Abstract

In response to comments by Borovoi [J. Opt. Soc. Am. A 19, 2517 (2002)] on my earlier work [J. Opt. Soc. Am. A 18, 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

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References

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  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.”

2002 (2)

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]

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]

2001 (3)

X. Lei, B. J. Ackerson, P. Tong, “Settling statistics of hard sphere particles,” Phys. Rev. Lett. 86, 3300–3303 (2001).
[CrossRef] [PubMed]

A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
[CrossRef]

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]

2000 (1)

A. B. Kostinski, A. R. Jameson, “On the spatial Distribution of cloud particles,” J. Atmos. Sci. 57, 901–915 (2000).
[CrossRef]

1998 (1)

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]

1995 (1)

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]

1984 (1)

A. G. Borovoi, “Radiative transfer in inhomogeneous media,” Dokl. Akad. Nauk SSSR 276, 1374–1378 (1984); in Russian.

1975 (1)

L. Romanova, “Radiative transfer in a horizontally inhomogeneous scattering medium,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 11, 509–513 (1975).

Ackerson, B. J.

X. Lei, B. J. Ackerson, P. Tong, “Settling statistics of hard sphere particles,” Phys. Rev. Lett. 86, 3300–3303 (2001).
[CrossRef] [PubMed]

Borovoi, A. G.

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]

A. G. Borovoi, “Radiative transfer in inhomogeneous media,” Dokl. Akad. Nauk SSSR 276, 1374–1378 (1984); in Russian.

Cahalan, R.

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]

Davis, A.

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]

Eisberg, R.

R. Eisberg, R. Resnick, Quantum Physics, 2nd ed. (Wiley, New York, 1985).

Fovell, R. G.

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]

Friedlander, S. K.

S. K. Friedlander, Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics, 2nd ed. (Oxford, New York, 2000).

Jameson, A. R.

A. B. Kostinski, A. R. Jameson, “On the spatial Distribution of cloud particles,” J. Atmos. Sci. 57, 901–915 (2000).
[CrossRef]

Kostinski, A. B.

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]

A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
[CrossRef]

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]

A. B. Kostinski, A. R. Jameson, “On the spatial Distribution of cloud particles,” J. Atmos. Sci. 57, 901–915 (2000).
[CrossRef]

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.

Lanterman, D. D.

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]

Lei, X.

X. Lei, B. J. Ackerson, P. Tong, “Settling statistics of hard sphere particles,” Phys. Rev. Lett. 86, 3300–3303 (2001).
[CrossRef] [PubMed]

Lew, J. K.

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]

Marshak, A.

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]

Newman, W. I.

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]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1991).

Ochi, M. K.

M. K. Ochi, Applied Probability and Stochastic Processes in Engineering and Physical Sciences (Wiley, New York, 1990).

Resnick, R.

R. Eisberg, R. Resnick, Quantum Physics, 2nd ed. (Wiley, New York, 1985).

Romanova, L.

L. Romanova, “Radiative transfer in a horizontally inhomogeneous scattering medium,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 11, 509–513 (1975).

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.

Shaw, R. A.

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]

A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
[CrossRef]

Siscoe, G. L.

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]

Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.

Tong, P.

X. Lei, B. J. Ackerson, P. Tong, “Settling statistics of hard sphere particles,” Phys. Rev. Lett. 86, 3300–3303 (2001).
[CrossRef] [PubMed]

van Kampen, N. G.

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).

Weinberg, A. M.

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactions (University of Chicago Press, Chicago, Ill., 1958).

Wigner, E. P.

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactions (University of Chicago Press, Chicago, Ill., 1958).

Wiscombe, W.

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]

Dokl. Akad. Nauk SSSR (1)

A. G. Borovoi, “Radiative transfer in inhomogeneous media,” Dokl. Akad. Nauk SSSR 276, 1374–1378 (1984); in Russian.

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

L. Romanova, “Radiative transfer in a horizontally inhomogeneous scattering medium,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 11, 509–513 (1975).

J. Atmos. Sci. (2)

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]

A. B. Kostinski, A. R. Jameson, “On the spatial Distribution of cloud particles,” J. Atmos. Sci. 57, 901–915 (2000).
[CrossRef]

J. Fluid Mech. (1)

A. B. Kostinski, R. A. Shaw, “Scale-dependent droplet clustering in turbulent clouds,” J. Fluid Mech. 434, 389–398 (2001).
[CrossRef]

J. Geophys. Res. (1)

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]

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transfer (1)

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]

Phys. Rev. Lett. (1)

X. Lei, B. J. Ackerson, P. Tong, “Settling statistics of hard sphere particles,” Phys. Rev. Lett. 86, 3300–3303 (2001).
[CrossRef] [PubMed]

Other (11)

S. K. Friedlander, Smoke, Dust and Haze: Fundamentals of Aerosol Dynamics, 2nd ed. (Oxford, New York, 2000).

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.”

M. K. Ochi, Applied Probability and Stochastic Processes in Engineering and Physical Sciences (Wiley, New York, 1990).

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

A. M. Weinberg, E. P. Wigner, The Physical Theory of Neutron Chain Reactions (University of Chicago Press, Chicago, Ill., 1958).

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.

R. Eisberg, R. Resnick, Quantum Physics, 2nd ed. (Wiley, New York, 1985).

E. L. O’Neill, Introduction to Statistical Optics (Dover, New York, 1991).

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).

N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992).

S. M. Rytov, Y. A. Kravtsov, V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, Berlin, 1989), Vol. 3.

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Figures (3)

Fig. 1
Fig. 1

Simple example of a distribution that causes linear (faster than exponential) extinction. Black squares indicate perfect absorbers. The absorbers are distributed completely at random, except for the constraint that no two obstacles can be aligned in the propagation direction (thus introducing correlations by forcing new absorbers to “avoid shadows”). While this distribution is anisotropic, one can readily construct more-realistic and isotropic distributions that result in extinction rates between the linear regime (a lower limit) and the exponential regime (e.g., see Fig. 3).

Fig. 2
Fig. 2

Schematic depiction of the chain of reasoning in Refs. 2 and 16. The left column shows that a random distribution of absorbers more uniform than Poisson (e.g., electrostatic or hydrodynamic repulsion) can yield superexponential extinction. Similarly, the right column demonstrates that subexponential extinction can occur in positively correlated (clustered) media; the middle column for perfectly random absorbers is included for reference. The first row shows a characteristic pair-correlation function that can be used to generate corresponding distributions of absorbers. The second row (adapted from Ref. 16) shows a typical thin slice from each distribution. Note that all three distributions have the same concentration and obstacle characteristics. The number of absorption-event series shown in the third row can be illustrated as follows. Imagine a “neutrino” (a particle unaffected by the absorbers) traversing a given depth of a distribution and counting the number of absorbers encountered. The numbers “reported” by different neutrinos are plotted in the third row. The fourth row is the corresponding probability distribution. The probability of (photon) transmission through the layer is the probability of encountering no absorbers. Again note that the mean number of encounters is the same for all three distributions (fixed depth and concentration), so the entire argument is about the change in variance while the mean is held fixed.

Fig. 3
Fig. 3

Illustration of correlation effects in “shadow overlap.” These are “end on” samples of transversally infinite distributions of obstacles, with the propagation direction being out of the page and toward the reader. The medium is dilute (volume fraction less than 10-4 in all three media, but the dots in the upper “thin-slice” panel are larger than actual size for better visibility). Left to right, we have a negatively correlated distribution, a perfectly random distribution, and a positively correlated (clustered) distribution. The bottom panels are the entire distributions (318 of the respective upper-panel slices). Despite seemingly minor visual differences among the three upper (thin) slices, the cumulative anti-correlated distribution is strikingly less transparent (absorber amount and obstacle cross section remaining equal for the three cases). All three absorber distributions yield intensity-versus-depth curves that begin at the same point and have equal initial slopes: then correlations set in. Therefore at least a transitional nonexponential regime is implied.

Equations (5)

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ptr=(1-βdx1)(1-βdx2)(1-βdxm),
dI(l)=-cσI(l)dl,
P(1, 2)=k¯2dV1dV2[1+η(l)],
(δK)2¯=K¯+η¯K¯2,
p(K)=0p(K|K¯)p(K¯)dK¯=0 K¯Kexp(-K¯)K! p(K¯)dK¯,

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