Please see attached pdf

This study attempts to determine which sliding law of the literature is more adapted to describe the basal condition of Pine Island Glacier in Antarctica. The authors introduce a novel method which use acoustic impedance observations at the base of the ice sheet to evaluate the effective pressure field retrieved from inverted τ_b and u_b and for a given sliding law. They adopt a Bayesian framework to rank the different sliding law as a function of the effective pressure likelihood estimated from the misfit between the modeled and observed impedance, and prior probability density function (PDF) of the model parameters. They found that plastic or quasi-plastic sliding laws lead to the more probable effective pressure field given the observed acoustic impedance and knowledge about critical parameters such as basal porosity and grain size.

I think that any novel method to constrain basal friction law should be welcome and using acoustic impedance to constrain a very unknown variable such as the effective pressure is original and may be useful. This is why I think the paper deserve publication in The Cryosphere. However, in the current manuscript, I doubt that Bayesian approach provides a reliable ranking of the different considered sliding laws. As any effective pressure field can explain the impedance data given the appropriate parameters values, the ranking is largely influenced by the design of the posterior PDF calculation.  I believe the current study lacks elements demonstrating the extent to which the seismic data constrain the problem and support the conclusion. The manuscript requires major revisions before it can be published.

General comments

• Bayesian approaches are generally used to determine the posterior probability density function (PDF) of model parameters, given prior information and constraining observations. For each sliding law, you thus obtain a posterior PDF as a function of the three chosen varying parameters. To obtain a probability for a sliding law, you integrate the posterior PDF over the three-dimensional parameter space (if I understand correctly). This last step is not justified at all in the manuscript while it is critical as all conclusion are based on this. It is not clear to me that a higher integrated probability over the whole parameter space makes a sliding law more likely than another. For example, a model with high but localized maximum PDF can have a lower score than smaller maximum PDF spread on a larger domain of the parameter space. The way the sliding law probability is calculated clearly needs theoretical background. This is critical for the paper as the data does not bring significant difference in misfit and thus data-based likelihood.
• I do not understand why you are limiting your parameter space to three varying parameters. I suspect this is because you do an exhaustive grid search to build the posterior PDF. There are simple methods such as the Monte Carlo algorithm, that can be used to efficiently calculate the posterior PDF in cases where the parameter space is large. This would be easy to implement in your case where the forward model is fast to compute. This limitation forces you to calculate two different probabilities for some sliding laws (Schoof and Zoet-Iverson), where you arbitrarily fix one of the sliding parameters. This makes no sense to me, especially when you assume μ=C_max=0.5 without justification (when varying ut or Cs). The PDF should be built with varying all relevant parameters together.
• I do not agree with the claim you are testing the Weertman law. You are simply testing the hypothesis of uniform effective pressure which as nothing to do with the Weertman law. If you want to say that the Weertman law is not appropriate you should show that the inverted τ_b as a function of u_b does not match a power law. A figure showing the inverted τ_b as a function of u_b is missing in the manuscript in any case.
• I do not think you are able to distinguish which of the stress bounded sliding law perform better when the result is so dependent of the design of the Bayesian approach. You also hide that the Schoof law is almost the exact same law as the Zoet-Iverson law. You can indeed write the equation (7) of the manuscript in this form:

This is very similar to Zoet and Iverson with p=1/m, μ=C_max and u_t= (C_max/C_s  N)^(1/m). The only difference is that u_t is a function of N^(1/m) in the Schoof formulation and a function of N in Zoet-Iverson.

• Posterior PDF are not shown, it would be usefull to have them in some figures to discuss the influence of prior PDF.
• Given the resolution of Bedmap-2, the estimation of C_max based on basal topography observation does not make any sense. Even if the inversion is performed at the kilometer scale, the relevant scale at which to estimate C_max is the meter scale, as this is the scale at which shear resistance is built. Also, the impedance model is based on the assumption of a sediment layer, which is inconsistent with the estimation of C_max based on the hard-bed theory. I do not see why μ and C_max should have different priors, given that they play the same role in the friction law. Doing so favours one sliding law based on unjustified choices.
• The title is a too strong statement compared to what you are actually able to infer. Furthermore you focus only on Pine Island glacier, not all Antarctica. I would propose instead: “Evidence of stress bounded friction law at Pine Island Glacier (Antarctica) inferred from seismic observations.”

Specific comments

You will find a list of corrections and specific comments embedded in the annotated PDF in attachment. Some are redundant with my general comments but may help to clarify them.