Abstract

In Part I of this paper we calculated depth profiles and polarization characteristics of airborne lidar return signals by the Monte Carlo method. Here we calculate the polarization characteristics of lidar return signals for different types of water. We demonstrate the feasibility of polarization lidar application to the detection of underwater inhomogeneities of different origins. It is shown that simultaneous analysis of depth profiles of the lidar return signal power and signal depolarization ratio substantially increases the information content of airborne lidar sensing of seawater. We compare calculated results with the data of airborne lidar measurements for λ = 0.53 μm.

© 1998 Optical Society of America

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  1. G. M. Krekov, M. M. Krekova, V. S. Shamanaev, “Laser sensing of a subsurface oceanic layer. I. Effect of the atmosphere and wind-driven sea waves,” Appl. Opt. 36, 1589–1595 (1997).
  2. I. E. Penner, I. V. Samokhvalov, V. S. Shamanaev, “Optical sensing of the marine frontal zone,” Atmos. Opt. 1, 60–66 (1988).
  3. E. P. Zege, M. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043–1049 (1985).
  4. A. P. Vasil’kov, T. V. Kondranin, E. V. Myasnikov, “Polarization characteristics of lidar return signals in case of pulsed sensing of the ocean with a narrow light beam,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 873–881 (1988).
  5. G. W. Kattawar, G. N. Plass, “Radiation and polarization of multiple scattered light from haze and clouds,” Appl. Opt. 7, 1519–1527 (1968).
    [CrossRef] [PubMed]
  6. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, Oxford, 1950).
  7. G. I. Marchuk, G. M. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, B. S. Elexov, Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), Chap. 5.
    [CrossRef]
  8. V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Study of the polarization characteristics of backscattered signals in laser sensing of clouds,” in Laser Sensing of the Atmosphere (Nauka, Novosibirsk, Russian Federation, 1976), pp. 29–45.
  9. V. V. Belov, G. M. Krekov, G. A. Titov, “Some methods for increasing the efficiency of numerical calculation in the laser sounding of the atmosphere,” in Aspects of Remote Sensing of the Atmosphere, V. E. Zuev, ed. (Institute of Atmospheric Optics, Siberian Branch of the USSR Academy of Sciences, Tomsk, Russian Federation, 1982), pp. 102–113.
  10. K. J. Voss, E. S. Fry, “Measurement of the Muller matrix for ocean water,” Appl. Opt. 23, 4427–4439 (1984).
    [CrossRef] [PubMed]
  11. E. A. Kadyshevich, “Phase matrices of light scattering by waters of the Baltic Sea,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 108–110 (1977).
  12. R. S. Thompson, J. R. Bottiger, E. S. Fry, “Scattering matrix measurements of oceanic hydrosols,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 43–48 (1978).
    [CrossRef]
  13. Yu. S. Lyubovtseva, G. S. Moiseenko, I. N. Plakhina, “Effect of the scattering particle shape on the phase matrices of scattering by artificial hydrosols,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 1097–2000 (1977).
  14. A. I. Abramochkin, V. V. Zanin, I. E. Penner, A. A. Tikhomirov, V. S. Shamanaev, “Airborne polarization lidars for investigation of the atmosphere and hydrosphere,” Opt. Atmos. 1, 92–96 (1988).
  15. I. E. Penner, V. S. Shamanaev, “Experiments on simultaneous sea sounding using shipboard and airborne lidars,” Atmos. Oceanic Opt. 6, 65–67 (1993).
  16. V. E. Zuev, G. M. Krekov, M. M. Krekova, “Polarization structure of signal backscattered by water droplet and crystal clouds,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 19, 595–602 (1983).
  17. O. V. Kopelevich, “Experimental data on the optical properties of sea water in ocean optics,” in Ocean Optics, A. S. Monin, ed. (Nauka, Moscow, 1983), Vol. 1, pp. 166–201.

1997

G. M. Krekov, M. M. Krekova, V. S. Shamanaev, “Laser sensing of a subsurface oceanic layer. I. Effect of the atmosphere and wind-driven sea waves,” Appl. Opt. 36, 1589–1595 (1997).

1993

I. E. Penner, V. S. Shamanaev, “Experiments on simultaneous sea sounding using shipboard and airborne lidars,” Atmos. Oceanic Opt. 6, 65–67 (1993).

1988

A. I. Abramochkin, V. V. Zanin, I. E. Penner, A. A. Tikhomirov, V. S. Shamanaev, “Airborne polarization lidars for investigation of the atmosphere and hydrosphere,” Opt. Atmos. 1, 92–96 (1988).

I. E. Penner, I. V. Samokhvalov, V. S. Shamanaev, “Optical sensing of the marine frontal zone,” Atmos. Opt. 1, 60–66 (1988).

A. P. Vasil’kov, T. V. Kondranin, E. V. Myasnikov, “Polarization characteristics of lidar return signals in case of pulsed sensing of the ocean with a narrow light beam,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 873–881 (1988).

1985

E. P. Zege, M. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043–1049 (1985).

1984

1983

V. E. Zuev, G. M. Krekov, M. M. Krekova, “Polarization structure of signal backscattered by water droplet and crystal clouds,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 19, 595–602 (1983).

1977

E. A. Kadyshevich, “Phase matrices of light scattering by waters of the Baltic Sea,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 108–110 (1977).

Yu. S. Lyubovtseva, G. S. Moiseenko, I. N. Plakhina, “Effect of the scattering particle shape on the phase matrices of scattering by artificial hydrosols,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 1097–2000 (1977).

1968

Abramochkin, A. I.

A. I. Abramochkin, V. V. Zanin, I. E. Penner, A. A. Tikhomirov, V. S. Shamanaev, “Airborne polarization lidars for investigation of the atmosphere and hydrosphere,” Opt. Atmos. 1, 92–96 (1988).

Belov, V. V.

V. V. Belov, G. M. Krekov, G. A. Titov, “Some methods for increasing the efficiency of numerical calculation in the laser sounding of the atmosphere,” in Aspects of Remote Sensing of the Atmosphere, V. E. Zuev, ed. (Institute of Atmospheric Optics, Siberian Branch of the USSR Academy of Sciences, Tomsk, Russian Federation, 1982), pp. 102–113.

Bottiger, J. R.

R. S. Thompson, J. R. Bottiger, E. S. Fry, “Scattering matrix measurements of oceanic hydrosols,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 43–48 (1978).
[CrossRef]

Chaikovskaya, M. I.

E. P. Zege, M. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043–1049 (1985).

Chandrasekhar, S.

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

Darbinyan, R. A.

G. I. Marchuk, G. M. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, B. S. Elexov, Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Elexov, B. S.

G. I. Marchuk, G. M. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, B. S. Elexov, Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Fry, E. S.

K. J. Voss, E. S. Fry, “Measurement of the Muller matrix for ocean water,” Appl. Opt. 23, 4427–4439 (1984).
[CrossRef] [PubMed]

R. S. Thompson, J. R. Bottiger, E. S. Fry, “Scattering matrix measurements of oceanic hydrosols,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 43–48 (1978).
[CrossRef]

Kadyshevich, E. A.

E. A. Kadyshevich, “Phase matrices of light scattering by waters of the Baltic Sea,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 108–110 (1977).

Kargin, B. A.

G. I. Marchuk, G. M. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, B. S. Elexov, Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Kattawar, G. W.

Kondranin, T. V.

A. P. Vasil’kov, T. V. Kondranin, E. V. Myasnikov, “Polarization characteristics of lidar return signals in case of pulsed sensing of the ocean with a narrow light beam,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 873–881 (1988).

Kopelevich, O. V.

O. V. Kopelevich, “Experimental data on the optical properties of sea water in ocean optics,” in Ocean Optics, A. S. Monin, ed. (Nauka, Moscow, 1983), Vol. 1, pp. 166–201.

Krekov, G. M.

G. M. Krekov, M. M. Krekova, V. S. Shamanaev, “Laser sensing of a subsurface oceanic layer. I. Effect of the atmosphere and wind-driven sea waves,” Appl. Opt. 36, 1589–1595 (1997).

V. E. Zuev, G. M. Krekov, M. M. Krekova, “Polarization structure of signal backscattered by water droplet and crystal clouds,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 19, 595–602 (1983).

V. V. Belov, G. M. Krekov, G. A. Titov, “Some methods for increasing the efficiency of numerical calculation in the laser sounding of the atmosphere,” in Aspects of Remote Sensing of the Atmosphere, V. E. Zuev, ed. (Institute of Atmospheric Optics, Siberian Branch of the USSR Academy of Sciences, Tomsk, Russian Federation, 1982), pp. 102–113.

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Study of the polarization characteristics of backscattered signals in laser sensing of clouds,” in Laser Sensing of the Atmosphere (Nauka, Novosibirsk, Russian Federation, 1976), pp. 29–45.

Krekova, M. M.

G. M. Krekov, M. M. Krekova, V. S. Shamanaev, “Laser sensing of a subsurface oceanic layer. I. Effect of the atmosphere and wind-driven sea waves,” Appl. Opt. 36, 1589–1595 (1997).

V. E. Zuev, G. M. Krekov, M. M. Krekova, “Polarization structure of signal backscattered by water droplet and crystal clouds,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 19, 595–602 (1983).

Lyubovtseva, Yu. S.

Yu. S. Lyubovtseva, G. S. Moiseenko, I. N. Plakhina, “Effect of the scattering particle shape on the phase matrices of scattering by artificial hydrosols,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 1097–2000 (1977).

Marchuk, G. I.

G. I. Marchuk, G. M. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, B. S. Elexov, Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Matvienko, G. G.

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Study of the polarization characteristics of backscattered signals in laser sensing of clouds,” in Laser Sensing of the Atmosphere (Nauka, Novosibirsk, Russian Federation, 1976), pp. 29–45.

Mikhailov, G. M.

G. I. Marchuk, G. M. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, B. S. Elexov, Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Moiseenko, G. S.

Yu. S. Lyubovtseva, G. S. Moiseenko, I. N. Plakhina, “Effect of the scattering particle shape on the phase matrices of scattering by artificial hydrosols,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 1097–2000 (1977).

Myasnikov, E. V.

A. P. Vasil’kov, T. V. Kondranin, E. V. Myasnikov, “Polarization characteristics of lidar return signals in case of pulsed sensing of the ocean with a narrow light beam,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 873–881 (1988).

Nazaraliev, M. A.

G. I. Marchuk, G. M. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, B. S. Elexov, Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

Penner, I. E.

I. E. Penner, V. S. Shamanaev, “Experiments on simultaneous sea sounding using shipboard and airborne lidars,” Atmos. Oceanic Opt. 6, 65–67 (1993).

I. E. Penner, I. V. Samokhvalov, V. S. Shamanaev, “Optical sensing of the marine frontal zone,” Atmos. Opt. 1, 60–66 (1988).

A. I. Abramochkin, V. V. Zanin, I. E. Penner, A. A. Tikhomirov, V. S. Shamanaev, “Airborne polarization lidars for investigation of the atmosphere and hydrosphere,” Opt. Atmos. 1, 92–96 (1988).

Plakhina, I. N.

Yu. S. Lyubovtseva, G. S. Moiseenko, I. N. Plakhina, “Effect of the scattering particle shape on the phase matrices of scattering by artificial hydrosols,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 1097–2000 (1977).

Plass, G. N.

Popkov, A. I.

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Study of the polarization characteristics of backscattered signals in laser sensing of clouds,” in Laser Sensing of the Atmosphere (Nauka, Novosibirsk, Russian Federation, 1976), pp. 29–45.

Samokhvalov, I. V.

I. E. Penner, I. V. Samokhvalov, V. S. Shamanaev, “Optical sensing of the marine frontal zone,” Atmos. Opt. 1, 60–66 (1988).

Shamanaev, V. S.

G. M. Krekov, M. M. Krekova, V. S. Shamanaev, “Laser sensing of a subsurface oceanic layer. I. Effect of the atmosphere and wind-driven sea waves,” Appl. Opt. 36, 1589–1595 (1997).

I. E. Penner, V. S. Shamanaev, “Experiments on simultaneous sea sounding using shipboard and airborne lidars,” Atmos. Oceanic Opt. 6, 65–67 (1993).

I. E. Penner, I. V. Samokhvalov, V. S. Shamanaev, “Optical sensing of the marine frontal zone,” Atmos. Opt. 1, 60–66 (1988).

A. I. Abramochkin, V. V. Zanin, I. E. Penner, A. A. Tikhomirov, V. S. Shamanaev, “Airborne polarization lidars for investigation of the atmosphere and hydrosphere,” Opt. Atmos. 1, 92–96 (1988).

Thompson, R. S.

R. S. Thompson, J. R. Bottiger, E. S. Fry, “Scattering matrix measurements of oceanic hydrosols,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 43–48 (1978).
[CrossRef]

Tikhomirov, A. A.

A. I. Abramochkin, V. V. Zanin, I. E. Penner, A. A. Tikhomirov, V. S. Shamanaev, “Airborne polarization lidars for investigation of the atmosphere and hydrosphere,” Opt. Atmos. 1, 92–96 (1988).

Titov, G. A.

V. V. Belov, G. M. Krekov, G. A. Titov, “Some methods for increasing the efficiency of numerical calculation in the laser sounding of the atmosphere,” in Aspects of Remote Sensing of the Atmosphere, V. E. Zuev, ed. (Institute of Atmospheric Optics, Siberian Branch of the USSR Academy of Sciences, Tomsk, Russian Federation, 1982), pp. 102–113.

Vasil’kov, A. P.

A. P. Vasil’kov, T. V. Kondranin, E. V. Myasnikov, “Polarization characteristics of lidar return signals in case of pulsed sensing of the ocean with a narrow light beam,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 873–881 (1988).

Voss, K. J.

Zanin, V. V.

A. I. Abramochkin, V. V. Zanin, I. E. Penner, A. A. Tikhomirov, V. S. Shamanaev, “Airborne polarization lidars for investigation of the atmosphere and hydrosphere,” Opt. Atmos. 1, 92–96 (1988).

Zege, E. P.

E. P. Zege, M. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043–1049 (1985).

Zuev, V. E.

V. E. Zuev, G. M. Krekov, M. M. Krekova, “Polarization structure of signal backscattered by water droplet and crystal clouds,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 19, 595–602 (1983).

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Study of the polarization characteristics of backscattered signals in laser sensing of clouds,” in Laser Sensing of the Atmosphere (Nauka, Novosibirsk, Russian Federation, 1976), pp. 29–45.

Appl. Opt.

Atmos. Oceanic Opt.

I. E. Penner, V. S. Shamanaev, “Experiments on simultaneous sea sounding using shipboard and airborne lidars,” Atmos. Oceanic Opt. 6, 65–67 (1993).

Atmos. Opt.

I. E. Penner, I. V. Samokhvalov, V. S. Shamanaev, “Optical sensing of the marine frontal zone,” Atmos. Opt. 1, 60–66 (1988).

Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana

E. P. Zege, M. I. Chaikovskaya, “Approximate solutions of the polarized radiation transfer equation in media with highly anisotropic scattering,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 21, 1043–1049 (1985).

V. E. Zuev, G. M. Krekov, M. M. Krekova, “Polarization structure of signal backscattered by water droplet and crystal clouds,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 19, 595–602 (1983).

E. A. Kadyshevich, “Phase matrices of light scattering by waters of the Baltic Sea,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 108–110 (1977).

Yu. S. Lyubovtseva, G. S. Moiseenko, I. N. Plakhina, “Effect of the scattering particle shape on the phase matrices of scattering by artificial hydrosols,” Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 13, 1097–2000 (1977).

Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana

A. P. Vasil’kov, T. V. Kondranin, E. V. Myasnikov, “Polarization characteristics of lidar return signals in case of pulsed sensing of the ocean with a narrow light beam,” Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana 24, 873–881 (1988).

Opt. Atmos.

A. I. Abramochkin, V. V. Zanin, I. E. Penner, A. A. Tikhomirov, V. S. Shamanaev, “Airborne polarization lidars for investigation of the atmosphere and hydrosphere,” Opt. Atmos. 1, 92–96 (1988).

Other

R. S. Thompson, J. R. Bottiger, E. S. Fry, “Scattering matrix measurements of oceanic hydrosols,” in Ocean Optics V, M. B. White, R. E. Stevenson, eds., Proc. SPIE160, 43–48 (1978).
[CrossRef]

O. V. Kopelevich, “Experimental data on the optical properties of sea water in ocean optics,” in Ocean Optics, A. S. Monin, ed. (Nauka, Moscow, 1983), Vol. 1, pp. 166–201.

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

G. I. Marchuk, G. M. Mikhailov, M. A. Nazaraliev, R. A. Darbinyan, B. A. Kargin, B. S. Elexov, Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, Berlin, 1980), Chap. 5.
[CrossRef]

V. E. Zuev, G. M. Krekov, G. G. Matvienko, A. I. Popkov, “Study of the polarization characteristics of backscattered signals in laser sensing of clouds,” in Laser Sensing of the Atmosphere (Nauka, Novosibirsk, Russian Federation, 1976), pp. 29–45.

V. V. Belov, G. M. Krekov, G. A. Titov, “Some methods for increasing the efficiency of numerical calculation in the laser sounding of the atmosphere,” in Aspects of Remote Sensing of the Atmosphere, V. E. Zuev, ed. (Institute of Atmospheric Optics, Siberian Branch of the USSR Academy of Sciences, Tomsk, Russian Federation, 1982), pp. 102–113.

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Figures (6)

Fig. 1
Fig. 1

Angular dependence of the normalized components of the scattering phase matrices S 1 (solid curves) and S 2 (dotted curves) used in the calculations.

Fig. 2
Fig. 2

Depth profiles of the depolarization ratio. Curves 1–4 were calculated with g 4(θ) and S 1(θ); curves 1′ and 2′, with g 4(θ) and S 2(θ); and curves 3′ and 4′, with g 2(θ) and S 1(θ); ϕ d = 10 (curves 1, 1′, 3, and 3′) and 35 mrad (curves 2, 2′, 4, and 4′); curve 5 shows the experimental data15 at ϕ d = 26 mrad; curve 5′ shows results of calculations at ϕ d = 17 mrad.

Fig. 3
Fig. 3

Depth profiles of the lidar return signal power P(h) (curves 4–6) and depolarization ratio δ(h) (curves 1–3) calculated for the depth profiles of the extinction coefficient σ w (h) shown by curves 1′ and 2′. Curves 3 and 6 show the results of our calculation in the presence of a stratified inhomogeneity.

Fig. 4
Fig. 4

Experimental depth profiles P(h) (solid curves) and δ(h) (dashed curves) measured in different regions of the Barents Sea.

Fig. 5
Fig. 5

Depth profiles P(h) (curves 1–6) and δ(h) (curves 1′–6′) in the presence of inversion layers having different stratifications and optical properties obtained for ϕ d = 10 (curves 1, 3, and 5) and 20 mrad (curves 2, 4, and 6).

Fig. 6
Fig. 6

Sensing of water in the presence of maritime fog. (a) Profiles P calculated by the Monte Carlo method (curves 2 and 3). The abscissa is labeled by altitude z to the left of the origin and depth h to the right, with different scales. The lidar was located at altitude H s = 1000 m above the water surface; the thickness of the fog layer was Δz = 200 m. The calculations were performed for ϕ d = 10 (curve 2) and 35 mrad (curve 3). In the fog layer σ a (z) = 5 km-1, in the underwater inversion layer σ w (h) = 600 km-1, and beyond this layer σ w (h) = 200 km-1. (b) Experimental profile P(t) measured with airborne lidar. Flight altitude, H s = 150 - 180 m.

Equations (3)

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

F ω ,   r = M ω j ,   ω ,   r j F ω j ,   r j ,
M ω j ,   ω ,   r j = π - α 2 S ω j ,   ω ,   r - i 1 .
S = S 11 S 21 0 0   S 12 S 22 0 0   0 0 S 33 S 43   0 0 S 34 S 44 ,

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