Using an extensive database of in situ observations we present a model that estimates the particle backscattering coefficient as a function of the total chlorophyll concentration in the open-ocean (Case-1 waters). The parameters of the model include a constant background component and the chlorophyll-specific backscattering coefficients associated with small (<20μm) and large (>20μm) phytoplankton. The new model performed with similar accuracy when compared with a traditional power-law function, with the additional benefit of providing information on the role of phytoplankton size. The observed spectral-dependency (γ) of model parameters was consistent with past observations, such that γ associated with the small phytoplankton population was higher than that of large phytoplankton. Furthermore, γ associated with the constant background component suggests this component is likely attributed to submicron particles. We envisage that the model would be useful for improving Case-1 ocean-colour models, assimilating light into multi-phytoplankton ecosystem models and improving estimates of phytoplankton size structure from remote sensing.
© 2012 OSA
Spectral variations in visible sea-surface reflectance (R) form the basis of ocean colour radiometry. By analysing these variations, concentrations of optically-distinct water constituents such as phytoplankton, detritus and coloured dissolved organic matter may be estimated. In Case-1 waters , typically encountered in the open ocean, and comprising ∼60% of the world ocean , spectral variations in R are mainly driven by the abundance of phytoplankton, with a co-varying influence from detritus and coloured dissolved organic matter.
At a given wavelength (λ), R can be related to the inherent optical properties (IOPs) of absorption (a) and backscattering (bb) according to:3], and λ represents wavelength. The inherent optical properties a and bb can be further partitioned according to the different water constituents influencing them. The absorption coefficient a(λ) can be expressed as
Whereas phytoplankton influence aph directly, the influence of phytoplankton on bbp in Case-1 water is thought to be indirect, caused by factors that covary with phytoplankton. Phytoplankton are thought to contribute only a small fraction of bbp as they have a refractive index close to that of seawater, and lower than inorganic particles , and are large with respect to visible wavelengths. However, recent evidence has suggested that phytoplankton-sized particles may contribute more to bbp than previously believed [5–8]. Our understanding of the relationship between phytoplankton and bbp has to be improved, to enhance further development of ocean-colour models in Case-1 waters.
Phytoplankton are the primary component responsible for changes in the ocean colour detected by satellites over Case-1 waters , and are recognised as an Essential Climate Variable by the Global Climate Observing System. Thus, a better understanding of the relationship between chlorophyll (a common measure of phytoplankton concentration) and bbp is required to improve satellite-based chlorophyll estimates .
Typically, in many Case-1 ocean-colour models, the non-water components of a and bb are parameterised as functions of chlorophyll (e.g. [9, 11–16]). Power-law functions have proven to be useful predictors of aph (e.g. [17–22]) and bbp (e.g. [15, 23, 24]). However, the interpretation of the parameters of a power-law is difficult from an analytical perspective [25, 26]. Recently, aph has been expressed as the sum of contributions of phytoplankton populations that are optically-distinct, typically linked with size structure [16,25,27,28]. Such an approach ensures biological and bio-optical interpretation of model parameters and has allowed for the extraction of additional information, such as phytoplankton community composition, from ocean-colour observations [29–31].
Considering that the spectral-characteristics of bbp have been related to phytoplankton size [32–34], the same modelling approach could be extended to bbp conceptually, bearing in mind that bbp may be influenced only indirectly by phytoplankton, even in Case-1 waters. Understanding how changes in phytoplankton size structure influence bbp would be useful for satellite detection of phytoplankton size classes [30, 31, 35–42] and for improving the assimilation of ocean-colour data into multi-phytoplankton ecosystem models .
In this paper, we develop a simple model that captures the relationship between chlorophyll and bbp, with a similar performance when compared with a traditional power-law model, but offering model parameters that are readily explained and the capability of estimating size-fractionated bbp. The model is parameterised using a large dataset of chlorophyll and bbp observations in the open ocean at two wavelengths (470 nm and 526 nm), validated using independent data, compared with current satellite models, and fitted to a multi-spectral database of chlorophyll and bbp to investigate spectral-dependency in model parameters beyond the two wavelengths used in model development.
1.1. Statistical tests
Model performance was quantified using the Pearson linear correlation coefficient (r) and the centre-patterned (or unbiased) root mean square error (Δ) between the modelled and measured values, the latter expressed as
In this study, data sampled in surface waters were used, after partitioning into four separate databases for model development (database A), validation (database B), comparison with satellite models (database C) and analysis of spectral-dependency in model parameters (database D).
1.2.1. Database A: model development
For model development, in situ particle backscattering coefficients at two wavelengths (bbp(470) and bbp(526)) and chlorophyll concentrations were downloaded from the NASA SeaBASS website (http://seabass.gsfc.nasa.gov/, ) inclusive of a variety of geographic locations and trophic conditions in the open ocean (Case-1 waters, ).
This included surface flow-through measurements taken in the Equatorial Pacific (April–May 2007, R/V Ka’imi Moana, [7, 45]) and the North Atlantic (May 2008, R/V Knorr, ), representing 1 minute averages sub-sampled at 10 minute intervals. Optical measurements were collected and processed with the same experimental set-up on both the Equatorial Pacific and the North Atlantic cruises. Periodic comparisons were made between optical flow-through data and optical data measured from surface seawater collected with a traditional CTD rosette . Flow-through absorption measurements were made with a hyperspectral absorption-attenuation meter (WET Labs ACs) and backscattering measurements with a multispectral backscattering sensor (WET Labs ECO-BB3). Every 10 minutes seawater was diverted through a 0.2 μm nylon cartridge filter providing a baseline for dissolved substances, calibration uncertainties and short-term bio-fouling. Calibration independent particulate optical quantities, with an uncertainty close to instrument precision [7,45], were computed by subtracting the baseline from the bulk measurements. For additional details of the flow-through set-up, the reader is referred to Dall’Olmo et al.  and Westberry et al. .
Each through-flow dataset provided measurements of bbp(470), bbp(526) and absorption coefficient of particulate matter (ap) at 650 nm, 676 nm and 714 nm. Any measurements less than or equal to zero were removed and the chlorophyll concentration was estimated from absorption using the approach of Boss et al. , such that chlorophyll = (ap(676) − (0.6ap(650) + 0.4ap(714)))/0.014. This approach uses the line-height of the chlorophyll absorption peak at red wavelengths and it was selected as it performed with high accuracy when compared with discrete in situ High Performance Liquid Chromatography (HPLC) chlorophyll data for both the Equatorial Pacific and North Atlantic datasets (see Fig. 3 of Westberry et al. ). This resulted in 633 and 217 samples from the Equatorial Pacific and North Atlantic respectively. HPLC data taken during the North Atlantic cruise were also downloaded from the NASA SeaBASS website, and an additional HPLC dataset from the Equatorial Pacific, taken as part of the Tropical Atmosphere Ocean project in 2006.
In addition to Equatorial Pacific and North Atlantic samples, data were downloaded from the NASA NOMAD database (Version 2.0, 18/07/2008, [47, 48]), which provided point in situ samples of remote-sensing reflectances, backscattering coefficients (bb(λ)), chlorophyll concentrations and HPLC data. To remove all Case-2 samples from the NOMAD data, we used the method of Lee and Hu  which partitioned Case-1 and 2 waters using remote-sensing reflectances. Only Case-1 NOMAD samples with original bb(470) were used in database A, to be consistent with the flow-through measurements. As original bb(526) data were unavailable in NOMAD, bb(526) values were estimated by application of a power-law spectral fit to nearby wavelengths [49, 50], fitted individually for each sample. This approach was validated by comparing original bb at available green wavelengths (514, 532 and 589 nm) with estimated bb at corresponding wavelengths assuming a power-law spectral fit. The average difference was ∼2.7 %. Note also that for these 163 measurements, the difference in spectral slopes between 470 and 526 nm using original bb(470) and estimated bb(526) and the slopes using original bb(470) and original bb at green wavelengths (514, 532 or 589 nm) was <9%.
This resulted in 163 NOMAD samples that could be used in database A. For these samples, where available, chlorophyll concentrations were taken from HPLC, and where not available, from fluorometric measurements or estimated from remote-sensing reflectances using the OC4v6 algorithm . Corresponding measurements of HPLC and OC4v6 estimated chlorophyll in NOMAD (Case-1 waters ) are in good agreement (r = 0.937, Δ = 0.197, δ = −0.059, 544 log10-transformed samples), as are corresponding measurements of HPLC and fluorometry estimated chlorophyll (r = 0.967, Δ = 0.118, δ = 0.037, 235 log10-transformed samples), supporting our combining of chlorophyll measured using these different methods. The bbp values were estimated from the NOMAD data according to bbp = bb − bbw, and the backscattering coefficient of pure sea-water (bbw) was computed from Zhang and Hu  and Zhang et al. , consistent with the analysis on flow-through measurements. Figure 1 shows the geographical distribution of database A.
1.2.2. Database B: independent validation
For model validation, a large dataset of surface flow-through measurements, representing 1 minute averages, were taken from the Atlantic Meridional Transect (AMT) cruise 19 (35,961 samples, October–November 2009, RRS James Cook), from the Mediterranean Sea (5618 samples, June–July 2008, R/V L’Atalante [45, 54]) and from the North East Pacific (4760 samples, August 2009, CCGS John P. Tully ). Data from these cruises were collected and processed using the same methods as flow-through data used in database A. Only data after the 26th of August 2009 from the North East Pacific were used, as prior to this date conditions were very rough and flow-through measurements may have been contaminated by bubbles . Database B included bbp taken at two wavelengths (470 and 526 nm) and corresponding hyperspectral measurements of particle absorption (see  for methodological details). The AMT continuous flow-through measurements of bbp were verified against discrete overlapping in situ bbp sampled using an optical profiler, and found to be in good agreement (Δ = 0.028, δ = −0.042 at 470 nm, and Δ = 0.025, δ = −0.022 at 526 nm) consistent with the flow-through measurements from the Equatorial Pacific, the Mediterranean and the North Atlantic [7,45]. For each sample, the chlorophyll concentration was estimated from absorption [45, 46] and found to be in good agreement with discrete HPLC estimated chlorophyll sampled from the flow-through system (Δ = 0.163 and δ = −0.030 in log10 units). Of the AMT samples, only those with chlorophyll concentrations >0.02 mg m−3 were used, which corresponded to the minimum values of the discrete HPLC extracted measurements. Database B was used for independent model validation, and its geographical distribution is shown in Fig. 1.
1.2.3. Database C: satellite data
To compare models developed using database A with models that estimate bbp(470) and chlorophyll from satellite remote sensing reflectance (Rrs(λ)) data, we downloaded monthly, Level 3, mapped SeaWiFS composites of Rrs(λ) for May 2006 from the NASA website (http://oceancolor.gsfc.nasa.gov/). Using Rrs(λ), bbp(443) and aph(443) were estimated for each pixel using the model of Smyth et al.  (hereafter denoted SMHA following the first initial of each author’s surname). In addition, monthly SeaWiFS composites of bbp(443) and chlorophyll for the GSM model  and the GIOP model [58, 59] (default configuration), and bbp(443) and aph(443) for the QAA model , were downloaded from the NASA website. The GIOP model is designed as a test platform for algorithm development, offering freedom to specify various optimisation approaches and parameterisations. An initial default configuration for GIOP has been defined by a committee for the global ocean, this includes a fixed spectral slope for adg(λ) of 0.018, an assigned specific phytoplankton absorption coefficient following Bricaud et al. , a spectral backscattering dependency following the QAA and Levenburg-Marquardt optimisation. It is worth noting that this initial default configuration may be changed with time. For the QAA and SMHA models, chlorophyll was estimated using a power-law function of aph(443) with the coefficients presented in Bricaud et al. . Values of bbp(470) for each pixel were estimated from bbp(443), according to the manner by which each model treats the spectral dependency of bbp. The GSM and SMHA models assume a fixed spectral backscattering dependency (γ) of 1.0337 and 0.5 respectively, and the QAA and GIOP treats the spectral backscattering as a function of a ratio of remote-sensing reflectance just below the sea-surface at two wavelengths (443/555 nm), estimated here using monthly Rrs(λ) for each pixel. Using the method of Lee and Hu , all Case-2 pixels were removed. The geographical distribution of Database C is shown in Fig. 1.
1.2.4. Database D: spectral dependency in model parameters
For additional analysis of spectral-dependency in model parameters, the NOMAD dataset was again used. This included samples of remote-sensing reflectances, in situ backscattering coefficients at 20 wavelengths in the visible spectrum (bb(λ)), which were estimated by fitting a power-law spectral dependency to original data to remove moderate noise often resulting from instrument artifacts or calibration , and corresponding chlorophyll concentrations. Data were synthesized in the same manner as with the NOMAD samples used in database A to remove Case-2 samples. Some 210 concurrent spectra of bb(λ) at the 20 NOMAD wavelengths and chlorophyll concentrations (0.064 to 11.83 mg m−3, note only 12 samples <0.2 mg m−3 and therefore biased towards more eutrophic waters) were extracted. For these 210 concurrent spectra, when comparing original bb data at various wavelengths with estimated bb data at the corresponding wavelengths assuming a power-law spectral fit, we compute an average difference of <4%. The bbp values were computed from bb as described in database A. The geographical distribution of database D is shown in Fig. 1. Note that some of these measurements were not independent of the NOMAD samples used in database A. However, this dataset was not designed for validation, purely to investigate spectral variations in model parameters while fully acknowledging the limitations of the method by which such a multi-spectral database were developed . A flow chart illustrating the procedures used to partition the data into the four databases (A–D) and develop and validate the model is provided in Fig. 2.
2. Model development and Results
2.1. Model development
As our study focuses on Case-1 waters, not inclusive of coastal waters, bbp(λ) can be tied to the chlorophyll concentration (C) using a power function (e.g. ), such thatEquation (6) was fitted to database A using a standard least squares fit (Levenberg-Marquardt, IDL Routine MPFIT-FUN ) with relative weighting to give equal weights to all measurements, and is superimposed onto database A in Fig. 3(a) and 3(b). Retrieved parameters are provided in Table 1 and are comparable to those obtained by Huot et al.  and by Antoine et al. . Statistical results from fit are provided in Table 2. When fitting Eq. (6) to database A, as with subsequent models proposed in this study, we used the method of bootstrapping  to compute model parameters and their uncertainties. This involved randomly re-sampling (10,000 times) database A (1013 samples) and re-fitting equations for each iteration. From the resulting parameter distribution, mean values and 95% confidence intervals were computed.
To explore how bbp(λ) may be related to the phytoplankton size structure, we also investigate an alternative parameterisation, as follows: We first express bbp(λ) as the sum of the contribution to particle backscattering by pico-phytoplankton and their co-varying constituents (bbp,1(λ)), nano-phytoplankton and their co-varying constituents (bbp,2(λ)) and micro-phytoplankton and their co-varying constituents (bbp,3(λ)), such that38], such that Eq. (8) to (11), subscript 1,2 refers to combined pico-nano-phytoplankton and the subscript 1 picophytoplankton, represents the asymptotic maximum that Ci can reach, and the initial slope of the curve between C and Ci is denoted Si.
Using pigment data from Brewin et al.  (NOMAD Version 1.3.h, 22/02/2007 HPLC evaluation dataset ), the parameters of Eq. (8) and (9), as well as their uncertainties, were computed using bootstrapping (Table 1). The model was found to be in agreement with independent HPLC data from other cruises used in this study, as indexed by a low Δ and a δ close to zero, when comparing the fractional contribution of each size class to total chlorophyll estimated from the model and from independent HPLC data [36, 38, 64] (Fig. 4). Expanding Eq. (7) by inserting the three-component pigment model (Eq. 8 to 11) leads to the following formulationEq. (12) was fitted to database A, to derive where i = 1 to 3 (Table 1), using the same fitting procedure as for Eq. (6). The fitted relationship is superimposed onto database A in Fig. 3(a) and 3(b) and statistical results from fit are provided in Table 2. Uncertainty in pigment parameters ( , S1,2, and S1) were accounted for when estimating using Monte-Carlo error propagation.
Whereas Eq. (12) captures the relationship between bbp(λ) and chlorophyll at intermediate to high chlorophyll (Fig. 3(a) and 3(b)), it significantly underestimates both bbp(470) and bbp(526) at low chlorophyll (<0.1 mg m−3 chlorophyll, δ = −0.182 at 470 nm and δ = −0.174 at 526 nm). Therefore, an additional parameter was added to Eq. (12) leading to the following expressionEquation (13) was fitted to database A using the same fitting procedure as with Eq. (12), and the results are shown in Fig. 3(a) and 3(b) and Table 2. By adding a constant background component, Eq. (13) captures the relationship between bbp and chlorophyll at low concentrations (<0.1 mg m−3 chlorophyll, δ = −0.012 at 470 nm and δ = −0.009 at 526 nm) and also performed with better accuracy than Eq. (12) at higher chlorophyll concentrations, as indexed by statistically lower Δ (t-test, p < 0.05) at both wavelengths for chlorophyll concentrations >0.1 mg m−3. Table 1 provides the retrieved chlorophyll-specific particulate backscattering coefficients ( ) for the three size-classes of phytoplankton and the estimated at 470 and 526 nm, when fitting Eq. (13) to database A.
For both bbp(470) and bbp(526), the microphytoplankton chlorophyll-specific particle back-scattering coefficients ( ) retrieved from fitting Eq. (13) were statistically lower than the corresponding values for both pico- ( ) and nano-phytoplankton ( ) (see Table 1). However, 95% confidence intervals in and overlapped at 470 nm (Table 1). Furthermore, when increasing confidence intervals to 99%, and overlapped at both 470 nm and 526 nm (99% confidence intervals: , , and ) suggesting these retrieved parameters were not significantly different for database A. Therefore, Eq. (13) was simplified by combining the pico- and nano-phytoplankton populations, such thatEquation (14) was fitted to database A (see Table 1 for parameters) and is plotted in Fig. 3(a) and 3(b). This simplified model (Eq. (14)) performed with the same accuracy as Eq. (13), as indexed by statistically similar r values (z-test, p > 0.05), Δ (t-test, p > 0.05) and δ (t-test, p > 0.05) for both wavelengths (Table 2), yet the number of model parameters at each wavelength was reduced from four to three ( , and ). Note that and S1,2 were derived from independent HPLC data and held constant when fitting Eq. (14) to database A.
In Eq. (14), the parameters used to partition chlorophyll into the two size fractions ( and S1,2) were derived using NOMAD HPLC Version 1.3.h data [28, 36, 38, 47, 64]. Recently, modifications have been proposed to the pigment-based phytoplankton size classification to account for fucoxanthin associated with nano-phytoplankton [31, 40]. To investigate the effect this may have on the parameters obtained for Eq. (14), and S1,2 were re-computed using the fucoxanthin modification of Devred et al. , and then Eq. (14) was re-fitted to database A using the refined pigment parameters as input (see Table 1). Parameters for Eq. (14) that accounted for the fucoxanthin modification were found to be statistically similar to the original parameterisation (overlapping 95% confidence levels, Table 1) suggesting Eq. (14) is insensitive to the fucoxanthin refinement, at least for database A.
2.2. Model validation
Using chlorophyll from database B as input, bbp was estimated using Eq. (6) and Eq. (14). These values were then compared with bbp from database B (Fig. 3(c) and 3(d)). Both models are seen to perform reasonably, as indexed by high r values (>0.8), a low Δ and a δ close to zero for both wavelengths (470 nm and 526 nm, Table 2). Equation (14) obtained a δ closer to zero when compared with Eq. (6), for both wavelengths, whereas Eq. (6) obtained slightly higher r values and a slightly lower Δ for both wavelengths (Table 2).
Figure 3(c) and 3(d) shows higher bbp values in the NE Pacific data in comparison with the AMT and Mediterranean cruises at similar chlorophyll concentrations. Whereas all data prior to 26th August 2009 were removed from this cruise to reduce contamination from bubbles due to rough seas , some measurements may still have been affected. Analysis of HPLC data from the NE Pacific indicated a dominance of nano-phytoplankton (see Fig. 4) with high concentrations of 19′-Hexanoyloxyfucoxanthin, a diagnostic pigment of prymnesiophytes, indicating presence of coccolithophores or Phaeocystis (also confirmed using additional HPLC data ). During the cruise, samples for microscopy at two stations were taken at 5 m and both samples indicated presence of coccolithophores (∼10 and 250 cells ml−1 at station P26 and P4, respectively ). It is important to note that the models developed are designed for Case-1 water exclusive of coccolithophore blooms. When removing NE Pacific data from database B, model performance significantly increased for both Eq. (6) and Eq. (14) (see Table 2), with Eq. (14) obtaining slightly higher r values, a lower Δ and a δ closer to zero for both wavelengths.
Below a chlorophyll value of 0.05 mg m−3 the two models deviate and Eq. (14) captures better the trends in database B (Fig. 3(c) and 3(d)), as further evidenced when plotting a histogram of the log10 residuals of bbp between model estimates and database B below 0.05 mg m−3 chlorophyll (Fig. 5). By incorporating a constant background parameter ( ), Eq. (14) performs better than a power-law model (Eq. (6)) at very low chlorophyll (<0.05 mg m−3) when compared using database B.
2.3. Comparison with satellite models
Figure 6 show the results from the satellite retrieved bbp(470) plotted against satellite retrieved chlorophyll for the (a) GSM, (b) SMHA, (c) GIOP and (d) QAA models. Superimposed onto Fig. 6 are Eq. (6) and (14) for comparison. The QAA and the SMHA models display similar relationships between bbp(470) and chlorophyll which is expected to a certain degree, considering both models are conceptually similar whereby solutions are obtained through algebraic decomposition. The GIOP model also displays a similar relationship to that of the QAA, considering its preliminary configuration has some similar features to the QAA (such as the conversion of above to below reflectance and the spectral shape of the bbp), although the GIOP preliminary configuration uses Levenburg-Marquardt optimisation. Unlike the other satellite models, the GSM model shows more of a bilinear relationship between bbp(470) and chlorophyll .
The SMHA model, and to a lesser degree the QAA and GIOP model, appear to overestimate bbp(470) at low chlorophyll when compared with Eq. (6) and (14). Note that the SMHA model has a flatter spectral dependency than the GSM model (γ of 0.5 in comparison with 1.0337, respectively) as it was developed using a Case-2 database , which is likely to contribute to higher bbp(470) when extrapolating from bbp(443). The higher bbp(470) values at low chlorophyll for the satellite models may be partly explained by the fact that they are not accounting for Raman scattering. Lee et al.  found that derived bbp values from analytical algorithms can be elevated by as much as 60% for clear waters when not accounting for Raman effects. This possibility needs further investigation, but is beyond the scope of this paper.
Particle backscattering at 470 nm (bbp(470)) was computed from Eq. (6) and (14) using chlorophyll from the four satellite models as input. The retrieved bbp(470) values were then compared with the bbp(470) estimates from the four satellite models (Table 2). Higher r values were observed when comparing the results from Eq. (6) and (14) with bbp(470) estimated by the GIOP, QAA and SMHA models, in comparison with the GSM model (Table 2), indicating the relationship between bbp(470) and chlorophyll is less linear (in log10 space) for the GSM model. An δ value closer to zero was observed when comparing the results from Eq. (6) and (14) with the GSM model, in comparison with the GIOP, QAA and SMHA models (note that δ in Table 2 was computed assuming Xi,E = satellite bbp(470) and Xi,M = bbp(470) computed using Eq. (6) or (14), see Eq. (5)). Similar Δ values were observed for all four satellite models and are comparable with Δ values obtained using database B (Table 2). Equation (6) and (14) are in reasonable agreement with the models of Sathyendranath et al.  and Huot et al.  (Fig. 6(b) and 6(d)) at chlorophyll >0.1 mg m−3, but display higher estimates of bbp(470) at chlorophyll <0.1 mg m−3 and are in closer agreement with the satellite models.
2.4. Spectral dependency in model parameters
Assuming that the spectral shape of the particulate backscattering coefficient can be expressed using a power function (e.g. [31,56–58,60]), the spectral dependency of the chlorophyll-specific particulate backscattering coefficients ( ) associated with small and large phytoplankton, and the constant background particle backscattering component ( ), may be expressed asEquations (18) to (20) were applied to database A to derive γ1,2, γ3 and γk (Table 3 and Fig. 7(a) and 7(b)) together with their 95% confidence levels using the bootstrap method. Expanding Eq. (14) by inserting Eq. (15) to (17) leads to the following expression: 62], and therefore limitations of database A, we also estimated γi using database D. Equation (14) was first fitted to database D to compute , and at each of the 20 wavelengths (Fig. 7), then Eq. (15) to (17) were fitted to the spectral data to compute γ1,2, γ3 and γk (Table 3).
Estimated γ1,2 and γ3 using database D were similar to corresponding estimates from database A (overlapping confidence intervals in Table 3). However, γk derived from database D (∼3.4) was higher than γk from database A (∼1.9) and closer to that of pure water (e.g. ∼4.3 ). Considering only 12 samples in database D contained < 0.2 mg m−3 chlorophyll, with a minimum of 0.064 mg m−3, this may have been an artifact of uneven sample distribution. Nonetheless, γk estimates from both database A and D are consistent with satellite estimates in oligotrophic waters [32, 33]. Unlike database A, database D has the advantage of being a multi-spectral database. However, it is difficult to determine the extent to which the power-law spectral dependency assumption of bb in database D influences the spectral dependency in model parameters.
Table 3 highlights that γ3 is not statistically different from zero (95% confidence intervals overlap with zero, also see Fig. 7(a)). Therefore, Eq. (21) can be further simplified by setting the spectral dependency of to equal zero, such thatEq. (22) are easy to interpret in relation to the composition of phytoplankton size and allow for estimates of size-fractionated bbp(λ). Power-law models, such as that of Huot et al. , typically requires four parameters to describe bbp(λ) as a function of chlorophyll (e.g. see Eq. 8 of Huot et al. ), whereas, considering and S1,2 were derived independently from HPLC data, Eq. (22) requires five parameters ( , , , γ1,2 and γk). Yet the addition of another parameter appears justified with respect to the law of parsimony, given the increase in performance at low chlorophyll concentrations (Fig. 5).
Figure 7(c) shows the fractional contribution of each component population to bbp(470) for a given chlorophyll concentration estimated using Eq. (22). According to the model, the constant background component dominates bbp(470) when chlorophyll is low; as the chlorophyll increases beyond ∼0.2 mg m−3 the influence of small cells on bbp(470) increases, with its fractional contribution reaching a maximum of ∼0.7 at ∼1.0 mg m−3 chlorophyll; as the chlorophyll increases further, the influence from large cells to bbp(470) increases. The fractional contribution of each component to bbp varies with wavelength due to differences in the spectral dependency of each component population.
Figure 7(d) shows modelled γ using Eq. (21) as a function of chlorophyll (parameterised to database A), as well as the minimum and maximum of the model ensemble, calculated by varying model parameters between their confidence intervals (Table 3) in every possible permutation. The results from the model are superimposed onto the parameterisation data (database A) and the validation data (database B). Consistent with other in situ studies [15,62,69,70], modelled γ is inversely related to chlorophyll for both database A (r= −0.3, p<0.001) and database B (r= −0.4, p<0.001), with a large amount of scatter. Interestingly, one observes differences in the relationship between chlorophyll and γ in database B, when compared with database A with which the model was parameterised (nonetheless the minimum of the ensemble run captures the overall trend in database B). This may be indicative of regional differences , sensitivity to the wavelength range from which γ is computed and sensitivity to measurement noise . Comparisons must be cautiously interpreted considering: different ways in which bbp is determined by different investigations (e.g. NOMAD data and flow-through measurements), possible effects of absorption on bbp(470) retrievals, sensitivity of γ to measurement noise, and the limitations of determining γ from only two wavelengths .
3.1. Model parameters
3.1.1. Phytoplankton size-specific backscattering coefficients
The chlorophyll-specific backscattering coefficients associated with small cells (Table 3) are consistent with recent observations. Martinez-Vicente et al.  reported average of ∼0.005 m2(mgC)−1 in mesotrophic regions of the North Atlantic with an associated chlorophyll of ∼1.0 mg m−3, consistent with our derived (Table 1) and chlorophyll conditions where backscattering associated with small cells strongly influence total bbp (Fig. 7(c)). Lower obtained for large cells (Table 1 and 3) are consistent with Morel , who suggests larger cells with high intracellular pigment concentrations have a lower chlorophyll-specific scattering coefficient in comparison with small cells with lower intracellular pigment concentrations. However, it is important to note that the Morel  study focused on scattering and not backscattering. The observed spectral-dependency of and further support the interpretation and are in accordance with recent modelling studies [32, 33], such that γ associated with the small phytoplankton population, γ1,2, is higher than that of large phytoplankton, γ3 (Table 3). This is consistent with the assumption that small phytoplankton have enhanced backscattering of blue photons, whereas larger phytoplankton tend to backscatter light similarly at all wavelengths .
It is important to stress that these retrieved chlorophyll-specific backscattering coefficients are representative of not just the phytoplankton, but also their co-varying constituents (e.g. detrital matter, bacteria and viruses). It may be that the phytoplankton have a negligible direct influence on these coefficients . We also recognise that laboratory studies have highlighted large variability in chlorophyll-specific backscattering for different cultures , which may explain some of the observed variability in bbp(λ) when plotted as a function of chlorophyll in Fig. 3, though it still remains to be revealed to what extent such laboratory studies are representative of the natural environment.
3.1.2. Constant background parameter
By introducing into Eq. (13), a significant model improvement was obtained by decoupling bbp from chlorophyll in clear waters (Fig. 5). The parameter may be interpreted in two ways: (i) is due to a constant background of non-algal particles (e.g. heterotrophic bacteria, detritus, viruses, minerogenic particles) [4, 76], or (ii) is partly influenced by very small phytoplankton (e.g. prochlorophytes), in addition to non-algal particles . The high spectral dependency associated with is consistent with submicron particles , regardless of their living or non-living status.
The decoupling between bbp and chlorophyll in the most oligotrophic waters (Fig. 3) is however, in disagreement with data from the South Pacific subtropical gyre, where bbp and chlorophyll were correlated down to chlorophyll values of ∼0.02 mg m−3 . Whereas values are above instrument sensitivity, uncertainties in bbp and chlorophyll are expected to increase in oligotrophic waters . These higher levels of uncertainty, coupled to errors in bbw , suggest that the parameter should be cautiously interpreted.
3.2. Model formulation and applications
The formulation of Eq. (22), as depicted in Fig. 7(c), may be interpreted in the following manner. In oligotrophic waters (low chlorophyll), bbp is controlled by the parameter. In mesotrophic waters (intermediate chlorophyll), bbp is controlled by , and therefore small phytoplankton together with their associated covarying constituents. In eutrophic waters (high chlorophyll), the backscattering coefficient is controlled by large phytoplankton cells displaying low chlorophyll-specific backscattering. The low chlorophyll-specific backscattering in eutrophic waters (Case-1) implies that the backscattering is primarily driven by the phytoplankton, and to a lesser degree the covarying detrital matter .
Recently, methods have been proposed to detect phytoplankton size structure from satellite observations. These methods typically use either information on the satellite-derived absorption coefficient of phytoplankton, such as its shape or magnitude [30, 31, 37, 39], or information on the shape of the satellite-derived backscattering spectrum [32–34]. Recently, Fujiwara et al.  used information on both these variables empirically to derive satellite estimates of phytoplankton size-fractions in the Chukchi and Bering Sea shelf region. Considering that the reflectance observed by satellites is a function of both absorption and backscattering (Eq. (1)), accounting explicitly for information on size structure using both these variables may ultimately lead to better satellite retrievals of phytoplankton size. This may be achieved by combining Eq. (22) with a conceptually similar model of phytoplankton absorption (e.g. [16, 25, 28, 31]).
The retrieval of satellite-derived chlorophyll using empirical algorithms is highly dependent upon specific IOP values [10, 78] and phytoplankton community structure [29, 79]. Loisel et al.  highlighted that empirical chlorophyll algorithms are sensitive to , with higher-than-average values resulting in an overestimation of chlorophyll. The model presented here may be useful for incorporation into ocean-colour models, to constrain better the influence of and phytoplankton community structure on satellite chlorophyll estimates.
To advance our understanding of marine biogeochemical cycling, marine ecosystem models have developed more sophisticated representations of the phytoplankton community, typically partitioned according to size (e.g. [80–82]). There are clear benefits of coupling bio-optical models to ecosystems models [43, 83], such as a better representation of the underwater light field required for photosynthesis and photochemistry computations. Coupling bio-optical models to ecosystems models requires specifying IOPs for state variables that modify the light field, such as the phytoplankton communities. Our backscattering model may be useful in such studies, considering the parameters are specific to the phytoplankton size structure.
In comparison with the relationship between chlorophyll and phytoplankton absorption, the relationship between bbp and chlorophyll is likely to be more variable considering phytoplankton absorption and chlorophyll are influenced by both abundance and physiology, whereas bbp is influenced by non-algal particles as well as phytoplankton abundance. Equation (22) was fitted to data representative of the global ocean (Fig. 1), but we acknowledge that a global parameterisation is not likely to capture fully the spatio-temporal variability in the relationship between bbp and chlorophyll. Nonetheless, with adequate regional datasets, the model may be used as a tool to investigate such spatio-temporal variations (e.g. ).
A model has been presented that infers total and size-dependent particle backscattering as a function of chlorophyll in Case-1 waters. The model was parameterised using a database of chlorophyll and particle backscattering coefficients collected in various oceans and representative of oligotrophic to eutrophic waters (Case-1 water only). When compared with a large independent dataset, the model was found to perform with similar accuracy to a traditional power-law model, but with the added advantages of providing better performance at very low chlorophyll (<0.05 mg m−3), greater interpretation in model parameters and estimating size-fractionated particle backscattering. Spectral dependency in model parameters supports their interpretation and a simplified equation was presented to estimate backscattering as a function of chlorophyll over the entire spectral range. Model applications include: refining forward and inverse Case-1 ocean-colour models, ecosystem modelling and estimating phytoplankton size structure from remote sensing.
The authors thank SeaBASS, contributors to NOMAD and all principle investigators, scientists and crew involved with flow-through and pigment data. We thank Toby Westberry and Angelica Pena for comments on the NE Pacific data and Jeremy Werdell for comments on GIOP. We thank two anonymous reviewers for their comments on the paper. SeaWiFS data used in this publication were produced by the SeaWiFS project at the Goddard Space Flight Center. Use of this data is in accord with the SeaWiFS Research Data Use Terms and Agreements. This is a contribution to the Ocean Colour Climate Change Initiative of the European Space Agency. This study was supported by the NCEO, the UK Natural Environment Research Council and Oceans 2025. This is contribution number 219 of the AMT programme.
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