AMTAtmospheric Measurement TechniquesAMTAtmos. Meas. Tech.1867-8548Copernicus GmbHGöttingen, Germany10.5194/amt-8-1913-2015On the microwave optical properties of randomly oriented ice hydrometeorsErikssonP.patrick.eriksson@chalmers.sehttps://orcid.org/0000-0002-8475-0479JamaliM.MendrokJ.https://orcid.org/0000-0002-0032-2021BuehlerS. A.https://orcid.org/0000-0001-6389-1160Earth and Space Sciences, Chalmers University of
Technology, 41296 Gothenburg, SwedenDivision of Space Technology, Lulea University of Technology,
98128 Kiruna, SwedenMeteorological Institute, Center for Earth System Research
and Sustainability, University of Hamburg, Bundesstrasse 55,
20146 Hamburg, GermanyP. Eriksson (patrick.eriksson@chalmers.se)5May201585191319332December201421December201431March201511April2015This 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://amt.copernicus.org/articles/8/1913/2015/amt-8-1913-2015.htmlThe full text article is available as a PDF file from https://amt.copernicus.org/articles/8/1913/2015/amt-8-1913-2015.pdf
Microwave remote sensing is important for observing the mass of ice
hydrometeors. One of the main error sources of microwave ice mass retrievals
is that approximations around the shape of the particles are unavoidable. One
common approach to represent particles of irregular shape is the soft
particle approximation (SPA). We show that it is possible to define a SPA
that mimics mean optical particles of available reference data over narrow
frequency ranges, considering a single observation technique at the time,
but that
SPA does not work in a broader context. Most critically, the required air
fraction varies with frequency and application, as well as with particle
size. In addition, the air fraction matching established density
parameterisations results in far too soft particles, at least for frequencies
above 90 GHz. That is, alternatives to SPA must be found.
One alternative was recently presented by Geer and Baordo (2014). They used a
subset of the same reference data and simply selected as “shape model” the
particle type giving the best overall agreement with observations. We present
a way to perform the same selection of a representative particle shape but
without involving assumptions on particle size distribution and actual ice
mass contents. Only an assumption on the occurrence frequency of different
particle shapes is still required. Our analysis leads to the same selection
of representative shape as found by Geer and Baordo (2014). In addition, we
show that the selected particle shape has the desired properties at
higher frequencies as well as for radar applications.
Finally, we demonstrate that in this context the assumption on particle
shape is likely less critical when using mass equivalent diameter to
characterise particle size compared to using maximum dimension, but a better
understanding of the variability of size distributions is required to fully
characterise the advantage.
Further advancements on these subjects are presently difficult to achieve
due to a lack of reference data. One main problem is that most available
databases of precalculated optical properties assume completely random
particle orientation, while for certain conditions a horizontal alignment is
expected. In addition, the only database covering frequencies above
340 GHz has a poor representation of absorption as it is based on outdated
refractive index data as well as only covering particles having a maximum
dimension below 2 mm and a single temperature.
Introduction
Microwave techniques are gaining in importance for satellite observations of
hydrometeors, i.e. clouds and precipitation. The main measurement target of
microwave sensors is mass content estimates, possibly in the form of a
precipitation rate. The detection mechanism used (absorption or scattering)
depends on phase (liquid, ice or mixed), frequency and whether the
instrument is active or passive. For example, for non-precipitating liquid
droplets passive measurements rely on absorption, while radars rely on
backscattering. The signature of ice hydrometeors in passive data is a mix of
scattering and absorption features, whereas in general the scattering part
dominates (Sect. ).
The accuracy of the retrievals depends on technique applied and a number of
variables, including observational noise and limitations in the radiative
transfer code used. However, for ice mass retrievals the main
retrieval error sources are frequently uncertainties associated with the
microphysical state of the particles, i.e. phase, size, shape and orientation.
This study focuses on the impact of assumed ice particle shape, probably the microphysical quantity with the least hope of being retrievable
based on microwave data alone. Information on particle size can be obtained by
combining data from different frequencies
,
while the phase of the particles is largely determined by the atmospheric
temperature. Measuring horizontal and vertical polarisation simultaneously
reveals whether the particles have a tendency to horizontal alignment or their
orientation is completely random
e.g..
Shape is normally not a critical aspect for purely liquid particles, as they
are quasi-spherical throughout. The deviation from a strict spherical shape
increases with droplet size and fall speed. However, the shape of
frozen hydrometeors is highly variable, both as single crystals (needles,
plates, columns, rosettes, dendrites, etc.) and as aggregates see
reviews by. The shape is frequently
denoted as the habit. It is unlikely that the air volume sampled contains a
single ice particle shape, i.e. a habit mix can be expected. Furthermore, this
mix normally varies with particle size. In principle, the shape of each
particle should be known to avoid a related retrieval error, but this is not a
feasible goal. Instead some “shape model” must be applied and the main aim of
this study is to examine such models for microwave sounding of pure ice
hydrometeors.
Considering the ice particles to be solid spheres is probably still the main
microwave shape model. This approach is, for example, used in the standard
2B-CWC-O CloudSat retrievals by . It is also
applied in the Community Radiative Transfer Model (CRTM;
). Accordingly, retrieval systems
e.g. and radiance
assimilation based on CRTM inherit the assumption of solid spheres.
Furthermore, this particle type has throughout been assumed in cloud ice
retrievals based on limb sounding data
. A main reason for the
popularity of this shape model is that the single-scattering properties are
simply calculated by well-established Mie codes.
Another common model is the “soft particle approximation” (SPA) where the
particles are treated to consist of a homogeneous mix of ice and air. This
approach requires that the volume or mass fraction of air and the corresponding
refractive index of the ice–air mix are determined; see
Sects. and . SPA could in principle be
used with a range of simplified particle forms, but it seems that only spheres
and spheroids have been used so far. Spheroids are not treated by Mie theory
but are covered by the also computationally efficient T-matrix method
. One application of SPA for practical retrievals is
. A more recent example is ,
arguing for using a soft spheroid model for cloud radar inversions. In
addition, SPA has widely been used in studies to mimic measured radiances by
radiative transfer tests
e.g.
in which the air fraction (AF) is either set to be fixed or derived from some
parametric relationship between particle size and effective density.
Single-scattering properties for arbitrary particle shapes can be calculated e.g. by
the discrete dipole approximation (DDA; ). DDA is used for incorporating realistic particle shapes in the
retrievals presented by . This study is likely the most
ambitious microwave retrieval set-up with regards to particle shape, but it
deals only with a specific measurement campaign and it does not provide any
general conclusions. Publicly available databases of DDA results for common
particle shapes are reviewed in Sect. . These databases have been
used in different ways: for example, used one of the
databases to estimate snowfall detection limits of several observation systems.
Furthermore, the two main databases were used by to
test whether simulations could recreate some collocated radar and passive microwave
data when applying different particle shapes. In a similar study by
, only passive data were considered but a more wide set
of frequencies and atmospheric conditions were investigated. They found that a
sector-like snowflake model gave the smallest overall error for the simulations
performed. This choice will replace a SPA treatment as the default for the snow
hydrometeor category in the RTTOV-SCATT package
. In Sect. an alternative version of
the approach of is tested that does not involve any
assumption on particle size distribution (PSD) or actual ice masses.
DDA calculations have also been used in a more direct manner to investigate
shape aspects. For example, compared DDA and solid sphere
results and claimed that particle shape is less critical for size parameters
below 2.5 (see Eq. () for the definition), but only a few DDA
shapes and frequencies were considered, radar backscattering was ignored and
no quantitative error estimate was given. Comparisons between DDA and
corresponding SPA data are found in e.g. ,
and , but the results have
a limited scope throughout and we have found no comprehensive analysis of the
limitations of SPA. An important result was obtained by
, showing that an optimal “softness parameter”,
to be applied in a SPA framework, varies with frequency. The same conclusion
has also been reached in indirect ways by others, as pointed out by
.
From the perspective of mass retrievals it is most practical to characterise
the size of the particles through their equivalent mass ice sphere diameter,
de:
de=6mρiπ3,
where m is the particle mass and ρi is the density of (solid)
ice. We define the size parameter, x, correspondingly:
x=πdeλ,
where λ is the wavelength at which the measurement is performed.
In microwave sounding, the mass is inferred from estimated extinction or
backscattering coefficients. Any type of such coefficient, γ, can be
expressed as
γ=∫0∞N(de)σγ‾(de)dde,
where N(de) is the PSD and
σγ‾(de) is the local average cross section for
particles having a mass matching de. In its turn,
Eq. ()
implies that trying to estimate σγ‾(de) from
observed satellite data as done in requires a good knowledge of
both the mass of frozen hydrometeors and the PSD.
Another common way to express particle size is by the maximum diameter, dm.
We start the study by using de, because usage of dm demands that
the relation between dm and particle mass must be introduced. Such
relationships depend on particle shape, and for the basic purpose of this study
that is a problematic complication. By using de, particle shape only
influences σγ‾(de). However, as dm is probably
more frequently used than de, this alternative to characterise particle sizes
is the last step of the study (Sect. ).
In summary, our scope is the approximation of particle shape in microwave
retrievals of the mass of pure ice hydrometeors. Focus is put on SPA and the
basic conclusion of . Both passive and active
measurements are considered, because merging information from different sensor types
is already in use e.g., and
such synergies should in the future just grow in importance. The practical aim
can be seen as finding a shape model that gives a good estimate of
σγ‾(de), for relevant optical properties, over a
large range of particle sizes, frequencies, measurement techniques and possible
habit mixes. Existing DDA data are reviewed and used as reference. Only
complete random orientation is treated because most established publicly
available DDA databases are currently limited to this assumption on
orientation.
Furthermore, compared to earlier similar works, much higher attention is given to
frequencies above 200 GHz. Ice mass retrievals are already performed at
sub-millimetre wavelengths using limb sounding and airborne sensors .
Additionally, a strong motivation for this assessment is upcoming
sub-millimetre instruments: the European ISMAR (International Sub-Millimetre
Airborne Radiometer) airborne instrument and the ICI (Ice Cloud Imager) sensor
will be part of the next series of Metop satellites. ICI is a down-looking
sub-millimetre cloud ice sensor, a concept that has already been described in
several articles but for which so far no actual satellite sensor has
been available. This study is part of our overall effort to build the
scientific foundation for the analysis of first the ISMAR airborne and then
eventually the ICI satellite data.
Refractive index
Any calculation of single-scattering properties, i.e. independently whether Mie,
T-matrix or DDA calculations are performed, requires that the refractive index
is specified. Parameterisations and expressions related to the refractive index
of ice at microwave frequencies are reviewed in this section. Both the real
(n′) and imaginary (n′′) part of the complex refractive index n are
relevant. Some relationships are more easily expressed in terms of the
(relative, complex) dielectric constant, ϵ. Neglecting magnetic effects,
which is a good assumption here, this quantity is related to the (complex)
refractive index as
n=ϵ.
Pure ice models
Providing complex refractive index practically over the complete
electromagnetic spectrum in the form of data tables, ,
in the following referred to as W84, has been a long-term standard in
atmospheric science for the refractive index of pure water ice.
H91 developed a
parameterisation for microwave frequencies up to 1 THz based on Debye and
Lorenz theories with parameters fitted from measured data. The parameterisation
was incorporated in the MPM93 atmospheric propagation model by
. Compared to W84, H91 generally predicts
lower n′′ for frequencies < 350 GHz (see Fig. ,
right panel). Consistent with measurements it predicts a stronger increase of
n′′ with temperature than W84 at sub-millimetre frequencies.
Real (left) and imaginary (right) part of the refractive index of
pure ice as a function of frequency, according to
,
,
, ,
and . The
temperature is set to 266 K.
In the advent of sub-millimetre observation techniques,
J04 added a higher-order frequency term to H91, as
suggested by , in order to cover frequencies
up to 3 THz. Resulting n′′ agree well with H91 until about 1 THz, as shown
in Fig. . Beyond 1 THz, where H91 claims no validity,
J04's n′′ exhibits the frequency and temperature dependence pattern expected
from measured far-infrared behaviour of n′′.
Z01 did the first measurements of both
n′ and n′′ at sub-millimetre frequencies and atmospheric temperatures. For
n′′ they found a linear temperature dependence of about 1 % K-1. The
measurements agree quite well with the H91 and J04 models. However, the
Z01 model falls short of reproducing their own measurements – it predicts
very low values over all frequencies (see Fig. ) and all
temperatures.
M06 introduced a permittivity parameterisation
that consolidates most earlier models and measurements. Regarding the imaginary
part, it agrees well with the J04 model, particularly at microwave
frequencies, deviating by less than 5 % at higher frequencies and high
temperatures. The largely revised and updated version of W84, the
W08 data, incorporates the M06 model at
T=-7∘C and proposes M06 as the model of choice at wavelengths beyond
200 µm when temperature dependence should be considered.
Real (left) and imaginary (right) part of the effective refractive
index of a mixture of ice and air as a function of air volume fraction
according to some mixing rules. The refractive index of ice at 183 GHz
and 263 K, nice=1.7831+i0.0039, was used, and the refractive
index of air was set to nair=1+i0.
J04 found their model to be within 12 % of the imaginary
permittivity measurements for frequencies below 800 GHz and within
15–40 % above 800 GHz. M06
estimated the uncertainty of their model from the standard deviations of
measurements to 5 % at 270 K and 14 % at 200 K. W08
state the uncertainty of their n′′ data to be 10 % (T=-7∘C).
Compared to recent models, the once quasi-standard model W84 strongly
overestimates n′′ at millimetre wavelengths (up to a factor of 5),
underestimates it at sub-millimetre wavelengths (up to a factor of 2) and
overestimates it in the far-infrared (up to a factor of 2).
The impact on particle absorption due to selection of M06 or W84 is
exemplified below (Sect. ).
In contrast to n′′, n′ is generally considered to be known with higher
accuracy and to vary little to negligibly with both temperature and frequency.
The measurements by Z01 confirm the small frequency dependence (0.3 % over
250–1000 GHz) and do not show significant temperature dependence. The
different models mentioned above provide slightly different relations of n′
with frequency and temperature. However, we find refractivity (n′-1) from all
models to agree with M06 within 1.3 %, according to the left panel of
Fig. . Based on estimates on propagation of n′
uncertainty to optical properties, we conclude that the uncertainty in n′ is
not a limiting factor and choice of model is not critical.
In summary, M06 seems the best choice for microwave to far-infrared imaginary
refractive index data. In view of the effects that errors in imaginary
refractive index have on cloud optical properties (see
Sect. ), we strongly suggest no longer using the
data in future.
Mixing rules
The parameterisations reviewed above deal with solid ice, while in the soft
particle approximation (Sect. ) the particles are treated to
consist of a homogeneous mixture of ice and air. The standard procedure for
assigning a refractive index to the mixture is to apply a so-called mixing rule.
In this paper we compare some commonly used mixing rules from a purely practical
perspective; a more theoretical review of mixing rules is provided by
.
Throughout we will assume the refractive index of air to be 1+i0; in
other words we assume that the optical properties of air are like
those of a vacuum. Of course, for the radiative transfer problem as a
whole both absorption and refraction by air matter strongly. However, given
the much larger refractive index of ice we neglect this in the
calculation of the single-scattering properties, as commonly done
by other authors.
Three mixing rules are considered: “Maxwell-Garnett”
, “Bruggeman” and
“Debye” . All these formulas operate with dielectric
constants. The Debye mixing rule is
ϵe-1ϵe+2=f1v(ϵ1-1)ϵ1+2+(1-f1v)(ϵ2-1)ϵ2+2,
where ϵe is the “effective” dielectric constant of the mixture,
f1v is the volume fraction of medium 1 (f1v+f2v=1) and
ϵ1 and ϵ2 are ϵ for medium 1 and 2 respectively. The
expression for Bruggeman is
f1v(ϵ1-ϵe)ϵ1+2ϵe+(1-f1v)(ϵ2-ϵe)ϵ2+2ϵe=0.
The Debye and Bruggeman expressions are symmetric with respect to the two
media. Maxwell-Garnett differs in this respect by making a distinction
between the “matrix” (ϵ=ϵm) and the “inclusion”
(ϵ=ϵi):
ϵe=ϵm+3fivϵm(ϵi-ϵm)ϵi+2ϵm-fiv(ϵi-ϵm),
where is fiv is the volume fraction of the inclusion medium
(fmv+fiv=1). That is, for Maxwell-Garnett we have two cases, “air
in ice” and “ice in air” (below shortened to MGai and MGia
respectively),
that result in different ϵe depending on whether air is set to be the matrix
or inclusion medium.
Overview of considered DDA databases.
DatabaseFrequency rangeTemperaturesParticle sizesParticle shapes[GHz][K][µm, max. dim.]3.0–340233, 243, 253,50–12 454Columns, plates, rosettes,263 and 273sector and dendrite snowflakes10.65–183.31263200–12 584Three aggregate types,consisting of 200 and/or 400 µmsix-bullet rosettes90–8742432–2000Solid and hollow columns,plates, six-bullet rosettes,droxtals and one type of aggregate
For completeness, the effective density (ρe) matching f1v is
ρe=f1vρ1+(1-f1v)ρ2,
where ρ1 and ρ2 are the density of medium 1 and 2
respectively. In terms of mass fraction of medium 1 (f1m), ρe is
ρe=ρ1ρ2f1mρ2+(1-f1m)ρ1.
An example comparison between the mixing rules is shown in
Fig. . A first observation is that the Debye and the
“ice in air” version of Maxwell-Garnett (MGia) give identical results (for
ϵm=ϵair=1+i0 the two formulas are mathematically
identical). Hence, the Debye rule is not explicitly discussed below but is
represented by the identical MGia. MGia gives consistently the lowest
refractive index for both real and imaginary part. The difference from the other
two rules is highest at air fractions around 0.45. The highest values are
throughout found for MGai, and Bruggeman falls between the two Maxwell-Garnett
versions. Repeating the calculations for other frequencies and temperatures
e.g.Fig. 2, i.e. other ice refractive indices,
shows that these patterns are of general validity and are not specific to our
example.
The deviations between the mixing rules are significant. For example,
conducted a sensitivity analysis for frequencies
between 2.8 and 150 GHz regarding the choice of mixing rule. The differences
when using MGai or MGia were found to be ∼ 2 dB for radar
reflectivity and reach at least 10 K for brightness temperature.
Some mixing rule can be optimal for representing a true homogeneous ice–air
spherical particle, as studied by , but this is not
the crucial point in this context. In Sect. we
instead pragmatically test whether any of the mixing rules leads to a simpler
approximation of realistically shaped particles.
Existing DDA databases
The DDA is the most widely used method for
computing the scattering properties of arbitrarily shaped particles. In the DDA
method, a particle is represented by an array of dipoles in a cubic lattice
with a given inter-dipole spacing. This spacing must be adequately small
relative to the incident wavelength in order to obtain desired accuracy, which
requires large computer memory and long calculation time for large particles.
Despite the wide usage of the method, the publicly available DDA data for
microwave scattering of ice particles are limited. The three databases that are
used in this study are the ones of ,
and . The main properties
of these databases are summarised in Table .
The only other open source of microwave DDA data that we know about is
http://helios.fmi.fi/~tyynelaj/, where data used in
and some other publications were recently
made available. These data, covering frequencies up to 220 GHz, are not
included in this paper as they deal with partly oriented particles, while the
other databases all are valid for completely random orientation.
Liu
applied the DDA code of , denoted
as DDSCAT, and computed single-scattering properties (i.e. scattering cross
section, absorption cross section, backscattering cross section, asymmetry
parameter and phase function) of 11 types of ice particle crystal shapes,
at 22 frequencies (3, 5, 9, 10, 13.4, 15, 19, 24.1, 35.6, 50, 60, 70, 80, 85.5,
90, 94, 118, 150, 166, 183, 220 and 340 GHz) and for five different
temperatures. To not clutter the figures below, we include only 6 of the
11 particle types. The ignored shapes are: short column, block column, thin
plate and four- and five-bullet rosettes; included shapes are pointed out in the
figure legends. The included six particle types cover the full range of
variation in the optical properties found in the database.
The particles were treated to have random orientation. The phase function is
provided for 37 equally spaced scattering angles between 0 and 180∘. In
terms of the “phase matrix” required for vector radiative transfer, only the
(1,1) element is given. The refractive index of ice applied in the DDA
calculation was taken from .
Nowell
A new snowflake aggregation model is introduced in .
The six-bullet rosette is a frequently observed crystal shape and therefore was
selected by as constituent crystals of the simulated
snowflake aggregates. The aggregates were allowed to grow in three dimensions,
following an algorithm resulting in quasi-spherical snowflakes following the
diameter-density parameterisation of . The
representation of the bullet rosettes is somewhat coarse, based on cubic blocks
with a size of ≈50µm. Only particles with a maximum diameter above
1 mm are included in our figures to make sure that the aggregates consist of
a relatively high number of building blocks.
The single-scattering properties of an ensemble of randomly generated
aggregates were calculated by the DDSCAT code. Calculations
for 10 frequencies (10.65, 13.6, 18.7, 23.8, 35.6, 36.5, 89, 94, 165.5 and
183.31 GHz) and a single temperature (263 K) were performed, with
refractive index taken from . The phase
function is not included in this database; only the corresponding asymmetry
parameter is stored.
Hong
also used DDSCAT to compute the scattering
properties (extinction efficiency, absorption efficiency, single-scattering
albedo, asymmetry parameter and scattering phase matrix) of six randomly
oriented non-spherical ice particles at 21 frequencies (90, 118, 157, 166,
183.3, 190, 203, 220, 243, 325, 340, 380, 425, 448, 463, 487, 500, 640, 664,
683 and 874 GHz) for a temperature of 243 K. All six independent elements
of the phase matrix are reported in steps of 1∘ between 0 and
180∘.
The geometrical information of the six ice particle shapes is detailed in
Table 1 of . Refractive index of ice
was taken from , which according to
Sect. is not the optimal choice with respect to particle
absorption.
DDA-based single-scattering properties at 183 GHz from the
databases of , and
. Absorption, scattering and backscattering
efficiencies (Eq. ) and asymmetry parameter are displayed. The
combined legends are valid for all panels. The figure includes also data
of solid and soft spheres and spheroids, with refractive index following
. The soft particles have an air fraction of
0.75, with the effective refractive indices derived by the MGai mixing
rule. The spheroids are oblate with an aspect ratio of 1.67. All results
are valid for 183 GHz and 243 K except those of
for 263 K.
Comparison of the databases
Example DDA data are found in Fig. for one of the few
frequencies that is found in all three databases (183 GHz).
All three aggregate types in the Nowell database are
plotted with the same symbol. The abscissa of the figure is size parameter
according to Eq. (), implying that the radiative properties are
compared between particles having the same mass. Absorption, scattering and
backscattering are reported as the corresponding efficiency, Q, calculated
with respect to de as
Q=4σπde2,
where σ is the cross section of concern. Even though usage of Q
provides some normalisation of the data, compared to when cross sections were to be
plotted, the ordinates in the first three panels of Fig.
still span several orders of magnitude. In Fig. another
normalisation is applied that brings out differences at lower size parameters:
the optical cross sections are divided by the corresponding optical
cross section of the equivalent mass solid ice sphere with same ice refractive
index.
Absorption (left) and scattering (right) cross sections of DDA data
and soft spheres at 183 GHz. The cross sections are reported as the ratio
to the corresponding cross section of the equivalent mass sphere, with the
same refractive index as used for the preparation of the DDA data. That is,
the dotted straight line at r=1 represents solid ice spheres. Database
source and particle shapes of the DDA data are found in figure legends
(same as in Fig. ). The soft spheres have an air fraction
of 0.25, where results for three different mixing rules (MGia, Bruggeman
and MGia) are included (solid lines).
Although we discuss the soft particle approximation in depth only in Sect. , solid and soft spheroids are already included in the
figures here for reference. We also already make some remarks on their optical
properties here but postpone the explanation of how the soft-spheroid results
were generated to the dedicated section later.
The Hong data have systematically a higher absorption than Liu. This can be
discerned in Fig. and is expected due to the higher imaginary
part of the refractive index (n′′) in W84 (used by Hong) compared to M06
(used by Liu) for frequencies below 400 GHz, as shown in
Fig. . This deviation is removed in
Fig. as the normalisation is done with respect to Mie
calculations with the refractive index set to match the DDA data. Without this
adjustment of the refractive index there would be a much higher variability in
the absorption ratios in Fig. , with the different DDA
databases at different mean levels. Droxtal particles are quasi-spherical and
the fact that these particles obtain ratios very close to 1 in
Fig. confirms that a correct normalisation has been
applied. This similarity in shape explains also why the droxtals end up close
to the data for solid spheres for all quantities in Fig. .
It is well known that the impact of shape on the extinction efficiency
increases with particle size. Accordingly, for x below ∼0.5 there is
a comparably low spread between different particles for both absorption and
scattering. In terms of the ratio in Fig. , the data are
mainly inside 1.2±0.2.
However, at x=2 the difference between the lowest and highest
scattering, for particles having the same mass, is about a factor of five
(Fig. ). These remarks consider also 340 GHz (not shown)
where the same particles result in higher size parameters.
The backscattering efficiency shows a similar pattern as the scattering one,
but the variation above x=2 is considerably higher by about a factor of
10. This is the case because the backscattering depends on the phase function for
a particular direction, resulting in a higher sensitivity to the exact shape of
that function, while the overall scattering extinction corresponds to the
integrated phase function.
There is a significant spread in the asymmetry parameter (g) from about
x=0.5 and above. Above x≈1.5, the difference between
highest and lowest g is about 0.3, where the Nowell aggregates and the
Liu bullet rosettes throughout cause the highest and lowest values
respectively. The six-bullet rosettes in the Hong database show the same tendency
of low g for combinations of size and frequency resulting in
x>1.5. At lower x the Hong rosettes tend to
give the highest g among all of the particles, then also higher than the
corresponding Liu rosette. That is, the different six-bullet rosette models used
by Liu and Hong result in significantly different optical properties.
Figure was inspired by Fig. 7 of ,
comparing that database with solid and soft particle calculations
in the same way. used a higher air fraction for
their soft particles and it is not clear whether the MGai or MGia version of the
Maxwell-Garnett mixing rule was used, but there are still some clear deviations
between the two figures for soft particles. For example, the scattering
efficiency of soft particles in our Fig. is quite close to
the data from , while in their Fig. 7 the soft
particles give significantly lower scattering. In addition, we obtain basically
identical scattering efficiencies for soft spheres and spheroids, while
got lower scattering for spheroids. We have
carefully checked our calculations and our results seem to fit better with what
has been found elsewhere. For example, in Fig. 5 of
a good agreement with the aggregates of
is obtained by soft particles having a density of
0.2 gm-3 (the air fraction of 0.75 in Fig. matches
0.23 gm-3), and basically identical results are obtained between spheres and
both prolate and oblate spheroids.
Relevance of absorption and asymmetry
parameter
To judge the performance of a soft particle approximation or some “shape
model”, a basic consideration is the detail in which the single-scattering
properties must be compared. The main issue is that the phase function
(describing the angular redistribution of scattered radiation, also denoted as
the scattering function) can be very complex and is basically unique for all
particles where Rayleigh conditions do not apply. However, it is
normally not required to compare the phase function in full detail. For
example, it is in general only the direct backscattering that is of interest
for radar applications. This is valid until multiple scattering becomes
significant, when also the phase function starts to be relevant.
Test of importance of absorption and asymmetry parameter for passive
microwave radiative transfer. The brightness temperature deviation from
simulations with no cloud layer is reported, where a positive value in the
figure corresponds to a decrease in absolute brightness temperature. The
stated optical depths refer to the zenith extinction of the cloud layer.
For solid lines, the imaginary part of the refractive index was set to
0, resulting in no cloud particle absorption. The simulations are
described further in the text.
For passive measurements, the standard choice is to give an overall description
of the phase function by using the parameter g. The asymmetry parameter is
known to have a strong influence in radiative transfer of solar radiation
e.g.. The quantity is also frequently reported in
connection to passive microwave radiative transfer
e.g., but, to our best knowledge,
the actual influence of g for such applications has not been
investigated in a general manner. A simple test of this type is found in
Fig. . The calculations were done with the DOIT (Discrete
Ordinate ITerative method) scattering module of the ARTS radiative transfer
model . Satellite measurements at
150 GHz and an incidence angle of 45∘ were simulated. Temperature and
gas profiles were taken from a standard tropical scenario (Fascod), and a
2 km thick “cloud” layer, centred at 10 km, was added. The selection of
surface emission is not critical for these qualitative calculations and for
simplicity the surface was treated to act as a blackbody. A single particle
size (monodispersive PSD) was used for each simulation, and the number of
particles was adjusted to obtain the specified zenith optical depths. Spherical
particles with an intermediate air fraction (0.4) were assumed, and
de was varied to obtain a range of g. Solid ice particles could not
be used for this test as they don not give a monotonic increase of g with
particle size, and neither provide g above 0.7 (Fig. ).
The solid lines in Fig. show how the brightness temperature
changes with g when the cloud optical depth is kept constant and all
particle absorption is suppressed. The basic pattern is that the cloud impact
on measured radiance decreases with increasing g. This makes sense as high
g means that the up-welling emission from the lower troposphere is less
redirected compared to the case of more isotropic scattering at low g;
see for a schematic figure and discussion of the
radiative transfer for this measurement geometry. It is hard to see in the
figure, but there actually are some “wiggles” around g=0.65, showing
that the relationship to g is not completely monotonic. That is, several
values of g can result in the same radiance.
In Fig. the cloud impact for g=0 and g=0.6
differs by a factor of about 2. That is, changing g by 0.1
results in a ∼10 % change in cloud impact. For low optical thickness the
relationship between scattering cross section (σs) and radiance impact is
close to linear. Accordingly, a 10 % error in σs and a 0.1 error in
g are in rough terms equally important. The test displayed in
Fig. was repeated for other frequencies and cloud altitudes.
The absolute values of the cloud impact change, primarily following the
magnitude of the gas absorption at the altitudes around the cloud layer, but
the mentioned relation between σs and g was found to be relatively
constant.
Figure exemplifies also the importance of ice particle
absorption for passive measurements. It is well known that absorption is most
significant for smaller particles, i.e. the single-scattering albedo increases
with particle size e.g..
Figure confirms this as the difference between considering
absorption (dashed lines) and neglecting it (solid lines) is high for small
x for all cloud optical depths. This aspect is especially important for
limb sounding, because in this observation geometry focus is put on higher altitudes
where smaller particles are more frequent, and it has been shown that the
measured signal can even be dominated by absorption .
Figure also shows the less obvious fact that absorption
increases in importance with increasing cloud optical thickness. For an optical
thickness of 2.0 absorption is significant up to at least x=1.2, while
for small optical depths (such as 0.1) the absorption can be neglected for
x above ∼0.5. This is a consequence of the behaviour that
probability of absorption increases when multiple scattering becomes more
prominent. The changed conditions caused by multiple scattering implies that
the relevance of absorption can not be judged alone from the single-scattering
albedo parameter. In addition, the observation geometry matters for the
relative importance of absorption and scattering, as discussed in
.
In summary, it is confirmed that the quantities normally considered (σa,
σs, σb and g) are all relevant but to a varying degree. Most
importantly, the relevance of absorption decreases with size
parameter.
Absorption (left) and scattering (right) cross sections of soft
spheres (183 GHz and 243 K), normalised by the equivalent mass ice
sphere absorption or scattering cross section as in
Fig. , as a function of size parameter and air
fraction. The two top panels are calculated using the MGai (air in ice)
mixing rule, while the two lower panels are calculated using the MGia
(ice in air) mixing rule.
Approximation by soft particles
The soft particle approach (SPA) is based on two main simplifications. Firstly,
the particle is treated to consist of a homogeneous mix of air, ice and, when mixed-phase particles are
considered,
also water
e.g.. The air fraction of the mix is either set
to a constant value or is obtained by assuming an effective density of the
particle, likely varying with particle maximum size. A single refractive index
is assigned to the mix by applying a mixing rule (Sect. ).
Secondly, the particles must be set to have some specific shape, to allow
the single-scattering properties to be determined with a limited calculation
burden. As mentioned, the T-matrix method allows e.g. soft columns and
plates to be possible options, but the standard choices are to model the
particles as spheres or spheroids. A much more detailed description of SPA is
provided by .
Selection of mixing rule
As a first step, we examined whether the choice of mixing rule is critical in any
way for SPA. The difference between mixing rules can in general be compensated
by selecting different air fractions, but exceptions exist. This is most
clearly seen for the absorption and scattering cross section at smaller x,
as exemplified in Fig. . In the figure, the absorption of
soft particles when using Maxwell-Garnet with “ice in air” (MGia) is
throughout lower than the DDA results. This is in contrast to using the
Bruggeman or the “air in ice” version of Maxwell-Garnett mixing rule (MGai),
in which the soft particle absorption matches some of the DDA data points. The
same pattern is found also for the scattering cross section but for a smaller
range of x.
The low bias in Fig. of MGia, compared to DDA data, can
not be removed by modifying the air fraction, as shown in
Fig. . In this figure, the ratios of
Fig. are calculated for air fractions between 0 and 0.95
and the ratios obtained when using MGia are below 1 throughout. Ratios around
at least 1.2 are required to represent the average values of the DDA data in
Fig. . For x<0.5 such ratios, and even much higher
values, can be obtained by selecting the MGai mixing rule. Ratios when using
Bruggeman (not shown) reach 1.25 for absorption and 1.15 for scattering, which
is on the limit to fit the DDA data. For MGai the ratios switch from being
>1 to <1 around x=1, which in the following is shown to be the general behaviour
of DDA data.
The conclusion of Figs. and is
that using the MGia mixing rule leads to a systematic underestimation of the
absorption and scattering at some size parameter ranges. The same applies to
the Debye mixing rule as it is identical to MGia. The Bruggeman rule gives
higher values but is not capable of reproducing the highest DDA-based ratios
found in Fig. . The patterns seen in
Fig. are not specific to 183 GHz but are
representative of the complete microwave region (1 to 1000 GHz). In
addition, these remarks do not depend on whether soft spheres or
spheroids are used. Accordingly, MGai appears as the best choice in this
context, and only this mixing rule is considered below.
Selection of particle shape
Solid spheres are known to exhibit resonance features for x above
∼1, which are reflected as oscillations in the properties displayed in
Fig. (blue solid lines). An individual spheroid would give
similar oscillations, but the assumption of completely random orientation
partly averages out those patterns for the spheroids (see
Fig. , red solid lines). The resonance phenomena are dampened
when going to soft particles. This results in a marginal difference in
extinction (absorption and scattering cross sections) and g between soft
spheres and spheroids (dashed lines). However, there is a significant
difference for the backscattering, where soft spheres give even stronger
oscillations than solid spheres. The soft spheroids show a more smooth
variation with x, and should allow a better fit to the DDA data.
made the same observations for soft particles and
showed that extinction and g are basically unaffected by the aspect ratio
of the spheroids or whether they are oblate or prolate. They found also that the
oscillations in size dependence of the backscattering decrease when the aspect
ratio moves away from 1.
That is, soft spheroids are preferable over soft spheres, primarily
due to the difference with respect to backscattering. For complete random
orientation the selection of aspect ratio is not critical, and only oblate
spheroids with an axial ratio of 0.6 are considered below, following
. In terms of the nomenclature used in the T-matrix code,
this equals an aspect ratio of 1.67.
Selection of air fraction
As shown in the above figures, for a given size parameter there is a spread of
the particle optical properties over the different habits. Hence, it is not
possible to match all particle shapes at the same time with a soft
particle approximation. The ambition is instead to approximately mimic the
average properties. Figure shows that solid spheres and
spheroids do not meet this criterion because they (e.g. for x>4)
systematically underestimate scattering cross section and g.
It is stressed that in this work, air fraction is treated as a pure tuning
parameter. This implies that statements about low/high air fractions just refer
to values relatively close to 0/1; they are not judgements about the true
density of the particles.
Relevance of the reference data
A more detailed analysis requires some consideration of the occurrence frequency
of the different particles in the DDA databases. For example, the Hong database
contains droxtals having a maximum diameter, dm, up to 2 mm, while the
general view is that this shape is only representative for the smallest ice
crystals . In fact,
found that cloud ice particles with dm above 250 µm are mainly
the
aggregate type, implying that also single plates and columns having dimensions
above this size are relatively rare. The aggregate types discussed by
appear to be relatively similar to the
aggregates in the Hong database.
For the representation of particles of snow type, the Liu database offers two
shapes (dendrite and sector-like snowflakes) that both have high aspect
ratios, while the aggregates of the Nowell database (claimed to represent
snowflakes) have an aspect ratio close to 1. This difference in aspect ratio
results particularly in deviations in the scattering cross section and
g of those particles (Fig. ). If “snow” is
understood as everything from the classical single-crystal snowflake to
graupel, both these assumptions on aspect ratio are realistic, but it is clear
that particles having intermediate aspect ratios also exist and, hence, they are
not yet covered by the DDA databases. The Hong aggregates could potentially
also work as proxy for snow particles, but data for dm above 2 mm are
lacking.
Fit of single-scattering data
Figure exemplifies a fit of the available DDA data using soft
particles having an AF of 0.75. For the frequency of concern
(183 GHz) the soft particles, compared to the solid ones, give indeed a
better general fit of absorption and scattering efficiencies. For g at
x>1.5, the soft particles agree well with the Nowell aggregates, while
all other particle shapes are better approximated with solid spheres or a
comparably low AF. That is, for e.g. the Liu sector-like snowflake, the AF
that gives the best fit with scattering efficiency is not the same AF that is
needed to match g.
Normalised absorption (left) and scattering (right). The top row
includes 90 GHz Hong/Liu and 89 GHz Nowell data, while the bottom
row covers 874 GHz. The soft spheroids have either a fixed
air fraction (AF) or follow , denoted as H12.
Normalisation and
plotting symbols used for DDA data are as in Fig. .
Example results for other frequencies are found in Fig. .
This figure, together with Fig. covering 183 GHz,
indicate that the MGai mixing rule combined with an AF of 0.25 give an
acceptable fit of absorption and scattering for size parameters below 0.5,
independent of frequency. At higher x, AF =0.25 is not optimal; it
gives a fairly good fit at 874 GHz (Fig. ),
but at e.g. 90 GHz this AF results in too-high values for both absorption
and scattering. For x>1 and lower frequencies, a fit of absorption and
scattering efficiencies requires higher AFs. For example, mimicking the
scattering efficiency at 90 GHz requires an AF on the order of
0.75–0.9, depending on whether all or just snow-type particles should be
fitted.
These remarks show three facts. Firstly, there is in general not a single AF
that simultaneously gives a fit of all four optical property parameters.
Secondly, at least when operating at lower frequencies, the AF to apply in a
soft particle approximation must be allowed to vary with particle size.
Thirdly, the best AF has a frequency dependence (at least for larger x).
The third point is well known and has been shown by e.g.
.
Normalised backscattering of the Liu particles at two
frequencies. As in Figs. and , the normalisation is performed with respect to the
backscattering cross section of the solid sphere having the same mass.
For 94 GHz and x≈1.5 some data points have a ratio above
3 partly due to a minimum of the solid sphere backscattering at that
size parameter. Solid and dotted lines are the same as in
Fig. .
The soft particle AF is frequently set to follow some density parameterisation.
This gives the AF a variation with size in line with the second point. However,
the known fact that an optimal AF varies with frequency (point 3)
signifies that using true densities cannot work as a general approach with
respect to optical properties. This is the case because density-based AFs are
independent of frequency. In addition, for larger particles standard density
parameterisations result in much higher AFs than the ones giving a match of
single-scattering data around 100 GHz and above. As an example, the particle
model of is included in Fig. . This
particle model is based on the frequently used density parameterisation of
that leads to AFs close to 1 for the largest DDA
particles. In fact, the scattering efficiency at 90 GHz becomes too low
already at x≈0.5. Also, backscattering is underestimated at
x above ≈0.5 even at lower frequencies
(Fig. ). All other density parameterisations we have tested
show the same general feature: to produce, in this context, too-high AFs for
larger particles. For more recent parameterisations the density goes below
100 kgm-3 at dm≈800µmFig. 6.
was selected as it provides a clearly defined particle
model. However, it should be noted that treat the
spheroids to be aligned with the maximum dimension in the horizontal plane,
while we apply completely random orientation.
Test radiative transfer simulations
Absorption, scattering and asymmetry parameter interact for simulations of
passive observations, as shown in Sect. . Some test simulations
were performed in order to check whether the different tendencies for these
quantities combine in a positive or negative manner. These simulations, shown
in Fig. , were performed for the same scenario as used for
Fig. . Again, a monodispersive PSD was used, but here the
number of particles was adjusted to obtain a specified vertical column of ice
mass, or ice water path (IWP). The IWP for each frequency was selected to give
a maximum cloud-induced brightness temperature change of 5–10 K in
order to get a significant response but still avoid a high degree of
multiple scattering.
Radiative transfer simulations at different frequencies. General
conditions as in Fig. , i.e. a 2 km thick cloud
layer at 10 km is simulated. A single particle type is included in
each simulation where the number density was adjusted to obtain the
stated ice water paths. The black solid line is a high-order polynomial
fit of the DDA-based results, while other lines are results for soft
spheroids with constant air fraction (AF).
As mentioned, the radiative transfer simulations were done with ARTS-DOIT
(Sect. ). This scattering method requires the full phase
function and no results for the Nowell database could be generated. For
Fig. a more strict selection of the DDA particles was done,
roughly matching the discussion in Sect. in order
to remove the less physically plausible particles. Column, plates and
three-bullet rosettes having dm above 1 mm were excluded. For droxtals the
limit was set to 200 µm. The black solid lines show a polynomial fit (in
linear scale) of the simulations based on the remaining DDA particles.
As expected from the discussion above, the different DDA particles give little
spread of simulated brightness temperatures for x<0.5.
However, there is a strong variation at larger size parameters. For example, the
Liu dendrite snowflakes, as well the Hong six-bullet rosettes as around x=1,
have particularly low influences. This is a combined effect of relatively low
scattering efficiency and high g
(Fig. ). The same combination enhances also the
differences between the Liu sector-like and dendrite snowflakes compared to
the differences for scattering efficiency alone. The relative influence between
the particles is not the same for all frequencies. For example, the Hong
aggregates are found on the high side for 90 and 166 GHz but are rather on
the low side for 874 GHz.
Figure shows that the selection of the soft particle AF is
not highly critical for size parameters below 0.5. This is partly due to
compensating errors. A too-high AF gives an overestimation of both absorption
and scattering, but this is counteracted by an overestimation of g. At
higher size parameters, the frequency dependence of the “optimal” AF noted
above is also seen here. For example, at 340 GHz an AF of 0.25–0.50 is
required to match the fit of the DDA results (black line), while for 90 GHz
a suitable AF is above 0.75. For 874 GHz, only covered by the Hong database,
an AF around 0.25 gives best agreement. The systematic deviation between the
soft particle and the DDA-based results seen for 874 GHz and low x is
due to the refractive index differences discussed in
Sect. .
Approximation by a single representative shape
Based on poor experience of using the SPA at ECMWF (the European Centre for
Medium-Range Weather Forecasts), attacked the
representation of particle shape in microwave radiative transfer from another
angle. Their application is data assimilation for numerical weather prediction,
but the basic problem is the same as for direct retrieval of frozen
hydrometeors. Their approach is simple: to try to find a particle type, for
which DDA calculations are at hand, that minimises the average deviation to
actual observations. They compared to measurements from the TMI and SSMIS
sensors for frequencies between 10 and 190 GHz. The Hong database does not
cover the lower end of this frequency range, and only the Liu database was
considered.
They performed global simulations for latitudes between 60∘ S and
60∘ N. Simulated brightness temperatures were obtained with RTTOV-SCATT,
a radiative transfer tool making use of several approximations. The atmospheric
data were taken from the ECMWF 4D-Var assimilation system, likely having biases
in ice mass amounts varying between regions, land/ocean and the different
hydrometeor types. Furthermore, a PSD must be assumed for the simulations. The
tropical version of the PSD of was found to give the best
overall fit with observations among the three PSDs considered. These aspects
introduce problems for a clear identification of the best overall proxy
particle shape, as discussed in detail by .
The final recommendation of is to apply the Liu
sector-like snowflake for all classes of both cloud ice and snow. A somewhat
better fit could be obtained by some combinations involving the six-bullet
rosettes and dendrite snowflake particles, but the improvement was not
sufficiently large to motivate a more complicated particle shape model. Our
results corroborate the selection of the sector-like snowflake as the
general proxy shape particle. This particle type does not stand
out in any obvious way; it shows in general intermediate values. In fact, the
best match with the polynomial fit of the DDA-based simulations in
Fig. (black lines) is given by the sector-like snowflakes
for both 90 and 166 GHz. A good fit is also obtained by the Liu six-bullet
rosette, another particle type that had as a strong
candidate. The sector-like snowflake tends to be on the high end for x
around 0.7 but on the low side at higher x. If these happen to be true
biases, they are partly averaged out in PSD-weighted bulk properties.
The Liu sector-like snowflakes exhibit average properties also at 340 GHz,
above the frequency range considered by . The pattern
is very similar to the lower two frequencies, with some tendency to
“overshoot” around x=1. If the upper limit for plates and columns
would have been set to a lower value, such as 500 µm, the sector-like
snowflakes would even have shown outlier behaviour around x=1. This
results in that for 340 GHz an even better agreement with the polynomial fit
is obtained with the Hong aggregates. This particle type of the Hong
database is throughout below the fitting line at 874 GHz, but this result
depends heavily on a strong influence on the polynomial fit of columns and
plates with de≈500µm. The Hong bullet rosette seems not to be a
candidate for the role as general proxy shape because it has a very low
scattering efficiency, around x≈1.5, which is also reflected in
Fig. .
As in Fig. but using the maximum dimension
(dm) as characteristic size. That is, the size parameter is here
defined as x′=πdm/λ and absorption and scattering
efficiencies are defined as Q′=(4σ)/(πdm2). As in Fig. , the
frequency is 183 GHz.
Ice particle size distributions for 0.1 gm-3, according to
F07 and
MH97. The F07 PSD is converted
from dm to de for two combinations of a and b (Eq. ).
Scattering extinction as a function of ice water content at
183 GHz for some of the particles of the DDA
database. The particle size distribution applied in the upper and lower
panels is F07 and MH97
respectively. Dashed lines in the
upper panel show results for the F07 distribution with fixed a=0.069 and
b=2.
Figures 2 and 3 of complement the figures of this
paper by reporting bulk optical properties of the Liu particles as a function of
ice mass and frequency. A bit surprising is that the sector-like snowflake is
found to have the lowest bulk g, seemingly for all
frequencies and ice masses. This shape has also the lowest g among the
Liu particles in Fig. but only up to x=1.
Nevertheless, the SPA spheres applied are found to give very high bulk
g. explain this as a result of the Mie theory,
but according to our Fig. the high g is rather a result
of the density assumed. The snow hydrometeor class is set to have a density of
100 kgm-3, corresponding to an AF ≈0.9. We cannot easily judge the
exact impact of this relatively high AF (meaning being close
to 1, not that the true density is misjudged) for several reasons; e.g.
used a mixing rule not considered by us. Another
complication is that the PSD operates with dm. Hence,
also differences in the relationship between particle mass and dm between
the particles affect the data derived by . There is a
much more intuitive mapping of our findings to bulk properties whether the PSD is
based on de.
analysed only passive observations, while it would be
highly beneficial if the representative shape selected could also be applied for
radar measurements. Figure indicates that this is the case.
The backscattering of the sector-like snowflake follows its pattern for the
scattering efficiency (Fig. ). This shape has a ratio (as
defined in discussed figures) above 1 up to a somewhat higher size parameter
than the other particles and more pronounced at 35.6 than at 94 GHz;
besides this, its properties are of average character also with respect to
backscattering.
Is using maximum dimension a better option?
Up to this point, we have compared radiative properties between particles
having equal de (thus also having the same mass), for reasons discussed in
the introduction. The second main measure for the size of individual
particles is the maximum dimension, dm. In fact, there are likely more
PSDs using dm than de. Hence, it
is useful to also understand how the radiative
properties vary with dm, and such an overview for 183 GHz is given in
Fig. . This figure was produced as Fig. but
with de replaced by dm in the definition of size parameter and
absorption and scattering efficiencies.
The panels for asymmetry parameter in Figs. and
are quite similar besides the range of x being extended
when using dm. However, there are clear differences for both
absorption and scattering efficiencies. There is a much more compact
relationship between de and these radiative properties than what is found
for dm. In the case of using dm as the size measure, relatively compact
particles (droxtals, plates, columns and spheres) obtain especially high
absorption and scattering efficiencies, while particles having high aspect
ratios (dendrite and sector-like snowflakes) exhibit especially low
efficiencies. The stronger influence of particle morphology and aspect ratio
results in a ratio between highest and lowest efficiency of ∼ 100
(besides for smallest particles). This is in clear contrast to
Fig. , where the same ratios are around or below 10 when using
de (Sect. ).
However, the higher variability in absorption and scattering
efficiencies is not directly mapped to the same variability in bulk
optical properties, i.e. the optical properties of the distribution
as a whole. The reason for this is that particles with high aspect
ratio have a lower mass as a function of dm. This aspect deserves
careful examination, so we analyse it in the remainder of this
section. As a measure for the bulk optical properties we select the
scattering extinction coefficient.
As an example of a dm-based PSD we selected the tropical version of
the PSD by , below denoted as F07. The
extinction coefficients were derived with a set-up basically identical
to the one described by which also used the
F07 PSD: only particles with dm≥100µm were included (as
the PSD does not cover smaller particles), and the PSD was rescaled as
described in their Appendix C to compensate for the truncation in
particle size.
An additional aspect of the size distribution is that it
uses two additional input parameters, a and b. They originate from the
common way to express the relationship between dm and particle mass,
m, as
m=admb.
There are some issues around how to derive a and b for a particle type and
how to perform the numerical integrations of the PSD. These issues are
described carefully in Appendix B and C of and there
is no need to repeat all details. In short, we also selected to derive the a
and b parameters by performing a functional fit, but only considering
particles with dm≥100µm. A reason to ignore the smaller particles
in the fit is that for them Eq. () may result in dm<de
(corresponding to density higher than the one of solid ice) in cases where
b<3. As did, we ensured that the bulk mass is preserved
by a rescaling of the PSD. This rescaling used the actual particle masses from
the DDA database (i.e. assumed a and b were ignored in the rescaling).
However, not all dm-based PSDs take a and b into account. To also
investigate the impact of neglecting the dependence on a and b, bulk
scattering was also derived with F07 and applying fixed a and b values for
all particles, namely a=0.069 and b=2. These values were selected following
and .
Since we want to compare dm-based and de-based bulk extinction
coefficients, we also need an example of a de-based PSD. For this we selected
the PSD by below MH97. A comparison
between F07 and MH97 is found in Fig. , where F07 is rescaled to
de-basis. Two combinations of a and b are considered: the first
combination (0.0015/1.55) matches the sector-like snowflakes, having the lowest
b among all the particles in the Liu database; the second combination
(480/3) represents solid spheres and thus also the upper limit of b. The
rescaling to match specified ice water content has a marginal impact on F07. However, MH97 puts a much larger fraction of the mass below 100 µm
and the rescaling gives a small but not negligible change; therefore this PSD
is displayed both before and after the rescaling. Like F07, MH97 is also a PSD
targeting tropical conditions, and the agreement with F07 is relatively high
for a=0.0015,b=1.55, while for a=480,b=3 the two PSDs deviate strongly.
Using the discussed PSDs, bulk scattering extinction coefficients can be
calculated by adding up extinction coefficients for individual particles with
appropriate weights. Figure shows the results: total extinction
coefficients for the two different PSDs, the dm-based F07 at the top and the
de-based MH97 at the bottom. For F07, results for fitted and fixed a and
b parameters are shown separately by straight and solid lines respectively.
As the top panel with the F07 results clearly demonstrates, the general pattern
seen for the individual particles (Fig. ) persists in bulk
extinction: the more compact particles are at the upper end and the less
compact (more “snowflake-like”) particles are at the lower end. However, as
expected, the ratio between highest and lowest value decreased from
∼ 100 when considering individual particles to ∼10 when considering
bulk extinction. A similar spread in bulk extinction was obtained by
see their Figs. 2 and 3.
As in Fig. but for 340 GHz. The results for MH97
are not shown as they show the same general pattern as in
Fig. (just shifted in mean level in the same way as the
results for F07).
Still discussing the top panel of Fig. , we now turn to the
issue of using fitted or fixed a and b parameters for the F07 PSD. For
fixed parameters, the extinction obtained for particular shapes is changed, but
the spread of the values is roughly maintained. It should here be noted that
keeping a and b fixed only has the consequence that the PSD gets the same
basic shape for all particles. The rescaling to ensure that specified mass is
matched maintains the relative fraction between particles having different
dm.
While all the discussion so far related to the top panel of
Fig. , we will now turn to the bottom panel. It shows that
particle shape indeed has a much lower impact on bulk scattering for the
de-based MH97 PSD compared to the dm-based F07 PSD. The factor between
highest and lowest extinction in case of MH97 is ∼2.5. This cannot be a
consequence of MH97 putting the highest weight on completely different particle
sizes, as the extinction using MH97 ends up inside the range resulting from
F07. Furthermore, the relative variability over the different habits is close
to constant with ice water content, for both F07 and MH97, and already
Fig. showed that MH97 ends up inside the range covered by F07
when a and b are varied. All in all, F07 and MH97 do a quite similar
relative weighting between different particle size ranges. We therefore
conclude that the difference in spread seen between upper and lower panel of
Fig. is a direct consequence of the deviations revealed by
Figs. and ; i.e. the scattering cross section
is more closely linked to de than to dm.
The difference between de and dm, exemplified by Fig. ,
seems to be of general validity for frequencies up to ∼ 200GHz. If
anything, the difference increases when going down in frequency (not shown). At
higher frequencies a somewhat different pattern is found for the dm case,
as shown in Fig. . Here at 340 GHz, the spread in extinction
of the different DDA particles is overall lower compared to
Fig. and is particularly low at high ice water content, where
it is even smaller than when using de. However, the
dendrite snowflake particles constitute an exception, which is possibly an indication that this particular
shape should perhaps be avoided for higher frequencies. In any case, the deviating results
for the dendrite snowflake show that the low spread in extinction between the
other particles may be a coincidence, not necessarily indicating that using
dm ensures low uncertainty in extinction for high frequencies and high ice
water content. Comparing usage of de and dm at high frequencies is
presently complicated by the fact that the Hong database is limited to
dm≤2mm and this size truncation can easily cause artefacts in the
comparison.
At the end of this section, we want to briefly mention two more general
aspects of the problem of representing bulk particle optical properties.
Firstly, our analysis assumed that F07 and MH97 give an equally good
representation of the mix of particle sizes. If in situ probes provide
better data for either de or dm, this should result in higher
systematic errors for PSDs based on the more poorly measured particle
size.
Secondly, besides possible systematic errors in the PSDs, the
variability around average conditions must be considered. For example,
it could be the case that there is a lower PSD variation (between
locations, day-to-day, etc.) as a function of dm than
as a function of de. This situation would decrease, or reverse, the
advantage of using de. If the opposite were true, that PSDs tend to
be more stable in de, this would enhance the advantage of selecting
de in favour of dm. As far as we know, this important aspect of PSD
variability has not been studied so far.
Summary and conclusions
We have reviewed the two most established databases of DDA calculations for
microwave atmospheric radiative transfer, and
. is associated with the
Liu database and was also considered. All three databases assume completely
random particle orientation. The databases have different frequency coverage:
Nowell from 10 to 183 GHz, Liu from 3 to 340 GHz and Hong from 90 to
874 GHz. Liu is the only database providing data for more than one
temperature. Scripts to convert the Hong and Liu data to the format expected by
the ARTS forward model can be obtained by contacting the authors.
We noted clear systematic differences in absorption between the Hong and Liu
databases. The deviations are explained by the fact that the refractive indices
are based on different sources. used the data from
that now are considered to be outdated. That is, we
judge the only easily accessible DDA data above 340 GHz to be inaccurate on
particle absorption. In the update of , the
parameterisation of is recommended for the microwave
region, and this is also the source of refractive index used by
and . Another problematic
aspect of the Hong database is the restriction to dm≤2mm.
We mainly compared optical properties between particles having the
same mass and defined the size parameter (x) accordingly
(Eq. ). For small x below ≈0.3, the variation of
absorption and scattering between the particles is about 20 % (1.2±0.2 in
terms of the ratio used in Figs. , and ).
Going towards higher x, the variation increases
most quickly for backscattering, followed by scattering and finally most slowly for
absorption. At higher x, the ratio between lowest and highest value is
∼10, ∼5 and ∼2.5 for those three radiative properties
respectively. The range in scattering is in general generated by the fact that
particles of solid types have comparably high scattering, while shapes of
“snow” character result in low scattering. found that
solid spheres are representative up to x=2.5 (backscattering not
considered and clearly allowing some systematic errors), but we, using a
larger set of DDA calculations, find this limit somewhere around
x=0.5.
We also scrutinised the soft particle approximation. A first conclusion
was that the selection of mixing rule can lead to systematic errors at low
x. A mixing rule giving comparably high refractive index, for given air
fraction, is needed to avoid this systematic error. We selected the
Maxwell-Garnett with ice as the matrix and air as the inclusion media. With this
selection of mixing rule, combined with an air fraction of about 0.25, SPA is
applicable up to about x≈0.5 across the considered frequency
range. This gives for absorption and scattering cross sections a maximum
deviation to individual DDA calculations of ≈30 %.
However, usage of SPA at higher x seems problematic. Each
individual property calculated by DDA can likely be reproduced by adjusting the
air fraction, but it is in general not possible to achieve a fit with several
radiative properties simultaneously. In any case, even fitting a single property,
such as backscattering, requires that the air fraction is decreased when
moving to higher frequencies. Thus, selecting the air fraction based on some
standard density parameterisation can, in the best case, only work in a small
frequency range. Our results indicate that this frequency range then is found
below 35 GHz as this approach leads to high AFs, passing 0.9 at
dm∼1mm. At very high frequencies, such as 874 GHz, an air fraction
of 0.25 could potentially be applied for all particle sizes, but at lower
frequencies the air fraction must also be adjusted with size, from about 0.25
at low x (see paragraph above) to a higher value at higher x, for
example 0.7–0.9 at 90 GHz. That is, applying SPA across the microwave region
requires a model with a high number of tuning variables to give the air
fraction the needed variation with frequency and size, while at the same time
the resulting particle densities have no physical basis.
Inspired by , we also investigated a second way to
represent average radiative properties. The idea is simple: when any of the
particles covered by the DDA databases exhibit average properties, use this
particle shape to represent true habit mixes. compared
the particles of the Liu database using real passive observations, but they
were then forced to involve assumptions on particle size distribution and
ice mass concentration, while we mainly compared the basic radiative properties
directly. However, some radiative transfer calculations were required to assess
how differences in scattering cross section and asymmetry parameter combine in
simulations of downward-looking passive measurements, but these calculations
did not involve any additional assumptions. The critical part in our approach
is the judgement of how representative the different DDA particles are with
respect to the mean conditions in the atmosphere.
Due to the lack of reference data, we selected to not push the analysis too far
at this point and discussed only in general terms which particle shapes
show overall average properties. It is of course possible to use the same
methodology to e.g. select a representative shape separately for “cloud ice”
and “snow” or target different cloud types.
Interestingly, and we find that the sector-like
snowflake particles, among the shapes found in the Liu database, best represent
average properties. This was found valid also for higher frequencies than
considered by , as well as for application in radar
retrievals. For frequencies above 340 GHz, where the selection is restricted
to the Hong database, an aggregate model appears to be a suitable choice.
However, solid conclusions can not yet be drawn, as the number of reference
data so far is quite limited. More data of optical properties of aggregate and
snow-type particles are needed to get a more robust basis for studies like
this. In turn, this requires new algorithms for generating realistic
particle models to be used as input to DDA or similar calculation methods. If
new databases are created, the limitations of present databases in temperature,
particle size and frequencies should be avoided.
Besides the “shape model”, we also investigated the representation of
particle size. Most importantly, it is demonstrated that there is a much more
compact relationship between absorption and scattering properties with mass
equivalent diameter (de) than with maximum dimension (dm). With the
exception of small x, the spread of absorption and scattering efficiencies
is at least a factor of 10 higher when dm is used to define the size
parameter compared to when using de. The difference is decreased when
summing up individual values to obtain bulk properties, but using a dm-based
PSD still gives a higher uncertainty in the extinction for a given ice water
content compared to using a de-based PSD. Below 200 GHz, the uncertainties
are roughly a factor 10 and 3 for the dm and de case respectively.
Scattering extinction at 340 GHz shows a somewhat different pattern and
perhaps indicates that the difference between dm and de could vanish at
even higher frequencies. In any case, it would be highly beneficial if future
in situ measurement campaigns could target to provide PSDs in terms of
de;
such measurements seem to be much less frequent than ones of dm.
Finally, we stress that the entire study was performed assuming completely
random particle orientation. This is probably the main limitation of the
conclusions made above. It can not be ruled out that e.g. the spread of
scattering and the difference between using dm and de is highly
dependent on particle orientation. That is, a main consideration for future
databases of ice hydrometeor optical properties is to make it possible to study
the radiative properties when assuming different distributions of horizontal
orientation.
The Supplement related to this article is available online at doi:10.5194/amt-8-1913-2015-supplement.
Acknowledgements
Financial support for this study was provided by the Swedish National Space
Board. Furthermore, we want to express our great appreciation to the persons
behind the publicly available DDA databases. M. Kahnert, B. Rydberg,
M. Brath and A. Geer are thanked for helpful comments.
Edited by: A. Lambert
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