Abstract

Infrared radiances from water become partially polarized at oblique viewing angles through both emission and reflection. I describe computer simulations that show how the state of polarization for water varies with environmental conditions over a wavelength range of 3–15 µm with 0.05-µm resolution. Polarization at wavelengths longer than approximately 4 µm generally is negative (p, or vertical) and increases with incidence angle up to approximately 75°, beyond which the horizontally polarized reflected atmospheric radiance begins to dominate the surface emission. The highest p polarization (∼4–10%) is found in the atmospheric window regions of approximately 4–5 and 8–14 µm. In the 3–5-µm spectral band, especially between 3 and 4 µm, reflected atmospheric radiance usually is greater than surface emission, resulting in a net s polarization (horizontal). The results of these simulations agree well with broadband measurements of the degree of polarization for a water surface viewed at nadir angles of 0–75°.

© 1999 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. F. H. Murcray, D. G. Murcray, W. J. Williams, “Infrared emissivity of lunar surface features. 1. Balloon-borne observations,” J. Geophys. Res. 75, 2662–2669 (1970).
    [CrossRef]
  35. H. K. Kieffer, R. L. Wildey, “Establishing the moon as a spectral radiance standard,” J. Atmos. Oceanic Technol. 13, 360–375 (1996).
    [CrossRef]

1999

J. A. Shaw, “Glittering light on water,” Opt. Photon. News 10(3), 43–45, 68 (1999).
[CrossRef]

1997

J. W. Salisbury, A. Basu, E. M. Fischer, “Thermal infrared spectra of lunar soils,” Icarus 130, 125–139 (1997).
[CrossRef]

Y. Han, J. A. Shaw, J. H. Churnside, P. D. Brown, S. A. Clough, “Infrared spectral radiance measurements in the tropical Pacific atmosphere,” J. Geophys. Res. 102, 4353–4356 (1997).
[CrossRef]

X. Wu, W. L. Smith, “Emissivity of rough sea surface for 8–13 µm: modeling and verification,” Appl. Opt. 36, 2609–2619 (1997).
[CrossRef] [PubMed]

J. A. Shaw, J. H. Churnside, “Scanning-laser glint measurements of sea-surface slope statistics,” Appl. Opt. 36, 4202–4213 (1997).
[CrossRef] [PubMed]

1996

D. L. Jordan, G. D. Lewis, E. Jakeman, “Emission polarization of roughened glass and aluminum surfaces,” Appl. Opt. 35, 3583–3590 (1996).
[CrossRef] [PubMed]

P. D. Watts, M. R. Allen, T. J. Nightingale, “Wind speed effects on sea surface emission and reflection for the Along Track Scanning Radiometer,” J. Atmos. Oceanic Technol. 13, 126–141 (1996).
[CrossRef]

H. K. Kieffer, R. L. Wildey, “Establishing the moon as a spectral radiance standard,” J. Atmos. Oceanic Technol. 13, 360–375 (1996).
[CrossRef]

1995

J. A. Shaw, J. B. Snider, J. H. Churnside, M. D. Jacobson, “Comparison of infrared atmospheric brightness temperatures measured by a Fourier transform spectrometer and filter radiometer,” J. Atmos. Oceanic Technol. 12, 1124–1128 (1995).
[CrossRef]

1992

1988

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sens. Environ. 24, 313–329 (1988).
[CrossRef]

1981

1973

1970

F. H. Murcray, D. G. Murcray, W. J. Williams, “Infrared emissivity of lunar surface features. 1. Balloon-borne observations,” J. Geophys. Res. 75, 2662–2669 (1970).
[CrossRef]

1969

1968

1965

1954

Allen, M. R.

P. D. Watts, M. R. Allen, T. J. Nightingale, “Wind speed effects on sea surface emission and reflection for the Along Track Scanning Radiometer,” J. Atmos. Oceanic Technol. 13, 126–141 (1996).
[CrossRef]

Anderson, G. P.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Basu, A.

J. W. Salisbury, A. Basu, E. M. Fischer, “Thermal infrared spectra of lunar soils,” Icarus 130, 125–139 (1997).
[CrossRef]

Berk, A.

A. Berk, L. S. Bernstein, D. C. Robertson, “MODTRAN: a moderate resolution model for LOWTRAN7,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).

Bernstein, L. S.

A. Berk, L. S. Bernstein, D. C. Robertson, “MODTRAN: a moderate resolution model for LOWTRAN7,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1991), p. 40.

Brown, P. D.

Y. Han, J. A. Shaw, J. H. Churnside, P. D. Brown, S. A. Clough, “Infrared spectral radiance measurements in the tropical Pacific atmosphere,” J. Geophys. Res. 102, 4353–4356 (1997).
[CrossRef]

Chan, P. M.

A. W. Cooper, W. J. Lentz, P. L. Walker, P. M. Chan, “Infrared polarization measurements of ship signatures and background contrast,” in Characterization and Propagation of Sources and Backgrounds, W. R. Watkins, ed., Proc. SPIE2223, 300–309 (1994).
[CrossRef]

Chetwynd, J. H.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Chun, C.

F. Sadjadi, C. Chun, “Machine recognition of objects using IR polarimetry,” Automatic Object Recognition VI, F. A. Sadjadi, ed., Proc. SPIE2756, 53–59 (1996).
[CrossRef]

Chun, C. S. L.

C. S. L. Chun, F. A. Sadjadi, D. Ferris, “Automatic target recognition using polarization-sensitive, thermal imaging,” in Automatic Object Recognition V, F. A. Sadjadi, ed., Proc. SPIE2485, 353–364 (1995).
[CrossRef]

Churnside, J. H.

Y. Han, J. A. Shaw, J. H. Churnside, P. D. Brown, S. A. Clough, “Infrared spectral radiance measurements in the tropical Pacific atmosphere,” J. Geophys. Res. 102, 4353–4356 (1997).
[CrossRef]

J. A. Shaw, J. H. Churnside, “Scanning-laser glint measurements of sea-surface slope statistics,” Appl. Opt. 36, 4202–4213 (1997).
[CrossRef] [PubMed]

J. A. Shaw, J. B. Snider, J. H. Churnside, M. D. Jacobson, “Comparison of infrared atmospheric brightness temperatures measured by a Fourier transform spectrometer and filter radiometer,” J. Atmos. Oceanic Technol. 12, 1124–1128 (1995).
[CrossRef]

Clough, S. A.

Y. Han, J. A. Shaw, J. H. Churnside, P. D. Brown, S. A. Clough, “Infrared spectral radiance measurements in the tropical Pacific atmosphere,” J. Geophys. Res. 102, 4353–4356 (1997).
[CrossRef]

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Cooper, A. W.

P. L. Walker, W. J. Lentz, A. W. Cooper, “Atmospheric and sea state dependence of polarized infrared contrast,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, ed., Proc. SPIE2469, 393–403 (1995).
[CrossRef]

A. W. Cooper, E. C. Crittenden, E. A. Milne, P. L. Walker, E. Moss, D. Gregoris, “Mid and far infrared measurements of sun glint from the sea surface,” in Optics of the Air-Sea Interface, L. Estep, ed., Proc. SPIE1749, 176–185 (1992).
[CrossRef]

A. W. Cooper, W. J. Lentz, P. L. Walker, “Infrared polarization ship images and contrast in the MAPTIP experiment,” in Image Propagation through the Atmosphere, J. Daintyll, R. Bissonnette, eds., Proc. SPIE2828, 85–96 (1996).
[CrossRef]

A. W. Cooper, W. J. Lentz, P. L. Walker, P. M. Chan, “Infrared polarization measurements of ship signatures and background contrast,” in Characterization and Propagation of Sources and Backgrounds, W. R. Watkins, ed., Proc. SPIE2223, 300–309 (1994).
[CrossRef]

Cox, C.

Crittenden, E. C.

A. W. Cooper, E. C. Crittenden, E. A. Milne, P. L. Walker, E. Moss, D. Gregoris, “Mid and far infrared measurements of sun glint from the sea surface,” in Optics of the Air-Sea Interface, L. Estep, ed., Proc. SPIE1749, 176–185 (1992).
[CrossRef]

Egan, W. G.

W. G. Egan, Photometry and Polarization in Remote Sensing (Elsevier, New York, 1985), pp. 337–354.

Ferris, D.

C. S. L. Chun, F. A. Sadjadi, D. Ferris, “Automatic target recognition using polarization-sensitive, thermal imaging,” in Automatic Object Recognition V, F. A. Sadjadi, ed., Proc. SPIE2485, 353–364 (1995).
[CrossRef]

Fischer, E. M.

J. W. Salisbury, A. Basu, E. M. Fischer, “Thermal infrared spectra of lunar soils,” Icarus 130, 125–139 (1997).
[CrossRef]

Friedman, D.

Gregoris, D.

A. W. Cooper, E. C. Crittenden, E. A. Milne, P. L. Walker, E. Moss, D. Gregoris, “Mid and far infrared measurements of sun glint from the sea surface,” in Optics of the Air-Sea Interface, L. Estep, ed., Proc. SPIE1749, 176–185 (1992).
[CrossRef]

Hale, G. M.

Hall, F. F.

Han, Y.

Y. Han, J. A. Shaw, J. H. Churnside, P. D. Brown, S. A. Clough, “Infrared spectral radiance measurements in the tropical Pacific atmosphere,” J. Geophys. Res. 102, 4353–4356 (1997).
[CrossRef]

Irvine, W. M.

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

Jacobson, M. D.

J. A. Shaw, J. B. Snider, J. H. Churnside, M. D. Jacobson, “Comparison of infrared atmospheric brightness temperatures measured by a Fourier transform spectrometer and filter radiometer,” J. Atmos. Oceanic Technol. 12, 1124–1128 (1995).
[CrossRef]

Jakeman, E.

Jordan, D. L.

Kieffer, H. K.

H. K. Kieffer, R. L. Wildey, “Establishing the moon as a spectral radiance standard,” J. Atmos. Oceanic Technol. 13, 360–375 (1996).
[CrossRef]

Kneizys, F. X.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Lentz, W. J.

P. L. Walker, W. J. Lentz, A. W. Cooper, “Atmospheric and sea state dependence of polarized infrared contrast,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, ed., Proc. SPIE2469, 393–403 (1995).
[CrossRef]

A. W. Cooper, W. J. Lentz, P. L. Walker, “Infrared polarization ship images and contrast in the MAPTIP experiment,” in Image Propagation through the Atmosphere, J. Daintyll, R. Bissonnette, eds., Proc. SPIE2828, 85–96 (1996).
[CrossRef]

A. W. Cooper, W. J. Lentz, P. L. Walker, P. M. Chan, “Infrared polarization measurements of ship signatures and background contrast,” in Characterization and Propagation of Sources and Backgrounds, W. R. Watkins, ed., Proc. SPIE2223, 300–309 (1994).
[CrossRef]

Lewis, G. D.

Liou, K. N.

Masuda, K.

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sens. Environ. 24, 313–329 (1988).
[CrossRef]

Milne, E. A.

A. W. Cooper, E. C. Crittenden, E. A. Milne, P. L. Walker, E. Moss, D. Gregoris, “Mid and far infrared measurements of sun glint from the sea surface,” in Optics of the Air-Sea Interface, L. Estep, ed., Proc. SPIE1749, 176–185 (1992).
[CrossRef]

Moss, E.

A. W. Cooper, E. C. Crittenden, E. A. Milne, P. L. Walker, E. Moss, D. Gregoris, “Mid and far infrared measurements of sun glint from the sea surface,” in Optics of the Air-Sea Interface, L. Estep, ed., Proc. SPIE1749, 176–185 (1992).
[CrossRef]

Munk, W.

Murcray, D. G.

F. H. Murcray, D. G. Murcray, W. J. Williams, “Infrared emissivity of lunar surface features. 1. Balloon-borne observations,” J. Geophys. Res. 75, 2662–2669 (1970).
[CrossRef]

Murcray, F. H.

F. H. Murcray, D. G. Murcray, W. J. Williams, “Infrared emissivity of lunar surface features. 1. Balloon-borne observations,” J. Geophys. Res. 75, 2662–2669 (1970).
[CrossRef]

Nightingale, T. J.

P. D. Watts, M. R. Allen, T. J. Nightingale, “Wind speed effects on sea surface emission and reflection for the Along Track Scanning Radiometer,” J. Atmos. Oceanic Technol. 13, 126–141 (1996).
[CrossRef]

Partridge, M.

M. Partridge, R. C. Saull, “Three-dimensional surface reconstruction using emission polarization,” in Image and Signal Processing for Remote Sensing II, J. Desachy, ed., Proc. SPIE2579, 92–103 (1995).
[CrossRef]

Pollack, J. B.

W. M. Irvine, J. B. Pollack, “Infrared optical properties of water and ice spheres,” Icarus 8, 324–360 (1968).
[CrossRef]

Querry, M. R.

Rice, J. E.

T. J. Rogne, F. G. Smith, J. E. Rice, “Passive target detection using polarized components of infrared signatures,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 242–251 (1990).

Robertson, D. C.

A. Berk, L. S. Bernstein, D. C. Robertson, “MODTRAN: a moderate resolution model for LOWTRAN7,” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1989).

Rogne, T. J.

T. J. Rogne, F. G. Smith, J. E. Rice, “Passive target detection using polarized components of infrared signatures,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 242–251 (1990).

Sadjadi, F.

F. Sadjadi, C. Chun, “Machine recognition of objects using IR polarimetry,” Automatic Object Recognition VI, F. A. Sadjadi, ed., Proc. SPIE2756, 53–59 (1996).
[CrossRef]

Sadjadi, F. A.

C. S. L. Chun, F. A. Sadjadi, D. Ferris, “Automatic target recognition using polarization-sensitive, thermal imaging,” in Automatic Object Recognition V, F. A. Sadjadi, ed., Proc. SPIE2485, 353–364 (1995).
[CrossRef]

Salisbury, J. W.

J. W. Salisbury, A. Basu, E. M. Fischer, “Thermal infrared spectra of lunar soils,” Icarus 130, 125–139 (1997).
[CrossRef]

Sandus, O.

Saull, R. C.

M. Partridge, R. C. Saull, “Three-dimensional surface reconstruction using emission polarization,” in Image and Signal Processing for Remote Sensing II, J. Desachy, ed., Proc. SPIE2579, 92–103 (1995).
[CrossRef]

Shaw, J. A.

J. A. Shaw, “Glittering light on water,” Opt. Photon. News 10(3), 43–45, 68 (1999).
[CrossRef]

J. A. Shaw, J. H. Churnside, “Scanning-laser glint measurements of sea-surface slope statistics,” Appl. Opt. 36, 4202–4213 (1997).
[CrossRef] [PubMed]

Y. Han, J. A. Shaw, J. H. Churnside, P. D. Brown, S. A. Clough, “Infrared spectral radiance measurements in the tropical Pacific atmosphere,” J. Geophys. Res. 102, 4353–4356 (1997).
[CrossRef]

J. A. Shaw, J. B. Snider, J. H. Churnside, M. D. Jacobson, “Comparison of infrared atmospheric brightness temperatures measured by a Fourier transform spectrometer and filter radiometer,” J. Atmos. Oceanic Technol. 12, 1124–1128 (1995).
[CrossRef]

J. A. Shaw, “The impact of polarization on infrared sea-surface temperature sensing,” in IGARSS ’98 (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 496–498.

Shettle, E. P.

E. P. Shettle, “Models of aerosols, clouds and precipitation for atmospheric propagation studies,” in Atmospheric Propagation in the UV, Visible, IR and mm-Wave Region and Related Systems Aspects (Advisory Group for Aerospace Research and Development, Paris, 1989), pp. 15-3–15-13.

G. P. Anderson, S. A. Clough, F. X. Kneizys, J. H. Chetwynd, E. P. Shettle, “AFGL atmospheric constituent profiles (0–120 km),” (U.S. Air Force Geophysics Laboratory, Hanscom Air Force Base, Mass., 1986).

Sidran, M.

Smith, F. G.

T. J. Rogne, F. G. Smith, J. E. Rice, “Passive target detection using polarized components of infrared signatures,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 242–251 (1990).

Smith, W. L.

Snider, J. B.

J. A. Shaw, J. B. Snider, J. H. Churnside, M. D. Jacobson, “Comparison of infrared atmospheric brightness temperatures measured by a Fourier transform spectrometer and filter radiometer,” J. Atmos. Oceanic Technol. 12, 1124–1128 (1995).
[CrossRef]

Takano, Y.

Takashima, T.

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sens. Environ. 24, 313–329 (1988).
[CrossRef]

Takayama, Y.

K. Masuda, T. Takashima, Y. Takayama, “Emissivity of pure and sea waters for the model sea surface in the infrared window regions,” Remote Sens. Environ. 24, 313–329 (1988).
[CrossRef]

Tooley, R. D.

R. D. Tooley, “Man-made target detection using infrared polarization,” in Polarization Considerations for Optical Systems II, R. A. Chipman, ed., Proc. SPIE1166, 52–58 (1989).
[CrossRef]

Walker, P. L.

A. W. Cooper, E. C. Crittenden, E. A. Milne, P. L. Walker, E. Moss, D. Gregoris, “Mid and far infrared measurements of sun glint from the sea surface,” in Optics of the Air-Sea Interface, L. Estep, ed., Proc. SPIE1749, 176–185 (1992).
[CrossRef]

P. L. Walker, W. J. Lentz, A. W. Cooper, “Atmospheric and sea state dependence of polarized infrared contrast,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, ed., Proc. SPIE2469, 393–403 (1995).
[CrossRef]

A. W. Cooper, W. J. Lentz, P. L. Walker, P. M. Chan, “Infrared polarization measurements of ship signatures and background contrast,” in Characterization and Propagation of Sources and Backgrounds, W. R. Watkins, ed., Proc. SPIE2223, 300–309 (1994).
[CrossRef]

A. W. Cooper, W. J. Lentz, P. L. Walker, “Infrared polarization ship images and contrast in the MAPTIP experiment,” in Image Propagation through the Atmosphere, J. Daintyll, R. Bissonnette, eds., Proc. SPIE2828, 85–96 (1996).
[CrossRef]

Watts, P. D.

P. D. Watts, M. R. Allen, T. J. Nightingale, “Wind speed effects on sea surface emission and reflection for the Along Track Scanning Radiometer,” J. Atmos. Oceanic Technol. 13, 126–141 (1996).
[CrossRef]

Wildey, R. L.

H. K. Kieffer, R. L. Wildey, “Establishing the moon as a spectral radiance standard,” J. Atmos. Oceanic Technol. 13, 360–375 (1996).
[CrossRef]

Williams, W. J.

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Figures (11)

Fig. 1
Fig. 1

Radiative transfer model simulates the radiance seen by a radiometer at height h viewing a water surface at nadir angle θ. The scene radiance comprises surface emission L sfc, specular-angle atmospheric path emission L atm, which is reflected at the surface, and short-path atmospheric path emission L atm-sp. The total radiance is computed separately for s and p polarization (respectively, horizontal and vertical from the radiometer’s viewpoint).

Fig. 2
Fig. 2

Complex refractive index of water (n - ik).19 The imaginary part is shown here as k + 1 for graphical convenience.

Fig. 3
Fig. 3

Spectral emissivity of a smooth water surface viewed obliquely is partially polarized. As the angle increases, the p emissivity first rises and then falls below the nadir value while the s emissivity becomes steadily smaller.

Fig. 4
Fig. 4

Degree of polarization versus angle for water emission (bottom) and reflection (top) at the indicated wavelengths. Note the different vertical scales above and below 0%.

Fig. 5
Fig. 5

Spectral atmospheric radiance (thermal emission and scattered solar) calculated for a vertical atmospheric path viewed from the surface. Both water vapor and clouds increase the radiance in the atmospheric transmission window regions of approximately 3–5 and 8–14 µm (the bottom two curves are for clear atmospheres that differ only in water-vapor content).

Fig. 6
Fig. 6

Degree of polarization of the total radiance at different nadir angles for the USSA76 and sensors at (a) 10-m height and (b) top of atmosphere. A satellite-based sensor above the atmosphere sees a polarization signature that is similar to the near-surface value in the atmospheric transmission windows up to approximately 60°. Beyond this angle, the longer atmospheric path length for a satellite sensor greatly reduces the polarization from what is seen near the surface.

Fig. 7
Fig. 7

Spectral degree of polarization of the total radiance for a smooth surface viewed from a 10-m height through a tropical atmosphere [compare with Fig. 6(a)]. The high water-vapor content greatly reduces the p polarization, especially in the 8–14-µm band, by absorbing surface-emitted radiance and by contributing to a greater reflected atmospheric radiance.

Fig. 8
Fig. 8

Band-averaged degree of polarization for four (a) short-wave and (b) long-wave radiometer bands. The short-wave curves indicate dominant reflection polarization and the long-wave curves indicate dominant emission polarization. The magnitude of the long-wave polarization increases monotonically with angle up to approximately 75°, after which it decreases as the reflected atmospheric radiance begins to dominate the emitted surface radiance.

Fig. 9
Fig. 9

Spectral degree of polarization of the total radiance for a rough surface (5-m s-1 wind speed) viewed at the indicated angles from a 10-m height through a USSA76 [compare with the corresponding smooth-surface curves in Fig. 6(a)]. The primary difference in the results with a rough surface is that the polarization magnitude continues to increase with angle up to at least 80°, whereas the smooth-surface polarization decreases beyond 75°.

Fig. 10
Fig. 10

Spectral degree of polarization for a water surface viewed at 60° with (a) Sun glints and (b) full-moon glints. Each curve is for a different percentage of the radiometer beam that contains glints.

Fig. 11
Fig. 11

Two sets of measured polarization (circles and crosses) compared with calculations for a smooth surface (dashed curve) and a rough surface at 1-m s-1 wind speed (solid curve), both using the mid-latitude winter atmosphere model.

Equations (8)

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

Ls,p=τatm-spLsfcs,p+Latm-sp+τatm-spRsfcs,pLatm.
DP=Ls-LpLs+Lp.
Lsfcs,pλ, θ, T=sfcs,pλ, θLBBλ, T.
LBBλ, T=2×10-6 hc2λ51exphcλkT-1,
sfcs,pλ, θ=1-Rsfcs,pλ, θ.
Rsfcsλ, θ=cosθ-ñλcosθrcosθ+ñλcosθr2,
Rsfcpλ, θ=ñλcosθ-cosθrñλcosθ+cosθr2,
θrλ, θ=sin-1sinθñλ.

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