This manuscript describes a new approach to estimating the mesoscale and coherent baroclinic tidal

sea level anomaly from satellite altimetry. It uses a type of Gauss-Markov filtering

on a reduced basis of space-time functions which have been cleverly chosen to approximately diagonalize

the covariance functions. The approach is applied locally to a partitioning of the global oceans, and the

results are stitched together via linear interpolation. Although, the results include estimates of both

the time-dependent mesoscale sea level anomaly and the baroclinic tides, only the tides are examined

in detail.

This is an original approach and the paper does a good job of describing the context, methodology,

and results of the study. I like the paper a lot and I believe it will be of interest to many readers

of Ocean Science Discussions. I recommend publication after minor revisions to address the points below.

Larger comments:

No details seem to be provided about the Q matrices for both the mesoscale and tidal signals. Can you provide,

   say, a pseudo-spectrum of variance that shows how Q varies as a function of wavenumber or horizontal mode

   number; or do you feel that this is represented equivalently with the physical-space representers

   shown in Fig 5? Can you provide maps of the mesoscale and tidal variance (i.e., the diagonal elements of the

   Q matrix)? Did you perform any tuning to adjust the ratio of Q and R, or the ratio of the tidal Q and the

   mesoscale Q?

If I understand correctly, the estimate you compute using eqn (8) or (13) is biased towards zero. A symptom

   of this bias is the observation that the explained variance is larger than the signal variance (as

   you noted with regard to the bottom panel of Fig 8). Can you plot a map of this variance ratio and interpret it

   with regard to either the bias or the tuning of Q? Would you consider using a non-zero estimate of the

   tide or the mesoscale in order to reduce this bias?

No details are provided with regard to the time-dependence of the tides, except for equation (4). It appears

   that the nodal modulations have been omitted, but this is a substantial effect over a 25 year record.

   Properly accounting for this would probably further increase the explained variance of the tidal estimates

   with respect to both the validation data and the assimilated data; and it should furthermore reduce the

   (low) bias of the tidal amplitudes.

The English language usage is sometimes awkward or non-standard, especially with regard to capitalization.

   I am not evaluating it or going to list all the potential edits during this reading.

Smaller comments:

l14-15: Not sure where they get the 70% phase-modulated.

l35: "covariances" --> "spatial covariances"?

l49: specify, "x_i and x_j are uncorrelated for i \ne j"

l62: Here it is specified that x_1 refers to the time-series of a scalar.

     Aha. But after line 70, it is clear that x_2 is a two-component vector containing the

     harmonic constants of the high-frequency component.

l75: The reference to localization should either be dropped or explained precisely what

     is meant.

p4, last line: How does it differ from harmonic analysis? If it is identical, then say so.

l80: By "sequential estimation" do you mean that the low-frequency component is estimated by itself

     from the entire time series, and then this estimate is subtracted before estimating the high-

     frequency component? This usage of "sequential" is confusing since the term might also refer

     to sequential estimation (i.e., a Kalman filter) which sequentially processes the observations

     in time.

l95-l100: This is very good discussion of bias in this context.

l125: Can you support your assumption that no correlation exists between the components (l116) by saying

      that the \Gamma_k are chosen to approximately diagonalize the state covariance? If you could

      provide some observational data to support the choice of \Gamma_k, that would be even better!

      Aha: now I see the mention of this later, around l130.

l161: I believe the reference should be to Fig 4, not Fig 3? You will probably need a reference or

      short discussion to explain what is a "representer".

Fig 4: Was the representer shown in the right panels constructed from the 12-equiangular basis elements?

       I am surprised that it is as radially-symmetric as shown.

l173: This is a good compromise between domain size and degrees of freedom.

Fig 5: Did you subsample or average the observations in the along-track direction? Or did you use

       1 Hz data? Why not show the same lat/lon window in each panel?

l231: How were the diagnonals of the Q matrices chosen initially? What information was used to estimate

      the variances of the signals?

       

Fig 8 and discussion: Usually "signal variance" refers to the data, but I believe you are using it to

      refer to the variance of the estimated signal. Perhaps this could be clarified. My interpretation

      is based on the fact that you note the explained variance is larger than the "signal variance" in

      cases 2 and 3.

Maybe I missed it, but no where do I see discussion of what altimeter missions were used. It looks like CryoSat-2

      is in the post-2017 validation dataset, but is this the only mission used?

l235: Why are you using only a year for the validation period?

Fig 10: Please state the units of the comparisons (cm^2, I think?).

    Why does the bias problem not seem to be as large as suggested by Fig 5? Perhaps I do not

    understand your sequential estimates, and they differ more significantly from the approach

    used in HRET. Or, maybe your low-frequency solution obtained here is quite different from

    the Duacs/Ssalto-based mesoscale correction used in HRET. Based on your Fig 5, I would have expected your estimate to explain a lot more variance than HRET.

Table 1: What do the percentages refer to (is the decimal point placed correctly?)? Please label the

      sub-tables with M2 and K1.

This manuscript tackles an important question which is how to separate well the mesoscale variability from the internal tides in the altimetric record, with a focus on the internal tide component which is coherent. The approach is innovative, and the results are tested first on artificial fields, and then on the real altimetric sea level record, with a validation and estimation of the skill with recent data not used in estimating the solution (for the internal tides).

The pproach relies on a set of assumptions on the respective spectral characteristics of the meso-scale variability and the tidal characteristics. The tests are done assuming a certain spectral shape of the meso-scales and tides which follow the classical (linear) dispersion characteristics and are low order (1 or 2, depending on the tidal mode). The tests indicate that with these assumptions, the joint inversion approach (which is numerically rather heavy) performs better than separate approachs.

I wonder whether the authors could go further and estimate how much the gain depends on the spectral shape. After all, it originates from the overlay of the time-space spectra of the meso-scales and of the tides. One can also wonder how sensitive is it to the exact shape of spectrum. It could be interesting to test different shapes overlapping more or less.

What is the impact of the assumptions on spectral characteristics for the mesoscales, as well as for using a specified dispersion relationship with modeal decomposition, extending only to order 2 or 1 depending on the tidal component, of course compounded by the use of a (spatial) Hamming window. The width of this wndow has to have an impact. What fully motivates the choice?

Section 3.1.1 summarizes the choices made in Ubelmann et al (2021) in a few sentences. This is fine not to present in details what is in this paper, but one is left a little bit wondering about what is been done. I was in particular wondering whether the choice to fit the covariance on the altimetry mapping covariance, which filters out some of the smaller oceanic spatial scales has an impact on the internal tide solution. Also, when mentioning the full altimetric record, it should be indicated what is the data set. I assume that the adjustments between the different altimetric missions (and other corrections and filtering of the data along track, but that I am less sure) are performed before hand. Have these steps (if done) some implication on the internal tide characteristics that will be afterwards retrieved.

My other comments are minor and could easily be fixed:

1. 107: For each component k…
2. 115: index p should be explained.
3. 142 ‘only mode-1 is considered…’
4. 155: ‘(sources and sinks)’
5. 157: ‘not too large’
6. 162: I assume ‘(upper panel) …’
7. 182, I don’t understand the end of the sentence?
8. 203: ‘for each component’
9. 207: I am not sure I got the end of the sentence: why ‘supposedly’?
10. 210: ‘the stationary persists’ (word missing?)
11. 249: why is the Cryosat-2 mission specifically mentioned at this point (and not earlier)
12. 257: ‘could be an interesting next step…’