Abstract

Results of an extensive validation study of the new radiative transfer code SHARM-3D are described. The code is designed for modeling of unpolarized monochromatic radiative transfer in the visible and near-IR spectra in the laterally uniform atmosphere over an arbitrarily inhomogeneous anisotropic surface. The surface boundary condition is periodic. The algorithm is based on an exact solution derived with the Green’s function method. Several parameterizations were introduced into the algorithm to achieve superior performance. As a result, SHARM-3D is 2–3 orders of magnitude faster than the rigorous code SHDOM. It can model radiances over large surface scenes for a number of incidence-view geometries simultaneously. Extensive comparisons against SHDOM indicate that SHARM-3D has an average accuracy of better than 1%, which along with the high speed of calculations makes it a unique tool for remote-sensing applications in land surface and related atmospheric radiation studies.

© 2002 Optical Society of America

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References

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  1. A. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt. 40, 3495–3501 (2001).
    [CrossRef]
  2. A. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. II. Spatially heterogeneous anisotropic surface,” Appl. Opt. 41, 5600–5606 (2002).
    [CrossRef] [PubMed]
  3. K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
    [CrossRef]
  4. K. Stamnes, S. C. Tsay, W. 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]
  5. E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
    [CrossRef]
  6. H. Rahman, B. Pinty, M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model. 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data,” J. Geophys. Res. 98, 20,791–20,801 (1993).
    [CrossRef]
  7. L. Elterman, “UV, visible and IR attenuation for altitudes to 50 km,” Environmental Research Paper NTIS-AD 671933 (U.S. Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).
  8. T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 60, 393–404 (1999).
    [CrossRef]
  9. V. Kourganoff, Basic Methods in Transfer Problems, Radiative Equilibrium and Neutron Diffusion (Dover, New York, 1963).
  10. A. H. Karp, “Computing the angular dependence of the radiation of a planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 25, 403–412 (1981).
    [CrossRef]
  11. T. Nakajima, M. Tanaka, “Algorithm for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
    [CrossRef]
  12. R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
    [CrossRef]
  13. I. Laszlo, National Oceanic and Atmospheric Administration, and W. Wiscombe, NASA Goddard Space Flight Center (personal communications, 2001).
  14. A. Marshak, A. Davis, R. Cahalan, W. Wiscombe, “Bounded cascades as nonstationary multifractals,” Phys. Rev. E 49, 55–67 (1994).
    [CrossRef]
  15. A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
    [CrossRef]
  16. O. Engelsen, B. Pinty, M. M. Verstraete, J. V. Martonchik, “Parametric bidirectional reflectance factor models: Evaluation, improvements and applications,” European Report 16426 EN, (Space Application Institute, Ispra, Italy, 1996).
  17. D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

2002

2001

A. Lyapustin, Yu. Knyazikhin, “Green’s function method in the radiative transfer problem. I. Homogeneous non-Lambertian surface,” Appl. Opt. 40, 3495–3501 (2001).
[CrossRef]

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

1999

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 60, 393–404 (1999).
[CrossRef]

1998

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

1997

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

1994

A. Marshak, A. Davis, R. Cahalan, W. Wiscombe, “Bounded cascades as nonstationary multifractals,” Phys. Rev. E 49, 55–67 (1994).
[CrossRef]

1993

R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
[CrossRef]

H. Rahman, B. Pinty, M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model. 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data,” J. Geophys. Res. 98, 20,791–20,801 (1993).
[CrossRef]

1988

K. Stamnes, S. C. Tsay, W. 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]

T. Nakajima, M. Tanaka, “Algorithm for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

1981

A. H. Karp, “Computing the angular dependence of the radiation of a planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 25, 403–412 (1981).
[CrossRef]

Asrar, G.

R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
[CrossRef]

Borel, C.

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

Cahalan, R.

A. Marshak, A. Davis, R. Cahalan, W. Wiscombe, “Bounded cascades as nonstationary multifractals,” Phys. Rev. E 49, 55–67 (1994).
[CrossRef]

Davis, A.

A. Marshak, A. Davis, R. Cahalan, W. Wiscombe, “Bounded cascades as nonstationary multifractals,” Phys. Rev. E 49, 55–67 (1994).
[CrossRef]

Deuze, J. L.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

Diner, D. J.

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

Elterman, L.

L. Elterman, “UV, visible and IR attenuation for altitudes to 50 km,” Environmental Research Paper NTIS-AD 671933 (U.S. Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).

Engelsen, O.

O. Engelsen, B. Pinty, M. M. Verstraete, J. V. Martonchik, “Parametric bidirectional reflectance factor models: Evaluation, improvements and applications,” European Report 16426 EN, (Space Application Institute, Ispra, Italy, 1996).

Evans, K. F.

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

Gerstl, S. A. W.

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

Gordon, H. R.

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

Herman, M.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

Jayaweera, K.

Karp, A. H.

A. H. Karp, “Computing the angular dependence of the radiation of a planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 25, 403–412 (1981).
[CrossRef]

Knyazikhin, Yu.

Kourganoff, V.

V. Kourganoff, Basic Methods in Transfer Problems, Radiative Equilibrium and Neutron Diffusion (Dover, New York, 1963).

Laszlo, I.

I. Laszlo, National Oceanic and Atmospheric Administration, and W. Wiscombe, NASA Goddard Space Flight Center (personal communications, 2001).

Lyapustin, A.

Lyapustin, A. I.

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 60, 393–404 (1999).
[CrossRef]

Marshak, A.

A. Marshak, A. Davis, R. Cahalan, W. Wiscombe, “Bounded cascades as nonstationary multifractals,” Phys. Rev. E 49, 55–67 (1994).
[CrossRef]

Martonchik, J. V.

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

O. Engelsen, B. Pinty, M. M. Verstraete, J. V. Martonchik, “Parametric bidirectional reflectance factor models: Evaluation, improvements and applications,” European Report 16426 EN, (Space Application Institute, Ispra, Italy, 1996).

Mocrette, J.-J.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

Muldashev, T. Z.

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 60, 393–404 (1999).
[CrossRef]

Myneni, R.

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

Myneni, R. B.

R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
[CrossRef]

Nakajima, T.

T. Nakajima, M. Tanaka, “Algorithm for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

Pinty, B.

H. Rahman, B. Pinty, M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model. 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data,” J. Geophys. Res. 98, 20,791–20,801 (1993).
[CrossRef]

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

O. Engelsen, B. Pinty, M. M. Verstraete, J. V. Martonchik, “Parametric bidirectional reflectance factor models: Evaluation, improvements and applications,” European Report 16426 EN, (Space Application Institute, Ispra, Italy, 1996).

Rahman, H.

H. Rahman, B. Pinty, M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model. 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data,” J. Geophys. Res. 98, 20,791–20,801 (1993).
[CrossRef]

Stamnes, K.

Sultangazin, U. M.

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 60, 393–404 (1999).
[CrossRef]

Tanaka, M.

T. Nakajima, M. Tanaka, “Algorithm for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

Tanre, D.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

Tsay, S. C.

Vermote, E. F.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

Verstraete, M. M.

H. Rahman, B. Pinty, M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model. 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data,” J. Geophys. Res. 98, 20,791–20,801 (1993).
[CrossRef]

O. Engelsen, B. Pinty, M. M. Verstraete, J. V. Martonchik, “Parametric bidirectional reflectance factor models: Evaluation, improvements and applications,” European Report 16426 EN, (Space Application Institute, Ispra, Italy, 1996).

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

Wiscombe, W.

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

E. F. Vermote, D. Tanre, J. L. Deuze, M. Herman, J.-J. Mocrette, “Second simulation of the satellite signal in the solar spectrum, 6S: an overview,” IEEE Trans. Geosci. Remote Sens. 35, 675–686 (1997).
[CrossRef]

J. Atmos. Sci.

K. F. Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55, 429–446 (1998).
[CrossRef]

J. Geophys. Res.

H. Rahman, B. Pinty, M. M. Verstraete, “Coupled surface-atmosphere reflectance (CSAR) model. 2. Semiempirical surface model usable with NOAA advanced very high resolution radiometer data,” J. Geophys. Res. 98, 20,791–20,801 (1993).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

T. Z. Muldashev, A. I. Lyapustin, U. M. Sultangazin, “Spherical harmonics method in the problem of radiative transfer in the atmosphere–surface system,” J. Quant. Spectrosc. Radiat. Transfer 60, 393–404 (1999).
[CrossRef]

A. H. Karp, “Computing the angular dependence of the radiation of a planetary atmosphere,” J. Quant. Spectrosc. Radiat. Transfer 25, 403–412 (1981).
[CrossRef]

T. Nakajima, M. Tanaka, “Algorithm for radiative intensity calculations in moderately thick atmospheres using a truncation approximation,” J. Quant. Spectrosc. Radiat. Transfer 40, 51–69 (1988).
[CrossRef]

R. B. Myneni, G. Asrar, “Radiative transfer in three-dimensional atmosphere-vegetation media,” J. Quant. Spectrosc. Radiat. Transfer 49, 585–598 (1993).
[CrossRef]

A. I. Lyapustin, T. Z. Muldashev, “Solution for atmospheric optical transfer function using spherical harmonics method,” J. Quant. Spectrosc. Radiat. Transfer 68, 43–56 (2001).
[CrossRef]

Phys. Rev. E

A. Marshak, A. Davis, R. Cahalan, W. Wiscombe, “Bounded cascades as nonstationary multifractals,” Phys. Rev. E 49, 55–67 (1994).
[CrossRef]

Other

O. Engelsen, B. Pinty, M. M. Verstraete, J. V. Martonchik, “Parametric bidirectional reflectance factor models: Evaluation, improvements and applications,” European Report 16426 EN, (Space Application Institute, Ispra, Italy, 1996).

D. J. Diner, J. V. Martonchik, C. Borel, S. A. W. Gerstl, H. R. Gordon, Yu. Knyazikhin, R. Myneni, B. Pinty, M. M. Verstraete, “MISR level 2 surface retrieval algorithm theoretical basis,” NASA EOS-MISR Doc., JPL D-11401, Rev. D, NASA JPL1999).

I. Laszlo, National Oceanic and Atmospheric Administration, and W. Wiscombe, NASA Goddard Space Flight Center (personal communications, 2001).

V. Kourganoff, Basic Methods in Transfer Problems, Radiative Equilibrium and Neutron Diffusion (Dover, New York, 1963).

L. Elterman, “UV, visible and IR attenuation for altitudes to 50 km,” Environmental Research Paper NTIS-AD 671933 (U.S. Air Force Cambridge Research Laboratory, Bedford, Mass., 1968).

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Figures (9)

Fig. 1
Fig. 1

Aerosol scattering function and relative error of its Legendre expansion χ L (γ) for several orders L. The errors in the forward-scattering and backscattering directions are e(0°) = -65.1, -45.2, -27.8, -15.3, 4.5%; e(180°) = -101.9, -53.3, -20.7, -6.7, -0.656% for L = 32, 64, 128, 256, 900, respectively.

Fig. 2
Fig. 2

Comparison of path radiance: left, SHARM-DISORT; middle, SHARM-6S; and right, SHARM–SHDOM. Results are presented for (a.1)–(c.1) NIR Clear and (a.2)–(c.2) NIR Hazy atmospheric models. (a.3), (b.3) The Molec. Scatt. model; (c.3) the Vis. Clear model. Results are shown for 21 zenith view angles; 5 solar zenith angles; and 2 azimuthal angles, 0° (solid curves, filled symbols) and 180° (dashed curves, open symbols).

Fig. 3
Fig. 3

Comparison of TOA radiance over an anisotropic surface with BRDF of grasses in the NIR. Top, NIR Clear atmospheric model; and bottom, NIR Hazy conditions. For other details see the caption of Fig. 2.

Fig. 4
Fig. 4

SHDOM – SHARM-3D relative difference at VZA = 45° and SZA = 60° for the circle model of surface albedo. The albedo of the circle is 0.02, the background albedo is 0.4, the circle’s radius is 1.5 km, and the image resolution is 0.1 km. The xy coordinates are given in pixel numbers. The relative azimuth φ = 0° corresponds to the observer’s position at the east; the Sun is at the west. Two horizontal lines show the transsects for which a detailed radiance intercomparison is shown in Fig. 5.

Fig. 5
Fig. 5

Radiance along the transsects y = 3.2 km and y = 1.8 km (see Fig. 4) for NIR Clear (top) and NIR Hazy (bottom) conditions. Solid curves, SHARM-3D radiance; nearest dashed black curves, SHDOM data; horizontal lines, 1-D solution over the dark circle and bright background. Results are shown for SZA = 60°, φ = 0°, and two view angles, 45° (squares) and 0° (circles).

Fig. 6
Fig. 6

Inhomogeneous surface albedo rendered with the bounded cascade model (left), and a relative difference (SHDOM – SHARM)/SHDOM × 100% for NIR Clear and NIR Hazy conditions (right) for VZA = 45°, SZA = 60°, and φ = 0°.

Fig. 7
Fig. 7

Left, Landsat-7 normalized radiance in band 4 over the Oklahoma site on 4 April 2000. The square outlined in white shows the image area used to generate surface reflectance. Middle, SHARM-3D simulated radiance in NIR Clear conditions for the contour area. Right, the relative difference of SHARM-3D and IPA solutions, respectively, with SHDOM. The xy coordinates are given in pixel numbers.

Fig. 8
Fig. 8

Image of the surface albedo at SZA = 30° in NIR Clear conditions. The spatial distribution of the surface reflectance was generated by the cascade model (parameter ρ), and the BRDF shape was generated randomly within the limits measured over land or predicted by the BRDF models. The scale of the axes is in kilometers.

Fig. 9
Fig. 9

Histograms of the relative differences SHARM-3D – SHDOM (left, NIR Clear) and IPA – SHDOM (center and right for clear and hazy conditions, respectively). The histograms were obtained for a relatively high-contrast image (Fig. 8) at a 1-km resolution of the image, SZA = 30°, and a relative azimuth 0°. The data are presented for view zenith angles of 0° (top) and 63.4° (bottom).

Tables (1)

Tables Icon

Table 1 Time (seconds) for Path Radiance and TOA Radiance Calculations with a Non-Lambertian Surface for 210 Anglesa for Several Orders of Solution

Metrics