter (maximum value 3.1 g C m-2 day-1). Over the ocean, the prior uncertainty of air–sea fluxes has standard deviations at the 0.5∘ and 6 h scale equal to 0.2 g C m-2 day-1, an e-folding spatial correlation length of 500 km and temporal correlations similar to those for the prior uncertainties over land. Prior ocean and land flux uncertainties are not correlated.

Broquet et al. (2011) analyzed the periods of time during which the CHIMERE European configuration bears transport biases that are too high, so that measurements from ground-based stations such as ICOS sites should not be assimilated to avoid erroneously projecting such biases into the corrections to the fluxes. In agreement with common practice, they concluded that observations at low altitude sites (approximately below 1000 m above sea level, m a.s.l.; see Broquet et al., 2011, for the exact definition of the different types of sites used for the time selection of the data and the configuration of the observation error), which include almost all of the ICOS tall towers, should be assimilated during daytime (12:00–20:00) while the observations at high altitude stations (approximately above 1000 m a.s.l.) should be used only during the night (00:00–06:00). This generally yields larger uncertainty reduction during daytime than during nighttime (Broquet et al., 2011). However, this does not raise a potential bias related to a better constraint on daytime inverted NEE (when the ecosystems are generally a sink of CO2) than on nighttime inverted NEE (when the ecosystems are generally a source of CO2), since uncertainties in both nighttime and daytime prior NEE, transport and measurements are assumed to be unbiased, as supported by the results from Broquet et al. (2013).

The observational error covariance matrix R accounts for various sources of error when comparing the hourly data selected for assimilation and their simulation, which are not controlled by the inversion: measurement error, aggregation error, atmospheric model representativeness and transport error (as explained previously, uncertainties in the anthropogenic emissions and in the boundary conditions are assumed to be negligible). The first two terms are negligible compared to the model representativeness and transport error due to the high measurement standard and to solving for the fluxes at 6 h and 0.5∘ resolution during the inversion, respectively.

Broquet et al. (2011) derived a quantitative estimation of the model error (depending on the station height) including transport and representativeness errors based on comparisons between simulations and measurements of CO2 and 222Rn during summer. Broquet et al. (2013) extended this analysis using 1-year-long time series of simulated and measured CO2 and 222Rn, to provide the season-dependent estimates that are used here. The model error is much higher during the winter than that during the summer. It is given for each site in Table A1 for the 2 months (July, December) considered in this study. We assume that the errors for two different sites are independent and that they do not bear temporal autocorrelations. Thus, the observation error covariance matrix R is set diagonal. There is no evidence that such autocorrelations could be significant in the analysis of Broquet et al. (2011). The resulting budget of observation errors at daily to monthly resolution is reliable (Broquet et al., 2011, 2013). This suggests that the temporal autocorrelations of the actual observation errors are negligible. If the autocorrelations of the actual observation errors were not negligible, this would mean that the errors for hourly data are overestimated. In both cases, the assumption that the temporal autocorrelations of the observation error are negligible does not seem to need to be balanced by an artificial increase of the estimate of the observation errors for hourly averages provided by Broquet et al. (2013).

We use 12 iterations of minimization for each variational inversion of the Monte Carlo ensemble experiments. This number is similar to that from Broquet et al. (2011) where they considered a longer time period for the inversions but far smaller observation networks and a smaller inversion domain, which reduces the dimension of the minimization problem. However, here, 12 iterations were still found to be sufficient for converging toward the theoretical minimum of the cost function, i.e., the number of assimilated data divided by 2 (Weaver et al., 2003), with less than 10 % relative difference to this theoretical minimum except for a few cases (for these cases, 18 iterations were used to reach a relative difference to the theoretical minimum that is smaller than 10 %).

Similarly to Broquet et al. (2011), 60 members are used in each Monte Carlo ensemble experiment. This is also the typical number of members that Bousserez et al. (2015) used for their Monte Carlo simulations. Broquet et al. (2011) found a satisfactory convergence of the estimate of the uncertainties in Europe and 1-month-average NEE with an ensemble size of 60, which is confirmed here (the estimates using 50 and more members are within 6 % of the results with 60 members).

Three and five Monte Carlo ensembles of inversions are conducted for December and July, respectively. For each season, three ensembles using the default setup of B and R described above are conducted in order to give results for the three different ICOS network configurations and consequently the sensitivity to the network configuration. In July, two ensembles are also conducted with a change in R in one case and in B in the other case in order to test the sensitivity to these inversion parameters. Such sensitivity tests were conducted in July only, using only one configuration of the ICOS network (ICOS50 and ICOS66 for the test of sensitivity to R and B, respectively); a more exhaustive set of tests of sensitivity for the two seasons and for each ICOS network configuration was not expected to bring new insights, but would raise significant additional computation costs. The setup of the inversion for these two sensitivity tests is now described.

There is a steady increase in the resolution of the atmospheric transport models used for atmospheric inversions, with corresponding improvements of the simulation precision (e.g., Law et al., 2008). In this test we simulate the effect of potential future transport model improvement on the posterior flux uncertainties by reducing the default observation error standard deviations in R by a factor of 2. This factor roughly corresponds to the improvement of the misfits between the model and actual measurement at the site TRN (see Fig. 1 for its location), which was observed when bringing CHIMERE from the current 0.5∘ resolution down to a 2 km resolution using the configuration presented in Bréon et al. (2015). The underlying assumption would be that ∼ 1 km horizontal resolution atmospheric transport models could be used for inversions at the European scale in the near future. Hereafter, we denote by Rref the reference configuration of R and by Rred the one corresponding to reduced standard deviations.

The test of the sensitivity of the inversion system to the prior uncertainty is focused on that of the sensitivity to the spatial correlation length in B (Gerbig et. al. 2006) (which impacts the budget of uncertainty over large regions). The possible use of better prior flux fields based on the merging of both estimates from vegetation models and from large-scale inventories (such as forest and agricultural inventories) can be expected to generate smaller-scale uncertainties than when using vegetation models whereas it is not obvious that local uncertainties would be decreased when adding information from inventories (since inventories only measure long-term integrated NEE). Therefore, we tested the impact of reducing the spatial correlation length for the prior uncertainty in NEE from 250 to 150 km, denoting hereafter the corresponding configurations for the B matrix: B250 and B150, respectively.

Figure 2a and b show the uncertainty reduction for estimates of a 2-week average NEE at 0.5∘ resolution in July and December, respectively. This grid-scale uncertainty reduction reaches 65 % for areas in the vicinity of the ICOS sites and decreases smoothly with distance away from measurement sites. For most of the area around eastern France–western Germany, this grid-scale uncertainty reduction ranges from 35 to 50 % for July and from 20 to 40 % for December. This stems from the combination of the dense observation network over that region, and from the 250 km correlation scale for the prior uncertainties, which spreads the error reduction beyond the immediate vicinity of each station where near-field fluxes have a large influence on the mixing ratio at this station (Bocquet, 2005). For other parts of Europe that are not well sampled by ICOS, significant uncertainty reductions are generally seen around each site but there are large areas where the inversion has no impact at the grid scale: Scandinavian countries, the eastern part of Germany, Poland, the south of the Iberian Peninsula and almost all of eastern Europe.

The spatial structure of the uncertainty reduction and the underlying spatial extrapolation from a site is a complex combination of transport influence and of the structure of the prior uncertainty. Due to varying transport conditions, standard deviation of the prior uncertainty at the grid scale (which is larger in summer; see below the comments on Fig. 3), and observation error (which is larger in winter), the spatial distribution of uncertainty reduction is found to vary from summer to winter. Because the prior uncertainties are larger and the observation errors are smaller in July than in December, there is generally a larger uncertainty reduction in July (especially in western Europe). But variations in meteorology alter (limiting or enhancing) this general behavior. The lower vertical mixing (which strengthens the sensitivity of the near-ground measurements to the local fluxes) partly balances the higher observation error in December and the range of local uncertainty reductions overlaps between July and December. The observations from the Angus tall tower (tta site, Table A1) in Scotland or from Pallas (pal site, Table A1) in Finland contribute differently to the uncertainty reduction during July and December (using meteorological conditions from 2007), showing better performance at the grid scale during summer. This also comes from the different weather regimes, with different dominant wind directions, different average wind speed and different vertical mixing in summer and winter. Regions lacking stations in ICOS23 have an uncertainty reduction that is more sensitive to the atmospheric transport than regions with a dense network. The uncertainty reduction in December is significantly larger in the east and in the southeast part of domain compared to July, due to more occurrences of winds from the east during December than during July.

Complementing the uncertainty reduction, Fig. 3 shows prior and posterior uncertainty standard deviations at the grid scale in order to illustrate the precision of the estimates of NEE that should be achievable with the reference inversion using the ICOS23 network. As already stated, prior uncertainties are up to ∼ 3 g C m-2 day-1 (Fig. 3a) but the winter values are smaller than the summer ones (due to a weaker activity of the ecosystems; Fig. 3b). During both July and December, the uncertainties in a 2-week-mean NEE in the regions that are best covered by observations (most of western Europe) at 0.5∘ resolution are reduced by the inversion down to typical values of ∼ 1.5 g C m-2 day-1 (Fig. 3c, d).

Figure 4a and b show the uncertainty reduction for a 2-week- and country-mean NEE in July and December, respectively. The countries and corresponding estimates of prior and posterior uncertainties are listed in Table A2. The results suggest the ability of the mesoscale inversion framework to derive estimates of the NEE at the national scales with relatively low uncertainties. The uncertainty reduction is particularly large for countries such as Germany, France and the UK, e.g., more than 80 % for France during July. It is larger than 50 % for a large majority of the countries in western Europe and Scandinavia both in July and December.

The smallest uncertainty reduction applies to southeastern European countries where it can be smaller than 10 % (e.g., for Greece in July) indicating that the presence of stations very close to or within a given country is a requisite for bringing significant improvement to the estimates of NEE in this country. In general, the differences of the inversion skill between July and December look consistent with what has been analyzed at the pixel scale. In particular the uncertainty reduction is higher in July for western European countries but higher in December for eastern European countries for the same reasons as that given when analyzing the same behavior at the pixel scale (see Sect. 3.1.1).

Table 1 shows that the uncertainty in a 2-week-mean NEE in July averaged over the full European domain (6.8 ×106 km2 of land surface) is reduced by the inversion by 50 % down to a value of ∼ 43 Tg C month-1 (see Table 1 for details) using the default configuration. The uncertainty reduction for December is 66 %, resulting in a posterior uncertainty of ∼ 26 Tg C month-1. The uncertainty reduction for the whole European domain is thus higher in December than in July. More precisely, while easterly winds in December strongly favor this period in terms of uncertainty reduction in eastern Europe, the uncertainty reduction for NEE averaged over the reduced western European domain defined in Fig. 1c does not vary significantly with the season (66 and 64 % for July and December, respectively). This lack of seasonal variation in the uncertainty reduction at the scale of the western European domain (where most of the ICOS23 stations are located) seems to contrast with the grid-scale and national-scale estimations in this domain, which indicates that the uncertainty reduction is generally significantly higher during summer than during winter. This contrast will be analyzed and interpreted in Sect. 3.1.4.

In order to examine here the dependency of the NEE uncertainty reduction to increasing spatial scales of aggregation for the analyses in July and December, we choose five locations at which we define centered areas with increasing size for which uncertainties in the average NEE are derived. These stations are located using the green circles in Fig. 1c. The five locations correspond to three observing sites of ICOS23: Trainou (TRN), Ochsenkopf (OXK) and Plateau Rosa (PRS); one site of ICOS50: SMEAR II-ICOS Hyytiälä (HYY); and one point in Sweden, which does not correspond to any site of the ICOS networks tested here, called SW1 hereafter (Fig. 1c). We compute the uncertainty reductions of the 2-week-mean NEE for July and December over five squares centered around each site and of increasing size (in square degrees): 1.5∘ × 1.5∘, 2.5∘ × 2.5∘, 3.5∘ × 3.5∘, 4.5∘ × 4.5∘ and 10.5∘ × 10.5∘ (which corresponds to surfaces of different size in terms of km2). Depending on their location and on their size, the corresponding domains expand over areas of Europe that are more or less constrained by the inversion at the pixel scale. But the variations of the uncertainty reduction when increasing the size of these domains are also strongly driven by the spatial correlations in the prior and posterior uncertainty. The results are displayed in Fig. 5.

The five locations used for this analysis are representative of the diversity of the situation regarding the differences between grid scale uncertainty reduction in July and in December. While the uncertainty reduction is slightly larger in July than in December for TRN and much larger in July for PRS and HYY, it is slightly larger in December at OXK and much larger in December at SW1. Furthermore, the values for these grid scale uncertainty reductions range from 15 to 50 % in July and from 7 to 47 % in December at these locations (Fig. 5).

The maximum scores of uncertainty reduction occur for spatial scales of aggregation ranging from 105 to 106 km2 when considering the sites located in western Europe. These scales approximately correspond to the range of the sizes of the European countries and it is larger than the typical area of correlation of the prior uncertainty (as defined by prior correlation lengths of 250 km). Increasing the spatial resolution generally increases the uncertainty reduction since posterior uncertainties have generally smaller correlation lengths than prior uncertainties, due to the spatial attribution error when trying to link the measurement information to local fluxes despite the atmospheric mixing. This explains the increase of uncertainty reduction from the grid scale to the national scales. This also explains why, for a given regional density of the measurement network, larger countries bear larger uncertainty reductions (Fig. 4). However, above such national scales, the corresponding domains include parts of eastern Europe being poorly sampled by the ICOS23 network, which explains the decrease in uncertainty reduction.

The convergence between the results around TRN, PRS and OXK in December and July (which tend to nearly 65 % uncertainty reduction when the averaging area reaches the western European domain), between the results around all sites in December (which tend to 66 % uncertainty reduction when the averaging area reaches the whole of Europe) or between the results around all sites in July (which tend to nearly 53 % uncertainty reduction when the averaging area reaches the whole of Europe), starts between the 105 and 106 km2 (national scale) averaging areas. For smaller areas, the differences between results in July and December or between results for different spatial locations stay similar to what is seen at the 0.5∘ × 0.5∘ scale.

The similarity of the results for the western European domain despite differences at the grid scale in July and December can be explained by differences of correlations between areas at scales similar to or larger than the national scale in the posterior uncertainties (since the correlations of the prior uncertainties aggregated at the national scale or at larger scales are very close for July and December). Figure 6 illustrates the variations of such correlations of the posterior uncertainty at the national scale between July and December using the example of correlations between Germany and other countries. These correlations are usually more negative in December, which indicates a larger difficulty in December than in July to distinguish in the information from the measurement network the separate contributions of the different neighboring countries (or of different areas of larger size). This can be attributed to the stronger winds in December, which increase the extent of the flux footprints of the concentration measurements. Such an increase of the footprints in December limit the ability to solve for the fluxes in the vicinity of the measurement sites but increase the ability to solve for the fluxes at large scales.