## The use of equivalent size distributions of natural phytoplankton assemblages for optical modeling

Optics Express, Vol. 15, Issue 5, pp. 1995-2007 (2007)

http://dx.doi.org/10.1364/OE.15.001995

Acrobat PDF (478 KB)

### Abstract

The effective cell size is expected to be one of the principal causes of variability in the inherent optical properties (IOPs) of a phytoplankton population. However, establishing simple size descriptors is complicated by the typically complex particle size distributions of natural phytoplankton assemblages. This study compares the use of measured and equivalent particle size distributions on the modeled IOPs of a wide range of natural phytoplankton assemblages. It demonstrates that several equivalent size distributions, using simple parameterizations of complex size distributions based on the effective radius or diameter, are capable of modeling phytoplankton IOPs with sufficient accuracy for further use in marine bio-optical models. The results offered here are expected to be of use in bio-optical studies of phytoplankton dynamics e.g. harmful algal bloom oriented inverse reflectance models.

© 2007 Optical Society of America

## 1. Introduction

1. Z.V. Finkel and A.J. Irwin, “Modelling size-dependent photosynthesis: light absorption and the allometric rule,”. J. Theor. Biol. **204**, 361–369 (2000) [CrossRef] [PubMed]

2. A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. **28**, 1375–1393 (1981). [CrossRef]

3. D.J.S. Montagnes, J.A. Berges, P.J. Harrison, and F.J.R. Taylor, “Estimating carbon, nitrogen, protein, and chlorophyll a from cell volume in marine phytoplankton,” Limnol. Oceanogr. **39**,1044–1060 (1994). [CrossRef]

4. J. Rodriguez, J. Tintore, J.T. Allen, J.M. Blanco, D. Gomis, A. Reul, J. Ruiz, V. Rodriguez, F. Echevarria, and F. Jimenez-Gomez, “Mesoscale vertical motion and the size structure of phytoplankton in the ocean,” Nature **410**, 360–363 (2001). [CrossRef] [PubMed]

5. E. Boss, M.S. Twardowski, and S. Herring, “Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution,” Appl. Opt. **40**, 4885–4893 (2001). [CrossRef]

6. A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. **25**, 571–580 (1986). [CrossRef] [PubMed]

7. D. Risovic, “Two component model of sea particle size distribution,” Deep-Sea Res. **40**, 1459–1473 (1993). [CrossRef]

8. M. Jonasz and G. Fournier, “Approximation of the size distribution of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. **41**, 744–754 (1996). [CrossRef]

7. D. Risovic, “Two component model of sea particle size distribution,” Deep-Sea Res. **40**, 1459–1473 (1993). [CrossRef]

5. E. Boss, M.S. Twardowski, and S. Herring, “Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution,” Appl. Opt. **40**, 4885–4893 (2001). [CrossRef]

8. M. Jonasz and G. Fournier, “Approximation of the size distribution of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. **41**, 744–754 (1996). [CrossRef]

10. R.W. Sheldon, A. Prakash, and W.H. Sutcliffe, Jr., “The size distribution of particles in the ocean,” Limnol. Oceanogr. **17**, 327–340 (1972). [CrossRef]

10. R.W. Sheldon, A. Prakash, and W.H. Sutcliffe, Jr., “The size distribution of particles in the ocean,” Limnol. Oceanogr. **17**, 327–340 (1972). [CrossRef]

8. M. Jonasz and G. Fournier, “Approximation of the size distribution of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. **41**, 744–754 (1996). [CrossRef]

12. Y.X. Hu and K. Stamnes, “An accurate Parameterization of Cloud Radiative Properties Suitable for Climate Modeling,” J. Climate. **6**, 728–742 (1993). [CrossRef]

12. Y.X. Hu and K. Stamnes, “An accurate Parameterization of Cloud Radiative Properties Suitable for Climate Modeling,” J. Climate. **6**, 728–742 (1993). [CrossRef]

14. R. McGraw, S. Nemesure, and S. E. Schwartz, “Properties and evolution of aerosols with size distributions having identical moments,” J. Aerosol. Sci. **29**, 761–772 (1998). [CrossRef]

15. J.E. Hansen and L.D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. **16**, 527–610 (1974). [CrossRef]

*k*th radial (or diametric) moment of a size distribution is given by:

*r*is the particle radius (m), and

*F(r) d(r)*is the number of particles per unit volume in the size range

*r*± 1/2

*d(r)*. The best single parameter describing the optical properties of a size distribution is the effective radius

*r*or diameter

_{eff}*D*(

_{eff}*r*= 0.5

_{eff}*D*) – the ratio of the third to second moment (<

_{eff}*r*

^{3}>/<

*r*

^{2}>), or the mean volume to surface area ratio of the distribution [15

15. J.E. Hansen and L.D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. **16**, 527–610 (1974). [CrossRef]

*V*[(<

_{eff}*r*

^{4}><

*r*

^{2}>/<

*r*

^{3}>

^{2})-1)], which describes the width of the distribution [15

15. J.E. Hansen and L.D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. **16**, 527–610 (1974). [CrossRef]

## 2. Methods

- 1. Calculation of algal spectral refractive index data from absorption and size distribution measurements, in conjunction with an anomalous diffraction approximation model [6] and a refractive index dispersion model [11]. Principal measurements consisted of particulate absorption, particle size distributions and intra-cellular pigments, and were made on 34 surface samples in a variety of waters in the southern Benguela (Table 1).
6. A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt.

**25**, 571–580 (1986). [CrossRef] [PubMed] - 2. Derived refractive index data were used to model a suite of algal inherent optical properties (IOPs), using both measured size distribution data and several equivalent size distribution formulations. Equivalent size distributions were scaled to have the same total projected area and effective diameters as those measured. IOPs include the absorption, attenuation, scattering, and backscattering coefficients, and the package effect parameter.
- 3. An assessment of the potential errors associated with use of the equivalent size distribution formulations for the suite of IOPs was performed.

### 2.1 Particulate absorption

16. C.S. Yentsch, “Measurement of visible light absorption by particulate matter in the ocean,” Limnol. Oceanogr. **7**, 207–217 (1962). [CrossRef]

17. C.S. Roesler, “Theoretical and experimental approaches to improve the accuracy of particulate absorption coefficients derived from the quantitative filter technique,” Limnol. Oceanogr. **43**, 1649–1660 (1998). [CrossRef]

17. C.S. Roesler, “Theoretical and experimental approaches to improve the accuracy of particulate absorption coefficients derived from the quantitative filter technique,” Limnol. Oceanogr. **43**, 1649–1660 (1998). [CrossRef]

*a*data were then obtained by subtraction of detrital absorption

_{ϕ}(λ)*a*from total particulate absorption

_{d}(λ)*a*.Both the filter pad and null-point correction methods suffer from unknown errors associated with the poorly known scattering properties of marine particulates [19

_{p}(λ)19. D. Stramski and J. Piskozub, “Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from 3-D radiative transfer equations,” Appl. Opt. , **42**, 3634–46 (2003). [CrossRef] [PubMed]

### 2.2 Particle size distributions (PSD)

*Aureococcus anophagefferens*. Samples were diluted to keep coincidence levels below 10%, and 40 ml of sample was typically counted. A numerical technique was employed to fractionate measured size distributions into algal and non-algal components. The detrital component of the particle population was assumed to obey a Junge distribution (Eq. 8), with diameters ranging from 0.7 μm to 100 μm, in log-spaced bins. An inverse anomalous diffraction model was then used to fit measured detrital absorption

*a*, using an imaginary refractive index of

_{d}(λ)*n*́ =0.001066 exp(-0.007168

*λ*) [20

20. D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. **40**, 2929–2945 (2001). [CrossRef]

### 2.3 Pigments

21. R.G. Barlow, D. G. Cummings, and S. W. Gibb, “Improved resolution of mono- and divinyl chlorophylls a and b and zeaxanthin and lutein in phytoplankton extracts using reverse phase C-8 HPLC,” Mar. Ecol. Prog. Ser. **161**, 303–307 (1997). [CrossRef]

### 2.4 Refractive index determinations

*a*(

_{ϕ}*λ*) is the algal absorption coefficient (m

^{2}mg

^{-1}),

*Q*̅

_{a}(

*λ*) is the absorption efficiency factor (where the overbar signifies the mean efficiency factor of a particle population),

*r*is the particle radius,

*F(r)d(r)*is the number of particles per unit volume in the size range

*r*± 1/2

*d(r)*, and

*λ*denotes wavelength. Using the measured algal absorption coefficients and size distribution data, as described above, the mean absorption efficiency factor

*Q*̅

_{a}(

*λ*) can be calculated for natural algal assemblages [6

**25**, 571–580 (1986). [CrossRef] [PubMed]

*4αn*́, where

*α*is the Mie size parameter

*2rπ/λ*, and

*n*́(λ) is the imaginary part of the refractive index. Calculations of additional inherent optical properties, namely attenuation and scattering coefficients, may be made if appropriate values of the real part of the refractive index are also derived. The Kramers-Kronig relations [11,24] were employed to derive spectral variations in the real part of the index, typically denoted as

*Δn*(

*λ*) from the imaginary part of the index. The central value of

*Δn*around which

*Δn*(

*λ*) varies, denoted as 1+ε, was fixed at 1.05 for all samples, chosen as a representative value for phytoplankton [6

**25**, 571–580 (1986). [CrossRef] [PubMed]

25. A. Bricaud, A.L. Bedhomme, and A. Morel, “Optical properties of diverse phytoplanktonic species: experimental results and theoretical interpretation,” J. Plankton Res. **10**, 851–873 (1988). [CrossRef]

*n*(

*λ*) data should be considered as representative theoretical values generated for the sole purpose of comparing the optical properties of equivalent size distributions - accurate determinations of 1+ ε would require additional use of attenuation or scattering data, not available to this study [6

**25**, 571–580 (1986). [CrossRef] [PubMed]

*a*-specific data to negate the impact of varying biomass concentration upon model performance assessment. Whilst Chl

*a*-specific phytoplankton absorption is a common bio-optical parameter [25

25. A. Bricaud, A.L. Bedhomme, and A. Morel, “Optical properties of diverse phytoplanktonic species: experimental results and theoretical interpretation,” J. Plankton Res. **10**, 851–873 (1988). [CrossRef]

*a*-specific size distributions are rarely used. However, test results employing absorption and size data normalised to Chl

*a*concentrations both before and after optical modeling confirmed that such scaling made no impact upon the refractive index and efficiency factor analyses.

### 2.5 Equivalent size distributions

*SA*> of the measured algal size distribution, as given by:

*ASF*(Area Scaling Factor) term is introduced here as the total projected surface area scaling parameter, i.e. it is used to manipulate the magnitude of the equivalent distributions by matching the total projected surface area to that of the measured distribution. The following four size distribution functions were assessed, expressed in radial terms for algebraic simplicity:

- 1. The special -7/2 generalised inverse Gaussian distribution [13]
13. M.D. Alexandrov and A.A. Lacis, “A new three-parameter cloud/aerosol particle size distribution based on the generalized inverse Gaussian density function,” Appl. Math. Comput.

**116**, 153–165 (2000) [CrossRef] - 2. The Standard distribution [15
**16**, 527–610 (1974). [CrossRef] - 3. The 0
^{th}order log normal distribution [8**41**, 744–754 (1996). [CrossRef]**16**, 527–610 (1974). [CrossRef]where*r*=_{g}*r*+_{eff}/(l*v*and σ_{eff})^{5/2}^{2}_{g}= ln*(l*+*v*._{eff}) - 4. The Junge distribution [9]In addition to the above distributions, a further approximation was assessed:
- 5. A single size approximation of
*r*=*r*(or_{eff}*D*=*D*)_{eff}

5. E. Boss, M.S. Twardowski, and S. Herring, “Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution,” Appl. Opt. **40**, 4885–4893 (2001). [CrossRef]

**25**, 571–580 (1986). [CrossRef] [PubMed]

*r*of the Junge distribution matched that of the measured distribution. The single size approximation was analysed as a simple and computationally economic alternative. Two analyses were carried out with regard to the effective variance

_{eff}*r*: the first using the variable experimental

_{eff}*r*values determined for each sample, and the second using a constant

_{eff}*r*value of 0.63, the mean value for all samples.

_{eff}### 2.6 Optical modeling

*c*(

*λ*), the scattering coefficient

*b*(

*λ*), the backscattering coefficient

*b*(λ) and their relative efficiency factors. Additional detail on such models can be found in [22]. In addition, the dimensionless package effect parameter (

_{b}*Q*

_{a}^{*}) can be calculated from the following expression [2

2. A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. **28**, 1375–1393 (1981). [CrossRef]

*a*is the absorption of cellular material (m

_{cm}^{-1}) and is given by

*a*= 4

_{cm}*πń/λ*[

*ibid*.]. For each sample two calculations of the suite of IOPs were made. The first was made using the measured algal size distribution (calculated as described in section 2.2) with corresponding refractive index data (as described in section 2.4) – these are referred to as the “measured” properties. The second was made using the equivalent algal size distribution (calculated as described in section 2.5) using exactly the same refractive index data as for the “measured” calculations – these are referred to as the “equivalent” properties.

*a*is the modeled absorption of the equivalent size distribution, and

_{equiv}*a*is the modeled absorption of the measured size distribution. The above expressions are for absorption – analogous expressions are employed for other IOPs.

_{meas}26. A.L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. **22**, 1242–1246 (1951). [CrossRef]

27. O.B. Toon and T.P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. **20**, 3657–3660 (1981). [CrossRef] [PubMed]

## 3. Results and discussion

*a*-specific phytoplankton absorption, size distributions and refractive index data are displayed in Fig. 1. The absorption data (Fig. 1(a)) demonstrate the effects of varying assemblage size and pigmentation [2

2. A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. **28**, 1375–1393 (1981). [CrossRef]

25. A. Bricaud, A.L. Bedhomme, and A. Morel, “Optical properties of diverse phytoplanktonic species: experimental results and theoretical interpretation,” J. Plankton Res. **10**, 851–873 (1988). [CrossRef]

**10**, 851–873 (1988). [CrossRef]

### 3.1 Optical properties of equivalent size distributions

13. M.D. Alexandrov and A.A. Lacis, “A new three-parameter cloud/aerosol particle size distribution based on the generalized inverse Gaussian density function,” Appl. Math. Comput. **116**, 153–165 (2000) [CrossRef]

**16**, 527–610 (1974). [CrossRef]

*c*and total scattering

*b*to within 10%, absorption

*a*and package effect parameter

*Q*

_{a}^{*}to within 6%, and backscattering

*b*to within 20%. The Standard distribution, offering a simpler algebraic expression, gives similar performance with the exception of slightly higher maximum backscattering errors of ∼25 %.

_{b}28. A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 water),” J. Geophys. Res. , **93**, 10,749-10,768 (1988). [CrossRef]

29. A.M. Ciotti, M. R. Lewis, and J.J. Cullen, “Assessment of the relationships between dominant cell size in natural phytoplankton communities and the spectral shape of the absorption coefficient,” Limnol. Oceanogr. **47**, 404–417 (2002). [CrossRef]

**16**, 527–610 (1974). [CrossRef]

30. M.I. Mishchenko and A.A. Lacis, “Morphology-dependent resonances of nearly spherical particles in random orientation,” Appl. Opt. **42**, 5551–5556 (2003). [CrossRef] [PubMed]

30. M.I. Mishchenko and A.A. Lacis, “Morphology-dependent resonances of nearly spherical particles in random orientation,” Appl. Opt. **42**, 5551–5556 (2003). [CrossRef] [PubMed]

*V*appears to have had little adverse effect on the returned errors, allowing both distributions to be expressed through two parameters for potential inversion applications: the effective diameter and a scaling parameter. The relative lack of sensitivity to

_{eff}*V*would appear to be due at least in part to the relatively dispersed nature of the majority of the algal assemblages analysed, even in high biomass bloom scenarios. Thus, whilst the assumption of a relatively high constant

_{eff}*V*of 0.63 appears appropriate for natural algal assemblages in productive coastal systems, application either in truly oligotrophic waters or to very highly size-constrained mono-specific blooms or cultures may require further validation.

_{eff}*Q*

_{a}^{*}vs

*D*(or

_{eff}**28**, 1375–1393 (1981). [CrossRef]

*Q*

_{a}^{*}calculations (Eq. 9). Whilst

*Q*and

_{a}*Q*

_{a}^{*}are obviously both directly dependent upon

*Q*(675) when considering the entire data set – thus the assemblage-averaged effective diameter and imaginary refractive index data play a greater role in the package effect calculations relative to those of the other inherent optical properties. The good performance of the single-size package-effect derivation offers a potentially extremely useful formulation: the ability to robustly express the Chl a-specific phytoplankton absorption of natural assemblages through a single effective diameter parameter via the package effect [2

_{a}**28**, 1375–1393 (1981). [CrossRef]

**40**, 4885–4893 (2001). [CrossRef]

10. R.W. Sheldon, A. Prakash, and W.H. Sutcliffe, Jr., “The size distribution of particles in the ocean,” Limnol. Oceanogr. **17**, 327–340 (1972). [CrossRef]

^{th}order formulation of the log-normal distribution employed here [15

**16**, 527–610 (1974). [CrossRef]

**41**, 744–754 (1996). [CrossRef]

**116**, 153–165 (2000) [CrossRef]

*D*, and the scaling parameter

_{eff}*ASF*. Such data demonstrate both the utility of equivalent size distributions with regard to optical simulation, and the disadvantages of replicating size distributions by matching measured shape. Simulation of the measured volume size distribution shape in Fig. 3(a) would require a minimum of three discrete distributions, each described by two parameters [8

**41**, 744–754 (1996). [CrossRef]

## 4 Application and conclusions

*ASF*for the inverse Gaussian and Standard distributions reveals a close relationship with effective diameter

*D*for the samples analysed using a mean effective variance of 0.63 (Fig. 4). The

_{eff}*ASF*parameter is used to manipulate the magnitude of the equivalent distributions by matching the total projected surface area to that of the measured distribution. The ability to express the

*ASF*to

*D*relationship using a power law allows both distributions to be parameterised using a single variable, the effective diameter (Fig 4).

_{eff}*a*-specific algal size distributions to be expressed as single variable functions, assuming a mean effective variance of 0.63:

- 1. Inverse Gaussian Chl
*a*-specific algal size distribution - 2. Standard Chl
*a*-specific algal size distribution

## References and links

1. | Z.V. Finkel and A.J. Irwin, “Modelling size-dependent photosynthesis: light absorption and the allometric rule,”. J. Theor. Biol. |

2. | A. Morel and A. Bricaud, “Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton,” Deep-Sea Res. |

3. | D.J.S. Montagnes, J.A. Berges, P.J. Harrison, and F.J.R. Taylor, “Estimating carbon, nitrogen, protein, and chlorophyll a from cell volume in marine phytoplankton,” Limnol. Oceanogr. |

4. | J. Rodriguez, J. Tintore, J.T. Allen, J.M. Blanco, D. Gomis, A. Reul, J. Ruiz, V. Rodriguez, F. Echevarria, and F. Jimenez-Gomez, “Mesoscale vertical motion and the size structure of phytoplankton in the ocean,” Nature |

5. | E. Boss, M.S. Twardowski, and S. Herring, “Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution,” Appl. Opt. |

6. | A. Bricaud and A. Morel, “Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling,” Appl. Opt. |

7. | D. Risovic, “Two component model of sea particle size distribution,” Deep-Sea Res. |

8. | M. Jonasz and G. Fournier, “Approximation of the size distribution of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. |

9. | E.C. Junge, |

10. | R.W. Sheldon, A. Prakash, and W.H. Sutcliffe, Jr., “The size distribution of particles in the ocean,” Limnol. Oceanogr. |

11. | S. Bernard, T.A. Probyn, and R.G. Barlow, “Measured and modelled optical properties of particulate matter in the southern Benguela,” S. Afr. J. Sci. |

12. | Y.X. Hu and K. Stamnes, “An accurate Parameterization of Cloud Radiative Properties Suitable for Climate Modeling,” J. Climate. |

13. | M.D. Alexandrov and A.A. Lacis, “A new three-parameter cloud/aerosol particle size distribution based on the generalized inverse Gaussian density function,” Appl. Math. Comput. |

14. | R. McGraw, S. Nemesure, and S. E. Schwartz, “Properties and evolution of aerosols with size distributions having identical moments,” J. Aerosol. Sci. |

15. | J.E. Hansen and L.D. Travis, “Light scattering in planetary atmospheres,” Space. Sci. Rev. |

16. | C.S. Yentsch, “Measurement of visible light absorption by particulate matter in the ocean,” Limnol. Oceanogr. |

17. | C.S. Roesler, “Theoretical and experimental approaches to improve the accuracy of particulate absorption coefficients derived from the quantitative filter technique,” Limnol. Oceanogr. |

18. | M. Kishino, M. Takahashi, N. Okami, and S. Ichimura, “Estimation of the spectral absorption coefficients of phytoplankton in the sea,” Bull. Mar. Sci. |

19. | D. Stramski and J. Piskozub, “Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from 3-D radiative transfer equations,” Appl. Opt. , |

20. | D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. |

21. | R.G. Barlow, D. G. Cummings, and S. W. Gibb, “Improved resolution of mono- and divinyl chlorophylls a and b and zeaxanthin and lutein in phytoplankton extracts using reverse phase C-8 HPLC,” Mar. Ecol. Prog. Ser. |

22. | A. Morel and A. Bricaud, “Inherent properties of algal cells including picoplankton: theoretical and experimental results,” Can. Bull. Fish. Aquat. Sci. , |

23. | H.C. Van de Hulst,. |

24. | C.F. Bohren and D.R. Huffman, |

25. | A. Bricaud, A.L. Bedhomme, and A. Morel, “Optical properties of diverse phytoplanktonic species: experimental results and theoretical interpretation,” J. Plankton Res. |

26. | A.L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. |

27. | O.B. Toon and T.P. Ackerman, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. |

28. | A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case 1 water),” J. Geophys. Res. , |

29. | A.M. Ciotti, M. R. Lewis, and J.J. Cullen, “Assessment of the relationships between dominant cell size in natural phytoplankton communities and the spectral shape of the absorption coefficient,” Limnol. Oceanogr. |

30. | M.I. Mishchenko and A.A. Lacis, “Morphology-dependent resonances of nearly spherical particles in random orientation,” Appl. Opt. |

**OCIS Codes**

(010.4450) Atmospheric and oceanic optics : Oceanic optics

(290.4020) Scattering : Mie theory

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: September 13, 2006

Revised Manuscript: November 23, 2006

Manuscript Accepted: November 28, 2006

Published: March 5, 2007

**Virtual Issues**

Vol. 2, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

S. Bernard, F. A. Shillington, and T. A. Probyn, "The use of equivalent size distributions of natural phytoplankton assemblages for optical modeling," Opt. Express **15**, 1995-2007 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-1995

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### References

- Z. V. Finkel and A. J. Irwin, "Modelling size-dependent photosynthesis: light absorption and the allometric rule," J. Theor. Biol. 204, 361-369 (2000) [CrossRef] [PubMed]
- A. Morel and A. Bricaud, "Theoretical results concerning light absorption in a discrete medium, and application to specific absorption of phytoplankton," Deep-Sea Res. 28, 1375-1393 (1981). [CrossRef]
- D. J. S. Montagnes, J. A. Berges, P. J. Harrison and F. J. R. Taylor, "Estimating carbon, nitrogen, protein, and chlorophyll a from cell volume in marine phytoplankton," Limnol. Oceanogr. 39,1044-1060 (1994). [CrossRef]
- J. Rodriguez, J. Tintore, J. T. Allen, J. M. Blanco, D. Gomis, A. Reul, J. Ruiz, V. Rodriguez, F. Echevarria and F. Jimenez-Gomez, "Mesoscale vertical motion and the size structure of phytoplankton in the ocean," Nature 410, 360-363 (2001). [CrossRef] [PubMed]
- E. Boss, M. S. Twardowski and S. Herring, "Shape of the particulate beam attenuation spectrum and its inversion to obtain the shape of the particulate size distribution," Appl. Opt. 40, 4885-4893 (2001). [CrossRef]
- A. Bricaud and A. Morel, "Light attenuation and scattering by phytoplanktonic cells: a theoretical modeling," Appl. Opt. 25, 571-580 (1986). [CrossRef] [PubMed]
- D. Risović, "Two component model of sea particle size distribution," Deep-Sea Res. 40, 1459-1473 (1993). [CrossRef]
- M. Jonasz and G. Fournier, "Approximation of the size distribution of marine particles by a sum of log-normal functions," Limnol. Oceanogr. 41, 744-754 (1996). [CrossRef]
- E. C. Junge, Air chemistry and radioactivity (Academic Press 1963), pp.382.
- R. W. Sheldon, A. Prakash and W. H. Sutcliffe, Jr., "The size distribution of particles in the ocean," Limnol. Oceanogr. 17, 327-340 (1972). [CrossRef]
- S. Bernard, T. A. Probyn and R. G. Barlow, "Measured and modelled optical properties of particulate matter in the southern Benguela," S. Afr. J. Sci. 97, 410-420 (2001).
- Y. X. Hu and K. Stamnes, "An accurate Parameterization of Cloud Radiative Properties Suitable for Climate Modeling," J. Climate. 6, 728-742 (1993). [CrossRef]
- M. D. Alexandrov and A. A. Lacis, "A new three-parameter cloud/aerosol particle size distribution based on the generalized inverse Gaussian density function," Appl. Math. Comput. 116, 153-165 (2000) [CrossRef]
- R. McGraw, S. Nemesure and S. E. Schwartz, "Properties and evolution of aerosols with size distributions having identical moments," J. Aerosol. Sci. 29, 761-772 (1998). [CrossRef]
- J. E. Hansen and L. D. Travis, "Light scattering in planetary atmospheres," Space. Sci. Rev. 16, 527-610 (1974). [CrossRef]
- C. S. Yentsch, "Measurement of visible light absorption by particulate matter in the ocean," Limnol. Oceanogr. 7, 207-217 (1962). [CrossRef]
- C. S. Roesler, "Theoretical and experimental approaches to improve the accuracy of particulate absorption coefficients derived from the quantitative filter technique," Limnol. Oceanogr. 43, 1649-1660 (1998). [CrossRef]
- M. Kishino, M. Takahashi, N. Okami and S. Ichimura, "Estimation of the spectral absorption coefficients of phytoplankton in the sea," Bull. Mar. Sci. 37, 634-642 (1985).
- D. Stramski and J. Piskozub, "Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from 3-D radiative transfer equations," Appl. Opt., 42, 3634-46 (2003). [CrossRef] [PubMed]
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