Abstract

The asymptotic law for the radial distribution of radiance density from an isotropic point source placed in a slab of homogeneous absorbing and scattering material is obtained within the framework of diffusion theory. The exponential shape of the tail of the resulting Green function has been observed but was not theoretically explained until now. We derive formulas for both the steady-state and the time-dependent problems. The theoretical results are verified by comparison with Monte Carlo simulations.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. A. Marshak, A. Davis, R. F. Cahalan, W. J. Wiscombe, “Nonlocal independent pixel approximation: direct and inverse problems,” IEEE Trans. Geosci. Remote Sens. 36, 192–205 (1998).
    [CrossRef]
  39. I. N. Polonsky, M. A. Box, A. B. Davis, “Radiative transfer through inhomogeneous turbid media: implementation of the adjoint perturbation approach at the first order,” J. Quant. Spectrosc. Radiat. Transf. 78, 85–98 (2003).
    [CrossRef]
  40. M. A. Box, I. N. Polonsky, A. B. Davis, “Higher-order perturbation theory applied to radiative transfer in non-plane-parallel media,” J. Quant. Spectrosc. Radiat. Transf. 78, 105–118 (2003).
    [CrossRef]
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  42. I. N. Polonsky, M. A. Box, “General perturbation technique for the calculation of radiative effects in scattering and absorbing media,” J. Opt. Soc. Am. A 19, 2281–2292 (2002).
    [CrossRef]
  43. S. P. Love, A. B. Davis, C. A. Rohde, L. Tellier, C. Ho, “Active probing of cloud multiple scattering, optical depth, vertical thickness, and liquid water content using Wide-Angle Imaging Lidar,” in Atmospheric Radiation Measurements and Applications in Climate, J. A. Show, ed., Proc. SPIE4815, 129–138 (2002).
    [CrossRef]

2003 (3)

K. F. Evans, R. P. Lawson, P. Zmarzly, D. O’Connor, W. J. Wiscombe, “In situ cloud sensing with multiple scattering lidar: Simulations and demonstration,” J. Atmos. Oceanic Technol. 20, 1505–1522 (2003).
[CrossRef]

I. N. Polonsky, M. A. Box, A. B. Davis, “Radiative transfer through inhomogeneous turbid media: implementation of the adjoint perturbation approach at the first order,” J. Quant. Spectrosc. Radiat. Transf. 78, 85–98 (2003).
[CrossRef]

M. A. Box, I. N. Polonsky, A. B. Davis, “Higher-order perturbation theory applied to radiative transfer in non-plane-parallel media,” J. Quant. Spectrosc. Radiat. Transf. 78, 105–118 (2003).
[CrossRef]

2002 (4)

I. N. Polonsky, M. A. Box, “General perturbation technique for the calculation of radiative effects in scattering and absorbing media,” J. Opt. Soc. Am. A 19, 2281–2292 (2002).
[CrossRef]

C. von Savigny, A. B. Davis, O. Funk, K. Pfeilsticker, “Time-series of zenith radiance and surface flux under cloudy skies: radiative smoothing, optical thickness retrievals and large-scale stationarity,” Geophys. Res. Lett. 29, 1825–1828 (2002).
[CrossRef]

A. B. Davis, A. Marshak, “Space–time characteristics of light transmitted by dense clouds. A Green function analysis,” J. Atmos. Sci. 59, 2713–2727 (2002).
[CrossRef]

J. Hampel, E. Schleicher, R. Freyer, “Volume image reconstruction for diffuse optical tomography,” Appl. Opt. 41, 3816–3826 (2002).
[CrossRef] [PubMed]

2001 (3)

A. Davis, A. Marshak, “Multiple scattering in clouds: Insights from three-dimensional diffusion/P1 theory,” Nucl. Sci. Eng. 137, 251–280 (2001).

S. P. Love, A. B. Davis, C. Ho, C. A. Rohde, “Remote sensing of cloud thickness and liquid water content with Wide-Angle Imaging Lidar (WAIL),” Atmos. Res. 59–60, 295–312 (2001).
[CrossRef]

L. M. Romanova, “Narrow light beam propagation in a stratified cloud: Higher transverse moments,” Izv., Acad. Sci., USSR Atmos. Oceanic Phys. 37, 748–756 (2001).

2000 (1)

S. Stolik, J. A. Delgado, A. Perez, L. Anasagasti, “Measurement of the penetration depths of red and near infrared light in human “ex vivo” tissues,” J. Photochem. Photobiol. B 57, 90–93 (2000).
[CrossRef]

1999 (2)

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth 24, 757–765 (1999).
[CrossRef]

1998 (4)

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

T. Farrel, M. Patterson, M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. 37, 1958–1972 (1998).
[CrossRef]

A. Marshak, A. Davis, R. F. Cahalan, W. J. Wiscombe, “Nonlocal independent pixel approximation: direct and inverse problems,” IEEE Trans. Geosci. Remote Sens. 36, 192–205 (1998).
[CrossRef]

A. A. Kokhanovsky, E. P. Zege, “On remote sensing of water clouds from space,” Adv. Space Res. 21, 425–428 (1998).
[CrossRef]

1997 (4)

1996 (1)

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, J.-P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124, 216–221 (1996).
[CrossRef]

1995 (1)

A. Marshak, A. Davis, W. J. Wiscombe, R. F. Cahalan, “Radiative smoothing in fractal clouds,” J. Geophys. Res. 100, 26247–26261 (1995).
[CrossRef]

1994 (3)

B. D. Ganapol, D. E. Kornreich, J. A. Dahl, D. W. Nigg, S. N. Jahshan, C. A. Temple, “The searchlight problem for neutrons in a semi-infinite medium,” Nucl. Sci. Eng. 118, 38–53 (1994).

R. F. Cahalan, W. Ridgway, W. J. Wiscombe, S. Gollmer, Harshvardhan, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776–3790 (1994).
[CrossRef]

J. Li, J. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

1992 (1)

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: Temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

1989 (2)

1982 (1)

1964 (1)

G. Marchuk, “Equation for the value of information from weather satellite and formulation of inverse problems,” Kosm. Issled. 2, 462–477 (1964).

1956 (1)

1941 (1)

L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Aalders, M. C.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Anasagasti, L.

S. Stolik, J. A. Delgado, A. Perez, L. Anasagasti, “Measurement of the penetration depths of red and near infrared light in human “ex vivo” tissues,” J. Photochem. Photobiol. B 57, 90–93 (2000).
[CrossRef]

Arridge, S. R.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: Temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Avrillier, S.

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, J.-P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124, 216–221 (1996).
[CrossRef]

Box, M. A.

I. N. Polonsky, M. A. Box, A. B. Davis, “Radiative transfer through inhomogeneous turbid media: implementation of the adjoint perturbation approach at the first order,” J. Quant. Spectrosc. Radiat. Transf. 78, 85–98 (2003).
[CrossRef]

M. A. Box, I. N. Polonsky, A. B. Davis, “Higher-order perturbation theory applied to radiative transfer in non-plane-parallel media,” J. Quant. Spectrosc. Radiat. Transf. 78, 105–118 (2003).
[CrossRef]

I. N. Polonsky, M. A. Box, “General perturbation technique for the calculation of radiative effects in scattering and absorbing media,” J. Opt. Soc. Am. A 19, 2281–2292 (2002).
[CrossRef]

Buis, J. P.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Cahalan, R. F.

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth 24, 757–765 (1999).
[CrossRef]

A. Marshak, A. Davis, R. F. Cahalan, W. J. Wiscombe, “Nonlocal independent pixel approximation: direct and inverse problems,” IEEE Trans. Geosci. Remote Sens. 36, 192–205 (1998).
[CrossRef]

A. Davis, A. Marshak, R. F. Cahalan, W. J. Wiscombe, “The Landsat scale-break in stratocumulus as a three-dimensional radiative transfer effect: Implications for cloud remote sensing,” J. Atmos. Sci. 54, 241–260 (1997).
[CrossRef]

A. Marshak, A. Davis, W. J. Wiscombe, R. F. Cahalan, “Radiative smoothing in fractal clouds,” J. Geophys. Res. 100, 26247–26261 (1995).
[CrossRef]

R. F. Cahalan, W. Ridgway, W. J. Wiscombe, S. Gollmer, Harshvardhan, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776–3790 (1994).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

Chylek, P.

J. Li, J. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

Contini, D.

Cope, M.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: Temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Cross, F. W.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Dahl, J. A.

B. D. Ganapol, D. E. Kornreich, J. A. Dahl, D. W. Nigg, S. N. Jahshan, C. A. Temple, “The searchlight problem for neutrons in a semi-infinite medium,” Nucl. Sci. Eng. 118, 38–53 (1994).

Davis, A.

A. Davis, A. Marshak, “Multiple scattering in clouds: Insights from three-dimensional diffusion/P1 theory,” Nucl. Sci. Eng. 137, 251–280 (2001).

A. Marshak, A. Davis, R. F. Cahalan, W. J. Wiscombe, “Nonlocal independent pixel approximation: direct and inverse problems,” IEEE Trans. Geosci. Remote Sens. 36, 192–205 (1998).
[CrossRef]

A. Davis, A. Marshak, R. F. Cahalan, W. J. Wiscombe, “The Landsat scale-break in stratocumulus as a three-dimensional radiative transfer effect: Implications for cloud remote sensing,” J. Atmos. Sci. 54, 241–260 (1997).
[CrossRef]

A. Marshak, A. Davis, W. J. Wiscombe, R. F. Cahalan, “Radiative smoothing in fractal clouds,” J. Geophys. Res. 100, 26247–26261 (1995).
[CrossRef]

Davis, A. B.

I. N. Polonsky, M. A. Box, A. B. Davis, “Radiative transfer through inhomogeneous turbid media: implementation of the adjoint perturbation approach at the first order,” J. Quant. Spectrosc. Radiat. Transf. 78, 85–98 (2003).
[CrossRef]

M. A. Box, I. N. Polonsky, A. B. Davis, “Higher-order perturbation theory applied to radiative transfer in non-plane-parallel media,” J. Quant. Spectrosc. Radiat. Transf. 78, 105–118 (2003).
[CrossRef]

C. von Savigny, A. B. Davis, O. Funk, K. Pfeilsticker, “Time-series of zenith radiance and surface flux under cloudy skies: radiative smoothing, optical thickness retrievals and large-scale stationarity,” Geophys. Res. Lett. 29, 1825–1828 (2002).
[CrossRef]

A. B. Davis, A. Marshak, “Space–time characteristics of light transmitted by dense clouds. A Green function analysis,” J. Atmos. Sci. 59, 2713–2727 (2002).
[CrossRef]

S. P. Love, A. B. Davis, C. Ho, C. A. Rohde, “Remote sensing of cloud thickness and liquid water content with Wide-Angle Imaging Lidar (WAIL),” Atmos. Res. 59–60, 295–312 (2001).
[CrossRef]

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth 24, 757–765 (1999).
[CrossRef]

S. P. Love, A. B. Davis, C. A. Rohde, L. Tellier, C. Ho, “Active probing of cloud multiple scattering, optical depth, vertical thickness, and liquid water content using Wide-Angle Imaging Lidar,” in Atmospheric Radiation Measurements and Applications in Climate, J. A. Show, ed., Proc. SPIE4815, 129–138 (2002).
[CrossRef]

Davison, B.

B. Davison, Neutron Transport Theory (Clarendon, Oxford, UK, 1957).

Deirmendjian, D.

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

Delgado, J. A.

S. Stolik, J. A. Delgado, A. Perez, L. Anasagasti, “Measurement of the penetration depths of red and near infrared light in human “ex vivo” tissues,” J. Photochem. Photobiol. B 57, 90–93 (2000).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: Temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Doetsch, G.

G. Doetsch, Guide to the Application of the Laplace and Z-Transforms (Van Nostrand Reinhold, New York, 1971).

Doornbos, R. M. P.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Eck, T. I.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Essenpreis, M.

Evans, K. F.

K. F. Evans, R. P. Lawson, P. Zmarzly, D. O’Connor, W. J. Wiscombe, “In situ cloud sensing with multiple scattering lidar: Simulations and demonstration,” J. Atmos. Oceanic Technol. 20, 1505–1522 (2003).
[CrossRef]

Farrel, T.

Feshbach, H.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1952).

Freyer, R.

Funk, O.

C. von Savigny, A. B. Davis, O. Funk, K. Pfeilsticker, “Time-series of zenith radiance and surface flux under cloudy skies: radiative smoothing, optical thickness retrievals and large-scale stationarity,” Geophys. Res. Lett. 29, 1825–1828 (2002).
[CrossRef]

Ganapol, B. D.

B. D. Ganapol, D. E. Kornreich, J. A. Dahl, D. W. Nigg, S. N. Jahshan, C. A. Temple, “The searchlight problem for neutrons in a semi-infinite medium,” Nucl. Sci. Eng. 118, 38–53 (1994).

Geldart, J.

J. Li, J. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

Gélébart, B.

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, J.-P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124, 216–221 (1996).
[CrossRef]

Gollmer, S.

R. F. Cahalan, W. Ridgway, W. J. Wiscombe, S. Gollmer, Harshvardhan, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776–3790 (1994).
[CrossRef]

Gordon, H.

H. Gordon, “Interpretation of airborne oceanic lidar: Effects of multiple scattering,” Appl. Opt. 21, 2996–3001 (1982).
[CrossRef] [PubMed]

H. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983).
[CrossRef]

Greenstein, J.

L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hampel, J.

Harshvardhan,

R. F. Cahalan, W. Ridgway, W. J. Wiscombe, S. Gollmer, Harshvardhan, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776–3790 (1994).
[CrossRef]

Henyey, L.

L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Ho, C.

S. P. Love, A. B. Davis, C. Ho, C. A. Rohde, “Remote sensing of cloud thickness and liquid water content with Wide-Angle Imaging Lidar (WAIL),” Atmos. Res. 59–60, 295–312 (2001).
[CrossRef]

S. P. Love, A. B. Davis, C. A. Rohde, L. Tellier, C. Ho, “Active probing of cloud multiple scattering, optical depth, vertical thickness, and liquid water content using Wide-Angle Imaging Lidar,” in Atmospheric Radiation Measurements and Applications in Climate, J. A. Show, ed., Proc. SPIE4815, 129–138 (2002).
[CrossRef]

Holben, B. N.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Ivanov, A.

E. Zege, A. Ivanov, I. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).

Jacques, S.

Jahshan, S. N.

B. D. Ganapol, D. E. Kornreich, J. A. Dahl, D. W. Nigg, S. N. Jahshan, C. A. Temple, “The searchlight problem for neutrons in a semi-infinite medium,” Nucl. Sci. Eng. 118, 38–53 (1994).

Jankowiak, I.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Katsev, I.

E. Zege, A. Ivanov, I. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).

Kienle, A.

Kokhanovsky, A. A.

A. A. Kokhanovsky, E. P. Zege, “On remote sensing of water clouds from space,” Adv. Space Res. 21, 425–428 (1998).
[CrossRef]

Kornreich, D. E.

B. D. Ganapol, D. E. Kornreich, J. A. Dahl, D. W. Nigg, S. N. Jahshan, C. A. Temple, “The searchlight problem for neutrons in a semi-infinite medium,” Nucl. Sci. Eng. 118, 38–53 (1994).

Lang, R.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Lavenu, F.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Lawson, R. P.

K. F. Evans, R. P. Lawson, P. Zmarzly, D. O’Connor, W. J. Wiscombe, “In situ cloud sensing with multiple scattering lidar: Simulations and demonstration,” J. Atmos. Oceanic Technol. 20, 1505–1522 (2003).
[CrossRef]

Li, J.

J. Li, J. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

Love, S. P.

S. P. Love, A. B. Davis, C. Ho, C. A. Rohde, “Remote sensing of cloud thickness and liquid water content with Wide-Angle Imaging Lidar (WAIL),” Atmos. Res. 59–60, 295–312 (2001).
[CrossRef]

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth 24, 757–765 (1999).
[CrossRef]

S. P. Love, A. B. Davis, C. A. Rohde, L. Tellier, C. Ho, “Active probing of cloud multiple scattering, optical depth, vertical thickness, and liquid water content using Wide-Angle Imaging Lidar,” in Atmospheric Radiation Measurements and Applications in Climate, J. A. Show, ed., Proc. SPIE4815, 129–138 (2002).
[CrossRef]

Marchuk, G.

G. Marchuk, “Equation for the value of information from weather satellite and formulation of inverse problems,” Kosm. Issled. 2, 462–477 (1964).

Marshak, A.

A. B. Davis, A. Marshak, “Space–time characteristics of light transmitted by dense clouds. A Green function analysis,” J. Atmos. Sci. 59, 2713–2727 (2002).
[CrossRef]

A. Davis, A. Marshak, “Multiple scattering in clouds: Insights from three-dimensional diffusion/P1 theory,” Nucl. Sci. Eng. 137, 251–280 (2001).

A. Marshak, A. Davis, R. F. Cahalan, W. J. Wiscombe, “Nonlocal independent pixel approximation: direct and inverse problems,” IEEE Trans. Geosci. Remote Sens. 36, 192–205 (1998).
[CrossRef]

A. Davis, A. Marshak, R. F. Cahalan, W. J. Wiscombe, “The Landsat scale-break in stratocumulus as a three-dimensional radiative transfer effect: Implications for cloud remote sensing,” J. Atmos. Sci. 54, 241–260 (1997).
[CrossRef]

A. Marshak, A. Davis, W. J. Wiscombe, R. F. Cahalan, “Radiative smoothing in fractal clouds,” J. Geophys. Res. 100, 26247–26261 (1995).
[CrossRef]

Martelli, F.

McGill, M. J.

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth 24, 757–765 (1999).
[CrossRef]

Morel, A.

H. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983).
[CrossRef]

Morse, P.

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1952).

Moulton, J.

Nakajima, K. Y. J. T.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Nigg, D. W.

B. D. Ganapol, D. E. Kornreich, J. A. Dahl, D. W. Nigg, S. N. Jahshan, C. A. Temple, “The searchlight problem for neutrons in a semi-infinite medium,” Nucl. Sci. Eng. 118, 38–53 (1994).

O’Connor, D.

K. F. Evans, R. P. Lawson, P. Zmarzly, D. O’Connor, W. J. Wiscombe, “In situ cloud sensing with multiple scattering lidar: Simulations and demonstration,” J. Atmos. Oceanic Technol. 20, 1505–1522 (2003).
[CrossRef]

Ollivier, J.-P.

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, J.-P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124, 216–221 (1996).
[CrossRef]

Ostermeyer, M.

Patterson, M.

Patterson, M. S.

Perez, A.

S. Stolik, J. A. Delgado, A. Perez, L. Anasagasti, “Measurement of the penetration depths of red and near infrared light in human “ex vivo” tissues,” J. Photochem. Photobiol. B 57, 90–93 (2000).
[CrossRef]

Pfeilsticker, K.

C. von Savigny, A. B. Davis, O. Funk, K. Pfeilsticker, “Time-series of zenith radiance and surface flux under cloudy skies: radiative smoothing, optical thickness retrievals and large-scale stationarity,” Geophys. Res. Lett. 29, 1825–1828 (2002).
[CrossRef]

Polonsky, I. N.

I. N. Polonsky, M. A. Box, A. B. Davis, “Radiative transfer through inhomogeneous turbid media: implementation of the adjoint perturbation approach at the first order,” J. Quant. Spectrosc. Radiat. Transf. 78, 85–98 (2003).
[CrossRef]

M. A. Box, I. N. Polonsky, A. B. Davis, “Higher-order perturbation theory applied to radiative transfer in non-plane-parallel media,” J. Quant. Spectrosc. Radiat. Transf. 78, 105–118 (2003).
[CrossRef]

I. N. Polonsky, M. A. Box, “General perturbation technique for the calculation of radiative effects in scattering and absorbing media,” J. Opt. Soc. Am. A 19, 2281–2292 (2002).
[CrossRef]

Pomraning, G. C.

G. C. Pomraning, “Diffusion theory via asymptotics,” Transp. Theory Stat. Phys. 18, 383–428 (1989).
[CrossRef]

Reagan, J. A.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Richards, P. I.

Ridgway, W.

R. F. Cahalan, W. Ridgway, W. J. Wiscombe, S. Gollmer, Harshvardhan, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776–3790 (1994).
[CrossRef]

Rohde, C. A.

S. P. Love, A. B. Davis, C. Ho, C. A. Rohde, “Remote sensing of cloud thickness and liquid water content with Wide-Angle Imaging Lidar (WAIL),” Atmos. Res. 59–60, 295–312 (2001).
[CrossRef]

S. P. Love, A. B. Davis, C. A. Rohde, L. Tellier, C. Ho, “Active probing of cloud multiple scattering, optical depth, vertical thickness, and liquid water content using Wide-Angle Imaging Lidar,” in Atmospheric Radiation Measurements and Applications in Climate, J. A. Show, ed., Proc. SPIE4815, 129–138 (2002).
[CrossRef]

Romanova, L. M.

L. M. Romanova, “Narrow light beam propagation in a stratified cloud: Higher transverse moments,” Izv., Acad. Sci., USSR Atmos. Oceanic Phys. 37, 748–756 (2001).

Schleicher, E.

Setxer, A.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Slutsker, I.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Smirnov, A.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Spinhirne, J. D.

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth 24, 757–765 (1999).
[CrossRef]

Sterenborg, H. J. C. M.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Stolik, S.

S. Stolik, J. A. Delgado, A. Perez, L. Anasagasti, “Measurement of the penetration depths of red and near infrared light in human “ex vivo” tissues,” J. Photochem. Photobiol. B 57, 90–93 (2000).
[CrossRef]

Tar, D.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Tellier, L.

S. P. Love, A. B. Davis, C. A. Rohde, L. Tellier, C. Ho, “Active probing of cloud multiple scattering, optical depth, vertical thickness, and liquid water content using Wide-Angle Imaging Lidar,” in Atmospheric Radiation Measurements and Applications in Climate, J. A. Show, ed., Proc. SPIE4815, 129–138 (2002).
[CrossRef]

Temple, C. A.

B. D. Ganapol, D. E. Kornreich, J. A. Dahl, D. W. Nigg, S. N. Jahshan, C. A. Temple, “The searchlight problem for neutrons in a semi-infinite medium,” Nucl. Sci. Eng. 118, 38–53 (1994).

Tinet, E.

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, J.-P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124, 216–221 (1996).
[CrossRef]

Tualle, J.-M.

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, J.-P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124, 216–221 (1996).
[CrossRef]

Vemte, J. E.

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

von Savigny, C.

C. von Savigny, A. B. Davis, O. Funk, K. Pfeilsticker, “Time-series of zenith radiance and surface flux under cloudy skies: radiative smoothing, optical thickness retrievals and large-scale stationarity,” Geophys. Res. Lett. 29, 1825–1828 (2002).
[CrossRef]

Wilson, B.

Wiscombe, W. J.

K. F. Evans, R. P. Lawson, P. Zmarzly, D. O’Connor, W. J. Wiscombe, “In situ cloud sensing with multiple scattering lidar: Simulations and demonstration,” J. Atmos. Oceanic Technol. 20, 1505–1522 (2003).
[CrossRef]

A. Marshak, A. Davis, R. F. Cahalan, W. J. Wiscombe, “Nonlocal independent pixel approximation: direct and inverse problems,” IEEE Trans. Geosci. Remote Sens. 36, 192–205 (1998).
[CrossRef]

A. Davis, A. Marshak, R. F. Cahalan, W. J. Wiscombe, “The Landsat scale-break in stratocumulus as a three-dimensional radiative transfer effect: Implications for cloud remote sensing,” J. Atmos. Sci. 54, 241–260 (1997).
[CrossRef]

A. Marshak, A. Davis, W. J. Wiscombe, R. F. Cahalan, “Radiative smoothing in fractal clouds,” J. Geophys. Res. 100, 26247–26261 (1995).
[CrossRef]

R. F. Cahalan, W. Ridgway, W. J. Wiscombe, S. Gollmer, Harshvardhan, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776–3790 (1994).
[CrossRef]

Zaccanti, G.

Zege, E.

E. Zege, A. Ivanov, I. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).

Zege, E. P.

A. A. Kokhanovsky, E. P. Zege, “On remote sensing of water clouds from space,” Adv. Space Res. 21, 425–428 (1998).
[CrossRef]

Zmarzly, P.

K. F. Evans, R. P. Lawson, P. Zmarzly, D. O’Connor, W. J. Wiscombe, “In situ cloud sensing with multiple scattering lidar: Simulations and demonstration,” J. Atmos. Oceanic Technol. 20, 1505–1522 (2003).
[CrossRef]

Adv. Space Res. (1)

A. A. Kokhanovsky, E. P. Zege, “On remote sensing of water clouds from space,” Adv. Space Res. 21, 425–428 (1998).
[CrossRef]

Appl. Opt. (5)

Astrophys. J. (1)

L. Henyey, J. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Atmos. Res. (1)

S. P. Love, A. B. Davis, C. Ho, C. A. Rohde, “Remote sensing of cloud thickness and liquid water content with Wide-Angle Imaging Lidar (WAIL),” Atmos. Res. 59–60, 295–312 (2001).
[CrossRef]

Geophys. Res. Lett. (1)

C. von Savigny, A. B. Davis, O. Funk, K. Pfeilsticker, “Time-series of zenith radiance and surface flux under cloudy skies: radiative smoothing, optical thickness retrievals and large-scale stationarity,” Geophys. Res. Lett. 29, 1825–1828 (2002).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

A. Marshak, A. Davis, R. F. Cahalan, W. J. Wiscombe, “Nonlocal independent pixel approximation: direct and inverse problems,” IEEE Trans. Geosci. Remote Sens. 36, 192–205 (1998).
[CrossRef]

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

L. M. Romanova, “Narrow light beam propagation in a stratified cloud: Higher transverse moments,” Izv., Acad. Sci., USSR Atmos. Oceanic Phys. 37, 748–756 (2001).

J. Atmos. Oceanic Technol. (1)

K. F. Evans, R. P. Lawson, P. Zmarzly, D. O’Connor, W. J. Wiscombe, “In situ cloud sensing with multiple scattering lidar: Simulations and demonstration,” J. Atmos. Oceanic Technol. 20, 1505–1522 (2003).
[CrossRef]

J. Atmos. Sci. (4)

A. B. Davis, A. Marshak, “Space–time characteristics of light transmitted by dense clouds. A Green function analysis,” J. Atmos. Sci. 59, 2713–2727 (2002).
[CrossRef]

R. F. Cahalan, W. Ridgway, W. J. Wiscombe, S. Gollmer, Harshvardhan, “Independent pixel and Monte Carlo estimates of stratocumulus albedo,” J. Atmos. Sci. 51, 3776–3790 (1994).
[CrossRef]

A. Davis, A. Marshak, R. F. Cahalan, W. J. Wiscombe, “The Landsat scale-break in stratocumulus as a three-dimensional radiative transfer effect: Implications for cloud remote sensing,” J. Atmos. Sci. 54, 241–260 (1997).
[CrossRef]

J. Li, J. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
[CrossRef]

J. Geophys. Res. (1)

A. Marshak, A. Davis, W. J. Wiscombe, R. F. Cahalan, “Radiative smoothing in fractal clouds,” J. Geophys. Res. 100, 26247–26261 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Photochem. Photobiol. B (1)

S. Stolik, J. A. Delgado, A. Perez, L. Anasagasti, “Measurement of the penetration depths of red and near infrared light in human “ex vivo” tissues,” J. Photochem. Photobiol. B 57, 90–93 (2000).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf. (2)

I. N. Polonsky, M. A. Box, A. B. Davis, “Radiative transfer through inhomogeneous turbid media: implementation of the adjoint perturbation approach at the first order,” J. Quant. Spectrosc. Radiat. Transf. 78, 85–98 (2003).
[CrossRef]

M. A. Box, I. N. Polonsky, A. B. Davis, “Higher-order perturbation theory applied to radiative transfer in non-plane-parallel media,” J. Quant. Spectrosc. Radiat. Transf. 78, 105–118 (2003).
[CrossRef]

Kosm. Issled. (1)

G. Marchuk, “Equation for the value of information from weather satellite and formulation of inverse problems,” Kosm. Issled. 2, 462–477 (1964).

Nucl. Sci. Eng. (2)

B. D. Ganapol, D. E. Kornreich, J. A. Dahl, D. W. Nigg, S. N. Jahshan, C. A. Temple, “The searchlight problem for neutrons in a semi-infinite medium,” Nucl. Sci. Eng. 118, 38–53 (1994).

A. Davis, A. Marshak, “Multiple scattering in clouds: Insights from three-dimensional diffusion/P1 theory,” Nucl. Sci. Eng. 137, 251–280 (2001).

Opt. Commun. (1)

J.-M. Tualle, B. Gélébart, E. Tinet, S. Avrillier, J.-P. Ollivier, “Real time optical coefficients evaluation from time and space resolved reflectance measurements in biological tissues,” Opt. Commun. 124, 216–221 (1996).
[CrossRef]

Phys. Chem. Earth (1)

A. B. Davis, R. F. Cahalan, J. D. Spinhirne, M. J. McGill, S. P. Love, “Off-beam lidar: An emerging technique in cloud remote sensing based on radiative Green-function theory in the diffusion domain,” Phys. Chem. Earth 24, 757–765 (1999).
[CrossRef]

Phys. Med. Biol. (2)

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: Temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Remote Sens. Environ. (1)

B. N. Holben, T. I. Eck, I. Slutsker, D. Tar, J. P. Buis, A. Setxer, J. E. Vemte, J. A. Reagan, K. Y. J. T. Nakajima, F. Lavenu, I. Jankowiak, A. Smirnov, “AERONET–A federated instrument network and data archive for aerosol characterization,” Remote Sens. Environ. 66, 1–16 (1998).
[CrossRef]

Transp. Theory Stat. Phys. (1)

G. C. Pomraning, “Diffusion theory via asymptotics,” Transp. Theory Stat. Phys. 18, 383–428 (1989).
[CrossRef]

Other (10)

P. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1952).

A. Erdelyi, ed., Tables of Integral Transforms, Vol. 1 (McGraw-Hill, New York, 1954).

G. Doetsch, Guide to the Application of the Laplace and Z-Transforms (Van Nostrand Reinhold, New York, 1971).

D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).

H. Gordon, A. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery, Vol. 4 of Lecture Notes on Coastal and Estuarine Studies (Springer-Verlag, New York, 1983).
[CrossRef]

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

B. Davison, Neutron Transport Theory (Clarendon, Oxford, UK, 1957).

E. Zege, A. Ivanov, I. Katsev, Image Transfer Through a Scattering Medium (Springer-Verlag, Heidelberg, Germany, 1991).

S. P. Love, A. B. Davis, C. A. Rohde, L. Tellier, C. Ho, “Active probing of cloud multiple scattering, optical depth, vertical thickness, and liquid water content using Wide-Angle Imaging Lidar,” in Atmospheric Radiation Measurements and Applications in Climate, J. A. Show, ed., Proc. SPIE4815, 129–138 (2002).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Normalized flux density FH(r)/T as a function of radial distance r at the top of a slab of thickness H=0.3 km for an isotropic point source at the bottom. The extinction coefficient σe is 53.3 km-1 (optical thickness τH=σeH is 16), the single-scattering albedo ϖ0 is unity, and the numbers near the curves show the asymmetry parameter g of the assumed Henyey–Greenstein phase function.

Fig. 2
Fig. 2

Flux density F0(r) as a function of radial distance r at the bottom of the slab of thickness H=0.7 km illuminated on the same boundary by a collimated beam. The curves show the results of calculations by using Eq. (34); numerical (Monte Carlo) results are depicted by symbols. The numbers near the curves are the optical thickness τH of the slab. The single-scattering albedo ϖ0 is 1. Our diffusion-based expression requires the asymmetry parameter g of the Deirmendjian “Cloud.C1” phase function that we used in the Monte Carlo simulation, which is 0.85.

Equations (53)

Equations on this page are rendered with MathJax. Learn more.

I(t, r, z, n)=14π [J(t, r, z)+3n  F(t, r, z)],
J(t, r, z)=4πI(t, r, z, n)dn,
F(t, r, z)=4πnI(t, r, z, n)dn.
1cJt+  F+(σe-σs)J(t, r, z)
=E0δ(t)δ(r)δ(z-z0),
1cFt+13 J+(σe-σsg)F(t, r, z)
=0.
D=c3(σe-σsg)=clt3
u=σect,
ρ=σer,
τ=σez,
f(s)=0exp(-su)f(u)du,
f(k)=+-exp(-ik  ρ)f(ρ)dρ.
F(s, k, τ)=-13(1+s-ϖ0g)iJk, dJdτ,
d2Jdτ2-m2J(s, k, τ)
=-3cσe3E0(1+s-ϖ0g)δ(τ-τ0).
m=[k2+3(1+s-ϖ0)(1+s-ϖ0g)]1/2.
1-χ τJ(s, k, τ)|τ=0=0,
1+χ τJ(s, k, τ)|τ=τH=0,
χ=2/3(1+s-ϖ0g),
J(s, k, τ=-χ)=0,
J(s, k, τ=τH+χ)=0.
J(s, k, τ)=cσe3E0cosh[m(τH+2χ-|τ-τ0|)]-cosh{m[τH-(τ+τ0)]}χm sinh[m(τH+2χ)]
cosh[(b-a)m]sinh[bm]=[exp(-am)+exp(am-2bm)]×n=0exp(-2bn)=n=-exp(-|a+2nb|m),
if0<a<b,
exp(-ms)=0δ(x-s)exp(-mx)dx,
ifs>0.
J(s, ρ, τ)
=cσe3E02χπ×n=-+exp(-γρ2+[|τ-τ0|+2n(τH+2χ)]2)ρ2+[|τ-τ0|+2n(τH+2χ)]2-exp(-γρ2+[(τ+τ0)+2χ+2n(τH+2χ)]2)ρ2+[(τ+τ0)+2χ+2n(τH+2χ)]2,
γ=[3(1+s-ϖ0)(1+s-ϖ0g)]1/2.
J(s, ρ, τ)γ exp(-γρ)ρ2+exp(-γρ)ρ3.
n=-+f(αn)=1αn=-+f˜2nπα,
-exp(-γρ2+(a+x)2)ρ2+(a+x)2exp(-iξx)dx
=2 exp(iξa)K0(ρξ2+γ2),
J(ρ, τ)=E02σe2πχ(τH+2χ)n=1sinπn(τ+χ)τH+2χ×sinπn(τ0+χ)τH+2χ×K0ργ2+πnτH+2χ21/2.
J(ρ, τ)E02σe2πχ(τH+2χ)sinπ(τ+χ)τH+2χsinπ(τ0+χ)τH+2χ×π2ργ2+πτH+2χ21/21/2×exp-ργ2+πτH+2χ21/2,
J(ρ, τ)exp-ργ2+πτH+2χ21/2ρ.
22exp-ρπτH+2χ1,
ρτH+2χ.
τHρ3/8χ,
γs0[3(1+s-ϖ0)(1-ϖ0g)]1/2,
J(u, ρ, τ)=E02cσe3exp[-(1-ϖ0)u]τH+2χ12πχu×exp-ρ22χun=1sinπn(τ+χ)τH+2χ×sinπn(τ0+χ)τH+2χexp-χu2πnτH+2χ2,
J(u, τ)exp-(1-ϖ0)+π2χ2(τH+2χ)2u,
F0(s, k)=121+χ τJ(s, k, τ)|τ=0,
FH(s, k)=121-χ τJ(s, k, τ)|τ=τH.
F0(s, k)=J(s, k, 0),
FH(s, k)=J(s, k, τH).
T=11+τH/2χ,
R=τH/2χ1+τH/2χ
J(r, z)1rexp-rrc
rc=Hπ1+2χτH=HπR,
J(t, z)exp[-t/tc],
tc=2π2HcτHχR2=3(1-g)σec rc2.

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