Radiative transfer calculations through an aerosol cloud

J. Quant. Spectrosc. Radiat. Transfer Vol. 31, No. 1, pp. 63-70, 1984 Printed in Great Britain.

00224073/84 $3.00+.00 © 1984 Pergamon Press Ltd.

RADIATIVE TRANSFER CALCULATIONS T H R O U G H AN AEROSOL CLOUD S. A. WEISROSE and G. SHADMON Engineering Division, Israel Aircraft Industries Ltd., Lod, Israel

(Received 29 November 1982) Abstract--A radiative transfer calculation of the EM radiation through an aerosol cloud is

presented. The effects taken into account are elastic and inelastic scattering of the incident radiation and self-emission by the aerosol cloud. The elastic scattering effects include multiple scattering of the incident photons. We describe the model and calculational procedure based on the multiple scattering approach. The results of the present model are compared with other models for the two given aerosol clouds. 1. INTRODUCTION The propagation of EM radiation through the atmosphere at optical and i.r. frequencies is affected by the absorption and scattering of particulate matter such as dust, fog and water droplets suspended in the atmosphere.l Under conditions of low visibility, scattering of the incident photons by the aerosol particles becomes the important interaction. For conditions in which the dimensions (in terms of optical depth) are large, multiple scattering effects become dominantfl Consequently, in calculation of the transmission through optically thick aerosol clouds, these higher order effects must be taken into considerationfl Many procedures have been developed to include multiple scattering effects in radiative transfer calculations. These techniques include Monte Carlo methods 3,4 and the doubling technique) '6 However, the Monte Carlo method is based on a statistical approach and the underlying physics of the photon-particle interaction is hidden. In the doubling technique, the medium is divided into slabs so that in each slab only single scattering occurs. This method does not tackle the physics of the multiple scattering problem. Some time ago, Theissing developed a multiple scattering model for the passage of radiation through a dielectric m e d i u m ) However, Theissing's treatment only considers elastic scattering. Consequently, the application of this approach to radiation transmission through aerosol clouds requires generalizations. These include inelastic scattering and self-emission of the aerosol cloud. In this paper, these generalizations are introduced and the resulting technique is applied to aerosol cloud transmission predictions. In Section 2, we present the method of calculating radiative transfer through an aerosol cloud with inclusion of multiple and inelastic scattering effects. In Section 3, the model is applied to two types of aerosol clouds. The results are analysed and compared with other methods. 2. THE RADIATIVE TRANSFER MODEL In calculating the transmission through an aerosol cloud, it is of interest to consider only the radiation flux along a line-of-sight (LOS). Assuming axial symmetry, in the two-stream approximation 8 the equations for radiative transfer along the line-of-sight, are

Nffs fn z 30

-

-N(ao + a~)I+(z) + --g- | f(O)I (z, O)sin 0 dO + Na~B(T)

(1)

dI - (z) dz -

Na~ f(O)I'(z, O) sin 0 dO + Na.B(T), N(a. + a~)I-(z) + --2-.)o

(2)

dl+(z)

dz and

where z is the direction from the source to the observer, see Fig. 1, l+(z) and I - ( z ) are the radiation fluxes at the distance z along the LOS in the directions of increasing and 63

64

S.A. WEISROSEand G. SHADMON O0

Z=O

Z'Z A

I +(Z)

_

z'(z)

=

A

B

r-Z

LOS

~ ' I P '

Ira-

-00

Fig. 1. Geometry of the present model. decreasing z, respectively, tra is the absorption cross section, as is the scattering cross section, I'(z, O) is the radiation flux reaching the scatterer at the angle 0, with respect to LOS, riO) is the scattering angular distribution at the angle 0, B(T) is the blackbody emission at temperature T and N is the particle density of the cloud. Define the "scattering integral" of Eqs. (1) and (2) as

S =

f

~f(

d0

O)l'(z, O)sin 0 dO. •

(3)

We now divide the range of integration into two parts from 0 to rt/2 (forward scattering) and from n/2 to n (backward scattering). Averaging the intensity I'(z, O) over the forward and backward scattering angles, the integral S becomes

S = Is+(Z) l"/Zf(O) sin 0 dO + IT(z)

riO) sin 0 dO,

(4)

J~/2

dO

where I+(z) and I;-(z) are defined by:

I+(z)

I

n/2

do

t ~n/2

f(O) sin 0 dO =

o

f(O)I'(z, O) sin 0 dO

(5)

and

IT(z) f : f(O) sin 0 = I x f ( O)I'(z, O) sin 0 dO. /2

(6)

d n/2

In our model we make the assumption that

I+(z) = I+(z)

(7)

Radiative transfer calculations through an aerosol cloud

65

and

I ; ( z ) = l-(z).

(8)

Thus, applying Eqs. (4), (7) and (8) to Eq. (1), the equation of radiative transfer becomes dI+(z) dz

N ( G + G)I+(z) + - - f - I+(z) | f(O) sin 0 dO + I - ( z ) f(O) sin 0 dO ,1o d ~/2 + NGB(T)

(9)

and analogously Eq. (2) becomes dI-(z) dz

N ( G + G)I-(z) +

vE ; I-(z)

f(O) sin 0 dO + I+(z)

L,,0,

sin 0 dO

+ NffaO(T).

] (10)

Defining the forward and backward scattering parameters ~ and/3 respectively, as

= 2~

I

~/2

f(O) sin 0 dO

(11)

d0 and /3 = 2~z

I"f(

0) sin 0 dO,

(12)

d~/2 Eqs. (9) and (10) reduce to dI+(z) dz

N ( ~ + G)I+(z) + Nctafl+(z) + NflGI-(z) + N G ( B ) T

(13)

dI-(z) dz

N ( G + G)I-(z) + NotoJ-(z) + Nflaj+(z) + NaaB(T).

(14)

and

In the single sc~ittering approximation the scattering function f ( 0 ) of Eqs. (1) and (2) is described by the Mie theory. Based on the model of Theissing, the higher orders of scattering can be taken into consideration by extending the original definition o f f ( 0 ) to include multiple scattering (MS) effects. In a previous paper, 9 the total elastically scattered monochromatic radiation at any point in the aerosol cloud is assumed to be given by a summation of all orders of scattering. Our case is characterized by the addition of inelastic scattering of the incident photons and the self-emission of the particles in the aerosol cloud. Thus, by analogy to the treatment by Hartel, 9 the total radiation is given by

I(z) = Io(z) + Y, Ik(z),

(15)

k>_l

where Io(z) is the amount of nonscattered radiation at z, which is written as a sum of the two intensities directed in the forward and backward direction, i.e.,

Io(z) = l+(z) + Io(z);

(16)

Ik(z) is the amount of radiation at z which has been scattered k-times namely Ik(Z) = I~(Z) + I£(Z). QSRT Vol. 31, No. 1--E

(17)

66

S . A . WEISROSEand G. SHADMON

From these equations, it follows that: (i) for k = 0, the change in the radiation over a distance dz is die-(z) -

-

dz

--

N(¢7 a + as)I~(z) + N a a B ( T )

(18)

- N(cra + a s ) I o ( z ) -t- N a , , B ( T ) .

(19)

and did(z) - dz

(ii) For k >/1, the radiative transfer equations are die-(z) - dz

N ( a , + ¢L)I+(z) + NO(k(r,l~ l(Z) +

-

Nflk~Yslk-

1(2.)

(20)

and die-(z) - dz

-

N(a,, + a s ) I £ ( z ) + Na,,,aj~_ l(Z) + NflkasI£- l(Z)

(21)

where ~k and flk are the forward and backward scattering parameters, respectively, for kth order scattering. Extending the definitions of ~ and fl, as given by Eqs. (11) and (12) to include MS effects, these two parameters are given by 7 [* hi2

=2n | /

fk(0)sin0d0

(22)

J0

and flk = 2n

I" A(0) sin 0 dO,

(23)

d ,~12

where fk(O) is the angular scattering distribution for kth order scattering and is given by 9

fk(O) =

4 mZ (2m +a"l) k k - lPm(COS0).

(24)

Equations (18)-(21) present an infinite set of coupled differential equations which describe the interactions between the incident photons and the aerosol particles. However, in the transmission calculations it is possible to introduce a limit (kL) for the number of scattering orders required. For each order of scattering k, the intensity Ik has a maximum at approx. z = k, where z is measured along the LOS in units of optical depth. From the nature of the equations, it is clear that beyond the maximum value, the decay of the scattering intensity is of an exponential form and is characterized by the optical depth as its decay constant. Thus, to a good approximation kL=z

+5,

(25)

where z is the optical depth of the cl0.ud. The geometry of the model is shown in Fig. 1. The source of the radiation is situated at B. The LOS is in the direction along the z-axis to the observer's position at A. The length of the LOS is zA. In order to calculate the radiation reaching A, it is assumed that the cloud is sandwiched between 2 plane-parallel, semi-infinite plates positioned at B and A. Monochromatic radiation is incident normally onto the plate at B. Absorption, elastic

Radiative transfer calculations through an aerosol cloud

67

scattering and self-emission occur in the cloud between B and A. The derived quantity is the radiation flux at A. To solve Eqs. (18)-(21), it was assumed that the surfaces A and B were black bodies at temperatures TA and TB, respectively. The boundary condtions are I~(0) = B(TB),

(26)

I£ (z~) = B( TA).

(27)

For the scattered energy,

/~-(o)

= o,

k = ], 2, 3 . . . .

(28)

I ~ ( z A ) = O,

k = 1, 2, 3 . . . .

(29)

Thus, having reduced the order of scattering to a finite number and established the boundary conditions, the feasibility of the multiple scattering approach is established. 3. A P P L I C A T I O N

OF THE MODEL

Some time ago, Miller et al. predicted the cloud transmissions for two types of aerosol clouds using the doubling technique. 6 It is of interest to test the validity of the present model

i

o

°

~

• MILLER'S

DATA

PRESENT

DATA

161

z o t/) (/3

,6 2 nID

0

._J (..)

1

6

2

4

4

6

~

8

I0

12

14

OPTICAL DEPTH, 1:, Fig. 2. Comparison of cloud transmission predictions between the present model and that of Ref. 6 for a water cloud.

68

S.A. WEISROSEand G. SHADMON

by comparing the predictions of the two different procedures for the same cloud conditions. The first type of aerosol cloud, considered by Miller, was one which contained water drops. The number density of the droplets in the cloud was the Haze model H of Deirmendjian l° with droplet numbers between r and r + dr given by

(30)

n(r) dr = 4 × 10 5 r2[exp(-2Or)] dr cm 3 _ # m - ' .

The cloud contained particles with radii from 0.12 to 14 #m. The wavelength of the incident radiation was 3/~m. The complex refractive index of the droplets was m = 1.364-0.3059i. For the second example of Miller, an aerosol mixture was considered. The mixture was composed of two constituents with the following mixing ratios and complex refractive indices: 1° 3 0 ~ of material with m = 1.52-0.008i and 70~o of material with m = 1.52-0.017i. The number density of the cloud was that as given by McClatchey, namely, j2

(31)

n(r) = 2.373 r 6 e x p ( - 1.5r) cm -3/~m -l.

The range of the particle radii was from 0.02 to 5 #m. The wavelength of the incident radiation was 1.06/lm. The results of calculations with the present model for the given conditions are compared with the calculations of Miller et al. in Figs. 2 and 3. In Fig. 2, the calculated transmissions are presented for the cloud of water droplets. The figure demonstrates that, over the range of optical depths considered, the two models are in good agreement. Furthermore, for this type of interaction, the transmission of the aerosol cloud can be approximated by the exponential function e -t, where z is the optical depth. Thus, for highly absorbing materials, the multiple scattering approach predicts the same transmission as is obtained for single scattering. In Fig. 3, the results of the present approach are compared with predictions of Miller for I0

I

I

I

I

I

I

[

• MILLER' S DATA u PRESENT DATA - - - EXPONENTIAL CURVE

0.6

z

_ ,

9 0.4!-

I

0.2 -

\\ \\

\\

0.0

[

I

2

I

|

4

l

I

6

I

I

8

OPTICAL DEPTH, T.

Fig. 3. Comparison of cloud transmission predictions between the present model and that of Ref. 6 for an aerosol mixture.

Radiative transfer calculations through an aerosol cloud

69

the aerosol mixture. In this type of aerosol, the absorption factor in the complex refractive index is relatively small. Consequently, in the photon-particle interaction, elastic scattering dominates. Also shown in Fig. 3 is the exponential curve e-L For optically thin clouds, z < 0.1 the results of the models are in agreement. Thus, in the single scattering limit, the models predict the same transmission. As the optical depth is increased, the second and higher orders of scattering become important. Figure 3 shows, for multiple scattering, that the trend of the two curves is the same. Over the indicated range of optical depths, the results for the two methods differ between 20 and 30~o. Figure 3 confirms the inadequacy of single scattering for optically thick clouds. Furthermore, these results demonstrate that, in the single scattering limit, the present model is in closer agreement with the exponential curve than the model of Miller. Figures 4 and 5, show the polar diagrams for fk(O) for the given cloud parameters and values of k. Both diagrams demonstrate large forward scattering of these aerosols. As k is increased,the scattering function approached a spherical distribution. This value corresponds to

fk.~O = 1/47t.

(32)

The physical interpretation of Eq. (32) is that, for the higher orders of scattering, there is equal propability of scattering in all directions. This result is known to be correct.

°0

00 ~

*0

°0

t'~

__

*0

*

*o

*o

*o

*o

*o

\j/\

\

0

~50'

,

30*

j

J 160'

20*

170'

i0°

18o'

•2

.4

.6

.8

I.

0

fk ( e ) , ARBITARY UNITS Fig. 4. Variation of the angular scattering function for the water droplets with R = 1.0 p m and 2 = 3.0 pm.

"o ¢_

"o _~

"o ~

*o =

°o

"o

~

"o

"o

"o

"o

3o*

,ed

2o" to"

.I

.2

.3

.4

fK (0), ARBITARY UNITS Fig. 5. Variation of the angular scattering function for the mixture of particles with R = 1.0 p m and 2 = 1.06/~m.

70

S. A. WEISROSE and G. SHADMON

REFERENCES 1. E. P. Shettle and R. W. Fenn, "Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on their Optical Properties", Air Force Geophysical Laboratory, Hanscom AFB, Massachusetts, AFGL-TR-79-0214, 1979. 2. S. Chu and S. W. Churchill, J. Phys. Chem. 59, 855 (1955). 3. G. N. Plass and G. W. Kattawar, Appl. Opt. 7, 1129 (1968). 4. G. W. Kattawar and G. N. Plass, Appl. Op. 7, 1519 (1968). 5. W. M. Irvine, J. Astrophys. 142, 1563 (1965). 6. A. Miller, R. L. Armstrong, and C. W. Welch, "High Resolution Atmospheric IR Transmittance Prediction", Dept. Physics, New Mexico State University, New Mexico, NTIS Rep. No. ,4D A017026, 1975. 7. H. H. Theissing, JOSA 40, 232 (1950). 8. W. E. Meador and W. R. Weaver, J. Atmos. Sci. 37, 630 (1980). 9. W. Hartel, Das Licht 10, 141 (1940). 10. D. Deirmendijian, Electromagnetic Scattering on Spherical Polydispersions. Elsevier, New York, 1969. 11. A. Millar, private communication (1981). 12. R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, and J. S. Gating, "Optical Properties of the Atmosphere", Air Force Cambridge Research Laboratories, Hanscom AFB, Massachusetts, NTIS Rep. No. AD 754075, 1972.