Introduction
Microwave modeling of snow is commonly addressed within multilayer
approaches to account for the vertical layer structure of a
snowpack. Examples are HUT , MEMLS , DMRT-ML , and
DMRT-QMS . A similarity of the models is their
common approach of applying a one-dimensional radiative transfer
scheme to a layered medium. Each layer is assumed to be a
statistically homogeneous chunk of snow, in which electromagnetic
properties, such as effective permittivity, scattering and absorption
coefficient and phase function, are determined by the
microstructure. However, the models differ significantly in both
the calculation of the solution of the radiative transfer
equation and the relation between snow microstructure and the
electromagnetic properties. In-depth comparisons of
microwave models are hampered by these fundamental differences which
remain a source of uncertainty . The differences
in the representation of snow microstructure led to an open discussion
about the appropriate choice of structural metrics for microwave
modeling . Given the crucial role of a grain
size for scattering in snow, the difficulties in comparing different
metrics of grain size strongly hinder necessary developments to
improve current retrieval schemes .
Electromagnetic properties of single snow layers can be theoretically
obtained by homogenization methods, if snow is treated as a random,
two-phase medium which is statistically homogeneous. The microwave emission model
of layered snowpacks (MEMLS) is based on the improved Born approximation (IBA) which
expresses the scattering coefficient in terms of the Fourier transform
of the two-point correlation function. The characterization of the
structure in terms of the two-point correlation function C(r) is
appealing for random materials, since it naturally emerges in rigorous
approaches to effective material properties . The
connection between C(r) and the scattering coefficient dates back to
the seminal work by , which considered general aspects of
scattering in random materials. From that perspective, the
microstructure representation in IBA appears to be
generic. Practically, however, entire correlation functions can hardly
be used as “parameters” and further simplifications are commonly
required. In MEMLS, IBA is evaluated by assuming an exponential
functional form for the correlation function . As an
advantage of this simplification, the microstructure model reduces to
a single parameter, the exponential correlation length. This parameter
can be conveniently fitted from micro-computed tomography (μCT)
data . Though an exponential correlation function is
a reasonable first guess, in particular for depth hoar
, the validity of a single length scale approach
for snow must be generally questioned . In addition,
a rapid retrieval of the exponential correlation length from field
measurements is still difficult; only one method was hitherto put
forward by . It is not even clear that three-dimensional
microstructures with an exponential correlation function actually
exist, and how realizations of such a medium can be generated
. Generating realizations of a microstructure is
however mandatory to compute the scattering from numerical solutions
of Maxwell's equations in order to test the IBA assumptions.
Advantages and disadvantages of IBA have to be discussed at eye level
of those inherent to another electromagnetic approach to scattering in
snow, namely dense media radiative transfer (DMRT). Originally DMRT
was developed for random media consisting of spheres or
spheroids. Sphere models are attractive from various
perspectives. First, the scattering coefficient of sphere assemblies
can be calculated analytically in various approximation schemes, such
as the quasi-crystalline approximation (QCA), which can be optionally
improved by the so-called coherent potential (QCA-CP)
Ch. 5. Second, microstructure realizations of
spheres can be readily generated to compute the numerical solutions of
Maxwell's equations . Third, sphere models have
successfully been used for optical properties of snow for a long time
. An equivalent sphere can always be defined from
the specific surface area (SSA) via the optical equivalent diameter.
In addition, the SSA can be rapidly measured in the field by various
techniques .
However, even for a hypothetical material consisting
of perfect spheres, the sphere diameter and the volume fraction do not
characterize the medium completely. Relevant for the scattering of
waves are the relative positions of the spheres which determine the
relative phases in the superposition of the scattered waves from
individual scatterers. Of special importance for snow is the sticky
hard sphere (SHS) model due to its ability to change relative particle
positions while leaving diameter and volume fraction constant. The concept of SHS
was introduced in the context of molecular fluids
and consists of spherical particles interacting
via hard-core repulsion and an attractive surface adhesion. In the
thermodynamic framework, this competition of attractive and repulsive
forces gives rise to a minimal model for a liquid-gas phase
transition. For other applications it is appealing to use the
thermodynamic equilibrium states for the particle positions as a means
of generating microstructures with interesting structural properties;
the contact adhesion, which is inversely proportional to the so-called
stickiness parameter τ , gives rise to
clustering of the spheres. The relevance of clustering can be directly
observed for some snow types, e.g., clustered rounded grains
. For other snow types the strength of the adhesion
must be regarded as a parametric approach to subsume effects of
sintering which causes ice crystals to be sticky. The relevance of
sticking and the implication on structural properties for new snow has
been demonstrated by . The potential of stickiness
for snow microwave modeling to account for relevant length scales
beyond a “grain size” was initiated for DMRT by .
Objective means of estimating the stickiness parameter for a given snow
sample are hitherto missing. However, simply resorting to the
non-sticky case causes other difficulties. Recent DMRT-based microwave
emission modeling and
comparison with measurements indicate that the measured optical
equivalent diameter can only be used directly as input, if either a
“grain-size scaling” (with a factor ranging from 1 to 3) is
introduced in the non-sticky case or if the stickiness is used as a
free, unknown parameter . This issue could not be
further investigated as long as the stickiness parameter τ remains
inaccessible from field measurements. In the absence of objective
parameter estimates, a single value is commonly used for all snow
types in the simulations , although a
large sensitivity of scattering properties on τ was
acknowledged. Due to the lack of interpretation of stickiness for
snow, attempts to intercompare DMRT-ML with MEMLS such as
remain challenging, since different microstructural
models are in fact used in the respective microwave models.
It is the aim of the present paper to advance the understanding of
microstructural models in different electromagnetic models for
microwave modeling of snow by establishing a rigorous link between the
scattering formulations used in DMRT-ML and MEMLS. More precisely, for
the DMRT theory we consider the QCA-CP approximation as used in
DMRT-ML . For IBA, we follow the derivation based
on the internal field factor for spherical scatterers
which slightly differs from the default
implementation in MEMLS that is based on an empirical factor
. In addition, for both theories we restrict
ourselves to small sizes of scatterers relative to the wavelength, which
is referred to as a low frequency assumption in the following. By
restating the scattering coefficient in both electromagnetic theories
and applying established theoretical results for correlation functions
in random media, we obtain a rigorous relation between IBA and QCA-CP
for arbitrary hard sphere models. On the one hand, this yields an
objective method of estimating the stickiness parameter τ of the
SHS model from μCT images of snow. On the other hand, the
procedure allows exactly the same microstructural model to be used,
namely SHS, in IBA and QCA-CP, and compares the scattering coefficient
quantitatively using realistic values for the parameters.
The paper is organized as follows. In Sect. we rederive
the scattering coefficient of IBA and QCA-CP to contrast their
representation of snow microstructure in terms of different
correlation functions. In Sect. we give an explicit
expression for the two-point correlation function for SHS which allows
the IBA theory to be expressed in terms of the microstructure
representation traditionally used by DMRT. In addition, an objective
retrieval method of the SHS parameters (diameter and stickiness) from
measurements of the two-point correlation function of snow is
presented. In Sect. we use existing μCT images to
derive the SHS parameters and explore their behavior for different
snow types. We discuss our findings in Sect. in view of
the relevance of the results for the scattering coefficient in their
host models, MEMLS and DMRT-ML. As an application of our work besides
the microwave context, we also discuss implications of the present
results in view of the discrete element modeling (DEM) of snow for
mechanical applications . Presently, DEM also lacks
efficient means of objectively reconstructing snow samples in terms of
particle models. The reconstruction of snow microstructure in terms of
SHS provides a new link to such a granular viewpoint.
Scattering coefficient and relation to microstructure
Two-phase media and two-point correlation functions
Random two-phase media are a natural starting point to characterize
the morphology of air and ice in snow. We consider a two-phase medium
in a region Ω where phases 1 and 2 occupy the subregions
Ω1 ⊂ V with volume V1 and Ω2 ⊂ V with
volume V2, respectively. Both subvolumes add up to the total volume
V1 + V2 = V. We assume phase 1 to be the void phase (i.e., air) and phase 2
the inclusion phase (i.e., ice).
Following , a single realization of the
microstructure can be fully described by the phase indicator function
of either phase j = 1, 2:
ϕj(x)=1ifx∈Ωj0otherwise,
which provides the complete information if a point x ∈ Ω
is covered by ice or air. A binary image obtained by
processing μCT data is actually a discrete form of indicator
function. From a theoretical point of view, individual realizations
are not of particular interest for random media. It is rather the
statistical properties which emerge in the analytical derivation of
effective physical properties. Volume fractions ϕj of either
phase j = 1, 2 are the simplest statistical properties which are first-order quantities, i.e., they just contain the first moment of the indicator function
ϕj(x)‾=1V∫Vdrϕj(x)=VjV=ϕj,
where volume averaging is denoted by •‾. Snow
density is directly related to ϕj. In contrast, higher-order
moments of the indicator function, i.e., averages of certain products
of ϕj(x) for j = 1, 2, characterize spatial
fluctuations of the phases. The simplest moments are the two-point
correlation functions
Sj(r)=ϕj(x+r)ϕj(x)‾
of either phase. A closely related, second-order quantity is the phase covariance
C(r)=S1(r)-ϕ12=S2(r)-ϕ22,
which is symmetric under the exchange of ice and air. Equations ()
and () provide comparable information about a random field
that the second moment and the variance provide for a random
variable. In practice, second-order moments contain information about
the size of heterogeneities at the microscopic scale which can be used
to define different “grain sizes” for a given microstructure.
Though snow is known to be anisotropic , the IBA and
QCA-CP theory have only considered isotropic correlation functions and
randomly oriented snow particles up to now. In the following we thus
focus on isotropic media, where C(r) = C(r) with r = |r|.
Scattering coefficient in MEMLS: improved Born approximation
Scattering coefficient and phase function
Below we state the governing equations for the scattering coefficient
in the improved Born approximation . We slightly
adapt the notation and terminology to establish the connection to DMRT.
Within the improved Born approximation, the scattering coefficient is
derived from the phase function (or bistatic scattering function)
cf. Eq. 25 which is given by
γ(k^s,k^i)=ϕ21-ϕ2ε2-ε12K2k04sin2(χ)I|kd|,
where the dielectric constants of the two phases are denoted by
ε1 and ε2. The vectors k^i, k^s are the propagation directions of the
incident and scattered waves, respectively. The angle between the
incident electric field and the scattering direction k^s
is denoted by χ. The wavevector difference of incoming and scattered waves is denoted by
kd = keff(k^s - k^i) with magnitude
kd=2keffsin(Θ/2);
hence the angle Θ denotes the scattering angle, i.e., the angle
between k^s and k^i. In
Eq. (), keff is the effective propagation
constant of the wave in the medium which is related to the effective
dielectric constant εeff by
keff=k0εeff1/2,
where k0 is the vacuum wavenumber. The remaining quantities in
Eq. () to be specified are K and I. K denotes the mean
squared magnitude ratio of incident and internal field in the ice
phase. Various formulations for K are given by
. To make contact with hard spheres later, we focus
on spherical heterogeneities, for which K is given by
K2=2εeff,0IBA+ε12εeff,0IBA+ε22.
Here εeff,0IBA is an approximation
to the effective dielectric constant. It represents the dielectric
constant of the effective medium in the absence of scattering in the
very low frequency limit. It is assumed to be given by the Polder–van
Santen mixing formula
εeff,0IBA=2ε1-ε2+3ϕ2ε2-ε14+2ε1-ε2+3ϕ2ε2-ε12+8ε1ε24.
The most relevant quantity in Eq. () for the purpose of this paper
is I(|kd|) which contains the entire information about
the microstructure entering the scattering coefficient in IBA. From the definition of I(|kd|)
in , we can rewrite it in terms of the Fourier
transform of the correlation function Eq. () according to
I|kd|=14πC̃(|kd|)ϕ21-ϕ2.
Here and throughout we use the shorthand notation f̃ to
indicate the three-dimensional Fourier transform of a function f
which is defined by
f̃(k)=∫R3dxf(x)exp(-ix⋅k),
which implies the inverse transform
f(x)=(2π)-3∫R3dkf̃(k)exp(ix⋅k).
We close this section by commenting on the ambiguous notion of
“particles” in the original derivation of IBA. The field ratio
matrix K Eq. 7 does not follow on
from the theory but needs to be given a priori. As explained by
, it depends on snow density and grain shape,
stating “the shape dependence is relatively weak; therefore, the real
situation can be well-modeled with idealized particles.” It is
however questionable that the signature of local shape (K) and the
correlation function C(x) can be chosen independently,
as suggested in IBA, since both quantities are related.
A potential impact of shape is however irrelevant for the present
isotropic considerations, where we choose K corresponding to
spherical particles Eq. 28 to be consistent
with the DMRT description.
Scattering coefficient in the low frequency limit
The scattering coefficient κsIBA for IBA
in is obtained by integrating the phase function
Eq. () over the scattering directions k^s, viz
κsIBA=14π∫4πdΩsγ(k^s,k^i),
where dΩs is the solid angle element in the scattering
direction k^s. In general, not only sin2χ
but also I(|kd|) depends on the scattering angle in
Eq. () which requires numerical integration of
Eq. () as done in MEMLS. In the following, we focus on the
low frequency limit which allows I(|kd|) to be replaced by
I(0) in Eq. () as also done by . It
follows that the only dependence of the bistatic scattering
coefficient Eq. () on k^s remains in the
term sin2χ. Without loss of generality, choosing a coordinate
system where the z axis is aligned with the local field, the average
Eq. () of sinχ yields 2/3. In summary, the scattering
coefficient is
κsIBA=23k04ε2-ε12εeff,0IBA+ε12εeff,0IBA+ε22C̃(0)4π,
where we used Eq. () to obtain an expression in terms of the
correlation function.
Particle-based media and pair correlation functions
The original versions of DMRT assume that the microstructure comprises N
discrete particles of a given shape and size in a container of
volume V. A particular realization of the particle configuration can
be specified in terms of the particle centers
{ri}i=1…N. In contrast to IBA, which
describes the microstructure in terms of the phase indicator function
containing implicit information about shapes and sizes, DMRT requires
explicit information about particle shape and size of a particle at
locations {ri}. Often a fixed shape is chosen
(e.g., sphere) and the size is either left constant (referred to as
monodisperse) or drawn from a probability distribution (referred to as
polydisperse). Thus for a given shape and size, a particular realization
of the microstructure can be described by the number density field
n(x)=∑i=1Nδx-ri.
The first moment is the mean number density
n(x)‾=1V∫Vdxn(x)=NV=n.
From the number density field we define a second-order correlation
function by
c(r)=(n(x)-n)(n(x+r)-n)‾,
which characterizes local fluctuation of the number density. The well-known pair correlation function g(r) of a particle
assembly is then essentially a modified version of Eq. (), viz
c(r)=n2[g(r)-1]+nδ(r).
The structure factor S(k) is the Fourier transform of
the pair correlation function and given by
S(k)=1+n∫Vdr[g(r)-1]exp(-ir⋅k).
Scattering coefficient in DMRT-ML: quasi-crystalline approximation – coherent potential and low frequency limit
Several flavors of DMRT have been developed over the years
. Here we consider
dense packings of spheres in the low frequency limit, in accordance
with the approximations made in the derivation of IBA. The
response of a sphere on an electric field is well-known. To obtain the
scattering coefficient the main task is to estimate the collective
excitation of all spheres upon an incident plane wave. This can be
stated in terms of an integral equation Ch. 5
which allows the effective dielectric constant εeff of the medium to be computed. In the quasi-crystalline
approximation (QCA) with coherent potential (CP), the result for the
effective dielectric constant is given by
εeff=ε1+3εeffϕ2Λεeff1+i23εeff3/2a3ΛεeffS(0).
Equation () is a nonlinear equation for the complex-valued effective dielectric constant
εeff. It involves the structure factor S(k) from
Eq. () in the low frequency limit k → 0, the sphere radius a and the auxiliary function
Λεeff=ε2-ε13εeff+ε2-ε11-ϕ2.
Different strategies are possible to solve Eq. (). The
simplest strategy is to replace εeff on the right-hand side by
ε1. This approximation is known as QCA (without
coherent potential) Eq. 5.3.113a. The solution
strategy followed by is instead iterative. First,
Eq. () is solved in the non-scattering limit, i.e., for
a = 0. This yields a quadratic equation,
cf. Eq. 5.3.125 or Eq. (5).
We denote the solution of the equation by
εeff,0QCA-CP which is given by
εeff,0QCA-CP=ε1-ε2-ε131-4ϕ22+ε1-ε2-ε131-4ϕ22+4ε1ε2-ε131-ϕ22.
The extinction coefficient can be derived from Eq. () via
κe = k0I(εeff1/2). Following
Eqs. 46 and 90, the scattering coefficient is
given by
κsQCA-CP=29k04a3ϕ23εeff,0QCA-CPε2-ε13εeff,0QCA-CP+ε2-ε11-ϕ22S(0).
The expression Eq. () for the scattering coefficient is
generic since it involves the microstructure in terms of the structure
factor at low frequency S(0). This can be specified to arbitrary
particle systems as long as the structure factor S(0) can be computed.
Link between IBA and QCA-CP
The statistical characterization of particle systems by the number
density field Eq. () has a long tradition in the statistical
physics of liquids. A comparison with Sect. reveals that
particle-based media and two-phase media are described by similar,
though slightly different formalisms: Both are microscopically defined
by a microscopic quantity characterizing the local density – local
number density Eq. () vs. local volume fraction Eq. () – and
moments of the microscopic quantities (Eqs. , vs. Eqs. , ) for
the mean and fluctuations of the microscopic quantities. Differences
emerge due to the mathematical properties of the microscopic
quantities. While in the two-phase case, products
ϕj(r)2 of the indicator function with itself are
well-defined, products of the form δ(r)2 are
mathematically meaningless. This requires some caution when computing moments.
The interrelation of the statistical description of two-phase media
and particle-based media in terms of correlation functions was
detailed by . To relate the scattering
coefficient from IBA and QCA-CP, Eqs. () and ()
respectively, we note that one is determined by
the low frequency limit of the structure factor S(0), while the other
is determined by the low frequency limit C(0) of the Fourier
transform of the correlation function. While the former is only
defined for particle systems, the latter can be specified to arbitrary
two-phase media. Thus for particle-based media, both correlation
functions can be defined. The pair correlation function of the
particle centers and the two-point correlation function are not
independent, since knowledge of particle positions together with the
particle shape uniquely determines the spatial region which is covered
by the particle phase (phase 2). This is exactly the information
contained in the indicator function.
Relation between g(r) and C(r)
The link between corresponding correlation functions was established
under quite general assumptions for arbitrary
particle interactions. The specification of the general result to the
case of monodisperse hard spheres can be found in
Eq. 52. In the present notation, the result reads
C(r)=nvint(r,d)+n2vint(r,d)∗[g(r)-1],
where (∗) denotes the three-dimensional convolution and
vint is the volume of the intersection set of two
identical spheres with diameter d = 2a which are separated by
r. The intersection volume Eq. 3.51 is given by
vint(r,d)=v(d)1-32|r|d+12|r|3d3H(d-|r|),
in terms of the Heaviside step function H(x) and the volume of the
sphere v(d) = πd3/6. In view of the scattering coefficient
derivation, we take the Fourier transform of Eq. () and
use the definition of the structure factor Eq. () to obtain
C̃(k)=nṽint(k,d)S(k).
The Fourier transform of the intersection volume Eq. () can be
readily computed and written in the scaling form
ṽint(k,d)=v(d)2P(kd)
with
P(kd)=3(sin(kd/2)-kd/2cos(kd/2))(kd/2)32
and k = |k|. The function P(kd) is known as the
spherical form factor Eq. 55. The relation
between Fourier transforms of the phase covariance and correlation
function of the particle assembly, given by Eq. (),
constitutes the key result to compare the scattering coefficient in
IBA and QCA-CP. In addition, this relation would allow any particle-based model to be implemented in IBA, as long as the structure factor S(0)
and field ratio K are known. The relation Eq. () is
not limited to spheres; it can be generalized also to anisotropic
particles, provided the intersection volume vint can be computed.
Relation between scattering coefficient in IBA and QCA-CP
By means of the fundamental relation Eq. () and ϕ2 = nv(d)
we can now express the IBA scattering coefficient Eq. () in
terms of the structure factor, yielding
κsIBA=29k04a3ϕ2ε2-ε12εeff,0IBA+ε12εeff,0IBA+ε22S(0),
which is very close to the form of the QCA-CP scattering coefficient
in Eq. (). For a comparison we define rs as
the ratio of the IBA and QCA-CP scattering coefficient which is given by
rsϕ2=κsIBAκsQCA-CP=2εeff,0IBA+ε13εeff,0QCA-CP+ε2-ε11-ϕ22εeff,0IBA+ε23εeff,0QCA-CP2.
For the given phase permittivities ε1, ε2 the ratio
is only a function of the ice volume fraction ϕ2. The dependence
on the microstructure via S(0) dropped out completely in
Eq. () since both scattering coefficients contain exactly the
same factor. The ratio rs is also independent of the
wavelength (except implicitly through the dielectric constants) and
varies by no more than ≈ 30 % for relevant volume fractions
0 < ϕ2 < 0.5 (Fig. ). This has to be contrasted to the
ratio calculated and shown in Fig. 4, in which
a strong influence of the volume fraction on rs is
apparent. The reason is the use of different microstructural models
(overlapping spheres versus SHS) in the respective scattering
models. For completeness, we also show that the real and imaginary
parts of the dielectric constants of the effective medium in the
absence of scattering of either model, Eqs. () and ()
in Fig. . The maximum relative difference between the real parts is
only 1.5 %, and between the imaginary parts it is 8.8 %.
Ratio of scattering coefficient calculated with IBA and DMRT theory
as a function of ice volume fraction ϕ2 evaluated for
ε1 = 1 and
ε2 = 3.17 + 0.0022 i.
Structure factor for sticky hard spheres
The DMRT formulation used in many DMRT-based models use the sticky
hard sphere (SHS) model to represent the position of the
scatterers. In IBA it is now possible to consider the same SHS model
since the scattering coefficient Eq. () can be specified to an
arbitrary sphere assembly as long as the structure factor S(0) is
known. To this end we give an expression of the structure factor S(k)
for SHS and its zero k limit in order to obtain an IBA-based
formulation as close as possible to that of DMRT. We consider the
monodisperse SHS model by comprising N point
particles at positions {ri}i=1…N in a volume V
interacting via a pair potential
U(|r|)=1β∞,|r|<d-lnσ12τ(σ-d),d<|r|<σ0,|r|>σ.
Here β is the inverse temperature and the potential consists of
a hard-core repulsion, which prevents particles from overlap at
distances smaller than the sphere diameter d, and a square well
attraction, which tends to a contact adhesion force in the limit
σ → d. The strength of the adhesion is proportional to the
inverse of the stickiness parameter τ. In the Percus–Yevick
approximation, the structure factor of SHS can be written in closed
form Eqs. 8.4.19–8.4.22 in terms of the particle
phase volume fraction ϕ2, the stickiness parameter τ, and
sphere diameter d according to
SSHS(k)=A(X)2+B(X)2-1A(X)=ϕ21-ϕ21-tϕ23ϕ21-ϕ2Φ(X)+3-t1-ϕ2Ψ(X)+cos(X)B(X)=ϕ21-ϕ2XΦ(X)+sin(X)Φ(X)=3sin(X)X3-cos(X)X2Ψ(X)=sin(X)X,
with X = kd/2 and the parameter t, given by the smallest solution
of the quadratic equation
ϕ212t2-τ+ϕ21-ϕ2t+1+ϕ2/21-ϕ22=0
under the additional condition t < (1 + 2ϕ2)/(ϕ2(1 - ϕ2))
which guarantees SSHS(0) to be positive
. The structure factor S(k) is thus
only a function of the scaling variable kd. Using the limiting
values Ψ(0) = Φ(0) = 1 and B(0) = 0, the low frequency limit
S(0) required for the scattering coefficient can be readily computed
from Eq. () and is given by
S(0)=A(0)-2=1-ϕ221+2ϕ2-tϕ21-ϕ22.
This is in agreement with , and
The structure factor for non-sticky hard spheres can
be recovered by taking the limit τ → ∞, corresponding to
t → 0, yielding S(0) = (1 - ϕ2)4/(1 + 2ϕ2)2.
Real and imaginary parts of dielectric constant
εeff,0IBA and
εeff,0QCA-CP in the very low frequency limit
as a function of ice volume fraction ϕ2.
In summary, the Fourier transform of the correlation function of
sticky hard spheres can be written as
C̃SHSk|ϕ2,d,τ=ϕ2v(d)P(kd)SSHS(kd)
in terms of the sphere volume v(d) = πd3/6, the spherical form
factor P(kd) from Eq. () and the SHS structure factor
SSHS from Eq. (). For later purposes we made the
dependence of C̃SHS(k|ϕ2, d, τ) on the
involved parameters ϕ2, d, τ explicit.
SHS analysis of snow from μCT images
Micro-computed tomography data
For the following analysis we employ the data set of μCT images
described in which was used therein in a different
context. The entire data set comprises 167 snow samples which are
divided into several series of data: two time series of isothermal
experiments (ISO-1, ISO-5), four time series of temperature gradient
metamorphism experiments (TGM-2, TGM-17, DH-1, DH-2) and a set of
37 uncorrelated samples (DIV) comprising various snow types. A detailed
characterization including the IACS international classification of
seasonal snow on the ground of the snow samples is
given in the supplement to . In summary, the data
sets contains 62 samples of depth hoar (DH), 54 of rounded grains (RG),
33 of faceted crystals (FC), 10 of decomposing and fragmented
precipitation particles (DF), 5 of melt forms (MF) and 3 of
precipitation particles (PP).
Data processing and Fourier transform
Computing correlation functions as convolutions of μCT images is
commonly done using fast Fourier transform (FFT) of the indicator
function Eq. (). Fourier transforms of the data are thus
naturally available.
Snow is known to be anisotropic which was explicitly analyzed for the
present data set in . To interpret the anisotropic
μCT data in terms of the isotropic SHS model, orientational
averaging of the experimental data is required. To this end we cut out
maximal cubic subsets of linear size L from the original μCT
images to obtain a wavevector spacing Δk = 2π/L which is
independent of coordinate direction in Fourier space. Sample sizes L
of the cubic images vary from 4.8 to 10.7 mm, and voxelsizes
lvox from 10 to 54 µm, respectively. An a
posteriori inspection will confirm that L is approximately an
order of magnitude larger than the optical diameter for all
samples. The angular average C̃(k) is obtained by
radially binning the three-dimensional Fourier transform and averaging over points in
concentric wavevector shells. The angular average C̃(k)
constitutes the experimental data that will be compared to the SHS
model C̃SHS(k|ϕ2, d, τ) from Eq. ().
Comparison of measured and fitted Fourier transforms of the
correlation function for one snow sample (blue curves). Dependence of sticky
hard sphere model C̃SHS(k|ϕ2, d, τ) on
stickiness τ (cyan curves) and volume fraction ϕ2 (red curves)
around a reference (black dashed line). The dependence of
C̃SHS(k|ϕ2, d, τ) on sphere diameter d
is captured by using the dimensionless variable kd and normalizing with the
sphere volume v(d).
Properties of the SHS correlation function
Before turning to the parameter estimation, we illustrate the
parametric behavior of C̃SHS(k|ϕ2, d, τ)
for sticky hard spheres. To this end we picked one snow sample of
rounded grains and show the experimental data C̃(k)
together with the best fit of the model in Fig. . We also
indicated the behavior for large k as a guide to the eye. The
Fourier transform of the correlation function must asymptotically
decay as ∼ sk-4, where the prefactor s is the surface area
per unit volume of the sample .
To further demonstrate the impact of the parameters (ϕ2, τ, d)
on the SHS model C̃SHS(k|ϕ2, d, τ), we
additionally varied the parameters around some reference values
d = 1, τ = 1 and ϕ2 = 0.15 (black dashed line in Fig. ). One
family of curves (red colors) illustrates the impact of increasing
volume fraction ϕ2 by plotting the curves
d = 1, τ = 1 and ϕ2 = [0.2, 0.25, 0.3, 0.35, 0.4]. The other family of
curves (cyan colors) illustrates the impact of increasing stickiness
τ-1 by plotting the curves
d = 1, τ = [0.5, 0.3, 0.25, 0.2, 0.15] and ϕ2 = 0.15. The dependence on
sphere diameter is fully captured by using a non-dimensionalized
scaling plot C̃(k)/v(d) vs. kd as predicted by
Eq. (). The red and cyan curves were multiplied by an
arbitrary factor to vertically translate the curves for better visibility.
SHS parameter estimation for snow
Using the closed form expression
C̃SHS(k|ϕ2, d, τ) for the correlation
function from Eq. () we are now able to objectively
estimate optimal SHS parameters d^ and τ^ for a given
snow sample by fitting the expression to the angular-averaged
experimental data C̃(k). The volume fraction ϕ2 is
prescribed by the value obtained from the μCT image. We used the
same fit interval k = [0, kmax/3] for all samples. The
number of points in this interval and the maximum wavevector
kmax = 2π/lvox however varies from sample
to sample due to variations in voxelsize lvox and sample
size L of the cubic μCT image. The regression on d, τ was
carried out using MATLAB's nonlinear, least-squares fitting tools to
minimize the sum of squared differences between model and
measurement. Thereby, we maximize the quality of fit in the low k
regime, since the scattering depends only on the value at the origin
kd = 0. First we present overview results for the optimal parameters;
further details such as goodness of fits and characteristics of the
cost function are explored in the subsequent section.
Estimated parameter pairs (d^, τ^) for all
167 samples. Experiment series are indicated by colors (TGM-17: red,
TGM-2: green, DIV: blue, DH-1: magenta, DH-2: cyan, ISO-1: black,
ISO-5: brown) and snow types by markers (PP: ⊲, DF: ⊳, RG: ∘,
FC: □, DH: ∨, MF: ⋄).
For an overview, we show the optimal parameters (d^, τ^)
for all samples in Fig. . The fit parameters are
apparently unrelated; a linear regression
τ^ = 0.15d^ + 0.10 as an attempt to predict stickiness from
the sphere diameter (black line) yields a coefficient of determination
of R2 = 0.23.
Next we plot the optimal stickiness values τ^ for all
167 samples in Fig. as a function of ice volume fraction. This
plot corresponds to the thermodynamic phase diagram of the original SHS
model which completely determines the physical behavior of the system.
It allows the distribution of pairs
(ϕ2, τ^) to be assessed further. Two additional lines are added in the
Fig. . The dashed line indicates the lower limit of
physically admissible values of τ for a given volume
fraction. The line is given by
τminϕ2=11214ϕ22-4ϕ2-12ϕ22-ϕ2-1
as an implication of the condition on t in Eq. ().
Estimated stickiness values τ^ as a function of ice volume
fraction ϕ2. Experiment series are indicated by colors (TGM-17: red,
TGM-2: green, DIV: blue, DH-1: magenta, DH-2: cyan, ISO-1: black,
ISO-5: brown) and snow types by markers (PP: ⊲, DF: ⊳, RG: ∘,
FC: □, DH: ∨, MF: ⋄).
The second (full) line in Fig. , which separates the
parameter space into a white and a gray part, corresponds to the
underlying percolation transition of the SHS model. The percolation
line is given by
τpercϕ2=11219ϕ22-2ϕ2+11-ϕ22
cf. . The relevance of both lines in view of the
present application is further detailed in the discussion.
Finally we compare the estimated SHS diameter d^ with the
optical diameter dopt. The optical diameter of a snow
sample is defined by
dopt=6SSAρice
in terms of the SSA which was obtained from the μCT image. The
results are shown in Fig. . The relation between both
diameters will be further analyzed in the context of grain-size scaling below.
Goodness of fits
To illustrate differences in the performance of the fit for the SHS
model, we show the coefficient of determination R2 for the optimal
values τ^, d^. The results are shown in Fig.
(open symbols) and illustrate that the performance of the fit differs
significantly. For the discussion below, we also fitted the μCT
data in the same range to the Fourier transform of the exponential
correlation function
C̃exp(k)=4πϕ21-ϕ22ξ31+(kξ)22
which is a single parameter form which involves the exponential correlation
length ξ. The results are also shown in Fig. (filled
symbols). For all time series of temperature gradient metamorphism
(TGM-17, TGM-2, DH-1, DH-2), an intermediate drop in R2 for both
models is observed. The worst performance of the SHS model (FC sample
from the DIV series) is the sample with the highest density; however, no
obvious trend of R2 with snow type or density was found.
Scatter plot of the SHS diameter estimate d^ and the optical
diameter dopt. Experiment series are indicated by colors
(TGM-17: red, TGM-2: green, DIV: blue, DH-1: magenta, DH-2: cyan,
ISO-1: black, ISO-5: brown) and snow types by markers (PP: ⊲,
DF: ⊳, RG: ∘, FC: □, DH: ∨, MF: ⋄). The
black solid line indicates the 1 : 1 relation.
To further investigate goodness-of-fit differences of the SHS model, we
analyzed the behavior of the cost function
J(τ,d)=∑i=0n/3C̃ki-C̃SHSki|ϕ2,d,τ21/2
for the SHS model C̃SHS(ki|ϕ2, d, τ)
from Eq. () which is minimized in the least-squares
optimization to obtain the estimates τ^, d^. The contour
plot of log[J(τ, d)] is shown in Fig. for three
different snow types. The three snow samples in Fig. were
(a) the first sample of the ISO-1 isothermal metamorphism time series
(precipitation particles, PP), (b) the last sample of the ISO-1
isothermal metamorphism time series (large rounded grains, RG) and (c) the
last sample of the DH-2 temperature gradient metamorphism time series
(depth hoar, DH). The plots indicate apparent differences in view of
location and shape of the minimum with respect to snow type. For the
PP example, the minimum is located close to the boundary of admissible
τ values (cf. Eq. ). In contrast, for the RG example
the minimum is located well in the interior of admissible τ, d
values. It is however contained in a valley almost parallel to the
τ axis, indicating some degree of degeneracy of the optimal
values. For the DH example the residuals are higher in magnitude and
the minimum is shallower compared to the other examples. To complete
the analysis of the three examples from Fig. , we finally
plot μCT data for C̃(k) together with the SHS and the
exponential model evaluated for the optimal parameters in Fig. .
Analysis of grain-size scaling
In order to assess the relevance of the grain-size scaling raised
in we further elaborate the comparison of the
optimal SHS diameter with the optical diameter from Fig. .
Fit parameters and standard errors for a linear regression between the optimal
SHS diameter and the optical diameter according to Eq. () carried out
for subsets of the data: for individual experiment series (top) and for individual
snow types (bottom).
Subset
a1
a0 (mm)
Exp. series
ISO-1
1.05 ± 0.13
-0.08 ± 0.05
ISO-5
1.04 ± 0.16
-0.11 ± 0.05
DIV
1.44 ± 0.11
-0.09±0.04
TGM-2
2.40 ± 0.04
-0.34 ± 0.01
TGM-17
1.47 ± 0.03
-0.16 ± 0.01
DH-1
2.38 ± 0.12
-0.58 ± 0.05
DH-2
2.03 ± 0.21
-0.40 ± 0.09
Snow type
DF
0.11 ± 0.37
0.14 ± 0.07
RG
1.33 ± 0.07
-0.15 ± 0.02
FC
1.25 ± 0.04
-0.06 ± 0.02
DH
1.59 ± 0.17
-0.11 ± 0.06
MF
1.77 ± 0.19
-0.33 ± 0.10
Coefficient of determination R2 for the SHS model (open
symbols). In addition, R2 for the exponential model is shown (filled
symbols). Experiment series are indicated by colors (TGM-17: red,
TGM-2: green, DIV: blue, DH-1: magenta, DH-2: cyan, ISO-1: black,
ISO-5: brown) and snow types by markers (PP: ⊲, DF: ⊳, RG: ∘,
FC: □, DH: ∨, MF: ⋄).
As a quantitative measure, we fitted the entire data in
Fig. by a linear regression
d^=a1dopt+a0,
yielding a1 = 1.50 and a0 = -0.14. If the experiment series (DIV,
TGM-2, TGM-17, ISO-3, ISO-5, DH1, DH2) are individually fitted to
Eq. () we obtain the values shown in Table . We
also fitted Eq. () to each snow type class containing more than
three samples. The results are also shown in Table .
Contour plots of the root-mean-square error surface for different snow types. Colors
show the logarithm of the sum of the squared differences between measured and
parametric SHS form C̃(k) (cf. Eq. ) as a function of
(τ, d) for different examples. (a) Precipitation
particles (PP). (b) Large rounded grains (RG). (c) Depth
hoar (DH). The optimal values (d^, τ^) for the respective
sample are shown as white circles.
Comparison of the μCT-derived C̃(k) data to the SHS
model (Eq. ) and the exponential model (Eq. ). To
discern individual curves, the RG and DH data were displaced vertically (by
factors 102, 104, respectively).
In addition, we conducted a numerical experiment to reproduce the
situation from where different, but constant
stickiness values were used. To this end we prescribed the stickiness
parameters τ = 0.13, 0.44, 1, 10, 100 in the cost function
Eq. () and conducted only a one-dimensional regression of the
SHS model, now involving only the diameter as optimization
parameter. This yields an optimal diameter d^τ for each
sample which depends on the prescribed stickiness value τ. For
each τ we obtain 167 pairs (d^τ, dopt)
which are fitted to
d^τ=b1dopt+b0.
The results of the fit for the entire data set as a function of
prescribed τ are shown in Table . The regression
parameters are identical for τ = 10 and τ = 100 which indicates
convergence to the non-sticky hard sphere model.
Comparison of the scattering coefficient
With the set of optimal parameters (d^, τ^) from
Figs. and we can compare the scattering
coefficient and evaluate the differences between IBA and QCA-CP, when
both electromagnetic models are fed with the same microstructure of SHS.
The results are shown in Fig. . Relative to the 1 : 1 line
(full black line), a small offset is observed and the scattering
coefficient from IBA is always larger than the QCA-CP counterpart. The
apparent offset in the double logarithmic plot Fig. is
equivalent to an overall prefactor. To assess the prefactor and the
deviation from 1 : 1, we evaluated the theoretical result for the ratio
rs of the scattering coefficients from
Eq. (). By computing the average volume fraction
ϕ‾2 = 0.265 of all 167 analyzed μCT samples, we can
compute an average ratio r‾s between IBA and QCA-CP by
inserting ϕ‾2 into Eq. (), viz
r‾s: = rs(ϕ‾2). This yields a value of
r‾s = 0.77. The corresponding
prediction κsIBA = r‾sκsQCA-CP is
shown in Fig. as a dashed red line which fully explains the offset.
Scatter plot of the scattering coefficient of IBA and QCA-CP from
Eqs. () and (), respectively, evaluated for the
optimal SHS parameters (τ^, d^) retrieved from the μCT
images.
Discussion
Main results
Three main implications can be drawn from the present work.
The first is related to the comparison of the electromagnetic models
IBA used in MEMLS and QCA-CP used in DMRT-ML. By rederiving the
scattering coefficient in IBA and QCA-CP and extracting the dependence
on microstructure we have shown that both electromagnetic
approximations involve exactly the same microstructural
characteristic, namely the zero-wavevector component of the structure
factor S(0). This implies that the different metrics (correlation
length and sphere diameter) which hitherto limited the comparison of
DMRT-based and MEMLS models, is only a consequence of different
microstructure descriptions and not related to fundamental differences
in the respective electromagnetic theories. We derived an expression
for the k-dependent quantity
C̃SHS(k|ϕ2, d, τ) for monodisperse SHS
in terms of the parameters ϕ2, d, τ which allows to implement
exactly the same microstructure in IBA and QCA-CP. The theoretical
analysis showed that if both electromagnetic models are evaluated for
SHS (or any other hard sphere model), differences occur only in the
zeroth-order effective dielectric constant (i.e., in the absence of
scattering). The theoretical ratio of the scattering coefficient
rs was evaluated (Eq. , Fig. )
and reveals only a weak dependence (maximum 30 %) on volume
fraction. The theoretical ratio rs is well-suited to
explain the scatter plot for the scattering coefficient
(Fig. ) where order-of-magnitude variations are
predominantly caused by variations of snow microstructure (via volume
fraction, diameter and stickiness), and only marginally by the
difference in the electromagnetic theories.
The second implication of the present work is related to parameter
estimation itself. The closed form expression of the correlation
function (or its FFT) for the monodisperse SHS model allows
parameters (d^, τ^) to be objectively found for a given snow
sample from a μCT image. This is of considerable interest for the
stickiness since no other method to estimate this quantity from
measurements is presently available. The parameter estimation showed
that optimal stickiness values vary significantly
(Fig. ). This may be partially explained by the weak
determination of this parameter when compared to the diameter, as
illustrated by the behavior of the least-squares cost function in
Fig. . Nevertheless, given the large sensitivity of the
scattering coefficient on stickiness , these
variations have a significant impact on the modeled electromagnetic
response of snow. From the present analysis, the stickiness has to be
considered as an independent parameter, which can be neither expressed
in terms of the diameter (Fig. ) nor in terms of the ice
volume fraction (Fig. ). The present work only suggests
(Fig. ) that stickiness values of snow are essentially
bounded from above by the percolation line (Eq. ) and
bounded from below by the theoretical lower bound (Eq. )
where SHS becomes physically meaningless. The variations in estimated
parameters also show that the pragmatic approach of using the same
stickiness value for the entire snowpack is
questionable. This should be considered in future use of DMRT models.
Fit parameters and standard errors for a linear regression
for all samples between the optimal SHS diameter and the optical diameter if the
optimization of d^τ is done for prescribed τ
according to Eq. ().
τ
b1
b0 (mm)
0.13
0.94 ± 0.07
0.04 ± 0.03
0.44
1.43 ± 0.09
0.04 ± 0.03
1.00
1.55 ± 0.10
0.03 ± 0.03
10.0
1.60 ± 0.10
0.04 ± 0.04
100.0
1.60 ± 0.10
0.04 ± 0.04
The third implication of the work is related to the applicability of
what has been termed the “short range limit” in microwave models
e.g., p. 504. To reveal the equivalence of the
scattering coefficient between IBA and QCA-CP
(Eqs. ,) we followed a common, but not
required assumption that all relevant length scales are small compared
to the wavelength. This allows to replace the k-dependent structure
factor S(k) by its value at the origin S(0) in the phase
function. On the other hand, the results of the fitting procedure show
(Fig. ) that the estimated stickiness values for some
samples are close to the line τmin(ϕ2) from
Eq. (). When approaching the line, the SHS structure factor
diverges . This is a consequence of the meaning of
τmin(ϕ2) as the coexistence line of the underlying first-order
liquid-gas-phase transition in the thermodynamic framework of
SHS near the critical point. Approaching the critical point is
accompanied by the occurrence of density fluctuations of increasing
spatial extent, causing maybe unrealistically large values of the scattering
coefficient of IBA and QCA-CP in the short range limit
(Fig. ). In other words, estimating stickiness parameters
close to the critical line renders the “short range” assumption
(i.e., that all relevant length scales are small compared to the
wavelength) invalid. The impact of the break-down of this assumption
can be principally analyzed, e.g., in IBA, by implementing the SHS
model in MEMLS where the integral over directions (Eq. )
is computed numerically. This would allow the full k
dependence of I(k) to be used in the phase function (Eq. ) and
the results to be compared to the short range limit I(0). Similarly, QCA
expressions under the long range assumption can also be obtained by
numerical integration Eq. 10.2.62. This is however
beyond the scope of the present work.
Using SSA measurements to run microwave models
One main motivation for the present work was the issue of grain-size
scaling raised by to relate the optical diameter
of snow to microwave simulations based on SHS.
The SSA, or optical radius, of snow can be easily measured in the
field and is an appealing first guess as the size parameter in DMRT-based
sphere models. However, all simulations using d = dopt
and τ = ∞, i.e., non-sticky spheres ,
underestimated the scattering coefficient. This was rectified by scaling
up the sphere diameter d by an empirical factor ranging from 2.3 to 3.5.
The scaling was suggested to replace the unknown dependence on
stickiness and distributions of grain sizes in snow
. The results from Fig. clearly show that
stickiness cannot be set to τ = ∞; and thus non-sticky hard
spheres are inadequate. This raises two important questions for
electromagnetic modeling when accepting the necessity of stickiness
and using SSA measurements. (1) Considering τ to be known, is d = dopt
an adequate approximation? (2) Are the parameters
(τ^, d^) found in the present work in agreement with the
grain-size scaling found in previous studies?
Figure and Table give a clear answer to question 1. We always observed an affine relation
d^ = a1dopt + a0 with a slope a1 which is (on
average for the entire data set) a1 ≈. 1.50. The slope, however
depends on snow type exemplified by the different subsets of snow
samples used in the present analysis. We observed that the temperature
gradient experiments TGM-17, TGM-2, DH-1, DH-2, which all include the
formation of depth hoar (cf. Supplement in ), lead
to an apparently stronger dependence on the optical diameter
(Table , top) when compared to the isothermal experiments,
where the slope is close to unity. This impact is confirmed by
restricting the relation d^ = a1dopt + a0 to
individual snow types (Table , bottom), or likewise by
Fig. where DH samples are predominantly located above the
1 : 1 line and RG samples below.
When stickiness is not known in advance, but set to a fixed,
prescribed value, our numerical experiment (Table ) has also
shown that the slope between the optical diameter and SHS diameter
depends on the prescribed stickiness. A simple scaling d^=Φdopt with a grain-size scaling factor Φ is
insufficient. In addition, the fit coefficient b1 from
Eq. () is always lower than the values for Φ found in
. This is valid, even for the non-sticky case which can
be identified with τ=10 or τ = 100 in Table , as
signaled by the convergence of the respective coefficients. However, a
value b1 = 1.6 tends to confirm a hypothesis put forward by
: they proposed that the polydisperse nature of snow
would justify a scaling of 1.6 based on , while the
stickiness would add an extra scaling by another factor of 1.6 which
is the value of b1 found here. If both effects are assumed to be
independent (which is unlikely), this yields a scaling close to 2.5
found by and . Further quantitative
analysis of the polydisperse SHS model is required to test this
hypothesis. However, a qualitative assessment of the impact of
polydispersity can be yet obtained from literature.
Monodisperse vs. polydisperse SHS
A peculiar feature of the monodisperse SHS correlation function
C̃(k) are the oscillations in the tail for large k
(Fig. ), which originate from the form factor
Eq. (). These features are clearly missing in the experimental
data shown in Fig. , but also for any other snow
sample. The oscillations are a manifestation of the monodisperse
nature of SHS used here. Using polydisperse SHS with a distribution of
diameters, these oscillatory features are smeared out as shown by
, leading to a smooth tail of C̃(k) and
a more realistic appearance of the model when compared to the
measurements. Furthermore, a comparison of the present results with
also reveals that an increase of stickiness
(increase of τ-1 in Fig. ) has a similar enhancing
effect on the scattering intensity for low k as the increase of
polydispersity (Fig. 4 in ), given that the mean
sphere diameter is held constant. This further supports the hypothesis
on the superposition of effects on grain scaling from the previous
section. This is intuitively reasonable, since both effects,
polydispersity and stickiness, essentially increase the variability
(i.e., fluctuations of the microscopic density) in the sample, and
thereby the scattering efficiency.
The monodisperse version of SHS bears another peculiarity. In general,
the Fourier transform of the correlation function must reveal the SSA
in its large k limit. This is known as the Porod law
which relates the large k asymptotics,
limk→∞k4C̃(k) = s/2, to the interfacial
area per unit volume s. This is generally valid for any two-phase
system with a smooth interface, and also present in the experimental
data (Fig. ). The Porod law is mathematically equivalent to
the existence of a cusp singularity of the real-space correlation
function C(r) at the origin, i.e., the existence of a small-r
expansion of the form C(r)/C(0) = 1 - sr/4 + O(r3). However
the limit limk→∞k4C̃(k) does not exist for
Eq. (). This is a known, subtle problem and a consequence
of monodispersity . Thus the monodisperse SHS model
bears some mathematical peculiarities which have to be examined with care.
If, however, the present method of μCT-based parameter estimation
were to be generalized to polydisperse SHS, some additional effort
would be required. In the polydisperse case, a system of coupled
quadratic equations must be solved to calculate the structure factor
S(k) according to Ch. 8.4.2. This can be done
only numerically in contrast to the closed-form solution for the
single quadratic Eq. () in the monodisperse case. In
addition, a generalization of Eq. () would be required. A
similar route was suggested by in the context of
small angle scattering.
Performance of the SHS model
To assess the goodness-of-fit of monodisperse SHS, we have evaluated
the coefficient of determination R2. Large differences are observed
(Fig. ), which poses the question for which snow type SHS
might be a suitable model. We also included a comparison of the R2
for SHS with the R2 for the exponential model which is commonly
used in MEMLS. At first sight, the differences between the models seem
to be smaller that sample to sample variations (Fig. ). A
discussion of small R2 differences is subject to caution, but
interestingly the temperature gradient time series (DH1, DH2, TGM2,
TGM17), which evolve from decomposing particles or rounded grains into
depth hoar, undergo a crossover in the relative performance of the
exponential and the SHS model; while initially the SHS model is
slightly superior or comparable to the exponential model, this order
changes at the end of the time series. This crossover is absent for
the isothermal metamorphism. This difference in performance depending
on snow type is also indicated by the behavior of the cost function in
Fig. or by the comparison of the μCT correlation
functions with both models in Fig. . The observed
goodness-of-fit differences, depending on correlation function models
and snow types, requires a more in-depth analysis in the future.
The performance of any correlation function model has to be assessed
against microwave measurements which eventually decide about the
quality of a particular model. We have shown that in both scattering
formulations, IBA and QCA-CP, it comes down to a single
microstructural quantity which must be well-predicted to describe
scattering correctly in the low frequency limit. This quantity is the
integral of the correlation function, or likewise, the zero-wavevector
component C̃(0) of the Fourier transform. A special name
has been coined for this type of parameter; it is referred to as the
coarseness of the medium . The coarseness
is a single number which quantifies the residual amplitude of volume
fraction fluctuations at the largest length scales.
Reinventing the wheel: small angle scattering
In view of the future task of finding the best microstructural model
for microwave modeling, we suggest building on the exhaustive work on
small angle scattering used for molecular systems. Our reanalysis has
stressed that the relevant quantity in IBA and QCA-CP in the
scattering coefficient for microwave modeling of snow is the Fourier
transform of the correlation function which must be well-matched. This
task is well-known and completely analogous to small angle scattering (SAS)
of molecular systems. SAS from X-ray or neutron sources has
become a standard technique to characterize microstructures by fitting
Fourier data . Indeed, the effective propagation
constant and the involved length scales in molecular systems are
entirely different, but the task of fitting the Fourier transform of
the correlation function to a parametric model to best match the
measured scattering intensity is exactly equivalent to the microwave
problem in snow at low frequencies. Libraries of microstructure models
(in terms of form factors, structure factors and generic forms of
correlation functions for bicontinuous media) are available in free
software packages, e.g., SASfit . In principle, these
packages can be applied directly to the present problem after
re-interpreting (i.e., rescaling) the length scales of the k axis.
Relevance for discrete element modeling of snow
The results about snow as a particle-based (granular) medium gained
from the present work can be exploited even beyond the context of
microwaves. As an example, discrete element modeling (DEM) is of
special interest for snow mechanics due to the
advantages in handling bond failure and the formation of new contacts
under large deformations; thereby, DEM faces the same difficulty as
microwave models i.e, mapping the real snow structure onto a particle-based microstructure which is in some sense equivalent to
snow. Deterministic approaches, which aim to recover the exact
grain structure, are very time-consuming
. Here DEM might also benefit from a
stochastic reconstruction of snow in terms of SHS, where the
computational effort for the parameter estimation is in the order of
seconds. For the given parameters, different realizations of the SHS model
can be generated with the Monte Carlo approach described in
. Our analysis revealed that essentially all
samples lie in the percolating regime of the SHS-phase diagram
(Fig. ). This implies that the corresponding SHS structures
have a static stiffness (elastic modulus) due to the percolating
cluster, a prerequisite for a meaningful granular model. The
interpretation of snow as a granular system via the SHS model allows,
for the first time, an objective definition of a coordination number
for snow as e.g., employed for a long time in snowpack models
. For the given parameters ϕ2, τ and d, we can use
the Percus–Yevick result from to obtain an average
coordination number nc = 2ϕ2t(ϕ2, τ) in terms of the
solution t of Eq. (). By means of the coordination
number, contact with other granular approaches, e.g., for the thermal
conductivity or optical properties
can be made to cross-correlate different SHS-based physical quantities.