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The development of numerical techniques to model general electromagnetic systems has paralleled the advancement of computational performance.
We provide a formal analytic solution of the radiative transfer equation for a gray moving atmosphere in a plane parallel geometry. A formal solution in the diffusion and the free-streaming limit is also provided in the case of a spherically extended atmosphere. The formal solutions are written explicitly for scattering atmospheres in which the density and the velocity fields are given by a power law. A self-consistent temperature profile accurate to O(Beta = v/c) is provided for the case in which the absorption or the scattering are temperature independent. The gray extinction temperature profile is considerably simplified in the case of a scattering atmosphere. Steady state flow and homologous expansion are special cases that are considered in detail.
Spectral reflectance of snow under diffuse illumination is studied using the two-stream approximation of the radiative transfer equation. The scattering and absorption parameters of the radiative transfer equation - the single scattering albedo, the optical depth, and the integrated phase function are obtained from the grain size and density of snow. Analytical expressions for the intensity within the snowpack, the reflectance, and the asymptotic flux extinction coefficient, are given. Good agreement is shown between the theory and available experimental data on visible and near-infrared reflectance, and the asymptotic flux extinction coefficient. The theory may also be used to explain the observed effect of aging on the snow reflectance.
In this paper, three approximate physical microwave radiometry models have been used to calculate brightness temperatures from a forest canopy at L-band. These models are (1) tau-omega model (zero order scattering approximation to radiative transfer equations), (2) successive order of scattering model up to first order (first order scattering approximation to the radiative transfer equations), and (3) Peak technique utilizing the active solution obtained from the Distorted Born Approximation (DBA). These models are physically-based and treat vegetation as a layer of discrete scatterers over a rough surface. Vegetation components within the canopy are represented by canonical shapes such as dielectric discs and cylinders. The tau-omega model is based on a zero-order solution to the radiative transfer (RT) equations. The model ignores scattering except for the effect of the scatterers in the attenuation of the emission through the vegetation. Application of the tau-omega model to data acquired during airborne and ground-based campaigns over the years has solidified scientific understanding of microwave interactions with different landscapes. In particular, shrubland, grasslands, agricultural crops, and light to moderate vegetation have been investigated. Its applicability to areas with a significant tree fraction is unknown. The first order scattering model is based on an iterative solution of the RT equation up to the first order. The first order solution is obtained by substituting the zeroth-order solution into the scattering source term and then solving the resulting radiative transfer equations. This formulation adds a new scattering term to the tau-omega model. It represents emission by particles in the layer and emission by the ground that is scattered once by particles in the layer. The resulting model represents an improvement over the standard zero-order solution (the tau-omega model) since it accounts for the scattered vegetation and ground radiation that can have a pronounced effect on the observed brightness temperature. The third model is based on the Peake formulation in conjunction with the DBA. The procedure for calculation of forest emission is accomplished by first calculating the bistatic scattering cross section for each type of scatterer, then by using the DBA to calculate specular albedo of the ground and the diffused albedo of the layer. Once the albedos are determined, Peake s principle relating active and passive problems can be used to determine the effective emissivity of the forest layer.
The effect of tropospheric aerosols on atmospheric infrared cooling rates is investigated by the use of recent models of infrared gaseous absorption. A radiative model of the atmosphere that incorporates dust as an absorber and scatterer of infrared radiation is constructed by employing the exponential kernel approximation to the radiative transfer equation. Scattering effects are represented in terms of a single scattering albedo and an asymmetry factor. The model is applied to estimate the effect of an aerosol layer made of spherical quartz particles on the infrared cooling rate. Calculations performed for a reference wavelength of 0.55 microns show an increased greenhouse effect, where the net upward flux at the surface is reduced by 10% owing to the strongly enhanced downward emission. There is a substantial increase in the cooling rate near the surface, but the mean cooling rate throughout the lower troposphere was only 10%.
The reaction concept is employed to formulate an integral equation for radiation and scattering from plates, corner reflectors, and dielectric-coated conducting cylinders. The surface-current density on the conducting surface is expanded with subsectional bases. The dielectric layer is modeled with polarization currents radiating in free space. Maxwell's equation and the boundary conditions are employed to express the polarization-current distribution in terms of the surface-current density on the conducting surface. By enforcing reaction tests with an array of electric test sources, the moment method is employed to reduce the integral equation to a matrix equation. Inversion of the matrix equation yields the current distribution, and the scattered field is then obtained by integrating the current distribution. The theory, computer program and numerical results are presented for radiation and scattering from plates, corner reflectors, and dielectric-coated conducting cylinders.
Numerical integration based on definition of Riemann integral, describing application to nonlinear integral equations of isotropic scattering of radiation in plane parallel atmosphere
A new method of solving the radiative transfer equation is developed in which the scattering and absorption coefficients may have arbitrary variations with depth, and in which both internal (thermal) emission and incident radiation are allowed. Specular and diffuse reflection at both boundaries also is taken into account. The method begins by forming a paired set of coupled first-order differential equations for the symmetric and antisymmetric parts of the radiation field after writing the scattering integral as a numerical quadrature. These differential equations are broken into finite difference form, in which the symmetric and antisymmetric parts of the radiation field are found on alternate grid points. Numerical results for a number of test problems are shown, demonstrating that the method is very fast, that it returns specific intensities and fluxes that are accurate to at least a percent, and that it can be applied to optically thick problems.
The piecewise-sinusoidal reaction technique is applied to low frequency radiation and scattering from noncircular cylinders with perfect or imperfect conductivity. This report presents the theory, computer programs and numerical results for these two-dimensional problems with the TE polarization.
The quasi-linear transport equation of energetic charged particles is derived, including scattering by Alfven waves propagating parallel and antiparallel to a uniform magnetic field, losses by synchrotron radiation, and acceleration of the cold background fluid supporting the waves. As in the comoving frame equations of radiative transfer, scattering, and loss terms are evaluated in the frame in which the fluid is locally at rest. The diffusion approximation is applied to the resulting equation, yielding a transport equation for the isotropic part of the distribution function and a first-order approximation to the anisotropy. The conditions under which this approximation may be applied are discussed. In the nonrelativistic regime, the standard equation of the diffusion approximation is recovered. As contributions previously investigated, new effects appear from the spatial and temporal variations of the fluid velocity as well as the synchrotron radiation terms. These are discussed in detail.
Solar radiative transfer through a coupled system of atmosphere and plant canopy is modeled as a multiple-scattering problem through a layered medium of random scatterers. The radiative transfer equation is solved by the discrete-ordinates finite-element method. Analytic expressions are derived that allow the calculation of scattering and absorption cross sections for any plant canopy layer form measurable biophysical parameters such as the leaf area index, leaf angle distribution, and individual leaf reflectance and transmittance data. An expression for a canopy scattering phase function is also given. Computational results are in good agreement with spectral reflectance measurements directly above a soybean canopy, and the concept of greenness- and brightness-transforms of Landsat MSS data is reconfirmed with the computed results. A sensitivity analysis with the coupled atmosphere/canopy model quantifies how satellite-sensed spectral radiances are affected by increased atmospheric aerosols, by varying leaf area index, by anisotropic leaf scattering, and by non-Lambertian soil boundary conditions. Possible extensions to a 2-D model are also discussed.
To date there have been multiple satellite missions to observe and retrieve cloud top properties and the liquid in, and precipitation from, clouds. There are currently a few missions that attempt to measure cloud ice properties as a byproduct of other observations. However, we do not yet quantitatively understand the processes that control the water budget of the upper troposphere where ice is the predominant phase, and how these processes are linked to precipitation processes and the radiative energy budget. The ice in clouds either melts into rain or is detrained, and persists, as cirrus clouds affecting the hydrological and energy cycle, respectively. Fully modeling the Earth's climate and improving weather and climate forecasts requires accurate satellite measurements of various cloud properties at the temporal and spatial scales of cloud processes. The uncertainty in knowledge of these ice characteristics is reflected in the large discrepancies in model simulations of the upper tropospheric water budget. Model simulations are sensitive to the partition of ice between precipitation and outflow processes, i.e., to the parameterization of ice clouds and ice processes. This presentation will describe the Submillimeter-wave InfraRed Ice Cloud Experiment (SIRICE) concept, a satellite mission designed to acquire global Earth radiance measurements in the infrared and submillimeter-wave region (183-874 GHz). If successful, this mission will bridge the measurement gap between microwave sounders and shorter-wavelength infrared and visible sensors. The brightness temperatures at submillimeter-wave frequencies are especially sensitive to cirrus ice particle sizes (because they are comparable to the wavelength). This allows for more accurate ice water path estimates when multiple channels are used to probe into the cloud layers. Further, submillimeter wavelengths offer simplicity in the retrieval algorithms because they do not probe into the liquid and near surface portions of clouds, thus requiring only one term of the radiative transfer equation (ice scattering) to relate brightness temperatures to ice. Scientific justification and the SIRICE approach to measuring ice water path and particle size that span a range encompassing both the hydrologically active and radiatively active components of cloud systems will be presented.
The radiative transfer equation is solved by the S-N discrete ordinates method in one-dimensional radial and two-dimensional r-z coordinates systems. The walls of the enclosures are diffuse, and the participating medium absorbs, emits and anisotropically scatters the radiative energy. Diffuse wall incidence, isothermal medium emission and collimated incidence problems are considered. Effects of the scattering phase functions on average incident radiation and net radiative heat fluxes are studied. In addition, the effects of scattering albedo, optical thickness and the wall emissivity are briefly discussed.
Numerical solutions for radiative transport in a class of anisotropically scattering materials are presented. Conditions for convergence and divergence of the iterative method are given and supported by computed results. The relation of two flux theories to the equation of radiative transfer for isotropic scattering is discussed. The adequacy of the two flux approach for the reflectance, radiative flux and radiative flux divergence of highly scattering media is evaluated with respect to solutions of the radiative transfer equation.
There are many satellite observations of cloud top properties and the liquid and rain content of clouds, however, we do not yet quantitatively understand the processes that control the water budget of the upper troposphere where ice is the predominant phase, and how these processes are linked to precipitation processes and the radiative energy budget. The ice in clouds in the upper troposphere either melts into rain or is detrained, and persists, as cirrus clouds affecting the hydrological and energy cycle, respectively. Fully modeling the Earth's climate and improving weather and climate forecasts requires accurate satellite measurements of various cloud properties at the temporal and spatial scales of cloud processes. These properties include cloud horizontal and vertical structure, cloud water content and some measure of particle sizes and shapes. The uncertainty in knowledge of these ice characteristics is reflected in the large discrepancies in model simulations of the upper tropospheric water budget. Model simulations are sensitive to the partition of ice between precipitation and outflow processes, i.e., to the parameterization of ice clouds and ice processes. One barrier to achieving accurate global ice cloud properties is the lack of adequate observations at millimeter and submillimeter wavelengths (183-874 GHz). Recent advances in instrumentation have allowed for the development and implementation of an airborne submillimeter-wave radiometer. The brightness temperatures at these frequencies are especially sensitive to cirrus ice particle sizes (because they are comparable to the wavelength). This allows for more accurate ice water path estimates when multiple channels are used to probe into the cloud layers. Further, submillimeter wavelengths offer simplicity in the retrieval algorithms because they do not probe into the liquid and near surface portions of clouds, thus requiring only one term of the radiative transfer equation (ice scattering) to relate brightness temperatures to ice. The next step is a satellite mission designed to acquire global Earth radiance measurements in the submillimeter-wave region, thus bridging the measurement gap between microwave sounders and shorter-wavelength infrared and visible sensors. This presentation provides scientific justification and an approach to measuring ice water path and particle size from a satellite platform that spans a range encompassing both the hydrologically active and radiatively active components of cloud systems.
The direct method for the solution of the spherical harmonics approximation to the equation of transfer of radiation is applied to the cases of (1) scattering of the solar radiation in the atmosphere with the Lambertian boundary and (2) thermal radiation transfer.
Integration of induced current distribution equations yields the scattered radiation field from a hyperboloidal reflector antenna in a cassegrainian feed system
Spectral reflectance of snow under diffuse illumination is studied using the two-stream approximation of the radiative transfer equation. The scattering and absorption within the snowcover due to the randomly distributed ice grains are characterized by the single scattering albedo and anisotropic phase function. Geometric optics calculations are used to relate the scattering and absorption parameters to grain size and density of snow. Analytical expressions for the intensity within the snowpack and the asymptotic flux extinction coefficient are also obtained. Good agreement is shown between the theory and available experimental data on visible and near-infrared reflectance and asymptotic flux extinction coefficient. The theory also may be used to explain the observed effect of aging on the snow reflectance.