Alan P. Wang

Nonstationary multiple scattering

This paper is devoted to the abstract formulation of the general nonstationary multiple scattering problem in radiative transfer. We consider the linear operators on curved surfaces. Therefore, the problem is attacked from the unified and general point of view. Special geometric considered are spherical shells and slabs. Special cases lead to stationary, instantaneous, and time invariant cases. The extension from stationary to nonstationary involves the distribution theory and the concept of nonpredictive operators. Many well−known physical problems in astrophysics are solved in the unified way.

Operator theory on the WKB method and Bremmer series

The well‐known WKB method and Bremmer series in mathematical physics have various modifications and applications. Basically, they are all limited to the finite‐dimensional case. Such methods have been extended to the infinite‐dimensional case in this paper, i.e., operators on a general normed linear space. This enables us to apply the methods to solve the problem of transport of radiation through a layer of cloud. The derivation is based on input‐output states and linear operators. Some elementary properties are exploited, along with physical interpretations. The convergence of the series is established. An example is given to demonstrate the methods and the rate of convergence.

Solutions of a dissipative diffusion equation in a Hilbert space

Abstract Two types of dissipation of a diffusion process on an abstract space are discussed. The main results are the bound, critical length, and extension of solutions of a dissipative Riccati equation on a Hilbert space. The bound result enables the local solution to be extended to a global solution. The analysis of critical length provides computational and theoretical information about the maximal interval of existence of solutions. It is shown that no bounded solution can be extended beyond the critical length.

Invariant imbedding and searchlight problem in turbid slab

Abstract In this paper, with the aid of invariant imbedding, we discuss the mathematical model of a searchlight on a target of turbid plane-parallel atmosphere, bounded below by a diffuse reflector. In the real physical situation, the faint background light illuminating the top is considered. However in this paper, for the sake of simplicity, the angular distribution of radiation emergent from the top with a point source is considered, by imbedding the problem within a family of parameters.

Searchlight on a target with diffuse background. I

A mathematical model of a searchlight on a target with diffused background is constructed based on three‐dimensional radiative transfer and integral operators. The main result is the recovery of the true target reflection from the measured reflection which consists of background light and multiscattering between atmospheric layer and the earth surface. An exact solution is solved and approximations are obtained by use of the assumption that the searchlight is strong and focused and the background light is weak and uniform in space in a given narrow frequency band of intensity.

Basic equations of three‐dimensional radiative transfer

Chandrasekhar developed one‐dimensional mathematical models of radiative transfer in 1949 [Radiative Transfer (Oxford U.P., New York, 1950)]. This paper is a systematic extension of Chandrasekhar’s work to three dimensions, including discussions of specular and diffused parts, reciprocity, solutions, and approximation.

Linear operators and transfer of radiation with spherical symmetry

This paper is concerned the construction of linear‐operator equations for transfer of radiation taking place on a spherical shell. A complete set of equations is obtained for inhomogeneous, anisotropically scattering media with internal or external illumination and with an arbitrary reflecting core. In application, linear‐operator equations are reduced to a class of familiar functional equations. Our general result provides answers to a set of well‐known problems in astrophysics.

Scattering operators and propagation solutions for a non-stationary transfer equation

Abstract Physically meaningful “mild” scattering operator solutions for non-stationary radiative transfer equations, including diffuse reflection and transmission by finite optical thickness plane parallel or discrete angular redistribution curved geometries, are presented. Our method is based on the construction of scattering matrices from a C 0 -propagation group. The difficulty of this problem is the unbounded generator occurring in the non-stationary case. In an energy dissipative case, the local existence of scattering operators is extended to global existence.

Invariant imbedding and inverse problems with applications to radiative transfer

Abstract In this paper, we developed two methods to solve the inverse problem of a nonlinear integro-differential equation. Both methods are based on the principle of invariant imbedding. The first method involves two auxiliary integro-differential equations. The inverse problem is solved by a sequence of approximation solutions of linear equations. The second method involves algebraic equations of scattering matrices under the so-called star-product. An application to a radiative transfer problem such as correction of atmospheric effects on remote sensing is discussed.

An inverse problem of reflection

Abstract This paper developed an analytic method to solve the inverse problem of reflection. The direct problem is to determine the reflection at the top of a known stratified medium bounded below by a reflector. The inverse problem is to determine the reflection of a reflector from the measured reflection at the top of the medium. The main results are based on the iteration of solutions of two initial value differential equations. They are suitable for numerical computation. A mathematical model is constructed on an abstract space for broad applications which include radiative transfer with inhomogeneous and anisotropic scattering. The solution is general is not unique. A condition has been established for a unique solution. Non-unique solutions are also constructed.