Extended discrete-ordinate method considering full polarization state

This paper presents an extension to the standard discrete-ordinate method (DOM) to consider generalized sources including: beam sources which can be placed at any (vertical) position and illuminate in any direction, thermal emission from the atmosphere and angularly distributed sources which illuminate from a surface as continuous functions of zenith and azimuth angles. As special cases, the thermal emission from the surface and deep space can be implemented as angularly distributed sources. Analytical-particular solutions for all source types are derived using the infinite medium Green's function. Radiation field zenith angle interpolation using source function integration is developed for all source types. The development considers the full state of polarization, including the sources (as applicable) and the (BRDF) surface, but the development can be reduced easily to scalar problems and is ready to be implemented in a single set of code for both scalar and vector radiative transfer computation.     

Approximate polarization calculations with scalar radiative transfer theory

The Pomraning phase function can be used to perform approximate polarized Rayleigh transfer calculations with a scalar radiative transfer equation. The approximation is numerically tested for the albedo problem consisting of azimuthally independent radiation incident on a homogeneous semi-infinite atmosphere. The numerical tests were carried out with the same approach used by Viik (JQSRT 68 (2000) 319-326) to numerically test the approximate phase function for solving the Milne problem. Away from the surface the Pomraning phase function gives marginally better results for the diffuse radiation than the usual scalar Rayleigh phase function because it was derived from an asymptotic limit more appropriate for deeper locations in an atmosphere. For optical depths less than unity, though, the scalar Rayleigh approximation is better than the Pomraning approximation.     

Radiative transfer solutions for coupled atmosphere ocean systems using the matrix operator technique

Accurate radiative transfer models are the key tools for the understanding of radiative transfer processes in the atmosphere and ocean, and for the development of remote sensing algorithms. The widely used scalar approximation of radiative transfer can lead to errors in calculated top of atmosphere radiances. We show results with errors in the order of±8% for atmosphere ocean systems with case one waters. Variations in sea water salinity and temperature can lead to variations in the signal of similar magnitude. Therefore, we enhanced our scalar radiative transfer model MOMO, which is in use at Freie Universität Berlin, to treat these effects as accurately as possible. We describe our one-dimensional vector radiative transfer model for an atmosphere ocean system with a rough interface. We describe the matrix operator scheme and the bio-optical model for case one waters. We discuss some effects of neglecting polarization in radiative transfer calculations and effects of salinity changes for top of atmosphere radiances. Results are shown for the channels of the satellite instruments MERIS and OLCI from 412.5 nm to 900 nm.     

Discrete-ordinate method with matrix exponential for a pseudo-spherical atmosphere: Vector case

The paper is devoted to the extension of the matrix-exponential formalism for the scalar radiative transfer to the vector case. Using basic results of the theory of matrix-exponential functions we provide a compact and versatile formulation of the vector radiative transfer. As in the scalar case, we operate with the concept of the layer equation incorporating the level values of the Stokes vector. The matrix exponentials which enter in the expression of the layer equation are computed by using the matrix eigenvalue method and the Padé approximation. A discussion of the computational efficiency of the proposed method for both an aerosol-loaded atmosphere as well as a cloudy atmosphere is also provided.     

Comparison between radiative transfer theory and the simplified spherical harmonics approximation for a semi-infinite geometry

In this study, the third-order simplified spherical harmonics equations (SP3), an approximation of the radiative transfer equation, are solved for a semi-infinite geometry considering the exact simplified spherical harmonics boundary conditions. The obtained Green's function is compared to radiative transfer calculations and the diffusion theory. In general, it is shown that the SP3 equations provide better results than the diffusion approximation in media with high absorption coefficient values but no improvement is found for small distances to the source.     

Properties of reflected sunlight derived from a Green's function method

The inference of optical depth and particle size of clouds and aerosols using remotely sensed reflected radiance at solar wavelengths has received much attention recently. The information these measurements provide is path integrated. However, very little is known about the vertical distribution of this weighting. To characterize it, we first solve the radiative transfer equation (RTE) by a Green's function approach, and then investigate the sensitivity of the weighting to vertical inhomogeneities in the extinction by introducing a function that is closely related to the Green's function, herein called the contribution function. This function calculates the contributions to the radiance at the upper boundary of the medium by underlying layers. Three hypothetical clouds of identical optical depth but exhibiting different extinction profiles were used in this study. The contribution function was found very sensitive to the extinction profile. The global reflection and transmission matrices used to construct the Green's function, derived using an eigenmatrix method, resulted in an efficient, stable, and accurate method for calculating the emerging radiances that can be extended to multi-layered media.     

A simple new radiative transfer model for simulating the effect of cirrus clouds in the microwave spectral region

The upwelling atmospheric radiation in the millimeter wave spectral range is influenced by the presence of cirrus clouds. A plane parallel radiative transfer model which can take into account the effect of multiple scattering by ice particles in the cirrus has been developed and is used to simulate the brightness temperatures as they would be measured by a satellite instrument. The model uses an iterative procedure to solve the radiative transfer equation. The formulation of the model is such that it can easily be adapted to treat the full specific intensity vector instead of just the scalar total intensity. A convergence test for the model is explained and two cirrus cloud scenarios are simulated. The results illustrate the linearity of microwave radiative transfer for not too strong cirrus clouds in this frequency region.     

Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system

A vector radiative transfer model has been developed for a coupled atmosphere-ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.     

Integration of Chandrasekhar's integral equation

We solve Chandrasekhar's integration equation for radiative transfer in the plane-parallel atmosphere by iterative integration. The primary thrust in radiative transfer has been to solve the forward problem, i.e., to evaluate the radiance, given the optical thickness and the scattering phase function. In the area of satellite remote sensing, our problem is the inverse problem: to retrieve the surface reflectance and the optical thickness of the atmosphere from the radiance measured by satellites. In order to retrieve the optical thickness and the surface reflectance from the radiance at the top-of-the atmosphere (TOA), we should express the radiance at TOA “explicitly” in the optical thickness and the surface reflectance. Chandrasekhar formalized radiative transfer in the plane-parallel atmosphere in a simultaneous integral equation, and he obtained the second approximation. Since then no higher approximation has been reported. In this paper, we obtain the third approximation of the scattering function. We integrate functions derived from the second approximation in the integral interval from 1 to ∞ of the inverse of the cos of zenith angles. We can obtain the indefinite integral rather easily in the form of a series expansion. However, the integrals at the upper limit, ∞, are not yet known to us. We can assess the converged values of those series expansions at ∞ through calculus. For integration, we choose coupling pairs to avoid unnecessary terms in the outcome of integral and discover that the simultaneous integral equation can be deduced to the mere integral equation. Through algebraic calculation, we obtain the third approximation as a polynomial of the third degree in the atmospheric optical thickness.     

Vector Diffraction Integrals for Solving Inverse Problems of Radio-Holographic Sensing of the Earth's Surface and Atmosphere

Vector relationships between the fields on a certain surface confining an inhomogeneous three-dimensional volume and the fields inside this volume are obtained by the Stratton–Chu method developed for the case of homogeneous media. The vector relationships allow us to solve the direct and inverse problems of determining the fields inside an inhomogeneous medium given the field on its boundary. The vector equations take into acount the polarization changes of direct and inverse waves propagated in an inhomogeneous medium. In the case of a two-dimensional homogeneous medium, the vector equations reduce to the previously obtained scalar equations used in the approximation of spherical symmetry to describe the process of backward wave propagation during the atmospheric and ionospheric radio-occultation monitoring. It is shown that the Green's function of the scalar wave equation in an inhomogeneous medium should be used as the reference signal for solving the inverse problem of radio-occultation monitoring. This validates the method of focused synthetic aperture previously used for high-accuracy retrieval of the vertical refractive-index profiles in the ionosphere and atmosphere. In this method, the reference-signal phase was determined from a model which describes with sufficient accuracy the radiophysical parameters of a refracting medium in the region of radio-occultation sensing. The obtained equations can be used for the high-accuracy solving of inverse problems of radio-holographic sensing of the Earth's atmosphere and surface by precision signals from radio-navigation satellites.     

A perturbation solution of the radiative transfer equation in a differentially moving atmosphere

We present a new technique for solving the radiative transfer equation in a differentially moving atmosphere. The method is based on a pertubation of the solution of the transfer problem in a static atmosphere. The perturbation technique may be applied with any method for solving the static atmosphere problem and leads to significant reductions in computer time and storage requirements.The method is flexible and may be used to solve problems involving depth dependence in any of the parameters of the transfer equation.     

Extension of the discrete-ordinate algorithm and efficient radiative transfer calculation

As an accurate and efficient algorithm, the discrete-ordinate method (DOM) has been used to solve the radiative transfer problem of plane-parallel scattering atmosphere illuminated by a parallel beam, an idealized case of the sun, from above the atmosphere. In this paper, we extend this algorithm so that radiative problems of more general sources, such as parallel surface sources that illuminate with a parallel beam in any direction from any vertical position, and general surface sources that illuminate continuously in a hemisphere, can be solved. For a problem where intensity distributions are sought for a number of different sources within the same atmosphere-surface system, the intrinsic properties of DOM are used so that the time required for the solution for extra sources is reduced to a substantially small amount. In the case of parallel surface sources, numerical testing has shown that the amount can be reduced to as little as 15% of a full solution. Examples of applications are presented.     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

Computation of Green's function for radiative transfer

In an accompanying paper, we develop the computational expressions for the higher order perturbation of the radiative transfer equation, and present some numerical results for typical cases. In this article, we discuss a number of issues regarding the implementation of the HOP computation: obtaining the Green's function, its expansion as a double series of Legendre polynomials, and obtaining the adjoint radiance of more general sources such as those for the fluxes at arbitrary altitudes. Examples of Green's function and its expansion coefficients are presented.     

Imaging theory of an aplanatic system with a stratified medium based on the method for a vector coherent transfer function

A simple formalism relating image fields to object fields, similar to that of the scalar and paraxial case, is presented for an aplanatic system obeying the sine condition, which shows that the vector plane-wave spectrum of image fields is equal to the product of the vector coherent transfer function due to the x- and y-polarized point electric field source and the scalar spectrum of the corresponding transverse object fields. Utilizing this formula and dyadic Green's function, a rigorous imaging theory of an aplanatic system for the point electric current source through a stratified medium is readily developed.     

Electromagnetic volume scattering in the half-space: a moment-method analysis without Sommerfeld integrals or their approximations

Traditionally, in moment-method analyses of electromagnetic scattering, the elements of the impedance matrix are calculated as convolutions of the basis elements with the appropriate dyadic Green's function. However, for scattering in the half-space, the vertical and azimuthal copolar terms of the Green's function require evaluation of Sommerfeld integrals which are computationally burdensome. In this paper, it is shown that, in populating the impedance matrix for the half-space problem, evaluation of Sommerfeld integrals is, in fact, not necessary. For monochromatic excitation, the plane-wave expansion of the scattered field constitutes a Fourier transform, in the horizontal plane, of a vector spectral function. This vector function results from the convolution, in the vertical dimension, of the respective angular spectra of the Green's function and the equivalent current. On application of the moment method, through the Weyl identity, the impedance-matrix elements corresponding to the singular terms of the Green's function are convolutions in the horizontal plane of spherical potentials, and Fourier transforms of scalar spectral functions. These scalar functions are derived from the basis elements and, with a judicious choice of basis, they are well behaved and of compact support, and consequently their Fourier transforms can be computed as FFTs.     

Transmission of radiation through an aerosol medium

The properties of radiation through an aerosol medium have been achieved. This has been done by employing Mie scattering theory to calculate the radiation transfer scattering parameters in the form of extinction, absorption and scattering efficiencies. The equation of radiative transfer for the heat flux through a plane parallel atmosphere of aerosol has been solved. The aerosol size distributions are found in practical systems. Average efficiencies over size distribution for spherical particles of complex refractive index are calculated. Therefore, the radiative properties of stratospheric aerosols have been done. The obtained results found to be in a good agreement with the previous work.     

A fast linearized pseudo-spherical two orders of scattering model to account for polarization in vertically inhomogeneous scattering-absorbing media

We calculate the reflection matrix for the first two orders of scattering in a vertically inhomogeneous, scattering-absorbing medium. We take full account of polarization and perform a complete linearization (analytic differentiation) of the reflection matrix with respect to both the inherent optical properties of the medium and the surface reflection condition. Further, we compute a scalar-vector correction to the total intensity due to the effect of polarization; this correction is also fully linearized. The solar beam attenuation has been computed for a pseudo-spherical atmosphere.Results from the two orders of scattering (2OS) model have been tested against scalar intensities for an inhomogeneous atmosphere, and against Stokes vector results for a homogeneous atmosphere. We have also performed backscatter simulations of reflected sunlight in the O₂A band for a variety of geometries, and compared our results with those from a full vector multiple scattering code. Our results are exact in the center of strong lines and most inaccurate in the continuum, where polarization is least significant. The s- and p-polarized radiances are always computed very accurately. The effect of gas absorption optical depth, solar zenith angle, viewing geometry, surface albedo and wind speed (in the case of ocean glint) on the intensity, polarization and corresponding weighting functions have been investigated. It is shown that the 2OS model provides fast and reliably accurate polarization corrections to the scalar-model radiance and weighting function fields. The model can be implemented in operational retrieval algorithms as an adjunct radiative transfer code to deal with polarization effects.     

Scalar Green's functions in an Euclidean space with a conical-type line singularity

In an Euclidean space with a conical-type line singularity, we determine the Green's function for a charged massive scalar field interacting with a magnetic flux running through the line singularity. We give an integral expression of the Green's function and a local form in the neighbourhood of the point source, where it is the sum of the usual Green's function in Euclidean space and a regular term. As an application, we derive the vacuum energy-momentum tensor in the massless case for an arbitrary magnetic flux.Supported by a grant from CNPq (Brazilian government agency FA)     

Layer-edge conditions for multislab atmospheric radiative transfer problems

We derive nonstandard layer-edge conditions for efficient solution of multislab atmospheric radiative transfer problems. We begin by defining a local radiative transfer problem on the lowermost layer of a multislab model atmosphere and we consider a standard discrete ordinates version of this local problem. We then make use of a recently developed computational method in order to derive layer-edge conditions involving incident, reflected and transmitted radiation. These layer-edge conditions for the lowermost layer are given in terms of inherent optical properties of the layer, the solar zenith angle and the quadrature set used in the discrete ordinates approach. They can be used to increase the efficiency of our computational method in solving practical problems in atmospheric radiative transfer. Moreover, they are amenable to incorporation into other discrete ordinates methods. To illustrate, we report numerical results for two atmospheric model problems.