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NASA NTRS1993

Observation of pressure ridges in SAR images of sea ice: Scattering theory and comparison with observations

Ridges and keels (hummocks and bummocks) in sea ice flows are important in sea ice research for both scientific and practical reasons. Sea ice movement and deformation is driven by internal and external stresses on the ice. Ridges and keels play important roles in both cases because they determine the external wind and current stresses via drag coefficients. For example, the drag coefficient over sea ice can vary by a factor of several depending on the fluid mechanical roughness length of the surface. This roughness length is thought to be strongly dependent on the ridge structures present. Thus, variations in ridge and keel structure can cause gradients in external stresses which must be balanced by internal stresses and possibly fracture of the ice. Ridging in sea ice is also a sign of fracture. In a practical sense, large ridges form the biggest impediment to surface travel over the ice or penetration through sea ice by ice-strengthened ships. Ridges also play an important role in the damage caused by sea ice to off-shore structures. Hence, observation and measurement of sea ice ridges is an important component of sea ice remote sensing. The research reported here builds on previous work, estimating the characteristics of ridges and leads in sea ice from SAR images. Our objective is to develop methods for quantitative measurement of sea ice ridges from SAR images. To make further progress, in particular, to estimate ridge height, a scattering model for ridges is needed. Our research approach for a ridge scattering model begins with a survey of the geometrical properties of ridges and a comparison with the characteristics of the surrounding ice. For this purpose we have used airborne optical laser (AOL) data collected during the 1987 Greenland Sea Experiment. These data were used to generate a spatial wavenumber spectrum for height variance for a typical ridge - the typical ridge is the average over 10 large ridges. Our first-order model radar scattering includes both the quasi-specular and Bragg resonant scatter mechanisms. This model is extended to include contributions from volume scatter and scatter from discrete objects. Geometrical characteristics from the AOL survey and model calculations imply that for radar wavelengths and observation geometries that are dominated by the quasi-specular scattering mechanism radar backscatter from a ridge is a measure of peak ridge height. We present scattering model results and compare them with ridges observed during the LEADEX experiment of March-April 1992 when both X, C, and L-band aircraft SAR and the ERS-1 satellite SAR observed a region in the Beaufort Sea near 86 deg N, 10 deg W. Data were also collected documenting ridge characteristics on the surface. The surface data are used to generate a SAR signature via the scattering model described above. The predicted SAR signatures compare well with the SAR observations.

Vesecky, J. F.↗

NASA NTRS1995

Polarization from scattering in blobs

We describe a Monte Carlo method for calculating polarization due to scattering in various geometries. We apply the method in this paper to the case of scattering in a spherical cloud illuminated by parallel rays. The polarization is largest in small optical depth blobs, where single scattering dominates. The polarization decreases as optical depth increases until a limit is reached where scattering occurs on the surface, and increasing optical depth has little effect on the results. For a given optical depth, decreasing the albedo leads to an increase of the degree of polarization of the scattered light because single scattering increases relative to multiple scattering. As the optical depth of the blob increases, the scattered flux becomes increasingly backward throwing. In the high albedo blobs, the maximum polarization becomes skewed towards the illuminated hemisphere at high optical depths. We discuss the applications to polarization in supergiants and R CrB stars, and scattering in a clumpy interstellar medium.

Code, Arthur D.↗

NASA NTRS1994

Errors induced by the neglect of polarization in radiance calculations for Rayleigh-scattering atmospheres

Although neglecting polarization and replacing the rigorous vector radiative transfer equation by its approximate scalar counterpart has no physical background, it is a widely used simplification when the incident light is unpolarized and only the intensity of the reflected light is to be computed. We employ accurate vector and scalar multiple-scattering calculations to perform a systematic study of the errors induced by the neglect of polarization in radiance calculations for a homogeneous, plane-parallel Rayleigh-scattering atmosphere (with and without depolarization) above a Lambertian surface. Specifically, we calculate percent errors in the reflected intensity for various directions of light incidence and reflection, optical thicknesses of the atmosphere, single-scattering albedos, depolarization factors, and surface albedos. The numerical data displayed can be used to decide whether or not the scalar approximation may be employed depending on the parameters of the problem. We show that the errors decrease with increasing depolarization factor and/or increasing surface albedo. For conservative or nearly conservative scattering and small surface albedos, the errors are maximum at optical thicknesses of about 1. The calculated errors may be too large for some practical applications, and, therefore, rigorous vector calculations should be employed whenever possible. However, if approximate scalar calculations are used, we recommend to avoid geometries involving phase angles equal or close to 0 deg and 90 deg, where the errors are especially significant. We propose a theoretical explanation of the large vector/scalar differences in the case of Rayleigh scattering. According to this explanation, the differences are caused by the particular structure of the Rayleigh scattering matrix and come from lower-order (except first-order) light scattering paths involving right scattering angles and right-angle rotations of the scattering plane.

Mishchenko, M. I.↗

NASA NTRS2000

Measurements and Modeling of Aerosol Absorption and Single Scattering Albedo at Ambient Relative Hum

Uncertainties in the aerosol single scattering albedo have been identified to be an important source of errors in current large-scale model estimates of the direct aerosol radiative forcing of climate. A number of investigators have obtained estimates of the single scattering albedo from a variety of remote sensing and in situ measurements during aerosol field experiments. During the Tropospheric Aerosol Radiative Forcing Observational Experiment (TARFOX, 1996) for example, estimates of the aerosol single scattering albedo were obtained (1) as a best-fit parameter in comparing radiative flux changes measured by airborne pyranometer to those computed from independently measured aerosol properties; (2) from estimates of the aerosol complex index of refraction derived using a combination of airborne sunphotometer, lidar backscatter and in situ size distribution measurements; and (3) from airborne measurements of aerosol scattering and absorption using nephelometers and absorption photometers. In this paper, we briefly compare the results of the latter two methods for two TARFOX case studies, since those techniques provide height-resolved information about the aerosol single scattering albedo. Estimates of the aerosol single scattering albedo from nephelometer and absorption photometer measurements require knowledge of the scattering and absorption humidification (i.e., the increase in these properties in response to an increase in ambient relative humidity), since both measurements are usually carried out at a relative humidity different from the ambient atmosphere. In principle, the scattering humidification factor can be measured, but there is currently no technique widely available to measure the absorption of an aerosol sample as a function of relative humidity. Frequently, for lack of better knowledge, the absorption humidification is assumed to be unity (meaning that there is no change in aerosol absorption due to an increase in ambient relative humidity). This assumption then enters the estimate of the single scattering albedo at ambient relative humidity. To investigate the validity of this assumption we have carried out modeling studies of the absorption humidification factor, assuming that the aerosols contain an insoluble soot core and a coating which determines its hygroscopic growth behavior. The aerosol optical properties are then computed on the basis of the shell/core particle morphology using a Mie-code for concentric shells. From basic physical principles, it is conceivable that aerosol absorption increases when an atmospheric aerosol particle collects a non-absorbing shell, since the soot core is then exposed to an increased (focused) electric field strength. Indeed, our preliminary modeling studies show that the absorption of an atmospheric aerosol particle composed of a soot core and an aqueous sulfuric acid shell may increase by a factor of 50% due to a change in ambient relative humidity from 30 to 95%. We will show how this increased absorption is a function of the initial particle size and soot mass fraction.

Redemann, J.↗

NASA NTRS1998

Fitting a Two-Component Scattering Model to Polarimetric SAR Data

Classification, decomposition and modeling of polarimetric SAR data has received a great deal of attention in the recent literature. The objective behind these efforts is to better understand the scattering mechanisms which give rise to the polarimetric signatures seen in SAR image data. In this Paper an approach is described, which involves the fit of a combination of two simple scattering mechanisms to polarimetric SAR observations. The mechanisms am canopy scatter from a cloud of randomly oriented oblate spheroids, and a ground scatter term, which can represent double-bounce scatter from a pair of orthogonal surfaces with different dielectric constants or Bragg scatter from a moderately rough surface, seen through a layer of vertically oriented scatterers. An advantage of this model fit approach is that the scattering contributions from the two basic scattering mechanisms can be estimated for clusters of pixels in polarimetric SAR images. The solution involves the estimation of four parameters from four separate equations. The model fit can be applied to polarimetric AIRSAR data at C-, L- and P-Band.

Freeman, A.↗

NASA NTRS2001

Inherent and Apparent Scattering Properties of Coated or Uncoated Spheres Embedded in an Absorbing Host Medium

The conventional Lorenz-Mie formalism is extended to the scattering process associated with a coated sphere embedded in an absorbing medium. It is shown that apparent and inherent scattering cross sections of a scattering particle, which are identical in the case of transparent host medium, are different if the host medium is absorptive. Here the inherent single-scattering properties are derived from the near-field information whereas the corresponding apparent counterparts are derived from the far-field asymptotic form of the scattered wave with scaling of host absorption that is assumed to be in an exponential form. The formality extinction and scattering efficiencies defined in the same manner as in the conventional sense can be unbounded. For a nonabsorptive particle embedded in an absorbing medium, the effect of host absorption on the phase matrix elements associated with polarization is significant. This effect, however, is largely reduced for strongly absorptive particles such as soot. For soot particles coated with water, the impurity can substantially reduce the single-scattering albedo of the particle if the size parameter is small. For water-coating soot and hollow ice spheres, it is shown that the phase matrix elements -P(sub 12)/P(sub 11) and P(sub 33)/P(sub 11) are unique if the shell is thin, as compared with the case for thick shell. Furthermore, the radiative transfer equation regarding a multidisperse particle system in an absorbing medium is discussed. It is illustrated that the conventional computation algorithms can be applied to solve the multiple scattering process if the scaled apparent single-scattering properties are applied.

Yang, P.↗

NASA NTRS2004

Time-Average Measurement of Velocity, Density, Temperature, and Turbulence Using Molecular Rayleigh Scattering

Measurement of time-averaged velocity, density, temperature, and turbulence in gas flows using a nonintrusive, point-wise measurement technique based on molecular Rayleigh scattering is discussed. Subsonic and supersonic flows in a 25.4-mm diameter free jet facility were studied. The developed instrumentation utilizes a Fabry-Perot interferometer to spectrally resolve molecularly scattered light from a laser beam passed through a gas flow. The spectrum of the scattered light contains information about velocity, density, and temperature of the gas. The technique uses a slow scan, low noise 16-bit depth CCD camera to record images of the fringes formed by Rayleigh scattered light passing through the interferometer. A kinetic theory model of the Rayleigh scattered light is used in a nonlinear least squares fitting routine to estimate the unknown parameters from the fringe images. The ability to extract turbulence information from the fringe image data proved to be a challenge since the fringe is broadened by not only turbulence, but also thermal fluctuations and aperture effects from collecting light over a range of scattering angles. Figure 1 illustrates broadening of a Rayleigh spectrum typical of flow conditions observed in this work due to aperture effects and turbulence for a scattering angle, chi(sub s), of 90 degrees, f/3.67 collection optics, mean flow velocity, u(sub k), of 300 m/s, and turbulent velocity fluctuations, sigma (sub uk), of 55 m/s. The greatest difficulty in processing the image data was decoupling the thermal and turbulence broadening in the spectrum. To aid in this endeavor, it was necessary to seed the ambient air with smoke and dust particulates; taking advantage of the turbulence broadening in the Mie scattering component of the spectrum of the collected light (not shown in the figure). The primary jet flow was not seeded due to the difficulty of the task. For measurement points lacking particles, velocity, density, and temperature information could reliably be recovered, however the turbulence estimates contained significant uncertainty. Resulting flow parameter estimates are presented for surveys of Mach 0.6, 0.95, and 1.4 jet flows. Velocity, density, and temperature were determined with accuracies of 5 m/s, 1.5%, and 1%, respectively, in flows with no particles present, and with accuracies of 5 m/s, 1-4%, and 2% in flows with particles. Comparison with hotwire data for the Mach 0.6 condition demonstrated turbulence estimates with accuracies of about 5 m/s outside the jet core where Mie scattering from dust/smoke particulates aided in the estimation of turbulence. Turbulence estimates could not be recovered with any significant accuracy for measurement points where no particles were present.

Mielke, Amy F.↗

NASA NTRS1973

Atmospheric Temperature from Raman Scattering

Raman scattering signatures are functions of the scattering-gas temperature, and are therefore of potential use for practical atmospheric temperature probes. The method described uses either ratios of pure rotational Raman scattering intensities for air utilizing various spectral bandpasses, or, alternatively, ratios of air-rotational to nitrogen-vibrational scattering intensities. Three aspects of work relating to the development of such probes are discussed in this presentation: (1) fundamental absolute Raman data, (2) construction of air spectra from these data, and (3) temperature-sensitivity of the signature. The fundamental data described are the absolute rotational and vibrational scattering cross sections. Recent measurements in this laboratory of rotational cross sections for N2, O2, and CO2 are emphasized, as are their use in predicting absolute magnitudes of Raman scattering signals. Next is described the computation of air rotational Raman spectra as a function of temperature calculated through use of the experimentally-measured cross sections. The spectra are based additively upon nitrogen and oxygen contributions, since pure rotational Raman scattering from water vapor is very weak. Finally, the sensitivity of the scattering intensity ratios to temperature is explored as a function of choice of spectral bandpass for the monitored rotational Raman scattering. Various compromises will be discussed which must be made in choosing bandpasses appropriate for specific purposes and experimental conditions.

Lapp, M.↗

NASA NTRS2016

First-Principles Modeling Of Electromagnetic Scattering By Discrete and Discretely Heterogeneous Random Media

A discrete random medium is an object in the form of a finite volume of a vacuum or a homogeneous material medium filled with quasi-randomly and quasi-uniformly distributed discrete macroscopic impurities called small particles. Such objects are ubiquitous in natural and artificial environments. They are often characterized by analyzing theoretically the results of laboratory, in situ, or remote-sensing measurements of the scattering of light and other electromagnetic radiation. Electromagnetic scattering and absorption by particles can also affect the energy budget of a discrete random medium and hence various ambient physical and chemical processes. In either case electromagnetic scattering must be modeled in terms of appropriate optical observables, i.e., quadratic or bilinear forms in the field that quantify the reading of a relevant optical instrument or the electromagnetic energy budget. It is generally believed that time-harmonic Maxwell's equations can accurately describe elastic electromagnetic scattering by macroscopic particulate media that change in time much more slowly than the incident electromagnetic field. However, direct solutions of these equations for discrete random media had been impracticable until quite recently. This has led to a widespread use of various phenomenological approaches in situations when their very applicability can be questioned. Recently, however, a new branch of physical optics has emerged wherein electromagnetic scattering by discrete and discretely heterogeneous random media is modeled directly by using analytical or numerically exact computer solutions of the Maxwell equations. Therefore, the main objective of this Report is to formulate the general theoretical framework of electromagnetic scattering by discrete random media rooted in the Maxwell- Lorentz electromagnetics and discuss its immediate analytical and numerical consequences. Starting from the microscopic Maxwell-Lorentz equations, we trace the development of the first principles formalism enabling accurate calculations of monochromatic and quasi-monochromatic scattering by static and randomly varying multiparticle groups. We illustrate how this general framework can be coupled with state-of-the-art computer solvers of the Maxwell equations and applied to direct modeling of electromagnetic scattering by representative random multi-particle groups with arbitrary packing densities. This first-principles modeling yields general physical insights unavailable with phenomenological approaches. We discuss how the first-order-scattering approximation, the radiative transfer theory, and the theory of weak localization of electromagnetic waves can be derived as immediate corollaries of the Maxwell equations for very specific and well-defined kinds of particulate medium. These recent developments confirm the mesoscopic origin of the radiative transfer, weak localization, and effective-medium regimes and help evaluate the numerical accuracy of widely used approximate modeling methodologies.

Michael I Mishchenko↗

NASA NTRS2022

Scattered Light in the Hinode/EIS and SDO/AIA Instruments Measured from the 2012 Venus Transit

Observations from the 2012 transit of Venus are used to derive empirical formulae for long- and short-range scattered light at locations on the solar disk observed by the Hinode Extreme ultraviolet Imaging Spectrometer (EIS) and the Solar Dynamics Observatory Atmospheric Imaging Assembly (AIA) instruments. Long-range scattered light comes from the entire solar disk, while short-range scattered light is considered to come from a region within 50” of the region of interest. The formulae were derived from the Fe XII 195.12 Å emission line observed by EIS and the AIA 193 Å channel. A study of the weaker Fe XIV 274.20 Å line during the transit and a comparison of scattering in the AIA 193 Å and 304 Å channels suggests the EIS scattering formula applies to other emission lines in the EIS wavebands. Both formulae should be valid in regions of fairly uniform emission such as coronal holes and quiet Sun, but may be less accurate for faint areas close (around 100”) to bright active regions. The formula for EIS is used to estimate the scattered light component of Fe XII λ195.12 for seven on-disk coronal holes observed between 2010 and 2018. Scattered light contributions of 56%–100% are found, suggesting that these features are dominated by scattered light, consistent with earlier work of Wendeln & Landi. Emission lines from the S X and Si X ions—formed at the same temperature as Fe XII and often used to derive the first ionization potential bias from EIS data—are also expected to be dominated by scattered light in coronal holes.

Transits↗

NASA NTRSDate not supplied

Velocity and Temperature Measurements in High-speed Flows with Naturally Present Dust Particles Using Rayleigh and Mie Scattering

Dust particles and occasional moisture condensations are unavoidable reality of all wind tunnels. On the path to pursue a goal of velocity and temperature measurements in high-speed wind tunnels we have created a tabletop, spectrally resolved, Rayleigh-Mie scattering setup around a small jet fed by ambient and air with different particle concentration to determine the accuracy of the technique. The other reality of a wind tunnel setup is the spurious reflection and scattering of the incident laser beam by solid surfaces (flare light), which contaminates the Rayleigh-Mie scattered light. This is simulated by backgrounds with different reflectivity towards the collection optics. Light from a CW laser is delivered via an optical fiber and the scattered light from a point on the laser path is spectrally resolved using a stabilized Fabry-Perot interferometer, followed by imaging on an EMCCD camera. A model of the combined background glare, Mie scattering, and the Rayleigh spectrum was fitted to the camera image using maximum likelihood estimation. It was observed that the present modeling approach can extract velocity and temperature with reasonable accuracy when the intensity of the Mie scattered light is less than or comparable to that of the Rayleigh scattered light; beyond that error in velocity measurement remains reasonable ±12m/s but the temperature measurement becomes progressively more inaccurate. Increasingly the flare light is found to cause a bias error in velocity. Similar trend is observed in the presence of both the flare light and the Mie scattered light. The setup has provided a set of data for the future improvement of the modeling procedure, and demonstrated that even in the case of large amount of dust particles and large flare light the present technique can provide measurement of velocity with reasonable accuracy.

Rayleigh scattering↗

NASA NTRS2026

Assessment of Near-Angle Scatter on Exo-Earth Coronagraphy

Near-Angle Scatter (NAS) of the host star’s light into the coronagraph dark hole may limit the ability of a potential Habitable Worlds Observatory (HWO) to detect and characterize an Earth-like planet around a Sun-like star. This paper summarizes a 5-year investigation of the impact of NAS on high-contrast coronagraphy. Science requirements were flowed down to a draft flux ratio noise ratio (FRN) error budget with allocations for NAS. The modeled sources of NAS include correlated (low and mid-spatial) surface structure, uncorrelated surface microroughness and coating structure, particulate contamination, micrometeoroid impacts and polarization leakage. The paper develops specifications for these sources of scattered light that meet their FRN error budget allocations, distinguishing between scatter that can and cannot be modulated. The development process utilizes an analytical expression that predicts scatter throughput into the dark hole based on BRDF/BSDF. Using this relationship, Photon Engineering performed a FRED® straylight analysis of the HWO EAC-1 (Exploratory Analytical Case #1) coronagraph to parametrically study predicted scattered light contrast in the dark hole. The model revealed that primary mirror microroughness or contamination is not a significant source of scattered light; the most critical surfaces are the smallest and those located closest to the coronagraph focal plane mask. The assessment found that uncorrelated microstructure, contamination, and micrometeoroid impacts require more investigation and potential significant sources of unmodulated scattered light, and highlights that slow thermal wavefront drift is an order of magnitude more impactful than fast mechanical jitter.

coronagraphy↗

NASA NTRS1972

Light scattering by ice clouds in the visible and infrared - A theoretical study.

Computations of the intensity and linear polarization for single scattering by ice clouds have been made based on the assumption that the particles in ice clouds can be approximated by long circular cylinders which are allowed to be polydispersive as well as arbitrarily oriented in space. The results of two models of optically thin ice clouds are presented and compared with those for polydisperse ice spheres. The two models for ice cylinders are assumed to be either uniformly or randomly oriented in a horizontal plane. Four different wavelengths, 0.7, 3, 3.5 and 6.05 microns, are employed in the light scattering computations. It is found that, compared to ice spheres, long ice cylinders scatter more light in the region with scattering angles near 90 deg, at the expense of scattering in both the forward and backward directions. The glory and cloudbows, which occur in light scattered by spherical particles, are either lost (the glory) or largely reduced and distorted (the cloudbows in the case of cylinders). It is probable that for more irregular particles the cloudbows would also disappear. These differences in scattering by spherical and non-spherical scatterers therefore provide useful information for the differentiation between the ice and liquid phase of cloud particles.

Liou, K.-N.↗

NASA NTRS1972

The inverse problem for radiation scattering

A brief survey of recent theoretical progress on the inverse scattering problem for radiation scattering from reflective boundaries and variable indices of refraction. The theory of radiation scattering is concerned primarily with the far-field relations between incident and scattered radiation in the presence of a scattering object. The primary, or direct, problem of this theory is to develop quantitative information about these scattering relations from a knowledge of the scattering object. The secondary, or inverse, problem, on the other hand, is to determine the nature of the object from an analysis of the scattering relations.

Prosser, R. T.↗

NASA NTRS1976

Multiple scattered radiation emerging from Rayleigh and continental haze layers. I - Radiance, polarization, and neutral points

The matrix operator method was used to calculate the polarization of radiation scattered on layers of various optical thicknesses, with results compared for Rayleigh scattering and for scattering from a continental haze. In both cases, there are neutral points arising from the zeros of the polarization of single scattered photons at scattering angles of zero and 180 degrees. The angular position of these Rayleigh-like neutral points (RNP) in the sky shows appreciable variation with the optical thickness of the scattering layer for a Rayleigh phase matrix, but only a small variation for haze L phase matrix. Another type of neutral point exists for non-Rayleigh phase functions that is associated with the zeros of the polarization for single scattering which occurs between the end points of the curve. A comparison of radiances calculated from the complete theory of radiative transfer using Stokes vectors with those obtained from the scalar theory shows that differences of the order of 23% may be obtained for Rayleigh scattering, while the largest difference found for a haze L phase function was of the order of 0.1%.

Kattawar, G. W.↗

NASA NTRS1977

Comparison of finite source and plane wave scattering from corrugated surfaces

The choice of a plane wave to represent incident radiation in the analysis of scatter from corrugated surfaces was examined. The physical optics solution obtained for the scattered fields due to an incident plane wave was compared with the solution obtained when the incident radiation is produced by a source of finite size and finite distance from the surface. The two solutions are equivalent if the observer is in the far field of the scatterer and the distance from observer to scatterer is large compared to the radius of curvature at the scatter points, condition not easily satisfied with extended scatterers such as rough surfaces. In general, the two solutions have essential differences such as in the location of the scatter points and the dependence of the scattered fields on the surface properties. The implication of these differences to the definition of a meaningful radar cross section was examined.

Levine, D. M.↗

NASA NTRS1978

Raman scattering in the atmospheres of the major planets

A technique is developed to calculate the detailed effects of Raman scattering in an inhomogeneous anisotropically scattering atmosphere. The technique is applied to evaluations of Raman scattering by H2 in the atmosphere of the major planets. It is noted that Raman scattering produces an insufficient decrease in the blue and ultraviolet regions to explain the albedos of all planets investigated. For all major planets, the filling-in of solar line cores and the generation of the Raman-shifted ghosts of the Fraunhofer spectrum are observed. With regard to Uranus and Neptune, Raman scattering is seen to exert a major influence on the formation and profile of strong red and near infrared CH4 bands, and Raman scattering by H2 explains the residual intensity in the cores of these bands. Raman scattering by H2 must also be taken into account in the scattering of photons into the cores of saturated absorption bands.

Cochran, W. D.↗

NASA NTRS1979

A general and computationally fast formulation for radiative transfer with scattering

A general formulation of monocromatic radiative transfer with scattering has been developed for plane-parallel geometry. The inhomogeneous and nonisothermal medium absorbs, emits, and anisotropically scatters radiation. Surfaces can emit and scatter radiation in any specified manner. The solution procedure uses the fact that phase incoherent scattering is linear in radiative sources. Certain basic scattering functions are then defined and calculated by an adding computer code using matrix algebra. These scattering functions are weighted by the temperature field and summed (superimposed) to obtain the solution for any specific problem. Numerical results for exiting intensities and one-sided heat fluxes from general media bound by one arbitrary surface are presented. These parametric studies demonstrate the effects of scattering particles and surfaces on radiative transfer from inhomogeneous and nonisothermal media. Application of the formulation to radiative equilibrium is also discussed. The conclusion is that all problems in plane-parallel radiative transfer with scattering can be solved by a common and computationally fast algorithm based on this formulation.

Cogley, A. C.↗