Comparisons between the small-angle approximation and the numerical solution for radiative transfer theory

Application of the small-angle approximation to ocean water types

The small-angle approximation to the radiative transport equation is applied to particle suspensions that emulate ocean water. A particle size distribution is constructed from polystyrene and glass spheres with the best available data for particle size distributions in the ocean. A volume scattering function is calculated from the Mie theory for the particles in water and in oil. The refractive-index ratios of particles in water and particles in oil are 1.19 and 1.01, respectively. The ratio 1.19 is comparable to minerals and nonliving diatoms in ocean water, and the ratio 1.01 is comparable to the lower limit for microbes in water. The point-spread functions are measured as a function of optical thickness for both water and oil mixtures and compared with the point-spread functions generated from the small-angle approximation. Our results show that, under conditions that emulate ocean water, the small-angle approximation is valid only for small optical thicknesses. Specifically, the approximation is valid only for optical thicknesses less than 3.

Limits of the small-angle approximation to the radiative transport equation

The small-angle approximation to the radiative transport equation is used extensively in imaging models in which the transport medium is optically thick. The small-angle approximation is generally considered valid when the particles are very large compared with the wavelength, when the refractive-index ratio of the particle to the medium is close to 1, and when the optical thickness is not too large. We report results showing the limits of the validity of the small-angle approximation as a function of particle size and concentration for a particle-to-medium fixed refractive-index ratio of 1.196. This refractive-index ratio is comparable with that of minerals or diatoms suspended in water.

Physics-Based Visualization of Dense Natural Clouds. II. Cloud-Rendering Algorithm

We discuss the representation of aerosol-scattering properties, boundary information, and the use of these results in line-of-sight rendering applications for visualization of a modeled atmosphere based on a discrete ordinates three-dimensional radiative-transport method. The outputs of the radiative-transfer model provide spatial and angular distributions of limiting path radiance, given an input density distribution and external illumination conditions. We discuss the determination of the direct attenuated radiance, integrated path radiance, and background radiance for each pixel in the rendered scene. Orthographic and perspective projection approaches for displaying these results are described, and sample images are shown.

Optical wave propagation in discrete random media with large particles: a treatment of the phase function

A treatment of the phase function in the radiative transfer equation is discussed for describing optical wave propagation in discrete random media with large particles. Unlike the conventional small-angle approximation, the phase function is normalized so that half of the scattered power is removed from a small angle in the forward direction for large particles with the refractive index not close to unity. With this normalization, an improved small-angle solution of the radiative transfer equation is given for the phase function adopted here. The validity of the proposed theory is confirmed by comparisons with both numerical solutions and experimental data on the attenuation of millimeter and optical waves in rain.

Multi-Gaussian phase function model for off-axis laser beam scattering

Within the small-angle approximation, a multi-Gaussian phase function model provides a new description of the multiple-scattering process and leads to improved agreement with experimental data. A general formula for the off-axis laser beam irradiance is given explicitly for a two-component Gaussian phase function. The predictions of the theory are compared with both the calculations based on the Mie scattering phase function and the results of measurements. It appears that the best fit of the model to experimental results requires an adjustment of the weight factors of the component phase functions. The solution, written in the form of a multiple-scattering series, allows one to interpret individual terms of the scattering series as scattering events of different orders.