Theoretical Base of Solar Irradiance and Radiance Calculation in the Earth Atmosphere - Short-Wave Solar Radiation in the Earth's Atmosphere - page 48

a random number generator or randomizer , is well known nowadays and we
are not dwelling upon its specific pointing only that we have been using here
the randomizer proposed in the study by Molchanov (1970). The totality of the
random numbers uniformly distributed over the interval [0, 1] is the basis of
the Monte-Carlo method. We are implying only these numbers using the term
“the random number”, specifying them by sign β , and at every appearance in

the text we mean a new randomnumber. For compact writing of the numerous
recurrent relations occurring in the method we are using the operation of the
assignment “:=” as it is accepted in the programming languages.
Let the probability of a certain discrete random event be equal to P .Choose
the random number and if

β ≤ P , then assume that the event has happened, in
the opposite case assume that it has not happened. The grounds of this approach
are evident: if the quantity of the simulating acts tends to the infinity then the
ratio of the quantity of the simulating acts when the event has happened
to the quantity of all acts is equal to the probability of the event, i. e. to P
due to the uniformity of the random numbers distribution. Note that for the
continuous random value simulating characterized with probability density
ρ ( u ) within the interval [ a , b ]theprobabilityvalue u within the interval [ a , u ]

is equal to P ( u ) = a ( u ) du according to the definition. The application of

the above-mentioned approach for the discrete random values leads directly
to the following equation for values u simulating:

ρ

u

ρ ( u ) du = β . (2.6)

a

As has been mentioned above, the process of radiative transfer in the Monte-
Carlo method is simulated as a photon motion. Coming to the atmosphere the
photon is moving along a certain trajectory, which finishes either with a photon
outgoingfromtheatmosphereorwithitsabsorptionintheatmosphereoratthe
surface. The positionof the photon in the atmosphere is determined completely
with three coordinates:

ϕ ,hence,thesimulationofthetrajectoryreduces
to the consequent counting of the coordinate data. Therefore, it is enough to
simulateonlythreeprocesses:thefreepathofaphoton(i.e.withoutinteraction
with the atmosphere), the interactionwith the atmosphere (the absorption and
scattering), and the interaction of a photon with the surface (the absorption
and reflection).
A free photon path is analogous to the transfer of solar direct radiation
throughout the atmosphere. Remember the formula of Beer's Law (1.42):

τ , µ ,

τ = µ 0 exp(− τ|µ 0 ). F d ( ) F 0

µ = Let K photons income to the top of the atmosphere, i. e. F 0 KE ,where E is

the energy of a single photon. Substituting KE to Beer's Law we obtain that the
quantity of photons reaching optical depth

0

τ|µ 0 ). However,
owing to the probability definition it means that the probability for a photon
to reach optical depth

τ τ = is K ( ) K exp(−

τ τ|µ 0 ). After replacing cosine of incident angle is exp(−