_{2}flux mesoscale inversions

_{2}mixing ratios have been used to constrain carbon sources and sinks using inversion methodologies. In this study, we performed sensitivity experiments using observation sites from the Mid Continent Intensive experiment to evaluate the required spatial density and locations of CO

_{2}concentration towers based on flux corrections and error reduction analysis. In addition, we investigated the impact of prior flux error structures with different correlation lengths and biome information. We show here that, while the regional carbon balance converged to similar annual estimates using only two concentration towers over the region, additional sites were necessary to retrieve the spatial flux distribution of our reference case (using the entire network of eight towers). Local flux corrections required the presence of observation sites in their vicinity, suggesting that each tower was only able to retrieve major corrections within a hundred of kilometres around, despite the introduction of spatial correlation lengths (~100 to 300 km) in the prior flux errors. We then quantified and evaluated the impact of the spatial correlations in the prior flux errors by estimating the improvement in the CO

_{2}model-data mismatch of the towers not included in the inversion. The overall gain across the domain increased with the correlation length, up to 300 km, including both biome-related and non-biome-related structures. However, the spatial variability at smaller scales was not improved. We conclude that the placement of observation towers around major sources and sinks is critical for regional-scale inversions in order to obtain reliable flux distributions in space. Sparser networks seem sufficient to assess the overall regional carbon budget with the support of flux error correlations, indicating that regional signals can be recovered using hourly mixing ratios. However, the smaller spatial structures in the posterior fluxes are highly constrained by assumed prior flux error correlation lengths, with no significant improvement at only a few hundreds of kilometres away from the observation sites.

_{2}mixing ratios and avoiding persistent errors from the atmospheric transport models (e.g. Gerbig et al.

_{2}fluxes show large diurnal patterns varying from negative values during the day to positive during the night (photosynthesis and respiration), resulting in a substantial loss of information at the daily time scale (Gerbig et al.,

_{2}trace gases,

*i.e.*those not affected by diurnal cycles, and limited by the resolution to extract the high time frequency atmospheric information from the observations (Gloor et al.,

_{2}mixing ratio tower sites that were deployed for the MCI experiment (Miles et al.,

_{2}fluxes are at 20 km resolution over the domain at a weekly time step. We solve for two flux components (one for daytime and one for nighttime). We also solve for boundary condition concentrations from the CarbonTracker system corrected by aircraft data (Lauvaux et al.,

**x**) that includes the three components described above (daytime fluxes, nighttime fluxes and boundary inflow) is obtained by the following equation:

**x**are the unknown fluxes and the boundary conditions we invert for,

**x**

_{0}the a priori flux and boundary estimates,

**y**the observations,

**H**the linearised transport matrix and

**R**and

**B**the error covariance matrices of the observations and the a priori fluxes, respectively.

*A*for fluxes given by the following expression:

*&sgr;*

_{ A }the posterior flux root mean square error (RMSE) and

*&sgr;*

_{ B }the prior flux RMSE. A value of 0 indicates no improvement of the initial prior errors. Between 0 and 1, the value is interpreted as a ratio of error reduction, referred in percentage in this study.

*L*, and second only by the correlation length

*L*(Lauvaux et al.,

*L*remains difficult to rigorously estimate but its impact on the retrieved fluxes can be large (Wu et al.,

_{2}concentration mismatch of the observation sites not used in the inversion.

*&lgr;*of the

*&khgr;*

^{2}test as follows:

*n*the degree of freedom of the state vector. A value close to one indicates reasonable estimates of prior errors in the inverse system, balancing the weight of the atmospheric observations and their related errors (

**y**and

**R**) compared with the initial uncertainties in the fluxes (

**x**

_{0}and

**B**) and the number of independent elements in the state vector. The values of

*&lgr;*range between 0.75 and 1.25 for all our tests, and the corresponding correlation lengths from 50 to 300 km, including both biome-dependent and non-biome-dependent structures. We increase (or decrease) the RMS (diagonal elements of

**B**) to compensate for changes in the correlation length based on the values of lambda for each case.

*i*(&Dgr;

_{ i }=

**y**−

**Hx**

_{ j }) is computed before (

**x**

_{ j }=

**x**

_{ 0 }) and after inversion (

**x**

_{ j }=

**x**). The mean of the mismatch represents the impact of the correction of weekly biases in the observation space (mixing ratios). The RMSE of hourly mismatches represents smaller-scale corrections (from hourly mixing ratios) produced by changing wind conditions at each site. These tests provide an assessment of the overall gain after inversion, gain from corrections on the weekly fluxes and in space around the validation site. Considering that most tower mixing ratio footprints do not overlap between sites, the LOOCV evaluates primarily the veracity of the spatial correlation in the prior flux errors.

_{2}fluxes over corn-dominated areas and non-corn-dominated areas to highlight the attribution of flux corrections over the domain in the two most distinct vegetated areas. Considering the MIN case, the density of the network is apparently not the main leverage to constrain the regional balance. Only two towers are used in this case, and the final balance and area averaged fluxes are close to the initial full network inversion result (about 30 TgC difference or less than 1-sigma from the posterior uncertainties). In the CORN case using three sites in the corn belt area, we observe that the correction is weaker (only −49 TgC instead of −84 TgC). The locations of the towers seem more important than the absolute number of sites. Considering the averaged fluxes over corn-dominated areas and grass-dominated areas, the complete network case (referred here as posterior) indicates a slight increase of the uptake in corn-dominated areas and an important increase elsewhere (cf.

^{−2})

^{−2})

*i.e.*extending the sink area to the North East. The most variable and important change compared to the initial setup occurs in northern Illinois where there is the largest uptake in the posterior fluxes [

*e.g.*the negative correction around Centerville in NON-CORN and MIN) or when two towers surround the area (Ozarks and WBI also decrease the Centerville area in SPARSE). The positive correction around Round Lake is produced in all cases. Otherwise, the corrections disappear if the closest tower is missing. As an illustration of the prior error correlation impact on the retrieved fluxes, the

*L*=300 km). The error covariances are based on model-data mismatch and correlation analysis using several eddy-flux sites over the domain (Lauvaux et al.,

*L*=100 km. The biome dependence, as defined here, reduces the initial correlation length (cf.

*L*=300 km). The simpler structure of the prior flux errors here induces the propagation of corrections in space from grass to corn dominated pixels for example. This assumption seems somewhat unrealistic as Net Ecosystem Exchange (NEE) for corn is driven by a different phenology and several human-driven processes such as irrigation or fertilisation. Corrections applied to corn-dominated pixels might not be applicable to grassland areas as vegetation responses and error sources might be highly variable across these ecosystems. The spatial distribution of the error reduction [

_{50}). We estimated the gain in terms of the final CO

_{2}concentration mismatch compared to the initial (a priori) model-data mismatch at the five remaining towers, in ppm. Over the 28 periods of inversions from June to December, the gain for the cases TR0 or TRD improves the initial mismatch by 0.823 and 0.861 ppm, respectively, compared to the case L

_{50}with only 0.561 ppm. On average, the simpler exponentially decaying model (=TRD) shows a larger gain compared to the more complex vegetation-based description TR0, but 4 of the 28 periods show small net degradations of the initial mismatch, against two for the TR0 case. Similarly, the DFS drops from 284 for the case L

_{50}, and 281 for the TR0 case, down to 59 for the case TRD, indicating an important increase of the apparent observational constraint due to the correlation length in the flux errors. This first analysis shows that the larger flux correlations of 300 km seems the most profitable assumption in terms of gain. But the presence of degradation of several periods (4 out of 28) indicates that more refinement is required, including temporal variability for example. The gain increasing with the correlation length might also correspond to the overall decrease of the regional flux bias. This overall gain remains valid at the regional scale, but the inherited structures in space in the posterior fluxes might be artificial, constrained by the assumed correlation length more than the data and their adjoint transport.

_{2}concentration mismatch. The principle of cross-validation relies on eight inversions using seven towers only out of the eight available concentration sites. The retrieved fluxes are then propagated through the influence function of the validation tower. The improvement in the concentration mismatch at the eliminated tower is a direct evaluation of the posterior fluxes. We computed both RMSEs and means for each of the inversions with a different validation tower. This analysis evaluates the assumptions made in the prior flux errors (spatial correlation) in terms of systematic error corrections and subweekly corrections (RMSE). The results are presented in

*L*~300 km) showed an improvement compared to the initial CO

_{2}concentration mismatch, and better results than smaller correlation lengths (

*L*~50 km). For the first term, the simulated atmospheric mixing drives primarily the size of the main area of influence on the concentrations. The model resolution might affect the dimensions of the concentration footprints noting that horizontal diffusion is related to model parameterisation optimised at given resolutions. Comparisons are needed to explore the sensitivity of the footprint size to the model configuration. Although the two terms might seem contradictory, they reflect two different facts. The first term represents directly observed flux signals in the atmospheric concentrations. The second term represents the common sources of errors in the fluxes. This term is problematic in the sense that corrections are distributed spatially, even though the observations alone were not able to constrain these areas initially. Chevallier et al. (

_{2}posterior fluxes over the corn belt of the US Midwest by subsampling the MCI tower network. Atmospheric inversions at 20-km resolution were performed for a 7-month period, with similar assumptions but variable observational constraints. These sensitivity tests correspond to different network configuration, including a sparser network of observations or ecosystem-specific networks. The four different subsampled networks showed consistent regional carbon balances despite tower removals (–178 TgC ± 13). The DFS showed that the posterior fluxes are constrained mainly by flux error correlation when the correlation length is larger than 150 km. The gain in the final concentration mismatch indicates an improvement of the overall regional fluxes with large correlation length (300 km or more) but might correspond to artificial extension of the regional bias correction rather than realistic spatial structures in the posterior fluxes. This preliminary study shows that the MCI campaign provides a sufficient number of observations to constrain the Corn Belt carbon balance over the 7-month period, but the spatial distribution of the inverse fluxes is still under-constrained with too little observational constraint compared to the assumed flux error structures.

_{2}growth from economic activity, carbon intensity, and efficiency of natural sinks

_{2}surface fluxes

_{2}fluxes: methods and perspectives

_{2}budgets at high spatial and temporal resolution

_{2}modeling framework to boundary conditions

_{2}emission fluxes for the United States

_{2}exchange

_{2}budget balance of the corn belt: exploring uncertainties from the assumptions in a mesoscale inverse system.

_{2}mole fractions detected with a tower-based network in the U.S. upper midwest

_{2}growth rate and airborne fraction

_{2}inversions based on interannually varying tracer transport

_{2}exchange across North America: results from the North American carbon program site synthesis