Basic Concepts

  • Harriet H. Natsuyama
  • Sueo Ueno
  • Alan P. Wang


We introduce basic concepts for the modeling of radiative transfer using the invariant imbedding approach. We show, for a one-dimensional reflection problem, how an initial value problem is formulated. We obtain a differential equation with the independent variable being the thickness, and an initial condition, for thickness zero. We describe the numerical procedure for integrating this equation. Tables of reflection functions are presented. Cauchy-initial value-problems for source and internal intensity functions are also treated. This chapter serves as an introduction to the more advanced concepts in Appendix A, as well as the remaining chapters of this book.


Radiative Transfer Source Function Scattered Radiation Single Scattering Scattered Particle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. W. Busbridge, The Mathematics of Radiative Transfer, Cambridge University Press, Cambridge, 1960.Google Scholar
  2. 2.
    S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960.Google Scholar
  3. 3.
    V. Kourganoff, Basic Methods in Transfer Problems, Dover Publications, New York, 1963.Google Scholar
  4. 4.
    V. V. Sobolev, Light Scattering in Planetary Atmospheres, Pergamon Press, New York, 1974.Google Scholar
  5. 5.
    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
  6. 6.
    R. E. Bellman, H. H. Kagiwada, R. E. Kalaba and M. C. Prestrud, Invariant Imbedding and Time-Dependent Transport Processes, American Elsevier Publishing Co., New York, 1964.Google Scholar
  7. 7.
    H. Kagiwada, R. Kalaba and S. Ueno, Multiple Scattering Processes: Inverse and Direct, Addison-Wesley Publishing Company, Reading, Mass., 1975.Google Scholar
  8. 8.
    H. H. Kagiwada and R. E. Kalaba, Integral Equations via Invariant Imbedding, Addison-Wesley Publishing Co., Reading, Mass., 1975.Google Scholar
  9. 9.
    A. P. Sage, “Invariant Imbedding in Control, Estimation, and System Identification,” Appl. Math. Comput., Vol. 45, 1991, p. 99.CrossRefGoogle Scholar
  10. 10.
    J. L. Calvet and G. Viargues, “Invariant Imbedding and Parallelism in Dynamic Programming for Feedback Control,” J. Optimiz. Theory Appl., Vol. 87, 1995, p. 121.CrossRefGoogle Scholar
  11. 11.
    J. Garnier, “Stochastic Invariant Imbedding. Application to Stochastic Differential Equations with Boundary Conditions,” Prob. Theory & Related Fields, Vol. 103, 1995, p. 249.CrossRefGoogle Scholar
  12. 12.
    I. S. Ayoubi, “The Riemann-Green Function and the Invariant Imbedding Equations for Hyperbolic Systems of First-order,” Appl. Math. Comp., Vol. 55, 1993, p. 101.CrossRefGoogle Scholar
  13. 13.
    M. E. Davison and R. C. Winther, “A General Approach to Splitting and Invariant Imbedding for Linear Wave Equations,” J. Math. Anal. Appl., Vol. 188, 1994, p. 158.CrossRefGoogle Scholar
  14. 14.
    A. J. Haines and M. V. deHoop, “An Invariant Imbedding Analysis of General Wave Scattering Problems,” J. Math. Phys., Vol. 37, 1996, p. 3854.CrossRefGoogle Scholar
  15. 15.
    Y. B. Band and I. Tuvi, “Quantum Rearrangement Scattering Calculations Using the Invariant Imbedding Method,” J. Chem. Phys., Vol. 100, 1994, p. 8869.CrossRefGoogle Scholar
  16. 16.
    J. Corones and Z. Sun, “Simultaneous Reconstruction of Material and Transient Source Parameters Using the Invariant Imbedding Method,” J. Math. Phys., Vol. 34, 1993, p. 1824.CrossRefGoogle Scholar
  17. 17.
    S. He and S. Strom, “The Electromagnetic Inverse Problem in the Time Domain for a Dissipative Slab and a Point Source Using Invariant Imbedding: Reconstruction of the Permittivity and Conductivity,” J. Comp. Appl. Math., Vol. 42, 1992, p. 137.CrossRefGoogle Scholar
  18. 18.
    S. K. Srinivasan and R. Vasudevan, “Particle Multiplicity Distribution à la Invariant Imbedding and Natural Scaling,” Comp. & Math. Appl., Vol. 22, 1991, p. 59.CrossRefGoogle Scholar
  19. 19.
    V. H. Weston, “Invariant Imbedding for the Wave Equation in Three Dimensions and the Applications to the Direct and Inverse Problems,” Inverse Problems, Vol. 6, 1990, p. 1075.CrossRefGoogle Scholar
  20. 20.
    G. Nadimuthu and E. S. Lee, “Invariant Imbedding Filter in the Modeling of Water Resources,” Comp. & Math. Appl., Vol. 21, 1991, p. 9.CrossRefGoogle Scholar
  21. 21.
    D. E. Womble, R. C. Allen, Jr. and L. S. Baca, “Invariant Imbedding and the Method of Lines for Parallel Computers,” Parallel Computing, Vol. 11, 1989, p. 263.CrossRefGoogle Scholar
  22. 22.
    M. I. Mischenko, “The Fast Invariant Imbedding Method for Polarized Light: Computational Aspects and Numerical Results for Rayleigh,” J. Quant. Spectrosc. Radiat. Transfer, Vol. 43, 1990, p. 163.CrossRefGoogle Scholar
  23. 23.
    A. P. Wang, “Basic Equations of Three Dimensional Radiative Transfer,” Journal of Mathematical Physics, Vol. 31, No. 10, 1990, p. 175.CrossRefGoogle Scholar
  24. 24.
    S. Ueno and A. P. Wang, “Invariant Imbedding and Order-of-Scattering Theory in Radiation Field,” Comp. & Math. Appl., Vol. 27, 1994, p. 175.CrossRefGoogle Scholar
  25. 25.
    B. Carnahan, H. Luther and J. Wilkes, Applied Numerical Methods, Wiley, New York, 1969.Google Scholar
  26. 26.
    E. G. Yanovitskij, Light Scattering in Inhomogeneous Atmospheres, Springer-Verlag, New York, 1996.Google Scholar
  27. 27.
    Jacqueline Lenoble, Atmospheric Radiative Transfer, A. Deepak Pub., 1993.Google Scholar
  28. 28.
    K. Y. Kondratyev, V. V. Kozoderov and O. I. Smokty, Remote Sensing of the Earth from Space: Atmospheric Correction, Springer-Verlag, New York, 1992.CrossRefGoogle Scholar
  29. 29.
    G. A. D’Almeida, P. Koepke and E. P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics, A. Deepak Pub., 1991.Google Scholar
  30. 30.
    W. Kalkofen, ed., Numerical Radiative Transfer, Cambridge University Press, Cambridge, 1988.Google Scholar
  31. 31.
    Jacqueline Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures, A. Deepak Pub., 1985.Google Scholar
  32. 32.
    H. C. Van de Hulst, Light Scattering by Small Particles, Dover, New York, 1982.Google Scholar
  33. 33.
    R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding, Soc. Indus. Appl. Math., 1962.Google Scholar
  34. 34.
    C. F. Bohren, ed., Selected Papers on Scattering in the Atmosphere, SPIE Optical Engineering Press, Bellingham, WA, 1989.Google Scholar

Copyright information

© Springer-Verlag Tokyo 1998

Authors and Affiliations

  • Harriet H. Natsuyama
    • 1
  • Sueo Ueno
    • 2
  • Alan P. Wang
    • 3
  1. 1.Yorba LindaUSA
  2. 2.KyotoJapan
  3. 3.TempeUSA

Personalised recommendations