Dear Nina,

I have now read your responses to the reviews and the new version of your paper, which I agree has been improved during the review process. If the idea beyond your work is interesting and deserve to be published, I think that the message is still not clear enough and that your work should be compared to other approaches (e.g. using an inversion of the basal friction). Moreover, the paper should emphasize much more the way the roughness parameters are estimated from the radar measurements and how the two approaches compare (how xi and xi_2 compare?).

I have moreover some minor remarks listed below.

Best regards
Olivier Gagliardini

(line and page numbers are for the version showing changes)

- page 3, line 25 down to page 4 line 12: you seem here to tell that using the roughness approach you will be able to solve prognostic problem and have an evolving friction with time. I don't think this is completely correct: you also need to invoke an hydrology model to make the basal friction evolves with time. And this is certainly the most difficult point. I guess, this could be done the same way using an inversion of beta following the procedure below:
(i) inverse for beta,
(ii) assume a friction law function of the effective pressure (e.g. Eq. (15) in Flowers et al., 2011 or your Eq. (15))
(iii) compute a water pressure distribution (assuming for example as in your approach that p_w = rho_w g (z_b-z_{sl})
(iv) compute the distribution of the friction parameters assuming the beta and p_w distribution (e.g. C_b in your Eq. (15)).
You should emphasize that in comparison to the inversion procedure alone, you use some physical information on the bedrock properties to evaluate the friction law parameters, but you still require to have an hydrological model to evaluate the distribution of the basal water pressure.

- page 4, line 2: i still think that a change in the grounding line position will change the basal conditions (i.e. the water pressure distribution)

- page 5, line 18: \mu is the effective viscosity (not ice viscosity)

- below Eq. (4): is a scalar invariant -> is the second invariant

- page 7, line 21: the only stress onto the ice is exerted by the water -> the only stress onto the ice is exerted by hydrostatic water pressure

- page 7, line 23: this is Eq. (11) which should then written here and only one time

- In 2.2.3 it is not clear if you account for the heat production induced by sliding (u_b . tau_b)

- page 11, lines 5-11: Boundary conditions should be defined as Dirichlet, Neumann or Robin boundary conditions. It is not clear from the text what type of boundary conditions are applied here and why it is not stable.

- page 15, line 29: xi should be noted xi_1.

- page 16, line 23: with N=5 (2^N=32)...

- Fig 3: the map of xi should also be added in Fig. 3 to allow the comparison with xi_2.

- page 17, line 30: N_b is the effective pressure (not the basal normal stress which is only the ice contribution). Change at other places.

- Eq. (15): should be (z_b - z_{sl}) and not z_b alone

- below Eq. (15): it is speculative in both way in fact: the effective pressure can be also lower where the hydrological network has higher water pressure than rho_w g (z_b-z_sl). But I agree that for PIG this is less probable.

- page 18, line 10: I think the expression for C_b as a function of xi should be given here.

- page 24, line 24: what do you mean by 'exact'? I found the previous part difficult to understand and to follow.

- page 27, line 13: cpu time constraints

- paragraph 4.3.2: I guess the analysis of the robustest of the method could be done an other way:
(i) invert for beta
(ii) knowing the roughness, T, N, obtain a map for C_L.
You would then have an information on its spatial variability.

- page 28, line 13: and than increasing -> and then increasing

- page 30, line 15: large parts below the glacier are below sea level -> large parts of the bed are below sea level

- page 32, line 5: two remarks regarding the erosion: (i) at which time scale it should start to be significant? and (ii) erosion will change the bed roughness and the initially estimated roughness should then be evolved?

- Fig. 2 is not very clear and I guess not so useful.

Dear Nina,

Thanks for this new version of your paper. I have read your reply as well as this new version and there are still some points that are not enough clear (see below).

Regards
Olivier Gagliardini

- page 3, line 17: 'we believe' doesn't sound very scientific? Especially when in the next sentence you use 'therefore'. I would suggest to rewrite these two sentences.

- I have some issue with the BC at the base. Can you confirm that the Dirichlet BC (8; u.n=0) is only applied where the ice is grounded (not under the ice shelf)? Then, my experience is that applying in the normal direction the hydrostatic pressure below the ice shelf is not stable. This is not stable as a very small shift from the hydrostatic equilibrium will induce very large normal velocity (if the bottom surface is too low, the whole ice-shelf is pushed up, and the reverse if it is to high). In our case, we have to had a 'viscous spring' which account for the fact that the sea water pressure is a function of the altitude (see for example Durand et al., 2009). I am therefore wondering if you are not applying u/n=0 below the ice shelf since you are solving a diagnostic problem?

- Equation 12: for a Stokes problem, the normal stress is not equal to the cryospheric pressure. To evaluate the overburden pressure you should use the normal stress from your Stokes solution.

- Equation 14: you should specify at which base is applied this equation. Grounded or floating. In any case this seems false for me. For the floating part, only the normal pressure should be applied (+ the viscous spring to make it stable) and for the grounded part, the BC are u.n=0 and the friction law... (so that the normal stress is not acting in the FE formulation, i.e. you can not have a Dirichlet and a Neumann BC applied at the same time in the same direction, the Dirichlet only will apply in that case).

- Equation 20: I am a bit annoyed here since you claim in the introduction that with your physical method you will avoid using inverse method, but at the end you need observed surface velocity to calibrate C_b, in the same way you have used them to 'inverse' \beta...

- page 18, line 25: should be C_b(\xi) as C_\xi has not yet been defined (next sentence).

- page 24, line 5: might also be due to the fact that you are solving for a diagnostic with no relaxation... and that the geometry (surf, bed, GL) is not consistent with the flow?

- Figure 7: how do you explain these features (patch of high T going up to the surface)? I suspect the diagnostic velocity field with a non relaxed observed surface topography to cause strange vertical velocity patterns. Did you check the vertical velocity field?

- Figures 8 and 10 are too small.

Dear Nina,

Thanks for the updated version. I guess we have converged to a publishable version of your paper.
Regards
Olivier