Microwave (1–

Passive microwave (MW) observations are nowadays a standard tool for
cloud observation. The ice-cloud-related sounding channels of passive
microwave sensors typically do not possess a fixed polarization or
they measure only at one polarization. Observations of polarization
in view of MW and submillimeter (SubMM) remote sensing of ice clouds
are still rare. Existing passive microwave sensors that measure polarization
are typically confined to frequencies below

Currently, Global Precipitation Measurement (GPM) Microwave Imager (GMI),

Existing single scattering databases of realistically shaped ice particles
for the microwave and submillimeter range, like the ones of

This paper aims to simulate the MW and SubMM scattering data of realistically
shaped ice crystals that possess arbitrary fixed orientations relative
to the zenith direction under the assumption that there is no preferred
orientation in azimuth direction. In reality, ice crystals have a myriad
of shapes, as shown, for example, by

To simulate the scattering properties, the scattering of ice crystals
from various incidence directions is simulated and consequently used
to calculate orientation-averaged scattering. Similar to the work
of

The text is structured as follows: in Sect.

Particle orientation refers to how the main axes of the particle are oriented with respect to the local horizon and the azimuthal reference. If the particle possesses a spherical symmetry, there is no particle orientation, as regardless of from which side the particle is viewed or how it is rotated it will always look the same. The particles considered in this paper are not spherically symmetric and therefore can be oriented.

In general, the orientation of a particle in a three-dimensional space
can be described by a set of three parameters. There is no unique
set of these parameters. Depending on the definition of the rotation
axes, there are different sets of these parameters. The three Euler
angles are one such parameter set. The Euler angles define the orientation
of the particle (coordinate) system relative to a fixed coordinate
system, hereafter called the laboratory system. The particle system is
the coordinate system that is attached to the particle. This means
that if a particle is rotated, the particle system is rotated the
same way. The laboratory system stays under the rotation of the particle,
whereas the particle system changes its orientation. The laboratory
system and particle system share the same origin. In this study, the
Euler angles, which are shown in Fig.

Euler angles.

In addition to the Euler angles, the orientation of the non-rotated
particle is needed. As there is no absolute coordinate system, the
orientation of the non-rotated particle is in general arbitrary. Therefore,
we define that the non-rotated particle lies with its center of gravity
at the origin of the laboratory system and all particle rotations
are relative to the origin of the laboratory system. Furthermore,
the non-rotated particle is defined as having its principal moments
of inertia axes aligned along the Cartesian coordinate axes, with
the maximum inertia axis along the

Within this study, we are not interested in the scattering of a single
oriented particle but in the scattering of an ensemble of particles
that are oriented differently but are otherwise identical. Generally,
the scattering properties of such an ensemble of oriented particles
are described by averaging the single scattering properties over the
three Euler angles. The scattering matrix

We distinguish between two basic states of particle orientation:

total random orientation (TRO),

azimuthal random orientation (ARO).

Totally randomly oriented particles are defined as the orientation
average over the three Euler angles in which the Euler angles are
uniformly distributed

Azimuthally randomly oriented particles with a specific orientation
to the horizon, also referred to as tilt or canting, are defined as
the orientation average over

Schematic of the difference between totally random (TRO) and azimuthally random orientation (ARO) for columnar particles.

Schematic showing that rotation results in a rotational symmetry around the flagpole (axis). The actual image that we see depends on the symmetry properties of the flag (object).

The scattering calculations are computationally demanding in view
of the computation time and the amount of data required. Therefore, we have
to compromise in terms of the accuracy of the resulting scattering
data. Considering the measurement errors of existing and upcoming
passive MW and SubMM sensors, which are on the order of

For the scattering calculations ADDA version 1.2 was used. ADDA is
a DDA implementation of

ADDA can simulate the scattering of totally randomly oriented particles
and the scattering of particles with a fixed but arbitrary orientation.
The internal averaging method of ADDA is not suitable for our approach
to simulate azimuthally oriented particles. Instead, we developed
an averaging approach that involves integration over a set of DDA
calculations at different angles, which is explained in Sect.

For DDA simulations it is important that the size of the dipoles is
small compared to the wavelength and to any structural length within
the scatterer

Following

plate type 1, which is a solid hexagonal plate-like single crystal, and

a large plate aggregate, which consists of several solid hexagonal plates aggregated to one particle.

In this work we follow the approach of

Overview of the selected habits:

Example scatterer shapes.

The frequencies of the scattering
calculations. Except for

In general, the scattering matrix

Schematic drawing of the calculation
of the single scattering properties.

The actual results of an ADDA calculation are the scattering amplitude matrix and the Mueller matrix for a desired incidence direction and a grid of scattering directions, whereas we are interested in the extinction matrix and scattering matrix. The relationship between the scattering amplitude matrix and the extinction matrix and between the Mueller matrix and the scattering matrix are explained in the following paragraphs. Difficulties arise from the fact that the matrices are defined in different coordinate systems. In the database, the scattering matrix and the extinction matrix are defined in the laboratory system. The extinction matrix that results from the scattering amplitude matrix and the Mueller matrix are defined in the coordinate system that is defined by the incidence direction and the particle system, from here on called wave reference system. Due to the relation to the particle system, the wave reference system rotates if the particle (particle system) rotates. Therefore, the main part of our averaging approach consists essentially of transformations from one coordinate system to another coordinate system.

The extinction matrix

Each Mueller matrix element

For each incidence direction, ADDA automatically calculates the Mueller
matrix for a desired regular grid of polar angles and azimuth angles.
A regular grid of polar and azimuth angles has the property that the
grid spacing at the pole is much finer than at the Equator. For the
incidence angles, a regular grid of polar angles and azimuth angles
are disadvantageous because an isotropic
sampling is needed for the incidence angle, but the distribution of the directions of a regular
grid of polar angles and azimuth angles is not isotropic. Therefore,
an icosahedral grid is used, which is shown in Fig.

Example of an icosphere grid with 162 vertices. Each grid point represents an incoming angle for which a DDA calculation is performed. This type of configuration ensures that the grid density is isotropic, making the overall calculations more efficient (a standard polar grid would be inefficient, since it yields an excessive amount of angles around the “north and south poles”).

The orientation-averaged Mueller matrix

The actual scattering calculations are done iteratively. For each
particle, the scattering calculation begins with

To test our approach, the scattering of azimuthally randomly oriented
prolate ellipsoids with an aspect ratio of

The methodology to calculate the scattering matrix and the extinction
matrix can be summarized as follows.

DDA calculations: a set of DDA runs is performed over an icosahedral
angle grid of incidence directions, as demonstrated in Fig.

Representation and truncation: represent the Mueller matrix elements of each ADDA run in a spherical harmonics series and truncate them to reduce the amount of data.

Averaging: azimuthally averaged Mueller matrices

Transformation: the averaged Mueller matrices are transformed to averaged
scattering matrices

In this section we give an overview of the scattering simulations
and show some example results. A total of

The orientation averaging is done for a finite set of incidence and
tilt angles. The incidence angles

The scattering database with the orientation-averaged data is publicly
available from Zenodo (

In the following analysis we will not address the absorption vector because it is derived directly from the extinction and scattering matrix and is only added to the database for convenience.

The orientation averaging (Eq.

Figures

For the large plate aggregate we skip the tilt angles

For the plate type 1 habit the effect of orientation and incidence
angle results in differences of up to

The

For

The asymmetry parameter describes the distribution between forward
scattering and backscattering and gives an overview of the scattering
behavior. For example,

Extinction matrix
elements

Extinction matrix
elements

The scattering matrix of a particle describes the angular distribution
of the scattered radiation in relation to the incidence direction
of the incoming radiation. For unpolarized incoming radiation, the

After the orientation averaging, the resulting scattering properties
possess a rotational symmetry relative to the laboratory

As an example, Fig.

Size distribution parameters and the scatterer
shape of the radiative transfer simulations. The size distribution
parameters were taken from the source code of the Milbrandt–Yau two-moment
bulk microphysics

For non-nadir and non-zenith incidence directions the

The data from the simulated scattering matrix can be used for simulations of passive and active observations. However, for simulations of horizontally scanning radars the scattering matrix in the backscattering direction has to be handled with care. In the spherical harmonics representation of the Mueller matrix, the polarization at the poles, which are in the forward and backward direction, is not well represented. This can result in errors for the polarization. Most of this is averaged out due to the orientation averaging and the transformation to the scattering matrix, but there can be some residual effects for the polarization at the backscattering direction. This will be revised for the next iteration of the database.

The upper-left block of the normalized
scattering matrix

The upper-left block of the normalized
scattering matrix

The upper-left block of the normalized
scattering matrix

In this section, we show radiative transfer simulations at

The simulations were done using the Atmospheric Radiative Transfer
Simulator (ARTS,

The Milbrandt–Yau two-moment microphysics

The scattering properties for the hydrometeors were taken from

For cloud ice and snow the azimuthally randomly oriented plate type
1 and the azimuthally randomly oriented large plate aggregate are
used, respectively. No averaging of the scattering data of the particles
with its mirrored version was done for the radiative transfer simulation.
Normally, this is done to assure that the scattering medium, in our
case ice clouds, are mirror symmetric to the incidence plane. Mirror
symmetric particles like the plate type 1 automatically fulfill this,
but unsymmetrical particles like the large plate aggregate generally
do not. Due to the orientation averaging and the random structure
of the large plate aggregate, the effect of the non-mirror symmetry
is so small that we neglected it for the radiative transfer simulations.
For the simulations the azimuthally randomly oriented particles are
orientation-averaged over Gaussian distributed

Figure

Additionally, Fig.

Additional tests show that the polarization difference and the brightness
temperature are mainly influenced by snow and graupel. For these tests
(not shown) one hydrometeor at a time was set to zero, while the others
were unchanged, and the simulations for the

Simulated brightness temperature
at

Same as Fig.

For the simulations shown in Fig.

The bell-like distribution of the polarization difference

Hydrometeor content profiles used for
the radiative transfer simulation in Fig.

Same as Fig.

In Fig.

As an additional scenario, the large plate aggregate habit for snow
was replaced by the plate type 1 habit and the simulations for the

The comparison of the three different scenarios with the observations
of

We provide microwave and submillimeter wave scattering simulations
of azimuthally randomly oriented ice crystals with a fixed but arbitrary
tilt angle. For the simulations, DDA simulations made with ADDA were
combined with a self-developed orientation averaging approach. The
scattering of

To give an example of the capabilities of the dataset, we conducted
radiative transfer simulations of polarized GMI measurements of differently
fluttering ice crystals at

Using the new scattering data, retrievals of polarized observations from GMI, MADRAS and especially the upcoming ICI can give us new insights for the understanding of clouds. For example, to the authors' knowledge none of the latest atmospheric weather and climate models handle orientation. Furthermore, polarization can give us additional information on the shape of the particle.

Before any orientation averaging can be performed, the initial orientation
of the particle has to be defined. The alignment algorithm is mainly
based on aligning the principal moments of inertia axes along the
Cartesian coordinate axes. Also, a number of special cases are treated
in order to make the alignment consistent between particles and not
dependent on small numerical differences. The result of the algorithm
is that the particle fulfills the following criteria: the principal
axis of the particle with the largest inertia is aligned along the

The algorithm involves a several steps. For particles that possess
no symmetries, one step can be skipped. The algorithm operates on
a coordinate grid and consists of the following steps.

First, the particle mass center coordinate

Next, the inertia matrix

It follows that

If the particle contains symmetries, then two or all of the principal
moments of inertia can be equal. This means that the rotation in the
previous step is unambiguous, i.e., several possible orientations
fulfill Eq. (

In the final step, it is determined whether the particle is aligned
upside down or upright. First, the minimum circumsphere of the particle
is calculated, with its corresponding center. If the center is found
to be below the mass center of the particle (with respect to the

The key point in our averaging approach is the rotation of the particle
for the averaging process. When rotating the particle, the wave reference
system rotates as well. The wave reference system is the coordinate
system that is defined by the incidence direction and the particle
system. The changed direction

Change of the polarization directions
under rotation.

On an icosahedral grid, any arbitrary point on the sphere is accompanied
by three nearest points that form a equilateral triangle. Within this
triangle the value at that point can be interpolated from the vertices
of the triangle. An illustration of the problem is shown in Fig.

Geometry of triangular barycentric interpolation.

Between the scattering matrix averaged

As defined in Sect.

The Stokes rotation matrices

Scattering geometry in the laboratory system

The Stokes rotation matrix

In the actual implementation each matrix element

Each matrix element

The scattering database of the azimuthally
randomly oriented particles is publicly available from Zenodo (

MB developed the orientation averaging approach, set up and conducted the scattering and the radiative transfer simulations, and wrote the article's text. RE prepared the scatterer shape data, designed the database structure and contributed to the text. PE acted as project leader, initiated the database and suggested the averaging approach. PE and SAB participated in the planning of the database and contributed to the text. OL helped set up and conduct the scattering simulations and prepared the data for publication.

The authors declare that they have no conflict of interest.

The authors would like to thank Christophe Accadia, study manager at EUMETSAT, for providing appreciated feedback and inspiration. The authors would also like to thank Howard Barker from Environment and Climate Change Canada for providing the GEM model simulations. Finally, our thanks go to the ARTS radiative transfer community for their help with using ARTS.

A large part of this work was produced during a study funded by EUMETSAT (contract no. EUM/COS/LET/16/879389). Robin Ekelund and Patrick Eriksson were further supported financially by the Swedish National Space Agency (SNSA) under grant no. 150/14. Stefan A. Buehler was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2037 “Climate, Climatic Change, and Society” – project no. 390683824, contributing to the Center for Earth System Research and Sustainability (CEN) of Universität Hamburg.

This paper was edited by Stefan Kneifel and reviewed by three anonymous referees.