# Assessing the Impact of a Two-Layered Spherical Geometry of Phytoplankton Cells on the Bulk Backscattering Ratio of Marine Particulate Matter

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Considerations

#### 2.1. Backscattering Cross Section for Polydisperse Particle Assemblages

#### 2.2. The Bulk Backscattering Ratio

#### 2.3. The Scattering Coefficient as Measured by In Situ Transmissometers

^{o}. If we want to compare, in a future study, our theoretical results to available in situ measurements, ${b}_{p}$ must be derived from the scattering cross section, rebuilt from the normalized phase function integrated between ${\theta}_{acceptance}$ and $\pi $ instead of 0 and $\pi $ [30]. To make a distinction, when ${C}_{sca}$ is calculated by integrating the scattering function between ${\theta}_{acceptance}$ and $\pi $, the symbols ${C}_{sca}^{{\theta}_{a}}$, ${b}_{p}^{{\theta}_{a}}$ and $\tilde{{b}_{bp}^{{\theta}_{a}}}$ (= ${b}_{bp}/{b}_{p}^{{\theta}_{a}}$) will be used. As in Twardowski et al. [5], we set the acceptance angle to 1

^{o}, which is consistent with acceptance angles of commercially available beam transmissometers such as the WETLabs C-Star (1.2

^{o}) or WETLabs ac9 (${0.93}^{o}$) ([30] and references therein).

## 3. Numerical Modeling of the Marine Particle Scattering

^{o}S2). Particular attention must be paid to the integration step to guarantee the accuracy of the numerical integration.

## 4. Abundance of the Various Particulate Components

## 5. Results

#### 5.1. Accuracy of Numerical Computations

^{o}M2, M3). The normalized phase function of polydisperse particles $\tilde{F}\left(\theta \right)$ exhibits a maximum around $\theta $ = 0

^{o}[26]. For small $\xi $ value, that is when the proportion of large-sized particles compared to smaller particles increases, the forward peak is sharper. Indeed, for particles with a large diameter as compared to the wavelength, $\tilde{F}(D,\theta )$ displays a sharp forward peak [26] due the concentration of light near $\theta $ = 0

^{o}caused by diffraction. The presence of the peak in $\tilde{F}\left(\theta \right)$ requires several integration points large enough to provide the desired numerical accuracy. The numerical integration over $\theta $ (Figure 1, N

^{o}M2) is performed using the “Trapz” function from the Numpy package with Python. The “Trapz” function performs an integration along the given axis using the composite trapezoidal rule. To test the accuracy of the integration and to find the correct integration step, $\Delta \theta $, we compare the result of the numerical integration of $\tilde{F}\left(\theta \right)$ between 0 and $\pi $ to its theoretical value (=2) (Figure 1, N

^{o}M3). When $\Delta \theta $ = 0.05

^{o}, corresponding to a total number of integration steps (N${}_{\theta}$) of 3600, the numerical integration value of $\tilde{F}\left(\theta \right)$ is in the range [1.999–2.000] for small $\xi $. For larger $\xi $, it is in the range [1.800–1.999]. When the value of the numerical integration is in the range [1.800–1.999], a renormalization factor is applied to $\tilde{F}\left(\theta \right)$ to ensure that the result of the numerical integration is exactly 2. We could also increase the number of integration points, but it would increase the computation time. Using a renormalization factor for large $\xi $ is a good compromise to guarantee the accuracy and save computation time.

^{o}S1), so the numerical integration over the particle diameter range (Equation (2)) is realized as a separate calculation with the Python “Trapz” function (Figure 1, N

^{o}S2). For monodisperse particles, the normalized phase function displays a forward peak as explained above but can also display a sequence of maxima and minima due to interference and resonance features [26,42]. The frequency of the maxima and minima over the range of $\theta $ increases with both increasing ${n}_{r}$ and size parameter (=$\pi \phantom{\rule{0.222222em}{0ex}}D/\lambda $). To test the accuracy of the numerical integration over the particle diameter range (Figure 1, N

^{o}S3), we ran the ScattnLay code for DS1 case studies and compared $\tilde{F}\left(\theta \right)$ and ${C}_{sca}$ rebuilt from Equation (2) with Lorentz-Mie computations as the Lorentz-Mie code provides the polydisperse phase functions and cross section as outputs (Figure 1, N

^{o}M1). Note that even a narrow polydispersion washes out the interference and resonance features, which explains why most natural particulate assemblages do not exhibit such patterns [26,42] (Figure 3). A perfect match is obtained between the ScattnLay-rebuilt-polydisperse and Lorentz-Mie-polydisperse $\tilde{F}\left(\theta \right)$ and ${C}_{sca}$ values when the integration step ($\Delta D$) is set to 0.01 $\mathsf{\mu}$m for D in the range [0.03, 2 $\mathsf{\mu}$m]; 0.1 $\mathsf{\mu}$m for D in the range [2, 20 $\mathsf{\mu}$m]; 2.0 $\mathsf{\mu}$m for D in the range [20, 200 $\mathsf{\mu}$m]; and 10.0 $\mathsf{\mu}$m for D in the range [200, 500 $\mathsf{\mu}$m].

^{o}(N${}_{\theta}$ = 750), the present results of the Lorentz-Mie calculations (solid lines in Figure 4) perfectly match those previously obtained by Twardowski et al. [5] (not shown). However, in this case (N${}_{\theta}$ = 750), the numerical integration is not accurate enough as the integration of Equation (1) gives values between 1.999 ($\xi $ = 4.9) and 1.04 ($\xi =2.5$). In the following, $\Delta \theta $ is set to 0.05

^{o}(N${}_{\theta}$ = 3600) and Figure 4 (dashed lines) will be the reference figure for homogeneous spheres.

#### 5.2. Impact of the Structural Heterogeneity of Phytoplankton Cells on the Bulk Particulate Backscattering Ratio

## 6. Concluding Remarks

^{o}(N${}_{\theta}$ = 3600) is required to obtain the required accuracy considering the inputs (refractive indices and size range) used in this study.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Preisendorfer, R.W. Hydrologic Optics, Volume 1: Introduction; Springfield National Technical Information Service; Office of Naval Research: Arlington, VA, USA, 1976. [Google Scholar]
- Morel, A. The Scattering of Light by Seawater: Experimental Results and Theoretical Approach; Translation by George Halikas of the paper published in French in AGARD Lecture Series; N
^{o}61; North Atlantic Treaty Organization: Neuilly-sur-Seine, France, 1973. [Google Scholar] - Boss, E.; Pegau, W.S.; Gardner, W.D.; Zaneveld, J.R.V.; Barnard, A.H.; Twardowski, M.S.; Chang, G.C.; Dickey, T.D. The spectral particulate attenuation and particle size distribution in the bottom boundary layer of a continental shelf. J. Geophys. Res.
**2001**, 106, 9509–9516. [Google Scholar] [CrossRef] - Boss, E.; Twardowski, M.S.; Herring, S. Shape of the particulate beam attenuation spectrum and its relation to the size distribution of oceanic particles. Appl. Opt.
**2001**, 40, 4885–4893. [Google Scholar] [CrossRef] [PubMed] - Twardowski, M.; Boss, E.; Macdonald, J.; Pegau, W.; Barnard, A.; Zaneveld, J. A model for estimating bulk refractive index from optical backscattering ratio and the implications for understanding particle composition in case I and case II waters. J. Geophys. Res.
**2001**, 106, 14129–14142. [Google Scholar] [CrossRef] - Boss, E.; Pegau, W.S.; Lee, M.; Twardowski, M.; Shybanov, E.; Korotaev, G.; Baratange, F. Particulate backscattering ratio at LEO 15 and its use to study particle composition and distribution. J. Geophys. Res.
**2004**, 109, C01014. [Google Scholar] [CrossRef] - Loisel, H.; Mériaux, X.; Berthon, J.F.; Poteau, A. Investigation of the optical backscattering to scattering ratio of marine particles in relation to their biogeochemical composition in the eastern English Channel and southern North Sea. Limnol. Oceanogr.
**2007**, 52, 739–752. [Google Scholar] [CrossRef][Green Version] - Nasiha, H.J.; Shanmugam, P. Estimating the Bulk Refractive Index and Related Particulate Properties of Natural Waters from Remote-Sensing Data. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens.
**2015**, 8, 5324–5335. [Google Scholar] [CrossRef] - Snyder, W.A.; Arnone, R.A.; Davis, C.O.; Goode, W.; Gould, R.W.; Ladner, S.; Lamela, G.; Rhea, W.J.; Stavn, R.; Sydor, M.; et al. Optical scattering and backscattering by organic and inorganic particulates in U.S. coastal waters. Appl. Opt.
**2008**, 47, 666–677. [Google Scholar] [CrossRef] - Sullivan, J.; Twardowski, M.; Donaghay, P.; Freeman, S. Use of optical scattering to discriminate particle types in coastal waters. Appl. Opt.
**2005**, 44, 1667–1680. [Google Scholar] [CrossRef] - Meyer, R.A. Light scattering from biological cells: Dependence of backscatter radiation on membrane thickness and refractive index. Appl. Opt.
**1979**, 18, 585–588. [Google Scholar] [CrossRef] - Bricaud, A.; Zaneveld, J.R.V.; Kitchen, J.C. Backscattering efficiency of coccolithophorids: Use of a three-layered sphere model. Proc. SPIE
**1992**, 1750, 27–33. [Google Scholar] - Kitchen, J.C.; Zaneveld, J.R.V. A three-layered sphere model of the optical properties of phytoplankton. Limnol. Oceanogr.
**1992**, 37, 1680–1690. [Google Scholar] [CrossRef][Green Version] - Stramski, D.; Piskozub, J. Estimation of scattering error in spectrophotometric measurements of light absorption by aquatic particles from three-dimensional radiative transfer simulations. Appl. Opt.
**2003**, 42, 3634–3646. [Google Scholar] [CrossRef] - Moutier, W.; Duforêt-Gaurier, L.; Thyssen, M.; Loisel, H.; Mériaux, X.; Courcot, L.; Dessailly, D.; Rêve, A.H.; Grégori, G.; Alvain, S.; et al. Evolution of the scattering properties of phytoplankton cells from flow cytometry measurements. PLoS ONE
**2017**, 12. [Google Scholar] [CrossRef] [PubMed] - Poulin, C.; Zhang, X.; Yang, P.; Huot, Y. Diel variations of the attenuation, backscattering and absorption coefficients of four phytoplankton species and comparison with spherical, coated spherical and hexahedral particle optical models. J. Quant. Spectrosc. Radiat. Transf.
**2018**, 217, 288–304. [Google Scholar] [CrossRef] - Quirantes, A.; Bernard, S. Light scattering by marine algae: Two-layer spherical and nonspherical models. J. Quant. Spectrosc. Radiat. Transf.
**2004**, 89, 311–321. [Google Scholar] [CrossRef] - Vaillancourt, R.D.; Brown, C.W.; Guillard, R.L.; Balch, W.M. Light backscattering properties of marine phytoplankton: Relationships to cell size, chemical composition and taxonomy. J. Plankton Res.
**2004**, 26, 191–212. [Google Scholar] [CrossRef] - Volten, H.; Haan, J.F.; Hovenier, J.W.; Schreurs, R.; Vassen, W.; Dekker, A.G.; Hoogenboom, H.J.; Charlton, F.; Wouts, R. Laboratory measurements of angular distributions of light scattered by phytoplankton and silt. Limnol. Oceanogr.
**1998**, 43, 1180–1197. [Google Scholar] [CrossRef][Green Version] - Witkowski, K.; Król, T.; Zielinski, A.; Kuten, E. A light-scattering matrix for unicellular marine phytoplankton. Limnol. Oceanogr.
**1998**, 43, 859–869. [Google Scholar] [CrossRef][Green Version] - Stramski, D.; Boss, E.; Bogucki, D.; Voss, K.J. The role of seawater constituents in light backscattering in the ocean. Prog. Oceanogr.
**2004**, 61, 27–56. [Google Scholar] [CrossRef] - Whitmire, A.L.; Pegau, W.S.; Karp-Boss, L.; Boss, E.; Cowles, T.J. Spectral backscattering properties of marine phytoplakton cultures. Opt. Express
**2010**, 18, 15073–15093. [Google Scholar] [CrossRef] - Robertson Lain, L.; Bernard, S.; Evers-King, H. Biophysical modelling of phytoplankton communities from first principles using two-layered spheres: Equivalent Algal Populations (EAP) model. Opt. Express
**2014**, 22, 16745–16758. [Google Scholar] [CrossRef] [PubMed] - Stramski, D.; Kiefer, D.A. Light scattering by microorganisms in the open ocean. Prog. Oceanogr.
**1991**, 28, 343–383. [Google Scholar] [CrossRef] - Stramski, D.; Bricaud, A.; Morel, A. Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community. Appl. Opt.
**2001**, 40, 2929–2945. [Google Scholar] [CrossRef] [PubMed] - Mishchenko, M.I.; Travis, L.D.; Lacis, A.A. Scattering, Absorption and Emission of Light of Small Particles; Cambridge University Press: Cambridge, UK, 2002; ISBN 9780521782524. [Google Scholar]
- Jonasz, M. Particle size distribution in the Baltic. Tellus
**1983**, B35, 346–358. [Google Scholar] [CrossRef] - Loisel, H.; Nicolas, J.M.; Sciandra, A.; Stramski, D.; Poteau, A. Spectral dependency of optical backscattering by marine particles from satellite remote sensing of the global ocean. J. Geophys. Res.
**2006**, 111, C09024. [Google Scholar] [CrossRef] - Morel, A.; Bricaud, A. Inherent optical properties of algal cells, including picoplankton. Theoretical and experimental results. Can. Bull. Fish. Aquat. Sci.
**1986**, 214, 521–559. [Google Scholar] - Boss, E.; Slade, W.H.; Behrenfeld, M.; Dall’Olmo, G. Acceptance angle effects on the beam attenuation in the ocean. Opt. Express
**2009**, 17. [Google Scholar] [CrossRef] - Dolman, V.L. Meerhoff Mie Program User Guide; Internal Report Astronomy Department, Free University: Amsterdam, The Netherlands, 1989. [Google Scholar]
- Peña, O.; Pal, U. Scattering of electromagnetic radiation by a multilayered sphere. Comput. Phys. Commun.
**2009**, 180, 2348–2354. [Google Scholar] [CrossRef] - Yang, W. Improved recursive algorithm for light scattering by a multilayered sphere. Appl. Opt.
**2003**, 42, 1710–1720. [Google Scholar] [CrossRef] - Aas, E. Refractive index of phytoplankton derived from its metabolite composition. J. Plankton Res.
**1996**, 18, 2223–2249. [Google Scholar] [CrossRef][Green Version] - Buonassissi, C.J.; Dierssen, H.M. A regional comparison of particle size distributions and the power law approximation in oceanic and estuarine surface waters. J. Geophys. Res.
**2010**, 115, C10028. [Google Scholar] [CrossRef] - Reynolds, R.A.; Stramski, D.; Wright, V.M.; Woźniak, S.B. Measurements and characterization of particle size distributions in coastal waters. J. Geophys. Res.
**2010**, 115, C08024. [Google Scholar] [CrossRef] - Reynolds, R.A.; Stramski, D.; Neukermans, G. Optical backscattering by particles in Arctic seawater and relationships to particle mass concentration, size distribution, and bulk composition. Limnol. Oceanogr.
**2016**, 61, 1869–1890. [Google Scholar] [CrossRef][Green Version] - Woźniak, S.B.; Stramski, D.; Stramska, M.; Reynolds, R.A.; Wright, V.M.; Miksic, E.Y.; Cichocka, M.; Cieplak, A.M. Optical variability of seawater in relation to particle concentration, composition, and size distribution in the nearshore marine environment at Imperial Beach, California. J. Geophys. Res.
**2010**, 115, C08027. [Google Scholar] [CrossRef] - Middleboe, M.; Brussaard, C.P.D. Marine Viruses: Key Players in Marine Ecosystems. Viruses
**2017**, 9, 302. [Google Scholar] [CrossRef] [PubMed] - Brotas, V.; Brewin, R.; Sá, C.; Brito, A.C.; Silva, A.; Mendes, C.R.; Diniz, T.; Kaufmann, M.; Tarran, G.; Groom, S.B.; et al. Deriving phytoplankton size classes from satellite data: Validation along a trophic gradient in the eastern Atlantic Ocean. Remote Sens. Environ.
**2013**, 134, 66–77. [Google Scholar] [CrossRef] - Brewin, R.J.W.; Sathyendranath, S.; Hirata, T.; Lavender, S.; Barciela, R.M.; Hardman-Mountford, N.J. A three-component model of phytoplankton size class for the Atlantic Ocean. Ecol. Model.
**2010**, 221, 1472–1483. [Google Scholar] [CrossRef] - Mishchenko, M.; Lacis, A. Manifestations of morphology-dependent resonances in Mie scattering matrices. Appl. Math. Comput.
**2000**, 116, 167–179. [Google Scholar] [CrossRef] - Bricaud, A.; Roesler, C.; Zaneveld, J.R.V. In situ methods for measuring the inherent optical properties of ocean waters. Limnol. Oceanogr.
**1995**, 40, 393–410. [Google Scholar] [CrossRef][Green Version] - Zaneveld, J.R.V.; Kitchen, J.C. The variation in the inherent optical properties of phytoplankton near an absorption peak as determined by various models of cell structure. J. Geophys. Res.
**1995**, 100, 309–313. [Google Scholar] [CrossRef] - Reynolds, R.A.; Stramski, D.; Mitchell, B.G. A chlorophyll-dependent semianalytical reflectance model derived from field measurements of absorption and backscattering coefficients within the Southern Ocean. J. Geophys. Res.
**2001**, 106, 7125–7138. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Composite PSD as derived from individual PSDs of the five considered particle groups for (

**a**) the oligotrophic-like water body and (

**b**) the phytoplankton bloom water body. N${}_{TOT}$ = 1.1262 × 10${}^{14}$ particles per m${}^{3}$ and $\xi $ = 4.

**Figure 3.**Interference and resonance features observed for the scattering phase function of monodisperse particles (light green). The major low-frequency maxima and minima are called the “interference structure”. The high-frequency ripples are resonance features. The interference and resonance feature are washed out for a polydisperse assemblage of particles (dark green).

**Figure 4.**Results of Lorentz-Mie calculations (DS1) of the particulate backscattering ratio $\tilde{{b}_{bp}^{{\theta}_{a}}}$ as a function of the hyperbolic slope, $\xi $, and different values of ${n}_{r}$ and N${}_{\theta}$. The imaginary part of the refractive index = 0.005 as in Twardowski et al. [5]. This figure can be compared to Figure 1 in Twardowski et al. [5].

**Figure 5.**(

**a**) Particulate backscattering ratio $\tilde{{b}_{bp}^{{\theta}_{a}}}$ as a function of the hyperbolic slope for the oligotrophic-like (red dashed line), phytoplankton bloom (green dashed line), and coastal-like (brown dashed line) water bodies as described in Section 4. Black and gray lines are for homogeneous reference cases. The gray solid line corresponds to ${n}_{r}$ = 1.045, ${n}_{i}$ = 9.93 × 10${}^{-4}$, the black dashed line to ${n}_{r}$ = 1.1043, ${n}_{i}$ = 1.36 × 10${}^{-3}$, and the black solid line to ${n}_{r}$ = 1.131, ${n}_{i}$ = 1.37 × 10${}^{-4}$, respectively. Phytoplankton cells are modeled as two-layered spheres with a relative volume of the cytoplasm of 20% (%cyt-%chl = 80–20). (

**b**) as in panel (

**a**) but for the real refractive index. (

**c**) as in panel (

**a**) but for the imaginary part of the refractive index.

**Figure 6.**Particulate backscattering ratio as a function of the hyperbolic slope for oligotrophic-like and phytoplankton bloom water bodies. Phytoplankton cells are modeled as two-layered spheres with a relative volume of the chloroplast of 20 % and 30 %, as indicated.

**Figure 7.**Particulate backscattering ratio as a function of the hyperbolic slope for oligotrophic-like and phytoplankton bloom water bodies. Phytoplankton cells are modeled as two-layered spheres (80%–20%) or three-layered spheres (80%–18.5%–1.5%), as indicated.

**Figure 8.**Contribution of the different particle groups the total bulk backscattering ratio for (

**a**) oligotrophic-like, (

**b**) phytoplankton bloom, and (

**c**) coastal-like water bodies. The phytoplankton cells are modeled as a two-layered sphere (80%–20%).

**Figure 9.**Backscattering cross sections, ${C}_{sca}^{bb}$, of the different particle groups. The phytoplankton cells are modeled as a two-layered sphere (80%–20%).

Component (j) | Sphere Model | ${\mathit{D}}_{\mathit{min}}$-${\mathit{D}}_{\mathit{max}}$ ($\mathsf{\mu}$m) | ${\mathit{n}}_{\mathit{r}}$ | ${\mathit{n}}_{\mathit{i}}$ |
---|---|---|---|---|

Viruses | homogeneous | 0.03–0.2 | 1.05 | 0 |

Heterotrophic bacteria | homogeneous | 0.2–2 | 1.05 | 1.0 × 10${}^{-4}$ |

Phytoplankton cells | two or three-layered | 0.3–40 | 1.044 * | 1.5 × 10${}^{-3}$ * |

Organic detritus | homogeneous | 0.05–500 | 1.04 | 2.3 × 10${}^{-5}$ |

Minerals | homogeneous | 0.05–500 | 1.18 | 1.0 × 10${}^{-4}$ |

**Table 2.**Refractive index (${n}_{r}$(chlp) + i${n}_{i}$(chlp)) of the sphere representing the chloroplast for two morphological models. The refractive index of the sphere representing the cytoplast is constant (1.02 + $i\phantom{\rule{0.222222em}{0ex}}$1.336 × 10${}^{-4}$). The equivalent refractive index of the cell is 1.044 + $i\phantom{\rule{0.222222em}{0ex}}$1.5 × 10${}^{-3}$.

Model * (%cyt-%chlp) | 80%–20% | 70%–30% | 80%–18.5%–1.5% |
---|---|---|---|

${n}_{r}$ | 1.140 | 1.100 | 1.144 |

${n}_{i}$ | 6.966 × 10${}^{-3}$ | 4.688 × 10${}^{-3}$ | 7.531 × 10${}^{-3}$ |

Relative Abundance N${}_{\mathit{j}}$ (%) | ||||||
---|---|---|---|---|---|---|

$\mathit{\xi}$ | $\tilde{{\mathit{n}}_{\mathit{r}}}$ | $\tilde{{\mathit{n}}_{\mathit{i}}}$ | VIR | BAC | PHY | DET |

2.5 | 1.040 | 4.280 × 10${}^{-4}$ | 78.85 | 5.349 | 0.4059 | 15.39 |

3 | 1.042 | 7.570 × 10${}^{-4}$ | 84.74 | 2.120 | 0.1002 | 13.04 |

3.5 | 1.043 | 1.034 × 10${}^{-3}$ | 88.50 | 0.8244 | 0.0281 | 10.64 |

4 | 1.045 | 9.931 × 10${}^{-4}$ | 91.15 | 0.3178 | 0.0084 | 8.528 |

4.9 | 1.047 | 6.718 × 10${}^{-4}$ | 94.35 | 5.651 × 10${}^{-2}$ | 0.0010 | 5.588 |

Relative Abundance N${}_{\mathit{j}}$ (%) | ||||||
---|---|---|---|---|---|---|

$\mathit{\xi}$ | $\tilde{{\mathit{n}}_{\mathit{r}}}$ | $\tilde{{\mathit{n}}_{\mathit{i}}}$ | VIR | BAC | PHY | DET |

2.5 | 1.041 | 6.195 × 10${}^{-4}$ | 51.96 | 3.760 | 1.995 | 42.29 |

3 | 1.041 | 1.048 × 10${}^{-3}$ | 61.91 | 1.599 | 0.6165 | 35.88 |

3.5 | 1.042 | 1.313 × 10${}^{-3}$ | 69.84 | 0.6575 | 0.1922 | 29.31 |

4 | 1.043 | 1.362 × 10${}^{-3}$ | 76.18 | 0.2650 | 0.0600 | 23.49 |

4.9 | 1.044 | 1.194 × 10${}^{-3}$ | 84.55 | 0.0499 | 7.367 × 10${}^{-3}$ | 15.40 |

Relative Abundance N${}_{\mathit{j}}$ (%) | |||||||
---|---|---|---|---|---|---|---|

$\mathit{\xi}$ | $\tilde{{\mathit{n}}_{\mathit{r}}}$ | $\tilde{{\mathit{n}}_{\mathit{i}}}$ | VIR | BAC | PHY | DET | MIN |

2.5 | 1.103 | 7.322 × 10${}^{-4}$ | 70.96 | 5.311 | 3.650 × 10${}^{-1}$ | 11.68 | 11.68 |

3 | 1.108 | 9.361 × 10${}^{-4}$ | 78.04 | 2.105 | 8.801 × 10${}^{-2}$ | 9.882 | 9.882 |

3.5 | 1.119 | 6.253 × 10${}^{-4}$ | 83.03 | 0.819 | 2.391 × 10${}^{-2}$ | 8.066 | 8.066 |

4 | 1.131 | 1.376 × 10${}^{-4}$ | 86.75 | 0.3155 | 6.902 × 10${}^{-3}$ | 6.462 | 6.462 |

4.9 | 1.145 | 9.794 × 10${}^{-6}$ | 91.47 | 5.607 × 10${}^{-2}$ | 7.782 × 10${}^{-4}$ | 4.23 | 4.23 |

**Table 6.**Comparisons between abundances defined in the present study and abundances defined by Stramski et al. [25]. The hyperbolic slope $\xi $ is 4 and N${}_{TOT}$ is 1.1262 × 10${}^{14}$ particles per m${}^{3}$.

Abundance (Particles per m${}^{3}$) | |||||
---|---|---|---|---|---|

Case Study | VIR | BAC | PHY | DET | MIN |

Oligotrophic-like | 1.0265 × 10${}^{14}$ | 3.5796 × 10${}^{11}$ | 9.4680 × 10${}^{9}$ | 9.6046 × 10${}^{12}$ | 0 |

Phytoplankton bloom | 8.5799 × 10${}^{13}$ | 2.9846 × 10${}^{11}$ | 6.7587 × 10${}^{10}$ | 2.6455 × 10${}^{13}$ | 0 |

Coastal-like | 9.7702 × 10${}^{13}$ | 3.5536 × 10${}^{11}$ | 7.7733 × 10${}^{9}$ | 7.2774 × 10${}^{12}$ | 7.2774 × 10${}^{12}$ |

Stramski et al. [25] | 2.5000 × 10${}^{12}$ | 1.0000 × 10${}^{11}$ | 2.4759 × 10${}^{10}$ | 8.2500 × 10${}^{13}$ | 2.7500 × 10${}^{13}$ |

Oligotrophic-Like | Phytoplankton Bloom | Coastal-Like | |
---|---|---|---|

$\mathit{\xi}$ | [Chla] | [Chla] | [Chla] |

3 | 8.35 | 11.51 | 7.497 |

3.5 | 0.773 | 1.580 | 0.6889 |

4 | 0.102 | 0.341 | 0.0884 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Duforêt-Gaurier, L.; Dessailly, D.; Moutier, W.; Loisel, H. Assessing the Impact of a Two-Layered Spherical Geometry of Phytoplankton Cells on the Bulk Backscattering Ratio of Marine Particulate Matter. *Appl. Sci.* **2018**, *8*, 2689.
https://doi.org/10.3390/app8122689

**AMA Style**

Duforêt-Gaurier L, Dessailly D, Moutier W, Loisel H. Assessing the Impact of a Two-Layered Spherical Geometry of Phytoplankton Cells on the Bulk Backscattering Ratio of Marine Particulate Matter. *Applied Sciences*. 2018; 8(12):2689.
https://doi.org/10.3390/app8122689

**Chicago/Turabian Style**

Duforêt-Gaurier, Lucile, David Dessailly, William Moutier, and Hubert Loisel. 2018. "Assessing the Impact of a Two-Layered Spherical Geometry of Phytoplankton Cells on the Bulk Backscattering Ratio of Marine Particulate Matter" *Applied Sciences* 8, no. 12: 2689.
https://doi.org/10.3390/app8122689