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.     

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.     

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.     

One-dimensional solar radiative transfer: Perturbation approach and its application to independent-pixel calculations for realistic cloud fields

The radiative transfer perturbation theory (RTPT), which has already been introduced in atmospheric radiative transfer several years ago, is applied to cloud related problems. The RTPT requires the solution of the radiative transfer equation in the forward and the adjoint mode. The basic principles of this technique are presented as well as its extensions to isotropic surface reflection and its conjunction with the Hermite interpolation. This set of methods is applied to different atmospheric conditions including realistic cloud scenes. The results are compared with the usual (forward) independent-pixel calculations with respect to errors of individual pixels and domain-averaged values. The RTPT turns out to be sufficiently accurate in the case the clouds’ internal vertical variations remain moderate. It is also shown that, depending on the specific radiative transfer problem, the RTPT can offer some advantages on computational speed. However, the limitations of the RTPT with regard to realistic clouds are addressed as well.     

Towards linearization of atmospheric radiative transfer in spherical geometry

We present a general approach for the linearization of radiative transfer in a spherical planetary atmosphere. The approach is based on the forward-adjoint perturbation theory. In the first part we develop the theoretical background for a linearization of radiative transfer in spherical geometry. Using an operator formulation of radiative transfer allows one to derive the linearization principles in a universally valid notation. The application of the derived principles is demonstrated for a radiative transfer problem in simplified spherical geometry in the second part of this paper. Here, we calculate the derivatives of the radiance at the top of the atmosphere with respect to the absorption properties of a trace gas species in the case of a nadir-viewing satellite instrument.     

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.     

Radiative transfer through inhomogeneous turbid media: implementation of the adjoint perturbation approach at the first order

Recently, the advantages of application of the perturbation technique which is based on joint use of both the direct and adjoint solutions of the radiative transfer equation to solve and analyze some 1D problems of atmospheric physics has been demonstrated. In this paper this technique is applied to problems of radiative transfer through spatially inhomogeneous scattering and absorbing media. This technique is shown to allow one both to obtain the solution with reasonable accuracy and to get physical insight into the problem under consideration. The accuracy of the perturbation technique is demonstrated through comparison with results from the SHDOM simulation code for one problem of cloud optics.     

Adjoint sensitivity analysis of radiative transfer equation: 2. Applications to retrievals of temperature in scattering atmospheres in thermal IR

An approach to formulation of inversion algorithms for thermal sounding in the case of scattering atmosphere based on the adjoint equation of radiative transfer (Ustinov, JQSRT 68 (2001) 195, referred to as Paper 1 in the main text) is applied to temperature retrievals in the scattering atmosphere for the nadir viewing geometry. Analytical expressions for the weighting functions involving the integration of the source function are derived. Temperature weighting functions for a simple model of the atmosphere with scattering are evaluated and convergence to the case of pure atmospheric absorption is demonstrated. The numerical experiments on temperature retrievals are carried out to demonstrate the validity of the expressions obtained.     

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.     

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.     

DIFFERENTIALLY MOVING MEDIA WITH MANY SPECTRAL LINES: STOCHASTIC APPROACH

Based upon the analytical solution of the radiative transfer equation for a given source function and a new approach to account for very many spectral lines contributing to the extinction, the connection between line properties and the emergent intensity is derived under the assumption that the wavelengths of the line centers follow a Poisson point process, whereas the other line parameters may have arbitrary distribution functions.A comparison with the widely used list of Kurucz shows that the Poisson distribution well describes deterministic “real” lines. The presentation by a Poisson point process requires only a modest number of parameters and is very flexible. It allows most operations to be carried out analytically and hence is very suitable to study the intricate influence of many lines on radiation fields in differentially moving media.We consider a simplified case of the solution of the radiative transfer equation in order to demonstrate the basic effects of the velocity field upon the emerging radiation field. Expressions for the expectation value of the intensity are derived, and examples are given for Lorentz line profiles and infinitely sharp lines, in particular as functions of the velocity gradient and the mean line density.     

Convergence properties of the accelerated Λ-iteration method for the solution of radiative transfer problems

We investigate the convergence properties and the computational speed of the accelerated Λ-iteration method for the solution of a wide variety of radiative transfer problems. The formal solution of the transfer equation is done using the short characteristics method. As the approximate Λ-operator we use bands of the discretized Λ-operator and derive the formulae for computing operators with arbitrary bandwidth. We define the optimum bandwidth of the approximate Λ-operator as the bandwidth for which the CPU time used for the solution of any particular radiative transfer problems is minimum. The radiative transfer equation is solved for a number of continuum and line transfer problems in spherical symmetry and for static to very rapidly expanding media. We find the optimum bandwidth is around 5–15 for workstation class computers whereas it is around 1–2 for supercomputers.     

Scattering in thick multifractal clouds, Part II: Multiple scattering

In Part I of this paper, we developed asymptotic approximations for single photon scattering in thick, highly heterogeneous, “Log-Lévy” multifractal clouds. In Part II, theoretical multiple scattering predictions are numerically tested using Monte Carlo techniques, which show that, due to long range correlations, the photon paths are “subdiffusive” with the corresponding fractal dimensions tending to increase slowly with mean optical thickness. We develop reasonably accurate statistical relations between N scatter statistics in thick clouds and single scatter statistics in thin clouds. This is explored further using discrete angle radiative transfer (DART) approach in which the radiances decouple into non-interacting families with only four (for 2-D clouds) radiance directions each. Sparse matrix techniques allow for rapid and extremely accurate solutions for the transfer; the accuracy is only limited by the spatial discretization.By “renormalizing” the cloud density, we relate the mean transmission statistics to those of an equivalent homogeneous cloud. This simple idea is remarkably effective because two complicating effects act in contrary directions: the “holes” which lead to increased single scatter transmission and the tendency for multiply scattered photons to become “trapped” in optically dense regions, thus decreasing the overall transmission.     

The shape gradient of the least-squares objective functional in optimal shape design problems of radiative heat transfer

Optimal shape design problems of steady-state radiative heat transfer are considered. The optimal shape design problem (in the three-dimensional space) is formulated as an inverse one, i.e., in the form of an operator equation of the first kind with respect to a surface to be optimized. The operator equation is reduced to a minimization problem via a least-squares objective functional. The minimization problem has to be solved numerically. Gradient minimization methods need the gradient of a functional to be minimized. In this paper the shape gradient of the least-squares objective functional is derived with the help of the shape sensitivity analysis and adjoint problem method. In practice a surface to be optimized may be (or, most likely, is to be) given in a parametric form by a finite number of parameters. In this case the objective functional is, in fact, a function in a finite-dimensional space and the shape gradient becomes an ordinary gradient. The gradient of the objective functional, in the case that the surface to be optimized is given in a finite-parametric form, is derived from the shape gradient. A particular case, that a surface to be optimized is a “two-dimensional” polyhedral one, is considered. The technique, developed in the paper, is applied to a synthetic problem of designing a “two-dimensional” radiant enclosure.     

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.     

Many polarized radiative transfer models

This note is an introduction to the reprint of the 1991 JQSRT article “A new polarized atmospheric radiative transfer model” by K.F. Evans and G.L. Stephens. We discuss the significance of the article, how our two plane-parallel polarized radiative transfer codes came about, how our codes have been used, and more recent developments in polarized radiative transfer modeling.     

Passive remote sensing of planetary atmospheres and retrievals of atmospheric macro- and microphysical parameters

In two recent papers, referred to as Paper 1 and Paper 2 in the main text, we have pursued an idea that atmospheric weighting functions for any geophysical parameters essentially consist of two kinds of entities: weighting functions for the atmospheric radiative parameters, which directly enter the radiative transfer equation, and partial derivatives of the atmospheric radiative parameters with respect to atmospheric geophysical parameters. Corresponding analysis was performed for non-scattering and scattering planetary atmospheres for nadir-viewing geometry and thermal spectral region. In the present paper we conduct a study of the second group of above entities, the partial derivatives of atmospheric radiative parameters with respect to atmospheric geophysical parameters. Along the way, we analyze the role of radiative parameters of individual atmospheric constituents, which serve as intermediaries between the total radiative parameters of the atmosphere and its geophysical parameters. The obtained expressions, combined with radiative weighting functions, are used in the end-to-end expressions for the weighting functions obtained for various specific geophysical parameters.