This paper reviews scattering theory required for analysis of light reflected by planetary atmospheres. Section 1 defines the radiative quantities which are observed. Section 2 demonstrates the dependence of single-scattered radiation on the physical properties of the scatterers. Section 3 describes several methods to compute the effects of multiple scattering on the reflected light.

Light scattering in planetary atmospheres

We present a review of methods for the forward and inverse problems in optical tomography. We limit ourselves to the highly scattering case found in applications in medical imaging, and to the problem of absorption and scattering reconstruction. We discuss the derivation of the diffusion approximation and other simplifications of the full transport problem. We develop sensitivity relations in both the continuous and discrete case with special concentration on the use of the finite element method. A classification of algorithms is presented, and some suggestions for open problems to be addressed in future research are made.

https://iopscience.iop.org/article/10.1088/0266-5611/15/2/022/pdf

Optical tomography in medical imaging

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

https://escholarship.org/content/qt6b0567k2/qt6b0567k2.pdf?t=oonkck

Boundary-conditions for the diffusion equation in radiative-transfer

Using the method of images, we examine the three boundary conditions commonly applied to the surface of a semi-infinite turbid medium. We find that the image-charge configurations of the partial-current and extrapolated-boundary conditions have the same dipole and quadrupole moments and that the two corresponding solutions to the diffusion equation are approximately equal. In the application of diffusion theory to frequency-domain photon-migration (FDPM) data, these two approaches yield values for the scattering and absorption coefficients that are equal to within 3%. Moreover, the two boundary conditions can be combined to yield a remarkably simple, accurate, and computationally fast method for extracting values for optical parameters from FDPM data. FDPM data were taken both at the surface and deep inside tissue phantoms, and the difference in data between the two geometries is striking. If one analyzes the surface data without accounting for the boundary, values deduced for the optical coefficients are in error by 50% or more. As expected, when aluminum foil was placed on the surface of a tissue phantom, phase and modulation data were closer to the results for an infinite-medium geometry. Raising the reflectivity of a tissue surface can, in principle, eliminate the effect of the boundary. However, we find that phase and modulation data are highly sensitive to the reflectivity in the range of 80-100%, and a minimum value of 98% is needed to mimic an infinite-medium geometry reliably. We conclude that noninvasive measurements of optically thick tissue require a rigorous treatment of the tissue boundary, and we suggest a unified partial-current--extrapolated boundary approach.

/pdf/boundary-conditions-for-the-diffusion-equation-in-radiative-2iwt63z8b7.pdf

Boundary conditions for the diffusion equation in radiative transfer

Linear transport theory

The equations of radiative transfer are systematically analyzed by asymptotic methods. To lowest order, the classical equilibrium diffusion approximation is recovered. The next order analysis leads to the equilibrium diffusion differential equations and initial condition, but with a boundary condition containing a linear extrapolation distance α. This quantity is related tothe solution of a canonical halfspace problem and is computed by deriving an appropriate variational principle. For the case of no scattering, an exact Wiener-Hopf solution is available. The FN solution technique is also applied to the problem of obtaining α with good results. Higher order asymptotic radiative transfer descriptions are discussed and, while not immediately constituting practical calculational techniques, do have implications for computing the parameters in the multiband treatment of the frequency variable.

Asymptotic analysis of radiative transfer problems

We give numerical benchmark results for particle transport in a randomly mixed binary medium, with the mixing statistics described as a homogeneous Markov process. A Monte Carlo procedure is used to generate a physical realization of the statistics, and a discrete ordinate numerical transport solution is generated for this realization. The ensemble averaged solution, as well as the variance, is obtained by averaging a large number of such calculations. Reflection and transmission results are given for several problems in both rod and planar geometry. In a separate development, two coupled transport equations are derived which formally described transport in a random binary mixture for arbitrary mixing statistics. Closing these equations by approximating their coupling terms in a low order and intuitive way leads to a model for stochastic transport previously obtained via the master equation. The present derivation, based upon approximating exact equations, allows in principle the opportunity to develop more accurate models by making higher order approximations in the coupling terms.

/pdf/benchmark-results-for-particle-transport-in-a-binary-markov-53d03fwanc.pdf

G. C. Pomraning

Papers

Linear transport theory

Asymptotic analysis of radiative transfer problems

Benchmark results for particle transport in a binary Markov statistical medium

Asymptotic and variational derivations of the simplified PN equations

Discretization methods for one-dimensional Fokker-Planck operators