We thank the referees for very helpful comments that we believe will significantly improve the manuscript. We respond to each review separately below.

Response to Review #1: We really appreciate this reviewer’s remarks concerning the high level of mathematics and the need to incorporate descriptions of the main results and methods that will be accessible to a broader audience. Adding such descriptions will make the paper more readable, particularly to the intended audience of researchers in the physics of the cryosphere and climate science more broadly. Indeed, in early applications and extensions of the analytic continuation method to sea ice and composites, which rely on the Bergman-Milton integral representation, reviewers for journals in geophysics and materials science made similar requests, which resulted in significantly improving the readability of the paper, and making it accessible to a much broader range of readers. Also, we are more than happy to include the references listed in the review, and are particularly interested to have a close look at the recent papers listed from 2020 and 2022.

Response to Review #2: We also thank reviewer #2 for several incisive comments and for targeting the key mathematical issue at the heart of the analytic continuation approach in any context, which is how the local geometry of the composite or velocity field is encoded into the spectral measure. In particular, this reviewer focuses on the fascinating relationship between connectivity of one of the components, say the brine inclusions in sea ice, and the properties of the spectral measure.

First, to address the noise question concerning Figure 3, we have not computed a measure of noise in this case, although this could be easily done. The noise level is more significant than need be due to resolution issues in the brine images used, where averaging over an ensemble generated through offset sampling can only be done over very small sample sizes. We will likely choose a different sequence of images with much higher resolution that will allow more averaging and less variance.

Concerning the formation of a Dirac delta component at the origin in the spectral variable as the percolation threshold is approached, numerically this is a rather subtle process which has been discussed in some detail in our 2015 Comm. Math. Sci. and 2017 Phys. Rev. Lett. papers. In a 1996 J. Phys.: Cond. Matt. paper by Day and Thorpe they also consider this issue and provide some analytical as well as numerical results that are echoed in our computations of the spectral measure. In the revision we will discuss these very interesting issues in more detail and explain more fully what is accounting for the behavior near the spectral endpoints, and address the excellent points brought up by the reviewer. To help illustrate in our spectral measure calculations the different roles played by the eigenvalues (locations) and the eigenvectors (weights), in the revision we will also display the eigenvalue densities on the spectral interval as the percolation threshold is approached. We believe this will help further explain the spectral measure behavior at the endpoints and address the reviewer’s remarks in this regard.    

In a separate paper currently in preparation we have explored in a more general graph-theoretic setting precisely the deep question raised by the reviewer – how does the spectral measure “signal” when a connection is formed. We will briefly mention our results in this general framework in the revised version of the paper in response to the excellent points brought up by the reviewer.

We agree that section 6 may appear to be somewhat disconnected from the rest of the manuscript. We believe, though, that we can easily revise the manuscript there to address the questions raised by the reviewer, which will significantly improve its readability. In fact, we will be including an alternative form for the Stieltjes representation of the effective diffusivity that puts it on a much more equal footing with the classic two component case.

Finally, the suggestion to add a paragraph or two on how the ACM results can contribute to climate models and to mention known limitations of the method in sea ice modeling, is an excellent one. We will certainly be adding a paragraph or two addressing these points in the revision.