Multiple-scattering radiative transfer

The propagation of light in multiple scattering media is quantified usually on the level of radiative transfer or particle diffusion. Scattering, absorption, and emission are considered as independent statistical processes, and the consequences of wave character are either ignored, like polarization, or added as an additional parameter, like the phase function P(ji n) that describes the angular distribution of scattered... 

Stamnes, K S.-C. Tsay, W. Wiscombe, and K. Jayaweera, Numerically Stable Algorithm for Discrete-Ordinate-Method Radiative Transfer in Multiple Scattering and Emitting Layered Media, Appl. Opt., 27, 2502-2509 (1988). 

According to the ground rules laid down at the beginning of this book, multiple scattering is excluded from consideration. But it is not always prudent to pretend that multiple scattering does not exist. Fortunately, it is almost trivial—the mathematical apparatus of radiative transfer theory is unnecessary—to extend our treatment of scattering and circular polarization to multiple scattering media, and in this instance it is worth the small amount of effort required to do so. 

Stammes K., Tsay S., W. Wiscombe and K. Jayaweera, Numerical stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media, Applied Optics 27, 2502-... 

Since 1992 the two Italian stations of Rome, urban site (latitude 41.9° N, longitude 12.5° E, altitude 60 m), and Ispra, semi-rural site (latitude 45.8° N, longitude 8.6° E, altitude 240 m), collect regular continuous measurements of spectral UV (290-325 nm) irradiance by means of Brewer Spectrophotometry. The measured data are compared with the output of the STAR model (System for Transfer of Atmospheric Radiation) [1], STAR is a multiple scattering radiative transfer model which considers all atmospheric factors modulating UV radiation at ground (ozone, aerosol, clouds, pollutants, albedo, pressure, temperature, humidity) [2], The model involves combination of a radiative transfer code, an initialisation procedure and an integration scheme. 

Solution of the Equation of Radiative transfer for Wavelengths Less than 3.5 /im Multiple Scattering... 

Several analytic methods have been proposed to solve the equation of radiative transfer in an absorbing and scattering atmosphere, but they can only be applied for the most simple cases. To obtain quantitative solutions, numerical methods are generally used, such as the Monte-Carlo method, DART method, iterative Gauss, discrete ordinate method, etc. A complete summary of these techniques is provided by Lenoble (1977), and a detailed discussion of multiple scattering processes in plane parallel atmospheres is given in the book by Liou (2002). 

Because multiple scattering, surface reflection, clouds, and aerosols have a significant effect on radiative intensities at photodissociative wavelengths and consequently on the composition of the middle atmosphere (see, e.g., Luther et al, 1978), simplified radiative transfer... 

The Beer law is similar to the Lambert law with the exception that the absorption coefficient is expressed as the product of a unit absorption coefficient and the concentration of particles. In general, this definition is more fundamental and appropriate for application to dispersed media, where the concentration can be directly measured. If there are particles in the medium, depending on their size with respect to the wavelength of the incident radiation, they scatter as well as absorb the incident radiation. With increasing concentration, the multiple scattering effect becomes significant, and the Beer law deviates from the experimental measurements, especially if the size of the particles is comparable to the wavelength of radiation. Under these conditions, the complete radiative transfer equation should be solved. For solid materials, the Beer and Lambert laws are identical. 

The second exception is if we are interested in the propagation of a collimated light source (i.e., a laser). In this case, since only one incident direction is to be considered, the problem can be modeled by direct simulation, even for multiple scattering media up to intermediate optical thicknesses (t = 1). In general, the direct simulation of the radiative transfer equation is to be chosen if a fundamental understanding of radiation-combustion or radiation-turbulence interactions is required. 

Analytic solutions to the radiative transfer equation (RTE) exist for simple cases however, for more realistic media with complex multiple scattering effects, numerical methods are required. The equation of radiative transfer simply states that as a beam of radiation travels, it loses energy to absorption, gains energy by emission and redistributes energy by scattering. The differential form of the equation for radiative transfer is ... 

Analysis of Multiple-Scattering Radiative Transfer Within Photobioreactors ... 

ANALYSIS OF MULTIPLE-SCATTERING RADIATIVE TRANSFER WITHIN PHOTOBIOREACTORS APPROXIMATE SOLUTIONS FOR THE RADIATION FIELD WITHIN ONE-DIMENSIONAL CARTESIAN PHOTOBIOREACTORS... 

A discussion of applications of these concepts to neutron transport theory can be found in the papers [4 5 6 7 8 9]. For the interested reader we also call attention to the applications to radiative transfer theory [1 10 11], to wave propagation, [12 13 14], to random walk and multiple scattering [15 16], and to problems with moving boundaries [9]. 

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