Chebyshev collocation method for solving the radiative transfer equation

A Chebyshev collocation method is presented for solving the spherical harmonics approximation to the equation of radiative transfer in a plane-parallel, homogeneous medium. As a result of test computations performed for Rayleigh and Henyey-Greenstein phase functions, it was found that the proposed method can be used to solve transfer problems stably and accurately, even for an optically thick case, and that the accuracy of computations is greatly improved by increasing the order of Chebyshev polynomials in expansion.     

Three-dimensional radiative transfer using a fourier-transform matrix-operator method

The three-dimensional equation of transfer for a scattering medium with planar geometry is solved by using a spatial Fourier transform and extending matrix-operator techniques developed previously for the one-dimensional equation. Doubling and adding algorithms were derived by means of an interaction principle for computing the fourier-transformed radiation field. The resulting expressions fully describe the radiative transfer process in a scattering medium, inhomogeneous in the x-, y- and z-directions, illuminated from above by an arbitrarily general intensity field and bounded from below by a surface with completely general reflection properties.     

Time shift and superposition method for solving transient radiative transfer equation

Due to the long-time transient response of pulse irradiation, the computational time required for solving transient radiative transfer (TRT) is often very long, especially for the case in which the boundary is subjected to continuous pulse train and the geometry is complicated. In addition, sometimes, before actual experiments are carried out, a suitable pulse shape or type often needs to be selected by numerical simulation and comparison. Because the numerical solution of TRT needs to be repeated many times, the selection processes is very time-consuming. In this paper, by considering that the TRT equation and its initial and boundary conditions are linear, a time shift and superposition method is developed for solving TRT equation in non-emitting media, in which only the transient response of a short square pulse needs to be solved, and the solution of TRT under any pulse shape can be constructed by time shift and then superposition using the basis solution of the short square pulse. Three numerical examples are studied to illustrate the peformance of the superposition method in solving TRT problems. The results show that the superposition is effective, accurate and very suitable for solving TRT in the medium subjected to a series of pulse train.     

Variational method for solving radiative transfer problems

The variational principle is used to solve two problems of radiative transfer. The first one is the temperature distribution and radiative heat flux for a plane layer of ceramic material. While the second is the calculation of the integral blankness degree of a sphere filled with dust of an arc steel-melting furnace. Numerical results obtained shows good agreement with the published data.     

A vector radiative transfer model of coupled ocean-atmosphere system using matrix-operator method for rough sea-surface

A vector radiative transfer model termed PCOART has been developed for the coupled ocean-atmosphere system, using the matrix-operator (or adding-doubling) method, which considers the rough sea-surface. The theoretical formulations of the solution of the vector radiative transfer equation of the coupled ocean-atmosphere system, and the reflection-transmission matrices and internal radiation sources for rough sea surface are described. The model intercomparison is performed for several radiative transfer problems in the atmosphere and ocean, and the results show that PCOART can exactly predict the radiance fields for both flat and rough sea surface. Also, the polarizing remote sensing data from POLDER is used to test the capacity of PCOART to simulate the polarization radiance at the top-of-atmosphere, which shows that PCOART can perfectly reproduce the linear polarization reflectance measured by POLDER. PCOART can not only simulate the total radiance field in the coupled ocean-atmosphere system with wind-induced rough sea surface but also predict the polarization radiance field both in the atmosphere and in the ocean, which can serve as a good tool for the ocean optics and ocean color remote sensing communities.     

Algorithm for solving the equation of radiative transfer in the frequency domain

We present an algorithm that provides a frequency-domain solution of the equation of radiative transfer (ERT) for heterogeneous media of arbitrary shape. Although an ERT is more accurate than a diffusion equation, no ERT code for the widely employed frequency-domain case has been developed to date. In this work the ERT is discretized by a combination of discrete-ordinate and finite-volume methods. Two numerical simulations are presented.     

Modified spherical harmonics method for solving the radiative transport equation

A new effective approach to solving the three-dimensional radiative transport equation with an arbitrary phase function is proposed. The solution depends on eigenvectors and eigenvalues of several symmetrical tridiagonal matrices of infinite size. The matrices must be truncated and diagonalized numerically. Then, given eigenvectors and eigenvalues of these matrices, the dependence of the solution on position and direction is found analytically. The approach is based on expanding the angular part of the specific intensity in q-dependent spherical functions for each spatial Fourier component characterized by the vector q. Apart from the truncation of the matrices, no other approximations are made.     

Modified spherical harmonics method for solving the radiative transport equation

A new effective approach to solving the three-dimensional radiative transport equation with an arbitrary phase function is proposed. The solution depends on eigenvectors and eigenvalues of several symmetrical tridiagonal matrices of infinite size. The matrices must be truncated and diagonalized numerically. Then, given eigenvectors and eigenvalues of these matrices, the dependence of the solution on position and direction is found analytically. The approach is based on expanding the angular part of the specific intensity in q-dependent spherical functions for each spatial Fourier component characterized by the vector q. Apart from the truncation of the matrices, no other approximations are made.     

Analytical approach for solving the radiative transfer equation in two-dimensional layered media

This study presents an analytical approach for obtaining Green's function of the two-dimensional radiative transfer equation to the boundary-value problem of a layered medium. A conventional Fourier transform and a modified Fourier series which is defined in a rotated reference frame are applied to derive an analytical solution of the radiance in the transformed space. The Monte Carlo method was used for a successful validation of the derived solutions.     

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.     

Multi-coupled single scattering method of solving vector radiative transfer equations

<正>A new method of multi-coupled single scattering(MCSS) for solving a vector radiative transfer equation is developed and made public on Internet.Recent solutions from Chandrasekhar’s X-Y method is used to validate the MCSS’s result,which shows high precision.The MCSS method is theoretically simple and clear,so it can be easily and credibly extended to the simulation of aerosol/cloud atmosphere’s radiative properties,which provides effective support for research into polarized remote sensing.     

Evaluation of numerical scattering in finite volume method for solving radiative transfer equation by a central laser incidence model

The numerical scattering caused by spatial discretization in finite volume method is discussed. Based on an analysis of the generation process of numerical scattering, a physical model of central laser incidence to a two-dimensional rectangle containing semitransparent medium is established to validate the numerical scattering, with Monte Carlo method as benchmark, in which numerical scattering does not exist. Numerical scattering will be affected by spatial grid number, spatial differential schemes and spectral absorption coefficient. With the spatial grid number increasing, numerical scattering will be decreased. The accuracy of the diamond scheme is the highest, and the exponential scheme is a bit lower, the lowest accuracy of the three schemes is the step scheme. The tendency of numerical scattering is reverse, i.e., the step scheme produces minimum numerical scattering, and exponential scheme produces more, while the diamond scheme produces maximum among three methods. When the bias of absorption efficient is high, the numerical scattering cannot be eliminated only by increasing the grid number. If we set the direction of laser incidence as central axis, it can be seen that numerical scattering distributed symmetry along the axis, which can be called as symmetrical cross-scattering. All of the three schemes show symmetrical cross-scattering.     

An application of the K-integrals for solving the radiative transfer equation with Fresnel boundary conditions

We introduce the reader to an approximate method of solving the transport equation which was developed in the context of neutron thermalisation by Kladnik and Kuscer in 1962 [Kladnik R, Kuscer I. Velocity dependent Milne's problem. Nucl Sci Eng 1962;13:149]. Essentially the method is based upon two special weighted integrals of the one-dimensional transport equation which are valid regardless of the boundary conditions, and any solution must satisfy these integral relationships which are called the K-integrals. To obtain an approximate solution to the transport equation we turn the argument around and insist that any approximate solution must also satisfy the K-integrals. These integrals are particularly useful when the problem under consideration cannot be solved easily by analytic methods. It also has the marked advantage of being applicable to problems where there is energy exchange in a collision and anisotropy of scattering. To establish the feasibility of the method we obtain a number of approximate solutions using the K-integral method for problems to which we have exact analytical solutions. This enables us to validate the method. It is then applied to a new problem that has not yet been solved; namely the calculation of the discontinuity in the scalar intensity at the boundary between two optically dissimilar materials.     

Discontinuous spectral element method for solving radiative heat transfer in multidimensional semitransparent media

A discontinuous spectral element method (DSEM) is presented to solve radiative heat transfer in multidimensional semitransparent media. This method is based on the general discontinuous Galerkin formulation. Chebyshev polynomial is used to build basis function on each element and both structured and unstructured elements are considered. The DSEM has properties such as hp-convergence, local conservation and its solutions are allowed to be discontinuous across interelement boundaries. The influences of different schemes for treatment of the interelement numerical flux on the performance of the DSEM are compared. The p-convergence characteristics of the DSEM are studied. Four various test problems are taken as examples to verify the performance of the DSEM, especially the performance to solve the problems with discontinuity in the angular distribution of radiative intensity. The predicted results by the DSEM agree well with the benchmark solutions. Numerical results show that the p-convergence rate of the DSEM follows exponential law, and the DSEM is stable, accurate and effective to solve multidimensional radiative transfer in semitransparent media.     

On the use of a meshless method for solving radiative transfer with the discrete ordinates formulations

A meshless method is presented for solving the radiative transfer equation in the discrete ordinates approach. It is shown that the primitive variables formulation is unstable for low values of the absorption coefficient while the even parity formulation is always stable and accurate.     

Direct solution of the radiative transfer equation for plane-parallel atmospheres

A method is described for solving the monochromatic radiative transfer equation for the case of inhomogeneous, plane-parallel scattering and absorbing atmospheres illuminated by external as well as internal sources. The solution procedure, which is based on a series expansion of the radiation intensity with respect to the angular and spatial coordinates, is analytical in nature and can thus be implemented on small computing facilites. Test calculations were performed for isotropic and Rayleigh scattering atmospheres of various optical thicknesses and single scattering albedos. The results coincide well with data from other methods given in the literature.     

A direct method for the integration of the equation of radiative transfer in a turbid atmosphere

In this paper we apply a numerical method on the equation of radiative transfer in a turbid atmosphere. The solution is obtained by means of direct integration of the equation of radiative transfer without any circuitous series development.     

Sinc-collocation method for solving the Blasius equation

Sinc-collocation method is applied for solving Blasius equation which comes from boundary layer equations. It is well known that sinc procedure converges to the solution at an exponential rate. Comparison with Howarth and Asaithambi's numerical solutions reveals that the proposed method is of high accuracy and reduces the solution of Blasius' equation to the solution of a system of algebraic equations.     

粗糙面上粒子层矢量辐射传输方程的二阶迭代解法

为了探究典型粗糙面上随机粒子层中能量传输的多次散射机制,提出了一种基于矢量辐射传输方程的建模二阶计算方法.该建模方法将建模场景(粗糙面上粒子层)在高度维(Z轴)划分为多个传输散射层,基于矢量辐射传输理论中的一阶迭代散射解,利用典型粗糙面的半经验半解析方法,求解出整个场景的二阶迭代散射解.同时,研究粒子层能量在粒子与粒子间的多次散射机制,以及粒子与地表粗糙面间的多次散射机制.数值结果表明,该二阶迭代解法相较于矢量辐射传输方程的一阶迭代散射解,能够更完整地探究互作用的散射特性,且可从能量传输角度解译建模场景中物体间的相干作用,从而可用于植被地物环境下的多次散射机制的解析以及散射系数变化趋势的预估.     

An efficient and robust reconstruction method for optical tomography with the time-domain radiative transfer equation

An efficient and robust method based on the complex-variable-differentiation method (CVDM) is proposed to reconstruct the distribution of optical parameters in two-dimensional participating media. An upwind-difference discrete-ordinate formulation of the time-domain radiative transfer equation is well established and used as forward model. The regularization term using generalized Gaussian Markov random field model is added in the objective function to overcome the ill-posed nature of the radiative inverse problem. The multi-start conjugate gradient method was utilized to accelerate the convergence speed of the inverse procedure. To obtain an accurate result and avoid the cumbersome formula of adjoint differentiation model, the CVDM was employed to calculate the gradient of objective function with respect to the optical parameters. All the simulation results show that the CVDM is efficient and robust for the reconstruction of optical parameters.