TCThe CryosphereTCThe Cryosphere1994-0424Copernicus GmbHGöttingen, Germany10.5194/tc-9-1277-2015Parameterization of single-scattering properties of snowRäisänenP.petri.raisanen@fmi.fihttps://orcid.org/0000-0003-4466-213XKokhanovskyA.https://orcid.org/0000-0001-7110-223XGuyotG.JourdanO.https://orcid.org/0000-0003-0890-3784NousiainenT.https://orcid.org/0000-0002-6569-9815Finnish Meteorological Institute, P.O. Box 503, 00101 Helsinki, FinlandEUMETSAT, Eumetsat Allee 1, 64295 Darmstadt, GermanyInstitute of Remote Sensing, University of Bremen, 28334 Bremen, GermanyLaboratoire de Météorologie Physique (LaMP) Université Blaise Pascal/CNRS/OPGC, 24 avenue des Landais, 63177 Aubiére CEDEX, FranceP. Räisänen (petri.raisanen@fmi.fi)23June2015931277130116December201413February201502June201505June2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/9/1277/2015/tc-9-1277-2015.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/9/1277/2015/tc-9-1277-2015.pdf
Snow consists of non-spherical grains of various shapes and
sizes. Still, in many radiative transfer applications,
single-scattering properties of snow have been based on the assumption
of spherical grains. More recently, second-generation Koch fractals
have been employed. While they produce a relatively flat phase
function typical of deformed non-spherical particles, this is still
a rather ad hoc choice. Here, angular scattering measurements for
blowing snow conducted during the CLimate IMpacts of Short-Lived
pollutants In the Polar region (CLIMSLIP) campaign at Ny Ålesund,
Svalbard, are used to construct a reference phase function for
snow. Based on this phase function, an optimized habit combination
(OHC) consisting of severely rough (SR) droxtals, aggregates of SR
plates and strongly distorted Koch fractals is selected. The
single-scattering properties of snow are then computed for the OHC as
a function of wavelength λ and snow grain volume-to-projected
area equivalent radius rvp. Parameterization equations are
developed for λ= 0.199–2.7 µm and
rvp= 10–2000 µm, which express the
single-scattering co-albedo β, the asymmetry parameter g and
the phase function P11 as functions of the size parameter and the
real and imaginary parts of the refractive index. The
parameterizations are analytic and simple to use in radiative transfer
models. Compared to the reference values computed for the OHC, the
accuracy of the parameterization is very high for β and
g. This is also true for the phase function parameterization, except
for strongly absorbing cases (β> 0.3). Finally, we consider
snow albedo and reflected radiances for the suggested snow optics
parameterization, making comparisons to spheres and distorted Koch fractals.
Introduction
Snow grains are non-spherical and often irregular in shape. Still, in
many studies, spherical snow grains have been assumed in radiative
transfer calculations due to the convenience of using Mie theory. In
fact, it has been shown that the spectral albedo of snow can be fitted
by radiative transfer calculations under the assumption of spherical
snow grains, when the effective snow grain size is considered an
adjustable parameter (i.e. determined based on albedo rather than
microphysical measurements) .
Snow albedo parameterizations used in climate models and numerical
weather prediction models are often semi-empirical and do not specify
the snow grain shape (for some examples, see ).
However, in most (if not all) physically based albedo
parameterizations that explicitly link the albedo to snow grain size,
spherical snow grains are assumed .
It is, however, well known that the single-scattering properties
(SSPs) of non-spherical particles, including the single-scattering
albedo ω, the phase function P11 and the entire phase
matrix P, can differ greatly from those of
spheres.
While symbols and abbreviations are introduced at
their first appearance, they are also listed in Table .
A consequence of this is that the assumed shape of snow grains has
a profound effect on the bidirectional reflectance distribution
function (BRDF) of snow . Furthermore,
showed that the modelled BRDF of snow agreed better
with observations when, instead of the actual phase function for
spheres, the Henyey–Greenstein (HG) phase function was
assumed. The HG phase function is very smooth, while that of spheres
features icebow and glory peaks not seen for real snow along with
very low sideward scattering. Based on a comparison of a few shape
models with phase function measurements for laboratory-generated
ice crystals ,
recommended, instead of spheres, the use of Gaussian random spheres
or Koch fractals ,
which both exhibit a relatively featureless phase function.
Since Gaussian random spheres have several free parameters while Koch
fractals have none (except for the degree of distortion for
randomized Koch fractals), Koch fractals were selected by
. further
demonstrated that the reflectance patterns computed for Koch
fractals agreed reasonably well with actual measurements for snow.
Subsequently, they have been used in several
studies related to remote sensing of snow grain size and snow albedo
.
Other snow grain shape models have also been considered.
suggested the use of non-spherical ice
particles with rough surfaces, specifically, cylindrical
particles for new snow and prolate ellipsoids for old granular
snow. These choices improved the agreement with observed
angular reflectance patterns compared to the use of spheres.
compared anisotropic reflectance factors computed using spheres,
hexagonal plates, hexagonal columns and aggregates of columns
with ground-based measurements in Antarctica, finding the best
agreement for the aggregate model and the largest discrepancies for
spheres. Furthermore, tested, in their retrieval
algorithm of snow grain size and soot concentration in snow,
a mixture of hexagonal columns and plates with rough surfaces.
Overall, while it is clear that spheres are not an ideal choice for
modelling the SSPs of snow, it is less clear which non-spherical
model should be used. noted that the
final decision of the shape model should be made when in situ
phase function measurements for snow become available. The
present paper makes a step towards this direction. We employ
angular scattering measurements for blowing snow performed with
a polar nephelometer during the CLimate IMpacts
of Short-Lived pollutants In the Polar region (CLIMSLIP)
campaign at Ny Ålesund, Svalbard , to
construct a reference phase function for snow grains at the
wavelength λ= 0.80 µm. This phase
function is used to select a new shape model for snow, an
“optimized habit combination” (OHC) consisting of severely
rough (SR) droxtals, aggregates of SR plates and strongly
distorted Koch fractals. The SSPs for the OHC are then computed
as a function of wavelength and snow grain size, and
parameterization equations are developed for the
single-scattering co-albedo β= 1 -ω, the asymmetry
parameter g and the phase function P11. Such
parameterizations are of substantial practical significance, as
they greatly facilitate the use of the OHC in radiative transfer
applications. We are not aware of any such previous
parameterizations for representing the snow SSPs.
The outline of this paper is as follows. First, in
Sect. , the models used to compute the SSPs of Koch
fractals, Gaussian spheres and spheres are introduced along with the
database of used for several other shapes. In
Sect. , the reference phase function for snow is
constructed. In Sect. , several shape models are
compared in terms of their ability to reproduce the reference phase
function, and the OHC is selected. In Sect. , the SSPs
for the OHC are computed as a function of wavelength and snow grain
size, and in Sect. parameterization equations are
developed. In Sect. , the snow SSP parameterization is
applied to radiative transfer computations, and comparisons are made
to spheres and Koch fractals. Finally, a summary is given in Sect. .
Shape models and single-scattering data
Here, several shape models are considered as candidates for
representing the SSPs of snow. These include (1) second-generation
Koch fractals, (2) Gaussian random spheres, (3) nine different crystal
habits in the single-scattering database and, for
comparison, (4) spheres. The snow grains are assumed to consist of
pure ice (i.e. no impurities such as black carbon are included). The
ice refractive index of is employed.
The SSPs (extinction cross section, single-scattering albedo, phase
function and asymmetry parameter) of Koch fractals are simulated using
the geometric optics code of see
also. Both regular and distorted Koch fractals are
considered. A regular second-generation Koch fractal has
144 equilateral triangular surface elements. Distortion is simulated using
a statistical approach, in which for each refraction–reflection event
the normal of the crystal surface is tilted randomly around its
original direction . The zenith (azimuth) tilt angle is
chosen randomly with equal distribution between [0, θmax]
([0, 360∘]), where θmax is defined using
a distortion parameter t=θmax/90∘. Values of t= 0
(regular), t= 0.18 (distorted) and t= 0.50 (strongly distorted) are
considered. The geometric optics solution consists of ray tracing and
diffraction parts, which are combined as in . For
diffraction, the Fraunhofer (far-field) approximation is
employed. Either 3 million (in Sect. ) or 1 million (in
Sect. ) rays per case (i.e. crystal size, wavelength
and degree of distortion) are used for the ray tracing part. Up to
p= 12 ray–surface interactions per initial ray are considered (see
Sect. 3A in ).
The SSPs of Gaussian random spheres are computed with the geometric
optics code of . Details of the Gaussian random
sphere shape model are discussed e.g. in . The
shape of the particles is described in terms of three parameters: the
relative SD of radius σ, the power-law index ν in the
Legendre polynomial expansion of the correlation function of radius
(the weight of the nth degree Legendre polynomial Pn being
cn∝n-ν), and the degree of truncation nmax for this
polynomial expansion. In broad terms, increasing σ increases
the large-scale non-sphericity of the particle, while decreasing ν
and increasing nmax adds small-scale structure to the particle
shape. Four values were considered for σ (0.15, 0.20, 0.25 and
0.30), four for ν (1.5, 2.0, 2.5 and 3.0) and three for
nmax (15, 25 and 35), which yields 48 parameter
combinations. A total of 1 million rays with 1000 realizations of
particle shape per case were employed in the ray tracing
computations. Diffraction was computed by applying the Fraunhofer
approximation to equivalent cross-section spheres.
Recently, published a comprehensive library of SSPs of
non-spherical ice crystals, for wavelengths ranging from the
ultraviolet to the far infrared, and for particle maximum dimensions
dmax ranging from 2 to 10 000 µm.
The library is based on the Amsterdam discrete dipole approximation
for small particles (size parameter smaller than
about 20) and improved geometric optics for large
particles. Here, single-scattering properties for nine ice particle
habits in the database are used: droxtals, solid and
hollow hexagonal columns, aggregates of 8 columns, plates, aggregates
of 5 and 10 plates, and solid and hollow bullet rosettes. For each
habit, the SSPs are provided for three degrees of particle surface
roughness: completely smooth (CS), moderately rough (MR) and severely
rough. The effect of roughness is simulated in a way that closely
resembles the treatment of distortion for Koch fractals: the surface
slope is distorted randomly for each incident ray, assuming a normal
distribution of local slope variations with a SD of 0, 0.03 and 0.50
for the CS, MR and SR particles respectively in Eq. (1) of
. In fact, this approach does not represent any specific
roughness characteristics but attempts instead to mimic the effects on
SSPs due to non-pristine crystal characteristics in general (both
roughness effects and irregularities).
For comparison, results are also shown for spheres. The SSPs of
spheres are computed using a Lorenz–Mie code .
Observation-based phase function for blowing snow
We employ as a reference an observation-based phase function for
blowing snow. The reference phase function was derived from
ground-based measurements conducted during the CLIMSLIP field
campaign at Ny Ålesund, Svalbard , on 23 and
31 March 2012. The blowing snow case on 23 March was preceded
by heavy snowfall on 22 March, ending during the night of
23 March. The last snowfall before 31 March blowing snow case
occurred on 29 March. Consequently, the case of 23 March
represents essentially new snow, while on 31 March some snow
metamorphism had occurred, and the snowpack was probably denser
(although snow density was not measured). The near-surface air
temperature ranged from -5 to -9 ∘C during
the 23 March case and from -18 to -20 ∘C
during 31 March. Correspondingly, the wind speeds ranged from
1 to 9 m s-1 on 23 March (median value 4 m s-1) and from
5 to 8 m s-1 on 23 March (median value 7 m s-1).
Mainly cloudy conditions prevailed on 23 March, while 31 March
was cloud free. The phase functions discussed below are
averages over the entire blowing snow events, which lasted for
approximately 10 h (8–18 UTC) on 23 March and 12 h (12–24 UTC) on 31 March.
The angular scattering coefficient
Ψ(θs) [µm-1 sr-1] of blowing
snow was measured with a polar nephelometer
PN; on 23 and 31 March 2012 at
31 scattering angles in the 15∘≤θs≤ 162∘ range at a nominal wavelength of
λ= 0.80 µm. The corresponding phase
function P11(θs) was obtained by normalizing
Ψ(θs) by the volume extinction coefficient σext:
P11θs=4πΨθsσext.
Here σext was estimated from the PN data following
, with a quoted accuracy of 25 %.
The derived phase functions are shown in Fig. a. There
are only minor differences between the 23 and 31 March cases. In
both cases P11 decreases sharply from 15 to 50∘, then
more gradually until 127∘. At larger scattering angles
P11 increases slightly with a local maximum around
145∘ (discussed below). Hereafter, the average over the two
cases is used as a reference for the modelled phase functions:
P11ref=0.5⋅P1123 March+P1131 March.
(a) Phase function of blowing snow as derived from the
CLIMSLIP data on 23 March 2012 (red) and on 31 March 2012 (blue). The
reference phase function P11ref (grey) was defined as the
average of the 23 and 31 March cases. (b) Comparison of
P11ref with phase functions for non-precipitating cirrus
(CIRRUS'98, black line) and glaciated Arctic nimbostratus (ASTAR clusters 6
and 7, red and blue lines).
In Fig. b, P11ref is compared with
three other phase functions: a non-precipitating cirrus case over
Southern France in the CIRRUS'98 experiment
(discussed in ) and two phase functions for
glaciated parts of nimbostratus over Svalbard in the ASTAR 2004
experiment, corresponding to clusters 6 and 7 in
. These phase functions were derived from raw PN data
using a statistical inversion scheme
. Perhaps as expected, the blowing snow
phase function P11ref is generally closer to the
glaciated nimbostratus phase functions than to the cirrus phase
function. In particular, at sideward angles between roughly
55 and 135∘, P11ref falls mostly
between the two nimbostratus phase functions, while the cirrus phase
function exhibits somewhat smaller values. The smallest P11 in
the cirrus and nimbostratus cases occurs at θs= 120∘,
as compared with θs= 127∘ for
P11ref. All four phase functions then increase until
θs≈ 140∘, after which the nimbostratus and cirrus
phase functions become quite flat. In contrast, P11ref
shows a local maximum around θs≈ 145∘.
The origin of the maximum at θs≈ 145∘ is not clear. While it may, in principle, be
caused by scattering by snow grains, this feature is neither
captured by any of the particle shapes considered in this
study nor present in phase functions measured for
laboratory-generated ice crystals in and .
Rather, it falls between the icebow peak for spherical
ice particles near 135∘ and a maximum seen for many
pristine hexagonal shapes at 150–155∘ (see
Fig. ). Curiously, this feature coincides
with the scattering maximum of small water droplets with
a ∼ 10 µm diameter at 140–145∘. However, water
droplets seem like an implausible explanation, since the
conditions at the measurement site were subsaturated with respect
to liquid water (the relative humidity being roughly 92–95 % on
23 March and 79–87 % on 31 March), and especially the 31 March case
was quite cold. Yet the 145∘ feature is clearly visible
in the measured phase function in both cases. Finally, we
cannot discount the possibility that inaccuracy in the PN
angular scattering measurements influences this
feature. report relative accuracy of
scattered intensities of 3–5 % between 15 and
141∘, but degrading to 30 % for 162∘, for an
experimental setup with low extinction. Thus the phase function
derived from the PN measurements is, overall, less reliable near the
backscattering direction than in near-forward and side-scattering directions.
Whether the phase function feature at 145∘ is an artifact or
a real feature caused by scattering by snow should be resolved
through further measurements, preferably using some alternative technique.
However, in either case, it has only a small impact on
the snow SSP parameterizations derived in this paper. This detail
cannot be captured by any of the shape models considered, so it is not
present in the parameterized phase functions. Its influence on the
asymmetry parameter is also modest. Even a complete elimination of the
maximum by linear interpolation of P11ref between the
minima at 127 and 155∘ would increase g by only ≈ 0.007.
(a) Examples of snow grains imaged by the CPI instrument on
31 March 2012 and (b) size distributions for both the 23 and
31 March cases.
The size distribution of blowing snow was measured with the Cloud
Particle Imager (CPI) instrument . The CPI registers
particle images on a solid state, one million pixel digital
charge-coupled device (CCD) camera by freezing the motion of the
particle using a 40 ns pulsed, high power laser diode. Each
pixel in the CCD camera array has an equivalent size in the sample
area of 2.3 µm. In the present study, the minimum size for
the CPI's region of interest is set up to 10 pixels. Therefore
particles with sizes ranging approximately from 25 µm to
2 mm are imaged.
Figure a shows examples of particles imaged by the CPI on
31 March 2012. While some needle-shaped crystals can be spotted, many
of the particles are irregular, which also applies to the 23 March 2012
case. It is also noted that many of the particles show rounded edges,
possibly related to sublimation during snow metamorphosis.
Size distributions derived from the CPI data are shown in
Fig. b. A lognormal distribution was fitted to the data
(averaged over the 23 and 31 March cases):
ndp=12πlnσgdpexp-lndp-lndp,022ln2σg.
Here, dp is the projected-area equivalent diameter of the
particles, dp,0= 187 µm is the median
diameter and σg= 1.62 the geometric SD. This
size distribution was used for all shape models when comparing the
modelled phase functions with P11ref. Since
absorption is weak at λ= 0.80 µm and the particles
are much larger than the wavelength, the modelled P11 is only
weakly sensitive to the size distribution employed if the shape
of the snow grains is independent of size. This holds true for
spheres, Gaussian spheres, Koch fractals, droxtals and the three
aggregate habits in the database. However, for solid and
hollow hexagonal columns, plates as well as solid and hollow bullet
rosettes, the crystal geometry is a function of size, with some
influence on P11 (see end of Sect. for more discussion).
Selecting a shape model for snow optics
The purpose of this section is to select a shape model of snow for use
in Sects. to . The phase function for
blowing snow from the CLIMSLIP campaign, as defined by
Eq. (), is used as a reference. It is emphasized that
the approach is deliberately pragmatic: we do not attempt to model the
scattering based on the shapes of the observed snow grains; rather we
try to develop an equivalent microphysical model for representing the
SSPs. Previously, the choice of Koch fractals for approximating the
scattering by snow was likewise based on phase
function data only. Furthermore, our approach is conceptually analogous
to the widely used practice of modelling the SSPs of irregular dust
particles. Instead of considering the actual dust particle shapes,
shape distributions of spheroids are used operationally in a variety
of applications , as they have
been found to reasonably mimic scattering by dust.
In contrast, current state-of-the-art models for ice cloud SSPs
include ice crystal habit distributions parameterized as a function of
crystal size, based on in situ microphysical observations
. In principle, it would be desirable
to use this approach also for snow to provide a more direct link
between the actual snow grain shapes and those assumed in the
parameterization and to account for changes in snow grain shape with
size, which we currently neglect. This would require, first, the
analysis and subsequent parameterization of snow grain shape
distributions as a function of size and, second, the computation and
parameterization of the respective SSPs. The main reason why we have
not attempted this approach in the current work is that a very large
fraction of the particles in blowing snow (and snow on ground)
are irregular, more than 80 % according to manual classification
of CPI images (see also Fig. a), and cannot be unambiguously
associated with habits considered e.g. in the database of .
To provide a quantitative measure for the agreement between the
modelled and reference phase functions (P11model and
P11ref respectively) we define a cost function as the
root-mean-square (rms) error of the logarithm of phase function:
cost=∫15∘162∘lnP11model-lnP11ref2sinθsdθs∫15∘162∘sinθsdθs.
To start with, the phase function for single crystal shapes is
compared with P11ref in Fig. . To be
consistent with the CLIMSLIP observations, the phase function is
computed at λ= 0.80 µm, and it is integrated
over the size distribution defined by Eq. (). Several
points can be noted.
First, unsurprisingly, the phase function for spheres agrees poorly
with the observations (Fig. a). In particular,
sideward scattering is underestimated drastically, and there is
a strong icebow peak at θs= 134∘, which is not seen
in P11ref.
Comparison of phase function for various shape models with the
reference phase function derived from CLIMSLIP data (P11ref
shown with grey dots in each panel). (a) Spheres,
(b) regular and distorted second-generation Koch fractals (with
distortion parameters t= 0.18 and t= 0.50), (c) four
realizations of Gaussian spheres and (d–l) nine habits
in the database. For each habit, the phase function was
averaged over the size distribution defined by Eq. (). In
the figure legends, the two numbers in parentheses give the asymmetry
parameter and the cost function defined by Eq. () respectively.
For the Gaussian spheres in (c), the notation indicates the shape
parameters (e.g. for 0.15_3.0, σ= 0.15 and ν= 3.0;
nmax was fixed at 15). For the habits
in (d–l), CS, MR and SR refer to particles with
completely smooth surface, moderate surface roughness and severe surface
roughness respectively.
Comparison of modelled phase functions with the reference phase
function (P11ref shown with grey dots in a–c).
(a) Selected single-habit cases: 1 = distorted Koch fractals
with t= 0.18; 2 = Gaussian spheres with σ= 0.30,
ν= 1.5 and nmax= 15; and 3 = aggregates of
eight severely rough (SR) columns. (b) Best combinations of two habits:
4 = aggregates of eight SR columns and SR hollow bullet rosettes
(weights 0.61 and 0.39); 5 = aggregates of eight SR columns and aggregates of
five SR plates (weights 0.61 and 0.39); and 6 = aggregates of eight SR columns
and SR hollow columns (weights 0.68 and 0.32). (c) Best combinations
of three habits: 7 = SR droxtals, SR hollow columns and distorted Koch
fractals (t= 0.50) (weights 0.32, 0.30 and 0.38); 8 = SR droxtals,
SR hollow bullet rosettes and distorted Koch fractals (t= 0.50)
(weights 0.26, 0.36 and 0.38); and 9 = SR droxtals, aggregates of 10 SR
plates and distorted Koch fractals (t= 0.50) (weights 0.36, 0.26
and 0.38). In the legends in (a–c), the two numbers in
parentheses give the asymmetry parameter and the cost function defined by
Eq. () respectively. (d–f) show the
corresponding differences from P11ref.
Second, for second-generation Koch fractals (Fig. b),
the agreement with P11ref is considerably better than
for spheres. The main features of the phase function are similar for
regular and distorted Koch fractals. However, the regular Koch
fractal's phase function exhibits several sharp features specific to
the tetrahedral geometry which are not observed in
P11ref. The distorted Koch fractals' versions are more
consistent with the measurements even though marked deviations from
P11ref are still present. Scattering is underestimated
between 15 and 30∘ and overestimated between 45 and
100∘. Also, the gradient of P11 in the backscattering
hemisphere is consistently negative, while P11ref
rather increases slightly between 127 and 162∘. Overestimated
sideward scattering by Koch fractals has been previously noted in the
context of cirrus clouds and in a comparison with
a measured phase function for laboratory-generated ice crystals
Fig. 3 in.
Third, for Gaussian spheres, the level of agreement with
P11ref depends on the shape parameters chosen. Four
cases out of the 48 considered are shown in Fig. c
(for all of these nmax= 15, but the general features for
nmax= 25 and nmax= 35 are similar). For example, for the
parameter values σ= 0.15 and ν= 3.0, which are close to those
estimated from shape analysis of small quasi-spherical ice crystals in
cirrus clouds in , the deviations from
P11ref are substantial. The phase function features
undesirable large-scale oscillations and, in particular, scattering at
θs≈ 45–75∘ is underestimated substantially.
Best agreement with P11ref is obtained in the case
σ= 0.30, ν= 0.15, which features both pronounced
large-scale non-sphericity and small-scale structure in the particle
shape. The sideward scattering is overestimated (mainly between 70 and
100∘), but the cost function (0.163) is clearly smaller than
that for distorted Koch fractals (0.284) and is, in fact, the
smallest among all single-habit shape models considered.
Fourth, regarding the habits in the database
(Fig. d–l), both visual inspection and the cost
function values indicate that the agreement with P11ref
improves with increasing particle surface roughness. While completely
smooth and, in many cases, moderately rough particles exhibit halo
peaks, for severely rough particles the phase function is quite smooth
and featureless, as is P11ref. It is further seen that,
in general, increasing the roughness increases sideward scattering and
reduces the asymmetry parameter. While none of the habits considered
provides perfect agreement with P11ref, the cost
function is smallest for the aggregate of eight columns (0.172).
Since none of the individual shape models agrees fully satisfactorily
with P11ref, we considered combinations of two or three
shapes. We thus use
P11model=∑j=1nwjP11j,
where n= 2 or n= 3 is the number of shapes in a combination and
P11j is the phase function for shape j, integrated over the
size distribution (Eq. ) for each shape
separately. Thus, the potential dependence of snow grain shapes on
their size is not considered here. For each combination of shapes
considered, the optimal weight factors wj were searched by
minimizing the cost function (Eq. ), subject to the
conditions that all wj are non-negative and their sum
equals 1. Since pristine particles and even moderately rough particles
feature halo peaks (or an icebow peak in the case of spheres), which
are absent in P11ref, the following groups of habits
are considered: distorted Koch fractals, Gaussian spheres and
severely rough particles in the database.
List of best three-habit combinations. w1, w2 and w3 are the weights
(i.e. fractional contributions to projected area) of each habit, “cost” is the
cost function, g the asymmetry parameter and ξ a non-dimensional absorption
parameter defined by Eq. (). SR refers to severely rough particles,
and t is the distortion parameter for second-generation Koch fractals. The “optimized
habit combination” (OHC) is highlighted with italic font.
Figure illustrates a comparison with
P11ref for three single-habit cases
(Fig. a and d) (the best Koch fractal case, the best
Gaussian sphere case and the best case with
particles), the best three two-habit cases (Fig. b
and e) and the best three three-habit cases (Fig. c
and f) as defined in terms of the cost function. As expected, the
agreement of P11model with P11ref
improves with increasing number of shapes in the combination. The
best three-habit cases follow P11ref quite faithfully,
though slightly underestimating P11ref in near-forward
directions and not capturing the details of
P11ref near θs= 145∘.
Furthermore, it is seen that the best three-habit combinations produce
nearly identical P11, agreeing even better with each other than
with P11ref.
The relationship between the asymmetry parameter g and the cost
function is considered in Fig. , where all
single-habit cases and combinations of two or three habits are
included. While high values of cost function can occur at any g, the
lowest values (< 0.10) always occur for three-habit combinations
with 0.775 <g< 0.78. This supports a best estimate of
g≈ 0.78 for snow at λ= 0.80 µm, of
course subject to the assumption that the measurements for blowing
snow used to construct P11ref are also representative
of snow on the ground.
A scatter plot of asymmetry parameter vs. cost function
(Eq. ) for single habits (black dots), combinations of two
habits (red dots) and combinations of three habits (blue dots). The
“optimized habit combination” selected for parameterization of snow
single-scattering properties is marked with an arrow. Note that some
single-habit cases fall outside the range plotted here. These include
spheres for which cost = 1.90 and g= 0.892.
The three-habit combinations with cost function below 0.1 are
listed in Table . All of them include SR
droxtals and either strongly distorted (t= 0.50) or distorted
(t= 0.18) Koch fractals, but the third habit included in the
combinations varies from case to case. The differences in cost
function and asymmetry parameter between the best habit
combinations are very small, which makes the choice of a single
“best” habit combination for representing the SSPs of snow
somewhat arbitrary. For further use in representing the SSPs as
a function of wavelength and size, we select the following habit
combination: 36 % of SR droxtals, 26 % of aggregates of
10 SR plates and 38 % strongly distorted second-generation
Koch fractals (t= 0.50), where the weights refer to
fractional contributions to the projected area. This habit
combination is represented with a blue line in
Fig. c and f and is marked with an arrow in
Fig. . Hereafter, this habit combination will
be referred to as the “optimized habit combination”.
The primary reason why we selected this OHC rather than either
of the first two habit combinations in Table ,
which have a marginally lower cost function, is that these habit
combinations include either hollow columns or bullet
rosettes. For these habits (unlike aggregates of plates), the
particle geometry assumed in the database depends
on particle size, with the aspect ratio of the crystals
increasing with their length. However, due to snow metamorphosis
on ground, size–shape relationships based on crystal growth in
ice clouds are most likely not valid for snow. Therefore, we
considered it better to use a crystal geometry that is
independent of size. This also helps to keep the SSP
parameterization simpler.
Comparison of single-scattering properties for spheres (black
lines), distorted Koch fractals with t= 0.18 (red) and the optimized
habit combination (blue), for rvp= 50 µm (solid
lines) and rvp= 1000 µm (dashed lines), for a
monodisperse size distribution. (a) Asymmetry parameter g;
(b) single-scattering co-albedo β= 1 -ω;
(c) non-dimensional absorption parameter ξ (Eq. );
and (d)ξ divided by the real part of refractive index squared.
In (c and d), the grey line represents
Eq. ().
Snow single-scattering properties as a function of size and wavelength
The SSPs, including the extinction efficiency Qext,
single-scattering co-albedo β, asymmetry parameter g and
scattering phase function P11(θs) were determined for the
OHC for 140 wavelengths between 0.199 and 3 µm and for
48 particle sizes between 10 and 2000 µm. Here, the size is
defined as the volume-to-projected area equivalent radius
rvp= 0.75 V/P. As stated above, the OHC consists of SR
droxtals, aggregates of 10 SR plates and strongly distorted Koch
fractals. The SSPs for droxtals and aggregates of plates were taken
from the database (interpolated to fixed values of
rvp) while those of Koch fractals were computed using the
geometric optics code of , as explained in
Sect. . Four caveats should be noted:
Due to problems associated with the truncation of numerical
results to a finite number of digits (P. Yang, personal communication,
2013), the values of β in the database are
unreliable in cases of very weak absorption. To circumvent this issue,
it was assumed that in cases of weak absorption (β< 0.001 for
Koch fractals), the values for droxtals and aggregates of plates may
be approximated asβdroxtalλ,rvp=0.943βfractalλ,rvp,βaggregateλ,rvp=0.932βfractalλ,rvp.Here the scaling factors were determined as
βdroxtal‾/βfractal‾ and
βaggregate‾/βfractal‾,
where the overbar refers to averages over the cases in which
0.001 <βfractal< 0.01 and the size parameter
x= 2πrvp/λ> 100.
While the largest maximum dimension for particles in the
database is 10 000 µm for all habits, the
corresponding maximum values of rvp are smaller and depend
on the habit. For droxtals, rvp,max= 4218 µm, while for the aggregates
of 10 plates, it is only rvp,max= 653 µm.
Thus, to extend the SSPs for the OHC to sizes up to
rvp= 2000 µm, we extrapolated the SSPs for the
aggregates of plates based on how the SSPs depend on size for Koch
fractals. See Appendix for details.
The SSPs for Koch fractals were computed using a geometric optics
code, which means that the accuracy deteriorates somewhat in cases
with smaller size parameters (typically for x< 100). This issue
pertains mainly to small snow grains at near-IR wavelengths
(e.g. for λ= 2.5 µm, x= 100 corresponds to
rvp≈ 40 µm).
Lastly but importantly, since the OHC was selected based on
measurements at a single wavelength λ= 0.80 µm for
only two cases, there is no guarantee that it represents the snow SSPs
equally well at other wavelengths, or for all snow grain sizes.
Figure compares wavelength-dependent SSPs for the OHC
with those for two shape assumptions previously used in modelling snow
optics: spheres and Koch fractals (distorted Koch fractals with
t= 0.18 were selected for this comparison; this is close though
not identical to the shape assumption used by
). Two monodisperse cases are considered, with
rvp= 50 µm and
rvp= 1000 µm. For all three
habits, the asymmetry parameter g (Fig. a) and the
single-scattering co-albedo β (Fig. b) show
well-known dependencies on particle size and wavelength. Thus, g is
largely independent of both λ and rvp in the
visible region where β is very small. In the near-IR region,
β increases with increasing imaginary part mi of the
refractive index and with increasing particle size. With increasing
β, the fractional contribution of diffraction to the phase
function increases, which results in larger values of ge.g.. The most
striking differences between the three shape assumptions occur for the
asymmetry parameter, especially in the visible region, where
g≈ 0.89 for spheres, g≈ 0.74 for distorted Koch
fractals and g≈ 0.77–0.78 for the OHC. The values of β
for the OHC are also intermediate between the two single-shape cases:
larger than those for spheres (except for
rvp= 1000 µm at the strongly absorbing
wavelengths λ> 1.4 µm) but slightly smaller than
those for distorted Koch fractals. The implications of these
differences for snow albedo are considered in Sect. .
While the co-albedo values in Fig. b are strongly
wavelength dependent through mi, the effects of shape on absorption
can be distinguished more clearly by considering the non-dimensional
absorption parameter ξ=CabsγV=QextPβγV,
where Cabs is the absorption cross section,
Qext the extinction efficiency, P the projected
area and V the particle volume, and γ= 4πmi/λ,
where mi is the imaginary part of ice refractive
index. Figure c displays ξ at the
wavelengths λ= 0.199–1.4 µm, where
absorption by snow is relatively weak. Consistent with the
co-albedo values (Fig. b) and previous studies
e.g., Fig. c indicates
that absorption is generally stronger for non-spherical than
spherical particles for the same rvp. The
difference is particularly clear in the visible region, where
ξ≤ 1.3 for spheres (except for some spikes that occur in
the Mie solution especially for
rvp= 50 µm), ≈ 1.7 for the
Koch fractals and slightly over 1.6 for the OHC.
At wavelengths beyond λ= 1.0 µm, ξ tends to
decrease especially for the larger particle size
rvp= 1000 µm considered, as absorption no
longer increases linearly with mi. Furthermore, in the UV region,
Koch fractals and the OHC show a distinct increase in ξ with
decreasing wavelength. This is related to the corresponding increase
of the real part of the refractive index
mr. Interestingly, it is found that for these shape
assumptions, absorption scales linearly with mr2;
furthermore, for Koch fractals ξ/mr2≈ 1 when
absorption is weak (Fig. d). For spheres, the
dependence of ξ on mr is weaker. Equation (4) in
provides the absorption efficiency of weakly
absorbing spheres in the limit of geometric optics, which can be
rewritten in terms of ξ as
ξ=mr3-mr2-13/2mr=mr2-mr2-13/2mr.
For rvp= 1000 µm, ξ for spheres follows
this approximation closely until λ≈ 1.0 µm
(Fig. c and d). However, it appears that for Koch
fractals, only the first term should be included.
It should be noted that ξ for the OHC is not independent of that
for Koch fractals (due to the scaling of co-albedo in
Eqs. () and ()). However, we found
that ξ also scales linearly with mr2 for Gaussian
spheres (this was tested for σ= 0.17, ν= 2.9,
nmax= 15), suggesting that this might apply more generally to
complex non-spherical particles.
Finally, it should be recalled that our choice of the OHC was
based on phase function observations at the wavelength
λ= 0.80 µm. At this wavelength, absorption is
so weak that it has very little impact on the phase
function. Therefore, these observations cannot be used
to constrain absorption by snow. In spite of this, we think
it is worth providing a co-albedo parameterization based on the
OHC (Eq. in Sect. ). The
reason for this is that snow grains are distinctly
non-spherical, and for non-spherical particles, ξ and
β are, in general, systematically larger than those for
spheres, as demonstrated by Fig. . In fact,
considering the wavelength λ= 0.80 µm, the
values of ξ integrated over the size distribution defined by
Eq. () are, for the large majority of the
non-spherical shapes considered, between 1.55 and 1.75, the
value for the OHC being ξ= 1.62 (Table ). The corresponding value for spheres is
substantially lower: ξ=1 .29. Thus, while we cannot constrain
ξ or β precisely, it is very likely that the actual
values for snow exceed those for spheres.
Parameterizations for the single-scattering properties of snow
In this section, parameterization equations are provided for the
computation of snow SSPs (extinction efficiency Qext,
single-scattering co-albedo β, asymmetry parameter g and
scattering phase function P11(θs)) for the OHC discussed
above. The parameterizations are provided for the size range
rvp= 10–2000 µm and wavelength range
λ= 0.199–2.70 µm. They are expressed in
terms of the size parameter x and real and imaginary parts of
refractive index (mr and mi). Here, the size parameter
defined with respect to the volume-to-projected area equivalent radius
is used:
x=xvp=2πrvpλ.
For the OHC, the size parameter defined with respect to the projected
area is xp≈ 1.535 xvp.
Extinction efficiency
The extinction efficiency Qext for the OHC is displayed in
Fig. . For most of the wavelength and size region
considered, Qext is within 1 % of the asymptotic value
Qext= 2 for particles that are large compared to the wavelength.
Note that the deviations from Qext= 2 are probably
somewhat underestimated because the OHC includes Koch fractals,
for which Qext≡ 2 due to the use of geometric optics.
For simplicity, we assume this value in our parameterization, while
acknowledging that the actual value tends to be slightly higher
especially for small snow grains in the near-IR region.
Single-scattering co-albedo
The single-scattering co-albedo is parameterized as
β=0.4701-exp-2.69xabs1-0.31minxabs,20.67,
where the size parameter for absorption is defined as
xabs=2πrvpλmimr2.
The general form of this parameterization was inspired by the ice
crystal optics parameterization of ; however, our
definition of xabs differs from theirs in that the factor
mr2 is included based on the findings of
Fig. c and d. The performance of this parameterization
is evaluated in Fig. a and c. In
Fig. a, the parameterized values (shown with contours)
follow extremely well the reference values computed for the OHC
(shading). The relative errors Δβ/β are mostly below
1 %; errors larger than 3 % (and locally even > 10 %)
occur only for small snow grains (rvp< 50 µm)
at wavelengths λ> 1.2 µm. The
rms value of the relative errors (computed over 125 values of
λ∈ [0.199, 2.7 µm] and 48 roughly logarithmically spaced
values of rvp∈ [10, 2000 µm]) is 1.4 %.
Extinction efficiency Qext for the optimized habit
combination as a function of wavelength (λ) and volume-to-projected
area equivalent radius (rvp).
Comparison of (a) parameterized single-scattering
co-albedo β (contours) with the reference values computed for the OHC
(shading) and (b) parameterized asymmetry parameter g (contours)
with the reference values (shading). (c) Relative errors (%) in the
parameterized co-albedo. (d) Absolute errors in the parameterized
asymmetry parameter.
Asymmetry parameter
The asymmetry parameter is parameterized as
g=1-1.146mr-10.8[0.52-β]1.051+8xvp-1.5,
where the parameter values were determined by trial and error, with
the aim of minimizing the rms error in g.
The form of this parameterization reflects how g decreases with
increasing mr, increases with increasing absorption
(i.e. increasing co-albedo β) and increases slightly with increasing
size parameter xvp even at non-absorbing wavelengths, in part
because the diffraction peak becomes narrower.
In practice, the co-albedo β plays the most important
role cf., which explains the general increase
of g with increasing rvp in the near-IR region
(Fig. b). The parameterized values of g (shown with
contours in Fig. b) follow the reference values
(shading) very well. Note that when producing these results,
parameterized rather than exact β was used in
Eq. (). The differences from the reference are mostly below 0.001
at the weakly absorbing wavelengths up to
λ= 1.4 µm, and while larger differences up to
|g|= 0.007 occur at the strongly absorbing wavelengths
(Fig. d), the overall rms error is only 0.0019.
Phase function
The phase function parameterization consists of three terms,
P11θs=wdiffPdiffθs+wrayPrayθs+Presidθs,
which represent contributions due to diffraction, due to the ray
tracing part and a residual that corrects for errors made in
approximating the former two parts.
The weight factors for diffraction
wdiff and ray tracing wray are given by
wdiff=1Qextω≈12ω,wray=Qextω-1Qextω≈2ω-12ω,
where the latter form assumes Qext= 2 e.g..
It should be noted that in practice, the division of the phase
function expressed by Eq. () is conceptual rather
than rigorous. The fitting was based on the total phase
function rather than the diffraction and ray tracing parts
separately, as these two parts are not separated in the
database. The general aim of the fitting was to
minimize the rms errors in lnP11.
For diffraction, the HG phase function is used:
Pdiffθs=PHGgdiff,θs.
The HG phase function is given by
PHGg,θs=1-g21+g2-2gcosθs3/2,
and the asymmetry parameter gdiff is approximated as
gdiff=1-0.60/xvp=1-0.921/xp,
where we have utilized the relation xp≈ 1.535 xvp
specific to the OHC. Compared to the parameterization derived by
, Eq. () yields somewhat
lower values of gdiff, which to some extent
compensates for the fact that the actual shape of the
diffraction peak deviates from the HG phase function. Overall,
this treatment of diffraction is a rough approximation and
clearly not ideal for studies of very near-forward scattering,
but it serves well the current purpose. On one hand, it improves
the accuracy compared to the assumption of a delta spike, and on
the other hand, the HG phase function has a very simple Legendre expansion
PHGg,θs=∑n=0∞(2n+1)gnPncosθs,
where Pn denotes the nth order Legendre polynomial. This
facilitates greatly the use of PHG in radiative transfer
models such as DISORT .
The phase function for the ray tracing part is approximated as
Prayθs=w1PHGg1,θs+1-w1,
where the latter term 1 -w1 is intended to emulate the nearly flat
behaviour of P11 in the near-backward scattering directions. The
weight factor for the HG part is parameterized as
w1=1-1.53⋅max0.77-gray,01.2,
where gray is the asymmetry parameter for the ray tracing
(i.e. non-diffraction) part. It is derived from the condition
g=wdiffgdiff+wraygray,
which yields
gray=g-wdiffgdiffwray.
The total asymmetry parameter g is computed using Eq. ()
above. Finally, the asymmetry parameter g1 needed in Eq. () is
g1=gray/w1.
Cost function for the phase function parameterization as defined by
Eq. () for (a) the full parameterization
(Eq. ) and (b) without the term Presid.
The black solid line indicates, for reference, a co-albedo value of
β= 0.3, which approximately corresponds to a spherical albedo
of 0.03 for an optically thick snow layer.
While the sum of the first two terms of Eq. () already
provides a reasonably good approximation of the phase function (see
below), the fit can be further improved by introducing the
residual Presid, which is represented as a Legendre
series. It turns out that, except for cases with strong absorption,
a series including terms only up to n= 6 yields very good results
Presidθs=∑n=06(2n+1)anPncosθs,
provided that δ-M scaling is applied, with
a truncated fraction f=a6. Thus,
Presidθs≈Presid*θs=2fδ1-cosθs+(1-f)∑n=05(2n+1)an-f1-fPncosθs=2a6δ1-cosθs+∑n=05(2n+1)an-a6Pncosθs,
where δ is Dirac's delta function. What remains to be
parameterized, then, are the coefficients a0 … a6. A rough
but useful approximation is to express them as a simple function of
the co-albedo β and the asymmetry parameter g:
an=c1n+c2nβ+c3ng+c4nβg.
The parameterization coefficients cmn were determined by
minimizing the rms errors of an with the LAPACK subroutine DGELS,
and they are given in Table . Note specifically
that the coefficients cm0 and cm1 are all 0. The formulation of
Pdiff and Pray ensures that the phase function
(Eq. ) is correctly normalized and that its asymmetry
parameter is consistent with Eq. () even without considering
Presid; therefore a0=a1= 0. Equivalently, the Legendre
expansion may be replaced by an ordinary polynomial. This yields
Presidθs≈Presid*θs=2a6δ1-cosθs+∑n=05bncosθsn,
where
bn=d1n+d2nβ+d3ng+d4nβg.
Here, the coefficients dmn were obtained directly based on
the coefficients cmn in Eq. () by
writing out the Legendre polynomials in Eq. ().
Their numerical values are given in Table . In summary,
the phase function parameterization reads
P11θs=wdiffPHGgdiff,θs+wrayw1PHGg1,θs+wray1-w1+Presidθs,
where Presid(θs) is given by
Eq. () or, equivalently, by Eq. ().
Finally, it is worth noting how this parameterization can be used in
DISORT, when applying a “δ-NSTR-stream” approximation for
radiative transfer, NSTR being the number of streams. In this case,
DISORT assumes by default a truncation factor f=aNSTR. If
NSTR > 6, the Legendre expansion for Presid in
Eq. () should be formally extended to n= NSTR,
with an=a6 for n= 7 … NSTR. Thus the Legendre
coefficients input to DISORT become
pn=1,forn=0wdiffgdiffn+wrayw1g1n+an,for1≤n≤6wdiffgdiffn+wrayw1g1n+a6,for7≤n≤NSTR,
where we have utilized the Legendre expansion of the HG phase function
in Eq. ().
Parameterization coefficients appearing in Eq. ().
To provide a compact view of how the phase function parameterization performs,
we define, analogously to Eq. (), a cost function as the rms
error of the natural logarithm of the phase function,
cost=∫0∘180∘lnP11param-lnP11OHC2sinθsdθs∫0∘180∘sinθsdθs,
where P11param is the parameterized phase function and
P11OHC is the reference value, defined here as the
“exact” phase function computed for the
OHC. Figure a shows the cost function for the full phase
function parameterization, and Fig. b shows that for a simpler
parameterization that includes only the first two terms of
Eq. () (i.e. Presid is excluded). Note
that the parameterized phase function is computed here using
parameterized (rather than exact) values of Qext, β and g.
Most importantly, Fig. a shows that in a large part of
the wavelength and size domain, the accuracy of the full
parameterization is very high, with cost function values ≤ 0.03.
This corresponds to a typical relative accuracy of 3 %
in the computed phase function, as compared with the reference values
for the OHC. The primary exception is that substantially larger errors
occur for large snow grains at the strongly absorbing wavelengths in
the near-IR region. In broad terms, the accuracy starts to degrade
appreciably when β> 0.3, that is, in cases in which snow
reflectance is quite low (β= 0.3 corresponds roughly to
a spherical albedo of 0.03 for an optically thick snow layer). At the
largest wavelengths considered (λ> 2.5 µm),
somewhat larger values of the cost function also occur for smaller
values of rvp and β. The cost function for the
simplified parameterization (Fig. b) shows mainly the
same qualitative features as the full parameterization in
Fig. a; however, the cost function values in the weakly
absorbing cases are ≈ 0.07, in contrast to the values of
≈ 0.03 for the full parameterization.
Parameterization coefficients appearing in Eq. ().
Examples of the reference phase function computed for the OHC (black
lines) and of the parameterized phase function for the full parameterization
(red lines), the simplified parameterization without the term
Presid in Eq. () (blue lines) and the
Henyey–Greenstein phase function with asymmetry parameter defined by
Eq. () (dashed green lines) for nine combinations of
wavelength λ and volume-to-projected area equivalent radius
rvp. For reference, the values of single-scattering
co-albedo β, asymmetry parameter g and cost functions for the full
parameterization (cost1), for the simplified parameterization (cost2) and for
the Henyey–Greenstein phase function (cost3) are listed in each
panel.
Albedo of a semi-infinite snow layer for direct incident radiation
with the cosine of zenith angle μ0= 0.5. (a) Reference
values computed for the OHC (shading) and values for the full snow optics
parameterization (contours). The difference (b) between the
parameterization and the reference, (c) between distorted Koch
fractals (t= 0.18) and the reference and (d) between spheres
and the reference. Note that the colour scale differs between the figure
panels.
Figure displays examples of phase function for nine
combinations of λ and rvp. In the weakly absorbing
cases in Fig. a–c, and also at the more strongly
absorbing wavelength λ= 1.50 µm for
rvp= 10 µm and
rvp= 100 µm (Fig. d and e), the
full parameterization follows extremely well the reference phase
function computed for the OHC, to the extent that the curves are
almost indistinguishable from each other. Even at
λ= 2.00 µm, the deviations from the reference
are generally small in the cases with relatively small snow grains
(rvp= 10 µm and
rvp= 100 µm; Fig. g and h),
although backward scattering is slightly overestimated in the latter
case. In contrast, in cases with very strong absorption and large snow
grains (rvp= 1000 µm for
λ= 1.50 µm and λ= 2.00 µm in
Fig. f and i) there are more substantial deviations from
the reference. Here, the parameterized phase function is generally
underestimated in the backscattering hemisphere and overestimated at
θs< 30∘ especially for
λ= 2.00 µm, rvp= 1000 µm. Furthermore, the Legendre
expansion in Presid leads to oscillations in the
backscattering hemisphere which do not occur in the reference phase
function. Again, it should be noted that the largest errors occur in
cases in which snow is very “dark”: the spherical albedo
corresponding to the cases in Fig. f and i is only ∼ 0.005.
In many respects, the simplified parameterization (i.e. without
Presid) produces quite similar phase functions as the full
parameterization. Two differences can be noted. First, the simplified
parameterization does not capture the slight increase in phase
function at angles larger than θs≈ 120–130∘, which is present in the reference and
full parameterization phase functions and was also suggested by
the CLIMSLIP data for blowing snow at λ= 0.80 µm,
along with the other phase functions in
Fig. b. Second, in the cases with very strong
absorption (Fig. f and i) the simplified phase function
avoids the oscillations seen in the full parameterization.
The utility of providing a phase function parameterization is further
demonstrated by showing in Fig. , for comparison, the HG
phase function computed using the asymmetry parameter from Eq. ().
The differences from the reference phase function are systematic.
The scattering in the diffraction peak is underestimated (although
this is not properly seen from Fig. ), but otherwise
forward scattering is overestimated until a scattering angle of
≈ 35–80∘, depending on the case. Conversely, at sideward
and backscattering angles, scattering is underestimated.
Consequently, the cost function values for the HG phase function given
in Fig. substantially exceed those for both the full
and simplified phase function parameterizations.
Radiative transfer applications
In this section, we consider the impact of snow optics assumptions on
snow spectral albedo A and reflected radiances L↑.The
purpose is, on one hand, to evaluate the accuracy of the proposed snow
SSP parameterization and, on the other hand, to compare the results
obtained with three shape assumptions: spheres, second-generation Koch
fractals (distorted with t= 0.18) and the OHC proposed
here. Throughout this section, the results for the OHC are used as the
reference, although it is clear that they cannot be considered an
absolute benchmark for scattering by snow. The radiative transfer
computations were performed with DISORT (with 32 streams,
δ-M scaling included), assuming an optically thick
(i.e. semi-infinite) layer of pure snow with a monodisperse size distribution.
Like most other solar radiative transfer studies involving snow,
close-packed effects are ignored in the
calculations. It has been shown by that,
at least as a first approximation, they do not have a pronounced
impact on the snow reflectance.
First, snow albedo as a function of λ and rvp is
considered in Fig. . Direct incident radiation with
a cosine of zenith angle μ0=cosθ0= 0.5 is
assumed. Figure a demonstrates the well-known
features of snow albedo: the values are very high in the UV and
visible region and decrease with increasing particle size in the
near-IR. The results computed using the parameterized snow optical
properties Qext, β, g and P11 are almost
indistinguishable from those obtained using the “exact” optical
properties for the OHC. The differences between these two are mostly
within 0.002 (Fig. b), although larger differences up
to 0.02 occur for very small snow grains (rvp≈ 10–20 µm)
at wavelengths with strong absorption by snow
(λ> 1.4 µm). These results are only weakly
sensitive to the assumed direction of incident radiation. Furthermore,
while the parameterized albedo values were computed using the full
phase function parameterization, the values for the simplified
parameterization (without Presid in Eq. ())
differed very little from them, mostly by less than 0.001.
For distorted Koch fractals, the albedo values are higher than those
for the OHC, but the difference is rather small, at most 0.017
(Fig. c). Conversely, for spheres the albedo values
are lower, with largest negative differences of -0.08 from the
reference (Fig. d). This stems from the higher
asymmetry parameter of spheres, which is only partly compensated by
their lower co-albedo (Fig. ). To put it another
way, for a given albedo A in the near-IR region, a smaller (slightly
larger) particle size is required for spheres (for distorted Koch
fractals) than for the OHC.
Root-mean-square errors in ln(radiance) (Eq. )
for (a)–(c) the full parameterization and
(d)–(f) the simpler parameterization without the term
Presid in the phase function, as compared with reference
calculations for the OHC, for three directions of incident radiation (cosine
of zenith angle μ0= 0.8, μ0= 0.4 and
μ0= 0.1). (g) and (h) show the
respective differences from the reference calculations for distorted Koch
fractals (t= 0.18) and spheres (for μ0= 0.4
only).
(a)–(c) Angular distribution of reflected
radiances for the OHC for a single wavelength
λ= 0.80 µm and a single particle size
rvp= 200 µm. The yellow sphere indicates the
cosine of zenith angle for the incident radiation (μ0= 0.8,
μ0= 0.4 and μ0= 0.1 for (a)–(c)
respectively). The azimuth angle for the incident radiation is
ϕ0= 0∘. (d)–(f) and
(g)–(i) show the fractional differences in reflected
radiances (in %) from the OHC for distorted Koch fractals with
t= 0.18 and for ice spheres respectively. (j)–(l)
show the differences from the OHC for the Henyey–Greenstein phase function
(with g computed using Eq. and β using
Eq. ), (m)–(o) for the full snow optics
parameterization and (p)–(r) for the simpler
parameterization without Presid in Eq. (). Note
that the colour scale in (m)–(r) differs from that
in (d)–(l).
To compare the simulated radiance distributions to the reference, we
next consider the root-mean-square error in the logarithm of reflected
radiances integrated over the hemisphere:
LOGRMSE=12π∫02π∫0π/2lnL↑(θ,ϕ)-lnLOHC↑(θ,ϕ)2sinθdθdϕ,
where θ and ϕ denote the zenith angle and azimuth angle
respectively and LOHC↑ is the radiance in the
reference computations for the OHC. Figure a–c show
LOGRMSE as a function of particle size and wavelength for the full
parameterization for three directions of incident radiation
(μ0= 0.8, μ0= 0.4 and μ0= 0.1,
corresponding to θ0= 36.9∘,
θ0= 66.4∘ and θ0= 84.3∘). For weakly absorbing wavelengths up to
λ= 1.4 µm, the performance of the
parameterization is extremely good for all particle sizes, with values
of LOGRMSE < 0.01 for μ0= 0.8 and μ0= 0.4 and
between 0.01 and 0.02 for μ0= 0.1. LOGRMSE ∼ 0.01
implies a typical relative accuracy of ∼ 1 % in the reflected
radiances. The accuracy in radiances at weakly absorbing wavelengths
is even higher than that in the phase function (Fig. a)
because strong multiple scattering diminishes the effect of phase
function errors. At wavelengths λ> 1.4 µm,
LOGRMSE increases, not only due to larger phase function errors but
also because multiple scattering is reduced due to stronger
absorption. Even here, LOGRMSE stays mainly below 0.05 for relatively
small snow grains (rvp< 100 µm), but
substantially larger errors occur in the cases with large and strongly
absorbing grains, consistent with the modest accuracy of the phase
function parameterization in these cases (Fig. a). These
errors depend only weakly on μ0. It should be noted that the
largest relative errors occur in cases where the reflected radiances
and radiance errors are small in an absolute sense and probably matter
little for practical applications.
Values of LOGRMSE obtained using the simplified phase function
parameterization are shown in Fig. d–f. Consistent
with the phase function errors (cf. Fig. a vs. b), the simplified
parameterization is slightly less accurate in simulating reflected
radiances than the full parameterization except for the most strongly
absorbing cases. Nevertheless, the accuracy is quite high for the
weakly absorbing cases; LOGRMSE ranging from ∼ 0.01 (or even less)
for μ0= 0.8 to ∼ 0.03 for μ0= 0.1.
For comparison, Fig. g and h show LOGRMSE computed
for distorted Koch fractals and spheres (for μ0= 0.4 only).
Unsurprisingly, LOGRMSE is generally smaller for Koch fractals than
for spheres (e.g. 0.05–0.10 in weakly absorbing cases compared
to ∼ 0.20 for spheres). In both cases, again excepting large
particles at strongly absorbing wavelengths, the values of LOGRMSE are
substantially larger than those associated with the snow SSP
parameterization. This indicates that in general, numerical fitting
errors in the parameterization are a minor issue in
comparison with the radiance differences associated with
different shape assumptions.
Examples of the angular distribution of reflected radiances are given
in Figs. and . Here, only a single
particle size rvp= 200 µm is considered, and
the azimuth angle for incident radiation is ϕ0= 0∘. In
Fig. , results are shown for three zenith angles of
incident radiation, corresponding to μ0= 0.8, μ0= 0.4 and
μ0= 0.1, for a single wavelength
λ= 0.80 µm. In Fig. , three
wavelengths are considered (λ= 0.30, 1.40 and 2.20 µm) but for
μ0= 0.4 only. In each figure, panels a–c display the
distribution of reflected radiances in the reference calculations for
the OHC, while the remaining panels show the relative differences from
the reference for distorted Koch fractals with t= 0.18 (panels d–f),
for spheres (g–i), for the Henyey–Greenstein phase function (j–l),
for the full snow SSP parameterization (m–o) and for the simpler
parameterization without Presid in Eq. () (p–r).
For brevity, only some main points are discussed.
First, it is seen, consistent with Fig. , that in
general the radiance distribution for spheres differs more from the
reference than the distribution for Koch fractals does. For example,
for λ= 0.80 µm and μ0= 0.4 both positive and
negative differences larger than 50 % occur for spheres
(Fig. h), while for Koch fractals the differences
exceed 10 % only locally (Fig. e). Furthermore, in
the same case, the radiance errors are < 1 % almost throughout
the (θ, ϕ) domain for the full parameterization
(Fig. n) and mostly < 2 % even for the simplified
parameterization (Fig. q). In contrast, when the HG
phase function is employed in the calculations, the differences from the
reference reach locally 30 and -40 % (Fig. k).
Second, while the results noted above for
λ= 0.80 µm and μ0= 0.4 are also
mostly valid for μ0= 0.8 and μ0= 0.1 and for
λ= 0.30, 1.40 and 2.20 µm, some quantitative
differences can be noted. When μ0 decreases from 0.8 to 0.1,
the pattern of reflected radiances becomes increasingly
non-uniform and more sensitive to both the assumed particle
shape and the errors in phase function parameterization. This
occurs because the relative role of first-order scattering
increases e.g.. For the same reason,
the sensitivity of the radiance pattern to the phase function
increases with increasing absorption. Thus, while the
qualitative features are mostly similar at all wavelengths
considered here, the relative differences are generally larger
at λ= 1.40 µm and
λ= 2.20 µm than at
λ= 0.30 µm and
λ= 0.80 µm. Especially at the
wavelength λ= 2.20 µm, at which snow
absorption is quite strong and the albedo for the OHC is only 0.11,
the radiance pattern is dominated by first-order
scattering and is thus very sensitive to the details of the phase
function. In a relative (though not absolute) sense, the errors
in parameterized radiances are also somewhat larger than at the
other wavelengths considered (Fig. o and r).
As Fig. but for three wavelengths
λ= 0.30, 1.40 and 2.20 µm for a single value of the
cosine of zenith angle for incident radiation μ0= 0.4 and a
single particle size rvp= 200 µm.
Third, even at weakly absorbing wavelengths, the role of first-order
scattering is clearly discernible: many differences in the pattern of
reflected radiances can be traced directly to phase function
differences. For example, considering the results for
λ= 0.80 µm for both μ0= 0.4 and
μ0= 0.1, we note the following.
Three regions appear in the radiance differences between distorted
Koch fractals and the OHC in Fig. e and f. Going from
left to right, negative radiance differences occur at large values
of θ and small values of ϕ (roughly for
θ> 65∘ and ϕ< 20∘), followed by a region
of positive differences and another region of negative differences
(roughly for θ> 40∘, ϕ> 140∘). These regions
occur because the phase function for Koch fractals is larger than
that for the OHC at intermediate scattering angles
(29∘≤θs≤ 134∘)
but smaller in the near-forward and near-backward directions.
For spheres in Fig. h and i, the reflected
radiances greatly exceed those for the OHC for roughly θ> 60∘,
ϕ< 40∘ because the phase function for spheres
is generally larger than that for the OHC for θs< 54∘.
Conversely, at larger θs the phase
function for spheres is (mostly) considerably smaller than that for
the OHC. This results in generally smaller reflected radiances for
spheres in most of the (θ, ϕ) domain with ϕ> 50∘. As an exception, the icebow feature for spheres at
θs≈ 135∘ results in an arc with larger
radiances for spheres than for the OHC.
For the HG phase function, the pattern of overestimated radiances
up to ϕ∼ 60∘ and underestimated radiances at larger
azimuth angles (Fig. k and l) arises because the HG
phase function exceeds that for the OHC for
θs< 80∘ and falls below it at larger scattering
angles (see also Fig. ).
Summary
In this work, measurements of angular distribution of scattering by
blowing snow made during the CLIMSLIP campaign in Svalbard were used
to select a shape model for representing the single-scattering
properties of snow. An optimized habit combination
consisting of SR droxtals, aggregates of SR plates
and strongly distorted Koch fractals was selected. The SSPs
(extinction efficiency Qext, single-scattering co-albedo β,
asymmetry parameter g and phase function P11)
were then computed for the OHC as a function of wavelength and snow
grain size. Furthermore, parameterization equations were developed for
the SSPs for the wavelength range
λ= 0.199–2.7 µm and for snow grain
volume-to-projected area equivalent radii rvp= 10–2000 µm. The parameterizations are expressed in terms
of the size parameter and real and imaginary parts of refractive
index. The relative accuracy of the parameterization, as compared with
the reference calculations for the OHC, is very high for the
single-scattering co-albedo and the asymmetry parameter. This is also
true for the phase function parameterization in weakly and moderately
absorbing cases, while in strongly absorbing cases (mainly for
β> 0.3) the accuracy deteriorates. Such strongly absorbing cases
are, however, associated with small values of snow albedo and
reflected radiances.
The SSPs and the resulting snow albedo and reflected radiances for the
OHC were compared with two previously used shape assumptions for snow
grains, spheres and second-generation Koch fractals. The asymmetry
parameter for the OHC is distinctly smaller than that for spheres but
slightly higher than that for Koch fractals. Consistent with this,
snow albedo for the OHC is generally substantially higher (slightly
lower) than that for spheres (Koch fractals) for a given snow grain
size rvp. Also for the distribution of reflected
radiances, spheres differ more from the OHC than Koch fractals do.
The main limitation of the current work is that the SSP parameterization
is based on a rather limited observational data set. The OHC was selected
using scattering measurements at a single wavelength
λ= 0.80 µm for only two cases with blowing snow.
This raises several potential issues:
The choice of the OHC based on scattering measurements only implies
that it most probably does not represent properly the actual
distribution of snow grain shapes in blowing snow (or snow on ground).
It also neglects the potential dependence of snow grain shapes
on their size. Therefore, there is no guarantee
that it represents the snow SSPs equally well at other wavelengths or
for all snow grain sizes.
Since absorption is very weak at λ= 0.80 µm, the
observations do not constrain properly absorption by snow.
Therefore, we cannot expect that our parameterization of β
(Eq. ) predicts precisely the actual values for snow.
However, we do expect that it captures reasonably the systematic
difference between non-spherical snow grains and spheres: in general
β is larger for non-spherical particles.
It is also possible that the snow grain shapes, and therefore the SSPs
of snow on ground, might differ from those of blowing snow, and they might
well vary from case to case, depending on how much metamorphosis
the snow has experienced.
All these issues point to the need for validation of the derived
parameterization against actual snow reflectance measurements in future work.
In spite of the concerns mentioned above, it seems reasonable to
assume that the OHC selected here provides a substantially better
basis for representing the SSPs of snow than spheres do. Moreover,
the parameterization equations provided in this paper are analytic and simple
to use. A Fortran implementation of the snow SSP parameterizations
is available at https://github.com/praisanen/snow_ssp.
To conclude, this paper describes a first-of-its-kind parameterization for
representing the SSPs of snow in the solar spectral region.
The parameterization is provided in hope that it will be useful,
especially to those researchers that still use spherical particles for
computing the radiative effects of snow. Nevertheless, it
should definitely not be viewed as the “final solution” to the treatment of
SSPs of snow. We hope that the present work will
inspire the future development of snow SSP parameterizations based
on more comprehensive data sets. Furthermore, at least in principle,
it would be desirable to replace the current approach (where the shape
distribution of snow grains is selected based on scattering measurements
only) with an approach that more directly links the snow grain shapes
to those actually observed. This would require, first, the
parameterization of the size–shape distribution of snow grains
based on observations and, second, the computation and parameterization
of their SSPs. The main challenge in such an approach is the treatment of
irregular grains, which are very common in snow.
Extrapolation of single-scattering properties
The largest value of volume-to-projected area equivalent radius for
which the SSPs are defined for aggregates of 10 plates in the
database is rvp,max= 653 µm, which falls below the
upper limit of 2000 µm considered for the OHC. Thus, to
extend the SSPs for the OHC to sizes up to
rvp= 2000 µm, we extrapolated the SSPs for the
aggregates of plates based on how the SSPs depend on size for Koch fractals:
Qext,aggregatervp=2+Qext,aggregatervp,lim-2⋅rvp,limrvp,βaggregatervp=βaggregatervp,lim⋅βfractalrvpβfractalrvp,lim,gaggregatervp=1-1-gaggregatervp,lim⋅1-gfractalrvp1-gfractalrvp,lim,P11,aggregatervp,θs=P11,aggregatervp,lim,θs⋅P11,fractalrvp,θsP11,fractalrvp,lim,θs.
Here, rvp,lim= 650 µm. While this
is an ad hoc approach, the resulting uncertainty in the SSPs for the
OHC (in which the aggregates of plates have a weight of 26 %) is
most likely small. When the extrapolation was based on droxtals
instead of Koch fractals, this changed the values of g by at most 0.0025
and β by at most 0.006 (or 1.4 % in relative terms).
List of abbreviations and symbols.
CLIMSLIPCLimate IMpacts of Short-Lived pollutants In the Polar regionCPIcloud particle imagerCScompletely smooth particles DISORTDiscrete Ordinates Radiative Transfer Program for a Multi-Layered Plane-Parallel Medium HGHenyey–Greenstein LAPACKLinear Algebra PackageLOGRMSEroot-mean-square error in the logarithm of reflected radiancesMRmoderately rough particles OHCoptimized habit combinationPNpolar nephelometerSSPssingle-scattering propertiesSRseverely rough particles βsingle-scattering co-albedo = 1 - single-scattering albedoδDirac's delta functionθzenith angleθ0zenith angle for incident radiationθsscattering angleλwavelengthμ0cosine of zenith angle for incident radiationνpower-law index in the Legendre polynomial expansion of the correlation function of radius for Gaussian random spheresξnon-dimensional absorption parameter (Eq. )σrelative SD of radius for Gaussian random spheresϕazimuth angleωsingle-scattering albedoftruncated fraction of phase function in δ-M scaling gasymmetry parameterg1asymmetry parameter for the Henyey–Greenstein part in Eq. (), defined by Eq. ()gdiffasymmetry parameter for diffraction (Eq. )grayasymmetry parameter for the ray-tracing part (Eq. )miimaginary part of refractive indexmrreal part of refractive indexnmaxdegree of truncation of the Legendre polynomial expansion of the correlation function of radius for Gaussian random spheresPprojected areaP11phase functionP11refreference phase function constructed from CLIMSLIP data (Eq. )P11OHCphase function for the optimized habit combinationPHGHenyey–Greenstein phase function (Eqs. , )Pdiffparameterized phase function for diffraction (Eq. )Prayparameterized phase function for the ray tracing part (Eq. )Presidresidual in the phase function parameterization (Eq. )Presid*residual in the phase function parameterization, truncated for δ-M scaling (Eqs. , )Pnnth order Legendre polynomialQextextinction efficiencyrvpvolume-to-projected area equivalent radiustdegree of distortion for Koch fractalsVvolumew1weight factor for the Henyey–Greenstein part in Eq. (), defined by Eq. ()wdiffweight factor for the diffraction part in the parameterized phase function (Eqs. , ), defined by Eq. ()wrayweight factor for the ray tracing part in the parameterized phase function (Eqs. , ), defined by Eq. ()xsize parameterxabssize parameter for absorption (Eq. )xpsize parameter defined with respect to the projected area equivalent radiusxvpsize parameter defined with respect to the volume-to-projected area equivalent radius (Eq. )Acknowledgements
P. Räisänen was supported by the Nordic Centre of Excellence
for Cryosphere-Atmosphere Interactions in a changing Arctic climate
(CRAICC) and the Academy of Finland (grants nos. 140915 and 254195) and
T. Nousiainen by the Academy of Finland (grant no. 255718) and the Finnish
Funding Agency for Technology and Innovation (Tekes; grant no. 3155/31/2009).
A. Kokhanovsky acknowledges the support of University
of Bremen and project CLIMSLIP funded by BMBF. The CLIMSLIP field
campaign was funded by the French Agence Nationale de la Recherche
(ANR) and the Institut Polaire Français Paul Emile Victor
(IPEV). We gratefully acknowledge the NILU and the Norsk
Polarinstitutt for their technical assistance during the field
campaign at Mount Zeppelin Station. Andreas Macke (Leibniz Institute
for Tropospheric Research, Germany) is thanked for making available
his ray tracing code. Ping Yang and Bingqi Yi (Texas AM University)
are thanked for providing the database.
Last but not least, Bastiaan van Diedenhoven and an anonymous referee
are thanked for their helpful comments on the original manuscript.
Edited by: P. Marsh
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