S. Ueno

Numerical results for Chandrasekhar's X and Y functions of radiative transfer

Abstract From many points of view the X and Y functions of Chandrasekhar may be considered to be the basic functions of radiative transfer. In this paper the authors present graphs and selected tables of these functions covering wide ranges of slab thicknesses and albedos for single scattering. The numerical procedure involves integrating the integro-differential equations satisfied by the X and Y functions. One of the great advantages of such extensive graphs and tables is that they can be used to test the accuracy of simple approximate expressions and procedures.

Inverse Problems in Radiative Transfer: Layered Media

Abstract A basic problem in radiative transfer is the estimation of the physical parameters of a scattering and absorbing atmosphere based on measurements of the diffusely reflected light. In this series of papers we shall show how such tasks may be viewed as nonlinear boundary-value problems which may be solved computationally using the technique of quasilinearization. In this paper we consider a stratified atmosphere consisting of two layers illuminated by a source of uniform parallel rays. Our aim is to determine the optical depth and the albedo for single scattering of each layer based on measurements of the angular dependence of the light diffusely reflected from the slab. The results of some numerical experiments are presented.

Reflection and transmission functions for finite isotropically scattering atmospheres with specular reflectors

Abstract A finitely thick isotropically scattering atmosphere is bounded below by a specular reflector. A Cauchy system for the scattering and transmission functions is obtained. Results of numerical experiments are presented.

Radiative Transfer in Spherical Shell Atmospheres with Radial Symmetry

In recent years, the need to allow for the effect of curvature on the formation of spectral lines has been recognized not only in the fields of planetary and stellar atmospheric physics but also in the field of neutron transport in spherical reactors. This paper is of interest in these and related fields. Initial value systems are obtained for the scattering and transmission functions and emergent intensities for inhomogeneous, anisotropically scattering spherical media both with internal and external sources and with various types of cores.

Invariant imbedding and time-dependent diffuse reflection of a pencil of radiation by a finite inhomogeneous flat layer—I

The principle of invariant imbedding is used to obtain integral equations for the scattering function of a pencil of radiation by a finite, plane- parallel, inhomogeneous, nonemitting, and isotropically scattering atmosphere with mono-directional illumination of the lower surface, allowing for the incident Dirac delta-function and unit stepfunction, time-dependent, net fluxes. Finally, the Laplace transform of the integral equation for the s-function is derived. (auth)

Invariant Imbedding and Diffuse Reflection from a Two-Dimensional Flat Layer

Abstract In the present paper the angular distributions of radiation diffusely refleted by an inhomogeneous, plane-parallel, nonemitting, and anisotropically scattering atmosphere of finite thickness are obtained exactly with the aid of the invariant imbedding technique, allowing for an albedo for single scattering which varies both with the depth and the horizontal distance. When the albedo depends only upon the depth and the scattering is isotropic, the solution reduces to that given by Bellman and Kalaba (1956) , Busbridge (1961) , and Ueno (1960) , respectively.

On the diffuse reflection of parallel rays by an inhomogeneous flat layer as a limiting process

The diffuse reflection of parallel rays by an inhomogeneous, plane- parallel, nonemitting, and isotropically scattering atmosphere of finite optical thickness is developed. The S-function for the standard diffuse reflection problem is derived. The standard diffuse reflection problem is considered as a limiting process of a collimated point source problem and as a limiting process of a transient diffuse reflection problem when the incident intensity is given in the form of the unit step function of time. (C.E.S.)

Numerical results for the estimation of source distributions from external radiation-field measurements

Abstract The authors consider a transport process in a slab bounded by two parallel planes. Radiation is both absorbed and scattered isotropically. The radiation is due to continuously distributed internal isotropic sources. The aim is to estimate the distribution of the internal sources based on experimental measurements of the angular dependence of the emergent radiation. First, a system of differential-integral equations for the emergent radiation is deduced. These are approximated by a system of ordinary differential equations, and some numerical results are given. Quasi-linearization is then used to solve the inverse problem numerically. The results of some computational experiments are given which indicate the sensitivity of the estimates of the source distribution to the observational errors in the emergent-radiation measurements. The methods can be generalized to the cases of anisotropic scattering and shell geometry.

A COMPUTATIONAL APPROACH TO CHANDRASEKHAR'S PLANETARY PROBLEM,

Abstract Chandrasekhars planetary problem (1) is reformulated using an initial-value approach. The problem concerns the diffuse reflection of radiation from a finite atmosphere with a reflecting surface at the bottom. In the direct problem, the angular distribution of multiple-scattered radiation is computationally obtained as the solution of an initial-value problem for ordinary differential equations for S, a generalization of the Chandrasekhar scattering function for an inhomogeneous atmosphere. In the inverse planetary problem, the properties of the atmosphere and the surface are estimated, given the angular distribution of scattered radiation. The quasilinearization technique for nonlinear multipoint boundary-value problems provides an effective method for obtaining a computational solution to the inverse problem.

Invariant imbedding and time-dependent diffuse reflection by a finite inhomogeneous atmosphere

Abstract An integral equation fo the scattering function for a finite, plane-parallel inhomogeneous, nonemitting and anisotropically scattering atmospheres is derived in the case of time-dependent monodirectional illumination of the lower boundary. For use in subsequent computational studies, the corresponding equation for the Laplace transform is also given. The derivations are based on the use of invariant imbedding techniques.