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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 19, Iss. 12 — Dec. 1, 2002
  • pp: 2521–2525

On the extinction of radiation by a homogeneous but spatially correlated random medium: reply to comment

Alexander B. Kostinski  »View Author Affiliations


JOSA A, Vol. 19, Issue 12, pp. 2521-2525 (2002)
http://dx.doi.org/10.1364/JOSAA.19.002521


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Abstract

In response to comments by Borovoi [J. Opt. Soc. Am. A19, 2517 (2002)] on my earlier work [J. Opt. Soc. Am. A18, 1929 (2001)], the kinetic approach to extinction is compared with the traditional radiative transfer formalism and advantages of the former are illustrated with concrete examples. It is pointed out that the basic differential equation dI(l)=-cσI(l)dl already implies perfect randomness (absence of correlations) on small scales. One of the consequences is that the extinction of radiation in a negatively correlated random medium cannot be treated within the traditional framework. This limits the usefulness of the Jensen inequality. Also, simple counterexamples to theorems given in the first reference above and in Dokl. Akad. Nauk SSSR, 276, 1374 (1984) are presented.

© 2002 Optical Society of America

OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(030.5290) Coherence and statistical optics : Photon statistics
(030.5620) Coherence and statistical optics : Radiative transfer
(030.6600) Coherence and statistical optics : Statistical optics

History
Original Manuscript: May 29, 2002
Revised Manuscript: July 22, 2002
Manuscript Accepted: July 22, 2002
Published: December 1, 2002

Citation
Alexander B. Kostinski, "On the extinction of radiation by a homogeneous but spatially correlated random medium: reply to comment," J. Opt. Soc. Am. A 19, 2521-2525 (2002)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-19-12-2521


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References

  1. A. G. Borovoi, “On the extinction of radiation by a homogeneous but spatially correlated random medium: comment,” J. Opt. Soc. Am. A 19, 2517–2520 (2002). [CrossRef]
  2. A. B. Kostinski, “On the extinction of radiation by a homo-geneous but spatially correlated random medium,” J. Opt. Soc. Am. A 18, 1929–1933 (2001). [CrossRef]
  3. A. G. Borovoi, “Radiative transfer in inhomogeneous media,” Dokl. Akad. Nauk SSSR 276, 1374–1378 (1984); in Russian.
  4. I find the connection in Ref. 1between average attenuation, cumulants, and correlation functions quite interesting, despite the use of abstract notions such as a characteristic functional (e.g., Ref. 5, pp. 63 and 405). However, the meaning of the various averages is not clearly delineated in concrete physical terms, but rather ergodicity is assumed instead. This renders the approach impractical, as one often has to deal with variability on many (sometimes all) scales, thus violating wide-sense stationarity, let alone ergodicity (e.g., whenever correlation length is comparable with the propagation distance or the medium dimensions; see Ref. 6, pp. 22–23).
  5. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).
  6. S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.
  7. W. I. Newman, J. K. Lew, G. L. Siscoe, R. G. Fovell, “Systematic effects of randomness in radiative transfer,” J. Atmos. Sci. 52, 427–435 (1995). [CrossRef]
  8. A. Marshak, A. Davis, W. Wiscombe, R. Cahalan, “Radiative effects of sub-mean free path liquid water variability observed in stratiform clouds,” J. Geophys. Res. 103, 19557–19567 (1998). [CrossRef]
  9. L. Romanova, “Radiative transfer in a horizontally inhomogeneous scattering medium,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 11, 509–513 (1975).
  10. I was particularly interested to learn from Ref. 3that the earliest application of the Jensen inequality to the transport equation occurred already in 1958.11
  11. A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactions (University of Chicago Press, Chicago, Ill., 1958).
  12. This may be a subtle point, as even texts containing thorough discussions of the topic seem to miss or omit it; e.g., see Ref. 13, pp. 48–50.
  13. R. Eisberg, R. Resnick, Quantum Physics, 2nd ed. (Wiley, New York, 1985).
  14. E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1991).
  15. X. Lei, B. J. Ackerson, P. Tong, “Settling statistics of hard sphere particles,” Phys. Rev. Lett. 86, 3300–3303 (2001). [CrossRef] [PubMed]
  16. R. A. Shaw, A. B. Kostinski, D. D. Lanterman, “Super-exponential extinction in a negatively correlated random medium,” J. Quant. Spectrosc. Radiat. Transfer 75, 13–20 (2002). [CrossRef]
  17. S. K. Friedlander, Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics, 2nd ed. (Oxford, New York, 2000).
  18. A. B. Kostinski, A. R. Jameson, “On the spatial Distribution of cloud particles,” J. Atmos. Sci. 57, 901–915 (2000). [CrossRef]
  19. M. K. Ochi, Applied Probability and Stochastic Processes in Engineering and Physical Sciences (Wiley, New York, 1990).
  20. A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001). [CrossRef]
  21. In the abstract of Ref. 1as well as in the conclusions, the word “extinction” is apparently reserved for attenuation of a given layer with depth rather than horizontally averaged attenuation for layers that are not very deep. It seems rather pedantic to insist on replacing “extinction” with “horizontally averaged transmittance dependence on depth.”

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