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Gayana (Concepción)

versión impresa ISSN 0717-652Xversión On-line ISSN 0717-6538

Gayana (Concepc.) v.68 n.2 supl.TIIProc Concepción  2004 


Gayana 68(2): 558-564, 2004



Tamara A. Sushkevich, Sergey A. Strelkov & Ekaterina V. Vladimirova Sveta V. Maksakova, Alexey K. Kulikov, Ekaterina I. Ignatijeva, Alex N. Volkovich

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 4, Miusskaya Ploschadj, Moscow, 125047 Russia E-mail:

Boundary-value problem of radiation transfer in optically thick layers is considered while describing ocean, cloudiness, aerosol particle outbreaks, dust traces and other specific effects resulted from large fires (forest, peat-lands, in steppe regions, anthropogenic). Space and angular distributions of radiation inside the relevant layer of these media as well as reflected and passed through the layer radiation are formed as a result of multiple scattering and absorption.

A new approach has been proposed to radiation transfer modeling in thick layers based on an influence function method. An internal border is inferred that separates the relevant layer on two regions. The influence function is calculated in the first region as a complete solution of the planar problem and as an asymptotic solution in the second region considering both cases of this last solution as that of a azimuth symmetric problem or as a complete solution of kinetic equation. The total radiation in the layer and outside it is found by a matrix functional known as the optical transfer operator the nuclei of which are given by the influence functions for each region. Forward and back hemispheres of the phase function serve as reflection and extinction coefficients of the internal border.


The different models and approximations of the optical transfer operator are employed in multidimentional problems of radiative correction for remote sensing of ocean, cloudiness, in theories of light and image propagation in turbin media, and in theoretical and computational fundamentals of optical-electronic remote sensing systems.

Boundary-value problem of radiation transfer in optically thick finite planar layers is considered which are represented, in particular, by:

· cloudiness;
· outbreaks of aerosol particles;
· dust plumes which are generated under influence of vast fires (forest, peat, steppe, anthropogenic);
· and the atmosphere with optically thick layers of trace gases.

Spatial and angular distributions of radiation inside such layer as well as reflected and transmitted radiation are formed for these media as a result of multiple scattering of radiation and its absorption [1, 2].

One-dimensional models of solar radiation transfer in optically thick and not so thick layers of the two-media systems of the types like

· "the atmosphere-ocean",
· "the atmosphere-cloudiness",
· "the atmosphere-hydrometeores",
· "the atmosphere-gas/aerosol outbreaks cloud",
· "the atmosphere-dust plume", i.e. in the atmosphere with admixtures resulted from vast fires influence (forest, peat, steppe, anthropogenic), for instance, which can result in an extended curtain, are elaborated by us using the following two approaches.

The related calculations are conducted in the first model by the iterative characteristics method (ICM) [3] with procedures of speeding up the iteration convergence for the planar system with two layers on altitude: the lower layer ­ cloudiness or the layer with admixtures, the upper layer ­ free atmosphere.

Radiation of the system in the second model is obtained by the functional that is called as the optical transfer operator (OTO) via the influence function (IF) of the atmosphere (IFA) and the layer with admixtures (IFLA) [3 ­ 5].

Optically thick layers

The first methodological investigations, algorithms and calculation programs were made by T.A. Sushkevich in 1965.

Methods of speeding up the iteration convergence of successive approximations had been actively developed in the 1960-th and 1970-th (T.A. Germogenova and T.A. Sushkevich, V.Ya. Goldin et al., G.I. Marchuk and V.I. Lebedev, Sh.S. Nikolaishvili, V.I. Zhuravlev, A.V. Volotschenko and other).

Miln's problem and asymptotical methods had also been developed that time (M.V. Maslennikov and M.G. Kuzmina, T.A. Germogenova and N.V. Konovalov, G.A. Mikhailov et al., G.V. Rosenberg, V.V. Inanov, I.N. Minin, E.G. Yanovitsky, I.N. Melnikova, I.Kuscer, M.King and other).

A natural aspiration was to use the known approximate analytical & asymptotical solutions:

- for modeling radiation transfer in two-media systems with the first layer being optically thin and the second optically thick;
- as an initial approximation for a new algorithm of speeding up the iteration convergence;
- for finding solution of the problems connected with remote sensing methods and techniques for cloudiness, dusts, admixture outbreaks, etc.

The asymptotical or diffuse radiation regime is known to be reached in any optically thick layer on a sufficiently large optical depth due to multiple scattering.

Properties of the diffuse regime are maintained from physical considerations to be in the following:

- the role of direct radiation (passed without scattering) is negligibly small comparing to the role of scattering radiation;
- the radiation intensity does not depend on azimuth;
- the relative angular distribution of the intensity does not depend on optical depth.

Analytical solutions are obtained for any scattering layer with large optical depth which are expressed by asymptotic formulas of radiation transfer theory. Authors of such solutions often directly indicate, but sometimes keep silence that the errors of these analytical solutions are higher than 10-20% and even more in the layer nearest to the upper boundary and are increasing as the radiation absorption is enhancing.

The errors due to not taking the azimuthal dependence into account have not yet been investigated. Theoretically and practically incorrect computation task is known to take the azimuthal dependence into account through short (3-5 terms) Fourier series.

The authors of paper [2] have succeeded in the realization of the azimuthal harmonics calculation to determine the angular structure of reflected radiation from cloudiness while having in mind that this part of radiation is anisotropic and its description through albedo is rather inaccurate.


In fact, our approach (the iterative characteristics method/ICM with procedures of speeding up the iteration convergence and with taking large anisotropy of scattering into account) to the boundary problem solution of radiation transfer theory in optically thick homogeneous and inhomogeneous layers with three variables (altitude, zenith angle and azimuth) without any preliminary expansion in azimuthal harmonics series is the only method that enables to calculate on updated computers with large resources of operative and external memory and study angular and spatial distributions not only reflected and transmitted radiation, but that on any depth inside the layer.

Speeding up procedures in the ICM:

- relaxation speeding up (convergence speeding up not more than in two time);
- for the expense of a small "absorption" starting from the third iteration in the successive approximation method through the algorithm of large anisotropy scattering taking into account;
- the similarity method is not used and a specific procedure of the large anisotropy scattering selection is applied for each inhomogeneous layer;
- control procedure of normalization of collision integrals serves to control the calculation accuracy in order not to allow "a false absorption" appearance and a multiplicity while using quadrature in calculations of the collision integrals on a unit sphere (this implies a control for balance and calculation accuracy preservation).

Radiation fields have been calculated by an iterative method of characteristics with speeding up procedures inside and outside of the layer. An external mono-directional flux is incident for the layer and the calculations have been conducted for a great number of optical models with different thickness of the layers, scattering phase functions, single scattering albedo ­ all are corresponding to conservative scattering without absorption and weak, medium and heavy absorption as well. Additional calculations have been also conducted for the same models using asymptotic solutions. An adjustment of asymptotic regime through the layer thickness for outgoing and incident fluxes together with angular distributions and symmetry on azimuth has been investigated using updated graphical and visualization techniques. Azimuth harmonics have been calculated in accordance with the angular distributions. Applicability and accuracy problems for asymptotic solutions of the transfer equation have been studied.

Numerical research has been conducted of how the asymptotical regime is established and what is the area of the asymptotical approach applicability.

Radiation fields have been calculated by the iterative characteristics method inside and outside the layer for a large set of optical models (more than 200 calculation models) while an external mono-directional flux is incident with different:

- directions of the solar flux (cosines are from 1.0 to 0.1),
- layers thickness (from 1 to 20),
- phase functions (isotropic, Rayleigh, Henyey-Greenstein with the anisotropy parameter values from 0.6 to 1),
- single scattering albedo for conservative scattering (without absorption), low, medium and high absorption (from 1 to 0.3).

Calculations have been carried out for a number of the models based on the asymptotical solutions (using formulas of I.N. Melnikova, I.N. Minin, E.G. Yanovitsky, N.V. Konovalov).


An adjustment of the asymptotical regime along the layer thickness for outgoing and down-welling fluxes as well as angular distribution and azimuthal symmetry has been studied by updated graphical and visualization means.

Azimuthal harmonics have been calculated using angular distributions inside the system and on its boundaries which are gained by the iterative method on the finite difference grid of large dimension (100 zenith angles, 91 azimuth, 50-100 layers).

Applicability and accuracy problems of the asymptotical solutions of radiation transfer equation have been studied.

Some illustrations of the conducted research are given at fig. (a ­ h). Model of layer: optical thickness t0 = 8.0, directions of the solar flux m0 = 0.5, single scattering albedo w0 = 0.98, phase functions anisotropy parameter g = 0.62.

Some important conclusions have been obtained based on the numerical studies:

1. about spatial and angular distributions of radiation intensity inside and on the boundaries of an optically thick layer;
2. about efficiency of the expansion procedure on azimuthal harmonics series;
3. about applicability and accuracy of asymptotic approximations for homogeneous and inhomogeneous optically thick layers;
4. about a new approach based on the optical transfer operator and influence function method.


1. About spatial and angular distributions of radiation intensity inside and on the boundaries of an optically thick layer:

- dependence of reflected radiation from the thickness layer, phase function, degree of absorption, direction of solar flux;
- dependence of angular structure not only from phase function and single scattering albedo, but from the depth as well;
- blurring and shift of a light spot (theory confirmation);
- dependence of reflected radiation from effective thickness of a layer under different phase functions, single scattering albedo, direction of solar flux; the dependence found enables to conduct calculations for the layers with a decreased optical thickness (calculations have been shown to be sufficient for the layers on the level on 10-20 while using known models of clouds with their optical thickness near to 64).

2. About efficiency of the expansion procedure on azimuthal harmonics series:

- practically all one-dimensional problems are solved by numerical and approximate analytical methods for single azimuthal harmonics with two variables (thickness and zenith angle);
- most tools of speeding up the iteration convergence have been elaborated and applied for azimuthal harmonics, i.e. for the problems with two variables (thickness and zenith angle);
- studies are absent of an angular distribution analysis inside the layer and no result exists concerning angular distributions comparison obtained without and with use of non-linear speeding up techniques (the most part of the relevant results is regarding to the integral on angle altitude profiles, for example, fluxes, density, etc.)
- techniques of regularization are needed in the summation procedure of the azimuthal harmonics series due to the problem is incorrect;
- the azimuthal dependence is greatly changeable through the layer depth and requires different number of harmonics be described being greatly dependent from phase function and the radiation absorption level.

3. About applicability and accuracy of asymptotic approximations for homogeneous and inhomogeneous optically thick layers;

- all known approaches to solution of the problems for inhomogeneous layers utilize an adding procedure for the layers with reflection R and transmission T coefficients for separate layers; these coefficients are calculated by approximate asymptotical formulas without azimuthal dependence and the approaches have a low accuracy;
- there is no operative effective techniques for asymptotical solutions of the problems with inhomogeneous layers; should these techniques be realized, and a negative result would be obtained since the accuracy of asymptotical approximations for finite layer is large, in particular, for reflected radiation and near to the source boundary.

4. A new approach to modeling the radiation fields in optically thick layers based on the radiation transfer model in two media systems with optical transfer operator and with influence function method. This new approach is proposed to modeling the radiation transfer in optically thick finite and semi-infinite layers by the influence function method [3-5].

- The overall problem is in dividing a thick layer on sub-layers known methods can be used to and after that the problem is in obtaining the summation solution.
- Internal boundary is inferred that divides the layer on two areas with different radiation regimes.
- The influence function is calculated in the first area as a total solution of the problem and in the second area as an asymptotical solution either as a solution of the azimuth-symmetrical problem or as a total solution of kinetic equation.
- The total radiation inside the layer and out of it is found by matrix functional known as the optical transfer operator the influence functions for each area are given by kernels of the operator.
- Phase functions are utilized as the reflection and transmission coefficients for the internal boundary.

This new original approach formulated within a universal linear-system approach is particular perspective for studying processes of solar radiation transformation in any two-media system, of short-wave radiation exchange between the atmosphere and any layer with admixtures as well as of light scattering regime in a layer with admixtures, in a free atmosphere, on the boundary "free atmosphere-layer with admixtures".

This approach is particular effective if two media are such different that for modeling the radiation transfer in them modelers have to use various methods and different approximations to solve kinetic equation.

The same approach is effective for modeling the angular and spatial radiation distributions in optically thick layers (clouds, dusts, etc.) which can be represented as two sub-layers with different radiation regimes.


About opportunities to use a new approach based on optical transfer operator and influence functions method:

- Optically thick layer can be homogeneous or inhomogeneous on altitude;
- Different regimes inside any homogenous layer which can be described by azimuth-dependent and azimuth-independent solutions of transfer equation, by asymptotical and non-asymptotical approximations, etc.;
- Inhomogeneous layer (for example, the atmosphere with a strata cloud, etc.) can contain an optically thin layer (free atmosphere) and an optically thick layer (cloud, outbreaks, dust, etc.), i.e. layers with different optical parameters when the known adding layers methods are not applicable;
- The total radiation of such system is found with the most realistic optical characteristics and angular profiles of radiation on the boundary of the layers taking into account;
- Possibility to calculate angular and spatial distributions of radiation on boundaries and inside the layers;
- Influence functions are inferred which are invariant characteristics of sub-layers not depending on particular structures of the angular radiation distribution on boundaries of the layers;
- Influence functions can be calculated in sub-layers by various methods in different approximations;
- Finite difference angular grids of different dimension can be in sub-layers while any transition is conducted through the collision integrals.

Our approach is completely different with the known adding layers method. In the adding layers method:

- Only planar models;
- Invariant and factorization principle (V.A. Ambartsumyan, S.Chandrasekhar, V.V. Sobolev, N.B. Engibaryan and others);
- R and T are coefficients of reflection and transmission of layers, but not characteristics of the media boundary;
- Layers with the same properties and the same R and T;
- The same R and T are used for outgoing and down-welling radiation;
- Dimension of the problem is decreased preliminary to two (thickness of the layer, zenith angle) by expansion of solution on azimuth in Fourier series;
- Solution is calculated only on external boundaries.

In IF and OTO method:

- Research and analysis, but not only simple calculation of the multi-layer system;
- Splitting on the problems for each layer with its boundary conditions;
- Total solution by factorization;
- R and T properties of reflection and transmission of the boundaries;
- The IF calculation while multiple scattering taking into account in the layer, on its upper and lower boundaries;
- Different initial approaches;
- Different iteration approaches;
- Calculation of "scenarios" of radiation distribution on the boundaries;
- Calculation of angular and spatial radiation distributions in any point of the system;
- Different methods of calculation in different layers;
- A generalization concerning polarization and without it;
- A generalization concerning planar and spherical shells;
- A generalization concerning multi-layer systems with reflecting bondary.

The influence functions method possesses an important priority: having available the IFA and IFLA set, it is possible:

- To make different structures of radiation filed in any system;
- To study in details mechanisms of its formation/ Optical properties of media can be greatly different:
- The first medium usually has a finite optical thickness;
- The second medium is considered as a thick finite and semi-infinite layer.

IFA and IFLA can be calculated besides that by various numerical and analytical methods with different degree of accuracy and approximation:

- The influence function is calculated in the first area as a total solution of the planar problem,
- This function is calculated in the second area as
- an asymptotical solution,
- as a solution of azimuth-symmetrical problem,
- as a total solution of kinetic equation.

This approach enables to model radiation field in a wide range of variations of optical characteristics for the layers, to analyze mechanisms of radiation characteristics formation inside and outside the layers as well as to estimate the influence of each area.

Reflected radiation and distribution of radiation characteristics inside any layer near to the boundary with an incident flux are calculated in the proposed approach with much higher accuracy. This statement is important to verify and improve remote sensing methods for clouds and outbreaks of different admixtures as well as methods of operative express-analysis of space data, in particular, in regions with natural environmental disasters including consequences of wide-scale fires.


This work has been supported by The Russian Foundation for Basic Research (projects 02-01-00135, 03-01-00132).



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Melnikova I.N., Dlugach Z.M., Nakajima T., Kawamoto K., 2000: On the reflection function calculation in case of optically thick scattering layer, for Henyey-Greenstein phase function // Applied Optics. Vol. 39, 24, 4195-4204. [2]

Sushkevich T.A., Strelkov S.A., Ioltukhovsky A.A., 1990: Characteristic Method to the Atmospheric Optics Problems. (in Russian) Moscow, Nauka. (Book review in Transport theory and statistical physics, 1993. Vol. 22, 4, 587-591.). [3]

Sushkevich T.A., 1996: A solution to the boundary-value problem of the transport theory for a planar layer with a horizontally inhomogeneous boundary of an interface between two media // Doklady Mathematics. Vol. 54, 2, 775-779. Translated from Doklady Akademii Nauk. 1996. Vol. 350, 4, 460-464. [4]

Sushkevich T.A., 2000: Linear-system approach and the theory of optical transfer operator // Atmospheric Oceanic Optics. V. 13, 8, 692-700. [5]


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