There is a branch of radiative transport theory that is customarily expressed with an integrodifferential equation or an integral equation. The new formulation in this article is, without approximation, expressed through partial differential equations in both the frequency and time domains. Its accuracy is demonstrated in the frequency domain by applying it to a problem solved long ago. It was expressed with the conventional integrodifferential equation. Confidence is bolstered in the new method by showing how the new method produces the identical analytical answer. This article also analyses a situation in the time domain in both the appropriate differential and integrodifferential equations and the identical results are again obtained.
I present, apparently, a new description of radiative transfer problems in the time domain. It appears that for the first time a simple physical picture emerges of the underlying essence of scattered radiance when dealing with isotropic axially-symmetric scattering in nonconservative linear media as attenuated travelling waves was by analogy. The method used a new differential equation approach. Initially its accuracy in the frequency domain was demonstrated by applying it to a solved problem, where in the literature it is dealt with using the conventional 95-year-old integro-differential equation description. Confidence in the differential equation method was bolstered by showing how this new method produces the same analytical answer. The new technique converts the integro-differential equation formulation of radiative transfer into a “pure” differential equation formulation, consisting here in a mixture of ordinary and partial derivatives, and solves that. This paper analyzes the situation in the time domain using the differential equation description and again yields a travelling wave description. However, this time it is not simply by analogy that such a description is obtained. It is exact. This result of attenuated travelling waves was demonstrated in a prior paper by solving the integro-differential equation for the classic problem of axially-symmetric scalar isotropic scattering in a nonconservative linear medium. In this paper we revisit the problem, this time solving it by the differential equation method and obtain the identical result, once again confirming the method.
The spectral signature of a material is the curve of power density vs. wavelength (λ) obtained from measurements of reflected light. It is used, among other things, for the identification of targets in remotely acquired images. Sometimes, however, unpredictable distortions may prevent this. In only a few cases have such distortions been explained. We propose some reasonable arguments that in a significant number of circumstances, atmospheric turbulence may contribute to such spectral signature distortion. We propose, based on this model, what appears to be one method that could combat such distortion.
We have presented apparently new points of view for both radiative transfer problems in the frequency and time domains. Apparently for the first time a simple physical picture has emerged of the inderlying essence of scattered radiance when dealing with isotropic axially-symmetric scattering as a single set of cutoff traveling waves in nonconservative linear media. First, it is shown in the frequency domain. Its accuracy is demonstrated by applying it to a solved problem in the frequency domain whose solution was published by Chandrasekhar some five decades ago. He determined it by dealing with the conventional 95- year-old, usually difficult to solve, integro-differential equation and we bolster confidence in our method by showing how the new method produces the same analytical answer. The new technique converts the integro-differential equation formulation of radiative transfer into a pure differential equation formulation, consisting here in a mixture of ordinary and partial derivatives, and solves that. This paper also analyzes the situation in the time domain in a different manner but the same type of result is obtained.
We present a new point of view for investigation radiative transfer problems by showing it involves the scattering of traveling evanescent waves. Its accuracy is demonstrated by applying it to a solved problem whose solution was published by Chandrasekhar. He determined it with a conventional method and we bolster confidence in our method by showing how the new method produces the same analytical answer. The new technique converts the 95-year-old, usually difficult to solve, integro-differential equation formulation of radiative transfer into a less formidable 'pure' differential equation formulation, consisting here in a mixture of ordinary and partial derivatives, and solves that. This paper focuses on a single class of cases. It also demonstrates surprising success at solving a narrowly defined class of nonlinear radiative transfer problems initially expressed as a nonlinear integro-differential equation formulation of the radiative transport problem.
Optical systems associated with imaging sensors and instruments typically distort the 'true' or object image, I(x), in a manner usually characterized by their point spread function (PSF). Determining I(x) from the measured image data, M(z), using the convolutional relation with the PSF is called deconvolution. This paper proposes what appears to be a new deconvolution technique by taking advantage of a remarkable coincidence. It is that for most optical systems of interest here the PSF is Gaussian, which is a zeroth order Hermite function. By expressing I(x) in an orthogonal representation using Hermite functions, which are to be distinguished from Hermite polynomials, the convolution integral can be evaluated exactly in analytical form, perhaps for the first time for the general case. This, in turn, leads to simple, precise linear relations between the coefficients of the Hermite representation of I(x) and that of M(x); while avoiding the common problem of division of noisy data by small quantities. The coefficients in those linear equations have precise values obtained from the nature of Hermite function interrelations rather than measured data. These values of I(x) may be more useful than M(x) as the initial iterate in the iteration techniques commonly used for deconvolution.
Most atmospheric correction codes are based in part or in whole on the radiative transfer equation (RTE), which is an integrodifferential equation. It is well known to be an ad hoc equation, which can and has produced incorrect answers. This paper initiates a new way of exploring where the RTE can produce unphysical answers in parameter-ratio ranges of genuine concern. The exploration begins with formulating a new technique for rigorously transforming the scalar RTE, without approximation, into a `pure' partial differential equation (PDE), i.e., one involving only partial derivatives of finite and relative small order. The virtue of this approach is that there are only a small number of analytical and numerical techniques for dealing with integrodifferential equations compared to the vast array of techniques for PDEs. A variety of tools are developed that are more powerful than needed for the particular physical problems to demonstrate the robustness of the technique. An atmosphere is then considered where Rayleigh scattering is dominant and its PDE derived, apparently for the first time. A class of nonlinear integrodifferential equations were also transformed into linear PDEs and solved for a multiplicity of solutions.
This paper discusses the nonuniform illumination of individual pixels in an array that is intrinsic to the scene viewed, as opposed to turbulence or platform motion as an error source in quantitative imagery. It describes two classes of algorithms to treat this type of problem. It points out that this problem can be viewed as a type of inverse problem with a corresponding integral equation unlike those commonly treated in the literature. One class allows estimation of the spatial variation of radiance within pixels using the single digital number irradiances produced by the measurements of the detectors within their instantaneous-fields-of-view (IFOVs). Usually it is assumed without discussion that the intrapixel radiance distribution is constant. Results are presented showing the improvements obtained by the methods discussed.
This paper discusses an aspect of spatially varying dissipative (extinction) and reflective (backscattering) losses that ultimately may allow a relatively quick appraisal of measured data quality, using a new figure of merit q(x), after some limited preprocessing. In order to do this, the paper presents the exact analytical, closed-form, finite solution for the inversion of the classical first-order nonlinear Riccati differential equation (RDE) which relates the reflection coefficient (R) and its derivative (dR/dx) to the otherwise arbitrary profiles of the media parameters [(epsilon) 1(X) equals (epsilon) 1(X) - j(epsilon) 2(X)] in the coefficients of the RDE. This synthesis assumes a deterministic medium which can be accurately represented by the usual media parameters.
This paper discusses apparent deficiencies of the reflection coefficient (R) for inhomogeneous media (IX) and single-mode nonuniform waveguides (NOW5) as used in current wave propagation theory for time-harmonic waves. A new formulation is proposed (Rn) which tracks energy transport spatially, unlike the classical Schelkunoff formulation (Re) which may not display, even approximately, this property. Although the paper is focused on electromagnetic waves, the new formulation is also applicable to acoustics and related wave types. The paper discusses why, after more than four decades of careful measurement and analysis, the deficiencies have apparently gone undetected.
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