A perturbation solution of the radiative transfer equation in a differentially moving atmosphere

We present a new technique for solving the radiative transfer equation in a differentially moving atmosphere. The method is based on a pertubation of the solution of the transfer problem in a static atmosphere. The perturbation technique may be applied with any method for solving the static atmosphere problem and leads to significant reductions in computer time and storage requirements.The method is flexible and may be used to solve problems involving depth dependence in any of the parameters of the transfer equation.     

A deficiency problem of the least squares finite element method for solving radiative transfer in strongly inhomogeneous media

The accuracy and stability of the least squares finite element method (LSFEM) and the Galerkin finite element method (GFEM) for solving radiative transfer in homogeneous and inhomogeneous media are studied theoretically via a frequency domain technique. The theoretical result confirms the traditional understanding of the superior stability of the LSFEM as compared to the GFEM. However, it is demonstrated numerically and proved theoretically that the LSFEM will suffer a deficiency problem for solving radiative transfer in media with strong inhomogeneity. This deficiency problem of the LSFEM will cause a severe accuracy degradation, which compromises the performance of the LSFEM too much and makes it not a good choice to solve radiative transfer in strongly inhomogeneous media. It is also theoretically proved that the LSFEM using the one dimensional linear element is equivalent to a second order form of radiative transfer equation discretized by the central difference scheme.     

Towards linearization of atmospheric radiative transfer in spherical geometry

We present a general approach for the linearization of radiative transfer in a spherical planetary atmosphere. The approach is based on the forward-adjoint perturbation theory. In the first part we develop the theoretical background for a linearization of radiative transfer in spherical geometry. Using an operator formulation of radiative transfer allows one to derive the linearization principles in a universally valid notation. The application of the derived principles is demonstrated for a radiative transfer problem in simplified spherical geometry in the second part of this paper. Here, we calculate the derivatives of the radiance at the top of the atmosphere with respect to the absorption properties of a trace gas species in the case of a nadir-viewing satellite instrument.     

One-dimensional solar radiative transfer: Perturbation approach and its application to independent-pixel calculations for realistic cloud fields

The radiative transfer perturbation theory (RTPT), which has already been introduced in atmospheric radiative transfer several years ago, is applied to cloud related problems. The RTPT requires the solution of the radiative transfer equation in the forward and the adjoint mode. The basic principles of this technique are presented as well as its extensions to isotropic surface reflection and its conjunction with the Hermite interpolation. This set of methods is applied to different atmospheric conditions including realistic cloud scenes. The results are compared with the usual (forward) independent-pixel calculations with respect to errors of individual pixels and domain-averaged values. The RTPT turns out to be sufficiently accurate in the case the clouds’ internal vertical variations remain moderate. It is also shown that, depending on the specific radiative transfer problem, the RTPT can offer some advantages on computational speed. However, the limitations of the RTPT with regard to realistic clouds are addressed as well.     

Finite element method for radiation heat transfer in multi-dimensional graded index medium

In graded index medium, ray goes along a curved path determined by Fermat principle, and curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectories, a finite element method based on discrete ordinate equation is developed to solve the radiative transfer problem in a multi-dimensional semitransparent graded index medium. Two particular test problems of radiative transfer are taken as examples to verify this finite element method. The predicted dimensionless net radiative heat fluxes are determined by the proposed method and compared with the results obtained by finite volume method. The results show that the finite element method presented in this paper has a good accuracy in solving the multi-dimensional radiative transfer problem in semitransparent graded index medium.     

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.     

Higher order perturbation theory applied to radiative transfer in non-plane-parallel media

Radiative transfer in non-plane-parallel media is a very challenging problem, which is currently the subject of concerted efforts to develop computational techniques which may be used to tackle different tasks. In this paper we develop the full formalism for another technique, based on radiative perturbation theory. With this approach, one starts with a plane-parallel ‘base model’, for which many solution techniques exist, and treat the horizontal variability as a perturbation. We show that under the most logical assumption as to the base model, the first-order perturbation term is zero for domain-average radiation quantities, so that it is necessary to go to higher order terms. This requires the computation of the Green's function. While this task is by no means simple, once the various pieces have been assembled they may be re-used for any number of perturbations—that is, any horizontal variations.     

A new inverse approach for the equivalent gray radiative property of a non-gray medium using a modified zonal method and the complex-variable-differentiation method

Non-gray radiative properties of an absorbing, emitting, non-gray participating medium significantly increase the difficulty of solving the radiative transfer equation. This paper presents a new inverse approach for the equivalent gray radiative property of a non-gray medium. In this approach, the unknown equivalent gray radiative properties are treated as the optimization variables, and the errors to be minimized are the differences between the calculated temperatures and the measured ones. The measured data are simulated by solving the direct problem, in which a modified zonal method together with the Edwards exponential wide-band model is employed. In the inverse problem, the sensitivity coefficients are first calculated by the complex-variable-differentiation method, and then the least-square method and the Newton-Raphson iterative method are employed to minimize the target function. The effectiveness and efficiency of the inverse problem are demonstrated in an example, and another case is given to show the accuracy and potential of the proposed algorithm. The effects of the measurement error and the number of measurement points on the accuracy of the inverse analysis are also investigated in detail.     

Application of variational iteration method and homotopy–perturbation method for nonlinear heat diffusion and heat transfer equations

Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this Letter; one is He's variational iteration method (VIM) and the other is the homotopy–perturbation method (HPM). Nonlinear convective–radiative cooling equations are used as examples to illustrate the simple solution procedures. These methods are useful and practical for solving the nonlinear heat diffusion equation, which is associated with variable thermal conductivity condition. Comparison of the results obtained by both methods with exact solutions reveals that both methods are tremendously effective.     

Prediction of radiative heat transfer in 2D irregular geometries using the collocation spectral method based on body-fitted coordinates

Recently, an efficient numerical method, which is called the collocation spectral method (CSM), for radiative heat transfer problems, has been proposed by the present authors. In this numerical method there exists the exponential convergence rate, which can obtain a very high accuracy even using a small number of grids. In this article, the CSM based on body-fitted coordinates (BFC) is extended to simulate radiative heat transfer problems in participating medium confined in 2D complex geometries. This numerical method makes simultaneously the use of the merits of both the CSM and BFC. In this numerical approach, the radiative transfer equation (RTE) in orthogonal Cartesian coordinates should be transformed into the equation in body-fitted nonorthogonal curvilinear coordinates. In order to test the efficiency of the developed method, several 2D complex irregular enclosures with curved boundaries and containing an absorbing, emitting and scattering medium are examined. The results obtained by the CSM are assessed by comparing the predictions with those in references. These comparisons indicate that the CSM based on BFC can be recommended as a good option to solve radiative heat transfer problems in complex geometries.     

Evaluation method for radiative heat transfer in polydisperse water droplets

Simplifications of the model for nongray radiative heat transfer analysis in participating media comprised of polydisperse water droplets are presented. Databases of the radiative properties for a water droplet over a wide range of wavelengths and diameters are constructed using rigorous Mie theory. The accuracy of the radiative properties obtained from the database interpolation is validated by comparing them with those obtained from the Mie calculations. The radiative properties of polydisperse water droplets are compared with those of monodisperse water droplets with equivalent mean diameters. Nongray radiative heat transfer in the anisotropic scattering fog layer, including direct and diffuse solar irradiations and infrared sky flux, is analyzed using REM². The radiative heat fluxes within the fog layer containing polydisperse water droplets are compared with those in the layer containing monodisperse water droplets. Through numerical simulation of the radiative heat transfer, polydisperse water droplets can be approximated by using the Sauter diameter, a technique that can be useful in several research fields, such as engineering and atmospheric science. Although this approximation is valid in the case of pure radiative transfer problems, the Sauter diameter is reconfirmed to be the appropriate diameter for approximating problems in radiative heat transfer, although volume-length mean diameter shows better accordance in some cases. The CPU time for nongray radiative heat transfer analysis with a fog model is evaluated. It is proved that the CPU time is decreased by using the databases and the approximation method for polydisperse particulate media.     

Computational transport methodology based on decomposition of a problem domain into transport and diffusive subdomains

A large class of radiative transfer and particle transport problems contain highly diffusive regions. It is possible to reduce computational costs by solving a diffusion problem in diffusive subdomains instead of the transport equation. This enables one to decrease the dimensionality of the transport problem. In this paper we present a methodology for decomposition of a spatial domain of a transport problem into transport and diffusion subregions. We develop methods for solving one-group problems in 1D slab geometry. To identify and locate diffusive regions, we develop metrics for measuring transport effects that are based on the quasidiffusion (Eddington) factor. We present the results of test problems that demonstrate the accuracy of the proposed methodology.     

Vector Green's function algorithm for radiative transfer in plane-parallel atmosphere

Green's function is a widely used approach for boundary value problems. In problems related to radiative transfer, Green's function has been found to be useful in land, ocean and atmosphere remote sensing. It is also a key element in higher order perturbation theory. This paper presents an explicit expression of the Green's function, in terms of the source and radiation field variables, for a plane-parallel atmosphere with either vacuum boundaries or a reflecting (BRDF) surface. Full polarization state is considered but the algorithm has been developed in such way that it can be easily reduced to solve scalar radiative transfer problems, which makes it possible to implement a single set of code for computing both the scalar and the vector Green's function.     

Exploiting the linearity of radiative transfer

A standard problem in radiative transfer is finding the external and internal radiative fields produced by uniform, parallel rays illuminating the top of a one-dimensional, scattering and absorbing medium of finite optical thickness. This problem has been solved in several ways with various physical restrictions. One approach is by finding the source function that represents the rate of production of scattered radiation per unit volume per unit solid angle at each point in the medium. The present paper develops and uses the idea that the standard source function is an influence function for a given medium. The linearity of radiative transfer is then used to find certain general source functions in terms of the standard one. The usefulness of the above concept is demonstrated by the following four problems: (1) derivation of Chandrasekhar's four principles of invariance from the radiative transfer equation, (2) derivation of the equations governing Chandrasekhar's X- and Y- functions without using the invariance principles or resolvent kernels, (3) finding the source function for a medium with a Lambert's-law bottom, and (4) finding the source function for a medium with a bottom that is a perfect specular reflector.     

Angular quadrature perturbations in radiative transfer theory

A new numerical method is presented for solving the general equation of radiative transfer. The approximation, which replaces the integral term over angle in the transfer equation by a quadrature sum, is studied; an estimate of the error involved is obtained and this error, which may be thought of as a further source or sink of photons (depending upon the sign), can then be used to evaluate a corection to the radiation field originally determined. This process may then be continued as a perturbation series. The method is found to give a final solution, when starting from the Eddington approximation, at least as accurate as that obtained using variable Eddington factors. Furthermore, the technique involves very little extra computing over that required using the Eddington approximation, and may be trivially generalized to any radiative transfer problem. It can also be used in conjunction with any of the existing methods for solving the equation of transfer. Examples are given in the context of spectral line formation in slab geometry.     

Boundary conditions in the radiative relaxation problem

The problem of the dissipation of temperature perturbations in a finite homogeneous atmosphere is solved for the situation in which the temperature at one boundary is maintained constant (that is, the temperature perturbation is zero for all times) while energy can be freely radiated to space through the other boundary. Exact solutions are shown for the exponential-sum fit to the kernel of the basic integral equation. These solutions constitute the set of radiative eigenfunctions. Also, approximate solutions in terms of the radiative eigenfunctions in the diffusion approximation (one exponential term in the expansion of the kernel) are obtained. These, in turn, are used in the solution of an initial value problem. The constant temperature boundary condition simulates the interface between two regions in one of which the relaxation processes are much more rapid than the purely radiative relaxation of the other.     

基于稳态辐射传输方程的小动物尺寸组织体扩散光学层析法

在Newton-Raphson逆模型框架下,发展了基于稳态辐射传输方程光子输运正模型和代数重建技术的扩散光学层析图像重建算法,并针对小尺寸组织体内部吸收和散射系数同时重建情况下的标准代数重建技术算法进行了改进,数值结果表明改进后的代数重建技术算法重建图像效果优于标准代数重建技术算法.     

An exact solution to the multi-dimensional line transfer equation

The theory of generalized analytic functions is used to obtain an exact closed form analytical solution to a transfer problem for spectral line radiation in a multi-dimensional atmosphere. The multi-dimensional full-space and half-space Green's functions so obtained are quite general and may be used, along with the corresponding orthogonality relationships, to obtain solutions to any general multi-dimensional radiative transfer problem involving model two-level atoms. An application of the method using perturbation techniques is illustrated.     

Finite element approach for radiative transfer in multi-layer graded index cylindrical medium with Fresnel surfaces

Because the optical plane defined by the incidence and reflection direction at a cylindrical surface has a complicated relation with the local azimuthal angle and zenith angle in the traditional cylindrical coordinate system, it is difficult to deal with the specular reflective boundary condition in the solution of the traditional radiative transfer equation for cylindrical system. In this paper, a new radiative transfer equation for graded index medium in cylindrical system (RTEGCN) is derived based on a newly defined cylindrical coordinate system. In this new cylindrical coordinate system, the optical plane defined by the incidence and reflection direction is just the isometric plane of the local azimuthal angle, which facilitates the RTEGCN in dealing with cylindrical specular reflective boundaries. A least squares finite element method (LSFEM) is developed for solving radiative transfer in single and multi-layer cylindrical medium based on the discrete ordinates form of the RTEGCN. For multi-layer cylindrical medium, a radial basis function interpolation method is proposed to couple the radiative intensity at the interface between two adjacent layers. Various radiative transfer problems in both single and multi-layer cylindrical medium are tested. The results show that the present finite element approach has good accuracy to predict the radiative heat transfer in multi-layer cylindrical medium with Fresnel surfaces.