a modified fourier transform method for multiple...

A Modified Fourier Transform Method for Multiple ScatteringCalculations in a Plane Parallel Mie Atmosphere

J. V. Dave and J. Gazdag

A method for evaluating characteristics of the scattered radiation emerging from a plane parallel atmo-sphere containing large spherical particles is described. In this method, the normalized phase functionfor scattering is represented as a Fourier series whose maximum required number of terms depends uponthe zenith angles of the directions of incident and of scattered radiation. Some results are presented toshow that this method can be used to obtain reliable numerical values in a reasonable amount of computertime.

1. Introduction

Reliable evaluation of the characteristics of the radi-ation scattered by a planetary atmosphere runs intoserious difficulties, especially when the atmosphericmodel under investigation is nonhomogeneous in thevertical and contains particles that are large comparedto the wavelength of interest. Basically, the computa-tional problem reduces to that of performing severaliterations and that of evaluating several thousands ofintegrals at each stage of iteration. When the normalscattering optical thickness of the atmosphere is of theorder of unity, one may have to deal with data sets con-sisting of several million words and may have to per-form anywhere from two to twenty iterations dependingupon the desired accuracy. Even though one mightobtain a reliable numerical solution for a single case atthe cost of several hundred hours of computer time, theresults so obtained are of very little practical value un-less further results are also obtained for several hundredother cases generated by varying input parameters.Thus, it is essential to develop and to use computationalprocedures which can provide results of highest reli-ability with a minimum of computer time. Because ofthis, considerable effort has been directed toward de-velopment of acceptable models in which the scatteringphase function and/or the vertical nonhomogeneity areso approximated as to permit some analytic reduc-tion.'- 3 However, such attempts have met with onlylimited success.

A direct numerical solution of the equation of radia-tive transfer for a plane parallel, nonhomogeneous Mie

The authors are with the IBM Scientific Center, Palo Alto,California 94304.

Received 3 December 1969.

atmosphere without any approximation to phase func-tion and/or vertical nonhomogeneity can be obtainedby appealing to Monte Carlo methods. However, be-cause of the excessive computer time requirements,published results4 -6 are of marginal reliability at themost and fail to throw much light on the effect of mul-tiple scattering on some salient features of the scat-tering phase function. In fact, it appears that muchmore reliable results can be obtained by using a straight-forward computational method in which all three inte-grations (over azimuth angle, over zenith angle, andover optical depth) are carried out numerically withrather crude integration increments, e.g., 300 overazimuth. 7 -9

The computational task is considerably simplified byexpressing the phase function in the form of a fourierseries in terms of the difference between azimuth anglesof the directions of incidence and of scattering. Whenthis can be done, integration over azimuth can be car-ried out analytically. This has been used with greatadvantage for obtaining numerical solutions to multiplescattering problems in plane parallel, molecular atmo-spheres where the fourier series can be terminated afterthe first three terms. When particles large comparedto the wavelength of interest are present (Mie scatteringif particles are assumed to be spherical), a meaningfulrepresentation of the phase function is obtained onlyafter including several tens to several hundreds of terms,depending upon the model. Hence, a Fourier seriesapproach to multiple scattering calculations in thepresence of large particles has been generally thought ofas unattractive (e.g., Ref 10).

Chandrasekhar" showed that a solution of the equa-tion of transfer for a plane parallel, homogeneous atmo-sphere can be obtained in the nth approximation by ex-pressing a general phase function in the form of a Fourierseries. Twomey et al.12 used the matrix method (or anequivalent of van de Hulst's doubling method) cou-

June 1970 / Vol. 9, No. 6 / APPLIED OPTICS 1457

pled with a Fourier representation of the phase functiontechnique to simplify the task of multiple scatteringcalculations for a homogeneous case. However, almostall of their published results 2 '4",5 are for the first termof the fourier series only, i.e., for the intensities of theemergent radiation averaged over azimuth. Because ofthis usefulness of their published results is very limited.Very recently, Hansen 6" 7 has published some results ofhis extensive computations of radiation emerging fromplane parallel, homogeneous Mie atmospheres usingVan de Hulst's doubling method. He has expanded thephase function in a fourier series with as many as 180terms for some models which contain fairly large par-ticles and has used all these terms in his computations.

The purpose of this paper is to show that multiplescattering calculations can be simplified still further byusing a modified form of the Fourier series in which ad-vantage is taken of the fact that the maximum requirednumber of terms in the series is also a function of zenithangles of the directions of incidence and scattering.The results of the scattered radiation emerging fromhomogeneous Mie atmospheres presented here are ob-tained by using an iterative procedure in which the nthapproximation amounts to using up to and includingnth orders of scattering. 8" 9

11. Modified Fourier Series for thePhase Function

A. General

Let a direction be represented by its zenith angle 0-[=cos-f,411 and its azimuth angle so referred to anarbitrary verticle plane. Let (p) and (i",so') be,respectively, the directional parameters of the scat-tered and of the incident radiations. The normalizedscattering phase function P[u,,,(o' - so)] for a unitvolume containing an arbitrary size distribution ofspherical particles of known refractive index can be ex-pressed in the form of a fourier series as follows:

N

P[1A'1,(s'P- so)] = E F,(ln',j) cos(n - 1)(so' - so). (1)

It should be noted that the coefficients F0 (,u',u) are alsofunctions of the size distribution parameters and the re-fractive index m of the material of the sphere with re-spect to its surroundings. N, the maximum number ofterms after which the series can be terminated, is ap-proximately 2x2 + 10, where x2 is the size parameter ofthe largest particle in the model (x = 27rr/X, r is radius,and X is the wavelength of the radiation under investi-gation). For Rayleigh scattering, N = 3 and the co-efficients are expressible in terms of ,z and "' as follows:

F(,(,u) = (3 - _ u" + 3jut'"), (2)

F2(A",)= E At'(j -2)i(l - g12)l, (3)

and

F3(A',A) =g(1-U2)(1 - (4)The highest number of terms ( . 2x2 + 10) in the

series should be required for the case A = .t' = 0, i.e.,

when one has to make use of the entire phase functioncurve from scattering angle e = O to e = 1800. Thephase function P [u, p' - so) ] is fully representable byonly one term of the series when A and/or A' = i 1, i.e.,N is equal to unity. For intermediate cases, all Fn(u",u) do exist, but F0(,A') decreases with n, and henceit should be possible to represent P (p' - ) ] withsome desired accuracy by N(,4',') number of terms suchthat N(,4',u) is less than N. Thus in the modifiedfourier method, we shall represent the normalized phasefunction for scattering as follows:

N(O',O)P[,,(so '- so)] = E F(O',O) cos(n - 1)(o' - so). (5)

n= 1

The parameters of N and F n are changed from gi' andu4 to 0' and 0, respectively, as subsequent multiple scat-tering calculations are performed with equal incrementsin zenith angle.

Computation of Fn(O',0) and N(6',0) were carried outfor two different size distribution models referred to ashaze M and cloud by Deirmendjian.' 0 The size distri-bution functions and lower (r,) and upper (r2) limits ofintegration used in these computations are as follows:

haze M1: n(r) = 5.33 X 104r exp[-8.944(r)il,

r = 0.001 u, and r2 = 7.0 p, (6)

and

cloud: n(r) = 2.373r6 exp(-1.5r),

r = 0. ju andr, = 14 .0p. (7)

B. Method of Computations

Values of the normalized phase function at 0.2°interval in 0 were first obtained by following a proce-dure outlined elsewhere.2 ' An integration increment of0.1 in x was used for this purpose. Values of Fn(O',O)and N(O',O) were then obtained from these values byfollowing the procedure outlined below:

(1) For a given set of g',j', values of P [uIA', (so' - so) ]for 2 + 1 equally spaced values of so' - so in the range0°-180° are obtained by linear interpolation.

(2) The values obtained from step (1) are then usedto compute 2J + 1 coefficients of the fourier series usingsubroutine HARM.2 2

(3) A procedure is then set up to determine howmany of the 2J + 1 coefficients are necessary to regen-erate 2i + 1 values of P [,L, ,(so' - so) ] with the desiredaccuracy. If this cannot be achieved by using the first20>-) coefficients, a message to that effect is printed out,and the computations are terminated.

(4) After completing the computations for all ucombinations, the data is rearranged so as to permit ac-cess to all coefficients for a given frequency n by issuingone READ command.

For the case haze M and X = 0.75 ,(x2 - 58.5 and

1458 APPLIED OPTICS / Vol. 9, No. 6 / June 1970

zo MODEL. HAZE M

108 _ \ X=0.751,Z -= 134-O.OiLU.Lii

LiI 10°

N

cr 10- I I I , I

0

-(2

0 40 80 120 160SCATTERING ANGLE

Fig. 1. Normalized phase function for scattering by a polydis-persed water spheres model haze M, X = 0.75 u, m = 1.34.

N *. 130) for which variations in phase function vsscattering angle are shown in Fig. 1, it was possible toregenerate P[,t',(so' - so) ] within 0.1% accuracy forall 0',0 combinations given by 0' = 0(20)1800 ando = 00(20)900. The CPU time for all these computa-tions using an IBM System/360 Model 91 computer wasabout 8.5 min when a value of 8 for j was used. Theaccuracy with which such regeneration can be achieveddecreases with X and also when one switches from thehaze M to the cloud model. For the case cloud, X =0.45 ,u (x 2 * 195, N . 400), P[u,',(so' - so)] could beregenerated within 4-.3% only. Such a decrease instability is partly due to errors inherent in the originalscattering function vs scattering angle table.

C. Results

F0 (6',6) for values of 6 greater than 90° can be obtainedby making use of the following relationships:

N(O',o) = N(180 0- O', 180 - 0), (8)

and

Fn(O',0) = F&(1800 - 0', 180° - ). (9)

D. Comparison with Results for a Single Particle

The results presented in Figs. 2 and 3 are importantenough to warrant a further investigation into thecauses which lead to such trends. Consequently, afterthis study was completed, a considerable amount ofwork was done to compute values of Fn(6',) and N-(6',6) directly for a single sphere. The results of thislater work are presented elsewhere.2 3' 24 It was foundthat as in Fig. 2, the value of F,(90',90') for a spherewith size parameter x = 100.0 is less than that of F2-(900, 900). However, a further increase in n results ina much less rapid decrease in Fn(900 ,900 ) than the oneshown in Fig. 2. This effect results from integrationover the size distribution.

Values of N(6',6) for a single sphere were obtaineddirectly from computed values of Fn(6',) by using asomewhat different but essentially equivalent criterion.Plots of these values of N(6',6) as a function of 6' do notshow two sharp peaks as is the case of polydispersedaerosol model haze M, X = 0.75 u (Fig. 3). For a singleparticle, the N(6',6) vs 6' curve depicts a broad max-imum around 6' = 900 whose width decreases withincrease of 0. The values of N(900,6) increase with0. This difference between the shapes of N(6',6') vs 6'curves for a single particle and those for polydispersedaerosol model can be partly understood when one takesinto account the fact that most of the ripple-likeMie scattering features disappear upon integrationover a size distribution. This then implies that the

Values of Fn (',6) for = 900 and for ' = 100, 400,600, 800, and 900 are plotted in Fig. 2 as a function ofthe subscript n. For the case haze M, X = 0.75 u, theentire phase function of Fig. 1 can be reproduced within-40.1% accuracy with not more than 101 terms. Thisdecrease in the value of N from 130 to 101 is due to in-significant contributions from extremely large particlesin the model under the adopted accuracy criterion.From Fig. 2, it can also be seen that N(6',90') de-creases from 101 to about 5 as 6' is decreased from 900to 100.

The variations of N(O',6) as a function of ' arepresented in Fig. 3 for 0 = 200, 40°, 600, and 800. For6 = 00, N(6',6) = 1 for all values of 6'. The strikingdependence of N(O',6) on the zenith angle of the inci-dent and of the scattered radiation is very evident. Allthe curves show two maxima; a strong one at 6' = 0, acase for which the scattering angle 0 varies from 0 to20, and a weak one at 6' = 1800 - , a case for which 0varies from 1800 to 180 -2 0. Values of N(6',6) and

l0

-

10

Fig. 2. Variations as a function of subscript n, of the coefficientsFn(0t',) of the fourier series for the normalized phase function in

Fig. 1.


102_

IC

6

6

4

2

8

,

2ii

4

2

2

0 20 40 60 80 100 120 140 160 180°ZENITH ANGLE OF THE DIRECTION OF THE SCATTERED RADIATION (0')

Fig. 3. N(0',0), the number of terms in the fourier series [Eq.(5)] necessary for reproducing a section of phase function curve(Fig. 1) between 0' - | and 0' + 0 within 40.1% accuracy.

shape of the N(',6) - 6' curve is also dependentupon size distribution parameters.

Ill. Multiple Scattering Calculations

A. Necessary ExpressionsThe equations for the transfer of solar radiation

through a plane parallel, nonabsorbing homogeneousatmosphere of finite optical extent in the vertical, butof infinite extent in the horizontal direction, has thefol owing form":

1 (+ 1 r2+ 4 j j P[py (' - )I(r;',&)dp'd&. (11)It is assumed that the atmosphere is illuminated atT = 0 by unidirectional solar radiation of 7rF unitsper unit area normal to the direction of incidencerepresented by -osoo. (F is taken to be unity forthe computations presented in this paper.) Theboundary conditions are

(12)I(r,; +±pso) a 0

which means that there is no scattered radiationilluminating the atmosphere from above its top(T = 0), or from below its bottom ( = 7r).

When the findings of Sec. II are substituted in Eq.(11), we have the following expression for the firstapproximation of the source function given by thefirst term on the right-hand side of Eq. (11):

N(Oo)

Ji(r; -l4p) = E J1(',)(r; 1 -i-Ao) cos(n - 1)( - o), (13)n=w1

where Jl(n)(r; +g, -go) = Fe-1/-Fn(180O -0,Oo) (14)

and

Ji(n)(r; -, -) = Fe -/izoFn(090o). (15)

N( 0o) is the maximum value of N(6,6o) vs. curvefor the given solar zenith angle Oo. From Fig. 3, wehave N(0o) = N(0 ,,0o). It is true that the numberof terms in the series of J [Eq. (13) ] is a function ofboth and Oo. However, even though it is possible togenerate high-frequency components upon multiplescattering, in no case can the number of terms ex-ceed the highest value N(6o) which is determined bythe position of the sun and the atmospheric modelunder investigation. The intensities of the radiationemerging after one scattering are then given byIPn)(r; lp, -o) = I,(&)(r Ar; dp, -o)e b

+ J,(n)(r; u, -po)(l - e /"). (16)

Equation (16) is arrived at after dividing the homo-geneous, nonabsorbing atmosphere into severallayers, each of scattering optical thickness ATr.J1(n) is the mean value of the source function in thelayer bounded by and - Ar or T + AT as the casemay be. From Eq. (12) we have,

dI(r;p,) = I(r; ) - J(T;g"P),dr (10)

where I(T;uso) represents the intensity of the scat-tered radiation emerging at the optical depth in thedirection whose parameters are ,u,so. A minus(plus) sign before A implies that the scattered radi-ation is propagating downwards (upwards) in theatmosphere. The source function J(T;,so) is givenby

J(r;.u,v) = Fec/PoP[g, -o,(vo - )]

Il(,)(O; -, -co) = IP)(rc; +p, -o) = 0. (17)

Evidently,

N(Oo)

I(r; 4p,ip) = E Il(')(r; 4±u, - o)n=1

X cos(n - 1)(Po - ). (18)

The values of I, given by Eq. (18) and those ofP [uAt',(so' - s) ] given by Eq. (5) are then substitutedin the double integral on the right-hand side of Eq. (11).


l I l I

j MODEL: HAZE M

'0 _ -

I \' I- \

~~~~~O ~ ~ ~ O

_ .. - S~~~=8o°

20._ -- ...........:0

10

4 - \,

J '. S~=40'

20 - --- *-------- ~~~~~~~~~~~~~...---.......

20 - 9/ 8=20 -

In *-----j--....... I I I ....... .. ...1 II

2

After integrating over s' from 0 to 27r, we have thefollowing result:

1 1 2 p -

f7 1 [Jo) (1 ± ( - )]Im._(r;A,p)d ', 0ddp

N(0o) ]4 Ad + 5ln) Fn(-A,,u)I._j1,(,r; -,u -A)d~ul

+ E (1 + n) Fn(PAP)Im._j()(;' uo)dAn=1 fOI

X cos(n - )(po v-). (19)

As discussed above, Eq. (19) is valid for m = 2 only.However, by repeated substitution, it can be shownthat Eq. (19) as well as all subsequent equations arevalid for m > 2. The quantity 1n is the Kroneckerdelta function given by a1 = 1 when n = 1 and other-wise zero.

For integration over ,A', the interval of integration isdivided into equal increments in 6'. This is advisablefor large particle scattering where the properties varyrapidly with scattering angle. This change in variablecan be accommodated by use of the trapezoidal rule ofintegration with unequal intervals in At'. We thereforehave

N(OO)

Jm(r; Jpolf) = E Jm.n)(r; h, -o)n= 1

X cos(n - )(,oo - so), (20)

where

J.1n)(,r; -A, -o) = Jn)(r; -U, -UO)1 i4(n,O)

+ 4 (1 + 5on) E (F(180o - 0'i,0)Im..L(n)(r; +Pi',i -u.))i= i3(nO)

1 iL(n,o)

X (i - 'i+l) + (1 + 3en) E (Fn(0'i,0)Imi(n)(r; -pli,i (no)

-toN.4 - P'ii+) (21)

and

Jm(n)(; ±,t - Po) = Jl(,)(r; +p, -0uo)1 Wi(no)

+ (1 + an) E (Fn(0i',0)Im.l(n)(T; +Ai, -,AO)) (i- Ali+' )i = il (n,O)

1 i4 (n.,O)+ (1 + d1n) E (Fn(180o - i',0)I..(i)((; -i'

i i(n,o)-Po))(i' - pj+1). (22)

The angle brackets around the quantities under thesummation sign represent the mean value of the quan-tity for the interval uj' - At'i+ 1. The subscript i is as-sumed to increase continuously from unity at zenith(or nadir) to its maximum value at the horizon. Thevalues of the limits, i.e., i(n,6), etc., can be found byconsulting variations of N(6',6) vs 6'.

Having obtained values of J2 (n) (; = ,, -o), thevalues of I2()(,r; ±au, -Ao) and12(r; 4A,so) can be ob-tained by making use of the equations obtained afterreplacing subscript 1 by 2 in Eqs. (16)-(18). I(T;-Atso) so obtained contains radiation which emergesafter suffering up to and including two scatterings.

Equations (20)-(22) and the appropriately modified

forms of Eqs. (16)-(18) can then be used to include fur-ther higher orders of scattering (m > 2).

B. Computational Details

The equations given in Sec. III.A were used to com-pute the intensity of the radiation emerging at variouslevels of a plane parallel, nonabsorbing, Mie atmo-sphere. For this purpose, the atmospheric model wasdivided into several layers, each having a thickness(Ar) of 0.01. For integration over At', the angular re-gion from zenith to nadir was divided into 20 intervals(AO). Values of the quantity Jm(n)(T; -4t, -o) wereobtained by taking the mean values of the appropriatesource function [Eqs. (20)-(22) ] at the top and at thebottom of the atmosphere. The values of Im(n)(7;0.0, -/uo) were obtained by linear extrapolation. Theiterative procedure outlined above and used in thesecomputations is referred to as a successive scatteringiterative procedure, since each higher approximationamounts to including one more order of scattering.The iteration procedure was terminated when suc-cessively iterated values of I(n) (,1; t, -to) and 1 _ (n)_

(7,; At-/o) converged within 0.1% for all values of a.The upper section of Fig. 4 shows as a function of

frequency (n) the number of iterations (m) required forachieving the desired convergence of the downwardradiation at the bottom of the atmosphere when r =1.00 and O = 600. The quantity m decreases rapidly

E 20

0

X 2 10

z WI- o

0ZW

2a:LIWZ

0

U_0

cnZ01-1(L2

R

0:

2

U)0ZRU,

0)

Z

WMI-

0 I .......0'0 20 40 60 80

FREQUENCY (n)

Fig. 4. Variations of the number of iterations, and CPU timefor computations for a given frequency as a function of frequency

n.


.... . . .................... . . .....I I . .....T................... ---.........

0 m= 1.34

0 - T '

C,

z

N~~~~~~~~~

0w

0

poyipesos Model hae , =0.75a .4-.i

z

pn =

Q001~~~~~~~

0 20 40 60 801NADIR ANGLE OF THE DIRECTION OF THE SCATTERED RADIATION (8)

Fig. 5. Variations of the intensity of the radiation backscatteredby a plane parallel, homogeneous atmosphere containing sphericalpolydispersions. Model haze M, X = 0.75 pA, m = 1.342 - 0.0i.The x axis represents the nadir angle of the emergent radiationin a vertical plane making an angle of 1800 with the sun's verticalplane. o = 60'. The broken curve (marked P) represents thescattering property of a unit volume. Diff erent curves are forthe atmospheric models with different values for their normalscattering optical thicknesses. Solar flux incident on the topof the atmosphere = r units per unit area normal to the direction

of incidence.

10.0

Z0

I-Z0

PW(n2:(La_jN

Ma:Z0

0:0

ZCf)WWZW

WCr

1.0

0.1

0.0 r , I I I I0 20 40 60 80°

ZENITH ANGLE OF THE DIRECTION OF THE SCATTERED RADIATION (e)

Fig. 6. Same as Figure 5 but for the downward radiation emerg-ing at the bottom of the atmosphere. The x axis represents thezenith angle of the emergent radiation in the sun's vertical plane.

from 15 at n = 1 to a value of about 6 at n = 10. Itshows very little change for further increase of n from10 to 55. For still higher values of n, m = 2. Suchdependence of m on n can be anticipated from similarresults for Rayleigh scattering calculations publishedby Dave and Walker. 19

The CPU time for computations at one frequency de-creases much more rapidly with increase of n (lowersection of Fig. 4). When computations are performedwith AO = 2 and AT = 0.01, about 98 see of CPU timeare required for computations at n = 1 on an IBVISystem/360 Model 91 computer. For n = 10, it isabout 20 sec. Because of a rapid decrease in numericalwork involved in the computations of the source func-tion, CPU time decreases rapidly from 20 sec to 4 seein the frequency domain where m is constant. For stillhigher values of n, it takes about 0.4 see of CPU timefor all computations for a given n.

The total CPU time required for the entire calcula-tion is about 3 min for 6o = 00, for which it is necessaryto compute at one frequency only, and about 13 minfor O = 600, for which computations have to be car-ried out for eighty-nine frequencies. This total CPUtime includes the following: (1) computations of up-ward and downward fluxes at 101 levels; (2) intensitiesof the upward and downward radiation at = 0, 71/2and T, for 0 = 0(20)900 and soo - so = 0(10)1800, and(3) execution of a very substantial number of printoutstatements.

It was noted that the time quoted above decreases byfactors of about eight and twenty when computationsare carried with 6 and 100 increment in the zenithangle, respectively. This time can be reduced stillfurther by relaxing the present convergence criterion.

C. Tests for ReliabilityThe reliability of the numerical results obtained with

the present method can be checked to some extent bycomparing the values obtained for a Rayleigh case withthose obtained with Chandrasekhar's method" (X andY functions) where integrations over both optical depthand azimuth are performed analytically. However, thepublished numerical results using Chandrasekhar'smethod are for a Rayleigh atmosphere in which thepolarization of the scattered radiation is also taken intoaccount, i.e., a phase matrix treatment. Apart fromthe significant differences between the values obtainedwith the phase matrix and phase function treatments,reliable comparison is hampered due to the fact thattabulated values are generally not available for the de-sired parameters. Because of these reasons, radiationemerging from an atmosphere scattering according to aRayleigh phase function was calculated by modifyinga similar Rayleigh phase matrix computer program dis-cussed elsewhere.25

Some results of a comparison are presented in TableI for the radiation emerging from the bottom of a Ray-leigh atmosphere with T = 1.0 and illuminated by thesun at 860 from the zenith. For a meaningful com-parison, it is desirable to select a case like this where thecontribution due to higher orders (m > 1) of scattering


Table 1. Intensity of the Radiation Emerging at the Bottom ofa Plane Parallel, Nonabsorbing, Rayleigh Atmosphere

as Obtained Using Three DifferentComputational Proceduresa

Method

Chandrasekhar's Modifiedmethod fourier

Phase Phase transform0 p matrix function method

0.0° 1.00 0.01571 0.01755 0.0174820.0 0.94 0.01785 0.01893 0.0188636.0 0.81 0.02150 0.02127 0.0212044.0 0.72' 0.02360 0.02254 0.0224648.0 0.67 0.02457 0.02308 0.0230060.0 0.50 0.02632 0.02360 0.0235064.0 0.44 0.02612 0.02307 0.0229570.0 0.34 0.02436 0.02098 0.0209472.0 0.31 0.02344 0.02002 0.0198978.0 0.21 0.01922 0.01594 0.0157682.0 0.14 0.01601 0.01305 0.0129586.00 0.07 0.01358 0.01103 0.01096

Note: Computations using Chandrasekhar's method werecarried out for the zenith variables u and Pu, while those using themodified Fourier method were performed for the zenith variables0 and 0o.

a ri = 1.0, o = 86.00, uo = 0.07, so - = 00.

is at least two to three times that due to primary scat-tering alone.26 From Table I it can be seen that nu-merical solutions of the radiative transfer equation as ob-tained using the Rayleigh phase matrix (column 3) andthe Rayleigh phase function (column 4) can differ by asmuch as 23% in some cases. Hence, for a meaningfulcomparison, it is essential to use exactly the same scat-tering law for all test cases. Because of the differ-ences in quadrature procedures, values given in Table Ifor Chandrasekhar's method are for the zenith dis-stances given by 4 and 4o, while those for the methoddescribed in this paper are for zenith distances given by0 and Oo. The values of 0 as obtained from cosiAt differby about ± 0.10 from the corresponding values given incolumn 1. A comparison of the values given in the lasttwo columns shows that the values for a Rayleigh caseas obtained from use of the modified fourier transformmethod are good to three significant figures when valuesof 0.01 and 20 for integration increments over and 0,respectively, are used.

The above finding may not hold true for computa-tions in Mie atmospheres where the phase functionchanges much more rapidly with the scattering angle.Some confidence in this direction can be gained by ex-amining the net flux c1(r) given by

¢(T) = rpoeFT/Y£ + 2v f I(0)(T; - A, -po)pdA

I- 27rf I(°)(r; +,u, -o),dA. (23)

For a nonabsorbing atmosphere @b(T) should be inde-

pendent of . For a zenith sun, the mean value of!(r) decreases from about 3.0 to 2.1 for the Rayleighatmosphere, and from about 3.1 to 2.9 for the Mie at-mosphere as rl is increased from 0.1 to 1.0. A decreasein the value of (r) with r is due to an increase in thereflecting power of the atmosphere with rl. Because ofthe strong forward peak in the phase function vs scat-tering angle diagram (Fig. 1), the reflecting power of aMlie atmosphere is somewhat smaller than an equiv-alent Rayleigh atmosphere. For 6 = 600, the meanvalue of P(T) decreases from about 1.4 to 0.8 for theRayleigh atmosphere, and from about 1.5 to 1.2 for theMie atmosphere as 7iis increased from 0.1 to 1.0.

In Table II, we have presented the difference be-tween the maximum and minimum values of (r) forri = 0.1 and 1.0, and 6o = 0 and 600 when computa-tions are performed with different integration incre-ments in r and . This difference can be thought of asan upper limit of absolute error. When the incrementATr is halved from 0.01 to 0.005,this difference remainsunaffected. Furthermore, when a value of 20 for A isused, the highest error of about one part in 300 is en-countered for case B, T = 1.0 and 6o = 00. Thus, itappears that an increase in value of Ar from 0.01 to0.02 should also give fairly reliable numerical results8 ifthe atmosphere is homogeneous and T is of the order ofunity or less.

An increase in the value of AO from 20 to 100 resultsin about a thirty-fold increase in the values of thisdifference. The worst case is encountered when a Mieatmosphere with Tr = 1.0 is illuminated by the sun atzenith. This is because of the appearance of a haloaround the sun. In the Mie case, errors decrease withincrease in Go as values of diffuse fluxes are obtained aftertaking normal components of the intensities accordingto Eq. (23).

From the results presented above, we see that use ofAx ` 0.01 and AO G 20 provides numerical results thatare reliable to within three significant figures. Thisfinding is valid when the scattering optical thickness isof the order of unity or less, and the phase function isnot much more strongly peaked than the one shown inFig. 1. For the cloud model [Eq. (7)], it will be neces-sary to use a value of AO of about 1 or less.

D. Discussion of Results

The expressions given in Sec. III.A were used to com-pute the intensity of the scattered radiation emerging atthe top, at the bottom and at the half-way level of ahomogeneous Mie atmosphere whose scattering prop-erties are described in Fig. 1. Computations were car-ried out for three different values of r (namely, 0.1,0.5, and 1.0) and for five different values of o given byG = 0(200)800. The phase function used here isvery similar to one of the phase functions used byHansen 7 who has recently published some results of hisextensive calculations of the radiation emerging fromhomogeneous, haze and cloud type atmospheres. Hehas given results for several values of Ti between 1 and32. Reference should also be made of other publishedresults of Hansen, 6 and of Hansen and Cheyney.'7 ' 8


Table II. Difference Between the Maximum and MinimumValues of <I(r) as Observed Using Different Integration

Increments in r and O'

Difference betweenmaximum and minimum

values of 'i(r) for

Case BAO in 0o in Case A haze M,

Ar degrees T7 degrees Rayleigh X = 0.75

0.01 2 0.1 0 0.007 0.00170.005 2 0.1 0 0.0006 0.00170.005 6 0.1 0 0.007 0.0140.005 10 0.1 0 0.014 0.0380.01 2 1.0 0 0.001 0.0130.005 2 1.0 0 0.001 0.0130.005 6 1.0 0 0.01 0.110.005 10 1.0 0 0.02 0.320.005 2 0.1 60 0.0008 0.00060.005 6 0.1 60 0.009 0.0070.005 10 0.1 60 0.02 0.0120.005 2 1.0 60 0.0014 0.00130.005 6 1.0 60 0.010 0.0060.005 10 1.0 60 0.016 0.014

a Maximum number iterations performed: 6 for r = 0.1, 15for r = 1.0.

Because of this, and as the primary purpose of thispaper is to put forward a new computational procedure,the discussion of the results will be kept to a bare min-imum.

The variations of the intensity of the radiationemerging at the top of the atmosphere are shown in Fig.5 for -r = 0.1, 0.5, and 1.0 and o = 600. For the ver-tical plane containing the antisolar point (oo - so =1800), scattering by a unit volume (broken curvemarked P) shows a maximum at 600 preceded as well asfollowed by a minimum and another equally strongmaximum. This is due to the presence of the so-called

Table Ill. Intensity of the Radiation Emerging at theTop of a Plane Parallel Atmosphere as Obtained Using20 and 100 Angular Increments for Integration over ZenithAngle. Solar Flux or Units per Unit Area Normal to the

Direction of the Incident Radiationa

Nadir 0.1 1.0angle 0 A = 2 AO= 10° AO = 20 AO = 100

0 0.00234 0.00225 0.0348 0.035810 0.00267 0.00259 0.0358 0.036020 0.00357 0.00349 0.0414 0.041430 0.00487 0.00478 0.0510 0.050940 0.00647 0.00634 0.0639 0.063850 0.00817 0.00797 0.0795 0.079760 0.0123 0.0120 0.105 0.10570 0.0157 0.0149 0.126 0.12780 0.0279 0.0257 0.145 0.145

aModel: haze M, X = 0.75 , m = 1.34 -0.0i, Go = 600,'O- = 180°.

glory feature in the original scattering diagram (Fig. 1).This glory feature stands out very clearly in the inten-sity distribution of the radiation backscattered by anatmosphere of optical thickness 0.1 in spite of a strongincrease in brightness from nadir to horizon. Rem-nants of the glory can be clearly seen for ri = 0.5 and1.0 in the forms of sharp changes in the slopes of inten-sity vs nadir angle curves. When the computations areperformed with AO = 100, the individual values so ob-tained (Tables III and IV) agree within 10% with moreexact values obtained with a 2 interval. However,because of the sparseness of the data points, it would bevery difficult to identify any glory feature with reason-able confidence when the entire computations are car-ried out with a 100 angular increment.

Similar results for the intensity of the radiationemerging at the bottom of the atmosphere, but for thesun's vertical plane (soo - so = 0) are presented in Fig.6. The important features are the changes in the slopein the forward peak region, and a strong increase inlimb darkening as 7, increases from 0.1 to 1.0. Boththese features are evidently due to the nature of thephase function. They are completely absent in thecharacteristics of the radiation scattered by a Rayleighatmosphere under otherwise similar circumstances.

E. Possible ImprovementsThe results present presented in Sec. III.C can be used

as a guide during thedevelopment of acomputerprogramaimed at obtaining results of desired accuracy with aminimum of computer time. One can further reducethe computational task by relaxing the convergencecriterion used in this paper and also by using gaussianquadrature.6",7 Further computations with r = 1.0and 2.0 showed that the present convergence criterionis not an ideal one. When the propagation of trunca-tion errors cannot permit the achievement of the de-sired convergence, further iterations lead to less reliableresults. This can be avoided by terminating the itera-

Table IV. Intensity of the Radiation Emerging at theBottom of a Plane Parallel Atmosphere as Obtained Using20 and 100 Angular Increments for Integration over ZenithAngle. Solar Flux r Units per Unit Area Normal to the

Direction of the Incident Radiationa

Zenith 7= 0.1 1.0angle0 AO = 2 AO= 100 AO= 2 AO = 10°

0 0.0107 0.0107 0.104 0.11010 0.0192 0.0191 0.164 0.16620 0.0383 0.0381 0.278 0.27930 0.0849 0.0846 0.509 0.51040 0.214 0.214 1.01 1.0150 0.690 0.690 2.29 2.2960 2.47 2.47 5.38 5.4470 1.24 1.24 2.74 2.7380 0.850 0.823 1.51 1.47

,Model: haze l, A = 0.75 , m = 1.34 -0.Oi, Oo = 600,'Po - P = 00.


Table V. Number of Iterations Needed for Obtaining aFour Significant Figure Convergence of the Diffuse

Downward Flux at the Bottom of a Nonabsorbing,Mie Atmospherea

Maximum number of iterations

SuccessiveGauss-Seidel scattering

Tr Oo in degrees method method

0.1 0 5 70.5 0 6 91.0 0 9 132.0 0 14 221.0 84 13 192.0 84 17 26

a Model: haze M, X = 0.75 p, AO = 2°, AT = 0.01.

tion when the successively iterated values of diffusedownward flux at the bottom of the atmosphere showconvergence in the fourth significant place. An upperlimit for the number of iterations can be thus obtainedduring computations for n = 1, which can be then usedduring computations for higher frequencies.

One can reduce the computation time still further bymaking use of a more efficient iteration procedure suchas Gauss-Seidel iteration, 2 9 used extensively by Hermanand Browning.8 This was done and it was found thatequally reliable results can be obtained with a signifi-cant reduction in computer time (Table V). It shouldbe noted that most of the time is spent in the evaluationof the source function given by Eqs. (20)-(22). Hence,in order to obtain best timings, it is necessary to defineJm(n) as the value of the source function at the centerof the layer. It should be further noted that the com-putations of the first approximation in the successivescattering method take very little time, while those inthe Gauss-Seidel method take a very significant amount.

Still another method of saving computer time is tocombine the modified fourier method with Van deHulst's doubling method." A considerable amount ofcomputer time can be saved, especially when one isinterested in homogeneous atmospheres only. Hansenand Pollack' 0 have also put forward such an idea in theappendix of one of their most recent papers.

Since one is forced to use a fairly large number of pointsduring integration over ', it would be worthwhile tolook into the spherical harmonic method" for solutionof the transfer equation for a given frequency n. Thisis a noniterative method and as such its successful usemay lead to saving of a substantial amount of computertime. It should be emphasized that all such time-saving alternatives are needed only during computa-tions of 1 (n) (7; hu A, -go) corresponding to the first 10-20 terms of the Fourier series (see Fig. 4).

IV. Conclusion

In the preceding sections an efficient computingmethod that can provide reliable values of the intensityof the scattered radiation emerging from a plane parallel,

I\'lie atmosphere is described in great detail. Some re-sults of computations are also presented to demonstratethe efficiency of this method and reliability of results ob-tained therefrom. This method is very flexible as themagnitude of computational load is automatically ad-justed by analyzing the scattering characteristics of aunit volume, and the position of the sun. The only twoparameters which an investigator has to vary forresults with a minimum of time are the integration in-crements used for numerical quadrature over opticaldepth and zenith angle. It should be now possible toperform reliable multiple scattering calculations forvarious models containing large particles without usingan excessive amount of time and money. It is proposedto extend the method described here for calculationsaimed at determining the polarization characteristics ofthe radiation scattered by a plane parallel, Mie atmo-sphere.

We would like to take this opportunity to express oursincere thanks to our colleague, B. H. Armstrong, for acareful reading of the original manuscript and for hisvery helpful comments.

References1. L. M. Romanova, Opt. Spectrosc. 13, 238 (1962).2. W. M. Irvine, Astrophys. J. 142, 1563 (1965).3. J. W. Chamberlain and M. B. McElroy, Astrophys. J. 144,

1148 (1966).4. D. G. Collins, K. Cunningham, and M. B. Wells, "Monte

Carlo Studies of Light Transport," Tech. Rep. ECOM-00240-2, Radiation Research Associates, Inc., Fort Worth,Texas (1967).

5. G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 1129 (1968).6. G. W. Kattawar and G. N. Plass, Appl. Opt. 7, 1519 (1968).7. B. M. Herman, J. Geophys. Res. 70, 1215 (1965).8. B. M. Herman and S. R. Browning, J. Atmos. Sci. 22, 559

(1965).9. B. M. Herman, Proceedings of the IBM Scientific Computing

Symposium on Environmental Sciences (IBM Data ProcessingDivision, White Plains, New York, 1967), pp. 211-237.

10. W. M. Irvine, J. Quant. Spectry. Radiative Transfer 8, 471(1968).

11. S. Chandrasekhar, Radiative Transfer (Clarendon Press, Ox-ford, 1950).

12. S. Twomey, H. Jacobowitz, and H. B. Howell, J. Atmos. Sci.23,289 (1966).

13. H. C. Van de Hulst, "A New Look at Multiple Scattering,"Institute for Space Studies, NASA, New York (1963).

14. S. Twomey, H. Jacobowitz, and H. B. Howell, J. Atmos. Sci.24, 70 (1967).

15. H. B. Howell, J. Atmos. Sci. 25, 1090 (1968).16. J. E. Hansen, Astrophys. J. 155, 565 (1969).17. J. E. Hansen, J. Atmos. Sci. 26, 478 (1969).18. J. V. Dave, Astrophys. J. 140, 1292 (1964).19. J. V. Dave and W. H. Walker, Astrophys. J. 144, 798 (1966).20. D. Deirmendjian, Appl. Opt. 3, 187 (1964).21. J. V. Dave, Appl. Opt. 8, 1161, 2153 (1969).22. "System/360 Scientific Subroutine Package (360A-CM-03X)

Version II," Doc. No. H20-0205-02, IBM Technical Publica-tions Dept., White Plains, New York (1966).

23. J. V. Dave and B. H. Armstrong, J. Quant. Spectry. RadiativeTransfer 10, No. 6 (1970).

24. J. V. Dave, Appl. Opt. 9, No. 8 (1970).


25. J. V. Dave and R. M. Warten, "Program for Computing theStokes Parameters of the Radiation Emerging from a Plane-Parallel Non-absorbing, Rayleigh Atmosphere," (Rep. 320-3248, IBM Scientific Center, Palo Alto, Calif., 1968).

26. J. V. Dave, J. Opt. Soc. Amer. 54, 307 (1964).

27. J. E. Hansen and H. Cheyney, J. Atmos. Sci. 25, 629 (1968).

28. J. E. Hansen and H. Cheyney, J. Geophys. Res. 74, 3337(1969).

29. F. B. Hildebrand, Introduction to Numerical Analysis (Mc-Graw-Hill Book Company, Inc., New York, 1956), Chap. 10.

30. J. E. Hansen and J. B. Pollack, J. Atmos. Sci. 27, 265 (1970).31. V. Kourganoff, Basic Methods in Transfer Problems (Claren-

don Press, Oxford, 1952).

Meetings Calendar continued from page 1452

4-9 XVI Colloq. Spectroscpicum Internat., HeidelbergW. Fritsche, Ges. Deut. Chem., 6 Frankfurt/Main 8,Postfach 119075, Germany

5-8 Optical Society of America, Chateau Laurier, OttawaJ. W. Quinn, OSA, 2100 Pa. Ave., N.W., Wash.,D.C. 20037

18-22 SAS, St. Louis, Mo. J. Westermeyer, TitaniumPigment Div., Nat. Lead Co., Carondelet Sta., St.Louis, Mo. 63111

May

7-11 Internat. Quantum Electronics Conf., Queen Eliza-beth Hotel, Montreal IEEE, 45 E. 47th St.,New York, N.Y. 10017

? Soc. for Exp. Stress Analysis, Olympic Hotel, Seattle,Wash. SESA, 21 Bridge Sq., estport, Conn.06880

September

24-29 SMPTE 112th Semiann. Conf., Los Angeles D. A.Courtney, 9 E. 41st St., New York, N.Y. 10017

OctoberNovember

? APS, NYC TV. W. Havens, Jr., 336 E. 45th St.,New York, N.Y. 10017

8-13 SAS, Dallas, Tex.

9-13 ICO, Santa Monica, Calif.Corp., Norwalk, Conn.

1972

January

31-Feb. 3 APS-AAPT, San Francisco W. W. Havens, Jr.,335 E. 45th St., New York, N.Y. 10017

March

1-3 Scintillation and Semiconductor Symp., ShorehamHotel, Washington, D.C. IEEE, 345 E. 47th St.,New York, N.Y. 10017

6-10 Pittsburgh Conf. on Analytical Chem. and Appl.Spectroscopy, Cleveland Conv. Ctr. E. E. Hodge,Mellon Inst., 4400 Fifth Ave., Pittsburgh, Pa. 15213

April

9-15 SMPTE 111th Semiann Conf Washington, D.C.D. A. Courtney, 9 E. 41st St., New York, N.Y. 10017

11-14 Optical Society of America, Statler Hilton, NYC J.W. Quinn, OSA, 2100 Pa. Ave., N.W., Washington,D.C. 20037

23-26 ISA 27th Ann. Conf. & Exhibit, New York400 Stanwix St., Pittsburgh, Pa. 15222

1973

October

1-5 SAS, Niagara Falls, N.Y.

1974

October

7-11 SAS, Indianapolis, Ind.

1976

October

8-12 SAS, Philadelphia, Pa.


R. Scott, Perkin-Elmer

17-20 Optical Society of America, 57th Ann. Mtg., Jack TarHotel, San Francisco J. W. Quinn, OSA, 2100Pa. Ave., N.W., Wash., D.C. 20037

ISA Q,

a modified fourier transform method for multiple...

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