Spectral element method for vector radiative transfer equation

A spectral element method (SEM) is developed to solve polarized radiative transfer in multidimensional participating medium. The angular discretization is based on the discrete-ordinates approach, and the spatial discretization is conducted by spectral element approach. Chebyshev polynomial is used to build basis function on each element. Four various test problems are taken as examples to verify the performance of the SEM. The effectiveness of the SEM is demonstrated. The h and the p convergence characteristics of the SEM are studied. The convergence rate of p-refinement follows the exponential decay trend and is superior to that of h-refinement. The accuracy and efficiency of the higher order approximation in the SEM is well demonstrated for the solution of the VRTE. The predicted angular distribution of brightness temperature and Stokes vector by the SEM agree very well with the benchmark solutions in references. Numerical results show that the SEM is accurate, flexible and effective to solve multidimensional polarized radiative transfer problems.     

Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system

A vector radiative transfer model has been developed for a coupled atmosphere-ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.     

Radiative transfer solutions for coupled atmosphere ocean systems using the matrix operator technique

Accurate radiative transfer models are the key tools for the understanding of radiative transfer processes in the atmosphere and ocean, and for the development of remote sensing algorithms. The widely used scalar approximation of radiative transfer can lead to errors in calculated top of atmosphere radiances. We show results with errors in the order of±8% for atmosphere ocean systems with case one waters. Variations in sea water salinity and temperature can lead to variations in the signal of similar magnitude. Therefore, we enhanced our scalar radiative transfer model MOMO, which is in use at Freie Universität Berlin, to treat these effects as accurately as possible. We describe our one-dimensional vector radiative transfer model for an atmosphere ocean system with a rough interface. We describe the matrix operator scheme and the bio-optical model for case one waters. We discuss some effects of neglecting polarization in radiative transfer calculations and effects of salinity changes for top of atmosphere radiances. Results are shown for the channels of the satellite instruments MERIS and OLCI from 412.5 nm to 900 nm.     

The method of lines solution of discrete ordinates method for non-grey media

A radiation code based on method of lines (MOL) solution of discrete ordinates method (DOM) for radiative heat transfer in non-grey absorbing-emitting media was developed by incorporation of a gas spectral radiative property model, namely wide band correlated-k (WBCK) model, which is compatible with MOL solution of DOM. Predictive accuracy of the code was evaluated by applying it to 1-D parallel plate and 2-D axisymmetric cylindrical enclosure problems containing absorbing-emitting medium and benchmarking its predictions against line-by-line solutions available in the literature. Comparisons reveal that MOL solution of DOM with WBCK model produces accurate results for radiative heat fluxes and source terms and can be used with confidence in conjunction with computational fluid dynamics codes based on the same approach.     

Radiation field of an optically finite homogeneous atmosphere with internal sources

The equation of radiative transfer in an optically finite homogeneous atmosphere with different internal sources is solved using the method of kernel approximation the essence of which is to approximate the kernel in the equation for the Sobolev resolvent function by a Gauss-Legendre sum. This approximation allows to solve the equation exactly for the resolvent function while the solution is a weighted sum of exponents. Since the resolvent function is closely connected with the Green function of the integral radiative transfer equation, the radiation field for different internal sources can be found by simple integration. In order to simplify the obtained formulas we have defined the x and y functions as the generalization of the well-known Ambarzumian-Chandrasekhar X and Y functions.For some types of internal sources the package of codes in Fortran-77 can be found at http://www.aai.ee/∼viik/HOMOGEN.FOR.     

On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media

Light transport in graded index media follows a curved trajectory determined by Fermat's principle. Besides the effect of variation of the refractive index on the transport of radiative intensity, the curved ray trajectory will induce geometrical effects on the transport of polarization ellipse. This paper presents a complete derivation of vector radiative transfer equation for polarized radiation transport in absorption, emission and scattering graded index media. The derivation is based on the analysis of the conserved quantities for polarized light transport along curved trajectory and a novel approach. The obtained transfer equation can be considered as a generalization of the classic vector radiative transfer equation that is only valid for uniform refractive index media. Several variant forms of the transport equation are also presented, which include the form for Stokes parameters defined with a fixed reference and the Eulerian forms in the ray coordinate and in several common orthogonal coordinate systems.     

The spatial polarization distribution over the dome of the sky for abnormal irradiance of the atmosphere

The paper deals with the polarized radiative transfer within a slab irradiated by a collimated infinitely wide beam of arbitrary polarized light. The efficiency of the proposed analytical solution lies in the assumption that the complete vectorial radiative transfer solution is the superposition of the most anisotropic and smooth parts, computed separately. The vectorial small-angle modification of the spherical harmonics method is used to evaluate the anisotropic part, and the vectorial discrete ordinates method is used to obtain the smooth one. The azimuthal expansion is used in order to describe the light field spatial distribution for the case of abnormal irradiance and to obtain some known neutral points in the sky especially useful for polarized remote sensing of the atmosphere.     

On vector radiative transfer equation in curvilinear coordinate systems

The differential operator of polarized radiative transfer equation is examined in case of homogeneous medium in Euclidean three-dimensional space with arbitrary curvilinear coordinate system defined in it. This study shows that an apparent rotation of polarization plane along the light ray with respect to the chosen reference plane for Stokes parameters generally takes place, due to purely geometric reasons. Analytic expressions for the differential operator of transfer equation dependent on the components of metric tensor and their derivatives are found, and the derivation of differential operator of polarized radiative transfer equation has been made a standard procedure. Considerable simplifications take place if the coordinate system is orthogonal.     

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.     

非球形生物气溶胶对偏振光的多次散射特性研究

基于回转椭球模型和有限长圆柱模型,采用T矩阵方法研究了非球形生物气溶胶的单次散射特性,计算了鼠疫耶尔森氏杆菌、土拉热杆菌二种生物气溶胶对氦氖激光的单次相矩阵、单次散射反照率以及不对称因子。根据矢量辐射传输理论,研究了激光在生物气溶胶中传输的偏振散射特性,基于累加-倍加法(adding-doubling method)求解矢量辐射传输方程,并计算了非球形生物气溶胶对激光多次散射的斯托克斯参量。计算结果表明,生物气溶胶的尺寸和形状对光的极化更为敏感,因此在利用激光进行生物气溶胶微观特性探测和反演时,利用激光的偏振散射特性为非常有效的方法。     

Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices

We present an algorithm for polarized radiative transfer in a vertically stratified system consisting of two plane-parallel media with different refractive indices. It is based on the discrete ordinate method and includes multiple elastic scattering, thermal radiation, Fresnel reflection and transmission, incident parallel-beam or isotropic radiation at the top of the upper medium and bidirectional reflection at the bottom of the lower medium. Comparisons with results from Monte Carlo simulations show that the discrete-ordinate code provides accurate results for all four elements of the Stokes vector (I, Q, U, and V) at a speed that is orders of magnitude faster.     

地表双向反射对天基矢量辐射探测的影响分析

矢量辐射传输方程定量描述了辐射在地表-大气耦合介质中的传输过程,是定量遥感的基础.在处理辐射和离散介质相互作用时,如何处理多次散射、辐射偏振效应和耦合地表模型是研究的重点,直接影响定量化遥感反演的精度.文中基于逐次散射近似法求解了矢量辐射传输方程,求解过程中耦合典型地表的非偏双向反射（BRDF）模型和偏振双面反射（BPDF）模型.采用相对误差因子定量分析了地表双向反射效应和大气偏振效应对天基矢量辐射的影响.为进一步研究地表-大气耦合介质系统的偏振特性及地表大气参数的定量遥感反演提供理论支持. 关键词： 矢量辐射传输方程 逐次散射近似 定量遥感 偏振遥感     

A linearized vector radiative transfer model for atmospheric trace gas retrieval

We present a plane parallel radiative transfer model for polarized light, that provides the intensity vector as well as the derivatives of the four Stokes parameters with respect to atmospheric trace gas profiles. These derivatives are essential for retrieval of height resolved trace gas information from satellite measurements of backscattered sunlight. The model uses the Gauss-Seidel iteration technique for solving the radiative transfer equation. For the first time, the forward-adjoint radiative perturbation theory is applied for the linearization of a radiative transfer model including polarization. The accuracy of the model is better than 0.025% for all four Stokes parameters and better than 0.03% for the derivatives.     

The non-equilibrium Marshak wave problem

An analytic solution to a particular Marshak wave problem is given. The radiative transfer model used is the gray, non-equilibrium diffusion approximation which allows the radiation and material fields to be out of equilibrium. This solution should be useful as a reference problem for validating time-dependent radiative transfer computer codes, as well as investigating the convergence, as a function of space and time step size, for such codes. The coupling of the radiation field to the material field in a multigroup code, a difficult numerical problem, can also be tested against this solution. Typical numerical results are given for surface quantities, integral quantities, and the distribution of radiative energy and material temperature as a function of space and time.     

Three-dimensional polarized Monte Carlo atmospheric radiative transfer model (3DMCPOL): 3D effects on polarized visible reflectances of a cirrus cloud

A polarized atmospheric radiative transfer model for the computation of radiative transfer inside three-dimensional inhomogeneous mediums is described. This code is based on Monte Carlo methods and takes into account the polarization state of the light. Specificities introduced by such consideration are presented. After validation of the model by comparisons with adding-doubling computations, examples of reflectances simulated from a synthetic inhomogeneous cirrus cloud are analyzed and compared with reflectances obtained with the classical assumption of a plane parallel homogeneous cloud (1D approximation). As polarized reflectance is known to saturate for optical thickness of about 3, one could think that they should be less sensitive to 3D effects than total reflectances. However, at high spatial resolution (80 m), values of polarized reflectances much higher than the ones predicted by the 1D theory can be reached. The study of the reflectances of a step cloud shows that these large values are the results of illumination and shadowing effects similar to those often observed on total reflectances. In addition, we show that for larger spatial resolution (10 km), the so-called plane-parallel bias leads to a non-negligible overestimation of the polarized reflectances of about 7–8%.     

A comparison of fast codes for the evaluation of the Voigt profile function

Several fast codes for the evaluation of individual Voigt profile functions are compared for accuracy and speed. An apparent error in one of these codes is discussed, along with a suggested correction. A general comment regarding the use of such codes in line-by-line radiative transfer programs is made.     

A probabilistic study of the influence of parameter uncertainty on solutions of the radiative transfer equation

The influence of uncertainty in the absorption and scattering coefficients on the solution and associated parameters of the radiative transfer equation is studied using polynomial chaos theory. The uncertainty is defined by means of uniform and log-uniform probability distributions. By expanding the radiation intensity in a series of polynomial chaos functions we may reduce the stochastic transfer equation to a set of coupled deterministic equations, analogous to those that arise in multigroup neutron transport theory, with the effective multigroup transfer scattering coefficients containing information about the uncertainty. This procedure enables existing transport theory computer codes to be used, with little modification, to solve the problem. Applications are made to a transmission problem and a constant source problem in a slab. In addition, we also study the rod model for which exact analytical solutions are readily available. In all cases, numerical results in the form of mean, variance and sensitivity are given that illustrate how absorption and scattering coefficient uncertainty influences the solution of the radiative transfer equation.     

Spin correlation coefficients for neutron-proton radiative capture

We use a theoretical model for deuteron photodisintegration below the pion production threshold, which has previously been used to calculate the observables for which experimental data is available, in order to calculate the spin correlation coefficients for radiative capture of a polarized neutron beam by a polarized proton target. We give results for the coefficients whose measurement is possible in the near future and explain how such experimental results could improve our understanding of the reaction mechanism. We also comment on exciting measurements of the neutron analyzing power at very low energies.     

O(αs) corrections to polarized top quark decay into a charged Higgs boson: $t(\uparrow)\to{H}^+ + b$

We calculate the O(α_s) radiative corrections to polarized top quark decay into a charged Higgs boson and a massive bottom quark in two variants of the two-Higgs-doublet model. The radiative corrections to the polarization asymmetry of the decay may become as large as 25%. We provide analytical formulae for the unpolarized and polarized rates for m_b≠0 and for m_b=0. For m_b=0 our closed-form expressions for the unpolarized and polarized rates become rather compact. PACS 12.38.Bx; 13.88.+e; 14.65.Ha; 14.80.Cp     

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%.