Forward And Inverse Particle Transport Problems
Norman J. McCormick
University of Washington, Mechanical Engineering Department, Seattle, WA 98195-2600
email: mccor@u.washington.edu
Transport theory is the study of problems involving the propagation of "particles" within a medium or vacuum. The particles may be electrostatically neutral, or charged as in a plasma in which ions and electrons interact with each other and any other surrounding charged or neutral particles. Or the particles may be gas molecules, in which case the problems involve the kinetic theory of gases. If any of the species of particles can interact with other particles of the same species, then the (Boltzmann) transport equation governing the transfer process is nonlinear.
For the study of neutral photons in radiative transfer applications and neutrons in the core of nuclear reactors, like-particle interactions can be ignored and the transport equation is linear, which makes the analysis much easier. But still, the mathematics of the governing equation is messy and somewhat unique to the discipline of transport theory because both a differential operator in configuration space and an integral operator in velocity space are involved. Thus in general one must solve an integro-differential equation involving three spatial coordinates, three velocity (or energy/wavelength and two directional) coordinates, and time. The differential operator is needed to account for the loss of particles traveling along a straight line in the medium that are either absorbed or scattered and the integral operator incorporates the effects of scattering events from particles initially traveling in other directions that get scattered into the direction of interest.
The presence of the differential operator in the transport equation means that boundary conditions are required to make a solution unique. Such boundary conditions can be applied only to the directions for particles entering the medium---the distribution of the particles within and leaving the medium is then obtained as the solution of a forward problem. In forward (or direct) transport problems, the objective usually is to estimate the particle distribution within and emerging from a prescribed medium with specific in-going boundary and initial conditions.
Many measurement problems are inverse to the forward problem in that the objective is to determine the absorption and scattering properties or the size of the medium or the characteristics of an obscured surface (i.e., the boundary condition) . Some imaging methods, such as computed tomography, take advantage of the fact that the medium is thin enough that only the attenuation of uncollided beams of photons need be measured; then it is possible to estimate spatially dependent density variations of the medium. As another application, an inverse solution of the integral transport equation can be used to estimate the spatially dependent temperature in an atmosphere from the Planck radiation source function provided the effects of multiple scattering can be neglected.
The inverse transport problems of interest to me are ones for which multiple scattering effects are important. Such inverse problems usually are poorly conditioned, i.e., unstable to small perturbations in the input data. This more likely occurs when the number of unknowns is large, as when estimating the coefficients in a series expansion describing highly anisotropic scattering, or the spatial variation of absorption and scattering properties within an inhomogenous medium. But this ill-posedness of the problem also occurs when experimental data are insensitive to variations in a single unknown.
Key considerations when undertaking an inverse transport problem are briefly described to in Tables I and II. More details can be found in N. J. McCormick, "Inverse Radiative Transfer Problems: A Review," Nuclear Science and Engineering 112, 185-198 (1992).