The authors present a new framework for performing atmospheric flux invesions. This framework is based on a four-dimensional local ensemble transform Kalman filter (4D-LETKF), with GEOS-Chem used as the transport model. The method is applied to version 10r column-averaged CO₂ retrievals from the OCO-2 satellite over the five year period from 2015 to 2020. The estimated fluxes were found to be broadly consistent with those from other flux inversion systems. This represents the first application of a 4D-LETKF algorithm to perform an atmospheric inversion using OCO-2 data.

I find the paper to be well written and clear in general. The method is novel because it combines a high-dimensional grid-based parameterisation with an ensemble Kalman filtering approach. I appreciate that the authors performed sensitivity experiments to better understand what is driving the results. My main critique is that I find the mathematical description of the method lacking. My secondary critique relates to the lack of a discussion of the advantages or disadvantages of the method in comparison to other inversion systems.

Major comments

• I found it difficult to understand the method based on Section 2.1. Here are my specific questions:
  • How are the ensemble members initialised?
  • The matrix B appears in equation (1) but not in equations (2)-(5). Is B the error structure for x^b that is mentioned in line 115? How does B affect the posterior state if it does not appear in the calculations?
  • What is contained within the vector x^b (and x^a, and so on)? Is it the control variables (scaling factors) for the whole assimilation period (7 days), or is it just for the first day? If it's just the first day, what are the implied values for the next 6 days, which will affect the modelled concentrations y^{b(i)}? Are these assumed to be equal to the prior mean? I looked at Figure~1 but I still could not understand what was happening.
  • Related to the last point, the calculation of \bar{x}^b is described as "the average optimized result from the two previous time steps and a fixed value of one". Does this calculation apply to the new day entering the assimilation period, or to all the days?
  • The modelled concentrations y^{b(i)} must also depend on state values from before the assimilation period. Are these set to the posterior mean, or are they different from each ensemble member? What is assumed exactly?
  • I find the notation regarding \bar{x} a little confusing. Is this the unweighted average of the ensemble members? I ask because \bar{x}^a and \bar{x}^b are not unweighted averages, so the notation is a little bit inconsistent.
  • How does the localisation length work? It is stated that "y^o contains the assimilated OCO-2 XCO₂ within the assimilation window and localization length". Since the state vector contains every grid cell for a day, how can any observations be excluded by the localization length?
• I think it would help for the authors to discuss how their method compares to other methods. For example, a convention al 4D-Var system has a similar state space and a similar cost function. What, in the authors view, are the advantages of their method? I think just a short discussion of the most common methods and how they compare qualitatively to the authors method would be enough.

Minor comments

Line 151, what does the word ``integrated'' mean here? Does it mean that the flux field was shifted to have annual mean zero? How was this done?

General comments.

The authors applied a flux inversion system based on LETKF algorithm to estimating the global and regional CO₂ fluxes based on OCO-2 satellite observations. They obtained useful results indicating ability to reconstruct the regional surface fluxes fitting within a spread of the recent global inverse modelling results by other modelling systems. On the other hand, there are deficiencies of the method that authors could not overcome and hope to improve in further developments. Paper is well written and illustrated and can be accepted after minor revisions.

Detailed comments.

1. The length of optimization window of 1 week limits the power of the remote observations to constrain the fluxes. One can see a difference between fluxes retrieved with Kalman smoother when applying 1-month and 3-month assimilation window (Bruhwiler et al, 2005). The deficiency has been noted in the abstract as ‘Four sensitivity experiments are performed herein to vary the prior fluxes and uncertainties in our inversion system, suggesting that regions that lack OCO-2 coverage are sensitive to the priors, especially over the tropics and high latitudes’, which authors hope to address in future.
2. There are visible problems with posterior fluxes, as shown on Fig. 4, the range of annual mean grid fluxes occasionally goes out of reasonable range, exceeding 100 gC/m2/year, pointing to a poor balance between large scale and grid scale uncertainties, lack of a spatial correlation constraint on flux correction gradients. The results with the presumably similar algorithm in Liu et al (2019) do not show such noise, which pint to having some important differences that must be documented. Similar flux noise problem was encountered by Miyazaki et al, (2011) and later studies. Can authors isolate the cause of the problem? Could it be a result of using random grid fields as ensemble flux perturbations, while there is an alternative of using smoother random fields?
3. Another issue related to the grid flux noise is the ensemble size. As shown by Chatterjee et al (2012), Chatterjee and Michalak, (2013) the inversion results are sensitive to ensemble size, and useful improvement are archived by increasing the ensemble size beyond 100. Miyazaki et al (2011) also obtained visible improvement of flux constraint by increasing the ensemble size from 48 to 96. Compared to those designs a system presented in this study relies on rather small ensemble size.

Despite of the visible success in weather forecast applications, LETKF use in carbon flux inversion has been tried in several studies but did not become widely used due to limitations, presumably not providing a better computational efficiency over adjoint-based variational or low rank inversion algorithms. In a revised manuscript it is advisable to mention the deficiencies of the LETKF system: limitations of small ensemble size and short window length (which may be reasonable for coupled weather-carbon cycle assimilation) and provide better arguments in support of this direction in comparison to other settings, for example Kalman filter approaches formulated by Feng et al, (2009).

Detailed comments

Line 72 In addition to Liu et al 2019, it is useful to mention the results by Miyazaki et al 2011 who also studied adding GOSAT satellite observations.

References

Bruhwiler, L. M. P., Michalak, A. M., Peters, W., Baker, D. F., and Tans, P.: An improved Kalman Smoother for atmospheric inversions, Atmos. Chem. Phys., 5, 2691–2702, https://doi.org/10.5194/acp-5-2691-2005, 2005

Chatterjee, A., A. M. Michalak, J. L. Anderson, K. L. Mueller, and V. Yadav (2012), Toward reliable ensemble Kalman filter estimates of CO2 fluxes, J. Geophys. Res., 117, D22306, doi:10.1029/2012JD018176

Chatterjee A., A. M. Michalak, Technical Note: Comparison of ensemble Kalman filter and variational approaches for CO2 data assimilation, Atmospheric Chemistry and Physics, 10.5194/acp-13-11643-2013, 13, 23, 11643-11660, 2013.

Miyazaki, K., Maki, T., Patra, P., and Nakazawa, T. (2011), Assessing the impact of satellite, aircraft, and surface observations on CO2 flux estimation using an ensemble-based 4-D data assimilation system, J. Geophys. Res., 116, D16306, doi:10.1029/2010JD015366.