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.     

A linearized vector radiative transfer model for atmospheric trace gas retrieval

We present a plane parallel radiative transfer model for polarized light, that provides the intensity vector as well as the derivatives of the four Stokes parameters with respect to atmospheric trace gas profiles. These derivatives are essential for retrieval of height resolved trace gas information from satellite measurements of backscattered sunlight. The model uses the Gauss-Seidel iteration technique for solving the radiative transfer equation. For the first time, the forward-adjoint radiative perturbation theory is applied for the linearization of a radiative transfer model including polarization. The accuracy of the model is better than 0.025% for all four Stokes parameters and better than 0.03% for the derivatives.     

Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. I. Analytic Jacobians from the linearized CAO-DISORT model

In this paper and the sequel, we investigate the application of classic inverse methods based on iterative least-squares cost-function minimization to the simultaneous retrieval of aerosol and ocean properties from visible and near infrared spectral radiance measurements such as those from the SeaWiFS and MODIS instruments. Radiance measurements at the satellite are simulated directly using an accurate coupled atmosphere-ocean-discrete-ordinate radiative transfer (CAO-DISORT) code as the main component of the forward model. For this kind of cost-function inverse problem, we require the forward model to generate weighting functions (radiance partial derivatives) with respect to the aerosol and marine properties to be retrieved, and to other model parameters which are sources of error in the retrievals.In this paper, we report on the linearization of the CAO-DISORT model. This linearization provides a complete analytic differentiation of the coupled-media radiative transfer theory, and it allows the model to generate analytic weighting functions for any atmospheric or marine parameter. For high solar zenith angles, we give an implementation of the pseudo-spherical (P-S) approach to solar beam attenuation in the atmosphere in the linearized model. We summarize a number of performance enhancements such as the use of an exact single-scattering calculation to improve accuracy. We derive inherent optical property inputs for the linearized CAO-DISORT code for a simple 2-parameter bio-optical model for the marine environment coupled to a 2-parameter bimodal atmospheric aerosol medium.     

Fast and accurate 4 and 6 stream linearized discrete ordinate radiative transfer models for ozone profile retrieval

The global and long-term measurement of ozone vertical and horizontal distributions is one of the most important tasks in the monitoring of the earth's atmosphere. A number of satellite instruments are capable of delivering ozone profile distributions from UV nadir backscatter measurements. Retrieval algorithms should be efficient enough to deliver profiles in real-time without compromising accuracy. Such algorithms require a radiative transfer model that can generate quickly and accurately both simulated radiances and Jacobian matrices of weighting functions. We develop fast and analytic 4 stream and 6 stream linearized discrete ordinate models designed to satisfy performance and accuracy requirements for such an algorithm. The models have the pseudo-spherical treatment of the direct beam attenuation. For anisotropic scattering we use the delta-M scaling method to deal with strong forward scattering peaks. We demonstrate that the accuracy of the models is improved greatly upon application of a single scatter correction based on an exact specification of the phase function. For wide-angle off-nadir viewing, a sphericity correction is developed to deal more precisely with attenuation in a curved atmosphere. Radiances and weighting functions for the 4 and 6 stream models are compared with 20 stream output from the LIDORT model. We show that for the UV range pertinent to ozone profile retrieval from space, the 4 stream model generates backscatter radiances to an accuracy >1.25% for all viewing situations in a clear sky Rayleigh and background aerosol reference atmosphere, and up to 1.75% for a number of special scenarios with optically thick particulate layers. Six stream radiances are accurate to the 0.25% level for clear sky situations, and 0.65% for the special cases; weighting functions for the 6 stream output are accurate to ±2% in all cases. We discuss the implications of these comparisons regarding the performance and accuracy of the radiative transfer forward model in the ozone profile retrieval context.     

Application of multidimensional scheme and the discrete ordinate method to radiative heat transfer in a two-dimensional enclosure with diffusely emitting and reflecting boundary walls

Radiative heat transfer is the dominant mode of heat transfer in many engineering problems, including combustion chambers, space, greenhouses, rocket plume sensing, among others. The aim of this study is to develop an efficient method capable of eliminating ray effects in complex 2D situations and to use the developed code for other problems including combined conduction and convection in connection with CFD codes. A complete genuinely multidimensional discretization in two-dimensional discrete ordinates method is formulated to solve radiative heat transfer in a rectangular enclosure composed of diffusely emitting and reflecting boundaries and containing homogeneous media that absorbs, emits and scatters radiation. A new genuinely multidimensional differencing scheme is used to solve the radiative transfer equation with S₄, S₆, S₈, T₆, T₇, T₈ and T₉ angular quadrature schemes. Different cases are analyzed and the results are compared when possible with those obtained by others researchers.     

Adjoint problem of radiative transfer for a pseudo-spherical atmosphere and general viewing geometries

We formulate the adjoint radiative transfer for a pseudo-spherical atmosphere and various retrieval scenarios. The single scattering radiance is computed in a spherical atmosphere by using the source integration technique, while for the multiple scattering radiance we formulate an one-dimensional adjoint radiative transfer equation in a plane-parallel atmosphere. The adjoint solution of the radiative transfer equation is obtained by employing the discrete ordinate method with matrix exponential. We provide an abbreviated derivation of our formalism as well as a discussion of the numerical implementation of the theory.     

On the use of a meshless method for solving radiative transfer with the discrete ordinates formulations

A meshless method is presented for solving the radiative transfer equation in the discrete ordinates approach. It is shown that the primitive variables formulation is unstable for low values of the absorption coefficient while the even parity formulation is always stable and accurate.     

Discrete-ordinate radiative transfer in a stratified medium with first-order rotational Raman scattering

Rotational Raman scattering (RRS) by air molecules in the Earth's atmosphere is predominantly responsible for the Ring effect: Fraunhofer and absorption-feature filling-in observed in UV/visible backscatter spectra. Accurate determination of RRS effects requires detailed radiative transfer (RT) treatment. In this paper, we demonstrate that the discrete-ordinate RT equations may be solved analytically in a multi-layer multiple scattering atmosphere in the presence of RRS treated as a first-order perturbation. Based on this solution, we develop a generic pseudo-spherical RT model LIDORT-RRS for the determination of backscatter radiances with RRS included; the model will generate output at arbitrary viewing geometry and optical thickness. Model comparisons with measured RRS filling-in effects from OMI observations show very good agreement. We examine telluric RRS filling-in effects for satellite-view backscatter radiances in a spectral range covering the ozone Huggins absorption bands. The model is also used to investigate calcium H and K Fraunhofer filling-in through cloud layers in the atmosphere.     

Finite element formulation of the first- and second-order discrete ordinates equations for radiative heat transfer calculation in three-dimensional participating media

Two finite element methods (FEMs), FEDOM1 and FEDOM2 (standing for the first and the second finite element discrete ordinates methods, respectively), are formulated and numerically tested. The reference second-order discrete equation is modified in its scattering terms and is applied to the problems of absorbing/emitting and anisotropically scattering media by using the FEM. Numerical features of the developed FEMs are compared with one of the discrete ordinates interpolation method (DOIM), which uses a finite difference scheme. Prediction results of radiative heat transfer by these two FEMs are compared with reference solutions and verified in three-dimensional enclosures containing participating media. The results of FEDOM1 and FEDOM2 agree well with exact solutions for the problem of absorbing/emitting medium with various range of optical thickness. Generally, the two FEMs show more accurate results than DOIM. And FEDOM1 shows more accurate results than FEDOM2 in most of the test problems. Both of the developed FEMs show reasonable results compared with published Monte Carlo solutions for the tested absorbing/emitting and anisotropically scattering media. Although the FEDOM2 is not as accurate as the FEDOM1, it shows its own advantages that it reduces CPU time and memory space of dependent variable to half.     

Fast approximate technique for the cumulative wavenumber model to modeling radiative transfer in a mixture of real gas media

In the cumulative wavenumber (CW) model, the total range of the absorption cross-section Cη is subdivided into the supplementary absorption cross-section of gray gases C_j, j=1,…,n, where n is the number of gray gases; and the wavenumber region is subdivided into intervals Δ_i=[η_i−1, η_i], i=1, 2,…,p, where p is the number of intervals. The intersection of the two spectral subdivisions is used to define the modeling of the fractional gray gas D_ij. In the CW model, we solve the radiative transfer equation (RTE) in every subinterval D_ij; then it is necessary to solve n x p times the spectral form of the RTE for complete spectral integration. In this work, the CW model is used with a numerical approximation technique based on additive properties of radiative intensity to reduce the solution of RTE to n new fractional gray gas D_j for complete spectral integration. The CW model was first coupled with the discrete ordinates method and the accuracy of the simplified technique and the algorithm was first examined for one-dimensional homogeneous media; results are compared with line-by-line calculations and it is found that the CW model with the simplified technique is exact for the homogeneous media examined. Also, the fast approach is tested in the diffuse reflecting boundaries case. The CW model is implemented in a bi-dimensional enclosure containing real gases in isothermal cases. Afterwards, this approximate technique is extended to non-isothermal and non-homogeneous cases; the results are compared with line-by-line calculations taken from literature and good agreement was found. The results obtained using the acceleration technique for the CW model agree with the results of original CW model. With this acceleration technique the CPU time decreases p times. Spectral database HITRAN and HITEMP are used to obtain the molecular absorption spectrum of the gases.     

Assessment of the re-ordered wide band model for non-grey radiative transfer calculations in 3D enclosures

The objective of the present study is to evaluate variations of the re-ordered wide band model for non-grey radiative transfer calculations in 3D enclosures using the discrete ordinates method. First, the performance of various angular and spatial discretisation schemes of the discrete ordinates method is investigated. Then, several formulations, averaging procedures, and scaling methods of the re-ordered wide band model are tested, and the results are validated against those of a statistical narrow band model. The grey gases formulation using three optimised absorption coefficient is found to be the most efficient method.     

Least-squares finite element formulations for one-dimensional radiative transfer

We present least-squares-based finite element formulations for the numerical solution of the radiative transfer equation in its first-order primitive variable form. The use of least-squares principles leads to a variational unconstrained minimization problem in a setting of residual minimization. In addition, the resulting linear algebraic problem will always have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We consider space-angle coupled and decoupled formulations. In the coupled formulation, the space-angle dependency is represented by two-dimensional finite element expansions and the least-squares functional minimized in the continuous space-angle domain. In the decoupled formulation the angular domain is represented by discrete ordinates, the spatial dependence represented by one-dimensional finite element expansions, and the least-squares functional minimized continuously in space domain and at discrete locations in the angle domain. Numerical examples are presented to demonstrate the merits of the formulations in slab geometry, for absorbing, emitting, anisotropically scattering mediums, allowing for spatially varying absorption and scattering coefficients. For smooth solutions in space-angle domain, exponentially fast decay of error measures is demonstrated as the p-level of the finite element expansions is increased. The formulations represent attractive alternatives to weak form Galerkin finite element formulations, typically applied to the more complicated second-order even- and odd-parity forms of the radiative transfer equation.     

Linearization of the interaction principle: Analytic Jacobians in the “Radiant” model

In this paper we present a new linearization of the Radiant radiative transfer model. Radiant uses discrete ordinates for solving the radiative transfer equation in a multiply-scattering anisotropic medium with solar and thermal sources, but employs the adding method (interaction principle) for the stacking of reflection and transmission matrices in a multilayer atmosphere. For the linearization, we show that the entire radiation field is analytically differentiable with respect to any surface or atmospheric parameter for which we require Jacobians (derivatives of the radiance field). Derivatives of the discrete ordinate solutions are based on existing methods developed for the LIDORT radiative transfer models. Linearization of the interaction principle is completely new and constitutes the major theme of the paper. We discuss the application of the Radiant model and its linearization in the Level 2 algorithm for the retrieval of columns of carbon dioxide as the main target of the Orbiting Carbon Observatory (OCO) mission.     

The method of lines solution of discrete ordinates method for radiative heat transfer in cylindrical enclosures

A radiation code based on method of lines solution of discrete ordinates method for radiative heat transfer in axisymmetric cylindrical enclosures containing absorbing-emitting medium was developed and tested for predictive accuracy by applying it to (i) test problems with black and grey walls (ii) a gas turbine combustor simulator enclosing a non-homogeneous absorbing-emitting medium and benchmarking its steady-state predictions against exact solutions and measurements. Comparisons show that it provides accurate solutions for radiative heat fluxes and can be used with confidence in conjunction with CFD codes based on the same approach.     

A linearized two-stream radiative transfer code for fast approximation of multiple-scatter fields

Performance is an issue for radiative transfer simulations in hyper-spectral remote sensing backscatter retrieval algorithms. 2-Stream models are often used to speed up flux and radiance calculations. Here we present a linearized 2-stream multiple-scatter code, with the ability to generate analytic weighting functions with respect to any atmospheric or surface property. We examine 2-stream accuracy for the satellite intensity diffuse field, and the corresponding Jacobians for total ozone column and surface albedo, for an application in the ozone UV Huggins bands.     

The forward and inverse problem in tissue optics based on the radiative transfer equation: A brief review

This note serves as an introduction to two papers by Klose et al. [2] and [3] and provides a brief review of the latest developments in optical tomography of scattering tissue. We discuss advancements made in solving the forward model for light propagation based on the radiative transfer equation, in reconstructing scattering and absorption cross sections of tissue, and in molecular imaging of luminescent sources.     

On the finite volume method and the discrete ordinates method regarding radiative heat transfer in acute forward anisotropic scattering media

The ability of the finite volume method (FVM) and the discrete ordinates method (DOM) to model radiative heat transfer in acute forward anisotropic scattering media has been investigated. The test case involves a purely scattering medium in a cubic enclosure, irradiated by one boundary with diffuse emission. Four phase functions have been considered: three of the Henyey-Greenstein type with respective asymmetry factors of 0.2, 0.8 and 0.93, and a Mie phase function with a strong forward scattering peak (computed for a size parameter of 245 and corresponding to an asymmetry factor of 0.93). Results obtained with the FVM are in good agreement with Monte Carlo reference solutions, whatever the level of acute anisotropic scattering (for asymmetry factors up to 0.93). The DOM combined with the renormalization procedures of the phase function proposed by Kim and Lee (Effect of anisotropic scattering on radiative heat transfer in two-dimensional rectangular enclosures. Int J Heat Mass Transfer 1988;31:1711-21. [1]) and Wiscombe (On initialization error and flux conservation in the doubling method. JQSRT 1976;18:637-58. [2]) provides accurate results only for the smallest asymmetry factor. As the asymmetry factor increases, the renormalization procedures induce strong modifications in the values of the discretized phase function resulting in an underestimation of the effective attenuation by scattering. This error has been found to increase with optical thickness. In fact, when using the DOM, results would be more accurate combining this method with a Delta-Eddington approximation of the phase function, instead of using the actual phase function which is altered too much by renormalization.     

On vector radiative transfer equation in curvilinear coordinate systems

The differential operator of polarized radiative transfer equation is examined in case of homogeneous medium in Euclidean three-dimensional space with arbitrary curvilinear coordinate system defined in it. This study shows that an apparent rotation of polarization plane along the light ray with respect to the chosen reference plane for Stokes parameters generally takes place, due to purely geometric reasons. Analytic expressions for the differential operator of transfer equation dependent on the components of metric tensor and their derivatives are found, and the derivation of differential operator of polarized radiative transfer equation has been made a standard procedure. Considerable simplifications take place if the coordinate system is orthogonal.