241
Atmospheric
Radiation
equation is quadratic, while the matrix Green’s function lead to a system of
linear equations of the Dyson type. That allows getting much more general re‑
sults in the derivation of kinetic equations in the approximation of geometrical
optics. A solution of the discrete RTE for the planar stratified slab with arbitrary
conditions at the boundary is found in the analytical matrix form [Budak et al.,
2011]. Numerical solution of the RTE is possible only based on its sampling that
demands the replacement of the scattering integral by the finite sum. Since the
physical basis of the transfer theory is the geometrical optics approximation, the
solution always contains angular singularities. Therefore, to sample RTE it is
necessary to eliminate the anisotropic
part of the solution,
including all of its
singularities, even approximately, but analytically, and to formulate the equation
for the regular part solution. The anisotropic part elimination in the RTE solu‑
tion in other geometries and numerical determination of its regular part is con‑
sidered in papers [Ilyushin and Budak, 2011a, 2011b, 2011c; Budak and Ilyush‑
in, 2011]. The generalization of the small-angle modification of the spherical
harmonics method (MSH) is proposed for the refinement of the scattered photon
path dispersion. MSH is the most general form of the small-angle approxima‑
tion, which permits to eliminate the solution anisotropic part and to formulate
the equation for the regular part, which solution is possible numerically [Budak
and Ilyushin, 2011]. Comparison of different algorithms for solving the RTE for
a slab by the efficiency and the estimation of the hardware and software impact
has been performed. It is shown that it is based on a unique analytic solution of
the discrete RTE. For sampling RTE, the scattering integral should be represent‑
ed by a finite sum, which is possible only after the angular singularity elimina‑
tion in the solution. Exist algorithms differ in the way of the solution anisotrop‑
ic part elimination. It is shown that the most effective method of such
elimination is MSH. Because the solution is reduced to a matrix expression, the
efficiency of the algorithm is determined by the effectiveness of matrix opera‑
tions implementation [Budak et al., 2012]. The fact that the basis of all algo‑
rithms for solving problems for the slab is one analytical solution of the discrete
RTE imposes significant limitations: the speed and accuracy of the solution are
inversely related. The calculation difficulties are determined by the dimension
of the solution matrix, which in turn depend on the accuracy of the solution
regular part representation. It is shown that the number of basis functions in the
solution regular part representation is determined by the analog of the Kotel‑
nikov-Shannon theorem: the sampling step should correspond to the smallest
angular details of the solutions. This defines a contradiction: the requirements
for algorithms of the inverse solution define the direct problem. Today we need
an algorithm solving the RTE with a precision of a uniform metric better than
1%, and the computation time is not more than 1 second for one wavelength. It
is obvious that satisfy both conditions at once in the framework of the
242
Yu. M. Timofeyev, E. M. Shulgina
traditional RTE solution impossible. To overcome this problem it is proposed
to use the synthetic iteration that was developed in neutron transport theory. It
is shown that it can satisfy the requirements of the inverse solution [Budak and
Shagalov, 2013; Budak et al., 2014]. An algorithm for the calculation of the light
field in a scene of arbitrary geometry with the multiple reflections from the
boundaries has been proposed. It is based on the double local estimation of the
Monte Carlo method for the calculation of the
decomposition of the desired
angular radiance distribution system of spherical functions that eliminates the
divergence of the double local estimation. The proposed solution of discrete
RTE for a slab is tested for the extreme case of a semi-infinite medium with a
single scattering albedo equal 1 by the comparison
of other algorithms and in
situ measurements. It is shown that, in this case, for an accurate determination
of the eigenvalues of the system matrix should use its Jordan form [Sokoletsky
et al., 2014]. The effect of coherent scattering impact on the formation of the
light field in a turbid medium has been investigated. For this purpose, it was
proposed the algorithm of the numerical solution of
the Mie theory for a system
of one, two and four particles. The calculation of multiple scatterings was based
on the Monte Carlo method. It has been shown that coherent scattering does not
affect the point spread function of the slab that defines the object visibility
through the thickness of a turbid medium [Fokina et al., 2014].
A rigorous approach to the solution of Maxwell equations for a monochro‑
matic field in a homogeneous uniaxial medium has been considered [Marakasov
and Troitskii, 2012]. The approach is based on using the tensor Green’s function.
The general solution satisfying arbitrarily specified boundary conditions is pre‑
sented. Various models of exponentially correlated random fields associated with
Poisson point ensembles, as well as algorithms for simulating radiative transfer
in
stochastic media of that type, are considered [Mikhailov, 2012].
A number of papers have been devoted to techniques for calculating the ra‑
diative transfer in the context of atmospheric and underlying surface remote
sensing. The light scattering in water-drop clouds for various distributions of
droplet size has been studied. Polarization and angular distributions are simulat‑
ed by Monte Carlo method for radiation reflected by cloud layers. Computation‑
al results make it possible to develop procedures for analyzing the microphysical
structure of clouds [Prigarin et al., 2014]. The passive retrieval technique for
investigation of the vertical structure of clouds and aerosols from satellites is
discussed [Fomin and Falaleeva, 2014]. This technique uses a combination of
cross-nadir polarimetry and high-resolution infrared (IR) spectroscopy. The real
potential of this technique is demonstrated with the help of a set of numerical
experiments, e. g. for the detection of cirrus clouds (Ci). In this regard, new vec‑
tor versions for the longwave and shortwave in the FLBLM (Fast Line-by-Line
Model) radiative transfer model are presented. These new versions are based on