This study explores geothermal heat flow in the Lambert-Amery sector of East Antarctica. A complex system of coupled models (shallow ice, full stokes, and subglacial hydrology) are initiated by six available geothermal heat flow maps to estimate basal temperature distribution across the sector. The basal temperatures are generally realistic and conform to expectation, with their differences being informative. Basal heat conduction is also presented and discussed, although the value of this field is less clear to me at the moment. Below, I provide some comments from this article.

Surface Accumulation – The surface accumulation field is used in the balance flux model, and presumably ultimately influences vertical velocity profile. I do not see any description, or citation, documenting the source of the surface accumulation field. It would be helpful to have better description of the accumulation field, including how possible biases in accumulation (or “recent” versus “steady” temporal variations) may manifest in the parameterization of vertical velocity field from the balance flux model, and ultimately in the simulated basal thermal state.

Topographic Effect – We discuss the topographic effect on geothermal heat flow in https://doi.org/10.1029/2020JF005598. Specifically, we highlight how subglacial topography in the subglacial Gamburtsev Mountains, East Antarctica, can strongly influence geothermal heat flow at kilometer scale. Whereby subglacial ridges can receive 50% less heat flow and subglacial valley can receive 50% more heat flow, in comparison to the regional average. I suspect that explicitly acknowledging the influence of subglacial topography on geothermal heat flow, by applying such a topographic correction field to the GHF input field used by the 2D SIA model, might further improve the realism of simulated hydrology.

Temperate Ice – I understand how the basal state is parameterized as either melting (Dirichlet; Eq 6) or freeing (Neumann; Eq 5), but the reader would benefit from knowing whether a thicker temperate basal ice layer is permitted. Temperate basal ice layers can form at the convergence of outlet glacier flow, even with tremendous downstream advection relatively cold inland ice (https://doi.org/10.3189/172756502781831322). The presence or absence of a temperate basal ice layer clearly influences the vertical temperature profile, which here seems critical to presented basal heat conduction (i.e. whether temperature gradient simply pinned to the pressure-melting point at the bottom, or the temperature gradient effectively becomes the Clausius–Clapeyron gradient). Allowing ice to become temperate general requires assumptions about liquid pore water content, which I do not see stated here.

Heat Conduction – I am confused by Figure 8. I would expect that, in the absence of basal hydrologic processes, basal heat conduction is effectively equivalent to geothermal heat flux. Yet inland areas, where basal hydrology is not active, have a very different heat conduction from the forcing geothermal heat flux. The sign of basal heat conduction is also negative, in comparison to the positive sign/direction of geothermal heat flux. Finally, should there not also be “opposing signed” pockets where the basal heat conduction is opposite over subglacial areas where basal water is refreezing (i.e. Vostok in https://doi.org/10.5194/tc-14-4021-2020)? Right now, the axis stops at zero. For these reasons, I find Figure 8 (and associated discussion) difficult to follow.

Subglacial Lakes – Here, subglacial lakes are being used as an indicator of basal ice at the pressure-melting point. The subglacial lake literature, however, now has suggestions that melting temperatures can be depressed significantly lower than pressure-melting point by salinity (https://doi.org/10.1126/sciadv.aar4353) and that radar-derived subglacial water indications can be false positives (https://doi.org/10.5194/tc-14-4495-2020). I think these caveats should be mentioned. The L511-515 discussion should also use the terminology “false negative” to describe the “one-sided” aspect constraint.

Consensus Map – Given the time and interest that the authors have clearly expended with this study, it would seem that they are in a very good position to produce a consensus geothermal heat flow map of the Lambert-Amery sector. My final thought would be asking why the authors simply stop with saying the Li and Martos geothermal heat flow maps are most suitable for this region, and do not provide an accompanying data product of a geothermal heat flow map that is self-consistent, or optimized, with an ice flow model (i.e. https://doi.org/10.20575/00000006)?

This paper describes a method that can be used to evaluate observations of geothermal heat flux,  and its application to a large region of Antarctica: the Lambert glacier and its drainage basin. A sophisticated model is used to estimate warm bedded regions given observations of ice geometry and surface velocity and several estimates of the geothermal heat flux field (GHF). The resulting warm bedded regions vary considerably depending on the choice of GHF, allowing the paper to rank them  by comparing those regions to known locations of sub-glacial lakes.  The Lambert glacier is representative of the majority of Antarctica since it is large and includes cold based ice, warm based ice that sees slow sliding, and warm based ice that sees fast sliding. That suggest that the method could be applied more widely, so both the method and its results should be of interest. The paper is generally well written and clear.

General Comments

The ice flow model (a Stokes flow model) is a complex one. It is certainly a better choice given unlimited resources than any of its common approximations (SIA,SSA,HOM…) and looks to have been applied correctly, but why is it necessary in this case? In which parts of the domain? It seems that in some parts you only use the direction from Elmer/Ice: how much does that differ from the direction of the surface gradient? The discussion says that this work ‘builds on the earlier inversion method employed by Wolovick et al’. (which is SIA based) but how important is that extra effort?

Equation 4 appears to be a ‘cold ice’ model, i.e one that assumes T < Tm. The manuscript should justify this choice, with reference to polythermal models e.g Aschwanden 2013.

Use consistent notation for vectors etc throughout.

Specific Comments

L17. Are abbreviations (GHF) permitted in the abstract?

L38. “Suggesting..”? How?

L41 ice penetrating radar *data*

L50, infers -> implies?

L67. comments on melt-water routing seem out of place in this paragraph

L73. ‘Ice sheet models are useful tools’ is a matter of opinion, and not connected to the rest of the paragraph.

L97 “Hence, we make inferences” –> We state / We determine?

Section 2

L 103 and fig2 – rephrase,  and draw the whole shelf /gl so that the reader can easily tell what is meant by ‘half’ . Related to this, in 3.3.3 explicitly state the boundary condition  at this segment of lateral boundary (I assume it is the same as the other boundaries) and give a justification.

L:156 ‘Inverse method’- no such thing. You are solving an inverse (that is, ill-posed) problem, using (most likely) some sort of gradient based optimization method. You are also not estimating ice flow velocity and stress, but inferring the basal friction such that the model velocity best fits observations.

L158-163 – Some rewording is needed here. You don’t describe the procedure that you hint at for some time, so provide a summary here (‘we will describe each model component in sections X and Y, then the coupling in Z’)

272, In general, this section need to be cleared up, how for example does ‘water input supply a large freezing rate’. 

L219: no need to say ‘taking six GHF datasets…’ or at least rephrase to be clear that you only use one at a time.

L329; Eq 15 is not the Weertman law, it is a linear viscous law which works satisfactorily  in inverse problems (because you are really finding Tb, not C) but not in general.

L347 Use subscripts consistently

L363 (and elsewhere) the conductive heat flux Fc =  -k dT/dz is positive (upward) when the bed is warmer than the ice above, so should you not have + k dT/dz (i.e – Fc ) if the bed “loses heat from upward heat conduction” .  What about the case where dT/dz is negative (pressure melting point reached above the ice bed). Does that simply never happen?

L400. This procedure seems important but is glossed over. If Bnew != Bold, then why does the modelled surface velocity not change?

L450 and fig 6. Why is the heat flux negative? Especially since you talk about magnitudes in the text.

L460 and fig 7 – it is difficult to tell the difference between these. Would it help to show differences relative to (a) Martos? That said, you don’t seem to depend much on these figures so are they really needed?

L487. It is not quite accurate to say that ‘The Li experiment gives the best fit’ (how is the fit quantified?). I suggest rephrasing along the lines of the following sentence which sums up the results more accurately, i.e it is only the Li experiment that results in a warm base that covers all observed lakes.  

506-507 (and elsewhere):  The datasets/fields should be referenced correctly ‘(Li et al 2021)’, rather than just ‘Li’. I also would prefer to see you write ‘our experiment using the Li et al 2021 GHF’  rather than ‘the Li experiment’, but I don’t think there is any real danger of the reader being misled by that.