OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 38, Iss. 6 — Feb. 20, 1999
  • pp: 932–936

Diffuse Reflection Coefficient of a Stratified Sea

Vladimir I. Haltrin  »View Author Affiliations

Applied Optics, Vol. 38, Issue 6, pp. 932-936 (1999)

View Full Text Article

Acrobat PDF (173 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A differential equation of a Riccati type for the diffuse reflection coefficient of a stratified sea is proposed. For a homogeneous sea with arbitrary inherent optical properties this equation is solved analytically. For an inhomogeneous sea it is solved approximately for any arbitrary stratification. The resulting equation expresses the diffuse reflection coefficient of the sea through vertical profiles of absorption and backscattering coefficients, bottom albedo, and sea depth. The results of calculations with this equation are compared with Monte Carlo computations. It was found that the precision of this approach is in the range of 15%.

[Optical Society of America ]

Vladimir I. Haltrin, "Diffuse Reflection Coefficient of a Stratified Sea," Appl. Opt. 38, 932-936 (1999)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. E. P. Zege, A. P. Ivanov, and I. L. Katsev, Image Transfer through a Scattering Media (Springer-Verlag, Berlin, 1991).
  2. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).
  3. C. D. Mobley, Light and Water (Academic, San Diego, Calif., 1994).
  4. C. D. Mobley, “Hydrolight 3.0 user’s guide,” Final Rep. 5632 (SRI International, Menlo Park, Calif., 1995).
  5. The approach outlined in Refs. 6 and 7 and in this paper differs from the approach of Aas22 in that the values Ed and Eu in Eqs. (2) correspond to downward and upward irradiances by renormalized components of light. In Ref. 22 irradiances Ed and Eu in two-flow equations (12) and (13) are total irradiances. This means that the coefficients in Eqs. (2) should not be compared with the coefficients of Eqs. (12) and (13) of Ref. 22.
  6. V. I. Haltrin and G. W. Kattawar, “Self-consistent solutions to the equation of transfer with elastic and inelastic scattering in oceanic optics. I. Model,” Appl. Opt. 32, 5356–5367 (1993).
  7. V. I. Haltrin, “Self-consistent approach to the solution of the light transfer problem for irradiances in marine waters with arbitrary turbidity, depth, and surface illumination. I. Case of absorption and elastic scattering,” Appl. Opt. 37, 3773–3784 (1998).
  8. V. I. Haltrin, “Theoretical and empirical phase functions for Monte Carlo calculations of light scattering in seawater,” in Proceedings of the Fourth International Conference on Remote Sensing for Marine and Coastal Environments (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 509–518.
  9. V. A. Timofeyeva, “Relation between the optical coefficients in turbid media,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 8, 654–656 (1972).
  10. The left-hand side of Eqs. (4) were derived by the author from the experimental results published by Timofeyeva.9
  11. K. Schwarzschild, “Über das Gleichgewicht der Sonnenatmosphere,” Gottingen Nachrichten 41, 1–24 (1906).
  12. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
  13. P. Kubelka and F. Munk, “Ein Beitrag zur Optik der Farbanstriche,” Z. Tech. Phys. (Leipzig) 12, 593–607 (1931).
  14. G. C. Pomraning, “An extension of the Eddington approximation,” J. Quant. Spectrosc. Radiat. Transfer 9, 407–422 (1969).
  15. V. I. Haltrin, “Exact solution of the characteristic equation for transfer in the anisotropically scattering and absorbing medium,” Appl. Opt. 27, 599–602 (1988).
  16. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
  17. H. R. Gordon and O. B. Brown, “Diffuse reflection of the ocean: some effects of vertical structure,” Appl. Opt. 14, 2892–2895 (1975).
  18. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vols. 1 and 2.
  19. V. I. Haltrin, “Apparent optical properties of the sea illuminated by Sun and sky: case of the optically deep sea,” Appl. Opt. 37, 8336–8340 (1998).
  20. The values of a and bB used here correspond to the 530-nm wavelength band. We should note that in the elastic radiative transfer the wavelength is a parameter. This means that use of values of a and bB at certain wavelengths to estimate possible errors does not restrict this theory to a certain wavelength.
  21. After the submission of this paper the resulting Eqs. (27)–(29) were tested with the Monte Carlo simulations with a modified23 J. T. O. Kirk’s code.24,25 The inherent optical properties were adopted from T. J. Petzold.8, 26 The results of simulations confirm the conclusion of Section 4 that the precision of calculation with Eqs. (27)–(29) is in the range of 15%.
  22. E. Aas, “Two-stream irradiance model for deep waters,” Appl. Opt. 26, 2095–2101 (1987).
  23. V. I. Haltrin, “Monte Carlo modeling of light field parameters in ocean with Petzold laws of scattering,” in Proceedings of the Fourth International Conference on Remote Sensing for Marine and Coastal Environments: (Environmental Research Institute of Michigan, Ann Arbor, Mich., 1997), Vol. I, pp. 502–508.
  24. J. T. O. Kirk, “Monte Carlo procedure for simulating the penetration of light into natural waters,” Division of Plant Industry Tech. Paper 36 (Commonwealth Scientic and Industrial Research Organization, Canberra, Australia, 1981).
  25. J. T. O. Kirk, “Characteristics of the light field in highly turbid waters: a Monte-Carlo study,” Limnol. Oceanogr. 39, 702–706 (1994).
  26. T. J. Petzold, Volume Scattering Functions for Selected Ocean Waters, SIO Ref. 72–78 (Visibility Laboratory, Scripps Institution of Oceanography, San Diego, Calif., 1972).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited