of the random numbers uniformly distributed over the interval [0, 1], the individual
for every photon, and if the inequality a P is true, consider that the photon passes
throughout the atmosphere, otherwise – it does not pass. Calculate number of all
passed photons: N(t 0 ,m 0 ). The incident flux at the atmosphere top F 0 m 0 recalculating
to one photon leads to the photon energy F 0 m 0 /N. The flux (irradiance) at the
atmosphere base is derived after multiplying this energy to the number of photons
passing throughout the atmosphere F
#
ðt 0 ; m 0 Þ ¼
F0m 0
N Nðt 0 ; m 0 Þ.
Unlike other methods in the Monte-Carlo method, it is appropriate not to divide
radiation to the direct, diffused and reflected from the surface.
Certainly the simple solution might be written without this consideration
(explain why
F0m 0
N Nðt 0 ; m 0 Þ ! F 0 m 0 expðÀt 0 =m 0 Þ while N ! 1 as an exercise).
However, it is convenient to obtain irradiance with real multiple scattering and
absorptions (i.e. to solve the general radiative transfer problem) if three processes
are successfully modelled: photon-atmosphere interaction (scattering and absorption), photon-surface interaction (reflection and absorption) and the photon free path.
13.2 Simulating Random Events and Values
Consider some mathematical definitions before considering the Monte-Carlo
method.
For the statistical simulation on computer, it is necessary to reproduce a process
that will play the role of random event. Special algorithms are elaborated for
random number choice that is called a random number digitizer (RND) or randomizer and there are special computer programs for generating different sequences of
random numbers including to programming languages. However, testing of such
ready algorithms shows no convenient distribution parameters of sequences (the
mean and dispersion) for scientific purposes. Hence improved algorithms of RND
have been elaborated for the Monte-Carlo method. This choice plays the role of
“blind chance” similar to roulette wheel or shuffling the cards. Just this analogue
causes the name of the method.
The totality of the random numbers uniformly distributed over the interval [0, 1]
is the base of the Monte-Carlo method. We are implying only these numbers using
the term “the random number”, specifying them by sign b, and at every its
appearance in the text we mean a new random number.
Let the probability of a certain discrete random event be equal to P. Choose the
random number and if b 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. Simulating random values is
needed aside from random events. Note that according the definition the probability
value u within the interval [a,u] is equal to PðuÞ ¼
R u
a
rðu
0
Þdu
0 for simulating the
130
13 Monte-Carlo Method for the Solar Irradiance Calculation
for every photon, and if the inequality a P is true, consider that the photon passes
throughout the atmosphere, otherwise – it does not pass. Calculate number of all
passed photons: N(t 0 ,m 0 ). The incident flux at the atmosphere top F 0 m 0 recalculating
to one photon leads to the photon energy F 0 m 0 /N. The flux (irradiance) at the
atmosphere base is derived after multiplying this energy to the number of photons
passing throughout the atmosphere F
#
ðt 0 ; m 0 Þ ¼
F0m 0
N Nðt 0 ; m 0 Þ.
Unlike other methods in the Monte-Carlo method, it is appropriate not to divide
radiation to the direct, diffused and reflected from the surface.
Certainly the simple solution might be written without this consideration
(explain why
F0m 0
N Nðt 0 ; m 0 Þ ! F 0 m 0 expðÀt 0 =m 0 Þ while N ! 1 as an exercise).
However, it is convenient to obtain irradiance with real multiple scattering and
absorptions (i.e. to solve the general radiative transfer problem) if three processes
are successfully modelled: photon-atmosphere interaction (scattering and absorption), photon-surface interaction (reflection and absorption) and the photon free path.
13.2 Simulating Random Events and Values
Consider some mathematical definitions before considering the Monte-Carlo
method.
For the statistical simulation on computer, it is necessary to reproduce a process
that will play the role of random event. Special algorithms are elaborated for
random number choice that is called a random number digitizer (RND) or randomizer and there are special computer programs for generating different sequences of
random numbers including to programming languages. However, testing of such
ready algorithms shows no convenient distribution parameters of sequences (the
mean and dispersion) for scientific purposes. Hence improved algorithms of RND
have been elaborated for the Monte-Carlo method. This choice plays the role of
“blind chance” similar to roulette wheel or shuffling the cards. Just this analogue
causes the name of the method.
The totality of the random numbers uniformly distributed over the interval [0, 1]
is the base of the Monte-Carlo method. We are implying only these numbers using
the term “the random number”, specifying them by sign b, and at every its
appearance in the text we mean a new random number.
Let the probability of a certain discrete random event be equal to P. Choose the
random number and if b 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. Simulating random values is
needed aside from random events. Note that according the definition the probability
value u within the interval [a,u] is equal to PðuÞ ¼
R u
a
rðu
0
Þdu
0 for simulating the
130
13 Monte-Carlo Method for the Solar Irradiance Calculation