I agree with you: the analysis increment lies indeed in the range of B, so its effective resolution is that of the coarse grid. Theoretically, we could perform the minimization at this coarse resolution, and then interpolate the analysis increment to the resolution of the first guess in order to add it to the first guess and produce the analysis. If the NICAS method is used as a localization operator in an EnVar framework, we could even store the ensemble perturbations on the coarse grid to save more on memory and CPU usage.

In practice, this option is difficult to implement because we must apply the linearized observation operator to the increment. Doing so on the coarse NICAS grid introduces a dependency between the background error covariance B and the linearized observation operator H that we would prefer to avoid. Moreover, the coarse NICAS grid is not column-based, but may be different for each selected vertical level. This means that horizontal interpolation would be necessary anyway to reconstruct vertical columns for vertically integrated observations (e.g. from satellites). It is much easier to keep this interpolation within B (in NICAS) and apply the linearized observation operator on the predefined, column-based grid of the first guess.

I recognize that this is not an entirely satisfactory answer, and I will take a look at the JEDI code to evaluate how much work would be needed to use the NICAS grid in the observation operator.

Thank you for your positive and constructive feedback. As you suggested, I have add in the revised manuscript:

- a first figure in section 2.1 illustrating the full grid and subgrid parameters,

- a second figure in section 3.1 illustrating the successive subsampling in vertical and horizontal directions,

- a paragraph in section 3.5 to clarify the distinction between the setup step and the application step.

Thank you for your positive and constructive feedback.

Regarding the very short section 4, it was a Latex typo: I defined subsection 4.1 as a "section" instead of a "subsection". As a consequence, section 5 should have been section 4.1, section 4.1 should have been 4.2, etc. In the revised manuscript, the whole section 4 is about performances, while section 5 is a discussion about the different similar methods and section 6 contains the conclusions, as explained in the Introduction. I am sorry for this mistake that I should have noticed earlier.

As for the relationships between the variables, this is a very interesting question, but I think it goes beyond the scope of this article. However, it should be addressed more clearly in the text. I have therefore added a paragraph to the introduction. This paragraph includes references to both correlation operators and localization operators.