Theoretical Base of Solar Irradiance and Radiance Calculation in the Earth Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 55

Ψ where function ( u ) is a certain function allowing the desired value calculation

through the source function [e. g. (1.52)]. Variable u specifies here and further
coordinates ϕ according to (1.52) and (1.6). The source function

in its turn is definedby the Fredholmintegral equationof the secondkind (1.54)
and (1.55) with kernel K and q as an absolute term.
The Monte-Carlo method has been primordially elaborated for computing
the integrals analogous to (2.20):

τ , or (and) µ ,

Ψ = Ψ ξ ( u ) B ( u ) du M ( ( )) , (2.21)

ξ

ξ where M simulated with probability

density B ( u ) as per (2.6). Therefore, (2.20) and the equation for the source
function (1.54) at the Monte-Carlo method are written for a single trajectory
and the desired value is computed over the totality of the trajectories as an
expectation according to (2.21). Applying (2.20) to the formal solution of the
Fredholm equation, i. e. to the Neumann series (1.56) we obtain:
Ψ

(. . .) is the expectation of random value

ξ

K 3 q + . . . . (2.22)
The computer scheme of theMonte-Carlomethod is reduced to consequent ap-
plying of (2.22). Term

= Ψ Ψ Ψ K 2 q + Ψ B q + Kq +

Ψ q is formed as follows: we are simulating randomvalue

ξ

(1) corresponded to probability density q and value

Ψ ξ

(1) )isbeingwritten

(

Ψ ξ

(1) random value

to the counter. Then the term Kq is forming: using value

ξ

(2) corresponded to density of the probability of the transition K (

ξ

(1) ,

ξ

(2) )

(2) ) is being written to the counter. The follow-
ing procedures are simulating analogously. Finally, the absolute term

Ψ ξ is simulating, and value (

Ψ K n q is

ξ ξ

( n ) we are simulating random value ( n +1) corresponded

forming: using value

ξ ξ Ψ ξ

( n +1) )
isbeingwrittentothecounter.Thephotontrajectoryinthephasespaceis
a chain of the pointed transitions, the simulation is accomplished over many
trajectories, and, in accordance with (2.22) the desired value is mean value
Ψ

( n ) , ( n +1) )andvalue

to density of the probability of the transition K ( (

ξ ( )overalltrajectories.

Now we are showing that the explicit form of operators q , K and

Ψ in

the above-described algorithms corresponds to their form in the equations of
radiativetransfertheorypresentedinSect.1.3.Furthermore,asdirectradiation
is not included in (1.54)-(1.56), operator Kq corresponds to q in (1.55) and
(1.56), the latter is specified as q . The phase space is specified with three
coordinates ( τ , µ , ϕ ). Operator q is evidently extraterrestrial solar radiation

µ 0 I 0 considered in Sect. 1.3
while (1.57) have been derived. Hence, to prove the correspondence of the
Monte-Carlo method algorithms to (1.54)-(1.56) it is enough to demonstrate
the correspondence of integral operators K to each other.
To begin with, consider the case without accounting for photon weights w ,
i. e. the radiation absorption is simulated explicitly. Let w

= µ 0 δ µ µ 0 ) δ ϕ q F 0 ( − ( ) that corresponds to operator

1inthelocal
estimation expressed by (2.18) and (2.19). The K operator describes, as has
beenmentioned above, the probability density of the photon path between two
points of the phase space, whose coordinates are specified as (

≡

τ , µ , ϕ )and