**Research article**| 24 Jan 2023

# Retrieval of terahertz ice cloud properties from airborne measurements based on the irregularly shaped Voronoi ice scattering models

Ming Li Husi Letu Hiroshi Ishimoto Shulei Li Lei Liu Takashi Y. Nakajima Dabin Ji Huazhe Shang and Chong Shi

^{1,2},

^{1},

^{3},

^{4},

^{4},

^{5},

^{1},

^{1},

^{1}

**Ming Li et al.**Ming Li Husi Letu Hiroshi Ishimoto Shulei Li Lei Liu Takashi Y. Nakajima Dabin Ji Huazhe Shang and Chong Shi

^{1,2},

^{1},

^{3},

^{4},

^{4},

^{5},

^{1},

^{1},

^{1}

^{1}State Key Laboratory of Remote Sensing Science, Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, 100101, China^{2}University of Chinese Academy of Sciences, Beijing, 100049, China^{3}Meteorological Research Institute, Japan Meteorological Agency (JMA), Nagamine 1-1, Tsukuba, 305-0052, Japan^{4}College of Meteorology and Oceanography, National University of Defense Technology, Changsha, 410073, China^{5}Research and Information Center (TRIC), Tokai University, 4-1-1 Kitakaname Hiratsuka, Kanagawa, 259-1292, Japan

^{1}State Key Laboratory of Remote Sensing Science, Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, 100101, China^{2}University of Chinese Academy of Sciences, Beijing, 100049, China^{3}Meteorological Research Institute, Japan Meteorological Agency (JMA), Nagamine 1-1, Tsukuba, 305-0052, Japan^{4}College of Meteorology and Oceanography, National University of Defense Technology, Changsha, 410073, China^{5}Research and Information Center (TRIC), Tokai University, 4-1-1 Kitakaname Hiratsuka, Kanagawa, 259-1292, Japan

**Correspondence**: Husi Letu (husiletuw@hotmail.com)

**Correspondence**: Husi Letu (husiletuw@hotmail.com)

Received: 27 Aug 2022 – Discussion started: 08 Sep 2022 – Revised: 13 Dec 2022 – Accepted: 14 Dec 2022 – Published: 24 Jan 2023

Currently, terahertz remote sensing technology is one of
the best ways to detect the microphysical properties of ice clouds.
Influenced by the representativeness of the ice crystal scattering (ICS)
model, the existing terahertz ice cloud remote sensing inversion algorithms
still have significant uncertainties. In this study, based on the Voronoi
ICS model, we developed a terahertz remote sensing inversion algorithm of
the ice water path (IWP) and median mass diameter (*D*_{me}) of ice clouds.
This study utilized the single-scattering properties (extinction efficiency,
single-scattering albedo, and asymmetry factor) of the Voronoi, sphere, and
hexagonal column ICS models in the terahertz region. Combined with 14 408
groups of particle size distributions obtained from aircraft-based
measurements, we developed the Voronoi, sphere, and column ICS schemes based
on the Voronoi, sphere, and column ICS models. The three schemes were applied
to the radiative transfer model to carry out the sensitivity analysis
of the top-of-cloud (TOC) terahertz brightness temperature differences
between cloudy and clear skies (BTDs) on the IWP and *D*_{me}. The
sensitivity results showed that the TOC BTDs between 640 and 874 GHz are
functions of the IWP, and the TOC BTDs of 380, 640, and 874 GHz are
functions of the *D*_{me}. The Voronoi ICS scheme possesses stronger
sensitivity to the *D*_{me} than the sphere and column ICS schemes. Based on
the sensitivity results, we built a multi-channel look-up table for BTDs.
The IWP and *D*_{me} were searched from the look-up table using an optimal
estimation algorithm. We used 2000 BTD test data randomly generated by the
RSTAR model to assess the algorithm's accuracy. Test results showed that the
correlation coefficients of the retrieved IWP and *D*_{me} reached 0.99 and
0.98, respectively. As an application, we used the inversion algorithm to
retrieve the ice cloud IWP and *D*_{me} based on the Compact Scanning Submillimeter-wave Imaging Radiometer (CoSSIR) airborne
terahertz radiation measurements. Validation against the retrievals of the
Bayesian algorithm reveals that the Voronoi ICS model performs better than
the sphere and hexagonal column ICS models, with enhancement of the mean
absolute errors of 5.0 % and 12.8 % for IWP and *D*_{me}, respectively.
In summary, the results of this study confirmed the practicality and
effectiveness of the Voronoi ICS model in the terahertz remote sensing
inversion of ice cloud microphysical properties.

Ice clouds account for about 20 %–30 % of the total global cloud mass (Liou, 1992; Rossow and Schiffer, 1991). They play an essential regulatory role in the global radiation balance and the water cycle (Hong et al., 2016; Stephens et al., 2012). Microphysical properties such as the ice water content, ice particle size, orientation, and shape are the main influencing factors of the scattering and radiative properties of ice clouds (Li et al., 2022; Yi et al., 2017; Zhao et al., 2018; Chen and Zhang, 2018) and, in turn, affect the radiation budget (Heymsfield et al., 2013, 2017; Rossow, 2014). The latest report of the 6th Intergovernmental Panel on Climate Change (IPCC6) (Forster et al., 2021) identifies cloud radiative properties and their feedback effects as the largest source of uncertainty in the overall climate feedback, with errors in ice clouds being one of the most significant factors (Zhang et al., 2021). Therefore, the accurate acquisition of microphysical properties of ice clouds is of great importance for studying global climate change and improving the accuracy of numerical model simulations (Baran, 2009, 2012). Remote sensing techniques are one of the most effective means of obtaining microphysical and radiative properties of ice clouds, mainly including ground-based (Cimini et al., 2006), aircraft-based (Fox et al., 2017), and satellite-based remote sensing observation technologies (Yang et al., 2015, 2018). Currently, large numbers of passive sensors (visible, infrared, and microwave detectors) have been developed, and related ice cloud retrieval algorithms have been reported in a substantial body of literature (Nakajima and King, 1990; Nakajima et al., 1991, 2019; Nakajima and Nakajima, 1995; Platnick et al., 2003, 2017; Fox et al., 2019; Brath et al., 2018). The visible and infrared wavelengths are sensitive to the visible optical depth and cloud top (Minnis et al., 1993a, b). The millimetre-wave ice cloud remote sensing technique is more suited to detect vertical cloud properties. Sensors such as the Millimeter-wave Imaging Radiometer (MIR) (Racette et al., 1992) and Special Sensor Microwave Water Vapor Sounder (SSM/T-2) have been used in several studies of ice water path (IWP) and particle size retrievals (Lin and Rossow, 1996; Liu and Curry, 1998, 1999). MIR channels at 89, 150, and 220 GHz have been used by Deeter and Evans (2000) and Liu and Curry (2000) to retrieve IWP and particle size in cirrus anvils over the tropical ocean. Compared to passive sensors, the cloud satellite radar (CloudSat), with an onboard millimetre-wavelength (94.05 GHz) radar, and the radar–lidar cloud product (DARDAR) (Ceccaldi et al., 2013) present new opportunities to infer the microphysical properties of ice clouds on a global scale. In practice, the current detectors and approaches are confined to a limited range of ice particle sizes (Cho et al., 2015). For example, visible and infrared sensors are only sensitive to particles smaller than 50 µm (Evans et al., 2005). Additionally, microwave detectors are mainly useful for particles larger than 500 µm (Fox, 2020; Fox et al., 2019). There is a pressing need to develop an effective frequency region for obtaining comprehensive information about ice particles. To bridge the gap between the technologies of visible–infrared and microwave measurement of ice clouds, the terahertz (THz) wavelength between the infrared and microwave regions has the potential advantage of complementing existing visible–infrared and microwave techniques (Gasiewski, 1992).

The terahertz wave is the sub-millimetre wave with frequencies in the range of 0.1–10 THz and corresponding wavelengths of 0.03–3 mm, comparable to the size of ice particles in ice clouds. Many theoretical studies (Evans and Stephens, 1995a, b; Evans et al., 1998) have shown that the passive terahertz wave has a higher detection capability and sensitivity to the IWP and particle size of ice clouds (Liu et al., 2020). Although terahertz waves were proposed for ice cloud remote sensing in the 1960s (Gao et al., 2016), the technology of terahertz radiometry of ice clouds lagged behind the theory (Evans et al., 2005). It is only within the last decade that terahertz radiometry has been applied to aircraft-based and satellite-based ice cloud remote sensing. With advances in terahertz detectors, researchers have successively developed aircraft-based terahertz radiometers (Gao et al., 2016), such as the Sub-millimeter Wave Cloud Ice Radiometer (SWCIR) (Evans et al., 2002), the Compact Scanning Submillimeter-wave Imaging Radiometer (CoSSIR) (Evans et al., 2012), and the International SubMillimeter Airborne Radiometer (ISMAR) (Fox et al., 2017). Also, research institutions have developed satellite-based terahertz radiometers, including the Superconducting Submillimeter-Wave Limb Emission Sounder (SMILES) (Inatani et al., 2000), Ice Cloud Imager (ICI) (Eriksson et al., 2020; Kangas et al., 2014), and IceCube (Wu et al., 2014), which were proposed in 2013 and are still under development.

Several methods have been reported to retrieve the ice clouds' IWP and
particle size from terahertz brightness temperature. Inversion methods can
be divided into physical and linear regression methods (Weng et al.,
2019). The linear regression method is efficient and convenient, but it
lacks a definite physical mechanism and is heavily dependent on the accuracy
of the pre-calculated database. The physical method includes the Bayesian
algorithm (Evans et al., 2005, 2002), the look-up table
(LUT) method (Li et al., 2017, 2018; Liu et al., 2021), and
the neural network algorithm (Jimenez et al., 2003).
Evans et al. (1998) modelled terahertz brightness
temperature using a polarized radiative transfer model based on eight ice
particle shapes calculated by the discrete dipole approximation (DDA)
method. The study found that the ice particle shape plays a vital role in
modelling ice cloud scattering in the terahertz region. Evans et
al. (2002) developed a Monte Carlo Bayesian integration (MCBI) algorithm to
retrieve ice clouds' IWP and median mass diameter (*D*_{me}) from simulated
SWCIR brightness temperatures. According to the validation results, the
total median retrieval error for clouds with IWP of more than 5 g m^{−2} is
roughly 30 % for IWP and 15 % for *D*_{me} (Evans et al.,
2002). Moreover, Evans et al. (2005) utilized the MCBI method to
retrieve the IWP and *D*_{me} of ice clouds from the CoSSIR brightness
temperatures (referred to as the CoSSIR-MCBI hereafter). During the
retrieval procedure, one of five particle shapes (spherical snow, aggregates
of frozen droplets, aggregates of hexagonal plates, and aggregates of plates
and hexagonal columns) developed by the DDA method (Evans
and Stephens, 1995a) was selected for each ice cloud retrieval. The
CoSSIR-MCBI results are validated by the Cloud Radar System data and showed
a good agreement of radar backscattering with errors smaller than 5 dB.
Later, Jimenez et al. (2007) used the neural network method
combined with the radiative transfer code to retrieve the IWP and *D*_{me} of
ice clouds. The ice cloud microphysical input is based on one of three
randomly oriented particle shapes (solid columns, hexagonal plates, and
four-bullet rosettes) simulated with the DDA method (Evans and Stephens,
1995a; Evans et al., 1998) in the microwave region. Results showed overall
median relative errors of around 20 % for IWP and 33 µm for
*D*_{me} for a mid-latitude winter scenario and 17 % for IWP
and 30 µm for *D*_{me} for a tropical scenario. Based on early studies and the
background, Buehler et al. (2007) proposed a formal scientific
requirement for a passive sub-millimetre-wave cloud ice mission. The
requirements are that the low IWP should be less than 10 g m^{−2}, the high
IWP should be less than 50 %, and the particle diameter should be less
than 50 µm. Li et al. (2016) investigated the effects of five
ice crystal scattering (ICS) models from the single-scattering property database of Hong
et al. (2009) at frequencies ranging from 100 to 1000 GHz, namely
aggregates, hollow columns, flat plates, rosettes, and spheres
(Van de Hulst, 1957), on the transmission characteristics of
terahertz radiation. Results showed that the ice particle shape is one of
the dominant factors affecting terahertz radiation. Lately, Fox (2020) evaluated seven ice particle habits from the ARTS scattering database
(Eriksson et al., 2018). They showed that the randomly oriented large
column aggregate performs best when simulating brightness temperatures
between 183 and 664 GHz.

In summary, the physical method is based on the radiative transfer principle and relies on the forward physical model simulation and effective ice crystal scattering (ICS) model. Different assumptions of ice cloud microphysical properties (shape, size, orientation, and particle size distribution of ice particles) in the forward physical model significantly affect the retrieval of the IWP and particle size of ice clouds. Therefore, a practical and representative ICS model is essential for the ice cloud remote sensing in the terahertz region.

Many aircraft observations have demonstrated that ice clouds comprise a
complex and diverse range of non-spherical ice particles (Lawson et al.,
2006, 2019; Liou, 1992). It is still challenging for one specific
light-scattering method to precisely calculate the single-scattering
properties for non-spherical particles with different size parameters
(SZPs). The SZP is the ratio of the equivalent-volume sphere's circumference
dimension (or *π* times particle maximum diameters) to the
incident wavelength (Baran, 2012; Nakajima et al., 2009). So far, the
light-scattering methods can be roughly divided into the approximation
method (AM) and the numerical simulation method (NM). The AM method is based
on ray-tracing techniques, including the geometrical optics method (GOM)
(Macke et al., 1996; Takano and Liou, 1989), which is suitable for large
particles. The NM includes the finite-difference time domain (FDTD) (Yang
and Liou, 1996a; Yee, 1966) and the DDA (Draine and Flatau, 1994; Yurkin
and Hoekstra, 2007) methods, which are appropriate for small particles.
Combined with the advantages of the AM and NM methods, several improved
geometrical optics approximation (GOA) methods, including the geometric
optics integral equation (GOIE) method (Yang and Liou, 1996b), have
been developed. Moreover, the boundary element method (Groth et al.,
2015; Kleanthous et al., 2022) has been recently applied to complex ice
particles. Based on the above-mentioned light-scattering calculation
methods, a series of regularly shaped ICS models have been designed, including
hexagonal columns, hexagonal plates, bullet rosettes, droxtals, and so on
(Yang et al., 2000a, 2013; Fu et al., 1999; Takano and Liou, 1989). Since
the regularly shaped ICS models are not fully representative of the scattering
properties of natural ice particles, researchers have developed complex ICS
models. Yang et al. (2013) developed the surface-roughened non-spherical
ICS models and applied them to the production of MODIS C6 ice cloud products
(Platnick et al., 2017). C.-Labonnote et al. (2000, 2001) and
Doutriaux-Boucher et al. (2000) developed an ICS database for the
inhomogeneous hexagonal monocrystal (IHM) containing embedded inclusions
(air bubbles and aerosols). The IHM database was applied for the ice cloud
retrievals from the French satellite Polarization and Directionality of the
Earth's Reflectance (POLDER) measurements (Deschamps et al., 1994).
Furthermore, Baran and Labonnote (2007) and Baran et al. (2014a)
developed an ensemble ice particle model made of hexagonal column ice
particles for use in the Met Office Unified Model Global Atmosphere 5.0
(GA5) configuration (Baran et al., 2014b). Unlike the
above-mentioned regularly shaped ICS models, Letu et al. (2012)
and Ishimoto et al. (2013) analysed ice particles of different shapes
and sizes from many NASA aircraft observations and developed an
irregularly shaped complex Voronoi ICS model. The single-scattering property
database of the Voronoi ICS model in the visible and infrared regions uses
a combined FDTD, GOIE, and GOM approach. The Voronoi ICS model has been
adopted for generating official ice cloud products for the Second Generation
gLobal Imager (SGLI)/Global Change Observation Mission – Climate (GCOM-C)
(Letu et al., 2012, 2016; Nakajima et al., 2019); AHI/Himawari-8 (Letu
et al., 2018); and the Multi-Spectral Imager (MSI)/Earth Cloud Aerosol and
Radiation Explorer (EarthCARE) satellite programmes (Illingworth et al.,
2015), which will be launched in 2023. Furthermore,
Li et al. (2022) showed the effectiveness of the
Voronoi ICS model in describing ice clouds' global visible and infrared
radiative properties in the Community Integrated Earth System Model (CIESM).
As a result, researchers have proven the effectiveness of the database of
the Voronoi ICS model in the visible and infrared regions in the ice cloud
satellite remote sensing and climate model applications (Letu et al.,
2016). Recently, Letu et al. (2012) and Ishimoto et al.
(2013) have successfully extended the spectral range of the
single-scattering property database of the Voronoi ICS model to the
terahertz wave region. The database of the Voronoi ICS model in the
terahertz region was adopted by Baran et al. (2018) as standard
data for the modelling and evaluation of the ISMAR (International SubMillimeter Airborne Radiometer), which the European Space Agency
(ESA) and Met Office jointly developed (Kangas et al., 2014). The
results showed good evaluation performance of the Voronoi ICS model.
However, the effectiveness of the Voronoi ICS model in the terahertz region
in a practical retrieval of ice cloud microphysical properties from
terahertz radiation has yet to be discovered.

Motivated by the above-mentioned situations, this study aims to apply the Voronoi ICS model to the ice cloud remote sensing retrieval of IWP and particle size from aircraft-based terahertz radiation measurements. To achieve this goal, we use the Voronoi ICS model to create a parameterization scheme (referred to as the Voronoi ICS scheme hereafter) for the ice cloud scattering property in the terahertz region. The Voronoi ICS scheme is employed in the RSTAR radiative transfer model (Nakajima and Tanaka, 1986, 1988) to build the LUT for upward terahertz brightness temperature differences between cloudy and clear skies (BTDs) at the top of the ice cloud (TOC) between multi-channel frequencies. The LUT based on the Voronoi ICS model is compared with that of the sphere and hexagonal column ICS models. Finally, the CoSSIR-MCBI evaluates the retrieval results searched iteratively from the LUTs for the Voronoi, sphere, and hexagonal column models. This paper is organized as follows: Sect. 2 introduces the data and radiative transfer model used in this study. Section 3 describes the ice cloud parameterization scheme, retrieval algorithm, and validation with the CoSSIR-MCBI algorithm. Section 4 presents the results of the retrieved IWP and particle size; a comparison of the Voronoi, sphere, and hexagonal column ICS models; and the validation of the retrieval algorithm. Section 5 presents the conclusions of this study.

## 2.1 Single-scattering property database for the Voronoi, sphere, and column ICS models

This study used the single-scattering property database of three ice crystal
habits (Voronoi, sphere, and hexagonal column) in the terahertz radiative
transfer forward simulation. For the Voronoi ICS model, the
single-scattering property database contains 31 ice particle sizes ranging
from 0.25 to 9300 µm and covers 20 terahertz channels with frequencies
ranging from 10 to 874 GHz (see Table 1), corresponding to wavelengths from
0.03 to 3 cm. The hexagonal column ICS model (referred to as the column ICS
model hereafter) is randomly oriented and was defined by Yang et
al. (2000b). For the column ICS model, the aspect ratios $a/L$ (defined as the
ratio of the semiwidth *a* of a particle to its length *L*) are defined as 0.35
and 3.48, respectively, when *L* is less than 100 µm and greater than or
equal to 100 µm. The single-scattering property database of the column
ICS model used in the study was developed by Hong (2007) and Hong et al. (2009) using the
DDA method. For the column ICS model, the single-scattering property
database includes 38 maximum particle dimensions ranging from 2 to
2000 µm and covers 21 terahertz channels with frequencies ranging from 90 to 874 GHz (see Table 1).

The single-scattering properties – including the extinction efficiency, single-scattering albedo, and asymmetry factor of the Voronoi, sphere, and column ICS models in the terahertz region – are used to calculate the terahertz scattering properties of ice clouds. For the Voronoi ICS model, the single-scattering properties are derived from the single-scattering property database developed by Letu et al. (2012, 2016) and Ishimoto et al. (2012) using a combination of FDTD and DDA methods. The DDA method is used to calculate the single-scattering properties of moderate ice particles (SZP>30). The FDTD method is used for small ice particles (SZP<30). As the particle size increases, the shape of the Voronoi ICS model changes and becomes complicated. The geometrical characteristics of Voronoi ICS model shape with increasing ice crystal size have been shown in Fig. 3 and Table 1 in Ishimoto et al. (2012). The mass–dimension and area–dimension power law relationships of the Voronoi ICS model are defined by Ishimoto et al. (2012) and are described in Eqs. (1)–(3) as shown below.

(in centimetre–gram–second system of units, cgs)

where *m* is the mass, *G* is the cross-sectional area, *A*_{r} is the area
ratio, and *D* is the maximum particle dimension of the Voronoi ICS model.
The single-scattering property database of the sphere ICS model is developed
based on the exact solution of the Lorentz–Mie theory (Van de
Hulst, 1957). The sphere ICS database contains the same ice particle sizes
and terahertz frequencies as the Voronoi ICS database. For the Voronoi and
sphere ICS models, calculations of their single-scattering properties
utilized the real and imaginary parts of ice from the newest library of the
refractive index provided by Warren and Brandt (2008). The refractive
indices of the Voronoi and sphere ICS models at frequencies of 10–874 GHz
are computed at a temperature of 266 K, according to Warren and
Brandt (2008). The refractive indices of the column ICS model at frequencies
of 90–340 GHz are derived from Warren (1984) at a temperature of −30 ^{∘}C.

## 2.2 Airborne measurements and auxiliary data

The measured terahertz brightness temperature from CoSSIR/ER-2 aircraft
during the TC4 mission on 17 and 19 July 2007 is utilized in this study
(available at https://espoarchive.nasa.gov/archive/browse/tc4/ER2, last access: 26 December 2021). During the
TC4 field campaign, the CoSSIR measured brightness temperatures in channels
from 183.3 to 873.6 GHz (183.3 ± 1.0, 3.0, 6.6, 220, 380.2 ± 1.8, 3.3, 6.2, and 640 V and 874 GHz), all with matched beamwidths of about
4^{∘} (Evans et al., 2012). According to the studies by
Li et al. (2016) and Liu et al. (2021), the atmospheric windows
are near 640 and 874 GHz, and the atmospheric absorption peaks are near 380 GHz. The leading absorbing gases are water vapour and ozone. Therefore, both
the 640 and 874 GHz brightness temperature are affected by ice clouds, while
the brightness temperature of 380 GHz is insensitive to ice cloud
microphysical properties. Hence, the 380 GHz minus 640 GHz brightness
temperature differences can highlight the brightness temperature depression
caused by ice clouds. And the 640 GHz minus 874 GHz brightness temperature
differences can reflect the difference in the scattering properties of
differently shaped ice clouds. According to Li et al. (2016), the
differences between 640 and 874 GHz also can offset the regional errors due
to different latitudes and atmospheric profiles. In this study, we choose
the centre frequencies of 380, 640, and 874 GHz.

The particle size distributions (PSDs) describe the relationship between ice
particle size and particle number concentration and are essential in the
calculation of the average scattering properties of ice clouds. In this
study, we select 14 408 groups of PSDs from aircraft observation sampling
data (available at https://www.ssec.wisc.edu/ice_models/microphysical_data.html, last access: 18 February 2022) (Heymsfield et al.,
2013), which are based on 11 field flight observation experiments covering
ice cloud observations in mid-latitude and tropical regions. The 11 field
programmes span a wide range of locations (ranging from 12^{∘} S to
70^{∘} N and from 148^{∘} W to 130^{∘} E) and encompass the temperature range 0 to
−86 ^{∘}C, with altitudes from near the surface to 18.7 km. This
dataset is representative of the wide range of conditions where ice clouds
are found in the troposphere and lower stratosphere on a near-global scale
(Li et al., 2022; Heymsfield et al., 2013). For the fitting of PSDs for
the Voronoi, sphere, and column ICS models, we adopted the gamma distribution
following Heymsfield et al. (2013) in the form of Eq. (4):

where *L* is the maximum particle dimension, *n*(*L*) is the particle
concentration per unit volume (e.g. cm^{−3}), *N*_{0} is the intercept,
*λ* is the slope, and *μ* is the dispersion. The physical meaning
of the PSDs is that *n*(*L*)×d*L* is the number of particles per unit area.

The total water vapour and ozone column data provided by the ERA5 reanalysis
data are used as input data for the radiative transfer model to simulate
clear-sky brightness temperature. The ERA5 reanalysis data are the fifth-generation reanalysis data product developed by the European Centre for
Medium-Range Weather Forecasts (ECMWF) (Hans et al., 2019). The
total water vapour and ozone column data used have a horizontal resolution
of $\mathrm{0.25}{}^{\circ}\times \mathrm{0.25}{}^{\circ}$ and a temporal resolution of 1 h. The retrieved results are validated against the IWP and *D*_{me}
(Evans et al., 2005) from the CoSSIR-MCBI algorithm during the same
period. The IWP and *D*_{me} are available at
https://espoarchive.nasa.gov/archive/browse/tc4/ER2 (last access: 20 May 2022).

## 2.3 Radiative transfer model

The RSTAR radiative transfer model is a set of numerical radiative transfer
models developed by Nakajima and Tanaka (1986, 1988) for the
plane-parallel atmosphere. The calculated wavelengths can cover from 0.17 to
1000 µm, and the assumed plane-parallel atmosphere could be divided
into 50 layers from sea level to a maximum altitude of 120 km. The RSTAR
model contains six atmospheric profiles (tropical, mid-latitude summer,
mid-latitude winter, high-latitude summer, high-latitude winter, and US
standard atmosphere), including vertical profiles of temperature, pressure,
water vapour, and ozone. Calculating gas absorption in RSTAR is based on the
K-distribution method developed by Sekiguchi and Nakajima (2008), which
considers important atmospheric radiative gases (H_{2}O, CO_{2}, O_{3}, N_{2}O, CO CH_{4},
etc.) and trace gases. The K-distribution method parameters were obtained
from the HITRAN-2004 database. The RSTAR radiative transfer model assumes
the simulated scene is composed of a homogeneous ice cloud layer.

Figure 1 shows the general flowchart for the inversion of the IWP and
*D*_{me} of ice clouds using the CoSSIR brightness temperature measurements.
For convenience, the difference between the 640 GHz BTD and the 874 GHz BTD
is simplified to BTD_{2–3}, and the difference between the 380 GHz BTD and
the 640 GHz BTD is simplified to BTD_{1–2}. We refer to the difference
between the BTD_{1–2} and BTD_{2–3} as BTD_{1–3}. Firstly, based on
the single-scattering property database of the Voronoi, sphere, and column
ICS models, the atmospheric radiative transfer model RSTAR is used to build
LUTs of top-of-cloud (TOC) BTD_{2–3} and BTD_{1–3} for three ICS models,
respectively. With the assumptions of an a priori value of IWP and *D*_{me}, the
initial TOC BTD_{2–3} and BTD_{1–3} simulations can be searched from the
LUTs for three ICS models. Secondly, we use the RSTAR to construct a
clear-sky LUT of TOC brightness temperature (BT) for 380, 640, and 874 GHz,
respectively. The inputs are different ozone and water vapour column values
under clear-sky conditions (see Sect. 3.2 for details). Then, the ERA5
reanalysis data are used to estimate the clear-sky TOC BT based on the
clear-sky LUT. The measured cloudy-sky TOC BTs of 380, 640, and 874 GHz are
obtained from the CoSSIR measurements. Consequently, the TOC BTD_{2–3} and
BTD_{1–3} measurements can be calculated using the measured cloudy-sky TOC
BT minus the interpolated clear-sky TOC BT. Finally, the cost function will
provide an estimate of the coherence between the measured and simulated TOC
BTD_{2–3} and BTD_{1–3}. The IWP and *D*_{me} are obtained by using the Gaussian–Newtonian non-linear optimization estimation method. The retrieved
results are validated against the IWP and *D*_{me} retrieved from the
CoSSIR-Evans algorithm. In the RSTAR forward simulation, the input parameter
of the ice particle size is *R*_{e}, which is defined by Eq. (5):

where *r* is the equivalent-volume sphere's particle radius, and *n*(*r*) is the
particle size distribution. According to Sieron et al. (2017), a
mass-weighted size *D*_{me} would be a more appropriate parameter size than
the area-weighted size for describing the PSDs. The *D*_{me} is given by Eq. (6):

where *D* is the maximum particle dimension of ice particles, *m* is the
particle mass, and *n*(*D*) is the particle size distribution. To validate the
retrieved *D*_{me} using the *D*_{me} from the CoSSIR-Evans algorithm, the
transformation from the *R*_{e} to the *D*_{me} is necessary due to the
different definitions of *R*_{e} and *D*_{me}. Hence, we developed a
conversion relationship between the *R*_{e} and *D*_{me} combined with
different particle size distributions. Based on the integration of *r* and
*D* over 14 408 PSDs, multiple groups of *R*_{e} and *D*_{me} are calculated
according to Eqs. (5) and (6). Numerical fitting was used to build a
relationship between the *R*_{e} (independent variable) and *D*_{me}
(dependent variable) over 14 408 PSDs. The conversion formula between
*R*_{e} and *D*_{me} is demonstrated as follows:

where *a*, *b*, and *c* are fitting coefficients obtained by numerical fitting and
provided as input. Based on this relationship, we have unified all the
*R*_{e} into *D*_{me} for comparability for the Voronoi, sphere, and column ICS
models.

## 3.1 Parameterization of ice cloud optical properties

To apply the Voronoi, sphere, and column ICS models to the RSTAR radiative transfer model for forwarding simulation, three parameterization schemes (referred to as the Voronoi, sphere, and column ICS schemes hereafter) for the scattering properties of ice clouds in the terahertz spectrum need to be constructed. The ice cloud optical properties depend on the single-scattering properties of the ICS and the PSDs. The effective ice cloud particle size measures the average ice cloud particle size for a given PSD. Different methods have been used to determine the effective ice cloud size, and according to Baum et al. (2005a, b), the effective particle diameter for a given PSD is determined by Eq. (8):

where *D*_{e} is the effective particle diameter, and *V* and *A* are the volume
and projected area of Voronoi and sphere models. Then, the spectral ice
cloud optical properties (mass-averaged extinction coefficients,
single-scattering albedo, asymmetry factor, and mass-averaged absorption
coefficients) for the Voronoi and sphere ICS schemes are calculated for all
PSDs given by Eqs. (9)–(12):

where *K*_{ext}(*λ*) represents spectral-mass-averaged extinction coefficients
(m^{2} g^{−1}), *ϖ*(*λ*) is spectral single-scattering albedo,
*g*(*λ*) is a spectral asymmetry factor, and *K*_{abs}(*λ*) represents
spectral-mass-averaged absorption coefficients (m^{2} g^{−1}). *Q*_{ext}, *g*, *Q*_{sca},
and *Q*_{abs} are extinction efficiency, asymmetry factor, scattering
efficiency, and absorption efficiency for Voronoi, sphere, and column models.
Based on the parameterized scattering properties of ice clouds and the
effective particle diameter of ice clouds, we developed a parameterization
scheme for the scattering properties of ice clouds by establishing the
spectral bulk-scattering properties of ice clouds as functions of the
effective particle diameter of ice clouds. The least squares method is used
to obtain first-order and third-order polynomial fitting equations according
to Eqs. (13)–(16):

where *a*_{i}, *b*_{j}, *c*_{j}, and *d*_{j} ($i=\mathrm{0},\mathrm{1}$; $j=\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3}$) are fitting
coefficients and are spectral functions of the terahertz wavelength. Values
of the above coefficients for the Voronoi ICS scheme are listed in Appendix A (Tables A.1–A.4).

## 3.2 Look-up table

Before constructing the LUT, the sensitivity of different terahertz
brightness temperatures to the IWP and *D*_{me} of ice clouds needs to be
analysed to build a representative and efficient LUT. According to the
sensitivity results (see Sect. 4.3), the BTDs between the cloudy and
clear-sky conditions at 380, 640, and 874 GHz frequencies are simplified to
BTD_{1}, BTD_{2}, and BTD_{3}, respectively, and the differences
between the two BTDs are represented by a dash. Based on the Voronoi, sphere,
and column ICS schemes, the RSTAR is used to construct three LUTs (Voronoi,
sphere, and column LUTs) of BTD_{2–3} and BTD_{1–3} for cloudy- and
clear-sky conditions. For the cloudy-sky LUT, the number of the IWP is set
to 12, and the range of values is defined in log10 space from 0 to 3.36 in
steps of 0.28. The number of *R*_{e}'s is set to six, and the range of values is
defined in log10 space from 1.6 to 3.1 in steps of 0.25. For the clear-sky
LUT, the number of total ozone columns is set to seven, and the range of values
is from 200 to 500 in steps of 50. To be consistent with aircraft
observations of terahertz brightness temperatures, BTD_{2–3} and
BTD_{1–3} are simulated at the mean altitude (20 km). The US standard
atmospheric profile is used in the simulation, and the cloud top temperature
is assumed to be the same as the atmospheric temperature at that level. The
surface is assumed to be a black body (emissivity equals 1) with a
temperature of 288.15 K.

## 3.3 Optimal estimation inversion method

Based on the terahertz BTD LUTs for the Voronoi, sphere, and column ICS
schemes established in the previous section, the IWP and *D*_{me} are
retrieved using an optimization method with Gaussian–Newtonian non-linear
iterations. Based on the atmospheric radiative transfer transmission in the
terahertz spectrum, if the background field under clear-sky conditions is
subtracted from the cloudy-sky condition, the TOC BTDs are only functions of
the cloud microphysical property parameters, which are given by
Eqs. (17) and (18):

where *X* is the vector matrix composed of the variables of the IWP and
*D*_{me} to be solved. *Y* is the vector composed of the two BTDs and the
uncertainty vector. The vector *ϵ* represents the uncertainties that
are attached to the measurements (i.e. instrumental accuracy) and to the
radiative transfer forward model (i.e.approximation errors in the radiative
transfer model). Following Marks and Rodgers (1993), a good convergence
can be obtained when the value of the cost function is lower than the size
of the measurement vector. Since there is no robust a priori value for IWP and
*D*_{me}, we selected an average value as the initial value for an a priori
value of the IWP and *D*_{me}. Assuming that *Y* is the value of two BTDs
measured by the aircraft, the inverse *X* is the minimum value for solving
Eq. (19):

where *X*_{a} is a vector consisting of the prior estimates of the IWP and
*R*_{e}, *γ* is a Lagrangian operator, min denotes the minimal value
of solving this function, and the value of *X* at the minimal value is
considered to be the retrieved result. For the solution method of the non-linear
problem in the inversion process, the Newton non-linear iterative method is
usually used. It has a faster inversion speed and higher solution accuracy,
and its iterative form follows Eq. (20):

where the *i* represents the number of iterations, and the superscripts
denote the first-order and second-order derivations. Then the iterations
start from the a priori initial estimate until the convergence criterion is
satisfied or when the number of iterations satisfies the required number of
iterations. Then the iteration is stopped, and the solution results are
obtained. The final inversion results are validated using the results from
the CoSSIR-MCBI algorithm.

## 4.1 Single-scattering property database for the Voronoi, sphere, and column ICS models

Figure 2 compares the extinction efficiency, single-scattering albedo, and asymmetry factor, varying with the SZP for the Voronoi, column, and sphere ICS models at 325 and 874 GHz. For small ice particles with SZPs less than 0.1, the single-scattering properties are small and barely influenced by the shape of the ice particles. This is because the single-scattering properties of ice particles are close to Rayleigh scattering when the SZP is small. As particle size increases, particle scattering is predominantly Mie scattering, and the sensitivity of the single-scattering properties to the ice crystal habits becomes pronounced so that the ice crystal shape contributes to the large differences for large particle sizes. The single-scattering properties of the Voronoi and column models vary more smoothly than those of the sphere model. As shown in Fig. 2, the smooth curve of the Voronoi model reflects the notion that the FDTD and DDA methods provide good continuity for the transition between different size parameters of the Voronoi model. For large particles (SZP>3) at 874 GHz, the Voronoi model has a higher extinction efficiency than the other two models. At 325 GHz, the single-scattering albedo of the Voronoi model is close to the sphere model and is higher than the column model. At 874 GHz, the single-scattering albedo of the Voronoi model is close to the column model and is higher than the sphere model. The Voronoi model has a higher asymmetry factor than the other two models at 325 and 874 GHz. On the one hand, the higher extinction efficiency and single-scattering albedo of the Voronoi ICS model for large particles are possibly due to the multifaceted shapes of the Voronoi ICS model, which can result in significant side and backward scattering and increase the scattered energy. On the other hand, the higher asymmetry factor of the Voronoi ICS model for large particles is possible because the scattered energy is dominated by diffraction. The diffracted energy is concentrated in the forward direction, leading to a large asymmetry factor of the Voronoi ICS model.

Figure 3 shows the contours of the single-scattering properties for the Voronoi, sphere, and column ICS models over 20 terahertz wavelengths (0.03–0.3 cm) and 31 particle sizes (1–270 µm). The sharp changes in the single-scattering albedo for the Voronoi, sphere, and column ICS models can be seen from small to large ice particles. For small particles in the low-frequency channels (long wavelengths), their scattering is mainly Rayleigh scattering with significant absorption effects. The absorption energy is proportional to the volume of ice particles and is barely affected by the ice particle shape. In the high-frequency channels (short wavelengths), the Mie scattering plays a leading role, and the scattering function plays a dominant role. Especially for large ice particles, the single-scattering albedo is close to one, and the influence of the ice particle shape becomes obvious. Figure 4 shows the differences in extinction efficiency, single-scattering albedo, and asymmetry factor between the Voronoi and the other two ICS models as functions of the effective radius and terahertz wavelength. Overall, the extinction efficiency of the Voronoi model is higher than the other two ICS models. The asymmetry factor of the Voronoi model is higher than the other two ICS models in the low-frequency channel for large ice particles.

Figure 5 displays the scattering phase functions of the Voronoi, sphere, and
column ICS models. The variation in the scattering phase function for the
Voronoi model tends to be more dramatic compared to the sphere and column
ICS models. The scattering phase function is axisymmetric about the
90^{∘} scattering angle for small ice particles, which can be
approximated as Rayleigh scattering. For small ice particles, the scattering
phase function shows extreme values in the forward (0^{∘}) and
backward (180^{∘}) directions and shows minimal values in both side
directions (90 and 270^{∘}). As the ice particle size
increases, the forward scattering is significantly larger than the side
and backward scattering and gradually tends to be more Mie scattering. For
the sphere model in the high-frequency channels, the forward scattering
increases with the increase in particle size. In the low-frequency channels,
the forward scattering remains almost constant with the increase in particle
size. Compared to the Voronoi and sphere ICS models, the column ICS model
shows smaller phase functions. For the Voronoi model, the low values of the
scattering phase function move towards large scattering angles with the
increase in particle size.

## 4.2 Bulk-scattering properties of ice clouds in the terahertz region

Based on the single-scattering properties of the Voronoi, sphere, and column ICS models, the Voronoi, sphere, and column ICS schemes are developed in the terahertz region with the integration over both PSDs and terahertz frequencies. The calculated bulk-scattering properties of ice clouds include the mass extinction coefficients, single-scattering albedo, asymmetry factor, and mass absorption coefficients. Figure 6 compares the calculated bulk-scattering properties of the Voronoi, sphere, and column ICS schemes over 14 408 PSDs at four terahertz frequencies (325, 448, 664, and 874 GHz). The bulk-scattering properties of ice clouds depend on the effective diameter and the terahertz frequency. The single-scattering albedo increases linearly with the effective diameter for the sphere ICS scheme. For the Voronoi ICS scheme, the single-scattering albedo increases for the effective diameter smaller than 100 µm and approaches 0.95 with effective diameters exceeding 100 µm. The column ICS scheme has the lowest single-scattering albedo compared to the other two schemes for small effective diameters and gradually exceeds the other two schemes with the increase in the frequency and effective diameter. The mass extinction coefficients obtained from the three schemes show a uniformly positive correlation with the effective diameter and the terahertz frequency. At four terahertz frequencies, the mass extinction coefficients of the Voronoi ICS scheme are close to the column ICS scheme for all effective diameters. The sphere ICS scheme has the highest mass extinction coefficients compared to the other two schemes for all effective diameters and four terahertz frequencies. The Voronoi ICS scheme has a similar asymmetry factor to the sphere ICS scheme at low frequency and becomes the highest compared to the other two schemes at high frequency. The column ICS scheme has the lowest asymmetry factor compared to the other two schemes for all effective diameters. For all effective diameters, the mass absorption coefficients of the sphere ICS scheme have a stronger absorption effect than the other two schemes and decrease with increasing effective diameter. The Voronoi ICS scheme has the lowest mass absorption coefficients compared to the other two schemes for effective diameter smaller than 100 µm and becomes higher than the column ICS scheme with increasing effective diameters. Overall, the bulk-scattering properties for all schemes increase with increasing frequencies. This is mainly because the scattering properties of ice clouds are more significant as the wavelength decreases.

## 4.3 Sensitivity results

We discuss the sensitivity of the TOC BTD_{2–3} and BTD_{1–3} to the IWP
and *D*_{me} based on the Voronoi, sphere, and column ICS schemes,
respectively, in the RSTAR model. Figure 7 shows that the BTD_{2–3} and
BTD_{1–3} are positively correlated with the IWP and *D*_{me}. For the
Voronoi and column ICS models, the BTD_{2–3} and BTD_{1–3} show
monotonically increasing relationships with the increase in IWP. For ice
clouds with the *D*_{me} larger than 200 µm, the 50 to 1000 µm
particle sizes can lead to approximately 0–20 K BTD_{2–3} and 0–30 K
BTD_{2–3}. The BTD_{2–3} and BTD_{1–3} increase with the increase in
the *D*_{me} and are close to a constant value for *D*_{me} larger than 600
µm. As shown in Fig. 8, the Voronoi ICS scheme has higher BTD_{2–3}
and BTD_{1–3} compared to the sphere and column ICS scheme. In summary,
the TOC BTD_{2–3} and BTD_{1–3} have a strong sensitivity to the IWP and
*D*_{me}, especially for moderate to large ice particles and large IWP. The
different ICS schemes vary considerably in their modelling of terahertz
brightness temperatures.

Figure 9 exhibits the variation in the TOC BTDs at different IWPs and
*D*_{me} for 380, 640, and 874 GHz frequencies based on the RSTAR model. The
BTD_{1–3} has a strong sensitivity to the IWP and increases with the
increase in the IWP, and the slope shows that the increase is pronounced
when the IWP is less than 600 g m^{−2}. The BTD_{2–3} has a strong
sensitivity to the *D*_{me} and increases with the increasing *D*_{me}. By
comparing the LUTs of the Voronoi, sphere, and column ICS models, it is found
that at the same BTD_{2–3}, the sphere LUT has the highest IWP compared to
the column and Voronoi LUT, while the Voronoi LUT has the lowest. The
*D*_{me} of the sphere LUT is smaller than that of the column and Voronoi LUT
for the same BTD_{1–3}. This is mainly due to the higher single-scattering
albedo and asymmetry factor of the Voronoi ICS model at low frequencies,
resulting in stronger forward-scattering energy and a larger BTD than the
sphere and column ICS model for the same IWP and *D*_{me}.

## 4.4 Inversion and validation results

To develop the inversion method from the CoSSIR measurements of the
terahertz brightness temperature, we need to perform a quantitative
simulation to test the accuracy of the algorithm. Figure 10
shows the
validation results of the inversion algorithm using 2000 groups of
BTD_{2–3} and BTD_{1–3} simulated by RSTAR using 2000 randomly generated
IWP and *D*_{me}. The results show that most of the validation results of the
red density lie on the 1:1 line. The correlation coefficients of the IWP and
*D*_{me} are 0.94 and 0.99, and the mean absolute errors (MAEs) are 35.46 g m^{−2} and 8.56 µm, respectively. The test results meet the formal
scientific mission requirement of ice cloud remote sensing retrieval,
according to Buehler et al. (2007). The high accuracy of the
validation results proves the effectiveness and accuracy of the inversion
algorithm.

We adopt the CoSSIR-MCBI results as the benchmark for evaluating the
retrieved IWP and *D*_{me} based on three schemes using the new inversion
algorithm. To quantify the differences in the retrieval results between the
three schemes and the CoSSIR-MCBI, we use the CoSSIR-MCBI results minus the
retrieved results of the Voronoi, column, and sphere ICS schemes,
respectively, to analyse their differences. Figure 11a and b show the
CoSSIR-MCBI results on 19 July 2007 overlaid with the retrieval results of
the IWP and *D*_{me} based on the Voronoi scheme. Red and blue dots,
respectively, represent the Voronoi ICS scheme and CoSSIR-MCBI results.
Figure 11a and b show that the overall performance shows good agreement
for the Voronoi ICS scheme compared to the CoSSIR-MCBI results. The matching
rates for the IWP and *D*_{me} are 80.3 % and 83.2 %, respectively.
Figure 11c and d show the joint histogram of differences in the
retrieved IWP and *D*_{me} between the Voronoi, sphere, and column schemes and
the CoSSIR-MCBI results, respectively. Overall, the retrieved IWP and
*D*_{me} parameters from the Voronoi ICS scheme agree well with the
CoSSIR-MCBI results for all cases. Figure 12 shows the inversion results of
IWP and *D*_{me} based on the optimal estimation algorithm from the CoSSIR
aircraft-measured brightness temperature. For the Voronoi, sphere, and column
ICS models, the correlation coefficients of the retrieved IWP are 0.87, 0.67,
and 0.74, with MAEs of 22.38, 23.55, and 22.91 g m^{−2} and RMSEs
of 30.45, 35.22, and 32.64 g m^{−2}, respectively. The correlation
coefficients of the *D*_{me} are 0.83, 0.64, and 0.76, with MAEs of 18.46,
21.19, and 19.91 µm and RMSEs of 24.57, 26.51, and 25.04 µm,
respectively. Overall, the accuracy for the IWP and *D*_{me} based on the
Voronoi ICS model is better than the other two ICS models compared to the
CoSSIR-MCBI results. The sphere ICS model overestimates IWP and *D*_{me}
compared with the validation data. According to the sensitivity results of
Figs. 7 and 8, the Voronoi ICS scheme has higher BTD_{2–3} and
BTD_{1–3} compared to the sphere and column ICS schemes, especially for
large particles and IWP. This characteristic can be explicitly explained by
the higher asymmetry factor of the Voronoi ICS model compared to the sphere
and column ICS models. Thus, stronger forward-scattering energy can be
detected for the Voronoi ICS model than for the other two models. The
look-up table of the Voronoi ICS model can cover more IWP and *D*_{me}. The
brightness temperature variations in the Voronoi-shaped ice clouds are more
prominent and sensitive to the IWP and *D*_{me}. Therefore, the Voronoi ICS
model results are better than the other two models. According to the look-up
table as shown in Fig. 9, there are overlapping lines when *D*_{me} is
small (*D*_{me}<40 µm) and large (*D*_{me}>140 µm). When BTD_{2–3} and BTD_{1–3} data fall under such overlapping
lines of the look-up table, this overlapping region can lead to obtaining
the same IWP and *D*_{me} when searching the look-up table.

In this study, we applied the irregularly shaped Voronoi ice crystal
scattering (ICS) model to the ice cloud remote sensing retrieval of the ice
water path (IWP) and particle size based on the Compact Scanning Submillimeter-wave Imaging Radiometer (CoSSIR) terahertz radiation
measurements. The bulk-scattering property parameterization (Voronoi ICS
scheme) in the terahertz region was developed based on the single-scattering
properties of the Voronoi ICS database and 14 408 groups of particle size
distributions from in situ observations. The Voronoi ICS scheme was applied
to the atmospheric radiative transfer model RSTAR and compared with the
sphere and column ICS schemes to carry out the sensitivity analysis. We
analysed the sensitivity of brightness temperature differences
between cloudy and clear skies (BTDs) at 380, 640, and 874 GHz to the
IWP and particle size. Based on the sensitivity analysis results, we built
three terahertz multi-channel BTD look-up tables (LUTs) for the Voronoi,
sphere, and column ICS schemes using the RSTAR atmospheric radiative transfer
model. Based on the three LUTs, we utilized the Gaussian–Newtonian non-linear
optimization estimation method to retrieve the IWP and particle size from
the CoSSIR terahertz radiation measurements. Finally, the retrieval results
were evaluated by the IWP and median mass diameter (*D*_{me}) derived from
the CoSSIR-MCBI algorithm. The main conclusions were obtained as follows.

The bulk-scattering properties of ice clouds in the terahertz region, including the mass extinction coefficients, single-scattering albedo, asymmetry factor, and mass absorption coefficients of the Voronoi ICS scheme, were applied to the RSTAR model and were compared with the sphere and column ICS schemes. The results showed that the Voronoi ICS scheme has a distinct feature of lower-absorption properties and higher asymmetry factors for larger particle sizes in the terahertz region. This feature could be related to the complex, multifaceted shape of the Voronoi ICS model and suggests that the Voronoi ICS scheme can produce relatively stronger forward scattering and fewer absorption effects compared with the sphere and column ICS schemes.

The sensitivity analysis showed that the BTDs between 640 and 874 GHz are
sensitive to the IWP variation, and the BTDs of 380, 640, and 874 GHz
are sensitive to the particle size. The atmospheric absorption peak near 380 GHz and the atmospheric window near 640 and 874 GHz can be effectively
used for the IWP in the range of 50–200 g m^{−2} and particle size of
50–300 µm.

A comparison of the results from the Voronoi, sphere, and column ICS models
shows that the results from the Voronoi ICS model are better than the sphere
and column ICS models. The IWP and *D*_{me} retrieved from the Voronoi ICS
scheme showed higher consistency with CoSSIR-MCBI results than the other two
schemes, with correlation coefficients of 0.87 and 0.83 for IWP and
*D*_{me}, respectively. For the Voronoi ICS model, the mean absolute errors
(MAEs) of the IWP and *D*_{me} are improved by 5.0 % and 12.8 %, and RMSE
is improved by 13.5 % and 7.3 %, respectively. The sphere ICS scheme
overestimates the IWP and *D*_{me} by up to the MAEs of 23.55 g m^{−2} and
21.19 µm, respectively. This is mainly due to the differences in the
absorption efficiency and asymmetry factor in the single-scattering
properties of ice particles, which have a significant impact on the
description of the scattering and radiative properties of ice clouds in the
terahertz region.

In conclusion, the analysis of terahertz BTDs of 380, 640, and 874 GHz exhibits obvious sensitivity to the IWP and particle size of ice clouds, which could complement visible–infrared and microwave spectra. The present work provides the potential utility of inferring the IWP and particle size of ice clouds using BTDs of 380, 640, and 874 GHz. With the LUT for BTDs of 380, 640, and 874 GHz, the retrieval of terahertz ice cloud properties from airborne measurements based on the irregularly shaped Voronoi ICS models is newly developed. We find that the retrieval results based on the Voronoi ICS scheme present a better agreement with the CoSSIR-MCBI algorithm than the sphere and column ICS schemes. This study confirmed that the Voronoi ICS model has ice cloud inversion capabilities in the terahertz region, which may provide a reference for future use in aircraft-based and satellite-based terahertz ice cloud remote sensing applications.

The CoSSIR/ER-2 aircraft data during the TC4 mission on 17 and 19 July 2007
are available at https://espoarchive.nasa.gov/archive/browse/tc4/ER2 (NASA, 2022). The
IWP and *D*_{me} from the CoSSIR-MCBI algorithm are available at
https://espoarchive.nasa.gov/archive/browse/tc4/ER2 (NASA, 2022). The 14 408 groups of
PSDs from 11 field flight observation experiments are available at
https://www.ssec.wisc.edu/ice_models/microphysical_data.html (SSEC, 2022).

ML developed the terahertz ice cloud remote sensing inversion algorithm for the IWP and particle size of ice clouds and evaluated the retrieval result by validating it against the results from the CoSSIR-MCBI algorithm. ML is also responsible for downloading auxiliary data and writing the initial draft of this paper. HL provided the single-scattering property database of Voronoi models in the terahertz region and assisted in developing the parameterization of ice cloud scattering properties in the RSTAR model. HL also designed the aims and structures of this study and guided the writing and revisions of the manuscript.

HI developed the single-scattering property database of Voronoi models in the terahertz region and helped in the writing and revisions of the manuscript.

TYN provided the atmospheric radiative transfer model RSTAR and is responsible for the optimization of the RSTAR model and assisted in the parameterization of ice cloud scattering properties.

SL and LL assisted in developing the ice cloud remote sensing retrieval algorithm of the IWP and particle size of ice clouds, provided the CoSSIR terahertz radiation measurement data, and helped with reviewing the manuscript.

DJ guided the development of the ice cloud remote sensing retrieval algorithm and helped with the review of the manuscript.

HS assisted in analysing the results, guided the flowchart of the study, and reviewed the manuscript. CS assisted in designing the structures of this study, guided the writing of the paper, and helped review the manuscript.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This paper was edited by Alexander Kokhanovsky and reviewed by five anonymous referees.

Baran, A., Ishimoto, H., Sourdeval, O., Hesse, E., and Harlow, C.: The applicability of physical optics in the millimetre and sub-millimetre spectral region. Part II: Application to a three-component model of ice cloud and its evaluation against the bulk single-scattering properties of various other aggregate models, J. Quant. Spectrosc. Ra., 206, 83–100, https://doi.org/10.1016/j.jqsrt.2017.10.027, 2018.

Baran, A. J.: A review of the light scattering properties of cirrus, J. Quant. Spectrosc. Ra., 110, 1239–1260, https://doi.org/10.1016/j.jqsrt.2009.02.026, 2009.

Baran, A. J.: From the single-scattering properties of ice crystals to climate prediction: A way forward, Atmos. Res., 112, 45–69, https://doi.org/10.1016/j.atmosres.2012.04.010, 2012.

Baran, A. J. and Labonnote, L. C.: A self-consistent scattering model for cirrus. I: The solar region, Q. J. Roy. Meteor. Soc., 133, 1899–1912, https://doi.org/10.1002/qj.164, 2007.

Baran, A. J., Cotton, R., Furtado, K., Havemann, S., Labonnote, L. C., Marenco, F., Smith, A., and Thelen, J. C.: A self-consistent scattering model for cirrus. II: The high and low frequencies, Q. J. Roy. Meteor. Soc., 140, 1039–1057, https://doi.org/10.1002/qj.2193, 2014a.

Baran, A. J., Hill, P., Furtado, K., Field, P., and Manners, J.: A Coupled Cloud Physics Radiation Parameterization of the Bulk Optical Properties of Cirrus and Its Impact on the Met Office Unified Model Global Atmosphere 5.0 Configuration, J. Climate, 27, 7725–7752, 2014b.

Baum, B. A., Heymsfield, A. J., Yang, P., and Bedka, S. T.: Bulk scattering properties for the remote sensing of ice clouds. Part I: Microphysical data and models, J. Appl. Meteorol., 44, 1885–1895, https://doi.org/10.1175/JAM2308.1, 2005a.

Baum, B. A., Yang, P., Heymsfield, A. J., Platnick, S., King, M. D., Hu, Y. X., and Bedka, S. T.: Bulk scattering properties for the remote sensing of ice clouds. Part II: Narrowband models, J. Appl. Meteorol., 44, 1896–1911, https://doi.org/10.1175/JAM2309.1, 2005b.

Brath, M., Fox, S., Eriksson, P., Harlow, R. C., Burgdorf, M., and Buehler, S. A.: Retrieval of an ice water path over the ocean from ISMAR and MARSS millimeter and submillimeter brightness temperatures, Atmos. Meas. Tech., 11, 611–632, https://doi.org/10.5194/amt-11-611-2018, 2018.

Buehler, S., Jimenez, C., Evans, K., Eriksson, P., Rydberg, B., Heymsfield, A., Stubenrauch, C., Lohmann, U., Emde, C., John, V., Tr, S., and Davis, C.: A concept for a satellite mission to measure cloud ice water path and ice particle size, Q. J. Roy. Meteor. Soc., 133, 109–128, https://doi.org/10.1002/qj.143, 2007.

C.-Labonnote, L., Brogniez, G., Doutriaux-Boucher, M., Buriez, J.-C., Gayet, J.-F., and Chepfer, H.: Modeling of light scattering in cirrus clouds with inhomogeneous hexagonal monocrystals. Comparison with in-situ and ADEOS-POLDER measurements, Geophys. Res. Lett., 27, 113–116, https://doi.org/10.1029/1999GL010839, 2000.

C.-Labonnote, L., Brogniez, G., Buriez, J.-C., Doutriaux-Boucher, M., Gayet, J.-F., and Macke, A.: Polarized light scattering by inhomogeneous hexagonal monocrystals: Validation with ADEOS-POLDER measurements, J. Geophys. Res.-Atmos., 106, 12139–12153, https://doi.org/10.1029/2000JD900642, 2001.

Ceccaldi, M., Delanoë, J., Hogan, R., Pounder, N., Protat, A., and Pelon, J.: From CloudSat-CALIPSO to EarthCare: Evolution of the DARDAR cloud classification and its comparison to airborne radar-lidar observations, J. Geophys. Res.-Atmos., 118, 7962–7981, https://doi.org/10.1002/jgrd.50579, 2013.

Chen, Q. and Zhang, H.: Effects of ice crystal habit weight on ice cloud optical properties and radiation, Acta Meteorol. Sin., 76, 279–288, https://doi.org/10.11676/qxxb2017.088, 2018.

Cho, H.-M., Zhang, Z., Meyer, K., Lebsock, M., Platnick, S., Ackerman, A. S., Di Girolamo, L., C.-Labonnote, L., Cornet, C., Riedi, J., and Holz, R. E.: Frequency and causes of failed MODIS cloud property retrievals for liquid phase clouds over global oceans, J. Geophys. Res.-Atmos., 120, 4132–4154, https://doi.org/10.1002/2015JD023161, 2015.

Cimini, D., Westwater, E., Gasiewski, A., Klein, M., Leuski, V., Mattioli, V., Dowlatshahi, S., and Liljegren, J.: Ground-Based Millimeter- and Submillimiter Wave Observations of the Arctic Atmosphere, in: 2006 IEEE MicroRad, San Juan, PR, USA, 28 February–3 March 2006, 247–251, https://doi.org/10.1109/MICRAD.2006.1677097, 2006.

Deeter, M. and Evans, K.: A Novel Ice-Cloud Retrieval Algorithm Based on the Millimeter-Wave Imaging Radiometer (MIR) 150- and 220-GHz Channels, J. Appl. Meteorol., 39, 623–633, https://doi.org/10.1175/1520-0450-39.5.623, 2000.

Deschamps, P. Y., Breon, F. M., Leroy, M., Podaire, A., Bricaud, A., Buriez, J. C., and Seze, G.: The POLDER Mission: Instrument Characteristics and Scientific Objectives, IEEE T. Geosci. Remote, 32, 598–615, https://doi.org/10.1109/36.297978, 1994.

Doutriaux-Boucher, M., Buriez, J.-C., Brogniez, G., C.-Labonnote, L., and Baran, A.: Sensitivity of retrieved POLDER directional cloud optical thickness to various ice particle models, Geophys. Res. Lett., 27, 109–112, https://doi.org/10.1029/1999GL010870, 2000.

Draine, B. T. and Flatau, P. J.: Discrete-Dipole Approximation for Scattering Calculations, J. Opt. Soc. Am. A, 11, 1491–1499, https://doi.org/10.1364/JOSAA.11.001491, 1994.

Eriksson, P., Ekelund, R., Mendrok, J., Brath, M., Lemke, O., and Buehler, S. A.: A general database of hydrometeor single scattering properties at microwave and sub-millimetre wavelengths, Earth Syst. Sci. Data, 10, 1301–1326, https://doi.org/10.5194/essd-10-1301-2018, 2018.

Eriksson, P., Rydberg, B., Mattioli, V., Thoss, A., Accadia, C., Klein, U., and Buehler, S. A.: Towards an operational Ice Cloud Imager (ICI) retrieval product, Atmos. Meas. Tech., 13, 53–71, https://doi.org/10.5194/amt-13-53-2020, 2020.

Evans, K., Wang, J., Racette, P., Heymsfield, G., and Li, L.: Ice Cloud Retrievals and Analysis with the Compact Scanning Submillimeter Imaging Radiometer and the Cloud Radar System during CRYSTAL FACE, J. Appl. Meteorol., 44, 839–859, https://doi.org/10.1175/JAM2250.1, 2005.

Evans, K. F. and Stephens, G. L.: Microwave Radiative Transfer through Clouds Composed of Realistically Shaped Ice Crystals. Part I. Single Scattering Properties, J. Atmos. Sci., 52, 2041–2057, https://doi.org/10.1175/1520-0469(1995)052<2041:MRTTCC>2.0.CO;2, 1995a.

Evans, K. F. and Stephens, G. L.: Microwave Radiative Transfer through Clouds Composed of Realistically Shaped Ice Crystals. Part II. Remote Sensing of Ice Clouds, J. Atmos. Sci., 52, 2058–2072, https://doi.org/10.1175/1520-0469(1995)052<2058:MRTTCC>2.0.CO;2, 1995b.

Evans, K. F., Walter, S. J., Heymsfield, A. J., and Deeter, M. N.: Modeling of Submillimeter Passive Remote Sensing of Cirrus Clouds, J. Appl. Meteorol., 37, 184–205, https://doi.org/10.1175/1520-0450(1998)037<0184:MOSPRS>2.0.CO;2, 1998.

Evans, K. F., Walter, S. J., Heymsfield, A. J., and McFarquhar, G. M.: Submillimeter-Wave Cloud Ice Radiometer: Simulations of retrieval algorithm performance, J. Geophys. Res.-Atmos., 107, AAC 2-1–AAC 2-21, https://doi.org/10.1029/2001JD000709, 2002.

Evans, K. F., Wang, J. R., O'C Starr, D., Heymsfield, G., Li, L., Tian, L., Lawson, R. P., Heymsfield, A. J., and Bansemer, A.: Ice hydrometeor profile retrieval algorithm for high-frequency microwave radiometers: application to the CoSSIR instrument during TC4, Atmos. Meas. Tech., 5, 2277–2306, https://doi.org/10.5194/amt-5-2277-2012, 2012.

Forster, P., Storelvmo, T., Armour, K., Collins, W., Dufresne, J. L., Frame, D., Lunt, D. J., Mauritsen, T., Palmer, M. D., Watanabe, M., Wild, M., and Zhang, H.: The Earth's Energy Budget, Climate Feedbacks, and Climate Sensitivity, in: Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change, edited by: Masson-Delmotte, V., Zhai, P., Pirani, A., Connors, S. L., Péan, C., Berger, S., Caud, N., Chen, Y., Goldfarb, L., Gomis, M. I., Huang, M., Leitzell, K., Lonnoy, E., Matthews, J. B. R., Maycock, T. K., Waterfield, T., Yelekçi, O., Yu, R., and Zhou, B., Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 923–1054, 2021.

Fox, S.: An Evaluation of Radiative Transfer Simulations of Cloudy Scenes from a Numerical Weather Prediction Model at Sub-Millimetre Frequencies Using Airborne Observations, Remote Sens.-Basel, 12, 2758, https://doi.org/10.3390/rs12172758, 2020.

Fox, S., Lee, C., Moyna, B., Philipp, M., Rule, I., Rogers, S., King, R., Oldfield, M., Rea, S., Henry, M., Wang, H., and Harlow, R. C.: ISMAR: an airborne submillimetre radiometer, Atmos. Meas. Tech., 10, 477–490, https://doi.org/10.5194/amt-10-477-2017, 2017.

Fox, S., Mendrok, J., Eriksson, P., Ekelund, R., O'Shea, S. J., Bower, K. N., Baran, A. J., Harlow, R. C., and Pickering, J. C.: Airborne validation of radiative transfer modelling of ice clouds at millimetre and sub-millimetre wavelengths, Atmos. Meas. Tech., 12, 1599–1617, https://doi.org/10.5194/amt-12-1599-2019, 2019.

Fu, Q., Sun, W. B., and Yang, P.: Modeling of scattering and absorption by nonspherical cirrus ice particles at thermal infrared wavelengths, J. Atmos. Sci., 56, 2937–2947, https://doi.org/10.1175/1520-0469(1999)056<2937:MOSAAB>2.0.CO;2, 1999.

Gao, T., Li, S., Liu, L., and Huang, W.: Development study of THz instruments for atmospheric sounding, Infrared and Laser Engineering, 45, 56–67, https://doi.org/10.3788/IRLA201645.0425002, 2016.

Gasiewski, A. J.: Numerical sensitivity analysis of passive EHF and SMMW channels to tropospheric water vapor, clouds, and precipitation, IEEE T. Geosci. Remote, 30, 859–870, https://doi.org/10.1109/36.175320, 1992.

Groth, S. P., Baran, A. J., Betcke, T., Havemann, S., and Smigaj, W.: The boundary element method for light scattering by ice crystals and its implementation in BEM plus, J. Quant. Spectrosc. Ra., 167, 40–52, 2015.

Hans, H., Bell, W., Berrisford, P., Andras, H., Muñoz-Sabater, J., Nicolas, J., Raluca, R., Dinand, S., Adrian, S., Cornel, S., and Dick, D.: Global reanalysis: goodbye ERA-Interim, hello ERA5, ECMWF Newsletter, 159, 17–24, https://doi.org/10.1002/qj.3803, 2019.

Heymsfield, A. J., Schmitt, C., and Bansemer, A.: Ice Cloud Particle Size Distributions and Pressure-Dependent Terminal Velocities from In Situ Observations at Temperatures from 0^{∘} to −86 ^{∘}C, J. Atmos. Sci., 70, 4123–4154, https://doi.org/10.1175/JAS-D-12-0124.1, 2013.

Heymsfield, A. J., Krämer, M., Luebke, A., Brown, P., Cziczo, D. J., Franklin, C., Lawson, P., Lohmann, U., McFarquhar, G., Ulanowski, Z., and Van Tricht, K.: Cirrus Clouds, Meteor. Mon., 58, 2.1–2.26, 2017.

Hong, G.: Parameterization of scattering and absorption properties of nonspherical ice crystals at microwave frequencies, J. Geophys. Res., 112, D11208, https://doi.org/10.1029/2006JD008364, 2007.

Hong, G., Yang, P., Baum, B. A., Heymsfield, A. J., Weng, F., Liu, Q., Heygster, G., and Buehler, S. A.: Scattering database in the millimeter and submillimeter wave range of 100–1000 GHz for nonspherical ice particles, J. Geophys. Res.-Atmos., 114, D06201, https://doi.org/10.1029/2008JD010451, 2009.

Hong, Y., Liu, G., and Li, J. L. F.: Assessing the Radiative Effects of Global Ice Clouds Based on CloudSat and CALIPSO Measurements, J. Climate, 29, 7651–7674, https://doi.org/10.1175/JCLI-D-15-0799.1, 2016.

Illingworth, A. J., Barker, H. W., Beljaars, A., Ceccaldi, M., Chepfer, H., Clerbaux, N., Cole, J., Delanoe, J., Domenech, C., Donovan, D. P., Fukuda, S., Hirakata, M., Hogan, R. J., Huenerbein, A., Kollias, P., Kubota, T., Nakajima, T., Nakajima, T. Y., Nishizawa, T., Ohno, Y., Okamoto, H., Oki, R., Sato, K., Satoh, M., Shephard, M. W., Velazquez-Blazquez, A., Wandinger, U., Wehr, T., and van Zadelhoff, G.-J.: The EarthCARE Satellite: The Next Step Forward in Global Measurements of Clouds, Aerosols, Precipitation, and Radiation, B. Am. Meteorol. Soc., 96, 1311–1332, https://doi.org/10.1175/BAMS-D-12-00227.1, 2015.

Inatani, J., Ozeki, H., Satoh, R., Nishibori, T., Ikeda, N., Fujii, Y., Nakajima, T., Iida, Y., Iida, T., Kikuchi, K., Miura, T., and Masuko, H.: Submillimeter limb-emission sounder JEM/SMILES aboard the Space Station, Proc. SPIE, 4152, 243–254, https://doi.org/10.1117/12.410604, 2000.

Ishimoto, H., Masuda, K., Mano, Y., Orikasa, N., and Uchiyama, A.: Irregularly shaped ice aggregates in optical modeling of convectively generated ice clouds, J. Quant. Spectrosc. Ra., 113, 632–643, https://doi.org/10.1016/j.jqsrt.2012.01.017, 2012.

Ishimoto, H., Masuda, K., Mano, Y., Orikasa, N., and Uchiyama, A.: Optical Modeling of Irregularly Shaped Ice Particles in Convective Cirrus, AIP Conf. Proc., 1531, 184–187, https://doi.org/10.1063/1.4804737, 2013.

Jimenez, C., Eriksson, P., and Murtagh, D.: First inversions of observed submillimeter limb sounding radiances by neural networks, J. Geophys. Res., 108, 4791, https://doi.org/10.1029/2003JD003826, 2003.

Jimenez, C., Buehler, S., Rydberg, B., Eriksson, P., and Evans, K.: Performance simulations for a submillimetre wave cloud ice satellite instrument, Q. J. Roy. Meteor. Soc., 133, 129–149, https://doi.org/10.1002/qj.134, 2007.

Kangas, V., D'Addio, S., Klein, U., Loiselet, M., Mason, G., Orlhac, J.-C., Gonzalez, R., Bergada, M., Brandt, M., and Thomas, B.: Ice cloud imager instrument for MetOp Second Generation, 2014 13th Specialist Meeting on Microwave Radiometry and Remote Sensing of the Environment (MicroRad), Pasadena, CA, 24–27 March 2014, 228–231, https://doi.org/10.1109/MicroRad.2014.6878946, 2014.

Kleanthous, A., Betcke, T., Hewett, D., Escapil-Inchauspé, P., Jerez-Hanckes, C., and Baran, A.: Accelerated Calderón preconditioning for Maxwell transmission problems, J. Comput. Phys., 458, 111099, https://doi.org/10.1016/j.jcp.2022.111099, 2022.

Lawson, R. P., Baker, B., Pilson, B., and Mo, Q. X.: In situ observations of the microphysical properties of wave, cirrus, and anvil clouds. Part II: Cirrus clouds, J. Atmos. Sci., 63, 3186–3203, https://doi.org/10.1175/JAS3803.1, 2006.

Lawson, R. P., Woods, S., Jensen, E., Erfani, E., Gurganus, C., Gallagher, M., Connolly, P., Whiteway, J., Baran, A. J., May, P., Heymsfield, A., Schmitt, C. G., McFarquhar, G., Um, J., Protat, A., Bailey, M., Lance, S., Muehlbauer, A., Stith, J., Korolev, A., Toon, O. B., and Kramer, M.: A Review of Ice Particle Shapes in Cirrus formed In Situ and in Anvils, J. Geophys. Res.-Atmos., 124, 10049–10090, https://doi.org/10.1029/2018JD030122, 2019.

Letu, H., Nakajima, T. Y., and Matsui, T. N.: Development of an ice crystal scattering database for the global change observation mission/second generation global imager satellite mission: investigating the refractive index grid system and potential retrieval error, Appl. Optics, 51, 6172–6178, https://doi.org/10.1364/AO.51.006172, 2012.

Letu, H., Ishimoto, H., Riedi, J., Nakajima, T. Y., C.-Labonnote, L., Baran, A. J., Nagao, T. M., and Sekiguchi, M.: Investigation of ice particle habits to be used for ice cloud remote sensing for the GCOM-C satellite mission, Atmos. Chem. Phys., 16, 12287–12303, https://doi.org/10.5194/acp-16-12287-2016, 2016.

Letu, H. S., Nagao, T. M., Nakajima, T. Y., Riedi, J., Ishimoto, H., Baran, A. J., Shang, H. Z., Sekiguchi, M., and Kikuchi, M.: Ice Cloud Properties From Himawari-8/AHI Next-Generation Geostationary Satellite: Capability of the AHI to Monitor the DC Cloud Generation Process, IEEE T. Geosci. Remote, 57, 3229–3239, https://doi.org/10.1109/TGRS.2018.2882803, 2018.

Li, M., Letu, H., Peng, Y., Ishimoto, H., Lin, Y., Nakajima, T. Y., Baran, A. J., Guo, Z., Lei, Y., and Shi, J.: Investigation of ice cloud modeling capabilities for the irregularly shaped Voronoi ice scattering models in climate simulations, Atmos. Chem. Phys., 22, 4809–4825, https://doi.org/10.5194/acp-22-4809-2022, 2022.

Li, S., Liu, L., Gao, T., Huang, W., and Hu, S.: Sensitivity analysis of terahertz wave passive remote sensing of cirrus microphysical parameters, Acta Physica Sinica, 65, 100–110, https://doi.org/10.7498/aps.65.134102, 2016.

Li, S., Liu, L., Gao, T., Hu, S., and Huang, W.: Retrieval method of cirrus microphysical parameters at terahertz wave based on multiple lookup tables, Acta Physica Sinica, 66, 78–87, https://doi.org/10.7498/aps.66.054102, 2017.

Li, S., Liu, L., Gao, T., Shi, L., Qiu, S., and Hu, S.: Radiation characteristics of the selected channels for cirrus remote sensing in terahertz waveband and the influence factors for the retrieval method, J. Infrared Millim. W., 37, 60–66, https://doi.org/10.11972/j.issn.1001-9014.2018.01.012, 2018.

Lin, B. and Rossow, W.: Seasonal Variation of Liquid and Ice Water Path in Nonprecipitating Clouds over Oceans, J. Climate, 9, 2890–2902, https://doi.org/10.1175/1520-0442(1996)009<2890:SVOLAI>2.0.CO;2, 1996.

Liou, K.-N.: Radiation and cloud processes in the atmosphere: theory, observation and modeling, Oxford monographs on geology and geophysics, 20, Oxford University Press, New York, ix, 487 pp., https://doi.org/10.1063/1.2809044, 1992.

Liu, G. and Curry, J.: Remote Sensing of Ice Water Characteristics in Tropical Clouds Using Aircraft Microwave Measurements, J. Appl. Meteorol., 37, 337–355, https://doi.org/10.1175/1520-0450(1998)037<0337:RSOIWC>2.0.CO;2, 1998.

Liu, G. and Curry, J.: Tropical Ice Water Amount and Its Relations to Other Atmospheric Hydrological Parameters as Inferred from Satellite Data, J. Appl. Meteorol., 38, 1182–1194, https://doi.org/10.1175/1520-0450(1999)038<1182:TIWAAI>2.0.CO;2, 1999.

Liu, G. and Curry, J.: Determination of Ice Water Path and Mass Median Particle Size Using Multichannel Microwave Measurements, J. Appl. Meteorol., 39, 1318–1329, https://doi.org/10.1175/1520-0450(2000)039<1318:DOIWPA>2.0.CO;2, 2000.

Liu, L., Weng, C., Li, S., Hu, S., Ye, J., Dou, F., and Shang, J.: Review of terahertz passive remote sensing of ice clouds, Advances in Earth Science, 35, 1211–1221, https://doi.org/10.11867/j.issn.1001-8166.2020.103, 2020.

Liu, L., Weng, C., Li, S., Letu, H., Hu, S., and Dong, P.: Passive Remote Sensing of Ice Cloud Properties at Terahertz Wavelengths Based on Genetic Algorithm, Remote Sens.-Basel, 13, 735, https://doi.org/10.3390/rs13040735, 2021.

Macke, A., Mishchenko, M. I., and Cairns, B.: The influence of inclusions on light scattering by large ice particles, J. Geophys. Res.-Atmos., 101, 23311–23316, https://doi.org/10.1029/96jd02364, 1996.

Marks, C. and Rodgers, C.: A retrieval method for atmospheric composition from limb emission measurements, J. Geophys. Res., 98, 14939–14953, https://doi.org/10.1029/93JD01195, 1993.

Minnis, P., Heck, P., and Young, D.: Inference of Cirrus Cloud Properties Using Satellite-observed Visible and Infrared Radiances. Part II: Verification of Theoretical Cirrus Radiative Properties, J. Atmos. Sci., 50, 1305–1322, https://doi.org/10.1175/1520-0469(1993)050<1305:IOCCPU>2.0.CO;2, 1993a.

Minnis, P., Liou, K.-N., and Takano, Y.: Inference of Cirrus Cloud Properties Using Satellite-observed Visible and Infrared Radiances. Part I: Parameterization of Radiance Fields, J. Atmos. Sci., 50, 1279–1304, https://doi.org/10.1175/1520-0469(1993)050<1279:IOCCPU>2.0.CO;2, 1993b.

Nakajima, T. and King, M. D.: Determination of the Optical-Thickness and Effective Particle Radius of Clouds from Reflected Solar-Radiation Measurements. 1. Theory, J. Atmos. Sci., 47, 1878–1893, https://doi.org/10.1175/1520-0469(1990)047<1878:Dotota>2.0.Co;2, 1990.

Nakajima, T. and Tanaka, M.: Matrix formulation for the transfer of solar radiation in a plane-parallel scattering atmosphere, J. Quant. Spectrosc. Ra., 35, 13–21, https://doi.org/10.1016/0022-4073(86)90088-9, 1986.

Nakajima, T. and Tanaka, M.: Algorithms for radiative intensity calculations in moderately thick atmospheres using truncation approximation, J. Quant. Spectrosc. Ra., 40, 51–69, https://doi.org/10.1016/0022-4073(88)90031-3, 1988.

Nakajima, T., King, M. D., Spinhirne, J. D., and Radke, L. F.: Determination of the Optical-Thickness and Effective Particle Radius of Clouds from Reflected Solar-Radiation Measurements. 2. Marine Stratocumulus Observations, J. Atmos. Sci., 48, 728–750, https://doi.org/10.1175/1520-0469(1991)048<0728:Dotota>2.0.Co;2, 1991.

Nakajima, T., Nakajima, T., Yoshimori, K., Mishra, S., and Tripathi, S.: Development of a light scattering solver applicable to particles of arbitrary shape on the basis of the surface-integral equations method of Müller type. I. Methodology, accuracy of calculation, and electromagnetic current on the particle surface, Appl. Optics, 48, 3526–3536, https://doi.org/10.1364/AO.48.003526, 2009.

Nakajima, T. Y. and Nakajima, T.: Wide-area determination of cloud microphysical properties from NOAA AVHRR measurements for FIRE and ASTEX regions, J. Atmos. Sci., 52, 4043–4059, https://doi.org/10.1175/1520-0469(1995)052<4043:Wadocm>2.0.Co;2, 1995.

Nakajima, T. Y., Ishida, H., Nagao, T. M., Hori, M., Letu, H., Higuchi, R., Tamaru, N., Imoto, N., and Yamazaki, A.: Theoretical basis of the algorithms and early phase results of the GCOM-C (Shikisai) SGLI cloud products, Prog. Earth Planet. Sc., 6, 1–25, https://doi.org/10.1186/s40645-019-0295-9, 2019.

NASA: ESPO Data Archive, NASA [data set], https://espoarchive.nasa.gov/archive/browse/tc4/ER2, last access: 26 December 2022.

Platnick, S., King, M. D., Ackerman, S. A., Menzel, W. P., Baum, B. A., Riedi, J. C., and Frey, R. A.: The MODIS cloud products: algorithms and examples from Terra, IEEE T. Geosci. Remote, 41, 459–473, https://doi.org/10.1109/TGRS.2002.808301, 2003.

Platnick, S., Meyer, K. G., King, M. D., Wind, G., Amarasinghe, N., Marchant, B., Arnold, G. T., Zhang, Z. B., Hubanks, P. A., Holz, R. E., Yang, P., Ridgway, W. L., and Riedi, J.: The MODIS Cloud Optical and Microphysical Products: Collection 6 Updates and Examples From Terra and Aqua, IEEE T. Geosci. Remote, 55, 502–525, https://doi.org/10.1109/Tgrs.2016.2610522, 2017.

Racette, P., Dod, L., Shiue, J., Adler, R., Jackson, D., Gasiewski, A., and Zacharias, D.: An Airborne Millimeter-Wave Imaging Radiometer for Cloud, Precipitation, and Atmospheric Water Vapor Studies, J. Atmos. Ocean. Tech., 13, 610–619, https://doi.org/10.1175/1520-0426(1996)013<0610:AAMWIR>2.0.CO;2, 1992.

Rossow, W.: Clouds in Weather and Climate: The International Satellite Cloud Climatology Project at 30: What Do We Know and What Do We Still Need to Know?, B. Am. Meteorol. Soc., 95, 441–443, https://doi.org/10.1175/BAMS-D-13-00138.1, 2014.

Rossow, W. B. and Schiffer, R. A.: Isccp Cloud Data Products, B. Am. Meteorol. Soc., 72, 2–20, https://doi.org/10.1175/1520-0477(1991)072<0002:ICDP>2.0.CO;2, 1991.

Sekiguchi, M. and Nakajima, T.: A k-distribution-based radiation code and its computational optimization for an atmospheric general circulation model, J. Quant. Spectrosc. Ra., 109, 2779–2793, https://doi.org/10.1016/j.jqsrt.2008.07.013, 2008.

Sieron, S., Clothiaux, E., Zhang, F., Lu, Y., and Otkin, J.: Comparison of using distribution-specific versus effective radius methods for hydrometeor single-scattering properties for all-sky microwave satellite radiance simulations with different microphysics parameterization schemes, J. Geophys. Res.-Atmos., 122, 7027–7046, https://doi.org/10.1002/2017JD026494, 2017.

SSEC: The 14408 groups of PSDs from 11 field flight observation experiments, SSEC [data set], https://www.ssec.wisc.edu/ice_models/microphysical_data.html, last access: 18 February 2022.

Stephens, G. L., Li, J., Wild, M., Clayson, C. A., Loeb, N., Kato, S., L'Ecuyer, T., Stackhouse, P. W., Lebsock, M., and Andrews, T.: An update on Earth's energy balance in light of the latest global observations, Nat. Geosci., 5, 691–696, https://doi.org/10.1038/ngeo1580, 2012.

Takano, Y. and Liou, K. N.: Solar Radiative-Transfer in Cirrus Clouds. 1. Single-Scattering and Optical-Properties of Hexagonal Ice Crystals, J. Atmos. Sci., 46, 3–19, https://doi.org/10.1175/1520-0469(1989)046<0003:SRTICC>2.0.CO;2, 1989.

Van de Hulst, H. C.: Light scattering by small particles, Wiley, New York,, xiii, 470 pp., https://doi.org/10.1063/1.3060205, 1957.

Warren, S.: Optical Constants of Ice from the Ultraviolet to the Microwave, Appl. Optics, 23, 1206, https://doi.org/10.1364/AO.23.001206, 1984.

Warren, S. G. and Brandt, R. E.: Optical constants of ice from the ultraviolet to the microwave: A revised compilation, J. Geophys. Res.-Atmos., 113, D14220, https://doi.org/10.1029/2007JD009744, 2008.

Weng, C., Liu, L., Gao, T., Hu, S., Li, S., Dou, F., and Shang, J.: Multi-Channel Regression Inversion Method for Passive Remote Sensing of Ice Water Path in the Terahertz Band, Atmosphere-Basel, 10, 437, https://doi.org/10.3390/atmos10080437, 2019.

Wu, D. L., Esper, J., Ehsan, N., Piepmeier, J. R., and Racette, P.: Icecube: Spaceflight Validation of an 874 GHz Submillimeter Wave Radiometer for Ice Cloud Remote Sensing, 2014/12/1, 2014 Earth Science Technology Forum. Location: Leesburg, Virginia, USA, 28–30 October 2014, A22F-02, https://doi.org/10.1117/12.2530589.

Yang, P. and Liou, K. N.: Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space, J. Opt. Soc. Am. A, 13, 2072–2085, https://doi.org/10.1364/JOSAA.13.002072, 1996a.

Yang, P. and Liou, K. N.: Geometric-optics-integral-equation method for light scattering by nonspherical ice crystals, Appl. Optics, 35, 6568–6584, https://doi.org/10.1364/AO.35.006568, 1996b.

Yang, P., Liou, K. N., Mishchenko, M. I., and Gao, B. C.: Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols, Appl. Optics, 39, 3727–3737, https://doi.org/10.1364/Ao.39.003727, 2000a.

Yang, P., Liou, K. N., Wyser, K., and Mitchell, D.: Parameterization of the scattering and absorption properties of individual ice crystals, J. Geophys. Res.-Atmos., 105, 4699–4718, 2000b.

Yang, P., Bi, L., Baum, B. A., Liou, K. N., Kattawar, G. W., Mishchenko, M. I., and Cole, B.: Spectrally Consistent Scattering, Absorption, and Polarization Properties of Atmospheric Ice Crystals at Wavelengths from 0.2 to 100 µm, J. Atmos. Sci., 70, 330–347, https://doi.org/10.1175/JAS-D-12-039.1, 2013.

Yang, P., Liou, K. N., Bi, L., Liu, C., Yi, B. Q., and Baum, B. A.: On the Radiative Properties of Ice Clouds: Light Scattering, Remote Sensing, and Radiation Parameterization, Adv. Atmos. Sci., 32, 32–63, https://doi.org/10.1007/s00376-014-0011-z, 2015.

Yang, P., Hioki, S., Saito, M., Kuo, C. P., Baum, B. A., and Liou, K. N.: A Review of Ice Cloud Optical Property Models for Passive Satellite Remote Sensing, Atmosphere-Basel, 9, 499, https://doi.org/10.3390/atmos9120499, 2018.

Yee, K. S.: Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media, IEEE T. Antenn. Propag., 14, 302–307, https://doi.org/10.1109/TAP.1966.1138693, 1966.

Yi, B. Q., Rapp, A. D., Yang, P., Baum, B. A., and King, M. D.: A comparison of Aqua MODIS ice and liquid water cloud physical and optical properties between collection 6 and collection 5.1: Cloud radiative effects, J. Geophys. Res.-Atmos., 122, 4550–4564, https://doi.org/10.1002/2016JD025654, 2017.

Yurkin, M. A. and Hoekstra, A. G.: The discrete dipole approximation: An overview and recent developments, J. Quant. Spectrosc. Ra., 106, 558–589, https://doi.org/10.1016/j.jqsrt.2007.01.034, 2007.

Zhang, H., Wang, F., Zhao, S., and Xie, B.: Earth's energy budget, climate feedbacks, and climate sensitivity, Adv. Clim. Chang. Res., 17, 691–698, 2021.

Zhao, W. J., Peng, Y. R., Wang, B., Yi, B. Q., Lin, Y. L., and Li, J. N.: Comparison of three ice cloud optical schemes in climate simulations with community atmospheric model version 5, Atmos. Res., 204, 37–53, https://doi.org/10.1016/j.atmosres.2018.01.004, 2018.