The quantitative evaluation of the electromagnetic(e.m.) radiance field is of major interest for variouspurposes and applications, as testified in the openliterature. The latter go from astrophysics such as thestudy of the transfer of solar radiation1–3 to remotesensing such as the retrieval of Earth and planetaryatmospheres by passive and active remote sensingtechniques4–7 and to telecommunications such as theeffect of Earth’s atmosphere on microwavesignals.8–10 In these applications the theory of radi-ative transfer plays a major role as a suitable modelto describe the e.m. interaction and propagation ofradiation intensity through a random medium suchas a planetary atmosphere and surface.11–14 In addi-tion to forward modeling, use of fast and fairly accu-
rate radiative transfer solutions can also be essentialto solve inverse problems by variational approacheswhere the solution is searched by minimizing a costfunction depending on the measured and simulatedradiance observables.15
In an absorbing and scattering random mediumthe radiative transfer equation (RTE) takes the gen-eral form of an integrodifferential equation for amonochromatic unpolarized radiance. Its full solu-tion can be accomplished by using several numericaltechniques that generally reduce the problem to a setof differential equations in a matrix form with pre-scribed boundary conditions.13,16,17,10,9 The interestand the importance in developing analytical approx-imate solutions of the radiative transfer problemarises from the need to have relatively fast algo-rithms and from the difficulty to derive precise solu-tions for many practical applications where themedium properties are not generally known with suf-ficient accuracy. In this respect, uncertainties in theinput parameter description within a given radiativetransfer problem tend to reduce the need of calcula-tions with a high degree of precision.18,19 As alreadymentioned, the precision of numerical RTE models isobtained at the expense of a long computational time,which is unwanted for applications that require a fast
F. S. Marzano ([email protected]) is with the Centro diEccellenza in Telerilevamento E Modellistica di Precipitazioni Se-vere (CETEMPS), Università dell’Aquila and the Dipartimento diIngegneria Elettronica, Università La Sapienza di Roma, Via Eu-dossiana 18, 00184 Roma, Italy. G. Ferrauto is with the Diparti-mento di Ingegneria Elettronica, Università La Sapienza di Roma,Via Eudossiana 18, 00184 Roma, Italy.
response such as iterative remote sensing tech-niques.5,6 (e.g., Kummerow5; Tanrè et al.6).
A number of fast analytical models, with variouseffective approximations, have been proposed for thesolution of the RTE, sacrificing a certain degree ofmathematical precision in favor of a closed-formrapid algorithm.3,12,18–24 The so-called second-orderEddington approximation is probably the most com-mon and effective between these, even though it isgenerally referred to as the azimuthally averagedform of the RTE for flux calculations.21,26,27 Azimuth-ally dependent radiance models represent valuableapproximated methods that can yield both irradianceand radiance results.3,19,28 In particular, these mod-els can be developed in extension of either the two-stream or the Eddington approach19,20 or are basedon the Sobolev approximation framework.12,28 Tomaintain consistency between their radiance and ir-radiance solutions, several approximate solutionsneed to resort to ad hoc numerical integrations sothat they could prove unsuitable for applications in-volving the integration of single RTE solutions overvarious angular, spectral, and time intervals.
In this paper a fast and fully analytical radiativetransfer solution used to calculate the unpolarizedmonochromatic radiation in a random medium is de-veloped using an Eddington-like approach combinedwith the delta phase-function transformation tech-nique.22 In Section 2 the Eddington approximationscheme, proposed by Shettle and Weinman,21 is ex-tended in a natural way, allowing us to unfold theazimuthal dependence of the radiance field. This ap-proximation, further refined to adequately describesingle-scattering phenomena, shows itself sufficientlyflexible to be adapted to the treatment of radiation ininhomogeneous random media, schematized through aseries of plane-parallel homogeneous adjacent layers.We try to avoid ad hoc correction procedures to furtherimprove the overall accuracy; this exercise would re-duce the generality of the proposed solution and canalways be played for specific applications.28 In Section3 results from the generalized Eddington model arethen compared over a wide range of optical parametersand observation geometries, with the correspondingsolutions generated by a fully numerical and fairlyaccurate algorithm such as the discrete ordinate radi-ative transfer (DISORT) method for backscattered ra-diance at the top of the medium.
2. Generalized Eddington Radiative Model
In this section, we will first introduce the notation ofRTE in a rigorous way, then we will illustrate thegeneralized Eddington model to take into accountazimuthal dependence of the radiance field, and fi-nally we will conclude by modifying the analyticalsolution by means of a first-order scattering correc-tion. The latter choice will also be numerically justi-fied in Section 3.
A. Radiative Transfer Theory
Even though some of the definitions and equations thatare given below are quite well known in the literature, we
will briefly overview the background concepts with thebasic aim to clarify the adopted notation.
The transfer of unpolarized monochromatic radia-tion in a plane-parallel homogeneous random me-dium is regulated by the integrodifferential RTE12–14:
�dI(�, �, �)
d�� �I(�, �, �) � J(�, �, �), (1a)
where I��, �, �� is the diffuse specific intensity (orradiance) along the direction ��, �� at the opticaldepth � and J��, �, �� is the source function, given by
J(�, �, �) ��
4� �0
2� ��1
1
P(�, �; ��, ��)
I(�, ��, ��)d��d��
��
4�F0P(�, �; �0, �0)exp(����0).
(1b)
The first term of J is sometimes referred to as themultiple-scattering source, and the second term rep-resents the coherent contribution due to an incidentcollimated e.m. wave (either a plane wave or a direc-tive beam) along the direction ��0, �0�. Thermal emis-sion is assumed to be negligible with respect to theincident beam.13
The notation in Eqs. (1a) and (1b) is in accordancewith a spherical coordinates system ��, , ��, with �the vertical optical thickness between 0 and �s and isas follows (see also Fig. 10 in Appendix B): � is thezenith (or nadir) angle, � � cos with � � 0 indicat-ing downward radiance and � � 0 indicating upwardradiance; � is the azimuth angle; � is the single-scattering volumetric albedo; �0 is the cosine of theincident radiation zenith angle 0; �0 is the azimuthangle of the incident radiation; and F0 is the powerflux density, or irradiance, of the incident radiation atz � 0. For brevity, the dependence on the wavelengthis omitted because, in a way, it is implicitly includedin the medium optical parameters �, �, and P.
It is worth mentioning that if the incident e.m.source is highly but not infinitively directive, that is,it has a finite field of view such as a pencil beam, Eq.(1a) can be generalized by replacing Eq. (1b) with
J(�, �, �) ��
4� �0
2� ��1
1
P(�, �; ��, ��)
I(�, ��, ��)d��d�� ��
4�
�0
2� ��1
1
P(�, �; ��, ��)I0(��, ��)
exp(����0)d��d�, (2)
where I0 is the incident radiance field at z � 0. If I0
has a sufficiently narrow beam width, then the sec-ond term of Eq. (2) can be approximated by the anal-ogous term in Eq. (1b), as in the case of radar andlaser sources. In the same perspective, once the ra-diance field is computed from Eqs. (1) at a givenposition, the received power of an antenna (or detec-tor) with a directive gain G can be derived from10,13
Wr � 2
4� �0
2� ��1
1
G(�, �; ��, ��)
I(�, ��, ��)d��d��, (3)
where � is the e.m. wavelength in the medium.The zenith opacity at height z can be related to the
volumetric extinction coefficient ke�z� as follows:
� ��0
z
ke(z�)dz�, (4)
where � is allowed to vary in the range �0, �s� because�s is the total optical depth (see Fig. 10). The irradi-ance F �Wm�2 Hz�1� at a given � is related to radiancethrough the following integral relationship:
F↓(�) ��0
2� �0
1
I(�, ��, ��)d��d��,
F↑(�) ��0
2� �0
1
I(�, ���, ��)d��d��, (5)
where F↓ and F↑ represent the downward and up-ward fluxes, and from now on we assume � as apositive quantity (i.e., � � |cos |) so that �� can beused to designate the upwelling direction. The diffuseupward and downward flux densities, then, are givenby Eqs. (5), integrating from 0 and 1 with respect to��� and ��, respectively.
In Eq. (1b), P��, �; ��, ��� is the so-called volumet-ric scattering phase function (normalized to 4�), de-fining the intensity of a radiation incident atdirection �� � ���, ��� that is scattered into direction� � ��, ��. In an absorbing and scattering mediumwith spherical particles, for example, it can be com-puted by means of Mie theory. The phase functioncan also be expressed in terms of the scattering angle� (i.e., angle between the incident and the scatteredradiances). A well-known expansion of the phasefunction is given by a N-term series of Legendre poly-nomials Pn (Ref. 13):
P(cos �) � �n�0
N
(2n � 1)bnPn(cos �)
� �n�0
N
(2n � 1)bnPn{��� � [(1 � �2)
(1 � ��2)]1�2 cos(� � ��)}, (6)
where Pn�cos �� exhibits orthogonality in a ��1, 1�interval. The coefficients bn are the moments of Pwith respect to the polynomials Pn, defined by
bn �12 �
�1
1
P(cos �)Pn(cos �)d cos �, (7)
with P0�cos �� � 1, P1�cos �� � cos � and recursivelyfor higher-order polynomials. Note that b0 � 1 be-cause of the normalization of the phase function,whereas, by definition, b1, the first moment of thephase function, is called the volumetric asymmetryfactor of the phase function.13,17 This parameter, usu-ally indicated as g, is zero for isotropic scattering andincreases as the diffraction peak of the phase functionbecomes increasingly sharpened. As an example, theHenyey–Greenstein phase function clearly showsmore noticeable diffraction peaks associated withhigher values of g.1
However, Eq. (6) is inadequate to representstrongly asymmetric phase functions. The sharp for-ward diffraction peaks that characterize phase func-tions for strongly scattering conditions (i.e., g � 0.5)are difficult to reproduce by low-degree polynomials.This implies the need to use higher values of N thatwould involve an increasing mathematical complex-ity.29
To adequately approximate P�cos �� with a lownumber of moments, a delta-function transformationtechnique can be adopted, introducing a Dirac func-tion describing the sharp peak of the phase functionand a series expansion representing the phase func-tion without the peak18,22:
P(cos �) � P�(cos �)� 2f�(1 � cos �) � P*(cos �)
� 2f�(1 � cos �) � �n�0
2M�1
(2n � 1)bPn(cos �),
(8)
where f is the fractional scattering into the forwardpeak. Joseph et al.22 demonstrated that the solutionof the RTE (i.e., calculation of the radiance I) usingthe actual phase function P is equivalent to the solu-tion of the same problem in which P is replaced withthe smoothed function P*, if one set f � b2M and theoptical parameters �, �, and g are modified as follows:
g* �g � f1 � f, �* �
1 � f1 � �f �, �* � (1 � �f)�.
(9)
The advantage in using P* is related to the fact thatits series expansion can be truncated to a small-orderLegendre term, since it represents only the smoothedpart of the original phase function. It is worth notingthat the choice of the peak fraction f is quite arbi-trary.
phase function, analytical complexities of the RTEsolution in absorbing and scattering random media,characterized by quite asymmetric phase functions,can be significantly reduced. In Subsections 2.B and2.C we will always assume that the optical parame-ters have been transformed by Eqs. (9) with f � g2,derived from imposing the second moment of P equalto that of the Henyey–Greenstein function.22 For sim-plicity of notation, later on we will indicate �*, g*,and �* by �, g, and �, if not otherwise stated.
B. Generalized Solution with an Eddington-LikeApproximation
We consider a scattering and absorbing homogeneousrandom medium with a plane-parallel geometry (i.e.,Fig. 10 with N � 1). The radiative transfer is gov-erned by Eqs. (1a) and (1b), where all the coefficientsare constant. It is clear, however, that a solution forI in a closed form can be derived only under givensimplifying approximations.
The generalized Eddington approximation consistsof expanding the diffuse radiance in the form of aFourier cosine series truncated to the first order, thatis,
I(�, �, �) � I0(�, �) � I1(�, �)cos �, (10a)
where the functions I0��, �� and I1��, �� are approxi-mated in accordance with the standard Eddingtonapproach21:
I0(�, �) � I00(�) � �I01(�), (10b)
I1(�, �) � I10(�) � �I11(�). (10c)
The approach, summarized by Eqs. (10), closely re-sembles the theoretical framework of the discrete or-dinate numerical method30 even though our objectivehere is to derive a closed-form solution, paying theprice of simplifying approximations that will reflectin a less overall accuracy.
From Eqs. (10), the source function J in Eq. (1b)can be rewritten as
with the phase function approximated by a delta-transformed Sobolev function12,28:
P � 1 � 3g cos �. (12)
Assuming �0 � 0 and integrating Eq. (1a) over �between 0 and 2�, after dividing by 2�, one obtains
12� �
0
2�
�dI(�, �, ��)
d�d�� � �
dd�
[I00(�) � �I01(�)]
� �[I00(�) � �I01(�)] � �[I00(�) � �gI01(�)]
��
4�F0(1 � 3g��0)exp(����0). (13)
Performing the same integration on Eq. (1a) multi-plied by cos �, we obtain
12� �
0
2�
�dI(�, �, ��)
d�cos ��d�� � �
dd�
[I10(�)
� �I11(�)]
� �[I10(�) � �I11(�)] �38 ��g(1 � �2)1�2I10(�)
�3�
4�F0g[(1 � �2)(1 � �0
2)]1�2exp(����0).
(14)
With the aim to find the solution for I00��� and I01���,it is convenient to integrate Eq. (13) over � between�1 and 1 and then do the same for Eq. (13) multipliedby �, obtaining
dI01(�)d�
� �3(1 � �)I00(�) �3�
4�F0 exp(����0),
(15a)
dI00(�)d�
� �(1 � g�)I01(�) �3�
4�F0g�0 exp(����0).
(15b)
The same procedure, carried out for Eq. (14), yieldsan analogous system of nonhomogeneous differen-tial linear equations for I10��� and I11���:
dI11(�)d�
� �3I10(�)1 �3�2
32 �g�9
16 F0�g(1
� �02)1�2 exp(����0), (16a)
dI10(�)d�
� �I11(�). (16b)
Solutions of the differential equation system, givenin Eqs. (15) are
Analogously, the solution of the differential equationsystem given by Eqs. (16) is
I10(�) � p1[C11 exp(�k1�) � C12 exp(k1�)] � �1
exp(����0), (18a)
I11(�) � C11 exp(�k1�) � C12 exp(k1�) � �1
exp(����0), (18b)
where
k1 � �31 �3�2
32 g��1�2
, (18c)
p1 �1k1
, (18d)
�1 �(9�16)��0F0g(1 � �0
2)1�2
1 � �02k1
2 , (18e)
�1 �(9�16)��0F0g(1 � �0
2)1�2
1 � �02k1
2 . (18f)
Substituting Eqs. (17) and (18) into Eqs. (10), it ispossible to obtain the general solution for azimuth-dependent diffuse radiance, rewritten for clarity inthe following way:
To completely determine the RTE solution, bound-ary conditions need to be imposed. For a nonreflectingsurface, the boundary conditions to compute the in-tegration constants C01, C02, C11, and C12 of Eqs. (17)and (18) can be written as
I(� � 0, �, �) � 0, (20a)
I(� � �s, ��, �) � 0, (20b)
that is, using the generalized Eddington expansion:
Trying to determine a solution for the previous sys-tem of equations that are valid for all � and for all �would require us to make equal to zero every termIij�0� of Eq. (21a) and every term Iij��s� of Eq. (21b),that is, to solve a system of eight equations with thefour unknown constants C01, C02, C11, and C12.
A possible procedure that allows to compute theabove-mentioned constants consists of fixing a properconstant value � � �c in Eqs. (21). This choice allowsus to reduce Eqs. (21) to the following system of fourequations with four unknown quantities:
I00(0) � �cI01(0) � 0,
I10(0) � �cI11(0) � 0,
I00(�s) � �cI01(�s) � 0,
I10(�s) � �cI11(�s) � 0. (22)
The constant value �c can be selected in a mannerthat, for the azimuthally independent radiance field[i.e., fixing cos � � 0 in Eq. (19)], the solution coin-cides with that obtained adopting the classical Ed-dington approximation.
To do this, considering that the downward and up-ward diffuse irradiance, respectively, at the top andthe bottom of the medium are zero, it is possible towrite from Eqs. (5) the following boundary conditionsin terms of downward and upward flux densities (ir-radiances):
F↓(0) ��0
2� �0
1
I(0, ��, ��)d��d��
� ��I00(0) �23 I01(0)�� 0,
F↑(�s) ��0
2� �0
1
I(�s, ���, ��)d��d��
� ��I00(�s) �23 I01(�s)�� 0, (23)
which suggests that we choose �c � 2�3. In AppendixA, a further detailed discussion on this issue from acomplementary point of view is provided.
After substituting Eqs. (17) and (18) into Eqs. (22),it is possible to solve the last ones for the unknownconstants C01, C02, C11, and C12 adopting the value�c � 2�3, yielding
For several applications, the calculation of the back-scattered (or reflected) radiance at the top of the me-dium is particularly interesting. We call this solutionthe azimuth-dependent Eddington radiative model(AERM) to distinguish it from that derived in Sub-section 2.C. The formal expression of the AERM so-lution is
where each term on the right-hand side is derivedfrom Eqs. (17), (18), and (24) together with Eqs. (9).As expected, the upwelling irradiance, computedfrom Eq. (25), is equal to the one given by the classicalEddington approximation:
F↑(�) ��0
2� �0
�1
I(�, ��, ��)��d��d��
� �[I00(�) � (2�3)I01(�)], (26)
as found by Shettle and Weinman.21
C. First-Order Scattering Correction
The RTE solution for azimuthally dependent radi-ance, as was shown in Subsection 2.B, is subject tovarious approximations regarding the form of thephase function and the imposition of the boundaryconditions. The truncation of the actual phase func-tion P and its consequential representation with low-order Legendre polynomials allows some essentialanalytical simplifications, but on the other hand it
tends to smooth the angular details of the P behaviorand modifies the scattering phenomena description.
By looking at the generalized solution given by Eq.(19) together with Eqs. (24), we realize that the ex-plicit expression of the phase function is no longertraceable. Indeed, from the scattering theory, it iswell known that the low orders of radiation scatter-ing, and in particular the single-scattering contribu-tion, are quite sensitive to the approximatedrepresentation of P.3,24,28
To remove this problem in our theoretical frame-work, a first-order scattering correction can be car-ried out. It simply consists of (i) canceling the single-scattering contribution I� (biased by the truncation ofthe phase function) from the radiance solution I ob-tained by adopting the Eddington-like scheme of ap-proximation and (ii) replacing I� with IFOSM, the exactweight of the single-scattering effects calculable bythe so-called first-order scattering model (FOSM).14
In Section 3 we will show that this improvement isnumerically significant because of this correction.
In the absence of multiple-scattering phenomena,the FOSM represents the exact solution for RTE. Forthe reflected radiance for a finite homogeneous me-dium bounded on two sides at � � 0 and � � �s, theFOSM solution is given by
IFOSM(0, ��, �) ��
4��0F0
P(��, �; �0, �0)� � �0
1 � exp���s1�
�1�0��, (27)
where, from now on, IFOSM will be indicated as a first-order scattering (FOS) solution. In the FOS correc-tion scheme, IFOSM is calculated using the actualphase function P and the original optical parameters
�, g, and � (i.e., not scaled to account for the deltatransformation). Note that only upward backscat-tered radiance is given here; for the transmitted ra-diance, similar expressions can be easily derived froman analogous approach.
The single-scattering contribution I� to the gener-alized solution I, given by Eq. (19), can be computedby removing the multiple-scattering terms within thegeneralized Eddington model itself. This means that,within the hypothesis of only a single process of scat-tering, the source function J in Eq. (11) can be ap-proximated by
J(�, �, �) � JFOS(�, �, �)
��
4�F0(1 � 3g{��0 � [(1 � �2)
(1 � �02)]1�2cos �})exp(����0). (28)
Repeating the same mathematical steps to obtain thesolution for I expressed by Eq. (19), one can simplyverify that in this case the general solution for radi-ance maintains the same form:
where I�0, ��, ��, I��0, ��, ��, and IFOSM�0, ��, ��are given, respectively, by Eqs. (25), (32), and (27).Equation (33) can be regarded as the closed-form so-lution of the generalized Eddington radiative model,herein referred to as GERM.
The GERM solution, as well the AERM solutiongiven in Eq. (25), can be simply adapted to the treat-ment of radiation in inhomogeneous media, schema-tized through a series of plane-parallel homogeneousadjacent layers, as shown in Appendix B.
3. Numerical Tests
In this section we will show the numerical evaluationof the accuracy of the generalized Eddington model,
for both the AERM and the GERM versions, by sup-posing an arbitrary medium characterized by a largeset of optical parameters and observed under severalangles.
A. Model Setup
Optical parameters represent the inputs to RTE andtheir considered variability the domain of the numer-ical validation test. As an example, Fig. 1 shows theoptical parameters in terms of albedo and asymmetryfactor as a function of extinction coefficients, as ob-tained from Mie scattering and absorption simulationfor spherical raindrops, graupel particles, ice crys-tals, and snow particles for microwaves between 3and 90 GHz and a rain rate between 0 and100 mm�h.9,10
To derive the optical parameters in Fig. 1, an in-verse exponential particle size distribution (PSD) hasbeen assumed with the slope parameter parameter-ized to a surface rain rate. The latter has been de-rived from the Marshall–Palmer PSD for raindrops,the Sekhon–Srivastava PSD for ice crystals and grau-pel, and the Gunn–Marshall PSD for snow aggre-gates.17 A Gamma PSD has been chosen for clouddroplets. Radius size ranges of cloud droplets, rain-drops, ice graupel, ice crystals, and snow have beenfixed to 0.001–0.01, 0.1–3.0, 0.1–5, 0.1–1.5, and0.1–5.0 mm, respectively. Density of ice graupel, icecrystals, and snow has been set to 0.5, 0.2, and
Fig. 1. Optical parameters in terms of albedo and asymmetry factor versus extinction coefficients, as obtained from Mie scattering andabsorption simulation for spherical raindrops (upper left panel), graupel particles (upper right panel), ice crystals (bottom left), and snowparticles (bottom right) for microwaves between 3 and 90 GHz and a rain rate between 0 and 100 mm�h.
0.1 g cm�3, respectively. Snow dielectric constant hasbeen derived by a second-order Maxwell–Garnett for-mula for inclusions of air in an ice matrix. Ambient
temperature has been set to 10 °C for cloud dropletsand raindrops, 0 °C for ice graupel, and �10° for icecrystals and snow aggregates.
Looking at Fig. 1 and analogous results for visibleand infrared radiance in the presence of aerosols andclouds,6 the variability of input optical parameters ofa homogeneous slab has been discretized as shown inTable 1. We have chosen the strategy to stress theapproximate solution by letting the optical parame-ters vary without any physical constraint and corre-lation. In this respect, we can consider the resultsbelow as the worst case in a test scheme. Emphasiswill be given to the analysis of backscattered (or re-flected) specific intensity at the top of the medium,even though tests on the transmitted specific inten-sity have also been performed obtaining similar re-sults.
The accuracy of the generalized Eddington solutioncan arise from the quantitative comparison with RTEnumerical solutions, such as the DISORT model.16
The discrete ordinate model gives highly accurateresults in the solution of the RTE so that it is usuallyconsidered as a reference.24,25,28 In the followingtreatment, then, deviations of the analytical modelresults from the DISORT solution, obtained with anumber M of streams set to 48, are regarded as er-rors. All simulations are normalized to the incident
Fig. 2. Accuracy of the AERM solution, given in Eq. (25), illustrated by means of contour plots showing the PFE given in Eq. (34) as afunction of optical thickness � � �s and albedo � for asymmetry factor g equal to 0 (right column), 0.4 (middle column), and 0.8 (rightcolumn). Incident angles are �0 � 0.6 and �0 � 0°, and the backscattering (reflection) angles are � � 0.2 (upper row), � � 0.6 (middle row),and � � 1.0 (bottom row) and � � 90°. Discrete values of �s and � are those prescribed in Table 1.
Table 1. Discrete Values Given to the Optical Parameters (albedo �,asymmetry factor g, and optical thickness �s) and Observation
radiance by setting F0 equal to 1, whereas �0 is set to0 in all tests without loss of generality. A delta-MHenyey–Greenstein phase function, as given in ap-proximation (8), is used for the DISORT phase func-tion.
Differences between the backscattered intensities,calculated by means of each proposed model and theDISORT algorithm, are evaluated for each of the540,000 test cases of Table 1. These differences areexpressed as percentage fractional errors (PFEs) asfollows:
�f �IGEdd � IDISORT
IDISORT100, (34)
where IGEdd refers to AERM, FOSM, or GERM solu-tions, and IDISORT refers to the DISORT radiance so-lution.
B. Numerical Results
To illustrate the various steps of the proposed mod-els, a comparison between AERM and DISORT isfirst discussed. Figure 2 shows the accuracy of theAERM solution, given by Eq. (25), in terms of contourof the PFE as a function of optical thickness �s andalbedo � for asymmetry factor g equal to 0, 0.4, and0.8. Incident angles are �0 � 0.6 and �0 � 0°, and thebackscattering angles are � � 0.2, � � 0.6, and �� 1.0 and � � 90°. Discrete values of �s, g, and � arethose prescribed in Table 1.
We note that for � � 0.2 there is a systematicunderestimation (up to �65%), which converts to anoverestimation for � � 0.6 (up to 100%) and �� 1.0 (larger than 300%). AERM also shows a sub-stantial sensitivity to variations of optical parame-ters g and �; this model tends to have its greatestaccuracy for thick atmospheres and for nearly con-servative scattering conditions �� � 0.9�, especiallyfor backscattering zenith angles far from nadir (val-ues of � much smaller than 1). For brevity, we do notshow the results for decreasing values of �0, but theyconfirm the trends commented as above. AERM per-forms worst in the single-scattering limit, that is, �� 0.1 (Ref. 23); in these conditions, FOSM is expectedto be one of the most accurate among the approxi-mated models.
Therefore, before we illustrate the results relativeto the GERM algorithm, Fig. 3 shows the same as inFig. 2, but for the FOSM solution given in Eq. (27)and with incident angles �0 � 0.6 and �0 � 0°. TheFOSM solution systematically underestimates theDISORT results, even if it exhibits a smaller � sen-sitivity than the AERM method. For a given � and �,for any � the percentage error tends to becomegreater as g increases. Furthermore, FOSM has beenconfirmed to be a suitable method in the single-scattering limit, especially for scattering conditionsfar from the conservative case (small values of �). Toa certain extent, FOSM errors exhibit an error trendopposite to that of AERM; it should not be a surprise
Fig. 3. Same as in Fig. 2, but for the FOSM solution given in Eq. (27) and incident angles �0 � 0.6 and �0 � 0°.
that the combination of the two closed-form solutionscan give better results.
Figure 4 shows the same as in Fig. 2, but for theGERM solution given in Eq. (33) and incident angles�0 � 0.6 and �0 � 0°. To complete the analysis of thisintercomparison, Fig. 5 shows the same as in Fig. 4,but for incident angles �0 � 1.0 and �0 � 0°, and Fig.6 refers to incident angles �0 � 0.2 and �0 � 0°. Notethat for �0 � 1.0 the RTE solution becomes indepen-dent from �.
The comparison of Fig. 4 with Figs. 2 and 3 clearlyreveals the significant improvement gained by adopt-ing the GERM solution. Maximum error is containedwithin 25%–30%, except for high albedo �� � 0.8�,small scattering zenith angles �� � 0.2�, and opticaldepths near unity. The latter behavior is dominatedby the AERM solution: It is known from the literaturethat the Eddington approximation works worst in theneighborhood of optical depth unity.29 For low scat-tering conditions, instead, FOSM is predominant.The analysis of Figs. 5 and 6, characterized by �0� 1 and �0 � 0.2, respectively, confirms the above-mentioned overall error trends for the GERM solu-tion. Furthermore, in the worst conditions (i.e., for� � 0.8 and � near unity), Figs. 5 and 6 point out thatmaximum errors can be found when � � �0. Thisresult can be explained by noting that, when �� �0, the medium opacity tends to increase so thatthe diminished transmittance tends to reduce theoverall fractional error.
Finally, Fig. 7 illustrates the GERM behavior forextreme values of � and g and for an intermediatezenith angle ��0 � 0.6�. In these conditions, classifi-able as the worst case from a medium scattering pointof view, the GERM accuracy decreases and the PFEgrows to values of �70%. Again, the GERM accuracybecomes particularly worse for values of � near 1.
The angular dependence of the overall error budgetis shown in Fig. 8, where the mean relative error plusand minus its standard deviation band of the PFE areplotted for each zenith cosine angle � and azimuthangle �, obtained by performing all possible testsprescribed in Table 1 as in Figs. 2–7. From Figs. 2–7it emerges that the overall mean error is containedwithin �20%, with underestimations for � less than�0.8 �37°� and overestimations for � larger than 0.8.The latter behavior can be noted for the azimuthdependence as well for � � 150° and � � 150°, re-spectively. In both cases the standard deviation of thePFE is �15%. Analogous figures for the FOSM algo-rithm (not shown) yield a mean PFE of approximately�30%, and for AERM the mean error ranges from�100% to �100% as the angle increases.
As a summary for the comparison of AERM,FOSM, and GERM, Fig. 9 shows the relative accu-racy of these different algorithms in terms of a his-togram of the PFE obtained by performing all testsprescribed in Table 1. The overall PFE mean andstandard deviation are 10.0% and 152.8% for AERM,�24.1% and 37.9% for FOSM, and �9.4% and 20.1%
Fig. 4. Same as in Fig. 2, but for the GERM solution given in Eq. (33) and incident angles �0 � 0.6 and �0 � 0°.
for GERM. If a subset of scattering conditions char-acterized by � ranging from 0.1 to 0.9 is considered,leaving unchanged all the other optical parametervalues of Table 1 (i.e., 270,000 test cases), the overall
PFE mean and standard deviation come down to7.9% and 16.6% for GERM, respectively.
Such a level of accuracy in the radiance computa-tion can be considered acceptable for several applica-tions since errors and uncertainties in the measure ofthe absorbing and scattering properties of the me-dium can produce comparable errors in the retrievalof the radiance field.19 Numerical experiments havebeen carried out by supposing uncertainties in theknowledge of input optical parameters uniformly dis-tributed in the range 0%–10%. The true and the bi-ased intensities have both been calculated by theDISORT model, the biased values being computedusing input optical parameters affected by uncer-tainty. An analysis of the results shows that the stan-dard errors are �15% due to the nonlinearity of theRTE and thus are substantially comparable withthose obtainable from GERM.
As a final comment, it is opportune to remark thatthe above analysis has been conducted over a widerange of absorbing and scattering conditions of themedium, considering an exhaustive number of ob-serving geometries. Once the particular applicationrequiring the retrieval of the radiance field is speci-fied, a better accuracy can be obtained by restrictingthe analysis to the corresponding range of opticalparameters and viewing geometries. In particular, atuning of the exponent of the �0 term in the numer-ator of both the �1 and the �1 coefficients given byEqs. (18e) and (18f), in a way similar to that proposed
Fig. 7. Same as in Fig. 4 for the GERM solution, but with extreme values of � and g.
Fig. 8. Top panel: Overall accuracy of the GERM solution interms of moments of the PFE given in Eq. (34) as a function ofazimuth angle � by performing all tests prescribed in Table 1,derived from all possible combinations of optical parameter dis-crete values. Mean relative error plus and minus its standarddeviation band are plotted for each zenith angle �. Bottom panel:Same as in top panel, but as a function of azimuth angle �.
by Xiang et al.,28 allows us to improve the accuracy forthe specified application by halving the obtained er-ror results.31
4. Conclusions
A fast analytical approximated method of a solutionfor the RTE, based on the generalization of the Ed-dington approximation and capable of unfolding theazimuthal dependence of the radiance field, has beendeveloped in this paper. Performances of this newradiative model, called GERM, in simulating thebackscattered (reflected) intensity field due to aplane-parallel homogeneous medium excited by a in-cident radiation, have been evaluated by comparisonwith the DISORT model.
The consequences of this approximation used toexpress the radiance field are mean percentage errorsin the intensity calculations less than �10% with astandard deviation of �20% over an extremely widerange (540,000 samples) of independent values ofsingle-scattering albedo, asymmetry factor, opticaldepth, and incident radiation direction. The signifi-cance of such errors is less relevant in problemswhere the input parameters are known with a certaindegree of uncertainty. In particular, it has beenshown that the related level of accuracy is sufficientwhen the uncertainties in the knowledge of the opti-cal parameters are uniformly distributed in the range0%–10%.
The overall accuracy is comparable, if not better,than available azimuthally dependent analyticalmodels such as those by Davies19 and Xiang et al.,28
based on the Eddington and Sobolev approximations,respectively. A thorough comparison might be possi-ble even though each analytical model tends to use adhoc corrections to improve the numerical accuracy. Itis worth mentioning that the results shown in Section3.B are obtained without any specific tuning of themodel parameters. This approach, justified by theneed to illustrate results depending as less as possi-ble on arbitrary choices, opens, indeed, a wide possi-bility in the refinements of GERM for specificapplications.31
The worse accuracy in reproducing the radiancefield, as compared with the DISORT model, is obvi-ously balanced by a more rapid calculation of theradiance itself. In this respect, we note that the com-putational time of the DISORT model is approxi-mately proportional to N2, with 2N the number ofstreams, whereas the generalized Eddington modelhas a constant computational complexity. This fea-ture makes the proposed GERM well suited to prob-lems that require iterations over finite spectraland�or time intervals. Moreover, as mentionedabove, for a specific application requiring the compu-tation of the radiance field, a better accuracy can beobtained by restricting the analysis to the corre-sponding range of optical parameters and viewinggeometries.
The main theoretical limitations of GERM are re-lated to the assumption of (i) stratified medium ge-
Fig. 9. Comparison of relative accuracy of AERM (top panel),FOSM (middle panel), and GERM (bottom panel) solutions interms of a histogram of the PFE given in Eq. (34) by performing alltests prescribed in Table 1 from all possible combinations of opticalparameter discrete values. Mean relative error and its standarddeviation are also indicated.
ometry and (ii) unpolarized radiation. With respect tothe first item, the plane-parallel geometry allows usto simplify the general expression of the RTE in athree-dimensional (3-D) space and to unfold the an-gular dependence of radiance through the general-ized Eddington approximation. A way to applyGERM to a 3-D problem is to resort to a one-dimensional (1-D) slant geometry along the viewingangle, that is to construct an equivalent 1-D problemfrom the given 3-D one.9,32 The limitation due to thehypothesis of unpolarized radiance can be removedonly if the elements of the rotated phase matrix canbe approximated by means of a Sobolev phase func-tion as in Eq. (12). The validity of this approximationwould, of course, depend on the properties of the par-ticle distribution.
Future work will be devoted to possibly extendGERM to a polarized radiation and to perform a sys-tematic comparison with other numerical solutionssuch the finite-element method10 and available ana-lytical approximated models. Application to the re-mote sensing of the atmosphere is also foreseen, andits tests could be carried out by following an approachsimilar to that of Smith et al.25
Appendix A: Boundary Conditions
In Subsection 2.B it was shown that computation ofthe integration constants C00, C01, C10, and C11, rela-tive to the radiance general solution, is subject tosome approximations. In this appendix we point outthe proposed approach in an explicit way.
The choice of the above-mentioned approximationscan be justified through a direct comparison withthose proposed by Xiang et al.28 The latter boundaryconditions, expressed in terms of irradiances, assumethe following form:
�0
2� ��1
1
I(0, ��, ��)d��d��
� �2 �0
2� ��1
1
I(0, ��, ��)��d��d��, (A1)
�0
2� ��1
1
I(�s, ��, ��)d��d��
� �2 �0
2� ��1
1
I(�s, ��, ��)��d��d��, (A2)
�0
2� ��1
1
I(0, ��, ��)(1 � ��2)1�2 cos ��d��d��
�23 �
0
2� ��1
1dI(�, ��, ��)
d� ���0
(1 � ��2)cos ��d��d��, (A3)
�0
2� ��1
1
I(�s, ��, ��)(1 � ��2)1�2 cos ��d��d��
� �23 �
0
2� ��1
1dI(�, ��, ��)
d� ����s
(1 � ��2)cos ��d��d��. (A4)
These expressions have a general validity, so theycan be applied to the generalized Eddington modelsolution, expressed by Eq. (19).
Thus, substituting into Eqs. (1) the expression ofradiance given by Eqs. (10), one obtains
�0
2� ��1
1
{[I00(0) � ��I01(0)] � [I01(0)
� ��I11(0)]cos ��}d��d��
� � 2 �0
2� ��1
1
{[I00(0) � ��I01(0)] � [I10(0)
� ��I11(0)]cos ��}��d��d��, (A5)
and then integrating, we obtain
I00(0) � �23 I01(0). (A6)
Starting from approximation (A2), with analogousmathematical passages, we obtain
I00(�s) �23 I01(�s). (A7)
Inserting now the expression of radiance into approx-imation (A3), the latter can be rewritten as follows:
�0
2� ��1
1
[I10(0) � ��I11(0)](1 � ��2)1�2 cos2 ��d��d��
�23 �
0
2� ��1
1dI1(�, ��)
d� ���0
(1 � ��2)cos2 ��d��d��,
(A8)
where the intensity terms, multiplied by cos �, havebeen omitted for brevity as they cancel when inte-grated over �.
Considering Eqs. (10c) and (18), it is simple to ver-ify that
In the same way, approximation (4) produces thefollowing relation:
I10(�s) �23 I11(�s). (A11)
Equations (A6), (A7), (A10), and (A11) are analogousto the approximations expressed by Eqs. (22),adopted in Subsection 2.B for the calculation of theconstants C01, C02, C11, and C12 and consequently forthe computation of the radiance field.
Appendix B: Generalized Solution for VerticallyInhomogeneous Media
Electromagnetic propagation in a vertically inhomo-geneous random medium is usually approached byapproximating the medium through a series of homo-geneous adjacent layers (e.g., Ishimaru,13 Stamnes etal.30).
In particular, media can be represented with anumber Nl of plane-parallel layers, as shown in Fig.10. Each layer is characterized by different values of�i and gi, with
d�i
d��
dgi
d�� 0, i � 1, 2, . . . , N. (B1)
Thus the solution of the RTE through the generalizedEddington approximation is appropriate within eachlayer. For the ith layer with optical thickness �s
i
� �i � �i�1, the RTE allows for the following analyticalsolution:
Ii(�, �, �) � I00i(�) � �I01
i(�) � [I10i(�)
� �I11i(�)]cos �, �i�1 � � � �i,
(B2)
where �0 � 0 and �N � �s, and the general solution isgiven by
I(�, �, �) � Ii(�, �, �), �i�1 � � � �i. (B3)
Furthermore, it holds that
I00i(�) � C01
i exp(�k0i�) � C02
i exp(k0i�)
� �0i exp(����0), (B4a)
I01i(�) � p0
i[C01i exp(�k0
i�) � C02i exp(k0
i�)] � �0i
exp(����0), (B4b)
I11i(�) � C11
i exp(�k1i�) � C12
i exp(k1i�) � �1
i
exp(����0), (B4c)
I10i(�) � C11
i exp(�k1i�) � C12
i exp(k1i�) � �1
i
exp(����0), (B4d)
where
k0i � [3(1 � �i)(1 � gi�i)]
1�2,
p0i � [3(1 � �i)�(1 � gi�i)]
1�2,
�0i �
3�i
4�F0�0
21 � gi(1 � �i)
1 � (k0i�0)
2 ,
�0i �
3�i
4�F0�0
1 � 3gi(1 � �i)�02
1 � (k0i�0)
2 ,
k1i � �31 �
3�2
32 gi�i�1�2
,
p1i � 1�k1
i,
�1i �
916 �i�0F0gi(1 � �0
2)1�2
1 � (k1i�0)
2 ,
�1i � �0�1
i.
Fig. 10. Representation of an inhomogeneous random mediumwith a plane-parallel geometry with N homogeneous scatteringlayers characterized by albedo � and phase function P.
Calculation of the four unknown constants C00i, C01
i,C10
i, and C10i for each of N layer needs 4N equations.
Four of these equations derive from the boundaryconditions given below:
I1(� � �0, �, �) � 0, (B5a)
IN(� � �N, �, �) � 0, (B5b)
where � is fixed to 2�3, while the remaining 4N� 4 are determined by requiring that I00
i, I01i, I10
i, andI11
i are continuous:
�I00
i(�i) � I00i�1(�i)
I01i(�i) � I01
i�1(�1)I10
i(�i) � I10i�1(�i)
I11i(�i) � I11
i�1(�1)
i � 1, 2, . . . , N � 1. (B6)
This work has been partially supported by the Ital-ian Space Agency; by the Italian National ResearchCouncil through the National Project on Preventionfrom Hydro-Geological Disasters (GNDCI); and bythe Italian Ministry of Education, University and Re-search. The GERM code is available from the authorsupon request.
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