30 Aug 2021
30 Aug 2021
Simultaneous estimation of Ocean mesoscale and coherent internal tide Sea Surface Height signatures from the global Altimetry record
 ^{1}OceanNext, 90 chemin du Moulin, 38660 La Terrasse, France
 ^{2}Collecte Localisation Satellite, 11 rue Hermès, Parc Technologique du Canal 31520 Ramonville SaintAgne France
 ^{3}Centre National d’Etudes Spatiales, 18 avenue Edouard Belin, 31401 Toulouse, France
 ^{4}LEGOS, 14 avenue Edouard Belin, 31400 Toulouse, France
 ^{1}OceanNext, 90 chemin du Moulin, 38660 La Terrasse, France
 ^{2}Collecte Localisation Satellite, 11 rue Hermès, Parc Technologique du Canal 31520 Ramonville SaintAgne France
 ^{3}Centre National d’Etudes Spatiales, 18 avenue Edouard Belin, 31401 Toulouse, France
 ^{4}LEGOS, 14 avenue Edouard Belin, 31400 Toulouse, France
Abstract. This study proposes an approach to estimate the Ocean Sea Surface Height signature of coherent internal tidesfrom 25 years of alongtrack altimetry record, with a single inversion over time, resolving both internal tide contributions andmesoscale eddy variability. The inversion is performed through reducedorder basis with conjugate gradient resolution. Theparticularity of this approach is to mitigate the potential aliasing effects between mesoscales and internal tide estimation fromthe uneven altimetry sampling (observing the sum of these components) by accounting of their statistics simultaneously, while other methods generally use a prior for mesoscales. The four major tidal components are considered (M2,K1,S2,O1) over theperiod 1992–2017 on a global configuration. From the solution, we use altimetry data after 2017 for an independent validation,to evaluate the benefits of the simultaneous inversion, and also to compare the skills with an existing model.
Clément Ubelmann et al.
Status: final response (author comments only)

RC1: 'Comment on os202180', Edward Zaron, 22 Sep 2021
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 GaussMarkov filtering
on a reduced basis of spacetime 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 timedependent 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 pseudospectrum 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 physicalspace 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 nonzero estimate of the
tide or the mesoscale in order to reduce this bias?No details are provided with regard to the timedependence 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 nonstandard, especially with regard to capitalization.
I am not evaluating it or going to list all the potential edits during this reading.Smaller comments:
l1415: Not sure where they get the 70% phasemodulated.
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 timeseries of a scalar.
Aha. But after line 70, it is clear that x_2 is a twocomponent vector containing the
harmonic constants of the highfrequency 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 lowfrequency 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.l95l100: 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 12equiangular basis elements?
I am surprised that it is as radiallysymmetric 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 alongtrack 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 CryoSat2
is in the post2017 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 lowfrequency solution obtained here is quite different from
the Duacs/Ssaltobased 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
subtables with M2 and K1. 
CC1: 'Comment on os202180', Zhongxiang Zhao, 19 Oct 2021
This is a nice work that will draw interest from the altimetry community. It provides another new approach to map internal tides from satellite altimetry. It is a very cool method. It should be accepted with some minor revisions. In particular, the writing has a large room to improve.
L1, 'Ocean Surface Height', capital letters!?
L3, 'reducedorder basis with conjugate gradient resolution ' is vague in abstract, please specify.
L6, should be Msub2 etc.
L8, 'benefits' better be 'performance'?
L10, 'Internal Tides', capital letters?
L11, '150 km ... and below' is not accurate, better say 'below 200 km '
l12, 'at first order'? Please explain or just cross out.
L13, 'seasonal' a convincing paper is Zhao JPO (2021)
L1718, 'O...G...C...M' no need in uppercase
L18, 'accurately' is not accurate
L21, ‘later’ or ‘latter’?
L47, EQ2, move R definition here from a later place
L90, here and later, why do you repeat it ‘100’ times? Any reason or criterion? Please give more details
L97, ‘precisely’?
L143, Did you test with mode2 S2 and O1?
L151, ‘3.5 m/s in the tropics’? I think higher speeds are at high latitudes, mainly due to the effect of f in EQ12.
L166, ‘c2=c/2’ is wrong! Please read Figures 3 and 4 in Zhao 2018 (JGR)
Figure 6, please give unit
Figure 8, based on the bottom panel, may we draw that MIOSTMSIT might underestimate internal tides, compared to MISOTIT?
Figure 9, for the bottom statistics, are you using the whole region above?
Table 1 is tedious: most of these numbers are very small. Please consider compress Table 1. These regions are defined in Carrere et al. (2021)? Many values are lower than 0.1%, may we say they are insignificant or within errorbar? ‘Cryosat data’ are from 20172018?
L262, I am not sure that ‘opens the door for solving uncoherent internal tides’. There are TWO existing methods (Zaron; Zhao).

RC2: 'Comment on os202180', Anonymous Referee #2, 20 Oct 2021
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 mesoscale variability and the tidal characteristics. The tests are done assuming a certain spectral shape of the mesoscales 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 timespace spectra of the mesoscales 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:
 107: For each component k…
 115: index p should be explained.
 142 ‘only mode1 is considered…’
 155: ‘(sources and sinks)’
 157: ‘not too large’
 162: I assume ‘(upper panel) …’
 182, I don’t understand the end of the sentence?
 203: ‘for each component’
 207: I am not sure I got the end of the sentence: why ‘supposedly’?
 210: ‘the stationary persists’ (word missing?)
 249: why is the Cryosat2 mission specifically mentioned at this point (and not earlier)
 257: ‘could be an interesting next step…’
Clément Ubelmann et al.
Clément Ubelmann et al.
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