A review on the theory of photon transport in leaf canopies

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/27867/0000280.pdf;sequence=1

Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II

We report on the development of an iterative image reconstruction scheme for optical tomography that is based on the equation of radiative transfer. Unlike the commonly applied diffusion approximation, the equation of radiative transfer accurately describes the photon propagation in turbid media without any limiting assumptions regarding the optical properties. The reconstruction scheme consists of three major parts: (1) a forward model that predicts the detector readings based on solutions of the time-independent radiative transfer equation, (2) an objective function that provides a measure of the differences between the detected and the predicted data, and (3) an updating scheme that uses the gradient of the objective function to perform a line minimization to get new guesses of the optical properties. The gradient is obtained by employing an adjoint differentiation scheme, which makes use of the structure of the finite-difference discrete-ordinate formulation of the transport forward model. Based on the new guess of the optical properties a new forward calculation is performed to get new detector predictions. The reconstruction process is completed when the minimum of the objective function is found within a defined error. To illustrate the performance of the code we present initial reconstruction results based on simulated data.

Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer.

In 1968, Bengt Carlson and Kaye Lathrop published a comprehensive review on the state of the art in discrete-ordinates (SN) calculations [10]. At that time, SN methodology existed primarily for reactor physics simulations. By today’s standards, those capabilities were limited, due to the less-developed theoretical state of SN methods and the slower and smaller computers that were then available. In this chapter, we review some of the major advances in SN methodology that have occurred since 1968. These advances, combined with the faster speeds and larger memories of today’s computers, enable today’s SN codes to simulate problems of much greater complexity, realism, and physical variety. Since 1968, several books and reviews on general numerical methods for SN simulations have been published [32, 46, 71], but none of these covers the advanced work done during the past 20 years.

Advances in Discrete-Ordinates Methodology

The authors present a discretization of the diffusion equation that can be used to accelerate transport iterations when the transport equation is spatially differenced by a discontinuous finite element (DFE) method. That is, they present a prescription for diffusion synthetic acceleration of DFE transport iterations. (The well-known linear discontinuous and bilinear discontinuous schemes are examples of DFE transport differencings.) They demonstrate that the diffusion discretization can be obtained in any coordinate system on any grid. They show that the diffusion discretization is not strictly consistent with the transport discretization in the usual sense. Nevertheless, they find that it yields a scheme with unconditional stability and rapid convergence. Further, they find that as the optical thickness of spatial cells becomes large, the spectral radius of the iteration scheme approaches zero (i.e., instant convergence). They give analysis results for one- and two-dimensional Cartesian geometries and numerical results for one-dimensional Cartesian and spherical geometries.

Diffusion Synthetic Acceleration of Discontinuous Finite Element Transport Iterations

A study of spatial discretization schemes for the multigroup discrete-ordinates transport equations in slab geometry is described. The purpose of the study is to determine the most computationally efficient method, defined as the one that produces the minimum error for a given cost. Cost is defined as the total amount of computer time required to complete one inner iteration, given a limit on storage, and three error norms are used to measure the accuracies of edge fluxes, cell average fluxes, and integral parameters. Three test problems are studied: the first is a model one-group problem examined in detail, while the second and third are more realistic multigroup problems. One conclusion is that a new method, labeled linear characteristic, significantly outperforms all other methods that have been implemented up to the present time. 15 references.

W. F. Miller

Papers

Computational Efficiency of Numerical Methods for the Multigroup, Discrete-Ordinates Neutron Transport Equations: The Slab Geometry Case