Grain shape is commonly understood as a morphological characteristic of snow that is independent
of the optical diameter (or specific surface area) influencing its physical properties. In this study we use tomography
images to investigate two objectively defined metrics of grain shape
that naturally extend the characterization of snow in terms of the
optical diameter. One is the curvature length

Linking physical properties of snow to the microstructure always
requires the identification of appropriate metrics of grain size. In this
regard the two-point correlation function has become a key quantity
for the prediction of various properties such as thermal conductivity,
permeability and electromagnetic properties of snow

A common practical way to characterize the two-point correlation function is a
fit to an exponential, such that the fit parameter, the so-called
exponential correlation length

The exponential correlation length is often inferred from measurements
of the optical equivalent diameter

A similar issue of grain shape emerges in the context of optical
measurements. Optical properties (e.g., reflectance) can be largely
predicted from the optical diameter or SSA

The two examples from microwave or optical modeling above reflect the
known fact that the optical diameter as a single metric of grain size
is not sufficient to characterize the microstructure for many physical
properties. It is thus necessary to account for additional grain size
metrics which implement the idea of grain shape. A key requirement for
potential new shape metrics is a well-defined geometrical
meaning. Present snowpack models

One appealing route to define shape is via curvatures of the ice–air
interface because curvatures (i) have already been used to comprehend
snow metamorphism via mean and Gaussian curvatures

Another appealing route to define shape is via chord length
distributions because they (i) naturally implement the idea of size
dispersity and (ii) have recently been put forward by

The motivation of the present paper is to investigate and interconnect
these two routes of (objectively) defining grain shape. First, we will
assess the curvature length in the expansion of the two-point correlation
function. We will be guided by the questions of whether, and how, the well-known
statistical relation Eq. (

The paper is organized as follows. In Sect.

The interaction of microwaves with snow is commonly interpreted as
scattering at permittivity fluctuations in the microstructure, which
can be described by the two-point correlation function

The value of the two-point correlation function

For small arguments

For a two-phase material with a smooth interface, the second-order
term

In snow optics the microstructural characterization within radiative
transfer theory

In contrast to the Born approximation for microwaves, where the
microstructure enters as the Fourier transform of the two-point correlation
function, the theoretical approach

For known chord length distribution, all optical quantities (phase
function, single scattering albedo, etc.) can be directly
computed from

Following the previous two sections, a link between optical and
microwave metrics of snow thus requires to establish a link between two-point
correlation functions and chord length distributions. To this end we
employ a relation between the two-point correlation function and chord length
distribution that was put forward in the early stages of small angle
scattering

Although Eq. (

For the following analysis we used an existing

Obtaining the normalized two-point correlation function

Since the snow samples in the data set are anisotropic

From the normalized two-point correlation function two types of parameter
fittings are performed. First, the exponential correlation length

To confirm the geometrical interpretation of

The measured total surface area is obtained by integrating (summing)
the surface area of the triangles over the surface and the estimate

Comparison between smoothing parameter

It is illustrative to note that even without smoothing for

An estimate for the curvature-length

Overall, the comparison provides reasonable confidence that the
geometrical interpretation of the two-point correlation function parameters is
correct, though uncertainties inherent to the smoothing operations
must be acknowledged. In the following we solely use the quantities
derived from the two-point correlation function,
viz.

To compute the ice chord length distribution from the binary images,

The main part of the following analysis comprises statistical
relations between the length scales derived from the chord length
distribution and the two-point correlation function in Sect.

As a starting point for the statistical analysis we revisit the
empirical relation

Here the adjusted correlation coefficient, accounting for the
inclusion of extra parameters, is

Scatterplots of

In the next step we include the curvature-length

To relate the chord length metrics to the Porod length and the curvature length, we first assessed the relation
between the chord length distribution

Comparison of the chord length distributions computed from Eq. (

Using the previous results we can derive an approximate relation
between the second moment of the chord length distribution and the
interfacial curvatures. To motivate a statistical model, we start from
Eq. (

Scatterplots of

In view of the inclusion of the curvature-length

We have heuristically found a possibility of improving
Eq. (

In the previous sections we found a statistical relation between the
exponential correlation length

Scatterplot of the exponential correlation length

The summary of all models is given in Table

Summary of statistical models.

As an application of the values obtained for the moments of the chord
length distribution we can now compute the “shape diagram” of the
optical parameters

Determination of the absorption coefficient

Scatterplot of (one minus) the asymmetry factor

The values in Fig.

Before turning to the discussion of physical implications of the
results, we first address methodological details. Retrieving
parameters from

The present analysis and cross validation of the curvature metric
imposes requirements on the smoothness of the interface. The subtle
influence of the smoothing parameter on the surface area

Resolution plays an important role in obtaining estimates for

The image resolution plays another important role in the
interpretation of the expansion of the two-point correlation function. As
pointed out by

The present data set was previously used to study the anisotropic properties of snow

Overall, we are confident that the method can be applied to arbitrary anisotropic samples to provide orientationally averaged length scales with the correct geometric interpretation with acceptable uncertainties due to image resolution.

Accepting the methodological uncertainties, we shall now discuss our findings of the statistical analysis and their relevance for the interpretation of snow microstructure.

By construction, the exponential correlation length

To discuss the statistical relations we will start with
recovering Mätzler's model

In addition we established a new statistical relation
Eq. (

All proposed statistical models show an improvement to Eq. (

This seems surprising at first sight. Why should local aspects of the
interface (

Overall, we conclude that both,

Hitherto no geometrical interpretation for the second moment

The relation Eq. (

To test the range of validity of the relation (

Overall, our analysis confirms that both approaches to microstructure
characterization, via two-point correlation functions (with metrics

The international classification for seasonal snow on the ground

Local curvatures are often regarded as shape parameters and used to
characterize snow on a more fundamental level. The relevance of the
mean curvature is described and analyzed in detail in

It is therefore natural to use objective measures such as the mean and
Gaussian curvature

Overall, we suggest that both parameters,

Thus far, the exponential correlation length

This equivalence of shape and size dispersity at the level of
two-point correlation functions can be further illustrated by an interesting
example. Consider a microstructure of polydisperse spherical
particles. The definition of grain shape from the classification

Finally, we turn to the implications of size dispersity or grain shape
on geometrical optics within the scope of

As pointed out by

Using the chord length distributions we were able to calculate the
shape factors

The predicted values for

Overall, our analysis indicates a smaller variation of optical
properties with shape via

We have analyzed different microstructural length scales
(

For the two-point correlation function, the length scale

We have argued that size dispersity is one possible route towards an
objective definition of grain shape, and thus both quantities
(

We have also used this interpretation of shape to assess the so-called
optical shape factor

Overall, defining grain shape via dispersity measures

The length-scale data required to reproduce all statistical models with corresponding figures (2, 3, 5, 6, 7) can be found in the Supplement. The underlying 3-D image data can be obtained through the corresponding author.

To derive an expression of the optical shape factor

To obtain an expression for

To explicitly reveal the correction of

The authors thank G. Picard for a constructive feedback on an
earlier version of the manuscript and S. Torquato for helpful
clarifications on the factor