EUM�CO�96�430�SAT
E�ect of Thin Cirrus Clouds
on
Meteosat Second Generation �MSG�
Observations
Final Report
Matthias Wiegner, Peter Seifert, and Peter Schl�ussel
Universit�at M�unchen, Meteorologisches Institut
Theresienstra�e 37, D-80 333 M�unchen
Germany
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Final Report for EUM�CO�96�430�SAT
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Final Report for EUM�CO�96�430�SAT
Contents
1 Introduction and Background
9
2 SEVIRI channels
11
3 Scattering Properties of Ice Crystals
13
3.1
3.2
3.3
3.4
3.5
3.6
Theoretical Approach . . . . . . . . . . . . . . . . .
Selection of Particles . . . . . . . . . . . . . . . . .
Selection of Size Distributions . . . . . . . . . . . .
Phase functions . . . . . . . . . . . . . . . . . . . .
Asymmetry Parameter and Single Scattering Albedo
Wavelength Dependence in SEVIRI Channels . . . .
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15
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4 Comparison with Mie-Calculations
35
5 Radiative Transfer Calculations �IR-Channels�
41
5.1 Discrete ordinate method . . . . . . . . . . . .
5.2 Expansion of the phase function . . . . . . . .
5.3 Radiances at Top of Atmosphere . . . . . . . .
5.3.1 Dependence on Cloud Optical Depth . .
5.3.2 Dependence on Size Distribution . . . .
5.3.3 Dependence on Cloud Height . . . . . .
5.3.4 Dependence on Satellite Zenith Angle .
5.3.5 In�uence of Phase Function Expansion .
5.4 Justi�cation of scattering . . . . . . . . . . . .
5.5 Discussion . . . . . . . . . . . . . . . . . . . . .
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6 Radiative Transfer Calculations �VIS-Channel�
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41
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6.1 Method of Successive Order of Scattering . . . . . . . . . . . . . . . . . . . 56
6.2 Radiances at the Top of the Atmosphere . . . . . . . . . . . . . . . . . . . . 57
7 Summary and Conclusions
64
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Final Report for EUM�CO�96�430�SAT
8 Acknowledgments
66
9 References
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Final Report for EUM�CO�96�430�SAT
5
List of Figures
1
Number densities of hexagonal crystals as derived from Heyms�eld and
Platt's size distributions HP1 and HP7. . . . . . . . . . . . . . . . . . . . . 18
2
Phase function vs. scattering angle � for wavelength ��0.635 �m. The
curves show di�erent particle sizes as indicated, cf. Tab. 3 . . . . . . . . . . 19
3
Same as Fig. 2, but ��3.8 �m. . . . . . . . . . . . . . . . . . . . . . . . . . 19
4
Same as Fig. 2, but ��8.7 �m. . . . . . . . . . . . . . . . . . . . . . . . . . 20
5
Phase function vs. scattering angle � for wavelength ��0.635 �m. The
curves show di�erent particle size distributions as indicated, cf. Tab. 4 . . . 21
6
Same as Fig. 5, but ��3.8 �m. . . . . . . . . . . . . . . . . . . . . . . . . . 21
7
Same as Fig. 5, but ��8.7 �m. . . . . . . . . . . . . . . . . . . . . . . . . . 22
8
Phase function vs. scattering angle � for ��0.635 �m for size distributions
HP1 and HP1ST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
9
Phase function vs. scattering angle � for ��3.8 �m for size distributions
HP1 and HP1ST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
10 Same as Fig. 9, but size distributions HP7 and HP7ST. . . . . . . . . . . . 25
11 Phase function vs. scattering angle � for ��8.7 �m for size distributions
HP1 and HP1ST. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
12 Same as Fig. 11, but size distributions HP7 and HP7ST. . . . . . . . . . . 26
13 Real part of the refractive index as a function of wavelength . . . . . . . . . 31
14 Imaginary part of the refractive index as a function of wavelength . . . . . . 31
15 Phase function vs. scattering angle � for several wavelengths between 3.4
�m and 4.2 �m and size distribution HP1ST . . . . . . . . . . . . . . . . . 32
16 Same as Fig. 15, but size distribution HP7ST . . . . . . . . . . . . . . . . . 33
17 Phase function vs. scattering angle � for several wavelengths between 8.3
�m and 9.1 �m and size distribution HP1ST . . . . . . . . . . . . . . . . . 33
18 Same as Fig. 17, but size distribution HP7ST . . . . . . . . . . . . . . . . . 34
19 Comparison of phase functions of spherical �dashed line� and non-spherical
particles �full line�; size distribtion is HP1ST, ��0.635 �m. . . . . . . . . . 36
20 Same as Fig. 19, but size distribtion HP7ST . . . . . . . . . . . . . . . . . . 36
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Final Report for EUM�CO�96�430�SAT
21 Comparison of phase functions of spherical �dashed line� and non-spherical
particles �full line�; size distribtion is HP1ST, ��3.8 �m. . . . . . . . . . .
22 Same as Fig. 21, but size distribtion HP7ST . . . . . . . . . . . . . . . . . .
23 Comparison of phase functions of spherical �dashed line� and non-spherical
particles �full line�; size distribtion is HP1ST, ��8.7 �m. . . . . . . . . . .
24 Same as Fig. 23, but size distribtion HP7ST . . . . . . . . . . . . . . . . . .
25 Phase function of size distribution HP1ST for 8.7 �m �full line� and reconstructed from a Legendre expansion with 1000 coe�cients . . . . . . . . . .
26 Phase function of size distribution HP1ST for 8.7 �m �full line� and reconstructed from a Legendre expansion with 1800 coe�cients . . . . . . . . . .
27 Phase function of size distribution HP1ST for 8.7 �m �dashed line� and the
reconstructed �full line, Delta-M-Method, M �100�. . . . . . . . . . . . . . .
28 Phase function of size distribution HP1ST for 8.7 �m �dashed line� and the
reconstructed �full line, Delta-M-Method, M �50�. . . . . . . . . . . . . . . .
29 Change of brightness temperature with �c �3:8�m�; size distribution HP1ST,
��3.8 �m �dashed line�, ��8.7 �m �full line�. . . . . . . . . . . . . . . . .
30 Change of brightness temperature as a function of cloud optical depth for
di�erent size distributions as indicated; ��3.8 �m. . . . . . . . . . . . . . .
31 Same as Fig. 31, but ��8.7 �m. . . . . . . . . . . . . . . . . . . . . . . . . .
32 Error in brightness temperature as a function of �c when Mie-phase functions are used; ��3.8 �m �see text�. . . . . . . . . . . . . . . . . . . . . . .
33 Same as Fig. 32, but ��8.7 �m. . . . . . . . . . . . . . . . . . . . . . . . . .
34 Vertical pro�le of brightness temperature for di�erent cloud heights; ��3.8
�m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35 Vertical pro�le of brightness temperature for di�erent cloud heights; ��8.7
�m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36 Change of brightness temperature with satellite zenith angle; �c �0.1. . . . .
37 Anisotropy function � ��; '� for size distribution HP1 �KRIPO model�;
�o �30� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38 Same as Fig. 37, but �o �60� . . . . . . . . . . . . . . . . . . . . . . . . . . .
39 Anisotropy function � ��; '� for size distribution HP1 �Mie calculations�;
�o �30� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
37
38
39
43
44
45
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49
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52
53
60
60
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Final Report for EUM�CO�96�430�SAT
7
40 Same as Fig. 39, but �o �60� . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
41 Anisotropy function � ��; '� for size distribution HP7ST �KRIPO model�;
�o �30� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
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Final Report for EUM�CO�96�430�SAT
List of Tables
1
List of wavelengths of SEVIRI channels, spectral interval ��1 , �2 � and central wavelength ��c � in �m . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2
Refractive index, nr and ni being real and imaginary part, respectively, for
selected wavelengths within SEVIRI channels . . . . . . . . . . . . . . . . . 12
3
Dimensions �in �m� of particles of the eight selected classes: c and a are
the large and small half-axes, respectively, rc is the radius of a sphere with
same cross section. cl and cu are the lower and upper boundaries of c for
each class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4
Selected size distributions from Heyms�eld and Platt and the corresponding
temperature range. �'s and � 's are parameters describing the size distribution �see text, Eq. 2�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5
Single scattering albedo and asymmetry parameter for eight size distributions HP1 to HP8 for ��0.635 �m. . . . . . . . . . . . . . . . . . . . . . . 27
6
Single scattering albedo for di�erent size distributions and six wavelengths
from SEVIRI channel IR 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7
Asymmetry parameter for di�erent size distributions and six wavelengths
from SEVIRI channel IR 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8
Single scattering albedo for di�erent size distributions and �ve wavelengths
from SEVIRI channel IR 8.7. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
9
Asymmetry parameter for di�erent size distributions and �ve wavelengths
from SEVIRI channel IR 8.7. . . . . . . . . . . . . . . . . . . . . . . . . . . 30
10 Asymmetry parameter for size distribution HP1ST and HP7ST: Mie and
KRIPO calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Single scattering albedo for size distribution HP1ST and HP7ST: Mie and
KRIPO calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 E�ective radii for di�erent size distributions. . . . . . . . . . . . . . . . . .
13 Planetary albedo in percent for a cirrus cloud �9.5 � 11 km�. ��0.635 �m,
�0 �30� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14 Same as Tab. 13, �0 �60� . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
40
48
58
59
Section 1: Introduction and Background
9
1 Introduction and Background
The use of data from operational meteorological satellites requires the development of
retrieval techniques for the interpretation of satellite-measured radiances in terms of geophysical parameters. Usually, the algorithm development is done by means of radiative
transfer simulations. In the frame of these simulations the interaction of atmospheric
parameters with the radiation �eld must be described. In the visible and near infrared
spectrum the radiative transfer is signi�cantly controlled by clouds. Clouds re�ect solar
radiation to the satellite's sensors and modify the radiation leaving the surface and the
lower atmosphere on its way to the satellite. Even thin cirrus clouds can modify upwelling radiance �elds to an extent that may not be neglected. Otherwise, interpretation
of satellite images with respect to properties of the surface or the lower atmosphere can
be erroneous.
As a consequence, one of the key tasks in using data from present, but in particular of
future instruments like the Spinning Enhanced Visible and Infra-Red Imager �SEVIRI�
to be �own as part of Meteosat Second Generation �MSG� programme, is cloud detection
and classi�cation. Many scene identi�cation schemes have been successfully applied to
detect and characterize clouds in visible and infrared satellite imagery, however, it is
still an unsolved problem to adequately detect optically thin clouds. Especially, thin
��subvisible"� cirrus clouds are di�cult to detect. This de�ciency is a severe drawback.
Therefore, it is necessary to thoroughly describe the interactions between cirrus clouds
and the radiation �eld by means of radiative transfer simulations. This �nally leads to
the development of mathematical models which allow the retrieval of properties of the
troposphere and the Earth's surface.
In the past cirrus clouds have not been adequately described in radiative transfer simulations due to numerical de�ciencies in modelling the non-spherical particle shapes which
are typical for ice crystals in cirrus clouds. Commonly spherical particles were assumed
and the Lorenz-Mie theory was applied to determine the single scattering properties. One
main reason was unavailable computing power for the calculation of scattering properties
of ice crystals. Early attempts to consider hexagonal structures as scattering particles
were made in the 1970'ties. �e.g., Wendling et al., 1979�. However, only with the recent
availability of enhanced computing power more complete calculations including absorption
and polarization were possible �Hess and Wiegner, 1994; Hess et al., 1996�. These studies
have shown that the non-spherical particle shapes cause optical properties that drastically
di�er from those known from spheres. In particular, the shape of the scattering function of
ice crystals di�ers from that of spheres. This in turn will likely be important in cases where
10
Section 1: Introduction and Background
the single scattering of radiation plays the major role, which is true for thin cirrus clouds.
Therefore, the detection of thin cirrus clouds by radiometers like SEVIRI will critically
depend on the knowledge about the impact of nonspherical particles on the atmospheric
radiation �eld.
It is the objective of this study to contribute to the development of an improved identi�cation of cirrus clouds by the SEVIRI radiometer. In this context two issues are discussed
in detail: the �rst deals with radiative properties of ice clouds, the second concerns the
induced modi�cations of radiative transfer.
11
Section 2: SEVIRI Channels
2 SEVIRI channels
To determine the e�ect of cirrus clouds on satellite measurements it is su�cient to restrict
oneself to the consideration of the spectral bands of the radiometer. The current de�nition
of SEVIRI is based on the outcome of the MSG Phase-A-Study and includes 12 channels.
The spectral bands are listed in Tab. 1.
Channel
VIS 0.6
VIS 0.8
IR 1.6
IR 3.8
IR 8.7
IR 10.8
IR 12.0
WV 6.2
WV 7.3
IR 9.7
IR 13.4
HRV
�1
�c
�2
0.560 0.635 0.710
0.740 0.830 0.880
1.500 1.615 1.780
3.400 3.800 4.200
8.300 8.700 9.100
9.800 10.800 11.800
11.000 12.000 13.000
5.350 6.260 7.150
6.850 7.350 7.850
9.460 9.700 9.940
13.040 13.400 13.760
0.500
0.900
Table 1: List of wavelengths of SEVIRI channels, spectral interval ��1 , �2 � and central
wavelength ��c � in �m
�c denotes the central wavelengths, �1 and �2 gives the lower and upper limit of the
channels, respectively. All numbers in Tab. 1 are given in �m.
Within this study we consider all channels below 10 �m with focus on channels IR 8.7
and IR 3.8. It is discussed in Sec. 3.6 whether the wavelength dependence of the optical
properties within the channel requires a subdivision into several spectral intervals or not.
If the latter is true the calculation of one wavelength for each channel su�ces.
In Tab. 2 the refractive indices of ice �Warren, 1984� are listed for the central wavelengths
of the SEVIRI channels. Furthermore, the refractive index is given for those wavelengths
which are used to investigate the variability of optical properties within certain channels
�cf. above�.
12
Section 2: SEVIRI Channels
Channel
�c
0.560
VIS 0.6 0.635
0.710
0.740
VIS 0.8 0.830
0.880
1.500
IR 1.6
1.615
1.780
3.400
3.500
3.600
IR 3.8
3.800
4.000
4.200
5.350
WV 6.2 6.260
7.150
6.850
WV 7.3 7.350
7.850
8.300
8.500
IR 8.7
8.700
8.900
9.100
IR 9.7
9.700
IR 10.8 10.800
IR 12.0 12.000
IR 13.4 13.400
nr
1.311
1.308
1.307
1.306
1.304
1.304
1.292
1.289
1.283
1.505
1.455
1.422
1.384
1.422
1.347
1.319
1.318
1.319
1.319
1.318
1.315
1.301
1.292
1.284
1.275
1.267
1.231
1.089
1.280
1.522
ni
�3.29�10,9
�1.13�10,8
�3.44�10,8
�4.92�10,8
�1.45�10,7
�3.35�10,7
�5.88�10,4
�3.32�10,4
�1.15�10,4
�3.87�10,2
�1.64�10,2
�8.97�10,3
�6.72�10,3
�9.62�10,3
�1.36�10,2
�1.62�10,2
�6.65�10,2
�5.51�10,2
�5.77�10,2
�5.21�10,2
�4.91�10,2
�3.75�10,2
�3.91�10,2
�4.00�10,2
�4.18�10,2
�4.41�10,2
�4.46�10,2
�1.83�10,2
�4.13�10,1
�3.44�10,1
Table 2: Refractive index, nr and ni being real and imaginary part, respectively, for selected
wavelengths within SEVIRI channels
Section 3: Scattering Properties of Ice Clouds
13
3 Scattering Properties of Ice Crystals
3.1 Theoretical Approach
Several methods have been proposed for the simulation of optical properties of nonspherical particles. They can be classi�ed in discrete dipole approximation �DDA�, extended boundary condition method �EBCM�, method of anomalous di�raction �MAD�,
method of suppression of resonances �MSR�, statistical approach �STA�, and geometrical
optics approach �GOA�.
DDA �Draine, 1988; Draine and Flateau, 1994� is suitable for particles of arbitrary shape
but limited to very small size parameters x � 2�r��. r is the particle radius and �
is the wavelength of the incident radiation. It is used where � � a, i.e., it can be
utilized for describing the interaction of millimetre and sub-millimetre waves with cirrus
particles �Evans and Vivekanadan, 1990�. In principle, it can also be used in the visible
or near infrared spectral range. However, for cirrus clouds size parameters larger than
100 frequently occur, resulting in an extremely large number of dipoles to be considered
in the model. As a consequence the requirements on computing resources �memory, time�
are far beyond any reasonable limit. A realistic upper limit of the application of DDA is
about x�5 �several hours to one day on a cluster of 20 RISC workstations�.
The EBCM, also referred to as T-matrix approach, covers the radiation-particle interaction
where the wavelength is of the same order as the particle size �Mugnai and Wiscombe,
1986; Mishchenko, 1991; Mishchenko and Travis, 1994; Mishchenko et al., 1996�. It has
been shown that it agrees well with results obtained from GOA up to size parameters of
approximately 60 �Macke et al., 1996�. However, it should be noted that the T-matrix
method can only be applied to particles which are rotationally symmetric; i.e., �relatively
smooth" particles such as spheroids or so called Chebychev particles. Thus, most of the
natural particles cannot be precisely modelled.
The MAD was developed by Van de Hulst �1957� for cases where jm , 1j � 1 and x is
large. Comparisons with the EBCM made by Latimer and Barber �1978� for ellipsoids of
m � 1:05 and m � 1:20 with x � 2:37 and x � 4:31 generally agreed well but disagreed at
large scattering angles. The results obtained for m � 1:05 were better than those obtained
for m � 1:20.
The MSR followed the detection by Chylek �1976, 1977� that resonances in the Mie coe�cients an and bn are responsible for the ripples superimosed on the general curve of
extinction e�ciency versus size parameter of spherical particles. According to Van de
Hulst �1957� the resonances are due to waves travelling along the surface of the sphere
14
Section 3: Scattering Properties of Ice Clouds
and spewing o� energy which interferes with the forward di�racted wave. Chylek et al.
�1976� conjectured that surface waves do not exist on aspherical particles. Hence, a good
approximation of non-spherical particles would be the removal of all resonances in the
Lorenz-Mie theory. Good agreement between this theoretical attempt and measurements
were obtained for the range of size parameter 3 � x � 30. The method does not work at
x � 30.
The STA was described by Nevitt and Bohren �1984� for randomly oriented anisotropic
oscillators named Rayleigh ellipsoids. This method is only adequate for x � �.
For particles being large compared to the wavelength only GOA gives reliable results
�Macke et al., 1995�. This is the size range relevant for cirrus clouds. Another advantage
of this approach is that there are no limitations with respect to the particle shape. Finally
it should be noted that the numerical resources required for GOA are reasonable � the
typical time scale for one run �one particle size, one wavelength� is in the order of a few
minutes on a typical desktop computer �DEC Alpha 255�.
The selection of a suitable method is governed by the wavelength of the radiation and
typical particle sizes and shapes. As already stated the spectral range is between 0.5 �m
and 10 �m �SEVIRI channels�, particle sizes are between approximately 10 �m and a few
millimetre, and typical shapes are hexagonal crystals. As a consequence, GOA seems to
be the best choice for modelling scattering processes in cirrus clouds.
We are aware that using GOA for small ice crystals at wavelengths near 10 �m might
be critical. However, errors caused by GOA at this size parameter range can hardly be
quanti�ed. On the one hand reference models do not exist, on the other hand comparisons
between di�erent approaches su�er from inconsistent particle shapes �either the shapes
are not exactly the same or the non-sphericity is only idealized�. Consequently, only
some rough estimates on the accuracy are available. For example, Macke et al. �1995�
state that GOA certainly is a better solution than Mie theory. They also found that the
single scattering albedo � being one of the most important parameters govering radiation
transfer � agrees within about one percent when T-matrix and GOA are compared, even
at size parameters as small as x�10 �spheroids�. The asymmetry parameter seems to be
overestimated by 5 � at maximum for x�20.
For this reason a GOA model �`KRIPO'� has been developed on the basis of ray-tracing
and Fraunhofer di�raction �Hess and Wiegner, 1994; 1995�. The model includes multiple
refraction, absorption, di�raction and polarization of radiation at hexagonal plates and
columns.
Results from the KRIPO model have already been included in radiative transfer sim-
15
Section 3: Scattering Properties of Ice Clouds
ulations for a comparison of modelled and groundbased radiance measurements. Good
agreement has been obtained �Hess, 1996�.
3.2 Selection of Particles
For this study hexagonal columns have been selected as typical particle shape for modelling
the optical properties of ice crystals; they represent the most fundamental crystal shape
relevant for ice clouds. This is true in particular for large particles which occur in `warm'
cirrus clouds �T � ,20� C�
Eight classes of columns are de�ned covering the full range of particle sizes. The classes
are de�ned by logarithmic intervals of the large axis of the crystal c as described in Tab. 3.
cl
cu Class
2
6
6
20
20
40
40
90
90 200
200 400
400 900
900 2000
1
2
3
4
5
6
7
8
c
a
rc
3.5
1.4 1.77
10.0
4.0 5.07
30.0 10.0 13.6
60.0 22.0 28.8
130.0 41.0 56.9
300.0 60.0 100.0
600.0 80.0 159.9
1300.0 110.0 270.7
Table 3: Dimensions �in �m� of particles of the eight selected classes: c and a are the
large and small half-axes, respectively, rc is the radius of a sphere with same cross section.
cl and cu are the lower and upper boundaries of c for each class.
Parameters c and a represent the length of the c-axis and the half-length of the a-axis of
the crystals, respectively. The parameter c is a mean value representative for each class
de�ned by its lower boundary cl and upper boundary cu . Dimension a is calculated from
prescribed aspect ratios �c��2a�� published by Auer and Veal �1970�; note that they vary
with size. The equivalent radius rc is the radius of a sphere with the same cross section
as the hexagonal column when averaged over all randomly distributed orientations. All
dimensions in Tab. 3 are given in �m.
The equivalent radius rc can be calculated from a and c by
v
�p
u
2
u
3
3
a
t
rc �
�
c
+
4 2a
!
�1�
16
Section 3: Scattering Properties of Ice Clouds
Size Dist. Tmin Tmax
�1
HP1
�20 �25 1.85�107
HP2
�25 �30 1.83�107
HP3
�30 �35 3.42�106
HP4
�35 �40 9.50�106
HP5
�40 �45 7.35�107
HP6
�45 �50 2.79�107
HP7
�50 �55 3.20�108
HP8
�55 �60 2.52�108
�1
�2.56
�2.51
�2.21
�2.29
�3.23
�3.15
�3.83
�3.85
�2
5.39�1010
7.67�1012
1.59�1011
1.67�1012
�2 c12
�3.74
�4.49
�3.94
�4.37
860
690
500
330
Table 4: Selected size distributions from Heyms�eld and Platt and the corresponding temperature range. �'s and � 's are parameters describing the size distribution �see text, Eq. 2�.
Large particles �c � 300 �m� frequently exist in horizontal orientation. This results in
phase functions which are not rotationally invariant with respect to the direction of the
incident radiation, i.e., p��; �� in lieu of p��� with � being the scattering angle and �
the scattering azimuth. In this case the scattering also depends on the solar zenith angle.
However, these two-dimensional phase functions cannot be used in present state radiative
transfer codes. For this reason it is assumed throughout this study that all particles are
randomly oriented in space.
3.3 Selection of Size Distributions
In real clouds always an ensemble of particles of di�erent sizes and shapes exist. In this
study we assume all particles to be hexagonal columns. The particle size spectra are taken
from measurements of Heyms�eld and Platt �1984� who introduced a classi�cation of size
distributions � referred to as HP1 till HP8 � in terms of the ambient air temperature. It
is given in Tab. 4.
The size distributions HP1 - HP4 are described by one power law each for small and large
particles according to
N � �j c�
j
for j � 1; 2
�2�
while for HP5 - HP8 one equation su�ces �j �1�. c is the largest dimension �length of
the c-axis� of the crystal. Parameters �1 ; �1 and �2 ; �2 are given in Tab. 4. They were
derived by Heyms�eld and Platt by �tting the measured size distributions for the given
temperature ranges separately below and above a threshold value c12 which is de�ned by
Section 3: Scattering Properties of Ice Clouds
� �1���1 ,�2 �
c12 � ��2
1
17
�3�
Note that Heyms�eld and Platt's measurements do not include particles with c � 20 �m
because no reliable measurements were available.
HP1 and HP7 are the most interesting distributions with respect to the natural variability
and the sensitivity of cirrus cloud optical properties: for HP1 �`warm cloud', �20� C to
�25� C� the relative contribution of small particles to the total particle number is the
smallest of all distributions, while for HP7 �`cold cloud', �50� C to �55� C� the relative
number of small particles is largest.
A graphical display of the size distributions HP1 and HP7 is given in Fig. 1. Note, that the
number densities for the �rst two classes �cf. Tab. 3� are amended according to measurements published by Strauss �1994�. To distinguish these `more complete' size distribution
from the original ones they are marked as HP1ST and HP7ST, respectively. The in�uence
of these small crystals on the phase functions is discussed in Sec. 3.4.
To easily describe the size distributions by one parameter it is common to de�ne an
e�ective radius reff . It is given by
reff �
P � Qi �3�2
Ni
�
P � Qi �
� Ni
with Qi being the mean cross section of the crystal, Ni the number density of the particle
and i denoting the number of the particle class as given in Tab. 3.
The e�ective radii of HP1 and HP7 are reff �91.7 �m and reff �30.7 �m, respectively.
3.4 Phase functions
Phase functions of a size distribution of cirrus ice crystals are calculated by superposition
of phase functions of eight columns according to
p�� � �
8
X
Qi
i�1
� N i p i �� �
�4�
The phase functions are normalized to unity, i.e.,
1 Z 2� Z 1 p��� d cos � d� � 1
4� 0 ,1
or
�5�
18
Section 3: Scattering Properties of Ice Clouds
Figure 1: Number densities of hexagonal crystals as derived from Heyms�eld and Platt's
size distributions HP1 and HP7.
Z
1
,1
p��� d cos � � 2
with � being the scattering angle and � the azimuth angle. Note, that all phase functions
are rotationally invariant, that means, they only depend on �.
For the individual particles described in Sec. 3.2, Qi and phase functions p��� are calculated utilizing the ray-tracing model KRIPO, the size distributions are taken from Tab. 4.
With this input, realistic phase functions of cirrus clouds are calculated with Eq. �4�.
In the following we show a few typical examples of phase functions and discuss their main
features. First, p��� for individual particles are shown �Figs. 2-4� then phase functions of
di�erent size distributions are presented �Figs. 5-7�.
As just mentioned, Figs. 2-4 show as examples the phase functions for ��0.635 �m, ��3.8
�m and ��8.7 �m of all eight ice column classes as de�ned in Tab. 3.
It can be seen that in the visible spectral range �Fig. 2� the di�erences of the scattering
characteristics are relatively small. The two well known haloes are clearly visible near
22� and 46� , also the pronounced forward scattering peak and the enhanced backward
Section 3: Scattering Properties of Ice Clouds
19
Figure 2: Phase function vs. scattering angle � for wavelength ��0.635 �m. The curves
show di�erent particle sizes as indicated, cf. Tab. 3
Figure 3: Same as Fig. 2, but ��3.8 �m.
20
Section 3: Scattering Properties of Ice Clouds
Figure 4: Same as Fig. 2, but ��8.7 �m.
scattering. In general, the forward scattering peak is the larger and narrower the larger
the particles are � curve labelled `8' denotes the largest crystal.
For the 3.8 �m wavelength �Fig. 3�, the haloes are much less pronounced. The position
of the �rst halo is shifted towards larger scattering angles and the second one has almost
disappeared. This is caused by the strong absorption and the slightly greater real part
of the refractive index at ��3.8�m when compared to the visible spectral range. The
smooth parts of the small-particle-curves ��1 and �2� for 5� � � � 30� are caused by
di�raction; this contribution becomes less important when the particle size increases. The
general pattern of the phase function is similar for all crystals, however, the di�erences
with particle size are larger than in the case of the short wavelength. The height of the
forward scattering peak in general is smaller compared to p��; � � 635�m� as expected
due to the smaller size parameter x.
For the third wavelength of interest, 8.7 �m �Fig. 4�, the general shape of phase functions
reveals features which are already known from the two previous cases. The position of the
haloes is similar to the visible case while the magnitude of the forward scattering peak
is reduced because of the reduced sizeparameter. As the particles become large, haloes
vanish due to absorption and the phase functions remains almost unchanged.
In Figs. 5-7 phase functions of the eight HP-size distributions �Tab. 4� are shown, the
Section 3: Scattering Properties of Ice Clouds
21
Figure 5: Phase function vs. scattering angle � for wavelength ��0.635 �m. The curves
show di�erent particle size distributions as indicated, cf. Tab. 4
Figure 6: Same as Fig. 5, but ��3.8 �m.
22
Section 3: Scattering Properties of Ice Clouds
Figure 7: Same as Fig. 5, but ��8.7 �m.
wavelengths are again 0.635 �m, 3.8 �m and 8.7 �m. Note that these size distributions
do not include ice crystals smaller than 20 �m. In view of the discussions concerning the
implementation of phase functions into the radiative transfer model �Sec. 5� it is worthwhile
to have a closer look at some details of the phase functions.
Fig. 5 �0.635 �m� shows only a very small in�uence of the size distributon on the phase
function; they can hardly be distinguished in the plot. The zero-angle scatting value �
lying between 8980 sr,1 � p�0� � 66000 sr,1 for HP7 and HP3, respectively � shows large
variations, however, for all cases the peak is extremely narrow and the di�erences become
smaller. At ��90� the phase function varies only within � 12�.
Fig. 6 reveals a somewhat larger in�uence of the size distribution: the forward scattering
peak p�0� varies between 2900 sr,1 for HP3, and 669 sr,1 for HP7, and the width of the
peak is also changing. From the eight phase functions, four can be clearly distinguished.
According to the lowest forward peak, HP7 is the uppermost curve in the sideward scattering range, followed by almost identical curves for HP5, 6, and 8. The next �coinciding�
curves represents HP1, 2, and 4, while HP3 is the lowest which is in agreement with the
quite large p�0�-value. The variability at ��90� is approximately � 50�.
For 8.7 �m, shown in Fig. 7, the results for the di�erent size distributions are the same as in
Fig. 6. Again, neither HP5, 6, and 8 nor HP1, 2, and 4 can be distinguished. HP3 exhibits
Section 3: Scattering Properties of Ice Clouds
23
the lowest sideward scattering and the most dominating forward scattering �p�0��590
sr,1 �, whereas HP7 shows just the opposite behaviour �p�0��196 sr,1�. The variability
of the phase function at 90� is again in the order of � 50�.
The general pattern re�ects the special features of the phase functions of individual particles discussed above. The most prominent properties can be summarized as follows: The
magnitude of the forward scattering peak decreases with wavelength while the width of
the peak increases. The haloes are present at all wavelengths but most pronounced in the
visible spectral region. And the sensitivity of the phase functions to variations in the size
distributions is large in the infrared domain but negligible at 0.635 �m.
As already mentioned in Sec. 3.3 it is interesting to investigate the role of small ice crystals
which often have been neglected because of measurement problems. For this purpose we
have compared HP-size distributions with the corresponding size distributions when small
crystals according to Strauss' measurements were added. A few typical results are brie�y
discussed.
Fig. 8 shows the in�uence of small particles on the phase function at ��0.635 �m. Phase
functions HP1 and HP1ST are compared. It is obvious that the e�ect is almost negligible;
as a consequence, further size distributions are not discussed here. However, in the infrared spectral region this is not true as is demonstrated by the following �gures.
Fig. 9 shows the comparison assuming HP1 with or without small particles �referred to
as HP1ST and HP1� for ��3.8 �m. Fig. 10 shows the corresponding comparison for size
distributions HP7 and HP7ST.
Both �gures elucidate signi�cant di�erences. The general shape of the phase functions did
not change, but adding small particles results in larger p��� for any scattering angle larger
than approximately 10� �full lines�. Accordingly, the forward scattering peak is somewhat
reduced. For ��3.8 �m, p�0� is reduced from 2840 sr,1 to 2370 sr,1 �HP1� and p�0� from
669 sr,1 to 533 sr,1 �HP7�.
For a second wavelength, the overall e�ect of small particles is the same, however, the
di�erences are even larger �Figs. 11-12� since particle size and wavelength are of the same
order of magnitude. In particular, the forward scattering peak is considerably reduced by
including the small particle fraction: from 590 sr,1 to 534 sr,1 �HP1� and from 196 sr,1
to 29 sr,1 �HP7�.
As a conclusion, small ice crystals signi�cantly in�uence the scattering function of cirrus
clouds: forward scattering is reduced whereas scattering in the backward hemisphere
increases. The variability of the phase function with the particle size is also enlarged
when compared to Figs. 6 and 7.
24
Section 3: Scattering Properties of Ice Clouds
Figure 8: Phase function vs. scattering angle � for ��0.635 �m for size distributions HP1
and HP1ST.
Figure 9: Phase function vs. scattering angle � for ��3.8 �m for size distributions HP1
and HP1ST.
Section 3: Scattering Properties of Ice Clouds
25
Figure 10: Same as Fig. 9, but size distributions HP7 and HP7ST.
Figure 11: Phase function vs. scattering angle � for ��8.7 �m for size distributions HP1
and HP1ST.
26
Section 3: Scattering Properties of Ice Clouds
Figure 12: Same as Fig. 11, but size distributions HP7 and HP7ST.
Consequently, changes of the asymmetry parameter are also expected.
3.5 Asymmetry Parameter and Single Scattering Albedo
The single scattering albedo !0 and the asymmetry parameter g are two optical properties
relevant for radiative transfer. The single scattering albedo is the ratio of the scattering
coe�cient and the extinction coe�cient. The smaller !0 the stronger is the absorption.
The asymmetry parameter is derived from the phase function and can be interpreted as
the expectation value of the cosine of the scattering angle. It is de�ned as
Z1
1
g�2
p��� cos � d�cos ��
�6�
,1
In many radiative transfer calculations g replaces the full information inherent in the phase
function p���. This reduces computing time signi�cantly at the expense of an information
loss on the angular distribution of the radiances. Nevertheless, comparison of g for di�erent
particle ensembles is a good indicator for the variability of the scattering characteristics
of the medium.
Asymmetry parameters and single scattering albedoes are listed for selected wavelengths
in the following tables. Tab. 5 gives the values for the eight HP-size distributions �partcles
27
Section 3: Scattering Properties of Ice Clouds
Size Dist.
HP1
HP2
HP3
HP4
HP5
HP6
HP7
HP8
��0.635�m
!0
g
0.999 983
0.999 984
0.999 982
0.999 985
0.999 990
0.999 989
0.999 994
0.999 990
0.797
0.796
0.801
0.792
0.780
0.782
0.773
0.780
Table 5: Single scattering albedo and asymmetry parameter for eight size distributions
HP1 to HP8 for ��0.635 �m.
larger than 20 �m� for the central wavelength of a SEVIRI channel in the visible spectral
range, ��0.635 �m. It can be seen, that absorption is extremely low, i.e., !0 �1 and that
g is in the range between 0.77 and 0.80, i.e., the variability of g is very small. This result
could already be anticipated from visual inspection of Fig. 5.
In the following we want to focus on the central wavelengths of the IR 3.8 and IR 8.7 � the
wavelength dependence within each channel which is of interest for the radiative transfer
calculations will be discussed later.
For the center of channel IR 3.8, single scattering albedoes and asymmetry parameters
are given in Tabs. 6 and 7, �fth column. Compared to the visible spectral range, !0 is
smaller and g is larger. The variability with the size distribution is considerable for both
parameters: 0.60 � !0 � 0.77, and 0.85 � g � 0.93. This example shows that a simple
relationship between p�0� �which was considerably larger in the visible; see above� and g
does not exist. The reason for this variation is that di�erent particle sizes contribute in
a di�erent way to the total scattering and extinction properties of the particle-ensemble
according to their relative number.
Both HP-ST size distributions reveal signi�cant di�erences compared to their corresponding counterparts. !0 is larger, g is smaller than in case of the original HP distributions.
The e�ect is particularly strong if the amount of relatively small particles is already large
�HP7ST�. Note that the e�ect of regarding or disregarding small particles �e.g., HP1 vs.
HP1ST� is larger than the variability between the di�erent HP distributions �e.g., HP1
vs. HP2 vs. HP3 etc.�.
For the central wavelength of SEVIRI channel IR 8.7, single scattering albedoes !0 and
28
Section 3: Scattering Properties of Ice Clouds
Size Dist.
HP1
HP1ST
HP2
HP3
HP4
HP5
HP6
HP7
HP7ST
HP8
��3.4�m
0.551
0.567
0.551
0.551
0.551
0.554
0.554
0.558
0.629
0.555
��3.5�m
0.563
0.596
0.563
0.557
0.561
0.584
0.581
0.602
0.733
0.585
��3.6�m
!0
0.592
0.632
0.592
0.580
0.590
0.633
0.628
0.667
0.809
0.635
��3.8�m
0.617
0.659
0.618
0.602
0.616
0.671
0.664
0.771
0.846
0.672
��4.0�m
0.594
0.635
0.594
0.581
0.592
0.639
0.633
0.675
0.817
0.640
��4.2�m
0.576
0.616
0.576
0.566
0.574
0.612
0.608
0.642
0.786
0.613
Table 6: Single scattering albedo for di�erent size distributions and six wavelengths from
SEVIRI channel IR 3.8.
Size Dist.
HP1
HP1ST
HP2
HP3
HP4
HP5
HP6
HP7
HP7ST
HP8
��3.4�m
��3.5�m
��3.6�m
0.943
0.923
0.943
0.945
0.943
0.937
0.938
0.933
0.856
0.937
0.940
0.905
0.940
0.945
0.941
0.924
0.926
0.911
0.796
0.923
0.925
0.884
0.925
0.943
0.927
0.895
0.899
0.873
0.759
0.895
g
��3.8�m
��4.0�m
��4.2�m
0.914
0.874
0.914
0.926
0.916
0.879
0.883
0.855
0.749
0.878
0.930
0.890
0.930
0.939
0.932
0.902
0.905
0.881
0.767
0.901
0.941
0.902
0.941
0.948
0.943
0.919
0.922
0.902
0.785
0.918
Table 7: Asymmetry parameter for di�erent size distributions and six wavelengths from
SEVIRI channel IR 3.8.
29
Section 3: Scattering Properties of Ice Clouds
Size Dist.
HP1
HP1ST
HP2
HP3
HP4
HP5
HP6
HP7
HP7ST
HP8
!0
��8.3�m ��8.5�m ��8.7�m ��8.9�m ��9.1�m
0.561
0.598
0.560
0.552
0.558
0.588
0.585
0.612
0.753
0.589
0.560
0.597
0.560
0.552
0.558
0.588
0.584
0.611
0.753
0.588
0.560
0.597
0.560
0.552
0.558
0.588
0.584
0.612
0.754
0.589
0.560
0.597
0.559
0.551
0.557
0.587
0.584
0.611
0.754
0.588
0.558
0.596
0.558
0.550
0.556
0.586
0.582
0.609
0.751
0.587
Table 8: Single scattering albedo for di�erent size distributions and �ve wavelengths from
SEVIRI channel IR 8.7.
asymmetry parameters g are given in Tabs. 8 and 9, fourth column. The single scattering
albedo is even lower and the asymmetry parameter is larger than at ��3.8 �m. Again,
di�erences among the eight size distributions occur, however, they are not as pronounced
as in the previous case. Typical values for g are 0.93, for !0 0.56.
Concerning the e�ect of particles smaller than 20 �m, the conclusions drawn from the
calculations at ��3.8 �m are con�rmed.
Summarizing, we strongly recommend to apply size distributions which include the full
spectrum of ice crystal sizes though small particles are di�cult to be detected. The uncertainty in estimating asymmetry parameters and single scattering albedoes for a speci�c
cloud due to the unknown size distribution certainly introduces errors, however, using
optical parameters calculated from uncomplete measurements of the size spectrum will
introduce even larger errors.
3.6 Wavelength Dependence in SEVIRI Channels
To investigate the variability of optical properties of cirrus clouds, we have calculated phase
functions p���, single scattering albedoes !0 and asymmetry parameters g for di�erent
wavelengths within selected channels. This answers the question about the number of
wavelengths for which optical properties should be provided for the radiative transfer
calculation. In this section the 3.8- and 8.7-�m-channels are discussed in more detail.
30
Section 3: Scattering Properties of Ice Clouds
Size Dist.
HP1
HP1ST
HP2
HP3
HP4
HP5
HP6
HP7
HP7ST
HP8
g
��8.3�m ��8.5�m ��8.7�m ��8.9�m ��9.1�m
0.947
0.905
0.947
0.953
0.948
0.927
0.929
0.911
0.780
0.926
0.948
0.907
0.948
0.954
0.949
0.928
0.930
0.913
0.782
0.927
0.948
0.907
0.949
0.955
0.950
0.929
0.931
0.913
0.782
0.928
0.949
0.908
0.949
0.955
0.950
0.929
0.923
0.914
0.783
0.929
0.950
0.909
0.950
0.956
0.951
0.931
0.933
0.916
0.784
0.930
Table 9: Asymmetry parameter for di�erent size distributions and �ve wavelengths from
SEVIRI channel IR 8.7.
A �rst indication of the spectral variability can be derived from the wavelength dependence
of the refractive index within the SEVIRI channels. Figs. 13 and 14 show the real nr and
imaginary part ni of the refractive index, respectively. The spectral windows of the SEVIRI
channels mentioned above are indicated by the dashed lines.
It can be clearly seen that the variation of both, nr and ni , is signi�cant in the 3.8 �m
region whereas it is much less pronounced around 8.7 �m. Accordingly, !0 and g also vary
with the spectral band of the SEVIRI channels.
Let us �rst consider the asymmetry parameter and the single scattering albedo.
In Tabs. 6 and 7 g and !0 for six wavelengths within this channel are shown. The wavelengths correspond to the lower limit of the channel �3.4 �m�, the central wavelength �3.8
�m� and the upper limit �4.2 �m�. The additional wavelengths 3.5 �m, 3.6 �m and 4.0
�m have been included to allow for the strong variation of the spectral properties in the
3.7-�m-window.
The corresponding numbers for the IR 8.7-channel are given in Tabs. 8 and 9. For this
channel, �ve wavelengths su�ce.
Tab. 6 reveals that the !0 varies considerably within the spectral band of the SEVIRI
�lter. This is caused by the wavelength dependence of the refractive index in this spectral
range which shows a non-monotonic behaviour �cf. Tab. 2�. There is also a variability
with the size distribution. This is a consequence of the di�erent relative contributions of
Section 3: Scattering Properties of Ice Clouds
Figure 13: Real part of the refractive index as a function of wavelength
Figure 14: Imaginary part of the refractive index as a function of wavelength
31
32
Section 3: Scattering Properties of Ice Clouds
Figure 15: Phase function vs. scattering angle � for several wavelengths between 3.4 �m
and 4.2 �m and size distribution HP1ST
small and large particles which exhibit di�erent absorption characteristics caused by the
photon path lengths in the crystals.
As expected from the variability of the asymmetry parameter, the phase functions also
show a wavelength dependence.
The wavelength dependence of p��� in the 3.8-�m-channel is shown in Figs. 15 and 16:
The �rst corresponds to size distribution HP1ST, the latter to HP7ST.
It can be seen that the phase functions are quite similar. The main di�erences are the
missing of haloes at 3.4 �m which is caused by the strong absorption, and the shift of the
haloes with wavelength. The latter is caused by the changing refractive index of ice which
in turn changes the refraction angles �Snell's law� of the photon paths inside the crystal
and hence the position of the haloes.
As a consequence of these �ndings, it seems to be necessary to use a certain number
of di�erent phase functions �or g� and di�erent !0 within the 3.8-�m-channel and to
interpolate for the wavelength steps used in the radiance sampling method.
Similar results are found for 8.7 �m �see Figs. 17 and 18, however, less pronounced than in
the previous case. The reason is obvious from inspection of nr ��� and ni ��� which do not
change as strong as in the 3.8 �m region. Nonetheless, di�erent phase functions should
Section 3: Scattering Properties of Ice Clouds
33
Figure 16: Same as Fig. 15, but size distribution HP7ST
Figure 17: Phase function vs. scattering angle � for several wavelengths between 8.3 �m
and 9.1 �m and size distribution HP1ST
34
Section 3: Scattering Properties of Ice Clouds
Figure 18: Same as Fig. 17, but size distribution HP7ST
be considered in this SEVIRI channel. We feel that �ve wavelengths as indicated in the
�gures are su�cient to account for the moderate wavelength dependence of the scattering
properties.
Section 4: Comparison with Mie-Calculations
35
4 Comparison with Mie-Calculations
Calculations of optical properties considering non-spherical particles require computationally expensive methods such as the ray-tracing approach which was used in this study.
Application of the Lorenz-Mie-theory � which is equivalent to the assumption of spherical
particles � reduces the amount of computer resources considerably however, it is expected
that non-negligible errors are introduced. This is especially true for short wavelengths,
whereas in the far infra-red the errors should be less signi�cant because the particles are
smaller with respect to the wavelength.
We have investigated the e�ect by performing �equivalent" Mie calculations for comparison. Two aspects must be considered in this context: an equivalent size and an equivalent
size distribution must be found. We have assumed that spheres of equal cross sections best
suit for a comparison. The size distribution has been approximated by the corresponding
eight individual sizes that have been used in the KRIPO-calculations. The advantage of
using the same � discrete � size distribution for the comparison has been rated higher than
the smoothness of the phase function which is only provided when a quasi continuous size
distribution is assumed.
Again, we focus on wavelengths 0.635 �m, 3.8 �m and 8.7 �m.
Analogously to Sec. 3 we have calculated phase functions of individual particles and of
size distributions. For the sake of brevity only comparisons between phase functions
determined from geometrical optics and Mie-theory are discussed in this Section in detail.
The comparison of Mie- and KRIPO-calculations for wavelength 0.635 �m are shown in
Figs. 19 and 20 for size distributions HP1ST and HP7ST, respectively.
It is obvious that signi�cant di�erences occur. In the case of spherical particles the haloes
disappear and the sideward scattering is lower. On the other hand the forward scattering
peak is of the same order of magnitude because it is dominated by di�raction. Note that
the oscillations of the Mie-phase function are caused by the small number of particle sizes
considered.
This result is valid for all size distributions used in this study � two are shown in the
previous �gures. The corresponding comparison for 3.8 �m is shown in Figs. 21 and 22.
Here, the di�erences between sphericity and non-sphericity are much less pronounced in the
phase function. The general shape of p��� the phase functions is quite the same. However,
no haloes are present and scattering at angles between 90� and 140� is again reduced but
less signi�cant than in the visible part of the spectrum. Furthermore, backward scattering
is reduced but only for angles very close to 180� .
36
Section 4: Comparison with Mie-Calculations
Figure 19: Comparison of phase functions of spherical �dashed line� and non-spherical
particles �full line�; size distribtion is HP1ST, ��0.635 �m.
Figure 20: Same as Fig. 19, but size distribtion HP7ST
Section 4: Comparison with Mie-Calculations
37
Figure 21: Comparison of phase functions of spherical �dashed line� and non-spherical
particles �full line�; size distribtion is HP1ST, ��3.8 �m.
Figure 22: Same as Fig. 21, but size distribtion HP7ST
38
Section 4: Comparison with Mie-Calculations
Figure 23: Comparison of phase functions of spherical �dashed line� and non-spherical
particles �full line�; size distribtion is HP1ST, ��8.7 �m.
For even longer wavelengths the di�erences between the two phase functions remain small.
The results for the third SEVIRI channel of interest � 8.7 �m � are shown in Figs. 23 and
24. The most signi�cant di�erence is the reduced scattering between ��50� and ��150�
and the missing backward peak.
Single scattering albedo and asymmetry parameter for size distributions assuming spherical particles are given in Tabs. 10 and 11. The values derived from the ray-tracing calculations are also included for comparison �taken from Tabs. 6 - 9�.
The di�erences of the asymmetry parameters in general are small; even the sign changes
with wavelength. The largest e�ect is visible for HP7ST at 8.7 �m which is consistent
with the previous discussion of Fig. 24. Here, Mie calculations result in an overestimate of
g by approximately 0.08. The extent to which g�0.78 and g�0.86 � equivalent to mean
scattering angles of 39� and 31� , respectively � will a�ect upward radiances is discussed
in Sec. 5.3.2.
It is emphasized that both g and !0 change simulataneously. Note that there are wavelengths and size distributions where an increase of g and !0 coincides, but there are also
cases where g increases while !0 decreases. Thus, one should beware of simple predictions
of the radiative e�ect.
39
Section 4: Comparison with Mie-Calculations
Figure 24: Same as Fig. 23, but size distribtion HP7ST
�
HP1ST
Mie KRIPO
0.635 �m 0.879 0.791
3.4 �m 0.926 0.923
3.5 �m 0.891 0.905
3.6 �m 0.877 0.884
3.8 �m 0.884 0.874
4.0 �m 0.895 0.890
4.2 �m 0.913 0.902
8.3 �m 0.928 0.905
8.7 �m 0.936 0.907
9.1 �m 0.942 0.909
HP7ST
Mie KRIPO
0.852 0.762
0.861 0.856
0.728 0.796
0.694 0.759
0.739 0.749
0.797 0.767
0.837 0.785
0.857 0.780
0.862 0.782
0.868 0.781
Table 10: Asymmetry parameter for size distribution HP1ST and HP7ST: Mie and KRIPO
calculations.
40
Section 4: Comparison with Mie-Calculations
�
HP1ST
Mie KRIPO
0.635 �m 1.000 1.000
3.4 �m 0.556 0.567
3.5 �m 0.581 0.596
3.6 �m 0.618 0.632
3.8 �m 0.664 0.659
4.0 �m 0.644 0.635
4.2 �m 0.631 0.616
8.3 �m 0.578 0.598
8.7 �m 0.576 0.597
9.1 �m 0.569 0.596
HP7ST
Mie KRIPO
1.000 1.000
0.603 0.629
0.684 0.733
0.779 0.809
0.851 0.846
0.841 0.817
0.827 0.786
0.734 0.753
0.714 0.754
0.686 0.751
Table 11: Single scattering albedo for size distribution HP1ST and HP7ST: Mie and
KRIPO calculations.
In conclusion it can be said that the di�erences in the angular distribution of scattered
radiation are signi�cant in the visible spectral range but decrease with wavelength. For
SEVIRI channels IR 3.8 and IR 8.7 the di�erences are so small that errors in radiative
transfer calculations are expected to be negligible in the infra-red region when Mie phase
functions are applied a lieu of phase functions accounting for non-sphericity.
Section 5: Radiative Transfer Calculations �IR Channels�
41
5 Radiative Transfer Calculations �IR-Channels�
The radiance sampling method �Tjemkes and Schmetz, 1997� is to be used for the simulation of SEVIRI measurements. While the original version of the method is designed
to handle absorption and emission but no scattering it must be updated with a multiple
scattering module that accepts optical and thermodynamic properties of the atmosphere
as pre-calculated by the radiance sampling method and the ray tracing technique. The required output from the multiple scattering module is a set of monochromatic radiances at
the top of the atmosphere for speci�ed user angles that can be passed back to the radiance
sampling method for convolution with the radiometer's �lter function and calculation of
brightness temperatures within the radiometer's channels.
5.1 Discrete ordinate method
The basis chosen here for calculating radiances in the scattering atmosphere is the discrete
ordinates method �DOM� as described by Stamnes et al. �1988�. The DOM is derived
from the radiative transfer equation �RTE� which is decomposed into Fourier components.
The ordinate �cosine of the zenith angle� is discretized using a double Gau� quadrature
and the vertically inhomogeneous atmosphere is subdivided into a set of homogeneous
layers, thus, giving a set of 2N coupled ordinary di�erential equations with constant
coe�cients where 2N is the number of discrete ordinates. The homogeneous equation
system is solved by �nding eigenvalues and eigenvectors numerically. Subsequently, the
order of the eigenvalue problem is halved by transformation of the equation system with
the square of the eigenvalues and particular solutions are found for thermal and solar
sources. Finally, the boundary conditions and atmospheric layers are incorporated. The
combination of layers leads to an equation system of the order �2N � L� � �2N � L�
where L is the number of layers. A scaling transformation is applied to remove positive
exponentials from the solution in order to avoid numerical ill-conditioning. The advantages
of DOM are: The solution of the RTE does not depend on the optical thickness, thus,
the computing time is independent of the optical thickness. The solution is numerically
stable. A disadvantage of the method is the di�cult to discretize ordinate in the case of
strongly peaked or strongly varying phase functions. Therefore, the Delta-M-method of
Wiscombe �1977� is incorporated in the model �cf. Sec. 5.2�.
The vertical coordinate of the model is the optical depth which can be layered arbitrarily.
Hence, the DOM program code can easily be incorporated in the radiance sampling method
to be used for the calculation of the gaseous absorption at infrared wavelengths. For each
wavenumber chosen by the radiance sampling method the DOM has to be run once for the
42
Section 5: Radiative Transfer Calculations �IR Channels�
entire atmosphere. The number of quadrature angles to be used depends on the number
of Legendre polynomials chosen to expand the phase function and must not be smaller
than the latter. For a good �t of the sharply peaked phase function of ice crystals to be
described here the number of Legendre polynomials must be at least 50 even when using
the Delta-M method.
Besides the optical depths the DOM requires the single scattering albedo, the phase function expressed as coe�cients of the Legendre polynomial expansion and temperature at
each level. Further input is required if solar radiation is considered. In this case the intensity of the incident parallel beam at the top boundary must be entered together with
the cosine of the zenith �or polar� angle of the incident beam and the azimuth angle. The
output angles are independent from the computational zenith angles and can be freely
chosen. The DOM returns radiances at the speci�ed user angles and user levels. Here, the
upwelling radiances at the top of the atmosphere is of concern only. The monochromatic
radiances are passed back to the radiance sampling program where they are convolved
with the radiometer's �lter functions.
5.2 Expansion of the phase function
The phase function p��� �or p�cos ��� is introduced into the radiative transfer equation in
terms of coe�cients �i of a series of Legendre polynomials Pi .
p�cos �� �
N
X
i�0
�2i + 1� �i Pi �cos ��
�7�
The Legendre polynomials Pi �x� can be calculated recursively according to
�
�
�
1 � P �x�
Pk+2 �x� � 2kk++23 x Pk+1 �x� , kk +
+2 k
with P0 � 1 and P1 � x.
The coe�cients �i are determined via
�i � 12
Z�
0
p�cos �� Pi �cos ��d�cos ��
In general, the number of coe�cients N in Eq. �7� to accurately describe a phase function
increases with increasing height of the forward scattering peak. Therefore, cirrus phase
functions, where p�0� � 100 quite frequently occur, require N of at least 1000. As a
consequence, memory and time resources of radiative transfer calculations are extremely
high and thus not very e�ective.
Section 5: Radiative Transfer Calculations �IR Channels�
43
Figure 25: Phase function of size distribution HP1ST for 8.7 �m �full line� and reconstructed from a Legendre expansion with 1000 coe�cients
As an example, the HP1ST phase function has been expanded in a Legendre series and
reconstructed using N �1000 or N �1800 coe�cients; the results in comparison to the
original phase function are shown in Figs. 25 and 26.
It can be seen that applying 1800 coe�cients almost perfectly represents the original p���,
N �1000 is still acceptable. The reason for this large number of terms is the magnitude
of the forward scattering peak which is in the range of 200 to 600 sr,1 as has been shown
in Sec. 3.4. For shorter wavelengths even larger N are required because of the increasing
di�raction peak.
This problem can be solved by applying the so called `Delta-Approximation' �Potter,
1970�. If most photons are scattered in forward direction scattered and unscattered photons are redistributed according to the idea that photons which are scattered exactly in
forward direction or under very small scattering angles �� � �0 � cannot be distinguished
from unscattered photons. Thus, forward scattered photons are added to the unscattered
photons while � for compensation � the extinction coe�cient and the optical depth are
reduced accordingly, and the phase function is substituted by a `truncated' one without
the extreme forward scattering peak of the original phase function. As a consequence, the
new `truncated' phase function requires a much smaller number of Legendre coe�cients
and the radiative transfer calculation becomes much more e�ective.
44
Section 5: Radiative Transfer Calculations �IR Channels�
Figure 26: Phase function of size distribution HP1ST for 8.7 �m �full line� and reconstructed from a Legendre expansion with 1800 coe�cients
The truncation of the phase function is arbitrary to a certain degree. To avoid this ambiguity Wiscombe �1977� proposed a special version of this approach called the 'Delta-MMethod'. Here, the truncated fraction of the phase function and the �remaining� number
M � N of Legendre coe�cients are related. In summary, the basic equations of the
Delta-M-Method read as
� 0 � �1 , !0 f ��
for the transformation of the optical depth,
!00 � 1 1,,! f f !0
0
for the single scattering albedo and
p0 �cos �� � 2f��1 , cos �� + �1 , f �
MX
,1
i�0
�2i + 1��i0 Pi �cos ��
for the `more di�use' �truncated� phase function p0 that substitutes the original p. The
advantage of p0 over p is that p0 does not require such a large number of Legendre coe�cients.
Section 5: Radiative Transfer Calculations �IR Channels�
45
Figure 27: Phase function of size distribution HP1ST for 8.7 �m �dashed line� and the
reconstructed �full line, Delta-M-Method, M �100�.
The truncated fraction of the phase function f is set to
f � �M
�8�
and the coe�cients �i0 are related to the previous ones according to
�i0 � �1i ,, ff
In contrast to the conventional Delta-Approximation the `Delta-M Method' does not explicitely require a �0 . By choosing an upper limit of the Legendre expansion, M , e.g.
M �50, factor f is determined `automatically' via Eq. 8.
Figs. 27 and 28 show the results when M �100 and M �50 are chosen, respectively. Though
100 coe�cients result in a much better retrieval of the original phase function we feel that
50 coe�cients are still su�cient. A corresponding sensitivity study is brei�y discussed in
Sec. 5.3.
It should be emphasized that in the visible spectral range M �100 will not be su�cient to
describe cirrus phase functions.
46
Section 5: Radiative Transfer Calculations �IR Channels�
Figure 28: Phase function of size distribution HP1ST for 8.7 �m �dashed line� and the
reconstructed �full line, Delta-M-Method, M �50�.
5.3 Radiances at Top of Atmosphere
The e�ect of cirrus clouds on the radiation �eld at the top of the atmosphere in the SEVIRI
channels is calculated after having completed the provision of input data �mainly phase
functions� and the modi�cation of the numerical tools as described in Sec. 5.1. The focus
is put again on channels IR 3.8 and IR 8.7. If not otherwise stated the zenith angle of
observation is assumed to be 0� .
Calculations are made for di�erent phase functions �i.e., size distributions�, di�erent optical depths of the clouds, di�erent cloud heights and di�erent climatological temperatureand water vapour pro�les �from the tropics to the mid-latitudes�. Furthermore, we have
studied the e�ect of the satellite's viewing angle for a limited number of cases.
In all calculations following we have expressed radiances in terms of brightness temperature
TB .
5.3.1 Dependence on Cloud Optical Depth
The �rst examples concern the e�ect of the cirrus optical depth �c on brightness temperature. Fig. 29 shows the change of brightness temperature �TB at the top of the atmosphere
Section 5: Radiative Transfer Calculations �IR Channels�
47
Figure 29: Change of brightness temperature with �c �3:8�m�; size distribution HP1ST,
��3.8 �m �dashed line�, ��8.7 �m �full line�.
de�ned as
�TB � TB ��c � , TB ��c � 0�
for 0.01 � �c � 10. A vertical temperature and humidity pro�le from 50.0� N, 20.2� E
�TIGR pro�le �469� has been taken. The cloud height is set to model-level �15, i.e.,
between 300 and 250 hPa �approximately 9.6-11 km�.
From Fig. 29 it can be seen that the absolute value of �TB increases with optical depth.
The optical depth shown is �c at 3.8 �m. For the purpose of this plot it can also be taken
for ��8.7�m � the relative di�erences of the optical depth at both wavelength is between
2 and 3 �; only slightly depending on the size distribution.
The larger �c, the lower is the brightness temperature TB , i.e, the stronger is the cloud
e�ect. For 8.7 �m the cloud e�ect is larger than for 3.8 �m; for an optical depth of
�c � 0:1, TB decreases by 1.48 K or 0.84 K, respectively, relative to the no-cloud reference
case.
For subvisible cirrus clouds with optical depth as low as 0.01, �TB is 0.15 K for 8.7 �m
and 0.08 K for 3.8 �m.
48
Section 5: Radiative Transfer Calculations �IR Channels�
Size Dist.
HP1
HP1ST
HP7
HP7ST
ST
reff
91.7
78.0
30.7
11.9
5.0
Table 12: E�ective radii for di�erent size distributions.
5.3.2 Dependence on Size Distribution
The e�ect of the size distribution on �TB is relatively small. Two examples are shown
here.
In Fig. 30 the change in brightness temperature is shown for �ve size distributions: two
realisitic including the full spectrum of small and large particles, �HP1ST and HP7ST�,
two without small particles below 20 �m �HP1 and HP7� and the size distribution with
only small particles �ST�. The wavelength is ��3.8 �m.
For illustration, the e�ective radii of the di�erent size distributions are displayed in Tab. 12.
In particular, size distributions ST consists of very small particles typical rather for contrails than for �natural" cirrus clouds. That was the reason why we did not account for
the ST-size distribution in the previous sections.
If size distributions HP1, HP1ST and HP7 are considered, di�erences are below 0.2 K as
long as �c � 0.2. For �c�1 the di�erences are still only approximately 0.7 K, and maximum
deviations occur at �c �2 with 1.2 K.
Including size distributions with very small e�ective radii �HP7ST and in particular ST�
the sensitivity of the irradiance to the microphysics is increasing. For optical depths of
�c�0.1, the variability is about 0.3 K, for �c �1 it is up to about 3 K, and more than 6 K
for �c�2.
In Fig. 31 the corresponding results for ��8.7 �m are plotted. The same conclusions hold.
Note, that in both �gures the cloud e�ect is interpolated between �c�2 and �c �10.
An interesting question already raised in Sec. 4 is the error of the radiance at the top of
the atmosphere when Mie-phase functions are applied instead of more realistic ones. The
error �TB is de�ned as
�TB � �TB�M � ��c� , �TB�K ���c�
Section 5: Radiative Transfer Calculations �IR Channels�
49
Figure 30: Change of brightness temperature as a function of cloud optical depth for
di�erent size distributions as indicated; ��3.8 �m.
Figure 31: Same as Fig. 31, but ��8.7 �m.
50
Section 5: Radiative Transfer Calculations �IR Channels�
Figure 32: Error in brightness temperature as a function of �c when Mie-phase functions
are used; ��3.8 �m �see text�.
The superscripts �K � and �M � mark brightness temperature di�erences calculated assuming KRIPO- and Mie-phase functions, respectively. Negative �TB means an underestimate
of the brightness temperature in the case of spherical particles, i.e., the reduction of TB
by the cloud is overestimated.
From Figs. 32 and 33 it can be seen that the error is small, but not negligible. For ��3.8
�m, �TB is as large as 0.22 K �for �c � 2�, for 8.7 �m it changes sign and the absolute
values reach 0.14 K. The e�ect of di�erent crystal size distributions �considering the full
spectrum of small and large crystals� is so small in comparison that it can hardly be
resolved in the �gures. That means that the errors caused by the assumption of spherical
particles are much larger than the errors which might occur if a wrong size distribution of
a natural cirrus cloud �e.g., HP1ST instead of HP7ST� is used. This conclusion might be
changed if contrails are considered.
5.3.3 Dependence on Cloud Height
The e�ect of the cloud height is shown in Fig. 34 ���3.8 �m� and Fig. 35 ���8.7 �m�.
The optical depth of the cloud is �c�1 in both cases. The horizontal lines show the layers
of the atmospheric model �TIGR �194, i.e., 1.2� N latitude, 172.6� E longitude�.
Section 5: Radiative Transfer Calculations �IR Channels�
51
Figure 33: Same as Fig. 32, but ��8.7 �m.
The cloud is shifted from layer �17 �200-150 hPa� to �20 �115-100 hPa�. It can be seen
that the upward radiance remains almost unchanged in SEVIRI channel IR 3.8, but not
in IR 8.7. At 3.8 �m, the cloud e�ect �TB is almost constant with height: �10.04 K if the
cloud is in layer �17; �10.13 K for layer �20. For 8.7 �m however �TB changes by almost
one degree, i.e., from �18.24 K to �19.32 K �layer �17 to �20�.
5.3.4 Dependence on Satellite Zenith Angle
The dependence of the radiances at the top of the atmosphere from the zenith angle of
observation was investigated between �s�0� and 60� . Results are shown in Fig. 36. A
cirrus cloud is assumed in layer �15 with an optical depth of 0.1 �TIGR pro�le �469�.
�TB is plotted for ��3.8 �m and ��8.7 �m. It is obvious that the in�uence of the cloud
increases as the zenith angle increases. This is caused by the longer photon paths through
the clouds.
5.3.5 In�uence of Phase Function Expansion
We have also investigated whether 50 terms of the Legendre expansion are su�cient for
describing the scattering properties of the cloud. For this purpose few runs are made with
M �100.
52
Section 5: Radiative Transfer Calculations �IR Channels�
Figure 34: Vertical pro�le of brightness temperature for di�erent cloud heights; ��3.8 �m.
Figure 35: Vertical pro�le of brightness temperature for di�erent cloud heights; ��8.7 �m.
Section 5: Radiative Transfer Calculations �IR Channels�
53
Figure 36: Change of brightness temperature with satellite zenith angle; �c�0.1.
The computer time required for one run �one optical depth� is approximately 55 minutes
on a DEC-Alpha-Station �128 MB RAM, 233 MHz� when the phase functions is approximated by 50 Legendre coe�cients. If M is set to 100, an Alpha-Station with 512 MB
RAM and 300 MHz must be used and though the computer time increases by a factor of
approximately 3.
Typical radiative transfer calculations have shown that increasing the computer resources
to consider more Legendre-terms is not necessary. The changes in the brightness temperature are virtually zero.
5.4 Justi�cation of scattering
The strong contribution of forward scattering to the totally scattered radiation and the
impossibility to distinguish photons scattered in forward direction from unscattered ones
that are transmitted in the same direction might suggest to ignore the scattering completely. This is tested by comparing calculations accounting for full scattering with those
where the optical depth is reduced by that of the scattering and where only absorption
and emission is considered. The optical depth has been �xed to 0.1, the cloud is inserted
in a topical atmosphere near 100 hPa.
The test calculations show that scattering becomes increasingly important when the single
54
Section 5: Radiative Transfer Calculations �IR Channels�
scattering albedo increases. At !0 � 0:6 the brightness temperature di�erences between
scattering and non-scattering case at the top of the atmosphere are less than �TB �
,0:08 K. For !0 � 0:7, 0.8, 0.955 we have maximum di�erences of �TB ��0.13, �0.84,
and �7.5 K, respectively. Hence, even in cases with strong forward scattering the scattering
cannot be ignored when !0 exceeds values of about 0.6.
5.5 Discussion
Brightness temperatures have been calculated for two SEVIRI channels in the infrared. It
was found that the decrease of TB in the presence of a cloud is smaller at 3.8 �m than at
8.7 �m. The reason is primarily based on the higher single scattering albedo and lower
imaginary part of the refractive index �lower absorption�emission�, respectively, at 3.8
�m.
TB are virtually the same if size distributions with reff � 30 �m are considered. Largest
� but still small � e�ects occur at moderate optical depths around 2. For very large
�c the di�erences become smaller again as the brightness temperature approaches the
thermodynamical temperature at the cloud's altitude.
In case of anthropogenic cirrus clouds �contrails� the size distributions are shifted signi�cantly towards smaller particles. Only such drastical changes lead to a clearly visible e�ect
in the brightness temperatures.
Summarizing, if size distributions measured by Heyms�eld and Platt and extended by
measurements of Strauss are accepted to be typical for natural cirrus clouds, the e�ect
of size distributions on TB is small. This is in part caused by the similarity of the size
distributions, i.e. they all cover the full range of sizes from a few micrometers to some
millimeters. Another reason for the small di�erences is the opposite change of g and !0
when � for example � size distributions HP1ST and HP7ST are considered �see Tabs. 10
and 11�. In the spectral region between 3.4 �m and 4.2 �m, g decreases from about 0.90 to
0.78 while !0 increases from 0.62 to 0.76. Both e�ects tend to cancel out each other: the
increase of !0 leads to a lower emission of the cloud and thus to a stronger contribution
from atmospheric layers and the surface. Hence, there is a tendency to increase TB . On the
other hand the decrease of the asymmetry parameter results in a stronger backscattering
of radiation coming from below the cloud so that there is a tendency of blocking the
radiation from the warm �in comparison with the cloud� surface.
Both e�ects act non-linearity so that a quantitative estimate of the overall e�ect "by eye"
is not possible. The radiative transfer calculations however have demonstrated that the
overall e�ect indeed is small.
Section 5: Radiative Transfer Calculations �IR Channels�
55
We have also compared our results with calculations performed by Betancor-Gothe and
Gra�l �1993�. They have investigated the radiative in�uence of thin cirrus clouds and
contrails in the 3.8 �m spectral region. Their study focussed on contrails so that they
selected phase functions C1 and C5 which only consists of particles less than 20 �m
and 40 �m, respectively. Furthermore, they used a HP size distribution similar to our
assumptions. Phase functions determined from Mie theory and a correction scheme were
used.
The e�ective radii of the three phase functions �C1, C5, HP� were 4 �m, 6 �m and 100
�m, respectively. Thus, comparisons with our ST- and HP1-size distribution are possible.
The general behaviour of their and our TB ��c �-curves is the same. In particular, there is
a clear separation between the size distributions with very small and large e�ective radii.
Furthermore, both studies show a smaller e�ect on TB in case of small particles �contrails�.
Calculating �TB ��c � 2� from Betancor-Gothe and Gra�l's paper we get values of approximately �18 K for their HP cloud, and �10 K for the C1 or C5 contrail. These values
agree very well with the corresponding values of our calculations �16.7 K and �10.9 K, respectively. Moreover, the decrease of the brightness temperatures relative to the no-cloud
reference case is in both studies approximately �32 K for �c �3.5 �interpolated value�.
Larger optical depths were not investigated by Betancor-Gothe and Gra�l �1993�.
56
Section 6: Radiative Transfer Calculations �VIS Channel�
6 Radiative Transfer Calculations �VIS-Channel�
We have also calculated the e�ect of cirrus clouds in the VIS 0.6 SEVIRI channel. The
concept resembles in many aspects the calculations of Sec. 5 for the IR spectral region,
however, another radiative transfer code was used. It is brie�y described in the following
section.
6.1 Method of Successive Order of Scattering
The radiative transfer calculations for wavelength ��0.635 �m were performed with a
successive order of scattering �SOS� model.
The SOS is based on the monochromatic radiative transfer equation including multiple
scattering. The radiances are expanded in a series of Fourier coe�cients � a typical
number is 20. For each coe�cient the radiative transfer equation is solved, separately
for the upward and downward looking hemispheres. The phase function is treated in the
same way as described in the previous section, i.e, it is expanded in a Legendre polynomial
series. Finally, a set of coupled di�erential equations is obtained which is solved iteratively.
As a consequence, the contribution of each scattering order to the radiance �eld can be
determined. Thus, each step of the SOS model can be attributed to a physical meaning.
The iteration is stopped when the di�erence between the irradiance and the irradiance of
the previous iteration step is smaller than a prescribed threshold. The convergence of the
iteration slows down with increasing optical depth, however, for � � 1, approximately 10
scattering orders are su�cient.
The SOS is a one-dimensional radiation code with optical depth � as variable. The vertical
layers �i.e., layers of � � are de�ned in terms of height and extinction coe�cients � if
optically thick layers occur they are subdivided into smaller layers. Scattering by air
molecules, aerosols and clouds is considered.
As already mentioned, cirrus phase functions require � caused by the pronounced forward
scattering peak � the application of �a version of� the Delta-approximation. In contrast
to the DOM �IR-calculations, Sec. 5.1� we apply the standard Delta-approximation.
Output of the model are radiances are determined at arbitrary levels of the atmosphere
as a function of azimuth and zenith angle of observation, for a �xed but arbitrary solar
zenith angle.
Section 6: Radiative Transfer Calculations �VIS Channel�
57
6.2 Radiances at the Top of the Atmosphere
Calculations of the radiance �eld at the top of the atmosphere were performed for
��0.635�m. The surface albedo was set to 15�. In the troposphere we assume continental aerosols, in the stratosphere background aerosols. The solar zenith angle �0 is set
to 30� or 60� .
31 coe�cients were considered for the radiance expansion, 181 for the truncated phase
function.
Results are expressed in terms of planetary albedo �pl . This is a more convenient measure
than radiances if the radiative impact of di�erent clouds on the radiation �eld should
be compared. On the other hand they can easily be transformed into irradiances and
radiances.
The conversion of planetary albedo to upward directed spectral irradiances S at the top
of the atmosphere is simply performed by
S � �0 �pl So
with So being the extraterrestial spectral solar radiation at the selected wavelength and �o
the cosine of the solar zenith angle. At ��0.635 �m So equals 1658 W m,2 �m,1 �WMO,
1986�.
Spectral radiances L are determined by division through � � provided the radiation �eld
is isotropic. Thus, Liso is proportional to the planetary albedo.
Liso � �o ��pl So
Though this is not exactly true in the case of cirrus clouds �as will be seen below� this
approximation is quite useful to describe cloud e�ects and their sensitivity to microphysical
properties. The reason is that the anisotropy function � ��; '� �bidirectional re�ection
distribution function; � and ' being the zenith angle and azimuth angles of observation,
respectively� does not change so much for a given viewing geometry in comparison to the
planetary albedo.
Tab. 13 gives the planetary albedo �pl for ��0.635 and a surface albedo of 15�. The cloud
base height is 9.5 km, cloud top height 11 km. Listed are values for di�erent cloud optical
depths and phase functions: four size distributions �namely HP1ST and HP7ST and their
�counterparts" without small particles� are considered and two particle shapes �hexagonal
columns, KRIPO; and spheres, Mie�.
58
Section 6: Radiative Transfer Calculations �VIS Channel�
�c
Mie-Theory
KRIPO �GOA�
HP1 HP1ST HP7 HP7ST HP1 HP1ST HP7 HP7ST
0.
16.8
0.05 16.9 17.0 16.8 17.0 17.1 17.1 17.1 17.2
0.1 17.0 17.1 17.0 17.2 17.4 17.5 17.5 17.5
0.5 18.0 18.2 18.3 19.2 19.9 20.0 20.3 20.4
1.0 19.5 19.8 19.9 21.8 23.1 23.3 23.9 24.2
Table 13: Planetary albedo in percent for a cirrus cloud �9.5 � 11 km�. ��0.635 �m,
�0 �30� .
In the cloudfree case, a planetary albedo of 16.8 � is derived. This corresponds to a
spectral radiance of 76 W m,2 sr,1 �m,1 .
It can be seen that the planetary albedo increases with cloud optical depth for all phase
functions. However, the increase is di�erent for di�erent phase functions: in case of
non-spherical particles �KRIPO algorithm� the radiation leaving the atmosphere is generally larger than in the case of spherical particles. The absolute di�erence is 4� for
the HP7-distribution and about 2.5� for HP7ST when the cloud optical depth is 1. A
1�-albedo change is equivalent to a change in �isotropic� spectral radiance of about 4.6
W m,2 sr,1 �m,1.
For thin cirrus clouds ��c � 0.1� the change of the albedo is less than 0.5 �. The variability
with size distribution �HP1ST vs. HP7ST etc.� is relatively small; even for �c�1 it is less
than 1� in case of KRIPO. For spherical particles the di�erences amount up to about 2�.
These �ndings are in accordance with the phase functions shown in Fig. 5 � small variability
of p��� with size distribution � and in Figs. 19 and 20 � large di�erences between Mie- and
KRIPO-phase functions. Furthermore, the enhanced sideward and backward scattering of
the non-spherical particles is obvious which corresponds to the larger planetary albedo.
In case of larger solar zenith angles ��0 �60� � shown in Tab. 14, the planetary albedo is
larger caused by the longer photon paths through the atmosphere, and the increase of �pl
with cloud optical depth is stronger. The cloudfree planetary albedo of 19.3 � corresponds
to an �isotropic� radiance of 51 W m,2 sr,1 �m,1 ; 1�-albedo to a radiance of about 2.6
W m,2 sr,1 �m,1.
Again, the planetary albedo of non-spherical particles is larger than in case of spheres.
That means that using Mie phase functions leads to an underestimate of �pl of 2-4�
��c �1�.
Section 6: Radiative Transfer Calculations �VIS Channel�
59
�c
Mie-Theory
KRIPO �GOA�
HP1 HP1ST HP7 HP7ST HP1 HP1ST HP7 HP7ST
0.
19.3
0.05 19.8 19.8 19.9 20.0 20.1 20.1 20.1 20.2
0.1 20.3 20.3 20.3 20.7 20.8 20.8 20.9 21.1
0.5 24.0 24.2 24.3 26.2 26.5 26.7 27.2 27.5
1.0 28.4 28.7 29.0 32.2 32.6 32.9 33.6 34.2
Table 14: Same as Tab. 13, �0 �60� .
These �ndings in general agree with results � integrated over the solar spetral range �
published by Kinne und Liou �1989�.
As already mentioned the radiance �eld is anisotropic in case of clouds. This anisotropy
must be taken into account when speci�c viewing geometries of satellite, ground pixel and
sun are of interest. The angular dependence of the radiance L��; '� is described by
L��; '� � � ��; '� Liso � � �0 ��pl So
The bidirectional re�ection function � is derived by normalizing the radiances obtained
from the SOS runs.
Typical distributions of � show large values �scattering stronger than in the isotropic
case� for large zenith angles of observation �. This is in particular true for small and
large �forward and backward scattering� azimuth angles. Over sea surfaces a pronounced
sunglint is visible ��o � �; ' � 0�. In case of clouds often a �hot spot" appears at
��o � �; ' � 180�.
We have plotted four typical examples of the anisotropy function � . On one hand we show
the changes of the overall pattern when the solar zenith angle changes, on the other hand
we show that the pattern signi�cantly changes when spherical particles are assumed a lieu
of hexagonal crystals �Mie vs. KRIPO�.
Figs. 37 and 38 show the normalized angular distribution of the radiances for ��30� and
��60� , respectively. The cloud optical depth is set to unity, and size distribution HP1 is
assumed.
Fig. 37 shows that near zenith �� � 0� � the sky appears relatively dark while for large
zenith angles of observation the sky becomes brighter. This is in particular true for
forward scattering with respect to the sun �small '-angles� as expected. In backward
60
Section 6: Radiative Transfer Calculations �VIS Channel�
Figure 37: Anisotropy function � ��; '� for size distribution HP1 �KRIPO model�; �o �30� .
Figure 38: Same as Fig. 37, but �o �60� .
Section 6: Radiative Transfer Calculations �VIS Channel�
61
Figure 39: Anisotropy function � ��; '� for size distribution HP1 �Mie calculations�;
�o �30� .
scattering direction at ��30� the hot spot can clearly be seen. This is a consequence of
the strong backward scattering of ice crystals as has already been shown in �e.g.� Fig. 5.
Similar scattering pattern are visible in Fig. 38 where the results for �o �60� are shown.
In general, the dark and bright regions of the sky appear at similar angles, however, the
�amplitude" of � is larger. The area of enhanced backscattering now has moved to larger
zenith angles according to the position of the sun.
Figs. 39 and 40 show a similar comparison, however, spherical particles have been assumed.
As a �rst approximation, the normalized radiances increase with the zenith angle of observation as has been the case in the KRIPO calculations. Two main di�erences occur:
�rst, a �ring" of large radiances around the position of the anti-point of the sun is visible
which corresponds to the relative maximum of the phase function at � � 135� �cf. e.g.
Fig. 19�. Second, the hot spot has vanished. This is caused by the reduced backscattering
of spheres in comparison with hexagonal columns.
The in�uence of di�erent crystal size distributions, HP1, HP1ST, HP7 and HP7ST, on the
angular distribution has also been investigated. Only one example is illustrated: Fig. 41
is compared to Fig. 37, i.e., HP1 and HP7ST.
It is obvious that the general pattern is very much alike. Di�erences appear in several
62
Section 6: Radiative Transfer Calculations �VIS Channel�
Figure 40: Same as Fig. 39, but �o �60� .
Figure 41: Anisotropy function � ��; '� for size distribution HP7ST �KRIPO model�;
�o �30� .
Section 6: Radiative Transfer Calculations �VIS Channel�
63
small-scale features which however are primarily based on the isoline routine of the plot
program. Same results are found for the other size distributions.
In summary, applying phase functions based on Mie theory introduces signi�cant errors in
the calculated angular distributions of upward radiances at ��0.635 �m. The di�erences
between variuos realistic cirrus particle size distributions however are very small.
64
Section 7: Summary and Conclusions
7 Summary and Conclusions
Radiation scattered by cirrus clouds in�uence radiances measured at the top of the atmosphere. It was the objective of this study to develop numerical tools to describe this
e�ect for three channels �at 0.635 �m and in the infrared spectral region at 3.8 �m and
8.7 �m� of SEVIRI, a radiometer which is planned to be �own as part of METEOSAT
Second Generation.
Since ice clouds consist of non-spherical crystals, application of the Lorentz-Mie theory to
describe scattering is not adequate. Thus, more complicated models are required and the
computations become more expensive.
In a �rst step we have adapted a ray tracing model �KRIPO�, based on the geometric optics
approach and developed by Hess, for the wavelengths of SEVIRI. KRIPO provides optical
properties of individual particles, in particular, phase function and the single scattering
albedo.
The resulting data sets for hexagonal columns serve as input for a second program which
determines phase function, asymmetry parameter g and single scattering albedo !0 for
di�erent size distributions known from literature.
With these two numerical tools the data base required for the radiative transfer calculations was established for several wavelengths within each SEVIRI channel under consideration. This was done to account for the spectral dependence of the optical properties of
ice particles.
The basis of the radiative transfer calculations was the radiance sampling method provided by EUMETSAT which, however, does not consider atmospheric scattering. As a
consequence, a more general model has been developed.
This task was performed by combining the discrete ordinate method and the radiance sampling technique. This approach results in 375 to 750 full multiple scattering calculations
for both SEVIRI channels.
Radiances at the top of the atmosphere were calculated for di�erent cirrus optical depths,
di�erent crystal size distributions, di�erent cloud heights and climatological pro�les of
temperature and water-vapour distributions. The e�ect of di�erent viewing geometries is
also brie�y outlined. Furthermore, the errors introduced by assuming spherical particles
instead of hexagonal crystals were estimated.
The main results for the infrared SEVIRI channels are the following:
� For the 3.8 �m channel, brightness temperatures at the top of the atmosphere are
Section 7: Summary and Conclusions
65
reduced by typically �0.08 K if a cirrus cloud of optical depth �c �0.01 is present,
�0.8 K and �8.2 K, if �c equals 0.1 or 1, respectively.
� For the 8.7 �m channel, brightness temperatures at the top of the atmosphere are
somewhat more reduced. Typical values are �0.15 K for �c �0.01, �1.5 K for �c �0.1
and �13.7 K for �c�1.
� The e�ect of di�erent crystal size distributions on the radiance at the top of the
atmosphere is small in the infrared if natural cirrus clouds are considered. The
inclusion of contrails with a higher concentration of small particles causes greater
derivations.
� Assuming spherical particles instead of hexagonal columns results in errors of approximately 0.2 K in brightness temperature.
For the 0.635 �m channel the main results are
� The e�ect of di�erent size distributions of natural ice clouds on upwelling radiances
is small.
� Application of phase functions derived from Mie theory leads to a signi�cant underestimate of upward radiances and to an erroneous angular distribution of the
radiation �eld.
If wavelengths of the order of 10 �m are considered, the application of the geometric optics
approach is critical because the basic physical concept of this method is violated in the
case of small crystals. Nevertheless, GOA is an acceptable approach as was outlined in
Sec.3.1. Discrete dipole approximation might become an alternative with the advent of
more powerful computers in the near future.
Another computational problem raises from the need to express any phase function by
a series of Legendre polynomials. In the shortwave spectral range a very large number
of coe�cients is required to accurately model the pronounced forward scattering peak.
This causes serious problems even if the delta approximation is applied. For wavelengths
between 4 and 10 �m this problem is diminished.
The e�ect of di�erent size distributions on the radiation �eld at the top of the atmosphere
is � according to our calculations � negligible. Consequently, the retrieval �or a �rst-order
estimate� of size distribution or particle shape seems not to be possible on the basis of the
two SEVIRI channels chosen. However, the discrimination of natural cirrus from contrails
should be feasibile.
66
Section 7: Summary and Conclusions
This conclusion is not in contradiction to other studies. One should not be confused by
recent papers which publish histograms of monthly mean e�ective particle sizes of cirrus
clouds with a resolution of �reff � 4 �m �Han et al., 1996�. These results are based on
AVHRR measurements in the framework of ISCCP and on an inversion scheme originally
developed for water clouds. Data evaluation uses a very simple cirrus cloud detection
algorithm and only �ve di�erent size distributions. Thus, the discrimination of 25 distinct
reff -classes is rather a mathematical consequence than a realistic retrieval of individual
particle sizes. An error analysis is missing in this paper.
We did not investigate the e�ect of particle shapes other than hexagonal crystals and the
e�ect of horizontally aligned crystals �oriented particles�. We do not feel that the �rst
will change our conclusions, however, the latter might be subject to further investigations.
Such investigations would not only require even more complex models for the scattering
characteristics of a particle. In addition, radiative transfer models must be extended
signi�cantly: �rst, the models must be capable to deal with a much larger number of
Legendre coe�cients �see above� to describe each angular pattern of the phase function,
and second, the radiative transfer code must account for two-dimensional phase functions
p��; '�. Present radiative transfer codes cannot treat this problem.
It was beyond the scope of this project to thoroughly investigate the potential of discriminating di�erent cirrus clouds by SEVIRI measurements. For this purpose the full number
of channels should be considered.
We recommend to investigate more deeply the di�erences between natural and anthropogenic ice clouds. Due to their very di�erent size distributions a discrimination might
be possible. A discrimination of di�erent types of natural cirrus clouds as we have done
in this study seems to be critical. On one hand the e�ects on the radiation �eld are small
caused by simultaneous changes of g and !0 . On the other hand spatial inhomogeneities
of a cirrus cloud �eld at the scale of a pixel will lead to an averaging of di�erent cirrus size
distributions which makes it di�cult to invert radiance measurements for microphysical
properties. As a consequence, only mean properties can be derived, and an interpretation
of very small di�erences in the received radiances is not necessary.
8 Acknowledgments
The authors want to thank Dr. Michael Hess for the supply of his program `KRIPO' and
many valuable comments.
Section 9: References
67
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