Inverse solutions to radiative-transfer problems based on the binomial or the Henyey-Greenstein scattering law

Analytical techniques are used to solve two inverse radiative-transfer problems, for a finite plane-parallel medium, that are (i) based on the binomial scattering law and (ii) based on the Henyey-Greenstein scattering law. In addition, previously reported analytical results (valid for isotropic scattering) that yield an analytical inverse solution for the unknown optical thickness of the medium are extended to the case of anisotropic scattering. The algorithms for the inversions are verified numerically, and some effects of noise on the simulated experimental data are observed.     

Discrete-ordinates solution for radiative-transfer problems

The discrete-ordinates method is used to solve radiative-transfer problems, in plane-parallel media. We present a generalized analytical discrete-ordinates formulation for solving single- and multi-region problems. Included in our model are internal sources, reflecting and emitting boundaries, incident distribution of radiation on each surface and a beam incident on one surface. Numerical results for three test problems are reported.     

A numerical scheme for the solution of axisymmetric radiative transfer problems in irregular domains filled by media with opaque and transparent diffuse and specular boundaries

This paper presents a new numerical scheme of the discrete ordinates method for the solution of axisymmetric radiative transfer problems in irregular domains filled by media with opaque and transparent diffuse and specular (Fresnel) boundaries and interfaces. New test problems of radiative transfer, which describe radiative transfer in domains with Fresnel interfaces, are proposed in this paper. These problems admit analytic solutions and can be used as benchmark ones. The proposed scheme is applied to the solution of the problems. Numerical results show that the presence of Fresnel interfaces leads to an appreciably larger error in numerical solution. This is connected with the “discontinuity” of the Fresnel reflectivity, which, through numerical diffusion, leads to the distortion of numerical solution. Modification of the scheme allows to reduce the numerical error.     

Regularization of inverse boundary design radiative heat transfer problems

Inverse boundary design heat transfer problems, including only radiation, are considered. Variational methods of regularization are used for solving these (mathematically ill-posed) problems, they give possibility to take into account various a priori information about the desired solution. For minimizing the discrepancy functional we use the adjoint problem method; however, we use it not only in iterative regularization, but in Tikhonov and parametric ones as well. We use all the available a priori information about desired solutions in all the techniques; this allows to obtain physically feasible solutions in all the cases.     

A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters

Several numerical and analytical solutions of the radiative transfer equation (RTE) were compared for plane albedo in a problem of solar light reflection by sea water. The study incorporated the simplest case—a semi-infinite one-dimensional plane—parallel absorbing and scattering homogeneous layer illuminated by a monodirectional light beam. Inelastic processes (such as Raman scattering and fluorescence), polarization and air-water surface refraction-reflection effects, were not considered. Algorithms were based on the invariant imbedding method and two different variants of the discrete ordinate method (DOM). Calculations were performed using parameters across all possible ranges (single-scattering albedo ω₀ and refracted solar zenith angle θ₁), but with a special emphasis on natural waters. All computations were made for two scattering phase functions, which included an almost isotropic Rayleigh phase function and strongly anisotropic double-peaked Fournier-Forand-Mobley phase function. Models were validated using quasi-single-scattering (QSSA) and exponential approximations, which represent the extreme cases of ω₀→0 and ω₀→1, respectively. All methods yielded relative differences within 1.8% for modeled natural waters. An analysis of plane albedo behavior resulted in the development of a new extended QSSA approximation, which when applied in conjunction with the extended Hapke approximation developed earlier, resulted in a maximum relative error of 2.7%. The study results demonstrated that for practical applications, the estimation of inherent optical properties from observed reflectance can best be achieved using an extended Hapke approximation.     

Two analytic methods applied to radiative transfer in the linear-anisotropic scattering medium with graded index and diffuse gray boundaries

The curved ray-tracing method is extended to radiative transfer in the graded index medium with diffuse gray boundary conditions instead of black boundary conditions and the pseudo-source adding method is extended to the case of the linear-anisotropic scattering medium with graded index from non-scattering medium. Furthermore, the equivalence of the two methods is verified by formulation derivation. As exact analytical solutions, both the methods have high accuracy and fast computational speed. The predicted temperature distributions and dimensionless radiative heat flux at radiative equilibrium are determined by the proposed methods, and the numerical results are compared with the data in references. The results show that the present methods have a good accuracy. Influences of various combinations of refractive index and boundary emissivities on the temperature distributions and dimensionless radiative heat flux are also investigated.     

On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media

The ADO (analytical discrete ordinates) method is used to establish a concise and accurate solution for a multi-layer radiative-transfer problem with Fresnel boundary and interface conditions. A finite plane-parallel medium composed of a number (K) of sub-strata with different material properties is considered to be illuminated by isotropically incident radiation. While a general result is obtained, emphasis in the numerical work is given to computing accurately the currents and the intensities that exit each of the two exterior surfaces. Monochromatic forms (with anisotropic scattering) of the radiative-transfer equation are used, and numerical results are given for several specific cases. The complications introduced by the Fresnel boundary and interface conditions are well resolved, so that the numerical results obtained are thought to define a very high standard.     

On the use of a nascent delta function in radiative-transfer calculations for multi-layer media subject to Fresnel boundary and interface conditions

The “pre-processing” procedure and the “break-point” analysis developed in a previous work based on the ADO (analytical discrete ordinates) method are used, along with a nascent delta function to describe the polar-angle dependence of an incident beam, to solve the classical albedo problem for radiative transfer in a plane-parallel, multi-layer medium subject to Fresnel boundary and interface conditions. As a result of the use of a nascent delta function, rather than the Dirac distribution, to model the polar-angle dependence of the incident beam, the computational work is significantly simplified (since a particular solution is not required) in comparison to an approach where both the polar-angle and the azimuthal-angle dependence of the incident beam are formulated in terms of Dirac delta distributions. The numerical results from this approach are (when a sufficiently small “narrowness” parameter is used to define the nascent delta) found to be in complete agreement with already reported (high-quality) results for a set of challenging multi-layer problems.     

A new approach to the inverse scattering and spectral problems for the Sturm-Liouville equation

New proofs of the known uniqueness theorems for the one-dimensional inverse spectral and scattering problems are given. Proof of the invertibility of all of the steps in the inversion procedures of Gelfand-Levitan and Marchenko is given. The proposed method of investigation yields some new results, for example, a Marchenko-type equation at x = 0 which holds on the whole axis, rather than on a half-axis, as usual for the scattering theory on half-axis. It also yields a new method, shorter and simpler than earlier published, for proving that the potential in the class L_1,1, obtained by the Marchenko reconstruction procedure, generates the scattering data from which it was reconstructed.     

Image reconstruction in diffuse optical tomography using the coupled radiative transport-diffusion model

The coupled radiative transport-diffusion model can be used as light transport model in situations in which the diffusion equation is not a valid approximation everywhere in the domain. In the coupled model, light propagation is modelled with the radiative transport equation in sub-domains in which the approximations of the diffusion equation are not valid, such as within low-scattering regions, and the diffusion approximation is used elsewhere in the domain. In this paper, an image reconstruction method for diffuse optical tomography based on using the coupled radiative transport-diffusion model is developed. In the approach, absorption and scattering distributions are estimated by minimising a regularised least-squares error between the measured data and solution of the coupled model. The approach is tested with simulations. Reconstructions from different cases including domains with low-scattering regions are shown. The results show that the coupled radiative transport-diffusion model can be utilised in image reconstruction problem of diffuse optical tomography and that it produces as good quality reconstructions as the full radiative transport equation also in the presence of low-scattering regions.     

Radiative transfer with internal reflection via the converged discrete ordinates method

In response to the challenge of establishing highly accurate solutions to the plane layer radiative transfer equation with the simplest of methods, the converged discrete ordinates method is presented. With this algorithm of only finite difference, quadrature and acceleration, we show how to obtain highly accurate intensities for radiative transfer in a finite layer with internal surface reflection. The method features angular smoothing and angular interpolation through “faux” quadrature. In addition, a manufactured solution demonstrates the high accuracy of the method for forward peaked scattering. We consider scattering in a heterogeneous medium as a final demonstration.     

Temperature field of radiative equilibrium in a two-dimensional graded index medium with gray boundaries

In this paper, a numerical method is presented for the study of the radiative transfer in a two-dimensional graded index semitransparent medium with diffuse gray boundaries. The numerical method is a combination of the linear refractive index bar model, the discrete curved ray-tracing technique and the pseudo source adding method (LRIB-CRTP). In the traditional ray-tracing technique, it is difficult to deal with the diffuse gray boundary while solving the radiative transfer. Using the pseudo source adding method, the diffuse gray boundary of the medium can be treated as a black boundary. We have also studied the radiative equilibrium temperature field of the medium and analyzed the influence of some parameters involved. The results show that the directional discrete number is important for the medium having a large absorption coefficient. The results also show that the refractive index distribution greatly influences the temperature field, whereas the linear absorption coefficient distribution has little influence on the temperature field.     

Relationship between different approaches to derive weighting functions related to atmospheric remote sensing problems

The paper is devoted to the investigation of the relationship between different methods used to derive weighting functions required to solve numerous inverse problems related to the remote sensing of the Earth's atmosphere by means of scattered solar light observations. The first method commonly referred to as the forward-adjoint approach is based on a joint solution of the forward and adjoint radiative transfer equations and the second one requires the linearized forward radiative transfer equation to be solved. In the framework of the forward-adjoint method we consider two approaches commonly used to derive the weighting functions. These approaches are referenced as the “response function” and the “formal solution” techniques, respectively. We demonstrate here that the weighting functions derived employing the formal solution technique can also be obtained substituting the analytical representations for the direct forward and direct adjoint intensities into corresponding expressions obtained in the framework of the response function technique. The advantages and disadvantages of different techniques are discussed.     

Three-dimensional radiative transfer in an anisotropically scattering, plane-parallel medium: generalized reflection and transmission functions

The problem of spatially varying, collimated radiation incident on an anisotropically scattering, plane-parallel medium is considered. A very general phase function is allowed. An integral transform is used to reduce the three-dimensional radiative transport equation to a one-dimensional form, and a modified Ambarzumian's method is applied to derive nonlinear integral and integro-differential equations for the generalized reflection and transmission functions. The integration is over the polar and azimuthal angles—this formulation is referred to as the double-integral formulation. The integral equations are used to illustrate symmetry relationships and to obtain single- and double-scattering approximations. The generalized reflection and transmission functions are important in the construction of the solutions to many multidimensional problems. Coupled integral equations for the interior and emergent intensities are developed and, for the case of two identical homogeneous layers, used to formulate a doubling procedure. Results for an isotropic and Rayleigh scattering medium are presented to illustrate the computational characteristics of the formulation.     

An inverse analysis to determine conductive and radiative properties of a fibrous medium

In present paper, a modified factor of extinction coefficient and an equivalent albedo of scattering were defined taking into account anisotropic scattering in fibrous insulation. An inverse conduction-radiation analysis in an absorbing, emitting and scattering medium was conducted for the simultaneous estimation of the conductive and radiative properties using the experimentally measured temperature responses for external temperatures up to 980 K. The estimated properties were validated by comparing the predicted and measured results under transient and steady-state condition. It was found that the calculated results corresponded well with the experimental data within an average of 3.1% under transient condition and 9.8% under steady-state condition. This confirms the good behavior of the model and the validity of results.     

Discrete ordinates solution of coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation

The coupled conductive radiative heat transfer in a two-layer slab with Fresnel interfaces subject to diffuse and obliquely collimated irradiation is solved. The collimated and diffuse components problems are treated separately. The solution for diffuse radiation is obtained by using a composite discrete ordinates method and includes the development of adaptive directional quadratures to overcome the difficulties usually encountered at the interfaces. The complete radiation numerical model is validated against the predictions obtained by using the Monte Carlo method.     

Fluctuation theory for radiative transfer in random media

We consider the effect of small scale random fluctuations of the constitutive coefficients on boundary measurements of solutions to radiative transfer equations. As the correlation length of the random oscillations tends to zero, the transport solution is well approximated by a deterministic, averaged, solution. In this paper, we analyze the random fluctuations to the averaged solution, which may be interpreted as a central limit correction to homogenization.With the inverse transport problem in mind, we characterize the random structure of the singular components of the transport measurement operator. In regimes of moderate scattering, such components provide stable reconstructions of the constitutive parameters in the transport equation. We show that the random fluctuations strongly depend on the decorrelation properties of the random medium.     

The Iterative-DRESOR method to solve radiative transfer in a plane-parallel, anisotropic scattering medium with specular-diffuse boundaries

Using the intensity with high directional resolution obtained by the Basic-DRESOR method as an initial guess, which is substituted into the integrated radiative transfer equation (IRTE), an iterative algorithm is proposed, called the Iterative-DRESOR method. This method can reduce the error levels of the intensity from several percent using the Basic-DRESOR method to a level of less than 1.0×10⁻⁶ with acceptable computation costs. The method is also validated against the exact heat flux in literature in some cases. It further clarifies some uncertain results for the reflectance in a pure, linearly anisotropic scattering medium with specular-diffuse boundaries. The directional distributions of intensity are obviously influenced by the reflecting modes of the boundary, especially in the zone near the boundary. The reflecting mode of an emitting boundary has little effect on the transmittance or reflectance. The reflecting mode of a non-emitting boundary also has little effect on the transmittance, but it obviously influences the reflectance. The difference between the reflectance for specular and diffuse boundaries increases at first, and then decreases, as the optical thickness of the medium increases. The difference will decrease as the scattering albedo of the medium increases, and it is negligible when the medium is pure scattering. The effect of the scattering phase function of the medium on the difference can also not be ignored. The Iterative-DRESOR method is expected to strengthen the capability of the Monte Carlo method to produce accurate results and to validate the results of other methods to solve RTE.     

Non-stationary radiative-conductive heat transfer in a layer with partially absorbing boundaries

Here we consider one-dimensional heating of a layer of gray semitransparent medium by an outer source of radiation and convection. The sample boundaries reflect, absorb (radiate), and transmit radiation. It is shown that heating dynamics and character of temperature fields depend significantly on optic parameters of the boundaries. The work was financially supported by the President of the Russian Federation (MK-601.2008.8) and Russian Foundation for Basic Research (Grant No. 08-08-00527-a).