Abstract

We have developed a two-dimensional (2D) model for inhomogeneous cirrus clouds in plane-parallel and spherical geometries for the analysis of the transmission and backscattering of high-energy laser beams. The 2D extinction-coefficient and mean effective ice-crystal size fields for cirrus clouds can be determined from a combination of the remote sensing of cirrus clouds by use of the Advanced Very High Resolution Radiometer on board National Oceanic and Atmospheric Administration satellites and the vertical profiling of ice-crystal size distributions available from limited measurements. We demonstrate that satellite remote sensing of the position and the composition of high cirrus can be incorporated directly in the computer model developed for the transmission and backscattering of high-energy laser beams in realistic atmospheres. The results of laser direct transmission, forward scattering, and backscattering are analyzed carefully with respect to aircraft height, cirrus cloud optical depth, and ice-crystal size and orientation. Uncertainty in laser transmission that is due to errors in the retrieved ice-crystal size is negligible. But uncertainty of the order of 2% can be produced if the retrieved optical depth has errors of ±0.05. With both the aircraft and the target near the cloud top, the direct transmission decreases, owing to the propagation of the laser beam through the curved portion of the cloud top. This effect becomes more pronounced as the horizontal distance between the aircraft and the target increases.

© 2002 Optical Society of America

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References

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  1. K. N. Liou, Y. Takano, S. C. Ou, M. W. Johnson, “Laser transmission through thin cirrus clouds,” Appl. Opt. 39, 4886–4894 (2000).
    [CrossRef]
  2. P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite observed visible and infrared radiance. Part I: Parameterization of radiance field,” J. Atmos. Sci. 50, 1279–1304 (1993).
    [CrossRef]
  3. S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
    [CrossRef]
  4. L. M. Miloshevich, A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance, and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
    [CrossRef]
  5. K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1886 (1991).
    [CrossRef]
  6. J. D. Spinhirne, W. D. Hart, “Cirrus structure and radiative parameters from airborne lidar and spectral radiometer observations: the 28 October 1986 FIRE study,” Mon. Weather Rev. 118, 2329–2343 (1990).
    [CrossRef]
  7. G. G. Mace, T. Ackerman, P. Minnis, D. F. Young, “Cirrus layer microphysical properties derived from surface-based millimeter radar and infrared interferometer data,” J. Geophys. Res. 103, 23207–23216 (1998).
    [CrossRef]
  8. K. N. Liou, Y. Takano, S. C. Ou, A. Heymsfield, W. Kreiss, “Infrared transmission through cirrus clouds: a radiative model for target detection,” Appl. Opt. 29, 1886–1896 (1990).
    [CrossRef] [PubMed]
  9. S. C. Ou, K. N. Liou, M. D. King, S. C. Tsay, “Remote sensing of cirrus cloud parameters based on a 0.63–3.7 µm radiance correlation technique applied to AVHRR data,” Geophys. Res. Lett. 26, 2437–2440 (1999).
    [CrossRef]
  10. Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. II. Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci. 46, 20–36 (1989).
    [CrossRef]
  11. N. X. Rao, S. C. Ou, K. N. Liou, “Removal of the solar component in AVHRR 3.7-µm radiances for the retrieval of cirrus cloud parameters,” J. Appl. Meteorol. 34, 481–499 (1995).
    [CrossRef]
  12. K. N. Liou, N. X. Rao, “Radiative transfer in cirrus clouds. IV. Cloud geometry, inhomogeneity, and absorption,” J. Atmos. Sci. 53, 3046–3065 (1996).
    [CrossRef]

2000 (1)

1999 (1)

S. C. Ou, K. N. Liou, M. D. King, S. C. Tsay, “Remote sensing of cirrus cloud parameters based on a 0.63–3.7 µm radiance correlation technique applied to AVHRR data,” Geophys. Res. Lett. 26, 2437–2440 (1999).
[CrossRef]

1998 (1)

G. G. Mace, T. Ackerman, P. Minnis, D. F. Young, “Cirrus layer microphysical properties derived from surface-based millimeter radar and infrared interferometer data,” J. Geophys. Res. 103, 23207–23216 (1998).
[CrossRef]

1997 (1)

L. M. Miloshevich, A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance, and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
[CrossRef]

1996 (1)

K. N. Liou, N. X. Rao, “Radiative transfer in cirrus clouds. IV. Cloud geometry, inhomogeneity, and absorption,” J. Atmos. Sci. 53, 3046–3065 (1996).
[CrossRef]

1995 (2)

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

N. X. Rao, S. C. Ou, K. N. Liou, “Removal of the solar component in AVHRR 3.7-µm radiances for the retrieval of cirrus cloud parameters,” J. Appl. Meteorol. 34, 481–499 (1995).
[CrossRef]

1993 (1)

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite observed visible and infrared radiance. Part I: Parameterization of radiance field,” J. Atmos. Sci. 50, 1279–1304 (1993).
[CrossRef]

1991 (1)

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1886 (1991).
[CrossRef]

1990 (2)

J. D. Spinhirne, W. D. Hart, “Cirrus structure and radiative parameters from airborne lidar and spectral radiometer observations: the 28 October 1986 FIRE study,” Mon. Weather Rev. 118, 2329–2343 (1990).
[CrossRef]

K. N. Liou, Y. Takano, S. C. Ou, A. Heymsfield, W. Kreiss, “Infrared transmission through cirrus clouds: a radiative model for target detection,” Appl. Opt. 29, 1886–1896 (1990).
[CrossRef] [PubMed]

1989 (1)

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. II. Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

Ackerman, T.

G. G. Mace, T. Ackerman, P. Minnis, D. F. Young, “Cirrus layer microphysical properties derived from surface-based millimeter radar and infrared interferometer data,” J. Geophys. Res. 103, 23207–23216 (1998).
[CrossRef]

Baum, B.

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

Fu, Q.

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

Hart, W. D.

J. D. Spinhirne, W. D. Hart, “Cirrus structure and radiative parameters from airborne lidar and spectral radiometer observations: the 28 October 1986 FIRE study,” Mon. Weather Rev. 118, 2329–2343 (1990).
[CrossRef]

Heymsfield, A.

Heymsfield, A. J.

L. M. Miloshevich, A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance, and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
[CrossRef]

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

Johnson, M. W.

King, M. D.

S. C. Ou, K. N. Liou, M. D. King, S. C. Tsay, “Remote sensing of cirrus cloud parameters based on a 0.63–3.7 µm radiance correlation technique applied to AVHRR data,” Geophys. Res. Lett. 26, 2437–2440 (1999).
[CrossRef]

Kinne, S. A.

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

Kreiss, W.

Liou, K. N.

K. N. Liou, Y. Takano, S. C. Ou, M. W. Johnson, “Laser transmission through thin cirrus clouds,” Appl. Opt. 39, 4886–4894 (2000).
[CrossRef]

S. C. Ou, K. N. Liou, M. D. King, S. C. Tsay, “Remote sensing of cirrus cloud parameters based on a 0.63–3.7 µm radiance correlation technique applied to AVHRR data,” Geophys. Res. Lett. 26, 2437–2440 (1999).
[CrossRef]

K. N. Liou, N. X. Rao, “Radiative transfer in cirrus clouds. IV. Cloud geometry, inhomogeneity, and absorption,” J. Atmos. Sci. 53, 3046–3065 (1996).
[CrossRef]

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

N. X. Rao, S. C. Ou, K. N. Liou, “Removal of the solar component in AVHRR 3.7-µm radiances for the retrieval of cirrus cloud parameters,” J. Appl. Meteorol. 34, 481–499 (1995).
[CrossRef]

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite observed visible and infrared radiance. Part I: Parameterization of radiance field,” J. Atmos. Sci. 50, 1279–1304 (1993).
[CrossRef]

K. N. Liou, Y. Takano, S. C. Ou, A. Heymsfield, W. Kreiss, “Infrared transmission through cirrus clouds: a radiative model for target detection,” Appl. Opt. 29, 1886–1896 (1990).
[CrossRef] [PubMed]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. II. Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

Mace, G. G.

G. G. Mace, T. Ackerman, P. Minnis, D. F. Young, “Cirrus layer microphysical properties derived from surface-based millimeter radar and infrared interferometer data,” J. Geophys. Res. 103, 23207–23216 (1998).
[CrossRef]

Miloshevich, L. M.

L. M. Miloshevich, A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance, and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
[CrossRef]

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

Minnis, P.

G. G. Mace, T. Ackerman, P. Minnis, D. F. Young, “Cirrus layer microphysical properties derived from surface-based millimeter radar and infrared interferometer data,” J. Geophys. Res. 103, 23207–23216 (1998).
[CrossRef]

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite observed visible and infrared radiance. Part I: Parameterization of radiance field,” J. Atmos. Sci. 50, 1279–1304 (1993).
[CrossRef]

Ou, S. C.

K. N. Liou, Y. Takano, S. C. Ou, M. W. Johnson, “Laser transmission through thin cirrus clouds,” Appl. Opt. 39, 4886–4894 (2000).
[CrossRef]

S. C. Ou, K. N. Liou, M. D. King, S. C. Tsay, “Remote sensing of cirrus cloud parameters based on a 0.63–3.7 µm radiance correlation technique applied to AVHRR data,” Geophys. Res. Lett. 26, 2437–2440 (1999).
[CrossRef]

N. X. Rao, S. C. Ou, K. N. Liou, “Removal of the solar component in AVHRR 3.7-µm radiances for the retrieval of cirrus cloud parameters,” J. Appl. Meteorol. 34, 481–499 (1995).
[CrossRef]

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

K. N. Liou, Y. Takano, S. C. Ou, A. Heymsfield, W. Kreiss, “Infrared transmission through cirrus clouds: a radiative model for target detection,” Appl. Opt. 29, 1886–1896 (1990).
[CrossRef] [PubMed]

Rao, N. X.

K. N. Liou, N. X. Rao, “Radiative transfer in cirrus clouds. IV. Cloud geometry, inhomogeneity, and absorption,” J. Atmos. Sci. 53, 3046–3065 (1996).
[CrossRef]

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

N. X. Rao, S. C. Ou, K. N. Liou, “Removal of the solar component in AVHRR 3.7-µm radiances for the retrieval of cirrus cloud parameters,” J. Appl. Meteorol. 34, 481–499 (1995).
[CrossRef]

Sassen, K.

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1886 (1991).
[CrossRef]

Spinhirne, J. D.

J. D. Spinhirne, W. D. Hart, “Cirrus structure and radiative parameters from airborne lidar and spectral radiometer observations: the 28 October 1986 FIRE study,” Mon. Weather Rev. 118, 2329–2343 (1990).
[CrossRef]

Takano, Y.

K. N. Liou, Y. Takano, S. C. Ou, M. W. Johnson, “Laser transmission through thin cirrus clouds,” Appl. Opt. 39, 4886–4894 (2000).
[CrossRef]

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite observed visible and infrared radiance. Part I: Parameterization of radiance field,” J. Atmos. Sci. 50, 1279–1304 (1993).
[CrossRef]

K. N. Liou, Y. Takano, S. C. Ou, A. Heymsfield, W. Kreiss, “Infrared transmission through cirrus clouds: a radiative model for target detection,” Appl. Opt. 29, 1886–1896 (1990).
[CrossRef] [PubMed]

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. II. Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

Tsay, S. C.

S. C. Ou, K. N. Liou, M. D. King, S. C. Tsay, “Remote sensing of cirrus cloud parameters based on a 0.63–3.7 µm radiance correlation technique applied to AVHRR data,” Geophys. Res. Lett. 26, 2437–2440 (1999).
[CrossRef]

Young, D. F.

G. G. Mace, T. Ackerman, P. Minnis, D. F. Young, “Cirrus layer microphysical properties derived from surface-based millimeter radar and infrared interferometer data,” J. Geophys. Res. 103, 23207–23216 (1998).
[CrossRef]

Appl. Opt. (2)

Bull. Am. Meteorol. Soc. (1)

K. Sassen, “The polarization lidar technique for cloud research: a review and current assessment,” Bull. Am. Meteorol. Soc. 72, 1848–1886 (1991).
[CrossRef]

Geophys. Res. Lett. (1)

S. C. Ou, K. N. Liou, M. D. King, S. C. Tsay, “Remote sensing of cirrus cloud parameters based on a 0.63–3.7 µm radiance correlation technique applied to AVHRR data,” Geophys. Res. Lett. 26, 2437–2440 (1999).
[CrossRef]

J. Appl. Meteorol. (1)

N. X. Rao, S. C. Ou, K. N. Liou, “Removal of the solar component in AVHRR 3.7-µm radiances for the retrieval of cirrus cloud parameters,” J. Appl. Meteorol. 34, 481–499 (1995).
[CrossRef]

J. Atmos. Ocean. Technol. (1)

L. M. Miloshevich, A. J. Heymsfield, “A balloon-borne continuous cloud particle replicator for measuring vertical profiles of cloud microphysical properties: instrument design, performance, and collection efficiency analysis,” J. Atmos. Ocean. Technol. 14, 753–768 (1997).
[CrossRef]

J. Atmos. Sci. (4)

Y. Takano, K. N. Liou, “Radiative transfer in cirrus clouds. II. Theory and computation of multiple scattering in an anisotropic medium,” J. Atmos. Sci. 46, 20–36 (1989).
[CrossRef]

P. Minnis, K. N. Liou, Y. Takano, “Inference of cirrus cloud properties using satellite observed visible and infrared radiance. Part I: Parameterization of radiance field,” J. Atmos. Sci. 50, 1279–1304 (1993).
[CrossRef]

S. C. Ou, K. N. Liou, Y. Takano, N. X. Rao, Q. Fu, A. J. Heymsfield, L. M. Miloshevich, B. Baum, S. A. Kinne, “Remote sounding of cirrus cloud optical depths and ice crystal sizes from AVHRR data: verification using FIRE-II-IFO composite measurements,” J. Atmos. Sci. 52, 4143–4158 (1995).
[CrossRef]

K. N. Liou, N. X. Rao, “Radiative transfer in cirrus clouds. IV. Cloud geometry, inhomogeneity, and absorption,” J. Atmos. Sci. 53, 3046–3065 (1996).
[CrossRef]

J. Geophys. Res. (1)

G. G. Mace, T. Ackerman, P. Minnis, D. F. Young, “Cirrus layer microphysical properties derived from surface-based millimeter radar and infrared interferometer data,” J. Geophys. Res. 103, 23207–23216 (1998).
[CrossRef]

Mon. Weather Rev. (1)

J. D. Spinhirne, W. D. Hart, “Cirrus structure and radiative parameters from airborne lidar and spectral radiometer observations: the 28 October 1986 FIRE study,” Mon. Weather Rev. 118, 2329–2343 (1990).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Laser transmission through a 2D inhomogeneous cirrus cloud. A gray-scale contour inside the cirrus shows a distribution of relative extinction coefficient β e (km-1). s′ = EB, s - s′ = ET, u = BM, v = AT. (b) Laser backscattering through a 2D inhomogeneous cirrus cloud. A gray-scale contour inside the cirrus shows a distribution of effective particle size D e (µm). s - sb b = EM, sb = EA.

Fig. 2
Fig. 2

Schematic diagram (a) for laser transmission and (b) for laser backscattering through a spherical cirrus cloud. s* = T 1 T 2, s′ = ET 2, u = T 2 M, sb = ET 1, v = T 1 A.

Fig. 3
Fig. 3

Backscattered power that is due to the first-order scattering and the reflection from the target. (a) Upper panels show effects of aerosols and water vapor. (b) Lower panels show effects of cloud particle size. The assumed conditions are as follows: λ = 1.315 µm, F 0 = 106 W, r m = 3 m, R λ = 0.2, r a = 4 m, d = 100 km, z a = 11 km, z t = 9.5 km, and z b = 9 km.

Fig. 4
Fig. 4

Same as Fig. 3, except that the aircraft is within the cloud (z a = 9.25 km).

Fig. 5
Fig. 5

Backscattered power that is due to the first-order scattering and the reflection from the target. Each panel shows effects of cloud particle shape and orientation and aircraft height: (a) aircraft at 9.6 km and (b) aircraft at 9.25 km. λ = 1.315 µm, D e = 42 µm, F 0 = 106 W, r m = 3 m, R λ = 0.2, r a = 4 m, d = 100 km, z t = 9.5 km, and z b = 9 km.

Fig. 6
Fig. 6

Direct transmission, forward scattering, and backscattering in an inhomogeneous and a homogeneous cirrus clouds with the same optical depth. λ = 1.315 µm, D e = 74 µm, F 0 = 106 W, r m = 3 m, R λ = 0.2, r a = 4 m, d = 40 km, z a = 11.2 km, z t = 11.1 km, and z b = 7.5 km.

Fig. 7
Fig. 7

Percentage errors in the direct transmission and forward scattering that are due to uncertainties in the cloud optical depth (±0.05) and in the mean effective size (±5 µm) as functions of the target height. τ = 0.4. Other conditions are the same as for Fig. 6.

Fig. 8
Fig. 8

Percentage errors for the first-order backscattering and the reflection from the target that are due to uncertainties in the cloud optical depth (±0.05) and in the mean effective size (±5 µm) as functions of the target. The conditions are the same as for Fig. 6.

Fig. 9
Fig. 9

Direct transmission, forward scattering, and backscattering in a spherical atmosphere and a plane-parallel atmosphere. λ = 1.315 µm, D e = 42 µm, F 0 = 106 W, τ = 0.05, r m = 3 m, R λ = 0.2, r a = 4 m, z a = 12 km, z t = 11.5 km, and z b = 10.5 km.

Equations (28)

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

Fd=F0 exp-βnc,av+βnc,buexp-0s βeds =F0 exp-βnc,av+βnc,bu×exp-0sβair+βaer+kvρ+βcldds,
s=z-zb/μ,
s=zt-zb/μ.
F10, Ω=exp-βnc,av+βnc,bu0s J1s, Ω×exp-0s βesdsβesds.
J1s, Ω=ϖs2 F0s, Ω0Ψ1 PcldΘsin ΘdΘ,
Ψ1=tan-1rmu+s,
F0s, Ω=F0 exp-ss βesds.
J1sb, Ω=ϖs2 F0sb, Ωπ-Ψbπ PaveΘsin ΘdΘ,
Ψb=tan-1ra/sb,
F0sb, Ω=F0 exp-ssb βesds.
PaveΘ=hz-zb-hz-ztβs,cldPcldΘ+βs,aerPaerΘ+βairPairΘhz-zb-hz-ztβs,cld+βs,aer+βair,
βe=hz-zb-hz-ztβcld+βaer+βair+kvρ,
Fb10, Ω=0s J1sb, Ω×exp-ssb βesdsβesbdsb.
Fref=RλT2F0πra2/sb2,
Fd*=FdFs,
Fs=exp-0s* βesds,
F*10, Ω=F10, Ω+Fs10, Ω.
Fs10, Ω=0s* J1s, Ω×exp-0s βesdsβesds.
J1s, Ω=ϖs2 F0s, Ω0Ψ1* PΘsin ΘdΘ,
Ψ1*=tan-1rts+u
F0s, Ω=F0 exp-0s*-s βesds.
Fb*10, Ω=Fb10, Ω+Fsb10, Ω,
Fsb10, Ω=0s* Jb1sb, Ω×exp-0sb βesdsβesbdsb.
Jb1sb, Ω=ϖsb2 F0sb, Ωπ-Ψb*π PaveΘsin ΘdΘ,
Ψb*=tan-1rasb+v,
F0sb, Ω=F0 exp-ssb βesds.
Fref*=RλTs2F0πra2/sb2,
Ts=FdFs/F0.

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