Review of

DINCAE 2: multivariate convolutional neural network with error estimates to reconstruct sea surface temperature satellite and altimetry observations

The paper presents an update on the previous version of DINCAE, a convolutional autoencoder method for in-painting of sparse satellite data. DINACE2 presents some improvements over the previous version, most notably in performance (vs DINCAE1), speed (presumably due to being rewritten to Julia from Python) and an option to treat ungridded data like satellite altimetry observations. It also introduces an extra refinement step in the cost function to increase its depth, and an intermediate loss term is included in the total loss to compensate for the vanishing gradients of the deep network. When treating sparse data, the error variance estimation of DINCAE2 is more reliable than that from variational interpolation method DIVAnd. The results are solid, the paper is well written, the figures are clear. I recommend publication after minor revision. I do have some comments which might be worth discussing further.

Specific comments:

page 4, eq4: In the comment to equation 4, the authors state that CAE refinement leads to a deeper network and thus to potential worsening of the vanishing gradient problem. They attempt to mitigate this by including intermediate loss term into the total loss function. They state that by doing so, the vanishing gradient problem is reduced. Can the authors perhaps illustrate more clearly that this is indeed the case? Is there a way to say a bit more about this, so that the reader does not have to take the author’s word for this?

Also, wouldn’t an arbitrary number of refinements further exacerbate the vanishing gradient problem? Which term would be dominant in this case – adding further refinement steps versus including further intermediate losses to the total loss?

Page5: when handling missing data there is an interpretation throughout the text that setting the missing value to zero corresponds to an infinitely large error. This is undoubtedly true for variables which are normalized by their variance, as those in this paper. However there are a number of other scalings where variables are not normalized by their variance. In these cases, it seems to me, the authors interpretation is not the most appropriate. I would propose an independent interpretation that setting the missing values to zero simply numerically means that there will be no back-propagation of error from those missing data – thus the training can continue without any impact from the missing data. This interpretation does not have anything with the specific variable normalization at hand.

A cosmetic remark. The hyperparameters were tuned using Bayesian optimization, which seems adequate. Let’s say hyperparameter optimization gives you an optimal network. Separate instances of training this same optimal network (with the same fixed optimal hyperparameters) would provide separately trained versions of this same network. We can use these set of the same network to create a set of predictions. What is the error variance of this set of predictions? I would expect that this error variance is on the order of a 5%, and hence MUCH larger than the stated precision of the RMS errors (4 decimal places) in Table 2. Long story short, perhaps 4 decimal placed is an overkill of precision.

General comments

The paper by Barth et al. (2021) presents a modification of the DINCAE method for reconstructing missing data from satellite sources. The main advantages of this updated procedure lie on the ability to handle multivariate and non-gridded data, on the superior performance given by an innovative form of the skip connection and the addition of a refinement pass. Moreover, in order to prove the improvements with respect to the previous version, some applications are shown. The main result of the paper concerns the accuracy of the reconstructed error variance used in the refinement of the cost function when dealing with non-gridded datasets. The computational speed improvements are probably due to the change in the programming language.  Overall, the paper shows interesting results concerning the accuracy of reconstructed sparse data but more attention to the scientific rigor and the clarity of concepts and figures is needed. I recommend publication with minor revisions, although I suggest paying careful attention in resolving them.

Specific comments

• I think that you use the words “variance” and “standard deviation” in an equivalent manner. The standard deviation is the square root of the variance that you call both σ throughout the paper (see for example in Section 2.2 or Section 4). Also I think that sometimes the same variable is also mistaken for the error associated with that variance. Please, revise their definition and be consistent with the name of the variables. 
• Pg. 4: In the definition of the loss function  (Eq. 3) a term is missing, which I think is negligible for your purpose but needs to be there (or, at least, you should mention that you are neglecting it). Also it is a good practice to specify what represents each variable before or after every equation.
• Pg. 5: Equation (5) is incorrect. It should be:
  
  g(x,y) = ∫ w(x-x',y-y')f(x',y')dx'dy'.
• In Section 3.2 you explain that in preliminary studies you found that including the data from adjacent seas can help a neural network to better generalize and to prevent overfitting. Can you cite some valid reference or say a little bit more about that? Do you have some results that you can show or are you able to give a reasonable explanation on this topic?
• I strongly suggest to spend more words analyzing the results of Table 2 and all the figures thereafter, including (average, max or min) values of the calculated errors with respect to the original data in the region.
• The last sentences of Sec. 5.2 (lines 6-8 of page 21) seem to state that even if the SD of the DINCAE reconstruction is higher, the RMS is lower. I do not see the contradiction in that, in fact the reconstructed SD from the DINCAE method is on average closer to the observed one than the DIVAnd’s one. The point is not that the SD is higher or lower but simply more or less accurate with respect to the one derived from the DIVAnd method, and it would be nice to show a quantity that describes this improvement. Maybe you could further describe this improvement showing a whole map of the errors over the whole time period.

Technical corrections

• In general the paper presents a few errors (for example, several parentheses in citations are missing, singular/plural agreement in the sentences needs to be corrected and the adverbs position in several sentences is wrong). In particular, the word “data” is plural, please check the corresponding verb throughout the paper.
• Pg. 2: 
• At the beginning of line 2 it should be “which” instead of “who”.
• At line 13, since "deeper" is a comparative, the object of the comparison should be explained (deeper than what?). This is something that can be found several times in the paper.
• Pg. 3: 
• It is not recommended to start a sentence with an equation (line 20-21). Also, it is a good practice to explain each variable when showing an equation, saying what X, f and l stands for in Eq (1). Same for equations (6) and (7).
• At the end of Section 2.1 (line 28) I think “basis” should be “bias”.
• Line 6: The word “error” appears twice.
• Line 10: I think “expect” should be “except”.
• At the end of Section 2.2 (line 21) I think “gage” should be “gauge”.
• Pg. 6 (line 26): The dataset SEALEVEL_EUR_PHY_L3_REP_OBSERVATIONS_008_061 changed its name in SEALEVEL_EUR_PHY_L3_MY_OBSERVATIONS_008_061 after some corrections. Can you please correct the name or declare the date in which you downloaded the data?
• Pg. 10 (line 15): I would rather say “sine and cosine”.
• Pg. 11 (line 16): Either you are missing a word when you say “as explained in the following…” or you should change it to “as described further on” or similar.
• Pg. 17: In the caption of Figure 7 you state that the map for the expected error standard deviation by DIVAnd is without adjustment while the title of the panel suggests “Adjusted error std. dev.”. Could you correct either one of them? Also in both captions of Fig.s 7 and 8 you mention the panels (a), (b), (c) and (d) but those letters do not appear in the images.
• In Figures 9 and 10 there are no units. Also, the title of the left panels suggest “Number of data points” which you never explain nor comment in the text. Moreover, there is no explanation of the colors or the dashed lines in both plots.