Two-Stream Approximation to the Radiative Transfer Equation:A New Improvement and Comparative Accuracy with Existing Methods

Mathematical modeling of the interaction between solar radiation and the Earth's atmosphere is formalized by the radiative transfer equation(RTE), whose resolution calls for two-stream approximations among other methods. This paper proposes a new two-stream approximation of the RTE with the development of the phase function and the intensity into a third-order series of Legendre polynomials. This new approach, which adds one more term in the expression of the intensity and the phase function, allows in the conditions of a plane parallel atmosphere a new mathematical formulation of γparameters. It is then compared to the Eddington, Hemispheric Constant, Quadrature, Combined Delta Function and Modified Eddington, and second-order approximation methods with reference to the Discrete Ordinate(Disort) method(δ –128 streams), considered as the most precise. This work also determines the conversion function of the proposed New Method using the fundamental definition of two-stream approximation(F-TSA) developed in a previous work. Notably,New Method has generally better precision compared to the second-order approximation and Hemispheric Constant methods. Compared to the Quadrature and Eddington methods, New Method shows very good precision for wide domains of the zenith angle μ 0, but tends to deviate from the Disort method with the zenith angle, especially for high values of optical thickness. In spite of this divergence in reflectance for high values of optical thickness, very strong correlation with the Disort method(R ≈ 1) was obtained for most cases of optical thickness in this study. An analysis of the Legendre polynomial series for simple functions shows that the high precision is due to the fact that the approximated functions ameliorate the accuracy when the order of approximation increases, although it has been proven that there is a limit order depending on the function from which the precision is lost. This observation indicates that increasing the order of approximation of the phase function of the RTE leads to a better precision in flux calculations. However, this approach may be limited to a certain order that has not been studied in this paper.

Temperature and Water Vapor Weighting Functions from Radiative Transfer Equation with Surface Emissivity and Solar Reflectivity

Linearization of Radiative Transfer Equation (RTE) is the key step in physical retrieval of atmospheric temperature and moisture profiles from InfRared (IR) sounder observations. In this paper, the successive forms of temperature and water vapor mixing ratio component weighting functions are derived by applying one term variation method to RTE with surface emissivity and solar reflectivity contained. Retrivals of temperature and water vapor mixing ratio profiles from simulated Atmospheric Infrared Sounder (AIRS) observations with surface emissivity and solar reflectivity are presented.

An Easy Algorithm for Solving Radiative Transfer Equation in Clear Atmosphere

An accurate and rapid method for solving radiative transfer equation is presented in this paper. According to the fact that the multiple scattering component of radiance is less sensitive to the error of phase function than the single scattering component is,we calculate the multiple scattering component by using delta-Eddington approximation and the single scattering component by solving radiative transfer equation. On the ground, when multiple sattering component is small, for example, when the total optical depth T is small, the accurate radiance can be obtained with this method. For the need of the space remote sensing, the upward radiance at the top of the atmosphere is mainly studied, and an approximate expression is presented to correct the multiple scattering component. Compared with the more precise Gauss-Seidel method.the results from this method show an accuracy of better than 10% when zenith angle 0 < 50 掳 and T < 1. The computational speed of this method is, however, much faster than that of Gauss-Seidel method.

3D Radiative Transfer Equation Coupled with Heat Conduction Equation with Realistic Boundary Conditions Applied on Complex Geometries

This paper presents the solution of coupled radiative transfer equation with heat conduction equation in complex three-dimensional geometries. Due to very different time scales for both physics, the radiative problem is considered steady-state but solved at each time iteration of the transient conduction problem. The discrete ordinate method along with the decentered streamline-upwind Petrov-Galerkin method is developed. Since specular reflection is considered on borders, a very accurate algorithm has been developed for calculation of partition ratio coefficients of incident solid angles to the several reflected solid angles. The developed algorithms are tested on a paraboloid-shaped geometry used for example on concentrated solar power technologies.

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper,we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes.Different from the former scheme [J.Comput.Phys.285（2015）,265-279] on uniform meshes,in this paper,in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme（UGKS）,we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations.We can prove that the scheme is asymptotic preserving,and especially for the distorted quadrilateral meshes,a nine-point scheme [SIAM J.SCI.COMPUT.30（2008）,1341-1361] for the diffusion limit equations is obtained,which is naturally reduced to standard five-point scheme for the orthogonal meshes.The numerical examples on distorted meshes are included to validate the current approach.

On the combined heat transfer of natural convection, radiation and conduction in the non-gray semitransparent thermal fibrous insulation

A detailed numerical modeling is performed to investigate heat transfer in high-porous, high-temperature non-gray semitransparent silica insulation materials. Radiation between fibers, conduction within fibers and convection from the fibers to the surrounding fluid are considered. Macroscopic (porous media) modeling is used to determine the velocity, pressure and temperatures fields for fibrous insulation with a random packing geometry under natural convection. Based on a non-gray application of the solution to the radiative transfer equation, the value of the refractive index(n,m)is used to generate macroscopic average radiative properties such as extinction coefficient, scattering albedo and phase function. Key features of the macroscopic model include two-dimensional effects,non-gray radiative exchange, and the relaxation of the local thermodynamic non-equilibrium. The effectiveness of this numerical model is validated by the previous experimental data.

METHOD OF SOLVING THE RADIATIVE TRANSFER EQUATION FOR ATMOSPHERE-OCEAN SYSTEM AND ASYMPTOTIC STATE OF RADIANCE IN SEA WATERS

Urder the condition of regarding the atmosphere and sea waters as homogeneous mediaseparately and the interface between them as a flat, a complete method of solvnig the radi-ative transfer equation (RTE) for an atmosphere-ocean system is presented. Based on the de-veloped general solution to the RTE, the asymptotic state of the radiance and its propertiesare deduced. A computation example is provided, which is in good agreement with the the-oretical results. Some questions about practical applications of the method are discussed.

Iteration Method for Solving the Radiative Transfer Equation of an Atmosphere-Ocean System

Taking the atmosphere and ocean water as homogeneous respectively and regarding the ocean surface as a plane, a new method, the successive iteration method, for solving the radiative transfer equation (RTE) of the atmosphere-ocean system is presented. As a result, an expression of the azimuth integrated radiance in this system is developed.

PERTURBATION APPROACH TO SOLVING THE RADIATIVE TRANSFER EQUATION FOR OCEAN

A new analytical method, the perturbation approach, is presented, which can widely beused to solve the radiative transfer equation in Case 1 sea waters. A computation example isprovided, whose results are in good agreement with those obtained with other methods.Based on the general solution obtained in this way, an analytical relationship is found be-tween the shape of the asymptotic state of the radiance and the inherent optical propertiesof sea waters.

ANALYSIS OF A NUMERICAL METHOD FOR RADIATIVE TRANSFER EQUATION BASED BIOLUMINESCENCE TOMOGRAPHY

In the bioluminescence tomography （BLT） problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal＇s body surface. The BLT problem is ill-posed and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter to balance the accuracy and stability of approximate solutions. In this paper, a parameter-dependent coupled complex boundary method （CCBM） based Tikhonov regularization is applied to the BLT problem governed by the radiative transfer equation （RTE）. By properly adjusting the parameter in the Robin boundary condition, we achieve one important property： the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy. The discrete-ordinate finite-element method is used to compute numerical solutions. Numerical results are provided to illustrate the performance of the proposed method.

Quantitative Evaluation of Bioluminescence Imaging Based on Radiative Transfer Equation

B iolum inescence in aghg is a khd ofem erghg detection technology at cellular, m olecu lar and genetic level. The most popular b io lum m inescence in aghg model is diffusion approxin ation （DA）. However, because of the ill-posedness of the D A -based hverse problem and the instability of reconstruction algorithm s, the location accuracy of the reconstucted sources is low. Radiative transfer equation （RTE）, which considers the direction of the photon m igration and the effect of absorption and scattering in tissues, can accurately express the transmission ofbiolum hescent photns through the tissues. In this paper, we studied the biolum hescence inaging based on the RTE. 2D sinuiations were performed, and quantitative evaluation was given by the absolute source position error, the relative source area error and the m h in um bound hg box. The resu Its of the experin ent showed that the in aging quality based on R TE was beer than thatone based on D A.

Discrete Duality Finite Volume Discretization of the Thermal-P_(N) Radiative Transfer Equations on General Meshes

The discrete duality finite volume method has proven to be a practical tool for discretizing partial differential equations coming from a wide variety of areas of physics on nearly arbitrary meshes.The main ingredients of the method are:(1)use of three meshes,(2)use of the Gauss-Green theorem for the approximation of derivatives,(3)discrete integration by parts.In this article we propose to extend this method to the coupled grey thermal-P_(N) radiative transfer equations in Cartesian and cylindrical coordinates in order to be able to deal with two-dimensional Lagrangian approximations of the interaction of matter with radiation.The stability under a Courant-Friedrichs-Lewy condition and the preservation of the diffusion asymptotic limit are proved while the experimental second-order accuracy is observed with manufactured solutions.Several numerical experiments are reported which show the good behavior of the method.

The Radiation Loss of a Cylindrical Methane-Argon Plasma Arc

Calculation of the net radiation emitted by a CH4-Ar mixture, in a temperature range of 5,000-30,000 K with the assumption of local thermodynamic equilibrium （LTE）, is conducted. Continuum and line emissions are taken into account. The radiative transfer of each line is calculated by means of an escape factor depending on the shape and broadening of the line. Assuming a cylindrical, homogeneous, and isothermal plasma, the net emission coefficient is calculated for different pressures between 1 atm and 10 atm and arc radia of 0 mm to 1 mm. Results show that the argon presence in the CH4-Ar mixture has a significant effect on the total radiation emitted for the temperature above 17,000 K and the results for pure argon agree with those of BAUDER and EVANS.

天津市热环境时空演变及影响因素分析

随着城市化进程加快,城市热环境发生显著变化,对人类活动造成很大影响。为探究天津市城市热环境时空变化及其影响因素,基于2005—2020年四期Landsat遥感数据,利用辐射传输方程法反演地表温度,计算城市热岛比例指数,用标准差椭圆法分析城市热环境发展和布局,利用地表温度并结合土地利用分类研究城市地物覆盖类型与地表温度的响应关系,并采用地理探测器探究高程、绿地、建筑用地和水体对地表温度变化的影响差异。结果表明:天津市热岛足迹沿主干道由市内六区向环城四区发展,天津城市热环境呈现先增高后降低的趋势;由标准差椭圆分析可知,城市热岛呈现发展主轴维持在东北—东南方向,整体空间格局呈现放射状分布;天津市中心城区建筑用地面积占比逐年增加,水体面积占比逐年降低,各类型地物平均温度排序为:建筑用地>绿地>水体;地理探测器分析表明:NDBI和NDVI是影响地表温度变化的主要因子,NDVI和MNDWI对LST交互作用最强,是城市热环境驱动作用最强的因子组合。

基于Landsat-8影像的高原湖泊水体表面温度反演

目前,在高原环境下的水体表面温度反演成果相对较少,本文结合Landsat-8影像和辐射传输方程法对纳木错湖水体表面温度进行反演并验证其有效性。首先,计算研究区的地表比辐射率;其次,计算同温度下黑体的辐射亮度;最后,利用普朗克函数计算水体表面温度并基于MODIS地表温度产品对反演结果进行验证。实验结果表明,湖泊水体表面温度反演绝对误差最小值为0.449℃、最大值为1.685℃,均方根误差最小值为1.269℃、最大值为1.781℃,反演结果与MODIS温度产品的日均温结果较接近,夏季湖泊水体表面温度变化特征基本与纳木错气温一致。该方法可为后续高原湖泊水体水表温度反演研究提供一定的参考。

Recent progress in computational thermal radiative transfer

The equation describing the transfer of radiant energy in semitransparent media is radiative transfer equation. In three-dimensional semitransparent media, radiative intensity is a function of 7 dimensions, which can only be solved through the numerical method in most circumstances. Numerical simulation has become an important way in the study and application of the theory of thermal radiative transfer in semitransparent media. This paper reviews the recent progress of Chinese scholars in the field of computational thermal radiative transfer, and proposes some important subjects in this field for future study.

Four-stream Radiative Transfer Parameterization Scheme in a Land Surface Process Model

Accurate estimates of albedos are required in climate modeling. Accurate and simple schemes for radiative transfer within canopy are required for these estimates, but severe limitations exist. This paper developed a four-stream solar radiative transfer model and coupled it with a land surface process model. The radiative model uses a four-stream approximation method as in the atmosphere to obtain analytic solutions of the basic equation of canopy radiative transfer. As an analytical model, the four-stream radiative transfer model can be easily applied efficiently to improve the parameterization of land surface radiation in climate models. Our four-stream solar radiative transfer model is based on a two-stream short wave radiative transfer model. It can simulate short wave solar radiative transfer within canopy according to the relevant theory in the atmosphere. Each parameter of the basic radiative transfer equation of canopy has special geometry and optical characters of leaves or canopy. The upward or downward radiative fluxes are related to the diffuse phase function, the G-function, leaf reflectivity and transmission, leaf area index, and the solar angle of the incident beam. The four-stream simulation is compared with that of the two-stream model. The four-stream model is proved successful through its consistent modeling of canopy albedo at any solar incident angle. In order to compare and find differences between the results predicted by the four- and two-stream models, a number of numerical experiments are performed through examining the effects of different leaf area indices, leaf angle distributions, optical properties of leaves, and ground surface conditions on the canopy albedo. Parallel experiments show that the canopy albedos predicted by the two models differ significantly when the leaf angle distribution is spherical and vertical. The results also show that the difference is particularly great for different incident solar beams. One additional experiment is carried out to evaluate the simulations of the BATS land surface model coupled with the two- and four-stream radiative transfer models. Station observations in 1998 are used for comparison. The results indicate that the simulation of BATS coupled with the four-stream model is the best because the surface absorbed solar radiation from the four-stream model is the closest to the observation.

Discrete unified gas kinetic scheme for multiscale anisotropic radiative heat transfer

In this work,a discrete unified gas kinetic scheme(DUGKS)is developed for radiative transfer in anisotropic scattering media.The method is an extension of a previous one for isotropic radiation problems[1].The present scheme is a finite-volume discretization of the anisotropic gray radiation equation,where the anisotropic scattering phase function is approximated by the Legendre polynomial expansion.With the coupling of free transport and scattering processes in the reconstruction of the flux at cell interfaces,the present DUGKS has the nice unified preserving properties such that the cell size is not limited by the photon mean free path even in the optical thick regime.Several one-and two-dimensional numerical tests are conducted to validate the performance of the present DUGKS,and the numerical results demonstrate that the scheme is a reliable method for anisotropic radiative heat transfer problems.

TWO-DIMENSIONAL THERMAL RADIATIVE TRANSFER FROM A LAYER OF SPATIALLY INHOMOGENEOUS RANDOM MEDIUM

An approach to solve the two-dimensional vector thermal radiative transfer equation foralayer of spatially inhomogeneous random scattering medium is developed. By using theFourier transformation and matrix expression, the one-dimensional radiative transfer equationof the spectrum of the Stokes vector is derived. Employing an iterative approach in theapproximation of small albedo, the zeroth-and first-order solutions to the integral radiativetransfer equation are obtained. Numerical results of polarized brightness temperature ofatmospheric precipitation, cloud, and vegetation canopy with a Gaussian fractional volumes,respectively, are obtained. The functional dependence of brightness temperature on the in-homogeneous scale, frequency, polarization, dielectric properties of tlie underground medium,etc. are discussed.

RADIATIVE TRANSFER THEORY FOR STRONGLY FLUCTUATING, CONTINUOUS RANDOM MEDIA

Employing the strong fluctuation theory, the radiative transfer equation for strongly fluctuating, continuous random media; and the associated phase matrix and scattering coefficient are obtained. By using the Gaussian quadrature and the eigenvalue-eigenvector approaches, the vector thermal radiative transfer equation for a layer of random medium is solved and is favorably matched with the experimental data of snowfield in remote sensing. The comparison with the conventional theory for weak fluctuation is discussed.