Particle size distribution at Ocean Station Papa from nanometers to millimeters constrained with intercomparison of seven methods

Mediated by food web processes, inorganic carbon is converted photosynthetically into organic matter and transferred from the surface to the deep ocean. The combination of biological, physical, and chemical processes that contribute to and control this export is collectively known as the biological pump (Eppley and Peterson, 1979; Falkowski et al., 1998; Ducklow et al., 2001). Current estimates of the global magnitude of this export range between 5 and 12 Pg C yr⁻¹ (Laws et al., 2000; Boyd and Trull, 2007; Henson et al., 2011). This wide range of estimates highlights our poor knowledge of this major carbon pathway. To develop a predictive understanding of the biological pump and contribute to constraining the uncertainty, the Export Processes in the Ocean from Remote Sensing (EXPORTS) field campaign (Siegel et al., 2021) was proposed to sample the ocean carbon pathways over a range of contrasting ecosystem/carbon cycle states (Siegel et al., 2016).

The dominant processes involved in the biological pump (Burd et al., 2016) include primary production of particulate organic carbon by phytoplankton, alteration of these phytoplankton particles in both size and composition by biological processes such as consumption by zooplankton and remineralization by bacteria and by physical processes such as coagulation and fragmentation, and subsequent settling of both unaltered and altered particles (Stemmann et al., 2004b). In addition to passive sinking of particulate carbon, the downward mixing of picoplankton (Richardson and Jackson, 2007; Omand et al., 2015) and dissolved organic carbon (Hansell et al., 2009; Carlson et al., 2010) from the surface is also a key component of export below the euphotic zone. Size is an important particle property affecting these processes (Stemmann et al., 2004a; Woodward et al., 2005; Stemmann and Boss, 2012). For example, the size of phytoplankton influences their ecology and physiology (Kiørboe, 1993; Siegel, 1998). Different phytoplankton size classes (Uitz et al., 2008) have been incorporated into primary productivity (e.g., Uitz et al., 2010) and food web (e.g., Buesseler and Boyd, 2009) models. Settling speeds of particles vary with size (Khelifa and Hill, 2006), and coagulation rate scales with size (Jackson, 1990).

During the 2018 EXPORTS field campaign in the Northeast Pacific near Ocean Station Papa (OSP; Siegel et al., 2021), we deployed a suite of seven instruments, including three light-scattering instruments, a ViewSizer, a Coulter Counter, an imaging-in-flow cytometer, and an underwater imaging system, each based on different approaches to size and/or identify particles within the instrument/technique-specific detection ranges. Some of the instrument size ranges overlap and together extend each other, allowing us not only to compare and validate the results but also to construct quantitatively consistent marine particle size distributions extending over an unprecedented size range from approximately 0.02 µm to 2000 μm.

In this study, we focused on comparing the particle size distributions estimated by various methods. This comparative analysis is the first comprehensive evaluation of the particle size distributions measured by these instruments. The results provide an estimate of uncertainty and pave the way for estimating mass and carbon flux from the measured size distributions of particles.

The experiment was conducted near Ocean Station Papa (50°N, 145°W) from August 9 to September 13, 2018, as part of the EXPORTS field campaign in the Northeast Pacific. The water was very clear during the experiment, with a mean chlorophyll-a (Chl-a) concentration of 0.25 µg L⁻¹ and particulate organic carbon concentration of 2.5 µmol L⁻¹ in the upper mixed layer (average depth of 23 m), considerably less than the typical August values of the area (Siegel et al., 2021). The data reported here were collected on board the R/V Sally Ride, which conducted a series of small (approximately 900 km²) and large (approximately 7700 km²) scale surveys within the boundaries of 49.5 to 50.9°N and 144.5 to 145.8°W.

Water samples were collected at 47 stations, each at 7 or 8 depths, on the ascending path of a CTD-rosette profiling the water column down to a depth of 300–500 m and occasionally to 1000–3000 m. In addition to the typical temperature, salinity and pressure measurements, the rosette was also equipped with an Underwater Vision Profiler (UVP). A total of 315 water samples were collected. For each water sample, approximately 10 L of water was analyzed by the suite of instruments operating in the ship’s lab. A brief background on the instruments and their respective particle-sizing approach is provided below and summarized in Table 1.

The LISST-100X Type B (Laser In-Situ Scattering and Transmissometer, Sequoia Scientific) measures the optical volume scattering function (VSF; m⁻¹ sr⁻¹) at 532 nm at 32 scattering angles from 0.08° to 13° (Agrawal and Pottsmith, 2000; Agrawal, 2005; Agrawal et al., 2008; Agrawal and Mikkelsen, 2009). The relatively clear waters at OSP, especially at depth, presented a challenge for the LISST-100X, which has a 5-cm path length and is not especially sensitive to low particle concentrations. In addition, the power output of the 532-nm laser in our LISST drifted significantly over time, which caused issues with baseline subtraction: in some cases, the signal of the baseline or reference water was larger than the sample signal. To address these issues, we developed a protocol for both baseline subtraction and laser drift. The LISST sample chamber was rinsed three times, filled with approximately 100 mL of sample, and a set of 500 measurements were recorded while the water sample was stirred constantly. For samples collected at three of the depths at each station—the surface, the depth of the chlorophyll fluorescence maximum, and the deepest, we also pumped a portion of the sample continuously through a 0.2-µm filter for 20 minutes and then recorded another 200 measurements of the filtered water to use as the background reference. We then found the filtered sample with the lowest overall average signal on the LISST ring detector, defined simply as the sum of the digital counts over all 32 ring detectors. This sample, which was usually but not always the deepest sample, was then used as the background reference for this station. Comparing to using pure water, using filtered seawater as the background file helps to address the index of refraction effect, especially evident at low particle concentrations. Using the data with the lowest overall signal allowed the inclusion of stations that would otherwise be unusable due to the background data having larger signal than the unfiltered data.

We also found that the data were highly susceptible to laser drift. In our LISST, when the laser is first turned on, the laser power (measured from the LISST reference sensor) starts high, falls to a minimum after 2–3 minutes, then slowly increases toward an asymptote, reaching a stable value after 15–20 minutes. This difference between the maximum and minimum power levels can vary 10–20% about the mean, and these variations also affect the baseline subtraction. To avoid the minimum in laser power, we computed a series of averaged LISST ring data in each measurement set, starting with the last measurement and consecutively adding more measurements working toward the first. We averaged an increasing subset of samples from the total until the coefficient of variation in each subset began to increase, generally as the laser power began to approach the minimum. We then chose as our most reliable data the subset of samples with the overall lowest coefficient of variation. The goal of this procedure was to include as many samples in our measurement as possible while avoiding the laser minimum and minimizing the effects of random laser noise. By using both our optimal background estimation and drift minimization methods in combination, we were able to include approximately 80% of the data that would otherwise be rejected due to low signal-to-noise ratio.

After subtracting the reference, the average values of sample measurements were processed as outlined in Zhang et al. (2012) to generate the VSF due to particles. We denote this particulate VSF as β_LISST-100X(θ), where θ represents the scattering angle. The measured β_LISST-100X(θ) values were in turn used to estimate the particle size distribution (PSD) as a function of cross-sectional area equivalent circular diameter following Agrawal and Pottsmith (2000). PSDs inferred from the LISST-100X are denoted as P_LISST-100X(d_cs-area), where d_cs-area represents cross-sectional area equivalent circular diameter and ranges approximately from 1 µm to 200 μm.

The LISST-VSF (Sequoia Scientific) measures the VSF at 517 nm from 0.08° to 155° (Koestner et al., 2018; Hu et al., 2019). It consists of two optical units: an array of ring detectors (similar to the one installed in the LISST-100X) measures the VSF at 32 angles from 0.08° to 14.4°, and a rotating eyeball detector measures the VSF from 15° to 155° at 1° intervals. The LISST-VSF was turned on for 30 minutes for the laser to stabilize before taking any measurements. For each sample, approximately 1.8 L of water was slowly pumped peristaltically to the sampling enclosure of the LISST-VSF. The sample was visually inspected for residual bubbles, which were then removed with a de-bubbler. Thirty repeated measurements (4–5 seconds for each measurement) were taken for each sample and the median values were processed following Hu et al. (2019) to produce the bulk VSFs. The background scattering, which was subsequently removed from the bulk VSFs to estimate the VSFs due to particles, were estimated in two ways. For scattering as angles <15°, which is measured by the ring detectors, the background was filtered seawater; for scattering at larger angles, which is measured by the eyeball detector, the background was computed using a pure seawater scattering model (Zhang et al., 2009). For reference, the scattering by pure seawater can account for up to 50% of the scattering measured at backward angles at the experiment site (Zhang et al., 2020). The resulting particulate VSFs, denoted β_LISST-VSF, were inverted to obtain the size and the refractive index of particles based on the VSF-Inversion method described in detail in Zhang et al. (2011) and refined later in Twardowski et al. (2012) and Zhang et al. (2012). We denote the final PSDs derived from the VSF-Inversion as P_LISST-VSF(d_surface-area), where d_surface-area represents surface-area equivalent sphere diameter and ranges from approximately 0.02 µm to 2000 µm.

We did not use the theoretical seawater scattering as baseline consistently for all the sensors because this “inconsistency” is dictated by the sensitivity of the sensor. The detectors used in LISST-100X and in LISST-VSF for measuring scattering at angles <15° are photodiode, whose dynamic range is relatively narrow and chosen to measure forward scattering by typical oceanic and coastal waters that is several orders of magnitude greater than pure seawater scattering. Even for measuring filtered water or seawater, their signal would appear elevated far above theoretical pure water/seawater values. This elevated signal is not real signal and represents the baseline value of the instrument. On the other hand, the eyeball detector used in LISST-VSF for measuring scattering at angles from 15° to 155° uses a photomultiplier tube (PMT), which has a much wider dynamic range that can measure the scattering at levels of pure seawater.

The Multi-spectral Volume Scattering Meter (MVSM) is a prototype instrument that measures the VSF at 8 wavelengths (443, 490, 510, 532, 555, 565, 590, and 620 nm) at nominal scattering angles from 0.5° to 179° at 0.25° intervals (Zhang et al., 2012). The MVSM uses a prism that rotates 360° and produces two measurements of the VSF (90° to 180° in ascending angular order, and 180° to 360° in descending angular order) in approximately 1 minute at each wavelength. The MVSM can operate in both multi-spectral mode and single wavelength mode at 532 nm. The sampling volume of the MVSM is approximately 1.5 L. However, because of the relatively low sampling rate of the instrument and to avoid possible sedimentation, we slowly pumped approximately 3 L of water peristaltically through the sample chamber while measurements occurred, letting excess water simply overflow out of the chamber. Due to time constraints during the experiment, we operated MVSM in full spectral mode only for the near-surface samples and occasionally for the deeper water samples, and at one wavelength of 532 nm for the others. Also, because of this time constraint, only one 360° scan was taken for each wavelength in full spectral mode and six 360° scans were taken when using only 532 nm. Therefore, the final VSF data for each water sample was an average of two measurements for each wavelength in multispectral mode, and an average of 12 measurements in single wavelength mode. The scattering by pure seawater at the corresponding wavelength, temperature and salinity was subtracted from the VSFs measured by MVSM to derive the particulate VSFs. We denote the particulate VSFs from MVSM as β_MVSM(θ).

We found that the MVSM has issues with the measurement at angles <12° (Zhang et al., 2012). As the LISST-100X operates at 532 nm, we combined 532 nm data from β_MVSM at θ > 13° (by dropping those at θ < 13°) with β_LISST-100X to form a dataset called β_LISST+MVSM(θ). We applied the same VSF-Inversion as for the LISST-VSF to β_LISST+MVSM(θ) to estimate the size distribution of particles, which are denoted as P_LISST+MVSM(d_surface-area). As in the case of the LISST-VSF, d_surface-area ranges approximately from 0.02 µm to 2000 µm.

For particles undergoing Brownian motion, their diffusivity is inversely proportional to the size of the particle (Einstein, 1956). ViewSizer 3000 (Horiba Scientific) tracks movement of particles illuminated with three laser beams of different wavelengths and estimates the diffusivity and hence the size of each particle. During the experiment, we placed the ViewSizer 3000 on a vibration-isolation table to mitigate the potential effect of ambient vibration caused by the engine or the generators in the ship. The sample chamber of ViewSizer 3000 is a quartz cuvette (3 mL) with a magnetic stir bar. The sampling volume, approximately 2.5 nL, is inside the cuvette and formed by the field of view of the CCD camera taking videos of particles undergoing Brownian motion. For each sample, one hundred 10-second videos at a 30-Hz frame rate were recorded. Each 10-second video was analyzed individually to derive the PSD, and the final result was the average of the 100 individual PSDs each from a single 10-second video.

Xiong et al. (2022) conducted a comprehensive analysis using Monte Carlo simulation, lab experiments with beads of standard sizes, and field experiments to examine the performance of ViewSizer 3000, which was estimated to have an average uncertainty of 38% in measuring the concentration of particles. The size of particles derived from ViewSizer 3000 represents the diameter of an equivalent sphere that would undergo the same Brownian motion as the particle in the same liquid. We denote the PSD derived from ViewSizer 3000 as P_ViewSizer(d_diffusion), where d_diffusion ranges from 0.3 µm to 2 μm.

The Coulter Counter (Multisizer 3e, Beckman) measures particle sizes and concentrations through resistive pulse sensing (DeBlois and Bean, 1970), which quantifies changes in electrical resistance produced by particles suspended in seawater as they pass through an aperture (20–500 µm). Measurements are reported within the 2–60% range of the aperture opening, which was 100 µm during the EXPORTS cruise. Four to six replicates, each with a volume varying between 4 and 10 mL depending on the sample depth (4 mL for depths 5–50 m; 6 mL for 50–200 m; 10 mL for >300 m), were run for each sample and averages were computed. Furthermore, for each CTD cast one blank was measured (2 replicates of 10 mL) with 0.2-μm-filtered seawater from the deepest sample (between 330 m and 1000 m). The size obtained from the instrument is the diameter of a sphere of the same volume as the particle. We denote the PSD derived from the Coulter Counter as P_Coulter(d_volume), where d_volume is the diameter of an equivalent sphere and ranges from 2 µm to 40 µm.

The IFCB (McLane Research Laboratories, Inc.) is an imaging-in-flow cytometer optimized for analysis of particles including nano- and microplankton (Olson and Sosik, 2007). IFCB measures laser-based light scattering and chlorophyll fluorescence of individual particles, and corresponding image acquisition (10X objective) is triggered when those signals exceed user-configured trigger thresholds. Pixel resolution is approximately 1 μm, and trigger thresholds were set to reliably image particles larger than approximately 5 μm. For the samples reported here, images were acquired if either the chlorophyll or light-scattering trigger level (or both) was exceeded. As in conventional cytometry, particles are hydrodynamically focused in the IFCB flow cell so images are consistently in focus. Five-mL water samples were run on IFCB, with duplicate or triplicate samples analyzed when time permitted. For each run, the volume of sample effectively imaged by IFCB ranged from approximately 1 mL to 4 mL depending on trigger rate, with typically 3000–5000 images per sample.

IFCB images were processed with the methods described in Sosik and Olson (2007) and updated in Sosik et al. (2020). The distance map algorithm of Moberg and Sosik (2012) was used to determine the biovolume of each imaged target from its two-dimensional boundary. The diameter of a volume equivalent sphere was then computed from the calculated volume and used in this study. We denote the PSD derived from the IFCB as P_IFCB(d_volume), where d_volume ranges from 5 µm to 100 µm.

The UVP-5, described in detail in Picheral et al. (2010), uses a 1.3 megapixel camera and two synchronized red (625 nm) LED light sources to acquire images of a 1.02-L volume at a frequency up to 6 Hz. The camera takes images within a 22 × 18 cm domain that is roughly 40 cm from the camera. Particle cross-sectional areas are quantified by assessing the contiguous pixels for a given image brightness level. Calibration of the optimal brightness level for each UVP-5 instrument is evaluated in situ by comparing PSD determinations with a “gold” standard instrument as detailed in Picheral et al. (2022) and in Picheral et al. (2010). The UVP-5 used here was calibrated in this manner before and after the cruise. The number of pixels each particle occupies in an image is converted to the cross-sectional area using a power-law relationship determined through the calibration. The area is then used to estimate the equivalent circular diameter enabling the aggregate PSD to be determined for 25 bins with center bin sizes ranging from 90 µm to 23,160 µm. Due to rarity of very large aggregates and the approximate 1-L sampling volume of the UVP-5, the largest aggregate sizes quantified were 2,300 µm (bin center), and then only rarely.

During the experiment, the UVP-5 was mounted on the bottom of the R/V Sally Ride CTD rosette, sampling at a frequency of 6 Hz. PSD data from the UVP-5 were binned vertically to 5-m intervals to improve statistics of the rarest large particles. Given typical CTD frame lowering rates this corresponds to nearly 100 individual scans making up each 5-m vertical average. We matched the UVP data with other data measured in the ship’s lab using the depth of water samples collected. We denote the PSDs derived from the UVP as P_UVP(d_cs-area), where d_cs-area ranges from 50 µm to 2000 µm.

We conducted two types of comparison to evaluate the consistency in PSDs determined by the various techniques examined in this study. The first type of comparison was made for the VSFs measured by the LISST-100X, LISST-VSF and MVSM. The purpose of this comparison is to test the consistency among light-scattering instruments. The second comparison was made for the PSDs inferred from the VSF data from the LISST-100X, LISST-VSF and MVSM and derived from the ViewSizer, Coulter Counter, IFCB, and UVP. For comparison of PSDs, several aspects should be considered. To illustrate these, the PSDs derived from these seven instruments are presented in Figure 1.

The particle-sizing techniques that we used in this study fall into two general categories: (i) methods that measure all particles simultaneously (ensemble methods) and (ii) methods that measure particles individually (individual-particle methods). Particle counting techniques used in the ViewSizer, Coulter Counter, IFCB, and UVP belong to the individual-particle category (Figure 1a and b), whereas the techniques of inverting PSDs from the LISST-100X, LISST-VSF and LISST+MVSM belong to the ensemble category (Figure 1c–e). Each of the techniques has advantages as well as shortcomings (Syvitski et al., 1991; Jonasz and Fournier, 2007, pp. 301–376). Examining the technical merit of each technique is beyond the scope of this study. Instead, we focused on comparing results without holding a particular approach as the standard or “truth.” In principle, particles of all sizes contribute to the scattering. Therefore, the ensemble methods based on light scattering tend to retrieve particles of wider size ranges. In contrast, individual particle-based methods retrieve particles of sizes dictated by the specific technique. For example, the particles measured by the ViewSizer are limited to submicrometer sizes for which particles undergo Brownian motion. And particles measured by the Coulter Counter or IFCB are of sizes dependent on the size of the aperture through which the particles flow.

The ensemble method must meet the additional constraint that there be at least one particle in the size interval in the sampling volume to estimate the PSD (Jackson et al., 1997). This constraint means that the PSDs by the ensemble approach need to be truncated beyond the minimum particle concentration size, above which the total number of particles predicted by the derived PSD in the sampling volume would be less than 1. The actual cutoff size depends on the particle concentration in each size bin as well as the sampling volume. For LISST-VSF, MVSM and LISST-100X, with sampling volumes of approximately 1.8 L, 1.5 L and 100 mL, the ranges of cutoff sizes for the largest particles were 98–1747 μm, 52–1872 μm, and 21–183 μm, respectively. These cut-off PSDs are shown as red lines in Figure 1c–e. We noticed very low signal-to-noise level on the outer rings of the LISST-100X, indistinguishable from the background values. As they are mostly responsible for the smaller particle sizes, we discarded LISST-100X PSDs at sizes <3 µm.

Also discarded were PSDs from the ViewSizer at sizes <0.3 μm, where the signal-to-noise ratios of these smaller particles were significantly degraded due to the presence of larger particles, and hence their concentrations are underestimated (Xiong et al., 2022).

We only considered particles >5 µm from IFCB data because smaller particles are not reliably above the light-scattering or fluorescence trigger levels and thus are underestimated in concentration. The IFCB was configured with this cutoff to optimize acquisition of images of plankton and particles in the size range where the approximately 1-µm–resolution images contain useful detail (Olson and Sosik, 2007).

The Coulter Counter reports PSDs of sizes within 2–60% of the aperture diameter. With a 100-µm aperture used, the size range was 2–60 µm. However, we found the PSDs above 40 µm were very noisy after removing the corresponding background PSDs. Therefore, we only used Coulter Counter data from 2 µm to 40 µm.

Inconsistencies were found between the UVP-5 PSD observations from the two systems used on the R/V Revelle and R/V Ride during the EXPORTS North Pacific campaign. One issue is that the smallest bin size was not sampled consistently by both systems. Hence, the smallest bin (90 µm center) was removed from consideration in the present analyses. The second issue is that in both of the UVP units light sources would occasionally fail to illuminate due to a fleetwide hardware issue (M Picheral, personal communication, 2020). This failure to illuminate happened between 2% and 10% of the images collected from both units during the EXPORTS North Pacific campaign and was considerably worse in the data from the UVP-5 system on the R/V Revelle than on the R/V Ride (the system used here). This issue led to a bimodal distribution of particle counts from individual images, and an undercounting of the true particle concentrations. The problem was corrected by removing the few, obviously low signal images by digital filtering before the vertical binned PSD data were constructed.

For individual-particle methods, the occurrence of particles in the sample volume can be assumed to be Poisson process if the particles are distributed randomly and not correlated spatially. For a Poisson process, the relative uncertainty in particle concentration is $1/N$⁠, where N is the number of particles counted. As replication is equivalent to increasing sampling volume, N should be treated as the total, not the mean, count of particles of all replications. As the minimum number of particles detected is 1, the maximum uncertainty associated with sampling particles using individual-particle methods is 100%. For ViewSizer, number of particles counted during EXPORTS ranged from 3 to 20 for sizes from approximately 2 µm down to 0.3 µm, which translated to an uncertainty of 22–58%, the mean value of which is very close to the 38% uncertainty in measuring the concentration of beads of standard sizes (Xiong et al., 2022). For Coulter Counter, particles averaged approximately 700 counts at 2-µm-size bin to 1 at 40-µm-size bin, with a mean count of 19 over all the size bins. The corresponding uncertainty ranged from 4% to 100%, with an average of 24%. For IFCB, particles averaged approximately 700 counts at 5-µm-size bin and to 1 count at 100-µm-size bin, with a mean count of 10. The corresponding uncertainty ranged from 4% to 100%, with an average of 32%. For UVP, counts of particles averaged 450 at 90-µm-size bin to 3–4 at 2.3-mm-size bin, with an average of 35. The corresponding uncertainty ranged from 5% to 55%, with an average of 17%.

For ensemble methods, we estimated the uncertainty associated with sampling particles using the LISST-VSF data. LISST-VSF took 30 measurements for each water sample, and the median values were used for the rest of analysis. If we applied VSF-inversion to estimate the PSD from each of 30 measured VSFs, the coefficient of variation is 0.74, meaning the random variation in particle population would translate to approximately 75% uncertainty in derived PSDs. We assumed the same uncertainty for LISST-100X and LISST+MVSM derived PSDs.

For particles of non-spherical shapes, including most oceanic particles, the size and resulting PSD depend on the metric used to represent size. In this study, the equivalent spherical diameter (ESD) is used as the metric of size for all instruments. ESD can be obtained based on different measurements of the particle, and its value will depend on which measurement is used. For example, an ESD based on the particle volume will be different from the ESD based on its cross-sectional area for all but spherical particles. The definition of particle size for each instrument/method is listed in Table 1. Because volume is generally considered the least controversial geometric parameter of irregularly shaped, solid particles (Bader, 1970) and can be linked directly to mass, we converted all the sizes to the volume ESD, i.e.,

where c is the conversion factor and the subscript x represents the size metric, such as cross-sectional area, surface area, diffusion, settling, etc. The value of the conversion factor depends on the shape of the particles. Two particle shapes are tested herein: cube and spheroid, resembling roughly mineral (with edges) and biogenic (without edges) particles. For spheroids, we used the conversion developed by Jennings and Parslow (1988) (Figure 2a). The exact conversion factor varies with the shape of the spheroids (oblate versus prolate) and the aspect ratio of the spheroids. While the error of using an incorrect correction factor increases with decreasing aspect ratio, for aspect ratio >0.8, all the conversion factors can be assumed to be unity. Errors are much smaller between prolate and oblate spheroids. The aspect ratios estimated from the IFCB images by fitting an ellipse to each particle showed a bio-modal distribution with a global mean approximately 0.6 (Figure 2b). Using Feret diameters showed a very similar result (not shown). Conversion of one size to another also requires scaling the PSD to ensure the total number of particles is conserved (e.g., Jonasz, 1987):

where P represents the size distribution and dd_x/dd_volume denotes the derivative of d_x with respect to d_volume.

In comparing PSDs obtained by a Coulter Counter and an imaging device, Jackson et al. (1997) used conserved-volume ESD (d_c) to account for the porous nature of marine aggregates (Suzuki and Kato, 1953; Riley, 1963). Conserved volume is the volume of the solid constituents of an aggregate and is proportional to the solid mass when the constituents are of constant density. For marine aggregates, d_c is calculated as (e.g., Jackson et al., 1997):

where d₀ is the diameter of monomers that make up the aggregates and D is the fractal dimension. For a solid particle with no porosity, D = 3 and d_c = d_volume. Also, for particles of sizes smaller than d₀, d_c = d_volume. Because the electrical resistance of a particle is approximately proportional to its solid volume that excludes any water between the solid parts of aggregates, the sizes of particles measured by the Coulter Counter correspond to d_c (e.g., Li and Logan, 1995; Stemmann et al., 2008); i.e., d_c = d_coulter. On the other hand, Jackson et al. (1995) argued that d_coulter is similar to but not exactly the same as d_c and derived $dc=2−13dCoulter$ based on the observation of Archie (1942) relating particle electrical resistivity to that of the surrounding fluid when the particle has a porosity. Note that this derivation implies that the dry matter volume (∝d_c³) of the aggregate measured by a Coulter Counter is always half of its apparent volume (∝d_coulter³). In this study, we assumed d_c = d_coulter.

In contrast to the Coulter Counter, the size of an aggregate measured by the IFCB and UVP corresponds to its outer diameter that encloses both the solid parts and the pores between them.

The optical properties, including the angular scattering, of aggregates vary with the characteristics of the aggregates and differ from the particles that comprise them and from a solid particle of the same size (Latimer, 1985; Boss et al., 2009). Therefore, the sizes of PSDs derived based on the relationship between the light-scattering property and particle sizes also correspond to the overall size of the aggregates instead of the size of solid mass (Berry and Percival, 1986). For comparison with Coulter Counter data, different sizes need to be converted to d_c using Equation 3, which in turn needs values for d₀ and D. The fractal nature of particles, which varies with the characteristics of particle aggregates and dominant processes governing the formation of the aggregates (Jackson, 1990), affects light scattering (Sorensen, 2001) as:

where q = (4π/λ)sin(θ/2) is called the scattering wave vector, θ is the scattering angle, and λ is the wavelength of light in the medium. The inverse of q represents the length scale, or the probe length, of the scattering experiment. Scattering by particles of different sizes relative to q⁻¹ scale differently with respect to q⁻¹, and the scaling factor is related to the fractal dimension of the particles (Sorensen, 2001). Risoviá and Martinis (1996) applied Equation 4 to the VSFs measured by Petzold (1972) and found two fractal dimensions. The first was estimated for q values <2.5 µm⁻¹, representing relatively large particles due to cluster-cluster aggregation, and the second for 2.5 µm⁻¹ < q < 20 µm⁻¹, representing relatively small particles due to particle-cluster aggregation. We followed the Risoviá and Martinis (1996) approach and estimated the fractal dimensions using both LISST-VSF and LISST+MVSM data (Figure 3). The fractal dimensions estimated from LISST-VSF and LISST+MVSM are consistent with each other. For relatively large particles, D varied from 1 to 2 with a mean value of 1.5, similar to the range measured for marine snow (Alldredge and Gotschalk, 1988; Logan and Wilkinson, 1990). For relatively small particles, D varied from 2.8 to 3.5 with a mean value of 3.3. As D cannot be greater than 3, obtaining values above 3 means that the small particles we measured during the experiment were not very porous. This finding is also an indication of uncertainty associated with either the measurements or the Risoviá and Martinis (1996) approach or both. Unfortunately, while this approach can be used to estimate D in Equation 3, it cannot find the value for d₀.

Unlike the conversion of PSDs between the ESD metrics, accounting for particle porosity alters the shape of the PSD. To illustrate this change, assume that a PSD of oceanic particles can be represented by a simple power law (Bader, 1970), i.e.,

where a is a scaling factor representing the number concentration of particles per unit size interval at d_volume = 1 µm and b is often called the Junge slope. The conversion of the PSD from d_volume to d_c is:

The slope of the PSD versus d_c is 3(1+b)/D − 1, which is always less than or equal to b as long as b < −1, which typically applies to oceanic particles. For example, if D = 3 (i.e., no porosity), the slope value does not change; if D = 2 and b = −3, the new slope would be −4. In other words, a particle size distribution would become steeper if plotted as a function of conserved volume ESD by accounting for the porosity of particles.

The sizes of particles investigated in this study span approximately 5 orders of magnitude, and the PSD amplitudes span approximately 20 orders of magnitude (Figure 1). For comparison of such datasets, Seegers et al. (2018) suggested the use of median difference (MD) and median absolute difference (MAD), where

and

MD and MAD provide measures of bias and precision between two PSD measurements P₁ and P₂ (Seegers et al., 2018; McKinna et al., 2021). For example, MD = 1.20 or 0.80 and MAD = 1.5 between P₁ and P₂ would indicate that P₁ is on average 20% greater and smaller, respectively, than P₂ (a bias) and the precision between P₁ and P₂ is 50%. However, as defined, MAD contains overlapping information with MD, and sometimes the values of MD and MAD can be the same. For example, if P₁ is always greater than P₂ by a factor of 2, then both MD and MAD would be equal to 2, with MAD providing no additional information. To overcome this issue, we subtracted MD from the calculation of MAD and refer to the modified parameter as de-biased MAD (DMAD), i.e.,

DMAD would be equal to 1 for the previous example, indicating that after the bias is removed, P₁ and P₂ would match perfectly. DMAD measures the spread of differences between two datasets, except that the difference is calculated first in logarithmic space before transforming back to linear space. DMAD also suits our study better because we are interested in pair-wised comparisons of PSDs that were measured using different methods without assuming a particular one as the truth. Note that MD can be less than 1 but DMAD (or MAD) is always greater than 1. In addition to MD (Equation 7) and DMAD (Equation 9), we also used two more quantitative measures for evaluating comparisons: Pearson correlation coefficient (r) and type II linear regression slope (m), both computed on the logarithmic values of the two PSDs being compared.

All measurements were made with water samples from the same Niskin bottles except for the UVP which samples in situ. Therefore, two PSDs from two different instruments sampling the same particle population are expected to be highly correlated. If the r value is low (<0.5), the comparison would be considered poor regardless of what the other evaluation metrics indicate. If r > 0.5, we examine the m value, which links the two PSDs as P₁ ∼ P₂ ^m. If m deviates significantly from 1, it indicates that the two PSDs exhibit a nonlinear relationship. If r > 0.5 and m = 0.8–1.2, then the two PSDs can be considered linearly correlated, and MD and DMAD values provide additional details on how these two PSDs compare to each other.

Among the seven instruments used to derive PSDs, three are light-scattering devices. For better understanding of how PSDs obtained with different instruments compare with each other, we first evaluated comparison among the VSFs measured by the three light-scattering devices. The LISST-100X operates at 532 nm (β_LISST-100X), the LISST-VSF operates at 517 nm (β_LISST-VSF), and the MVSM operates at 8 wavelengths (β_MVSM) that include 532 nm but not 517 nm. Here, we interpolated β_MVSM to 517 nm and compared the interpolated results with β_LISST-100X and β_LISST-VSF (Figure 4). Because water was clear during the experiment, the VSFs due to particles at angles >90° were on par or less than the scattering by pure seawater (Figure 4a).

The VSFs at scattering angles <15° were measured by three instruments (LISST-100X, LISST-VSF, and MVSM). The overall comparison between β_LISST-100X and β_LISST-VSF (Figure 4b) indicates that the LISST-100X and LISST-VSF are consistent with each other in measuring the VSFs at angles <15°, with a Pearson correlation coefficient r = 0.96, type-II regression slope m = 1.08, a bias of 6%, and an absolute difference (100% *(DMAD–1)) of 32%. The LISST-VSF and LISST-100X data compared in Figure 4b differ slightly in wavelength (517 versus 532 nm). The measurements of ACS showed no significant spectral variation for the total particulate scattering coefficient (not shown); therefore, we do not expect this slight spectral difference to cause systematic bias between the two datasets. There is contamination by stray light from the lamp and optical surfaces affecting the measurements by MVSM at angles <10° (Zhang et al., 2012), which is manifested by β_MVSM departing from β_LISST-100X and β_LISST-VSF increasingly as scattering angle decreases (red line versus the blue and orange lines on the left side of Figure 1a). For this reason, we discarded MVSM data at angles <13° and replaced them with LISST-100X data to form a combined dataset LISST+MVSM at wavelength 532 nm (see Section 2.1.2).

The VSFs at scattering angles >15° were measured by two instruments (LISST-VSF and MVSM). At scattering angles near and greater than 90°, β_MVSM appeared noisy (Figure 4a). We sometimes noticed negative values in β_MVSM at these angles after the water contribution was removed. While these negative values were discarded, their appearance suggests the signal level at these angles approaches the sensitivity level of the MVSM. Also, β_MVSM shown in Figure 4 was interpolated from the multispectral data by the MVSM for which only two measurements were averaged for each sample (see Section 2.1.3), whereas the LISST-VSF data represented the median of 30 repeated measurements. Therefore, β_MVSM was more susceptible to random noise than β_LISST-VSF. β_LISST-VSF exhibits a sudden increase at scattering angles from 145° to 155°. These values were an artifact caused by internal reflection and were discarded. The overall comparison (Figure 4c) indicates that the particulate VSFs derived from the two instruments corresponded with each other very well (r = 0.99, m = 1.0), but β_MVSM is on average 30% smaller than β_LISST-VSF. Because our testing with microbeads of standard sizes also showed that the MVSM measurements at low bead concentrations tend to be lower than the predicted values (results not shown), we believe this systematic difference is caused mainly by the low signal-to-noise ratio of the MVSM in this environment, which had very low particle concentrations. Ignoring this bias, the two instruments would measure VSFs that agree with each other with an absolute difference of only 11%.

In the following, we examine pairwise comparison of PSDs measured by the instruments with detection size ranges that overlap. As detailed in Section 2.2.3, we used volume ESD (d_volume) as the common size parameter for all the comparisons. To convert to d_volume, we assumed particles are of spheroidal shape, except for IFCB data where shape is determined from the images and considered explicitly to compute d_volume according to Moberg and Sosik (2012). Analysis of particle images acquired by the IFCB showed a mean aspect ratio of approximately 0.64 (Figure 2b). Using 0.6 as the average aspect ratio for spheroids to represent oceanic particles, the conversion factor c (Equation 1) is c_surface-area ≈ c_cs-area ≈ c_diffusion ≈ 0.98 (Figure 2a). The term c_surface-area was used to convert the PSDs measured by LISST-VSF and LISST+MVSM, c_cs-area for LISST-100X and UVP, and c_diffusion for Viewsizer. For comparison with the Coulter Counter, we made additional comparisons with conserved volume ESD (d_c) as the common size parameter (Equation 3). The comparison of PSDs using spheroids with other aspect ratios and using cubes to calculate d_volume will be presented later. The statistical measures for evaluating the comparisons are listed in Tables 2 and 3. Different instruments have different size bins. The size bins of the instrument to which the other instruments are compared were chosen as the common grid, and PSDs from the other instruments were interpolated to this common grid for comparison.

LISST-VSF and LISST+MVSM measured PSDs from 0.2 µm to 2000 µm (Figure 5). Even though d_volume was used as the common size parameter, converting to d_volume barely affected the comparison because both instruments use the same VSF-inversion technique with the same definition of size. PSDs derived from LISST-VSF and LISST+MVSM had r = 0.97 and m = 0.99. The LISST+MVSM PSDs were on average 46% higher than the LISST-VSF PSDs (MD = 1.46). Correcting this mean bias, the PSDs by the two instruments would agree with each other with an absolute difference within a factor of 2.5 (DMAD = 2.33). In Section 2.2.3, we estimated that the uncertainty in PSDs derived by LISST-VSF due to random sampling was approximately 75%. Clearly, the DMAD value for PSDs between LISST-VSF and LISST+MVSM is greater than this uncertainty, suggesting that the difference between the two instruments caused additional uncertainty. Despite this uncertainty, the overall agreement between the PSDs derived by LISST-VSF and LISST+MVSM is reasonably good, considering that the comparison was evaluated over a size range spanning 4 orders of magnitude over which the PSD changed by nearly 15 orders of magnitude (Figure 5). This reasonable agreement is also not surprising because the same VSF-inversion algorithm was applied to both LISST-VSF and LISST+MVSM, and VSF data were found to be very consistent with each other (Figure 4).

For natural oceanic particle assemblages, submicrometer particles generally contribute significantly to the backward scattering but negligibly to the near forward scattering (Stramski and Kiefer, 1991). In the study area, Zhang et al. (2020) found that approximately 20–30% of the particulate backscattering was due to particles <0.2 μm. In other words, the scattering signal of these small particles — and therefore the information to infer their size — was contained mainly in the backward angles of the VSFs. The water was relatively clear and its backward scattering signal, while still well detected by the LISST-VSF, reached the detection limit of the MVSM (see the noisiness of MVSM data at backward angles in Figure 4a). Therefore, few LISST+MVSM data could infer any particles of sizes <0.2 µm (also see Figure 1e). As a result, even though the VSF-inversion algorithm retrieves particles over the size range 0.02 µm to 2000 µm, we can only compare PSDs by LISST-VSF and LISST+MVSM over the sizes >0.2 μm.

ViewSizer, LISST-VSF and LISST+MVSM estimate PSDs that overlap from 0.3 µm to 2 µm (Figure 6). PSDs by both LISST-VSF and ViewSizer have reasonably good linear correlation (r = 0.71 and m = 1.19). The PSDs by LISST+MVSM and ViewSizer have relatively poor linear correlation (r = 0.56 and m = 1.41), likely due to the noisiness of MVSM data at backward angles (e.g., see red line in Figure 4a) that affects the retrieval of submicrometer particles. There is greater uncertainty in retrieving particles from scattering data at this submicrometer size range, which is in the optical transition domain (van de Hulst, 1981) where the sizes of particles are of similar length as the wavelengths of visible light. For relatively smaller particles, the relative shape of angular scattering or the phase function only varies with the size and does not depend on the refractive index or the shape of particles; for relatively larger particles, the phase function only varies with the refractive index and the shape of particles and not with sizes (Zhang et al., 2011). At this transition domain, however, all these variables affect the phase function, making VSF-inversion in this size range more challenging. This challenge is also reflected in Figure 1d and e, where PSDs derived from LISST-VSF and LISST+MVSM in this size range exhibited greater variability than those at other size ranges. The PSDs by the LISST-VSF and LISST+MVSM were larger than the ViewSizer by a factor of approximately 2.5 (MD = 2.62) and 4 (MD = 4.11), respectively. These differences between ViewSizer and LISST-VSF or LISST+MVSM are greater than the sampling uncertainties for ViewSizer (38%) and LISST-VSF or LISST+MVSM (70%) (Section 2.2.1).

At the Coulter Counter’s size range of 2 µm to 40 μm, there were four additional estimates of PSDs by IFCB, LISST-100X, LISST-VSF, and LISST+MVSM. We made two comparisons using d_volume and d_c, respectively, as the common size parameter (Figures 7 and 8). Because a Coulter Counter responds to the solid fraction of particles, i.e., d_volume = d_c for Coulter, in both comparisons, the PSDs from Coulter Counter remained unchanged, whereas the PSDs derived from IFCB and the optical instruments were scaled accordingly. Within this Coulter Counter size range, where PSDs varied approximately 6 orders of magnitude from 10⁴ to 10¹⁰ m⁻³ μm⁻¹, IFCB and all the optical inversion results correlated strongly (r > 0.93) with the Coulter estimate regardless of which size parameter was used. With d_volume as the common size parameter (Figure 7), IFCB, LISST-100X, and LISST+MVSM correlated linearly with Coulter Counter with m = 0.91, 1.06 and 1.14, respectively, whereas LISST-VSF showed a nonlinear relationship with Coulter Counter with m = 1.32, which is caused by overestimation by LISST-VSF at smaller sizes (with greater particle concentration). The three optical estimates had biases against the Coulter Counter ranging from an approximate 30% underestimation (MD = 0.66) by LISST-100X to 70% overestimation (MD = 1.76) by LISST+MVSM, with LISST-VSF having an approximate 10% underestimation (MD = 0.92). IFCB overestimated Coulter by approximately 30% (MD = 1.31). After correcting these biases, IFCB, LISST-100X and LISST+MVSM would agree within approximately 50% with Coulter, which is only moderately greater than the average sampling uncertainty of 24% that we estimated for the Coulter Counter. On the other hand, the difference between LISST-VSF and Coulter was approximately a factor of 2 (DMAD = 2.2).

Using d_c as the common size parameter (Figure 8) altered the PSDs measured by IFCB and the optical instrument such that their comparisons with the Coulter Counter deteriorated with increasing m and DMAD values. Judging from the changes in m and DMAD values, the PSDs by the IFCB and the optical instruments seem to align better with the Coulter Counter when using d_volume rather than d_c as the common size parameter. In converting d_volume to d_c using Equation 3, we assumed d₀ = 5 µm, which was used in Stemmann et al. (2008) for the South Pacific Gyre, and used the fractal dimensions estimated in Figure 3b for each measurement. Using different d₀ value in the range of 1 µm to 20 µm would change the numerical values of m, MD and DMAD, but with all indicating a degraded comparison in using d_c instead of d_volume as a common size parameter. Because Coulter Counter responds to the solid fraction of particles, this degradation in comparison from using d_volume to d_c could suggest that the particles in the size range of 2 µm to 40 µm were not porous. This absence of porosity is also consistent with the IFCB images showing that particles in this size range were typically dominated by living cells.

Reynolds et al. (2010) measured PSDs using an LISST-100X and a Coulter Counter in the coastal waters of California and found that the volumes of particles between the two estimates had a correlation coefficient > 0.9 and median difference of 13.5% without accounting for particle porosity. The volume concentrations estimated from our LISST-100X and Coulter Counter measurements had r = 0.93 and median difference of 29%, which are similar to their results.

At the IFCB size range from 5 µm to 100 μm, there were three additional estimates of PSDs by LISST-100X, LISST-VSF, and LISST+MVSM (Figure 9). All three optical estimates correlated very well with IFCB data (r = 0.96). However, only LISST-100X and LISST+MVSM showed a linear correlation with IFCB (m is within 0.8–1.20), with a debiased difference of approximately 50% (DBMD = 1.56 and 1.58, respectively), which is only moderately greater than the estimated 32% sampling uncertainty for IFCB. On average, LISST-100X underestimated by 40% (MD = 0.58) while LISST+MVSM overestimated by 70% (MD = 1.70) the PSDs relative to the IFCB estimates. Even though having an overall low bias against IFCB (MD = 1.09), LISST-VSF data overestimated IFCB at the lower size range, resulting in a skewed correlation with IFCB (m = 1.42) and a factor of approximately 3 (DBMB = 2.79) debiased difference.

At the LISST-100X size range of 3 µm to 180 µm, there were two additional estimates of PSDs by LISST-VSF and LISST-MVSM (Figure 10). Both LISST-VSF and LISST+MVSM estimates correlated extremely well and linearly with the LISST-100X data (r > 0.96, m within 1.0 ± 10%); however, both overestimated the LISST-100X data by a factor of approximately 2 (MD = 1.99) and 2.5 (MD = 2.79), respectively. Ignoring the biases, LISST+MVSM (DMAD = 1.6) agreed better with LISST-100X than LISST-VSF (DMAD = 2.3). The approximately 60% agreement between LISST+MVSM and LISST-100X after removing the bias is within the 75% uncertainty due to random sampling. This agreement is probably because LISST+MVSM data included the measurements by LISST-100X and hence the LISST-100X data were used in both inversions and not completely independent. This agreement also attests to the efficacy of the inversion algorithms used in LISST-100X and LISST+MVSM, respectively, in retrieving particle size distribution from the VSF data.

At the UVP size range of 100–2000 µm, there were three additional estimates of PSDs that cover parts of this range: LISST-100X (up to 200 µm), LISST-VSF (up to 2000 µm) and LISST+MVSM (up to 2000 µm). As mentioned in Section 2.2.5, we expect the porosity of particles to affect the measurements by both UVP and the light-scattering devices. However, the comparison results would be similar using either d_volume or d_c because UVP and light-scattering PSDs would be subjected to the same conversion from d_volume to d_c. We chose to use d_volume as the common size parameter because it avoids the additional conversion. The comparisons of LISST-VSF and LISST+MVSM with UVP were reasonably good (Figure 11), both with r ≈ 0.7 and m within 1.0 ± 10%. However, both overestimated UVP measurements by a factor of approximately 8. On the other hand, LISST-100X and UVP did not correlate with each other (r = 0.04), with the former overestimating the latter by a factor of approximately 16 (MD = 16.16). The poor comparison between LISST-100X and UVP was mainly because the two size ranges barely overlap, resulting in a small sample number for evaluation (N = 76). All three optical estimates of PSDs were much higher than the UVP estimates. Ignoring the biases, the three optical inversion estimates agree with UVP data within a factor of approximately 3, all of which are significantly greater than the UVP’s 17% average sampling uncertainty.

All PSDs compared in this study, regardless of the techniques used to derive them, are reported with reference to an ESD. Because ESDs can be defined differently, we converted different ESDs to the volume ESD. The exact conversion, however, depends on the shape of particles. The results shown above were based on solid spheroids with an aspect ratio of 0.6, which represents the global mean value of aspect ratios derived from the IFCB images (Figure 2b). As the IFCB images showed a bi-modal distribution in the aspect ratios of particles in the study area, we also tested spheroids with two different aspect ratios, 0.3 and 1.0. We recognize that these two values are close to but not the same as the two modes in Figure 2b, but their range covers >93% of aspect ratios from the IFCB images. We also emphasize that these aspect ratios were estimated for particles in the 5–100 µm IFCB size range and the distribution may be different for different size ranges. For this study, we have assumed one aspect ratio for the entire size range. We also tested using a cube for particle shape, which can serve as an end member for particles with edges. An aspect ratio = 1 is equivalent to representing particles using solid spheres, with the corresponding conversion factor in Equation 1 being 1. For aspect ratio = 0.3, the conversion factors (Equation 1) are c_surface-area ≈ c_cs-area ≈ 0.89 and c_diffusion ≈ 0.88; for cubes, c_surface-area ≈ 0.90 and c_cs-area ≈ 1.10. We do not know c_diffusion for cubes but assume c_diffusion = 1.

Using different particle shapes to estimate d_volume is a linear process (Equation 2) that does not change the shape of PSDs; therefore, it only affects the MD values of the comparison (i.e., relative difference). Because the conversion factors from surface-area ESD, cross-sectional area ESD, or diffusion ESD to volume ESD respond in roughly the same way for different aspect ratios (Figure 2), using a different aspect ratio will only affect the comparison of the light-scattering instruments (need to convert to d_volume) against either Coulter Counter or IFCB, where size is already in d_volume and hence no conversion is needed. Table 3 lists the comparison of three light-scattering-based PSDs against Coulter Counter and IFCB assuming particles of spheroidal shape with three different aspect ratios. We cannot generalize which aspect ratio produces the best comparison among all the instruments listed in Table 3. For example, for LISST-VSF, the best comparison (judging by MD values) was obtained with aspect ratio = 1.0 (MD = 0.99) against Coulter Counter but with aspect ratio = 0.5 (MD = 1.09) against IFCB. For LISST+MVSM, however, the best comparison was obtained with aspect ratio = 0.3 against both Coulter Counter and IFCB.

Table 2 lists the comparison results of using spheroids with aspect ratio = 0.6 and using cubes at all size ranges. With two exceptions (LISST-VSF versus Coulter and LISST-VSF versus IFCB), using cubes to represent particle shape generally produced better comparison between different instruments (as measured by closeness of MD values to unity) than using spheroids with aspect ratio = 0.6. With spheroids representing particle shape, the PSDs by LISST-VSF and LISST+MVSM tend to overestimate the PSDs by the other instruments (except LISST-VSF versus Coulter Counter). With cubes, these overestimations were reduced, leading to an overall better comparison. Also, using cubes instead of spheroids to represent particle shape alleviated the underestimation of the PSD by LISST-100X as compared to Coulter Counter and IFCB.

The concentration of oceanic particles varies strongly with particle size. As a result, the bias of the PSD comparison is very sensitive to the uncertainty in the size. To illustrate this sensitivity, assume two different instruments measuring PSDs of the same water sample, P₁(d₁) and P₂(d₂), and that these two instruments work perfectly, i.e., if d₁ = d₂, then P₁(d₁) = P₂(d₂). Further assume that both PSDs follow the power law (Equation 5), with b = −4. If there is an uncertainty in estimating the size, then the bias between the two PSDs, P₁(d₁)/P₂(d₂) = (d₁/d₂)⁻⁴. Therefore, a 20% over- or under-estimation in size, i.e., d₁/d₂ = 1.2 or 0.8, would translate to a factor of 2 underestimation or a factor of 2.5 overestimation in bias. Even though we converted different sizes to a common size parameter for comparing different PSDs, the uncertainty remains because the exact conversion depends on the shapes of the particles, which we do not know, as well as on the original definition of the sizes, which differs among different instruments.

How size is defined for each instrument or technique is listed in Table 1. The inversion technique used in LISST+MVSM and LISST-VSF assumes a hexahedral shape for particles (Zhang et al., 2012) and that the size derived is surface-area equivalent spherical diameter (Bi et al., 2010). The inversion technique used by LISST-100X assumes particles are spherical and that the size derived is the cross-sectional area equivalent diameter. Sizes derived from ViewSizer are the diameter of spheres that would undergo the same Brownian motion; Coulter Counter uses volume equivalent diameter; IFCB estimates volume equivalent diameter from 2-D images using a distance map algorithm; and UVP converts the area of a particle to equivalent circle diameter. Relating these sizes or knowing which PSD represents the true size distribution is difficult because oceanic particles are seldom spherical and are very diverse in shape. All the comparisons (except LISST-VSF versus Coulter Counter) indicated that the LISST+MVSM and LISST-VSF overestimated the PSDs by the other instruments, with degree of overestimation varying for different instruments and for assumptions of particle shape (Table 2). Using cubes to estimate the conversion factor was found to produce an overall better comparison of PSDs by LISST-VSF and LISST+MVSM with the other instruments, probably because both a cube and a hexahedron have six sides with sharp edges. Of interest is that using cubes also resulted in better comparison between LISST-100X and Coulter Counter than using spheroids (Table 2). One possible explanation for this result is that some of the particles in the phytoplankton size range are diatoms that can be elongated with sharper edges than other cells or detrital particles.

The biases of LISST-VSF and LISST+MVSM against UVP were significantly higher (on the order of a factor of 8 for spheroidal shape and a factor of 6 for cubes) than the other comparisons (less than a factor of 3). UVP was deployed in situ, while the other instruments measured the collected water samples in the lab. One possible source of the bias was the breakup of large particles during either collection or measurement of the water samples, resulting in elevated concentration of smaller particles measured by LISST-VSF and LISST+MVSM. Another source could be due to possible underestimation by some imaging-based approaches at the smaller end of the size range or identification biases where only part of the particle is identified during processing. We learned from analyzing the ViewSizer data that particles of sizes less than 0.3 µm in the natural samples were underestimated even though in the lab the instrument can accurately count those small particles. This discrepancy is because the ViewSizer uses the images of particles from the scattered light to identify and track particles but the presence of larger particles in the natural environment elevates the background light field in a way that overwhelms the scattering by the smaller particles, making them undistinguishable in the images (Xiong et al., 2022). UVP differs from ViewSizer in the operating principle as well as the size range of particles to be counted, but both utilize scattered light to form images of particles, suggesting that the issue we observed with ViewSizer may also apply to UVP. This elevated-background effect, however, does not affect IFCB data, in which each particle is imaged individually.

The bias between LISST-VSF and LISST+MVSM (Figure 5) is less likely due to the size definition because the two instruments use the same algorithm to derive the PSD. That PSDs derived from LISST+MVSM overestimated those from LISST-VSF is counter-intuitive, given that VSF values measured by LISST+MVSM were 30% less than VSF values by LISST-VSF at angles >15° (Figure 3c). LISST-VSF and LISST+MVSM differ in three areas (Figure 3a). First, the two instruments have different angular range and angular resolution (up to 145° at 1° resolution for LISST-VSF versus up to 179° at 0.25° resolution for LISST+MVSM). As LISST+MVSM has greater angular range and finer angular resolution, we sub-sampled LISST+MVSM data to match the LISST-VSF angular setup and ran the VSF-inversion on both original and sub-sampled LISST+MVSM data. The PSDs derived from the sub-sampled LISST+MVSM data were on average very similar to the PSDs from the original LISST+MVSM data (MD = 0.96), even though there were subtle differences between the two (DMAD = 1.6). Therefore, while instrumental difference in angular range and resolution does affect the details of PSDs derived from VSF-inversion as expected, it is unlikely to cause systematic bias as observed in the PSDs between LISST-VSF and LISST+MVSM. Second, the LISST+MVSM data were noisy at angles >90° (Figure 3a). We smoothed the LISST+MVSM data at angles >90° using a median pass with a 2° window (9 data points). The PSDs derived from the smoothed LISST+MVSM data were almost identical to the PSDs from the original LISST+MVSM data (MD = 0.99 and DMAD = 1.03). Therefore, the noisiness of LISST+MVSM data in this study does not cause the systematic bias between the PSDs derived from the two instruments. Third, the MVSM had low signal-to-noise ratios at angles >90°, which was reflected in noisiness of the LISST+MVSM data; in contrast, the LISST-VSF has sufficient sensitivity to resolve the backscattering in this environment (Zhang et al., 2020). As discussed in Sections 3.2.1 and 3.2.2, the signal from submicrometer particles, particularly those of sizes <0.2 µm, is relatively strong at angles >90°. The low signal-to-noise ratio at backward angles degraded this signal from submicrometer particles and hence degraded their retrieval from LISST+MVSM. On the other hand, the signal from submicrometer particles was still present at angles <90° where MVSM has sufficient signal-to-noise ratio to detect them, though signals were relatively weak. This signal, if not ascribed to submicrometer particles, would be imputed to particles of greater sizes and in turn lead to overestimation of larger particles by LISST+MVSM versus LISST-VSF.

If systematic biases are ignored or corrected, all the instruments would agree with each other within a factor of 3, as measured by the DMAD values. The best agreements are found among LISST+MVSM, LISST-100X, IFCB, and Coulter Counter, which would agree with each other within approximately 50% if d_volume is used as the common size parameter (Table 2). This level of agreement is on par with the average uncertainties associated with random sampling (Table 1), i.e., 75% for optical inversion, 24% for Coulter, and 32% for IFCB, which suggests that the PSDs by these instruments are consistent with each other if the systemic biases are removed.

PSDs of oceanic particles are often approximated by a power-law distribution (Equation 5). The key parameter in this idealized representation is the Junge slope, i.e., the parameter b in Equation 5. We estimated b values from the PSDs measured by the seven instruments (Figure 1) over their respective size ranges with d_volume as the common size parameter (Figure 12a). We used cubes to compute d_volume here, but the same distribution of the Junge slope would be obtained using the other particle shapes because conversion from other ESDs to d_volume (Equation 1) is a linear process and does not change PSD shape. The slope values estimated from PSDs by different instruments over their respective size ranges differed, indicating deviation of actual PSDs from the idealized power law function as reported in previous studies (Reynolds et al., 2010; Reynolds and Stramski, 2021). The PSD slopes were all consistently confined within a range from −4.5 to −3.5 except for UVP data. The mean slope values were approximately −3.8 for Coulter Counter, LISST-VSF, LISST+MVSM, IFCB, and ViewSizer, with a slightly steeper value of −4.2 for LISST-100X. However, the mean slope value for UVP is −2.6, significantly different from the other instruments.

The UVP PSD data differ in terms of both shape (Figure 12a) and magnitude (Figure 8) when comparing to the PSDs obtained by other instruments. Unlike the smaller particle size assessments, we do not have independent measurements to corroborate the validity of the UVP PSD determinations. To reconcile the difference shown in Figure 12a between UVP and the other instruments, we recalculated the slopes after converting d_volume to d_c using Equation 3 with d₀ = 60 µm and D values taken from Figure 3b. Using d_c as a common size parameter, the slopes of UVP data became consistent with the other instruments (Figure 12b), with a mean UVP slope = −4. The slopes of PSDs derived from LISST-VSF and LISST+MVSM were also affected because their size distributions extend significantly beyond 60 μm. Comparison with the Coulter Counter (Figure 7), which measures d_c directly, was better when not considering porosity, suggesting that the particles in the Coulter Counter size range of 2–40 µm were probably not porous, consistent with this size range being dominated by living cells as revealed in the IFCB images. Therefore, here we chose d₀ = 60 μm. Other values for d₀ were considered, such as 100 μm, which produced an average UVP slope = −3.8 but still improved consistency with the other instruments (results not shown).

As illustrated in Equation 6, using conserved volume size, d_c, by accounting for porosity of particles alters the shape of a PSD. This effect bears an important implication when comparing or reporting particle distributions measured by light-scattering or imaging techniques, as both are affected by the porosity of particles. Previously reported global PSDs that showed an average slope value of −4 were based on the Coulter Counter measurements (Sheldon et al., 1972; Jonaszz, 1983) and hence corresponded to the conserved volume.

There were 87 of a total of 315 water samples that were measured by all seven instruments, with depths of these samples restricted to the upper 75 m. We scaled the PSDs estimated by each instrument to d_c with d₀ = 60 µm and merged the scaled PSDs over the size range from 0.02 µm to 2000 µm (Figure 13a). At sizes with multiple estimates of the PSD by different instruments, we took the geometric mean of those estimates. To the best of our knowledge, this work provides the first reconstruction of the PSDs of oceanic particles from multiple instruments over 5 orders of magnitude from nanometer to millimeter. The reconstructed PSDs exhibited a rather consistent shape, all having Junge slope values between −3.7 and −4.3, with no discernible change over depth in the upper 75 m (Figure 13b).

For each merged PSD, we also calculated cumulative distributions of the number, cross-sectional area, and volume of particles as well as the depth distributions of total number, area, and volume of particles (Figure 14). The total number of particles within the size range of 0.02 µm to 2000 µm was predominately determined by particles of sizes <1 µm (Figure 14a), which is not surprising given the general shape of the PSDs for the oceanic particles that increases rapidly with decreasing size. The total cross-sectional area of particles is contributed mainly by particles of sizes <10 µm (Figure 14b). Despite their numerical abundance, submicrometer particles only account for <15% of total volume of particles (Figure 14c), whereas particles of sizes between 1 µm and 100 µm accounted for 70–90% of the total volume of particles.

The concentrations of number (Figure 14d), area (Figure 14e) and volume (Figure 14f) of particles all decreased with depth, even though the values at near-surface depths (<10 and 10–25 m) exhibited greater variability than at deeper depths. For the size range explored here (0.02–2000 µm), on average, there were approximately 10–20 trillion particles (10¹²) particles per m³ at depths <25 m, occupying an area of 1–1.2 m² and a volume of 0.8–1.0 mL. At depths between 50 m and 75 m, there were approximately 6 trillion particles per m³, occupying an area of approximately 0.5 m² and a volume of 0.6 mL. The areas and volumes mentioned above were computed from the PSDs based on the conserved volume and hence represent the solid matter of the particles. This general decrease of particle concentration with depth also imposed a detection limit on small particles. The merged PSDs show continual increase in the concentration of particles at sizes <0.2 µm in the upper 25 m (the blue and cyan lines in Figure 13 extend below 0.2 µm), with virtually no particles of sizes <0.2 µm detected at greater depths. We emphasize that particles of sizes <0.2 µm are expected to exist at depth but our instruments could not detect them in this study.

We measured particle size distributions using a suite of seven instruments that together cover the size spectrum from 20 nm to 20 mm. The techniques we used can be generalized into two categories: ensemble-based inversion from light-scattering measurements by LISST-100X, LISST-VSF, and LISST+MVSM, and individual particle-based inversion from ViewSizer, Coulter Counter, IFCB, and UVP. We applied a minimum particle concentration criterion to discard the ensemble-based estimates of the PSD that would predict <1 particle in the instrument-specific sampling volume. Different principles in sizing particles lead to different definitions of size, even though all are given in equivalent sphere diameter (ESD). For the instruments used in this study, these ESDs include volume ESD (Coulter Counter and IFCB), surface area ESD (LISST-VSF, LISST+MVSM), cross-sectional area ESD (LISST-100X and UVP), and diffusion ESD (ViewSizer). We converted different ESDs to volume ESD (d_volume) or a conserved volume ESD (d_c). The former requires knowledge of particle shape, and the latter further assumed those particles are porous. For the shape of particles, we tested spheroids with aspect ratios of 0.3, 0.6 and 1.0 (Figure 2) and cubes, and for porosity, we estimated the fractal dimensions using the measured VSFs (Figure 3). We converted the PSDs to the common size parameter, either d_volume (Equations 1 and 2) or d_c (Equations 3 and 6), and compared the converted PSDs between each pair of estimates across their overlapping size ranges, which include 0.2–2000 µm between LISST-VSF and LISST+MVSM (Figure 5), 0.3–2 µm among ViewSizer, LISST-VSF and LISST+MVSM (Figure 6), 2–40 µm among Coulter Counter, IFCB, LISST-100X, LISST-VSF and LISST+MVSM (Figures 7 and 8), 5–100 µm among IFCB, LISST-100X, LISST-VSF and LISST+MVSM (Figure 9), 3–180 µm among LISST-100X, LISST-VSF and LISST+MVSM (Figure 10), and 50–2000 µm among UVP, LISST-VSF and LISST+MVSM (Figure 11). The comparison was evaluated using four statistical parameters: Pearson correlation coefficient (r), Type-II linear regression slope (m), median difference (MD; Equation 7), and debiased median absolute difference (DMAD; Equation 9), all based on the logarithmic values of the PSDs. The pair of r and m and the pair of MD and DMAD evaluate the overall agreement in terms of the shape and the magnitude, respectively, of the PSDs. Here are the major findings and conclusions.

1. With d_volume as the common size parameter, very strong linear correlation (r > 0.9 and m within 1.0 ± 10%) was found among the PSDs by the optical inversion (LISST-100X, LISST-VSF and LISST+MVSM), Coulter Counter and IFCB. Good linear correlation (r > 0.7 and m within 1.0 ± 30%) was found between LISST-VSF and ViewSizer, and among LISST-VSF, LISST+MVSM, and UVP. Relatively poor correlation (r < 0.6 and m beyond 1.0 ± 40%) was found for comparison between ViewSizer and LISST+MVSM and between LISST-100X and UVP. The former was probably due to the low signal-to-noise ratio in the MVSM data at scattering angles >90°, where up to 60% of the signal was due to submicrometer particles (Zhang et al., 2020). The latter was caused by a small sample size.

2. There are systematic differences between PSDs measured by different instruments because different sizing techniques define size differently. Converting different sizes to a common size parameter, for which we used volume ESD, requires knowledge of particle shapes, which for oceanic particles are diverse and impossible to represent by a single shape. Despite this challenge, we found that using cubes to represent particle shape for scaling PSDs reduces the systematic biases between the ensemble-based optical estimates and the individual particle-based estimates as compared to using spheroids. The single particle methods we tested do not overlap in sizes except Coulter and IFCB, and therefore we cannot compare their results to test particle shape. Between Coulter and IFCB, using either spheroids or cubes produced almost identical comparison, but mainly because both methods derived d_volume, hence were not sensitive to particle shape.

3. If the systematic biases could be corrected, the differences of the PSDs, as measured by DMAD, would generally be within a factor of 2 among the optical inversions, between the optical inversions and the Coulter Counter, and between the optical inversion and the IFCB; and within a factor of 3 between the optical inversions and the ViewSizer and between the optical inversions and the UVP. These values can be considered as the precision of different instruments in quantifying natural oceanic particles.

4. When using d_c as the common size parameter, the comparison degraded between the optical inversion and the Coulter Counter and between IFCB and the Coulter Counter, indicating that the particles in the Coulter Counter size range (2–40 μm) were not porous. However, accounting for the porosity for particles in the UVP size range (100–2000 μm) would lead to slopes of PSDs estimated from the UVP that agree better with the slopes by the other instruments. This effect may indicate that large particles were porous, which is also consistent with the general understanding that the porosity of oceanic particles increases with the particle size (Logan and Wilkinson, 1990; Khelifa and Hill, 2006).

5. PSDs from these seven different instruments were merged as a function of d_c from 0.02 µm to 2000 μm, where a geometric mean was used when there were multiple estimates of PSDs at a particular size. The merged PSDs showed that the total concentration, area, and volume of particles decrease with depth, with no clear change in the Junge slope values. While submicrometer particles are numerically dominant, particles of sizes 1 µm to 100 µm account for 70–90% of the solid volume of particles.

Many properties of particles in the ocean, including sinking rate and carbon content, depend on the particle size distribution. The PSDs examined in this study and their merged products provide critical information for estimating late-summer carbon export through the biological pump and constraining its uncertainty near Ocean Station Papa in the Northeast Pacific Ocean.

The data used for this study have been submitted to NASA SeaBASS (http://seabass.gsfc.nasa.gov/cruise/EXPORTSNP).

We are gratefully appreciative of the captain and crew of the R/V Sally Ride and the scientists on boards. We thank two anonymous reviewers for their comments that have helped to improve the manuscript.

XZ, YH, DG, YX, GP: NASA 80NSSC17K0656 and 80NSSC20K0350. YH, LH, XZ: NASA 80NSSC18M0024. XZ: NSF 1917337. DS, AM: NASA 80NSSC17K0654. HMS, ETC, CR: NASA 80NSSC17K0700. HMS: the Woods Hole Oceanographic Institution’s Ocean Twilight Zone Project, funded as part of The Audacious Project housed at TED. HMS: the Simons Foundation 561126.

The authors have declared that no competing interests exist.

Contributed to conception and design: XZ, YH, DG, DS, and HMS.

Contributed to acquisition of data: All authors.

Contributed to analysis and interpretation of data: LH, YX, XZ, DG, YH, GP, HMS, AM, and DS.

Drafted and/or revised the article: XZ, YH, DG, HMS, DS, LH, YX, GP, ETC, AM, and CR.

Approved the submitted version for publication: All authors.

How to cite this article: Zhang, X, Huot, Y, Gray, D, Sosik, HM, Siegel, D, Hu, L, Xiong, Y, Crockford, ET, Potvin, G, McDonnell, A, Roesler, C. 2023. Particle size distribution at Ocean Station Papa from nanometers to millimeters constrained with intercomparison of seven methods. Elementa: Science of the Anthropocene 11(1). DOI: https://doi.org/10.1525/elementa.2022.00094

Domain Editor-in-Chief: Jody W. Deming, University of Washington, Seattle, WA, USA

Associate Editor: Laurenz Thomsen, Department of Marine Sciences, University of Gothenburg, Gothenburg, Sweden

Knowledge Domain: Ocean Science

Part of an Elementa Special Feature: Accomplishments from the Export Processes in the Ocean from Remote Sensing (EXPORTS) Field Campaign to the Northeast Pacific Ocean