Abstract

Exponential extinction of incoherent radiation intensity in a random medium (sometimes referred to as the Beer–Lambert law) arises early in the development of several branches of science and underlies much of radiative transfer theory and propagation in turbid media with applications in astronomy, atmospheric science, and oceanography. We adopt a stochastic approach to exponential extinction and connect it to the underlying Poisson statistics of extinction events. We then show that when a dilute random medium is statistically homogeneous but spatially correlated, the attenuation of incoherent radiation with depth is often slower than exponential. This occurs because spatial correlations among obstacles of the medium spread out the probability distribution of photon extinction events. Therefore the probability of transmission (no extinction) is increased.

© 2001 Optical Society of America

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References

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  1. L. Romanova, “Radiative transfer in a horizontally inhomogeneous scattering medium,” Izv., Acad. Sci. USSR Atmos. Oceanic Phys. 11, 509–513 (1975).
  2. J. Weinman, Harshvardhan, “Solar reflection from a regular array of horizontally finite clouds,” Appl. Opt. 21, 2940–2944 (1982).
    [CrossRef]
  3. G. L. Stephens, P. M. Gabriel, S. C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20, 139–175 (1991).
    [CrossRef]
  4. 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]
  5. 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]
  6. A. Davis, “Radiation transport in scale-invariant optical media,” Ph.D. dissertation (McGill University, Montreal, 1992).
  7. A. Davis, A. Marshak, “Levy kinetics in slab geometry: scaling of transmission probability,” in Fractal Frontiers, M. Novak, T. Dewey, eds. (World Scientific, Singapore, 1997), pp. 63–72.
  8. Y. Knyazikhin, J. Kranigk, R. B. Myneni, O. Panfyorov, G. Gravenhorsi, “Influence of small-scale structure on radiative transfer and photosynthesis in vegetation canopies,” J. Geophys. Res. 103, 6133–6144 (1998).
    [CrossRef]
  9. G. C. Pomraning, Linear Kinetic Theory and Particle Transport in Stochastic Mixtures (World Scientific, Singapore, 1991).
  10. R. Goody, Y. Yung, Atmospheric Radiation (Oxford U. Press, Oxford, UK, 1989).
  11. L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, New York, 1980).
  12. James Jeans, The Dynamical Theory of Gases (Dover, New York, 1954).
  13. A. Burshtein, Introduction to Thermodynamics and Kinetic Theory of Matter (Wiley, New York, 1995).
  14. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  15. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).
  16. W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968), Vol. 1.
  17. A. Davis, A. Marshak, H. Gerber, W. Wiscombe, “Horizontal structure of marine boundary layers from centimeter to kilometer scales,” J. Geophys. Res. 104, 6123–6144 (1999).
    [CrossRef]
  18. A. Kostinski, A. Jameson, “Fluctuation properties of precipitation. Part I: Deviations of single size drop counts from the Poisson distribution,” J. Atmos. Sci. 54, 2174–2186 (1997).
    [CrossRef]
  19. A. Kostinski, A. Jameson, “On the spatial distribution of cloud particles,” J. Atmos. Sci. 57, 901–915 (2000).
    [CrossRef]
  20. R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, UK, 1983).
  21. L. Mandel, E. C. D. Sudarshan, E. Wolf, “Theory of photoelectric detection of light fluctuations,” Proc. Phys. Soc. 84, 435–444 (1964).
    [CrossRef]
  22. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995).
  23. A. Pumir, B. Shraiman, E. D. Siggia, “Exponential tails and random advection,” Phys. Rev. Lett. 66, 2984 (1991).
    [CrossRef] [PubMed]
  24. K. Pfeilsticker, “First geometrical pathlength distribution measurements of skylight using the oxygen a-band absorption technique. Part 2: Derivation of the Levy-index for skylight transmitted by mid-latitude clouds,” J. Geophys. Res. 104, 4101–4116 (1999).
    [CrossRef]

2000 (1)

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

1999 (2)

A. Davis, A. Marshak, H. Gerber, W. Wiscombe, “Horizontal structure of marine boundary layers from centimeter to kilometer scales,” J. Geophys. Res. 104, 6123–6144 (1999).
[CrossRef]

K. Pfeilsticker, “First geometrical pathlength distribution measurements of skylight using the oxygen a-band absorption technique. Part 2: Derivation of the Levy-index for skylight transmitted by mid-latitude clouds,” J. Geophys. Res. 104, 4101–4116 (1999).
[CrossRef]

1998 (2)

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]

Y. Knyazikhin, J. Kranigk, R. B. Myneni, O. Panfyorov, G. Gravenhorsi, “Influence of small-scale structure on radiative transfer and photosynthesis in vegetation canopies,” J. Geophys. Res. 103, 6133–6144 (1998).
[CrossRef]

1997 (1)

A. Kostinski, A. Jameson, “Fluctuation properties of precipitation. Part I: Deviations of single size drop counts from the Poisson distribution,” J. Atmos. Sci. 54, 2174–2186 (1997).
[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]

1991 (2)

G. L. Stephens, P. M. Gabriel, S. C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20, 139–175 (1991).
[CrossRef]

A. Pumir, B. Shraiman, E. D. Siggia, “Exponential tails and random advection,” Phys. Rev. Lett. 66, 2984 (1991).
[CrossRef] [PubMed]

1982 (1)

1975 (1)

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

1964 (1)

L. Mandel, E. C. D. Sudarshan, E. Wolf, “Theory of photoelectric detection of light fluctuations,” Proc. Phys. Soc. 84, 435–444 (1964).
[CrossRef]

Burshtein, A.

A. Burshtein, Introduction to Thermodynamics and Kinetic Theory of Matter (Wiley, New York, 1995).

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. Davis, A. Marshak, H. Gerber, W. Wiscombe, “Horizontal structure of marine boundary layers from centimeter to kilometer scales,” J. Geophys. Res. 104, 6123–6144 (1999).
[CrossRef]

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]

A. Davis, A. Marshak, “Levy kinetics in slab geometry: scaling of transmission probability,” in Fractal Frontiers, M. Novak, T. Dewey, eds. (World Scientific, Singapore, 1997), pp. 63–72.

A. Davis, “Radiation transport in scale-invariant optical media,” Ph.D. dissertation (McGill University, Montreal, 1992).

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968), Vol. 1.

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]

Gabriel, P. M.

G. L. Stephens, P. M. Gabriel, S. C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20, 139–175 (1991).
[CrossRef]

Gerber, H.

A. Davis, A. Marshak, H. Gerber, W. Wiscombe, “Horizontal structure of marine boundary layers from centimeter to kilometer scales,” J. Geophys. Res. 104, 6123–6144 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Goody, R.

R. Goody, Y. Yung, Atmospheric Radiation (Oxford U. Press, Oxford, UK, 1989).

Gravenhorsi, G.

Y. Knyazikhin, J. Kranigk, R. B. Myneni, O. Panfyorov, G. Gravenhorsi, “Influence of small-scale structure on radiative transfer and photosynthesis in vegetation canopies,” J. Geophys. Res. 103, 6133–6144 (1998).
[CrossRef]

Harshvardhan,

Jameson, A.

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

A. Kostinski, A. Jameson, “Fluctuation properties of precipitation. Part I: Deviations of single size drop counts from the Poisson distribution,” J. Atmos. Sci. 54, 2174–2186 (1997).
[CrossRef]

Jeans, James

James Jeans, The Dynamical Theory of Gases (Dover, New York, 1954).

Knyazikhin, Y.

Y. Knyazikhin, J. Kranigk, R. B. Myneni, O. Panfyorov, G. Gravenhorsi, “Influence of small-scale structure on radiative transfer and photosynthesis in vegetation canopies,” J. Geophys. Res. 103, 6133–6144 (1998).
[CrossRef]

Kostinski, A.

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

A. Kostinski, A. Jameson, “Fluctuation properties of precipitation. Part I: Deviations of single size drop counts from the Poisson distribution,” J. Atmos. Sci. 54, 2174–2186 (1997).
[CrossRef]

Kranigk, J.

Y. Knyazikhin, J. Kranigk, R. B. Myneni, O. Panfyorov, G. Gravenhorsi, “Influence of small-scale structure on radiative transfer and photosynthesis in vegetation canopies,” J. Geophys. Res. 103, 6133–6144 (1998).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, New York, 1980).

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]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, New York, 1980).

Loudon, R.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, UK, 1983).

Mandel, L.

L. Mandel, E. C. D. Sudarshan, E. Wolf, “Theory of photoelectric detection of light fluctuations,” Proc. Phys. Soc. 84, 435–444 (1964).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995).

Marshak, A.

A. Davis, A. Marshak, H. Gerber, W. Wiscombe, “Horizontal structure of marine boundary layers from centimeter to kilometer scales,” J. Geophys. Res. 104, 6123–6144 (1999).
[CrossRef]

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]

A. Davis, A. Marshak, “Levy kinetics in slab geometry: scaling of transmission probability,” in Fractal Frontiers, M. Novak, T. Dewey, eds. (World Scientific, Singapore, 1997), pp. 63–72.

Myneni, R. B.

Y. Knyazikhin, J. Kranigk, R. B. Myneni, O. Panfyorov, G. Gravenhorsi, “Influence of small-scale structure on radiative transfer and photosynthesis in vegetation canopies,” J. Geophys. Res. 103, 6133–6144 (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]

Panfyorov, O.

Y. Knyazikhin, J. Kranigk, R. B. Myneni, O. Panfyorov, G. Gravenhorsi, “Influence of small-scale structure on radiative transfer and photosynthesis in vegetation canopies,” J. Geophys. Res. 103, 6133–6144 (1998).
[CrossRef]

Pfeilsticker, K.

K. Pfeilsticker, “First geometrical pathlength distribution measurements of skylight using the oxygen a-band absorption technique. Part 2: Derivation of the Levy-index for skylight transmitted by mid-latitude clouds,” J. Geophys. Res. 104, 4101–4116 (1999).
[CrossRef]

Pomraning, G. C.

G. C. Pomraning, Linear Kinetic Theory and Particle Transport in Stochastic Mixtures (World Scientific, Singapore, 1991).

Pumir, A.

A. Pumir, B. Shraiman, E. D. Siggia, “Exponential tails and random advection,” Phys. Rev. Lett. 66, 2984 (1991).
[CrossRef] [PubMed]

Reif, F.

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

Romanova, L.

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

Shraiman, B.

A. Pumir, B. Shraiman, E. D. Siggia, “Exponential tails and random advection,” Phys. Rev. Lett. 66, 2984 (1991).
[CrossRef] [PubMed]

Siggia, E. D.

A. Pumir, B. Shraiman, E. D. Siggia, “Exponential tails and random advection,” Phys. Rev. Lett. 66, 2984 (1991).
[CrossRef] [PubMed]

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]

Stephens, G. L.

G. L. Stephens, P. M. Gabriel, S. C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20, 139–175 (1991).
[CrossRef]

Sudarshan, E. C. D.

L. Mandel, E. C. D. Sudarshan, E. Wolf, “Theory of photoelectric detection of light fluctuations,” Proc. Phys. Soc. 84, 435–444 (1964).
[CrossRef]

Tsay, S. C.

G. L. Stephens, P. M. Gabriel, S. C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20, 139–175 (1991).
[CrossRef]

Weinman, J.

Wiscombe, W.

A. Davis, A. Marshak, H. Gerber, W. Wiscombe, “Horizontal structure of marine boundary layers from centimeter to kilometer scales,” J. Geophys. Res. 104, 6123–6144 (1999).
[CrossRef]

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]

Wolf, E.

L. Mandel, E. C. D. Sudarshan, E. Wolf, “Theory of photoelectric detection of light fluctuations,” Proc. Phys. Soc. 84, 435–444 (1964).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995).

Yung, Y.

R. Goody, Y. Yung, Atmospheric Radiation (Oxford U. Press, Oxford, UK, 1989).

Appl. Opt. (1)

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. (3)

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. Kostinski, A. Jameson, “Fluctuation properties of precipitation. Part I: Deviations of single size drop counts from the Poisson distribution,” J. Atmos. Sci. 54, 2174–2186 (1997).
[CrossRef]

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

J. Geophys. Res. (4)

A. Davis, A. Marshak, H. Gerber, W. Wiscombe, “Horizontal structure of marine boundary layers from centimeter to kilometer scales,” J. Geophys. Res. 104, 6123–6144 (1999).
[CrossRef]

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]

Y. Knyazikhin, J. Kranigk, R. B. Myneni, O. Panfyorov, G. Gravenhorsi, “Influence of small-scale structure on radiative transfer and photosynthesis in vegetation canopies,” J. Geophys. Res. 103, 6133–6144 (1998).
[CrossRef]

K. Pfeilsticker, “First geometrical pathlength distribution measurements of skylight using the oxygen a-band absorption technique. Part 2: Derivation of the Levy-index for skylight transmitted by mid-latitude clouds,” J. Geophys. Res. 104, 4101–4116 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

A. Pumir, B. Shraiman, E. D. Siggia, “Exponential tails and random advection,” Phys. Rev. Lett. 66, 2984 (1991).
[CrossRef] [PubMed]

Proc. Phys. Soc. (1)

L. Mandel, E. C. D. Sudarshan, E. Wolf, “Theory of photoelectric detection of light fluctuations,” Proc. Phys. Soc. 84, 435–444 (1964).
[CrossRef]

Transp. Theory Stat. Phys. (1)

G. L. Stephens, P. M. Gabriel, S. C. Tsay, “Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere,” Transp. Theory Stat. Phys. 20, 139–175 (1991).
[CrossRef]

Other (12)

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, UK, 1983).

A. Davis, “Radiation transport in scale-invariant optical media,” Ph.D. dissertation (McGill University, Montreal, 1992).

A. Davis, A. Marshak, “Levy kinetics in slab geometry: scaling of transmission probability,” in Fractal Frontiers, M. Novak, T. Dewey, eds. (World Scientific, Singapore, 1997), pp. 63–72.

G. C. Pomraning, Linear Kinetic Theory and Particle Transport in Stochastic Mixtures (World Scientific, Singapore, 1991).

R. Goody, Y. Yung, Atmospheric Radiation (Oxford U. Press, Oxford, UK, 1989).

L. D. Landau, E. M. Lifshitz, Statistical Physics, 3rd ed. (Pergamon, New York, 1980).

James Jeans, The Dynamical Theory of Gases (Dover, New York, 1954).

A. Burshtein, Introduction to Thermodynamics and Kinetic Theory of Matter (Wiley, New York, 1995).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965).

W. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968), Vol. 1.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995).

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Equations (15)

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pn(x)=n(x)¯nexp[-n(x)¯]n!,
p0(x)=exp[-n(x)¯]=exp(-βx),
NtrNinc=exp(-βx),
ptr=(1-β dx1)(1-β dx2)  (1-β dxm).
ln(ptr)=-β dx1-β dx2--β dxm=-βx,
P(1, 2)=k¯2dV1dV2[1+η(l)],
η(l)[K(l)K(0)¯-K¯]2K¯2=K(l)K(0)¯K¯2-1,
(δK)2¯K¯-1=K¯VVη dV,
(δK)2¯=K¯+η¯K¯2,
p(n)=0p(n|n¯)p(n¯)dn¯=0n¯nexp(-n)n! p(n¯)dn¯,
p(n¯)=1μexp-n¯μ,
p(n)=1μ+1μμ+1n.
(δn)2¯=μ+μ2,
p0(x)=1μ+1=1βx+1,
NtrNinc=11+βx.

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