Inhomogeneous Plane-Parallel Atmospheres
This chapter builds upon the physical descriptions of scattering processes of Chapter 1. It develops via invariant imbedding techniques effective mathematical and computational models of diffuse reflection and transmission due to multiple scattering in vertically stratified media. It treats the determination of internal diffuse intensities without use of the unstable equations of transfer and the computation of source functions without having to solve their ill-conditioned integral equations. Furthermore, internal and external intensity fields as well as source functions due to vertically inhomogeneous distributions of emitting sources are obtained. The emphasis here is to obtain exact Cauchy problems which are well solved computationally and to present samplings of the extensive numerical results that have been obtained. Cauchy problems are initial value problems for systems of differential equations and are attractive for computational solution.
KeywordsCauchy Problem Incident Angle Optical Thickness Source Function Scattered Radiation
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- 1.M. Abramowitz and I. Stegun (eds.), Handbook of Mathematical Functions, National Bureau of Standards, 1964.Google Scholar
- 2.R. Bellman, R. Kalaba and M. Prestrud, Invariant Imbedding and Radiative Transfer in Slabs of Finite Thickness, American Elsevier Publishing Co., New York, 1963.Google Scholar
- 3.R. Bellman, H. Kagiwada, R. Kalaba and M. C. Prestrud, Invariant Imbedding and Time-Dependent Transport Processes, American Elsevier Publishing Co., New York, 1964.Google Scholar
- 4.H. Kagiwada, R. Kalaba and S. Ueno, Multiple Scattering Processes: Inverse and Direct, Addison-Wesley Publishing Co., Reading, Mass., 1975.Google Scholar
- 5.B. Carnahan, H. Luther and J. Wilkes, Applied Numerical Methods, Wiley, New York, 1969.Google Scholar
- 6.W. Press, et al., Numerical Recipes in C, Cambridge University Press, Cambridge, 1992.Google Scholar
- 7.J. V. Dave, J. Quant. Spect. Rad. Transfer, Vol. 8, 1968, p. 25.Google Scholar
- 15.H. Kagiwada and R. Kalaba, “Numerical Results for Internal Intensities in Atmospheres Illuminated by Isotropic Sources,” The Rand Corporation, RM-4958-PR, 1966.Google Scholar
- 19.R. Bellman, H. Kagiwada and R. Kalaba, “Invariant Imbedding and a Reformulation of the Internal Intensity Problem in Radiative Transfer Theory,” Monthly Notices Royal Astronomical Soc., Vol. 132, 1966, 183–191.Google Scholar