Review of EGUSphere-2024-4124, Propagating Information Content: An Example with Advection, by D. Turner et al., submitted for publication in Atmospheric Measurement Techniques.

This manuscript describes how to propagating uncertainties and determine the information content of quantities derived from multiple instruments, and it illustrates this technique with an example. It is well written and will be a valuable addition to this topic. I recommend that it be published after minor revisions. I have listed comments/suggestions below that I hope the authors consider to make the manuscript a bit more accessible to those not proficient in these topics.

As a general comment coming from a reader who is not as familiar in this way of thinking as the authors, terms such information content and DFS are not in my daily vocabulary, and some of the results are not intuitive. I would urge the authors to assist us in making these concepts, which I believe is important ones, more accessible, and to provide more physical explanations where possible. For instance, the definition of information content is not given until line 55, and DFS not until line 110. Only on line 194 did I see it clearly stated that the variability in these quantities is due to instrument differences (also stated on line 324). For instance, is it possible to show or discuss the contributions of the various measurements to DFS? These comments should not be construed as criticisms of the manuscript, which is well written and reads nicely, but as an appeal to make challenging and non-intuitive concepts easier to comprehend.

Specific comments:

Line 130: This would be a good place to start a new section.

Line 145: It is not obvious from the figures that the retrieved winds are larger above 1 km.

Line 177: Addition of another equation or a bit more explanation of how Eq. 5 becomes Eq. 6 would help the reader who is not as familiar with this topic as the authors are.

Line 179: The phrase “where the superscript T in this content represents matrix transpose” should be place near Eq. 1 where it first appears.

Lines 180-195: These would be better places before line 150.

Line 194: The explanation “due to the differences in the instrument uncertainties at the different locations” should appear earlier.

Line 200: The statement “the cold air advection … has a lot of uncertainty” is vague; is it the magnitude of the cold air advection, the timing, the direction, or what? Perhaps (likely?) that is my lack of understanding, but “cold air advection” sounds like a process, not a quantity (such as temperature advection or water vapor advection); thus, it is unclear what the uncertainty would refer to.

Line 203: Show the location of the other two sites in Fig. 4.

Line 210: The statement that the information content is approximately 5 seems impossible from the statement on line 112 that the information content is between 0 and 1, and those shown in Fig. 5 are less than unity. I may be (likely am?) confusing different quantities, but that merely demonstrates that a typical reader may be confused here, and that a bit more explanation would be useful.

Line 234: Perhaps move the title of the section to line 245, as that seems to be where the example actually starts.

Line 244: The statement that the information content on one quantity can be near unity even though the information content of an instrument can be low is crucial and should be more strongly emphasized.

Line 244: “in information content” should be “an information content”

Line 244: Panels b1, b2, and b3 of Fig. 5 show DFS of water vapor exceeding 0.05 at heights greater than 50 m.

Line 245: The end of this sentence is a great location to remind the reader that this is due to the instrument.

Line 259: I had a “why?” after the statement that doubling the covariance matrices by a factor of 2 had little effect on the DFS, and found myself desiring a more physical explanation of this result.

Figure 5: In the caption, perhaps label that columns 1, 2, and 3 refer to E37, C1, and E39 so they don’t have to look back to Figure 2 to find this information. The rest of the information on this panel (e.g., T, WV, U, V) are labeled, but the locations aren’t.

Line 260: “diagonal” or “diagonal elements”?

Line 298: A bit more discussion would be helpful here. What happends at those altitudes, where the standard deviation is greater than the mean?

Line 324: This statement should appear much earlier in the manuscript.

Line 331: “strong function height” should be “strong function of height”

Line 336: “derived from remote sensors” or “retrieved from remote sensors” or “obtained from remote sensors” 

I suggest to accept the present manuscript with minor revisions that are addressed in the attached document.