Atmospheric inversions inform us about the magnitude and variations of greenhouse gas (GHG) sources and sinks from global to local scales. Deployment of observing systems such as spaceborne sensors and ground-based instruments distributed around the globe has started to offer an unprecedented amount of information to estimate surface exchanges of GHG at finer spatial and temporal scales. However, all inversion methods still rely on imperfect atmospheric transport models whose error structures directly affect the inverse estimates of GHG fluxes. The impact of spatial error structures on the retrieved fluxes increase concurrently with the density of the available measurements. In this study, we diagnose the spatial structures due to transport model errors affecting modeled in situ carbon dioxide (

Atmospheric carbon dioxide (

Atmospheric inversions of greenhouse gases (GHG) are now widely used to infer surface fluxes from natural

The increased density in existing tower networks and the availability of fine-scale satellite retrievals raised concerns about spatial and temporal structures in transport model errors

Ensemble approaches are useful to describe flow-dependent errors

In this study, we apply the filter of variances and the covariance localization developed in

We generate an ensemble using the Weather Research and Forecasting (WRF) model version 3.5.1

The ensemble was calibrated over the Midwest US using the available meteorological observations and the 10 km model simulation as described in

Here, we will compare the different ensembles generated in

Filtering of variances and covariances is made necessary because of the finite size of the sample ensembles, which can generate significant sampling errors. The sampling errors can be filtered out by applying a linear filter to the variances and covariances. The most general linear filter is of the form

The theory proposed in

The first one consists in requiring that the residual sampling error be minimal. Assume that we have an estimator

If

The second ingredient is to exploit the structure relationships that bind the moments of sample estimators of the reference distribution.
For any reference distribution (referred to as the non-Gaussian case in the following), the second-order moments of the sample covariances

In spite of the above key ideas, some local spatial averaging will additionally be needed to obtain robust estimators for the filters and their correlation lengths. Such averaging can be justified by ergodic assumptions regarding the statistics of the errors.

In the following, we make the difference between the cases where the true distribution is assumed to be Gaussian or not, since we saw it has an impact on the structure function such as Eq. (

It turns out that it is more convenient to filter the variance and the correlation independently, in particular using a general linear filter for the variances and a Schur filter for the correlation

We denote

In the non-Gaussian case, the structure relationship incorporates a term that depends on the fourth-order moments

Again, but in the non-Gaussian case, if we regularize the covariances with a Schur filter, i.e.,

For the localization of the covariances and hence the correlations, Eqs. (

We note that all these formulae still depend on some statistical expectation, such as

There is an alternative to using the optimality condition (Eq.

For instance, assuming Schur regularization, we obtain the following Wiener filter:

Both Wiener and Schur filters will be applied to sub-domains defined around instrumented tower locations measuring continuously

We want to explore the relationships between the different variables especially in situ mole fractions of

We computed the sample variances over the domain from the 5-, 8-, 10- and 25-member ensembles as shown in Fig.

Variances of

We show here the values of the length scales in our filter resulting from the optimality criteria, applying both Gaussian (see Eq.

For

Length scale (in km) of the variance filter for in situ

For

Length scale (in km) of the variance filter for

Length scale (in km) of the variance filter for planetary boundary layer heights using Gaussian (solid lines) and non-Gaussian (dash lines) equations from 27 June to 21 July 2008.

The filtered variances shown in Fig.

Monthly averages of the filtered variances of in situ

We show in Fig.

Filtered variances using the calibrated 25-member ensemble of in situ

Error covariances in

Hourly raw

Raw

We show in Fig.

Filtered correlations (using Schur product only) of

Filtered correlations (using Wiener and Schur) of

We discussed in Sect.

We have explored the impact of the calibration process on the error variances and covariances by filtering non-calibrated sub-ensembles of 8 and 10 members (see Fig.

Length scale (in km) of the Gaussian variance filter for 10-member random (in gray) and calibrated (in black) ensembles, and 8-member random (in light blue) and calibrated (royal blue) ensembles, for in situ

In Fig.

Characteristic length scales

This study presents a methodology to filter the noise in error structures from a small-size ensemble. The evaluation of the filtered structures would benefit from dense measurement campaigns sampling spatial structures across large domains, such as the Atmospheric Carbon and Transport (ACT)-America campaigns.

We have diagnosed the error variances and the spatial error structures from our mesoscale transport models at daily and monthly timescales. Applied to both

We have discussed the potential use of meteorological error structures such as the mean horizontal wind to approximate spatial error correlations of in situ

The code is accessible under request by contacting the corresponding author (thomas.lauvaux@lsce.ipsl.fr).

The model simulation outputs are available under request by contacting the corresponding author (thomas.lauvaux@lsce.ipsl.fr).

In this study, we generate a reference ensemble to evaluate the sampling noise in the ensembles of smaller sizes. The original 45-member ensemble, uncalibrated, cannot be used as a reference as it underestimates model errors. Yet, the reference ensemble needs to include enough members to limit the sampling noise. Based on the same original ensemble of 45 members as in

Rank histograms of the 25-member ensemble after calibration for wind speed

The WRF-Chem simulations were performed by LIDI and TL; the filtering technique was coded by MB and TL based on the work of Benjamin Ménétrier; the concept and ideas were designed by MB, TL and NB; the paper was prepared by TL, LIDI, NB and MB.

The authors declare that they have no conflict of interest.

This research was supported by National Aeronautics and Space Administration (NASA) Terrestrial Ecosystem and Carbon Cycle Program (grant no. NNX11AE79G), by the NASA’s Earth Venture Program Atmospheric Carbon and Transport (ACT) – America (grant no. NNX15AG76G), by the Alfred P. Sloan Graduate Fellowship, by the NASA Carbon Monitoring System program (grant no. NNX13AP34G), and by the National Oceanic and Atmospheric Administration (grant no. NA14OAR4310136). CEREA is a member of Institut Pierre-Simon Laplace (IPSL).

This research has been supported by NASA (grant nos. NNX15AG76, NNX13AP34G, NNX15AI42G and NNX11AE79G) and NOAA (grant no. NA14OAR4310136).

This paper was edited by Christoph Gerbig and reviewed by Benjamin Ménétrier and one anonymous referee.