This paper introduce what is referred to a DA in parametric space for a simple 1-D eddy model.  The paper is not well constructed with the use of informal language throughout as well as sweeping statements that are not justified by the results.  I believe that the manuscript has potential but needs tidying up as well as better description of the work but also the significance of it.

Major Comments:

1) You refer to non-Gaussian methods but fail to mention the work at U. Reading and Colorado State University on this topic.

2) Again referring to non-Gaussian DA but not actually performing any of it needs to be addressed.

3) Present the cost function for the two different approaches so that we can relate how the parameters are solved for.

4) It is very confusing how you are calculating the observations for both set of experiments.

5) Figure 1 make no sense to me I cannot understand how each component are related to each other 

6) The remaining figures also do not make sense.  The first thing is that the true solution should be plotted or the true errors.  These plots do not tell us anything if we do not know that they are supposed to be approximating.  Finally I do not understand why you have the standard deviations  and what are they supposed to be telling us?

Minor comments:

1) Line 7: There should be a however at the start of the sentence.

2) Line 8: remove the

3) Line 14: Do not use the phrase we are not interested! It should be something like in  this study we shall not be considering the full state but ...

4) Line 33: expectation and covariance with respect to what random variable and which PDF?

5) Paragraph at line 34: Rephrase.

6) Line 40: define what you mean by standard detection algorithms

7) Line 45: This equation is confusing for a couple of different reasons:1) what is lon? Is it a vector of the values of longitude? 2) Is this used as an observation operator or is it the numerical model? lower case h is usually reserved for the nonlinear observation operator.

8) Lines 52 and 54: Define the error covariance matrices B and R and use their correct names, also state which PDF type they are associated with.

9) Line 57: Again what is meant by detection algorithm?

10) Line 68 Define what Pa is.

11) Line 72: This paragraph is not very well written.  We do not use the phrase perfectly known.  I believe that you are referring to the true values which is the correct way to describe what is happening here.

12) Line 87: Should be Figs. are you are referring to more that one figure.

13) Line 108: geostrophic current field is not defined.

14) Line 109: Eddy Kinetic Energy is not defined mathematically of even explained why it is important or how it is calculated.

15) Line 120: This phrasing is incorrect and has misleading statements, almost to the point where I would recommend rejection.

16) Line 125: This statement is also incorrect and is too sweeping.  You have only shown this for the very simple test case shown.

17) Line 127: Why is a machine learning algorithm required? Justify this!

This manuscript presents a clear and pedagogically useful illustration of an  issue in geophysical DA: when uncertainty is dominated by displacement/shape of coherent structures (e.g., eddies), Gaussian/linear updates in gridded space can distort features and derived dynamics. The proposed reduced parametric representation for a 1D Gaussian eddy is intuitive and the figures effectively communicate the mechanism.



However, as written, the current comparison between “grid-space DA” and “parametric-space DA” appears asymmetric. Parameter-space DA is given direct access to observations and predictions in parameter space with simple Gaussian errors, leading to a near-textbook low-dimensional Gaussian update. On the other hand, grid-space DA applies a mean/covariance (Kalman/EnOI-style) update to a distribution over gridded states that becomes inherently non-Gaussian, and then parameters are recovered from the analyzed grid field via a nonlinear profile fit.

It seems to me therfore that the experiments primarily demonstrate a representation/coordinate effect, (namely, Gaussian updates behave better in coordinates where the state distritbuions are actually Gaussian), but do not yet support broader claims. 

To support the stronger claims, I recommend major revisions that include at least one “symmetric observation” experiment: start from the same noisy gridded observations, explicitly estimate parameters from these observations (including a realistic, likely correlated error model for the fitted parameters), and then perform parameter-space DA. Alternatively, the manuscript should be reframed explicitly as a conceptual/didactic demonstration and claims about generality and computational savings should be softened and supported quantitatively. 



A more detailed review is attached.