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.     

Absorption and emission line profile coefficients of multilevel atoms—I. Atomic profile coefficients

The line profile coefficients for absorption and emission appearing in the radiative transfer equation are formulated in terms of atomic line profile coefficients and velocity distribution functions. In order to derive the atomic profile coefficients of a multilevel atom, one defines generalized atomic redistribution functions that describe the correlations between photons involved in consecutive radiative transitions of the atom. Besides their dependence on the radiation field, the atomic line profile coefficients of a multilevel atom depend on the velocity distributions of the atoms in the various excitation states, in contrast to the case of a two-level atom where only the radiation intensity but not the velocity distributions affect the atomic emission profile. Closed expressions of the atomic profile coefficients in terms of generalized redistribution functions are obtained if stimulated emissions are neglected, and one is led to an iterative approximation scheme if stimulated emissions are taken into account. The possibility of a nonlocal character of the atomic profile coefficients is pointed out, and the effect of elastic, velocity-changing collisions with excited atoms is discussed. A major aim of this paper is to draw attention to the fact that ordinary redistribution functions that describe only the correlations between the absorbed and reemitted photons in the same spectral line are not sufficient to formulate the line profile coefficients of a multilevel atom.     

Simultaneous derivation of intensities and weighting functions in a general pseudo-spherical discrete ordinate radiative transfer treatment

The retrieval of atmospheric constituents from measurements of backscattered light requires a radiative transfer forward model that can simulate both intensities and weighting functions (partial derivatives of intensity with respect to atmospheric parameters being retrieved). The radiative transfer equation is solved in a multi-layer multiply-scattering atmosphere using the discrete ordinate method. In an earlier paper dealing with the upwelling top-of-the-atmosphere radiation field, it was shown that a full internal perturbation analysis of the plane-parallel discrete ordinate solution leads in a natural way to the simultaneous generation of analytically-derived weighting functions with respect to a wide range of atmospheric variables. In the present paper, a more direct approach is used to evaluate explicitly all partial derivatives of the intensity field. A generalization of the post-processing function is developed for the derivation of weighting functions at arbitrary optical depth and stream angles for both upwelling and downwelling directions. Further, a complete treatment is given for the pseudo-spherical approximation of the direct beam attenuation; this is an important extension to the range of viewing geometries encountered in practical radiative transfer applications. The numerical model LIDORT developed for this work is able to generate intensities and weighting functions for a wide range of retrieval scenarios, in addition to the passive remote sensing application from space. We present a number of examples in an atmosphere with O₃ absorption in the UV, for satellite (upwelling radiation) and ground-based (downwelling radiation) applications. In particular, we examine the effect of various pseudo-spherical parameterizations on backscatter intensities and weighting functions with respect to O₃ volume mixing ratio. In addition, the use of layer-integrated multiple scatter output from the model is shown to be important for satellite instruments with wide-angle off-nadir viewing geometries.     

Generalized form of the direct and adjoint radiative transfer equations

The main goal of this paper is to give a rigorous derivation of the generalized form of the direct (also referenced as forward) and adjoint radiative transfer equations. The obtained expressions coincide with expressions derived by Ustinov [Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. JQSRT 2001;68:195-211]. However, in contrast to [Ustinov EA. Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR. JQSRT 2001;68:195-211] we formulate the generalized form of the direct radiative transfer operator fully independent from its adjoint. To illustrate the application of the derived adjoint radiative transfer operator we consider the angular interpolation problem in the framework of the discrete ordinate method widely used to solve the radiative transfer equation. It is shown that under certain conditions the usage of the solution of the adjoint radiative transfer equation for the angular interpolation of the intensity can be computationally more efficient than the commonly used source function integration technique.     

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.     

Application of adaptive multi-frequency grids to three-dimensional astrophysical radiative transfer

We present a method to solve the three-dimensional (3D) radiative transfer equation for astrophysical applications using adaptive photon transport grids. Contrary to earlier treatments, they are calculated for each frequency separately. Generated minimizing the first-order discretization error in the scattered radiation intensity, they provide global error control for solutions of radiative transfer problems on the grid. We discuss minimization of the grid point number in regions where the optical depth becomes large and show that the method allows for treating applications with optical depth of any value using the concept of penetration depth. The proposed grid generation algorithm is easy to implement, allows pre-calculation of the grids and storage in integer arrays, making a fast solution of the 3D radiative transfer equation possible. The grid generation algorithm is suitable for optimization in cases where simple radiation source distributions are given. Besides discussing application to simple density distribution commonly occurring in astrophysical objects, we illustrate the capabilities of the method by generating grids for an accretion disk around a young star.     

Monte Carlo ray-tracing simulation for radiative heat transfer in turbulent fluctuating media under the optically thin fluctuation approximation

The Monte Carlo ray-tracing method (MCRT) based on the concept of radiation distribution factor is extended to solve radiative heat transfer problem in turbulent fluctuating media under the optically thin fluctuation approximation. A one-dimensional non-scattering turbulent fluctuating media is considered, in which the mean temperature and absorption coefficient distribution are assumed and the shape of probability density function is given. The distribution of the time-averaged volume radiation heat source is solved by MCRT and direct integration method. It is shown that the results of MCRT based on the concept of radiation distribution factor agree with these of integration solution very well, but results of MCRT based on the concept of radiative transfer coefficient do not agree with these of integration solution. The solution of time-averaged radiative transfer equation by the concept of radiative transfer coefficient should be treated with caution.     

Radiation field in a semi-infinite homogeneous atmosphere with internal sources

The equation of radiative transfer in a semi-infinite homogeneous atmosphere with different internal sources is solved by the method of kernel approximation—the kernel in the equation for the Sobolev resolvent function is approximated by a Gauss-Legendre sum. Then the obtained approximate equation can be solved exactly and the solution is a weighted sum of exponentials. All the necessary coefficients of the solutions may be easily found. 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. For the considered cases the formulas for the radiation field are obtained and the respective accuracy estimated. The package of codes in Fortran-77 is given at http://www.aai.ee/∼viik/homogen.for.     

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.     

The exact solution of the time-dependent equation of radiative transfer in the interior of a semi-infinite atmosphere

An exact solution to the time-dependent radiative transfer equation for the interior intensity in a semi-infinite atmosphere exhibiting temporal capture is obtained. The intensity is constructed from a solution found in neutron transport theory and can be expressed in terms of quadratures of elementary functions. The origin of the form of the emergent intensity found by Matsumoto is given, as well as a similarity relation and the relationship of the intensities in absorbing and nonabsorbing media.     

管内高温介质层流入口段中的热辐射作用

数值研究了高温介质密度随温度变化时,管内层流入口段耦合换热中的热辐射作用。采用离散坐标法、控制容积法耦合求解辐射传递方程、能量方程及N-S方程。考察了中等大小光学厚度下,热辐射作用对介质内速度分布、温度分布以及换热的影响。结果表明,即使在不大的光学厚度下,热辐射作用对管内高温介质层流入口段耦合换热的速度场与换热强度都有明显影响。     

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.     

Time-dependent aerodynamic non-LTE radiative transfer

A method is developed for solving simultaneously in one dimension the equation of transfer for non-LTE spectral line radiation and the time-dependent equations specifying conservation of mass, energy and linear momentum. In particular, we illustrate the method on a ‘simple’ time-dependent problem in which a pulsating disturbance at some point in a model homogeneous atmosphere propagates towards the surface and steepens into a shock. The resulting emergent intensities show rather dramatic changes over very small time intervals due to the effect of the velocity, density and temperature distributions on the radiative absorption properties of the gas, and thus emphasises the need to solve the above-mentioned four basic equations if one is to obtain physically realistic model atmospheres experiencing initial disturbances.     

The velocity-dependent source function in radiative transfer theory

We consider the effect of velocity fields upon the transfer of line radiation by two-level atoms. We show that a simultaneous solution of the radiation transfer equation and the time-dependent rate equations leads to an equation for the source function which contains the Lagrangian derivative. We discuss a physical interpretation of the derivative term and present a method for solving this type of problem.We exhibit calculations which show that, for quite reasonable velocity fields, large errors can be produced if the derivative terms in the rate equations are neglected.     

Derivation and application of the reciprocity relations for radiative transfer with internal illumination

A Green's function formulation is used to derive basic reciprocity relations for planar radiative transfer in a general medium with internal illumination. Reciprocity (or functional symmetry) allows an explicit and generalized development of the equivalence between source and probability functions. Assuming similar symmetry in three-dimensional space, a general relationship is derived between planar-source intensity and point-source total directional energy. These quantities are expressed in terms of standard (universal) functions associated with the planar medium, while all results are derived from the differential equation of radiative transfer.     

Radiative transfer theory with time delay for the effect of pulse entrapping in a resonant random medium: General transfer equation and point-like scatterer model

Abstract

A pulse propagation of a vector electromagnetic wave field in a discrete random medium under the condition of Mie resonant scattering is considered on the basis of the Bethe–Salpeter equation in the two-frequency domain in the form of an exact kinetic equation which takes into account the energy accumulation inside scatterers. The kinetic equation is simplified using the transverse field and far wave zone approximations which give a new general tensor radiative transfer equation with strong time delay by resonant scattering. This new general radiative transfer equation, being specified in terms of the low-density limit and the resonant point-like scatterer model, takes the form of a new tensor radiative transfer equation with three Lorentzian time-delay kernels by resonant scattering. In contrast to the known phenomenological scalar Sobolev equation with one Lorentzian time-delay kernel, the derived radiative transfer equation does take into account effects of (i) the radiation polarization, (ii) the energy accumulation inside scatterers, (iii) the time delay in three terms, namely in terms with the Rayleigh phase tensor, the extinction coefficient and a coefficient of the energy accumulation inside scatterers, respectively (i.e. not only in a term with the Rayleigh phase tensor). It is worth noting that the derived radiative transfer equation is coordinated with Poynting's theorem for non-stationary radiation, unlike the Sobolev equation. The derived radiative transfer equation is applied to study the Compton–Milne effect of a pulse entrapping by its diffuse reflection from the semi-infinite random medium when the pulse, while propagating in the medium, spends most of its time inside scatterers. This specific albedo problem for the derived radiative transfer equation is resolved in scalar approximation using a version of the time-dependent invariance principle. In fact, the scattering function of the diffusely reflected pulse is expressed in terms of a generalized time-dependent Chandrasekhar H-function which satisfies a governing nonlinear integral equation. Simple analytic asymptotics are obtained for the scattering function of the front and the back parts of the diffusely reflected Dirac delta function incident pulse, depending on time, the angle of reflection, the mean free time, the microscopic time delay and a parameter of the energy accumulation inside scatterers. These asymptotics show quantitatively how the rate of increase of the front part and the rate of decrease of the rear part of the diffusely reflected pulse become slower with transition from the regime of conventional radiative transfer to that of pulse entrapping in the resonant random medium.     

Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

Absorption and emission line profile coefficients of multilevel atoms—II. Velocity-averaged profile coefficients

Starting from the atomic profile coefficients of a multilevel atom derived in the previous first part of this paper, we consider the velocity-averaged line profile coefficients appearing in the radiative transfer equation for the important special case that the velocity distribution of atoms in the ground state is Maxwellian and that the streaming of excited atoms is negligible. Elastic velocity-changing collisions of excited atoms with other particles are taken into account in the framework of a strong-collision model. Neglecting stimulated emissions, we obtain explicit, albeit in some cases approximate, expressions for the line profile coefficients of a three-level atom in terms of the specific radiation intensity locally present. The emission and absorption profile coefficients are written in a form that exhibits the various physical effects responsible for deviations of these profiles from complete redistribution. The case of two-level atoms in the presence of elastic collisions with the excited atoms is also treated.     

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.