Dear authors,

thank you for the revised manuscript and the responses to the reviewers. The manuscript is now acceptable for publication. Please see below the new comments by reviewer #2. The comments include seven suggestions for improving the language of the manuscript. Please implement these improvements when submitting the final version of the manuscript.

New comments by reviewer #2:

The new version of the manuscript is much better. The results of both simulated and real-data retrievals are now much easier to understand, and some descriptions of the key methods have been added. The generation of keograms used for simulated retrievals have been described in detail. Error analysis was also clarified. I very much appreciate these changes. It seems that the authors had given a lot of thought to many of the aspects of this work that I had questioned in my original review, but the original version of the manuscript was not submitted in a very complete and comprehensive state.

Generally, I am satisfied to all the answers to my questions except for one. In the general comment 3 in my original review, I noted that the algorithm proposed by the authors can retrieve zonal and meridional wavelengths of the observed gravity wave from spatially disjoint regions, thus highly increasing the probability that zonal and meridional wavelengths do not characterize the same wave packet. The key argument in the authors response to my comment was the following: "Our central assumption is that the wave has a large scale, allowing it to traverse a significant portion of the images and persist over time. Small-scale waves more frequently pass through only a portion of the image, while it is rarer for medium-scale waves to do so". This is indeed a reasonable assumption, which I would not have questioned, if the internal logic of the algorithm would be fully consistent with it. In my opinion, it is not, and that was the main point of my comment. To determine the slope of of the phase lines, the algorithm selects a small subsection of image row/column, where the phase lines are most linear. In other words, the algorithm looks for small regions of coherent waves, rather than looking for a linear change in phase present in the whole row/column, as the authors assumption would require. For example, Figure 8b of the revised manuscript shows phase lines that have several regions with mostly linear phase change but different slopes. One of these regions was selected to determine the "correct" slope. If the data contains features that prevent the determination of the phase slope across the whole row/column (e.g. the Milky Way, small scale GWs), then one could try to remove these features by ignoring the problematic parts of the image entirely and/or using low pass filters to remove small scale features.

However, since the authors fully acknowledge the possibility that parameters of several different waves may end up being fitted by the algorithm, the paper can be recommended for publication. I have several minor remarks, suggestions and typo corrections for the newly added material:

1) L28: poorly -> poor
2) L29: With less than two samples per period (wavelength), the wave cannot be detected at all (Nyquist theorem), let alone reliably. Reliable estimation of wave parameters normally requires more samples.
3) L63: What is meant by "phase lag"? Is this the normal change of phase as the wave propagates, or something else?
4) L64: simulation -> simulated
5) L285: they were -> their errors were
6) L350: "enhancing operator clarity and care": I do not quite understand what is meant by this, consider rephrasing.
7) L352: keograms components -> keogram components.