The authors analyse a commonly used finite-difference method for estimating thermal diffusivity of supraglacial debris using vertical debris-temperature profile. The aim of the study is to understand the effects of temporal and spatial discretisation on the uncertainty in estimated diffusivity. Due the importance of debris thermal properties in computing the dynamics of debris-covered glaciers, this topic is of importance. However, the study appears to  have serious weaknesses in terms of the experimental design and the underlying assumptions. unless these issues are addressed with suitable (and doable) modification of the methods, the results and conclusions will remain weak.

Major comments

1. I see a few possible careless mathematical errors and inappropriate assumptions, which could have been avoided. For example,
1. The expansion given in equation 20 is wrong.  (1+x)^-2=1-2x+3x²-4x³+5x⁴+….. 
2. Please check Eq 9. For example, according to https://en.wikipedia.org/wiki/Propagation_of_uncertainty,  ifσ_f=a/b, then
3. In section 3.5, equation 19, you assume that the temperature values correspond to equispaced sensors and thus errors in the numerator vanishes. However, the spacings are not equal anymore due to the random shifts (both in a real experiment and your simulations). The errors in the numerator must be considered.

1. A couple of recent publications (Laha et al. & Petersen et al.) went beyond the assumptions of a homogenous, source-free, purely conductive heat flux, as being considered here. You need to provide more compelling an argument about the motivation behind and significance of the present study than you do in L107. (Of course it is a different matter, if you were to actually analyse the existing data of thermal diffusivities reported in the literature, but that is not something you attempt here.)
2. While the space- and time-discretisation steps are important for setting up actual measurements, it is difficult to judge whether your analysis can actually lead to useful insights about real experiments, due to the idealisations involved in your study design. You have totally ignored the sources that are present due to the horizontal inhomogeneities and temperature gradients, water content, advected heat, latent heat transport, convection etc, and vertical variation in Kappa. If one were to incorporate the noise due to these effects in the forward model, which are present in the real system anyway, will your results hold?
3. Since you are solving the forward problem numerically, it is easy to create an ensemble of experiments where all the parameters and variables in your model are perturbed by appropriate space and time dependent noise, and the corresponding mean values are drawn from a distribution (eg, a range of values).  If one does that, then all the lines in your plot will have an associated uncertainty band. When such uncertainty is considered, the differences between different curves that you have discussed throughout your manuscript may become insignificant. Till you do this exercise and demonstrate that your results/conclusions are robust against such the  inherent variability and measurement noise, your conclusions are not on a firm footing.

I am not going into the details of your results and conclusion, because of the limitations of your methods mentioned in major comments 1, 3 &4 above.

Other comments

I think there is there is scope and need for improving the writing. There are several sentences which either lacks justification, or are vague, or even incorrect. Also, it may be better to avoid subjective discussion when you set up/introduce the equations and symbols, eg, at the beginning of your methods section. Please just state the standard definitions, and provide units. Please revise/reconsider the sentences/phrases listed below.

L18: The regional-scale debris-covered effect was discussed in several papers eg, Scherler et al, Nature Geosci., 2011, Banerjee & Shankar, 2013. Even though Hock et al. 2019 ignored such effect, it was not previously thought to be unimportant - only there was ready-made way to incorporate it in large-scale models.

L33: Not sure how attenuation of daily signal controls heat flow. Of course, it is a consequence of the diffusive evolution of T(x,t).

L34: what is “thermal instability” in a conductive system?

L47: What processes? "The supply of melt energy” can never be “represented” by an effective thermal conductivity.

L57: Despite the long, general introduction, the question addressed in this paper is not motivated at all.

L87: Does this paragraph belong to methods? Seems to be more suitable for the introduction section.

L135: please demonstrate that it is enough to consider the intercept, to incorporate all the errors/uncertainties mentioned in major comment 3 and 4, for example.

L197: Limiting values are still well defined.

L206: Please provide mathematical justification or some relevant reference where such justification has been provided, for this average method (method 2). What you are calling skipping, is just a standard finite-difference method as explained in any textbook. 

I like the idea of this paper. I believe not enough attention is placed on error slinked to both study design and data collection, and numerical approaches. This paper discusses both these issues extensively concerning the calculation of the thermal diffusivity, a key parameter in calculating the diffusion heat flux to the surface, and the subsequent ice melt. Specifically, the influence of temporal and spatial temperature sampling intervals in the data collection, as well as the influence of truncation errors in the calculations, is explored along a gradient of idealized data to observed data. While this approach has limitations, like assuming a homogeneous debris layer, I think drawing attention to often ignored sources of uncertainty is valuable.

However, there are some flaws in the manuscript that I think should be addressed. First, the manuscript would benefit from closer attention to the writing. In many places, the writing is highly informal and imprecise. While I value directness and simplicity to convey a clear message, in some cases, the manuscript reads more like a blog post, with vague statements and a lack of structure.

The text would benefit from being streamlined to allow for a clearer flow. There are also some sections of methods that read like introduction and discussions that also read like methods.  The discussion lacks a section on the limitations of this study, and there is no conclusion, simply a long list of broad best practice guidelines that are very general. Citations are formatted wrong and many places and there is a general lack of commas throughout the text. Overall, I found the text hard to follow and found it hard to see the key points in the study. Therefore, despite my support of this study, I think it needs a lot of clarification and rewriting before it can be published.

The authors present a theoretical experiment to estimate thermal diffusivity within the debris layer of a debris-covered glacier. The topic of thermal properties within debris-cover is timely and relevant given the many other studies that have been published in the past few years. Overall, the paper is well thought through and lays out the structure and approach used. I think that the writing style could be improved to be more formal as is typical in a journal article, and attention to detail is needed for many sentences throughout. There are many minor edits that I suggest throughout, and in general the text should be made much clearer as it is difficult to follow in places. The entire paper should be read with an eye towards grammar and citation formatting, as there are many (beyond the comments I made) that are incorrect. I think the overall suggestions are useful for future work, and should be an informative paper, but the structure and cleanliness of the paper needs to be improved before publication.

I think that the paper clearly states the justification to focus its analysis on the Conway and Rasmussen (2000) paper, which I feel accounts for many of the comments of a prior reviewer. To address this, it might be useful to provide more of a comparison between the newer Laha et al (2022) paper and the Conway and Rasmussen (2000) paper to further support the need for this study. I think that the value of this paper comes from its theoretical approach instead of using entire seasons of existing data from other papers.

The methods used are well outlined overall and supported by the figures and online tool provided in the paper. There are a couple paragraphs in the methods that could be more useful in the introduction or discussion (L87, L110) Please see my in-line comments regarding minor edits to certain paragraphs and sentences.

It is important for readers to understand that this is a theoretical model and will of course not account for every eventuality that might occur in the “real debris” layer on a glacier, but I think a major benefit of this study is the online tool to explore the errors interactively. This tool was easy to use and provides the theoretical knowledge to explore thermal diffusivity within a debris-layer. The assumptions that are made regarding density, conduction, and specific heat capacity are reasonable considering the constraints of a numerical study. The results show that this model holds up against specific instances of real data which helps validate the model being presented here. However, there should be more in the discussion or conclusion about the limitations of this study and more specific reasoning for the best practice guidelines that it provides.

As it is intended, this paper focuses its analysis on the Conway and Rasmussen paper and identifies limitations and error sources based on that method of estimating thermal diffusivity values. This paper provides value to the literature because of how frequently the Conway and Rasmussen method is used and oftentimes, without deeper consideration for its error sources and limitations. This paper does not aim to analyze other new methods as discussed in Laha et al., and Petersen et al., and I think this is okay due to the relative lack of use of these newer methods. While the guidelines and suggestions that this paper makes are based on a theoretical model, there needs to be more consistency in the field-based methods used in this discipline, so this paper provides some suggestions for how to do that.

The best practice guidelines provided are helpful considerations for future field studies that aim to measure thermal diffusivity within the debris-layer. The debris-covered glacier literature needs to have more consistent methods of measuring thermal diffusivity to compare findings across different field sites. The guidelines this paper presents should be used in the future by other field-based studies and these future studies should consider the limitations and error sources that are discussed here. Please just add to these guidelines and explain the reasoning for these guidelines in a clear and structured manner.

Line by Line comments – also in line on PDF.

L57: Specify vertical spacing and provide slightly more reasoning for why you are exploring these variables in the sentences before. Also discuss why horizontal spatial variability is not focused on in this study.

L87: This paragraph could be better in the intro or discussion sections.

L93: Grammatical issues. A period or comma is needed, and the spacing is strange.

L103:  Be more specific in terms of “they.”

L103-109: Also adding another sentence here to improve the justification of this approach would help convince readers this is a valuable approach you are taking. Provide more comparison from Laha et al. and Petersen et al.

EQ 3, 4, 5: The O in these equations is not defined and is not consistent across these three Eqs.

L177: Figures need to be cited in line, and when cited need to be clearly referencing that given figure.

Figure 3 caption: data 3, 4, 5 are not shown on the graphic – I think I know which data you are referring to, but please clarify and provide the same description as the values in the figure.

L182: Another figure citation missing.

Figure 4 caption: “timeseries” will need a space.

L184: Figure 6 is being referenced here and it seems a bit out of order if you are indeed citing that figure. Please make sure every figure is referenced.

L206-208: Provide a little more clarity on these two resampling methods. Method 1 is clear, but method 2 is less so, provide more detail here to avoid confusion.

L208: Figure reference missing.

L218: Make this entire paragraph more clear. T

L272: “purly” to purely

Eqs. 16-20: Why are these equations in the results? Shouldn’t these be in the methods section and discussed there?

L290: This paragraph in discussion needs to be re-written or at least made more clear. It is confusing and doesn’t read smoothly.

L316: remove comma after “true, “