Journal of Quantitative Spectroscopy &
Radiative Transfer 69 (2001) 635}650
E!ectiveness of the MS-method for computation of the intensity
"eld re#ected by a multi-layer plane-parallel atmosphere
Chiara Levoni , Elsa Cattani , Marco Cervino , Rodolfo Guzzi *,
Walter Di Nicolantonio
Instituto di Scienze dell+ Atmosfera e dell+Oceano, Consiglio Nazionale delle, Ricerche, Via Gobetti 101,
40129, Bologna, Italy
Carlo Gavazzi Space, Milano, Italy
Received 25 October 1999; accepted 8 December 1999
Abstract
A radiative transfer code based on the coupling of the currently labeled MS method (MS refers to the
separation of the multiply and singly scattered radiation), with the reliable and widely used radiative transfer
package DISORT is presented. We show that this code can be used to compute the intensity "eld re#ected by
a plane-parallel, non-emitting, aerosol loaded atmosphere with the same accuracy as a non-approximate
model but maintaining a high computational speed. Results obtained for a two-layer atmosphere show that
the single scattering features are clearly visible in the radiative "eld in the range from moderate to high aerosol
optical thicknesses. Tests carried out in a reasonable range of viewing geometries, restricted to dark surfaces,
and considering a signi"cant set of aerosol optical properties, have shown that the present code is capable of
attaining the same accuracy as DISORT but using a greatly decreased number of angular discretizations
(streams), thereby reducing the computational time by a factor of between 2 and 10 with respect to DISORT,
depending on the complexity of the scenario. 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Radiation model; Radiative transfer equation solution
1. Introduction
The need to simulate the radiance re#ected by the atmosphere}surface system for the purpose
of retrieving atmospheric constituents (for instance the aerosol loading) from space-based
* Correspondence address: Instituto di Scienze dell' Atmosfera e dell' Oceano, Consiglio Nazionale delle Ricerche,
Via Gobetti 101, 40129, Bologna, Italy. Tel.: #39-051-639-8004; fax: #39-051-639-8132.
E-mail address: [email protected] (R. Guzzi).
0022-4073/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 1 2 1 - 7
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C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
measurements (mainly in nadir viewing satellite) is in general met by using approximate radiative
transfer models or by interpolating from a pre-computed table of re#ectance values. Both these
approaches are subject to large errors: simpli"ed models do not allow all the relevant phenomena
(for instance the aerosol absorption) to be properly taken into account. On the other hand, when
using look-up tables of pre-computed radiances or re#ectances, investigators are forced to drastically reduce the number of allowed combinations of atmospheric parameters, thereby reducing the
capability of representing the real status of the atmosphere. Most of the methods currently used for
retrieving aerosol content from space-based measurements "t measured radiances (or re#ectances)
to spectra computed by means of a suitable radiative transfer model [1}4].
In order to simplify the radiative transfer problem, relevant to aerosol retrieval, when the
spectral resolution of the instrument is su$ciently high, the spectral measurements are selected in
the most transparent region of the atmospheric absorption spectrum, i.e. outside the gas absorption
bands. Several issues have to be dealt with if this approach is to be adopted. These are:
(1) A set of standard aerosol models must be selected or the properties of one `universala model
varied in order to represent the natural variability of the aerosol "eld.
(2) The e!ect of the underlying surface must be accounted for, including the interaction with the
atmosphere.
(3) All the relevant radiative transfer processes must be properly accounted for, including multiple
scattering and coupling between scattering and absorption.
(4) The adopted radiative transfer model must be su$ciently #exible to easily allow the introduction of new components (for instance more aerosol layers in order to represent tropospheric
and stratospheric aerosol over the same scenario) or new physical phenomena (for instance the
absorption properties of the atmospheric aerosol) without the need to re-analyze the principles
of the adopted approximations or to re-compute a large set of atmospheric scenarios.
Based on these consideration we have developed a fast and accurate radiative transfer model
(RTM), called DOWNSTREAM, coupling the MS-method by Nakajima et al. [5] with the reliable
and widely used package DISORT [6] based on the discrete ordinate method.
Although a very high accuracy is not really required for remote-sensing applications, in the light
of the errors and uncertainties inherent in the measurements, we consider that an accuracy within
1% in computing the intensity "eld must be guaranteed by the radiative transfer model, and this
accuracy must be as stable as possible when varying the input parameters. This will allow the
model to be applied in the widest range of atmospheric conditions and will avoid misinterpreting
the measurements under particular circumstances, for instance observing geometry or high aerosol
loadings.
DOWNSTREAM allows to reach the required accuracy because it analytically computes the
singly scattered radiation using the original phase function instead of its Legendre expansion, and
because the multiply scattered part of the intensity "eld is computed by means of the DISORT
without any modi"cation, thereby maintaining the original reliability of the package itself. In this
paper we present the mathematical problems and computational issues together with the numerical
validation of the presented code in a reasonable range of viewing geometries for dark underlying
surface and for a signi"cant set of aerosol optical properties. The appendix contains an equivalence
proof of the delta-M approximation.
C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
637
2. The DOWNSTREAM model
2.1. RTM for strongly peaked phase functions
The basic equation describing the transfer of monochromatic radiation in a plane parallel
atmosphere is given by
dI(; , )
"I(; , )!
P(, ; , )I(; , ) d d
!Q(; , ),
d
\ (1)
where I(; , ) is the di!use intensity at the optical thickness in the direction (, ), "cos is
the cosine of the nadir angle , and is the azimuth angle (see Fig. 1). Eq. (1) will be represented by
the functional notation J(I, P, ), according to Nakajima et al. [5]. The dependence on the
wavelength is omitted, for brevity. is the single scattering albedo, P(, ; , )"P(cos ) is the
phase function * normalized to 1 * of the particles involved in the scattering; both depend on . is the scattering angle de"ned by
cos "#(1!(1! cos(
!
).
(2)
For a solar beam F incident on a non-emitting atmosphere, from direction ( , ) to direction
(, ), we have
Q(; , )"FP(cos ) exp !
,
(3)
Fig. 1. Observation geometry for a nadir viewing sensor mounted on-board a satellite. The picture schematically
represents a plane parallel atmosphere of total optical thickness composed of ¸ layers or, equivalently, ¸#1 levels.
is the vertical integration variable.
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C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
where cos is given by relation (2) by setting the Sun coordinates ( , ) in place of (, ). In
order to solve the problem J(I, P, ) using the discrete ordinate method it is necessary to expand
the phase function P(cos ) in a N-term series of Legendre polynomials P (cos ),
L
1 ,
(2n#1) P (cos ),
P(cos )"
L L
4
L
(4)
where the coe$cients are the moments of P with respect to the P and take the form:
L
L
"2
L
\
P(cos )P (cos )d cos ,
L
(5)
with "1.
Strongly asymmetric phase functions such as aerosol and cloud phase functions computed by
means of the Mie theory cannot be represented by low-degree polynomials. These functions are
characterized by sharp forward di!raction peaks and by a number of secondary features (such as
the glory and the bows) that are di$cult to reproduce by a sum of low-degree polynomials.
Nevertheless, increasing N negatively a!ects radiative transfer computations because, as a consequence, quadrature rules, series expansions, and matrix operations all consume an increasing
amount of computer time and storage space [7,8].
The technique called delta-M approximation (described in depth by Wiscombe [9] and Joseph et
al. [10]) applies pro"tably to radiative transfer problems where the phase function of the medium is
described by a Dirac delta-function (representing the sharp peak) and a series expansion representing the phase function without the peak. We now look at the delta-M technique in detail as it is of
crucial importance to have a through understanding of the functional equivalence established by
this approximation in order to fully apprehend the MS-technique.
2.2. The delta-M approximation
In order to adequately approximate the sharp peak with a "nite sum of Legendre polynomials,
a very high number of terms is required. The delta-M approximation allows a lower number of
moments to be used to represent such phase functions. This objective is achieved by approximating
the phase function P(cos ) with PH(cos ), where the forward peak is represented by a delta function
as follows:
f
PH(cos )" (1!cos )#(1!f )PI (cos ).
2
(6)
The factor 1/(2) arises from normalization of the phase function and f is the truncation fraction. It
is well known that this approximate form of the phase function allows the solution of the problem
represented by Eq. (1) to be reduced to the solution of the equation J(IH, HPI , H) with the
advantage that PI requires a smaller number of Legendre terms because it represents only the
`smootha part of the phase function. This reduces the computational workload. All the details that
C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
639
prove the equivalence between J(IH, PH, ) and J(IH, HPI , H) are given in the appendix, along
with the meaning of the various symbols and variables.
2.3. The MS-method
Although Eq. (6) is a suitable form that allows computational problems to be overcome, PH is not
a good representation of the original phase function because using a small number of Legendre
moments gives rises to #uctuations in the truncated phase function PI .
It is well known from the successive order of scattering theory [11] that the intensity "eld of an
individual order of scattering becomes more isotropic and smoother when the scattering order
increases, i.e. multiple-scattering events tend to mask the details of the phase function. In other
words it is quite probable that hardly any of the details of the phase function will survive the
smoothing caused by more than one scattering in succession. So the #uctuations produced in the
truncated phase function mainly a!ect the singly scattered radiation while the higher orders of
scattering are very accurate even using a truncated phase function.
Note that hereinafter the singly scattered radiation has to be considered purely from direct
sunlight scattered once in the atmosphere, without contribution from surface re#ection, whatever
the surface albedo may be.
The idea underlying the MS-method [5] is to remove the unrealistic features introduced by
the delta-M method. This is done by simply subtracting from the solution IH the singly
scattered radiation IH a!ected by the oscillating truncated function PI and adding the single
scattering radiation I corresponding to the original phase function P(cos ). The "nal intensity is
therefore
I+IH!IH#I .
(7)
As reported by Nakajima et al. [5], the singly scattered radiation of the truncated problem
J(IH, HPI , H) is characterized by J (IH, (1!f )PI , ). A detailed explanation of this is given in
Section 2.4.
The e!ectiveness of the MS-method in improving simulation results is linked to the
actual fraction of the singly scattered radiation with respect to the total radiation: higher is the
fraction more signi"cant is the correction. The singly scattered fraction is a complicated function
due to several conditions: observing geometry, surface albedo, atmospheric absorption
and extinction optical depth. One can note that it is, however, out of the scope of this paper to test
the e!ectiveness of MS-method over all di!erent conditions. The fraction of singly scattered
(re#ected) spectral intensity has been computed at 400 nm over a set of observing geometry
from space, at two di!erent aerosol optical thickness values (0.1 and 2, from clean to
turbid conditions), to show that it is relevant at least for aerosol-loaded atmospheres over
dark surface. Results are depicted in Fig. 2, for two di!erent aerosol classes. Note that the
singly scattered radiation is relevant even for the lowest aerosol optical thickness value due
to the presence of molecular scattering (at 400 nm). As the aerosol optical thickness increases, the
fraction due to the single-scattering decreases but remains still relevant: the contribution of the
multiple scattered radiation is here partly limited by the aerosol absorption (Fig. 2 left-bottom, see
also Table 1 for the current aerosol classes single scattering albedo value) and by the presence of the
dark surface.
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C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
Fig. 2. Polar plot of the percentage of single scattered re#ectent to space intensity with respect to the total intensity. The
selected aerosol classes uniformly mixed with molecules at the bottom of a two layer atmosphere are Desert (Shettle 51
left side) and Maritime (right side) aerosol, for two di!erent aerosol optical thickness values; computation performed at
"700 nm: the sun zenith angle is 453. As a guide to the polar plot, proceeding outward along a radius of the
semicircle (relative azimuth angle) remains constant and (viewing zenith angle) increases, whereas moving along
a dotted circle varies and remains "xed.
Table 1
Parameters describing the phase functions of the aerosol adopted in the test scenarios. "400 nm, "780 nm. The
, i"1, 2, 100 are the Legendre coe$cients (not given here for the sake of brevity)
G
Henyey}Greenstein
Maritime 90% RH
Desert (Shettle 84)
g
( )
P(cos ; )
( )
P(cos ; )
0.2 0.4 0.6 0.8 0.9
0.7 0.8 0.85
0.998 0.999
( ) ( )
G G 0.886 0.915
( ) ( )
G G 2.4. Single-scatter components of I and IH
For solar radiation singly scattered in a non-emitting parallel atmosphere I , the radiative
transfer equation is
dI (; , )
"I (; , )!Q(; , ),
d
(8)
whose functional notation is J (I , P, ). For this problem the solution can be written in an
analytical form even in the presence of a non-uniform atmosphere. In the case of the upward
C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
641
intensities at the optical thickness the solution is
( !)
I>(; , )"I ( ; , ) exp ! O d
(!)
()P(; cos ) exp !
!
,
(9)
O
where is the total optical thickness. If we substitute the approximated phase function (Eq. (6))
into Eq. (9) we obtain
#F
( !)
IH>(; , )"IH( ; , ) exp ! O d
(!)
() (1!f )PI (; cos ) exp !
!
(10)
O
because for nadir observations we can let (1!cos )"0.
Eq. (10) means that IH> is really the solution of the J (IH, (1!f )PI , ) problem. In particular,
for the upward intensity singly scattered at the top of the atmosphere ("0) in the case of no
upward di!use intensity at the bottom of the atmosphere (i.e. dark surface where
I ( ; , )"IH( ; , )"0) we have
O d
()P(; cos ) exp[!
],
(11)
I>(0; , )"F
where "1/#1/ .
The solution for a vertically non-uniform atmosphere consisting of ¸ layers in which the
constituents are uniformly mixed together is straightforward. Each layer consists of di!erent
components (for instance molecules and aerosol particles or molecules and cloud droplets)
uniformly mixed. The optical properties of each layer can be written as
#F
PG G (s)
G(s)
G(s, e)" G (s, e),
PM G" A A A ,
G"
,
(12)
A
G (s)
G(e)
A A
A
where G(e) and G(s) are the extinction and scattering optical thicknesses of the ith layer. The
subscript c indicates the components present in that layer. Thus, the integral in Eq. (11) can be
rewritten as
F *
OG >
I>(0; , )"
GPM G
exp[!
] d,
OG
G
F *
GPM Gexp[!
G] (1!exp[!
G] ),
(13)
"
G
where exp[!
G] is the two-way transmittance at the level which accounts for the layers above
G
the ith one:
G\
for ¸5i'1, ¹ "1.
¹G"exp !
H(e)
H
Using these assumptions it is straightforward to compute IH> by Eq. (10).
(14)
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C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
3. Results and numerical validation
The accuracy of DOWNSTREAM and DISORT was examined for the case in which the number
of streams (2M, see Eq. (16)) decreases with respect to the reference model. Since the computational
time is approximately proportional to M in the discrete ordinate method and M in the matrix
operator method [5,9] and storage requirements are proportional to M, the possibility of
reducing the number of streams become very attractive.
Tests were performed using two di!erent types of phase functions (see Fig. 3 and Table 1). We
considered the Henyey}Greenstein phase function for several values of the asymmetry factor
(customary symbolized as g and de"ned by g"3 ) and the Mie phase function for two selected
Fig. 3. Left panels (a) and (c): three Henyey}Greenstein phase function (g"0.7, 0.8, 0.85) as a function of cos (a) and of
(c). Right panels: Mie phase functions at 400 nm for the Desert (Shettle 84) class and for the Maritime class at 90%
humidity level as a function of cos (b) and of (d). The reported asymmetry factors for the Mie phase functions are for
comparison with the Henyey}Greenstein.
C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
643
aerosol classes: a Maritime class [12] with a relative humidity level of 90% and a Desert class [13].
These two aerosol classes have been selected to represent opposite, and to some degree extreme
conditions, one being a dry, absorbing, mineral aerosol and the other a non-absorbing, very wet
aerosol class, typical of maritime environments.
The optical properties of the selected classes were computed at two wavelengths "400 and
780 nm using the software tool described by Levoni et al. [14]. The Mie and Henyey}Greenstein
phase functions were chosen because they are at somewhat opposite extremes in terms of structure:
the Mie function has about as much structure as one is likely to encounter in a polydispersed cloud
or aerosol, while by contrast the Henyey}Greenstein functions are very smooth. They have
considerably lower and broader forward peaks, and are shaped more like phase functions for
smaller, moderate to highly absorbing particles, while the steepness of their forward peaks
analytically increases with the asymmetry factor g [11].
To assess the accuracy of the two models we chose as reference values the intensities (I )
0#$
produced by DISORT with 2M"100 streams (hereafter the number of streams will be called nstr
as in the DISORT input) in order to account for the phase function features. Computations were
performed in a cloudless, horizontally homogeneous atmosphere vertically divided into two layers:
a uniform mixture of aerosol and molecules below a pure molecular layer, bounded at the bottom
by a dark lambertian surface. The nstr reference value has been selected by observing that, for all
the simulations carried out, the DISORT's results do not change beyond nstr"100 (but in many
cases at nstr"68 the results are already stabilized).
This simple vertical structure represents a clear atmosphere with an aerosol present in the
boundary layer. Several values of aerosol optical thickness were selected for the test to represent
atmosphere from slightly to heavily aerosol loaded. Since our interest focuses on space-based,
nadir-viewing observations of the intensity "eld di!used by the atmosphere, solar zenith angles
and emergent angles (, ) are taken in the range of variability typical of these observations (see
Table 2).
The intensities I
(nstr) were computed in double precision with both DISORT and
2#12
DOWNSTREAM using a decreasing number of streams (nstr3[68, 32, 24, 20, 18, 16, 14, 12, 10, 8] )
for the set of scenarios described above. The relative percentage di!erences between the reference
and the test intensities were evaluated at each number of streams as follows:
I !I
(nstr) 2#12
;100.
I ( , , ; , P(cos ), ; nstr)" 0#$
H I
0#$
(15)
The index j runs from 1 to 19,200 for the simulations involving the Henyey}Greenstein phase
functions. This "gure comes from the combination of 32 observing geometries, 4 optical thickness
Table 2
Observation geometries and optical thicknesses used to realize test scenarios. For the Mie phase function the optical
thickness refers to 500 nm
03 203 403 603
153 503
03 503 1003 1503
0.1 0.5 1.0 2.0
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C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
Fig. 4. DOWNSTREAM(DSTREAM) and DISORT(DS)
accuracy as a function of number of streams (NSTR).
Computations are carried out for Henyey}Greenstein
phase functions with g"0.85, 0.80, 0.7. The black-shaded
areas contain all accuracy values between IM # M and
'
IM ! M , where M is the standard deviation about the
'
'
mean IM . The complete set of accuracy values, for a given
aerosol class, is contained in the envelope covered by the
gray-shaded area (which includes the inner 1 set).
Fig. 5. As in Fig. 4 but for spherical particles whose optical
properties have been computed by Mie theory. Data refers
to Maritime (relative humidity RH"90%, asymmetry factor g"0.79, "400 nm) and Desert (Shettle 84; asymmetry factor g"0.72, "400 nm) aerosol classes.
values (see Table 2), 5 scattering albedo values, 3 asymmetry parameter values (see Table 1) and
considering that simulations are repeated for 10 nstr values. For the simulations considering Mie
aerosol optical properties, the number of scenarios is reduced to 5120 (see Tables 1 and 2).
C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
645
Fig. 6. The relative speed of DOWNSTREAM and DISORT with respect to DISORT, computed with 68 streams, as
a function of the number of streams.
The dependence of I
and I
upon the parameters of the scenario is the same as that of I
2#12
0#$
and has been omitted for conciseness. I will be called `accuracya in this context. The aerosol
phase function P(cos ) depends only upon one parameter (g) for the Henyey}Greenstein phase
function; for the Mie phase function the Legendre moments were computed by using the already
cited software [14]. For each phase function and for each number of streams the mean IM and the
standard deviation M of I were computed for both DISORT and DOWNSTREAM and plotted
H
'
against nstr (Figs. 4 and 5).
It can be observed (Figs. 4 and 5) that DISORT gradually loses its accuracy with a decreasing
number of streams, whereas DOWNSTREAM maintains its high accuracy with respect to the
reference model for all the cases tested. The two adopted types of phase functions show two
di!erent behaviours: for close values of the asymmetry factor (compare the case with g"0.8 in
Fig. 4 with the Maritime case of Fig. 5) the bene"t gained from using DOWNSTREAM is greater
for the Mie phase function. Furthermore, with a growing asymmetry factor (compare the cases of
Fig. 4) DISORT gradually loses its accuracy because a small number of streams is not su$cient to
properly reconstruct the forward peak, which depends on the scattering angle.
Fig. 6 depicts the gain in speed when decreasing the number of streams nstr. Because the time
used for the two evaluations of the single scattering is negligible, the time spent by a single call to
DISORT is, for a "xed number of streams, equal to the time for a call to DOWNSTREAM. Then
this graph is common to both DISORT and DOWNSTREAM. From this "gure we see that a single
call with nstr"8 is around 12 times faster that a call with 68 streams. From our tests it is also
evident that, contrary to what is in general reported [5,9], the execution time scales as nstr 2.5}3
only for nstr greater than 40. For smaller values of nstr the proportionality ranges between
nstr and nstr. A possible explanation is that the time used in the linear algebra parts of
DISORT becomes less signi"cant when the dimension of the matrix that has to be inverted
(proportional to nstr) is smaller.
4. Concluding remarks
We have presented the mathematics of a newly developed radiative transfer code and its
performances in simulating the intensity "eld re#ected by a plane parallel, non-emitting, multilayer atmosphere, in which a component such as aerosol or cloud is present in one layer. A code of
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C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
this type is able to compute the re#ected intensity "eld 2 to 10 times faster than DISORT, making it
suitable for processing satellite data, at least for scienti"c applications. In this connection
DOWNSTREAM has been used for the purpose of simulating the GOME spectral re#ectance in
order to derive the aerosol content [3,4]. While being based on classic techniques and approaches
already presented in literature, DOWNSTREAM o!ers several new capabilities that make it
reliable and accurate. The code uses the well known and widely used subroutine DISORT to
compute the multiply scattered intensity: no modi"cations have been made to DISORT itself, so its
high level of reliability has been maintained. The MS-approach has been developed to reduce the
computational workload without losing accuracy. This fact has proved to be important in the task
of processing remote sensing data to extract information on atmospheric constituents [4] from
space-based measurements where DOWNSTREAM has been used to produce the spectra during
the retrieval procedure. The code has been interfaced with an existing aerosol properties data-base
[14] so the properties of the aerosol (or cloud) present in the atmosphere can be easily varied.
Finally, the code has been tested in a wide set of scenarios in an attempt to cover the larger range of
variability of the input quantities: sun-observer geometries, aerosol loadings and phase functions
were varied so as to encompass the relevant conditions experienced by a space-based nadir viewing
instrument. In all these conditions the code proved to be able to compute the re#ected intensity
"eld with high execution speed but without losing its accuracy.
Acknowledgements
The authors intend to thank Warren Wiscombe for his friendly suggestions and comments. This
work was partly funded by ESA contract 11572/95/NL/CN. We thank also C. Readings, A. Hahne
and J. Callies for their valuable support and suggestions.
Appendix
In this appendix we outline the steps involved in the demonstration of the equivalence between
the two problems J(IH, PH, ) and J(IH, HPI , H). This demonstration makes use of the integral
properties of the delta function and adopts the associated Legendre polynomials as a basis for
expanding the Legendre polynomials themselves. In this way, PI represents the phase function
without the peak and can be written in the form
1 +\
(2n#1)I P (cos ).
(A.1)
PI (cos )"
L L
4
L
M is essentially the order of the approximation and f and I are the unknown quantities that can
L
be found by matching the moments of P H(cos ) to those of P(cos ) (Eq. (4)). When the delta
function is expanded in a series of Legendre polynomials, we obtain (see, for instance, Morse and
Feshbach [15])
(1!cos )" (2n#1)P (1)P (cos )" (2n#1)P (cos ),
L
L
L
L
L
(A.2)
C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
647
knowing that, for whatever n, P (1)"1. By substituting Eq. (A.2) into Eq. (6) PH(cos ) takes the
L
following form:
1 1 +\
(2n#1)P (cos ) [ f#(1!f )I ]#
f (2n#1)P (cos ).
(A.3)
PH(cos )"
L
L
L
4
4
L
L+
By solving the equation P(cos )"PH(cos ), the coe$cient I and the fraction f are determined.
L
Therefore,
f#(1!f )I
L
"
L
f
for n42M!1,
(A.4)
for n52M,
from which we obtain
I "( !f )/(1!f ) for n42M!1,
L
L
(A.5)
f"
for n52M.
L
Note that the solution for f is not unique. The selection of f appears somewhat indeterminate,
but as Wiscombe [9] and other authors suggest, we select
f" .
(A.6)
+
This choice is particularly successful when the are small and they become equal for n52M.
L
With PH(cos ) in this form (Eq. (6)) the radiative transfer equation (1) becomes
dIH(; , )
f
"IH(; , )! D (; , )
d
2 B
!(1!f )
PI (cos )IH(; , ) d d
!G (; , , , )
B
(A.7)
d
(1!cos )IH(; , )
(A.8)
,
!F(1!f )PI (cos ) exp !
where
D (; , )"
B
\
d
and
f
.
(A.9)
G (; , , , )" F(1!cos ) exp !
B
2
By the properties of the Dirac-delta function, it is possible to reduce Eq. (A.7). Infact, since the sun
and observer direction are not coincident in nadir viewing satellite geometry, then cos O1, and
hence expression (A.9) is zero. Furthermore as far as regards the calculation of the integral in
Eq. (A.8), we can expand the intensity IH in Fourier cosine N-series in azimuth, i.e.,
,
IH(; , )" I (; ) cos m
.
K
K
(A.10)
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C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
Using the de"nition in the form given by Eq. (A.2), Eq. (A.8) becomes
,
D (; , )" (2n#1) dI (;) d
P (cos ) cos m
.
(A.11)
B
K
L
L
K \
Expanding the Legendre polynomials P (cos ) of Eq. (A.11) in an n-series of Legendre associated
L
functions PI () and cos k
(k"0, 1, 2, n) in the following way (see, for instance, Liou [16, p.
L
369]). We obtain
L
(n!k)!
PI ()PI () cos k(
!
),
P (cos )" (2! )
L
L
I (n#k)! L
I
where is the Kronecker delta. If we put Eq. (A.12) into the Eq. (A.11) we have
I
,
D (; , )" (2n#1)
d I (;)
B
K
\
L K
L
(n!k)!
PI ()PI ()
d
cos k(
!
) cos m
,
; (2! )
L
I (n#k)! L
I
where the integral in the azimuth angle is equal to zero for any kOm. In particular,
(A.12)
(A.13)
if k"m"0,
2
d
cos k(
!
) cos m
" cos m
for k"mO0,
for kOm.
0
So we obtain
1
,
(n!m)!
PK()PK().
(A.14)
D (; , )"2 n# cos m
d I (; )
L
B
K
2
(n#m)! L
\
L K
Now we must consider that the Dirac-delta function can be expanded in any orthogonal set of
eigenfunctions (see Morse and Feshbach [15]). For "xed m, the Legendre functions PK(x) become
L
a set of n-eigenfunctions with the properties
0
dx PK(x)PK(x)"
(n#m)!
L
I
(n#)\
(n!m)!
Thus, in particular,
if nOk,
if n"k.
1 (n!m)!
PK(x)PK(x)"(x!x)
n#
L
2 (n#m)! L
L
which is valid for any m. Using Eq. (A.15) in Eq. (A.14), Eq. (A.8) is reduced to
,
,
D (; , )"2 cos m
d (!)I (; )"2 I (; ) cos m
B
K
K
\
K
K
and by Eq. (A.10)
D (; , )"2IH(; , ).
B
(A.15)
(A.16)
(A.17)
C. Levoni et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 69 (2001) 635}650
649
Inserting Eq. (A.17) in Eq. (A.7) and considering G (; cos )"0 the initial problem
B
J(IH, PH, ), given by Eq. (A.17), becomes
dIH(; , )
"(1!f)IH(; , )!(1!f )
d
PI (cos )IH(; , ) d d
!F(1!f )PI (cos ) exp !
and, dividing by (1!f),
dIH(; , )
(1!f )
"IH(; , )!
d(1!f)
1!f
PI (cos )IH(; , ) d d
(1!f )
PI (cos ) exp !
.
1!f
The last equation is exactly the problem J(IH, HPI , H) with
!F
(1!f )
,
H"
1!f
(A.18)
(A.19)
(A.20)
d H"(1!f) d.
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