Abstract

We compare Monte Carlo (MC) and discrete-ordinate radiative-transfer (DISORT) simulations of irradiances in a one-dimensional coupled atmosphere-ocean (CAO) system consisting of horizontal plane-parallel layers. The two models have precisely the same physical basis, including coupling between the atmosphere and the ocean, and we use precisely the same atmospheric and oceanic input parameters for both codes. For a plane atmosphere-ocean interface we find agreement between irradiances obtained with the two codes to within 1%, both in the atmosphere and the ocean. Our tests cover case 1 water, scattering by density fluctuations both in the atmosphere and in the ocean, and scattering by particulate matter represented by a one-parameter Henyey-Greenstein (HG) scattering phase function. The CAO-MC code has an advantage over the CAO-DISORT code in that it can handle surface waves on the atmosphere-ocean interface, but the CAO-DISORT code is computationally much faster. Therefore we use CAO-MC simulations to study the influence of ocean surface waves and propose a way to correct the results of the CAO-DISORT code so as to obtain fast and accurate underwater irradiances in the presence of surface waves.

© 2003 Optical Society of America

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  3. K. Stamnes, W. Li, B. Yan, A. Barnard, W. S. Pegau, J. J. Stamnes, “Accurate and self-consistent ocean color algorithm: simultaneous retrieval of aerosol optical properties and chlorophyll concentrations,” Appl. Opt. 49, 939–951 (2003).
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  5. S. Jiang, K. Stamnes, “Enhancement of the downward solar irradiance across the atmosphere-sea ice interface,” J. Geophys. Res., submitted for publication.
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    [CrossRef]
  7. K. Stamnes, S. C. Tsay, W. J. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
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    [CrossRef] [PubMed]
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  15. G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to the Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2003

K. Stamnes, W. Li, B. Yan, A. Barnard, W. S. Pegau, J. J. Stamnes, “Accurate and self-consistent ocean color algorithm: simultaneous retrieval of aerosol optical properties and chlorophyll concentrations,” Appl. Opt. 49, 939–951 (2003).
[CrossRef]

B. Yan, K. Stamnes, “Fast yet accurate computation of the complete radiance distribution in the coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 76, 207–223 (2003).
[CrossRef]

2001

1999

Z. Jin, J. J. Simpson, “Bidirectional anisotropic reflectance of snow and sea ice in AVHRR channel 1 and 2 spectral regions—Part I: theoretical analysis,” IEEE Trans. Geosci. Remote Sensing 37, 543–554 (1999).
[CrossRef]

1998

1994

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

Z. Jin, K. Stamnes, W. F. Weeks, S. C. Tsay, “The effect of sea ice on the solar energy budget in the atmosphere-sea ice-ocean system: a model study,” Geophys. Res. 99, 25281–25294 (1994).
[CrossRef]

1993

1991

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[CrossRef]

1989

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to the Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

1988

1983

1975

1974

1973

G. Kattawar, G. N. Plass, J. A. Quinn, “Monte Carlo calculation of polarization of radiation in the earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

1972

G. Plass, G. Kattawar, “Monte Carlo calculations of radiative transfer in the Earth’s atmosphere ocean system: I. Flux in the atmosphere and ocean,” Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

D. G. Collins, W. G. Blattner, M. B. Wells, H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of the radiation field emerging from spherical shell atmospheres,” Appl. Opt. 11, 2684–2705 (1972).
[CrossRef] [PubMed]

1969

1968

G. N. Plass, G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. 7, 669–704 (1968).
[CrossRef]

1954

1941

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

Adams, C.

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to the Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

Barnard, A.

K. Stamnes, W. Li, B. Yan, A. Barnard, W. S. Pegau, J. J. Stamnes, “Accurate and self-consistent ocean color algorithm: simultaneous retrieval of aerosol optical properties and chlorophyll concentrations,” Appl. Opt. 49, 939–951 (2003).
[CrossRef]

Blattner, W.

Blattner, W. G.

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge University, Cambridge, UK, 1980).

Bosch, J. J. T.

Brown, O.

Chen, B.

Collins, D.

Collins, D. G.

Cox, C.

Elterman, L.

L. Elterman, “UV, visible, and IR attenuation for altitude to 50 km,” AFCRL-68-0153 (Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).

Erga, S. R.

Fenn, R. W.

E. P. Shettle, R. W. Fenn, Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on their Optical Properties (Air Force Geophysics Laboratory, Hanscomb AFB, Mass. 01731, 1979).

Ferwerda, H. A.

Frette, Ø.

Gentili, B.

Gordon, H.

Gordon, H. R.

Greenstein, J. L.

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

Groenhuis, R. A. J.

Guinn, J.

Henyey, L. C.

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

Horak, H.

Horak, H. G.

Jacobs, M.

Jayaweera, K.

Jiang, S.

S. Jiang, K. Stamnes, “Enhancement of the downward solar irradiance across the atmosphere-sea ice interface,” J. Geophys. Res., submitted for publication.

Jin, Z.

Z. Jin, J. J. Simpson, “Bidirectional anisotropic reflectance of snow and sea ice in AVHRR channel 1 and 2 spectral regions—Part I: theoretical analysis,” IEEE Trans. Geosci. Remote Sensing 37, 543–554 (1999).
[CrossRef]

Z. Jin, K. Stamnes, W. F. Weeks, S. C. Tsay, “The effect of sea ice on the solar energy budget in the atmosphere-sea ice-ocean system: a model study,” Geophys. Res. 99, 25281–25294 (1994).
[CrossRef]

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

Kattawar, G.

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to the Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

G. Plass, G. Kattawar, J. Guinn, “Radiative transfer in the Earth’s atmosphere and ocean: influence of ocean waves,” Appl. Opt. 14, 1924–1936 (1975).
[CrossRef] [PubMed]

G. Kattawar, G. N. Plass, J. A. Quinn, “Monte Carlo calculation of polarization of radiation in the earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

G. Plass, G. Kattawar, “Monte Carlo calculations of radiative transfer in the Earth’s atmosphere ocean system: I. Flux in the atmosphere and ocean,” Phys. Oceanogr. 2, 139–145 (1972).
[CrossRef]

G. Plass, G. Kattawar, “Radiative transfer in an atmosphere-ocean system,” Appl. Opt. 8, 455–466 (1969).
[CrossRef]

Kattawar, G. W.

Li, W.

K. Stamnes, W. Li, B. Yan, A. Barnard, W. S. Pegau, J. J. Stamnes, “Accurate and self-consistent ocean color algorithm: simultaneous retrieval of aerosol optical properties and chlorophyll concentrations,” Appl. Opt. 49, 939–951 (2003).
[CrossRef]

Mobley, C.

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

Mobley, C. D.

Morel, A.

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

A. Morel, “Light and marine photosynthesis: a model with geochemical and climatological implications,” Prog. Oceanogr. 26, 263–306 (1991).
[CrossRef]

A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov, E. S. Nielsen, eds. (Academic, New York, 1974), pp. 1–24.

Munk, W.

Pegau, W. S.

K. Stamnes, W. Li, B. Yan, A. Barnard, W. S. Pegau, J. J. Stamnes, “Accurate and self-consistent ocean color algorithm: simultaneous retrieval of aerosol optical properties and chlorophyll concentrations,” Appl. Opt. 49, 939–951 (2003).
[CrossRef]

Plass, G.

Plass, G. N.

G. Kattawar, G. N. Plass, J. A. Quinn, “Monte Carlo calculation of polarization of radiation in the earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

G. N. Plass, G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. 7, 669–704 (1968).
[CrossRef]

Quinn, J. A.

G. Kattawar, G. N. Plass, J. A. Quinn, “Monte Carlo calculation of polarization of radiation in the earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

Reinersman, P.

Shettle, E. P.

E. P. Shettle, R. W. Fenn, Models for the Aerosols of the Lower Atmosphere and the Effects of Humidity Variations on their Optical Properties (Air Force Geophysics Laboratory, Hanscomb AFB, Mass. 01731, 1979).

Simpson, J. J.

Z. Jin, J. J. Simpson, “Bidirectional anisotropic reflectance of snow and sea ice in AVHRR channel 1 and 2 spectral regions—Part I: theoretical analysis,” IEEE Trans. Geosci. Remote Sensing 37, 543–554 (1999).
[CrossRef]

Stamnes, J. J.

Stamnes, K.

K. Stamnes, W. Li, B. Yan, A. Barnard, W. S. Pegau, J. J. Stamnes, “Accurate and self-consistent ocean color algorithm: simultaneous retrieval of aerosol optical properties and chlorophyll concentrations,” Appl. Opt. 49, 939–951 (2003).
[CrossRef]

B. Yan, K. Stamnes, “Fast yet accurate computation of the complete radiance distribution in the coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 76, 207–223 (2003).
[CrossRef]

Ø. Frette, S. R. Erga, J. J. Stamnes, K. Stamnes, “Optical remote sensing of waters with vertical structure,” Appl. Opt. 40, 1478–1487 (2001).
[CrossRef]

B. Chen, K. Stamnes, J. J. Stamnes, “Validity of diffusion approximation in bio-optical imaging,” Appl. Opt. 40, 6356–6366 (2001).
[CrossRef]

Ø. Frette, J. J. Stamnes, K. Stamnes, “Optical remote sensing of marine constituents in coastal waters: a feasibility study,” Appl. Opt. 37, 8318–8326 (1998).
[CrossRef]

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

Z. Jin, K. Stamnes, W. F. Weeks, S. C. Tsay, “The effect of sea ice on the solar energy budget in the atmosphere-sea ice-ocean system: a model study,” Geophys. Res. 99, 25281–25294 (1994).
[CrossRef]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

K. Stamnes, S. C. Tsay, W. J. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

S. Jiang, K. Stamnes, “Enhancement of the downward solar irradiance across the atmosphere-sea ice interface,” J. Geophys. Res., submitted for publication.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, Cambridge, UK, 1999).
[CrossRef]

Stavn, R.

Stavn, R. H.

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge University, Cambridge, UK, 1999).
[CrossRef]

Tsay, S. C.

Z. Jin, K. Stamnes, W. F. Weeks, S. C. Tsay, “The effect of sea ice on the solar energy budget in the atmosphere-sea ice-ocean system: a model study,” Geophys. Res. 99, 25281–25294 (1994).
[CrossRef]

K. Stamnes, S. C. Tsay, W. J. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering—Tables, Formulas and Applications (Academic, New York, 1980).

Weeks, W. F.

Z. Jin, K. Stamnes, W. F. Weeks, S. C. Tsay, “The effect of sea ice on the solar energy budget in the atmosphere-sea ice-ocean system: a model study,” Geophys. Res. 99, 25281–25294 (1994).
[CrossRef]

Weidemann, A.

Wells, M.

Wells, M. B.

Wiscombe, W. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge University, Cambridge, UK, 1980).

Yan, B.

B. Yan, K. Stamnes, “Fast yet accurate computation of the complete radiance distribution in the coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 76, 207–223 (2003).
[CrossRef]

K. Stamnes, W. Li, B. Yan, A. Barnard, W. S. Pegau, J. J. Stamnes, “Accurate and self-consistent ocean color algorithm: simultaneous retrieval of aerosol optical properties and chlorophyll concentrations,” Appl. Opt. 49, 939–951 (2003).
[CrossRef]

Appl. Opt.

Ø. Frette, J. J. Stamnes, K. Stamnes, “Optical remote sensing of marine constituents in coastal waters: a feasibility study,” Appl. Opt. 37, 8318–8326 (1998).
[CrossRef]

Ø. Frette, S. R. Erga, J. J. Stamnes, K. Stamnes, “Optical remote sensing of waters with vertical structure,” Appl. Opt. 40, 1478–1487 (2001).
[CrossRef]

K. Stamnes, W. Li, B. Yan, A. Barnard, W. S. Pegau, J. J. Stamnes, “Accurate and self-consistent ocean color algorithm: simultaneous retrieval of aerosol optical properties and chlorophyll concentrations,” Appl. Opt. 49, 939–951 (2003).
[CrossRef]

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

B. Chen, K. Stamnes, J. J. Stamnes, “Validity of diffusion approximation in bio-optical imaging,” Appl. Opt. 40, 6356–6366 (2001).
[CrossRef]

K. Stamnes, S. C. Tsay, W. J. Wiscombe, K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502–2509 (1988).
[CrossRef] [PubMed]

D. G. Collins, W. G. Blattner, M. B. Wells, H. G. Horak, “Backward Monte Carlo calculations of the polarization characteristics of the radiation field emerging from spherical shell atmospheres,” Appl. Opt. 11, 2684–2705 (1972).
[CrossRef] [PubMed]

G. N. Plass, G. W. Kattawar, “Monte Carlo calculations of light scattering from clouds,” Appl. Opt. 7, 669–704 (1968).
[CrossRef]

C. D. Mobley, B. Gentili, H. R. Gordon, Z. Jin, G. W. Kattawar, A. Morel, P. Reinersman, K. Stamnes, R. H. Stavn, “Comparison of numerical models for computing underwater light fields,” Appl. Opt. 32, 7484–7504 (1993).
[CrossRef] [PubMed]

H. Gordon, O. Brown, M. Jacobs, “Computed relationships between the inherent and apparent optical properties of a flat homogeneous ocean,” Appl. Opt. 14, 417–427 (1975).
[CrossRef] [PubMed]

G. Plass, G. Kattawar, J. Guinn, “Radiative transfer in the Earth’s atmosphere and ocean: influence of ocean waves,” Appl. Opt. 14, 1924–1936 (1975).
[CrossRef] [PubMed]

A. Morel, B. Gentili, “Diffuse reflectance of oceanic waters: its dependence on Sun angle as influenced by the molecular scattering contribution,” Appl. Opt. 30, 4427–4438 (1991).
[CrossRef] [PubMed]

W. Blattner, H. Horak, D. Collins, M. Wells, “Monte Carlo studies of the sky radiation at twilight,” Appl. Opt. 13, 534–547 (1974).
[CrossRef] [PubMed]

R. Stavn, A. Weidemann, “Optical modeling of clear oceanlight fields: Raman scattering effects,” Appl. Opt. 27, 4002–4011 (1988).
[CrossRef] [PubMed]

G. Plass, G. Kattawar, “Radiative transfer in an atmosphere-ocean system,” Appl. Opt. 8, 455–466 (1969).
[CrossRef]

R. A. J. Groenhuis, H. A. Ferwerda, J. J. T. Bosch, “Scattering and absorption of turbid materials determined from reflected measurements,” Appl. Opt. 22, 2456–2462 (1983).
[CrossRef] [PubMed]

Astrophys. J.

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

Geophys. Res.

Z. Jin, K. Stamnes, W. F. Weeks, S. C. Tsay, “The effect of sea ice on the solar energy budget in the atmosphere-sea ice-ocean system: a model study,” Geophys. Res. 99, 25281–25294 (1994).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing

Z. Jin, J. J. Simpson, “Bidirectional anisotropic reflectance of snow and sea ice in AVHRR channel 1 and 2 spectral regions—Part I: theoretical analysis,” IEEE Trans. Geosci. Remote Sensing 37, 543–554 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. Oceanogr.

G. Kattawar, G. N. Plass, J. A. Quinn, “Monte Carlo calculation of polarization of radiation in the earth’s atmosphere-ocean system,” J. Phys. Oceanogr. 3, 353–372 (1973).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

B. Yan, K. Stamnes, “Fast yet accurate computation of the complete radiance distribution in the coupled atmosphere-ocean system,” J. Quant. Spectrosc. Radiat. Transfer 76, 207–223 (2003).
[CrossRef]

Limnol. Oceanogr.

G. Kattawar, C. Adams, “Stokes vector calculations of the submarine light field in an atmosphere-ocean with scattering according to the Rayleigh phase matrix: effect of interface refractive index on radiance and polarization,” Limnol. Oceanogr. 34, 1453–1472 (1989).
[CrossRef]

C. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind-roughened surfaces,” Limnol. Oceanogr. 34, 1473–1483 (1989).
[CrossRef]

Phys. Oceanogr.

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Figures (15)

Fig. 1
Fig. 1

Photon travels from the point, start, to the point, absorption. The intersection points between the layers and the points of refraction, scattering, and absorption are marked by circles.

Fig. 2
Fig. 2

Irradiances computed by the CAO-DISORT code (solid and dashed curves) and the CAO-MC code (circles and asterisks) for an atmosphere with IOPs as given in Table 3 and an ocean with a chlorophyll concentration of 0.02 mg/m3. Assumptions: isotropic scattering in the atmosphere and no Rayleigh scattering in the ocean. Upper panels, downward and upward direct and diffuse irradiances in the atmosphere; lower panels, downward direct and diffuse and upward diffuse irradiances in the ocean.

Fig. 3
Fig. 3

Percentage deviation as defined in Eq. (9) between results obtained with the CAO-MC and CAO-DISORT codes for irradiances obtained with the same atmosphere as in Fig. 2. Assumptions, isotropic scattering in the atmosphere and no Rayleigh scattering in the ocean. Curves marked with stars, a chlorophyll concentration of 1 mg/m3 and an asymmetry parameter g of 0.90; curves marked with crosses, squares, circles, and diamonds, a chlorophyll concentration of 10 mg/m3 and asymmetry parameters g of 0.80, 0.85, 0.90, and 0.95, respectively. Left column, direct beams; right columns, diffuse light. Upper row, downward irradiances in the atmosphere; second row, downward irradiances in the ocean; third row, upward irradiances in the atmosphere.

Fig. 4
Fig. 4

Percentage deviation [as defined in Eq. (9) with E DISORT - E MC replaced by |E DISORT - E MC| between upward diffuse irradiances in the atmosphere for the same case as in Fig. 2 obtained with the CAO-MC and the CAO-DISORT codes. Note that the deviation is reduced as the number of streams in the CAO-DISORT code is increased.

Fig. 5
Fig. 5

Irradiances in the atmosphere and the ocean versus optical depth obtained with CAO-MC and CAO-DISORT codes. The model includes Rayleigh scattering from molecules and particulate scattering from aerosols in the atmosphere as well as Rayleigh scattering from pure water in the ocean. Upper left panel, downward direct E d,dir and diffuse E d,diff irradiances; upper right panel, downward total scalar E 0d irradiances; lower left panel, upward total E u irradiances; lower right panel, upward total scalar E 0u irradiances. The horizontal lines between optical depths of 0 and 1 indicate the atmosphere-ocean interface.

Fig. 6
Fig. 6

Percentage deviation [as defined in Eq. (9)] between results with the CAO-MC and the CAO-DISORT codes. The horizontal lines between optical depths of 0 and 1 indicate the atmosphere-ocean interface.

Fig. 7
Fig. 7

Irradiances in the atmosphere and in the ocean versus optical depth obtained with the CAO-MC and CAO-DISORT codes. The models include Rayleigh scattering from molecules and particulate scattering from aerosols in the atmosphere as well as Rayleigh scattering from pure water and particulate scattering from chlorophyll in the three uppermost layers in the ocean. Upper left panel, downward direct E d,dir and diffuse E d,diff irradiances; upper right panel, downward total scalar E 0d irradiances; lower left panel, upward total E u irradiances; lower right panel, upward total scalar E 0u irradiances. The horizontal lines between optical depths of 0 and 1 indicate the atmosphere-ocean interface.

Fig. 8
Fig. 8

Percentage deviation [as defined in Eq. (9)] between results with the CAO-MC and the CAO-DISORT codes. The horizontal lines between optical depths of 0 and 1 indicate the atmosphere-ocean interface.

Fig. 9
Fig. 9

Normal vector of a rough sea surface.

Fig. 10
Fig. 10

Right, photons refracted through a rough ocean surface in the CAO-MC code and spread into a cone. The dashed arrows show the average direction of the refracted photons. Left, in the CAO-DISORT code the water surface is plane. When the refractive index of the ocean is changed from n 2 to n 2′ = χn 2, the direction of the refracted beam is changed from a solid arrow to a dashed arrow, the latter being parallel to the average direction of the refracted photons in the CAO-MC code with surface waves present.

Fig. 11
Fig. 11

Percentage deviations between irradiances obtained with the CAO-MC and CAO-DISORT codes. The CAO-MC code was run with a wind speed of 12 ms-1. Solid curves, unchanged CAO-DISORT code; solid curves with dots, refractive index of the CAO-DISORT code modified in accordance with Eq. (25); left column, direct beams; right column, diffuse light; upper row, irradiances in the downward direction in the atmosphere; second row, irradiances in the downward direction in the ocean; third row, irradiances in the upward direction in the atmosphere; bottom row, irradiance in the upward direction in the ocean.

Fig. 12
Fig. 12

Modification factor χ as a function of wind speed V and zenith angle θ0: left, zenith angle fixed at θ0 = 45°; right, wind speed fixed at V = 6 ms-1.

Fig. 13
Fig. 13

Approximations for the average values θ̅ t of the angle of refraction in the case of a rough atmosphere-ocean interface. Here θ̅ t is approximated by a second-degree polynomial of two variables, wind speed V and solar zenith angle θ0. Left, true and approximate values when the wind speed is fixed at V = 6 ms-1; right, true and approximate values when the zenith angle is fixed at θ0 = 45°.

Fig. 14
Fig. 14

Approximations for the average values θ̅ r of the angle of reflection in the case of a rough atmosphere-ocean interface. Here θ̅ r is approximated by second-degree polynomials of two variables, wind speed V and solar zenith angle θ0. Left, true and approximate values when the wind speed is fixed at V = 6 ms-1; dashed curve, Eq. (27); solid curve, Eq. (28); right, true and approximate values when the zenith angle is fixed at θ0 = 45°.

Fig. 15
Fig. 15

K d (z) and R(z) versus depth for the CAO system. The solar zenith angle θ0 was 0° for the solid curve and triangles and 80° for the dashed curve and squares. The atmosphere-ocean interface was plane for the solid and dashed curves but rough due to a wind speed of 7.5 ms-1 for the triangles and squares.

Tables (4)

Tables Icon

Table 1 Reflection, Absorption, and Transmission through a Homogeneous Single-Layer Slab

Tables Icon

Table 2 Overview over Test Cases for the CAO Systema

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Table 3 Inherent Optical Properties of the Atmosphere Used for the Computations in Fig. 2 a

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Table 4 Inherent Optical Properties of the Atmosphere Used for the Computations in Figs. 5 8

Equations (35)

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η=bRaybPart+bRay=bRayb,
cos Θ= 1+g22g- 1-g222g2gρ+1-g2.
0Θ02π pRayΘsinΘdϕdΘ=ρ
pcos Θ3+3cos Θ+3+p1-2ρ=0,
= 12sin2θi-θtsin2θi+θt+ tan2θi-θttan2θi+θt,
Lsolτ, u, ϕ=Fsδu-μ0δϕ-ϕ0exp-τ/μ0.
Esolτ=μ0Fs exp-τ/μ0, Ed,diffτ=2π 01dμμLd,diffavτ, μ,
Eu,diffτ=2π 01dμμLu,diffavτ, μ,
Esolτ=μ0Fs exp-τa/μ0-τ/μt, Ed,diffτ=2π 01dμμLd,diffavτ, μ,
Eu,diffτ=2π 01dμμLu,diffavτ, μ,
ΔE= EDISORT-EMCEDISORT×100%
pPartcos Θ= 1-g24π1+g2-2g cos Θ3/2,
pRayΘ= 34π3+p1+p cos2 Θ
patmΘ= bRaybatm pRayΘ+ bPartbatm pPartΘ,
pocnΘ= bwbocn pwΘ+ bchlbocn pchlΘ,
aocnλ=awλ+0.06AλChl0.65+ay440exp-0.014λ-440
ay440=0.2aw440+0.06A440Chl0.65,
bocnλ=bwλ+550/λ0.3Chl0.62,
nx=cos α sin β, ny=sin α sin β, nz=cos β,
α=2πρ1,
pβ= 2σ2exp-tan2 βσ2tan β sec2 β,
σ=0.003+0.00512V1/2
0π/2 pβdβ=0exp-udu=1
ρ2=Pu=0uexp-xdx=1-exp-u.
u=-ln1-ρ2= tan2 βσ2.
β=tan-1-σ2 ln1-ρ21/2=tan-1-σ2 ln ρ21/2,
n2=χn2; χ= n1 sin θ0n2 sin θ¯t,
χV, θ0a1V2+a2θ02+a3V+a4θ0+a5,
θ¯t-9.722×10-3V2-2.148×10-3θ02+1.882×10-3Vθ0+0.2694V+0.7993θ0+0.6769 for θ00, 80°,
θ¯r-8.050×10-3V2+1.951×10-2θ02-4.819×10-2Vθ02+1.712V-6.661×10-2θ0+8.516 for θ00, 35°,
θ¯r3.988×10-2V2+2.094×10-3θ02-6.937×10-3Vθ0-0.5372V+0.7655θ0+2.582 for θ035°, 80°,
Ed=Ediexp-Kdz,
Kd=- 1zlnEdEdi,
Kdzi=- 1zi+1-zilnEdzi+1Edzi.
R=Eu/Ed.

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