Review to Shahateet 

General comment:

The paper by Shahatet et al. aims to provide an improved ice-thickness map of the APIS. As their introduction to previous studies highlights, such an improved product is necessary to reduce the uncertainties of this area's mass loss/volume estimates or even provide better maps for, e.g., ice modelling. The approach to arriving at the thickness maps is reasonable. However, I need help following the novelty or improvement of the manuscript compared to previous studies (see major comments). The manuscript should be organized and concise in a better way.  I have several concerns about the current manuscript (methods and limitations, presentation, interpretation, terminology) in addition to the report by reviewer#1.

I am in a lucky position to read the (very detailed and constructive) review of reviewer#1, and I agree with almost all his/her comments. I even had similar comments in my report, but I dropped them not to duplicate it. So, my comments are rather an addition to his/her comments. I generally think the work is worth publishing, but some work must be invested in the revised version.

Major comments

•    I don’t understand if your result is more trustworthy than, e.g.  BedMachine (thickness map) or Huss & Farinotti (ice discharge). Do you really improve ice discharge or ice thickness distribution? Is your SIA/PP method giving better ice thickness results than BedMachine ‘diffusion’ or BedMap ‘interpolation’? This is not well answered or discussed. As reviewer#1 mentioned, your results and error estimates are questionary (Figure 4). Could you also compile Fig. 4 for other products to show that your method works better? I mean, that is the main result you aim to arrive at.

•    Interpolation of tau_d and B is weird. Improve or discuss/mention limitations (see comments by rev#1)

•    I am also not okay with the terminology used. In the introduction, you say you would overcome the limitation of SIA/PP, but afterwards, you use SIA and PP wording for your results (Line 217, Caption of Table 2,f). Step 1 says SIA or PP is applied. The second step introduced ‘an improvement’. Later in the text, every result is mentioned as SIA or PP (sometimes with the addition that step 2 is applied).  I found that very misleading as you aim to overcome this limitation (Line 38). Sentences like Line 220  ‘In general, the first step of the SIA and PP approaches ..’ are very misleading. I expect the ‘second step of SIA’. By the way, discussing the first step over the whole domain is odd because it’s not valid.

•    Line 57 says that mass conservation was used in the second step, which improved the second step (if understood correctly). However, MC was also used in step 1 (as shown in Figure 2). I think MC is used in every step (Eq. 1) (except for PP), but the flux is computed differently. I am a bit lost in all this terminology. I think some clarification is needed here, and clearer wording for the final results (e.g. SIA/PP + step 2) is needed. 

•    You should include more maps to highlight areas of fast vs slow-flow i.e. where step 1 or step 2 is used. Input fields like slope, velocity field, dh/dt, and SMB, would be worth showing. 

Minor Comments

Line 14/15: acceleration of what?

Line 21, 31 change AP to APIS

Line 24: The sentence read weird. Please rewrite ‘… mass loss … mass change …’

Line 29: “it suggest”. Not clear to me. What suggests an important contribution?

Line 31 For the unknown reader, please explain briefly how ice discharge is related to mass loss. It’s one component of the overall mass balance.

Line 31-33: Please add e.g. percentage values for differences in outlet and entire inferred discharge values between the five products. Just to support ‘large uncertainties’.

Line 33: “five models”: Do you mean models or products (of a physical model)?

Lines 31-38. I am a bit lost without looking into Shahateet et al. 2023: ‘Three were interpreted …’. Please clarify which products are underestimated in terms of ice discharge and why they are underestimated. Maybe specify methods used to relate to your study and approach. “The remaining two ice-thickness maps …”. Which maps products please give a citation. The sentence reads weirdly (I understand physically based models are not applicable), so please rephrase. Also, remember that you use SIA and PP. According to this sentence, you shouldn’t.

Line 37/38: I don’t understand the limitations. Maybe add something like “ … by replacing the SIA/PP model in areas where they fail with an improved model’. 

Line 38: Move ‘aim’ to paragraph starting in Line 48. Concise the aims of our study.

Line 60. Section data. For some products, you specify years/time periods used, and for some not (e.g. DEM and Glacier outline). See comment “2.6. Input data” by Rev 1.

Line 95: A Pixel is equal to 7.5x7.5 km?

Line 118 replace ‘shallow ice approximation’ -> SIA

Line 122: I disagree. SIA applies to flat regions with small slopes. I think it’s a problem how SIA (and PP) is used by rearranging the equation so that the slope is in the dominator and the equation diverges when the slope approaches Zero. 

Lien 123: This is a weird argumentation. You say SIA and PP are not valid in flat areas; therefore, both models are replaced by an improved model in regions with reliable velocities. This doesn’t make sense because the problem of flat regions isn’t solved. 

Around Lines 125: I don’t understand the averaging; I thought PP and SIA were not applicable in some regions. Well, you overcome it by prescribing a slope threshold. Please clarify. There is a velocity threshold (200m/a) where you use SIA/PP (1st step) or the 2nd step. Is MC mass conservation? It isn’t explained in the text or caption. I am confused with the most righthand side grey boxes saying ‘final SIA/PP’ result. I thought SIA and PP were only applied in the first step. Please clarify.  

Around Line 155, ‘interpolation of B’. I agree with reviewer#1 that the viscosity interpolation is weird, and the resulting field would have almost no physical meaning or even large errors. This should be at least mentioned. I would use an iterative process between thickness and B to overcome the first B-interpolation: use the first generated thickness map (with initial B-guess) -> updated B -> redo (maybe do some iterations).

Line 168: “tau_d is the driving stress, a function of material-specific yield stress (tau_0)”. I think, some clarification is needed. Equation 5 is the driving stress (Greve & Blatter 2009).  See also the comment of reviewer#1 to Line 168. Similar to the B-interpolation, the interpolation of tau_d could be overcome by an interactive process, I think.

Line 175: “reliable velocities”. It remains unclear how reliable is defined/estimated. The 200m/a threshold appears arbitrary and poorly supported. Are observed velocities <200m/a (generally) not reliable? I suggest rewriting it to something like “… step 2 is applied in regions where SIA and PP are expected to fail.”. Then, provide a well-supported criterion for the threshold. Based on your Introduction (Lines 37/38), it should be a function of the slope.

Line 176 surface velocity -> observed?

Line 185: I don’t understand ‘No boundary condition’. You need a BC in the numerical model. Maybe neutral BC? So, zero gradients? Fürst et. al. (2017) impose a free boundary condition at marine fronts.

Line 188: I don’t understand ‘since we cannot ensure that the input data are consistent regarding mass balance’. Please rephrase.

Line 202: What is meant by ‘ensemble`? Also, I don’t understand the averaging. You mentioned that SIA and PP are not applicable in some regions? I could imagine this gives smoother results in the transition zone from step 1 to step 2 (hard-cut area separation at a surface velocity of 200m/a). Why not smoothly average 1st and 2nd step products between two velocity thresholds? Something like 

if(v_obs>200)

results=second_step;

elseif(v_obs>200 & v_obs<50)

results=second_step+(first_step-second_step)*(1-cos(pi*(200-v_obs)/(200-50)))/2;

elseif(v_obs<50)

results=first_step;

end

This would give a smooth transition. You can also argue that basal sliding is gently increased within this window (the abrupt onset of sliding from step 1 to step 2 was criticised by reviewer#1; I agree with his/her argument).

Line 215: ‘Similar result’. This is not true for SIA-step1 and PP-step1 ice discharge. By the way how is ice discharge computed for PP step1? You do not have any velocities computed for this step, I guess.

Line 225. You lost me. Figure 3 displays PP and SIA's average ice thickness map (1st and second step), right? In Figure 3 (right), you show the difference between both. How do you conclude that SIA overestimates ice thickness due to (absence ) of sliding compared to PP? Also, (1) the second step assumes 100% sliding (2) fast-flowing regions are not indicated in the figure. Also: “In contrast, the introduction of the second step decreases the ice-thickness values in fast-flowing regions when the SIA approach is used” ->  I thought you replaced SIA in the fast-flowing regions (second step).

Line 295: What does that mean? Are the methods used in BM v2 and v3 different? I don’t understand why BM v3 agrees better with your study.

Line 297. Please explain super-resolution.

Line 398 I haven’t found the two-step approach in Fürst et al. (2017). I thought that the novelty of this manuscript to overcome the limitations of previous studies.

Line 417: ‘All models showed large errors when validated against independent ice-thickness data. “. I found this a very critical comment. Is that really the case? What do you mean by ‘all models’? 

Line 574: doi is missing

Line 633: why do you use a segment of 200m? Your model resolution is 100m.

Figure 1 needs minor improvements: Coordinate grid lines, and OIB data appear at black lines. Please use another colour, either for OIB or the grid. Ice shelf and land appear as a uniform grey colour. You may use a darker grey colour for land or ice shelf.

Figure 3. I don’t see the optical image mentioned in the caption. Please add (a) and (b) to the panel.