Abstract

Scattering phase functions derived from measured (volume-scattering meter, VSM) volume-scattering functions (VSFs) from Crimean coastal waters were found to have systematic differences in angular structure from Fournier–Forand (FF) functions with equivalent backscattering ratios. Hydrolight simulations demonstrated that differences in the angular structure of the VSF could result in variations in modeled subsurface radiance reflectances of up to ±20%. Furthermore, differences between VSM and FF simulated reflectances were found to be nonlinear as a function of scattering and could not be explained with the single-scattering approximation. Additional radiance transfer modeling demonstrated that the contribution of multiple scattering to radiance reflectance increased exponentially from a minimum of 16% for pure water to a maximum of 94% for turbid waters. Monte Carlo simulations demonstrated that multiple forward-scattering events were the dominant contributors to the generation of radiance reflectance signals for turbid waters and that angular structures in the shape of the VSF at forward angles could have a significant influence in determining reflectance signals for turbid waters.

© 2006 Optical Society of America

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2006 (2)

M. Chami, E. Marken, J. J. Stamnes, G. A. Khomenko, and G. K. Korotaev, "Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles," J. Geophys. Res. 111, C05013, doi: (2006).
[CrossRef]

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, "Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment," Appl. Opt. 45, 3605-3619 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (1)

2003 (3)

2002 (4)

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

S. Johnson, "Moderate resolution imaging spectroradiometer (MODIS)," Technical Specification Sheet, RTSC MS 3/05 DTS00-0830 (Raytheon Space and Airborne Systems, Santa Barbara Remote Sensing, Calif., 2002).

C. D. Mobley, L. K. Sundman, and E. Boss, "Phase function effects on oceanic light fields," Appl. Opt. 41, 1035-1050 (2002).
[CrossRef] [PubMed]

S. Maritorena, D. A. Siegel, and A. R. Peterson, "Optimization of a semianalytical ocean color model for global-scale applications," Appl. Opt. 41, 2705-2714 (2002).
[CrossRef] [PubMed]

2001 (2)

1999 (1)

G. Fournier and M. Jonasz, "Computer-based underwater imaging analysis," in Airborne and In-Water Underwater Imaging, G. Gilbert, ed., Proc. SPIE 3761, 62-77 (1999).
[CrossRef]

1998 (4)

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, "Laboratory measurements of angular distributions of light scattered by phytoplankton and silt," Limnol. Oceanogr. 43, 1180-1197 (1998).
[CrossRef]

K. J. Voss, W. M. Balch, and K. A. Kilpatrick, "Scattering and attenuation properties of Emiliania huxleyi cells and their detached coccoliths," Limnol. Oceanogr. 43, 870-876 (1998).
[CrossRef]

International Ocean-Colour Coordinating Group (IOCCG), "Minimum requirements for an operational ocean colour sensor for the open ocean," Vol. 1 Reports of the International Ocean-Colour Coordinating Group (1998).

A. Morel and H. Loisel, "Apparent optical properties of oceanic water: dependence on the molecular scattering contribution," Appl. Opt. 37, 4765-4776 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

J. R. V. Zaneveld, "A theoretical derivation of the dependence of the remotely sensed reflectance of the ocean on the inherent optical properties," J. Geophys. Res. 100, 13135-13142 (1995).
[CrossRef]

1994 (2)

R. A. Barnes, W. L. Barnes, W. E. Esaias, and C. R. McClain, "Prelaunch acceptance report for the SeaWiFS radiometer," in SeaWiFS Technical Report Series, S. B. Hooker, E. R. Firestone, and J. G. Acker, eds., NASA Technical Memorandum 104566, 22 (1994).

G. Fournier and J. L. Forand, "Analytic phase function for ocean water," in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE 2258, 194-201 (1994).
[CrossRef]

1993 (1)

1990 (1)

W. W. Gregg and K. L. Carder, "A simple spectral solar irradiance model for cloudless maritime atmospheres," Limnol. Oceanogr. 35, 1657-1675 (1990).
[CrossRef]

1988 (1)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. D. 93, 10909-10924 (1988).
[CrossRef]

1982 (1)

1981 (1)

1958 (1)

1957 (1)

K. M. Case, "Transfer problems and the reciprocity principle," Rev. Mod. Phys. 29, 651-663 (1957).
[CrossRef]

Baker, K.

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. D. 93, 10909-10924 (1988).
[CrossRef]

Balch, W. M.

K. J. Voss, W. M. Balch, and K. A. Kilpatrick, "Scattering and attenuation properties of Emiliania huxleyi cells and their detached coccoliths," Limnol. Oceanogr. 43, 870-876 (1998).
[CrossRef]

Barnes, R. A.

R. A. Barnes, W. L. Barnes, W. E. Esaias, and C. R. McClain, "Prelaunch acceptance report for the SeaWiFS radiometer," in SeaWiFS Technical Report Series, S. B. Hooker, E. R. Firestone, and J. G. Acker, eds., NASA Technical Memorandum 104566, 22 (1994).

Barnes, W. L.

R. A. Barnes, W. L. Barnes, W. E. Esaias, and C. R. McClain, "Prelaunch acceptance report for the SeaWiFS radiometer," in SeaWiFS Technical Report Series, S. B. Hooker, E. R. Firestone, and J. G. Acker, eds., NASA Technical Memorandum 104566, 22 (1994).

Berseneva, G. A.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, "Optical properties of the particles in the Crimea coastal waters (Black Sea)," J. Geophys. Res. 110, C11020, doi: (2005).
[CrossRef]

Bézy, J.-L.

J.-L. Bézy, S. Delwart, and M. Rast, "MERIS-A new generation of ocean-colour sensor onboard Envisat," ESA Bulletin 103 (2000), http://www.esa.int/esapub/bulletin/bullet103/besy103.pdf.

Boss, E.

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. D. 93, 10909-10924 (1988).
[CrossRef]

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. D. 93, 10909-10924 (1988).
[CrossRef]

Carder, K. L.

Z. P. Lee, K. L. Carder, and K. P. Du, "Effects of molecular and particle scatterings on the model parameter for remote-sensing reflectance," Appl. Opt. 43, 4957-4964 (2004).
[CrossRef] [PubMed]

W. W. Gregg and K. L. Carder, "A simple spectral solar irradiance model for cloudless maritime atmospheres," Limnol. Oceanogr. 35, 1657-1675 (1990).
[CrossRef]

Case, K. M.

K. M. Case, "Transfer problems and the reciprocity principle," Rev. Mod. Phys. 29, 651-663 (1957).
[CrossRef]

Chami, M.

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, "Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment," Appl. Opt. 45, 3605-3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. Marken, J. J. Stamnes, G. A. Khomenko, and G. K. Korotaev, "Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles," J. Geophys. Res. 111, C05013, doi: (2006).
[CrossRef]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, "Optical properties of the particles in the Crimea coastal waters (Black Sea)," J. Geophys. Res. 110, C11020, doi: (2005).
[CrossRef]

M. Chami, R. Santer, and E. Dilligeard, "Radiative transfer model for the computation of radiance and polarization in an ocean-atmosphere system: polarization properties of suspended matter for remote sensing," Appl. Opt. 40, 2398-2416 (2001).
[CrossRef]

Charlton, F.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, "Laboratory measurements of angular distributions of light scattered by phytoplankton and silt," Limnol. Oceanogr. 43, 1180-1197 (1998).
[CrossRef]

Churilova, T. Y.

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, "Optical properties of the particles in the Crimea coastal waters (Black Sea)," J. Geophys. Res. 110, C11020, doi: (2005).
[CrossRef]

Clark, D. K.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. D. 93, 10909-10924 (1988).
[CrossRef]

Craig, S.

Cunningham, A.

de Haan, J. F.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, "Laboratory measurements of angular distributions of light scattered by phytoplankton and silt," Limnol. Oceanogr. 43, 1180-1197 (1998).
[CrossRef]

Dekker, A. G.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, "Laboratory measurements of angular distributions of light scattered by phytoplankton and silt," Limnol. Oceanogr. 43, 1180-1197 (1998).
[CrossRef]

Delwart, S.

J.-L. Bézy, S. Delwart, and M. Rast, "MERIS-A new generation of ocean-colour sensor onboard Envisat," ESA Bulletin 103 (2000), http://www.esa.int/esapub/bulletin/bullet103/besy103.pdf.

Dilligeard, E.

Du, K. P.

Esaias, W. E.

R. A. Barnes, W. L. Barnes, W. E. Esaias, and C. R. McClain, "Prelaunch acceptance report for the SeaWiFS radiometer," in SeaWiFS Technical Report Series, S. B. Hooker, E. R. Firestone, and J. G. Acker, eds., NASA Technical Memorandum 104566, 22 (1994).

Evans, R. H.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. D. 93, 10909-10924 (1988).
[CrossRef]

Forand, J. L.

G. Fournier and J. L. Forand, "Analytic phase function for ocean water," in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE 2258, 194-201 (1994).
[CrossRef]

Fournier, G.

G. Fournier and M. Jonasz, "Computer-based underwater imaging analysis," in Airborne and In-Water Underwater Imaging, G. Gilbert, ed., Proc. SPIE 3761, 62-77 (1999).
[CrossRef]

G. Fournier and J. L. Forand, "Analytic phase function for ocean water," in Ocean Optics XII, J. S. Jaffe, ed., Proc. SPIE 2258, 194-201 (1994).
[CrossRef]

Fry, E. S.

Gentili, B.

Gordon, H. R.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. D. 93, 10909-10924 (1988).
[CrossRef]

Gregg, W. W.

W. W. Gregg and K. L. Carder, "A simple spectral solar irradiance model for cloudless maritime atmospheres," Limnol. Oceanogr. 35, 1657-1675 (1990).
[CrossRef]

Højerslev, N. K.

Hoogenboom, H. J.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, "Laboratory measurements of angular distributions of light scattered by phytoplankton and silt," Limnol. Oceanogr. 43, 1180-1197 (1998).
[CrossRef]

Hovenier, J. W.

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, "Laboratory measurements of angular distributions of light scattered by phytoplankton and silt," Limnol. Oceanogr. 43, 1180-1197 (1998).
[CrossRef]

Johnson, B.

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

Johnson, S.

S. Johnson, "Moderate resolution imaging spectroradiometer (MODIS)," Technical Specification Sheet, RTSC MS 3/05 DTS00-0830 (Raytheon Space and Airborne Systems, Santa Barbara Remote Sensing, Calif., 2002).

Jonasz, M.

G. Fournier and M. Jonasz, "Computer-based underwater imaging analysis," in Airborne and In-Water Underwater Imaging, G. Gilbert, ed., Proc. SPIE 3761, 62-77 (1999).
[CrossRef]

Khomenko, G. A.

M. Chami, E. Marken, J. J. Stamnes, G. A. Khomenko, and G. K. Korotaev, "Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles," J. Geophys. Res. 111, C05013, doi: (2006).
[CrossRef]

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, "Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment," Appl. Opt. 45, 3605-3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, "Optical properties of the particles in the Crimea coastal waters (Black Sea)," J. Geophys. Res. 110, C11020, doi: (2005).
[CrossRef]

Kilpatrick, K. A.

K. J. Voss, W. M. Balch, and K. A. Kilpatrick, "Scattering and attenuation properties of Emiliania huxleyi cells and their detached coccoliths," Limnol. Oceanogr. 43, 870-876 (1998).
[CrossRef]

Korotaev, G. K.

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, "Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment," Appl. Opt. 45, 3605-3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. Marken, J. J. Stamnes, G. A. Khomenko, and G. K. Korotaev, "Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles," J. Geophys. Res. 111, C05013, doi: (2006).
[CrossRef]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, "Optical properties of the particles in the Crimea coastal waters (Black Sea)," J. Geophys. Res. 110, C11020, doi: (2005).
[CrossRef]

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

Kullenberg, G.

G. Kullenberg, "Observed and computed scattering functions," in Optical Aspects of Oceanography, N. G. Jerlov and E. S. Nielsen, eds. (Academic, 1974), pp. 25-49.

Lee, M. E.

M. E. Lee and M. R. Lewis, "A new method for the measurement of the optical volume scattering function in the upper ocean," J. Atmos. Ocean. Technol. 20, 563-571 (2003).
[CrossRef]

Lee, M. E.-G.

M. Chami, E. B. Shybanov, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, and G. K. Korotaev, "Spectral variation of the volume scattering function measured over the full range of scattering angles in a coastal environment," Appl. Opt. 45, 3605-3619 (2006).
[CrossRef] [PubMed]

M. Chami, E. B. Shybanov, T. Y. Churilova, G. A. Khomenko, M. E.-G. Lee, O. V. Martynov, G. A. Berseneva, and G. K. Korotaev, "Optical properties of the particles in the Crimea coastal waters (Black Sea)," J. Geophys. Res. 110, C11020, doi: (2005).
[CrossRef]

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

Lee, Z. P.

Lewis, M.

X. Zhang, M. Lewis, M. E.-G. Lee, B. Johnson, and G. K. Korotaev, "The volume scattering function of natural bubble populations," Limnol. Oceanogr. 47, 1273-1282 (2002).
[CrossRef]

Lewis, M. R.

M. E. Lee and M. R. Lewis, "A new method for the measurement of the optical volume scattering function in the upper ocean," J. Atmos. Ocean. Technol. 20, 563-571 (2003).
[CrossRef]

Loisel, H.

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H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, "Laboratory measurements of angular distributions of light scattered by phytoplankton and silt," Limnol. Oceanogr. 43, 1180-1197 (1998).
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H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, "Laboratory measurements of angular distributions of light scattered by phytoplankton and silt," Limnol. Oceanogr. 43, 1180-1197 (1998).
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[CrossRef]

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[CrossRef]

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[CrossRef]

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H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, "A semianalytic radiance model of ocean color," J. Geophys. Res. D. 93, 10909-10924 (1988).
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J.-L. Bézy, S. Delwart, and M. Rast, "MERIS-A new generation of ocean-colour sensor onboard Envisat," ESA Bulletin 103 (2000), http://www.esa.int/esapub/bulletin/bullet103/besy103.pdf.

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Figures (11)

Fig. 1
Fig. 1

Distribution of scattering angles for 2900 MERIS scenes recorded over the bay of Villefranche-sur-Mer, in the Mediterranean (French Riviera). The broad range of scattering angles observed for a single site illustrated the need to consider bidirectional effects for remote sensing applications.

Fig. 2
Fig. 2

Three particulate VSFs recorded with the VSM in coastal waters off the Crimean coast. These samples were selected as they cover the range of b b p / b p observed during the 2004 sampling campaign.

Fig. 3
Fig. 3

Solid curves represent the particulate scattering phase functions for three values of b b p / b p derived from data recorded with the VSM (see Fig. 2). Dashed curves represent FF phase functions with the same backscattering ratio. There are significant differences between VSM and FF phase functions at intermediate and large scattering angles.

Fig. 4
Fig. 4

Percentage differences between VSM measured and FF scattering phase functions for angles between (a) 0.7° and 100° (logarithm scale for the x axis), (b) 90° to 180° (linear scale for the x-axis).

Fig. 5
Fig. 5

Phase functions of the particles obtained from FF approximation (solid lines and curve and dashed curves) and from VSM data at three scattering angles (100°, 125°, and 150°) (black symbols). The VSM measurements at each angle were integrated over ± 9 ° to reproduce the measurements made by an ECO–VSF instrument (Wetlabs Inc.). Simulated ECO–VSF data points (black symbols) matched the equivalent FF phase functions well for two samples: b b p / b p = 0.010 (black circles, solid line) and b b p / b p = 0.018 (black squares, dashed curve). The match between simulated ECO–VSF data and the equivalent FF phase function was not as good for b b p / b p = 0.025 (black triangles, dash-dot curve). The white symbols representing selected VSM data points at other angles demonstrated that data from additional scattering angles (notably 170°) would be helpful for determining how close an analytic phase function resembles the true phase function.

Fig. 6
Fig. 6

Differences between simulated radiance reflectances for VSM and FF phase functions for scattering angles between 92.5° and 180° and a broad range of scattering values. Nonlinearity of the plots and negative values for scattering angles where β ˜ VSM ( θ ) > β ˜ FF ( θ ) could not be explained with the single-scattering approximation.

Fig. 7
Fig. 7

Differences between VSM and FF simulated radiance reflectances expressed as a percentage of VSM reflectance. The differences reached as high as ± 20 % and tended toward negative values between 0% and 15 % for high turbidity levels. Differences of this magnitude could be significant for remote sensing applications.

Fig. 8
Fig. 8

Contribution of multiple scattering to radiance reflectance as a function of the ratio b b / a . The contribution of multiple scattering to radiance reflectance at 90°, 135°, and 170° increased exponentially as the level of scattering increased. The multiple-scattering contribution was calculated using a radiative transfer model based on the successive order of scattering method to calculate total radiance ( L tot ) and radiance from single scattering ( L 1 ) at several different angles for VSM and FF phase functions with b b p / b p = 0.018 .

Fig. 9
Fig. 9

(a) Angular distribution of scattering events contributing to radiance at 170° as derived from a Monte Carlo simulation using a VSM phase function with b b p / b p = 0.018 and b b / a = 0.01 . For low turbidity water, single scattering (the peak at 170°) provided the dominant contribution. (b) Schematic of the multiple-scattering photon trajectory (black arrow) in the water body as suggested by (a). The single-scattering regime dominates. The particles are represented by circles. The diagram is drawn for a solar zenith angle of 0° and a signal measured at a scattering angle of 170° relative to the sun. θ f is the scattering angle at forward angles.

Fig. 10
Fig. 10

(a) Angular distribution of scattering events contributing to radiance at 170° as derived from a Monte Carlo simulation using a VSM phase function with b b p / b p = 0.018 and b b / a = 0.5 . For high turbidity water, single-scattering events (the peak at 170°) were massively outnumbered by scattering at other angles, particularly toward the forward direction. (b) Schematic of the multiple-scattering photon trajectory (black arrow) in the water body as suggested by (a). The multiple-scattering regime dominates. The diagram is drawn for a solar zenith angle of 0° and a signal measured at a scattering angle of 170° relative to the sun. θ f is the scattering angle at forward angles.

Fig. 11
Fig. 11

(a) Same as Fig. 9(a) except for addition of the case where a FF phase function is used. In low turbidity waters angular distributions of scattering events contributing to radiance at 170° were almost identical for equivalent VSM and FF scattering phase functions and were dominated by single scattering at 170°. (b) Same as Fig. 10(a) except for addition of the case where a FF phase function is used. Angular distributions of scattering events in high turbidity waters were dominated by multiple scattering toward forward angles, and there were significant differences between VSM and FF simulations for intermediate angles between 10° and 100°, closely resembling the differences previously observed between these two scattering phase functions.

Equations (2)

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R L = f Q b b a ,
R L = l 1 b b a + b b + l 2 ( b b a + b b ) 2 ,

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