1. Introduction

The recent focus on Phytoplankton Functional Types (PFTs) in ocean colour applications and optical signals [1] has exposed the need to better understand the relative contributions of a water body’s various optical properties to the water-leaving signal, and where IOP and radiative transfer approximations may be used appropriately. The scattering phase function is central to the radiative transfer calculation but frequently approximated due to the lack of quantitative knowledge about systematic variability, and also for ease of computation. The impact of choice of phase function in the radiative transfer calculation has been found to be significant [2], and determining the errors associated with the use of approximated or estimated phase functions is crucial to obtaining closure between radiometric and IOP measurements [3]. Modelling has an important role to play in providing appropriate tools to do so, and hence in understanding and quantifying second order assemblage effects on water colour across a range of water types.

A water body’s whole suite of total IOPs and the entire angular structure of the light field can be completely determined from just the absorption coefficient aλ and the Volume Scattering Function (VSF) βλ [4]. The variability in intensity of scattered light over all angles in a water body is therefore fundamental to the determination of all the other Inherent Optical Properties (IOPs). The VSF describes the angular variation of scattered light intensities [5]; the integrated VSF therefore gives the total scattering coefficient bλ. Normalising the VSF to the scattering coefficient bλ, gives the scattering phase function $\tilde{\beta }\lambda$ [5]. Precise knowledge of the VSF and hence $\tilde{\beta }\lambda$ is critical to the accurate description of radiative transfer in the aquatic environment. Spectral measurements of βλ or $\tilde{\beta }\lambda$ are extremely challenging, and furthermore when transforming individual-angle VSF measurements into estimates of the phase functions, assumptions must be made about the spectral shape of angular particular scatter (i.e. the chi factor), which results in additional uncertainties [2, 3, 6]. Modelling-based analyses of the water-leaving signal are the only practical way in which to examine the impact of the inherent angular variability systematically.

For ease of use in radiative transfer models, three approaches to approximating phase functions are generally employed [4]. Firstly, a phase function can be derived from a measured VSF by integrating and normalising as described above (e.g. Petzold, 1972). As all the IOPs can be calculated from $\tilde{\beta }\lambda$ and the absorption coefficient aλ (and it is known that IOPs are unique to varying water body types), it follows that even if aλ is well known, the use of a phase function measured in another water body will likely result in considerable errors.

In the second approach, a simple functional form of the phase function (e.g. Henyey-Greenstein in [4]) can be used, and is attractive due to its mathematical simplicity. However, these approximations do not adequately describe scattering at the small (close to 0°) and large (close to 180°) scattering angles [4]. The simple functional forms have also been shown to be particle- and water-type specific, and are not suitable for generalised use in the radiative transfer problem [4]. It is expected that due to biophysical differences observable in species ultrastructure and hence spectrally in absorption and backscattering, PFTs will display distinct optical behaviour under different water type conditions.

In the third instance, a particle modelling approach such as Mie or Aden-Kerker theory can be used to calculate the phase function numerically. This method requires information about the complex refractive index n and the size distribution parameter µ [4], quantities which are not frequently exactly known. A commonly used example of such an approach is the Fournier Forand formulation [7]. The phase functions of the Equivalent Algal Populations (EAP) model [8] are presented here as representative of an alternative particle modelling approach: an assemblage-based simulation of phytoplankton IOPs accounting for differences in cell size and assemblage size distribution, dominant pigments, cell composition and ultrastructure [9]. With the ability to vary these parameters systematically, better understanding of the contribution of the consituent IOPs to the phase function form can be gained. While it is appreciated that increasingly complex non-spherical models may represent the scattering properties of particle populations more accurately, this simplified model adequately simulates the optical properties of phytoplankton while allowing a quantitative relationship with cellular Chlorophyll a and refractive indices.

When the EAP model is coupled to a radiative transfer model, the effects on reflectance of systematically varying the IOPs can be assessed. This study is primarily designed as a validation of the EAP two-layered phase functions, with the secondary aim of isolating the magnitude of the contribution of the phase function to uncertainty in R_rs calculated numerically.

2. Methods

2.1. Modelling outline

The modelled assemblages for this study are a first order representation of two broad functional types: A generalised eukaryotic group (e.g. diatoms, dinoflagellates) made up of chloroplast-containing cells with pigment-related spectral characteristics (carotenoid, Chl a, Chl c) [9], and a generalised prokaryotic group (e.g. cyanobacteria) comprising cells dominated by a vacuole [10]. Phytoplankton IOPs are generated from the real and imaginary parts of the refractive index [9] for each PFT. The populations are modelled with a Standard normal size distribution and other parameters consistent with those described in [8].

First, the EAP phytoplankton phase functions of these two generalised groups are presented here with their respective population effective diameter and phytoplankton backscatter fraction, at each wavelength. Eukaryote and prokaryote are distinguished in terms of their IOPs and their respective phase functions are compared with their corresponding Fournier Forand equivalents (chosen for the same backscatter fraction). Then, these phytoplankton IOPs are combined with non-algal particle scatter and gelbstof and detrital absorption in a 4-component Hydrolight 5.2 model to simulate naturally occuring water types, and an exercise in radiometric closure using modelled and measured reflectance data [8] is presented.

The modelled data are then explored further with a simulation of the contribution to reflectance due only to the phytoplankton component of each the two PFT groups, as biomass increases. The study concludes with an analysis of the sensitivity of modelled R_rs to the choice of phase function and a discussion of the causal variability in the phase functions themselves.

As the EAP model is run primarily in conjunction with Hydrolight 5.2 to solve the radiative transfer equation, R_rs resulting from the use of the two-layered EAP phase functions (hereafter ${\tilde{\beta }}_{EAP}$) are compared with the most comprehensive Hydrolight option of a backscatter fraction ${\tilde{b}}_b\lambda \text{-specific}$ Mobley-parameterised Fournier Forand phase function (hereafter ${\tilde{\beta }}_{FF}$). A description of the derivation of each of these functions follows.

2.2. Fournier Forand parameterisation used in Hydrolight 5.2

Mobley [4] determined that the use of a parameterised ${\tilde{\beta }}_{FF}$ for known particulate ${\tilde{b}}_b$ can provide a satisfactory substitute for a measured $\tilde{\beta }$. The necessity of using a $\tilde{\beta }$ for the correct backscatter fraction is clearly emphasised, as is the significance of the shape of the phase function (especially at intermediate angles). This is important because there are theoretically an infinite number of functions for an individual ${\tilde{b}}_b$, and not all of them will be appropriate for marine particles.

A fully analytical particle modelling approach is most desirable in terms of accuracy for individual water types. It is likely the only way of systematically producing phase functions across a wide range of particle and water types, and examining the variability in the dependent IOPs. The Fournier-Forand approach [7] is an approximate formulation of the computationally-intensive full Mie calculation, using a Jungian particle distribution, with each particle scattering according to the anomalous diffraction approximation of the exact Mie theory. This formulation still requires knowledge of the real refractive index n and the Junge size parameter µ (slope of the hyperbolic function). If this formulation is integrated [5], it results in a relationship between n, µ and particulate backscatter fraction ${\tilde{b}}_b$. Because the n and µ are infrequently known for individual samples, Mobley proposed a further parameterisation of this formulation in order that a phase function dependent only on a known particulate ${\tilde{b}}_b$ could be calculated [4]. That ${\tilde{b}}_b$ is combined with a simple linear relationship between n and µ, determined from the measured Petzold phase functions, to produce a ${\tilde{b}}_b\text{-specific}$ ${\tilde{\beta }}_{FF}$.

It should be noted that the ${\tilde{b}}_b$ backscatter fraction parameter describes the probability of scattering into the backward direction during a single scattering process and does not consider the additional effects of multiple scattering. So the choice of a single phase function according to this parameter is likely limiting in highly scattering waters, where multiple scattering has an increased optical influence. The Fournier Forand approach (with its origins in Mie modelling) describes angular scattering for homogenous, non-absorbing, spherical particles in populations with a Junge size distribution. This is rarely the case in natural waters, e.g. phytoplankton are strongly absorbing particles, and problems associated with simulating their optical properties in this way are described in [11] and [12].

2.3. Calculation of EAP phase functions

The EAP model simulates the total IOPs of a chosen algal population as described by its overall effective diameter (representing a full size distribution [13]) as well as its component PFTs: that is, characteristics of particle ultrastructure in terms of distinguishing diatom, dinoflagellate, cryptophyte etc. Particles are modelled as two-layered spheres, where both the real and imaginary parts of the spectral refractive indices of both core and shell spheres [8] are specified, together with the chlorophyll density of cells, proportion of core to shell sphere volume, and the optional representation of a vacuole [10]. This approach allows for the mathematical derivation of the full set of wavelength-dependent, spectrally variable phase functions at 0.1° resolution. The full mathematical description is available in [9].

${\tilde{\beta }}_{EAP}$ are generated at 0° to 180° at 0.1° resolution from the individual βθ functions for discrete particle sizes as described in [9] and then integrated across the whole size distribution:

In this equation, the pre-computed βθ functions for each discrete size (as functions of the intensity parameter i and scattering efficiency parameter Q_b) are weighted according to the selected size distribution, written as function F of α, the relative size i.e. the geometric size relative to the wavelength of the surrounding medium (water). α_M is the modal value corresponding to the maximum frequency in the distribution and m is the relative index of refraction. Please refer to Equation 20 in [14] for further details on notation and derivation of these quantities.

The importance of the contribution of the spectral shape of backscatter to the R_rs in the EAP model was established in [8]. It has previously been determined that there is a relationship between the spectral shape of particulate backscatter ${\tilde{b}}_b$ and the shape of the phase function $\tilde{\beta }$, and that the shape of $\tilde{\beta }$ is important [6,15–17]. Due to the highly spectrally variable nature of EAP particulate (especially phytoplankton) backscatter, it is expected that the simple phase function forms may not be adequate as biomass increases and therefore the contribution of phytoplankton to the bulk IOPs increases [2].

The EAP model provides an additional dimension of complexity with respect to the Fournier Forand method, as the resulting phase functions are independently related to both the ${\tilde{b}}_b$ (through the population effective diameter, D_eff) and wavelength λ, as well as to the PFT-related parameters. They are unconstrained in shape (i.e. angular distribution of scattering) as they are not inherently dependent on the assumption of a Jungian size distribution, which may not be appropriate in highly productive waters where reduced species diversity is generally observed with increasing biomass [13]. Mobley [4] acknowledges the need for λ-dependent phase functions as particulate ${\tilde{b}}_b$ varies with wavelength. However due to the parameterised Fournier Forand formulation which constrains the relationship between the size distribution parameter and the particle refractive index, the phase function used at different wavelengths but for the same ${\tilde{b}}_b$ will be identical. In the EAP model this is not the case.

3. Results

3.1. Comparing the Fournier Forand and EAP phase functions: eukaryote group

In [16], Chami et al. compared measured coastal VSFs and derived phase functions with their appropriate Fournier Forand counterparts chosen for their equivalent backscatter fractions. Following their findings of significant differences between both the actual structure of the phase functions themselves, and between the respective resulting R_rs, this study presents a similar comparison. It should be emphasised that the Fournier Forand functions are intended for generalised use with the combined phytoplankton and non-algal particulate components of a water sample, whereas the EAP phase functions are phytoplankton specific, and applicable only when it is known that phytoplankton dominate the IOPs. For an example eukaryote population of mixed dinoflagellates, ${\tilde{\beta }}_{EAP}$ for selected wavelengths and D_eff are shown in Fig. 1, together with their corresponding ${\tilde{b}}_b$. They share the most notable features of $\tilde{\beta }\mathrm{s}$ derived from VSF measurements in coastal waters, described in [16] as typical of natural waters. These features include strong forward peaks, minima at intermediary angles, and elevated levels of scattering towards reverse angles (160°–180°) [16]. This increase past 150° is also noted by Harmel et al. [18] in measured VSFs of phytoplankton cultures at 515 nm, and Fig. 1(D) shows a comparable EAP phase function at 515 nm (sample population with D_eff 10 µm). It can additionally be observed that this elevation is not present in the ${\tilde{\beta }}_{EAP}$ at shorter wavelengths (e.g. 440 nm), where there are strong phytoplankton absorption features, and that there is some variability in the shape and magnitude of this feature at different wavelengths: at much smaller particle sizes (e.g. 2 µm), the opposite effect is observed at 440 nm. This is in some sense compensated for by the overall comparably greater magnitude of the $\tilde{\beta }$ at intermediate angles. This emphasizes the complexity of the relationships between ${\tilde{b}}_b$, D_eff and λ: the shape of ${\tilde{\beta }}_{EAP}$ changes with ${\tilde{b}}_b$ as well as D_eff and λ.

 

Fig. 1 EAP phase functions for the phytoplankton component only of generalised eukaryote populations at selected λ, for three different D_eff 2, 10 and 50 µm, with corresponding phytoplankton backscatter fraction ${\tilde{b}}_{b\varphi }$ (A–C). (D) illustrates the differences in EAP phase functions due only to wavelength, for constant ${\tilde{b}}_{b\varphi }$, compared with a single corresponding Fournier Forand phase function which does not vary with wavelength.

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${\tilde{\beta }}_{EAP}$ features compare well to Chami’s measured $\tilde{\beta }$ in terms of structural features which are not evident in the Fournier Forand equivalents, calculated according to [4][Fig. 1(D)], and the shape of the relative percentage differences $({\tilde{\beta }}_{EAP}-{\tilde{\beta }}_{FF})$ between them agreeing well with Chami et al. (Fig. 4 in [16], here Fig. 2), although varying somewhat in magnitude at very small scattering angles. Also notable are the interference structures most evident at small D_eff. Their prominence is reduced with larger D_eff as the high-angular resolution discrete particle size phase functions are integrated over a greater range of particle sizes, where the correspondingly larger core spheres absorb more, reducing reflected light at the outer/core sphere interface. The resulting phase functions at larger D_eff are shapewise more consistent with Chami’s data. These features are not seen in measured oceanic VSF data, as natural waters generally contain both phytoplankton and non-algal sources of scattering, and also have unique particle size distributions.

 

Fig. 2 Percentage differences between ${\tilde{\beta }}_{EAP}$ for a generalised eukaryote assemblage with D_eff = 12 µm, and corresponding ${\tilde{\beta }}_{FF}$. Percentages are calculated by $({\tilde{\beta }}_{EAP}-{\tilde{\beta }}_{FF})/{\tilde{\beta }}_{EAP}*100$. Selected phase functions for different wavelengths and phytoplankton backscatter fractions are shown in the top panel. The differences between them are shown in the bottom two panels: 0.1° to 100° are shown on a log scale, and 90° to 180° below. These phase functions were selected to illustrate variability in wavelength and backscatter fraction, and how the differences between ${\tilde{\beta }}_{EAP}$ and ${\tilde{\beta }}_{FF}$ vary with both of these parameters.

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3.2. Comparing the Fournier Forand and EAP phase functions: prokaryote (vacuolate) group

Figure 3 shows the analagous comparison of vacuolate ${\tilde{\beta }}_{EAP}$ with the equivalent ${\tilde{\beta }}_{FF}$ as determined by the appropriate particulate backscatter fraction. Both eukaryotic (dinoflagellate) and prokaryotic (cyanobacterial) ${\tilde{\beta }}_{EAP}$ have reduced small angle maxima with respect to to their corresponding ${\tilde{\beta }}_{FF}$, but the cyanobacterial ${\tilde{\beta }}_{EAP}$ is reduced to a much greater extent. The reduction in forward light scatter is caused by the lowering of the overall real refractive index of the cell by the gas vacuole [19]. The reduction of scatter at small angles is compensated for by a noticeable increase in scatter compared to the ${\tilde{\beta }}_{FF}$ at intermediate angles of 20° to about 100°. The high perpendicular light scatter was also attributed to gas vacuoles in cyanobacteria, observed experimentally [20]. This is therefore not evident in the case of the dinoflagellates. The contrasting IOPs of these two populations are presented in Figs. 4(A)–4(C), together with the 550 nm phase function for each population and its Fournier Forand equivalent in Fig. 4(D).

 

Fig. 3 Selected EAP phase functions for a generalised prokaryote (vacuolate) assemblage with D_eff = 5 µm, are shown in the top panel together with their corresponding ${\tilde{\beta }}_{FF}$, and the percentage differences between them are shown in the panels below.

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Fig. 4 IOPs for generalised eukaryote and prokaryote assemblages, with effective diameters of 12 and 5 µm respectively. The elevated total scatter by the prokaryotes (B), and correspondingly elevated backscatter probability (C), is reflected in the comparatively higher phase function values (D), with unique angular variability (D). The angular variability in the phase function of the highly scattering prokaryote assemblage is markedly different in shape from the equivalent ${\tilde{\beta }}_{FF}$.

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3.3. High biomass validation of EAP phase functions in terms of resulting modelled R_rs

In Case 1 bloom conditions (Chl a > 100 mg.m⁻³) it can be assumed that the phytoplankton IOPs dominate the IOP budget, and Benguela blooms therefore offer an ideal opportunity to compare the use of the EAP and FF phase functions in terms of their impact on the R_rs. Mean measured R_rs are shown for three samples in Fig. 5, with the standard deviation representing the variability in the measurements. The measurements are all from a 2005 Southern Benguela bloom (see [8]) dominated by Prorocentrum triestinum, a small dinoflagellate approximately 18–22 µm in length and 6–11 µm in width. Size was measured by Coulter Counter (see [21]), with a resulting effective diameter of 13.1 µm ± 1.2.

 

Fig. 5 Demonstration of measured/modelled R_rs closure for 3 high biomass examples from a 2005 bloom in the Southern Benguela. The corresponding FF R_rs are much brighter than their EAP counterparts.

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The modelled R_rsλ were generated using a 4-component Hydrolight 5.2 model with non-algal particle (NAP) scatter b_napλ, combined gelbstof and detrital absorption a_gdλ and phytoplankton components as detailed in [8]. This method describes the use of the combined spectrally variant particulate ${\tilde{b}}_b\lambda$ to determine an appropriate ${\tilde{\beta }}_{FF}$. For this exercise, for comparison with the combined ${\tilde{b}}_b$ Fournier Forand approach, the appropriate ${\tilde{\beta }}_{EAP}$ were used for the phytoplankton component, combined with a 1% ${\tilde{b}}_b\text{-determined}$ ${\tilde{\beta }}_{FF}$ for the non-algal particulate component. Because the ${\tilde{\beta }}_{EAP}$ vary with both ${\tilde{b}}_b$ and λ, it was only possible to model this in Hydrolight using 101 single-wavelength runs (400 to 900 nm at 5 nm resolution) and collating the results to get the full spectral R_rs. This means that the effects of fluorescence could unfortunately not be included, so deviation from the measured R_rs is to be expected in the 683 nm Chl a fluorescence region and will be ignored in this discussion.

Forward modelled populations of D_eff of both 12 and 16 µm are presented, to illustrate how the influence of particle size is inextricable from that of the choice of scattering functions unless size information is known. The corresponding FF R_rs are much brighter than their EAP counterparts. This is consistent with the differences in phase functions as described earlier, and also with previously published results [2, 16]. While a good fit in terms of shape and magnitude is attainable with ${\tilde{\beta }}_{FF}$ combined with a larger D_eff, the proper assessment of uncertainty due to choice of phase function must be measured between R_rs modelled with identical size parameters. When it comes to model closure, this ambiguity in contributions to brightness of the R_rs must be satisfactorily resolved. To be clear, in the absence of more detailed assemblage information from this bloom, no definitive conclusion can be made about any advantage of the ${\tilde{\beta }}_{EAP}$ at 12 µm over the ${\tilde{\beta }}_{FF}$ at 16 µm in terms of validation of either model with respect to the measurements. But when evaluating the impact of choice of phase function in modelling R_rs, the comparison must be made between FF and EAP R_rs resulting from the use of the same population effective diameter - these differences indicate the uncertainty attributable only to the phase function.

In both versions of the modelled R_rs there is a notable deviation from the measured data from about 730 nm onwards, although the two modelled versions agree well. Mobley et al. [4] describe a drop off in instrument performance (resulting in underestimated upwelling radiance L_u) as the signal approaches the instrument noise level with depth - the TSRB measures at a depth of 0.66 m, which in such high biomass equates to many optical depths in the NIR, leading to greater uncertainties at these wavelengths due to rapid light attenuation and instrument self-shading, particularly in the red region where absorption by water is significant [22]. In addition, the measured data were not corrected for temperature-dependent absorption by water in the NIR. Pending further measurements the authors are unable to comment further on the deviation of both modelled R_rs from the measurements in this region.

Importantly, the 709 nm region (carrying critical biomass and assemblage-related signal [23]) is sensitive to the use of the comparative phase functions in the modelled data. Likewise, large differences are noticeable in the 550 – 650 nm region. Both of these regions (550 to 650 nm, and 709 nm peak) contain absorption features of diagnostic accessory pigments useful in resolving for example, trophic status and the presence of diagnostic features of cyanobacteria, including phycocyanin pigment [24], and have previously been identified as sensitive to size-related assemblage variability as well [21]. Information in these critical spectral regions is vital for PFT algorithm development including inversions and the retrieval of IOPs.

Figure 6 shows the R_rs generated from Hydrolight, resulting from the use of the ${\tilde{\beta }}_{EAP}$ and ${\tilde{\beta }}_{FF}$ respectively for idealised prokaryote and eukaryote populations. These modelled examples were run for Chl a values of 1, 10, 100 and 500 mg.m⁻³ in order to show the progressive effect of the difference in phase function as trophic status increases. In order to isolate the effect of the phytoplankton absorption and scattering, additional optical components are not included here (i.e. absorption by gelbstoff and/or tripton, and additional small particle/sediment/detrital scattering). This is not the case in natural waters and these examples are intended as illustrative.

 

Fig. 6 Modelled R_rs for prokaryotes (cyanobacteria) (A), with D_eff = 5 µm, and eukaryotes (mixed dinoflagellates) (B), with D_eff = 12 µm, for increasing Chl a = 1, 10, 100 and 500 mg.m⁻³.

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When examining the R_rs systematically in this way, the resulting effect of the differences in shape and magnitude of the phase functions becomes apparent. Overall, the cyanobacterial R_rs (as representative of a prokaryotic assemblage) using the ${\tilde{\beta }}_{EAP}$ are brighter than the R_rs using the corresponding ${\tilde{\beta }}_{FF}$. However this is not the case with the mixed dinoflagellates (representing eukaryotes). Here, the R_rs resulting from the use of the ${\tilde{\beta }}_{FF}$ are brighter overall. This appears to indicate that the greatly elevated scatter seen in the intermediate angles of the vacuolate ${\tilde{\beta }}_{EAP}$ contributes significantly to the overall magnitude of the spectral R_rs. This observation is consistent with Chami et al’s 2006 Monte Carlo experiment which determined, among other results, that in highly scattering conditions brighter R_rs is due to the phase function being greater at angles 10° to 100° [16]. Differences observed in the R_rs in a single scattering environment (where differences in small angle scattering are important) are magnified in a multiple scattering environment, where increased variability in scattering at the intermediate angles is seen.

The EAP model has enormous potential as a tool for evaluating the PFT signal across different biomass and IOP ranges. The large differences in the phase functions themselves have been presented above, and their impact on model closure explored. Detailed modelling of the interaction of biophysical relationships is now possible with some understanding of causality. Figures 7 and 8 show that differences in phase function shape are translated into significant uncertainties in the R_rs even at quite low biomass where total particulate scatter is small, such as in an oceanic environment. The contribution of non-algal scatter is modelled with b_bnap550 = 0.0005m⁻¹ [21]. ’Model uncertainty’ denotes the difference in R_rs resulting from the use of either ${\tilde{\beta }}_{EAP}$ or ${\tilde{\beta }}_{FF}$ in the radiative transfer calculation. For a eukaryote (e.g. dinoflagellate) phytoplankton population with effective diameter of 6 µm (Fig. 7), the absolute magnitude of the model uncertainty is in the order of 10⁻⁴ from a Chl a concentration of 1 mg.m⁻³, which is significant given that the instrument threshhold resolution of measured reflectance is about 1×10⁻⁴ sr⁻¹ [16]. Figure 9 shows the actual comparative R_rs at selected wavelengths of the large and small absolute and percentage differences discussed with respect to Figs. 7 and 8.

 

Fig. 7 Differences between R_rs EAP and R_rs FF, R_rs expressed per steradian (above) and as unsigned percentages of R_rs FF (below), shown for a generalised 6µm population against a background of low NAP backscatter (b_bnap550 = 0.0005m⁻¹). Comparative EAP and FF phase functions for 3 different wavelengths and backscatter fractions are presented in the bottom row.

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Fig. 8 Differences between R_rs EAP and R_rs FF, R_rs expressed per steradian (above) and as unsigned percentages of R_rs FF (below), shown for a generalised 16 µm population, low NAP (b_bnap550 = 0.0005m⁻¹) conditions. Comparative EAP and FF phase functions for 3 different wavelengths and backscatter fractions are presented in the bottom row.

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Fig. 9 R_rs EAP and R_rs FF, R_rs shown for generalised 6 and 16 µm populations in low NAP (b_bnap550 = 0.0005m⁻¹) conditions. The black lines indicate the wavelengths for which the phase functions are shown in Figs. 7 and 8. The grey boxes identify three example regions of high absolute and low percentage error (shown in Figs. 7 and 8) and vice versa.

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The spectral variability of the absolute differences in R_rs means that the comparative ${\tilde{\beta }}_{EAP}$ or ${\tilde{\beta }}_{FF}$ R_rs yield notably different shapes in many spectral regions. The absolute differences translate into percentages of over 10% across much of the spectrum, and notably in diagnostically important regions for PFT analysis. The effect of an increased population effective diameter can be seen in Fig. 8, showing that at the same biomass, both the absolute and percentage differences are reduced for a 16 µm population with respect to the 6 µm example. This is due to the elevated total scatter by smaller cells - which means pronounced spectral variability in the backscatter, translating into large differences between the shapes of the Fournier Forand and EAP phase functions. Harmel et al. observe that the most notable source of inaccuracy of the Fournier Forand approximation is in the region from 150° upwards [18], and these results appear to confirm this. In the bottom panel of each figure, three wavelengths have been selected to examine this effect further: 440 nm is an area of low resulting difference in R_rs (due to large absorption by phytoplankton and thus a reduced number of photons available for scattering), and the differences between ${\tilde{\beta }}_{EAP}$ and ${\tilde{\beta }}_{FF}$ are the smallest of the three cases - negligible for the 16 µm example, which results in comparably small R_rs error. 515 nm is an area of intermediate difference in the R_rs which corresponds to some divergence in the phase functions, while 650 nm represents an area of large divergence in the R_rs and shows sizeable differences between ${\tilde{\beta }}_{EAP}$ and ${\tilde{\beta }}_{FF}$ post 150°.

Both figures show Chl a from 0 to 15 mg.m⁻³, but it should be considered that in practice it is likely that a 16 µm population of eukaryotes (e.g. diatoms) would occur at the higher end of this Chl a scale while a 6 µm population would be observed at the lower end [25], and so the magnitude of differences in R_rs are comparable when considering the respective ecological niches of these populations. At low biomass, the impact on R_rs due to choice of phase function is greater for small cells than for a large celled population because small cells scatter proportionally more: in certain spectral regions differences of 30% are seen as low as Chl a = 1.5 mg.m⁻³ for a 6 µm population. At high biomass the uncertainty due to the phase function grows because of the increased number of cells.

4. Conclusions

It is evident that given the impracticalities of VSF and phase function measurements, only by modelling can we generate phytoplankton-specific phase functions across wide ranges of assemblages and water types. The need for angularly resolved VSFs both measured and modelled has been articulated elsewhere [27, 28] and confirmed here. The EAP model presents a full physics-based calculation of the directional scattering characteristics of various modelled phytoplankton populations, unveiling a new opportunity to further understanding of the importance and impact of the highly variable spectral and angular scattering properties of phytoplankton. It is clear that to properly address the PFT question, detailed variability in phytoplankton phase functions must be represented.

In both forward and inverse models in use by the ocean colour community, reflectance is generally simplified in one of two ways: either by approximating the phase functions themselves, or by avoiding the radiative transfer calculation altogether and instead representing the bidirectional character of the upwelling light field using an f/Q parameter [26]. These are powerful and necessary tools but it is important to properly understand the implications of these approximations in a PFT context, given that the magnitude of the uncertainty in R_rs due to choice of phase functions has been shown here to be significant. A previous study on accuracy and ambiguity in inversion models making these approximations (including a Fournier Forand version of the EAP model) describes the difficulties encountered by these models in highly scattering waters [21]. It has been shown here that even for intermediate levels of phytoplankton scatter (whether due to elevated biomass or small particle sizes), the question of an appropriate phase function must be adequately addressed if measurement/model closure is to be attained.

It is clear that there is considerably more variability in the spectral and angular scattering properties of phytoplankton that is either explicity or implicity acknowledged by the ocean colour community. It is also clear that this variability has considerable effect on the spectral nature of the water-leaving radiance. This fully quantitative model allows us to explore the causal biophysical relationships central to the PFT question. It is hoped that this study is a starting contribution to a full radiative transfer based modelling study of PFT sensitivity in the ocean colour signal.