MONTE CARLO METHODS IN RADIATIVE TRANSFER

George W. Kattawar

Center for ‘Theoretical T(ysics

Department of TIjsics, Thas &94( Vuiversity

Coltege Station, ‘Tea.c 77843 -4242

I. Introduction

Let us begin our scenario with a visualization of our ultimate goal

i.e., the solution of the equation of transfer for any arbitrary geometry

of source, medium, and detector. We should state at the outset that the

power of the Monte Carlo method is usually not demonstrated for plane-

parallel systems with no lateral variations since we have more accurate

methods for obtaining solutions for this situation. The power of the

method is borne out when one wishes to solve such problems as those

involving spherical atmospheres, laser beam propagation (LIDAR), search

light beam propagation, and broken cloud problems, just to mention a few.

The basic idea behind the Monte Carlo method is the following: if

one knows the probability of each in a sequence of events then one can

obtain an estimate of the final outcome. Let us analyze this statement

in the context of radiation incident upon a scattering and absorbing

medium. When radiation impinges on a medium we know that the path lengths

are distributed with an exponential probability density function. At an

interaction point the photon is either scattered or absorbed. If it is

scattered then the probability that it is scattered through some angle

is governed by another density function which we normally call the phase

function. This process is continued until the particle encounters some

obstacle, for example the ground, which again has some known scattering

-2—

and absorbing probability. Now this process is repeated enough times until

we have statistics which are sufficient for our purposes. Now this type

of straight forward simulation is called “brute force” or ‘crude” Monte

Carlo. The real finesse comes when we learn to play the game with

“loaded dice”. Such schemes fall under the general classification of

variance reduction techniques and involve such things as Russian roulette,

sequential sampling, importance sampling, forced collisions, etc. Some

of these methods will be described later in the sequel.

II. Random Numbers

In any Monte Carlo simulation we must have an adequate supply of

random numbers, actually they are called pseudorandom numbers since they

are produced by a precise algorithm. Most random numbers generators use

a variation of a method first proposed by Lehmer in 1951 called con

gruential methods. One such method is the following:

x E c x1 + (modulo m) (1)

where m is usually taken as 2(bit length 1) for a binary computer where

the first bit is the sign bit and c, , and x. are integers between 0

and m-l. The numbers x/m are then used as the pseudorandom sequence.

Prob. 1 Let m = 16, = 3, = 1 and = 2 and compute the sequence

of numbers and the period of this generator.

-3-

F or the algorithm of eqn (1) the full period of m can always be achieved

provided that

a) 6 and m have no comon divisor

b) . 1 (modulo p) for every prime factor p of m

c) 1 (modulo 4) if m is a multiple of 4.

Now let us state the fundamental principle of Monte Carlo calculations.

Let r be a random number which is uniformly distributed on 0 < r < 1 and let

p(x)dx be the probability of x lying between x and x ÷ dx with a<x < b, and

ajb p()dn = 1 (2)

then

r = P(x) = afX p()d (3)

determine x uniquely as a function of r and x falls with frequency p(x)dx

in the interval (x, x+dx). Let us now consider an application of the

above method as it pertains to radiative transfer theory. The probability

that a photon makes a collision between optical depth t and t + dt is

—tp(t) dr = e dt (4)

Therefore r ft e_tdt = 1 - e (5)

and t = - n (l-r) (6)

but since l-r is also uniformly distributed we can just as well use r leaving

us with

-4-

— 2n r (7)

Prob. 2 Use equation (7) to generate a sequence of -r5 and plot their

density on semi-logarithmic paper

It should be clear from our example that if a medium is optically thin

then most photons will escape the medium. This of course can be quite

wasteful and costly in terms of computer time. To avoid this we will

use the method of forced collisions and introduce the use of a statisti

cal weight. Instead of considering each photon as a single entity we

will regard it as a packet of photons. As an example let the injected

photons have initial Weight W0 (usually set equal to unity). The fraction

of photons escaping a first collision and passing through the medium is

e_tm where Tm is the optical thickness of the medium. Therefore the

fraction remaining is 1 - e_Tm. We will now force a collision by sampling

from the density function

e dtp(t) dt = , 0 < < r 81 - e_tm

Using eqn (3) we get

1 - eTr= (9)1 - e Tm

and

=- n[l - r(l - eTm)] (10)

To remove the “bias” in our sampling we must multiply the weight W0 by

—5—

1 - etm which gives the fraction of our packet of particles which remain.

We will illustrate the generalization of the method just used. Con

sider the expectation value of some function f(x)

<f(x)> = I f(x) p(x)dx (11)

where p(x) is the probability density and f(x) is called the estimator.

Now suppose that instead of sampling from p(x) we sample from p(x) then

= I f(x) p(x)dx = I f(x) p(x)dx (12)p(x)

so the estimator must be multiplied by p(x)/p(x).

Let us use this method to analyze the converse of the case for thin

media, namely media of large optical thickness >> 1. If we are inter

ested in transmission through the medium it is clear that sampling from

the exponential density function will place very few particles deep in

side the medium. We can remedy this by sampling uniformly in t i.e.,

r=1— , (13)tm

where now p(t) = and p(t) = e_t therefore the weight is multiplied by

Tm e

We have now seen how eqn. (3) can be effectively used to sample from

a given probability density; however, it is clear that this equation can

result in very difficult implicit problems since x must be determined

from r. An example of such a complication arises when one wishes to

sample a highly asymmetric phase function which is encountered from aero

-6-

sols, or clouds. In these cases we may have only tabular data to work

with. Let us assume that in the interval (a,b) we have accurate values

of P(x) P1 stored for points

x0=a<x1<. . . <x=b

Let i be the first value for which r - P. < 0 where r is a random

number, then x can be determined from one of the following formulas

x =- (x - x11) , (14)

x = [x2 -(x2 - x1)]2 (15)

x = [x2-

p’ (x2 -x1)2]2 (16)

Eqn (14) distributes x uniformly on (x1, x) and is correct only when p(x)

is a step function. Eqns (16) and (17) are appropriate when P(x) is concave

up or down, respectively.

Another very useful scheme which can be used when p(x) is easy to com

pute was introduced by Von Neumann and is called the rejection technique.

To illustrate consider p(x) defined on the interval [a,b] and let

p(x) = p(x) / sup p(x)a<x<b

y —7—

1

px)

a

Think of now throwing points at the board bounded by x = a, x = b,

y = 0, y = 1 and rejecting points lying above the curve

y = p(x)

and letting x = whenever (,n) falls below the curve. For a large

number of throws, the ratio of the number of points retained with 5 between

xand x+dx to the number of points retained altogether will be the ratio of

areas i.e.,

b bPiX)dX/af p()ds = P(X)dX/af p()d = p(x)dx (17)

Prob. 3 In the rejection method we could also retain points (,n) above

the curve y = p(x), but the weight will have to be adjusted.

Determine the factor which multiplies the weight for such a

scheme.

III. Direction Cosine Routines

We have now discussed ways of sampling path length and scattering angle

relative to the incident photon direction. We now need a set of equations

to give us the new photon direction in our fixed coordinate system. Let

(u, v, w) be the direction cosines

. (,Ti)

x

z

x

Y

of the photon before scattering relative to the (x,y,z) axes respectively.

Let be the scattering angle and the angle representing the rotation of

the scattering plane. These angles will be selected from the phase function

p(c)dc2 = p( e , ) sine d d. Also let (u, v, wj be the direction

cosines of the scattered photon relative to the fixed coordinate system

(x,y,z). Now let

a = cos C

b = sine

C = COS

d = sin

-r’r

With this notation we have

= (bcwu - bdv)/A - w2 ÷ au (18)

= (bcwv + bdu)/A - w2 + av

1 2w = - bc l - w + aw

(19)

I (u,v’,w’)I (u,v,w)

(20)

-9-

It should be noted these equations are indeterminate for wi = 1 and are

inaccurate for wi 1 and should be bypassed in favor of

= bc (21)

v = bd (22)

w = aw (23)

IV. Isotropic Scattering

The phase function for isotropic scattering is simply given by

= = dd(24)

where = cose

Prob. 4 Derive an algorithm which will generate the variables .i and for

isotropic scattering. (eqn (24))

It should now be noted that for isotropic scattering it is totally un

necessary to use eqns (18) - (20), since a uniform distribution of points

does not depend on a fiduciary axis. Therefore our technique for generating

(u,v,w) is quite simple and goes as follows:

w = cos e

and p(w)dw = , - 1 < w < 1

wtherefore r

= .1.1 = (25)

-10-

Now the azimuthal angle is obtained from

r = p()d = = (26)

It is now easy to compute u and v, namely

u= l—w cosp .27)

v = A - w2 sin • (28)

V. Ground and Other Surface Reflections

Let us now assume that our photon encounters some surface. The

simplest type of surface we can consider is the so called Lambert surface.

What is meant here is that the radiance emanating from this surface is a

constant in all directions. This implies that the probability density

function for photon reflection from the surface is

p(3.i,) = - ddp 0 < < 1, 0 < < 2Tr (29)

(Convince yourself that this density function will produce constant radiance

in all directions.)

Prob. 5 Develop a simple algorithm using the point rejection method to

sample the normalized density function p() = 2 used in eqn. (29).

—11—

If the Lambert surface has albedo A then upon reflection the weight is

multiplied by A. It should now be clear that by carrying multiple photon

weights that any number of surface albedos can be handled in a single

computer run.

If the surface is a dielectric, such as water, then one needs the

Fresnell reflection coefficients as a function of angle. Let R(i) denote

the reflection coefficient, then a random number can be generated and

tested against R(). If it is less than R() then specular reflection

will occur, and if it is greater then the photon is transmitted into the

medium and its angle of entrance appropriately transformed by Snell ‘S law.

It should now be clear that as long as we know the reflection properties

of any surface, whether in tabular or analytic form, it can be successfully

simulated by Monte Carlo techniques.

VI. Selection of Type of Scattering

Suppose the photon has made a collision in some layer of the medium.

Also let

SR = Rayleigh scattering coefficient

SM = Mie scattering coefficient

= total extinction coefficient

The single scattering albedo= sR sM)/ST. At each collision the

weight is multiplied by A random number r is then generated and tested

—12-

against the ratio SR/(SSR + If r is less than this ratio then the

Rayleigh scattering phase function is used to select the scattering angle.

If it is greater than this ratio then the Mie scattering phase function is

used.

VII. Backward Monte Carlo

In the forward Monte Carlo method we start photons at the source and

track them to the detector. In the backward Monte Carlo we use time re

versal and track photons from the detector back to the source.

We will demostrate the efficacy of the backward Monte Carlo method by

considering the following simple example. Consider a semi-infinite, plane-

parallel, homogeneous, conservative, isotropically scattering medium. Let

us assume that the medium is illuminated vertically by a flux F(r,p) (see

figure below). Let us now compute a scalar irradiance (ia) from the radiance

I at point 0 on the top of the medium (see figure below). Now

(30)10 = i 1(2)dc2

Incident SourceF(r,Ø)

z

Interaction Point

-13-

where the integral is taken over the upper hemisphere and the quantity

is also proportional to the energy density of the scattered radiation

at point 0. The contribution to I from singly scattered photons can be

written as

F’ -6p —6Z10 1 f f 6 e rdrdz, (31)

0 0 0 4irp

where 6 is the extinction coefficient. Now we immediately see a problem

using forward Monte Carlo, i.e., source coordinates. We must select some

maximum radius to terminate the selection of the r coordinate. A poor choice

of this cutoff will lead to sampling in regions which produce no significant

contribution to the final result. Also the presence of the term in the

estimator can cause large statistical fluctuations in the results for small

p. These problems can be eliminated by going to backward Monte Carlo, i.e.,

we will transform (3l)from source coordinates to detector coordinates.

This can easily be done with the following coordinate relationships (see the

Figure on the preceeding page).

r = p sin e,

(32)

z = p cos 8.

Evaluating the Jacobian for the transformation we get

-14-

2ir 7T/2 F ‘-

= I J .1. r,p,e e 8d8dd.0

000 (33)

This integral can easily be evaluated by the Monte Carlo method by first

sampling o from the normalized density function sinede, sampling from

the normalized density function dp/2rr, and sampling p from the normalized

density function The estimator is now F(r,)e e72

should be noted that the two basic problems encountered with forward Monte

Carlo have now been eliminated.

VIII. Inclusion of Polarization

Up to the present we have only demonstrated the technique to generate

the radiance which is only one component of the more general Stokes Vector.

We will adopt the I, Q, U, V representation where I Ir + I and Q = I.z - Ir

where r and 2. refer to components parallel and perpendicular respectively to

the reference plane. A clockwise rotation of the axes through an angle

(looking into the direction of propogation) produces the following transforma

tion matrix

1 0 0 010 cos2 sin2cp 0

L() = I (34)0 -sin2c cos2 0

0 0 1

This respresentation of the rotation of the axes is not the one normally found

in the literature (see Chandrasekhar ref 4 pg 35) since earlier researchers

-15—

used the I, Ir, U, V representation. The linear transformation connecting

the two can easily be derived as follows

1 1 0 0

1 -l 0 0

0 0 1 0

0 0 0 1

1

½ -½ 0 0

0 0 10

0 0 01

0 in the dia

the angle of

let I

Jr Q1 ‘2

U U

V V

where I = I + Ir, and Q = LQ - Jr. Therefore

where

(35)

(36)

(37)and

Now let us consider that a scattering has occured at point

gram below. Let I. be the incident direction of the particle,e

scattering, and i1 the orientation of the scattering plane

I

.0•

—j

- 16 -

relative to the meridian reference plane. Then the Stokes vector I’ after

scattering is related to the Stokes vector I before scattering by

I’ L(—12) (e) L(—i1) (38>

where

_____

M2-M1 a a2 2

M2-M1 M2+M1 a a2 2 (39

R(e) =

o a - D21

o o 1321 S21

The four quantities in the scattering matrix M1, M2, S21, and are

consistent with the definition used by van de Huist (Ref.5, pg. 44).

Now if L1 and are in the 12., Ir, U, ‘I representation and L2 and R2 are in

the I, Q, U, V representations then from eqn. (38)

= i (-i2) l (-i1) (40)

multiplying eqn. (40) from the left by T we get

1 (-ia) T T R () T T L1 (i1) T T ‘1 (41)

= 2 12’) R2(e)L2(-i1) .2 (42)

In the Monte Carlo method, the scattering angle C is selected by

a random process from the cumulative distribution of the scattering

function ½ (M1 + M2); similarly the angle i1 is chosen from a uniform dis

tirbution between 0 and 271. These distributions are a first approximation

to the correct distributions fore and i1. The calculation allows for the

— 17 —

difference between the actual distribution and the approximate one by correct

ing the components of the Stokes’ vector after collision by a method described

below. It should be emphasized that the procedure would yield the correct

result for any initial distribution function for e and i1, but the statistical

fluctuations are less if the initial density functions are reasonably close

to the actual ones. Once the angles e and i1 have been selected, the angle

is computed from the equations of spherical trigonometry. It should be

noted that it is not necessary to sample a and i1 from a bivariate density

function but instead a biased density may be used

To see the bivariate nature of the density function, let us expand eqn.

(38) without the final rotation L C-i2). We find

(M1 + M2) (M2 - M1)I’ (e, jl = I 2 + 2 C cos 2i1 + U sin 2 i1) (43)

it is now clear that I’ not only depends on I but on Q and U as well.

At this juncture we will now show how to sample a bivariate density funtion

p (x,y), when x and y are functionally independent variables i.e., x may

change arbitrarily holding y fixed and vice versa. The conditional probability

p (xy) can be expressed as

p (xy) = p (x,y) / p(y) (44)

whereb

p (y) f P (x,y) dx (45)

a

is called the unconditional probability. Let us apply the above theorem

(eqn. (44)) to eqn. (43). Since we now have the machinery to sample highly

anisotropic phase functions, we will first sample the scattering angle

= cos 9 by

- 18 -

2 (M1 () + 112 (.))p () = I p ( , i) d i = 2 , and f1p() d u 1 (46)

0 2 -1

Mi + 112where ( ) 2ir is the phase function used for unpolarized light and

2

p () has been normalized by dividing by I. Therefore

1 M -Mp Ci ) = { 1 + 2 1 C cos 2i1 + U sin 2i1)} (47)

M2 +

We can nowuse the rejection method to sample eqn (47), however, to do this we must

have the maximum value of the function over the interral therefore i ) =

diM -M 1

2 1) (Q sin 2i1 + U cos 211))M + M1

Setting di1 0 gives

tan 2i1 = U/Q

using this result in eqn. (47) gives

max (i) =

+ : : : (/Q2 ÷u2 (48)

We can now use the rejection method to sample eqn.(47) for i1. We should

now make a very important point and that is if we are doing

backward Monte Carlo we can not use the method just described since we

do not know the initial state of the photon. It should be remembered

that the last collision in the backward Monte Carlo is the first collision in the

forward Monte Carlo and vice versa. We are therefore forced to use biased

sampling i.e., we first sample . from and i1 randomly between 0

and 2’rr. Now since we are performing time reversal the matrix product in

eqn. (38) now becomes

- 19 —

M +M= 2j2 2 2(-i1) 21(

12)(49)

M +Mwhere the factor 12 2 is used to remove the bias.

Prob. 6. Use the transformations (eqn (37)) and show that

cos2 sin ½ sin 2 0

sin Co5 -½ sin 2 0L1(e) =

-sin 2p sin 2 cos 2 0

0 0 0 1

which is in the I, Ir, U, V representation can be transformed into eqn. (34).

VIII. Sampling the Henyey-Greenstein Phase Function

The Henyey-Greenstein phase function is given by

1 2p() =

-

4i(l -

where = cos e and e is of course the usual scattering angle and the quantity

g is called the asymmetry factor and is just the first moment of the density

function. Let P() be the distribution function of defined by

2= f .1 p(u,)dud

-l 0

Setting r = p(j) and solving for in terms of the random number r we get

- 20 -

= {l + g2 - [(1 - g2)/(g(2r - 1) + l)]2}/2g

It should be noted -l when r = 0 and = 1 when r = 1. In Fig. 1 below

we show how good the above algorithm is for sampling this density function.

r r rr T r

• -o7 FxAcr• qO7 M \E AL

IgO5 EXAC’+O5 MCNECRLD

IDo,occ SOR ES

Ziâz_

D‘.a.. .

LJJCl) -

L—

LAJ

H L

H

0 08 060402 0O02Q406O80Cos

I I ii \[ nh ( in’I inni plnnig1Inni n—( irennntfnnn nhae linnin Inn n rh t h

I nI(nnIaI rn.

- 21 —

Biblography

1. Hammersley, J. M. and Handscomb, D. C., “Monte Carlo Methods’

(1964) John Wiley & Sons, Inc., New York.

2. Spanier, Jerome and Gelbard, Ely M., ‘Monte Carlo Principles and

Neutron Transport Problems”, (1969) Addison-Wesley Publishing Co.

Reading, Mass.

3. Cashwell, E. D. and Everett, C. J., “A Practical Manual on the

Monte Carlo Method for Random Walk Problems”, (1959) Pergamon Press,

New York.

4. Chandrasckhar, S. “Radiative Transfer” (1960) Dover Publ. Inc.,

New York.

5. Van de Hulst, H. C., “Light Scattering by Small Particles”, (1957)

John Wiley & Sons, Inc., New York.

Backward Monte Carlo Comparisons

for Conservative Rayleigh Scattering

Figure 1 Comparison of plane parallel calculation for intensity with

the backward spherical Monte Carlo for = 66.9, t = 0.25,

• = 0 and 1800 for ground albedos of A = 0 and 0.8. Solid

curves are taken from Coulson, et. al.

Figure 2 Same as Fig. 1 except the ordinate is the degree of polariza

tion.

Figure 3 Same as Fig. 1 except for • = 30 and 150°.

Figure 4 Same as Fig. 2 except for • = 30 and 1500.

Figure 5 Same as Fig. 1 except for • = 60 and 120°.

Figure 6 Same as Fig. 2 except for • = 60 and 120°.

LU

0,9

0.8

U.S

0.4

0.3

0.2

0.1

0.0

90

0:0

0

0:6

6.9

°

TA

U:Q

,25

0:1

800

A=

0.8

>_

0.1

F— ,—

_-

0.6

(f)

2:

u-i

F 2:

A:Q

.0

6030

030

6090

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DIR

ANGL

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z 60

F—

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80 10 U90

6030

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60

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Fig

ure

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1.0

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II

II

II

0=

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ø13O

0

O=

66.9

°

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-T

AU

=O

.25

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E::: LU 1_0.I4

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(1.3 :

44

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60300306090

Figure6