Radiative Transfer in Inhomogeneous Atmospheres. I

This series of papers is devoted to multiple scattering of light in plane parallel, inhomogeneous atmospheres. The approach proposed here is based on Ambartsumyan's method of adding layers. The main purpose is to show that one can avoid difficulties with solving various boundary value problems in the theory of radiative transfer, including some standard problems, by reducing them to initial value problems. In this paper the simplest one dimensional problem of diffuse reflection and transmission of radiation in inhomogeneous atmospheres with finite optical thicknesses is considered as an example. This approach essentially involves first determining the reflection and transmission coefficients of the atmosphere, which, as is known, are a solution of the Cauchy problem for a system of nonlinear differential equations. In particular, it is shown that this system can be replaced with a system of linear equations by introducing auxiliary functions P and S. After the reflectivity and transmissivity of the atmosphere are determined, the radiation field in it is found directly without solving any new equations. We note that this approach can be used to obtain the required intensities simultaneously for a family of atmospheres with different optical thicknesses. Two special cases of the functional dependence of the scattering coefficient on the optical thickness, for which the solutions of the corresponding equations can be expressed in terms of elementary functions, are examined in detail. Some numerical calculations are presented and interpreted physically to illustrate specific features of radiative transport in inhomogeneous atmospheres.     

Linear fREDHOLM integral equations for radiative transfer problems in finite plane parallel media. I. Imbedding in an infinite medium

Linear FREDHOLM integral equations are derived for the STOKES vector of the radiation emerging from a scattering plane parallel medium of finite optical thickness. The integral equations are obtained by means of imbedding the slab in an infinite medium. They are formulated in terms of GREEN 's function matrices and renormalized for the asymptotic eigenmode. Explicitly, linear integral equations are given for the reflection and transmission matrices. The reciprocity principle is employed to obtain integral equations also for the mean intensity of the inner radiation field in the case of the slab albedo problem.     

Energy-dependent transport problem with generalized boundary conditions

The slowing-down Boltzmann equation for generalized boundary conditions is considered and transformed to one-speed equation in Laplace space. Exact relations between energy reflection and transmission coefficients for a problem with diffuse reflecting boundary conditions and the albedos for the problem with isotropic boundary conditions are obtained. The Galerkin method is used to calculate the energy reflection coefficient for a finite slab for different thicknesses at different mass ratiosA, target to projectile mass, at different synthetic-scattering kernels. The results for partial heat fluxes for isotropic and anisotropic-scattering dispersive medium are given. The results obtained for isotropic boundary conditions are compared well with the exact results.     

The first three orders of scattering in vertically inhomogeneous scattering-absorbing media

In order to facilitate the computations of the intensities of radiation reflected and/or transmitted by plane-parallel, vertically inhomogenous, scattering-absorbing media, we carry out the optical thickness integrations of the Cauchy systems (normally referred to as Invariant Imbedding Equations) for reflection and transmission functions originating from the first three orders of scattering: the medium in question is represented by a stack of a certain number of homogeneous slabs, each of which is characterized by a constant single scattering albedo and a constant phase function together with the optical thickness.The results are a set of recurrence formulae involving only the angular intergrations, a convenient feature for numerical computations, and should prove useful particularly for finding approximate values of the high frequency Fourier coefficients of reflection and transmission functions of inhomogeneous media or efficiently initializing the solution for a thin layer to perform rigorous multiple scattering computations by means of other techniques such as the Doubling-Adding method.     

Investigation of numerical properties of Hovenier's Exit Function equation for multiple scattering of light

We show that Hovenier's Exit Function equation describing reflection and transmission by a plane-parallel layer can be obtained from the Invariant Imbedding equations. As an immediate extension we obtain a similar equation for an Exit Function defined in terms of reflection and transmission functions for successive orders of scattering. These equations allow the reflection and transmission functions of a homogeneous atmosphere of arbitrary optical thickness to be obtained from angle integrations of only one function.A technique based on successive iterations is developed to solve Hovenier's equation. The numerical behavior of this equation is then investigated employing a few representative (i.e., isotropic, Rayleigh, and Henyey-Greenstein) phase functions with the following conclusions. (i) As long as the deviation from isotropy is small (cos 0.15), the Exit Function equation can be numerically solved with an efficiency comparable to that of the standard Doubling technique, which is one of the fastest algorithms available. (ii) The reflection function generated from the Exit Function is usually more accurate than the corresponding transmission function, particularly in the case of large optical thickness. (iii) As the degree of anisotropy increases, so does the difficulty in obtaining the numerical solution for the Exit Function. The solution of the equation depends sensitively on the treatment of the numerical singularities which arise from the integrands and also on the initial approximation employed for the iteration. An improved scheme is required for numerically obtaining the Exit Function in order for this method to yield accurate reflection and transmission functions for strongly anisotropic scattering.     

Exact solution of the problem of diffuse reflection and transmission in Rayleigh scattering by use of the Laplace transform and the theory of linear singular operators

We have considered six scalar equations which are obtained from the vector transport equation for radiative transfer to the problem of diffuse reflection and transmission in finite plane-parallel Rayleigh scattering atmosphere. By use of the Laplace transform and the theory of linear singular operators these equations have been solved exactly to get the angular distribution of the intensity diffusely reflected from the surface and diffusely transmitted below the surface.     

Time-dependent scattering and transmission function in an anisotropic two-layered atmosphere

In this paper we consider the time-dependent diffuse reflection and transmission problems for a homogeneous anisotropically-scattering atmosphere of finite optical depth and solve it by the principle of invariance. Also we consider the time-dependent diffuse reflection and transmission of parallel rays by a slab consisting of two anisotropic homogeneous layers, whose scattering and transmission properties are known. It is shown how to express the time-dependent reflected and transmitted intensities in terms of their components. In a manner similar to that given by Tsujita (1968), we assumed that the upward-directed intensities of radiation at the boundary of the two layers are expressed by the sum of products of some auxiliary functions depending on only one argument. Then, after some analytical manipulations, three groups of systems of simultaneous integral equations governing the auxiliary functions are obtained.     

Scattering and transmission functions of radiation by finite atmospheres with reflecting surfaces

In the present paper, with the aid of invariance principles in connection with the scattering matrix, we get the exact solution of diffuse reflection and transmission problems by finite inhomogeneous, anisotropically scattering atmospheres bounded by reflecting sufaces. On making use of the reflection and transmission integral operators, we show how to obtain the non-linear integro-differential equations for these operators, which do not depend on the initial condition. Then, we have a system of the required integro-differential equations for the scattering and transmission functions. The obtained result is new, so far as we know. Finally, using the scattering matrix, we reduce the diffuse reflection and transmission problems for planetary atmospheres with reflecting surfaces to the standard diffuse reflection and transmission problems.Supported by the National Science Foundation under Grant No. GP29049 and the Atomic Energy Comminission, Division of Research under Contract No. AT(40-3)-133, Project 19.     

Scattering of an Electromagnetic Wave on a Dielectric Layer Bounded on Two Sides by Two Different Homogeneous, Semi-Infinite Media

The problem of the passage of a plane electromagnetic wave through an arbitrary, inhomogeneous dielectric layer bounded on two sides by two different homogeneous, semi-infinite media is considered. Algebraic relations are obtained between the amplitudes of transmission and reflection (the scattering amplitudes) for the problem under consideration and the wave scattering amplitudes when the layer is bounded on both sides by a vacuum. It is shown that for s and p polarized fields the scattering problem (a boundary-value problem) can be formulated as a Cauchy problem directly for the s and p wave equations. It is also shown that the problem of finding the field inside the layer also reduces to a Cauchy problem in the general case.     

The interaction of an obliquely incidents-polarized plane electromagnetic wave at a warm moving magnetized plasma half-space

The interaction of ans-polarized plane electromagnetic wave incident from a dielectric (or vacuum) region on awarm moving magnetized plasma half-space is considered. The external magnetic field is assumed to be normal to the direction of the wave normal and the velocity of the moving medium. Using the first three moment equations, together with Maxwell's electromagnetic equations, we construct the constitutive relations in the rest frame of the moving medium. The constitutive relations are then transformed to the laboratory frame by invokingMinkowski's equations for the moving plasma medium, and the dispersion relation for the propagating ordinary mode in the moving medium is derived. Expressions are obtained for the phase and group velocities and the index of refraction for the ordinary mode, as also for power reflection and transmission coefficients. It is found that in contrast to the case of a cold magnetized plasma, the ordinary electromagnetic mode excited in the warm magnetoplasma medium getsmodified due to the presence of an external magnetic field. In addition, the various reflection and transmission characteristics for a warm magnetoplasma depend on the velocity of the moving plasma as well as on the strength of the applied magnetic field, as against the case for a cold moving magnetized plasma. Numerical results on the reflection coefficient are presented for several values of the parameters characterizing the electron-plasma temperature, the velocity of the moving medium and the strength of the applied magnetic field.     

Diffuse reflection and transmission of radiation for Rayleigh phase function

Diffuse reflection and transmission coefficients in a plane parallel medium are calculated for a Rayleigh phase-function averaged over polarization and Rayleigh polarized phase-function. This is calculated by imbedding the finite medium into a semi-infinite scattering and absorbing medium. Numerical calculations for semi-infinite albedo are compared with Pomraning results. The albedos for finite medium are calculated via the imbedding equations which converge for large τ to the value of semi-infinite medium.     

Average photon path-length of radiation emerging from finite atmospheres

The determination of the average path-length of photons emerging from a finite planeparallel atmosphere with molecular scattering is discussed. We examine the effects of polarisation on the average path-length of the emergent radiation by comparing the results with those obtained for the atmosphere where the scattering obeys the scalar Rayleigh function. Only the axial radiation field is considered for both cases.To solve this problem we have used the integro-differential equations of Chandrasekhar for the diffuse scattering and transmission functions (or matrices). By differentiation of these equations with respect to the albedo of single scattering we obtain new equations the solution of which gives us the derivatives of the intensities of the emergent radiation at the boundaries.As in the case of scalar transfer the principles of invariance by Chandrasekhar may be used to find an adding scheme to obtain both the scattering and transmission matrices and their derivatives with respect to the albedo of single scattering. These derivatives are crucial in determining the average path length.The numerical experiments have shown that the impact of the polarisation on the average pathlength of the emergent radiation is the largest in the atmospheres with optical thickness less than, or equal to, three, reaching 6.9% in the reflected radiation.     

Reflection function from a double layer — A new approach

To improve the accuracy of calculations for the reflection and transmission functions, doubling techniques are in use. The central theme of this method is to derive the total reflection function when two portions of a medium are adjoined together. The synthesis is done by the use of star product techniques which require a knowledge of theS andT functions of both parts of the medium. Infinite series expansion is necessary to compute the total reflection. The method developed in this article splits the total reflected beam into two parts, one relating to that flux which suffers no scattering at all in the first portion of the medium and the second portion that undergoes at least one scattering in the first portion. The first part can be evaluated simply by knowing the reflection function of the second portion of the medium. The other part of the total reflection for variations of the thickness of the first portion of the medium is found to obey a simple Riccati type integro-differential equation with zero initial conditions. Knowledge of theT functions is not necessary and integration has to be performed over the interval corresponding to thickness of the first portion of the medium. Order-of-scattering analysis is also carried out.     

Albedo-shift method in the problem of anisotropic light scattering in a plane atmosphere

A standard problem of radiative transfer theory — calculating the diffuse reflection and transmission of radiation by a plane scattering atmosphere — is considered. The recently proposed albedoshift method is used to calculate the X and Y functions (and the H function) for the case of anisotropic scattering with a Henyey-Greenstein indicatrix. The method enables one to “suppress” scattering and obtain iterative solutions of high accuracy in only a few iterations, even when the mean number of photon scatterings in the atmosphere is very large. Translated from Astrofizika, Vol. 41. No. 4, pp. 623–646, October–December, 1998.     

Particle escape probability for inhomogeneous medium

The particle escape probability from a non-multiplying slab containing an internal source is defined in terms of a functional relation for the scattering function for the diffuse reflection problem. The Padé approximation technique is used to obtain numerical results for particle escape probability for inhomogeneous medium. Numerical results for homogenous and inhomogeneous media are given.     

Exact solution of equation for radiative transfer in semi-infinite Rayleigh scattering atmosphere using the laplace transform and the Wiener-Hopf technique

We have considered six scalar transport equations which are obtained from the vector transport equation to determine four Stokes's parameters to the problem of diffuse reflection in the semi-infinite plane parallel Rayleigh scattering atmosphere. By use of the Laplace transform and the Wiener-Hopf technique, these equations have been solved exactly to obtain the emergent intensity and the intensity at any optical depth and to reconstruct the Stokes's parameters. Solutions for emergent distribution so obtained are identical with the results of Chandrasekhar (1950).     

Diffuse reflection and transmission of solar radiation by a finite atmosphere bounded by a reflecting surface

The problem of diffuse reflection and transmission of solar radiation through a planetary atmosphere bounded from below by a reflecting surface is solved. The solution method based on rewriting the solution of the proposed problem in terms of the well known standard problem solution, where the planetary surface does not reflect. The solution of the standard problem can be found elsewhere or as we did by using the maximum entropy method. Numerical results for the angular radiation intensity and for the reflection and transmission coefficients are presented and compared with those obtained by Chandrasekhar's method.     

Energy albedo for multiregion slabs

An asymptotic solution for the equation of radiative transfer in an inhomogeneous medium was obtained on the basis of the corresponding solutions for homogeneous sub-layers in the slowing down region. Function relations between the reflection and transmission coefficients for the whole slab and those of the sublayers are given. The invariant embedding concepts are used to get the reflection and transmission coefficients for the sub-layers. We assumed different models for the slowing-down kernels. Laplace transform was used to transform the Boltzmann equation to one velocity approximation with re-scaled mean-free path and single-scattering albedo. Numerical results are given for energy albedo as a function of the mass number of the host medium.     

Some differential equations for problems of wave scattering in a one-dimensional medium

A system of linear differential equations determining the amplitude of reflection R and the amplitude of transmission T for a plane wave (or an electron) and for an arbitrary medium (or a one-dimensional potential of an arbitrary type) is obtained. It is shown that the problem of determining the scattering parameters R and T reduces, in general, to a Cauchy problem for a stationary wave equation (or for the Schrödinger equation).     

Radiation transfer in a diffuse and specular reflecting slab with Rayleigh scattering

A method of analysis is presented for solving the radiative transfer problem in an absorbing, emitting, inhomogeneous, and anisotropically scattering plane-parallel medium with specular and diffuse reflecting boundaries and internal source (problem 1). Exact relations for the radiation heat flux at the boundaries of problem 1 are obtained in terms of the radiation density and albedos of the corresponding source-free medium with specular reflecting boundaries (problem 2). Two coupled integral equations for the radiation density and the second moment of the radiation intensity for problem 2 with Rayleigh phase functions are obtained. The Galerkin method is used to solve these equations. Albedos of problem 2 are compared with theF _n method. Numerical results for radiation heat fluxes at the boundaries of problem 1 are tabulated for different forms of the internal source.