Three-dimensional Radiative Transfer Equation Research Articles

An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres

The three-dimensional (3D) diffusion radiative transfer equation, which utilizes a four-term spherical harmonics expansion for the scattering phase function and intensity, has been efficiently solved by using the full multigrid numerical method. This approach can simulate the transfer of solar and thermal infrared radiation in inhomogeneous cloudy conditions with different boundary conditions and sharp boundary discontinuity. The correlated k-distribution method is used in this model for incorporation of the gaseous absorption in multiple-scattering atmospheres for the calculation of broadband fluxes and heating rates in the solar and infrared spectra. Comparison of the results computed from this approach with those computed from plane-parallel and 3D Monte Carlo models shows excellent agreement. This 3D radiative transfer approach is well suited for radiation parameterization involving 3D and inhomogeneous clouds in climate models.

An Approach of Three-Dimensional Vector Radiative Transfer (3-D–VRT) Equation for Inhomogeneous Scatter Media

To solve a three-dimensional vector radiative transfer (3-D-VRT) equation for the model of spatially inhomogeneous scatter media, the finite enclosure of the scatter media is geometrically divided, in both the vertical z and horizontal (x,y) directions, to form very thin multiboxes. The zeroth-order emission, first-order Mueller matrix of each thin box, and an iterative approach of high-order radiative transfer are applied to deriving high-order scattering and emission of whole inhomogeneous scatter media. Numerical results of polarized brightness temperature at microwave frequency from an inhomogeneous scatter model such as vegetation canopy are calculated and discussed.

A three-dimensional vector radiative transfer (3D-VRT) equation for inhomogeneous scatter media *

Abstract To solve 3D-VRT equation for the model of spatially inhomogeneous scatter media, the finite enclosure of the scatter media is geometrically divided, in both the vertical z and horizontal (x, y) directions, to form very thin multi-boxes. The zero-th order emission, first-order Mueller matrix of each thin box and an iterative approach of high-order radiative transfer are applied to deriving high-orderscattering and emission of whole inhomogeneous scatter media. Numerical results of polarized brightness temperature at microwave frequency from inhomogeneous scatter model such as vegetation canopy are calculated and discussed.

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.

Three-dimensional vector radiative transfer in a semi-infinite, Rayleigh scattering medium exposed to spatially varying polarized radiation: generalized reflection matrix

Three-dimensional vector radiative transfer in a semi-infinite medium exposed to spatially varying, polarized radiation is studied. The problem is to determine the generalized reflection matrix for a multiple scattering medium characterized by a 4×4 scattering matrix. A double integral transform is used to convert the three-dimensional vector radiative transfer equation to a one-dimensional form, and a modified Ambarzumian's method is then applied to derive a nonlinear integral equation for the generalized reflection matrix. The spatially varying backscattered radiation for an arbitrarily polarized incident beam can be found from the generalized reflection matrix. For Rayleigh scattering and normal incidence and emergence, the generalized reflection matrix is shown to have five non-zero elements. Benchmark results for these five elements are presented and compared to asymptotic results. When the incident radiation is polarized, the vector approach used in this study correctly predicts three-dimensional behavior, while the scalar approach does not. When the incident radiation is unpolarized, both the vector and scalar approaches predict a two-dimensional distribution of the intensity, but the error in the scalar prediction can be as high as 20%.

Three-dimensional numerical cosmological radiative transfer in an inhomogeneous medium

A numerical scheme is proposed for the solution of the three-dimensional radiative transfer equation with variable optical depth. We show that time-dependent ray tracing is an attractive choice for simulations of astrophysical ionization fronts, particularly when one is interested in covering a wide range of optical depths within a 3D clumpy medium. Our approach combines the explicit advection of radiation variables with the implicit solution of local rate equations given the radiation field at each point. Our scheme is well suited to the solution of problems for which line transfer is not important, and could, in principle, be extended to those situations also. This scheme allows us to calculate the propagation of supersonic ionization fronts into an inhomogeneous medium. The approach can be easily implemented on a single workstation and also should be fully parallelizable.

3D Radiative Transfer in Weakly Inhomogeneous Medium. Part I: Diffusive Approximation

The solution of the three-dimensional radiative transfer equation in weakly horizontally inhomogeneous medium has been obtained in the diffusion approximation using the expansion of the three-dimensional delta-Eddington approximation. The solution approach, referred as the gradient correction (GC) method, expands the horizontal fluxes and the source function in terms of the horizontal gradient of the extinction coefficient and/or the cloud-top boundary. In the transfer equation, only the zeroth- and first-order gradient terms are retained and hence the following limitations apply. First, the length of the horizontal variations of optical properties of the medium should be large in comparison to the mean radiative transport length. Second, the ratio of the vertical to horizontal scales should be small enough so that fluxes from boundaries may be neglected. Since there are no restrictions to the amplitude of the optical properties variations, this method may even be applicable to a medium with strong horizontal variations of optical properties, as long as scales of the variations are large enough in comparison to the radiative transport length. The analytical solutions are in excellent agreement with the more accurate numerical solutions. The solution also shows the solar zenith angle dependence of the albedo, similar to that observed in analyses of satellite imagery. The GC approach may be useful as a fast and computationally inexpensive method both for the correction of the independent pixel approximation used for extraction of cloud fields from satellite imagery and possibly for the calculation of the radiation fluxes in climate models.

Multidimensional radiative transfer in absorbing, emitting, and linearly anisotropic scattering cylindrical medium with space-dependent properties

The integral form of three-dimensional radiative transfer equation for an absorbing, emitting, and linear-anisotropic scattering medium with space-dependent properties is formulated. A product-integration method is subsequently applied to develop a numerical scheme for solving the corresponding integral transfer equations in a two-dimensional, axisymmetric and nonhomogeneous medium subjected to externally incident radiation or bounded by emitting and diffusely-reflecting walls. The numerical solutions for cases of constant, continuous, and stepwise variations of scattering albedo are presented to illustrate its accuracy and flexibility, and validated by comparing with results available in the literature.

Perturbation Solution for 3D Radiative Transfer in a Horizontally Periodic Inhomogeneous Cloud Field

For a vertical homogeneous plane-parallel layer with horizontal cosinusoidal periodic variations of the extinction coefficient, k = k0{1 + ε[cos(ax) + cos(by)]}, the first-order perturbation solution of the three-dimensional radiative transfer equation has been obtained. The first-order perturbation correction in cloud albedo cancels when a horizontal domain averaging is done. A correspondence exists between the distribution of the extinction coefficient and the distribution of the upwelling intensity. However, under certain conditions, the distribution of the upwelling intensity is opposite to the distribution of the extinction coefficient. If the solar zenith angle is large, shifts in the configurations of the distribution of the upwelling intensity may appear. The single scattering parameters can influence the distribution of the diffuse radiative intensity. The distribution of the heating rate inside the cloud and the distribution of the extinction coefficient are nearly coincident with each other.

Radiative transfer in three-dimensional atmosphere-vegetation media

The problem of radiation transfer in three-dimensional atmosphere-vegetation media is formulated in this paper. The atmosphere is treated as a horizontally homogeneous medium but with spatially varying anisotropic lower boundary. The boundary condition is defined in terms of vegetation canopy bidirectional reflectance factors to be evaluated from a numerical solution of the three-dimensional canopy radiative transfer equation. Numerical results obtained with the discrete ordinates method are in good agreement with the Monte Carlo results of Pearce and field soybean reflectance data of Ranson et al.

A finite-difference discrete-ordinate solution to the three-dimensional radiative transfer equation

Abstract A finite-difference discrete-ordinate iterative method developed to solve the three-dimensional (3D) radiative transfer equation is described. The method is applicable to a volume of ocean with position dependent volumetric absorption and scattering coefficients. Input quantities include sun position and sky radiance distribution, scattering phase function, and absorbing, reflecting, or emitting objects within the ocean volume. A solution of the one-dimensional radiative transfer equation is used to provide boundary values for the 3D solutions. Radiance distributions are calculated which model recent ocean radiance measurements showing ship shadow effects. The results are compared.

Effects of Multiple Scattering on Light Pulses Reflected by Turbid Atmospheres

Multiple scattering contributions to lidar returns from turbid atmospheres are derived by means of an analytical theory. It is assumed that scattering takes place mainly at small angles except for one event that scatters the light backward. The phase functions are approximated by the sum of Gaussian functions of the scattering angle in both the forward and backward directions. The three-dimensional radiative transfer equation is transformed to a one-dimensional problem by means of Fourier transforms. Neumann solutions to the transformed equation of radiative transfer are then found. A number of examples are presented for cloud, fog and haze models. The results are found to be in satisfactory agreement with results obtained from the Monte Carlo analysis of Kunkel (1974) and the theory of light pulses doubly scattered by turbid atmospheres which was developed by Eloranta (1972).