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
Phytoplankton size is of considerable importance to phycologists, exerting a strong influence upon metabolic rates and physiological behaviour , algal optical properties , algal biomass and carbon concentrations , and physical behaviour . There is thus a need to provide simple descriptors of algal size, and formulations for algal size distributions, that can account for both morphological variations and the polydispersity typically displayed by phytoplankton populations. Such schemes, by reducing algal size descriptors to one or more proxy parameters, can allow a greater understanding of cultured and natural algal populations from optical, physiological and ecological perspectives. In addition, size distribution formulae for marine particle populations allow the rapid and semi-continuous assessment of marine particle size variability through the application of inversion techniques to marine optical measurements .
A variety of distribution formulations have been employed to represent both phytoplankton and total marine particle distributions, typically from fitting such functions to measured size distribution data. Monospecific phytoplankton cultures display expectedly narrow size distributions, and can be represented by log-normal  and gamma  distributions. However, natural phytoplankton populations are rarely monospecific, and the use of log-normal or gamma distributions has been restricted to application with total marine particle distributions, either decomposition techniques to describe measured distributions , or distribution functions suitable for oligotrophic waters . The most commonly used distribution type for marine particulate, with specific regard to modeling optical properties, is the power law or Junge distribution [5,9]. Particle size distributions in phytoplankton-dominated waters typically display complex shapes, with variable intermediate maxima associated with the presence of dominant phytoplankton groups or species [8,10]. Such complex size distributions deviate markedly from the smooth exponential representation of a Junge distribution . Thus, whilst a Junge distribution offers a simple scheme conforming to general trophic and dynamic paradigms , and allows robust application to optical inversion, there is some doubt as to whether it is the most suitable function to describe algal optical properties, particularly in productive waters .
An alternative approach, from an optical perspective, to finding best-fit functions to measured size distribution data is to consider simply parameterised equivalent size distributions: formulations that, whilst dissimilar in distribution shape, can reproduce the optical properties of polydispersed particle populations. The approach has been successfully employed by atmospheric physicists to describe the optical properties of aerosol and cloud size distributions for the purposes of radiative transfer inversions [12–14]. Central to the concept of equivalent distributions is that dissimilar size distributions with identical moments display the same optical characteristics [12,14,15], where the kth radial (or diametric) moment of a size distribution is given by:
where r is the particle radius (m), and F(r) d(r) is the number of particles per unit volume in the size range r ± 1/2d(r). The best single parameter describing the optical properties of a size distribution is the effective radius reff or diameter Deff (reff = 0.5 Deff) – the ratio of the third to second moment (<r 3>/<r 2>), or the mean volume to surface area ratio of the distribution . Another parameter of importance is the effective variance Veff [(<r 4><r 2>/<r 3>2)-1)], which describes the width of the distribution .
This study seeks to establish whether a moment-based approach can be employed to define single parameter equivalent size distributions for the simulation of algal optical properties through a wide range of algal assemblage types. A functional size distribution form that is based upon optical equivalence rather than the replication of sometimes complex observed phytoplankton size distributions would have application in bio-optical and ocean colour modeling and inversion techniques. Such techniques are of particular relevance to the bio-optical monitoring and detection of harmful algal blooms.
Bio-optical measurements were employed in conjunction with Mie models to assess the ability of several common size distribution formulations to simulate the optical properties of algal populations calculated with measured size distributions. The following steps were taken:
- 1. Calculation of algal spectral refractive index data from absorption and size distribution measurements, in conjunction with an anomalous diffraction approximation model  and a refractive index dispersion model . 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).
- 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.
The principal focus of this study is a comparison between two modeling approaches using different size data, and the same refractive index data, as input. It is not the purpose of this study to provide an entire optical modeling technique that could be used to simulate the absolute optical properties of natural phytoplankton assemblages. It should be realised that both the optical models used (simple, spherical, homogeneous geometry) and the input refractive index data (similar real refractive index) are somewhat simplistic, although adequate for the purpose of this preliminary study.
2.1 Particulate absorption
Particulate absorption data were measured with the quantitative filter pad technique [16,17] using a Shimadzu UV-2501 spectrophotometer equipped with an ISR-2200 internal integrating sphere. Discrete seawater samples were filtered under less than 10 mm mercury pressure using 25 mm Whatman GF/F filters, which were placed at the entrance port of the sphere, and scanned from 350 nm to 750 nm using an air reference and baseline. In cases where filters could not be read immediately, they were stored frozen in liquid nitrogen. Blank filter pads were prepared by filtering several hundred ml of Milli-Q water through fresh GF/F filters, which were then read in the same manner as the samples. Filtration volumes were adjusted to maintain a large amount of material on the filter while avoiding excessive clogging. Absorption coefficients were calculated using the pathlength amplification factor of Roesler , assuming a null-point correction at 750 nm. Detrital measurements were made using methanol extraction on the filter, followed by re-reading in the spectrophotometer . Phytoplankton absorption aϕ(λ) data were then obtained by subtraction of detrital absorption ad(λ) from total particulate absorption ap(λ).Both the filter pad and null-point correction methods suffer from unknown errors associated with the poorly known scattering properties of marine particulates . However, these errors are unlikely to significantly impact this study, as the principal focus is a comparison between two modeling approaches using representative refractive index data as input.
2.2 Particle size distributions (PSD)
Particle size measurements were made using a 128 channel Coulter Multisizer II in manometer mode, using freshly prepared 0.2 μm filtered seawater as both blank and electrolyte. Choice of aperture size was dictated by size of the dominant algal species, with 140 μm used for all samples other than those dominated by the small pelagophyte 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 ad (λ), using an imaginary refractive index of ń =0.001066 exp(-0.007168λ) . The detrital size distribution resulting from the fit was then subtracted from the total measured size distribution to obtain the algal particle size distribution. Finally, data for optical analyses were re-sampled to linear size bins of 1 μm, with a range of 1 μm to 100 μm, through calculation of the spectral density.
Pigments were analysed using High Performance Liquid Chromatography (HPLC). Seawater was filtered through 25 mm GF/F filters and filters stored frozen in liquid nitrogen until analysis. Filtration volumes were adjusted to maintain approximately constant amounts of material on the filter and ranged from 0.1 l to 2 l. Analysis of pigments followed the reverse-phase HPLC procedure outlined by  using a 3 μm Hypersil MOS2 C8 column (100 × 4.6 mm), a Varian ProStar pump, a Thermo Separations AS3000 autosampler, a Thermo Separations UV6000 diode array absorbance detector, and ChromQuest chromatography software.
2.4 Refractive index determinations
The absorption coefficient of a phytoplankton population can be described by the following expression :
where aϕ (λ) is the algal absorption coefficient (m2 mg-1), Q̅a (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 Q̅a (λ) can be calculated for natural algal assemblages . Employing the anomalous diffraction approximation ,
dimensionless absorption efficiency factors can be expressed in terms of the optical thickness = 4αń, where α is the Mie size parameter 2rπ/λ, and ń(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]. Note that these 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 .
All refractive index determinations, and subsequent optical modeling runs, were made with Chl 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 , Chl 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
Equivalent distributions were calculated using the same effective radius as the corresponding measured size distribution and scaled to the total projected surface area <SA> of the measured algal size distribution, as given by:
The 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 
- 2. The Standard distribution 
where rg = reff/(l + veff)5/2 and σ2 g = ln(l + veff).
- 4. The Junge distribution 
In addition to the above distributions, a further approximation was assessed:
- 5. A single size approximation of r = reff (or D = Deff)
The first two distributions were chosen due to their proven ability as moment-matched equivalent distributions in atmospheric physics. In addition, these distributions can be conveniently expressed analytically as a function of their effective radii and variance. The second two distributions were chosen due to their common use in the marine bio-optical field [5,6]. Note that the Junge distribution is the only formula that is not explicitly expressed in terms of effective radius and variance, and equivalent distributions were calculated by iteratively adjusting the Junge slope ξ until reff 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 reff: the first using the variable experimental reff values determined for each sample, and the second using a constant reff value of 0.63, the mean value for all samples.
2.6 Optical modeling
Analogous expressions to Eq. 2 can be used to express the relationships between the attenuation coefficient c(λ), the scattering coefficient b(λ), the backscattering coefficient bb(λ) and their relative efficiency factors. Additional detail on such models can be found in . In addition, the dimensionless package effect parameter (Qa *) can be calculated from the following expression :
where acm is the absorption of cellular material (m-1) and is given by acm = 4πń/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.
The ability of the equivalent size distribution to match the IOPs calculated from the measured size distribution were assessed using the mean and standard deviations (SD) of the RMS errors (in percent) of the entire data set for each wavelength, as given by
where aequiv is the modeled absorption of the equivalent size distribution, and ameas is the modeled absorption of the measured size distribution. The above expressions are for absorption – analogous expressions are employed for other IOPs.
The Aden-Kerker  formulations, as provided by the Fortran code of Toon & Ackerman  were employed for all forward optical modeling, in a combined Matlab/Fortran environment. The Aden-Kerker formulations allow the absorbing and scattering properties of a two-layered particle to be calculated using size and refractive index data as input, in a way analogous to Mie theory use for a homogeneous particle. The suitability of the Aden-Kerker formulations for homogenous particle geometry was confirmed by testing sample results against the Fortran code of , which produced equivalent results.
3. Results and discussion
All Chl 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 [2,25]; the effects of which can also seen in the imaginary refractive index data (Fig. 1(c)), the magnitudes of which compare well with algal culture data . The multimodal phytoplankton volume distributions (Fig. 1(b)) are typical of the productive, phytoplankton-dominated waters of the southern Benguela, and demonstrate the difficulties of simulating the complex shapes of natural phytoplankton distributions using simple distribution functions.
3.1 Optical properties of equivalent size distributions
The two equivalent size distributions previously used for analogous atmospheric work, the inverse Gaussian  and Standard , perform well in simulating the optical properties of algal assemblages. Maximal RMS errors and their standard deviations are reported in Table 2 and Fig. 2 for all distributions, calculated in percentage terms for all optical properties relative to the derived optical properties of the full, i.e. measured, distributions. Realistic values of the maximum expected errors can be obtained by summing the maximal RMS error and standard deviation – the single values in Table 2 represent the largest spectral errors seen in Fig. 2. The best performing inverse Gaussian distribution therefore appears capable of simulating beam attenuation c and total scattering b to within 10%, absorption a and package effect parameter Qa * to within 6%, and backscattering bb 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 %.
Both these distributions appear capable of reproducing salient bio-optical variables with sufficient accuracy for inversion application through a wide range of algal assemblage types. Of particular importance is the ability of the inverse Gaussian and Standard distributions to reproduce spectral absorption coefficients and package effect parameters accurately (< 6 %), as phytoplankton absorption is an often dominant determinant of light attenuation in the sea , and offers an optical signal with the ability to provide algal assemblage descriptors .
The relatively high errors associated with the backscattering coefficient (∼20%) relative to the other coefficients (∼5 % to ∼10 %) are presumed to be a result of the greater sensitivity to small size changes of the scattering phase function relative to integrated variables such as the attenuation or absorption coefficient [15,30]. Both interference phenomena and morphology-dependent resonances, at their most pronounced in the phase function , are likely to result in much larger size-dependent variations in dependent variables such as the backscattering coefficient.
Previous studies considering hypothetical particle polydispersions  have demonstrated that the relationship between optical efficiency and particle size is considerably more complex for backscattering than those for attenuation, scattering or absorption, which tend to limiting values in the size range under consideration.
Forcing both the inverse Gaussian and Standard distributions to a constant Veff 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 Veff 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 Veff 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.
The single size approximation would appear to offer poor returns for all optical variables (∼25% to ∼50% errors), with the important exception of the package effect parameter (∼7% error) which offers comparable performance to the inverse Gaussian and Standard distributions (∼6% to ∼7% errors). The acceptable performance of the single-size package effect derivation, in comparison to the poorer performance with regard to the other optical coefficients, is likely to result both from the monotonic nature of the Qa * vs Deff (or ) relationship , and the high relative impact of the optical thickness upon direct Qa * calculations (Eq. 9). Whilst Qa and Qa * are obviously both directly dependent upon , the average deviation of (675) is four times higher than Qa(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 .
The Junge size distribution, perhaps the most commonly used in marine optics, performed markedly less well, with errors ranging from 35% to 62% (Table 2). Given the peaked nature of the measured phytoplankton size distributions, and the noted limitations of the Junge distribution for phytoplankton-dominated waters [5,10], the poor performance of the Junge distribution is not unexpected. The log-normal distribution, while offering more accurate returns than the Junge distribution, offers a mixed performance in comparison to the inverse Gaussian and Standard distributions. The log-normal returns for the attenuation and scattering coefficients (10% to 14% errors) are comparable to those of the inverse Gaussian and Standard distributions. The log-normal returns for the absorption and backscattering coefficients, and the package effect parameter, are however noticeably poorer (Table 2). Whilst this may be partially due to the particular 0th order formulation of the log-normal distribution employed here , it is also likely to be due to the less desirable asymptotic characteristics of the log-normal distribution at higher sizes [8,13].
Figure 3 details the measured and equivalent algal size distributions, and associated optical properties of a mixed dinoflagellate and diatom bloom in the southern Benguela. The example station is specifically chosen to illustrate the close simulation of optical properties using equivalent size distributions very different in shape to the highly multimodal measured size distribution. The equivalent size distributions in Fig. 3(a) are calculated using the mean effective variance of 0.63, and can therefore be parameterised using a maximum of two variables: the effective diameter Deff, and the scaling parameter 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 . With regard to the optical properties of the example assemblage, the close replication of all optical properties by the inverse Gaussian and Standard distributions can be observed. The approximate 7% difference between the measured phytoplankton absorption and that reproduced from the Mie modeling (Fig 3(c)) can be attributed to the errors associated with the anomalous diffraction approximation, used to derive the spectral refractive index data.
4 Application and conclusions
Further consideration of the scaling parameter ASF for the inverse Gaussian and Standard distributions reveals a close relationship with effective diameter Deff for the samples analysed using a mean effective variance of 0.63 (Fig. 4). The 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 Deff relationship using a power law allows both distributions to be parameterised using a single variable, the effective diameter (Fig 4).
This further simplifies potential inversion of the inverse Gaussian and Standard equivalent distributions by allowing Chl 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
It should also be noted that it is possible to use the above expressions as simple weighting formulae without the scaling parameters shown in Fig. 4, for example to calculate optical efficiency factors given refractive index data.
The equivalent distribution approach demonstrates that by shifting paradigm from attempting to simulate or reconstruct the shape of a marine particle distribution to simulating the optical properties of a size distribution, simple parameterizations of common size distribution formulae can be employed for optical purposes. In particular, such an approach circumvents the necessity to consider the detailed structure of complex particle size distributions in biologically dominated waters, even in the case of intense algal blooms.
The current study has several limitations: those of a methodological nature with regard to the derivation of algal size distributions, subsequent refractive index determinations, and the simplifications arising from the use of spherical, homogenous particle geometries in optical calculations. In this regard, the study is preliminary in nature, and it is to be hoped that a more comprehensive evaluation of the equivalent size distribution approach can be carried out, using more sophisticated optical models and independent measurements of algal inherent optical properties. Nevertheless, the study demonstrates the utility of the equivalent size distribution approach, providing the ability to simulate the primary inherent optical properties of a wide range of algal assemblage types. In addition, the distributions described here allow simple size descriptions of the chlorophyllous particle population, as opposed to the more general size distribution formulations used in the past for the total marine particle population.
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