Chapter 13
Monte-Carlo Method for the Solar Irradiance
Calculation
Abstract The base of the Monte-Carlo method is considered for atmospheric
optics application. The algorithm of calculating hemispherical fluxes and radiative
divergence is discussed. The description of the practice is presented.
13.1 The Basic Idea of Monte-Carlo Method
The Monte-Carlo method (more strict name is the method of statistical modelling) is a
most powerful method of the radiative transfer theory. It allows to solve the problems
concerned the radiance calculation with taking into account spherical geometry,
polarization, heterogeneity of the atmosphere and surface, etc. Here we are applying
this method for solving rather simple (comparing with above-mentioned) problem of
the solar irradiance calculation in the horizontally homogeneous and plane parallel
atmosphere. The approach allows simulations of the physical processes of radiative
transfer in the atmosphere, when it is not needed to attract a body of the transfer theory.
The Monte-Carlo method main idea is the interpretation of photon–atmosphere
interaction as the random process: the motion of light conditional particle called
“photon”, the computer simulation of the process, and the calculation of desired
characteristics as a mathematical expectation of random numbers appearing during
the simulation. It is to be mentioned that here the photon is a mathematical object (not
physical particle) and it might be divided to parts in further consideration. Desired
radiative characteristics (radiance and irradiance) are obtained as average values over
a multiplicity of photon trajectories sequent simulated.
The simplest example is considered for clarity, namely the plane atmosphere
with the optical thickness t 0 is illuminated by incident solar flux F 0 and cosine
of the incident angle is m 0 . The transmission function defining the solar energy
extinction in the atmosphere according to Beer’s law is P ¼ exp(Àt 0 /m 0 ). The P is
possible to treat as the probability of single photon passes throughout the atmosphere (actually, 0 P 1). Consider N photons and simulate their motion
throughout the atmosphere as follows: take the random number a from the totality
I. Melnikova et al., Remote Sensing of the Environment and Radiation Transfer,
DOI 10.1007/978-3-642-14899-6_13, # Springer-Verlag Berlin Heidelberg 2012
129
Monte-Carlo Method for the Solar Irradiance
Calculation
Abstract The base of the Monte-Carlo method is considered for atmospheric
optics application. The algorithm of calculating hemispherical fluxes and radiative
divergence is discussed. The description of the practice is presented.
13.1 The Basic Idea of Monte-Carlo Method
The Monte-Carlo method (more strict name is the method of statistical modelling) is a
most powerful method of the radiative transfer theory. It allows to solve the problems
concerned the radiance calculation with taking into account spherical geometry,
polarization, heterogeneity of the atmosphere and surface, etc. Here we are applying
this method for solving rather simple (comparing with above-mentioned) problem of
the solar irradiance calculation in the horizontally homogeneous and plane parallel
atmosphere. The approach allows simulations of the physical processes of radiative
transfer in the atmosphere, when it is not needed to attract a body of the transfer theory.
The Monte-Carlo method main idea is the interpretation of photon–atmosphere
interaction as the random process: the motion of light conditional particle called
“photon”, the computer simulation of the process, and the calculation of desired
characteristics as a mathematical expectation of random numbers appearing during
the simulation. It is to be mentioned that here the photon is a mathematical object (not
physical particle) and it might be divided to parts in further consideration. Desired
radiative characteristics (radiance and irradiance) are obtained as average values over
a multiplicity of photon trajectories sequent simulated.
The simplest example is considered for clarity, namely the plane atmosphere
with the optical thickness t 0 is illuminated by incident solar flux F 0 and cosine
of the incident angle is m 0 . The transmission function defining the solar energy
extinction in the atmosphere according to Beer’s law is P ¼ exp(Àt 0 /m 0 ). The P is
possible to treat as the probability of single photon passes throughout the atmosphere (actually, 0 P 1). Consider N photons and simulate their motion
throughout the atmosphere as follows: take the random number a from the totality
I. Melnikova et al., Remote Sensing of the Environment and Radiation Transfer,
DOI 10.1007/978-3-642-14899-6_13, # Springer-Verlag Berlin Heidelberg 2012
129