The paper applies the formulation developed by TK 21 (Torquato and Kim, Physical Review X 2021) to scattering by fresh snow. The merit of the non-local formulation is to extend the previous quasistatic model (Torquato 2002) to higher frequency so that the attenuation due to scattering is accounted for.  The attenuation in scattering is included in the imaginary part of the effective permittivity.  The TK21 method is supposedly valid up to ka =6 where k is the wavenumber and a is grain radius. The value ka=6 means the gran radius is 1 wavelength

There are questions whether the TK21model is applicable to snow

a.    A microscopic picture of snow shows that the ice grans have irregular shapes and that they also have “stickiness” that the grains adhere together.   A limitation of classical mixture formula as given in the book by Sihvola (1999) is that the mixture formula depends on the shape of the scatterers. The mixture formula is developed for simple shapes such as spheres, ellipsoids, disks etc. The problem with irregular shape is that the solutions of Maxwell equations are “discontinuous “across the boundaries between the scatterers and the background. Boundaries conditions indicate that although the tangential component of electric field is continuous, but the normal component electric field is discontinuous with the normal component in air that is 3.2 times that of in ice.  For well-defined shapes such as ellipsoids, such boundary value problems are solved by ellipsoid coordinates. But such problems cannot be solved analytically for irregular shapes The TK21 model is strongly dependent on the choice of exclusion volume which is analogous to particle shape.  The examples in the paper TK21 are limited to regular shapes such as spheres, disks etc.  In TK21, the beta pq in equation (33) with d=3 indicates the shape of a sphere. 

b.    In computational electromagnetics, such boundary value problems of irregular shape/boundary have been handled by more accurate numerical techniques such as edge elements in vectorial finite element method, RWG basis function in the method of moment, and Nystrom method for volume integral equations. The popular FDTD method which is used in TK21 is not accurate This is because FDTD uses rectangular grids in the discretization. It is unclear that the formulation of TK21 can handle the irregular shape of ice grains to correctly obey the boundary conditions on the surface of a scatterer. 

c.    For TK21, the methodology is based on point geometries and correlation functions associated with point geometries.  In principle, the point geometry has correlation functions of infinite order which makes the solution “exact”. But in practice to infinite order is not possible.  Only the second order correlation function is used which means that the solutions have inaccuracies.

d.    For ka extended beyond 0.5, there is incoherent field that contributes to radiative transfer equation. In addition to the attenuation due to scattering, there also is the phase matrix.  This part of phase matrix has not been treated in TK21. Recently, the cross-polarization of the phase matrix at C band, X band and Ku band have drawn significant interests in microwave remote sensing of snow. 

e.    In Mie scattering, there are two series with two sets of coefficients: one is “electric” and the other is “magnetic”. For ka<<1, the electric series dominate. However, when ka gets larger such as in TK21, the magnetic series contribute.  It is unclear that TK21 include the magnetic series if the model is applied to dense Mie scattering

Although I have the above reservations about the applicability of TK21 to snow, I do think it is worthwhile for this paper to compute the results of attenuation of TK21 in snow and compare with other models.

 I recommend the following revisions of this paper. 

1, The authors should discuss the 5-bullet points a, b, c, d, and e that are raised above.

2. In figure 1, the frequency dependence power law should be extracted and tabulated. For Rayleigh scattering, it is frequency to the 4th power.  The power law dependence makes the model comparisons easier to digest and remember.

3. The Mie-DMRT in figure 1 may not be correct. This is because when ka exceeds1, more terms in both the electric series and the magnetic series should be included. It is unlikely that Mie-DMRT go off like that as in figure 1.  The Mie-DMRT has weaker frequency dependence than the power law of 4. I suggest that the author delete the Mie-DMRT unless their results are correct. 

 

4. The results in figure 2 should be done for larger grain size.  At least, a new figure with a=0.4 mm should be added at 19GHz. The use a=0.2mm is too small.  The scattering coefficient is only a fraction per meter which is too small for real life problem. The measured volume radar backscattering at 19 Ghz is much larger for a snow depth of 1 meter. 

5. In equation (1), the summation over index n is up to infinity. However, in practice only second order, n=2, is used. The results in this paper are based on A2.  The expression for A2 should be explicitly given so that readers can write a computer code for A2 readily

This is a welcome short communication that addresses an important issue. The authors proposed to apply a new method for non-local strong contrast expansion, developed by Torquato and Kim (2021) to compute wave propagation in a two-phase media, to the full range of snow density and microstructure found throughout cryspheric regions. The intention is to help to address problems of grain size, shape, and snow density in electromagnetic theory, particularly related to the computation of the microwave scattering coefficient. The paper is well-written and concise. There are several corrections recommended for clarity, and a suggestion for expansion of the discussion. Several other minor corrections are recommended, listed sequentially.

General comments

• Can the authors comment on the general applicability for active microwave modelling? Or is this applicable only to passive microwave solutions?
• We have often observed enhanced scattering where substantial depth hoar (DH) is present.  Does the work of TK12/SCE support/explain this behaviour wrt DH given the distinct microstructure? Given the strong scattering from DH, it would be nice to see some comments on suitability for DH. The authors state in the abstract that this method should be applicable for coarse-grained snow and again in the conclusion but specific reference to DH would be welcome.

Specific comments

• on Line 22, the authors state that existing theories "prevent consistent modelling of snow". Can the authors clarify what they mean by consistent?
• Line 21-24. Related to the above point, perhaps the authors can cite specific studies that
• In the study by TK21, they do not refer to  Strong Contrast Expansion (SCE). While the authors' descriptor is useful, SCE in our community generally refers to snow cover extent which might be confusing for some readers. By way of a suggestion, perhaps the authors could use an alternative descriptor?
• Line 35. In the paper (TK21), they did not specify a snow medium but a 2-phase medium more generally. The authors make it seem as though TK21 had derived it explicitly for snow. Perhaps re-phrase “….which can be used to express the dielectric polarizability….”
• Equation 1, and Line 47 –  As is not defined.  It is sort of defined on Lines 51-52 but still not very clear. Please can the authors define this term as it is in equation 1.
• Line 41 - Is this the same as the scaled SCE in Figure 1? If so, be sure to use consistent nomenclature. If not, then can the authors explain the difference more clearly?
• Line 46 – it is not clear what the italicized symbol is in the in-text definition of Ke. This should be defined.
• Line 57 - what, specifically, would be needed in terms of ‘increasingly more detailed information of microstructure?’ What does this mean? The authors should provide the reader with more clarity since field experimentalists will be interested.
• Lines 56 – 59 – What is the benefit of n > 2 should it be a feasible computation? Or, in other words, what are we missing out on by using n = 2? What are the practical implications?
• Figure 1 and lines 70 – 71, Line 80 (and discussion throughout paper) – RT08 does not appear in Figure 1. Do the authors mean SCE R08? Similarly, TK21 in Figure 1. Do you mean Scaled SymSCE T21?  The authors should check that the text matches the figure descriptors, here, and throughout the paper.  Otherwise, it is confusing.
• Line 78 – what is the FDTD method ? Please expand this acronym.
• Line 80 – The authors imply that there is some spread between Scaled SymSCE T21 and Mie DMRT and IBA for koa>~1. What are the implications, or is this unimportant?
• Line 122 – the authors write “Let choose a linear combination…”  This should be “Let us choose…”
• Line 125 – you introduce SymSCE. Is this the same as Scaled SymSCE T21 shown in Figure 1?  This gets a little confusing.
• Line 149 – 151 - Is there work going forward on this? There must be some existing suite of in situ measurements that could satisfy the requirements, no?
• Line 155 – the authors call for more precise in-situ microwave and snow measurements than what is available at present.  What do you mean by more precise?  Please be more specific here – what measurements, exactly, are needed?  How does 'SCE' relate to physical field measurements?  What field measurements are needed?