Abstract

Recently it has been shown that the perturbation technique, based on joint use of both the direct and the adjoint solutions of the radiative transfer equation, is a powerful tool to solve and analyze various time-independent one-dimensional problems of atmospheric physics such as the calculation of weighting functions, prediction of radiative effects, and development of retrieval algorithms. Our primary goal is to obtain a general formulation of the perturbation technique for the most general case of the radiative transfer problem: time-dependent problems, with regard to polarization, and any possible external sources of radiation such as laser beams and solar illumination. Possible areas of application of the perturbation technique are discussed, and several examples to illustrate them are provided. The accuracy of this technique is discussed by considering the particular examples.

© 2002 Optical Society of America

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  3. E. P. Zege, A. P. Ivanov, I. L. Katsev, Image Transfer through a Scattering Medium (Springer-Verlag, Heidelberg, 1991).
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  5. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1952).
  6. G. I. Marchuk, Adjoint Equations and Analysis of Complex System (Kluwer Academic, Dordrecht, The Netherlands, 1994).
  7. L. M. Romanova, “Radiation transfer in a horizontally inhomogeneous scattering medium,” Izv., Acad. Sci. USSR Atmos. Oceanic Phys. 11, 509–513 (1975).
  8. J. Li, J. W. Geldart, P. Chylek, “Perturbation solution for 3D radiative transfer in a horizontally periodic inhomogeneous cloud field,” J. Atmos. Sci. 51, 2110–2122 (1994).
    [CrossRef]
  9. J. Li, J. W. Geldart, P. Chylek, “Second order perturbation solution for radiative transfer in clouds with a horizontally arbitrary periodic inhomogeneity,” J. Quant. Spectrosc. Radiat. Transfer 53, 445–456 (1995).
    [CrossRef]
  10. R. J. D. Spurr, T. P. Kurosu, K. V. Chance, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
    [CrossRef]
  11. G. I. Marchuk, “Equation for the value of information from weather satellite and formulation of inverse problems,” Kosm. Issled. 2, 462–477 (1964).
  12. S. A. W. Gerstl, “Application of modern neutron transport methods to atmospheric radiative transfer,” in Volume of Extended Abstracts, International Radiation Symposium, Fort Collins, Colorado, August 11–16, 1980, pp. 500–502.
  13. M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).
  14. M. A. Box, M. Keevers, B. H. J. McKellar, “On the perturbation series for radiative effects,” J. Quant. Spectrosc. Radiat. Transfer 39, 219–223 (1988).
    [CrossRef]
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  16. M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).
  17. E. A. Ustinov, “Inverse problem of thermal sounding: to recover the vertical profile of the absorption coefficient of an optically active component of a planetary atmosphere from observing emitted radiation,” Cosmic Res. 28, 347–355 (1990).
  18. E. A. Ustinov, “Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR,” J. Quant. Spectrosc. Radiat. Transfer 68, 195–211 (2001).
    [CrossRef]
  19. E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. Mathematical methods and weighting functions of linearized inverse problem,” Cosmic Res. 29, 519–532 (1991).
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  21. E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. 3. Remote sensing of minor gaseous constituents and an atmospheric aerosol,” Cosmic Res. 30, 170–181 (1992).
  22. J. Landgraf, O. Hasekamp, M. A. Box, T. Trautmann, “A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach,” J. Geophys. Res. 106, 27291–27306 (2001).
    [CrossRef]
  23. J. Landgraf, O. Hasekamp, T. Trautmann, “Linearization of radiative transfer with respect to surface properties,” J. Quant. Spectrosc. Radiat. Transfer 72, 327–339 (2001).
    [CrossRef]
  24. C. Sendra, M. A. Box, “Retrieval of the phase function and scattering optical thickness of aerosols: a radiative perturbation theory application,” J. Quant. Spectrosc. Radiat. Transfer 64, 499–515 (2000).
    [CrossRef]
  25. I. N. Polonsky, M. A. Box, “Perturbation technique to retrieve scattering medium stratification,” J. Atmos. Sci. 59, 758–768 (2002).
    [CrossRef]
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  27. Y. Tian, M. A. Box, “Radiative perturbation theory for polarized radiance,” J. Quant. Spectr. Radiat. Transfer 72, 789–8002 (2002).
    [CrossRef]
  28. E. A. Ustinov, Earth and Space Sciences Division, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, Calif. 91109-8099 (personal communication, 2001).
  29. E. P. Zege, L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Amer. A 16, 1430–1438 (1999).
    [CrossRef]
  30. V. V. Rozanov, T. Kurosu, J. P. Burrows, “Retrieval of atmospheric constituents in the uv-visible: a new quasi-analytical approach for the calculation of weighting functions,” J. Quant. Spectrosc. Radiat. Transfer 60, 277–299 (1998).
    [CrossRef]
  31. J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).
  32. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, London, 1950).
  33. I. Kus̆c̆er, M. Ribaric̆, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
    [CrossRef]
  34. M. I. Mischenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).
  35. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1981).
  36. K. Stamnes, S.-C. Tsay, W. Wiscombe, K. Jayaweera, “A numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
    [CrossRef] [PubMed]
  37. J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
    [CrossRef]
  38. Y. Qin, D. L. B. Jupp, M. A. Box, “Extension of the discrete-ordinate algorithm and efficient radiative transfer calculation,” J. Quant. Spectrosc. Radiat. Transfer 74, 767–781 (2002).
    [CrossRef]
  39. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).
  40. G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Heidelberg, 1980).
  41. P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns—procedure and results,” Appl. Phys. Lett. 60, 325–329 (1995).
  42. L. R. Bissonnette, “Multiple-scattering lidar equation,” Appl. Opt. 35, 6449–6465 (1996).
    [CrossRef] [PubMed]
  43. D. Deirmendjian, Electromagnetic Scattering on Spherical Polydispersions (American Elsevier, New York, 1969).
  44. E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
    [CrossRef]
  45. P. Bruscaglioni, School of Physics, University of Florence, via S. Marta, 3, 50139, Firenze, Italy (personal communication, 1994).
  46. E. P. Zege, I. L. Katsev, I. N. Polonsky, “Effects of multiple scattering in laser sounding of a stratified scattering medium. 2. Peculiarities of sounding of the atmosphere from space,” Izv. Atmos. Ocean. Phys. 34, 227–234 (1998).
  47. K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
    [CrossRef]
  48. V. L. Galinsky, V. Ramanathan, “3D radiative transfer in weakly inhomogeneous medium. Part I: Diffusive approximation,” J. Atmos. Sci. 55, 2946–2959 (1998).
    [CrossRef]
  49. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]

2002

I. N. Polonsky, M. A. Box, “Perturbation technique to retrieve scattering medium stratification,” J. Atmos. Sci. 59, 758–768 (2002).
[CrossRef]

Y. Tian, M. A. Box, “Radiative perturbation theory for polarized radiance,” J. Quant. Spectr. Radiat. Transfer 72, 789–8002 (2002).
[CrossRef]

Y. Qin, D. L. B. Jupp, M. A. Box, “Extension of the discrete-ordinate algorithm and efficient radiative transfer calculation,” J. Quant. Spectrosc. Radiat. Transfer 74, 767–781 (2002).
[CrossRef]

2001

J. Landgraf, O. Hasekamp, M. A. Box, T. Trautmann, “A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach,” J. Geophys. Res. 106, 27291–27306 (2001).
[CrossRef]

J. Landgraf, O. Hasekamp, T. Trautmann, “Linearization of radiative transfer with respect to surface properties,” J. Quant. Spectrosc. Radiat. Transfer 72, 327–339 (2001).
[CrossRef]

E. A. Ustinov, “Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR,” J. Quant. Spectrosc. Radiat. Transfer 68, 195–211 (2001).
[CrossRef]

R. J. D. Spurr, T. P. Kurosu, K. V. Chance, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
[CrossRef]

2000

C. Sendra, M. A. Box, “Retrieval of the phase function and scattering optical thickness of aerosols: a radiative perturbation theory application,” J. Quant. Spectrosc. Radiat. Transfer 64, 499–515 (2000).
[CrossRef]

1999

E. P. Zege, L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Amer. A 16, 1430–1438 (1999).
[CrossRef]

1998

V. V. Rozanov, T. Kurosu, J. P. Burrows, “Retrieval of atmospheric constituents in the uv-visible: a new quasi-analytical approach for the calculation of weighting functions,” J. Quant. Spectrosc. Radiat. Transfer 60, 277–299 (1998).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Effects of multiple scattering in laser sounding of a stratified scattering medium. 2. Peculiarities of sounding of the atmosphere from space,” Izv. Atmos. Ocean. Phys. 34, 227–234 (1998).

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

V. L. Galinsky, V. Ramanathan, “3D radiative transfer in weakly inhomogeneous medium. Part I: Diffusive approximation,” J. Atmos. Sci. 55, 2946–2959 (1998).
[CrossRef]

1996

1995

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns—procedure and results,” Appl. Phys. Lett. 60, 325–329 (1995).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

J. Li, J. W. Geldart, P. Chylek, “Second order perturbation solution for radiative transfer in clouds with a horizontally arbitrary periodic inhomogeneity,” J. Quant. Spectrosc. Radiat. Transfer 53, 445–456 (1995).
[CrossRef]

1994

J. Li, J. W. 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

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. 3. Remote sensing of minor gaseous constituents and an atmospheric aerosol,” Cosmic Res. 30, 170–181 (1992).

1991

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. Mathematical methods and weighting functions of linearized inverse problem,” Cosmic Res. 29, 519–532 (1991).

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. 2. Numerical aspects and requirements on the observation geometry,” Cosmic Res. 29, 785–800 (1991).

1990

E. A. Ustinov, “Inverse problem of thermal sounding: to recover the vertical profile of the absorption coefficient of an optically active component of a planetary atmosphere from observing emitted radiation,” Cosmic Res. 28, 347–355 (1990).

1989

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

1988

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

M. A. Box, M. Keevers, B. H. J. McKellar, “On the perturbation series for radiative effects,” J. Quant. Spectrosc. Radiat. Transfer 39, 219–223 (1988).
[CrossRef]

K. Stamnes, S.-C. Tsay, W. Wiscombe, K. Jayaweera, “A numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

1983

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

1975

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

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

1964

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

1959

I. Kus̆c̆er, M. Ribaric̆, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

1941

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

Bell, G. I.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinholt, New York, 1970).

Bissonnette, L. R.

Box, M. A.

Y. Tian, M. A. Box, “Radiative perturbation theory for polarized radiance,” J. Quant. Spectr. Radiat. Transfer 72, 789–8002 (2002).
[CrossRef]

Y. Qin, D. L. B. Jupp, M. A. Box, “Extension of the discrete-ordinate algorithm and efficient radiative transfer calculation,” J. Quant. Spectrosc. Radiat. Transfer 74, 767–781 (2002).
[CrossRef]

I. N. Polonsky, M. A. Box, “Perturbation technique to retrieve scattering medium stratification,” J. Atmos. Sci. 59, 758–768 (2002).
[CrossRef]

J. Landgraf, O. Hasekamp, M. A. Box, T. Trautmann, “A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach,” J. Geophys. Res. 106, 27291–27306 (2001).
[CrossRef]

C. Sendra, M. A. Box, “Retrieval of the phase function and scattering optical thickness of aerosols: a radiative perturbation theory application,” J. Quant. Spectrosc. Radiat. Transfer 64, 499–515 (2000).
[CrossRef]

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

M. A. Box, M. Keevers, B. H. J. McKellar, “On the perturbation series for radiative effects,” J. Quant. Spectrosc. Radiat. Transfer 39, 219–223 (1988).
[CrossRef]

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

Bruscaglioni, P.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns—procedure and results,” Appl. Phys. Lett. 60, 325–329 (1995).

P. Bruscaglioni, School of Physics, University of Florence, via S. Marta, 3, 50139, Firenze, Italy (personal communication, 1994).

Burrows, J. P.

V. V. Rozanov, T. Kurosu, J. P. Burrows, “Retrieval of atmospheric constituents in the uv-visible: a new quasi-analytical approach for the calculation of weighting functions,” J. Quant. Spectrosc. Radiat. Transfer 60, 277–299 (1998).
[CrossRef]

Chaikovskaya, L. I.

E. P. Zege, L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Amer. A 16, 1430–1438 (1999).
[CrossRef]

Chance, K. V.

R. J. D. Spurr, T. P. Kurosu, K. V. Chance, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
[CrossRef]

Chandrasekhar, S.

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

Chylek, P.

J. Li, J. W. Geldart, P. Chylek, “Second order perturbation solution for radiative transfer in clouds with a horizontally arbitrary periodic inhomogeneity,” J. Quant. Spectrosc. Radiat. Transfer 53, 445–456 (1995).
[CrossRef]

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

Croke, B.

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

Darbinjan, R.

G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Heidelberg, 1980).

Dave, J. V.

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

Deirmendjian, D.

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

Elepov, B.

G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Heidelberg, 1980).

Evans, K. F.

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

Feshbach, H.

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

Galinsky, V. L.

V. L. Galinsky, V. Ramanathan, “3D radiative transfer in weakly inhomogeneous medium. Part I: Diffusive approximation,” J. Atmos. Sci. 55, 2946–2959 (1998).
[CrossRef]

Geldart, J. W.

J. Li, J. W. Geldart, P. Chylek, “Second order perturbation solution for radiative transfer in clouds with a horizontally arbitrary periodic inhomogeneity,” J. Quant. Spectrosc. Radiat. Transfer 53, 445–456 (1995).
[CrossRef]

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

Gerstl, S. A. W.

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

S. A. W. Gerstl, “Application of modern neutron transport methods to atmospheric radiative transfer,” in Volume of Extended Abstracts, International Radiation Symposium, Fort Collins, Colorado, August 11–16, 1980, pp. 500–502.

Glasstone, S.

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinholt, New York, 1970).

Greenstein, J. L.

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

Hasekamp, O.

J. Landgraf, O. Hasekamp, T. Trautmann, “Linearization of radiative transfer with respect to surface properties,” J. Quant. Spectrosc. Radiat. Transfer 72, 327–339 (2001).
[CrossRef]

J. Landgraf, O. Hasekamp, M. A. Box, T. Trautmann, “A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach,” J. Geophys. Res. 106, 27291–27306 (2001).
[CrossRef]

Henyey, L. G.

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

Hovenier, J. W.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

M. I. Mischenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).

Ishimaru, A.

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

Ismaelli, A.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns—procedure and results,” Appl. Phys. Lett. 60, 325–329 (1995).

Ivanov, A. P.

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

Jayaweera, K.

Jupp, D. L. B.

Y. Qin, D. L. B. Jupp, M. A. Box, “Extension of the discrete-ordinate algorithm and efficient radiative transfer calculation,” J. Quant. Spectrosc. Radiat. Transfer 74, 767–781 (2002).
[CrossRef]

Kargin, B.

G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Heidelberg, 1980).

Katsev, I. L.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Effects of multiple scattering in laser sounding of a stratified scattering medium. 2. Peculiarities of sounding of the atmosphere from space,” Izv. Atmos. Ocean. Phys. 34, 227–234 (1998).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

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

Keevers, M.

M. A. Box, M. Keevers, B. H. J. McKellar, “On the perturbation series for radiative effects,” J. Quant. Spectrosc. Radiat. Transfer 39, 219–223 (1988).
[CrossRef]

Kurosu, T.

V. V. Rozanov, T. Kurosu, J. P. Burrows, “Retrieval of atmospheric constituents in the uv-visible: a new quasi-analytical approach for the calculation of weighting functions,” J. Quant. Spectrosc. Radiat. Transfer 60, 277–299 (1998).
[CrossRef]

Kurosu, T. P.

R. J. D. Spurr, T. P. Kurosu, K. V. Chance, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
[CrossRef]

Kus?c?er, I.

I. Kus̆c̆er, M. Ribaric̆, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Landgraf, J.

J. Landgraf, O. Hasekamp, M. A. Box, T. Trautmann, “A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach,” J. Geophys. Res. 106, 27291–27306 (2001).
[CrossRef]

J. Landgraf, O. Hasekamp, T. Trautmann, “Linearization of radiative transfer with respect to surface properties,” J. Quant. Spectrosc. Radiat. Transfer 72, 327–339 (2001).
[CrossRef]

Li, J.

J. Li, J. W. Geldart, P. Chylek, “Second order perturbation solution for radiative transfer in clouds with a horizontally arbitrary periodic inhomogeneity,” J. Quant. Spectrosc. Radiat. Transfer 53, 445–456 (1995).
[CrossRef]

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

Marchuk, G.

G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Heidelberg, 1980).

Marchuk, G. I.

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

G. I. Marchuk, Adjoint Equations and Analysis of Complex System (Kluwer Academic, Dordrecht, The Netherlands, 1994).

McKellar, B. H. J.

M. A. Box, M. Keevers, B. H. J. McKellar, “On the perturbation series for radiative effects,” J. Quant. Spectrosc. Radiat. Transfer 39, 219–223 (1988).
[CrossRef]

Mikhailov, G.

G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Heidelberg, 1980).

Mischenko, M. I.

M. I. Mischenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

Morse, P. M.

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

Nazarliev, M.

G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Heidelberg, 1980).

Polonsky, I. N.

I. N. Polonsky, M. A. Box, “Perturbation technique to retrieve scattering medium stratification,” J. Atmos. Sci. 59, 758–768 (2002).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Effects of multiple scattering in laser sounding of a stratified scattering medium. 2. Peculiarities of sounding of the atmosphere from space,” Izv. Atmos. Ocean. Phys. 34, 227–234 (1998).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

Qin, Y.

Y. Qin, D. L. B. Jupp, M. A. Box, “Extension of the discrete-ordinate algorithm and efficient radiative transfer calculation,” J. Quant. Spectrosc. Radiat. Transfer 74, 767–781 (2002).
[CrossRef]

Ramanathan, V.

V. L. Galinsky, V. Ramanathan, “3D radiative transfer in weakly inhomogeneous medium. Part I: Diffusive approximation,” J. Atmos. Sci. 55, 2946–2959 (1998).
[CrossRef]

Ribaric?, M.

I. Kus̆c̆er, M. Ribaric̆, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Romanova, L. M.

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

Rozanov, V. V.

V. V. Rozanov, T. Kurosu, J. P. Burrows, “Retrieval of atmospheric constituents in the uv-visible: a new quasi-analytical approach for the calculation of weighting functions,” J. Quant. Spectrosc. Radiat. Transfer 60, 277–299 (1998).
[CrossRef]

Sendra, C.

C. Sendra, M. A. Box, “Retrieval of the phase function and scattering optical thickness of aerosols: a radiative perturbation theory application,” J. Quant. Spectrosc. Radiat. Transfer 64, 499–515 (2000).
[CrossRef]

Simmer, C.

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

Spurr, R. J. D.

R. J. D. Spurr, T. P. Kurosu, K. V. Chance, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
[CrossRef]

Stamnes, K.

Tian, Y.

Y. Tian, M. A. Box, “Radiative perturbation theory for polarized radiance,” J. Quant. Spectr. Radiat. Transfer 72, 789–8002 (2002).
[CrossRef]

Y. Tian, “Perturbation theory for polarized radiative transfer computation,” Ph.D. thesis (University of New South Wales, Sydney, Australia, 2000).

Trautmann, T.

J. Landgraf, O. Hasekamp, M. A. Box, T. Trautmann, “A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach,” J. Geophys. Res. 106, 27291–27306 (2001).
[CrossRef]

J. Landgraf, O. Hasekamp, T. Trautmann, “Linearization of radiative transfer with respect to surface properties,” J. Quant. Spectrosc. Radiat. Transfer 72, 327–339 (2001).
[CrossRef]

Travis, L. D.

M. I. Mischenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).

Tsay, S.-C.

Ustinov, E. A.

E. A. Ustinov, “Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR,” J. Quant. Spectrosc. Radiat. Transfer 68, 195–211 (2001).
[CrossRef]

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. 3. Remote sensing of minor gaseous constituents and an atmospheric aerosol,” Cosmic Res. 30, 170–181 (1992).

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. Mathematical methods and weighting functions of linearized inverse problem,” Cosmic Res. 29, 519–532 (1991).

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. 2. Numerical aspects and requirements on the observation geometry,” Cosmic Res. 29, 785–800 (1991).

E. A. Ustinov, “Inverse problem of thermal sounding: to recover the vertical profile of the absorption coefficient of an optically active component of a planetary atmosphere from observing emitted radiation,” Cosmic Res. 28, 347–355 (1990).

E. A. Ustinov, Earth and Space Sciences Division, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, Calif. 91109-8099 (personal communication, 2001).

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Zaccanti, G.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns—procedure and results,” Appl. Phys. Lett. 60, 325–329 (1995).

Zege, E. P.

E. P. Zege, L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Amer. A 16, 1430–1438 (1999).
[CrossRef]

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Effects of multiple scattering in laser sounding of a stratified scattering medium. 2. Peculiarities of sounding of the atmosphere from space,” Izv. Atmos. Ocean. Phys. 34, 227–234 (1998).

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

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

Appl. Opt.

Appl. Phys. B

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Analytical solution to lidar return signals from clouds with regard to multiple scattering,” Appl. Phys. B 60, 345–353 (1995).
[CrossRef]

Appl. Phys. Lett.

P. Bruscaglioni, A. Ismaelli, G. Zaccanti, “Monte Carlo calculations of lidar returns—procedure and results,” Appl. Phys. Lett. 60, 325–329 (1995).

Astron. Astrophys.

J. W. Hovenier, C. V. M. van der Mee, “Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere,” Astron. Astrophys. 128, 1–16 (1983).

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[CrossRef]

Beitr. Phys. Atmos.

M. A. Box, S. A. W. Gerstl, C. Simmer, “Application of the adjoint formulation to the calculation of atmospheric radiative effects,” Beitr. Phys. Atmos. 61, 303–311 (1988).

M. A. Box, S. A. W. Gerstl, C. Simmer, “Computation of atmospheric radiative effects via perturbation theory,” Beitr. Phys. Atmos. 62, 193–199 (1989).

M. A. Box, B. Croke, S. A. W. Gerstl, C. Simmer, “Application of the perturbation theory for atmospheric radiative effects: aerosol scattering atmospheres,” Beitr. Phys. Atmos. 62, 200–211 (1989).

Cosmic Res.

E. A. Ustinov, “Inverse problem of thermal sounding: to recover the vertical profile of the absorption coefficient of an optically active component of a planetary atmosphere from observing emitted radiation,” Cosmic Res. 28, 347–355 (1990).

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. Mathematical methods and weighting functions of linearized inverse problem,” Cosmic Res. 29, 519–532 (1991).

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. 2. Numerical aspects and requirements on the observation geometry,” Cosmic Res. 29, 785–800 (1991).

E. A. Ustinov, “The inverse problem of the photometry of solar radiation reflected by an optically thick planetary atmosphere. 3. Remote sensing of minor gaseous constituents and an atmospheric aerosol,” Cosmic Res. 30, 170–181 (1992).

Izv. Atmos. Ocean. Phys.

E. P. Zege, I. L. Katsev, I. N. Polonsky, “Effects of multiple scattering in laser sounding of a stratified scattering medium. 2. Peculiarities of sounding of the atmosphere from space,” Izv. Atmos. Ocean. Phys. 34, 227–234 (1998).

Izv., Acad. Sci. USSR Atmos. Oceanic Phys.

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

J. Atmos. Sci.

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

I. N. Polonsky, M. A. Box, “Perturbation technique to retrieve scattering medium stratification,” J. Atmos. Sci. 59, 758–768 (2002).
[CrossRef]

J. V. Dave, “A direct solution of the spherical harmonics approximation to the radiative transfer equation for an arbitrary solar elevation. Part I: theory,” J. Atmos. Sci. 32, 790–798 (1975).
[CrossRef]

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[CrossRef]

V. L. Galinsky, V. Ramanathan, “3D radiative transfer in weakly inhomogeneous medium. Part I: Diffusive approximation,” J. Atmos. Sci. 55, 2946–2959 (1998).
[CrossRef]

J. Geophys. Res.

J. Landgraf, O. Hasekamp, M. A. Box, T. Trautmann, “A linearized radiative transfer model for ozone profile retrieval using the analytical forward-adjoint perturbation theory approach,” J. Geophys. Res. 106, 27291–27306 (2001).
[CrossRef]

J. Opt. Soc. Amer. A

E. P. Zege, L. I. Chaikovskaya, “Polarization of multiply scattered lidar return from clouds and ocean water,” J. Opt. Soc. Amer. A 16, 1430–1438 (1999).
[CrossRef]

J. Quant. Spectr. Radiat. Transfer

Y. Tian, M. A. Box, “Radiative perturbation theory for polarized radiance,” J. Quant. Spectr. Radiat. Transfer 72, 789–8002 (2002).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

V. V. Rozanov, T. Kurosu, J. P. Burrows, “Retrieval of atmospheric constituents in the uv-visible: a new quasi-analytical approach for the calculation of weighting functions,” J. Quant. Spectrosc. Radiat. Transfer 60, 277–299 (1998).
[CrossRef]

M. A. Box, M. Keevers, B. H. J. McKellar, “On the perturbation series for radiative effects,” J. Quant. Spectrosc. Radiat. Transfer 39, 219–223 (1988).
[CrossRef]

Y. Qin, D. L. B. Jupp, M. A. Box, “Extension of the discrete-ordinate algorithm and efficient radiative transfer calculation,” J. Quant. Spectrosc. Radiat. Transfer 74, 767–781 (2002).
[CrossRef]

J. Landgraf, O. Hasekamp, T. Trautmann, “Linearization of radiative transfer with respect to surface properties,” J. Quant. Spectrosc. Radiat. Transfer 72, 327–339 (2001).
[CrossRef]

C. Sendra, M. A. Box, “Retrieval of the phase function and scattering optical thickness of aerosols: a radiative perturbation theory application,” J. Quant. Spectrosc. Radiat. Transfer 64, 499–515 (2000).
[CrossRef]

E. A. Ustinov, “Adjoint sensitivity analysis of radiative transfer equation: temperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR,” J. Quant. Spectrosc. Radiat. Transfer 68, 195–211 (2001).
[CrossRef]

J. Li, J. W. Geldart, P. Chylek, “Second order perturbation solution for radiative transfer in clouds with a horizontally arbitrary periodic inhomogeneity,” J. Quant. Spectrosc. Radiat. Transfer 53, 445–456 (1995).
[CrossRef]

R. J. D. Spurr, T. P. Kurosu, K. V. Chance, “A linearized discrete ordinate radiative transfer model for atmospheric remote-sensing retrieval,” J. Quant. Spectrosc. Radiat. Transfer 68, 689–735 (2001).
[CrossRef]

Kosm. Issled.

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

Opt. Acta

I. Kus̆c̆er, M. Ribaric̆, “Matrix formalism in the theory of diffusion of light,” Opt. Acta 6, 42–51 (1959).
[CrossRef]

Other

M. I. Mischenko, J. W. Hovenier, L. D. Travis, Light Scattering by Nonspherical Particles: Theory, Measurements and Applications (Academic, San Diego, Calif., 2000).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1981).

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

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, San Diego, Calif., 1994).

G. Marchuk, G. Mikhailov, M. Nazarliev, R. Darbinjan, B. Kargin, B. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Heidelberg, 1980).

Y. Tian, “Perturbation theory for polarized radiative transfer computation,” Ph.D. thesis (University of New South Wales, Sydney, Australia, 2000).

E. A. Ustinov, Earth and Space Sciences Division, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, Calif. 91109-8099 (personal communication, 2001).

S. A. W. Gerstl, “Application of modern neutron transport methods to atmospheric radiative transfer,” in Volume of Extended Abstracts, International Radiation Symposium, Fort Collins, Colorado, August 11–16, 1980, pp. 500–502.

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

J. Lenoble, ed., Radiative Transfer in Scattering and Absorbing Atmospheres: Standard Computational Procedures (A. Deepak, Hampton, Va., 1985).

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

G. I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinholt, New York, 1970).

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

G. I. Marchuk, Adjoint Equations and Analysis of Complex System (Kluwer Academic, Dordrecht, The Netherlands, 1994).

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

P. Bruscaglioni, School of Physics, University of Florence, via S. Marta, 3, 50139, Firenze, Italy (personal communication, 1994).

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Figures (2)

Fig. 1
Fig. 1

Depolarization degree δd of the lidar signal reflected from a homogeneous cloud, as a function of the sounding depth z, computed by use of the perturbation approach (solid curve) and results of the Monte Carlo simulation (crosses and circles). The receiver field of view is (1) 0.005 and (2) 0.0005.

Fig. 2
Fig. 2

Variation δI(x) of the upwelling radiation (arbitrary units) at the top of the cloud as a function of σebx calculated by use of the perturbation approach (solid curve) and the SHDOM (crosses). The cosine μ0 of the solar zenith angle is (1) 1.0 and (2) 0.3.    

Equations (87)

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

1ct+n+σe(r)I(t, r, n)
=σs(r)4π4πZ(r, n, n)I(t, r, n)dn+Q(t, r, n).
Z(r, n, n)=L(π-χ1)P(r, cos β)L(-χ2).
Z(r, -n, -n)=TZ+(r, n, n)T.
T=diag[1, 1, -1, 1].
P(r, cos θ)=a1b100b1a20000a3b200-b2a4,
120πa1(cos β)sin βdβ=1.
I(t, r, n)=0fortt1.
I(t, r, n)=0for rS(r),nn>0,
I(t, r, n)=1πμ<0M(r, n, n)I(t, r, n)μdn
forrS(r),nn>0,
M(r, n, n)=A(r)000000000000000.
LˆI(t, r, n)=Q(t, r, n),
Lˆ=1ct+n+σe(r)-σs(r)4π4πdnZ(r, n, n) .
G(t, r, n)=GI(t,r,n)GQ(t,r,n)GU(t,r,n)GV(t,r,n),
H(t, r, n)=HI(t,r,n)HQ(t,r,n)HU(t,r,n)HV(t,r,n),
G+, H=ΞG+HdΞ=ΞjG¯j(t, r, n)×Hj(t, r, n)dtdrdn,j=I, Q, U, V.
G+, LˆH=(L˜ˆG)+, H.
L˜ˆI˜(t, r, n)=Q˜(t, r, n),
L˜ˆ=-1ct-n+σe(r)-σs(r)4π4πdnTZ(r, -n, -n)T
1c[I˜+(t2, r, n)I(t2, r, n)
-I˜+(t1, r, n)I(t1, r, n)]drdn+SI˜+(t, r, n)I(t, r, n)(nn)dSdtdn
=0.
I˜(t, r, n)=0fortt2,
I˜(t, r, n)=-1πμ>0TM(r, -n, -n)TI˜(t, r, n)μdn
forrS(r),nn<0.
L˜ˆ(t, r, n, n)=TLˆ(-t, r, -n, -n)T,
I˜(t, r, n)=TI(-t, r, -n)
Q˜(t, r, n)=TQ(-t, r, -n).
E=R+, I.
R(t, r, n)=1000δ(z-z0)cos θ,
R(t, r, n)=0.5000.5δ(r-r0)δ(n-n0)δ(t-t0).
E=R+, I=(L˜ˆI˜)+, I=I˜, LˆI=I˜+, Q.
L^p=L^b+δLˆ,L˜^p=L˜^b+δL˜ˆ,
δE=R+, Ip-R+, Ib.
δE=-I˜b+, δL˜Ib+δI˜+, L^pδI.
δLˆ=supF0δLˆFF 1,
I˜b+, δL˜IbI˜b+ δLˆIbI˜b+ Ib,
δI˜+, L^pδIδI˜+ L^pδI=δI˜+ δLˆIb2I˜b+ IbL^p.
δE-I˜b+, δLˆIb.
δLˆ=dLˆdp δp,
dEdp=-I˜b+, dLˆdpIb.
P(z, cos β)=Pb(z, cos β)+δP(z, cos β).
F(t)=Fb(t)+I˜b+(t, r, n), σs(z)4π4πδZ(z, n, n)×Ib(t, r, n)dn.
1ct+n+σe(r)Ib(t, r, n)
=σsb(r)4π4πZb(r, n, n)Ib(t, r, n)dn+Q0δ(t-t0)δ(r-r0)δ(n-ez),
ω0b(z)=α σs(z)σe(z)α,
Ib(t, r, n)=Ib(r, n)δt-t0-tc-zc,
ns+σe(r)Ib(r, n)
=σs(r)4π4πZbs(r, n, n)Ib(r, n)dn+Q0δ(r-r0)δ(n-ez).
Zbs(z, n, n)=a10000a2+a320000a2+a320000a4,
δd=I-QI+Q
F(t)=(1-α)W0Σrecσsb(z)exp[-2τ(z)]π(H+z)2×0b(k)exp[2ω0τ(z)p(k)]kdk,
σeb(z)=σe(r)dxdydxdy,δσe(r)=σe(r)-σeb(z),
σsb(z)=σs(r)dxdydxdy,δσs(r)=σs(r)-σsb(z),
Zb(z, n, n)=σs(r)Z(r, n, n)dxdyσs(r)dxdy,
δZ(r, n, n)=σs(r)σsb(z) [Z(r, n, n)-Zb(z, n, n)],
Q(t, r, n)=Q0δ(z-zt)δ(n-n0),
R(t, r, n)=R0δ(r-rr)δ(n-nr).
δLˆ=δσe(r)-14π4πdn[δσs(r)Zb(z, n, n)+σsb(z)δZ(r, n, n)].
E=Eb-I˜b+(r, n),δLˆIb(z, n).
δI(rr)=-Ib+(z, r-rr, n)δLˆ(z, r, n)dr×Ib(z, n)dzdn,
P(cos β)=1-g2[1+g2-2g cos(β)]3/2,
δσe(r)=σebsin(k0σebx),
I˜+, LˆI-(L˜ˆI˜+)+,I=ΞI˜+(t, r, n)1ctI(t, r, n)+1ctI˜+(t, r, n)I(t, r, n)dΞ+Ξ {I˜+(t, r, n)nI(t, r, n)+[nI˜+(t, r, n)]I(t, r, n)}dΞ+Ξσs(r)4π4π [TZ(r, -n, -n)×TI(t, r, n)]+I(t, r, n)dndΞ-ΞI˜+(t, r, n) σs(r)4π×4πZ(r, n, n)I(t, r, n)dndΞ.
ΞI˜+(t, r, n)σs4πZ(r, n, n)I(t, r, n)dndΞ
=Ξ4πσs[Z+(r, n, n)I(t, r, n)]+I(t, r, n)dndΞ
=Ξ4πσs[TZ(r, -n, -n)TI(t, r, n)]+×I(t, r, n)dndΞ.
ΞI˜+(t, r, n)1ctI(t, r, n)
+1ctI˜+(t, r, n)I(t, r, n)dΞ=Ξ1ct [I˜+(t, r, n)I(t, r, n)]dΞ=1c[I˜+(t2, r, n)I(t2, r, n)-I˜+(t1, r, n)I(t1, r, n)]drdn,
Wn[I˜+(t, r, n)I(t, r, n)]dr
=SI˜+(t, r, n)I(t, r, n)(nn)dS,
Ξ {I˜+(t, r, n)nI(t, r, n)
+[nI˜+(t, r, n)]I(t, r, n)}dΞ
=Ξn[I˜+(t, r, n)I(t, r, n)]dΞ
=SI˜+(t, r, n)I(t, r, n)(nn)dSdtdn.
 
1c[I˜+(t2, r, n)I(t2, r, n)
-I˜+(t1, r, n)I(t1, r, n)]drdn+SI˜+(t, r, n)I(t, r, n)(nn)dSdtdn
=0,
L^pIp=L^bIb=Q,
L˜^pI˜p=L˜^bI˜b=R.
δE=R+, Ip-R+, Ib=R+, δI=(L˜^bI˜b)+, δI=I˜b+, L^bδI,
L^pIp-L^bIp=L^bIb-L^bIp,
δLˆIp=-L^bδI.
δE=-I˜b+, δL˜Ip=-I˜b+, δL˜Ib-I˜b+, δLˆδI=-I˜b+, δLˆIb-(δL˜ˆIb)+, δI=-I˜b+, δLˆIb-(L˜^pIb)+, δI+(L˜^bI˜b)+, δI=-I˜b+, δLˆIb+(L˜^pδI˜)+, δI-(L˜^pI˜p)+, δI+(L˜^bI˜b)+, δI.
δE=-I˜b+, LˆIb+δI˜+, L^pδI.

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