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VOLUME 51
Study of Horizontally Oriented Ice Crystals with CALIPSO Observations
and Comparison with Monte Carlo Radiative Transfer Simulations
CHEN ZHOU, PING YANG, AND ANDREW E. DESSLER
Department of Atmospheric Sciences, Texas A&M University, College Station, Texas
YONGXIANG HU
Climate Science Branch, NASA Langley Research Center, Hampton, Virginia
BRYAN A. BAUM
Space Science and Engineering Center, University of Wisconsin—Madison, Madison, Wisconsin
(Manuscript received 3 November 2011, in final form 14 February 2012)
ABSTRACT
Data from the Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP) indicate that horizontally
oriented ice crystals (HOIC) occur frequently in both ice and mixed-phase clouds. When compared with the
case for clouds consisting of randomly oriented ice crystals (ROIC), lidar measurements from clouds with
HOIC, such as horizontally oriented hexagonal plates or columns, have stronger backscatter signals and
smaller depolarization ratio values. In this study, a 3D Monte Carlo model is developed for simulating the
CALIOP signals from clouds consisting of a mixture of quasi HOIC and ROIC. With CALIOP’s initial
orientation with a pointing angle of 0.38 off nadir, the integrated attenuated backscatter is linearly related to
the percentage of HOIC but is negatively related to the depolarization ratio. At a later time in the CALIOP
mission, the pointing angle of the incident beam was changed to 38 off nadir to minimize the signal from
HOIC. In this configuration, both the backscatter and the depolarization ratio are similar for clouds containing HOIC and ROIC. Horizontally oriented columns with two opposing prism facets perpendicular to the
lidar beam and horizontally oriented plates show similar backscattering features, but the effect of columns is
negligible in comparison with that of plates because the plates have relatively much larger surfaces facing the
incident lidar beam. From the comparison between the CALIOP simulations and observations, it is estimated
that the percentage of quasi-horizontally oriented plates ranges from 0% to 6% in optically thick mixed-phase
clouds, from 0% to 3% in warm ice clouds (.2358C), and from 0% to 0.5% in cold ice clouds.
1. Introduction
In current satellite-based retrievals of ice cloud optical
thickness and effective particle size, bulk single-scattering
properties of ice clouds are used to simulate the reflectance and transmission characteristics over a range of cloud
microphysical conditions. These bulk properties are used
to build a static lookup table that is used with satellite
observations in the implementation of a retrieval algorithm (Platnick et al. 2003; Minnis et al. 2011). In the forward light scattering computation of the bulk ice cloud
Corresponding author address: Prof. Ping Yang, Department of
Atmospheric Sciences, Texas A&M University, College Station,
TX 77843.
E-mail: pyang@tamu.edu
DOI: 10.1175/JAMC-D-11-0265.1
Ó 2012 American Meteorological Society
optical properties, predefined size and habit (shape) distributions of ice particles are assumed, although some
efforts have been carried out to explore the retrieval of ice
cloud properties without a priori assumption of the shape/
size distribution of ice particles (Kokhanovsky and Nauss
2005). Furthermore, the particles are typically assumed to
be randomly oriented for computational and theoretical
simplicity.
In environments with relatively gentle updraft velocities, ice particles with certain shapes may become oriented as a result of the balance between gravitational
settling and the vertical updraft velocity with relevant
aerodynamic effects on a population of particles (Magono
1953; Ono 1969). The orientation of an ice particle depends on its shape, size, and ambient aerodynamic conditions. It is generally recognized that planar and columnar
JULY 2012
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ZHOU ET AL.
crystals tend to exhibit preferred orientations; in particular, ice plates with maximum dimensions greater
than 100 mm are likely to be quasi-horizontally oriented
(Sassen 1980). Furthermore, oriented small planar particles tend to have a range of tilt angles between 108 and
208 (Klett 1995). It has been previously shown that oriented ice particles are observed in more than 40% of ice
clouds (Chepfer et al. 1999).
However, our understanding of the impact of oriented
ice particles on light scattering calculations and subsequent application to satellite-based cloud property
retrievals is poor, resulting in a potential significant bias
in the inferred properties (Masuda and Ishimoto 2004).
The present study is intended to investigate the impact
of horizontally oriented ice crystals (HOIC) on the interpretation of lidar backscatter and depolarization
signals associated with ice clouds, and furthermore to
infer the percentage of HOIC in ice clouds.
Quasi-horizontally oriented hexagonal ice crystals are
the most dominant oriented ice crystal habit. As oriented
planar particles act like little mirrors, their presence can be
inferred in depolarization lidar data from a strong increase
in the backscattering signal g9 and a low depolarization
ratio d, and occurs when the incident beam points along
the local zenith or nadir for a spaceborne or ground-based
lidar. With the Facility for Atmospheric Remote Sensing
scanning lidar datasets, Noel and Sassen (2005) confirmed
the presence of quasi HOIC. Noel and Sassen (2005) also
suggested that the derivation angles of horizontally oriented planar crystals are smaller in high-level cold clouds
(,2308C) than in midlevel warmer clouds.
The Cloud–Aerosol Lidar with Orthogonal Polarization
(CALIOP) measurements (Winker et al. 2003) provide an
unprecedented opportunity to study oriented plates from
a global perspective. When the lidar pointing angle was
0.38 off nadir, as it was originally from the beginning of
operations through November 2007, the backscattering signals associated with quasi-horizontally oriented plates were
strong and had low depolarization (Hu et al. 2007). By
further analyzing the CALIOP depolarization-attenuated
backscatter relationship, Noel and Chepfer (2010) estimated the contribution of oriented ice crystals with a simple single-scattering model (Sassen and Benson 2001) and
suggested that most horizontally oriented plates could
be found in optically thick clouds. However, the singlescattering model used in the previous study is not optimal
for the case of optically thick clouds, where multiple
scattering contribution dominates.
To more accurately simulate the CALIOP measurements, a 3D Monte Carlo model is developed for clouds
containing quasi-horizontally oriented plates in both ice
clouds and mixed-phase clouds. This model is able to
simulate the lidar signals in both optically thin and thick
clouds. The simulated d–g9 relationships are then compared with CALIOP observations, and the percentage of
HOIC is inferred.
This paper is organized into four sections. Section 2
describes the datasets and method used in this study.
Section 3 presents various sensitivity studies based on
numerical simulations. In section 3, we also compare the
simulated results with Cloud–Aerosol Lidar and Infrared
Pathfinder Satellite Observation (CALIPSO) data to estimate the percentage of HOIC. The scientific findings of
this study are summarized in section 4.
2. Data and methodology
Three instruments are on the CALIPSO platform,
one of which is the CALIOP (Winker et al. 2003). The
CALIOP provides observations at two wavelengths:
532 and 1064 nm. The 532-nm channel has a polarization
capability that is critical to the detection of quasi-horizontally oriented plates. We will therefore focus on data
at this wavelength in this study, specifically from the
version-3 CALIPSO level-2 cloud-layer products with
a 5-km spatial resolution.
Two fundamental lidar parameters, namely, the layerintegrated depolarization ratio d and the total-layerintegrated attenuated backscatter g9, are defined as
follows:
ð cloud
d5
top
cloud base
ð cloud top
cloud base
ð cloud
g95
cloud
top
b9? (z) dz
and
(1)
b9k (z) dz
b9?(z)
dz 1
base cosuoff-nadir
ð cloud
top
b9k(z)
cloud base cosuoff-nadir
dz,
(2)
where b9k (z) and b9? (z) are the parallel and perpendicular components of attenuated backscatter as functions
of altitude, respectively; uoff-nadir is the angle of CALIOP
lidar beam with respect to the nadir. This study also uses
the cloud-layer temperature information provided by the
Goddard Earth Observing System, version 5 (GEOS-5),
reanalysis. The off-nadir pointing angle of CALIOP,
which is the angle between the incident laser beam and
nadir, was set to 0.38 before November 2007 and afterward permanently changed to 38. The backscattering
signal of horizontally oriented plates is quite sensitive to
the off-nadir angle; thus the CALIPSO data used in
this study are for two periods: July–December 2006 and
July–December 2010.
Before conducting the Monte Carlo simulations, we
computed the single-scattering properties of HOIC and
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VOLUME 51
FIG. 1. Coordinate systems for quasi-horizontally oriented plates. (a) The particle coordinate
system O–Xp, –Yp, –Zp. (b) The laboratory coordinate system O–Xf, –Yf, –Zf. (c) Euler angles
for particle orientation. (d) Incident angles in laboratory coordinate system and the related
scattering coordinates.
randomly oriented ice crystals (ROIC). The singlescattering properties of ice particles have been studied
extensively (e.g., Macke 1993; Muinonen 1989; Liu et al.
2006; Borovoi and Grishin 2003; Borovoi et al. 2005;
Baran 2009). Of relevance to this study, Takano and Liou
(1989), Baran et al. (2001), and Borovoi and Kustova
(2009) have investigated the optical properties of ice
particles with preferred orientations. In this study, we
use a physical-geometric optics hybrid (PGOH) method
discussed in detail in Bi et al. (2011) to simulate the single-scattering properties of HOIC. The PGOH method
does not artificially separate the contributions of diffraction and geometric optical rays and avoids numerous
shortcomings in the conventional ray-tracing technique.
Additionally, the latest version of the Improved Geometric Optics Method (IGOM; Yang et al. 2005; Bi et al.
2010) is used to obtain the single-scattering properties of
ROIC. The habit distribution for ROIC is taken from
Baum et al. (2010), which includes droxtals, solid columns
and plates, hollow columns, bullet rosettes, and hexagonal
aggregates. Following Baum et al. (2011), we also take
moderate roughness into consideration in the case of
hexagonal aggregates. Furthermore, a gamma distribution
is used to describe the particle size distribution (PSD):
n(D) 5 N0 Dm exp(2lD),
(3)
where D is the maximum dimension, frequently referred
to as the particle size; n(D) is the particle concentration;
and N0 is the total number of plate particles in a unit
volume. For illustrative purposes, a set of coefficients is
chosen for the gamma distribution, namely, m 5 1.026
and l 5 8.7 mm21. Note that N0 needs not be specified
because this factor cancels out in the simulation of the
bulk optical properties involved in this study [see Eqs.
(4)–(6) in the following discussion]. Simulations with
numerous sets of habit and size distributions for ROIC
were performed during the course of this study, with results similar to the one chosen for illustration. For the
mixed-phase cloud simulations, the Lorenz–Mie theory is
applied to calculate the scattering properties of liquid
droplets with an effective radius assumed to be 4 mm.
JULY 2012
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ZHOU ET AL.
The phase matrix of randomly oriented particles depends only on the scattering angle us, which is obtained
by integrating over the PSD:
Pr (us )
ðD
max
5
M
å
[Pr,hb (us , D) fhb (D)sscat,hb (D)]n(D) dD
Dmin hb51
ðD
,
M
max
å
Dmin hb51
[fhb (D)sscat,hb (D)]n(D) dD
(4)
and the corresponding average scattering and absorption cross sections are given in the form
ðD
max
sr,scat 5
M
å
Dmin hb51
[fhb (D)sscat,hb (D)]n(D) dD
ðD
max
(5)
M
å
Dmin hb51
fhb (D)n(D) dD
and
ðD
max
sr,abs 5
M
å
Dmin hb51
ðD
[fhb (D) sabs,hb (D)]n(D) dD
max
,
M
å
Dmin hb51
FIG. 2. Illustration of the scattering coordinates and scattering
angles.
(6)
fhb (D)n(D) dD
where Dmax is the maximum size of the particles, Dmin is
the minimum size, M is the number of habits, sscat,hb is
the scattering cross section for a given habit of randomly
oriented cloud particles, sabs,hb is the absorption cross
section, fhb is the particle fraction of habit hb, and P is
the phase matrix.
The calculations for HOIC are more complicated than
for ROIC because the phase matrix of oriented particles
depends on both the incident direction with respect to the
particle and scattering zenith and azimuthal angles. The
coordinate systems for HOIC in the case of hexagonal
plates are shown in Fig. 1. The particle coordinate
0
1 2
xf
cosg
By C 6
@ f A 5 4 sing
zf
0
2sing
cosg
0
32
0
cosb
76
0 54 0
0
1
1
0 cosb
2sinb
For quasi-horizontally oriented plates, a and g follow
the uniform distribution from 0 to 2p (note that for
a hexagonal particle, g needs to be considered only for
0–p/3 because of the particles’ symmetry), and b satisfies the normal distribution:
2
N(b) 5 NHOP,0 e2b
/ 2s2b
,
system, O–Xp, –Yp, –Zp (Fig. 1a), rotates as the particle
rotates. The laboratory coordinate system, O–Xf, –Yf,
–Zf (Fig. 1b), is a fixed coordinate system with its Z axis
pointing in the zenith direction.
Euler angles (a, b, g) are used to describe the orientation of quasi-horizontally oriented ice particles (Fig. 1c).
The line of nodes is the intersection line of coordinate
planes O–Xp, –Yp and O–Xf, –Yf. Here a is the angle between the O–Xf axis and the line of nodes, b is the angle
between axis O–Zf and O–Zp, and g is the angle between
the line of nodes and O–Yp axis. The transformation between the particle coordinates and the laboratory coordinates is given by
(8)
32
sinb
cosa
76
0 54 sina
0
2sina
cosa
0
30 1
xp
0
7B y C
0 5@ p A.
zp
1
(7)
where sb is the standard derivation of b, NHOP,0 is a
constant, and N(b) is the probability distribution function. Figure 1d shows the configuration of the incident
coordinate system with respect to the fixed coordinate
system.
The overall nonnormalized phase matrix for quasihorizontally oriented plates in the laboratory coordinate
system is calculated from
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JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY
ðD
max
Ph (uif , us , us ) 5 Ph (uif , uif , us , us ) 5
"ð ð ð
p 2p p/3
0 0
Dmin
#
P(uif , uif , us , us , a, b, g, D) dg da N(b) sinb db n(D) dD
0
ðD
max
Sh(uif , 0, 0) 5 Sh(uif , uif , 0, 0) 5
S11
S21
S12
5
S22
, (9)
dg da N(b) sinb db n(D) dD
0 0
where (a, b, g) are the Euler angles of a particle, uif and
uif are the incident angles in the laboratory coordinates
(Fig. 1d), us and us are the scattering zenith and azimuthal
angles (Fig. 2), and P(uif , uif , us , us , a, b, g, D) is the phase
matrix of HOIC. The aspect ratio of plates is given by
2a/L 5 0.8038a0.526 based on in situ measurements
(Pruppacher and Klett 1997), where a and L are the
semiwidth and length of a hexagonal ice crystal. The size
max
#
"ð ð ð
p 2p p/3
Dmin
ðD
VOLUME 51
0
distribution of HOIC is the same as that in ROIC, except
that only plates with a size greater than 100 mm are considered to be quasi-horizontally oriented since smaller
plates can have large tilt angles (Klett 1995). Subsequently,
the phase matrix is normalized before being used in the
Monte Carlo radiative transfer simulations.
The forward amplitude matrix is used to derive the
extinction matrix of oriented crystals:
"ð ð ð
p 2p p/3
Dminx
0 0
#
S(uif , uif , 0, 0, a, b, g, D) dg da N(b) sinb db n(D) dD
0
ðD
max
Dmin
"ð ð ð
p 2p p/3
#
.
dg da N(b) sinb db n(D) dD
0 0
0
(10)
Because the distribution of the plates’ orientations is
axial symmetric with respect to Zf axis, S21 and S12 are
zero and the extinction matrix can be calculated from
(Mishchenko 1991)
Kh 5 s
~ ext kh ,
s
~ ext 5
(11)
2p
Re(S11 1 S22 ),
k21
2
1
6
6
6 Re(S 2 S )
11
22
6
6 Re(S 1 S )
6
11
22
kh 5 6
6
6
0
6
6
4
0
and
(12)
Re(S11 2 S22 )
Re(S11 1 S22 )
0
1
0
0
1
0
Im(S11 2 S22 )
2
Re(S11 1 S22 )
where k1 is the wavenumber in free space; Re and Im
denote the real and imaginary part, respectively; and s
~ ext
is the differential extinction cross section and equals the
extinction cross section of the particle when the incident
beam is not polarized. In Eqs. (11) and (13), kh is the
normalized extinction matrix.
The absorption cross section of HOIC is integrated
similarly as ROIC. With the phase matrices, extinction
0
3
7
7
7
7
0
7
7
,
Im(S11 2 S22 ) 7
7
7
Re(S11 1 S22 ) 7
7
5
1
(13)
matrices, and absorption cross section for ROIC and
HOIC, the backscattering signals received by the lidar
can be simulated with the Monte Carlo model.
The 3D Monte Carlo model used in this study is based
on the modification of a model developed by Hu et al.
(2001) for ROIC. In the new model, clouds are assumed to be homogenous layers in which ROIC and
HOIC are mixed. The fraction of HOIC in terms of
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ZHOU ET AL.
particle numbers is Fh and the weight of HOIC can be
calculated from
Wh 5
Fh s
~ h0,ext
Fh s
~ h0,ext 1 (1 2 Fh )sr,ext
,
(14)
where s
~ h0,ext is the extinction cross section of HOIC
when the incident angle is zero with respect to nadir and
sr,ext is the extinction cross section of ROIC. The weight
of the particles is presented as a percentage in the following discussion.
Unlike the extinction matrix of ROIC that is diagonal,
the extinction matrix of HOIC is not diagonal when the
direction of transmission is neither perpendicular nor
parallel to the plate’s basal faces. In this case, the polarization ratio changes as light transfers within the medium.
The radiative transfer equation along a path is in the form
(Mishchenko et al. 2002)
dI(~t )
~
5 2kI(~t) 1 J(~t ), d~t 5 n~
sext ds 5 bds,
d~t
(15)
where k is the normalized extinction matrix, ds is the
~ is the differential extinction coefficient, and
distance, b
J is the source representing multiple scattering. To adjust the Stokes vector when a photon travels along a path
in the medium, we use
~ s
b(0)
~ ext (u)
,
s
~ ext (0)
~ h,ext (u),
s
~ ext (u) 5 sr,ext 1 s
Fh s
~ h,ext (u)kh (u) 1 (1 2 Fh )sr,ext E
,
Fh s
~ h,ext (u) 1 (1 2 Fh )sr,ext
~
~t 5 b(u)kr
2 r0 k 5 2log(1 2 j),
(18)
~
where b(u)
and s
~ ext (u) are the differential extinction
coefficient and scattering cross section, respectively.
When ROIC and HOIC are both present in clouds,
the normalized extinction matrix becomes
(20)
where r 2 r0 is parallel to the direction vector, and j is a
random number with uniform distribution between 0 and
~
1. The extinction coefficient b(u)
is calculated with Eq. (17).
The Stokes vector is adjusted using the equations
5
(17)
D~t 5
exp[2k(u)~t ]
(I, Q, U, V)T
exp(2~t )
exp[2kh (u)D~t]
(I, Q, U, V)T ,
exp(2D~t)
Fh s
~ h,ext (u)
Fh s
~ h,ext (u) 1 (1 2 Fh )sr,ext
~t ,
(21)
(22)
where the superscript T indicates the transpose of a
matrix, ~t is the optical depth for along the photon’s path,
and D~t is the equivalent optical depth of oriented plates.
Note here that the exponent of the extinction matrix remains a 4 3 4 matrix. Equation (21) is based on Eqs. (16)
and (19).
If the interaction is within the field of view, we estimate the possibility that the photon travels directly into
the lidar aperture after the interaction, and collect the
signal received by the lidar detector in the form
~
k(u )b(u
rcv )Dz
3 L(u9)L(p 2 F2 )P(ui , us , us )L(2F1 )(I0 , Q0 , U0 , V0 )T ,
(i, q, u, y)T 5 vDV exp 2 rcv
cos(urcv )
where P is the 4 3 4 phase matrix given by
(19)
where E is the unit diagonal matrix, and Fh is the fraction
of HOIC.
The Monte Carlo simulation begins with the phase
matrices of randomly oriented particles P(us) and quasihorizontally oriented plates P(uif, us, us), the absorption
cross section, and the extinction matrix. Subsequently,
the model begins tracing the transmission of a group of
photons. The initial state of a new photon is decided by
assuming lidar parameters. For CALIOP simulations,
the directional vector for a new photon is set to be
(u, u)5(uoff-nadir, 0), and the Stokes vector for the linearly polarized photons is (I0, I0, 0, 0).
The first step of the ray-tracing loop is to find the location of the next photon–particle interaction with
(16)
The spatial distribution of particles is considered to
be totally random, and the extinction coefficient can be
expressed as follows:
~
b(u)
5
5
F K 1 (1 2 Fh )Kr
K
5 h h
s
~ ext
s
~ ext
(I9, Q9, U9, V9)T 5
ð
I(~t ) 5 e2k~t I0 (~t) 1 e2k(~t2t9) J(t9) dt9.
k5
(23)
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FIG. 3. Phase function of horizontally oriented plates (sb 5 08) for (a)–(f) ui 5 08, 28, 58, 108, 308, and 608,
respectively.
P(ui , us , us ) 5
Pr (us )(1 2 Fh ) sr,scat 1 Ph (ui , us , us )Fh sh,scat (ui )
,
(1 2 Fh )sr,scat 1 Fh sh,scat (ui )
sh,scat (u) 5 sh,ext (u) 2 sh,abs (u)
Q
U
5s
~ h,ext (u) k11 1 0 k12 1 0 k13
I0
I0
V
(25)
1 0 k14 2 sh,abs (u);
I0
L(u) is the rotation matrix; v is the single-scattering
albedo (virtually equal to 1 for ice clouds at 532 nm),
and DV is the solid angle of the lidar aperture viewing
from the scattering location; Dz is the vertical distance
between the scattering location and the upper surface of
the medium; us and us denote the scattering polar and
azimuthal angles for the photon entering the lidar aperture after the scattering; and urcv is the viewing zenith
angle of the scattered light. In Eq. (23) the Stokes vector
is first rotated from the meridional plane of the incident
beam to the scattering plane, where F1 is the angle between the two planes. After scattering, the Stokes vector
is rotated to the meridional plane of the incident beam
and
(24)
by p 2 F2, and subsequently rotated back to the initial
reference coordinates by u9.
If the interaction between a photon and the medium is
predicted to occur outside of the medium, the photontracing loop in the Monte Carlo computation is ended and
the tracing process of a new photon begins. If the interaction is predicted to occur inside the medium, the
model samples the scattering angle (uscat, uscat) with the
phase function P11 (ui , us , us ). The scattering angle uscat is
obtained with the random number j1, which has uniform
distribution from 0 to 1:
uscat 5 F 21 (j1 ),
(26)
where F is the accumulative probability function:
ð u ð 2p
scat
F(uscat ) 5
0
0
P11 (ui , us , us ) sinus dus dus
4p
;
(27)
JULY 2012
ZHOU ET AL.
1433
FIG. 4. As in Fig. 3, but for the 2P12/P11 component of phase matrix.
then uscat is calculated with a third random number j2 :
ðu
scat
G(uscat ) 5 ð02p
0
P11(ui , uscat , us ) dus
and
(28)
P11 (ui , uscat , us ) dus
uscat 5 G21 (j2 ).
(I, Q, U, V)T 5 vL(p 2 F2 )
(29)
Although the determination of the photon propagation
direction is based on the phase function, the effect of the
polarization state on the transfer of radiation is accounted
for in the Monte Carlo radiative transfer model by incorporating the polarization information into the estimation of the probability of a photon scattering in a certain
direction (Hu et al. 2001). With the directional vector of
the incident light and the scattering angles, the directional
vector of the scattered beam is calculated, and the new
Stokes vector is obtained as follows:
P(ui , uscat , uscat )
L(2F1 )(I0 , Q0 , U0 , V0 )T .
P11 (ui , uscat , uscat )
The ray-tracing loop does not end until the photon is
outside of the medium. Upon completion, the values of
g9 and d are calculated from the Stokes vector of the
lidar-collected signals.
3. Simulation results
Figures 3–5 show three nonzero elements of the phase
matrix for horizontally oriented plates (sb 5 0). The
(30)
phase function has a peak at (us 5 180 – 2ui, us 5 0)
because of direct reflection from the prism facet (Fig. 3).
The size parameter of HOIC is very large at 532 nm
(.103) so the forward scattering is very strong because
of diffraction. There are also many small peaks as a result of multiple reflections. When the incident angle is
small, P22 is approximately equal to P11, and the depolarization of the backscattering signal is close to zero. The
backscattering is very strong when the incident angle is
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FIG. 5. As in Fig. 3, but for the P22/P11 component of phase matrix.
zero, but decreases rapidly as the incident angle increases.
It is evident from Figs. 3–5 that the phase matrix elements
are strongly dependent upon the azimuthal angle of the
scattering plane if the incident angle is large.
The d–g9 relationship is useful for the discrimination
of cloud phase (Hu et al. 2007) and for the estimation of
the percentage of HOIC. Before carrying out the Monte
Carlo simulations, a simple model is implemented to
derive a formula for the d–g9 relationship of horizontally
oriented plates when the incident angle is close to zero.
When a small fraction of horizontally oriented plates is
present in a cloud composed primarily of randomly oriented particles, the lidar signal can be approximated by
I 5 Wh Ih 1 (1 2 Wh )Ir ’ Wh Ih 1 Ir ,
(31)
where Ih is the lidar-received Stokes vector from clouds
with only horizontally oriented crystals, Wh is the weight
of the horizontally oriented crystals, and Ir is for clouds
without horizontally oriented crystals.
For the backscattering beam, the P12 (Fig. 4) and P21
components are equal, and P22 ’ P11 when the incident
angle is less than 108 (Fig. 5). When the incident angle is
close to zero, the backscatter signal is strong and has
a small depolarization ratio; thus the relationship between the I and Q components of the Stokes vector is
Ih 5 Qh .
(32)
The layer-integrated depolarization ratio is then given by
d5
I 1 Ih Wh 2 (Qr 1 Qh Wh )
I2Q
5 r
Ir 1 Ih Wh 1 Qr 1 Qh Wh
I1Q
2dr
1 1 dr
dr
1 1 dr
5
5
.
dr
2
1 2Wh Ih
g9 2
g9r
1 1 dr
1 1 dr
Ir
g9r
(33)
From Eq. (33) we get
dr g9r
1
.
11
g9 5
d
1 1 dr
(34)
Equation (34) shows the d–g9 relationship for clouds
with oriented crystals. This simple model also implies
that the fraction of oriented crystals and g9 are linearly
related. The Monte Carlo model provides a way to test
these relationships. In the Monte Carlo simulations, an
ice cloud is assumed to form a homogeneous layer that
contains a small fraction of horizontally oriented plates.
More than 10 simulations are run to simulate the lidar
JULY 2012
1435
ZHOU ET AL.
TABLE 1. Input parameters of Monte Carlo simulations (the percentage of quasi-horizontally oriented plates is Wh).
Cloud type
Percentage of
water droplets
Percentage of
ice crystals
Off-nadir
angle (8)
Extinction
coef (km21)
Type of
oriented crystals
Deviation angle
of HOIC (8)
Ice cloud A
Ice cloud B
Ice cloud C
Mixed-phase cloud A
Mixed-phase cloud B
Mixed-phase cloud C
0
0
0
0.25
1 – 4Wh
1 – 4Wh
1
1
1
0.75
4Wh
4Wh
0.3, 3
0.3
0.3
0.3, 3
0.3, 3
0.3, 3
0.5
0.5
0.5
2
6
6
Plates
Plates
Columns
Plates
Plates
Plates
sb 5 0.8
sb 5 0
sb 5 sg 5 0
sb 5 0.8
sb 5 0.8
sb 5 1.5
returns from the ice cloud layer. In each simulation, the
ice cloud layer is given a different percentage of oriented
plates, and more than 108 photons are traced.
As ice particles flutter in the atmosphere, the plates
will likely have a range of tilt angles depending on the
particle size and meteorological conditions. The standard
deviation of tilt angle sb for oriented plates is important
for backscattering properties. Relative to horizontally
oriented plates with sb 5 08, horizontally oriented plates
with sb 5 1.58 display weaker backscatter at ui 5 08 and
stronger backscatter when the incident angle deviates
from zero. The sb values for the Monte Carlo simulations
in this study can be seen in Table 1.
Figure 6 shows the d–g9 relationship for ice clouds with
quasi-horizontally oriented plates. When the off-nadir
angle is 0.38, the backscattering signal for clouds with
oriented plates is strong and the depolarization ratio is
small relative to ice clouds with solely ROIC. The dashed
line shows that the d–g9 relationship agrees very well with
Eq. (34). Figure 6b shows the weight of horizontally oriented plates and the value of g9 are approximately linearly related when the weight of oriented plates is small,
and the relationship provides a convenient way to estimate the percentage of oriented plates. When the offnadir angle is 38, the d and g9 values associated with
HOIC are very close to that of ROIC. Figures 6c,d show
the relationships for transmissive cloud (t 5 1); note that
the d–g9 relationship in Eq. (34) also holds.
For mixed-phase clouds, simulations are performed
where the clouds are considered to be a homogeneous
layer containing water droplets, ROIC, and quasihorizontally oriented plates. In the first set of simulations
(mixed-phase cloud A), the ratio of ROIC to water cloud
droplets is fixed, while the percentage of oriented
FIG. 6. (a),(c) The d–g9 relationship for ice clouds with HOIC (sb 5 0.88) when the off-nadir
angle is 0.38 and 38. (b),(d) Relationship between the percentage of HOIC and g9. Optical
depths are (top) t 5 4 and (bottom) t 5 1.
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VOLUME 51
FIG. 7. As in Fig. 6, but for mixed-phase clouds with HOIC. The optical depth of the cloud
layer is 4.
plates varies. In the second and third sets of simulations
(mixed-phase clouds B and C), the ratio of ROIC to
HOIC is fixed, but the total percentage of ice particles
varies (Table 1).
Figure 7a shows the d–g9 relationship of mixed-phase
clouds with quasi-horizontally oriented plates. When the
components for ROIC are fixed (mixed-phase cloud A),
the d–g9 relationship satisfies Eq. (34). Compared to ice
clouds with same percentage of oriented plates, the g9
value of mixed-phase clouds is slightly weaker because
of the difference in the multiple scattering factor. The
forward scattering by large ice particles is much stronger
than that by water cloud particles. At the 0.38 off-nadir
angle, after being scattered by a randomly oriented
particle, a photon propagating in the nadir direction has
a greater chance of entering the lidar aperture. Thus,
lidar signals associated with ice clouds with oriented
plates show slightly stronger backscatter compared with
mixed-phase clouds.
Figures 7c–f show the simulation results for mixed-phase
clouds B and C, which represent conditions in which water
cloud droplets dominate. When compared with mixedphase cloud B, the sb value of mixed-phase cloud C is much
larger and its backscatter at uoff 5 0.38 is much weaker.
However, when the off-nadir angle is 38, the backscattering signal is stronger for clouds with a larger sb value.
While the high backscatter–low depolarization feature
of ice clouds is often ascribed to the presence of platelike
ice particles, columns with two prism facets parallel to
horizontal plane may also have such an effect. We compared the two effects on backscattering signals from oriented columns with Monte Carlo simulations. For oriented
columns, the Euler angle g also satisfies the normal distribution, so they have an additional tilt angle compared
JULY 2012
ZHOU ET AL.
with oriented plates. In the simulations for comparison,
both sb and sg for HOIC are set to zero.
Figure 8a shows the value of P11 at the backscattering
angle for oriented plates and columns. When incident
angle is less than 18, the backscatter from plates is much
larger than from columns. Figure 8b shows the simulated
d–g9 relationship for optically thick (t 5 4) ice clouds
with horizontally oriented crystals and the off-nadir angle
set to 0.38. The circles denote simulation results for clouds
with horizontally oriented plates and the diamonds are
values for clouds with horizontally oriented columns. The
effect of oriented columns on the d–g9 relationship is
much smaller than for the counterpart of horizontally
oriented plates. Therefore, it is expected that the high
backscatter–low depolarization signals from CALIOP
are generated primarily from platelike particles.
Figure 9 shows the d–g9 relationship of opaque singlelayer clouds, where the color of each pixel represents the
frequency of occurrence. Figure 9a is the d–g9 relationship for clouds warmer than 2208C when the off-nadir
angle is set to 0.38. The branch with a positive slope represents water clouds with spherical particles while the
branch with a negative slope represents clouds with HOIC.
The HOIC branch is above the water cloud branch when
the off-nadir angle is 0.38, and the points are much closer
to the water cloud region when the off-nadir is 38, similar
to the simulations for mixed-phase clouds. Statistically,
the distribution function of the percentage of HOIC
should be continuous. Thus, it is concluded that singlelayered clouds with HOIC at temperatures warmer than
2208C could reasonably be considered as mixed-phase
clouds. This conclusion is consistent with Cho et al. (2008),
who also indicate that oriented plates are present in mixedphase clouds. From the comparison of the d–g9 distribution for HOIC in Fig. 9a with our simulation results, the
percentage of quasi-horizontally oriented plates ranges
from 0% to 6% in optically thick mixed-phase clouds.
Figures 9b,e show results for clouds at temperatures
between 2208 and 2358C as provided in the CALIOP
product. The HOIC and water cloud branches overlap
when the off-nadir angle is 0.38. The data tend to concentrate in the region representing ROIC when the
off-nadir angle changes to 38, implying that many of the
near-nadir CALIOP measurements were of ice particles
rather than water at temperatures between 2208 and
2358C. Comparison of the CALIPSO products with simulations suggests that the percentage of quasi-horizontally
oriented plates in these ice clouds ranges from 0% to 3%.
In Figs. 9c,f, the water cloud branch does not exist,
implying the nonexistence of supercooled water clouds
at temperatures colder than 2358C. The equivalent
percentage of HOIC in these cold clouds ranges from
0% to 0.5%.
1437
FIG. 8. (a) Backscattering P11 of oriented plates and columns as
a function of incident angle. (b) Simulated d–g9 relationship for
optically thick ice clouds with horizontally oriented plates and
columns. The deviation angles for oriented crystals are set to zero.
(c) Relationship between the percentage of HOIC and g9.
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VOLUME 51
FIG. 9. Frequency of occurrence for single-layer opaque clouds as a function of d and g9 observed by CALIPSO
lidar. Observations are for (top) 0.38 off-nadir angles and (bottom) 38 off-nadir angles with (left) T . 2208C,
(middle) 2358 , T , 2208C, and (right) T , 2358C.
From the Monte Carlo simulations, it would be appropriate to fit the downward branch with
drm g9rm
1
,
(35)
g9 5
11
d
1 1 drm
where drm and g9rm are values at maximum frequency of
occurrence for clouds with only randomly oriented particles as inferred from observations at 38 off-nadir angle. The
equation fits well to the observations shown in Fig. 9. For
both ice clouds and mixed-phase clouds with horizontally
oriented plates, the results of our Monte Carlo simulations
are consistent with the CALIOP observations.
From the CALIOP measurements and our simulation
results, oriented plates are found most frequently in
warm mixed-phase clouds, less frequently in ice clouds at
warmer temperatures, and infrequently in very cold ice
clouds.
4. Summary
In this study, we developed a Monte Carlo radiative
transfer model to simulate lidar returns from clouds
consisting of ROIC and HOIC, with a full consideration
of the extinction matrix. An advantage of this model is in
its ability to simulate both optically thin and optically
thick clouds. The model is used to simulate the CALIOP
backscattering and depolarization signals of water clouds,
mixed-phase clouds, and ice clouds.
A series of case studies for ice clouds and mixed-phase
clouds with HOIC are investigated. The results of the
Monte Carlo simulations are found to be consistent with
CALIOP observations. Comparison of the model results
and observations shows that most oriented plates exist in
mixed-phase clouds warmer than 2208C and are very
infrequent in cold high clouds at temperatures below
2408C. Specifically, it is estimated that the percentage of
horizontally oriented plates are 0%–6%, 0%–3%, and
0%–0.5% in optically thick mixed-phase clouds, warm
ice clouds (.2358C), and cold ice clouds, respectively.
This finding is consistent with previous studies (e.g.,
Noel and Chepfer 2010).
In addition to horizontally oriented plates, the effect
of oriented columns was also investigated, but the effect
of oriented columns was found to be much smaller than
that of horizontally oriented plates. The distribution of
d–g9 is quite different when the off-nadir angle for the
CALIOP measurements changes from 0.38 to 38 and
suggests the existence of large plates with horizontal
JULY 2012
ZHOU ET AL.
orientations (note that small plates tend to be randomly
oriented).
Acknowledgments. This study is supported by NASA
Grant NNX10AM27G, and partly by NNX11AF40G
and NNX11AK37G. The authors are grateful to Lei Bi
and Meng Gao for their help in the light scattering and
radiative transfer simulations involved in this study.
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