To compare variables, we used the Pearson linear correlation coefficient (r), the squared Pearson linear correlation coefficient (r ²), the centre-patterned (or unbiased) root mean square difference (Δ) and the bias (δ). The latter two statistics representing an index of precision and accuracy of a model, respectively. The root mean square difference ( Ψ ), which contains information on both accuracy and precision, can be reconstructed from Δ and δ, according to, Ψ = Δ 2 + δ 2 . The value of Δ and δ were computed according to

where X is the variable and N is the number of samples. The subscripts 1 and 2 represent different estimates of the same variable, with 1 typically representing the estimated variable and 2 the measured variable. In many cases, statistical tests were performed in log₁₀ space, depending on whether the distribution of the data was closer to log-normal. Linear models relating two variables (e.g., Secchi depth and Chl-a) were fitted (often after log₁₀-transformation of one or both variables, depending on their distribution) using a outlier-resistant fitting function (IDL function ROBUST_LINEFIT.pro) and non-linear models were fitted using least-square minimisation (Levenberg-Marquardt, IDL function MPFITFUN.pro (Moré, 1978; Markwardt, 2008), MatLab function fit.m).

Data used in this study were collected at 127 stations on four AMT cruises ( Figure 1 ) that took place onboard the RRS James Clark Ross ( Figure 2 ), each one departing the UK and arriving in Port Stanley, Falkland Islands, including: AMT cruise 23 (AMT23) that took place between the 7th October and 8th November 2013 (25 stations sampled); AMT cruise 25 (AMT25) that took place between the 11th September and 4th November 2015 (35 stations sampled); AMT cruise 26 (AMT26) that took place between the 20th September and 4th November 2016 (37 stations sampled); and AMT cruise 28 (AMT28) that took place between the 23rd September and 30th October 2018 (30 stations sampled). Stations were sampled primarily around local noon, with a few stations sampled on AMT26 around mid-morning, so as to align with the passing of ESA’s Sentinel 3A satellite.

Table 1 lists all datasets collected and used in the study, providing acronyms, units and number of stations sampled.

Figure 4 shows a comparison of optical measurements collected on the four AMT cruises. Consistent with previous understanding (e.g., Wernand, 2011), we see tight inverse relationships between Secchi depth (z_SD ) and Forel-Ule colour (F), with F explaining between 73-83% of the variance in z_SD , depending on the colour scale used (see Figures 4A–D ). For the colour of the disk (at 1/2 the Secchi depth, F_D,L and F_D,N ), and over the range of data collected, there appears to be a log-linear relationship between variables ( Figures 4A, B ). For infinite depth, the LaMotte scale (F_D,L ) also show a log-linear relationship ( Figure 4C ), but the Novoa et al. (2014) scale (F_I,N ) is closer to a log-log relationship, consistent with earlier models (Wernand, 2011, see Figure 4D ), albeit with differences in parameters. Figure 5 overlays the AMT data collected using the Novoa et al. (2014) scale onto historical datasets from NODC and CalCOFI. We find the relationship between z_SD and infinite colour (F_I,N ) on AMT to be in closer agreement with relationships between z_SD and F seen in both historical datasets, when compared with data on z_SD and the colour of the disk at 1/2 the Secchi depth (F_D,N ). Particularly for lower F values (1-3). This result suggests that the majority of the Forel-Ule historical data may have been collected by looking directly at the colour of the water, rather than at the colour of the water above the disk at 1/2z_SD , as other literature has implied (Wernand and van der Woerd, 2010a). Further investigation is needed to ascertain if this is in fact correct, perhaps by reviewing historical information on the protocols used for collecting these earlier Forel-Ule measurements.

On AMT, relationships between the maximum band ratio (R_rs (MB)/R_rs (G)) and F ( Figures 4E–H ) are remarkably consistent with the relationships seen between z_SD and F ( Figures 4A–D ), with F explaining between 69-80% of the variance in R_rs (MB)/R_rs (G). This is due to a tight relationship observed between z_SD and R_rs (MB)/R_rs (G) ( Figure 4I ). Inverse relationships between both diffuse and beam attenuation and z_SD were also observed ( Figures 4J, K ), with the former (K_d =1.3/z_SD ) in good agreement with the relationship proposed by Lee et al. (2018c), ( Figure 4J , K_d = 1.48/z_SD ), with differences possibly related to environmental conditions, considering the Lee et al. (2018c) relationship is for a fixed solar zenith angle (θ) of 30 degrees, and a fixed wind speed (w_s ) of 5 ms⁻¹. We also see good agreement between the euphotic depth (z_p ) and z_SD ( Figure 4L , z_p =3.67z _SD ), with no significant difference to the relationship proposed by Lee et al. (2018c) (where z_p is equal to 3.55 (± 0.15) times z_SD ).

Figure 6 shows a comparison of Chl-a and optical measurements collected on the four AMT cruises. We find z_SD to be tightly correlated with Chl-a, explaining around 85% of its variance (r²=0.85, on log₁₀-transformed variables). The relationship between z_SD and Chl-a is in good agreement with published models ( Table 2 ; Morel et al., 2007; Boyce et al., 2012; Lee et al., 2018c). Systematic differences (accuracy, δ) are close to zero for the models of Morel et al. (2007) and Lee et al. (2018c), with the a small bias for the model of Boyce et al. (2012) ( Table 2 ). The models of Boyce et al. (2012) and Lee et al. (2018c) have slightly better precision (Δ, Table 2 ) than the Morel et al. (2007) model. Fitting a power-law model to the data [ Table 2 , the same mathematical model to that of Lee et al. (2018c) and Boyce et al. (2012)] reduces the bias (δ) to zero, with no improvement in precision (Δ), and obtained parameters are found not to be significantly different to those of Lee et al. (2018c) ( Table 2 ). Statistical tests indicate that the retrieval of surface Chl-a from z_SD is comparable, in precision and accuracy, to retrievals of Chl-a using satellite ocean colour algorithms in the Atlantic Ocean (Brewin et al., 2016; Tilstone et al., 2021). Residuals in log₁₀(Chl-a), between the fitted power model and Chl-a data, were positively correlated with wind speed and R_rs (MB)/R_rs (G), and inversely correlated with log₁₀(Chl-a) ( Table 3 ). The latter two (which are inversely related, see Figure 6B ) perhaps suggesting the mathematical formulation of the relationship (power function) may need further consideration, with there being a slight tendency for the model to overestimate log₁₀(Chl-a) at lower concentrations and underestimate it at higher concentrations. Other fitting functions were explored, including log-linear and polynomial fits (data not shown), but they also showed the same tendency, suggesting this may be related to the distribution of the dataset. The positive correlation between residuals and wind speed ( Table 3 ), suggested that for the same Chl-a, as the wind speed increases, the observer sees a shallower Secchi depth. This is consistent with theory on the impact of wind speed on the apparent contrast of the disk (Preisendorfer, 1986). However, multi-linear regression (not shown) of log₁₀(Chl-a) as a function of both log₁₀(z_SD ) and wind speed, yielded no significant increase in r ² over log₁₀(z_SD ) alone (Z-test, p>0.05), suggesting any such effect is minor. No relationship was found between the VGI and VCI, and the residuals in log₁₀(Chl-a), between the fitted power model and Chl-a data.

Of all the optical proxies tested, R_rs (MB)/R_rs (G) was found to explain the greatest variance in Chl-a ( Figure 6B , r ²= 0.89, on log₁₀-transformed variables). The NASA OC4v6 algorithm was found to have a tendency to underestimate Chl-a in the Atlantic ( Table 2 , δ = −0.109), consistent with earlier work (Szeto et al., 2011). A retuning of the OC4v6 algorithm removed this bias, and showed improvements in both precision (Δ) and r ² ( Table 2 ). The retuning was also found to show a strong linear dependency (on log₁₀-transformed data) between variables, suggesting a simpler (more parsimonious) linear model to be more appropriate than a 4th order polynomial on this AMT dataset, which yielded statistically similar results to the retuned polynomial (OC4 Table 2 ). We found a small dependency in the residuals of this linear model and log₁₀(Chl-a), similar to the z_SD fits and likely related to data distribution, but no other dependency between residuals and other environmental variables were observed ( Table 3 ). R_rs (MB)/R_rs (G) was shown to produce the highest r ² and lowest Δ of all variables tested ( Table 2 ). R _rs (MB)/R_rs (G) only performs marginally better as a predictive variable of Chl-a, than z_SD ( Table 2 ), on the AMT.

All four Forel-Ule datasets (F_D,L , F_D,N , F_I,L and F_I,N ) were positively correlated with Chl-a, explaining between 71-81% of the variance in log₁₀-transformed Chl-a ( Figures 6C–F ). The Novoa et al. (2014) scale data (F_D,N , F_I,N ) were found to correlate more tightly to Chl-a (explaining around 81% of the variance) than the LaMotte scale data (explaining around 71% of the variance), with the Novoa et al. (2014) scale data predicting Chl-a with the highest accuracy using a power function (linear-function in log₁₀ space), consistent with the model of Boyce et al. (2012), with the LaMotte scale data best described using a log-linear relationship ( Figures 6C–F ; Table 2 ). For Forel-Ule data greater than 2, the lower limit of the range of data in which the Boyce et al. (2012) model was trained on, the Boyce et al. (2012) model is seen to overestimate Chl-a when using F_D,L and F_D,N as input ( Figures 6C, D ; Table 2 ), with better agreement (biases (δ) closer to zero) when using F_I,L and F_I,N as input ( Figures 6E, F ; Table 2 ). This is consistent with the Forel-Ule infinite colour data collected on AMT being in closer agreement with historical datasets than data collected on the colour of the water above the disk at 1/2z_SD ( Figure 5 ), considering the Boyce et al. (2012) model was parameterised on these historical data. As with the z_SD and R_rs (MB)/R_rs (G) models, there is a dependency in Forel-Ule model residuals (log₁₀(Chl-a) model minus data) on log₁₀(Chl-a) ( Table 3 , positive for F_D,L and negative for F_D,N ), likely related to data distribution. The LaMotte model residuals were also correlated with L_i (750)/E_s (750), R_rs (MB)/R_rs (G) and z_SD ( Table 3 ). Residuals in both models were also found to be correlated with solar zenith angle ( Table 3 ). This may suggest the sky conditions had some impact on data collection, possibly influencing the apparent colour of the disk, although no relationship was found between residuals and the VGI and VCI. Whereas measuring Forel-Ule of infinite water directly is useful for testing the Pitarch (2017) conversion, and more in-line with the historical observations ( Figure 5 ), it is difficult to observe subtle variations in Chl-a at low concentrations, since the Forel-Ule saturates at the lowest value ( Figure 3C ). In fact, this is a key reason Boyce et al. (2012) excluded Forel-Ule data less than two in their algorithm. Instead, by measuring the colour of the disk at 1/2z_SD , the observer can track these subtle changes in colour at very low Chl-a ( Figure 6 ), as the scale has not saturated at its lowest value.

Whereas it is clear from this analysis that modern optical tools for estimating Chl-a (R_rs (MB)/R_rs (G)) performed the best ( Table 2 ), it is nonetheless remarkable how well the historical tools performed, with some (e.g., z_SD ) only marginally lower in performance than the modern method, and with a higher number of observations (less effected by environmental constrains in data collection). These results supports the merging of historical data with modern methods (e.g., satellite remote sensing) in search of long term trends in Chl-a, providing systematic differences can be identified, and acknowledging the challenges in bridging data collected at very different temporal and spatial scales (Boyce et al., 2010; Rykaczewski and Dunne, 2011; Boyce et al., 2012; Boyce et al., 2014). Models proposed here are nonetheless limited to the range of data they have been calibrated with, to case-1 open-ocean waters, and to the Atlantic Ocean. Caution is also needed when applying such models, developed between 2013-2018, to data collected in the past, since the relationships among Secchi depth, Forel-Ule colour, R_rs , and Chl-a, could themselves be sensitive to climate change.

The AMT is, and has been, widely used as a platform for evaluating Chl-a remote-sensing algorithms (O’Reilly et al., 1998; Brewin et al., 2016; Tilstone et al., 2021), owing to the fact the transect samples such a wide variety of biogeochemical provinces. Here, we extend the range of remote-sensing variables evaluated on AMT, to include the Secchi depth and Forel-Ule colour.

Figure 7A and Table 4 show results from a statistical comparison between Forel-Ule data derived using the R_rs -based methods of van der Woerd and Wernand (2015) and Novoa et al. (2014) and that from the four in-situ datasets collected on AMT. The R_rs -based method performs well in the comparison, with r ² values ranging from 0.64 to 0.70, and unbiased root mean square differences (Δ) ranging from 0.48 to 0.71 ( Table 4 ). As expected, the R_rs -based method is in better agreement with the infinite colour in-situ datasets, with biases (δ) closer to zero for F_I,L , F_I,N and F_I,A , than for F_D,_L , F_D,N ( Table 4 ). Linear regression between data shows the biases to vary systematically over the range of Forel-Ule data, with the F_I,L and F_I,N in better agreement with the R_rs -based method at lower values (<4), but deviating significantly at higher values (>4, Figure 7A ; Table 4 ), with differences (residuals) between the R_rs -based method and F_I,N data inversely correlated with changes in Forel-Ule ( Table 5 ). These residuals were also found to be correlated with wind speed, the pitch and roll of the vessel, the Secchi depth and log₁₀(Chl-a) ( Table 5 ). Interestingly, the slope of the regression in the statistical comparison between the R_rs -based method and F_I,A (phone app) data was closer to one, and the bias (δ) closer to zero, when compared with other data ( Table 4 ). Furthermore, differences (residuals) between the R_rs -based method and F_I,A data were not significantly correlated with any environmental variables (data not shown), acknowledging that only one cruise (AMT26) of F_I,A data were available in the analysis. Figure 3C shows the transect of F data for AMT26. All infinite F data are in good agreement at low values, but at higher values towards the end of the cruise (in the South Atlantic), the F_I,L and F_I,N are significantly higher than both the F_I,A (phone app) data and the R_rs -based method, with the latter two in very good agreement. It may be that the conversion between the colour of the disk and infinite colour used here (Pitarch, 2017) to derive F_I,L and F_I,N , which performs very well at lower values (<4) ( Figure 3C ), requires revaluation at higher F values (>4).

A statistical comparison between in-situ z_SD and the R_rs -based algorithms of Lee et al. (2015) and Jiang et al. (2019) are shown in Figures 7B, C . Within the range of in-situ z_SD collected on the AMT cruises (8.5 - 51.8 m), both algorithms perform reasonably at retrieving z_SD from R_rs , with r ² values ranging from 0.77 to 0.78, unbiased root mean square differences (Δ) ranging from 4.4 to 4.5 m, and linear regression slopes close to one. Both R_rs -based algorithms show slightly higher estimates than the in-situ z_SD data (positive biases (δ)), but biases for the Lee et al. (2015) algorithm are closer to zero (δ = 2.0) than for the Jiang et al. (2019) algorithm, where there is a systematic overestimation of z_SD of around five meters (δ = 5.4). Interestingly, there was no significant change in model performance when removing stations with overcast conditions (VCI = 1). However, it is important to recognise both models were calibrated with Hydrolight simulations using cloudless skies (Lee et al., 2002, 2005, 2009).

Differences (residuals) between the Lee et al. (2015) and Jiang et al. (2019) algorithms, and the in-situ z_SD data, were both found to be positively correlated with R_rs (MB)/R_rs (G) and inversely correlated with log₁₀(Chl-a) ( Table 5 ). Residuals between the Lee et al. (2015) algorithm and the in-situ z_SD data were inversely correlated with solar zenith angle (θ), but not in the case of the Jiang et al. (2019) algorithm ( Table 5 and Figure 8 ). Both algorithms incorporate a dependency on θ in the conversion of IOPs [derived analytically from R_rs following Lee et al. (2002, 2005, 2009)] to K_d (λ), and both algorithms subsequently derive z_SD according to

where R r s p c represents remote sensing reflectance at the corresponding wavelength to min(K_d (λ)), C t r is the threshold contrast for sighting a white Secchi disk, and ω converts min(K_d (λ)) to the sum of downwelling diffuse attenuation ( K d t r ) and upwelling diffuse attenuation ( K T t r ), in the transparent window. The Lee et al. (2015) algorithm fixes ω at 2.5, whereas the Jiang et al. (2019)algorithm has a variable ω, itself dependent on θ according to

where u is the ratio of the backscattering coefficient to the sum of absorption and backscattering coefficients (derived from R_rs following Lee et al. (2002; 2005; 2009)), and n_w is the refractive index of water (1.34). For the same backscattering and absorption coefficient, as θ increases, K_d (λ) increases in model of Lee et al. (2005) (used in both Jiang et al. (2019) and Lee et al. (2015) models), which subsequently decreases z_SD in Eq. 3. However, in the Jiang et al. (2019) algorithm, as θ increases, ω decreases (Eq. 4), which counterbalances the increase in (K_d (λ)) with θ. The resulting effect is that z_SD estimated using the Jiang et al. (2019) algorithm is less dependent on θ than for the Lee et al. (2015) model.

Differences (residuals) between the Lee et al. (2015) and Jiang et al. (2019) algorithms, and the in-situ z_SD data, were also found to be positively correlated with wind speed ( Table 5 and Figure 8 ). At very low wind speed (<3 ms⁻¹), there is better agreement between models and in-situ data, with δ closer to zero (δ = −1.03 for Lee et al. (2015) and δ = 2.79 for Jiang et al. (2019)). These biases increased with increasing wind speed ( Figure 8 ) Neither models incorporate a dependency of z_SD on wind speed, but our results are broadly consistent with theory of Preisendorfer (1986) that assumes an increase in wind speed causes a reduction in the apparent contrast of the disk at the surface, and a subsequent reduction in z_SD .

Though the Preisendorfer (1986) and Lee et al. (2015) Secchi disk theory define contrast differently, the Preisendorfer (1986) approach could be used to introduce some relationship between contrast and wind speed in the theory of Lee et al. (2015). The inherent contrast (C ₀) is defined not as a relative difference in irradiance reflectance between the disk and the surrounding water (as in Preisendorfer, 1986), but as an absolute difference between the radiance reflectance of the disk (r ₀) and the surrounding water ( r ∞ ), such that

The derived theory states that the contrast is modified as the distance (z) from the observer to the disk increases, such that

Following Preisendorfer (1986), a contrast reduction factor ( τ 0 - ) could be introduced, such that Eq. 6 becomes

If we make the assumption that τ 0 - is controlled by wind speed (w_s ), acknowledging that other factors (which may or may not covary with w_s ) are likely to impact τ 0 - , we could relate τ 0 - to w_s empirically. To do that, we define x(z)=C ₀/C(z), and for z=z_SD , it follows that C(z_SD )=0.013. Then, as the measurement is made from above the surface, it has to include another contrast reduction factor due to surface reflection effects, leading to x ( z S D ) = x S D = | 0.14 − R rs | / 0.013 (see Section 4 of Lee et al., 2015). To have an operational expression to solve for τ 0 - , we make the ratio of Secchi depth with (z_SD,ws ) and without (z_{SD,ws =0}) wind, such that

Rearranging the equation following some algebra, we obtain

and derive τ 0 - by making the assumption z_{SD,ws =0} is equal to the Jiang et al. (2019) algorithm output, and z_SD,ws represents the in-situ data. Next we model τ 0 - as a function of wind speed (w_s ) and z_{SD,ws =0} (Jiang 531 et al. (2019) algorithm), according to

with the expression chosen carefully to follow the observed trend in which wind explains most of the variation, but it is modulated by z_{SD,ws =0}. Fitting of Eq. 10 to the data resulted in q 0 = 0.901  ( 0.805 ↔ 0.998 ) , q 1 = 0.022  ( − 0.007 ↔ 0.051 ) , and q 2 = 0.03  ( − 0.045 ↔ 0.106 ) . Once τ 0 - is known, z_SD,ws can be estimated from

Using the modelled z_SD,ws (Jiang et al., 2019) removes the significant positive correlation between wind speed and model residuals (r=−0.135, p=0.179) apparent in original (Jiang et al., 2019) algorithm (z_{SD,ws= 0}, Figure 8D ). Additional datasets are need to independently validate this approach, but it certainly offers a way to incorporate the influence of wind speed into the Secchi depth theory of Lee et al. (2015), and could be useful for standardising in-situ z_SD data to a common wind speed.

Our experience of collecting Secchi depth and Forel-Ule colour data on the Atlantic Meridional Transect revealed a number of insights. With the exception of a few sceptical scientists on-board the ship, there were many (scientists and members of the ship’s crew) that loved visually inspecting the colour and clarity of the water using the Secchi disk and Forel-Ule colour scales (see acknowledgements). It was the only time of the day where these participants had the opportunity to visually connect and interact with the ocean. The AMT transect was the perfect cruise to do this, as it transects through a ranges of conditions, with the clarity and colour of the water changing (sometimes rapidly) over the duration of the cruise. Visually inspecting the water also resulted in an increased number of sightings of marine wildlife and oceanic phenomena, including (and to name a few) observations of dolphins, sharks, whales, dolphinfish, rays, penguins, squid, turtles, icebergs, all manner of seabirds, as well as some of the more negative features (e.g., marine litter). For some, this had a positive influence on mental health, important to consider when on-board a research vessel for an extended period of time. Considering the nature of the measurement of Secchi depth and Forel-Ule colour, in that the observer is intrinsically part of the measurement process, it was a useful activity for teaching the concepts of ocean colour and clarity to non-scientists, and scientists working on other aspects of oceanography, on the AMT cruises.

Despite these positives, it was evident that measuring the Secchi depth in oligotrophic waters is challenging. After unsuccessfully attempting to deploy a Secchi disk using the traditional technique (rope and weighted Secchi disk), hindered by the disk drifting quickly away from the vessel (which was kept stationary), we followed the recommendations of the late Marcel Wernand, who in his 2010 paper states “…the author recommends a reintroduction of the Secchi disc to expand the historical Secchi depth database to facilitate climate change research. One option is to mount a Secchi disc on an instrumental or CTD frame…” (Wernand, 2010, p5). Once the disk was mounted on the optics rig ( Figure 2A ) it became feasible to do the measurements, as part of routine data collection.

A major challenge with conducting Secchi depth measurements in oligotrophic waters, is the fact that the disk disappears at a depth far greater than in mesotrophic and eutrophic waters. The angular subtense of the disk’s radius ( Φ = r S D / z S D , r_SD is the disk’s radius, the first order approximation of the MacLaurin expansion of the arctan function), for a 30 cm diameter disk at a z_SD of 50 m, is 0.003. To put this into perspective (and something that could easily be tested in inland water), the target viewed is equivalent to lowering a 1 cm diameter disk to 1.7 m. One solution could be to increase the disk size for oligotrophic waters. Although a 30 cm diameter disk is standard for measuring the Secchi depth in the ocean, Angelo Secchi himself used disks as big as 2.5 m in diameter in his early work (Pitarch et al., 2021), and others have modified the disk size to be smaller in more turbid waters (e.g., Brewin et al., 2019a). However, the exact impact of disk size on Secchi depth is still a matter of research (Hou et al., 2007). Another possible artefact of a very low angular subtense ( Φ ) in oligotrophic waters, could be an increasing impact of environmental factors on the detection threshold of the human eye (in air) seeing the disk, since the target is so small. Additionally, there may be other optical effects that occur, that are not currently accounted for in the theory. For example, with a very small target in a moving ocean, there may be some kind of adjacency effect, caused by the spectrum near the edge of the disk being a mixture of reflectance of the disk and the infinitely deep waters.

The dataset collected on AMT is freely available (Brewin et al., 2023), courtesy of the British Oceanographic Data Centre (BODC), for use in future research on the topic. In this paper, we have compared and evaluated broad relationships between historic and modern optical techniques for monitoring phytoplankton biomass. Future work could use these data to examine in greater detail the influence of varying concentrations of different optically-active constituents on the relationships between Secchi depth, Forel-Ule colour, R_rs , and Chl-a. For example, variations in coloured dissolved organic matter (CDOM) will have an impact on water colour, and consequently the relationships between bulk variables (Van der Woerd and Wernand, 2018). For some AMT cruises, underway CDOM absorption data (Dall’Olmo et al., 2017), as well as underway and profile particulate absorption and scattering measurements (Dall’Olmo et al., 2012; Brewin et al., 2016; Tilstone et al., 2021), have been collected, and could be useful for studying relationships between Secchi depth, Forel-Ule colour, R_rs , and Chl-a.

Measuring PAR on a profiling package and the Secchi depth from a large oceanographic vessel could lead to cases of ship shadow. For PAR, this was somewhat minimised by only using profiles where depth and log(PAR) were tightly correlated and (where possible) deploying the profiling package on the sunny side of the vessel. Our K_d values for oligotrophic waters (varying between 0.02 – 0.08) are consistent with values from satellite data in the region (e.g., Son and Wang, 2015) and consistent with BGC-Argo float data (see Demeaux and Boss (2022)). AMT K_d data also agreed well with K_d estimates from a BGC-Argo float WMO:3902121 in the South Atlantic gyre, (results not shown).

Another factor that may influence relationships between Secchi depth, Forel-Ule colour, R_rs , and Chl-a, are shifts in the community composition of phytoplankton. It is well know that the chl-specific absorption and backscattering coefficients vary with phytoplankton community composition (e.g., Ciotti et al., 2002; Sathyendranath et al., 2004; Kostadinov et al., 2010; Brewin et al., 2011; Devred et al., 2011; Brewin et al., 2012a; Brewin et al., 2019b). Although the median ratio of Secchi-depth to mixed-layer depth (computed using temperature criterion of de Boyer Montégut et al. (2004)) was found to be 0.49 (standard deviation 0.44), suggesting on average the mixed-layer depth was twice that of the Secchi depth, there were a few cases where the Secchi depth was shallower than the mixed-layer depth. Vertical variability in Chl-a and phytoplankton community structure in the region (Mojica et al., 2015), and within the Secchi depth layer, may exist (e.g., in very clear waters, with very shallow mixed-layers) which may complicate relationships between Secchi depth, Forel-Ule colour, R_rs , and Chl-a.

We chose to use surface Chl-a, rather than depth-averaged Chl-a within the Secchi depth layer, in our analysis, as it has been recommended by the community to use surface Chl-a when developing surface optical models (Sathyendranath et al., 2019; Lee et al., 2020), and it allowed us to compare relationships in our dataset with earlier algorithms (Boyce et al., 2012; Lee et al., 2018c). However, datasets on vertical variations in Chl-a and community composition (e.g. through flow cytometry and HPLC analysis) have been collected on AMT cruises, and could be useful for studying impact of vertical variability in water constituents on proposed models. Future efforts in collecting vertically-resolved inherent optical properties, alongside Secchi depth and FU measurements, would help further. Considering issues with direct observations of Forel-Ule colour of infinite waters not resolving subtle variations in low Chl-a in oligotrophic waters, we recommend measuring Forel-Ule colour in oligotrophic waters using the colour of the Secchi disk at 1/2z_SD .