Abstract

Absorption by non-algal particles (NAP, ad) and colored dissolved organic matter (CDOM, ag) are frequently modeled by exponential functions of wavelength, either separately or as a sum. We present a new representation of NAP-plus-CDOM absorption adg based on the stretched exponential function adg(λ) = A exp{−[s(λλo)]β}, whose parameter β can be considered a measure of optical heterogeneity. A double exponential representation of adg can be fit extremely well by a stretched exponential for all plausible parameter combinations, despite having one fewer free parameter than a double exponential. Fitting two published compilations of in situ adg data – one at low spectral resolution (n = 5, λ = 412–555 nm) and one at high spectral resolution (n = 201, λ = 300–700 nm) – the stretched exponential outperforms the single exponential, double exponential, and a power law. We thereby conclude that the stretched exponential is the preferred model for adg absorption in circumstances when NAP and CDOM cannot be separated, such as in remote sensing inversions.

© 2017 Optical Society of America

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References

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  1. C. S. Roesler, M. J. Perry, and K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive inland marine waters,” Limnol. Oceanogr. 34(8), 1510–1523 (1989).
    [Crossref]
  2. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  3. C. S. Roesler and M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 10013279–13294 (1995).
    [Crossref]
  4. S. A. Garver and D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
    [Crossref]
  5. Z.P. Lee, K. L. Carder, and R. Arnone, “Deriving inherent optical properties from water color: a multi-band quasianalytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002).
    [Crossref] [PubMed]
  6. P. J. Werdell, B. A. Franz, S. W. Bailey, G. C. Feldman, E. Boss, V. E. Brando, M. Dowell, T. Hirata, S. J. Lavender, Z. P. Lee, H. Loisel, S. Maritorena, F. Mélin, T. S. Moore, T. J. Smyth, D. Antoine, E. Devred, O. Hembise Fanton d’Andon, and A. Mangin, “Generalized ocean color inversion model for retrieving marine inherent optical properties,” Appl. Opt. 52(10) 2019–2037 (2013).
    [Crossref] [PubMed]
  7. M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Marine Chemistry 89(1), 69–88 (2004).
    [Crossref]
  8. K. S. Shifrin, Physical Optics of Ocean Water (Springer Science & Business Media, 1988).
  9. R. Kohlrausch, “Theorie des elektrischen Rückstandes in der Leidner Flasche,” Annalen der Physik un Chemie 9156–82 (1854).
    [Crossref]
  10. D. C. Johnston, “Stretched exponential relaxation arising from a continous sum of exponential decays,” Phys. Rev. B 74184430 (2006).
    [Crossref]
  11. K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: Their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34(1), 68–81 (1989).
    [Crossref]
  12. 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(C7), 3211 (2003).
    [Crossref]
  13. Z. P. Lee, ed. “Reports of the International Ocean-Colour Coordinating Group, No. 5, Remote sensing of inherent optical properties: fundamentals tests of algorithms, and applications,” IOCCG, Dartmouth, NS, Canada (2006).
  14. A. Bricaud, M. Bavin, H. Claustre, J. Ras, and F. Tieche, “Light absorption properties and absorption budget of South East Pacific waters,” J. Geophys. Res. 115(C08) 009 (2010).
    [Crossref]
  15. “ac-s In Situ Spectrophotometer Datasheet,” Sea-bird Scientific (WET Labs), Philomath, OR, USA (2016).

2013 (1)

2010 (1)

A. Bricaud, M. Bavin, H. Claustre, J. Ras, and F. Tieche, “Light absorption properties and absorption budget of South East Pacific waters,” J. Geophys. Res. 115(C08) 009 (2010).
[Crossref]

2006 (1)

D. C. Johnston, “Stretched exponential relaxation arising from a continous sum of exponential decays,” Phys. Rev. B 74184430 (2006).
[Crossref]

2004 (1)

M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Marine Chemistry 89(1), 69–88 (2004).
[Crossref]

2003 (1)

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(C7), 3211 (2003).
[Crossref]

2002 (1)

1997 (1)

S. A. Garver and D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[Crossref]

1995 (1)

C. S. Roesler and M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 10013279–13294 (1995).
[Crossref]

1989 (2)

C. S. Roesler, M. J. Perry, and K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive inland marine waters,” Limnol. Oceanogr. 34(8), 1510–1523 (1989).
[Crossref]

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: Their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34(1), 68–81 (1989).
[Crossref]

1854 (1)

R. Kohlrausch, “Theorie des elektrischen Rückstandes in der Leidner Flasche,” Annalen der Physik un Chemie 9156–82 (1854).
[Crossref]

Antoine, D.

Arnone, R.

Babin, 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(C7), 3211 (2003).
[Crossref]

Bailey, S. W.

Bavin, M.

A. Bricaud, M. Bavin, H. Claustre, J. Ras, and F. Tieche, “Light absorption properties and absorption budget of South East Pacific waters,” J. Geophys. Res. 115(C08) 009 (2010).
[Crossref]

Boss, E.

Brando, V. E.

Bricaud, A.

A. Bricaud, M. Bavin, H. Claustre, J. Ras, and F. Tieche, “Light absorption properties and absorption budget of South East Pacific waters,” J. Geophys. Res. 115(C08) 009 (2010).
[Crossref]

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(C7), 3211 (2003).
[Crossref]

Carder, K. L.

Z.P. Lee, K. L. Carder, and R. Arnone, “Deriving inherent optical properties from water color: a multi-band quasianalytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002).
[Crossref] [PubMed]

C. S. Roesler, M. J. Perry, and K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive inland marine waters,” Limnol. Oceanogr. 34(8), 1510–1523 (1989).
[Crossref]

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: Their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34(1), 68–81 (1989).
[Crossref]

Claustre, H.

A. Bricaud, M. Bavin, H. Claustre, J. Ras, and F. Tieche, “Light absorption properties and absorption budget of South East Pacific waters,” J. Geophys. Res. 115(C08) 009 (2010).
[Crossref]

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(C7), 3211 (2003).
[Crossref]

Devred, E.

Donaghay, P. L.

M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Marine Chemistry 89(1), 69–88 (2004).
[Crossref]

Dowell, M.

Feldman, G. 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(C7), 3211 (2003).
[Crossref]

Franz, B. A.

Garver, S. A.

S. A. Garver and D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[Crossref]

Harvey, G. R.

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: Their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34(1), 68–81 (1989).
[Crossref]

Hembise Fanton d’Andon, O.

Hirata, T.

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(C7), 3211 (2003).
[Crossref]

Johnston, D. C.

D. C. Johnston, “Stretched exponential relaxation arising from a continous sum of exponential decays,” Phys. Rev. B 74184430 (2006).
[Crossref]

Kohlrausch, R.

R. Kohlrausch, “Theorie des elektrischen Rückstandes in der Leidner Flasche,” Annalen der Physik un Chemie 9156–82 (1854).
[Crossref]

Lavender, S. J.

Lee, Z. P.

Lee, Z.P.

Loisel, H.

Mangin, A.

Maritorena, S.

Mélin, F.

Mobley, C. D.

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

Moore, T. S.

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(C7), 3211 (2003).
[Crossref]

Ortner, P. B.

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: Their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34(1), 68–81 (1989).
[Crossref]

Perry, M. J.

C. S. Roesler and M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 10013279–13294 (1995).
[Crossref]

C. S. Roesler, M. J. Perry, and K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive inland marine waters,” Limnol. Oceanogr. 34(8), 1510–1523 (1989).
[Crossref]

Ras, J.

A. Bricaud, M. Bavin, H. Claustre, J. Ras, and F. Tieche, “Light absorption properties and absorption budget of South East Pacific waters,” J. Geophys. Res. 115(C08) 009 (2010).
[Crossref]

Roesler, C. S.

C. S. Roesler and M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 10013279–13294 (1995).
[Crossref]

C. S. Roesler, M. J. Perry, and K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive inland marine waters,” Limnol. Oceanogr. 34(8), 1510–1523 (1989).
[Crossref]

Shifrin, K. S.

K. S. Shifrin, Physical Optics of Ocean Water (Springer Science & Business Media, 1988).

Siegel, D. A.

S. A. Garver and D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[Crossref]

Smyth, T. J.

Steward, R. G.

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: Their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34(1), 68–81 (1989).
[Crossref]

Stramski, D.

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(C7), 3211 (2003).
[Crossref]

Sullivan, J. M.

M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Marine Chemistry 89(1), 69–88 (2004).
[Crossref]

Tieche, F.

A. Bricaud, M. Bavin, H. Claustre, J. Ras, and F. Tieche, “Light absorption properties and absorption budget of South East Pacific waters,” J. Geophys. Res. 115(C08) 009 (2010).
[Crossref]

Twardowski, M. S.

M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Marine Chemistry 89(1), 69–88 (2004).
[Crossref]

Werdell, P. J.

Annalen der Physik un Chemie (1)

R. Kohlrausch, “Theorie des elektrischen Rückstandes in der Leidner Flasche,” Annalen der Physik un Chemie 9156–82 (1854).
[Crossref]

Appl. Opt. (2)

J. Geophys. Res. (4)

C. S. Roesler and M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 10013279–13294 (1995).
[Crossref]

S. A. Garver and D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation 1. Time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[Crossref]

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(C7), 3211 (2003).
[Crossref]

A. Bricaud, M. Bavin, H. Claustre, J. Ras, and F. Tieche, “Light absorption properties and absorption budget of South East Pacific waters,” J. Geophys. Res. 115(C08) 009 (2010).
[Crossref]

Limnol. Oceanogr. (2)

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: Their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr. 34(1), 68–81 (1989).
[Crossref]

C. S. Roesler, M. J. Perry, and K. L. Carder, “Modeling in situ phytoplankton absorption from total absorption spectra in productive inland marine waters,” Limnol. Oceanogr. 34(8), 1510–1523 (1989).
[Crossref]

Marine Chemistry (1)

M. S. Twardowski, E. Boss, J. M. Sullivan, and P. L. Donaghay, “Modeling the spectral shape of absorption by chromophoric dissolved organic matter,” Marine Chemistry 89(1), 69–88 (2004).
[Crossref]

Phys. Rev. B (1)

D. C. Johnston, “Stretched exponential relaxation arising from a continous sum of exponential decays,” Phys. Rev. B 74184430 (2006).
[Crossref]

Other (4)

Z. P. Lee, ed. “Reports of the International Ocean-Colour Coordinating Group, No. 5, Remote sensing of inherent optical properties: fundamentals tests of algorithms, and applications,” IOCCG, Dartmouth, NS, Canada (2006).

“ac-s In Situ Spectrophotometer Datasheet,” Sea-bird Scientific (WET Labs), Philomath, OR, USA (2016).

K. S. Shifrin, Physical Optics of Ocean Water (Springer Science & Business Media, 1988).

C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).

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Figures (2)

Fig. 1
Fig. 1 a) Example of a stretched exponential (dashed line) fit to a double exponential (solid line). b) Residual difference (solid line) between the two curves in Fig. 1(a).
Fig. 2
Fig. 2 Histogram of β estimates from the regression to the high-resolution (BIOSOPE) data and to the low-resolution (IOCCG) data.

Tables (2)

Tables Icon

Table 1 Summary statistics for stretched exponential fits to the sum of two exponential functions. Across fits to all parameter combinations, the median, minimum, and maximum r2 and RMSE are reported, as well as the median, minimum, and maximum of s and β.

Tables Icon

Table 2 Summary statistics for exponential fits to in situ data. First column corresponds to data compilation; ‘High-resolution’ data are those compiled in Ref. [14], and ‘Low-resolution’ data are those compiled in Ref. [13]. Second column corresponds to which function the stretched exponential is being compared: single exponential, double exponential, and power law. Third column reports the number of cases where the stretched exponential had a higher r2. Fourth column reports the number of cases where the stretched exponential had a lower RMSE.

Equations (4)

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a ( λ ) = a w ( λ ) + a ϕ ( λ ) + a d ( λ ) + a g ( λ )
a d ( λ ) = A d ( λ o ) exp [ s d ( λ λ o ) ] , a g ( λ ) = A g ( λ o ) exp [ s g ( λ λ o ) ]
a d g ( λ ) = A exp { [ s ( λ λ o ) ] β } , β [ 0 , 1 ] .
a i , j , k ( λ ) = A n exp [ s i ( λ λ o ) ] + A n R j exp [ s k ( λ λ o ) ]

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