A numerical comparison of algorithms for solving the radiative transfer equation by the Monte-Carlo method is performed for the direct simulation and local estimations. The problems of radiative transfer through a turbid medium slab in the scalar and vector case is considered. The case of reflections from the boundaries of the medium is analyzed. The calculations are performed in a wide variation of parameters of the medium. It is shown that the calculation time with the same accuracy for the local estimation method is less than one - two orders of magnitude.
For the numerical solution of the radiative transfer equation (RTE) it is required its discretization based on the elimination of the solution anisotropic part and the replacement of the scattering integral by a finite sum. Regardless of the particular method of the RTE sampling the boundary value problem for a slab is transformed into the boundary value problem for the matrix inhomogeneous linear differential equation of the first order. The solution of this problem can be represented both through the solution of the homogeneous equation (propagator), and through the scatterers, possessing the property of invariance that leads to the Ambartsumian invariance principle. It is shown that the equivalence of all these approaches can improve the efficiency of the numerical radiative transfer modeling in the turbid medium slab. Significant acceleration of the RTE solution convergence can be achieved by using the method of synthetic iterations.
In this work we suggest a precise calculation method for GOSAT program devoid of shortcomings of existing methods.
The algorithm was tested on the plain unidirectional source which already has an exact solution to compare with. In that
way the radiuses of mesh cylinders were a much larger than their height. Then we tested an algorithm on finite cylinders
mesh and compared the results with Monte-Carlo simulation to be sure of its appropriateness.
As a results of our work we got a fast and precise method for the calculation angular radiance with consideration of
polarization.
The method to define the form of particles according to remote sensing is given in this work. The method consists in
calculation of a phase matrix of environment, on the basis of the data of the solution of a direct scattering problem, by
the formulation and the solution of a inverse problem of the transport theory by the least squares method. The form of
particles is defined on the received scattering matrix. The solution of a direct scattering problem is spent on the basis of
the representation of required spatial distribution of Stokes vector-parameter in the form of superposition of an
anisotropic part and rather smooth not small-angular additive. The function of some expansion coefficients from
harmonic number is represented by the slowly varying monotonous function owing to strongly marked anisotropy of
scattering indicatrix that allows us to limit the number of members in expansion of coefficients at the generalised
spherical functions. It greatly simplifies the expressions for the coefficients of the vector equation solution of radiative
transfer in the form of matrix exponent - vector small-angular modification of spherical harmonics method (VMSH). In
calculation of the smooth additive the anisotropic part received by means of VMSH represents itself as a function of
sources.
The paper deals with the polarized radiative transfer within a slab irradiated by a collimated infinity-wide beam of arbitrary
polarized light. The efficiency of the proposed analytical solution lies in the assumption that the complete vectorial
radiative transfer solution is the superposition of the most anisotropic and a smooth parts, computed separately. The vectorial
small angle modification of the spherical harmonics method is used to evaluate the anisotropic part and the vectorial
discrete ordinates method is used to obtain the smooth one. The azimuthal expansion is used in order to describe the
light field spatial distribution for the case of abnormal irradiance and to obtain some known neutral points in the sky especially
useful for polarized remote sensing of the atmosphere.
In this paper the mathematical model of the visibility range determination of a laser beacon in a navigational complex is
offered. The model includes model of the optical properties of a coastal atmospheric haze, algorithm of the light field
calculation in a turbid medium and the statistical model of the image perception by the given level of detection
probability. Calculation of laser signal atmospheric attenuation is conducted with use small angle approximation method
of spherical harmonics.
Remote sensing inverse problem solution is carry out in independent pixel approximation (IPA) assumption on the base
of the radiative reflection from turbid media slab with arbitrary phase function and Lambertian surface numerical model.
The paper deals with the vectorial radiative transfer equation (VRTE) problem for a homogeneous strongly anisotropic scattering
slab illuminated by a plain unidirectional source of light with an arbitrary angle of irradiance and polarization state.
The problem is a theoretical base for the polarized satellite remote sensing (POLDER, PARASOL and others). The VRTE
boundary problem decomposition allows reducing to the nonreflecting bottom with subsequent including its polarization
properties. We give the complete analysis for the solution smooth non-small angle part for the vectorial small angle modification
of the spherical harmonics method (VMSH) built upon the smoothness of the spatial spectrum of the light field distribution
vector-function caused by mathematical singularities of the top-boundary condition for the VRTE boundary problem
and the anisotropy of many natural scattering media (clouds, ocean). The VMSH itself is described as well.
Proc. SPIE. 6522, Thirteenth Joint International Symposium on Atmospheric and Ocean Optics/ Atmospheric Physics
KEYWORDS: Mathematical modeling, Optical transfer functions, Scattering, Photons, 3D modeling, Finite element methods, Picosecond phenomena, Spherical lenses, Lawrencium, Radiative transfer
It is typical for the arbitrary turbid media the local compactness disturbance. It leads to appearing singularities in the solution
of the radiative transfer equation for the space-limited sources. Thus the classical solution methods are inefficient.
The solution for the point isotropic (PI) source has the singularities not only angular dependence but spatial variable as
well. For elimination such effect the solution may be rewritten in the form of sum of the small angle part and the rest is a
smooth function. The small angle approximation (SAA) contains all the singularities of the exact solution. The further
solution is lead only for the smooth function. It allows using the finite elements method. The system is solved by the iteration
method. Using of the discrete ordinates method allows to consider arbitrary boundary conditions. The suggested
method considers the scattering photons ways dispersion. It allows estimating the (SAA) application range for the different
three-dimensional medium parameters.
We offer the generalization of the vectorial small angle modification of the spherical harmonics method (VMSH) for an
arbitrary angle and polarization sate of irradiance of a slab. Non diagonal elements of an aerosol scattering matrix were
admitted. The smooth addition part for the VMSH is given. Thus we obtain a complete and accelerated solution of the
vectorial radiative transfer equation.
The analytical expression for solving the radiative transfer problem of the arbitrary polarized light in scattering and absorption
media in small angle approximation for aerosol scattering matrix is offered. The comparison of the results obtained
by using this analytical formula and by the direct matrix computation is realized.
Practically all the real problems of the atmospheric and oceanic optics are based on the solution of the radiative transfer
equation (RTE) for three-dimensional (3D) medium area with a strong anisotropic scattering. In case of the flat medium
geometry the most effective method of the RTE solution is the approach, when the difference between the exact solution
and the solution of RTE in the small angle approximation (SAA) is determined. As SAA contains all the singularities of
the exact solution, the indicated difference is a smooth angle function that essentially simplifies its determination by any
numerical method. The most general SAA form is the small angle modification of the spherical harmonics method
(MSH), an analytic form of which is a series on surface harmonic. It determined the choice of the spherical harmonics
(SH) method for the definition of the indicated difference.
However at the transition to 3D medium geometry the SH method loses its efficiency because of the huge difficulties of
the statement of the boundary conditions. At the same time an analytic form of SAA allows easily to calculate the scattering
integral in RTE and to determinate the source function. In this case the rest can be determined by the discrete ordinates
method (DOM). In DOM RTE is exchanged to a set of the ordinary differential equations for the directions fixed
in space (rays) that essentially simplifies a statement of both arbitrary boundary conditions, and accommodation to the
complex 3D medium geometry. Such approach is similar to SHDOM, but exceeds it at the convergence rate.
In this paper the problem of the light reflection from a turbid medium slab is considered. The method of the single
scattering separation on a scattering leading to the radiative propagation direction veering concerning a surface normal,
and on a scattering not leading to that is offered. The reflectance factor is represented as a series on the scattering
multiplicities with a single change of the direction. For each multiplicity the precise linear integro-differential equation
with homogeneous boundary conditions is arrived. The application of the discrete ordinates method brings to the linear
matrix equations. The solution of these equations without using the small angle approximation in the form of the matrix
exponential curves is obtained. The application range of the quasi-single scattering approximation is uniquely
determined.
The solution of the radiative transfer equation for a point unidirectional source is offered using the transformation of
spherical geometry to flat. The allocation of the small angle part allows taking into account the solution singularities.
In this paper the problem of the light reflection from a turbid medium slab is considered. The method of the single scattering separation on a scattering leading to a radiative transfer veering concerning a surface normal, and on a scattering not leading to that is offered. The reflectance is represented as a series on the scattering multiplicities with a single change of the direction. For each multiplicity the precise linear integro-differential equation with homogeneous boundary conditions is obtained. The application of the discrete ordinates method brings to the linear matrix equations. The solution of these equations without usage of the small angle approximation in the matrix exponential curves form is found. The application range of the quasi-single scattering approximation is uniquely determined.
Proc. SPIE. 5979, Remote Sensing of Clouds and the Atmosphere X
KEYWORDS: Mathematical modeling, Solar radiation models, Scattering, Light scattering, Clouds, 3D modeling, Numerical analysis, Finite element methods, Chlorine, Radiative transfer
The peculiarities of the light fields forming in the clouds are their complicated geometrical three-dimensional shape and strongly anisotropic light scattering. In these conditions the solution of the radiative transfer equation becomes mathematically ill-conditioned. The problem has a fundamental character and is concerned with a physical model of the radiation transfer-ray approximation, which determines the presence of the angular singularities in the solution. For the elimination of this problem S.Chandrasekhar has offered to subtract nonscattered radiation from the solution and to state the equations for the rest smooth part. However in the conditions of the strong anisotropic scattering the similar approach loses its efficiency. The most optimal approach in the solution of this problem is the determination of the analytically simple approximate solution including all the angular singularities of the exact solution. An example of such an approach is a small-angle approximation in the Goudsmit-Saunderson form, which however is obtained only for the flat medium geometry at an almost normal incident of the solar radiation.
The registration of the reflected radiation polarization at the remote sensing allows gaining all the information available to optical methods about the observed object. Mathematically it gives a boundary-value problem of the vectorial radiative transfer equation (VRTE). The natural media of the radiative transfer have strongly anisotropic light scattering. Because of their singularities the solution of the boundary-value problem of VRTE for such media is a mathematically illconditioned problem. The classical method (S.Chandrasekhar) of the elimination of this problem is based on the subtraction of the nonscattered component from the solution. However under the conditions of strong anisotropy a diffusion part is not distinguished enough from the nonscattered part that gives heavy oscillations in the numerical solution. In this paper it is offered to subtract from the required solution of VRTE its solution in a small angle approximation (SAA), which besides nonscattered component contains all the anisotropic part. The rest of the solution is a smooth function, which can be easily found by any numerical method. As SAA it is offered to take a small angle modification of a spherical harmonics method (MSH), presenting the generalization of Goudsmit-Saunderson's solution for the case of VRTE.
The analytical complexities of the radiative transfer equation (RTE) have allowed formulating only superficial, qualitative estimations of the precision and application range of the small angle approximation (SAA). Another approach based on the comparison of the calculations in SAA with the similar numerical solutions of RTE is possible. However numerical methods of the RTE solution in media with strong anisotropic scattering are ineffective. The numerical solution of RTE for turbid media with the arbitrary anisotropy of scattering irradiated by a plane monodirectional (PM) source suggested in<sup>3</sup> gives the opportunity of solving this problem. SAA was offered in paper<sup>4</sup> and is based on the approximate equality of the trajectory path of photons by the anisotropic scattering to the nonscattered path. The analytical complexities of the RTE solution have established three forms of SAA. Comparisons with the numerical solution<sup>3</sup> have shown, that all three forms use extra assumptions to<sup>4</sup> except the case of normal irradiation by PM source in<sup>8</sup>. The analysis of SAA has allowed offering the development<sup>8</sup> completely relevant to the ideas of<sup>4</sup>, from which all three forms follow at various assumptions. The generalization of this approach for cases of arbitrary boundary conditions doesn’t present any troubles.
Proc. SPIE. 5396, Tenth Joint International Symposium on Atmospheric and Ocean Optics/Atmospheric Physics. Part I: Radiation Propagation in the Atmosphere and Ocean
The model of a plane-parallel slab of a turbid medium is accepted, where there are particles of the spherical form scattering light separately from each other. Mathematically the problem is reduced to the solution of a boundary-value problem of a radiative transfer equation in turbid media with strong anisotropic scattering. In this case the computation of a backscattering is mathematically an ill-conditioned problem by using any numerical method of the solution. In the suggested method the backscattering radiance is determined by a SH method as a difference between the exact solution and SAA that essentially reduces the order of a system and smoothes the solution at any degree of an anisotropy scattering. As SAA the small angle modification of spherical harmonics method is chosen, which has a form of series of spherical harmonics that also enables to present a backscattering analytically. The calculation of medium parameters in model is made according to the Mie theory in the form of series of spherical harmonics, that allows to take into account the effect of physical properties of the medium on the radiation reflection.
Proc. SPIE. 5026, Ninth Joint International Symposium on Atmospheric and Ocean Optics/Atmospheric Physics. Part I: Radiation Propagation in the Atmosphere and Ocean
In actual mediums with anisotropic scattering in small angle approximation (SA) is possible simply and accurately to calculate light field radiance in forward from a source directions. The backscattering calculation in these mediums by any known method is complex and has no sufficient precision for practical problems: spherical harmonics (SH) method has strong oscillation, Monte-Carlo method has a large variance, concerned with improbable event with large weight, and the first iteration from SA has uncontrollable precision and validity range. In suggested solution the refinement of SA is found by SH method. The SA solution is taken in small angle modification of SH method. This allows receiving analytical solution in the form of an exponential of matrix. Last circumstance requires numerical calculation under the obtained expression, however in the most unfavorable case the amount of terms in the refinement does not exceed 21 harmonics. The solution is practically smooth at any sharpness of a scattering phase function. Calculation time on the obtained algorithm is almost equivalent to time of calculation on SA. In addition, algorithm improving radiance distributions in a forward hemisphere.
Generalization of small angle modification method of spherical harmonics in a case of a point monodirectional (PM) light source in an infinite turbid medium with an anisotropic scattering is carried out. Within the framework of the deduced generalization, a solution of the vector radiation transfer equation for an unpolarized PM-radiant is obtained. The expressions featuring the state of polarization of scattered light are given in a form convenient for use in engineering practice. The obtained expressions are analyzed and it is shown, that the minimum of polarization coincides with the direction of sighting on the maximum of radiance, which corresponds to the Umov law.