General comments

I have been following the developments by many of the authors in the area of Gaussian processes and Bayesian deconvolution for a few years now, and I greatly enjoyed this paper as an application of those ideas to deconvolving incoherent scatter radar data. I found the description of this hierarchical convolution technique to be clear and well-organized, and I have high confidence that I could implement the technique based on reading the paper. I think this is an exciting area of development for processing radar data and, in particular, incoherent scatter radar data, and I look forward to future developments. I have some specific comments that follow, but they mainly touch on areas where I think additional information or clarification would improve the paper.

Specific comments

1) In the paragraph containing Equation (7), it is introduced as, "In order to reach resolutions better than the elementary pulse length". I found this slightly confusing on the first read-through because I initially failed to recognize that Equation (7) is a discretization of Equation (5) since my attention had been directed to the resolution issue. The quoted clause implied to me that the form of the following Equation (7) is specialized in order to achieve increased resolution, but in truth the equation would look similar in all cases and is necessary just for discretization. I suggest removing the quoted clause and placing discussion of how to achieve resolutions better than the elementary pulse length to after the description of Equation (7).

2) Using a mean of 0 for the Gaussian Process prior for P is described as a "convenience", and I appreciate from my own experience with GPs that it is indeed such. Are there other justifications you can provide for why that is an appropriate assumption in this case?

3) Similarly, can you provide additional justification for why a Matérn covariance with v=1.5 is chosen? Including a quick statement in the text will help readers who are less familiar with Gaussian Processes so they don't have to turn to one of the references to find the answer.

4) It is noted that L_l is a tridiagonal matrix with reference to Roininen et al. 2014. I suggest adding a quick statement saying why this is the case (e.g. finite differences approximating the derivative) and why it is useful (e.g. efficient computation especially as the problem size scales up). Providing an explicit expression of L_l as a function of l_i here would also be good for clarity, although I do note that it appears in-text later in lines 204 and 205.

5) The Figure 3 labels and text discussing the figure refer to u as the "length-scale function". I think it would be clearer to note that this is the log of the underlying length scale, so that statements like "by factor 3-5 large in smooth parts of the profile" can more easily be associated with the log scale under discussion. Better yet would be to reference the physical units associated with the underlying length scale values.

6) The alpha tuning parameters were optimized to minimize the mean squared error between P and P_hat, and the resulting estimates all underestimate the peak power of the sporadic E layer. Presumably this is because the length scale would need to reach a smaller value at those altitudes in order to permit the large gradient that exists there. Did you test higher values for the alpha parameters, and does that end up fitting the sporadic E peak better? What does that do for the quality of the estimates at other altitudes for the background ionosphere? In other words, if one was more interested in the highest quality estimates of either a narrow feature or the background ionosphere at the expense of the other, how does that affect the decision for setting the alpha parameters?

7) Following on from the previous comment: did you test any other prior distributions (i.e. not Cauchy or TV/Laplace) for the length scale difference that might be better suited to really sharp gradients? If not, can you point to directions for future work in this area?

8) How specifically did you choose the tuning parameter values for the PMWE and PMSE results? (i.e. What "performance" [line 296] is being optimized?)

I look forward to future work that would include non-zero lags of the autocorrelation function for analysis of the full ionospheric incoherent scatter spectrum, and I also look forward to future work that would apply the technique to the data before it has been matched filtered.

Technical corrections

(line 313) "from in TV prior" -> "from the TV prior"?

(line 334) "Cauchy difference TV" -> "Cauchy and difference TV"?

Review comments for the manuscript entitled” Hierarchical Deconvolution for Incoherent Scatter Radar Data” by Ross et al.

This paper describes a methodology of 1-dimensional adaptive deconvolution based on hierarchical Bayesian probability models with the aid of MCMC. When a deconvolution, or super-resolution is concerned, it is always an issue to make balance among resolution, signal to noise ratio, stability (robustness), and accuracy. The authors have achieved a good balance of these factors using a Bayesian regularization model which is hierarchically structured. The results are excellent. The proposed technique is developed and described particularly for IS radar purpose, but the concept is quite versatile and it is applicable to many other deconvolution problem as well as other radars.

While the model and the algorithm are quite complicated, the authors made a large effort to make them as clear as possible in the text. In addition, no serious errors and flaws were found. On the other hand, it is very hard to overview the whole structure of the model at a glance. For the sake of readers, I give some minor comments in what follows.

The current version of Fig. 2 simply shows the relationship between the parameters, and the model structure is described in detail part by part throughout the sections 2-4. However, the current structure requires readers to go back and forth in the text until the model is understood and this is rather painful. In my opinion, Fig. 2, or perhaps better to add another figure, should also include the model structure itself to grasp the whole structure at a glance. More specifically, it should illustrate relationship of the Gaussian Process and Matern covariance, the additive epsilon and Gaussian pdf and so on in the diagram, as well as MCMC and MAP.

In addition to the logical relationship of the model parts mentioned above, it is recommended if possible schematically to show the sequence (in time) of the procedures to show which part of the model and how to start the calculation from.

Some other points including questions:

- In L.152, which is “Here p(P_m| P, L) is the likelihood…” (L is intentionally capitalized for readability purpose in this communication), can p(P_m| P, L) be p(P_m|P)? It is because P_m is presumably conditionally independent from L given P.

- Equation (12) indicates the name of prior PDFs (Cauchy & Laplace) but does not show their mathematical forms. While this is accepted in case actual expressions are not concerned, it is recommended to show them in this paper because the definitions of ALPHA_C/TV are needed in the following discussions.

- In Figure 5, what is the reason by which the sidelobe of the left plot (PMWE) is wavy while the other (PMSE) is quite smooth?

- In Figures 6 and 8, what is the reason by which u_C and u_TV are quite different where they are higher than 4.0?

- On p. 17, the authors discuss the difference between the results from Cauchy and Laplace priors, but its underlying reason is not mentioned at all. Since the difference is very curious and interesting, it is preferable to mention some of your ideas about it if you have any.

Other minor points below.

L169: Roininen et al. (2014) corresponds to two papers in the reference list. Please identify which one it is.

L169: Is “partical differential equation” correct? (10) and (11) look like ordinary differential equations.

L177: Roininen et al. (2014) corresponds to two papers in the reference list too.

L313: out from in -> out in

L334: difference TV -> difference and TV

L395: STEL -> ISEE

Figures 3, 4, 6, & 8: Is the "unit" of length-scale function [km] or [log km]?