Abstract

Field observations and theoretical studies have shown that shapes of the volume scattering functions (VSFs) of oceanic particles in the backward directions, i.e., VSFs normalized by the total backscattering coefficient, exhibit a surprisingly low variability at angles near 120 degree, which is also confirmed by measurements of VSFs in coastal waters around the US. To investigate what this minimum variability angle (θ*) represents, we estimated mean values of the VSFs in the backward angles using four mean value theorems: mean value for integral, weighted mean value for integral, classic mean value for differentiation and Cauchy’s mean value. We also estimated the angles corresponding to the minimum values of the VSFs. We found θ* to be very close to the angles representing the classic mean values for differentiation of the VSFs. The low variability is due to the fact that the classic mean values vary little with the composition and sizes of particles.

© 2014 Optical Society of America

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References

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  1. R. W. Preisendorfer, Hydrologic Optics: Introduction (Pacific Mar. Environ. lab/NOAA, 1976), Vol. 1, p. 218.
  2. 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(C7), 13135–113142 (1995).
    [Crossref]
  3. G. L. Clarke, G. C. Ewing, and C. J. Lorenzen, “Spectra of Backscattered Light from the Sea Obtained from Aircraft as a Measure of Chlorophyll Concentration,” Science 167(3921), 1119–1121 (1970).
    [Crossref] [PubMed]
  4. A. Morel, “Optical properties of pure water and pure sea water,” in Optical Aspects of Oceanography, N. G. Jerlov and E. S. Nielsen, eds. (Academic Press, 1974), pp. 1–24.
  5. X. Zhang and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express 17(3), 1671–1678 (2009).
    [Crossref] [PubMed]
  6. X. Zhang and L. Hu, “Scattering by pure seawater at high salinity,” Opt. Express 17(15), 12685–12691 (2009).
    [Crossref] [PubMed]
  7. X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: Effect of salinity,” Opt. Express 17(7), 5698–5710 (2009).
    [Crossref] [PubMed]
  8. X. Zhang, L. Hu, M. S. Twardowski, and J. M. Sullivan, “Scattering by solutions of major sea salts,” Opt. Express 17(22), 19580–19585 (2009).
    [Crossref] [PubMed]
  9. T. J. Petzold, “Volume scattering function for selected ocean waters,” SIO Ref. 72–78 (Scripps Institute of Oceanography, 1972).
  10. G. Kullenberg, “Scattering of light by Sargasso Sea water,” Deep-Sea Res. 15, 423–432 (1968).
  11. T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120 °,” Appl. Opt. 29(31), 4658–4665 (1990).
    [Crossref] [PubMed]
  12. 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(4), 563–571 (2003).
    [Crossref]
  13. J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48(35), 6811–6819 (2009).
    [Crossref] [PubMed]
  14. 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(15), 3605–3619 (2006).
    [Crossref] [PubMed]
  15. J.-F. Berthon, E. Shybanov, M. E. G. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46(22), 5189–5203 (2007).
    [Crossref] [PubMed]
  16. A. L. Whitmire, W. S. Pegau, L. Karp-Boss, E. Boss, and T. J. Cowles, “Spectral backscattering properties of marine phytoplankton cultures,” Opt. Express 18(14), 15073–15093 (2010).
    [Crossref] [PubMed]
  17. E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40(30), 5503–5507 (2001).
    [Crossref] [PubMed]
  18. X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51(21), 5085–5099 (2012).
    [Crossref] [PubMed]
  19. R. A. Maffione and D. R. Dana, “Instruments and methods for measuring the backward-scattering coefficient of ocean waters,” Appl. Opt. 36(24), 6057–6067 (1997).
    [Crossref] [PubMed]
  20. G. Dall Olmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6(6), 947–967 (2009).
    [Crossref]
  21. C. F. Bohren and S. B. Singham, “Backscattering by Nonspherical Particles: A Review of Methods and Suggested New Approaches,” J. Geophys. Res. 96(D3), 5269–5277 (1991).
    [Crossref]
  22. R. A. Meyer, “Light scattering from biological cells: Dependence of backscatter radiation on membrane thickness and refractive index,” Appl. Opt. 18(5), 585–588 (1979).
    [Crossref] [PubMed]
  23. D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
    [Crossref]
  24. X. Zhang, M. Twardowski, and M. Lewis, “Retrieving composition and sizes of oceanic particle subpopulations from the volume scattering function,” Appl. Opt. 50(9), 1240–1259 (2011).
    [Crossref] [PubMed]
  25. H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 µm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
    [Crossref]
  26. M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
    [Crossref]
  27. X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10(9), 6029–6043 (2013).
    [Crossref]
  28. X. Zhang, R. H. Stavn, A. U. Falster, D. Gray, and R. W. Gould., “New insight into particulate mineral and organic matter in coastal ocean waters through optical inversion,” Estuar. Coast. Shelf Sci. 149, 1–12 (2014).
    [Crossref]
  29. B. Epstein, “The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution,” J. Franklin Inst. 244(6), 471–477 (1947).
    [Crossref]
  30. F. S. Lai, S. K. Friedlander, J. Pich, and G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39(2), 395–405 (1972).
    [Crossref]
  31. J. W. Campbell and C. M. Yentsch, “Variance within homogeneous phytoplankton populations, I: Theoretical framework for interpreting histograms,” Cytometry 10(5), 587–595 (1989).
    [Crossref] [PubMed]
  32. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc., 1981).
  33. M. Jonasz and G. R. Fournier, Light Scattering by Particles in Water: Theoretical and Experimental Foundations (Academic Press, 2007), p. 704.
  34. G. R. Fournier, “Backscatter corrected Fournier-Forand phase function for remote sensing and underwater imaging performance evaluation,” in Current Research on Remote Sensing, Laser Probing, and Imagery in Natural Waters, L. M. Levin, G. D. Gilbert, V. I. Haltrin, and C. C. Trees, eds. (SPIE, 2007), pp. 66150N:66151–66157.
  35. G. R. Fournier and J. L. Forand, “Analytical phase function for ocean water,” SPIE Ocean Optics XII 2258, 194–201 (1994).
    [Crossref]
  36. E. Lamarre and W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature 351(6326), 469–472 (1991).
    [Crossref]
  37. S. A. Thorpe and P. N. Humphries, “Bubbles and breaking waves,” Nature 283(5746), 463–465 (1980).
    [Crossref]
  38. S. C. Ling and H. Pao, P., “Study of micro-bubbles in the North Sea,” in Sea Surface Sound, B. R. Kerman, ed. (Kluwer Academic Publishers, 1988), pp. 197–210.
  39. H. Medwin, “In situ acoustic measurements of microbubbles at sea,” J. Geophys. Res. 82(6), 971–976 (1977).
    [Crossref]
  40. T. J. O Hern, L. d Agostino, and A. J. Acosta, “Comparison of holographic and Coulter counter measurement of cavitation nuclei in the ocean,” J. Fluids Eng. 110(2), 200–207 (1988).
    [Crossref]
  41. L. Bi, P. Yang, G. W. Kattawar, and R. Kahn, “Modeling optical properties of mineral aerosol particles by using nonsymmetric hexahedra,” Appl. Opt. 49(3), 334–342 (2010).
    [Crossref] [PubMed]
  42. E. Aas, “Refractive index of phytoplankton derived from its metabolite composition,” J. Plankton Res. 18(12), 2223–2249 (1996).
    [Crossref]
  43. J. Sullivan, M. Twardowski, J. R. V. Zaneveld, and C. Moore, “Measuring optical backscattering in water,” in Light Scattering Reviews 7, A. Kokhanovsky, ed. (Springer, 2013), pp. 189–224.
  44. E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
    [Crossref]

2014 (1)

X. Zhang, R. H. Stavn, A. U. Falster, D. Gray, and R. W. Gould., “New insight into particulate mineral and organic matter in coastal ocean waters through optical inversion,” Estuar. Coast. Shelf Sci. 149, 1–12 (2014).
[Crossref]

2013 (1)

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10(9), 6029–6043 (2013).
[Crossref]

2012 (2)

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51(21), 5085–5099 (2012).
[Crossref] [PubMed]

2011 (2)

X. Zhang, M. Twardowski, and M. Lewis, “Retrieving composition and sizes of oceanic particle subpopulations from the volume scattering function,” Appl. Opt. 50(9), 1240–1259 (2011).
[Crossref] [PubMed]

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 µm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
[Crossref]

2010 (2)

2009 (7)

2007 (1)

2006 (1)

2004 (1)

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[Crossref]

2003 (1)

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(4), 563–571 (2003).
[Crossref]

2001 (1)

1997 (1)

1996 (1)

E. Aas, “Refractive index of phytoplankton derived from its metabolite composition,” J. Plankton Res. 18(12), 2223–2249 (1996).
[Crossref]

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(C7), 13135–113142 (1995).
[Crossref]

1994 (1)

G. R. Fournier and J. L. Forand, “Analytical phase function for ocean water,” SPIE Ocean Optics XII 2258, 194–201 (1994).
[Crossref]

1991 (2)

E. Lamarre and W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature 351(6326), 469–472 (1991).
[Crossref]

C. F. Bohren and S. B. Singham, “Backscattering by Nonspherical Particles: A Review of Methods and Suggested New Approaches,” J. Geophys. Res. 96(D3), 5269–5277 (1991).
[Crossref]

1990 (1)

1989 (1)

J. W. Campbell and C. M. Yentsch, “Variance within homogeneous phytoplankton populations, I: Theoretical framework for interpreting histograms,” Cytometry 10(5), 587–595 (1989).
[Crossref] [PubMed]

1988 (1)

T. J. O Hern, L. d Agostino, and A. J. Acosta, “Comparison of holographic and Coulter counter measurement of cavitation nuclei in the ocean,” J. Fluids Eng. 110(2), 200–207 (1988).
[Crossref]

1980 (1)

S. A. Thorpe and P. N. Humphries, “Bubbles and breaking waves,” Nature 283(5746), 463–465 (1980).
[Crossref]

1979 (1)

1977 (1)

H. Medwin, “In situ acoustic measurements of microbubbles at sea,” J. Geophys. Res. 82(6), 971–976 (1977).
[Crossref]

1972 (1)

F. S. Lai, S. K. Friedlander, J. Pich, and G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39(2), 395–405 (1972).
[Crossref]

1970 (1)

G. L. Clarke, G. C. Ewing, and C. J. Lorenzen, “Spectra of Backscattered Light from the Sea Obtained from Aircraft as a Measure of Chlorophyll Concentration,” Science 167(3921), 1119–1121 (1970).
[Crossref] [PubMed]

1968 (1)

G. Kullenberg, “Scattering of light by Sargasso Sea water,” Deep-Sea Res. 15, 423–432 (1968).

1947 (1)

B. Epstein, “The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution,” J. Franklin Inst. 244(6), 471–477 (1947).
[Crossref]

Aas, E.

E. Aas, “Refractive index of phytoplankton derived from its metabolite composition,” J. Plankton Res. 18(12), 2223–2249 (1996).
[Crossref]

Acosta, A. J.

T. J. O Hern, L. d Agostino, and A. J. Acosta, “Comparison of holographic and Coulter counter measurement of cavitation nuclei in the ocean,” J. Fluids Eng. 110(2), 200–207 (1988).
[Crossref]

Behrenfeld, M. J.

G. Dall Olmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6(6), 947–967 (2009).
[Crossref]

Berthon, J.-F.

Bi, L.

Bogucki, D.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[Crossref]

Bohren, C. F.

C. F. Bohren and S. B. Singham, “Backscattering by Nonspherical Particles: A Review of Methods and Suggested New Approaches,” J. Geophys. Res. 96(D3), 5269–5277 (1991).
[Crossref]

Boss, E.

A. L. Whitmire, W. S. Pegau, L. Karp-Boss, E. Boss, and T. J. Cowles, “Spectral backscattering properties of marine phytoplankton cultures,” Opt. Express 18(14), 15073–15093 (2010).
[Crossref] [PubMed]

G. Dall Olmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6(6), 947–967 (2009).
[Crossref]

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[Crossref]

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40(30), 5503–5507 (2001).
[Crossref] [PubMed]

Campbell, J. W.

J. W. Campbell and C. M. Yentsch, “Variance within homogeneous phytoplankton populations, I: Theoretical framework for interpreting histograms,” Cytometry 10(5), 587–595 (1989).
[Crossref] [PubMed]

Chami, M.

Clarke, G. L.

G. L. Clarke, G. C. Ewing, and C. J. Lorenzen, “Spectra of Backscattered Light from the Sea Obtained from Aircraft as a Measure of Chlorophyll Concentration,” Science 167(3921), 1119–1121 (1970).
[Crossref] [PubMed]

Cowles, T. J.

Czerski, H.

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 µm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
[Crossref]

d Agostino, L.

T. J. O Hern, L. d Agostino, and A. J. Acosta, “Comparison of holographic and Coulter counter measurement of cavitation nuclei in the ocean,” J. Fluids Eng. 110(2), 200–207 (1988).
[Crossref]

Dall Olmo, G.

G. Dall Olmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6(6), 947–967 (2009).
[Crossref]

Dana, D. R.

Epstein, B.

B. Epstein, “The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution,” J. Franklin Inst. 244(6), 471–477 (1947).
[Crossref]

Ewing, G. C.

G. L. Clarke, G. C. Ewing, and C. J. Lorenzen, “Spectra of Backscattered Light from the Sea Obtained from Aircraft as a Measure of Chlorophyll Concentration,” Science 167(3921), 1119–1121 (1970).
[Crossref] [PubMed]

Falster, A. U.

X. Zhang, R. H. Stavn, A. U. Falster, D. Gray, and R. W. Gould., “New insight into particulate mineral and organic matter in coastal ocean waters through optical inversion,” Estuar. Coast. Shelf Sci. 149, 1–12 (2014).
[Crossref]

Forand, J. L.

G. R. Fournier and J. L. Forand, “Analytical phase function for ocean water,” SPIE Ocean Optics XII 2258, 194–201 (1994).
[Crossref]

Fournier, G. R.

G. R. Fournier and J. L. Forand, “Analytical phase function for ocean water,” SPIE Ocean Optics XII 2258, 194–201 (1994).
[Crossref]

Freeman, S.

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

Friedlander, S. K.

F. S. Lai, S. K. Friedlander, J. Pich, and G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39(2), 395–405 (1972).
[Crossref]

Gilbert, S.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Gould, R. W.

X. Zhang, R. H. Stavn, A. U. Falster, D. Gray, and R. W. Gould., “New insight into particulate mineral and organic matter in coastal ocean waters through optical inversion,” Estuar. Coast. Shelf Sci. 149, 1–12 (2014).
[Crossref]

Gray, D.

X. Zhang, R. H. Stavn, A. U. Falster, D. Gray, and R. W. Gould., “New insight into particulate mineral and organic matter in coastal ocean waters through optical inversion,” Estuar. Coast. Shelf Sci. 149, 1–12 (2014).
[Crossref]

Gray, D. J.

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10(9), 6029–6043 (2013).
[Crossref]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51(21), 5085–5099 (2012).
[Crossref] [PubMed]

Gundersen, K.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Hawley, N.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

He, M.-X.

Hidy, G. M.

F. S. Lai, S. K. Friedlander, J. Pich, and G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39(2), 395–405 (1972).
[Crossref]

Hu, L.

Humphries, P. N.

S. A. Thorpe and P. N. Humphries, “Bubbles and breaking waves,” Nature 283(5746), 463–465 (1980).
[Crossref]

Huot, Y.

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10(9), 6029–6043 (2013).
[Crossref]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51(21), 5085–5099 (2012).
[Crossref] [PubMed]

Janzen, C.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Johengen, T.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Kahn, R.

Karp-Boss, L.

Kattawar, G.

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

Kattawar, G. W.

Khomenko, G. A.

Korotaev, G. K.

Kullenberg, G.

G. Kullenberg, “Scattering of light by Sargasso Sea water,” Deep-Sea Res. 15, 423–432 (1968).

Lai, F. S.

F. S. Lai, S. K. Friedlander, J. Pich, and G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39(2), 395–405 (1972).
[Crossref]

Lamarre, E.

E. Lamarre and W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature 351(6326), 469–472 (1991).
[Crossref]

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(4), 563–571 (2003).
[Crossref]

Lee, M. E. G.

Lee, M. E.-G.

Lewis, M.

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(4), 563–571 (2003).
[Crossref]

Lorenzen, C. J.

G. L. Clarke, G. C. Ewing, and C. J. Lorenzen, “Spectra of Backscattered Light from the Sea Obtained from Aircraft as a Measure of Chlorophyll Concentration,” Science 167(3921), 1119–1121 (1970).
[Crossref] [PubMed]

Maffione, R. A.

Martynov, O. V.

Medwin, H.

H. Medwin, “In situ acoustic measurements of microbubbles at sea,” J. Geophys. Res. 82(6), 971–976 (1977).
[Crossref]

Melville, W. K.

E. Lamarre and W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature 351(6326), 469–472 (1991).
[Crossref]

Meyer, R. A.

O Hern, T. J.

T. J. O Hern, L. d Agostino, and A. J. Acosta, “Comparison of holographic and Coulter counter measurement of cavitation nuclei in the ocean,” J. Fluids Eng. 110(2), 200–207 (1988).
[Crossref]

Oishi, T.

Pegau, W. S.

Pich, J.

F. S. Lai, S. K. Friedlander, J. Pich, and G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39(2), 395–405 (1972).
[Crossref]

Purcell, H.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Rhea, W. J.

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10(9), 6029–6043 (2013).
[Crossref]

Robertson, C.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Schar, D. W. H.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Shybanov, E.

Shybanov, E. B.

Singham, S. B.

C. F. Bohren and S. B. Singham, “Backscattering by Nonspherical Particles: A Review of Methods and Suggested New Approaches,” J. Geophys. Res. 96(D3), 5269–5277 (1991).
[Crossref]

Slade, W. H.

G. Dall Olmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6(6), 947–967 (2009).
[Crossref]

Smith, G. J.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Stavn, R. H.

X. Zhang, R. H. Stavn, A. U. Falster, D. Gray, and R. W. Gould., “New insight into particulate mineral and organic matter in coastal ocean waters through optical inversion,” Estuar. Coast. Shelf Sci. 149, 1–12 (2014).
[Crossref]

Stramski, D.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[Crossref]

Sullivan, J.

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

Sullivan, J. M.

Tamburri, M. N.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Taylor, L.

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Thorpe, S. A.

S. A. Thorpe and P. N. Humphries, “Bubbles and breaking waves,” Nature 283(5746), 463–465 (1980).
[Crossref]

Twardowski, M.

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 µm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
[Crossref]

X. Zhang, M. Twardowski, and M. Lewis, “Retrieving composition and sizes of oceanic particle subpopulations from the volume scattering function,” Appl. Opt. 50(9), 1240–1259 (2011).
[Crossref] [PubMed]

Twardowski, M. S.

Vagle, S.

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 µm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
[Crossref]

Voss, K. J.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[Crossref]

Weidemann, A.

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10(9), 6029–6043 (2013).
[Crossref]

Westberry, T. K.

G. Dall Olmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6(6), 947–967 (2009).
[Crossref]

Whitmire, A. L.

Yang, P.

Yentsch, C. M.

J. W. Campbell and C. M. Yentsch, “Variance within homogeneous phytoplankton populations, I: Theoretical framework for interpreting histograms,” Cytometry 10(5), 587–595 (1989).
[Crossref] [PubMed]

You, Y.

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51(21), 5085–5099 (2012).
[Crossref] [PubMed]

Zaneveld, J. R. V.

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(C7), 13135–113142 (1995).
[Crossref]

Zhang, X.

X. Zhang, R. H. Stavn, A. U. Falster, D. Gray, and R. W. Gould., “New insight into particulate mineral and organic matter in coastal ocean waters through optical inversion,” Estuar. Coast. Shelf Sci. 149, 1–12 (2014).
[Crossref]

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10(9), 6029–6043 (2013).
[Crossref]

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51(21), 5085–5099 (2012).
[Crossref] [PubMed]

X. Zhang, M. Twardowski, and M. Lewis, “Retrieving composition and sizes of oceanic particle subpopulations from the volume scattering function,” Appl. Opt. 50(9), 1240–1259 (2011).
[Crossref] [PubMed]

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 µm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
[Crossref]

X. Zhang, L. Hu, M. S. Twardowski, and J. M. Sullivan, “Scattering by solutions of major sea salts,” Opt. Express 17(22), 19580–19585 (2009).
[Crossref] [PubMed]

X. Zhang and L. Hu, “Scattering by pure seawater at high salinity,” Opt. Express 17(15), 12685–12691 (2009).
[Crossref] [PubMed]

X. Zhang and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express 17(3), 1671–1678 (2009).
[Crossref] [PubMed]

X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: Effect of salinity,” Opt. Express 17(7), 5698–5710 (2009).
[Crossref] [PubMed]

Zibordi, G.

Appl. Opt. (10)

J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48(35), 6811–6819 (2009).
[Crossref] [PubMed]

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(15), 3605–3619 (2006).
[Crossref] [PubMed]

J.-F. Berthon, E. Shybanov, M. E. G. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46(22), 5189–5203 (2007).
[Crossref] [PubMed]

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40(30), 5503–5507 (2001).
[Crossref] [PubMed]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51(21), 5085–5099 (2012).
[Crossref] [PubMed]

R. A. Maffione and D. R. Dana, “Instruments and methods for measuring the backward-scattering coefficient of ocean waters,” Appl. Opt. 36(24), 6057–6067 (1997).
[Crossref] [PubMed]

T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120 °,” Appl. Opt. 29(31), 4658–4665 (1990).
[Crossref] [PubMed]

R. A. Meyer, “Light scattering from biological cells: Dependence of backscatter radiation on membrane thickness and refractive index,” Appl. Opt. 18(5), 585–588 (1979).
[Crossref] [PubMed]

X. Zhang, M. Twardowski, and M. Lewis, “Retrieving composition and sizes of oceanic particle subpopulations from the volume scattering function,” Appl. Opt. 50(9), 1240–1259 (2011).
[Crossref] [PubMed]

L. Bi, P. Yang, G. W. Kattawar, and R. Kahn, “Modeling optical properties of mineral aerosol particles by using nonsymmetric hexahedra,” Appl. Opt. 49(3), 334–342 (2010).
[Crossref] [PubMed]

Biogeosciences (2)

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10(9), 6029–6043 (2013).
[Crossref]

G. Dall Olmo, T. K. Westberry, M. J. Behrenfeld, E. Boss, and W. H. Slade, “Significant contribution of large particles to optical backscattering in the open ocean,” Biogeosciences 6(6), 947–967 (2009).
[Crossref]

Cytometry (1)

J. W. Campbell and C. M. Yentsch, “Variance within homogeneous phytoplankton populations, I: Theoretical framework for interpreting histograms,” Cytometry 10(5), 587–595 (1989).
[Crossref] [PubMed]

Deep-Sea Res. (1)

G. Kullenberg, “Scattering of light by Sargasso Sea water,” Deep-Sea Res. 15, 423–432 (1968).

Estuar. Coast. Shelf Sci. (1)

X. Zhang, R. H. Stavn, A. U. Falster, D. Gray, and R. W. Gould., “New insight into particulate mineral and organic matter in coastal ocean waters through optical inversion,” Estuar. Coast. Shelf Sci. 149, 1–12 (2014).
[Crossref]

J. Atmos. Ocean. Technol. (1)

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(4), 563–571 (2003).
[Crossref]

J. Colloid Interface Sci. (1)

F. S. Lai, S. K. Friedlander, J. Pich, and G. M. Hidy, “The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime,” J. Colloid Interface Sci. 39(2), 395–405 (1972).
[Crossref]

J. Fluids Eng. (1)

T. J. O Hern, L. d Agostino, and A. J. Acosta, “Comparison of holographic and Coulter counter measurement of cavitation nuclei in the ocean,” J. Fluids Eng. 110(2), 200–207 (1988).
[Crossref]

J. Franklin Inst. (1)

B. Epstein, “The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution,” J. Franklin Inst. 244(6), 471–477 (1947).
[Crossref]

J. Geophys. Res. (5)

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 µm with optical and acoustical methods,” J. Geophys. Res. 116, C00H11 (2011).
[Crossref]

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. 117, C00H17 (2012).
[Crossref]

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(C7), 13135–113142 (1995).
[Crossref]

C. F. Bohren and S. B. Singham, “Backscattering by Nonspherical Particles: A Review of Methods and Suggested New Approaches,” J. Geophys. Res. 96(D3), 5269–5277 (1991).
[Crossref]

H. Medwin, “In situ acoustic measurements of microbubbles at sea,” J. Geophys. Res. 82(6), 971–976 (1977).
[Crossref]

J. Plankton Res. (1)

E. Aas, “Refractive index of phytoplankton derived from its metabolite composition,” J. Plankton Res. 18(12), 2223–2249 (1996).
[Crossref]

Limnol. Oceanogr. Methods (1)

E. Boss, L. Taylor, S. Gilbert, K. Gundersen, N. Hawley, C. Janzen, T. Johengen, H. Purcell, C. Robertson, D. W. H. Schar, G. J. Smith, and M. N. Tamburri, “Comparison of inherent optical properties as a surrogate for particulate matter concentration in coastal waters,” Limnol. Oceanogr. Methods 7, 803–810 (2009).
[Crossref]

Nature (2)

E. Lamarre and W. K. Melville, “Air entrainment and dissipation in breaking waves,” Nature 351(6326), 469–472 (1991).
[Crossref]

S. A. Thorpe and P. N. Humphries, “Bubbles and breaking waves,” Nature 283(5746), 463–465 (1980).
[Crossref]

Opt. Express (5)

Prog. Oceanogr. (1)

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61(1), 27–56 (2004).
[Crossref]

Science (1)

G. L. Clarke, G. C. Ewing, and C. J. Lorenzen, “Spectra of Backscattered Light from the Sea Obtained from Aircraft as a Measure of Chlorophyll Concentration,” Science 167(3921), 1119–1121 (1970).
[Crossref] [PubMed]

SPIE Ocean Optics XII (1)

G. R. Fournier and J. L. Forand, “Analytical phase function for ocean water,” SPIE Ocean Optics XII 2258, 194–201 (1994).
[Crossref]

Other (8)

S. C. Ling and H. Pao, P., “Study of micro-bubbles in the North Sea,” in Sea Surface Sound, B. R. Kerman, ed. (Kluwer Academic Publishers, 1988), pp. 197–210.

H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, Inc., 1981).

M. Jonasz and G. R. Fournier, Light Scattering by Particles in Water: Theoretical and Experimental Foundations (Academic Press, 2007), p. 704.

G. R. Fournier, “Backscatter corrected Fournier-Forand phase function for remote sensing and underwater imaging performance evaluation,” in Current Research on Remote Sensing, Laser Probing, and Imagery in Natural Waters, L. M. Levin, G. D. Gilbert, V. I. Haltrin, and C. C. Trees, eds. (SPIE, 2007), pp. 66150N:66151–66157.

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

R. W. Preisendorfer, Hydrologic Optics: Introduction (Pacific Mar. Environ. lab/NOAA, 1976), Vol. 1, p. 218.

T. J. Petzold, “Volume scattering function for selected ocean waters,” SIO Ref. 72–78 (Scripps Institute of Oceanography, 1972).

J. Sullivan, M. Twardowski, J. R. V. Zaneveld, and C. Moore, “Measuring optical backscattering in water,” in Light Scattering Reviews 7, A. Kokhanovsky, ed. (Springer, 2013), pp. 189–224.

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Figures (5)

Fig. 1
Fig. 1 (a) The backward portion of VSFs measured in Chesapeake Bay (CB), Mobile Bay (MB, Alabama), Monterey Bay (MT, California) and the LEO-15 site. (b) The χ factor estimated for the VSFs shown in (a). The three values are, respectively, the angle at which χ exhibits the minimum variability, the mean value of χ at this angle, and its coefficient of variation (std/mean).
Fig. 2
Fig. 2 A measured VSF (solid black line) is partitioned into contributions by small (green line) and large (red) particles using the inversion technique. The sum of VSFs due to small and large particles (dashed black line) matched the observed VSF nicely.
Fig. 3
Fig. 3 The variability of χ factor estimated for two different size groups and for different experiment sites. The bulk χ is estimated using Eq. (3). The three values in each panel are, respectively, the angle at which χ exhibits the minimum variability, the mean value of χ at this angle, and its coefficient of variation (std/mean).
Fig. 4
Fig. 4 Comparison of four mean value angles of θ1, θ2, θ3, and θ4 and minimum angle θ5 (circles with various colors) with the minimum variability angle θ* (black line) for small, large and bulk (small + large) particle populations.
Fig. 5
Fig. 5 Variations of θ3 estimated for the Fournier and Forand phase functions with the exponent (or Junge slope) of the power law size distribution and the refractive index of particles.

Equations (5)

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

b b =2π π/2 π β(θ)sinθdθ
β(θ)= β s (θ)+ β l (θ) b b = b b,s + b b,l
χ(θ)= χ s (θ) β s (θ) β(θ) + χ l (θ) β l (θ) β(θ) =y χ s (θ)+(1y) χ l (θ)
β(θ)1+ 1δ 1+δ cos 2 θ
β(θ) | r 1 | 2 + | r 2 | 2 + | r 1 | 2 (1 | r 1 | 2 ) 2 + | r 2 | 2 (1 | r 2 | 2 ) 2

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