Abstract

With radiative transfer simulations it is suggested that stable estimates of the highly anisotropic direct beam spectral albedo of snow surface can be derived reciprocally under a variety of overcast skies. An accuracy of ±0.008 is achieved over a solar zenith angle range of θ0 ≤ 74° for visible wavelengths and up to θ0 ≤ 63° at the near-infrared wavelength λ = 862 nm. This new method helps expand the database of snow surface albedo for the polar regions where direct measurement of clear-sky surface albedo is limited to large θ0’s only. The enhancement will assist in the validation of snow surface albedo models and improve the representation of polar surface albedo in global circulation models.

© 2003 Optical Society of America

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2003 (1)

2001 (2)

K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
[CrossRef]

X. Zhou, S. Li, K. Morris, “Measurement of all-wave and spectral albedos of snow-covered summer sea ice in the Ross Sea, Antarctica,” Ann. Glaciol. 33, 267–274 (2001).
[CrossRef]

2000 (2)

T. Aoki, T. Aoki, M. Fukabori, A. Hachikubo, Y. Tachibana, F. Nishio, “Effects of snow physical parameters on spectral albedo and bidirectional reflectance of snow surface,” J. Geophys. Res. 105D, 10219–10236 (2000).
[CrossRef]

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

1999 (2)

W. Han, K. Stamnes, D. Lubin, “Remote sensing of surface and cloud properties in the Arctic from AVHRR measurements,” J. Appl. Meteorol. 38, 989–1012 (1999).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

1998 (1)

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25789–25807 (1998).
[CrossRef]

1996 (1)

E. Leontieva, K. Stamnes, “Remote sensing of cloud optical properties from ground-based measurements of transmittance: a feasibility study,” J. Appl. Meteorol. 35, 2011–2022 (1996).
[CrossRef]

1995 (1)

D. Rind, R. Healy, C. Parkinson, D. Martinson, “The role of sea ice in 2 X CO2 climate model sensitivity. I: The total influence of sea ice thickness and extent,” J. Clim. 8, 449–463 (1995).
[CrossRef]

1994 (2)

T. C. Grenfell, S. G. Warren, P. C. Mullen, “Reflection of solar radiation by the Antarctic snow surface at ultraviolet, visible, and near-infrared wavelengths,” J. Geophys. Res. 99, 18669–18684 (1994).
[CrossRef]

D. K. Perovich, “Light reflection from sea ice during the onset of melt,” J. Geophys. Res. 99, 3351–3359 (1994).
[CrossRef]

1993 (1)

D. A. Robinson, K. F. Dewey, R. R. Heim, “Global snow cover monitoring: an update,” Bull. Am. Meteorol. Soc. 74, 1689–1696 (1993).
[CrossRef]

1992 (1)

T. C. Grenfell, “Radiative transfer model for sea ice with vertical variations,” J. Geophys. Res. 96, 16991–17001 (1992).
[CrossRef]

1991 (1)

S. Manabe, R. J. Stouffer, M. J. Spelman, K. Bryan, “Transient response of a coupled ocean-atmosphere model to gradual changes of atmospheric CO2. I: Annual mean response,” J. Clim. 4, 785–818 (1991).
[CrossRef]

1987 (2)

R. E. Dickinson, G. A. Meehl, W. M. Washington, “Ice albedo feedback in a CO2-doubling simulation,” Clim. Change 10, 241–248 (1987).
[CrossRef]

S. Li, Z. Wan, J. Dozier, “A component decomposition model for evaluating atmospheric effects in remote sensing,” J. Electromagn. Waves Appl. 1, 323–347 (1987).

1986 (1)

W. M. Washington, G. A. Meehl, “General circulation model CO2 sensitivity experiments: snow-sea ice albedo parameterizations and globally averaged surface air temperature,” Clim. Change 8, 231–241 (1986).
[CrossRef]

1984 (1)

T. C. Grenfell, D. K. Perovich, “Spectral albedos of sea ice and incident solar irradiance in the southern Beaufort Sea,” J. Geophys. Res. 89, 3573–3580 (1984).
[CrossRef]

1981 (1)

J. J. Carroll, B. W. Fitch, “Effects of solar elevation and cloudiness on snow albedo at the south pole,” J. Geophys. Res. 86C, 5271–5276 (1981).
[CrossRef]

1980 (3)

W. Wiscombe, S. Warren, “A model for the spectral albedo of snow. I: Pure snow,” J. Atmos. Sci. 37, 2712–2733 (1980).
[CrossRef]

S. Manabe, R. J. Stouffer, “Sensitivity of a global climate model to an increase of CO2 concentration in the atmosphere,” J. Geophys. Res. 85, 5529–5554 (1980).
[CrossRef]

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

1977 (1)

T. C. Grenfell, G. Maykut, “The optical properties of ice and snow in the Arctic Basin,” J. Glaciol. 18, 445–463 (1977).

1976 (2)

W. J. Wiscombe, “Extension of the doubling method to inhomogeneous sources,” J. Quant. Spectrosc. Radiat. Transfer 16, 477–489 (1976).
[CrossRef]

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 635–658 (1976).
[CrossRef]

1969 (1)

I. P. Grant, G. E. Hunt, “Discrete space theory of radiative transfer: I. Fundamentals,” Proc. R. Soc. London Ser. A 313, 183–197 (1969).
[CrossRef]

1968 (1)

G. Weller, “Heat-energy transfer through a four-layer system: air snow, sea ice, sea water,” J. Geophys. Res. 73, 1209–1220 (1968).
[CrossRef]

1941 (1)

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

Ackerman, T. P.

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

Aoki, T.

T. Aoki, T. Aoki, M. Fukabori, A. Hachikubo, Y. Tachibana, F. Nishio, “Effects of snow physical parameters on spectral albedo and bidirectional reflectance of snow surface,” J. Geophys. Res. 105D, 10219–10236 (2000).
[CrossRef]

T. Aoki, T. Aoki, M. Fukabori, A. Hachikubo, Y. Tachibana, F. Nishio, “Effects of snow physical parameters on spectral albedo and bidirectional reflectance of snow surface,” J. Geophys. Res. 105D, 10219–10236 (2000).
[CrossRef]

Brandt, R. E.

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25789–25807 (1998).
[CrossRef]

Bryan, K.

S. Manabe, R. J. Stouffer, M. J. Spelman, K. Bryan, “Transient response of a coupled ocean-atmosphere model to gradual changes of atmospheric CO2. I: Annual mean response,” J. Clim. 4, 785–818 (1991).
[CrossRef]

Campbell, W.

P. Gloersen, W. Campbell, D. J. Cavalieri, J. C. Comiso, C. L. Parkinson, H. J. Zwally, “Arctic and Antarctic sea ice, 1978–1987: satellite passive-microwave observations and analysis,” (National Aeronautics and Space Administration, Washington, D.C., 1992).

Campbell, W. J.

C. L. Parkinson, J. C. Comiso, H. J. Zwally, D. J. Cavalieri, P. Gloersen, W. J. Campbell, “Arctic sea ice 1973–1976 from satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1987).

H. J. Zwally, J. C. Comiso, C. L. Parkinson, W. J. Campbell, F. D. Carsey, P. Gloersen, “Antarctic sea ice, 1973–1976: satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1983).

Carroll, J. J.

J. J. Carroll, B. W. Fitch, “Effects of solar elevation and cloudiness on snow albedo at the south pole,” J. Geophys. Res. 86C, 5271–5276 (1981).
[CrossRef]

Carsey, F. D.

H. J. Zwally, J. C. Comiso, C. L. Parkinson, W. J. Campbell, F. D. Carsey, P. Gloersen, “Antarctic sea ice, 1973–1976: satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1983).

Cavalieri, D. J.

C. L. Parkinson, J. C. Comiso, H. J. Zwally, D. J. Cavalieri, P. Gloersen, W. J. Campbell, “Arctic sea ice 1973–1976 from satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1987).

P. Gloersen, W. Campbell, D. J. Cavalieri, J. C. Comiso, C. L. Parkinson, H. J. Zwally, “Arctic and Antarctic sea ice, 1978–1987: satellite passive-microwave observations and analysis,” (National Aeronautics and Space Administration, Washington, D.C., 1992).

Clothiaux, E. E.

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

Comiso, J. C.

P. Gloersen, W. Campbell, D. J. Cavalieri, J. C. Comiso, C. L. Parkinson, H. J. Zwally, “Arctic and Antarctic sea ice, 1978–1987: satellite passive-microwave observations and analysis,” (National Aeronautics and Space Administration, Washington, D.C., 1992).

C. L. Parkinson, J. C. Comiso, H. J. Zwally, D. J. Cavalieri, P. Gloersen, W. J. Campbell, “Arctic sea ice 1973–1976 from satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1987).

H. J. Zwally, J. C. Comiso, C. L. Parkinson, W. J. Campbell, F. D. Carsey, P. Gloersen, “Antarctic sea ice, 1973–1976: satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1983).

Croll, J.

J. Croll, Climate and Time in Geologic Relations: A Theory of Secular Change of the Earth’s Climate (Isbister, London, 1875).

Curry, J.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Dewey, K. F.

D. A. Robinson, K. F. Dewey, R. R. Heim, “Global snow cover monitoring: an update,” Bull. Am. Meteorol. Soc. 74, 1689–1696 (1993).
[CrossRef]

Dickinson, R. E.

R. E. Dickinson, G. A. Meehl, W. M. Washington, “Ice albedo feedback in a CO2-doubling simulation,” Clim. Change 10, 241–248 (1987).
[CrossRef]

Dlugach, J. M.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

Dong, X.

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

Dozier, J.

S. Li, Z. Wan, J. Dozier, “A component decomposition model for evaluating atmospheric effects in remote sensing,” J. Electromagn. Waves Appl. 1, 323–347 (1987).

T. H. Painter, J. Dozier, “Measurements of the bidirectional reflectance of snow at fine spectral and angular resolution,” in Proceedings of the 70th Annual Western Snow Conference, available online at http://www.westernsnowconference.org/2002/PDF/2002PainterAndDozier.pdf .

Fahnestock, M.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Fitch, B. W.

J. J. Carroll, B. W. Fitch, “Effects of solar elevation and cloudiness on snow albedo at the south pole,” J. Geophys. Res. 86C, 5271–5276 (1981).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

Fukabori, M.

T. Aoki, T. Aoki, M. Fukabori, A. Hachikubo, Y. Tachibana, F. Nishio, “Effects of snow physical parameters on spectral albedo and bidirectional reflectance of snow surface,” J. Geophys. Res. 105D, 10219–10236 (2000).
[CrossRef]

Ginsberg, I. W.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (U.S. Department of Commerce, Washington, D.C., 1977).

Gloersen, P.

P. Gloersen, W. Campbell, D. J. Cavalieri, J. C. Comiso, C. L. Parkinson, H. J. Zwally, “Arctic and Antarctic sea ice, 1978–1987: satellite passive-microwave observations and analysis,” (National Aeronautics and Space Administration, Washington, D.C., 1992).

C. L. Parkinson, J. C. Comiso, H. J. Zwally, D. J. Cavalieri, P. Gloersen, W. J. Campbell, “Arctic sea ice 1973–1976 from satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1987).

H. J. Zwally, J. C. Comiso, C. L. Parkinson, W. J. Campbell, F. D. Carsey, P. Gloersen, “Antarctic sea ice, 1973–1976: satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1983).

Grant, I. P.

I. P. Grant, G. E. Hunt, “Discrete space theory of radiative transfer: I. Fundamentals,” Proc. R. Soc. London Ser. A 313, 183–197 (1969).
[CrossRef]

Greenstein, J.

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

Grenfell, T. C.

T. C. Grenfell, S. G. Warren, P. C. Mullen, “Reflection of solar radiation by the Antarctic snow surface at ultraviolet, visible, and near-infrared wavelengths,” J. Geophys. Res. 99, 18669–18684 (1994).
[CrossRef]

T. C. Grenfell, “Radiative transfer model for sea ice with vertical variations,” J. Geophys. Res. 96, 16991–17001 (1992).
[CrossRef]

T. C. Grenfell, D. K. Perovich, “Spectral albedos of sea ice and incident solar irradiance in the southern Beaufort Sea,” J. Geophys. Res. 89, 3573–3580 (1984).
[CrossRef]

T. C. Grenfell, G. Maykut, “The optical properties of ice and snow in the Arctic Basin,” J. Glaciol. 18, 445–463 (1977).

Hachikubo, A.

T. Aoki, T. Aoki, M. Fukabori, A. Hachikubo, Y. Tachibana, F. Nishio, “Effects of snow physical parameters on spectral albedo and bidirectional reflectance of snow surface,” J. Geophys. Res. 105D, 10219–10236 (2000).
[CrossRef]

Han, W.

W. Han, K. Stamnes, D. Lubin, “Remote sensing of surface and cloud properties in the Arctic from AVHRR measurements,” J. Appl. Meteorol. 38, 989–1012 (1999).
[CrossRef]

Healy, R.

D. Rind, R. Healy, C. Parkinson, D. Martinson, “The role of sea ice in 2 X CO2 climate model sensitivity. I: The total influence of sea ice thickness and extent,” J. Clim. 8, 449–463 (1995).
[CrossRef]

Heim, R. R.

D. A. Robinson, K. F. Dewey, R. R. Heim, “Global snow cover monitoring: an update,” Bull. Am. Meteorol. Soc. 74, 1689–1696 (1993).
[CrossRef]

Henyey, L.

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

Hinton, P. O.

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25789–25807 (1998).
[CrossRef]

Howell, J. R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (Hemisphere, Washington, D.C., 1981).

Hsia, J. J.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (U.S. Department of Commerce, Washington, D.C., 1977).

Hunt, G. E.

I. P. Grant, G. E. Hunt, “Discrete space theory of radiative transfer: I. Fundamentals,” Proc. R. Soc. London Ser. A 313, 183–197 (1969).
[CrossRef]

Jeffries, M. O.

K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
[CrossRef]

Kennicutt, M. C.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Kokhanovsky, A. A.

A. A. Kokhanovsky, A. Macke, “The dependence of the radiative characteristics of optically thick media on the shape of particles,” J. Quant. Spectrosc. Radiat. Transfer 63, 393–407.

Leontieva, E.

E. Leontieva, K. Stamnes, “Remote sensing of cloud optical properties from ground-based measurements of transmittance: a feasibility study,” J. Appl. Meteorol. 35, 2011–2022 (1996).
[CrossRef]

Li, S.

X. Zhou, S. Li, K. Stamnes, “New geometrical-optics code for computing the optical properties of large dielectric spheres,” Appl. Opt. 42, 4295–4306 (2003).
[CrossRef] [PubMed]

X. Zhou, S. Li, K. Morris, “Measurement of all-wave and spectral albedos of snow-covered summer sea ice in the Ross Sea, Antarctica,” Ann. Glaciol. 33, 267–274 (2001).
[CrossRef]

S. Li, Z. Wan, J. Dozier, “A component decomposition model for evaluating atmospheric effects in remote sensing,” J. Electromagn. Waves Appl. 1, 323–347 (1987).

S. Li, “A model for the anisotropic reflectance of pure snow,” M.A. thesis (University of California at Santa Barbara, Santa Barbara, Calif., 1982).

Liljegren, J. C.

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

Limperis, T.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (U.S. Department of Commerce, Washington, D.C., 1977).

Long, C. N.

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

Lubin, D.

W. Han, K. Stamnes, D. Lubin, “Remote sensing of surface and cloud properties in the Arctic from AVHRR measurements,” J. Appl. Meteorol. 38, 989–1012 (1999).
[CrossRef]

Mace, G. G.

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

Macke, A.

A. A. Kokhanovsky, A. Macke, “The dependence of the radiative characteristics of optically thick media on the shape of particles,” J. Quant. Spectrosc. Radiat. Transfer 63, 393–407.

Manabe, S.

S. Manabe, R. J. Stouffer, M. J. Spelman, K. Bryan, “Transient response of a coupled ocean-atmosphere model to gradual changes of atmospheric CO2. I: Annual mean response,” J. Clim. 4, 785–818 (1991).
[CrossRef]

S. Manabe, R. J. Stouffer, “Sensitivity of a global climate model to an increase of CO2 concentration in the atmosphere,” J. Geophys. Res. 85, 5529–5554 (1980).
[CrossRef]

Martinson, D.

D. Rind, R. Healy, C. Parkinson, D. Martinson, “The role of sea ice in 2 X CO2 climate model sensitivity. I: The total influence of sea ice thickness and extent,” J. Clim. 8, 449–463 (1995).
[CrossRef]

Maykut, G.

T. C. Grenfell, G. Maykut, “The optical properties of ice and snow in the Arctic Basin,” J. Glaciol. 18, 445–463 (1977).

McGuire, A. D.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Meehl, G. A.

R. E. Dickinson, G. A. Meehl, W. M. Washington, “Ice albedo feedback in a CO2-doubling simulation,” Clim. Change 10, 241–248 (1987).
[CrossRef]

W. M. Washington, G. A. Meehl, “General circulation model CO2 sensitivity experiments: snow-sea ice albedo parameterizations and globally averaged surface air temperature,” Clim. Change 8, 231–241 (1986).
[CrossRef]

Mendenhall, W.

W. Mendenhall, D. D. Wackerly, R. L. Scheaffer, Mathematical Statistics with Applications, 4th ed. (Duxbury, Belmont, Calif., 1990).

Minnis, P.

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

Morris, K.

X. Zhou, S. Li, K. Morris, “Measurement of all-wave and spectral albedos of snow-covered summer sea ice in the Ross Sea, Antarctica,” Ann. Glaciol. 33, 267–274 (2001).
[CrossRef]

K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
[CrossRef]

Mullen, P. C.

T. C. Grenfell, S. G. Warren, P. C. Mullen, “Reflection of solar radiation by the Antarctic snow surface at ultraviolet, visible, and near-infrared wavelengths,” J. Geophys. Res. 99, 18669–18684 (1994).
[CrossRef]

Nicodemus, F. E.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (U.S. Department of Commerce, Washington, D.C., 1977).

Nishio, F.

T. Aoki, T. Aoki, M. Fukabori, A. Hachikubo, Y. Tachibana, F. Nishio, “Effects of snow physical parameters on spectral albedo and bidirectional reflectance of snow surface,” J. Geophys. Res. 105D, 10219–10236 (2000).
[CrossRef]

Painter, T. H.

T. H. Painter, J. Dozier, “Measurements of the bidirectional reflectance of snow at fine spectral and angular resolution,” in Proceedings of the 70th Annual Western Snow Conference, available online at http://www.westernsnowconference.org/2002/PDF/2002PainterAndDozier.pdf .

Parkinson, C.

D. Rind, R. Healy, C. Parkinson, D. Martinson, “The role of sea ice in 2 X CO2 climate model sensitivity. I: The total influence of sea ice thickness and extent,” J. Clim. 8, 449–463 (1995).
[CrossRef]

Parkinson, C. L.

P. Gloersen, W. Campbell, D. J. Cavalieri, J. C. Comiso, C. L. Parkinson, H. J. Zwally, “Arctic and Antarctic sea ice, 1978–1987: satellite passive-microwave observations and analysis,” (National Aeronautics and Space Administration, Washington, D.C., 1992).

H. J. Zwally, J. C. Comiso, C. L. Parkinson, W. J. Campbell, F. D. Carsey, P. Gloersen, “Antarctic sea ice, 1973–1976: satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1983).

C. L. Parkinson, J. C. Comiso, H. J. Zwally, D. J. Cavalieri, P. Gloersen, W. J. Campbell, “Arctic sea ice 1973–1976 from satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1987).

Perovich, D. K.

D. K. Perovich, “Light reflection from sea ice during the onset of melt,” J. Geophys. Res. 99, 3351–3359 (1994).
[CrossRef]

T. C. Grenfell, D. K. Perovich, “Spectral albedos of sea ice and incident solar irradiance in the southern Beaufort Sea,” J. Geophys. Res. 89, 3573–3580 (1984).
[CrossRef]

D. K. Perovich, “The optical properties of sea ice,” in Physics of Ice-Covered Seas, M. Lepparanta, ed. (Helsinki University Printing House, Helsinki, 1998), pp. 195–230.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

Richmond, J. C.

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (U.S. Department of Commerce, Washington, D.C., 1977).

Rind, D.

D. Rind, R. Healy, C. Parkinson, D. Martinson, “The role of sea ice in 2 X CO2 climate model sensitivity. I: The total influence of sea ice thickness and extent,” J. Clim. 8, 449–463 (1995).
[CrossRef]

Robinson, D. A.

D. A. Robinson, K. F. Dewey, R. R. Heim, “Global snow cover monitoring: an update,” Bull. Am. Meteorol. Soc. 74, 1689–1696 (1993).
[CrossRef]

Rossow, W. B.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Scheaffer, R. L.

W. Mendenhall, D. D. Wackerly, R. L. Scheaffer, Mathematical Statistics with Applications, 4th ed. (Duxbury, Belmont, Calif., 1990).

Siegel, R.

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (Hemisphere, Washington, D.C., 1981).

Spelman, M. J.

S. Manabe, R. J. Stouffer, M. J. Spelman, K. Bryan, “Transient response of a coupled ocean-atmosphere model to gradual changes of atmospheric CO2. I: Annual mean response,” J. Clim. 4, 785–818 (1991).
[CrossRef]

Stamnes, K.

X. Zhou, S. Li, K. Stamnes, “New geometrical-optics code for computing the optical properties of large dielectric spheres,” Appl. Opt. 42, 4295–4306 (2003).
[CrossRef] [PubMed]

W. Han, K. Stamnes, D. Lubin, “Remote sensing of surface and cloud properties in the Arctic from AVHRR measurements,” J. Appl. Meteorol. 38, 989–1012 (1999).
[CrossRef]

E. Leontieva, K. Stamnes, “Remote sensing of cloud optical properties from ground-based measurements of transmittance: a feasibility study,” J. Appl. Meteorol. 35, 2011–2022 (1996).
[CrossRef]

Steele, M.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Stouffer, R. J.

S. Manabe, R. J. Stouffer, M. J. Spelman, K. Bryan, “Transient response of a coupled ocean-atmosphere model to gradual changes of atmospheric CO2. I: Annual mean response,” J. Clim. 4, 785–818 (1991).
[CrossRef]

S. Manabe, R. J. Stouffer, “Sensitivity of a global climate model to an increase of CO2 concentration in the atmosphere,” J. Geophys. Res. 85, 5529–5554 (1980).
[CrossRef]

Tachibana, Y.

T. Aoki, T. Aoki, M. Fukabori, A. Hachikubo, Y. Tachibana, F. Nishio, “Effects of snow physical parameters on spectral albedo and bidirectional reflectance of snow surface,” J. Geophys. Res. 105D, 10219–10236 (2000).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

Vorosmarty, C. J.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Wackerly, D. D.

W. Mendenhall, D. D. Wackerly, R. L. Scheaffer, Mathematical Statistics with Applications, 4th ed. (Duxbury, Belmont, Calif., 1990).

Walsh, J. E.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Wan, Z.

S. Li, Z. Wan, J. Dozier, “A component decomposition model for evaluating atmospheric effects in remote sensing,” J. Electromagn. Waves Appl. 1, 323–347 (1987).

Warren, S.

W. Wiscombe, S. Warren, “A model for the spectral albedo of snow. I: Pure snow,” J. Atmos. Sci. 37, 2712–2733 (1980).
[CrossRef]

Warren, S. G.

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25789–25807 (1998).
[CrossRef]

T. C. Grenfell, S. G. Warren, P. C. Mullen, “Reflection of solar radiation by the Antarctic snow surface at ultraviolet, visible, and near-infrared wavelengths,” J. Geophys. Res. 99, 18669–18684 (1994).
[CrossRef]

Washington, W. M.

R. E. Dickinson, G. A. Meehl, W. M. Washington, “Ice albedo feedback in a CO2-doubling simulation,” Clim. Change 10, 241–248 (1987).
[CrossRef]

W. M. Washington, G. A. Meehl, “General circulation model CO2 sensitivity experiments: snow-sea ice albedo parameterizations and globally averaged surface air temperature,” Clim. Change 8, 231–241 (1986).
[CrossRef]

Weller, G.

G. Weller, “Heat-energy transfer through a four-layer system: air snow, sea ice, sea water,” J. Geophys. Res. 73, 1209–1220 (1968).
[CrossRef]

Wharton, R.

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

Wiscombe, W.

W. Wiscombe, S. Warren, “A model for the spectral albedo of snow. I: Pure snow,” J. Atmos. Sci. 37, 2712–2733 (1980).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
[CrossRef] [PubMed]

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 635–658 (1976).
[CrossRef]

W. J. Wiscombe, “Extension of the doubling method to inhomogeneous sources,” J. Quant. Spectrosc. Radiat. Transfer 16, 477–489 (1976).
[CrossRef]

Yanovitskij, E. G.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

Zakharova, N. T.

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

Zhou, X.

X. Zhou, S. Li, K. Stamnes, “New geometrical-optics code for computing the optical properties of large dielectric spheres,” Appl. Opt. 42, 4295–4306 (2003).
[CrossRef] [PubMed]

X. Zhou, S. Li, K. Morris, “Measurement of all-wave and spectral albedos of snow-covered summer sea ice in the Ross Sea, Antarctica,” Ann. Glaciol. 33, 267–274 (2001).
[CrossRef]

X. Zhou, “Optical remote sensing of snow on sea ice: ground measurements, satellite data analysis, and radiative transfer modeling,” Ph.D. dissertation (University of Alaska, Fairbanks, Alaska, 2002).

Zwally, H. J.

H. J. Zwally, J. C. Comiso, C. L. Parkinson, W. J. Campbell, F. D. Carsey, P. Gloersen, “Antarctic sea ice, 1973–1976: satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1983).

C. L. Parkinson, J. C. Comiso, H. J. Zwally, D. J. Cavalieri, P. Gloersen, W. J. Campbell, “Arctic sea ice 1973–1976 from satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1987).

P. Gloersen, W. Campbell, D. J. Cavalieri, J. C. Comiso, C. L. Parkinson, H. J. Zwally, “Arctic and Antarctic sea ice, 1978–1987: satellite passive-microwave observations and analysis,” (National Aeronautics and Space Administration, Washington, D.C., 1992).

Ann. Glaciol. (2)

X. Zhou, S. Li, K. Morris, “Measurement of all-wave and spectral albedos of snow-covered summer sea ice in the Ross Sea, Antarctica,” Ann. Glaciol. 33, 267–274 (2001).
[CrossRef]

K. Morris, M. O. Jeffries, “Seasonal contrasts in snow cover characteristics on Ross Sea ice floes,” Ann. Glaciol. 33, 61–68 (2001).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (1)

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

Bull. Am. Meteorol. Soc. (1)

D. A. Robinson, K. F. Dewey, R. R. Heim, “Global snow cover monitoring: an update,” Bull. Am. Meteorol. Soc. 74, 1689–1696 (1993).
[CrossRef]

Clim. Change (2)

W. M. Washington, G. A. Meehl, “General circulation model CO2 sensitivity experiments: snow-sea ice albedo parameterizations and globally averaged surface air temperature,” Clim. Change 8, 231–241 (1986).
[CrossRef]

R. E. Dickinson, G. A. Meehl, W. M. Washington, “Ice albedo feedback in a CO2-doubling simulation,” Clim. Change 10, 241–248 (1987).
[CrossRef]

J. Appl. Meteorol. (2)

E. Leontieva, K. Stamnes, “Remote sensing of cloud optical properties from ground-based measurements of transmittance: a feasibility study,” J. Appl. Meteorol. 35, 2011–2022 (1996).
[CrossRef]

W. Han, K. Stamnes, D. Lubin, “Remote sensing of surface and cloud properties in the Arctic from AVHRR measurements,” J. Appl. Meteorol. 38, 989–1012 (1999).
[CrossRef]

J. Atmos. Sci. (1)

W. Wiscombe, S. Warren, “A model for the spectral albedo of snow. I: Pure snow,” J. Atmos. Sci. 37, 2712–2733 (1980).
[CrossRef]

J. Clim. (2)

S. Manabe, R. J. Stouffer, M. J. Spelman, K. Bryan, “Transient response of a coupled ocean-atmosphere model to gradual changes of atmospheric CO2. I: Annual mean response,” J. Clim. 4, 785–818 (1991).
[CrossRef]

D. Rind, R. Healy, C. Parkinson, D. Martinson, “The role of sea ice in 2 X CO2 climate model sensitivity. I: The total influence of sea ice thickness and extent,” J. Clim. 8, 449–463 (1995).
[CrossRef]

J. Electromagn. Waves Appl. (1)

S. Li, Z. Wan, J. Dozier, “A component decomposition model for evaluating atmospheric effects in remote sensing,” J. Electromagn. Waves Appl. 1, 323–347 (1987).

J. Geophys. Res. (10)

G. Weller, “Heat-energy transfer through a four-layer system: air snow, sea ice, sea water,” J. Geophys. Res. 73, 1209–1220 (1968).
[CrossRef]

T. C. Grenfell, D. K. Perovich, “Spectral albedos of sea ice and incident solar irradiance in the southern Beaufort Sea,” J. Geophys. Res. 89, 3573–3580 (1984).
[CrossRef]

T. C. Grenfell, “Radiative transfer model for sea ice with vertical variations,” J. Geophys. Res. 96, 16991–17001 (1992).
[CrossRef]

X. Dong, P. Minnis, T. P. Ackerman, E. E. Clothiaux, G. G. Mace, C. N. Long, J. C. Liljegren, “A 25-month database of stratus cloud properties generated from ground-based measurements at the Atmospheric Radiation Measurement Southern Great Plain Site,” J. Geophys. Res. 105, 4529–4539 (2000).
[CrossRef]

J. J. Carroll, B. W. Fitch, “Effects of solar elevation and cloudiness on snow albedo at the south pole,” J. Geophys. Res. 86C, 5271–5276 (1981).
[CrossRef]

T. C. Grenfell, S. G. Warren, P. C. Mullen, “Reflection of solar radiation by the Antarctic snow surface at ultraviolet, visible, and near-infrared wavelengths,” J. Geophys. Res. 99, 18669–18684 (1994).
[CrossRef]

S. G. Warren, R. E. Brandt, P. O. Hinton, “Effect of surface roughness on bidirectional reflectance of Antarctic snow,” J. Geophys. Res. 103, 25789–25807 (1998).
[CrossRef]

D. K. Perovich, “Light reflection from sea ice during the onset of melt,” J. Geophys. Res. 99, 3351–3359 (1994).
[CrossRef]

S. Manabe, R. J. Stouffer, “Sensitivity of a global climate model to an increase of CO2 concentration in the atmosphere,” J. Geophys. Res. 85, 5529–5554 (1980).
[CrossRef]

T. Aoki, T. Aoki, M. Fukabori, A. Hachikubo, Y. Tachibana, F. Nishio, “Effects of snow physical parameters on spectral albedo and bidirectional reflectance of snow surface,” J. Geophys. Res. 105D, 10219–10236 (2000).
[CrossRef]

J. Glaciol. (1)

T. C. Grenfell, G. Maykut, “The optical properties of ice and snow in the Arctic Basin,” J. Glaciol. 18, 445–463 (1977).

J. Quant. Spectrosc. Radiat. Transfer (4)

W. J. Wiscombe, “Extension of the doubling method to inhomogeneous sources,” J. Quant. Spectrosc. Radiat. Transfer 16, 477–489 (1976).
[CrossRef]

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiat. Transfer 16, 635–658 (1976).
[CrossRef]

M. I. Mishchenko, J. M. Dlugach, E. G. Yanovitskij, N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transfer 63, 409–432 (1999).
[CrossRef]

A. A. Kokhanovsky, A. Macke, “The dependence of the radiative characteristics of optically thick media on the shape of particles,” J. Quant. Spectrosc. Radiat. Transfer 63, 393–407.

Proc. R. Soc. London Ser. A (1)

I. P. Grant, G. E. Hunt, “Discrete space theory of radiative transfer: I. Fundamentals,” Proc. R. Soc. London Ser. A 313, 183–197 (1969).
[CrossRef]

Other (13)

R. Siegel, J. R. Howell, Thermal Radiation Heat Transfer, 2nd ed. (Hemisphere, Washington, D.C., 1981).

F. E. Nicodemus, J. C. Richmond, J. J. Hsia, I. W. Ginsberg, T. Limperis, “Geometrical considerations and nomenclature for reflectance,” (U.S. Department of Commerce, Washington, D.C., 1977).

W. Mendenhall, D. D. Wackerly, R. L. Scheaffer, Mathematical Statistics with Applications, 4th ed. (Duxbury, Belmont, Calif., 1990).

X. Zhou, “Optical remote sensing of snow on sea ice: ground measurements, satellite data analysis, and radiative transfer modeling,” Ph.D. dissertation (University of Alaska, Fairbanks, Alaska, 2002).

T. H. Painter, J. Dozier, “Measurements of the bidirectional reflectance of snow at fine spectral and angular resolution,” in Proceedings of the 70th Annual Western Snow Conference, available online at http://www.westernsnowconference.org/2002/PDF/2002PainterAndDozier.pdf .

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1988).

S. Li, “A model for the anisotropic reflectance of pure snow,” M.A. thesis (University of California at Santa Barbara, Santa Barbara, Calif., 1982).

J. E. Walsh, J. Curry, M. Fahnestock, M. C. Kennicutt, A. D. McGuire, W. B. Rossow, M. Steele, C. J. Vorosmarty, R. Wharton, Enhancing NASA’s Contribution to Polar Science (National Academics Press, Washington, D.C., 2001).

D. K. Perovich, “The optical properties of sea ice,” in Physics of Ice-Covered Seas, M. Lepparanta, ed. (Helsinki University Printing House, Helsinki, 1998), pp. 195–230.

H. J. Zwally, J. C. Comiso, C. L. Parkinson, W. J. Campbell, F. D. Carsey, P. Gloersen, “Antarctic sea ice, 1973–1976: satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1983).

C. L. Parkinson, J. C. Comiso, H. J. Zwally, D. J. Cavalieri, P. Gloersen, W. J. Campbell, “Arctic sea ice 1973–1976 from satellite passive microwave observations,” (National Aeronautics and Space Administration, Washington, D.C., 1987).

P. Gloersen, W. Campbell, D. J. Cavalieri, J. C. Comiso, C. L. Parkinson, H. J. Zwally, “Arctic and Antarctic sea ice, 1978–1987: satellite passive-microwave observations and analysis,” (National Aeronautics and Space Administration, Washington, D.C., 1992).

J. Croll, Climate and Time in Geologic Relations: A Theory of Secular Change of the Earth’s Climate (Isbister, London, 1875).

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Figures (13)

Fig. 1
Fig. 1

Reciprocity between the clear-sky direct beam spectral albedo and the hemispherical-directional spectral reflectance when the incident radiance is uniform.

Fig. 2
Fig. 2

(a) Snow model and (b) coupled cloud‐snow model used in the RT simulation.

Fig. 3
Fig. 3

Direct beam spectral albedo as a function of solar incidence angle at wavelength (a) 415, (b) 500, (c) 862, and (d) 2250 nm. The snow grain size is given as a parameter.

Fig. 4
Fig. 4

Direct beam spectral albedo as a function of solar incidence angle with snow grain sizes (a) 0.2 and (b) 1.0 mm. Wavelength is given as a parameter.

Fig. 5
Fig. 5

Representative BRF patterns, including patterns with vertical incidence for (a) r s = 0.2 mm and (b) r s = 1.0 mm at three wavelengths, patterns along the principal plane for (c) λ = 500 nm and r s = 0.2 mm and (d) λ = 500 nm and r s = 1 mm under four different solar incidence (inc) angles, and full BRF patterns for (e) λ = 500 nm for θ0 = 30° and r s = 1.0 mm and (f) θ0 = 65° and r s = 1.0 mm at six individual viewing zenith angles.

Fig. 6
Fig. 6

Representative R λ0; θ v ) patterns for (a) θ0 = 30°, (b) θ0 = 80°, (c) three-dimensional view for λ = 500 nm and r s = 1.0 mm, and (d) three-dimensional view for λ = 2250 nm and r s = 0.2 mm. In (a) and (b), the symbols are given with the numerical values for λ and the letters M and L for medium and large snow grain sizes.

Fig. 7
Fig. 7

Patterns of normalized Lλ i , ϕ i ) as a function of ϕ i with θ i (expressed as Zi in the figures) as a parameter at λ = 862 nm for (a) τ c = 5 and θ0c = 30°, (b) τ c = 5 and θ0c = 80°, (c) τ c = 10 and θ0c = 30°, (d) τ c = 20 and θ0c = 30°. The cloud droplet size is 8 µm and the snow grain size is 1.0 mm.

Fig. 8
Fig. 8

Patterns of normalized Lλ i ) as a function of θ i at (a) λ = 500, (b) λ = 862, (c) λ = 2250, (d) λ = 2250 nm for various sky conditions (τ c = 5–60, r c = 4–14 µm, and θ0c = 30°–80°) over snow pack with r s = 1.0 mm. The continuous curves represent mean values. The different colors represent different cloud optical thicknesses with blue for thin (τ c = 5), green for moderately thin (τ c = 10), and red for others (τ c = 15–60). Different shades of blue and green curves denote different θ0c s in the thin and moderately thin cloud cases. The exception is (d) where only a τ c range from 15 to 60 is involved and different colors represent different r c terms. inc, incidence; cr, cloud droplet radius.

Fig. 9
Fig. 9

Comparison of direct beam spectral albedos derived from direct simulation (dir simu) and from a reciprocal approach for various cases: (a) λ = 500 nm, r s = 0.2 mm; (b) λ = 610 nm, r s = 1.0 mm; (c) λ = 862 nm, r s = 0.2 mm; (d) λ = 862 nm, r s = 1.0 mm; (e) λ = 2250 nm, r s = 0.2 mm; (f) λ = 2250 nm, r s = 1.0 mm. The green curve provides the direct beam spectral albedo from direct simulation. The other curves are from the reciprocal approach with different sky conditions.

Fig. 10
Fig. 10

Reciprocally derived direct beam spectral albedos versus the directly simulated albedo for all experimental cases except for thin clouds (τ c = 5) with small θ0c (30°). The rectangles encircle points for extremely large θ0’s (86.1°, 88.4°, and 89.7°).

Fig. 11
Fig. 11

Errors of direct beam spectral albedos [Δα b0 = θ v )] for various cases: (a) λ = 500 nm, r s = 0.2 mm; (b) λ = 610 nm, r s = 1.0 mm; (c) λ = 862 nm, r s = 0.2 mm; (d) λ = 862 nm, r s = 1.0 mm; (e) λ = 2250 nm, r s = 0.2 mm; (f) λ = 2250 nm, r s = 1.0 mm.

Fig. 12
Fig. 12

rms of errors under selected sky conditions for (a) the full θ0 range and (b) reduced θ0 range (θ0 ≤ 83°). L, large; S, small; X, extra large; WV, wavelength; r, snow grain radius.

Fig. 13
Fig. 13

Details and statistics of Δα b0 = θ v ) patterns for τ c ≥ 10 for (a) r s = 0.2 mm, (b) r s = 1.0 mm, (c) maximum deviation for both r s ’s, (d) maximum relative deviation for both r s ’s. Wavelength is given as a parameter.

Tables (1)

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Table 1 Specifications of Snow and Cloud‐Snow Coupled Models Used in the Simulation

Equations (11)

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Rλθv, ϕv=αb,λθ0, ϕ0, for θv=θ0, ϕv=ϕ0
Rλθv, ϕv=π  Lλθi, ϕifr,λθi, ϕi; θv, ϕvcos θi sin θidθidϕiEdif,λ,
Edif,λ= Lλθi, ϕicos θi sin θidθidϕi.
Rλθv=Rλ¯θv=02π Rλθv, ϕvdϕv2π =π2πEdif,λ  Lλθi, ϕifr,λθi, ϕi; θv, ϕv×cos θi sin θidθidϕidϕv.
Rλθi; θv=π 02π fr,λθi, ϕi; θv, ϕvdϕv-ϕi2π=π 02π fr,λθi, ϕi; θv, ϕvdϕv2π=π 02π fr,λθv, ϕv; θi, ϕidϕi2π=Rλθv; θi.
Lλθi=02π Lλθi, ϕidϕi2π
αb,λθ0=θv= fr,λθv, ϕv; θi, ϕi×cos θi sin θidθidϕi.
Δαb,λθ0=θv=Rλθv-αb,λθ0=θv =2πEdif,λ  LλθiRλθi; θvcos θi sin θidθi- fr,λθv, ϕv; θi, ϕicos θi sin θidθidϕi, =2πEdif,λ  LλθiRλθv; θicos θi sin θidθi-2  Rλθv; θicos θi sin θidθi.
Δαb,λθ0=θv=2πEdif,λ0π/2Lλ¯+ΔLλθi×Rλθv; θicos θi sin θidθi-2 0π/2 Rλθv; θicos θi sin θidθi =2πEdif,λ0π/2 ΔLλθiRλθv; θicosθi sin θidθi =2πEdif,λ0π/2 ΔLλθiΔRλθv; θicosθi sin θidθi+2παb,λθvEdif,λ0π/2 ΔLλθicosθi sin θidθi.
0π/2 ΔLλθicos θi sin θidθi =0π/2 Lλθicos θi sin θidθi-0π/2Lλ¯ cos θi sin θidθi =Edif,λ2π-Edif,λ2π=0.
Δαb,λθ0=θv=2πEdif,λ0π2 ΔLλθiΔRλθv; θicos θi sin θidθi.

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