The Scattering of Infrared Radiation from Clouds

Ernest Bauer

The scattering of infrared radiation from clouds composed of spherical water droplets has been calculatedby Mie theory. For wavelengths in the range from 2.5 to 6 M and for large angle scattering, the effectivereflectivity is due to single rather than multiple scattering by water droplets. This is a consequence of therather large absorption cross section and of the forward peak in the differential scattering cross section.The effective diffuse reflectivity under these conditions is of the order of 27o at X = 2.7 and 157o at =4.3 Ao for scattering through angles 0 > 600.

1. Introduction

The scattering of solar infrared radiation from cloudsis a problem of importance in a variety of applications,ranging from the over-all atmospheric radiation balanceto the background observed when the surface of theearth is viewed with an infrared sensor mounted on anearth satellite. To solve any of these problems calls forinformation on the altitude and size distribution ofclouds, information on the effective transmission lossesof the atmosphere as a function of wavelength, cloudheight, and angle of view, and finally for information onthe reflectivity of clouds in the infrared.

The third of these problems is treated here. The pri-mary problem discussed is the scattering of infraredradiation in the 2.7 ,u H20 and CO2 absorption band byclouds made up of water droplets. Calculations werealso made for longer wavelengths out to 6.3 a. In fact,the blackbody radiation from the earth or from cloudspredominates from the scattered solar radiation forwavelengths greater than 3.5 or 4 p. In the frequencyrange considered here, the absorptivity of water dropletsis sufficiently high so that single scattering provides theeffective reflection mechanism except for very smallscattering angles. The results for large-angle scatteringcan be represented approximately in terms of an effec-tive diffuse reflectivity, but it must be stressed that thephysical mechanism discussed here does not apply forwavelengths in the visible or very near infrared (X < 2.5A'), where the absorptivity of liquid water is so low thatmultiple scattering is important and thus Lambert'slaw holds by virtue of multiple scattering.

The author is with Aeronutronic, Ford Motor Company, New-port Beach, California.

Received 31 January 1963.This work was supported by an Advanced Research Projects

Agency contract.

II. Some Physical Properties of CloudsClouds are assemblies of water droplets or of ice crys-

tals with a water content of 0.1-1 g/m3 of air for higheraltitude clouds ( 2 km) and up to 10 g/m3 of precipi-table water near sea level. 1 A reasonably adequate par-ticle size distribution in clouds of water droplets is givenby the expression'

n(r) = No r e-ar, (1)

where n(r) dr is the number of particles per unit volumehaving radius between r and (r + dr). The functionn(r) has its maximum at r = n/a. Deirmendjian et al.2have used the set of values n = 6, a = 1.5 ,w- (givingr = 4 ). Here this distribution is employed, as areothers with r = 2 and 8 and the same effectivewidth, and also distributions having r = 4 but signifi-cantly smaller and larger effective widths. All the dis-tributions employed here are enumerated in Table I andshown in Fig. 1. The distribution of Deirmendjianet al.2 is denoted by the code letter C in Table I; thisdistribution will be regarded as a standard for compari-son, since it provides the best fit to the experimentaldata.

Clouds found above several kilometers in altitude will,in general, be made up at least partly of ice crystalsrather than of water droplets because of the low effectivecloud temperature. * If the ice particles are simplyfrozen water droplets, which appears to be a fairly com-mon situation, the present analysis applies. There are,however, situations in which ice crystals up to 100 insize occur in clouds. These large crystals which will be

* The physics underlying this statement is quite complex sinceclouds are not necessarily in temperature equilibrium with theambient atmosphere and also since water droplets of 1-10,u radiusfreeze at -350 to -40'C rather than at C because of surfacetension effects. See ref. 1.

February 1964 / Vol. 3, No. 2 / APPLIED OPTICS 197

Table I. Scattering Cross Section from Various Particle Size Distributions for 2.7-1a Radiation

Distribution code A B C D E

Particle mode radius r, = n/a () 2 4 4 4 8

Width of distribution Regular Narrow Regular Wide Regular

n 2 24 6 2 24

a(A-) 1 6 1.5 0.5 3

Mean mass per particle (10 -"g) 2.51 3.40 6.25 20.1 27.2

Density of water (g/m 3 ) for N = 100

particles/cc 0.0251 0.0340 0.0625 0.2011 0.2723

a,,bs (1O-6 cm2) 0.234 0.358 0.562 1.247 2 009

udc,,t(O 1 6 cm 2) 0.735 1.349 1.466 2.151 3.064

Cnr,/scatt 0.318 0.265 0.383 0.580 0.656

x, = 2 r rc/x 4.65 9.3 9.3 9.3 19.6

aligned parallel to one another due to aerodynamic dragforces give rise to very striking specular reflectionphenomena such as undersuns for the appropriate angles

of incidence and reflection. These specular reflectioneffects are not discussed here.

The refractive index of water in the infrared has beentabulated by Centeno.3 The data refer, in general, to

20'C; there is no detailed information on the tempera-ture variation of the refractive index of water in the in-frared, nor is there any information on the refractive in-

dex of ice in this frequency range. The most significantqualitative feature for the present application is thatwater absorbs significantly only for wavelengths greater

than 2.6 /i. The physical reason for this is that thiswavelength corresponds to the highest effective molecu-

lar vibration frequency of the H20 molecule in the liquidphase.

xE

V

ct

Fig. 1.

100000

10000

1000

-

10000

1000

-

10

(a)

80°

(b)

= -r--'0 - 300 600 90, 1200 1500 [Udu

I Fig. 2. Differential scattering cross section for a spherical water

0 2 6 8 10 12 14 droplet at = 2.7 u, averaged over the size distributions of

radius rip) Fig. 1 and Eq. (1). (a) Effect of varying the distribution width

Assumed particle size distributions: the code letters are for constant mode radius, r, = 4 p. (b) Effect of varying the

defined in Table I. distribution mode radius r, at constant width.

198 APPLIED OPTICS / Vol. 3, No. 2 / February 1964

- - ___1I

I

10'

-\\ . 2.7/i

" \\-- -- 6.3p

'> \ \s- r H I I I

X251 50 75 100 1 2~5ISSCATTERING ANGLE, 0°

Differential scattering cross section for a spherical waterusing Deirmendjian et al. particle size distribution C for

various wavelengths.

Ill. Scattering by a Single DropThe scattering of electromagnetic radiation of wave-

length from a sphere of radius r is characterized by theparameter

x = perimeter/wavelength = 2 r r/X. (2)

In the geometrical optics region one has x >> 1, whileRayleigh scattering is characterized by x << 1. How-ever, for x - 1, which is the situation here, it is neces-sary to use the full "Mie scattering" analysis.4 This in-volves solving Maxwell's equations for a plane incidentwave and spherical outgoing scattered wave subject to

the appropriate boundary conditions on the surface ofthe scatterer. Expressions for the differential and totalcross sections are given in the literature'; they have tobe evaluated numerically by a high-speed computer.

For the present purpose the computations were car-ried out on the Aeronutronic IBM 709 computer using aprogram called CLOUD locally. This program com-putes the scattering and the total (scattering plus ab-sorption) cross sections and also the differential scat-tered intensity per particle averaged over the particlesize distribution (1), expressed in units of (2 r R/X) 2,where R is the distance from scatterer to observer. Theintensity iav(O) shown in Fig. 2 has been averaged overthe scattered intensities for polarization parallel andperpendicular to the plane of scattering. The resultsof Fig. 2 and Table I have been averaged over the2.7-A band with the refractive index data of Centeno.'Calculations have also been carried out for otherwavelengths out to X = 6.3 A using Deirmendjianet al. particle size distribution C. These results arecompared with the data for X = 2.7 in Fig. 3 andTable II.

The results of these calculations show a number ofinteresting features:

(1) The effect of averaging over a particle sizedistribution is to smooth out the rapid variations in theangular scattering distribution. For illustrations, seethe work of Deirmendj ian et al.'

(2) The angular distribution is always heavilypeaked in the forward direction and the ratio offorward to backward scattering is very insensitive tothe details of the distribution. This may be understoodbecause the complex refractive index N, - i N is notvery different from one, so that the reflectivity atnormal incidence from an infinite plane slab

(N, - 1)2 + N22(N: + 1)2 + N22 (3)

is quite small, actually of the order of 1-3 % for thepresent situation. This may be compared with therelative amount of energy scattered through angles

Table II. Mie Scattering Calculations for Various Wavelengths using Deirmendjian's Size Distribution C of Table I

Wavelength () 2.7 3.5 4.3 6.3N, 1.213 1.423 1.343 1.352N2 0.0252 0.0128 0.0108 0.07680_bs(10' cm2) 0.562 0.385 0.232 0.755Ofscatt(l- cm2) 1.466 1.532 1.778 1.435ffabs/Uscatt 0.383 0.251 0.130 0.526z = 2 r/X 12.4 9.6 7.7 5.3J(0), cf. Eq. (9c) 745 479 390 141.5Reflection coefficients:

Effective diffuse reflectivity (900)from Eq. (15b) 0.020 0.109 0.157 0.0268

K(90'), cf. Eq. (4) 0.0157 0.0709 0.0555 0.0228R, cf. Eq. (3) 0.00939 0.0305 0.0214 0.0234

February 1964 / Vol. 3, No. 2 / APPLIED OPTICS 199

10

102

101

100

10 1

Fig. 3.droplet

II

l l

greater than 90°,rE°°f i(O) sino do

K(90°) 1801 (4)f iK(°) sinG dowhich is of the same order of magnitude. The twoquantities R and K(900) are compared in Table II for

various wavelengths and agree to within a factor of

2 at all wavelengths. The comparison between thesequantities has some meaning because the scattering

parameter xc = 2 r rc/X is of order 10, indicating an

approach to the geometrical optics regime.(3) The different particle size distributions do show

some difference in their behavior at the intermediateangles (100 < 0 < 120°), but even here the differencesare not very large. This is a consequence of thesmoothing effect of the particle size distribution.

(4) In the 2 .7 -A' band, the total or "extinction"cross section

cxt = scatt + Onbs (5)

increases with increasing r but not strictly propor-

tionately to r, 2. This is due partly to the size distri-bution and partly to the oscillatory behavior of Oext (x).

(5) The small maximum in i av (6) at 0 = 170° is a

residual rainbow2 4 that is essentially a refractionphenomenon arising from internal reflection combined

with scattering. Increasing the absorption will reducethe importance of this phe-ion-enon.

To sum up the results of tl e Mie calculation, most

of the scattering from a single drop is in the forward

direction. The scattering is essentially isotropic for

9 600, with an effective large angle reflectivity[defined by Eq. (4)] of 1-10% depending on wave-length. Finally, Oabs/0scatt 0.3, which representsa very significant degree of absorption for the mul-tiple scattering problem.

IV. Scattering of Infrared Radiation by a Cloud

Single Versus Multiple Scattering

The general problem of radiative transfer withpartial absorption and an anisotropic angular distri-bution has not been solved on account of the difficultiespresented by multiple scattering effects.5 7 However,in the present situation of predominantly forwardscattering with a significant degree of absorption the

effect of multiple scattering is important only forrelatively small scattering angles.

A specific, example of this general result has beenobtained by Goldstein6 by explicit numerical evalua-tion. Goldstein compares first- and second-orderscattering from a planar cloud by using representativeMie scattering results of Deirmendjian et al.2 andfinds that single scattering far outweighs double

scattering except for very small scattering angles 0.It should be pointed out that, while the expansion indifferent orders of scattering can be written downformally (see ref. 5, for example), yet for anisotropicangular distributions the relative importance of dif-ferent orders of scattering can hardly be estimatedwithout explicit numerical evaluation.

The general result may be understood qualitativelyin terms of the following argument. Most of thescattering is in the forward direction; at X = 2.7 '

for the distribution C of Table I and Fig. 1, 80% of thesingle scattered energy goes into scattering angles 9

below 250, and 38% of the energy incident on a drop isabsorbed. Thus after n = 6 collisions, the forwardscattered energy defined in terms of the 80% criterionhas an intensity (0.8 X 0.62)" = 0.015, spread throughan angle n'/l 25° = 61°. This may be compared withthe fraction of 0.02 of the energy which is single-scat-tered through angles greater than 900.

The predominance of single over multiple scatteringmakes possible the present calculation of the scat-tering of infrared radiation from clouds. This resulthas a number of interesting consequences.

(1) The absorptivity of water droplets is fairlyhigh throughout the wavelength region from 2.5 4 to10 u. Thus the present analysis applies throughoutthis frequency range with appropriate quantitativemodifications.

(2) In the visible region the absorptivity of wateris very low. Thus in this region a large cloud may berepresented at least roughly as a diffuse Lambert'slaw reflector on account of the large amount of multiplescattering. However, a detailed justification for thisprocedure is by no means trivial, and, in particular, itis not known how wavelengths from 1 to 2.5 1A should be

treated.(3) For wavelengths greater than 4 A', the tempera-

ture radiation emitted by a cloud acting as a gray-bodyemitter predominates over the scattered solar radiation.

(4) It follows that, except possibly in the visible,the description of a cloud as a Lambertian diffusereflector is a very poor one on physical grounds. How-ever, the concept of "diffuse reflectivity" is much usedfor an empirical description of cloud scattering and isuseful since the effective diffuse reflection coefficientdoes not vary strongly with scattering angle 6 for

0 > 600.(5) Under most normal circumstances, the extinc-

tion of a beam of infrared radiation in a cloud is dueessentially to the droplets rather than to absorptionby H2 0 or CO2 vapor within the cloud. Further, onaccount of the sharply forward peaked character of thedifferential scattering cross section of a water droplet,the effective extinction of the beam is obtained by sub-tracting the forward scattering cross section from the

extinction cross section,

200 APPLIED OPTICS / Vol. 3, No. 2 / February 1964

FROM SOURCE

Fig. 4. Geometry for scattering from a large planai

TO DE

r clou

aeff = ext - a (forward scattering)aro,. + a (large angle scattering).

Now,

(large angle scattering) K(90') 0 scatt

and because K(900 ) - 0.01-0.1 and 0abs /scatt0.3, the effective extinction distance f-1 within a c]having N absorbers per unit volume is given byrelation

N ojff N 0-abs.

For a cloud having the particle size distribution Ccontaining 0.1-g precipitable water per m, the nurof particles per cm3 is 160, and thus A' = 110 rrX = 2 .7 and 270 m at X = 4.3 A. Evidently the exttion distance is smaller than this for clouds haya higher density of precipitable water.

(6) The concept of extinction distance -1 permiquantitative distinction to be made between "thand "thick" clouds. Consider a cloud of illuminaarea A and thickness L. For L << p3-1 one is dealwith a thin cloud composed of NLA droplets all scatiing independently, giving a total scattering crsection

2cattjtlis) = NLA Osatt.

On the other hand, for a thick cloud defined in termsL>>0-1, the effective cloud volume accessible toradiation is not LA but A/10, since the incident be:only penetrates a distance of the order of -1 iithe cloud. Thus the total scattering cross sectiona thick cloud is

2seatt(thick) = iK4 6-1 acatt = A (ascatt/aa).

In other words, the total scattering cross section olarge cloud is proportional to its illuminated surfarea A multiplied by an "albedo" factor (scatt/u,

but is independent of the number density of scatterN or of the scattering cross section per scatterascatt. Of course, the differential (angular) cross stion does depend on the differential cross section 1scatterer, iav(0) of Sec. III and Figs. 2 and 3.

Scattering from a Planar CloudConsider the single scattering from a large slab of

CECTOR material containing N scatterers per unit volume andhaving a plane upper surface of illuminated area Sill,which is the plane z = 0 of the system of coordinatesused here. The geometry is shown in Fig. 4. Alight ray is incident on the slab at a point I, goes intothe medium to a point A where it is scattered throughan angle 0, and leaves the slab at a point E. The anglesof incidence and exit are respectively i and e measuredfrom the local normal, and z increases going into thescattering medium.

d. A volume dVA located at the point A has a totalscattering cross section NdVA aseatt, and the totalamount of single-scattered radiation sent into unit

(6a) solid angle in the direction 0 relative to the incidentray isHj(0)Sjj = Fo N .f dVAe (r1A + AE) [8b/b9lo,

1C)0'A)Q/a = scattiav(O)/J(O),

J(0) = 2 r i-(O) sin do.IJo

(9a)

(9b)

(9c)

H10 is the radiance of single scattering, measured inwatt cm-2 sterad-1 and F, which is measured in watt/cm2 , is the flux of incident radiation; both are re-ferred to the same frequency band. The factorexp{ j-/(rIA + rAE) } gives the effective extinctionof the ray which is scattered through an angle 0 at thepoint A.

The integration over dVA is carried out by extendingif d dy over Sill and letting z go from 0 to . Thisgives the result

H,(O) = FoN[a-/ŽloU d ef(ie) = FoN[a-/lQ0oJo /3 f(i,e)

f(i,e) = see i + see e.

(lOa)

(lob)

Finally, using Eq. (6) for , one obtains the result

F___________ ircat a (0 )(7) H(0) = FoQp (i,O) = = F,! (atjf(ie) C-abs J(O) f(i,e)

(11)

the The quantity Q (i,0) is the differential scattering,*am cross section of the cloud per unit illuminated areanto defined for unit solid angle about the scattering direc-of tion 0, and parametrized in the case of a planar cloud

by the local angle of incidence i. The physical mean-ing of Eq. (11) is intuitively clear in terms of the

(8) discussion of Eq. (8).

f a The differential scattering cross section Q (i, 0) isice shown in Fig. 5 for X = 2.7 At in the important case ofLbs) coplanar scattering, when the incident ray, the localers normal, and the exit ray all lie in the same plane, so that

i + e + = rn

f(i,e) = se i - see (0 + i).

(12a)

(12b)

In Fig. 5 is also shown a scattering cross-section curve

February 1964 / Vol. 3, No. 2 / APPLIED OPTICS 201

i = e = 0,

i = e = 5,

i = e = 450,

i = 0, e = 0 = 900

i = e = = (iO

'O. 90° 120o _

Fig. 5. Scattering from a large planar cloud at X = 2.7 ,u. The

incident ray, reflected ray, and local normal all lie in the same

plane. The differential scattering cross section i.,(O) correspond-

ing to Deirmendjian,.et al. distribution C of Table I have been

used. The parameter i gives the angle of incidence. The

quantity plotted is iv(O)/f(ie) = 285 Q (io), sinceJ(O) abs/asctt = 285 at X = 2.7 ,.

for a spherical cloud, Q,(maX) (0). The effective scat-

tering cross section for a spherical cloud is not char-

acterized by the angle i, since different scattering sur-

face elements correspond to different values of i, so that

this angle is automatically integrated over when the

volume integral of Eq. (9a) is evaluated for spherical

geometry. It has not proved possible to carry out this

evaluation exactly for spherical geometry, but instead

an upper bound for Q, (0) can be given in terms of the

effective minimum value of f(i,e). For coplanar scat-

tering, the function of f(i,e) ranges over the following

limits in the integration

f(O) = 2/sin(O/2) < f(i,e) < o (for i = 7r/2). (13)

Thus it is possible to set the following upper bound on

Q, (0)

Ws(0) <1 Ws-mn)(0t) = ) (14)

The good agreement between Q, (i,0) for appropriate

values of i and Qjmax) (0) that is shown in Fig. 5 indi-

cates that Q3(max) (0) is a useful quantity for over-all

comparison with experiment.For comparison with any experimental data it may

also be useful to express the result of Eq. (11) in terms

of Lambert's law diffuse scattering with an effective

diffuse reflection coefficient -q = 77 (i,0):

Q,(i,o) = 77 (iO) ir t cosi cose, (15a)

ri(i,O) = 7(alb)o [cosi coseo-r-n,,f(i,e)]~l (15b)

Some typical numerical values of q (i,0) are given here

for X 2.7 ,4.

0 = 180;r = 0.010. (16a)

o = 170 7 = 0.017.

o = 900; v = 0.024.

; = 0.033.

; = 0.060.

(16b)

(16c)

(16d)

(16e)

These values, like the results of Fig. 5, refer to the

coplanar case of Eq. (12), so that they give maximum

values for the scattering cross section and reflection

coefficient. Representative values of Xq (900) are given

in Table II for various other wavelengths.As yet little experimental information exists with

which to compare the present results. Calculations

have been made by McDonald,8 who replaces the water

droplets by an equivalent layer of liquid water and ob-

tains an effective reflectivity R from Eq. (3). His

calculations have been extended by the work of Her-

man,9 who obtains total Mie cross sections oscatt and

oext. McDonald's result of a net diffuse reflectivity of

0.04 is in order of magnitude agreement with the present

prediction for X = 2.7 tz. At longer wavelengths the

scattering parameter x = 2 r r/X decreases so that

McDonald's model of replacing the water droplets in a

cloud by an equivalent thickness of liquid water will be-

come less satisfactory at longer wavelengths. The

more detailed calculations of Havard,' 0 who combines

considerations of radiative transfer with Mie scatteringcalculations, give lmax- 0.20 for X = 3-6 A', which is in

general agreement with the present analysis.

A number of people have contributed materially to

this work. J. A. Jamieson, P. J. Wyatt, D. Deirmend-j ian, G. N. Plass, A. J. Ruhlig, and V. R. Stull have con-tributed materially to the formulation and solution of

the problem, and many discussions with M. H. Johnsonhave been invaluable in providing a physical under-

standing of the problem as a whole.

References1. B. J. Mason, The Physics of Clouds (Oxford Univ. Press,

London, 1957).2. D. Deirmendjian, R. Clasen, and W. Viezee, J. Opt. Soc.

Am. 51, 620 (1961). See also RAND Reports R-393-PR and

RM-3228-PR and earlier publications.3. M. Centeno, J. Opt. Soc. Am. 31, 244 (1941).

4. H. C. Van de Hulst, Light Scattering by Particles (Wiley,

New York, 1957).5. J. S. Goldstein, Astrophys. J. 132, 473 (1960).

6. J. S. Goldstein, Numerical Solutions of the Auxiliary Equa-

tion for an Inhomogeneous Planetary Atmosphere, talk pre-

sented at the 11th International Astrophysical Symposium,

July 1962.7. G. N. Plass and V. R. Stull, J. Opt. Soc. Am. 50, 121 (1960).

8. J. E. McDonald, J. Meteorol. 17, 232 (1960).9. B. M. Herman, Quart. J. Roy. Met. Soc. 88, 143 (1962).

10. J. B. Havard, Ph.D. Thesis, Univ. of Washington (1960).

Unpublished. ASTIA No. AD-238 268.

202 APPLIED OPTICS / Vol. 3, No. 2 / February 1964

u,

0.c.a11

.a