Three-dimensional Radiative Transfer Equation Research Articles

An Angular Discretization Method Using Repulsive Particles for the Three-Dimensional Radiative Transfer Equation

An Angular Discretization Method Using Repulsive Particles for the Three-Dimensional Radiative Transfer Equation

A comparative study of the delta-Eddington and Galerkin quadrature methods for highly forward scattering of photons in random media

A versatile and accurate treatment for the highly forward-peaked phase function in the three-dimensional (3D) radiative transfer equation (RTE) based on the discrete ordinates method (DOM) is crucial for biomedical optics. Our first objective was to compare the delta-Eddington (dE) and Galerkin quadrature (GQ) methods. The dE method decomposes the phase function into a purely forward-peaked component and the other component, and expands the other component by Legendre polynomials as well as the finite order Legendre expansion (FL) method does. The GQ method conducts the weighting procedure in addition to the Legendre expansion. Although it was reported that both methods can provide the accurate results for calculations of the RTE, the versatility of both methods is still unclear. The second objective was to examine a possibility of a conjunction of the GQ method with the dE method, called as the GQ-dE method, which has the advantages of both methods. We examined numerical errors in the moment conditions of the phase function using the FL, dE, GQ, and GQ-dE methods at various types and orders of the quadrature sets, mainly in the region of the errors induced by the angular discretization using the DOM. The errors were reduced by the dE method from those by the FL method, however the error reduction depended on the types and orders of the quadrature sets. Meanwhile, the errors were significantly reduced by the GQ and GQ-dE methods, regardless of the quadrature sets. We also verified the numerical calculations of the time-dependent 3D RTE by the analytical solution of the RTE for homogeneous media in the region of the scattering length scale, where the highly forward-peaked phase function strongly influences the RTE-results. The errors in the RTE-results were similar to those in the moment conditions. Our results suggest the higher versatility and accuracy of the GQ and GQ-dE methods than those of the FL and dE methods.

Light propagation in biological tissue

We developed a new parallel computational method for solving fluorescence and elastically scattered light propagation through a biological tissue illuminated by a collimated incident beam. The three-dimensional steady state radiative transfer equation was solved using a Modified Finite Volume Method with a cell-vertex formulation. An Exponential spatial differencing scheme was used to efficiently lessen the false scattering. Several test problems are presented to assess the performance and accuracy of the numerical Method. We show that it is possible to obtain a very good level of accuracy. Relative differences less than 1.5% were obtained in comparison with the Monte Carlo reference solution for the selected problems. This study shows the potential of the proposed computational method to be used as an accurate deterministic forward solver in Optical Tomography.

Effects of the approximations of light propagation on quantitative photoacoustic tomography using two-dimensional photon diffusion equation and linearization

Quantitative photoacoustic tomography (QPAT) employing a light propagation model will play an important role in medical diagnoses by quantifying the concentration of hemoglobin or a contrast agent. However, QPAT by the light propagation model with the three-dimensional (3D) radiative transfer equation (RTE) requires a huge computational load in the iterative forward calculations involved in the updating process to reconstruct the absorption coefficient. The approximations of the light propagation improve the efficiency of the image reconstruction for the QPAT. In this study, we compared the 3D/two-dimensional (2D) photon diffusion equation (PDE) approximating 3D RTE with the Monte Carlo simulation based on 3D RTE. Then, the errors in a 2D PDE-based linearized image reconstruction caused by the approximations were quantitatively demonstrated and discussed in the numerical simulations. It was clearly observed that the approximations affected the reconstructed absorption coefficient. The 2D PDE-based linearized algorithm succeeded in the image reconstruction of the region with a large absorption coefficient in the 3D phantom. The value reconstructed in the phantom experiment agreed with that in the numerical simulation, so that it was validated that the numerical simulation of the image reconstruction predicted the relationship between the true absorption coefficient of the target in the 3D medium and the reconstructed value with the 2D PDE-based linearized algorithm. Moreover, the the true absorption coefficient in 3D medium was estimated from the 2D reconstructed image on the basis of the prediction by the numerical simulation. The estimation was successful in the phantom experiment, although some limitations were revealed.

Asymptotic theory of circular polarization memory.

We establish a quantitative theory of circular polarization memory, which is the unexpected persistence of the incident circular polarization state in a strongly scattering medium. Using an asymptotic analysis of the three-dimensional vector radiative transfer equation (VRTE) in the limit of strong scattering, we find that circular polarization memory must occur in a boundary layer near the portion of the boundary on which polarized light is incident. The boundary layer solution satisfies a one-dimensional conservative scattering VRTE. Through a spectral analysis of this boundary layer problem, we introduce the dominant mode, which is the slowest-decaying mode in the boundary layer. To observe circular polarization memory for a particular set of optical parameters, we find that this dominant mode must pass three tests: (1)this dominant mode is given by the largest, discrete eigenvalue of a reduced problem that corresponds to Fourier mode k=0 in the azimuthal angle, and depends only on Stokes parameters U and V; (2)the polarization state of this dominant mode is largely circular polarized so that |V|≫|U|; and (3)the circular polarization of this dominant mode is maintained for all directions so that V is sign-definite. By applying these three tests to numerical calculations for monodisperse distributions of Mie scatterers, we determine the values of the size and relative refractive index when circular polarization memory occurs. In addition, we identify a reduced, scalar-like problem that provides an accurate approximation for the dominant mode when circular polarization memory occurs.

The performances of the discrete ordinates-collocation spectral method for the three-dimensional radiative transfer equation

ABSTRACTThe discrete ordinates-collocation spectral method (DO-CSM) is successfully applied to solve the three-dimensional radiative transfer equation (RTE). The RTE is angular discretized by the DOM first and then solved by the CSM. Two examples are adopted to identify the performances of the DO-CSM for 3D RTE. The results that are obtained from the DO-CSM and the discrete ordinates method with a diamond scheme (DOM-DS), respectively, are compared with those of the exact solutions or benchmarks. These comparisons indicate that the DO-CSM can give more accurate intensity distribution in both space and direction than the DOM-DS. The false scattering of the DO-CSM is much less than that of the DOM-DS. The errors of the DO-CSM are primarily due to the “ray-effect” and the errors caused by the false scattering can even be ignored when the resolutions are fine enough. The errors due to the ray-effect are significant especially for an optical thin medium and the ray-effects are largely suppressed by the finer angular discretization. The results of the DO-CSM improved more quickly than those of the DOM-DS with the increase in the number of directions because the errors caused by the false scattering of the DO-CSM are much less than those of the DOM-DS.

Analytical solution of beam spread function for ocean light radiative transfer.

We develop a new method to analytically obtain the beam spread function (BSF) for light radiative transfer in oceanic environments. The BSF, which is defined as the lateral distribution of the (scalar) irradiance with increasing depth in response to a uni-directional beam emanating from a point source in an infinite ocean, must in general be obtained by solving the three-dimensional (3D) radiative transfer equation (RTE). By taking advantage of the highly forward-peaked scattering property of the ocean particles, we assume, for a narrow beam source, the dependence of radiance on polar angle and azimuthal angle is deliberately separated; only single scattering takes place in the azimuthal direction while multiple scattering still occurs in the polar direction. This assumption enables us to reduce the five-variable 3D RTE to a three-variable two-dimensional (2D) RTE. With this simplification, we apply Fourier spectral method to both spatial and angular variables so that we are able to analytically solve the 2D RTE and obtain the 2D BSF accordingly. Using the relations between 2D and 3D solutions acquired during the process of simplification, we are able to obtain the 3D BSF in explicit form. The 2D and 3D analytical solutions are validated by comparing with Monte Carlo radiative transfer simulations. The 2D analytical BSF agrees excellently with the Monte Carlo result. Despite assumptions of axial symmetry and spike-like azimuthal profile of the radiance in deriving the 3D BSF, the comparisons to numerical simulations are very satisfactory especially for limited optical depths (< O(5)) for single scattering albedo values typical in the ocean. The explicit form of the analytical BSF and the significant gain in computational efficiency (several orders higher) relative to RTE simulations make many forward and inverse problems in ocean optics practical for routine applications.

Solving the atmospheric scattering optical transfer function using the multi-coupled single scattering method

The atmospheric scattering optical transfer function (OTF) is solved by applying the multi-coupled single scattering (MCSS) method to the three-dimensional radiative transfer equation (RTE) under the periodic ground condition. This approach is a direct hit to the atmospheric scattering OTF using the same original context of modulation transfer function (MTF) measurement, i.e., images of sinusoidal grating at different spatial frequencies. Both the amplitude and phase shift of the OTF at various zenith and azimuth angles can be obtained at an arbitrary spatial frequency.

Adjoint methods for adjusting three-dimensional atmosphere and surface properties to fit multi-angle/multi-pixel polarimetric measurements

This paper derives an efficient procedure for using the three-dimensional (3D) vector radiative transfer equation (VRTE) to adjust atmosphere and surface properties and improve their fit with multi-angle/multi-pixel radiometric and polarimetric measurements of scattered sunlight. The proposed adjoint method uses the 3D VRTE to compute the measurement misfit function and the adjoint 3D VRTE to compute its gradient with respect to all unknown parameters. In the remote sensing problems of interest, the scalar-valued misfit function quantifies agreement with data as a function of atmosphere and surface properties, and its gradient guides the search through this parameter space. Remote sensing of the atmosphere and surface in a three-dimensional region may require thousands of unknown parameters and millions of data points. Many approaches would require calls to the 3D VRTE solver in proportion to the number of unknown parameters or measurements. To avoid this issue of scale, we focus on computing the gradient of the misfit function as an alternative to the Jacobian of the measurement operator. The resulting adjoint method provides a way to adjust 3D atmosphere and surface properties with only two calls to the 3D VRTE solver for each spectral channel, regardless of the number of retrieval parameters, measurement view angles or pixels. This gives a procedure for adjusting atmosphere and surface parameters that will scale to the large problems of 3D remote sensing. For certain types of multi-angle/multi-pixel polarimetric measurements, this encourages the development of a new class of three-dimensional retrieval algorithms with more flexible parametrizations of spatial heterogeneity, less reliance on data screening procedures, and improved coverage in terms of the resolved physical processes in the Earth׳s atmosphere.

Application of the multigrid method in a deterministic solution scheme for the three-dimensional radiative transfer equation

An improved solution scheme is developed for the three-dimensional radiative transfer equation (RTE) in inhomogeneous cloudy atmospheres. This solution scheme is deterministic (explicit) and utilizes spherical harmonics series expansion and the finite-volume method for discretization of the RTE. The first-order upwind finite difference is modified to take into account bidirectional flow of radiance in spherical harmonics space, and an iterative solution method is applied. The multigrid method, which is generally employed to achieve rapid convergence in iterative calculation, is incorporated into the solution scheme. The present study suggests that the restriction and prolongation procedure for the multigrid method must be also modified to account for bidirectional flow, and proposes an efficient bidirectional restriction/prolongation procedure that does not increase the computational effort for coarser grids, resulting in a type of wavelet low-pass filter. Several calculation examples for various atmosphere models indicate that the proposed solution scheme is effective for rapid convergence and suitable for obtaining adequate radiation fields in inhomogeneous cloudy atmospheres, although a comparison with the Monte Carlo method suggests that the radiances obtained by this solution scheme at certain angles tends to be smoother.

Spatially modulated light source obliquely incident on a semi-infinite scattering medium

The three-dimensional radiative transfer equation is solved in the spatial frequency domain for modeling the light propagation due to a spatially modulated light source obliquely incident on a semi-infinite uniform medium. The dependence of the derived solution on the spatial frequencies as well as on position and direction is found analytically. The main computational procedure arises from the determination of several constants obtained by a system of linear equations. The obtained equations are verified and illustrated by comparisons with Monte Carlo simulations and the diffusion approximation, respectively.

Light transport in three-dimensional semi-infinite scattering media

The three-dimensional radiative transfer equation is solved for modeling the light propagation in anisotropically scattering semi-infinite media such as biological tissue, considering the effect of internal reflection at the interfaces. The two-dimensional Fourier transform and the modified spherical harmonics method are applied to derive the general solution to the associated homogeneous problem in terms of analytical functions. The obtained solution is used for solving boundary-value problems, which are important for applications in the biomedical optics field. The derived equations are successfully verified by comparisons with Monte Carlo simulations.

Canopy spectral invariants. Part 1: A new concept in remote sensing of vegetation

The concept of canopy spectral invariants expresses the observation that simple algebraic combinations of leaf and canopy spectral reflectance become wavelength independent and determine two canopy structure specific variables – the recollision and escape probabilities. These variables specify an accurate relationship between the spectral response of a vegetation canopy to incident solar radiation at the leaf and the canopy scale. They are sensitive to important structural features of the canopy such as forest cover, tree density, leaf area index, crown geometry, forest type and stand age. This paper presents the mathematical basis of the concept which is linked to eigenvalues and eigenvectors of the three-dimensional radiative transfer equation.

Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems II The hybrid matrix operator--Monte Carlo method

A hybrid method is developed to solve the vector radiative transfer equation (VRTE) in a three-dimensional atmosphere-ocean system (AOS). The system is divided into three parts: the atmosphere, the dielectric interface, and the ocean. The Monte Carlo method is employed to calculate the impulse response functions (Green functions) for the atmosphere and ocean. The impulse response function of the dielectric interface is calculated by the Fresnel formulas. The matrix operator method is then used to couple these impulse response functions to obtain the vector radiation field for the AOS. The primary advantage of this hybrid method is that it solves the VRTE efficiently in an AOS with different dielectric interfaces while keeping the same atmospheric and oceanic conditions. For the first time, we present the downward radiance field in an ocean with a sinusoidal ocean wave.

Impulse response solution to the three-dimensional vector radiative transfer equation in atmosphere-ocean systems I Monte Carlo method

We have developed a powerful 3D Monte Carlo code, as part of the Radiance in a Dynamic Ocean (RaDyO) project, which can compute the complete effective Mueller matrix at any detector position in a completely inhomogeneous turbid medium, in particular, a coupled atmosphere-ocean system. The light source can be either passive or active. If the light source is a beam of light, the effective Mueller matrix can be viewed as the complete impulse response Green matrix for the turbid medium. The impulse response Green matrix gives us an insightful way to see how each region of a turbid medium affects every other region. The present code is validated with the multicomponent approach for a plane-parallel system and the spherical harmonic discrete ordinate method for the 3D scalar radiative transfer system. Furthermore, the impulse response relation for a box-type cloud model is studied. This 3D Monte Carlo code will be used to generate impulse response Green matrices for the atmosphere and ocean, which act as inputs to a hybrid matrix operator-Monte Carlo method. The hybrid matrix operator-Monte Carlo method will be presented in part II of this paper.

A Finite-Volume Solution with a Bidirectional Upwind Difference Scheme for the Three-Dimensional Radiative Transfer Equation

Abstract A new calculation scheme is proposed for the explicitly discretized solution of the three-dimensional (3D) radiation transfer equation (RTE) for inhomogeneous atmospheres. To separate the independent variables involved in the 3D RTE approach, the spherical harmonic series expansion was used to discretize the terms, depending on the direction of the radiance, and the finite-volume method was applied to discretize the terms, depending on the spatial coordinates. A bidirectional upwind difference scheme, which is a specialized scheme for the discretization of the partial differential terms in the spherical harmonic-transformed RTE, was developed to make the equation determinate. The 3D RTE can be formulated as a simultaneous linear equation, which is expressed in the form of a vector–matrix equation with a sparse matrix. The successive overrelaxation method was applied to solve this equation. Radiative transfer calculations of the solar radiation in two-dimensional cloud models have shown that this method can properly simulate the radiation field in inhomogeneous clouds. A comparison of the results obtained using this method with those using the Monte Carlo method shows reasonable agreement for the upward flux, the total downward flux, and the intensities of radiance.

Parallelizing the Discrete Ordinates Method (DOM) for Three-Dimensional Radiative Heat Transfer Calculations using a Priority Queuing Technique

A parallelizing algorithm based on a priority queuing system is developed for solving the three-dimensional radiative transfer equation (RTE) using the discrete ordinates method (DOM) in an absorbing-emitting medium. The classical iterative method of initializing intensity values at processor boundaries and conducting iterations until convergence is very inefficient, and hence a better technique is desired. The method developed here is called the staged technique and relies on the careful organization and prioritization of the multiple inherently serial calculations present in the classical algorithm. A parallel performance analysis is presented for this technique, and measured values of performance metrics are compared to theoretical values. Although theoretical values exhibit possibilities of efficiencies up to 50%, the measured values are lower due to parallel logic overhead. Overall, better performance is obtained for larger problem sizes solved on a relatively small number of processors, but performance can be low outside this configuration.

Multiple-Scattering Formulation of Pulsed Beam Waves in Hydrometeors and Its Application to Millimeter-Wave Weather Radar

This letter deals with the backscattering of millimeter pulsed beam waves from hydrometeors. A new approach is presented for a solution of time-dependent three-dimensional vector radiative transfer equation for the Stokes vectors to study the multiple-scattering effects of beam waves on radar echoes. General solutions for beam waves are derived in an integral form without any approximation. They are given in numerically tractable forms representing the scattering process in the space and time domain. Time-dependent second-order solutions for radar echoes of pulsed beam waves are straightforwardly obtained to predict multiple-scattering effects depending on the variation of an incident beam size. It is shown that the inhomogeneity of the radial direction of beam waves causes the mode coupling of waves between the azimuth directions in the scattering matrix, and that the mode coupling depends on the ratio of the incident beam size to the total mean free path length of the medium

An Assessment of the Parameterization of Subgrid-Scale Cloud Effects on Radiative Transfer. Part II: Horizontal Inhomogeneity

Abstract The role of horizontal inhomogeneity in radiative transfer through cloud fields is investigated within the context of the two-stream approximation. Spatial correlations between cloud optical properties and the radiance field are introduced in the three-dimensional radiative transfer equation and lead to a two-stream model in which the correlations are represented by parameterizations. The behavior of the model is examined using simple single-layer single-column atmospheres. Positive correlations between extinction or scattering and the radiance field are shown to decrease transmission, increase reflection, and increase absorption within inhomogeneous media. The parameterization is used to evaluate the characteristics of inhomogeneous cloud fields observed by radar and lidar over a number of different locations and seasons, revealing that shortwave transfer is generally characterized by negative correlations between extinction and radiance, while longwave transfer is characterized by positive correlations. The results from this characterization are applied to the integration of an atmospheric general circulation model. Model surface temperatures are significantly affected, largely in response to changes in downwelling radiative fluxes at the surface induced by changes in cloud cover and water vapor distributions.

Scattering and emission from inhomogeneous vegetation canopy and alien target beneath by using three-dimensional vector radiative transfer (3D-VRT) equation

To solve the 3D-VRT equation for the model of spatially inhomogeneous scatter media, the finite enclosure of the scatter media is geometrically divided, in both vertical z and transversal ( x , y ) directions, to form very thin multi-boxes. The zeroth order emission, first-order Mueller matrix of each thin box and an iterative approach of high-order radiative transfer are applied to derive high-order scattering and emission of whole inhomogeneous scatter media. Numerical results of polarized brightness temperature at microwave frequency and under different radiometer resolutions from inhomogeneous scatter model such as vegetation canopy and alien target beneath canopy are simulated and discussed.