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]

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.

 

Fig. 1. Experiment stations during (a) CHB09 (black dots) and SABOR14 (blue dots); (b) MOB09 (black dots) and GOM06 (blue dots); (c) MTB10; (d) LP17 and LP18 (black dots) and EXPORTS18 (blue dots).

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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.

Table 1. Summary of Field VSF Measurements Used in This Study and Ranges (Median Values) of the Particulate Backscattering Coefficient $({{b}_{{bp}}})$ and ${\chi _p}(\pi )$ Derived Using the Inverse-Forward Modeling Method with Three Inversion Kernels: Homogenous Sphere (HS), Coated Sphere (CS), and Homogenous Hexahedra (HH)

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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.,

 

Fig. 2. Phase functions built for three inversion kernels for the MVSM data using (a) homogenous sphere (HS) and coated sphere (CS) particle models, and (b) homogenous hexahedra (HH) particle model. For better visualization, the $y$ and $x$ axes are in log-log scale at angles $ \lt {15}^\circ $ and in log-linear scale at angles $ \ge {15}^\circ $.

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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.

 

Fig. 3. (a) Particulate VSF (${\beta _p}$) measured by the MVSM (gray line) at 532 nm at one coastal station during MTB10 [red star in Fig. 1(c)]. The black, blue, and red dotted lines are reconstructed ${\beta _p}$ with homogenous sphere (HS), coated sphere (CS), and homogenous hexahedra (HH) kernels, respectively. The inserted plot highlights the comparison at backward scattering angles. The gray transparent vertical bars highlight the angular range in which the scattering measurements were unreliable and not used in the inversion. (b) The same with (a) but for one LISST-VSF measurement collected at an open water station during LP18 [red star in Fig. 1(d)].

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Fig. 4. (a) Mean percentage difference (MPD) between measured- and reconstructed-VSFs at all stations for the MVSM (top) and the LISST-VSF data (bottom) using homogenous sphere (HS), coated sphere (CS), and homogenous hexahedra (HH) inversion kernels. The gray transparent vertical bars highlight the angular range in which the scattering measurements were unreliable and not used in the inversion and comparison. (b) Comparison of particulate backscattering coefficient (${b_\textit{bp}}$) calculated using reconstructed VSFs among the three inversion kernels: ${b_{bp,HS}}$ vs. ${b_{bp,CS}}$ (blue dots) and ${b_{bp,HS}}$ vs. ${b_{bp,HH}}$ (red dots). Note that the blue and red points overlap each other. The number of measurements (N) and MPD are also shown.

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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.

 

Fig. 5. Scatter plots between particulate VSF at 180° (${\beta _p}(\pi )$) and particulate backscattering coefficient (${b_\textit{bp}}$) estimated from the VSFs measured in (a) CHB09, (b) MOB09, (c) MTB10, (d) SABOR14, (e) GOM16, (f) LP17, (g) LP18, and (h) EXPORTS18. Lines of black, blue, and red colors are robust linear regression lines forcing through the origin for the data derived using homogenous sphere, coated sphere, and homogenous hexahedra inversion kernels, respectively.

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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}\% $.

 

Fig. 6. Histograms of ${\chi _p}(\pi )$ estimated with homogenous sphere (HS, black color), coated sphere (CS, blue color), and homogenous hexahedra (HH, red color) inversion kernels for all measurements. For comparison, the linearly extrapolated ${\chi _p}(\pi )$ from Sullivan and Twardowski [32] (ST09) and lidar estimated ${\chi _p}(\pi )$ from Hair et al. [22] (H16) are also shown.

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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.

 

Fig. 7. Spectral variation of ${\chi _p}(\pi )$ (${\rm mean}\;{\pm }$ ${\rm one}\;{\rm standard}\;{\rm deviation}$) derived with homogenous sphere (HS), coated sphere (CS), and homogenous hexahedra (HH) inversion kernels using the VSFs measured in CHB09 (black lines), MOB09 (red lines), and MTB10 (blue lines). For better visualization, the wavelength values for CHB09 and MTB10 data were shifted by 1 nm toward shorter and longer wavelength, respectively.

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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 ${\chi _p}(\pi )$.

 

Fig. 8. Median values of ${\chi _p}(\pi )$ derived from the measured VSFs using coated sphere kernels with various thicknesses (${{D}_c}$) and relative refractive indices (${{n}_c}$) of the cell membrane.

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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.