## Introduction

Microscopic organic particles in the open ocean support a suite of processes, known as the biological carbon pump, which is ultimately responsible for transferring atmospheric CO2 from the sunlit surface to the abyss. The biological pump thus modulates atmospheric CO21,2 and supports deep water ecosystems and the fisheries upon which humans depend. Understanding this pump is of the utmost importance.

To achieve this understanding, measurements of the light scattered by marine particles are crucial3,4,5,6,7,8,9. Optical scattering is quantified by the volume scattering function (VSF) that measures the fraction of incident light deflected by particles in a given direction per unit distance10. By integrating the VSF in the forward and backward directions, one obtains the forward- and backscattering coefficients that are the most commonly measured optical proxies for particle concentration. Although the particulate backscattering coefficient (bbp) is only a small fraction (1–2%) of the total scattering (or of the particulate beam attenuation coefficient cp11), it has the great advantage of being directly linked to space-based measurements12. In addition, bbp can be measured in situ by miniaturised instruments installed on autonomous robotic platforms that sample at high vertical and temporal resolution from the surface to the mesopelagic zone (200–1000 m)13. Thus, optical backscattering measurements can be exploited to better understand the biological carbon pump.

The particulate optical backscattering coefficient can be used to estimate, through empirical relationships, the concentration of particulate organic carbon (POC)14,15,16 and phytoplankton carbon17,18, from which algal physiological parameters can be derived19. The bbp spectrum provides a proxy of the particle size20 and can be used to analyse the phytoplankton community structure and derive group-specific carbon biomass21. Yet, the biogeochemical quantities derived from these algorithms suffer from large uncertainties. For example, the performance of POC and phytoplankton carbon empirical models can vary widely15,16,17,18.

Difficulties in interpreting bbp arise due to an incomplete mechanistic understanding of which particles are detected by bbp. The bbp coefficient has been mainly modelled using Mie theory22,23,24,25 assuming that marine particles can be represented as homogeneous spheres. Although this assumption is sufficient to model the forward scattering of particles25,26,27, this approach severely underestimates the phytoplankton backscattering28,29,30 and the particulate backscattering in the open ocean31. This mismatch between theory and observations is known as the missing backscattering enigma32. To explain this missing backscattering, submicron detrital particles have been suggested as the major source of oceanic particulate backscattering26,27,33,34. Hence, it is currently assumed by many that particulate forward- and backscattering coefficients detect different oceanic particles.

Alternative approaches have been proposed to model marine particles as heterogeneous coated spheres35,36,37,38,39,40,41,42 or as non-spherical particles42,43,44,45,46. Heterogeneous coated spheres represent the simplest way to model the bulk structural complexity of marine particle populations. Coated spheres can reasonably represent the external and internal cellular structures (e.g., cell wall and chloroplasts) of marine spherical and non-spherical phytoplankton, and account for differences in their chemical nature35,36,37,38,39,40,41,42. Coated spheres can also be used to simulate aggregates of living and detrital particles47,48 and algal colonies49. Coated-sphere models produce backscattering signals that can be an order of magnitude higher than those predicted by Mie theory for the same overall diameter and volume-averaged particle composition (i.e., bulk refractive index38). Coated models, however, require parameters that quantify the thickness and chemical composition of each part of the particle and the choice of these parameters is critical32. Non-homogeneous spherical models thus have not been applied to study oceanic backscattering due to the high variability in the size and nature of natural particles. Prudence has been suggested for these more complex approaches32.

Here we show that the open-ocean missing backscattering can be found in the structural complexity of marine particles. We have applied a coated-sphere model to estimate the backscattering of particle populations across the Atlantic Ocean and through mid-ocean gyres (Supplementary Fig. 1). We based our analysis on coincident measurements of particle size distributions (PSDs; 0.59–60 µm) and optical measurements at the wavelength 532 nm (see Methods). We performed the analysis only at 532 nm because it allows us to minimise the second-order effect of particulate light absorption on scattering25, as well as to directly compare our results with previous results based on the homogeneous-sphere assumption27. We then examined the contributions to modelled scattering of different size fractions and confirmed the findings by additional optical measurements collected through independent size-fractionation experiments. Our results show that particles larger than 1 µm contribute the majority of the bbp measured across the Atlantic Ocean and that backward and forward scattering therefore detect different characteristics of particles within similar size ranges. We anticipate our results will open the door to a new way of interpreting marine optical backscattering measurements.

## Results

### Homogeneous spherical particles cannot entirely explain bbp

The homogeneous-sphere model (Mie theory) could not reproduce simultaneously the measured particulate beam attenuation (cp) and backscattering (bbp) coefficients (Fig. 1). Modelled cp and bbp at the ocean surface and at the level of the deep chlorophyll maximum (DCM) were tightly correlated with measured values regardless of the refractive index (n) assumed (Fig. 1; Supplementary Table 1). However, biases were evident and varied with n. When n was set equal to 1.06, the homogeneous-sphere model accurately reproduced cp (Fig. 1a) but severely underestimated bbp (Fig. 1b). Modelled bbp values could be forced to match observations by increasing n to 1.11 (Fig. 1d), but this increase in n enhanced the bias in cp predictions by about 10 times (Fig. 1c). These results were robust to variations in the refractive index within the PSD (Supplementary Fig. 2), as well as to the presence of absorbing particles (Supplementary Fig. 3). The accurate prediction of cp obtained for n equal to 1.06 suggested that Atlantic open-ocean particle populations were composed by phytoplankton-like organic material50.

### A coated-sphere model for open-ocean particles

In contrast to the results obtained using Mie theory, both cp and bbp coefficients could be accurately and simultaneously reproduced by a coated-sphere model (Fig. 2). No missing backscattering was observed for various combinations of the thickness of the outer coat (tk2) with assigned refractive index (n2) (Fig. 2b). The ensemble of model parameterisations was selected among the tk2n2 combinations that returned the lowest errors in bbp prediction (Fig. 3, Supplementary Fig. 4). This selection was achieved by excluding tk2n2 combinations that yielded values of the refractive index of the core of the sphere (n1) too low for marine particles (i.e., n1 < 1.02)26,27,51 or unrealistic (n1 < 1). Combinations with n2 equal to 1.22 were also disregarded because they were too high for algal cellular constituents50. The selected parameterisations thus included 12 combinations of 4% ≤ tk2 ≤ 15% and 1.12 ≤ n2 ≤ 1.20, with n2 decreasing as a function of tk2 according to a power law (Fig. 3). Corresponding values of n1 varied between 1.022 and 1.042 (Fig. 3). These 12 combinations had n2 values systematically higher than for n1, which implies that the outer coat of the spheres must always be composed of more refracting matter than the inner core. The shapes of the VSFs we obtained from these selected parameterisations were similar to existing measurements (Supplementary Fig. 5).

### The coated-sphere model is validated with independent data

The selected parameterisations could also reproduce field cp and bbp coefficients for an independent dataset (Fig. 4). These surface data were also collected across the Atlantic Ocean (Supplementary Fig. 1) but covered a broader range of cp and bbp values than the one used for model training. Outside of the range of values used for the model parameterisation (i.e., bbp(532) > 1.56 × 10−3 m−1), bbp predictions were less accurate (Fig. 4b). This suggests that the particulate optical backscattering may be more sensitive than cp to changes in particle composition and structural characteristics.

### cp and bbp detect particles within similar size ranges

The selected coated-sphere parameterisations predict that both the beam attenuation and backscattering coefficients measured across the Atlantic Ocean are generated by particles in approximately the same size range (Fig. 5; Supplementary Fig. 6). In agreement with previous studies23,27, 90% of cp was generated from particles with diameters between 0.59 and 7 µm, and <10% was produced by measured submicron particles. However, in stark contrast to previous studies based on the homogeneous-sphere model, when using the coated-sphere model we found that 90% of the backscattering signal came from particles between approximately 0.59 and 10 µm, whereas submicron particles between 0.59 and 1 µm contributed <20% of the signal (Fig. 5b).

### Independent confirmation of our findings

The smaller-than-expected contribution of submicron particles to surface optical backscattering was independently confirmed by additional measurements of 1-µm filtered water samples, which also accounted for particles with diameters <0.59 µm (Fig. 6). Tests on the efficiency of these filtrations indicated that the filter retained the majority of particles for diameters closest to and above its nominal pore size limit (i.e., 1 µm), whereas progressively increasing amounts of particles with diameters smaller than 1 µm passed through the filter (Fig. 6a). These measurements demonstrated that even when 90 ± 10% of particles with diameters larger than 1 µm were removed (Fig. 6a), on average only 40 ± 6% of the measured bbp signal was generated by the submicron particles (Fig. 6b). Our size-fractionation experiment further confirmed that particles < 1 µm contributed on average 20 ± 2% of the cp signal23,27. These size-fractionation results are also in agreement with previous studies that suggested that the bbp generated by particles with diameters smaller than 0.2 µm is negligible31,52.

### The coated-sphere model reproduces in-situ bbp-to-cp ratios

The bbp-to-cp ratio approximates, when the light absorption by particles is negligible (see Methods), to the ratio between bbp and total scattering (i.e., backscattering ratio), which measures the efficiency with which light is backscattered by particles25. The bbp-to-cp ratios for a coated sphere can be 40 times higher than for homogeneous spheres for particles with diameters between 1 and 12 µm, and 8 times for submicron particles (Fig. 7a). Estimated ratios using the coated-sphere model reproduced the ratios measured in situ. For the samples used to parameterise and validate the coated-sphere model, the modelled median value was 1.27 ± 0.09% (N = 125). This modelled value was consistent with those derived from our (1.27 ± 0.30%, N = 125; Fig. 7b) and other field optical measurements14,31,53, though with a standard deviation significantly smaller than observed in situ. This agreement between model results and experimental observations further highlights that, at the very least, a coated-sphere model is needed to reproduce oceanic bbp.

## Discussion

Experimental28,29,30,42 and theoretical43,44,45 studies have demonstrated that the optical backscattering of marine phytoplankton is severely underestimated when cells are modelled as homogeneous spheres. The homogeneous-sphere model also underestimates the optical backscattering generated by phytoplankton and other marine organic particles in the open ocean31. Here we showed that by modelling open-ocean particles as a population of coated spheres, we simultaneously predicted the measured beam attenuation and backscattering coefficients with considerably smaller biases than those obtained using the homogeneous-sphere model, over a wide range of trophic conditions across the sunlit Atlantic Ocean. We can thus provide an interpretation of in-situ oceanic particulate backscattering measurements that is not based on the homogeneous-sphere model. In contrast with previous interpretations based on the homogeneous-sphere model, our theoretical and experimental results suggest that the majority of the particulate backscattering coefficient in the studied area is due to particles with equivalent diameters between 1 and 10 µm, and that submicron particles generate <40% of the measured signal (Figs. 5b, 6b). This contribution by submicron particles is half that predicted by existing modelling studies for homogeneous spheres with theoretical power-law PSDs with an exponent of −4 and having a low refractive index (i.e., 1.05)27. Our modelling results may have underestimated the contribution of submicron particles because we used an integration limit of 0.59 µm for measured PSDs23. However, independent size-fractionation experiments confirmed the reduced optical contribution for all submicron particles (i.e., 0.001–1 µm; Fig. 6b) regardless of the lower size limit of our PSDs. According to our results, therefore, beam attenuation and backscattering coefficients most likely sense different characteristics of particles within similar size ranges (Fig. 5). Consistent with published results, the beam attenuation and thus forward scattering are sensitive to the average characteristics of particles54. Backscattering, on the other hand, reflects the characteristics of the outer shell of the coated sphere35 or, in more general terms, of the particle structure. We consider the coated sphere as a simple but efficient way to model the structural complexity of marine particles36,41. A coated sphere may account for the effects on the optical backscattering of external and internal phytoplankton cell structures such as the cell wall, cell membrane, chloroplasts, and other organelles35,36,39,40,41. The coated sphere may also reasonably reproduce the scattering of quasi-spherical organisms36,42 and can be used to model the optical properties of amorphous agglomerations of many particles47,48 or algal colonies49. Thus, the coated sphere can account for various levels of structural complexity that characterise particles in the marine environment. More in-depth studies are needed to identify exactly what particle characteristics are responsible for the backscattering effects reproduced by the coated sphere.

To understand what might be the consequences of using the homogeneous-sphere model to simulate the scattering properties of complex natural particles (e.g., phytoplankton), it is instructive to attempt to model the scattering of a complex particle by only using homogeneous spheres. Figure 8 demonstrates that, to reproduce the VSF of a coated sphere, multiple homogeneous particles of different sizes are needed. Specifically, as expected from previous studies25, one homogeneous sphere of the same size and volume-averaged refractive index as the coated sphere can reproduce the forward scattering. However, to also reproduce the backscattering of the coated sphere, we must add the scattering intensities of multiple small spheres to that of the largest sphere, as they are the only homogeneous particles with a high backscattering efficiency25. We thus postulate that previous conclusions regarding the sources of oceanic backscattering based on the homogeneous-sphere model might have been distorted by the same limitations illustrated in the example of Fig. 8. In summary, an optical model that is too simplistic to represent the complexity of marine particles might have considerably altered our interpretation of the sources of oceanic backscattering.

Some marine particles might, however, be represented by homogeneous models45,46 that could modify the contribution to the optical backscattering by the different size fractions. In our analysis, we assumed all marine particles to be coated spheres with similar characteristics because we did not have data to distinguish coated from homogeneous particles and to characterise their morphology. However, we performed a sensitivity analysis to understand how many homogeneous and coated spherical particles within the same population could still accurately reproduce the in-situ backscattering measurements. We found that modelled backscattering coefficients reproduced measured values until homogeneous particles represented up to 25% of the total number of particles measured (Supplementary Fig. 7). We therefore conclude that at least 75% of the marine particles sampled were too morphologically or structurally complex to be represented as homogeneous spheres. These complex spheres likely also represented agglomerations of both living and detrital material47,48,49, in addition to freely dispersed individual particles36,41,42.

The coated-sphere model that was parameterised using surface data underestimated the backscattering in the mesopelagic region of the Atlantic Ocean. Figure 9 shows that the selected parameterisations for the coated-sphere model predicted 61 ± 20% of the measured backscattering in the mesopelagic zone. Although in-situ bbp measurements in the clearer mesopelagic waters may suffer from higher uncertainties55, the differences between modelled and measured backscattering coefficients could be due to particles with thicker and harder coats or to denser and more structurally complex aggregates. Moreover, the volume-averaged refractive index of mesopelagic particles may be different from the 1.06 value found for surface particles. For example, a few modelled beam attenuation coefficients at 532 nm for particles collected between 150 and 200 m (N = 15) were accurately predicted when the volume-averaged refractive index was set equal to 1.07 (Bias = −0.01).

A coated-sphere model might therefore disclose additional information on particle characteristics from optical data. This hypothesis is also supported by the higher uncertainties in the modelled backscattering coefficients that were found for the AMT22 samples when bbp was higher than 1.56 × 10−3 m−1 (Fig. 4b). These AMT22 data were collected in surface waters that were not sampled in the AMT26 dataset used to parameterise the coated-sphere model. These waters were generally rich in large cells such as diatoms, as confirmed by the two-time higher relative pigment contribution than observed along the AMT26 transect (Supplementary Fig. 8). Algal populations encountered on AMT26 were mainly characterised by cyanobacteria and picoeukaryotes mixed with some coccolithophores, cryptophytes and other nanoeukaryotes (Supplementary Fig. 9). Thus, the observed vertical and regional differences predicted by the coated-sphere model suggest that oceanic bbp might be better estimated by parameterising the thickness and composition of the coat according to depth and trophic regime. In turn, we hypothesise that if this variability in the parameterisation could be inferred from combinations of optical measurements (e.g., bbp/cp), then new information of particle characteristics might be extracted from optical measurements. This new information could also help explain why predictions of POC are less noisy when using algorithms based on cp rather than on bbp14,16, and why performance of empirical algorithms for phytoplankton carbon varies widely17,18.

Understanding which particle characteristics determine optical scattering coefficients in natural waters may also shed light on why community production is better predicted from cp than bbp56. Particle and phytoplankton growth and loss rates can be derived from cp diel cycles57, and the correlation between cp and bbp observed in oceanic waters31,53,58 may suggest the same retrievals could also be possible by satellite observations of bbp. However, cp and bbp diel cycles are out of phase59 and diel phytoplankton carbon-specific backscattering and attenuation coefficients exhibit different behaviours60. Understanding the contribution of phytoplankton to in-situ measurements of cp and bbp might thus increase our capability to predict oceanic carbon biogeochemical rates. We therefore suggest that a potential new avenue of research could be to identify what changes in cell ultrastructures and composition govern diel variations of optical scattering properties.

The coated-sphere model predictions compared favourably to in-situ observations only when a specific set of parameters was selected. These selected parameters represent the known metabolite composition of oceanic particles50. The volume-averaged refractive index equal to 1.06 reveals that particle populations across the Atlantic Ocean are mainly of organic origin50. Although we may compare modelled coated-sphere thicknesses and compositions (Fig. 3) with published values for oceanic phytoplankton, a similar comparison would not be possible with detrital matter and particle aggregates. When considering only phytoplankton, the modelled high refractive coats of the spheres could represent cell walls made of calcite or cellulose and other polysaccharides50. However, coats with high refractive indices could also represent cellular and plastidial membranes composed by a bilayer of lipids with embedded proteins50. Moreover, the relative volume of the coat parameterised for coated spheres is similar to the relative cellular volume that chloroplasts and thylakoids generally occupy41, whereas the cell wall contributes <2% of the cellular volume41. Previous studies have suggested that cellular organelles may have a significant influence on bbp28. Highly refractive cell walls enhance the oceanic backscattering61. The outer shell of a coated sphere therefore most likely combines the effects that both external and internal cellular structures have on bbp, though what is the main driver of bbp still remains unclear. Finally, how the modelled coated spheres relate to oceanic detrital particles remains to be understood.

There is still much to do to mechanistically understand open-ocean particle backscattering observations. We need to characterise better the internal and external structures of marine organisms (e.g., cell wall and membrane thickness; number, size and position of intracellular organelles and starch inclusions) and how they vary with depth and latitude. We also need to characterise better the detrital matter and particle aggregates. For example, it is possible that detritus constitutes a large fraction of oceanic particles62 but its abundance, structural complexity and distribution along the full size range are still largely unknown. To characterise particle structural characteristics, integrated approaches based on microscopic and imaging analysis, coupled with optical measurements in situ and in the laboratory, such as polarised light scattering36,63, are needed. For example, by comparing simulated (see Methods) and measured64 polarised components of the Mueller scattering matrix65, the range of selected coated-sphere parameterisations may be further constrained (Supplementary Fig. 10). The comparison of our simulated polarised elements with in-situ measurements for a range of oceanic waters64 suggests indeed that oceanic particles may be better represented by high refractive indexes and low thicknesses of the outer shells, and further shows the inadequacy of the homogeneous-sphere assumption (Supplementary Fig. 10). Characterising the complexity of marine particles in situ is thus the necessary step towards identifying the main drivers of the optical backscattering. This step will also tell us if a coated sphere is indeed sufficient to explain the sources of the open-ocean backscattering.

Our findings clearly show that the structural complexity of particles is key to explain mechanistically the sources of open-ocean backscattering. In addition to measuring the enigmatic open-ocean submicron particles32, future efforts should also be directed to test and constrain complex optical models with in-situ measurements of particle and cellular characteristics. Only when we will understand the sources of oceanic bbp, we will be able to exploit the full potential of optical backscattering observations for investigating the biological carbon pump. Our study opens a new direction towards achieving this ultimate goal.

## Methods

### Cruise location and sampling strategy

Measurements were collected during two Atlantic Meridional Transect (AMT) cruises carried out in October–November 2012 (AMT22) and in September–November 2016 (AMT26). Both cruises covered the Atlantic Ocean from the United Kingdom to South America, by encompassing a wide range of oceanic conditions, from sub-polar to tropical and from eutrophic systems to mid-ocean oligotrophic gyres (Supplementary Fig. 1). During both cruises, we collected measurements of PSD (i.e., the number of particles per unit volume per unit of particle size; units of particles m−3 µm−1), and particulate beam attenuation and optical backscattering coefficients at the wavelength 532 nm (cp(532) and bbp(532), respectively). Here, we use the particulate beam attenuation coefficient as a measure of the total scattering because these two quantities are numerically close at 532 nm due to negligible particulate light absorption25.

During AMT22, PSD and optical measurements were collected at 134 stations in flow-through mode from the 5-m underway clean seawater supply of the ship (Supplementary Fig. 1). On AMT26, a rosette system equipped with 24 × 20-l Niskin bottles and sensors for hydrological and optical measurements was deployed in the upper 500 m at 23 stations (Supplementary Fig. 1). Up-cast seawater collection for PSD analysis was performed at five depths (5, 150/200, 300, 400/450, and 500 m) in addition to the DCM as detected by an AquaTracka III fluorometer (Chelsea Technologies Group Ltd). Temperature and salinity measurements were acquired during the cast by a Sea-Bird Scientific SBE 9 Conductivity-Temperature-Depth (CTD) sensor.

A size-fractionation experiment was conducted during the AMT26 cruise to assess the contribution of submicron particles (i.e., <1 µm) to optical scattering measurements. A total of 22 additional stations were sampled, at the same time as the ship’s CTD casts, in flow-through mode from the 5-m underway ship’s seawater supply (Supplementary Fig. 1). Optical measurements were taken, and seawater for PSD analysis collected, before and after filtration through a pre-rinsed (with ultra-pure water) large surface-area cartridge filter (Cole Parmer) with nominal pore size of 1 µm (i.e., AMT26_1µm). Filter back-flushing using ultra-pure water before filtrations minimised filter clogging. Filtering in flow-through mode minimised particle aggregation and precipitation. During filtration, the flow rate of the underway system was set to 2 l min−1.

AMT26 data collected at the ocean surface (5 m) and at the DCM were used to test the homogeneous-sphere assumption for marine particles and parameterise the coated-sphere model. AMT22 and AMT26_1µm samples were used as independent datasets to validate model results. AMT26 samples collected in the mesopelagic region of the Atlantic Ocean were used to discuss coated-sphere model advantages and limitations.

### Particle size distribution (PSD)

During AMT22, a total of 134 PSDs were measured on board, in multiple 2-ml replicates (from 2 to 26), through a Multisizer III Coulter counter (Beckman Coulter) fitted with a 70-μm aperture. The measured size range of equivalent spherical diameters (ESDs) was 1.4–42 μm and distributed in 200 logarithmically spaced size bins. During AMT26, seawater samples (N = 134 from casts + 44 from underway) were directly collected in 500-ml acid-washed amber bottles and immediately prepared under a vertical flow hood for the analysis. Samples were kept in a cold and dark place during the analysis. Measurements were conducted with a Multisizer III Coulter counter (Beckman Coulter) fitted with 20-μm and 100-μm apertures, in multiple 50-µl and 1-ml replicates, respectively. The ESD size ranges were 0.588–12 μm and 2–60 μm for the 20-μm and 100-μm apertures, respectively, each distributed in 256 logarithmically spaced size bins. The number of replicates was selected, sample by sample, to achieve an overall error <15% in a given size range (~3 µm and 6 µm for 20-μm and 100-μm apertures, respectively), which corresponded to a detection limit of 45 counted particles (i.e., from three repetitions at the DCM up to 15 for samples in the mesopelagic zone).

For both AMT22 and AMT26 samples, the PSD was calculated for each aperture by summation of all the replicates. The total analysed volume was 4–52 ml with the 70-µm aperture, 150–600 µl with the 20-µm aperture, and 3–15 ml for the 100-µm aperture. The accuracy of measurements within each size bin was calculated through the standard law of propagation of uncertainty66 considering the number of counted particles, the blank reference and the Multisizer III volumetric pump accuracy (0.5%) as sources of error. Specific to the AMT26, the PSDs obtained from the 20-µm and 100-µm apertures were combined in a single PSD by merging the measurements from the two apertures at around 2.14 µm, where they presented similar bin width and upper/lower limits. The resulting PSD spanned from 0.588 to 60 µm by 360 logarithmically spaced size bins with associated uncertainties.

Before each cruise, the Coulter counter apertures were calibrated using suspensions of recommended Beckman Coulter calibration spheres. During AMT26, the instrument performance was checked using suspensions of standardised spheres of 3.6 µm of diameter (Beckman Coulter) and provided counting accuracy within 10% for both 20-µm and 100-μm apertures. Blank references of 500-m seawater filtered through 0.1-μm Millex sterile polyvinylidene difluoride syringe filters (Millipore) were recorded daily for both 20-μm and 100-μm apertures. During AMT22, blanks were acquired through repeated (three times) filtration of surface seawater on 0.2-µm syringe filters.

Examples of PSDs from AMT22, AMT26, and AMT26_1µm with combined uncertainties are shown in Supplementary Figure 11. Supplementary Figure 12 shows examples of PSDs simultaneously collected during AMT26 from the ship’s underway and using Niskin bottles at 5 m depth. No significant differences in PSDs were observed between the two sampling platforms.

### In-situ particulate optical scattering

Bulk beam attenuation coefficients at 532 nm (c(532)) were measured with an ac-s spectrophotometer (WETLabs, Seabird-Scientific) for underway samples (AMT22 and AMT26_1µm), or through 0–250 m casts of an ac-9 spectrophotometer (WETLabs, Seabird-Scientific) at the same time of CTD measurements and water sampling. The ac-s and ac-9 had the same optical configuration and beam acceptance angle67, and collected data along a 25-cm path length. Both underway and cast measurements were repeated after filtration through a 0.2-µm filter in order to determine the contribution of the coloured dissolved organic matter, as well as calibration drifts31. The beam attenuation coefficient of particles (cp(532)) was obtained by subtraction of the 0.2-µm filtered signal from c(532), following established protocols52.

The angular scattering coefficient at 532 nm was measured at a central angle of 124°, β(124, 532), with three ECO-BB3 backscattering sensors (WETLabs, Seabird-Scientific). During AMT26, the ECO-BB3 was installed on the rosette sampling system and provided 0–500 m β(124, 532) measurements. For AMT22 and AMT26_1µm underway samples, β(124, 532) coefficients were acquired in a flow-through chamber. Data were collected and processed following established protocols31,52. The relative uncertainty associated with the extrapolation of the optical backscattering coefficient from a single angle measurement is <10%68,69.

All cp(532) and bbp(532) measurements were quality checked31,52. Underway data were binned into 1-min intervals and the median value was calculated for the timing corresponding to PSD water sampling. The cp(532) and bbp(532) coefficients acquired during upward casts were binned in 1-m intervals and the median was calculated for values within a 5-m window centred at the depth of the Niskin bottles fired for the PSD samples. Before 1 m binning, bbp(532) profiles were smoothed with a moving-median filter (five-point window). To assess the robustness of the optical measurements used in the analysis, we checked consistency through inter-comparisons among them and versus independent simultaneous measurements when available (e.g., HobiLabs HydroScat-6P; Supplementary Figs. 13 and 14).

### Modelling particulate optical scattering properties

The beam attenuation and optical backscattering coefficients of particles were modelled at the wavelength 532 nm as follows:27

$$c_{\mathrm{p}}\left( n \right) = \frac{{\mathrm{\pi }}}{4}{\int_{D_{{\mathrm{min}}}}^{D_{{\mathrm{max}}}}} Q_{\mathrm{c}}\left( {D,n} \right)D^2F\left( D \right){\mathrm{d}}D$$
(1)

and

$$b_{{\mathrm{bp}}}\left( n \right) = \frac{{\mathrm{\pi }}}{4}{\int_{D_{{\mathrm{min}}}}^{D_{{\mathrm{max}}}}} Q_{{\mathrm{b}}_{\mathrm{b}}}\left( {D,n} \right)D^2F\left( D \right){\mathrm{d}}D$$
(2)

where D is the particle diameter (units of m) with minima and maxima as defined from PSD measurements (i.e., (0.588–60) × 10−6 m for AMT26 and (1.4–42) × 10−6 m for AMT22 samples). We did not extrapolate the PSDs for particles with diameters <Dmin that consequently were not modelled. For each individual particle, Qc(D, n) and $$Q_{{\mathrm{b}}_{\mathrm{b}}}(D,\,n)$$ are the dimensionless efficiency factors for beam attenuation and optical backscattering respectively, at 532 nm, and n is the volume-averaged refractive index38, which represents the characteristics of the coated sphere with refractive indices (relative to that of seawater54) of its core and coat, n1 and n2, respectively, and fractional thickness of the coat tk2; the term πD2/4 indicates the particle cross-sectional area (units of m2); and F(D) is the concentration of particles for a given diameter as obtained from the PSD measurements (units of particles m−4). The values of n2 were varied between 1.02 and 1.2250, whereas those of tk2 varied between 2 and 30% of the radius of the particle. The volume-averaged refractive index of the particle was forced to be equal to 1.06, which was the single value that best fitted the beam attenuation measurements through the homogeneous-sphere assumption (see Results). As a consequence, the refractive index of the core n1 varied as a function of n2 and tk2. A total of 609 combinations of tk2 and n2 were tested with the coated-sphere model. Homogeneous spheres were represented as a coated sphere with tk2 = 0 and n1 values between 1.02 and 1.2250. Particles counted within each size bin of the measured PSD were considered as monodispersed and non-absorbing populations at the selected wavelength25. We expect the results of this modelling analysis to be more sensitive to the structural complexity of the spheres than to the impact of light absorbing material on bbp38,39, and thus to be representative also of other portions of the visible spectrum.

Look-Up-Tables of Qc(D, n) for homogeneous and coated spheres were generated by using the scattnlay software70, for angles between 0 and 180° (0.1° increments) and logarithmically spaced ESDs between 0.4 and 70 µm. The Qc(D, n) values both for homogeneous and coated spheres were corrected for the acceptance angle effect to be consistent with WETLabs ac-9 and ac-s in-situ measurements67.

The values of $$Q_{{\mathrm{b}}_{\mathrm{b}}}(D,\,n)$$ were computed as:

$$Q_{{\mathrm{b}}_{\mathrm{b}}}(D,n) = Q_{\mathrm{b}}(D,n)\frac{{\mathop {\int }\nolimits_{{\mathrm{\pi }}/2}^{\mathrm{\pi }} S_{11}(\theta ,D,n)\sin \theta {\mathrm{d}}\theta }}{{\mathop {\int }\nolimits_0^{\mathrm{\pi }} S_{11}(\theta ,D,n)\sin \theta {\mathrm{d}}\theta }}$$
(3)

where $$Q_{\mathrm{b}}$$ is the dimensionless efficiency factor for the total scattering (output of the freely available scattnlay software70) and $$S_{11}(\theta ,D,n)$$ is the top left component (i.e., 11) of the scattering matrix computed as described below. To compute the VSF and the other non-normalised components of the Mueller matrix we proceeded as follows70. First, for each set of particle parameters (i.e., D and n for the wavelength 532 nm) we computed the following elements of the scattering matrix:

$$S_{11}\left( {\theta ,D,n} \right) = 0.5\left[ {\left| {S_2\left( {\theta ,D,n} \right)} \right|^2 + \left| {S_1\left( {\theta ,D,n} \right)} \right|^2} \right]$$
(4)
$$S_{12}\left( {\theta ,D,n} \right) = 0.5\left[ {\left| {S_2\left( {\theta ,D,n} \right)} \right|^2 - \left| {S_1\left( {\theta ,D,n} \right)} \right|^2} \right]$$
(5)
$$\begin{array}{c}S_{33}\left( {\theta ,D,n} \right) = 0.5\left[ {S_1\left( {\theta ,D,n} \right)S_2^ \ast \left( {\theta ,D,n} \right)} \right.\\ \left. { + S_1^ \ast \left( {\theta ,D,n} \right)S_2\left( {\theta ,D,n} \right)} \right]\end{array}$$
(6)
$$\begin{array}{c}S_{34}\left( {\theta ,D,n} \right) = 0.5{\mathrm{i}}\left[ {S_1\left( {\theta ,D,n} \right)S_2^ \ast \left( {\theta ,D,n} \right)} \right.\\ \left. { - S_1^ \ast \left( {\theta ,D,n} \right)S_2\left( {\theta ,D,n} \right)} \right]\end{array}$$
(7)

where $$S_1\left( {\theta ,D,n} \right)$$ and $$S_2\left( {\theta ,D,n} \right)$$ are the complex scattering amplitude outputs of the scattnlay software70 and the starred quantities indicate complex conjugates.

Each $$S_{{\mathrm{ij}}}(\theta ,D,n)$$ component was then normalised by the integral of the $$S_{11}(\theta ,D,n)$$ component to compute normalised elements of the scattering matrix:

$$\overline {S_{{\mathrm{ij}}}} \left( {\theta ,D,n} \right) = \frac{{S_{{\mathrm{ij}}}(\theta ,D,n)}}{{2{\mathrm{\pi }}\mathop {\int }\nolimits_0^{\mathrm{\pi }} S_{11}(\theta ,D,n)\sin \theta {\mathrm{d}}\theta }}$$
(8)

We then computed the Mueller matrix components of particle populations composed of modelled coated spheres and the measured PSDs, $$F(D)$$, as:

$$M_{{\mathrm{ij}}}(\theta ,n) = \frac{{\mathrm{\pi }}}{4}{\int_{D_{{\mathrm{min}}}}^{D_{{\mathrm{max}}}}} \overline {S_{{\mathrm{ij}}}} \left( {\theta ,D,n} \right)Q_{\mathrm{b}}\left( {D,n} \right)D^2F(D){\mathrm{d}}D$$
(9)

with $$M_{11}(\theta ,n)$$ resulting in the modelled VSFs.

To compare the shape of the VSFs with existing measurements, we normalised $$M_{11}(\theta ,n)$$ by its value integrated between 0 and $${\mathrm{\pi }}$$ (i.e., total scattering) to obtain the corresponding phase functions as:

$${\mathrm{Normalized}}\,{\mathrm{VSF}}(\theta ,n) = \frac{{M_{11}(\theta ,n)}}{{2{\mathrm{\pi }}\mathop {\int }\nolimits_0^{\mathrm{\pi }} M_{11}(\theta ,n)\sin \theta {\mathrm{d}}\theta }}$$
(10)

Finally, to compare the polarised components with existing datasets, we normalised each component of the scattering matrix by $$M_{11}(\theta ,n)$$ as follows:

$$P_{{\mathrm{ij}}}\left( {\theta ,n} \right) = \frac{{M_{{\mathrm{ij}}}(\theta ,n)}}{{M_{11}(\theta ,n)}}$$
(11)

where now i and j are not simultaneously equal to 1.

### Statistics

The combined uncertainty of modelled cp(532) and bbp(532) coefficients was calculated by propagation of the errors associated with PSD measurements66. Modelled cp(532) and bbp(532) coefficients were then compared with in-situ measurements through match-up analysis. The model performance was evaluated by calculating the systematic error, in logarithmic space, as follows:

$${\mathrm{Bias}} = {\mathrm{median}}\left( {{\mathrm{log}}_{10}(\bar x_{\mathrm{i}}) - {\mathrm{log}}_{10}(x_{\mathrm{i}})} \right)$$
(12)

where $$\bar x_{\mathrm{i}}$$ and $$x_{\mathrm{i}}$$ are the modelled and measured values, respectively. The Pearson’s correlation coefficient (r) was also calculated between log-transformed quantities and the significance was verified by a two-tailed Student’s t-test (confidence level equal to 99%; N-2 degrees of freedom).

The cumulative percent contributions to cp(532) and bbp(532) of the size fraction from Dmin to D were defined, respectively, as:27

$$C_{c_{\mathrm{p}}}\left( D \right) = 100{\int_{D_{{\mathrm{min}}}}^D} Q_{\mathrm{c}}\left( {D,n} \right)D^2F\left( D \right){\mathrm{d}}D\left( {{\int_{D_{{\mathrm{min}}}}^{D_{{\mathrm{max}}}}} Q_{\mathrm{c}}\left( {D,n} \right)D^2F\left( D \right){\mathrm{d}}D} \right)^{ - 1}$$
(13)

and

$$C_{b_{{\mathrm{bp}}}}\left( D \right) = 100{\int_{D_{{\mathrm{min}}}}^D} Q_{{\mathrm{b}}_{\mathrm{b}}}\left( {D,n} \right)D^2F\left( D \right){\mathrm{d}}D\left( {{\int_{D_{{\mathrm{min}}}}^{D_{{\mathrm{max}}}}} Q_{{\mathrm{b}}_{\mathrm{b}}}\left( {D,n} \right)D^2F\left( D \right){\mathrm{d}}D} \right)^{ - 1}$$
(14)