MONTE CARLO CALCULATIONS OF REFLECTED

INTENSITIES FOR REAL SPHERICAL

ATMOSPHERES

APPROVED: ' > "

'U Major Professor

Minor Professor

Director of the Department of Physics

MONTE CARLO CALCULATIONS OP REFLECTED

INTENSITIES POR REAL SPHERICAL

ATMOSPHERES

THESIS

Presented'to the Graduate Council of the

North Texas State University In Partial

Fulfillment of the Requirements

For the Degree of

MASTER OP SCIENCE

By-

John A. Montgomery, B»A.

Denton, Texas

January, 1969

TABLE OF CONTENTS

Page LIST OF ILLUSTRATIONS IV

Chapter

I. INTRODUCTION . . . . . . . . . . . . . 1

Motivation Problem Definition Previous Research Methodology

II. THE MODEL ATMOSPHERE . 10

Geometry Parameters

III. THE ALGORITHM . . . . . . . . . . . . 17

Photon Selection Pathlength Selection Tau Calculation Determination of the Scattering Point The Scattering Event Post-scattering Direction Cosines Detector and Local Coordinate System Flux Calculation Determination of Angular Dependence Intensity Calculation

IV. COMPARISON OF RESULTS AND CONCLUSION 35

Comparison of Results Conclusion

APPENDIX » . 40

BIBLIOGRAPHY . . 51

ill

LIST OF ILLUSTRATIONS

Figure

1. The Earth-Atmosphere System to Scale .

2. The Coordinate Systems

3. Photon Selection

Flowchart of the Scattering Simulation 4.

5.

6.

7.

8*

9.

10.

Results of Spherical Case Compared to Those of Coulson et al. forX« .1, 0 . <

Results of Spherical Case Compared to Those of Coulson et al. for X = 1, V 0

Results of Spherical Case Compared to Those of Coulson et al. for "C = 1, .8 . . .

Results of Cumulative Rayleigh Case Compared to Those of Coulson et al, for X * .1, A.," 0. ,

Results of Spherical Case Compared to Those of Adams for t • 1, 0

Results of Spherical Case Compared to Those of Adams for T e l , .8 . . . . .

Page

41

42

43

44

45

46

, 47

. 48

, 49

, 50

iv

CHAPTER I

INTRODUCTION

Motivation

There are several motivating factors leading to a study

of light scattering in spherical planetary atmospheres. These

are briefly discussed in this section, and are based upon the

importance and applicability of such a study.

Importance

A long recognized need in radiative transfer theory has

been the development of a solution to the problem of light

scattering in spherical planetary atmospheres. This need is

discussed by 0. I. Smoktli (13). A successful technique must,

in effect, solve the transfer equations for a spherical model

sufficiently complex as to approach reality. It should

accurately predict and explain phenomena arising from atmos-

pheric sphericity.

Applicability

Such a solution to the spherical problem would have

application to a myriad of atmospheric scattering phenomena.

An extension of the understanding of the earth"s atmosphere

might become possible due to the greater realism of the

model considered—a model capable of involving numerous

light-scattering processes and atmospheric sphericity.

A solution to the inverse problem becomes feasible If

one has a means of calculating emergent radiation as a

function of atmospheric parameters. The inverse problem

is the determination of meteorological quantities such as

temperature from external measurements of emergent radiation,

and might have utility in the solution of the Venus problem.

Problem Definition

Before proceeding, the spherical problem as considered

herein must be defined. The intended quantitative determi-

nation, parameterization, and problem restrictions are

expressly delineated.

Desired Determination

To calculate the emergent radiation field, a realistic

atmospheric model and algorithm must be developed. The radi-

ation field may be characterized by the emergent intensities

of scattered light. This is possible only if the algorithm

determines these intensities as dependent upon atmospheric

and angular parameters.

Parameters

The parameters to be considered are as follows:

1. The total optical thickness of the atmosphere.

2. The altitude-dependent extinction coefficients.

3. The altitude-dependent cross-sections.

4. The albedo of the planet surface.

5. The angular dependence of the emergent

intensities.

Restrictions

This solution to the spherical problem is restricted by

several simplifying assumptions.

Atmospheric parameters are considered constant over radial

distanoes that are small compared to the atmospheric thickness.

The ground is taken as a homogeneous Lambert surface, although

different albedoes may be used. Uniform incident solar flux

is assumed, where the flux is normalized to unity over an area

perpendicular to the direction of incidence. Further simplifi-

cations are made by assuming azimuthal symmetry with respect

to the incident direction, and by disregarding polarization

effects.

Previous Research

There are many references in the literature to research

on light scattering in planetary atmospheres. These reports

are primarily concerned, analytically and numerically, with

the plane-parallel approximation. To date, most attempts to

formulate and solve the spherical problem have been largely

unsuccessful. The two approaches are considered in turn.

Plane-Parallel Approach

In the plane-parallel approach the scattering atmosphere *

Is approximated by stratification into parallel planes.

Usually all physical parameters are considered constant within

each stratum, although they may vary from one planar level to

the next. This provides great simplification of the transfer

equations, yet their solutions can yield very useful infor-

mation.

Chandrasekhar (2), and Coulson, Dave and Sekera (4) using

Chandrasekhar's methods, have successfully treated the problem,

within limitations, of light transfer in plane-parallel atmos-

pheres. More recently Adams (1) has successfully applied

invariant imbedding techniques to the problem. Using powerful

Monte Carlo techniques, Collins and Wells (3), and Kattawar

and Plass (6, 7, 10, 11, 12) developed numerical procedures

to simulate light propagation in the earth's atmosphere.

Spherical Approach

Many atmospheric phenomena require express consideration *

of sphericity before they can be mathematically described. The

plane-parallel model is inadequate In this situation. Hence

in the past several years, researchers have begun to address

the spherical problem. This is a problem of no mean difficulty,

Smoktii (13) developed an approximate spherical atmosphere

and devised an approximate solution, but his approach appeared

to be cumbersome and of limited utility. A valuable algorithm

for two classes of problems has been developed by Marchuk and

Mlkhailov (8, 9) at Novosslblrsk.

Comparison of the Two Approaches

The plane-parallel and spherical approaches have certain

relative advantages.

The plane-parallel approach has the advantage of producing

a simpler mathematical representation with correspondingly

simpler solutions. However, the plane-parallel model is

representative of reality only over short lateral distances,

has singularities in the equations for unit albedo, and is

unable to describe some of the more interesting facets of

atmospheric radiation, such as twilight phenomena.

The spherical Approach provides a more realistic model,

capable of including most atmospheric phenomena. If Monte

Carlo techniques are applied the complexity of the model is

limited primarily by practicality. Even using the Monte Carlo

method the spherical equations of transfer are very difficult

to solve.

Methodology

In this section the methods used, together with the reasons

for their selection, will be discussed.

Model

The purpose iB to develop a realistic spherical model

which utilizes commonly used atmospheric data. This model

has several strong likenesses to the plane-parallel one

after sphericity is discounted, providing a means of com-#

parison to the analogous plane-parallel case.

The model includes two basic types of atmospheric

scattering—Rayleigh scattering, which is molecular, and

Mie scattering from spherical aerosols. For Mie scattering

it is assumed that the dielectric constants of the Individual

particles are known; hence the scattering indicatrioes may

be determined.

Algorithm

The algorithm developed relies heavily on Monte Carlo

simulation of the scattering processes. These processes are

essentially a finite Markov Chain. The method computes the

emergent intensities of light which experiences multiple

scattering and reflection. Rayleigh, Mie, and Lambert events

are considered.

A general discussion of Monte Carlo techniques and their

development is given by Hammersley and Handscomb (5), but is

beyond the scope of this paper. Some important considerations

are discussed, however.

The Monte Carlo technique is applicable to a problem

which consists of a sequence of events where the probability

for the occurrence of each event is known. The sequence may

be simulated by Monte Carlo and the total probability for all

events determined. Light scattering in an atmosphere is such

a sequence; hence the Monte Carlo method is applicable.

As discussed by Plass and Kattawar (11, p. 415), the

Monte Carlo method applied to radiative transfer problems

has these advantages:

1. The atmospheric parameters may vary with

height in any desired manner.

2. Any scattering function may be considered

regardless of the anisotropy,

3. The angular distribution of surface-scattered

radiation may be varied.

4. The number of detectors and angles where

intensities are determined Is limited only

by practicality.

However, a disadvantage of the method is that the standard

deviation of the results is inversely proportional to the

square root of the computing time.

CHAPTER BIBLIOGRAPHY

1. Adams, Charles, "Solutions of the Equations of Radiative Transfer by an Invariant Imbedding Approach," unpublished master's thesis, Department of Physics, North Texas State University, Denton, Texas, 1968.

2. Chandrasekhar, S., Radiative Transfer, New York, Dover Publications, Inc., I960.

3. Collins, D. G. and M, B. Wells, Monte Carlo Codes for Study of Light Transport in the Atmosphere, Vol. I, 2 volumes, Fort Worth, Radiation Research Associates, Inc., 1965.

4. Coulson, K. L., J. V. Dave, and Z. Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Raylelgh Scattering. Berkeley. University of California Press, 19<5<n

5. Hammersley, J. M. and D. C. Handscomb, Monte Carlo Methods, New York, John Wiley and Sons, Inc., 1964.

6. Kattawar, G. W. and G. N. Plass, "Electromagnetic Scattering from Absorbing Spheres," Applied Opticss VI (August, 1967). 1377-1382.

7. Kattawar, G. W. and G. N. Plass, "Resonance Scattering from Absorbing Spheres," Applied Optics, VI (September, 1967). 1549-1554.

8. Marchuk, G. I. and G. A. Mlkhailov, "Results of the Solution ' of Certain Problems of Atmospheric Optics by the Monte Carlo Method," Izv,, Atmospheric and Oceanic Physics, III (1967), 227-231.

9« Marchuk, G. I. and G. A. Mikhailov, "The Solution of Problems of Atmospheric Optics by a Monte Carlo Method," Izv., Atmospheric and Oceanic Physics, III (1967), 147-155.

10. Plass, G. N. and G. W. Kattawar, "influence of Single Scattering Albedo on Reflected and Transmitted Light from Clouds," Applied Optloa. VII (February, 1968), 361-367.

8

11. Plass, G. N. arid G. W. Kattawar, "Monte Carlo Calculations of Light Scattering from Clouds," Applied Optica, VII (March, 1968), 415-419.

12. Plass, G. N. and G. W. Kattawar,"Radiant Intensity of

Light Scattered from Clouds," Applied Optics, VII (April, 1968), 699-704.

13. Smoktli, 0. I., "Multiple Scattering of Light in a 1(

Homogeneous Spherically Symmetrical Planetary Atmosphere, Izv., Atmospheric and Oceanic Physics, III (1967)# 140-146.

CHAPTER II

THE MODEL ATMOSPHERE

The model atmosphere considered herein la a spherical

shell concentric with the planet, and of finite extent. The

ratio of atmospheric radius to planetary radius is sixty-five

sixty-fourths, corresponding to 6500 kilometers and 6400

kilometers respectively for the earth. Arbitrary units may

be used, as in this calculation, as long as the proper ratio

is preserved. In Figure 1 the model cross-section is shown

to scale. The comparative thinness of the atmosphere empha-

sizes the fact that its curvature must be considered in a

realistic model.

Geometry

Figure 2 is a representation of the atmosphere-earth

system with the atmospheric thickness exaggerated for clarity.

Although only the upper hemisphere Is shown, the model includes

the entire sphere.

All coordinates and direction cosines are determined relative

to a coordinate system fixed in the planet, as shown in Figure 2.

Points are specified in either cartesian or spherical coordinates

as dictated by ease of determination*

10

11

The solar flux is Incident on the negative-x hemisphere

in a direction parallel to the x axis. It is assumed parallel,

unpolarized, and of unit strength.

Zone Structure

The atmosphere is divided into ten concentric spherical

zones which are analogous to the parallel planes of the plane-

parallel model.

Each .zone is referenced by the index of the outermost

radius of the zone. That is, the 1-th zone oontalns all points

of radius r such that

Ri-1* r i Ri*

The planet surface is considered the inner boundary of the

first zone.

All zones have nearly the same optical thickness

but different physical thicknesses. For a total atmospheric

optical thickness X , it is required that

i -i - i

The atmospheric parameters are considered constant within

each zone, but may vary from one zone to the next.

Detector Bands

The outer rim of the atmosphere is divided into five

detector bands. These bands are of equal width in ,

where JJ^is the cosine of the nadir angle . Equal spacing

12

in ensures that the detector bands have equal areas. The

assumption of azlmuthal symmetry relative to the incident

direction requires that all points with the same f L in a

given detector band are equivalent. Although those with

different are not equivalent within a given band, practi-

cality requires that they be treated as such.

The detector bands allow determination of the angular

dependence of emergent intensities on the planet nadir

angle ©0, where ©o is shown in Figure 2. Any reasonable

number of bands may be used.

Local Coordinate System

To determine the angular dependence of the emergent

intensities at the points of emergence, local coordinate

systems are constructed at these points. In these local

systems the angular dependencies may be ascertained.

A local coordinate system with origin at the detector

point is shown in Figure 2. The local x'y1 plane is tangent

to the atmospheric rim, and the z1 axis is outwardly normal.

The x' axis is required to be in a plane containing the radius

vector to the point of emergence, and the planet x axis. This

x1 axis is so directed that the incident flux makes a local ft

angle of zero relative to it.

As it is impossible to determine the angular dependence

in the local system on all possible angles of emergence, the

13

local system Is divided into a finite number of angular

bins. All emergent photons passing through a given bin

are treated as equivalent.

The bins are determined by the cosine of the local

zenith angle 9, while the & bins are determined by the local

azimuth angle,0, As azimuthal symmetry is assumed for the

model atmosphere, no distinction need be made as to which

side of the x® axis the 0 bins lie.

Parameters

It is now apropos to discuss the means of determining

the atmospheric parameters which describe the model atmosphere.

Tau

The total atmospheric d e p t h m a y be specified as any

reasonable value. For the calculation they range from one-

tenth to ten.

Extinction Coefficients

The atmospheric extinction coefficients are determined

by the method of Collins and Wells (1/ p. 22) „ Once ti is

selected, the extinction coefficient of each zone may be

computed in the following manner. Let h be the height

above the planet surface, and £(h) the extinction coefficient

at h. The dependence of C on h is given by

C(h) « oce~Bh,

where ocis determined by ^ , and B is a constant•determining

14

the height dependence of atmospheric density. Beta is taken

as 0.0125 for the earth, hence

E(h) = C C e " °, 0 1 2 5 h .

Alpha is determined by the requirement that

where h1 is the maximum height of the atmosphere, and B

is positive. This has the solution

^ » { <VB) e~ B h * «/k.

Taking the limit as h increases without bound, and solving

for ce 3 we find that

CL - 0.0125^ v

Hence,

E(h) - 0.0125Ue"*0,0125h.

The extinction coefficient of each zone, may be ' i

determined by picking the S ( h ) within eaoh such that the

relationship 10

t ( E i t t) X S5 *"

is satisfied, where t 1 is the physical thickness of each zone,

Proas-sections

The cross-sections for Rayleigh and Mie total scattering

and absorption are determined by physical measurements.

15

Scattering Functions

The scattering functions for aerosols are those computed

by Kattawar and Plass (2). Their computational method is

discussed in the article.

Albedo

The earth's surface albedo may be varied according to

calculatlonal need. However, an albedo of eight-tenths is

usually assumed for the earth.

CHAPTER BIBLIOGRAPHY

1, Collins, D. G. and M. B. Wells, Monte Carlo Codes for Study of Light Transport in the Atmosphere, Vol. II, 2 volumes, Port Worth, Radiation Research Associates, Inc., 1965.

2. Kattawar, G. W. and G. N. Plass, "Electromagnetic Scattering from Absorbing Spheres," Applied Optics, VI (August, 1967), 1377-1382.

16

CHAPTER III

THE ALGORITHM

The algorithm which simulates the light scattering

process Is described In this chapter. Implementation of

the algorithm Is effected by a program written for an

International Business Machines 360/50 Computer.

The separate computations required are presented

approximately in their order of performance in the simu-

lation process. This is done because each computation is

logically dependent on those preceding It.

Rejection techniques used to model the theoretical

distribution functions have been tested separately before

inclusion In the program. These techniques use the multi-

plicative congruentlal random-number generator provided in

the International Business Machines Scientific Subroutine *

Package.

Several variance-reducing techniques are used in the

algorithm. Each technique has two parts. These are sampling

from a biased distribution, then removing the bias. The

distribution Is biased in a way which increases the number

of events occurring In regions of Interest. Such a bias effec-

tively Increases the population of events in the desired region,

IT

18

thus decreasing the variance. This bias must be removed

after the desired computations have been performed in

order to preserve reality. The bias removal is effected

by assigning to each event a statistical weight determined

by the true probability of its occurrence. Examples of

this may be seen in the photon and pathlength selections.

Photon Selection

The first step in simulating the scattering sequence

is the selection of a photon to be followed. Both its

direction and coordinates must be chosen.

Each photon is incident in the positive x direction

in the planet system. Hence If we let S be a unit vector

in the photon direction* A ^ Jf% A S « & ^ ^ly + ®

where a, b, c are the direction cosines of § in the planet

system. For the inoident photon b and c are zero while

a is unity.

Assuming uniform incident flux and azimuthal symmetry,

a photon has uniform probability of incidence on any element

of a plane area perpendicular to S. As the photon Is actu-

ally incident on the atmosphere, it must be incident on the

rim of a disk of radius cr , perpendicular to S. This radius

must be determined as shown in Figure 3•

For Increased flexibility it is desired to be able

to bias the ©"distribution so that <r may be selected in

19

the range

R ' £ <r £ RA,

where R! Is in the range

0 £ R' < RA,

and R a is the radius of the atmosphere. Requiring that

A

n 2<rd<r - 1 , 'R'

one finds that , 2 o -1

N * (Ra - R' ) .

Hence, we may require that

CT • (CR2- R I 2)(R a2 - R*2)"1,

where CCis a random number in (0,1). Solving this

expression for 0* , the distribution-modelling formula is

found to be

0" 88 (R1 + &(RA2 " R'2) )*.

This reduces to the formula of Marchuk and Mikhailov (3, p.152)

when R® is zero. #

For R$ not zero, sampling <f in this manner introduces

bias into the calculation. The bias may be removed by a

statistical weight factor, w. If P( 0*) is the probability

of choosing 0* in (0, R A) and P'(0~) is the probability of

choosing CTln (R',RA), then the weight factor is given by

W - P( C)/P ' (Cr ) . :

20

Now

P (<r) dor a 2<t/ra2

and P'(<T) dflT- 2<T/(RA2 - R«2),

hence W * 1 - R,2/Ra2*

For R1 equal to zero the weight is unity. Not only does this

allow a shifting of <T as desired, but it also has the effeot

of reducing the variance in certain regions of interest.

To completely specify the point of incidence, an

azimuthal angle must be chosen. If T is the angle between O" and

the planet y axis, then T must be uniformly distributed in (0,21T).

Gamma is specified by

sinT * sin (2TT<£ ),

cos Tf - cos (2*trc),

where <C is a random number in (0,1). The coordinates of the point

of incidence (x,y,z), are given by

x • - ( r a 2 ~ < r 2 ) K

y " <rcos*f ,

z • qrsinf . «

This is shown in Figure 3.

Pathlength Selection

To decrease the computational time it is necessary to

require that no photon escape the atmosphere. If this is not

done a significant number of incident photons esoape before

experiencing a oolllsion. The requirement is satisfied In the

determination of P, the optical pathlength traversed by a photon

21

before collision. Two exponential distributions In optical

pathlength are used, one of which is truncated to ensure

that the photons remain within the atmosphere.

If a photon moving in direction § sees the planet along

its path, the unbiased exponential distribution is used. In

this case the photon cannot escape without scattering either

in the atmosphere or from the ground. Solving the equation

.P

flt • I -p« e dP®,

o

where <£ is a random number, gives a means of sampling from

this distribution. The solution for P is

P » -ln(l-<).

If £ is distributed uniformly in (0,1), then (1- (t ) is

distributed uniformly in (0,1). Hence P » -ln( <£).

A means of sampling from the truncated distribution

is needed when the photon does not see the planet along its

path, and could thus escape. This is accomplished by solving

the equation P ~PS

e 1 - e-T max

<£ . ^ e dPs

where is the total optical pathlength through the v ulc&X

A

atmosphere along S. The solution for P is

p . - m (1 - <t(l - e - W ) )

which, in the limit as X max i n c r e a s e a without bound, yields P . -In (1 - * )

as before.

22

Sampling from the truncated distribution introduces a

removable bias. The bias is removed by assigning an initial

statistical weight, w equal to unity, to the incident photons.

Each time P is sampled from the truncated exponential distri-

bution, the photon's statistical weight is reduced by the mX

factor (1 - © m a x ) . This is the probability that a collision

will occur before the photon escapes the atmosphere.

The above method is described by Cashwell and Everett (l).

Tau Calculation

The optical pathlength T , traversed in passing through

the atmosphere zones, is used in the selection of P, determi-

nation of the statistical weight, computation of scattering -

point coordinates, and calculation of emergent fluxes. A

large proportion of' the total computing time is required for

the tau calculations; hence they must be performed in the

optimal manner.

The calculations fall into two classes. These classes

involve the strictly outgoing photon and the incoming photon.

Only -the most general method within each class is discussed.

Let R a be the radius of the point where a photon begins

moving in the § direction, and let £ a be a unit vector along R g.

A photon is said to be strictly outgoing If

cos^to >

where A f*

COS • Rg* S.

Otherwise, the photon la said to be incoming.

23

Outgoing Photon

In this case t is computed by a method given by

Mateer (4), Let H * RS - RT ,

5 • cos"*1 (Rj* §),

K » (1 • H/R8 )sin5 .

The distance from R| to the i-th zone boundary along S is

given by

DT - (R? - K 2 R | )i - ( R | - K%2 ) * ,

where i is the index of any zone whose R^ is greater than

R S . If j is the index of the innermost zone whose Rj is a

greater than the total optical pathlength along S is

given by

X * fri , <Di " Di-i)Ei+DjEr max l *** J *

Incoming Photon

The treatment of incoming photons is more complicated,

as in this case the photon may penetrate each zone more than

once.

Let D be the distance of closest approach of an ca

incoming photon to the origin of the planet system, where

D c a * R$ I sir.5 1.

Further, let D H e within the j-th zone. Define Pc as ca

the point in the k-th zone where the photon began moving

in the S direction, and L to be the distance between

2k

and PQ such that

t - (R2 - D§a)i.

It is desired to begin at Pn and move towards D„Q w C o

A along S, computing theti traversed in each zone. The

optical pathlength.traversed in the k-th zone is given by

2 .4

k* (L - (Rfj-l - D o a)

5)E

D is the distance traversed in the \ th zone, where « P L

D| = L - (H^ 1 - D c a)2.

Thus D 1E 1 .

This procedure is followed until the incoming photon

reaches the D c a. On the outgoing side, Mateex's method (4)

may be used to compute the remainder of "£max.> Hence m a x

ia the sum of the T! 's on the inooming side and those on the

outgoing side.

Determination of the Scattering Point

Once the optical pathlength to collision, P, has been

determined, the coordinates of the scattering point may be

calculated. This is effected by computing the the photon

sees along beginning at PQ. At each zone the cumulative

is compared to P and the distance, D, accumulated. If TJ

is less than P the calculation continues until a value is

reached such that % is greater than, or equal to, P. Hence

if X becomes greater than, or equal to, P in the i-th zone,

25

the total physical distance along S to the scattering-point

is DT = D - { % - P)/Ci.

The coordinates of the scattering-point are given by

X • x + aQ,, ,

Y • y ^ bQj. ,

Z = z * 0 % ,

where (x, y, z) are the coordinates of P0, and (a, b, c) are

the direction cosines of

The Scattering Event

Three types of scattering events may occur in the model

atmosphere. The means of determining the scattering type,

and of modelling the scattering event is discussed in this

section.

Determination of Scattering Type

The three types of events fall into two scattering

classifications—ground scattering and atmospheric scattering.

If during the computation of the scattering-point coordinates

the photon strikes the ground while TJ is less than P, a ground-

scattering event is said to occur. Ground-scattering events

are treated as Lambert scattering events, while atmospheric

scattering may be either Rayleigh or Mie.

The means of selecting the type of atmospheric scattering

is described by Collins and Wells (2, p. 38). To determine

26

the type, the ratio of the Rayleigh scattering cross-section

to total cross-section is input for each zone. Suppose that

the scattering point lies within the i-th zone and is

this ratio. Generate a random number, & . 'Then for

( R p i - c t ) > 0

Rayleigh scattering Is selected, while for

i r k -<0 i o

Mle scattering Is selected.

Lambert Scattering

Before cne models Lambert's law of diffuse reflection,

x>rdinates of the scattering point on the earth's surface

le computed. In the % calculation the distance to the

, C^,is determined. The coordinates of the scattering

are computed in terms of the point of origin (x, y, z)

the c

must

plane

point

by

X * x * aD^p

Y - y 4- bD-yp, s

Z * z + cDtp «

The radial distance to the point R$ is

| - (X2^ Y 2* z2)h

The scattering angle given by Lambert's law is defined

with respect to the surface normal, which must be determined.

The direction cosines of the normal are given by

a - X/R'a ,

b • Y/R^ >

c . Z/Rl .

27

To model Lambert's distribution, let cC^, <£2 be random

numbers. The cosine of the scattering angle, jjL , is given by

JJL • max ( <t1, <t2),

as used by Marchuk and Mikhailov (3, p. 152).

As non-absorption is required, the photon's statistical

weight must be adjusted by the albedo of the surface, X , .

The new weight becomes W \ #, where Is the probability that

the photon is reflected.

Rayleigh Scattering

Rayleigh's law is modelled by a rejection technique

used by Marchuk and Mikhailov (3, p. 1^9). Let JL be the

scattering angle the photon makes with its incident direction,

and let oCg, b e random numbers. T h e n i s given by

( C 8 / 3 K 1 -1, for 0 £ «L 1 3 A

\[slgn (c£1 -7/8)]'[Max (* 2> diy for 3 / W ^ 1.

Hence, the modelled distribution function is

F (JUL ) " 3/ 8 U + >A2).

For each zone, the ratio of the Rayleigh scattering cross-

section to the total Rayleigh cross-section is input. This

ratio, R, is the probability that the photon is not absorbed.

The weight is adjusted by R and becomes WR.

Mle Scattering

The method of Collins and Wells (2, p. 4l) is adopted

for the Mle scattering event. For this spherical model it Is

28

possible to have as many Mie functions as zones. The index

of the zone in which the Mie scattering occurs is the index

of the Mie function to be used. The variety of Mie functions

allows the inclusion of different types of Mie scattering

within the same mo£el. For example one may have different

types of clouds at various layers in the atmosphere.

The cumulative Mie scattering angular distribution is

input for each Mie function. The scattering angle is selected

from this distribution. For each zone there are gs| possible

values k j. These are computed such that there is equal

probability between^JLk j and Jl T h e l n d e x i s determined

such that

(J-l)/ N k £ <£ <, J/Kk,

where cC is a random number. The cosine , / x , of the scattering

angle relative to the incident direction is determined by inter-

polating linearly between JJL and ^ ^ j.

For the zone in which the photon is scattered, the ratio

of the Mie scattering cross-section to the total Mie cross-

section is input. This ratio, R M , is the probability that

the photon is not absorbed. The statistical weight is adjusted

by and becomes WR^.

Post-Scattering Direction Cosines

After collision an azlmuthal angle must be selected and

new direction cosines determined.

29

Aa azimuthal symmetry Is assumed the azimuthal angle

must be selected In the interval (0, 2TT) In a plane perpen-

dicular to the Incident direction. Note however, that for

Lambert scattering the incident direction is taken to be the

outward normal to the scattering surface. The azimuthal

angle,p , is specified by

sin$f • sin (2TT<t),

cos0 a cos (2TTQL),

where & is a random number.

The new direction cosines may now be computed using the

method of Marchuk and Mikhailov (3, p. 149). Let S be the

pre-scattering direction denoted by

S 2£ • bly ^ f

and the new direction specified by

S1 = a'lx* b«ly4-c»lz.

L e t b e the cosine of the scattering angle. Define

(1 - } 1 2 F CO3 0 ,

(1 -p?)1 strip ,

k s as,.- bgg,. The new direction cosines are now given by

a8 = - K/(14 lcl)3V

b« s b[jJL - K/(l + lolfl

c« j. • k sign (c).

3°

Detector and Local Coordinate System

If D Is the distance along S to the rim of the atmosphere,

then the coordinates of the detector point are given by

x 0 * x aD,

y 0 = y + hD,

Z p S SS cD,

where (x, y, z) is the scattering point and (a, b, o) are the

new direction cosines.

The local coordinate system has origin at the detector

point and is subject to previously discussed requirements.

It is necessary to specify mathematically the local x1, y', z1

directions in the planet system. The 1 . direction is known z

to be normal to the atmospheric surface and is in the &

direction, where 1 is the unit vector towards the point R

y#J z§). Now

i ly

and

* ^ R * 1X^

A A

V - V* V hence

a

Sc* ~ x XR *> /* ^ v« 1v' " ^1R* l±' R*

where * , A A LH

Performing this, one finds that *

lp • ^D^x ^ V * *D

ix. = (yp • zl > - ( x

Dyp) - t ' p V V

p2 ®2 «2 RD RD KD

31

A ^

In general 1„, Is not a unit vector. Normalizing, 1„. is

found to be (yj * z d ^ i* - (xpyp) V - (xpyp) V

rD Rp(yp zp)^ Rp(yp+ zp)^

A \ A, The coefficients of lx„ ly, lz are the direction cosines

, . A

(d|, b|, C|), of lxi in the planet system. Hence the local

coordinate system is completely specified.

Flux Calculation

After each collision where the photon is directed outward,

a flux contribution to the detector penetrated is computed.

This raw flux, FR, is computed by

pH - w e ,

A where If is the optical pathlength to the detector along S and

-t

W is the statistical weight. The factor £ is the probability

that the photon reaches the detector without scattering.

Petermination of Angular Pependence

The angular dependence of the emergent flux in the local

system Is now determined. The cosine,p. , of the angle the

emergent flux makes with the local zenith is given by H A

)k = S • lzt, ' *

where ya is the oosine of 9, The projection of S on the x«

32

axis Is

Sx, » t* lx, = aa% «• bb% *• oc%, A

where S Is the direction of the emergent flux» However,

sx« Is also given by

1§1 sin 9 cos

where

sin 9 = (1 -)l2)K

The cosine of the local azimuthal angle, 0, is given by

cos 0 = (aa 4» bb * cc^)/sln 9,

for sin 9 not zero. If sin 9 is zero, cos # is taken to

be unity. The angle $ is given by

$ a cos [(aa^ * bb^ • CC| )/sin 9^ .

The cosine,^ , of the planet nadir angle is determined

by comparing the x coordinate of the detector point with the

x boundaries of the detector bands.

Once the angular dependence of the fluxes has been

determined, the fluxes are recorded In bins according to their

angular dependence.

Termination

The photon continues this scattering cycle, as shown by

the flow chart given in Figure 4, until its weight drops below

a termination criterion. When this occurs the photon is

terminated, a new one selected, and the process begins anew*

33

Intensity Calculation

After the desired number of photons have been processed,

the Intensities of the emergent radiation are computed from

the fluxes,

Let Fr be the total raw flux. The average flux

contribution per photon is given by

P * Fr/2.Nh*)Ao,

where the width of the detector bands. The effective

number of photons considered,N Is

N /(I - R'2/RI ),

as defined for photon selection and N is the number of

histories processed.

The intensity may now be computed from the flux.

Solving the equation

P . yidjjLcty

For I, which is assumed constant over a interval, yields

I - 2F/ [(02 ~ *

The 0 values only cover the interval (0, IT )] hence

X - F/Kfc, -0l)(>jf2 - M* )3 •

With little extra effort the intensity Is determined

for the case when the albedo Is zero, regardless of the

albedo assumed for the planet. This Is effected by separately

recording the flux contributions of photons that have not yet

experienced a Lambert reflection.

CHAPTER BIBLIOGRAPHY

1. Cashwell, E. D. and C. J. Everett, Practical Manual on the Monte Carlo Method for Random Walk Problems, New York, Pergamon Press, Inc., 1959.

2. Collins, D. G. and M. B. Wells, Monte Carlo Codes for Study of Light Transport In the Atmosphere, Vol. I, 2 volumes, Port Worth, Radiation Research Associates, Inc., 1965.

3• Marchuk, G. I. and G, A. Mikhailov, "The Solution of Problems of Atmospheric Optics by a Monte Carlo Method,' Izv., Atmospheric and Oceanic Physios, III (1967)* 147-155.

4, Mateer, C. L.', 'Laboratory of Atmospheric Sciences, Boulder, Colorado, letter, March, 1968,

3 4

CHAPTER IV

COMPARISON OP RESULTS AND CONCLUSION

Comparison of Results

No spherical results have been given in the literature

that are suitable for comparison to the values obtained

from this calculation. However, f o r b e t w e e n 0.40 and

0.80, the results obtained from this spherical model should

have very nearly the same values as those obtained from the

similar plane-parallel model. Comparisons will be made to the

well substantiated results of Coulson et aL (2) and Adams (1).

These results are determined within different 0 bins, As the

values of the reflected intensities are not strongly 0 depen-

dent, all results are averaged over 0 .

Results Compared to Those of Coulson et al.

Even though Coulson1s results (2) are for the vector

case, the results of the spherical calculation should closely

approximate them.

The Coulson and spherical results are compared in Figure 5

for the case » 0.10 and 0. For * 0.80, the histogram

showing the spherical results is much lower than the Coulson

curve for JL near the looal horizon, becoming slightly higher

35

36

near the zenith. A lower intensity is expected for the

spherical model in the region^JL* a s this includes

a much smaller volume of space than does the plane-parallel

model for JJL = 0.10. For |JL • 0.40, the plane-parallel

intensity is again larger for jx • 0.10 and slightly smaller

for towards the zenith. The overall agreement of the two

models is excellent.

Figure 6 shows the results of Coulson et_ aL compared

to the spherical results for the case X • 1.00 and ^ • 0.0.

The s 0.80 histogram shows excellent agreement with the

Coulson curve for yi greater than 0.20, and the expected lower

intensity near the horizon. For JJl » 0.40, the histogram

again shows the same type of agreement with Coulson's values.

The spherical results are compared to Coulson8s in

Figure 7* for the case X « 1.00 and « 0.80. The JUL - 0.80

histogram shows good agreement with Coulson's curve, especially

in the range from - 0.20 to JJL - 0.80. The elevated

value for jx » 0.80 to 0.90 is probably just a statistical

fluctuation, as it has not appeared consistently in the results.

For ji r 0.40, the plane-parallel case shows a slightly

larger reflected intensity up to Jl * 0.70, but matches very

closely for JL closer to the zenith.

In Figure 8 the results of the cumulative Rayleigh case

of Mie scattering are compared to those of Coulson et_ al. for

X - 0 .10 and X • 0.0. If the Mie scattering routine works

37

properly the cumulative Rayleigh distribution should show

the same emergent, intensities as pure Rayleigh scattering. *

Por 0,B0, the spherical intensities are quite close to

those of the plane-parallel case, especially for JL towards

the zenith. The case where * 0,40 shows a similar likeness

to Coulson's results.

Por all the above cases, it is noted that the spherical

case is in exoellent agreement with the plane-parallel within

the limitations imposed by the disparity between the two

models.

Results Compared to Those of Adams

The results of the algorithm are now compared to those

of the plane-parallel model devised by Adams (1). The values

that are used are for Adams1 solar case, where the incident

solar beam is assumed unpolarized. The 0 -averaged values to

be compared are those for 0.80 and 0.807 for Adams', since

he uses different values. *

Figure 9 shows the plane-parallel results compared to

the spherical results for % - 1.0 and ^ « 0.0. The spherical

model yields results that are, again, in excellent agreement

with those of the plane-parallel case. The spherical intensity

values are slightly lower than the plane-parallel values towards

the horizon, and in better agreement towards the zenith. Figure 10 shows the comparison for the case *£ = 1.0 and

JA = 0.80. The agreement is very good for Li between 0.20

38

and 0,80. For smaller the spherical model yields

lower intensities than the plane-parallel case, as expected.

The general agreement between Adams' results and those

of this algorithm are excellent.

Conclusion

The results of the algorithm show excellent agreement

with those of the analogous plane-parallel cases of Coulson et al>

and Adams, which are well substantiated.

On the basis of these comparisons it is concluded that

the spherical model and algorithm developed herein provide

the best representation of the spherical scattering atmosphere

developed to date. This method should have numerous important

applications, and may be extended to explain phenomena occur-

ring in many types of spherical atmospheres.

CHAPTER BIBLIOGRAPHY

1. Adams, Charles, "Solutions of the Equations of Radiative Transfer by an Invariant Imbedding Approach," unpublished master's thesis, Department of Physics, North Texas State University, Denton, Texas, 1968.

2. Coulson, K. L., J. V. Dave, and Z, Sekera, Tables Related to Radiation Emerging from a Planetary Atmosphere with Raylelgh Scattering, Berkeley, University of California Press, 19557

39

APPENDIX

40

41

A T M 0 8 P H E R I 0

THI0KNE88

IEAWTH

Fig. 1—The earth-atmosphere system to eoale

42

• X -

LOOAL SYSTEM

/^PLANET ¥ N

^ SYSTEM

Pig. 2—The coordinate systems

43

I

ATMOSPHERE

INCIDENT

PHOTON

/

Fig. 3—Photon selection

44

©

8ELE0T A

PHOTON

© DETERMINE

© S E L E C T

IP 9T8EE8 YE8^ THE

PLANET

J RHO

COMPUTE

t MAX

COMPUTE INITIAL

PLUX

C O M P U T E

W E I i H T

8 E L E 0 T

RHO

1

.AMBER T

8 E L E 0 T

ANOLE

DETERMINE 80ATTERIN8 TYPE

MIE

SELECT

AN8LE

DETERMINE NEW DIRECTION COSINES

RAYLEISH

YES DETERMINE IP IT 8EES

THE PLANET

COMPUTE t TO RIM A PLUX

OOMPUTE INTENSITY

SELECT

ANSLE

Pig. 4—Flowchart of the scattering simulation

45

I • 0 6 -> s~ '05

co z Ul I- ' 0 4

a uj *03

o UJ -J n- ,Q2 UJ

•Oi

— 0 0 U L 8 0 N i \ Pg « \ • SPHERICAL

— 0 0 U L 8 0 N

\\ •E-IPHERIOAL

\ Q

IJL**4 ' 0

A -

J- L J L

2 3 .4 »5 -6 •? -8 -9 10

JJL >

Pig. 5—Results of spherical oase compared to those of Coulson et al for X *"

46

t CO z til h-

Q su H O til

I&. til ac

•2 h

8PHERI0AL,JlP>»

— 00UL80N,U».« */©

* 8PHERiOAL^JL»*4

OOULSOIMJM*4 *0

J JL J L

'Z "3 *4 • 5 *6 »7 • 8 °9 1*0 • — • >

Fig. 6—Results of spherical case oompared to those of Coulson et al for T " 1* X 9 0.

47

I >-I-

(0 z fijj H Z

o m H o HI -I II. Ill

•flbcaeslte r - U _

8PHCRICAL,jl«*i

— 00UL8OM,U**9 'O

•#§* 8PHEfllOALtJ^*4

-- 00ULS0N,ll««4 0

Pig. 7—Results of spherical case compared to those of Coulson efe al for " 1* % 0

m .8.

•06

1 >

t •05 CO z III i - •04 z

a 1*1 ' 0 3 o Li -J 1*. yj •02 a:

•01 - # •

8PHERI0AL,UL«*S 'O

— OOUL8ON|U«a0 'O

-*• 8PHERI0AL^-4

Q0UL80N,p-4

' » 8 fi I I t i •I °2 -3 -4 -5 -6 -7 -8 -9 1 0

j i >

Pig. 8—Results of cumulative Rayleigh case compared to those of Coulson et al for s . 1 , ° <

I > I-

*© z III H z

a ui H o yi -§

ii. ui a:

•#*8PHCRI0AL|li**f '0

— ADAM9«|l«»t07

I I I ii i t

I 2 3 -4 -5 -6 T -8 -9 1*0

Pig. 9—Results of spherical case compared to those of Adams for t * 1, " 0.

> t CO

liJ I—

Q til

i— o UJ _J u. yj

§c

* 8PHERI0Al,JLU*8

— ADAM8|1L* *807 'o

M % i M

r r - m

, m K

i e i i _L L

I 2 3 4 5 6 7 8 9 80 p- >

Pig. 10—Results of spherical case compared to those of Adams for *X " 1* A # " ,8.

BIBLIOGRAPHY

Books

Cashwell, E. D. and C. J. Everett, Practical Manual on the ' Monte Carlo Method for Random Walk Problems, New York, Pergamon Press, Inc., 1959.

Chandrasekhar, S., Radiative Transfer, New York, Dover Publications, Inc., I960.

Coulson, K.L., J. V. Dave, and Z. Sekera, Tables Related to Radiation Emerging from a PI ane t ary At mo s'phe r e w ft h Raylelgh Scattering, Berkeley, University of California Press, I960.

Hammersley, J. M. and D. C. Handscomb, Monte Carlo Methods, New York, John Wiley and Sons, Inc., 196^

Articles

Kattawar, G. W. and G. N. Plass, "Electromagnetic Scattering from Absorbing Spheres," Applied Optics, VI (August, 1967)# 1377-1382.

Kattawar, G. W. and G. N. Plass, "Resonance Scattering from Absorbing Spheres," Applied Optics, VI (September, 1967)> 15119-1555.

Marchuk, G. I. and G. A. Mlkhailov, "Results of the Solution of Certain Problems of Atmospheric Optics by the Monte Carlo Method," Izv., Atmospheric and Oceanic Physics, III (1967). 227-231.

Marohuk, G. I. and G. A. Mlkhailov, "The Solution of Problems of Atmospheric Optics by a Monte Carlo Method," Izv., Atmospheric and Oceanic Physics, III (1967)? 147-155.

Plass, C-. N. and G. W. Kattawar, "Influence of Single Scattering Albedo on Refleoted and Transmitted Light from Clouds," Applied Optica, VII (February, 1968), 361-367.

51

52

Plass, G. N, and G. W. Kattawar, "Monte Carlo Calculations of Light Scattering from Clouds," Applied Optics. VII (March, 1968); 415-419.

Plass, G. N. and G. W. Kattawar, "Radiant Intensity of Light 6o9t?o5ed f r° m c l o u d s>" Applied Optics, VII (April, 1 9 6 8 ) ,

Smoktii, 0. I., "Multiple Scattering of Light in a Homogeneous Spherically Symmetrical Planetary Atmosphere," Izv., Atmospheric and Oceanic Physics, III ( 1 9 6 7 ) , HO-146.

Reports

Collins, D. G. and M. B, Wells, Monte Carlo Codes for Study 21 Light Transport in the Atmosphere, VoIi7"l"and"TT^ Port Worth, Radiation Research Associates, Inc., 1965 .

Unpublished Materials

Adams, Charles, "Solutions of the Equations of Radiative Transfer by an Invariant Imbedding Approach," unpublished master's thesis, Department of Physics, North Texas State University. Denton, Texas, 1968 .

Private Communications

Mateer, C. L., Laboratory of Atmospheric Sciences, Boulder, Colorado, letter, March, 1968 .