Rudolph W. Preisendorfer

Radiative Transfer on Discrete Spaces

Author(s): Preisendorfer, Rudolph W | Abstract: SIO Reference 59-53. We shall be concerned here with a study of radiative transfer processes on discrete rather than continuous optical media. That is, we shall explore the consequences of replacing the usual geometric setting for radiative transfer processes — namely some continuum of points in three dimensional euclidean space £ — by a spatially bounded set of points, finite in number, each of which is located in E3 in accordance with some explicit rule of selection. Each point of the set is then assigned certain scattering and absorbing properties with respect to impinging radiant energy, and each is generally allowed to interact radiometrically in a specified way with a certain preselected subset of the given collection. The main object of the present study is to formulate and solve, within this context ;ind on a phenomenological level, the problem of the steady state radiance distribution at each point of the set, under prescribed boundary conditions.

Markov chains and radiative transfer

This chapter discusses Markov chains and radiative transfer. A Markov chain is a steady-state Markov process, and a Markov process is a special stochastic process in which the object undergoing the process can remember at most one stage of the process back in time. The simpler the internal structure of an object undergoing a stochastic process, the more closely the process can be Markovian. Radiative transfer processes on discrete spaces can essentially be described by Markov chains—or, at most, Markov processes. The first and simplest of all connections that are established between radiative transfer and Markov chains are for the case of monochromatic radiative transfer on a discrete space where the radiance is governed by the scattering functions for elastic scattering. The transpectral scattering function is the concept within the continuous formulation of the theory that describes the details of heterochromatic radiative transfer, that is, the transfer of radiant energy from one frequency to another at a given point within a medium.

A COMPUTER STUDY OF RADIATIVE TRANSFER ON A CUBIC LATTICE

This chapter presents a numerical computation of the light field in an extended cubic lattice. The extended cubic lattice simulates a real optical medium whose boundary light fields and whose volume attenuation and volume scattering functions are known. The end result of the computation is a 26-component light vector at each point of a cubic lattice 50 monolayers deep. The main purpose of the computation is to see if the point-level interpretation of discrete-space theory yields a reasonably accurate model of natural light fields. The discrete-space model is a numerically tractable model. The discrete-space procedure yields a reasonably good model of light fields in plane-parallel media.

TWO METHODS OF POINT-SOURCE PROBLEMS IN DISCRETE SPACES

This chapter discusses two methods of point-source problems in discrete spaces. Some problems of the first class are the solution of problems of underwater or aerial visibility, television, and photography, given the inherent optical properties of the source of radiant flux and the properties of the optical targets. The solution of the point-source problem can be used to resolve basic research problems that require a determination of the inherent optical properties of a medium, given the radiometric response of the medium to a particular point source. These are problems of the second class. Another example of the point-source problem arises when it is required to determine the transient light field generated by powerful pulse-emitting light sources within a given medium. Point-source problems in radiative transfer theory are the most difficult of the standard problems encountered in the theory because of the relatively greater number of parameters required in their formulation and solution. In the case of a general point source within the medium, three additional parameters are needed at the very outset to locate a point in the medium relative to the source and then two more to specify the radiance distribution at that point. The chapter describes iteration method and categorical analysis method.

THEORY OF POLARIZED LIGHT FIELDS IN DISCRETE SPACES

This chapter discusses the theory of polarized light fields in discrete spaces. Discrete-space radiative transfer theory can supply the polarized context with relatively tractable discrete counterparts to the continuous-space concepts. The polarized form of the local interaction principle leads to a set of vector equations. The polarized form of the local interaction principle leads to a complete phenomenological description of radiative transfer processes on discrete spaces without adding essentially new mathematical problems to the discrete-space transfer theory. A phenomenological definition of a physical concept is based on the use of a given real apparatus and prescribed set of operations with that apparatus. The polarizer can be manufactured from some natural polarizing material such as tourmaline or from some man-made polarizing material, such as polaroid sheets. The chapter explains how the discrete-space theory can be uniformly elevated from the unpolarized to the polarized case.

THE INTERACTION PRINCIPLE

This chapter discusses the interaction principle. The point-level interpretation of the interaction principle subsists when the subset S is interpreted in the statement of the principle as either a point or as a scattering volume in a continuous optical medium X or as a point of a general space X. The surface-level interpretation of the interaction principle subsists when the subset S is interpreted in the statement of the principle as a subset of a space X that has one less dimension than X. The sequence of ascending interpretations of the interaction principle is interrupted by the apparent omission of a line-level interpretation that can normally occur at this point. The space-level interpretation of the interaction principle subsists when the subset S is interpreted in the statement of the principle as a subset of a space X that has the same dimension as X. For example, one particular space-level interpretation can center around a finite set of nonoverlapping spherical regions in Euclidean three-space.

RADIATIVE TRANSFER THEORY: DISCRETE FORMULATION

This chapter discusses the discrete formulation of radiative transfer theory. The interaction principle always uniquely determines the light field generated within a given location space by a given incident field when that location space is absorptive. The local interaction principle uniquely determines the radiance function when given a suitable absorption function, scattering function, and source function on the discrete optical medium. The local interaction principle for special discrete spaces is directly related to the basic transfer equation. By simple limiting processes, the discrete-space formulation on special discrete spaces can return to the continuous formulation. Thus, the local interaction principle is capable of tying together the fundamental concepts and laws of classical radiative transfer theory. For all its abstract power and comprehensiveness, the local interaction principle has a commonplace origin in the everyday procedures of radiative transfer.

RADIATIVE TRANSFER THEORY: CONTINUOUS FORMULATION

This chapter discusses the continuous formulation of radiative transfer theory. Beam transmittance function describes the fractional amount of radiance of a beam of light transmitted along a path through an optical medium. Beam-transmitted radiance is a nonincreasing function of distance along a path of sight. The first property of beam transmittance, the multiplication property of beam transmittance, is an important conceptual tool in radiative transfer theory. The multiplication property of beam transmittance holds under general lighting conditions in the atmosphere or the seas. It is physically impossible to distinguish between the directly transmitted and scattered components of a beam for the case ξ = ξ′. The path function describes the contribution to the radiance from the scattering of light into the direction of the path without a change in frequency. The emission function describes the contribution from either scattering into the path with a change of frequency or from the process of conversion of nonradiant energy to radiant energy of some given frequency.

CONNECTIONS WITH THE MAINLAND

This chapter discusses the connections of radiative transfer theory with the mainland. The rules of radiative transfer theory can be changed by filling in the gap between it and electromagnetic theory or some other point along the mainland of physics. The chapter describes a relation between the electromagnetic field and the radiance function used in the subsequent derivation of the interaction relation. In natural radiometric environments, such as in the atmosphere or seas of the earth, the Poynting vector, at some point, changes size and direction many times a second. For a particular frequency in the middle of the visible spectrum, the electric and magnetic vectors at a point complete about 6 × 10 11 cycles every second. The Poynting vector can have a relatively fixed direction extending over thousands of such cycles and can abruptly change to some new direction and magnitude for the next few thousand cycles. In a natural optical medium, such as in a natural hydrosol or aerosol, the light field changes relatively slowly over distances comparable to the size of a man.

RADIATIVE TRANSFER THEORY: AXIOMATIC FORMULATION

This chapter discusses the axiomatic formulation of radiative transfer theory. The static transition relation for radiative measures is the most primitive form of the equation of transfer on a set level within a given optical medium. It is the radiative counterpart to the Chapman-Kolmogorov transition relation for probabilities in the abstract theory of stochastic processes. In this way, the axiomatic formulations are conceptually linked to the diffusion process formulations by Feller. Dynamic transition relations are distinguished from their static correspondents by the presence of time derivatives of the measure or density. The interaction principle can be used as an algebraic foundation for the multitude of invariant imbedding relations. A key tactic in converting a conclusion of the general axiomatic formulations into the discrete context is to replace every integral over the space by a finite sum over that space. The chapter describes axiomatic basis for the theory of polarized radiance.