General comments:

Priergaard Zinck and Grinsted present a comparison of two models to estimate ice thickness distribution of the Müller Ice Cap in the Canadian Arctic. The thickness information is further used to select a best site for drilling an ice core. The brief communication paper is generally well structured and written and discusses a relevant topic. There are however some fundamental issues that make that I can at present not recommend this article for publication. My major concerns are related to the methods and concern both the simple SIA inversion and the PISM-based inversion. In my opinion, certain modelling choices are not justified and the comparison of the two methods in its current form does not give too much useful insight. Thorough revisions would be required to make the study worthwhile and would require a change of strategy. My major and minor comments are further explained below.

Major comments:

SIA inversion:

The only difference of this method and other SIA-based inversion methods (see e.g. Farinotti et al. 2017, 2021) is that this method takes more freedom to calibrate the inversion method. But by individually calibrating a, b and k, which are normally dependent on another as they all are a function of exponent n, the model is no longer really the SIA model after calibration. So the separate calibration of a, b and k in practice leads to a model that no longer follows the same physics as the SIA and it is unclear what physics it does follow instead. Even when a relatively good fit with observed bed data can be found with this tuning of a, b and k, it does not give much confidence in good performance of the same model with these parameter values elsewhere. It would make much more sense to only calibrate parameters like Glen’s exponent n or rate factor A, which would not change anything to the physics of the model.

Mass balance uncertainty:

In the SIA inversion it is claimed that the mass balance has not much impact on the reconstructed ice thickness. This in reality is of course not the case since the mass balance is a major factor that affects ice extent and average thickness. The reason that it does not play a major role here is that any biases in the mass balance (as clearly seen in Fig. 1) are indirectly compensated for by the tuning of parameters a, b and k. But this does not mean that ice thickness is not sensitive to mass balance, any offsets are simply calibrated away. From Figure 1 and the Discussion section it becomes clear that there is a major overestimation of mass balance in the HIRHAM product. This bias is ”calibrated away” / corrected for in the SIA inversion but the same is not done in the PISM-based inversion, where only parameters affecting ice flow speeds are used for tuning. It is hence not surprising that the PISM-based inversion provides worse fits to the bed data as the degree of tuning is much less. In other words, the degree of tuning of the two models used in this study is very different and makes it hard to draw any strong conclusions from the current results.

PISM-based inversion & factor K:

In the Methods it is mentioned that the PISM-based inversion uses a factor K that varies between 0 and 0.5 from 500 m a.s.l. to 1000 m a.s.l. This implies that below 500 m a.s.l. elevation K=0 and the bed is not modified at all after every iteration. Effectively, that means that at these elevations the bed remains at the initial height which, correct me if I am wrong, is taken from the global estimates by Farinotti et al. (2019). In the Discussion it is argued that the overestimation of ice thickness of the outlet glaciers (which are mostly below 500 m a.s.l.) is a result of the lack of a calving criterion, but I do not think the calving criterion plays any role here since the bed is fixed anyway in these lower areas of the outlet glaciers. The large deviations of bed heights between the PISM inversion and the observations, as shown in Fig. 1, nearly all happen in areas below 500 m a.s.l. Effectively, we are looking at a comparison of the Farinotti et al. (2019) bed and the observations for a large part of the domain. My suspicion is that the choice of variable K values with altitude was based on problems to make the PISM based thickness inversion converge. The lack of conversion should however not have been solved by choosing a variable K value, but rather by correcting a bias of the mass balance. A strategy with combined calibration of a mass balance correction and ice flow parameters, and a fixed value of K, would probably have yielded much more reasonable results. It is good to realize that a more detailed inversion technique, i.e. with a more accurate decription of ice motion and boundary conditions (mass balance, calving), should theoretically yield better thickness estimates than simplified approaches as long as the input data (DEM, mass balance and/or ice velocity) is of sufficient quality.

Specific and minor comments:

L5-6: See the second major comment above. The SIA inversion is not insensitive to mass balance (or at least it should not be), but mass balance biases are indirectly calibrated away through tuning of a, b and k.

L12-13: Large ice thickness does not always mean very old ice at great depths, only if it is close to an ice divide.

L17-18: “However, field work constraints…” This does not connect well to the previous sentence, more to the one before that.

L18: Please replace “to be clever” with “to be selective”.

L21: Please remove the obsolete bracket

L22: Remove the obsolete “in”

Figure 1: UTM x and y should be replaced with UTM Easting and Northing instead and the UTM zone should be mentioned in the caption. Furthermore, exponents in the SMB units are missing.

L27: “differs” --> “differ”

L29-31: The new method is not necessarily less sensitive to mass balance, steady state assumptions and ice flow physics. It just collectively calibrates any biases due to these factors away. But the problem is that by doing so the physics of the model are also changed, which is hard to justify. See my first major comment above.

L36: “why” --> “and”

Section 2 (or Introduction): I am missing some information on ice velocities of the ice cap. I suppose these data could for example be extracted from online resources. A source like Its_Live (https://its-live.jpl.nasa.gov/) could potentially be useful. It would give an idea on potential sliding rates which is relevant to know because the non-sliding SIA is used in the SIA inversion, which may be a poor assumption of sliding is significant.

L84: The slope threshold for ice thickness inversion is a critical parameter for SIA based inversions. The chosen value is however somewhat arbitrary which could be acknowledged as a source of uncertainty.

L88: ”why” --> ”which is why”

L98-100: Under normal circumstances, the mass balance and its distribution in space should have a large impact on the thickness distribution. I suppose better fits could be achieved wíth a better spatial representation of mass balance.

L107: The PISM inversion is known to work best when a variable climate forcing is applied, i.e. when the modelled ice cap does not reach steady state. The fact that a fixed climate forcing is used here hence adds to uncertainty of the PISM inversion, which needs to be acknowledged.

L118: See my third major comment above. Setting K to zero below 500 m a.s.l. effectively shuts down the inversion process in these areas. This is probably not desired.

L122: ”combinations” --> ”parameter ranges”

L144: See also my first and second major comment above. In the SIA inversion the entire mass continuity equation, including ice dynamics and mass balance, is tuned through a, b and particularly k. In the PISM inversion, no tuning of for example the mass balance is done, only of a sliding coefficient and enhancement factor. That makes the comparison somewhat odd, since the SIA inversion is much more widely tuned to match the available thickness data.

L151-152: The larger RMSE for the PISM based method is not surprising. Based on Figure 2 this seems to be completely dominated by the errors below 500 m a.s.l. where the bed is not allowed to change in the PISM inversion. Above 500 m a.s.l. there does not seem to much difference between the two approaches (?). An additional mass balance tuning of the PISM method could help to make a better justified comparison of both methods.

L178-179: See also the third major comment above. I cannot imagine that the calving criterion is of much influence, since the bed is not allowed to change below 500 m a.s.l. (K=0), which means that a too large extent due to a lack of calving does not really play a role.

L183: See also the second major comment above. There may be an overestimation of the SMB in HIRHAM5, but right now the overestimation of the ice thickness is a direct result of fixing the bed under these outlet glaciers to the Farinotti et al. initial bed. The apparent problems with the HIRHAM5 SMB data are exactly the reason why also in the PISM approach the mass balance should have been included in the tuning process, like is also done in the SIA inversion.

L188-189: Again, the SIA inversion is not insensitive to mass balance, it just removes mass balance (and other) biases by tuning a, b and k. Similarly, mass balance biases could (and should) have been tuned also in the PISM inversion to enable a direct comparison.

Figure 3: ”UTM x” --> ”UTM Easting”; ”UTM y” --> ”UTM Northing”

Zinck et al are presenting two approaches to estimate ice thicknesses over the Müller ice cap in the Canadian Arctic.  The authors are inverting ice thicknesses using (1) an inversion based the shallow ice approximation, and (2) an inversion based on the PISM model, and compare the results. With the present version of the manuscript, it is not clear to me why the authors choosed to compare these two approaches. Furthermore, there is not much description of the PISM model, which prevent the readers from understanding the physic of this approach without referring to other manuscript. Since we have here a manuscript that is solely based on a method comparison, I think that more details are needed regarding the PISM inversion. Concerning the results, it seems that the inversion based on the SIA is dominated by highly frequent variations on the surface slope. Indeed, to choice of appropriate smoothing parameters needs to addressed before being able to compare this method to the PISM inversion (see comments below). Similarly, the ice flux in the SIA inversion was calculated, using strong assumptions on the surface mass balance. Furthermore, the SMB value for the calculation seems to have been taken quite in an arbitrary way. All these approximation, are adding-up to the SIA assumptions, and could have been avoided by using a formulation of the SIA relative to the surface and basal flow velocity (Zorzut et al., 2020). Since this method is more straightforward, and that the authors have all the data needed (ice velocity can be downloaded here https://its-live.jpl.nasa.gov/), I would suggest to add this method as a comparison. This would indeed provide an assessment of the differences between two SIA formulations, in addition to the PISM inversion. Finally, the overall aim of this paper was to choose an appropriate drilling location. There is very few details on how this location was choosen. I think that having a paragraph within the result and discussion section about the choice of drilling location would make the story around this paper much more interesting.

 Specific comments.

L20-25. Something needs to be stated here about the Consensus estimate of 2019. How does the different model outputs agree over Müller ice cap.

L28-30. It is not clear to me how using the shallow ice approximation to calculate the ice thickness is new (cf. ITMIX; Zorzut et al., 2020)

L40. Which dates where the OIB data acquired ?  How does it compare to the date of the Arctic DEM ? How do you account for the offset between the different surface dataset ?

L49-50. The misinterpretation at the ice divides is not due to the fact that Farinotti provide a global product, but to the flowline approach that is used, as it was mentioned earlier in the paragraph.

L51. Why do you choose a different resolution for the SIA and PISM ? The Farinotti et al. data do not have missing values, but really thin ice thicknesses near the ice divides. How did you choose to mask out the dataset of Farinotti ? Can you provide a Figure of the interpolated bedrock topography product?

L55. Why did you choose to solve for a non-sliding version of the SIA ? Using ice velocity measurements from ITS_LIVE would allow you to better calculate ice flux, and indirectly accounts for the amount of deformation within the ice (i.e what portion of the flow is caused by either sliding or deformation). See Zorzut et al., 2020 for more details. Finally it is not clear how you perform the inversion. Do you calculate ice thickness at the basin scale or just along the line?

L70. How does the a, b and c values varies along the Icebridge line ? Can you provide us with an histogram, and statistical analysis on the variations of these parameters?

L82. Why did you choose a 250 m as a smoothing parameter ? Can you provide a map of the unsmoothed and smoothed map ? Why not choosing a lower resolution version of the Arctic DEM?

L95-103. This section is unclear to me. Why do you take a uniform SMB instead of using average In-situ data value or HIRHAM outputs?

L105-106. Can you provide more insights on what physics is solved into PISM, how does it differ from the SIA, and why would that be useful to use it in addition to the SIA?

L159-170. As it was suggested by Bamber et al 2000, the SIA suggests that the flow is linked to surface slope at the scale of multiple ice thicknesses. Hence the smoothing distance of the surface slope should carefully choosen. Slope processing is a well-known SIA limitation (Zorzut et al., 2020, Farinotti et al., 2009; ITMIX-1; ITMIX-2), hence I would recommend the authors to run their model again with better slope smoothing parameters, before comparing with PISM. It is obvious from Figure 2 and Figure 3, that the spatial distribution in ice thicknesses is completely dominated by artefacts in the surface slopes.

L173-175. This sentence is not clear here. Do you mean that information on ice velocity will only provide limited additional information ? Please rephrase the sentence to make it clearer. Moreover, despites what the author says, I will argue that the ice velocity will provide crucial information on the distribution of the ice thicknesses. Indeed, using a formulation of the SIA that includes the ice velocity (Zorzut et al., 2020), will allow you to easily account for the amount of internal deformation vs sliding, which can be important to mitigate for with Müller ice cap  (Copland et al., 2017). This will also reduce the strong hypothesis that are made here for the calculation of the flux Q, which depends on the SMB values, that was defined in quite an arbitrary way.