Three-dimensional Vector Radiative Transfer Research Articles

Asymptotic theory of circular polarization memory.

We establish a quantitative theory of circular polarization memory, which is the unexpected persistence of the incident circular polarization state in a strongly scattering medium. Using an asymptotic analysis of the three-dimensional vector radiative transfer equation (VRTE) in the limit of strong scattering, we find that circular polarization memory must occur in a boundary layer near the portion of the boundary on which polarized light is incident. The boundary layer solution satisfies a one-dimensional conservative scattering VRTE. Through a spectral analysis of this boundary layer problem, we introduce the dominant mode, which is the slowest-decaying mode in the boundary layer. To observe circular polarization memory for a particular set of optical parameters, we find that this dominant mode must pass three tests: (1)this dominant mode is given by the largest, discrete eigenvalue of a reduced problem that corresponds to Fourier mode k=0 in the azimuthal angle, and depends only on Stokes parameters U and V; (2)the polarization state of this dominant mode is largely circular polarized so that |V|≫|U|; and (3)the circular polarization of this dominant mode is maintained for all directions so that V is sign-definite. By applying these three tests to numerical calculations for monodisperse distributions of Mie scatterers, we determine the values of the size and relative refractive index when circular polarization memory occurs. In addition, we identify a reduced, scalar-like problem that provides an accurate approximation for the dominant mode when circular polarization memory occurs.

Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems II The hybrid matrix operator--Monte Carlo method

A hybrid method is developed to solve the vector radiative transfer equation (VRTE) in a three-dimensional atmosphere-ocean system (AOS). The system is divided into three parts: the atmosphere, the dielectric interface, and the ocean. The Monte Carlo method is employed to calculate the impulse response functions (Green functions) for the atmosphere and ocean. The impulse response function of the dielectric interface is calculated by the Fresnel formulas. The matrix operator method is then used to couple these impulse response functions to obtain the vector radiation field for the AOS. The primary advantage of this hybrid method is that it solves the VRTE efficiently in an AOS with different dielectric interfaces while keeping the same atmospheric and oceanic conditions. For the first time, we present the downward radiance field in an ocean with a sinusoidal ocean wave.

Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems I Monte Carlo method

We have developed a powerful 3D Monte Carlo code, as part of the Radiance in a Dynamic Ocean (RaDyO) project, which can compute the complete effective Mueller matrix at any detector position in a completely inhomogeneous turbid medium, in particular, a coupled atmosphere-ocean system. The light source can be either passive or active. If the light source is a beam of light, the effective Mueller matrix can be viewed as the complete impulse response Green matrix for the turbid medium. The impulse response Green matrix gives us an insightful way to see how each region of a turbid medium affects every other region. The present code is validated with the multicomponent approach for a plane-parallel system and the spherical harmonic discrete ordinate method for the 3D scalar radiative transfer system. Furthermore, the impulse response relation for a box-type cloud model is studied. This 3D Monte Carlo code will be used to generate impulse response Green matrices for the atmosphere and ocean, which act as inputs to a hybrid matrix operator-Monte Carlo method. The hybrid matrix operator-Monte Carlo method will be presented in part II of this paper.

Multiple-Scattering Formulation of Pulsed Beam Waves in Hydrometeors and Its Application to Millimeter-Wave Weather Radar

This letter deals with the backscattering of millimeter pulsed beam waves from hydrometeors. A new approach is presented for a solution of time-dependent three-dimensional vector radiative transfer equation for the Stokes vectors to study the multiple-scattering effects of beam waves on radar echoes. General solutions for beam waves are derived in an integral form without any approximation. They are given in numerically tractable forms representing the scattering process in the space and time domain. Time-dependent second-order solutions for radar echoes of pulsed beam waves are straightforwardly obtained to predict multiple-scattering effects depending on the variation of an incident beam size. It is shown that the inhomogeneity of the radial direction of beam waves causes the mode coupling of waves between the azimuth directions in the scattering matrix, and that the mode coupling depends on the ratio of the incident beam size to the total mean free path length of the medium

Scattering and emission from inhomogeneous vegetation canopy and alien target beneath by using three-dimensional vector radiative transfer (3D-VRT) equation

To solve the 3D-VRT equation for the model of spatially inhomogeneous scatter media, the finite enclosure of the scatter media is geometrically divided, in both vertical z and transversal ( x , y ) directions, to form very thin multi-boxes. The zeroth order emission, first-order Mueller matrix of each thin box and an iterative approach of high-order radiative transfer are applied to derive high-order scattering and emission of whole inhomogeneous scatter media. Numerical results of polarized brightness temperature at microwave frequency and under different radiometer resolutions from inhomogeneous scatter model such as vegetation canopy and alien target beneath canopy are simulated and discussed.

An Approach of Three-Dimensional Vector Radiative Transfer (3-D–VRT) Equation for Inhomogeneous Scatter Media

To solve a three-dimensional vector radiative transfer (3-D-VRT) equation for the model of spatially inhomogeneous scatter media, the finite enclosure of the scatter media is geometrically divided, in both the vertical z and horizontal (x,y) directions, to form very thin multiboxes. The zeroth-order emission, first-order Mueller matrix of each thin box, and an iterative approach of high-order radiative transfer are applied to deriving high-order scattering and emission of whole inhomogeneous scatter media. Numerical results of polarized brightness temperature at microwave frequency from an inhomogeneous scatter model such as vegetation canopy are calculated and discussed.

A three-dimensional vector radiative transfer (3D-VRT) equation for inhomogeneous scatter media *

Abstract To solve 3D-VRT equation for the model of spatially inhomogeneous scatter media, the finite enclosure of the scatter media is geometrically divided, in both the vertical z and horizontal (x, y) directions, to form very thin multi-boxes. The zero-th order emission, first-order Mueller matrix of each thin box and an iterative approach of high-order radiative transfer are applied to deriving high-orderscattering and emission of whole inhomogeneous scatter media. Numerical results of polarized brightness temperature at microwave frequency from inhomogeneous scatter model such as vegetation canopy are calculated and discussed.

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to a polarized laser beam: spatially varying reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to a polarized, Gaussian laser beam directed perpendicular to the surface is studied. The focus of this investigation is the 4×4, spatially varying reflection matrix that can be used to determine the normally backscattered radiation when the polarization of the incident radiation is specified. An inverse integral transform is used to construct the spatially varying reflection matrix from the generalized reflection matrix found in a previous study. The elements of this matrix depend on location specified by optical radius and azimuthal angle. The azimuthal variation is found by performing part of the inverse transform analytically, while the radial variation is described by five functions that are calculated numerically via an inverse Hankel transform. Benchmark numerical results for these five functions are presented, and the effects of beam radius and particle concentration are discussed. Expressions that describe the behavior of the reflection functions at small and large optical radii are developed, and comparisons are made to the one-dimensional and scalar situations. The scalar approximation fails to predict the three-dimensional effects produced by the polarized beam, and even when the incident radiation is unpolarized, the error in the scalar reflection function can be as high as 20%.

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to spatially varying polarized radiation: generalized reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite medium exposed to spatially varying, polarized radiation is studied. The problem is to determine the generalized reflection matrix for a multiple scattering medium characterized by a 4×4 scattering matrix. A double integral transform is used to convert the three-dimensional vector radiative transfer equation to a one-dimensional form, and a modified Ambarzumian's method is then applied to derive a nonlinear integral equation for the generalized reflection matrix. The spatially varying backscattered radiation for an arbitrarily polarized incident beam can be found from the generalized reflection matrix. For Rayleigh scattering and normal incidence and emergence, the generalized reflection matrix is shown to have five non-zero elements. Benchmark results for these five elements are presented and compared to asymptotic results. When the incident radiation is polarized, the vector approach used in this study correctly predicts three-dimensional behavior, while the scalar approach does not. When the incident radiation is unpolarized, both the vector and scalar approaches predict a two-dimensional distribution of the intensity, but the error in the scalar prediction can be as high as 20%.