Abstract

An innovative instrument dedicated to the multispectral measurements of the directional and polarized scattering properties of the hydrosols, so-called POLVSM, is described. The instrument could be used onboard a ship, as a benchtop instrument, or at laboratory. The originality of the POLVSM concept relies on the use of a double periscopic optical system whose role is (i) to separate the plane containing the light source from the scattering plane containing the sample and the receiver and (ii) to prevent from any specularly reflected light within the sample chamber. As a result, a wide range of scattering angle, namely from 1° to 179°, is covered by the detector. Another originality of the instrument is to measure the Mueller scattering matrix elements, including the degree of polarization. A relevant calibration procedure, which could be of great interest as well for other instruments, is proposed to convert the raw data into physical units. The relative uncertainty in POLVSM data was determined at ± 4.3%. The analysis of measurements of the volume scattering function and degree of polarization performed under controlled conditions for samples dominated either by inorganic hydrosols or phytoplankton monospecific species showed a good consistency with literature, thus confirming the good performance of the POLVSM device. Comparisons of POLVSM data with theoretical calculations showed that Mie theory could reproduce efficiently the measurements of the VSF and degree of polarization for the case of inorganic hydrosols sample, despite the likely non sphericity of these particles as revealed by one of the element of the Mueller matrix. Our results suggested as well that a sophisticated modeling of the heterogeneous internal structure of living cells, or at least, the use of layered sphere models, is needed to correctly predict the directional and polarized effects of phytoplankton on the oceanic radiation. The relevance of performing angularly resolved measurements of the Mueller scattering elements to gain understanding on the mechanisms processes involved in the scattering of light by marine particles, which has important implications for ocean color remote sensing studies, is demonstrated.

© 2014 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Laboratory experiments for inter-comparison of three volume scattering meters to measure angular scattering properties of hydrosols

T. Harmel, M. Hieronymi, W. Slade, R. Röttgers, F. Roullier, and M. Chami
Opt. Express 24(2) A234-A256 (2016)

Measurement of the Scattering Properties of Hydrosols*

John E. Tyler
J. Opt. Soc. Am. 51(11) 1289-1293 (1961)

References

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2013 (4)

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[Crossref]

M. W. Matthews and S. Bernard, “Using a two-layered sphere model to investigate the impact of gas vacuoles on the inherent optical properties of M. aeruginosa,” Biogeosci. Discuss. 10(6), 10531–10579 (2013).
[Crossref]

G. W. Kattawar, “Genesis and evolution of polarization of light in the ocean,” Appl. Opt. 52(5), 940–948 (2013).
[Crossref] [PubMed]

H. Tan, R. Doerffer, T. Oishi, and A. Tanaka, “A new approach to measure the volume scattering function,” Opt. Express 21(16), 18697–18711 (2013).
[Crossref] [PubMed]

2012 (4)

2011 (1)

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

2010 (2)

2009 (2)

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]

A. Gogoi, A. K. Buragohain, A. Choudhury, and G. A. Ahmed, “Laboratory measurements of light scattering by tropical fresh water diatoms,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1566–1578 (2009).
[Crossref]

2008 (1)

M. E. Zugger, A. Messmer, T. J. Kane, J. Prentice, B. Concannon, A. Laux, and L. Mullen, “Optical scattering properties of phytoplankton: Measurements and comparison of various species at scattering angles between 1° and 170°,” Limnol. Oceanogr. 53(1), 381–386 (2008).
[Crossref]

2007 (5)

2006 (2)

2005 (1)

H. Volten, O. Munoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. Van der Zande, and L. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transf. 90(2), 191–206 (2005).
[Crossref]

2004 (1)

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26(2), 191–212 (2004).
[Crossref]

2003 (2)

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211 (2003).

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 (2)

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14129–14142 (2001).
[Crossref]

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

1998 (1)

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(6), 1180–1197 (1998).
[Crossref]

1996 (1)

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

1993 (1)

K. Witkowski, L. Wolinski, Z. Turzynski, D. Gedziorowska, and A. Zielinski, “The investigation of kinetic growth of Chlorella vulgaris cells by the method of integral and dynamic light scattering,” Limnol. Oceanogr. 38(7), 1365–1372 (1993).
[Crossref]

1992 (1)

1989 (1)

M. S. Quinby-Hunt, A. J. Hunt, K. Lofftus, and D. Shapiro, “Polarized-light scattering studies of marine Chlorella,” Limnol. Oceanogr. 34(8), 1587–1600 (1989).
[Crossref]

1986 (1)

1985 (1)

E. S. Fry and K. J. Voss, “Measurement of the Mueller matrix for phytoplankton,” Limnol. Oceanogr. 30(6), 1322–1326 (1985).
[Crossref]

1984 (1)

1973 (1)

A. Morel, “Diffusion de la lumière par les eaux de mer. Résultats expérimentaux et approche théorique,” Opt. Sea 63, 1–76 (1973).

1958 (1)

Aas, E.

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

Ahmed, G. A.

A. Gogoi, A. K. Buragohain, A. Choudhury, and G. A. Ahmed, “Laboratory measurements of light scattering by tropical fresh water diatoms,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1566–1578 (2009).
[Crossref]

Ahmed, S.

Aursland, K.

J. K. Lotsberg, E. Marken, J. J. Stamnes, S. R. Erga, K. Aursland, and C. Olseng, “Laboratory measurements of light scattering from marine particles,” Limnol. Oceanogr. Methods 5, 34–40 (2007).
[Crossref]

Babin, M.

Balch, W. M.

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26(2), 191–212 (2004).
[Crossref]

Barnard, A. H.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14129–14142 (2001).
[Crossref]

Bernard, S.

M. W. Matthews and S. Bernard, “Using a two-layered sphere model to investigate the impact of gas vacuoles on the inherent optical properties of M. aeruginosa,” Biogeosci. Discuss. 10(6), 10531–10579 (2013).
[Crossref]

Bi, L.

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]

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]

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14129–14142 (2001).
[Crossref]

Bourrin, F.

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[Crossref]

Bressac, M.

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[Crossref]

Bricaud, A.

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211 (2003).

Brown, C. W.

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26(2), 191–212 (2004).
[Crossref]

Brown, J. F.

Buragohain, A. K.

A. Gogoi, A. K. Buragohain, A. Choudhury, and G. A. Ahmed, “Laboratory measurements of light scattering by tropical fresh water diatoms,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1566–1578 (2009).
[Crossref]

Cairns, B.

Cao, W.

Chachisvilis, M.

Chami, M.

M. Chami, “Importance of the polarization in the retrieval of oceanic constituents from the remote sensing reflectance,” J. Geophys. Res. 112, C05026 (2007).
[Crossref]

M. Chami, D. McKee, E. Leymarie, and G. Khomenko, “Influence of the angular shape of the volume-scattering function and multiple scattering on remote sensing reflectance,” Appl. Opt. 45(36), 9210–9220 (2006).
[Crossref] [PubMed]

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(6), 1180–1197 (1998).
[Crossref]

Choudhury, A.

A. Gogoi, A. K. Buragohain, A. Choudhury, and G. A. Ahmed, “Laboratory measurements of light scattering by tropical fresh water diatoms,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1566–1578 (2009).
[Crossref]

Chowdhary, J.

Claustre, H.

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211 (2003).

Concannon, B.

M. E. Zugger, A. Messmer, T. J. Kane, J. Prentice, B. Concannon, A. Laux, and L. Mullen, “Optical scattering properties of phytoplankton: Measurements and comparison of various species at scattering angles between 1° and 170°,” Limnol. Oceanogr. 53(1), 381–386 (2008).
[Crossref]

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]

de Haan, J. F.

H. Volten, O. Munoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. Van der Zande, and L. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transf. 90(2), 191–206 (2005).
[Crossref]

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

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(6), 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(6), 1180–1197 (1998).
[Crossref]

Desboeufs, K.

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[Crossref]

Doerffer, R.

Doxaran, D.

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[Crossref]

E. Leymarie, D. Doxaran, and M. Babin, “Uncertainties associated to measurements of inherent optical properties in natural waters,” Appl. Opt. 49(28), 5415–5436 (2010).
[Crossref] [PubMed]

Erga, S. R.

O. Svensen, O. Frette, and S. R. Erga, “Scattering properties of microalgae: the effect of cell size and cell wall,” Appl. Opt. 46(23), 5762–5769 (2007).
[Crossref] [PubMed]

J. K. Lotsberg, E. Marken, J. J. Stamnes, S. R. Erga, K. Aursland, and C. Olseng, “Laboratory measurements of light scattering from marine particles,” Limnol. Oceanogr. Methods 5, 34–40 (2007).
[Crossref]

Esener, S. C.

Ferrari, G. M.

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211 (2003).

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]

Frette, O.

Fry, E. S.

E. S. Fry and K. J. Voss, “Measurement of the Mueller matrix for phytoplankton,” Limnol. Oceanogr. 30(6), 1322–1326 (1985).
[Crossref]

K. J. Voss and E. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. 23(23), 4427–4439 (1984).
[Crossref] [PubMed]

Gedziorowska, D.

K. Witkowski, L. Wolinski, Z. Turzynski, D. Gedziorowska, and A. Zielinski, “The investigation of kinetic growth of Chlorella vulgaris cells by the method of integral and dynamic light scattering,” Limnol. Oceanogr. 38(7), 1365–1372 (1993).
[Crossref]

Gilerson, A.

Gogoi, A.

A. Gogoi, A. K. Buragohain, A. Choudhury, and G. A. Ahmed, “Laboratory measurements of light scattering by tropical fresh water diatoms,” J. Quant. Spectrosc. Radiat. Transf. 110(14–16), 1566–1578 (2009).
[Crossref]

Guieu, C.

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[Crossref]

Guillard, R. R. L.

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26(2), 191–212 (2004).
[Crossref]

Harmel, T.

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

M. A. Yurkin, K. A. Semyanov, V. P. Maltsev, and A. G. Hoekstra, “Discrimination of granulocyte subtypes from light scattering: theoretical analysis using a granulated sphere model,” Opt. Express 15(25), 16561–16580 (2007).
[Crossref] [PubMed]

Hoepffner, N.

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211 (2003).

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(6), 1180–1197 (1998).
[Crossref]

Hovenier, J. W.

H. Volten, O. Munoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. Van der Zande, and L. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transf. 90(2), 191–206 (2005).
[Crossref]

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

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(6), 1180–1197 (1998).
[Crossref]

Hu, S.

Hunt, A. J.

K. D. Lofflus, M. S. Quinby-Hunt, A. J. Hunt, F. Livolant, and M. Maestre, “Light scattering by Prorocentrum micans: a new method and results,” Appl. Opt. 31(15), 2924–2931 (1992).
[Crossref] [PubMed]

M. S. Quinby-Hunt, A. J. Hunt, K. Lofftus, and D. Shapiro, “Polarized-light scattering studies of marine Chlorella,” Limnol. Oceanogr. 34(8), 1587–1600 (1989).
[Crossref]

Ibrahim, A.

Jaffe, J. S.

Kane, T. J.

M. E. Zugger, A. Messmer, T. J. Kane, J. Prentice, B. Concannon, A. Laux, and L. Mullen, “Optical scattering properties of phytoplankton: Measurements and comparison of various species at scattering angles between 1° and 170°,” Limnol. Oceanogr. 53(1), 381–386 (2008).
[Crossref]

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.

Laux, A.

M. E. Zugger, A. Messmer, T. J. Kane, J. Prentice, B. Concannon, A. Laux, and L. Mullen, “Optical scattering properties of phytoplankton: Measurements and comparison of various species at scattering angles between 1° and 170°,” Limnol. Oceanogr. 53(1), 381–386 (2008).
[Crossref]

Leblond, N.

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[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]

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]

Leymarie, E.

Liu, L.

Livolant, F.

Lofflus, K. D.

Lofftus, K.

M. S. Quinby-Hunt, A. J. Hunt, K. Lofftus, and D. Shapiro, “Polarized-light scattering studies of marine Chlorella,” Limnol. Oceanogr. 34(8), 1587–1600 (1989).
[Crossref]

Lotsberg, J. K.

J. K. Lotsberg, E. Marken, J. J. Stamnes, S. R. Erga, K. Aursland, and C. Olseng, “Laboratory measurements of light scattering from marine particles,” Limnol. Oceanogr. Methods 5, 34–40 (2007).
[Crossref]

Macdonald, J. B.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14129–14142 (2001).
[Crossref]

Mackowski, D. W.

Maestre, M.

Maltsev, V. P.

Marken, E.

J. K. Lotsberg, E. Marken, J. J. Stamnes, S. R. Erga, K. Aursland, and C. Olseng, “Laboratory measurements of light scattering from marine particles,” Limnol. Oceanogr. Methods 5, 34–40 (2007).
[Crossref]

Matthews, M. W.

M. W. Matthews and S. Bernard, “Using a two-layered sphere model to investigate the impact of gas vacuoles on the inherent optical properties of M. aeruginosa,” Biogeosci. Discuss. 10(6), 10531–10579 (2013).
[Crossref]

McKee, D.

Messmer, A.

M. E. Zugger, A. Messmer, T. J. Kane, J. Prentice, B. Concannon, A. Laux, and L. Mullen, “Optical scattering properties of phytoplankton: Measurements and comparison of various species at scattering angles between 1° and 170°,” Limnol. Oceanogr. 53(1), 381–386 (2008).
[Crossref]

Mishchenko, M. I.

Morel, A.

A. Morel, “Diffusion de la lumière par les eaux de mer. Résultats expérimentaux et approche théorique,” Opt. Sea 63, 1–76 (1973).

Muinonen, K.

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

Mullen, L.

M. E. Zugger, A. Messmer, T. J. Kane, J. Prentice, B. Concannon, A. Laux, and L. Mullen, “Optical scattering properties of phytoplankton: Measurements and comparison of various species at scattering angles between 1° and 170°,” Limnol. Oceanogr. 53(1), 381–386 (2008).
[Crossref]

Munoz, O.

H. Volten, O. Munoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. Van der Zande, and L. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transf. 90(2), 191–206 (2005).
[Crossref]

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

Nousiainen, T.

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

Obolensky, G.

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211 (2003).

Oishi, T.

Olseng, C.

J. K. Lotsberg, E. Marken, J. J. Stamnes, S. R. Erga, K. Aursland, and C. Olseng, “Laboratory measurements of light scattering from marine particles,” Limnol. Oceanogr. Methods 5, 34–40 (2007).
[Crossref]

Pegau, W. S.

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]

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14129–14142 (2001).
[Crossref]

Prentice, J.

M. E. Zugger, A. Messmer, T. J. Kane, J. Prentice, B. Concannon, A. Laux, and L. Mullen, “Optical scattering properties of phytoplankton: Measurements and comparison of various species at scattering angles between 1° and 170°,” Limnol. Oceanogr. 53(1), 381–386 (2008).
[Crossref]

Quinby-Hunt, M. S.

K. D. Lofflus, M. S. Quinby-Hunt, A. J. Hunt, F. Livolant, and M. Maestre, “Light scattering by Prorocentrum micans: a new method and results,” Appl. Opt. 31(15), 2924–2931 (1992).
[Crossref] [PubMed]

M. S. Quinby-Hunt, A. J. Hunt, K. Lofftus, and D. Shapiro, “Polarized-light scattering studies of marine Chlorella,” Limnol. Oceanogr. 34(8), 1587–1600 (1989).
[Crossref]

Reynolds, R. A.

Richardson, W. H.

Ridame, C.

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[Crossref]

Rol, E.

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

Schreurs, R.

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(6), 1180–1197 (1998).
[Crossref]

Semyanov, K. A.

Shao, B.

Shapiro, D.

M. S. Quinby-Hunt, A. J. Hunt, K. Lofftus, and D. Shapiro, “Polarized-light scattering studies of marine Chlorella,” Limnol. Oceanogr. 34(8), 1587–1600 (1989).
[Crossref]

Spinrad, R. W.

Stamnes, J. J.

J. K. Lotsberg, E. Marken, J. J. Stamnes, S. R. Erga, K. Aursland, and C. Olseng, “Laboratory measurements of light scattering from marine particles,” Limnol. Oceanogr. Methods 5, 34–40 (2007).
[Crossref]

Stramski, D.

M. Babin, D. Stramski, R. A. Reynolds, V. M. Wright, and E. Leymarie, “Determination of the volume scattering function of aqueous particle suspensions with a laboratory multi-angle light scattering instrument,” Appl. Opt. 51(17), 3853–3873 (2012).
[Crossref] [PubMed]

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211 (2003).

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.

Sun, Z.

Svensen, O.

Tan, H.

Tanaka, A.

Tonizzo, A.

Turzynski, Z.

K. Witkowski, L. Wolinski, Z. Turzynski, D. Gedziorowska, and A. Zielinski, “The investigation of kinetic growth of Chlorella vulgaris cells by the method of integral and dynamic light scattering,” Limnol. Oceanogr. 38(7), 1365–1372 (1993).
[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]

Twardowski, M. S.

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. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14129–14142 (2001).
[Crossref]

Tyler, J. E.

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]

Vaillancourt, R. D.

R. D. Vaillancourt, C. W. Brown, R. R. L. Guillard, and W. M. Balch, “Light backscattering properties of marine phytoplankton: relationships to cell size, chemical composition and taxonomy,” J. Plankton Res. 26(2), 191–212 (2004).
[Crossref]

Van der Zande, W. J.

H. Volten, O. Munoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. Van der Zande, and L. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transf. 90(2), 191–206 (2005).
[Crossref]

Vassen, W.

H. Volten, O. Munoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. Van der Zande, and L. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transf. 90(2), 191–206 (2005).
[Crossref]

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

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(6), 1180–1197 (1998).
[Crossref]

Videen, G.

Volten, H.

H. Volten, O. Munoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. Van der Zande, and L. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transf. 90(2), 191–206 (2005).
[Crossref]

H. Volten, O. Munoz, E. Rol, J. F. de Haan, W. Vassen, J. W. Hovenier, K. Muinonen, and T. Nousiainen, “Scattering matrices of mineral aerosol particles at 441.6 nm and 632.8 nm,” J. Geophys. Res. 106(D15), 17375–17401 (2001).
[Crossref]

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(6), 1180–1197 (1998).
[Crossref]

Voss, K. J.

E. S. Fry and K. J. Voss, “Measurement of the Mueller matrix for phytoplankton,” Limnol. Oceanogr. 30(6), 1322–1326 (1985).
[Crossref]

K. J. Voss and E. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. 23(23), 4427–4439 (1984).
[Crossref] [PubMed]

Wang, G.

Waters, L.

H. Volten, O. Munoz, J. W. Hovenier, J. F. de Haan, W. Vassen, W. J. Van der Zande, and L. Waters, “WWW scattering matrix database for small mineral particles at 441.6 and 632.8 nm,” J. Quant. Spectrosc. Radiat. Transf. 90(2), 191–206 (2005).
[Crossref]

Whitmire, A. L.

Witkowski, K.

K. Witkowski, L. Wolinski, Z. Turzynski, D. Gedziorowska, and A. Zielinski, “The investigation of kinetic growth of Chlorella vulgaris cells by the method of integral and dynamic light scattering,” Limnol. Oceanogr. 38(7), 1365–1372 (1993).
[Crossref]

Wolinski, L.

K. Witkowski, L. Wolinski, Z. Turzynski, D. Gedziorowska, and A. Zielinski, “The investigation of kinetic growth of Chlorella vulgaris cells by the method of integral and dynamic light scattering,” Limnol. Oceanogr. 38(7), 1365–1372 (1993).
[Crossref]

Wouts, R.

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(6), 1180–1197 (1998).
[Crossref]

Wright, V. M.

Xu, Z.

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]

Yurkin, M. A.

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transf. 112(13), 2234–2247 (2011).
[Crossref]

M. A. Yurkin, K. A. Semyanov, V. P. Maltsev, and A. G. Hoekstra, “Discrimination of granulocyte subtypes from light scattering: theoretical analysis using a granulated sphere model,” Opt. Express 15(25), 16561–16580 (2007).
[Crossref] [PubMed]

Zaneveld, J. R. V.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, A. H. Barnard, and J. R. V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106(C7), 14129–14142 (2001).
[Crossref]

Zhang, X.

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]

Zhao, J.

Zhou, W.

Zielinski, A.

K. Witkowski, L. Wolinski, Z. Turzynski, D. Gedziorowska, and A. Zielinski, “The investigation of kinetic growth of Chlorella vulgaris cells by the method of integral and dynamic light scattering,” Limnol. Oceanogr. 38(7), 1365–1372 (1993).
[Crossref]

Zugger, M. E.

M. E. Zugger, A. Messmer, T. J. Kane, J. Prentice, B. Concannon, A. Laux, and L. Mullen, “Optical scattering properties of phytoplankton: Measurements and comparison of various species at scattering angles between 1° and 170°,” Limnol. Oceanogr. 53(1), 381–386 (2008).
[Crossref]

Appl. Opt. (9)

K. J. Voss and E. S. Fry, “Measurement of the Mueller matrix for ocean water,” Appl. Opt. 23(23), 4427–4439 (1984).
[Crossref] [PubMed]

R. W. Spinrad and J. F. Brown, “Relative real refractive index of marine microorganisms: a technique for flow cytometric estimation,” Appl. Opt. 25(12), 1930–1934 (1986).
[Crossref] [PubMed]

K. D. Lofflus, M. S. Quinby-Hunt, A. J. Hunt, F. Livolant, and M. Maestre, “Light scattering by Prorocentrum micans: a new method and results,” Appl. Opt. 31(15), 2924–2931 (1992).
[Crossref] [PubMed]

M. Chami, D. McKee, E. Leymarie, and G. Khomenko, “Influence of the angular shape of the volume-scattering function and multiple scattering on remote sensing reflectance,” Appl. Opt. 45(36), 9210–9220 (2006).
[Crossref] [PubMed]

O. Svensen, O. Frette, and S. R. Erga, “Scattering properties of microalgae: the effect of cell size and cell wall,” Appl. Opt. 46(23), 5762–5769 (2007).
[Crossref] [PubMed]

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]

E. Leymarie, D. Doxaran, and M. Babin, “Uncertainties associated to measurements of inherent optical properties in natural waters,” Appl. Opt. 49(28), 5415–5436 (2010).
[Crossref] [PubMed]

M. Babin, D. Stramski, R. A. Reynolds, V. M. Wright, and E. Leymarie, “Determination of the volume scattering function of aqueous particle suspensions with a laboratory multi-angle light scattering instrument,” Appl. Opt. 51(17), 3853–3873 (2012).
[Crossref] [PubMed]

G. W. Kattawar, “Genesis and evolution of polarization of light in the ocean,” Appl. Opt. 52(5), 940–948 (2013).
[Crossref] [PubMed]

Biogeosci. Discuss. (2)

M. W. Matthews and S. Bernard, “Using a two-layered sphere model to investigate the impact of gas vacuoles on the inherent optical properties of M. aeruginosa,” Biogeosci. Discuss. 10(6), 10531–10579 (2013).
[Crossref]

M. Bressac, C. Guieu, D. Doxaran, F. Bourrin, K. Desboeufs, N. Leblond, and C. Ridame, “Quantification of the lithogenic carbon pump following a dust deposition event,” Biogeosci. Discuss. 10(8), 13639–13677 (2013).
[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. Geophys. Res. (5)

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Figures (7)

Fig. 1
Fig. 1

A schematic representation of the measurement principle of the Mueller scattering matrix by POLVSM instrument: (a) side view, (b) view from above. Note that the prism P2 will be better illustrated in Fig. 2.

Fig. 2
Fig. 2

Characteristics of the prism P2 of the optical periscope of the POLVSM instrument. The red beam represents the direct incident light within the sample. The prism P2 has been designed to prevent the specular reflection of the incident beam onto the prism surfaces to enter the basin. An air gap is created inside the prism to avoid the light reflected onto the top surface of the prism to go back to the basin.

Fig. 3
Fig. 3

Schematic representation (from above) of the geometry of acquisition of the scattered light by the receiver at small angles position. The optical system is aligned to make the incident direct beam touching the edge of the prism P2. The distance between the edge of the prism and the direct incident beam is noted e. The distance between the two prisms P1 and P2 is noted D. The angle θmin is defined in Eq. (6).

Fig. 4
Fig. 4

Top view schematic representation of the POLVSM instrument configuration. All the geometrical parameters are defined as well.

Fig. 5
Fig. 5

Comparison of the nine Mueller matrix elements measured using the POLVSM instrument during a calibration experiment at 532 nm with Mie theory calculations. Microsphere beads of diameter 3.0 µm were used (see Table 1 for the specifications of the beads). The term M11 has been normalized to the scattering coefficient to get the phase function. The other matrix elements were divided by the term M11 to help the interpretation of the data. For example, the term M12/M11 corresponds to the degree of polarization of the beads.

Fig. 6
Fig. 6

Experimental results of three Mueller scattering matrix elements at 440 nm for two different polydispersed suspensions, namely mineral dust-like particles (brown dots) and the phytoplankton species Pseudo-nitzschia (green dots) under controlled laboratory conditions: (a) phase function (i.e., term M11/b), (b) degree of polarization (i.e., term M12/M11), (c) ratio M22/M11.

Fig. 7
Fig. 7

(a) Comparison between VSF measurements of inorganic dust-like hydrosols (POLVSM data, dotted line) with Mie theory (solid black line), (b) comparison of the degree of polarization (M12/M11) of inorganic dust-like hydrosols (POLVSM data, dotted colored line) with Mie theory (solid line), (c) same as Fig. 7(a) but for phytoplankton species Pseudo-nitzschia, (d) same as Fig. 7(b) but for phytoplankton species Pseudo-nitzschia.

Tables (3)

Tables Icon

Table 1 Specifications of the beadsa used for the calibration experiments of the POLVSM instrument.

Tables Icon

Table 2 Average value of the calibration coefficient C0mean (in digital counts*meter) and its standard deviation σ for each wavelength. The coefficient of variation (CV) is provided for each spectral bands. The mean value CVmean represents an estimation of the overall relative uncertainty in the POLVSM data.

Tables Icon

Table 3 Uncertainties in the main terms Mi,j/M11 of the Mueller matrix.

Equations (27)

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β( θ )= 2 Φ( θ ) EΩV
β( θ )= I( θ ) Er
b= Ω β( θ ) dΩ=2π 0 π β( θ )sinθdθ
b b =2π π/2 π β( θ )sinθdθ
I =( I Q U V )=MI=( m 11 m 12 m 13 m 14 m 21 m 22 m 23 m 24 m 31 m 32 m 33 m 34 m 41 m 42 m 43 m 44 )( I Q U V )
θ min =arctan( 2e D )=1°
I ˜ scat =A*M*P* I ˜ inc
A(α)=P(α)= 1 2 ( 1 cos2α sin2α cos2α cos 2 2α sin2αcos2α sin2α sin2αcos2α sin 2 2α )
{ L( θ )= d sinθ , if L( θ )<D L( θ )=D, otherwise.
I scat (s)= e c(s+r) A*M*P* I inc
r=R+( D 2 s )cosθ
I measured = s= D 2 L 2 D 2 + L 2 I scat ds = s= D 2 L 2 D 2 + L 2 e c( s+r ) A*M*Pds
I measured =A*M*P s= D 2 L 2 D 2 + L 2 e c( s+r ) ds =A*M*P*T
T= s= D 2 L 2 D 2 + L 2 e c( s+r ) ds = e c( R+ D 2 cosθ ) s= D 2 L 2 D 2 + L 2 e c( s( 1cosθ ) ) ds
T( θ,c )= 2 e c( R+ D 2 ) c( 1cosθ ) sinh( cL( 1cosθ ) 2 )
M( θ )=T ( θ,c ) 1 A -1 I scat ( θ ) P -1
M hydrosols ( θ )= M sample ( θ ) M water ( θ )
C 0 = 2π θ=0 π M 11 uncal ( θ ) sinθdθ b known
C 0 = 2π θ= θ min θ max M 11 uncal ( θ ) sinθdθ b known
b known =2π θ min θ max M 11 known ( θ )sinθdθ
M 11 (θ)= M 11 ( θ min )exp( S forward *θ) if θ< θ min
NRMS E p = 1 n k=1 n ( M ˜ i,j,measured ( θ k ) M ˜ i,j,theory ( θ k ) ) 2 M ˜ i,j,theory,max , M ˜ i,j,theory,min
{ I( θ a1 , θ p1 )=( 1 cos2 θ a1 sin2 θ a1 )M( ( 1cos2 θ p1 ) cos2 θ p1 ( 1cos2 θ p1 ) sin2 θ p1 ( 1cos2 θ p1 ) ) I( θ ai , θ pj )=( 1 cos2 θ ai sin2 θ ai )M( ( 1cos2 θ pj ) cos2 θ pj ( 1cos2 θ pj ) sin2 θ pj ( 1cos2 θ pj ) ) I( θ a N a , θ p N p )=( 1 cos2 θ a N a sin2 θ a N a )M( ( 1cos2 θ p N p ) cos2 θ p N p ( 1cos2 θ p N p ) sin2 θ p N p ( 1cos2 θ p N p ) )
[ I( θ a1 , θ p1 ) I( θ a1 , θ pj ) I( θ a1 , θ p N p ) I( θ ai , θ p1 ) I( θ ai , θ pj ) I( θ ai , θ p N p ) I( θ a N a , θ p1 ) I( θ a N a , θ pj ) I( θ a N a , θ p N p ) ]=[ a( θ a1 ) a( θ ai ) a( θ a N a ) ]M[ p( θ p1 ) p( θ pj ) p( θ p N p ) ]
I sys ( N a × N p )= A sys ( N a ×3 )M( 3×3 ) P sys ( 3× N p )
M ^ = ( A sys t A sys ) -1 A sys t I sys P sys t ( P sys P sys t ) -1
M ^ = A sys -1 I sys P sys -1

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