We generate an ensemble using the Weather Research and Forecasting (WRF) model version 3.5.1 , including the chemistry module modified in this study for CO2 . The ensemble consists of 45 members that were generated by varying the different physics parameterization and meteorological data. The land surface models, surface layers, planetary boundary layer schemes, cumulus schemes, microphysics schemes and meteorological data (i.e., initial and boundary conditions) are alternated in the ensemble . All the simulations use the same radiation schemes, both long- and shortwave. All simulations were run using the one-way nesting method, with two nested domains. The coarse domain uses a horizontal grid spacing of 30 km and covers most of the United States and part of Canada. The inner domain uses a 10 km grid spacing, is centered in Iowa and covers the Midwest region of the United States. The vertical resolution of the model is described with 59 vertical levels, with 40 of them within the first 2 km of the atmosphere. This work focuses on the WRF simulation with the higher resolution; therefore only the 10 km domain will be analyzed. Simulations were performed from 27 June to 21 July 2008, with a 10 d spin-up for initial conditions. The CO2 fluxes for summer 2008 were obtained from NOAA Global Monitoring Division's CarbonTracker version 2009 (CT2009) data assimilation system (; with updates documented at http://carbontracker.noaa.gov, last access: 30 January 2019). The different fluxes that CT2009 propagates into the models are fossil fuel burning, terrestrial biosphere exchange and the exchange with oceans. The CO2 lateral boundary conditions were obtained from CT2009 mole fractions. Only the meteorological transport fields vary between each model configuration or ensemble member.

The ensemble was calibrated over the Midwest US using the available meteorological observations and the 10 km model simulation as described in . The measurements used included balloon soundings collected over the Midwest region (http://weather.uwyo.edu/upperair/sounding.html, last access: 20 April 2017) for 14 rawinsonde stations. The ensemble was calibrated for three different meteorological variables: wind speed, wind direction and planetary boundary layer height (PBLH) in the late afternoon data (i.e., 00:00 UTC) from the different rawinsondes. Daytime data were used to represent well-mixed conditions, at the selected time when CO2 mole fractions are assimilated into atmospheric inversions to avoid stable conditions near the surface. The calibration algorithm is described in , selecting optimal ensembles of different sizes using simulated annealing and genetic algorithm techniques. The metric used in is the flatness of the rank histogram, which is a measure of the ensemble dispersion. By eliminating members with redundant information, smaller ensembles were able to better match the variability in the observations. We refer to for a full description of the calibration process and the final selection of optimal ensembles.

Here, we will compare the different ensembles generated in from 5 to 8 to 10 members. An additional ensemble was created for our study with a larger number of members in order to address the potential lack of representativeness of model errors with small-size ensembles. Therefore, we generated a 25-member ensemble and applied the same calibration process. This ensemble has not been documented in but is described here in Appendix . We compare the results of the filtering for small sizes to the 25-member calibrated ensemble instead of the original 45-member ensemble that was not calibrated.

have proposed a new theory for the optimal filtering of sample variances and covariances. These are defined by the following empirical second-order moment statistics. Assume we have an ensemble of states xk∈Rn for k=1,…,N of mean x‾, from which to infer the statistics. Define the associated anomalies δxk=xk-x‾, also called perturbations. Then, the sample covariance matrix is 1B̃=1N-1∑k=1NδxkδxkT, which is an unbiased estimator of the true covariance matrix B⋆; i.e., EB̃=B⋆ where E is the expectation operator over the reference distribution from which the xk are sampled. In the following, we denote by Bij̃ the entries of this covariance matrix.

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 2Bij^=∑klFijklBkl̃, where B^ is the filtered error covariance matrix. Typical examples are the application of a convolution to the vector of variances to smooth them out or the application of a Schur product with a nondegenerate, short-range correlation matrix to the sample covariance matrix. The linear filter often requires parameters, a correlation length typically, that must be tuned for the filter to be optimal.

The theory proposed in to achieve optimality of the filter is based on three key ingredients.

The first one consists in requiring that the residual sampling error be minimal. Assume that we have an estimator x̃ of some statistics of a reference distribution with true statistics x⋆, obtained from sampling from this distribution. We regularize x̃ with a linear filter F (a matrix here) in order to minimize the sampling error: x^=Fx̃. A typical criterion to minimize is3L(F)=E(x⋆-Fx̃)T(x⋆-Fx̃).The variation of this criterion with respect to a variation δF of F is δL(F)=-2TrδFTE(x⋆-Fx̃)x̃T, which implies, that, at the minimum, we have an optimality condition in the form of an orthogonality of random vectors:4E(x⋆-Fx̃)x̃T=E(x⋆-x^)x̃T=0.This is a linear equation in F, whose solution is5F⋆=Ex̃x̃T-1Ex⋆x̃T.

If F is a Schur filter, i.e., x^=f∘x̃, given by the Schur or Hadamart product (which is a subcase of the above problem) – hence F is now a vector f – then the solution has the form6f⋆=Ex⋆∘x̃Ex̃∘x̃,where the division of vectors is component-wise. Eqs. () and () can be applied to the filtering of B̃, storing the entries Bij in x. Hence, they provide optimality conditions for linear filtering of B̃. They are known in the signal scientific community as Wiener filters. We note that F⋆, or f⋆, still depends on the unknown true statistics x⋆

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. () and could yield distinct optimal filtering results.

We want to explore the relationships between the different variables especially in situ mole fractions of CO2, total-column XCO2 and PBLH. We will compare both the error variances and covariances to identify possible links between error structures in PBLH and CO2/XCO2 mole fractions. We will explore the spatial correlation lengths for CO2 mole fractions, mean horizontal wind (zonal and latitudinal components) and PBLH to quantify and possibly utilize error structures in meteorological fields to generate CO2 and XCO2 error structures. Most ensemble prediction systems (EPSs) provide spatial error correlations for meteorological variables which could be used to construct error covariances for CO2 and XCO2. Error covariances of CO2 mole fractions depend on the CO2 fluxes, but error structures in the atmospheric models should remain independent of the CO2 flux distribution. show that first-order discrepancies in PBLH seem related to large errors in CO2. Here, we investigate further the links between errors across different variables. We present the results in Sect.  for the variances and in Sect.  for the error correlations.

We computed the sample variances over the domain from the 5-, 8-, 10- and 25-member ensembles as shown in Fig. . The increase in variances and the presence of additional fine-scale structures are visible in small-size ensembles (5 to 10 members) compared to the 25-member ensemble. Fine-scale structures reflect the sampling noise in the small-size ensembles, reaching a maximum in the five-member ensembles (see Fig. d). These spurious structures appear with small-size ensembles and are to be filtered later. The range of values for error variances increases for small-size ensembles, independently of the calibration process. In addition, the variances in calibrated ensembles with more members are smaller because the inflation of the variance is a direct consequence of removing members. Hence, the calibration process better inflates the deviation of members from the mean for small ensembles. have shown that the calibration process yields smaller-size ensembles to better represent model errors. Here, the variance from the 25-member ensemble remains harder to inflate by the calibration process and also less affected by sampling noise, which is likely to be less representative of the actual transport model errors.

We show here the values of the length scales in our filter resulting from the optimality criteria, applying both Gaussian (see Eq. ) and non-Gaussian (see Eq. ) filters to the raw variances. We implemented the dichotomy algorithm proposed in to obtain the optimal length scales of the filter, dividing (or multiplying) the length scale by a factor of 2 until convergence. The algorithm solves for the optimal length scale of the filter by scanning the space of solutions iteratively (applying a multiplicative factor at each time step to minimize the cost function). For all our cases, we defined the upper bound of the diagnosed length scale at 750 km, to represent about half the size of our simulation domain (square of 1600 km wide). This large value means that the extent of noisy structures would encompass the entire domain and therefore would not be recoverable. Here, the sampling noise is characterized by length scales of sizes ranging from a few kilometers to several hundred kilometers. In practice, the algorithm always converges but length scale might be larger than the domain size, meaning that all spatial structures in the variances are regarded as noise. This situation happens for two main reasons: spatial structures in the noise are similar to spatial gradients in the true variances, or the noise is larger than the true variances. Hence, the length scale falls beyond the limit of our simulation domain. In the method described by , the ergodic assumption is necessary to diagnose robust estimators of the filter (see Sect. ).

For CO2 mole fractions (see Fig. ), the algorithm for the calibrated 25-member ensemble systematically converges to small length scales (<50 km), indicating that noise structures are very small in our optimal ensemble. When using the Gaussian filter, the algorithm systematically converges below our 750 km threshold for all cases except for 30 % of the days with the smallest ensemble (five members). In the non-Gaussian case, the filter converges to larger length scales beyond our threshold with the 5- and 10-member ensembles for less than 30 % of the days. Typically, length scales beyond our 750 km threshold are temporally coherent over periods of several days suggesting weather-related structures possibly inherited from synoptic-scale systems. These periods might be caused by high sampling noise compared to the true variances or by similar scales in spatial structures for both noise and true variances. Overall, the non-Gaussian filter shows a lower rate of convergence below 750 km compared to the Gaussian filter for CO2 mole fractions.

For XCO2 column mole fractions (see Fig. ), the optimal length scales are larger and the non-Gaussian filter converges beyond our threshold more frequently (about 50 % of the days for eight members or less). Even with the optimal 25-member ensembles, error structures of about 50 to 200 km are filtered out, significantly larger than for the CO2 mole fractions. We discuss in Sect.  the possible physical reasons behind these larger length scales, possibly due to large-scale structures in the free troposphere or to the complexity in noise structures as XCO2 data integrate noises from different altitudes. Figure  shows the results of the PBLH for which both filters converge beyond our threshold for half of the days in the Gaussian case. However, the filter converges more often below our threshold with the non-Gaussian filter applied to 10-member ensembles. Variance noise for PBLH present skewed distributions (not shown here) requiring the use of a non-Gaussian filter. We conclude here that 8- and 10-member ensembles are the minimum sizes with which sub-threshold convergence can be obtained on most days for most variables. Five-member ensembles will still be studied later on for covariances, as the localization of covariances does not depend on the filtered variances, but the low rate of convergence below 750 km might limit the use of the filtered variances.

The filtered variances shown in Fig. , here with the Gaussian filter, for the different ensemble sizes are in better agreement with the variances of the 25-member ensemble both in term of spatial patterns and magnitudes among the different ensembles. The filter successfully removed noisy structures, therefore decreasing the dependence on the number of members used in each case. Despite length scales beyond our threshold for 30 % of the days with the five-member ensemble, filtered variances at the monthly timescale show similar structures to 8- or 10-member ensembles, with nearly all of the noisy structures being removed by the filter. Compared to earlier results, the ensemble size does not seem to fundamentally limit the capacity of the filter to remove the noise, despite days with convergence beyond 750 km. The variance magnitude remains slightly larger for 10 members or less, with a relative overestimation of about 15 %. Our 5-member ensemble provided the best match with only 10 % higher than the 25-member filtered variances. The averaging over a whole month compensates for days with convergence beyond 750 km, producing reasonable estimates of the optimal variance even with five members. This result suggests that climatological error variances from small-size ensembles can be a good first approximation of the true variance when filtered correctly over most days.

We show in Fig.  the spatial distribution of error variances from the 25-member calibrated ensemble for in situ CO2 mole fractions in the planetary boundary layer (PBL; 100 m a.g.l.), in situ CO2 mole fractions in the free troposphere (about 5 km a.g.l.), total column of XCO2 dry air mole fractions and PBLH (in m a.g.l.). The four variables display very distinct spatial patterns. XCO2 variance spatial patterns (see Fig. c) exhibit distinct maximum values located in the eastern and southeastern part of the domain, whereas high CO2 variances are observed in the northeastern part of the domain for free-tropospheric CO2 (see Fig. b) or centrally located for CO2 variances in the PBL (see Fig. a). Finally, PBLH variances (see Fig. d) show no indication of direct relationship between large errors in the western part of the domain and the other three CO2 variables. We conclude here that no direct relationship can be utilized to construct CO2 variances based on PBLH. Similarly, maximum variances among the three CO2 variables are also significantly different in distribution and magnitudes.

Error covariances in CO2 mole fractions scale with the magnitude of the surface CO2 fluxes. They are therefore difficult to interpret. Instead, we present here the error correlations to highlight the spatial structures inherited from the transport models, independent of the magnitude of the underlying CO2 surface fluxes. We show in Fig.  the hourly correlation structures from the full 45-member ensemble (a–c) and our 25-member calibrated ensemble (d–f) at one of the instrumented towers in the US Midwest, i.e., Centerville, Iowa. We applied both Schur (see Fig. b, e) and Wiener (see Fig. c, f) filters to compare the impact of both filters on the raw correlations (see Fig. a, d). The Schur filter has less impact on the correlations compared to the Wiener filter which attenuates significantly the magnitude of the correlations for both ensembles. In Sect. , the local averaging of the optimal length scale assumes that the sampling noise is spatially homogeneous (third ingredient of the methods). This homogeneity assumption is required for the ergodicity assumption to apply, therefore yielding a domain-averaged filtering approach. The sub-domain used for the covariance filtering (here 400 km × 400 km) limits the spatial extent to 200 km around the observation location. The size of the domain was defined primarily for computational efficiency and based on the size of correlation structures, usually of about 100–200 km in length scale. To evaluate this assumption, we compared the size of the sub-domain to filter the covariances with a large area of 900 km × 900 km for the 45-member ensemble to a smaller area of 400 km × 400 km for the 25-member ensemble. Filtered correlations show similar results for Schur and slightly larger values for the smaller sub-domain when applying the Wiener filter. We conclude here that the spatial local averaging has a minor impact on the results and that our 25-member ensemble has similar spatial structures to the original 45-member ensemble, with larger correlations at short distances. We extend this analysis to the monthly timescale by showing monthly averaged error correlations, superimposed from different tower locations on the same map to aggregate the results at multiple locations (see Fig. ). When averaged over longer timescales (see Fig. ), the filtered correlations become isotropic, distributed around each location. The magnitudes remain larger with the Schur filter (see Fig. b) compared to the Wiener filter (see Fig. c) but the differences are noticeably smaller. The unfiltered correlations (see Fig. a) are noticeably larger due to noisy structures. After filtering, the spatial structures are distributed around the observation locations following a pseudo-Gaussian pattern. The magnitude of the error correlations, i.e., the length scale of the errors, is reduced in both cases compared to the raw correlations (see Fig. a). This result confirms that the sub-domain used here (400 km × 400 km) is sufficient to represent the error correlation structures around each measurement location and describes fully the error structures.

We show in Fig.  the results for the different ensemble sizes using the Schur filter. The 10- and 8-member ensembles show similar magnitude and patterns for the different sites, but the correlations are smaller than with the original ensemble. In comparison, the Wiener filter (see Fig. ) generates consistent patterns with 25-, 10- and 8-member ensembles. In both cases, the filters decrease significantly the correlations in the five-member ensemble, revealing the inability of the filter to separate the noise from the actual error correlations. We present the localized correlation length scales for each tower and for each day in Fig. a. For both Center and Mead, length scales are noticeably larger than for the other towers and decrease rapidly until 2 July, before converging back to the same values diagnosed for other measurement sites. The differences across towers suggest local differences in error correlations, even across the same region for a single day (up to 70 km across our sites). These differences correspond to the beginning of summer, when both weather and ecosystem fluxes vary rapidly especially in agricultural areas. We discuss in Sect.  the relationship between meteorological and CO2 variables for both error variances and error correlations.

We discussed in Sect.  the dependence of the success of the variance filtering on the ensemble size. From these results, an ensemble of at least 8 to 10 members seems required to reach convergence below our 750 km convergence and hence filter the error variances. However, the spatial representation of the averaged filtered variances using a calibrated five-member ensemble (see Fig. ) indicates a reasonable recovery of the error variances at the monthly timescale but not at the daily timescale. Theoretically, the minimum number of members for the covariance filtering is four, based on Eq. () with a factor (N-3) in the denominator, which shows that five-member ensembles are close to this limit and are not recommended in a more general context. The application here suggests 5-member ensembles are acceptable, but 8- to 10-member ensembles would be a minimum both in practice and in theory over different seasons and regions. Convergence beyond our threshold over single days has a limited impact on the monthly mean filtered variances. We conclude here that the filter produces satisfactory results to generate first-order estimates of the CO2 mole fraction errors. To achieve a systematic daily convergence below threshold, we recommend a larger number of members in the ensemble. One important point here is the calibration step performed before filtering, which optimizes the information content in each member relative to the other members. Therefore, a randomly generated ensemble may require additional members in order to represent the actual error variances. We tested the filtering technique on a random ensemble (i.e., uncalibrated) and found that beyond-threshold convergence is more frequent with 10 or less members (not shown here).

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. ). These random ensembles have no member in common with their calibrated counterpart and are composed of simulations using various physics configuration randomly selected among the 45 original model configurations. The optimal length scale of the variance filter is systematically lower for non-calibrated ensembles (see Fig. , in gray and light blue) compared to calibrated ensembles (see Fig. , in black and royal blue), suggesting lower levels of noise with larger spatial structures, similar to larger ensemble sizes (25 members or more). Because members of the calibrated ensembles were selected to maximize the information content, calibrated-ensemble members differ more from each other than non-calibrated ensemble members. As shown in , one member with higher or lower PBLH statistics (usually the monthly mean model estimate) is systematically selected in order to generate calibrated ensembles with enough variance, and therefore capture the spatial and temporal variability in observed PBLH's from 14 radiosondes. Calibration might suggest different combinations of model physics for every single time period. In future studies, we recommend a combination of several preselected model physics with added perturbations to produce a sufficient ensemble spread but not perform the calibration for every time period. The calibration procedure might still be applied to existing ensembles but remain insufficient to sample the full spatial and temporal variability as discussed in . Some of these members introduce different spatial structures compared to the original ensemble, increasing the spread significantly. However, the small size of our ensembles with a larger variance may affect the sampling of these structures and hence our ability to differentiate noise from actual error structures. The approach proposed by was initially designed for ensembles with independently distributed members, which is not the case here. Some true correlation structures in the calibrated ensemble may be regarded as noise by the filter. Another approach would generate an ensemble based on the localized ensemble-based error covariance matrix. This approach will be developed further in future studies but is beyond the scope of this paper.

In Fig. , significant differences in filtered error variances were observed between in situ CO2 mole fractions (at different altitudes), XCO2 mole fractions and PBLH. Therefore, we conclude here that transport errors of meteorological variables are not transferable to CO2 and XCO2 error variances. This finding is in agreement with , who found no direct relationship between errors in the meteorology and in situ CO2 mole fractions. However, considering the covariances, the spatial structures in CO2 mole fraction errors inherited from transport model errors exhibit well-defined patterns (e.g., Fig. ). By fitting a simple Gaussian function in the form of e-xL to the filtered covariance fields, we diagnosed the characteristic length scale of the spatial error structures for the different variables, here CO2 100 m-high mole fractions, the mean horizontal zonal wind component and PBLH. Figure  shows the daily length scales at the seven measurement locations over the simulation period (i.e., 27 June to 21 July). The length scales L for the three variables increase rapidly between 27 June and 4 July from less than 100 to 150 km or higher. As there is no long-term spin-up (re-initialization of the perturbations every 5 d), the asymptotic behavior is most likely due to the seasonality of the errors from early to late summer. The seasonal changes in the atmospheric dynamics impact the spatial structures in the errors for the three variables. Across the seven sites, the characteristic length scale of spatial error structures also vary significantly, in particular for PBLH with large differences across sites. Both CO2 mole fractions and the mean zonal wind component reach a maximum value over July: respectively, 140 and 120 km. The comparison of the mean length scales (Fig. d) highlights the differences between the three variables. Both the CO2 100 m high mole fractions and the mean horizontal zonal wind component show similar variations and converge towards the same values but differ significantly from 29 June to 10 July. We conclude here that first-order estimates for CO2 spatial error correlations may be derived from meteorological error structures, in particular from the mean horizontal wind speed. But these approximations may be valid only for specific time periods. As presented in Sect. , error variances in CO2 and XCO2 mole fractions are decoupled from PBLH errors. Here, we suggest that error correlations may be derived from wind errors, but error variances should still be computed independently.

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.

https://www-air.larc.nasa.gov/missions/ACT-America/, last access: 4 July 2018.