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

[CrossRef]

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

[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]

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

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

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

[CrossRef]
[PubMed]

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. 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]

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

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]

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

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

[CrossRef]

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]

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

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]

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

[CrossRef]
[PubMed]

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. 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]

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]

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

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

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

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

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

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]

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]

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

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

[CrossRef]
[PubMed]

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

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

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

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

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

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

[CrossRef]

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

[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]

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]

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

[CrossRef]
[PubMed]

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.”

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

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.