## Abstract

Properly interpreting lidar (light detection and ranging) signal for
characterizing particle distribution relies on a key parameter, ${\chi _p}(\pi
)$, which relates the particulate volume
scattering function (VSF) at 180° (${\beta _p}(\pi
)$) that a lidar measures to the
particulate backscattering coefficient (${b_\textit{bp}}$). However, ${\chi _p}(\pi
)$ has been seldom studied due to
challenges in accurately measuring ${\beta _p}(\pi
)$ and ${b_\textit{bp}}$ concurrently in the field. In this
study, ${\chi _p}(\pi
)$, as well as its spectral dependence,
was re-examined using the VSFs measured *in
situ* at high angular resolution in a wide range of waters. ${\beta _p}(\pi
)$, while not measured directly, was
inferred using a physically sound, well-validated VSF-inversion
method. The effects of particle shape and internal structure on the
inversion were tested using three inversion kernels consisting of
phase functions computed for particles that are assumed as homogenous
sphere, homogenous asymmetric hexahedra, or coated sphere. The
reconstructed VSFs using any of the three kernels agreed well with the
measured VSFs with a mean percentage difference $ \lt {5}\% $ at scattering angles $ \lt {170}^\circ
$. At angles immediately near or equal
to 180°, the reconstructed ${\beta _p}(\pi
)$ depends strongly on the inversion
kernel. ${\chi _p}(\pi
)$ derived with the sphere kernels was
smaller than those derived with the hexahedra kernel but consistent
with ${\chi _p}(\pi
)$ estimated directly from
high-spectral-resolution lidar and *in
situ* backscattering sensor. The possible explanation was that
the sphere kernels are able to capture the backscattering enhancement
feature near 180° that has been observed for marine particles. ${\chi _p}(\pi
)$ derived using the coated sphere
kernel was generally lower than those derived with the homogenous
sphere kernel. Our result suggests that ${\chi _p}(\pi
)$ is sensitive to the shape and
internal structure of particles and significant error could be induced
if a fixed value of ${\chi _p}(\pi
)$ is to be used to interpret lidar
signal collected in different waters. On the other hand, ${\chi _p}(\pi
)$ showed little spectral
dependence.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The volume scattering function (VSF, $\beta $) is a fundamental inherent optical property describing the angular distribution of the scattered light from an incident beam [1,2]. Operationally, the VSF is defined as the intensity ($I$) of scattered light at scattering angle ($\theta $) divided by the incident irradiance ($E$) per unit scattering volume ($\Delta v$):

A lidar measures signal corresponding to the VSF at or near 180° ($\beta (\pi )$) and the lidar attenuation coefficient ($\kappa $), both of which are usually unknown [10,18]. A typical lidar—elastic backscatter lidar—only measures return signal at the transmitted wavelength of laser pulses, and therefore the inversion of $\beta (\pi )$ or ${\beta _p}(\pi )$ (as ${\beta _w}(\pi )$ is known) is an ill-posed problem. Overcoming this issue often involves (1) estimating $\kappa $ separately using the relative change of lidar signal with depth [19,20]; or (2) relating $\beta (\pi )$ and $\kappa $ to a common variable, such as chlorophyll concentration [18]. High-spectral-resolution lidar (HSRL) can measure return signals at two wavelengths, one at the transmitted wavelength and the other at a nearby wavelength due to inelastic Brillouin scattering [21]. With two measurements, HSRL can independently determine ${\beta _p}(\pi )$ and $\kappa $ [11,22–24]. However, either using elastic or HSRL lidar, ${\beta _p}(\pi )$ derived from the lidar measurements still needs to be converted to ${b_\textit{bp}}$, a key biogeochemical parameter [14,22,24]. Generally, the conversion is expressed as [25]

and*et al.*[22] measured ${\beta _p}(\pi )$ with a HSRL over Gulf of Maine, open ocean near Bermuda, and coastal waters from Virginia to Rhode Island, and ${b_\textit{bp}}$ with an ECO-BB3 sensor deployed concurrently in the waters. The estimated ${\chi _p}(\pi )$ showed significant variability and its mean value of 0.5 has also been applied [23,24]. While either value of 1.06 or 0.5 has been used, their difference in terms of both value and variability is significant and requires further validation. Commercial backscattering sensors such as Hydroscat (HOBI Labs) and ECO scattering sensors (Sea-Bird/WET Labs) usually measure VSF at one backward angle (e.g., $\theta ={120}^\circ $ or 140°), which is then scaled to ${b_\textit{bp}}$ assuming a constant conversion factor of ${\chi _p}({120}^\circ )$ or ${\chi _p}({140}^\circ )$ [35,36]. For either simple elastic backscatter lidar or sophisticated HSRL, a single-scattering condition was assumed in inverting ${\beta _p}(\pi )$ from the lidar equation [10,37,38]. Single-scattering assumption is often violated in turbid waters or as the beam is propagating further away.

The technical challenges for measuring VSF at backward angles near 180°
come from (1) the finite physical size of the light source and detector
[39]; (2) the difficulty in
completely blocking specular reflection of the incident beam from the
instrument’s optical components (e.g., mirrors, splitters, prisms,
windows) [40–42]; and (3) the
difficulty in accurately determining scattering volumes [40]. From its definition [Eq. (1)], precise knowledge of
scattering volume ($\Delta v$) is needed to estimate a VSF. Scattering
volume can be well defined by the laser-receiver geometry except for
scattering at angles near 180° [40,42]. A slight
misalignment of the optical assembly will dramatically alter the
illuminated scattering volume, and thus affect the estimate of the VSF, at
these angles. The reflected stray light, if not properly shielded, would
spuriously elevate the scattering at the reflection angles [41–43]. To our best knowledge, Beta Pi was
the first and only *in situ* instrument that
was specifically designed and built to measure the VSF within a narrow
range of scattering angles from 178.8° to 180° at a fine angular interval
of $\sim 0.02^\circ$ [38,42]. However, few
measurements by this instrument have been reported. In addition, two
prototype VSF instruments, namely the multi-spectral volume scattering
meter (MVSM) [40,44] and the polarized volume scattering
meter [41], were capable of
measuring VSF up to 179°. However, both laboratory calibration [45] and field experiment (see below)
showed that the VSF measurements by the MVSM at angles $ \gt {173}^\circ
$ are unreliable due to stray light
contamination [45] and/or
difficulty in precisely calculating scattering volume. Laser In Situ
Scattering and Transmissometry VSF instrument (LISST-VSF), developed and
recently commercialized by Sequoia Scientific Inc., measures the VSF up to
155° [29,43], providing an opportunity for the ocean optics
community to routinely measure the VSF in different waters. However,
simple extrapolation to estimate ${\beta _p}(\pi )$ cannot be applied to LISST-VSF data,
which ends at 155°.

An alternative method to estimate ${\beta _p}(\pi )$ is to apply an inversion-forward modeling
approach. Zhang *et al.* [7] developed a VSF-inversion method to
simultaneously infer the particle size distributions (PSDs) and refractive
indices of individual particle subpopulations from a measured VSF. This
method has been validated successfully in deriving particle size
distribution [33], bubble and
sediments populations [44,46,47], mass concentrations of particulate inorganic and organic
matter [48], chlorophyll
concentration [49], and silica and
mineral sediments [50]. With
inferred PSDs and refractive indices of particle subpopulations, the VSF
can be reconstructed through forward modeling to estimate ${\beta _p}(\pi )$.

The objective of this study was to derive ${\beta _p}(\pi )$ using the Zhang *et al.* [7] VSF-inversion
method from the VSFs measured by the MVSM and the LISST-VSF in the coastal
and oceanic waters to investigate the variability of ${\chi _p}(\pi )$ and its spectral dependency.

## 2. DATA AND METHOD

#### A. Field Measurements

VSFs measured by the MVSM and the LISST-VSF in the field were used in
this study. The MVSM measures the VSF with an angular increment of
0.25° between 0.5° and 179° at eight spectral bands, each of 9 nm
bandwidth and centered at 443, 490, 510, 532, 555, 565, 590, and
620 nm, respectively [40,44,45]. The VSFs at approximately 1–2 m depth [44] were measured using the MVSM in
Chesapeake Bay 12–22 October 2009 (CHB09), Mobile Bay 17–26 February
2009 (MOB09), Monterey Bay 12–19 October 2010 (MTB10), Gulf of Mexico
16–27 March 2016 (GOM16), and Ship-Aircraft Bio-Optical Research
Experiment 17 July–7 August 2014 (SABOR14) (Fig. 1). During CHB09, MOB09 and MTB10, the MVSM was
operated at eight wavelengths but only operated at 532 nm during
SABOR14 and GOM16. One complete rotation from 0° to 360° for eight
wavelengths takes about 10 min and the measurements from 0° to 180°
and from 360° to 180° were averaged and processed following Berthon
*et al.* [45] to obtain the bulk VSF. As reported previously [45] and observed in this study,
MVSM-measured VSF at angles $ \gt {173}^\circ
$ were unreliable and discarded from
further analysis.

VSFs were measured by the LISST-VSF in three experiments: Line P cruise
during 4–20 June 2017 (LP17) and 18 February–8 March 2018 (LP18), and
EXport Processes in the Ocean from Remote Sensing during 14 August–10
September 2018 (EXPORTS18) (Fig. 1). The LISST-VSF measures the scattering from 0.1° to 14° in
32 log-spaced angles by a traditional LISST unit and from 15° to 155°
in 1° steps by an eyeball component at a nominal wavelength of 517 nm
[29,43]. In the three experiments, the LISST-VSF was
operated in benchtop mode measuring discrete water samples collected
by Conductivity-Temperature-Depth (CTD) casts at various depths. A
full angular scattering measurement took about 4 s and 30 measurements
were taken for one sample. The median values of these 30 repeated
measurements were used following the calibration and data processing
described in Hu *et al.* [43]. For this study, we only used the
VSF measured by the eyeball component because the experiment sites
were very clear where the forward scattering was generally too weak to
be detected by the LISST unit [43]. The VSF obtained by the LISST-VSF at scattering angles $ \gt {147}^\circ
$ appeared to be affected by specular
reflection from the receiver window [43] and discarded from further
analysis.

Scattering due to pure seawater was calculated using the Zhang *et al.* model [4] using the salinity and temperature measured concurrently
with the water samples and was subtracted from the bulk VSFs to obtain
particulate VSFs. Table 1
summarizes the field measurements.

#### B. VSF-Inversion Method

The details of the VSF-inversion method were described in Zhang *et al.* [7]
and refined in the follow-on studies [44,51]. For
completeness, the method is briefly reviewed here. A measured ${\beta _p}(\theta
)$, as an inherent optical property,
represents the cumulative contribution by individual particle
subpopulations [7], i.e.,

Three particle models were used to compute phase functions for different particle subpopulations expected to be present in the natural aquatic environments. Two of the particle models, homogenous spheres (HS) [7] and homogenous asymmetric hexahedra [44,47], have been used in previous studies. These two particle models represent two extremes in particle shape: symmetry versus asymmetry and smooth curvature versus sharp edge. However, neither accounts for the internal structure of marine particles. Studies have shown that phytoplankton populations with internal cellular structures can be modeled with coated spheres (CS) [52–56]. Compared to homogenous spheres, use of coated spheres has been shown to be able to reproduce the measured inherent optical properties, particularly the backscattering, reasonably well for phytoplankton particles [56,57]. The third particle model represents the phytoplankton-like particles as coated spheres and non-phytoplankton particles as homogenous spheres.

Corresponding to three particle models, three inversion kernels were
constructed as follows. First, angular scattering was simulated for
single particles with Mie theory for the homogenous
sphere, modified Mie theory for the coated sphere, and a combination
of the invariant-imbedding T-matrix method and physical-geometric
optics method [58–60]
for the homogenous hexahedra (HH). To represent optically active
particles in natural waters, sizes of particles are set to range from
0.001 to 200 µm and relative refractive index from 1.02 to 1.20 with a
fixed imaginary refractive index of 0.002 [44]. The phytoplankton-type particles were defined as
particles with a relative refractive index between 1.02 and 1.06 and
diameter between 1.0 and 20 µm. When simulating phytoplankton
particles using coated spheres, the thickness and relative refractive
index of the shell were assumed to be constant values of 0.1 µm and
1.10, respectively, following Poulin *et al.* [56]. Bubbles
were also included and modeled as spheres coated with a monolayer of
film with a constant thickness of 2 nm and a relative refractive index
of 1.20 [7,47]. Second, particle subpopulations were defined,
with each represented by a unique combination of a refractive index
and a lognormal size distribution. The values of mode size and
standard deviation of lognormal distribution for each particle
subpopulation were adopted following Zhang *et al.* [7]. Phase
functions were then computed for all particle subpopulations. Finally,
a rigorous sensitivity analysis was applied to the computed phase
functions to eliminate those that are similar to each other ($ \lt {10}\% $ in logarithmic scale). This ensures
only optically distinctive phase functions were maintained and
together they served as the inversion kernel. For each inversion
kernel, the least-squares with a non-negative constraint (i.e., ${b_{p,i}}\; \ge
\;{0}$) was applied to Eq. (4) for each measured ${\beta _p}(\theta
)$ to derive ${b_{p,i}}$. If ${b_{p,i}}\; \gt
\;{0}$, the corresponding particle
subpopulation was regarded being present. Otherwise, the particle
subpopulation was assumed to be absent. With ${b_{p,i}}$ inferred from the VSF-inversion, we
re-applied Eq. (4) to
reconstruct a VSF at angles not measured by the instruments [e.g., ${\beta _p}(\pi
)$] using the inferred particle
subpopulations.

In summary, the inversion was tested using three different inversion kernels: HS for all particles, HH for all particles, and CS for phytoplankton-type particles and homogenous sphere for non-phytoplankton particles. The HS and HH kernels were used to test the effect of particle shape on estimating ${\beta _p}(\pi )$ from measured VSFs, while the HS and CS kernels were used to test the effect of the internal structure.

For the MVSM, which measures VSFs at 691 angles, the three inversion kernels built using the HS, CS, and HH particle models comprise, respectively, 134, 137, and 107 optically distinctive phase functions. For the LISST-VSF, which measures VSFs at 133 angles, the three inversion kernels contain, respectively, 128, 125, and 100 optically distinctive phase functions. For illustration, the three kernels built for the MVSM are shown in Fig. 2. In the near forward scattering angles, phase functions computed with the HS, CS, and HH particle models are nearly identical, indicating the forward scatterings are not very sensitive to particle shapes or internal structure. On the other hand, distinct differences are observed in scattering at backward angles ($ \gt {90}^\circ $) among the three kernels. For example, phase functions in the CS kernel are general larger than those in the HS kernel [Fig. 2(a)] indicating an enhanced backscattering when taking the internal structure of phytoplankton cell into account [52–57]. Moreover, at or near 180°, phase functions in the HH kernel [Fig. 2(b)] appear relatively flat as compared to those in the spherical kernels (HS and CS).

## 3. RESULTS

#### A. VSF Inversion and Reconstruction

Two examples of VSF inversion and reconstruction are shown in Fig. 3(a) using the MVSM measurement at a location denoted as the red star in Fig. 1(c) and in Fig. 3(b) using the LISST-VSF measurement at a location denoted as the red star in Fig. 1(d). In both examples, the reconstructed VSFs with the HS, CS, and HH kernels agree with the measured VSFs at all scattering angles except at their respective end angles that are susceptible to stray light contamination. For example, the measured VSF shown in Fig. 3(a) exhibits a sharp increase toward 175° followed by a sharp decrease toward 178° [see the inserted image in Fig. 3(a)]. This behavior, which seems unrealistic, was also reported in other studies and was attributed to the long path length, wide acceptance angle, and possible contamination from stray light [45]. Similarly, the LISST-VSF data shown in Fig. 3(b) also exhibits a sharp increase at angles $ \gt {147}^\circ $ [see the inserted image in Fig. 3(b)], which is due to specular reflection by the optical window [43]. The spurious behavior shown in the measured VSFs toward end scattering angles manifests the challenges in directly measuring ${\beta _p}(\pi )$. Distinct behaviors of reconstructed VSFs are also observed at scattering angles $ \gt {170}^\circ $ [comparing blue and black lines with the red lines in Figs. 3(a) and 3(b)]. The reconstructed VSFs using the HS and CS kernels increase with the scattering angles toward 180°, while the reconstructed VSFs using the HH kernel appear flat. This is because some phase functions in the spherical kernels show enhanced scattering toward 180° [Fig. 2(a)], which is not observed in the hexahedral kernel [Fig. 2(b)]. The difference in the reconstructed VSFs at angles $ \gt {170}^\circ $ among the three inversion kernels indicates that applying simple extrapolation to estimate ${\beta _p}(\pi )$ could be problematic.

At angles not subject to stray light contamination, the VSF-inversion method worked very well in reproducing the measured VSFs regardless of which inversion kernel is used [Fig. 4(a)]. The mean percentage differences (MPDs) between the measured VSFs and the reconstructed VSFs at all stations using the HS, CS, and HH kernels are 3.2%, 3.9%, and 4.4%, respectively, for the MVSM data, and 2.9%, 2.9%, and 3.8%, respectively, for the LISST-VSF data, suggesting a good performance of the VSF-inversion method particularly considering the comparison was applied over 5–6 orders of magnitude change in the VSFs. Because of ${\sin}\theta $ weighting in Eq. (3), the differences at angles $(\theta )\; \gt \;{170}^\circ $ among the VSFs reconstructed with the HS, CS, and HH inversion kernels barely affect the estimation of ${b_\textit{bp}}$ [Fig. 4(b)] with a ${\rm MPD}\; \le \;{1.5}\% $.

#### B. $\chi_p(\pi)$

Figure 5 examines the relationship between ${\beta _p}(\pi )$ and ${b_\textit{bp}}$ derived from the MVSM [Figs. 5(a)–5(e)] and the LISST-VSF measurements [Figs. 5(f)–5(h)] using HS (black dots), CS (blue dots), and HH (red dots) kernels, respectively. The values of ${b_\textit{bp}}$ varied from 0.0003 to ${0.31}\;{{\rm m}^{ - 1}}$, suggesting that the dataset covered a wide range of water types. For each experiment, regardless in coastal water or open ocean, ${\beta _p}(\pi )$ and ${b_\textit{bp}}$ estimated using the HH kernel exhibit a very strong linear relationship with the coefficient of determination ${r^2}\; \gt \;{0.93}$. High correlations are also observed for ${\beta _p}(\pi )$ and ${b_\textit{bp}}$ estimated using the HH and CS kernels with the coefficient of determination ${r^2}\; \gt \;{0.90}$ in most cases. Also, values of ${\beta _p}(\pi )$ estimated using the HS and CS inversion kernels that assume spherical particles are generally greater than the result using the HH kernel that assumes non-spherical particles. This is mainly because spherical particles tend to exhibit enhanced scattering toward 180° [61,62]. ${\chi _p}(\pi )$ was estimated by applying a robust linear regression model [63] and forcing the regression line passing through the origin. The robust model weights points closer to the regression line more than more distant points to minimize the effects of the outliers.

${\chi _{p,HS}}(\pi )$ varied from 0.22 to 1.04 with a median of 0.56, ${\chi _{p,CS}}(\pi )$ from 0.19 to 0.94 with a median of 0.54, and ${\chi _{p,HH}}(\pi )$ from 0.51 to 1.45 with a median of 1.13 (Table 1 and Fig. 6). While each estimate showed approximately a factor of 5 variation, values of ${\chi _{p,HS\:}}(\pi )$ and ${\chi _{p,CS}}(\pi )$ based on spherical particle models are close to each other and both are lower than ${\chi _{p,HH}}(\pi )$ that was based on the hexahedral particle model. For comparison, ${\chi _p}(\pi )$ linearly extrapolated from the MASCOT measurements [32] exhibits a very limited variability with a mean value of 1.06, which is close to the median of ${\chi _{p,HH}}(\pi )$ estimated in this study. On the other hand, ${\chi _p}(\pi )$ determined by airborne HSRL-measured ${\beta _p}(\pi )$ and ship-based ECO-BB3-measured ${b_\textit{bp}}$ [22] shows a range of variability that is consistent with ${\chi _{p,HS\:}}(\pi )$ and ${\chi _{p,CS}}(\pi )$ estimated in this study. Large variations of ${\chi _p}(\pi )$ were also observed using a backscatter lidar and an ECO-BBFL2 sensor on the west coast of Florida [64].

#### C. Spectral Dependency of $\chi_p(\pi)$

${\chi _p}(\pi )$ has been assumed to be spectrally flat [12,18,22,34,64]. Measurements of ${b_\textit{bp}}$ in the ocean typically exhibited an inverse power law dependence on wavelength [65–67]. Similar spectral dependence was also observed in the VSF measurements [44,66]. Therefore, we expect a large portion of spectral variations would be canceled if taking their ratio to form ${\chi _p}(\pi )$. This assumption was examined using the MVSM multi-spectral measurements collected in the three coastal experiments, CHB09, MOB09, and MTB10 (Fig. 7). Though the exact values of spectral ${\chi _p}(\pi )$ are different between the HS, CS, and HH kernels, they all show a relative flat spectral dependence. Moreover, spectral variations are mostly within the uncertainty estimates for each wavelength.

## 4. DISCUSSION AND CONCLUSIONS

In this study, ${\chi _p}(\pi )$ and its spectral dependence were
re-examined by applying a VSF-inversion and forward modeling approach to
the VSFs measured in a wide range of waters. Basically, this method
involves three steps. First, the optical and size properties of particle
subpopulations were inverted from measured VSFs. Second, complete VSF at
all scattering angles was reconstructed through forward modeling using the
derived particle properties. Finally, ${\chi _p}(\pi )$ was calculated with ${\beta _p}(\pi )$ and ${b_\textit{bp}}$ obtained from the reconstructed VSF. In
essence, calculation of ${\beta _p}(\pi )$ from the reconstructed VSF was also a
kind of “extrapolation” but based on a physically sound, robust, and
well-validated VSF-inversion model. To test the effect of particle shape
and internal structure on the inversion method, homogenous sphere,
homogenous asymmetrical hexahedra, and coated sphere were used to build
three kernels of phase functions. The constructed VSFs using the three
kernels agree with measured VSFs very well with the ${\rm MPD}\; \lt \;{5}\%
$ at almost all scattering angles.
Relatively greater difference at angles $ \gt {170}^\circ
$ was observed. The possible explanations
included (1) inaccurate VSF measurements at angles $ \gt {170}^\circ
$ due to instrumental difficulty [40,41], and (2) high sensitivity of VSF with particle shape or
internal structure at angles $ \gt {170}^\circ
$ [7,68]. As discussed in
Zhang *et al.* [7], the particle shape and internal structure have less effect on
scattering for small particles and for scattering at forward angles but
are important for scattering by large particles at backward
directions.

The effect of particle shape on ${\chi _p}(\pi )$ estimated from the measured VSFs was
examined with homogenous sphere and hexahedra kernels. ${\chi _p}(\pi )$ estimated with the two sphere kernels was
in general less than those derived from the hexahedra kernel (Table 1, Figs. 5 and 6) because
spherical particles (including coated spherical particles) tend to show
enhanced scattering toward 180° than non-spherical particles. A sharp
increase in the scattering toward 180° has been observed in both the
laboratory experiments [41,67,69–72] and the natural environments
[7,25,38,42]. For example, Beta Pi observed 10% and 50% increases
in the VSF from 179° to 180° measured in the coastal waters of Monterey
Bay [42] and Gulf of Mexico [38], respectively. The enhanced
scattering toward 180° had been attributed to the medium turbulent
structure [73,74], Fraunhofer diffraction [75], multiple scattering [76,77], and coherent
backscattering [78–80].
The essential feature of these explanations is the wave interferences
[79]. The interference pattern
produced by a homogeneous sphere at angles near 180° is also known as
glory [81]. Although the glory
phenomenon still does not have a definitive explanation, it can be well
predicted with Mie theory [61,62,82]. The presence of glory does not necessarily mean that the
particles are spherical because particles of a shape with continuous
curvature theoretically could also trap the surface wave and generate
glory [7,82]. Using geometric-optics approximation, Shishko
*et al.* [80] showed that particles of irregular shapes would produce an
almost constant scattering from 170° to 180° [e.g., see Fig. 2(b)]. While hexahedral shape has been
shown to better represent the overall scattering by dusts [83] and marine minerals [44,47,51], our study showed
that at or near 180°, using spheres would likely better reproduce the
backscattering enhancement feature of marine particles. Therefore, at
least statistically, ${\chi _p}(\pi )$ calculated with sphere kernels was more
consistent with the HSRL result (Fig. 6).

The effect of particle internal structure on ${\chi _p}(\pi )$ estimated from the measured VSFs was examined by modeling phytoplankton-like particles as coated spheres. The thickness (${{D}_c}$) and relative refractive index (${{n}_c}$) of the shell (membrane) were assumed to be a constant value of 0.1 and 1.10 µm, respectively. With this assumption, ${\chi _p}(\pi )$ estimated with the coated sphere kernel was close to but slightly lower than ${\chi _p}(\pi )$ estimated with the homogenous sphere kernel (Table 1 and Fig. 5). However, both ${{D}_c}$ and ${{n}_c}$ values are expected to change for different phytoplankton species in different growth stages. For example, the thickness of the cell membrane can vary from 0.05 to 0.13 µm [52] and its relative refractive index can vary from 1.06 to 1.22 [84]. To examine the potential effect of these changes on estimating ${\chi _p}(\pi )$, additional coated sphere kernels were built by varying ${{D}_c}$ and ${{n}_c}$ and applied to the measured VSFs to estimate ${\chi _p}(\pi )$ (Fig. 8). ${\chi _p}(\pi )$ derived from coated sphere kernels is close to that from the homogenous sphere kernel (dashed line) when either the shell is thin ($ \le {0.05}\;{\unicode{x00B5}{\rm m}}$), the shell is of low refractive index values ($ \le {1.06}$), or both. With an increase in either ${{D}_c}$ or ${{n}_c}$, ${\chi _p}(\pi )$ decreases. Figure 8 suggests that the presence of phytoplankton-like particles would generally decrease ${\chi _p}(\pi )$.

The significant influence of particle shape and internal structure on ${\chi _p}(\pi )$ indicates that assuming a global ${\chi _p}(\pi )$ value for different natural waters could incur significant uncertainty. Cautions need to be taken when applying a global mean ${\chi _p}(\pi )$ value to interpret lidar signal. Indeed, our study, as well as lidar measurements, showed clearly that ${\chi _p}(\pi )$ varied with waters. On the other hand, the spectral variation of ${\chi _p}(\pi )$ is insignificant, at least in comparison to the uncertainty associated with estimating ${\chi _p}(\pi )$. More efforts are needed to build a reliable VSF instrument capable of measuring $\beta (\pi )$ directly.

## Funding

National Aeronautics and Space Administration (80NSSC17K0656, 80NSSC18M0024, 80NSSC19K0723); National Science Foundation (1917337); National Natural Science Foundation of China (61675187).

## Acknowledgment

We thank two anonymous reviewers for their insightful comments and suggestions that have helped us to improve the paper.

## Disclosures

The authors declare no conflicts of interest.

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