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NASA NTRS2011

Analysis of the Radiative Transfer Equation with Highly Asymmetric Phase Function

This paper considers a scalar radiative transfer problem with high scattering anisotropy, Two computational methods are presented based on decomposition of the diffuse light field into a regular and anisotropic part. The first algorithm (DOMAS) singles out the anisotropic radiance in the forward scattering peak using the Small-Angle Modification of RTE. The second algorithm (DOM2+) separates the single scattering radiance as an anisotropic part, which largely defines the fine detail of the total radiance in the backscattering directions. In both cases, the anisotropic part is represented analytically. With anisotropy subtraction, the regular part of the signal. which requires a numerical solution, is essentially smoothed as a function of angles. Further, the transport equation is obtained for the regular part that contains an additional source function from the anisotropic part of the signal. This equation is solved with the discrete ordinates method. A conducted numerical analysis of this work showed that algorithm DOMAS has a strong advantage as compared to the standard discrete ordinates method for simulation of the radiance transmission, and DOM2 + is the best of the three for the reflection computations. Both algorithms offer at least a factor of three acceleration of convergence of the azimuthal series for highly anisotropic phase functions.

Korkin, Sergey V.↗
NASA NTRS1984

Radiative transfer equations in broad-band, time-varying fields

A derivation of the equation of transfer is obtained by starting with Maxwell's equations in the 'slowly varying envelope' form. Particular attention is paid to characterizing the intensity that is 'seen' by the atom (which is found to be related to a Wigner distribution of the electric field). The equation of transfer is found to be valid for 'broad-band' slowly varying radiation fields.

Cooper, J.↗
NASA NTRS1991

Discrete ordinate solution of the radiative transfer equation in the 'polarization normal wave representation'

The transfer equations for normal waves in finite, inhomogeneous and plane-parallel magnetoactive media are solved using the discrete ordinate method. The physical process of absorption, emission, and multiple scattering are accounted for, and the medium may be forced both at the top and bottom boundary by anisotropic radiation as well as by internal anisotropic sources. The computational procedure is numerically stable for arbitrarily large optical depths, and the computer time is independent of optical thickness.

Kylling, A.↗
NASA NTRS1991

A Variational Formalism for the Radiative Transfer Equation: Prelude to Model 3

The MODEL III variational data assimilation model is the third of four general assimilation models designed to blend weather data measured from space based platforms in the meteorological data mainstream in a way that maximizes the information content of the satellite data. Because there are many different observation locations and there are many instruments with different measurement error characteristics, it is also necessary to require that the blending be done to maximize the information content of the data and simultaneously to retain a dynamically consistent and reasonably accurate description of the state of the atmosphere. This is ideally a variational problem for which the data receive relative weights that are inversely proportional to measurement error and are adjusted to satisfy a set of dynamical equations that govern atmospheric processes. The advantage of MODEL III over the previous two models is that radiance, the atmospheric variable measured by satellite, becomes a dependent variable. In the previous versions, mean layer temperatures that had been retrieved from the radiances by some method, were included in the assimilation by substituting them in place of the rawinsonde temperatures. Now both rawinsonde temperatures and satellite radiances are included independently in the assimilation.

Achtemeier, Gary L.↗
NASA NTRS1992

A fast operator perturbation method for the solution of the special relativistic equation of radiative transfer in spherical symmetry

A fast method for the solution of the radiative transfer equation in rapidly moving spherical media, based on an approximate Lambda-operator iteration, is described. The method uses the short characteristic method and a tridiagonal approximate Lambda-operator to achieve fast convergence. The convergence properties and the CPU time requirements of the method are discussed for the test problem of a two-level atom with background continuum absorption and Thomson scattering. Details of the actual implementation for fast vector and parallel computers are given. The method is accurate and fast enough to be incorporated in radiation-hydrodynamic calculations.

Hauschildt, P. H.↗
NASA NTRS1982

Solution of the equation of radiative transfer for remote sensing over nonuniform surface reflectivity

An understanding of radiative transfer in the earth's atmosphere is a necessity for the remote sensing of surface reflectivity from satellites and aircraft. The range of the adjacency effect, which represents the effect of bright areas on the radiance above dark areas, is the main parameter that distinguishes atmospheric radiative transfer over a nonuniform surface from that over a uniform one. A radiative transfer model which expresses this range correctly is, therefore, critical for developing remote sensing methods for the case of an atmosphere over a nonuniform surface. The present investigation is concerned with the development of a new approximate solution of the radiative transfer (RT) equation. The solution is not limited to nonabsorbing atmospheres, but it will still be limited to nadir observations. The results compare favorably with Monte Carlo simulations.

Kaufman, Y. J.↗
NASA NTRS2003

A New Look into the Effect of Large Drops on Radiative Transfer Process

Recent studies indicate that a cloudy atmosphere absorbs more solar radiation than any current 1D or 3D radiation model can predict. The excess absorption is not large, perhaps 10-15 W/sq m or less, but any such systematic bias is of concern since radiative transfer models are assumed to be sufficiently accurate for remote sensing applications and climate modeling. The most natural explanation would be that models do not capture real 3D cloud structure and, as a consequence, their photon path lengths are too short. However, extensive calculations, using increasingly realistic 3D cloud structures, failed to produce photon paths long enough to explain the excess absorption. Other possible explanations have also been unsuccessful so, at this point, conventional models seem to offer no solution to this puzzle. The weakest link in conventional models is the way a size distribution of cloud particles is mathematically handled. Basically, real particles are replaced with a single average particle. This "ensemble assumption" assumes that all particle sizes are well represented in any given elementary volume. But the concentration of larger particles can be so low that this assumption is significantly violated. We show how a different mathematical route, using the concept of a cumulative distribution, avoids the ensemble assumption. The cumulative distribution has jumps, or steps, corresponding to the rarer sizes. These jumps result in an additional term, a kind of Green's function, in the solution of the radiative transfer equation. Solving the cloud radiative transfer equation with the measured particle distributions, described in a cumulative rather than an ensemble fashion, may lead to increased cloud absorption of the magnitude observed.

Marshak, Alexander↗
NASA NTRS1993

A Discrete Probability Function Method for the Equation of Radiative Transfer

A discrete probability function (DPF) method for the equation of radiative transfer is derived. The DPF is defined as the integral of the probability density function (PDF) over a discrete interval. The derivation allows the evaluation of the PDF of intensities leaving desired radiation paths including turbulence-radiation interactions without the use of computer intensive stochastic methods. The DPF method has a distinct advantage over conventional PDF methods since the creation of a partial differential equation from the equation of transfer is avoided. Further, convergence of all moments of intensity is guaranteed at the basic level of simulation unlike the stochastic method where the number of realizations for convergence of higher order moments increases rapidly. The DPF method is described for a representative path with approximately integral-length scale-sized spatial discretization. The results show good agreement with measurements in a propylene/air flame except for the effects of intermittency resulting from highly correlated realizations. The method can be extended to the treatment of spatial correlations as described in the Appendix. However, information regarding spatial correlations in turbulent flames is needed prior to the execution of this extension.

Sivathanu, Y. R.↗
NASA NTRS1990

Radar polarimetry for geoscience applications

The present volume on radar polarimetry for geoscience applications discusses wave properties and polarization, scattering matrix representation for simple targets, scattering models for point and distributed targets, polarimetric scatterometer systems and measurements, polarimetric radar system design, and polarimetric SAR applications. Attention is given to plane waves in a lossless homogeneous medium-wave polarization, polarization synthesis and response, and coordinate system transformations. Topics addressed include high- and low-frequency scattering, rough-surface scattering models, radiative transfer theory and deficiencies thereof, solutions for the radiative transfer equation, and a radiative transfer model for a forest canopy. Also discussed are network analyzer-based polarimetric scatterometers, calibration of polarimetric scatterometers, synthesized polarization response of distributed targets, and measurement of the propagation parameters of a forest canopy.

Ulaby, Fawwaz T.↗
NASA NTRS1982

On redistribution and the equations for radiative transfer

The derivation of the equations of statistical equilibrium are outlined, starting from the quantum density-matrix equations, drawing particular attention to the approximations and assumptions used in the development of tractable expressions. Then, using the quantum-fluctuation-regression theorem, emission and absorption coefficients are obtained for multilevel atomic systems which are nondegenerate except for m-substates. These coefficients are valid to first order in the incident intensity. Possible extensions to higher intensity broadband incoherent fields are suggested.

Cooper, J.↗
NASA NTRS1987

A canopy reflectance model based on an analytical solution to the multiple scattering equation

An approximation to the radiative transfer equation for solar radiation in relatively full, homogeneous plant canopies is presented and solved analytically for solar zenith angles less than 60 deg. The model predicts reflectance at any depth in the canopy and in any direction and may be inverted with bidirectional reflectance measurements. The model was fit to data at two sun angles and two wavebands (visible and NIR) to within the assumed errors on the reflectance data. The calculated albedos are insensitive to achievable measurement errors. Some of the parameter values themselves found by the inversion agree reasonably well with independent measurements, but the uncertainties introduced by the data noise are rather large. However, the agreement is good enough to demonstrate that the model is physically realistic.

Camillo, Peter↗
NASA NTRS1976

Effects of multiple scattering on light pulses reflected by turbid atmospheres

Multiple scattering contributions to lidar returns from turbid atmospheres are derived by means of an analytical theory. It is assumed that scattering takes place mainly at small angles except for one event that scatters the light backward. The phase functions are approximated by the sum of Gaussian functions of the scattering angle in both the forward and backward directions. The three-dimensional radiative transfer equation is transformed to a one-dimensional problem by means of Fourier transforms. Neumann solutions to the transformed equation of radiative transfer are then found. A number of examples are presented for cloud, fog and haze models. The results are found to be in satisfactory agreement with results obtained from the Monte Carlo analysis of Kunkel (1974) and the theory of light pulses doubly scattered by turbid atmospheres which was developed by Eloranta (1972).

Weinman, J. A.↗
NASA NTRS1990

IR spectral characteristics of cirrus clouds

The recent focus of parameterization of the radiative properties of clouds has been to include the microphysical properties of the cloud. A variety of parameterization have been developed for both the shortwave and the longwave. In parameterizing the longwave properties of clouds, it is useful to consider the two stream solution of the radiative transfer equation appropriate for a thermal source. These radiative transfer equations are considered.

Ackerman, Steven A.↗
NASA NTRS1974

Radiative transfer in highly scattering materials - numerical solution and evaluation of approximate analytic solutions

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.

Weston, K. C.↗