Abstract

Lidar equations for a system with multiple-scattering beam spreading and pulse stretching are developed from an analytical model for the beam spread function. The resulting lidar equations are transparent to the physics and with some simple approximations for system transfer functions become mathematically simple engineering models for system studies. Application to and comparison with a variety of lidar applications in ocean environments (turbidity and bathymetry) and clouds (aerosol scattering) are presented. These examples provide model validation. The lidar model developed represents a significant extension beyond historical lidar models that exclude pulse stretching. Their mathematical simplicity should foster use in a broader class of problems involving light propagation in turbid media.

© 1999 Optical Society of America

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References

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  1. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992).
  2. V. I. Feigels, Y. I. Kopilevich, eds., Laser Remote Sensing in Natural Water: From Theory to Practice, Proc. SPIE2964 (1996).
  3. L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
    [CrossRef]
  4. D. Arnush, “Underwater light-beam propagation in the small-angle approximation,” J. Opt. Soc. Am. 62, 1109–1111 (1972).
    [CrossRef]
  5. I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and ocean sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
    [CrossRef]
  6. Y. I. Kopilevich, V. I. Feigels, “Theoretical model for backscattered pulse kinetics and interpretation of some anomalies in lidar remote sensing data,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 458–471 (1994).
    [CrossRef]
  7. A. Ishimaru, Wave Propagation and Scattering in a Random Media (Academic, N.Y., 1978).
  8. R. E. Walker, Marine Light Field Statistics (Wiley, N.Y., 1994).
  9. A. G. Luchinin, “Some properties of the backscattered signal in laser sounding of the upper ocean through a wavy surface,” Izv. Acad. Sci. Atmos. Oceanic Phys. 23, 725–729 (1987).
  10. T. V. Kondranin, D. V. Yurin, “Effect of waves and the aperture characteristics of a lidar on the statistical properties of the backscatter signal in laser sounding of the upper sea layer,” Izv. Acad. Sci. Atmos. Oceanic Phys. 27, 453–461 (1973).
  11. G. W. Kattawar, G. N. Plass, “Time of flight lidar measurements as an ocean probe,” Appl. Opt. 11, 662–666 (1972).
    [CrossRef] [PubMed]
  12. H. R. Gordon, “Interpretation of airborne oceanic lidar: effects of multiple scattering,” Appl. Opt. 21, 2996–3001 (1982).
    [CrossRef] [PubMed]
  13. J. W. McLean, J. D. Freeman, R. E. Walker, “Beam spread function with time dispersion,” Appl. Opt. 37, 4701–4711 (1998).
    [CrossRef]
  14. D. B. Rogozkin, “Propagation of light pulse in a medium with strongly anisotropic scattering,” Izv. Acad. Sci. Atmos. Oceanic Phys. 23, 275–281 (1987).
  15. J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
    [CrossRef]
  16. H. C. van de Hulst, G. Kattawar, “Exact spread function for pulsed collimated beam in a medium with small-angle scattering,” Appl. Opt. 33, 5820–5829 (1994).
    [CrossRef] [PubMed]
  17. B. D. Joelson, G. W. Kattawar, H. C. van de Hulst, “Multiple scattering metamorphosis of a non-Gaussian single-scatter phase function,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 234–246 (1994).
    [CrossRef]
  18. 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]
  19. Y. I. Kopilevich, V. I. Feigels, “Characteristics of light backscattered by sea water and lidar sounding of water column,” in Underwater Light Measurements, H. C. Eilertsen, ed., Proc. SPIE2048, 85–94 (1994).
    [CrossRef]
  20. R. A. Maffione, R. C. Honey, “Instrument for measuring the volume scattering function in the backward direction,” in Ocean Optics XI, G. Gilbert, ed., Proc. SPIE1750, 15–26 (1992).
    [CrossRef]
  21. V. I. Feigels, Yu. I. Kopilevich, “Applicability of lidar remote sensing methods for vertical structure investigation of ocean optic properties distribution,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 449–457 (1994).
    [CrossRef]
  22. R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
    [CrossRef] [PubMed]
  23. N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).
  24. B. Billard, “Remote sensing of scattering coefficient for airborne laser hydrography,” Appl. Opt. 25, 2099–2108 (1986).
    [CrossRef] [PubMed]
  25. R. H. Abbot, D. W. Lane, M. J. Sinclair, T. A. Spurling, “Lasers chart the waters of Australia’s Great Barrier Reef,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 72–90 (1996).
  26. O. Steinvall, K. Koppari, U. Karsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
    [CrossRef]
  27. O. Steinvall, K. Koppari, “Depth sounding lidar—an overview of Swedish activities and future prospects,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 2–25 (1996).
  28. J. H. Smart, K. H. Kwon, “Comparison between in situ and remote sensing estimates of diffuse attenuation profiles,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 100–109 (1996).
  29. G. D. Gilbert, M. H. North, “Studies of optical ringing in sea water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 472–479 (1994).
    [CrossRef]
  30. C. D. Mobley, Light and Water (Academic, N.Y., 1994).
  31. L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
    [CrossRef]
  32. G. D. Hickman, J. E. Hogg, “Application of an airborne pulsed laser for near-shore bathymetric measurements,” Remote Sens. Environ. 1, 47–58 (1969).
    [CrossRef]
  33. G. C. Guenther, R. W. L. Thomas, P. E. LaRaocque, “Design considerations for achieving high accuracy with the SHOALS bathymetric lidar system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 54–71 (1996).
  34. W. J. Lillycrop, L. E. Parson, J. L. Irish, “Development and operation of the SHOALS airborne hydrographic survey system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 26–37 (1996).
  35. A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).
  36. H. Krumboltz, “Experimental investigation of system attenuation coefficient for HALS,” (Naval Air Development Center, Warminster, Pa., 1979).

1998 (1)

1997 (1)

1995 (4)

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]

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
[CrossRef] [PubMed]

1994 (1)

1987 (2)

D. B. Rogozkin, “Propagation of light pulse in a medium with strongly anisotropic scattering,” Izv. Acad. Sci. Atmos. Oceanic Phys. 23, 275–281 (1987).

A. G. Luchinin, “Some properties of the backscattered signal in laser sounding of the upper ocean through a wavy surface,” Izv. Acad. Sci. Atmos. Oceanic Phys. 23, 725–729 (1987).

1986 (1)

1982 (1)

1973 (1)

T. V. Kondranin, D. V. Yurin, “Effect of waves and the aperture characteristics of a lidar on the statistical properties of the backscatter signal in laser sounding of the upper sea layer,” Izv. Acad. Sci. Atmos. Oceanic Phys. 27, 453–461 (1973).

1972 (3)

1969 (1)

G. D. Hickman, J. E. Hogg, “Application of an airborne pulsed laser for near-shore bathymetric measurements,” Remote Sens. Environ. 1, 47–58 (1969).
[CrossRef]

Abbot, R. H.

R. H. Abbot, D. W. Lane, M. J. Sinclair, T. A. Spurling, “Lasers chart the waters of Australia’s Great Barrier Reef,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 72–90 (1996).

Arnush, D.

Benayahu, Y.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Billard, B.

Bissonnette, L. R.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
[CrossRef]

Bruscaglioni, P.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Ciervo, A. P.

Cohen, A.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Egert, S.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Feigels, V. I.

Y. I. Kopilevich, V. I. Feigels, “Theoretical model for backscattered pulse kinetics and interpretation of some anomalies in lidar remote sensing data,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 458–471 (1994).
[CrossRef]

Y. I. Kopilevich, V. I. Feigels, “Characteristics of light backscattered by sea water and lidar sounding of water column,” in Underwater Light Measurements, H. C. Eilertsen, ed., Proc. SPIE2048, 85–94 (1994).
[CrossRef]

V. I. Feigels, Yu. I. Kopilevich, “Applicability of lidar remote sensing methods for vertical structure investigation of ocean optic properties distribution,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 449–457 (1994).
[CrossRef]

Freeman, J. D.

Gilbert, G. D.

G. D. Gilbert, M. H. North, “Studies of optical ringing in sea water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 472–479 (1994).
[CrossRef]

Gordon, H. R.

Guenther, G. C.

G. C. Guenther, R. W. L. Thomas, P. E. LaRaocque, “Design considerations for achieving high accuracy with the SHOALS bathymetric lidar system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 54–71 (1996).

Hall, G. J.

Hickman, G. D.

G. D. Hickman, J. E. Hogg, “Application of an airborne pulsed laser for near-shore bathymetric measurements,” Remote Sens. Environ. 1, 47–58 (1969).
[CrossRef]

Hogg, J. E.

G. D. Hickman, J. E. Hogg, “Application of an airborne pulsed laser for near-shore bathymetric measurements,” Remote Sens. Environ. 1, 47–58 (1969).
[CrossRef]

Honey, R. C.

R. A. Maffione, R. C. Honey, “Instrument for measuring the volume scattering function in the backward direction,” in Ocean Optics XI, G. Gilbert, ed., Proc. SPIE1750, 15–26 (1992).
[CrossRef]

Irish, J. L.

W. J. Lillycrop, L. E. Parson, J. L. Irish, “Development and operation of the SHOALS airborne hydrographic survey system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 26–37 (1996).

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in a Random Media (Academic, N.Y., 1978).

Ismalli, A.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Jerlov, N. G.

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

Joelson, B. D.

B. D. Joelson, G. W. Kattawar, H. C. van de Hulst, “Multiple scattering metamorphosis of a non-Gaussian single-scatter phase function,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 234–246 (1994).
[CrossRef]

Karsson, U.

O. Steinvall, K. Koppari, U. Karsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
[CrossRef]

Katsev, I. L.

I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and ocean sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

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]

Kattawar, G.

Kattawar, G. W.

G. W. Kattawar, G. N. Plass, “Time of flight lidar measurements as an ocean probe,” Appl. Opt. 11, 662–666 (1972).
[CrossRef] [PubMed]

B. D. Joelson, G. W. Kattawar, H. C. van de Hulst, “Multiple scattering metamorphosis of a non-Gaussian single-scatter phase function,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 234–246 (1994).
[CrossRef]

Kleinman, M.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Kondranin, T. V.

T. V. Kondranin, D. V. Yurin, “Effect of waves and the aperture characteristics of a lidar on the statistical properties of the backscatter signal in laser sounding of the upper sea layer,” Izv. Acad. Sci. Atmos. Oceanic Phys. 27, 453–461 (1973).

Kopilevich, Y. I.

Y. I. Kopilevich, V. I. Feigels, “Theoretical model for backscattered pulse kinetics and interpretation of some anomalies in lidar remote sensing data,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 458–471 (1994).
[CrossRef]

Y. I. Kopilevich, V. I. Feigels, “Characteristics of light backscattered by sea water and lidar sounding of water column,” in Underwater Light Measurements, H. C. Eilertsen, ed., Proc. SPIE2048, 85–94 (1994).
[CrossRef]

Kopilevich, Yu. I.

V. I. Feigels, Yu. I. Kopilevich, “Applicability of lidar remote sensing methods for vertical structure investigation of ocean optic properties distribution,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 449–457 (1994).
[CrossRef]

Koppari, K.

O. Steinvall, K. Koppari, “Depth sounding lidar—an overview of Swedish activities and future prospects,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 2–25 (1996).

O. Steinvall, K. Koppari, U. Karsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
[CrossRef]

Krumboltz, H.

H. Krumboltz, “Experimental investigation of system attenuation coefficient for HALS,” (Naval Air Development Center, Warminster, Pa., 1979).

Kwon, K. H.

J. H. Smart, K. H. Kwon, “Comparison between in situ and remote sensing estimates of diffuse attenuation profiles,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 100–109 (1996).

Lane, D. W.

R. H. Abbot, D. W. Lane, M. J. Sinclair, T. A. Spurling, “Lasers chart the waters of Australia’s Great Barrier Reef,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 72–90 (1996).

LaRaocque, P. E.

G. C. Guenther, R. W. L. Thomas, P. E. LaRaocque, “Design considerations for achieving high accuracy with the SHOALS bathymetric lidar system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 54–71 (1996).

Lillycrop, W. J.

W. J. Lillycrop, L. E. Parson, J. L. Irish, “Development and operation of the SHOALS airborne hydrographic survey system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 26–37 (1996).

Luchinin, A. G.

A. G. Luchinin, “Some properties of the backscattered signal in laser sounding of the upper ocean through a wavy surface,” Izv. Acad. Sci. Atmos. Oceanic Phys. 23, 725–729 (1987).

Lutomirski, R. F.

Maffione, R. A.

R. A. Maffione, R. C. Honey, “Instrument for measuring the volume scattering function in the backward direction,” in Ocean Optics XI, G. Gilbert, ed., Proc. SPIE1750, 15–26 (1992).
[CrossRef]

McLean, J. W.

Measures, R. M.

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992).

Mobley, C. D.

C. D. Mobley, Light and Water (Academic, N.Y., 1994).

Noormohammadian, M.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

North, M. H.

G. D. Gilbert, M. H. North, “Studies of optical ringing in sea water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 472–479 (1994).
[CrossRef]

Oppel, U. G.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

Parson, L. E.

W. J. Lillycrop, L. E. Parson, J. L. Irish, “Development and operation of the SHOALS airborne hydrographic survey system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 26–37 (1996).

Plass, G. N.

Polonsky, I. N.

I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and ocean sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

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]

Prikhach, A. S.

Rogozkin, D. B.

D. B. Rogozkin, “Propagation of light pulse in a medium with strongly anisotropic scattering,” Izv. Acad. Sci. Atmos. Oceanic Phys. 23, 275–281 (1987).

Schwendimann, P.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Shipley, S. T.

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
[CrossRef]

Sinclair, M. J.

R. H. Abbot, D. W. Lane, M. J. Sinclair, T. A. Spurling, “Lasers chart the waters of Australia’s Great Barrier Reef,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 72–90 (1996).

Smart, J. H.

J. H. Smart, K. H. Kwon, “Comparison between in situ and remote sensing estimates of diffuse attenuation profiles,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 100–109 (1996).

Spurling, T. A.

R. H. Abbot, D. W. Lane, M. J. Sinclair, T. A. Spurling, “Lasers chart the waters of Australia’s Great Barrier Reef,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 72–90 (1996).

Starkov, A. V.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Steinvall, O.

O. Steinvall, K. Koppari, U. Karsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
[CrossRef]

O. Steinvall, K. Koppari, “Depth sounding lidar—an overview of Swedish activities and future prospects,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 2–25 (1996).

Thomas, R. W. L.

G. C. Guenther, R. W. L. Thomas, P. E. LaRaocque, “Design considerations for achieving high accuracy with the SHOALS bathymetric lidar system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 54–71 (1996).

van de Hulst, H. C.

H. C. van de Hulst, G. Kattawar, “Exact spread function for pulsed collimated beam in a medium with small-angle scattering,” Appl. Opt. 33, 5820–5829 (1994).
[CrossRef] [PubMed]

B. D. Joelson, G. W. Kattawar, H. C. van de Hulst, “Multiple scattering metamorphosis of a non-Gaussian single-scatter phase function,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 234–246 (1994).
[CrossRef]

Walker, R. E.

Weinman, J. A.

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
[CrossRef]

Winker, D. M.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Yurin, D. V.

T. V. Kondranin, D. V. Yurin, “Effect of waves and the aperture characteristics of a lidar on the statistical properties of the backscatter signal in laser sounding of the upper sea layer,” Izv. Acad. Sci. Atmos. Oceanic Phys. 27, 453–461 (1973).

Zaccanti, G.

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

Zege, E. P.

I. L. Katsev, E. P. Zege, A. S. Prikhach, I. N. Polonsky, “Efficient technique to determine backscattered light power for various atmospheric and ocean sounding and imaging systems,” J. Opt. Soc. Am. A 14, 1338–1346 (1997).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

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. Opt. (6)

Appl. Phys. B (3)

L. R. Bissonnette, “Multiple scattering of narrow light beams in aerosols,” Appl. Phys. B 60, 315–323 (1995).
[CrossRef]

L. R. Bissonnette, P. Bruscaglioni, A. Ismalli, G. Zaccanti, A. Cohen, Y. Benayahu, M. Kleinman, S. Egert, P. Schwendimann, A. V. Starkov, M. Noormohammadian, U. G. Oppel, D. M. Winker, E. P. Zege, I. L. Katsev, I. N. Polonsky, “LIDAR multiple scattering from clouds,” Appl. Phys. B 60, 355–362 (1995).
[CrossRef]

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]

Izv. Acad. Sci. Atmos. Oceanic Phys. (3)

A. G. Luchinin, “Some properties of the backscattered signal in laser sounding of the upper ocean through a wavy surface,” Izv. Acad. Sci. Atmos. Oceanic Phys. 23, 725–729 (1987).

T. V. Kondranin, D. V. Yurin, “Effect of waves and the aperture characteristics of a lidar on the statistical properties of the backscatter signal in laser sounding of the upper sea layer,” Izv. Acad. Sci. Atmos. Oceanic Phys. 27, 453–461 (1973).

D. B. Rogozkin, “Propagation of light pulse in a medium with strongly anisotropic scattering,” Izv. Acad. Sci. Atmos. Oceanic Phys. 23, 275–281 (1987).

J. Geophys. Res. (1)

J. A. Weinman, S. T. Shipley, “Effects of multiple scattering on laser pulses transmitted through clouds,” J. Geophys. Res. 26, 7123–7128 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Remote Sens. Environ. (1)

G. D. Hickman, J. E. Hogg, “Application of an airborne pulsed laser for near-shore bathymetric measurements,” Remote Sens. Environ. 1, 47–58 (1969).
[CrossRef]

Other (20)

G. C. Guenther, R. W. L. Thomas, P. E. LaRaocque, “Design considerations for achieving high accuracy with the SHOALS bathymetric lidar system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 54–71 (1996).

W. J. Lillycrop, L. E. Parson, J. L. Irish, “Development and operation of the SHOALS airborne hydrographic survey system,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 26–37 (1996).

A. Papoulis, Signal Analysis (McGraw-Hill, New York, 1977).

H. Krumboltz, “Experimental investigation of system attenuation coefficient for HALS,” (Naval Air Development Center, Warminster, Pa., 1979).

N. G. Jerlov, Marine Optics (Elsevier, Amsterdam, 1976).

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Krieger, Malabar, Fla., 1992).

V. I. Feigels, Y. I. Kopilevich, eds., Laser Remote Sensing in Natural Water: From Theory to Practice, Proc. SPIE2964 (1996).

B. D. Joelson, G. W. Kattawar, H. C. van de Hulst, “Multiple scattering metamorphosis of a non-Gaussian single-scatter phase function,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 234–246 (1994).
[CrossRef]

Y. I. Kopilevich, V. I. Feigels, “Theoretical model for backscattered pulse kinetics and interpretation of some anomalies in lidar remote sensing data,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 458–471 (1994).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in a Random Media (Academic, N.Y., 1978).

R. E. Walker, Marine Light Field Statistics (Wiley, N.Y., 1994).

Y. I. Kopilevich, V. I. Feigels, “Characteristics of light backscattered by sea water and lidar sounding of water column,” in Underwater Light Measurements, H. C. Eilertsen, ed., Proc. SPIE2048, 85–94 (1994).
[CrossRef]

R. A. Maffione, R. C. Honey, “Instrument for measuring the volume scattering function in the backward direction,” in Ocean Optics XI, G. Gilbert, ed., Proc. SPIE1750, 15–26 (1992).
[CrossRef]

V. I. Feigels, Yu. I. Kopilevich, “Applicability of lidar remote sensing methods for vertical structure investigation of ocean optic properties distribution,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 449–457 (1994).
[CrossRef]

R. H. Abbot, D. W. Lane, M. J. Sinclair, T. A. Spurling, “Lasers chart the waters of Australia’s Great Barrier Reef,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 72–90 (1996).

O. Steinvall, K. Koppari, U. Karsson, “Airborne laser depth sounding. System aspects and performance,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 392–412 (1994).
[CrossRef]

O. Steinvall, K. Koppari, “Depth sounding lidar—an overview of Swedish activities and future prospects,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 2–25 (1996).

J. H. Smart, K. H. Kwon, “Comparison between in situ and remote sensing estimates of diffuse attenuation profiles,” in Laser Remote Sensing of Natural Waters: From Theory to Practice, V. I. Feigels, Y. I. Kopilevich, ed., Proc. SPIE2964, 100–109 (1996).

G. D. Gilbert, M. H. North, “Studies of optical ringing in sea water,” in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE2258, 472–479 (1994).
[CrossRef]

C. D. Mobley, Light and Water (Academic, N.Y., 1994).

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Figures (12)

Fig. 1
Fig. 1

Bistatic lidar geometry.

Fig. 2
Fig. 2

Lidar returns for coastal ocean water. Narrow and wide FOV receiver with narrow transmitter beam.

Fig. 3
Fig. 3

Lidar attenuation coefficient as a function of depth and receiver FOV for coastal ocean water. Receiver FOV varied as 2θrcvr = 0.00001 × 10 n rad with n = 0, 1, 2, 3, 4, 5. The transmitter beam is 2θxmtr = 0.00001 rad. Baseline narrow and wide FOV receiver are represented by bold curves. Symbols identify absorption coefficient, beam attenuation coefficient, and downwelling irradiance attenuation coefficient.

Fig. 4
Fig. 4

Lidar attenuation coefficient as a function of depth and receiver FOV for turbid harbor water. Variables are the same as in Fig. 3.

Fig. 5
Fig. 5

Effect of receiver FOV on lidar attenuation coefficient. The measurements are from Steinvall et al.26 (Fig. 23b). Model simulation used their measured values for water optical properties: a = 0.3m-1 and b = 0.7m-1. The platform altitude H = 300 m, and the transmitter beam width is 0.1 mrad (full width). The lidar attenuation coefficient was averaged over depth range z = 3–5 m, consistent with their experimental analysis.

Fig. 6
Fig. 6

Ratio of multiple-scattered to single-scattered lidar returns from a uniform cloud layer for 1-mrad receiver FOV. Symbols are nominal minima and maxima of several numerical experiments assembled by Bissonnette et al.3 (Fig. 2). The solid curve is predictions from our lidar model. See text for cloud and lidar parameters.

Fig. 7
Fig. 7

Same as Fig. 6 except for a 10-mrad receiver FOV. The symbols are nominal minima and maxima for various numerical experiments assembled by Bissonnette et al.3 (Fig. 1). The solid curve is our lidar model predictions.

Fig. 8
Fig. 8

Lidar returns for water (dashed curves) and bottom (solid curves) for coastal ocean water. Symbols identify the unscattered return for a 10-ps pulse width. Baseline lidar system with both narrow and wide FOV receiver are combined with narrow transmitter beam. Bottom reflectance R = 0.1.

Fig. 9
Fig. 9

Same as Fig. 8 except for turbid harbor water.

Fig. 10
Fig. 10

Matched filter signal-to-noise ratio (SNR) for bottom return in turbid harbor water. Baseline lidar system with wide FOV receiver (solid circles) and narrow FOV receiver (open circles).

Fig. 11
Fig. 11

Matched filter signal-to-noise ratio (SNR) for bottom return in coastal ocean water. Baseline lidar system with wide FOV receiver (solid circles) and narrow FOV receiver (open circles).

Fig. 12
Fig. 12

Lidar attenuation coefficient as a function of receiver FOV. The measurements (symbols) are from Krumboltz36 (Table II) and are based on bottom returns with the bottom depth ranging between z = 6.6 to 11 m. Platform altitude H = 166 m and transmitter beam width is 18 mrad (full width). Lidar model calculations (dashed curves) use experimentally measured beam attenuation a + b and diffuse attenuation coefficient K. The scattering coefficient is selected to provide agreement in K for large receiver FOV.

Tables (2)

Tables Icon

Table 1 Baseline Airborne Lidar System

Tables Icon

Table 2 Water Optical Properties

Equations (58)

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Lz, ρ, s, t=Qkz, ρ, s, τ
kz, ρ, s, τ=12π4    dκdqKz, κ, q, τ×exp-iκ·ρ+q·s, Kz, κ, q, τ=   dρdskz, ρ, s, τ×expiκ·ρ+q·s,
L0, ρ, s, ρ, s, t=Qδρ-ρδs-sδt,
Lz, ρ, s, ρ, s, t=Qkz, ρ, s, ρ, s, τQkz, ρ-ρ+zs, s-s, τ+ρ+zs22zc,
tτ+z/c+ρ+zs2/2zcτ+z/c.
Ez, ρ, t=Q  dskz, ρ, s, τ=Q2π2  dκKz, κ, 0, τexp-iκ·ρ.
Lz, ρ, s, tQ  dρπρkz, ρ-ρ, s, τQ2π4    dκdqKz, κ, q, τΠκ×exp-iκ·ρ+q·s.
Lz, ρ, s, t=Q/A  dρkz, ρ-ρ, s, τ=Q/A2π2    dqKz, 0, q, τ×exp-iq·s,
Lz, ρ, s, tQ  dsθskz, ρ-zs, s-s, τQ2π4    dκdqKz, κ, q, τ×Θq+zκ×exp-iκ·ρ+q·s,
Lz, ρ, s, tQ2π4    dκdqKz, κ, q, τ2π2×δq+zκexp-iκ·ρ+q·sQ2π2  dκKz, κ, -zκ, τ×exp-iκ·ρ-zs.
LxmtrH+z, ρ, s, t=QTatmTaw    dκ2π2dq2π2×exp-iκ·ρ+q·sKz, κ, q, τ×Θxmtrq/m+κH+z/m.
LbackH+z, ρ, s, t=Rπ  dsLxmtrH+z, ρ, s, t=Rπ ExmtrH+z, ρ, t,
Lrcvrρ, s, t=QTatm2Taw2m2Rπ  dκ2π2×exp-iκ·ρ-sH+z/mΘxmtrκH+z/m×Kz, κ, 0, τKz, κ, -zκ, τ.
Kz, κ, 0, τKz, κ, -zκ, τ= dτKz, κ, 0, τ×Kz, κ, -zκ, τ-τ.
Pt=ArcvrΩrcvr    dρdsLrcvrρ, s, tπrcvrρθrcvrs=Q ArcvrΩrcvrTatm2Taw2m2Rπ  dκ2π2×ΘrcvrκH+z/m×ΘxmtrκH+z/m×Kz, κ, 0, τKz, κ, -zκ, τ,
Pbottomζ=Q ArcvrΩrcvrTatm2Taw2m2Rπ  dκ2π2×ΘrcvrκH+D/mΘxmtrκH+D/m×KD, κ, 0, τKD, κ, -zκ, τ,
Pbottomt=Q ArcvrΩrcvrTatm2Taw2m2H+z/m2Rπ ×Kz, 0, 0, τKz, 0, 0, τ.
Kz, 0, 0, τ=   dρdskz, ρ, s, τ,
Pwatert=Q ArcvrΩrcvrTatm2Taw2βπm2×0c/2mt-2H/cdz  dκ2π2×ΘrcvrκH+z/mΘxmtrκH+z/m×Kz, κ, 0, τKz, κ, -zκ, τ.
PwaterζQ ArcvrΩrcvrTatm2Taw2βπm2× dκ2π2 ΘrcvrκH+ζ/m×ΘxmtrκH+ζ/mc/2m×02ζm/cdτKζ, κ, 0, τKζ, κ, -ζκ, τ,
PwaterζQ ArcvrΩrcvrTatm2Taw2βπm2× dκ2π2 ΘrcvrκH+ζ/m×ΘxmtrκH+ζ/mc/2m2ζ-Dm/c2ζm/c×dτKζ, κ, 0, τKζ, κ, -ζκ, τ.
kz, ρ, s, τ=δρδsδτexp-a+bz+1-exp-bzexp-az+cτ/mgz, τhz, ρ, s, τ.
gz, τ=μσ2Γμ2/σ2μτσ2μ2/σ2-1 exp-μτσ2,
μz/c/m=1-1-exp-bzvbzv14 bzΘ2-124bz2Θ22+,
σ2z/c/m2=23w2-3wvexp-bzv-1+bzv+2v2exp-bzw-1+bzwb2z2wv2w-v-1-exp-bzvbzv2112 bzΘ4+124bz2Θ22+,
hz, ρ, s, τ=34π2τc/m2×exp-s2-3s·ρ/z+3ρ2/z2τc/m/z
Hz, κ, q, τ=exp-τc/m/zq2+q·κz+13κ2z2.
Kz, κ, q, τ=exp-za+bδτ+1-exp-bzexp-az+τc/mgz, τHz, κ, q, τ.
Kz, κ1, 0, τKz, κ2, -zκ2, τ=×exp-2azexp-2bzδτ+exp-bz×1-exp-bzexp-aτc/mgz, τ×Hz, κ1, 0, τ+Hz, κ2, -zκ2, τ+1-exp-bz2 exp-aτc/m×gz, τHz, κ1, 0, τgz, τ×H(z, κ2, -zκ2,τ.
θks=exp-sx2/θk,x2-sy2/θk,y2/πθk,xθk,y
Θkq=exp-qx2θk,x2/4-qy2θk,y2/4.
Pwaterζ  Q ArcvrΩrcvrTatm2Taw2βππm2H+ζ/m2exp-2aζ×c2m02ζm/cdτAζ, τ,
Pbottomζ= QArcvrΩrcvrTatm2Taw2βππm2H+D/m2Rπ×exp-2aDAD, τ,
Az, τ=A11z, τ+A12z, τ+A21z, τ+A22z, τ,
A11z, τ=exp-2bzδτAz, 0Bz, 01/2,
A12z, τ=A21z, τ=exp-bz1-exp-bz×exp-aτc/mgz, τAz, τBz, τ1/2,
A22z, τ=1-exp-bz2×gz, τgz, τexp-aτc/mAz, τBz, τ}1/2,
Az, τ=43zτc/mH+z/m2+θxmtr,x2+θrcvr,x2
Bz, τ=43zτc/mH+z/m2+θxmtr,y2+θrcvr,y2.
gz, τgz, τ=μσ2Γ2μ2/σ2μτσ22μ2/σ2-1×exp-μτσ2.
PwaterζQ ArcvrTatm2Tawβπc/m2m2H+ζ/m2exp-ζa+b+exp-aζ1-exp-bζ×0dτ exp-aτc/mgζ, τ1+43ζτc/mθrcvr2H+ζ/m22.
PwaterζQ ArcvrTatmTaw2βπc/m2m2H+ζ/m2exp-2ζa+b,
PwaterζQ ArcvrTatm2Taw2βπc/m2m2H+ζ/m2exp-2ζKd,
exp-Kdζexp-ζa+b+exp-aζ×1-exp-bζ0exp-aτc/mgζ, τdτexp-ζa+b+exp-aζ1-exp-bζ1+aσ2c/m/μμ2/σ2
PwaterζQ ArcvrTatm2Taw2βπc/m2m2H+ζ/m2exp-2ζKlidar,
Klidarζ=12ddζlnH+ζ/m2Prcvrζ
PbottomζQ ArcvrTatm2Taw2R/πm2H+D/m2 exp-2aD×δτexp-2bD+2 exp-bD×1-exp-bDexp-aτc/mgD, τ+1-exp-bD2exp-aτc/m×gD, τgD, τ.
Pbottomζ=PbottomζTxmtrtQ ArcvrTatm2Taw2R/πm2H+D/m2exp-2DKdTrcvrτ
SNRmax2=ηhvc2mDPbottomζ2Pbottomζ+Pwater(ζ)dζ,
PbottomζQ ArcvrTatm2Taw2R/πm2H+D/m2exp-2KlidarDτxmtr,
Φz, κ=expiκ·ρxmtr-ρrcvr+sxmtr-srcvrH+z/m
Φxmtrz, κ1=expiκ1·ρxmtr+sxmtrH+z/m,
Φrcvrz, κ2=exp-iκ2·ρrcvr+srcvrH+z/m,
ΔLbackH+z, ρ, s, t=AtrgtRtrgtπ δρ-ρtrgt×ExmtrH+z, ρ, t,
ΔPtargetζ, ρtrgt=Q AtrgtArcvrΩrcvrTatm2Taw2m2Rtrgtπ×dκ12π2dκ22π2exp-iρtrgt·κ1-κ2×Θrcvrκ2H+z/mΘxmtrκ1H+z/m×KD, κ1, 0, τ  K(D, κ2,-Dκ2, τ
ΔPshadowζ, ρtrgt  Q AtrgtArcvrΩrcvrTatm2Taw2βπm2× dκ12π2dκ22π2×exp-iρtrgt·κ1-κ2×Θxmtrκ1H+ζ/m×Θrcvrκ2H+ζ/mc/2m×02ζ-Dm/cdτKζ, κ1, 0, τ Kζ, κ2, -ζκ2, τ.
ΔPtargetζ, ρ= dρtrgtΔPtargetζ, ρtrgtπtrgtρ-ρtrgt.
ΔPζ, ρ=ΔPtargetζ, ρ+ΔPshadowζ, ρ

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