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$):

$$\beta \left( \theta \right) = \mathop {\lim }\limits_{\Delta v \to 0} \frac{{I\left( \theta \right)}}{{E\Delta v}}.$$
In the aquatic environment, the VSF of seawater is commonly partitioned into the contributions from pure seawater (${\beta _w}$) and suspended particles (${\beta _p}$). Scattering of water molecules and associated sea salts is relatively well understood, with the models agreeing with the measurements within 2% [35]. However, our knowledge is limited on scattering by particles, which varies in a complex way with their concentration, size distribution, composition, shape, and internal structure [6,7]. Integrating a VSF over all scattering angles in the backward directions (i.e., 90°–180°) gives the backscattering coefficient (${b_b}$). While normally only a small fraction of total scattering, backscattering is of particular importance for interpreting ocean color imagery [8,9] or lidar signal [10,11]. Lidar, which stands for light detection and ranging, is an active remote sensing technique with distinct advantages over passive ocean color techniques, such as having the abilities of observing at nighttime and lower sun elevation conditions, penetrating thin clouds, and acquiring vertical structure of optically active constituents in the ocean [1013]. The recent progresses using lidar data from the Cloud-Aerosol Lidar with Orthogonal Polarization instrument, originally designed for atmospheric applications, have demonstrated the feasibility of mapping subsurface ocean optical properties and biogeochemical parameters on the globe scale [1417].

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 $ [11,2224]. 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]

$${\chi _{p}}\left( \theta \right) = {b_{{bp}}}{/2\pi }{\beta _{p}}\left( \theta \right),$$
and
$${b_{{bp}}}{= 2\pi }\int_{{{\pi } / 2}}^{\pi } {{\beta _{p}}\left( \theta \right)\sin \theta {\rm d}\theta } .$$
The challenge to accurately estimate ${\chi _p}(\pi )$ is that ${\beta _p}(\pi )$ and ${b_\textit{bp}}$ are seldom measured simultaneously. Most VSF instruments were able to measure the VSF at angles up to 170° very well [2631]. Because of weighting of sin $\theta $ [Eq. (3)], this type of measurement will not affect the estimate of ${b_\textit{bp}}$ very much. To estimate ${\beta _p}(\pi )$, however, requires extrapolation, which is often problematic. VSF measurements made over a variety of water types by the Multi-Angle Scattering Optical Tool (MASCOT) [32] at angles from 10° to 170° with a 10° increment were linearly extrapolated to estimate ${\beta _p}(\pi )$, which was then used to calculate ${\chi _p}(\pi )$. The estimated ${\chi _p}(\pi )$ exhibited little variation and its value of 1.06 has been used in several studies [12,13,17,18,33,34]. ${\chi _p}(\pi )$ has also been estimated using HSRL and in-water measurement of the VSF. Hair 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) [4042]; 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 [4143]. 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.

 figure: Fig. 1.

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.

Tables Icon

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)

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.,

$${\beta _{p}}\left( \theta \right) = \sum\limits_{i = 1}^{M} {{b_{{p},i}}{{\tilde \beta }_{{p,}i}}} \left( \theta \right),$$
where ${b_{p,i}}$ and ${\tilde \beta _{{p},i}}$ are the scattering coefficient and phase function of the $i$th particle subpopulation from a total of M potential particle subpopulations. The basic idea of VSF-inversion is to build a library of ${\tilde \beta _{{p},i}}$ serving as the kernel to infer ${b_{p,i}}$ from measured ${\beta _p}(\theta )$. With inferred ${b_{p,i}}$, we can re-apply Eq. (4) using ${\tilde \beta _{{p},i}}$ to reconstruct a VSF at angles not measured by the instruments [e.g., ${\beta _p}(\pi )$].
 figure: Fig. 2.

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) [5256]. 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 [5860] 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.

 figure: Fig. 3.

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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 figure: Fig. 4.

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.

 figure: Fig. 5.

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 [5257]. 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}\% $.

 figure: Fig. 6.

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 [6567]. 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.

 figure: Fig. 7.

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 [41,67,6972] 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 [7880]. 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 ${\chi _p}(\pi )$.

 figure: Fig. 8.

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.

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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References

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  1. J. T. O. Kirk, Light and Photosynthesis in Aquatic Ecosystems, 2nd ed. (Cambridge University, 1994).
  2. C. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic, 1994).
  3. X. Zhang and L. Hu, “Estimating scattering of pure water from density fluctuation of the refractive index,” Opt. Express 17, 1671–1678 (2009).
    [Crossref]
  4. X. Zhang, L. Hu, and M.-X. He, “Scattering by pure seawater: effect of salinity,” Opt. Express 17, 5698–5710 (2009).
    [Crossref]
  5. L. Hu, X. Zhang, and M. J. Perry, “Light scattering by pure seawater: effect of pressure,” Deep Sea Res. I 146, 103–109 (2019).
    [Crossref]
  6. D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. 40, 2929–2945 (2001).
    [Crossref]
  7. X. Zhang, M. Twardowski, and M. Lewis, “Retrieving composition and sizes of oceanic particle subpopulations from the volume scattering function,” Appl. Opt. 50, 1240–1259 (2011).
    [Crossref]
  8. A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
    [Crossref]
  9. H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
    [Crossref]
  10. J. H. Churnside, “Review of profiling oceanographic lidar,” Opt. Eng. 53, 051405 (2013).
    [Crossref]
  11. C. A. Hostetler, M. J. Behrenfeld, Y. Hu, J. W. Hair, and J. A. Schulien, “Spaceborne lidar in the study of marine systems,” Annu. Rev. Mar. Sci. 10, 121–147 (2018).
    [Crossref]
  12. J. H. Lee, J. H. Churnside, R. D. Marchbanks, P. L. Donaghay, and J. M. Sullivan, “Oceanographic lidar profiles compared with estimates from in situ optical measurements,” Appl. Opt. 52, 786–794 (2013).
    [Crossref]
  13. J. H. Churnside and R. D. Marchbanks, “Subsurface plankton layers in the Arctic Ocean,” Geophys. Res. Lett. 42, 4896–4902 (2015).
    [Crossref]
  14. M. J. Behrenfeld, Y. Hu, C. A. Hostetler, G. Dall’Olmo, S. D. Rodier, J. W. Hair, and C. R. Trepte, “Space-based lidar measurements of global ocean carbon stocks,” Geophys. Res. Lett. 40, 4355–4360 (2013).
    [Crossref]
  15. X. Lu, Y. Hu, C. Trepte, S. Zeng, and J. H. Churnside, “Ocean subsurface studies with the CALIPSO spaceborne lidar,” J. Geophys. Res. Oceans 119, 4305–4317 (2014).
    [Crossref]
  16. Y. Hu and P. Zhai, “Development and validation of the CALIPSO ocean subsurface data,” in International Geoscience and Remote Sensing Symposium (IGARSS) (2016), pp. 3785–3787.
  17. X. Lu, Y. Hu, J. Pelon, C. Trepte, K. Liu, S. Rodier, S. Zeng, P. Lucker, R. Verhappen, J. Wilson, C. Audouy, C. Ferrier, S. Haouchine, B. Hunt, and B. Getzewich, “Retrieval of ocean subsurface particulate backscattering coefficient from space-borne CALIOP lidar measurements,” Opt. Express 24, 29001–29008 (2016).
    [Crossref]
  18. J. H. Churnside, J. M. Sullivan, and M. S. Twardowski, “Lidar extinction-to-backscatter ratio of the ocean,” Opt. Express 22, 18698–18706 (2014).
    [Crossref]
  19. J. H. Churnside, “Polarization effects on oceanographic lidar,” Opt. Express 16, 1196–1207 (2008).
    [Crossref]
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2019 (5)

L. Hu, X. Zhang, and M. J. Perry, “Light scattering by pure seawater: effect of pressure,” Deep Sea Res. I 146, 103–109 (2019).
[Crossref]

P. Chen, D. Pan, Z. Mao, and H. Liu, “A feasible calibration method for type 1 open ocean water LiDAR data based on bio-optical models,” Remote Sens. 11, 172 (2019).
[Crossref]

P. Chen and D. Pan, “Ocean optical profiling in South China Sea using airborne LiDAR,” Remote Sens. 11, 1826 (2019).
[Crossref]

L. Hu, X. Zhang, Y. Xiong, and M.-X. He, “Calibration of the LISST-VSF to derive the volume scattering functions in clear waters,” Opt. Express 27, A1188–A1206 (2019).
[Crossref]

V. Shishko, A. Konoshonkin, N. Kustova, D. Timofeev, and A. Borovoi, “Coherent and incoherent backscattering by a single large particle of irregular shape,” Opt. Express 27, 32984–32993 (2019).
[Crossref]

2018 (4)

C. Poulin, X. Zhang, P. Yang, and Y. Huot, “Diel variations of the attenuation, backscattering and absorption coefficients of four phytoplankton species and comparison with spherical, coated spherical and hexahedral particle optical models,” J. Quantum Spectrosc. Radiat. Transfer 217, 288–304 (2018).
[Crossref]

E. Organelli, G. Dall’Olmo, R. J. W. Brewin, G. A. Tarran, E. Boss, and A. Bricaud, “The open-ocean missing backscattering is in the structural complexity of particles,” Nat. Commun. 9, 5439 (2018).
[Crossref]

S. Stamnes, C. Hostetler, R. Ferrare, S. Burton, X. Liu, J. Hair, Y. Hu, A. Wasilewski, W. Martin, B. van Diedenhoven, J. Chowdhary, I. Cetinić, L. K. Berg, K. Stamnes, and B. Cairns, “Simultaneous polarimeter retrievals of microphysical aerosol and ocean color parameters from the “MAPP” algorithm with comparison to high-spectral-resolution lidar aerosol and ocean products,” Appl. Opt. 57, 2394–2413 (2018).
[Crossref]

C. A. Hostetler, M. J. Behrenfeld, Y. Hu, J. W. Hair, and J. A. Schulien, “Spaceborne lidar in the study of marine systems,” Annu. Rev. Mar. Sci. 10, 121–147 (2018).
[Crossref]

2017 (7)

J. H. Churnside and R. D. Marchbanks, “Inversion of oceanographic profiling lidars by a perturbation to a linear regression,” Appl. Opt. 56, 5228–5233 (2017).
[Crossref]

J. A. Schulien, M. J. Behrenfeld, J. W. Hair, C. A. Hostetler, and M. S. Twardowski, “Vertically-resolved phytoplankton carbon and net primary production from a high spectral resolution lidar,” Opt. Express 25, 13577–13587 (2017).
[Crossref]

X. Zhang, G. R. Fournier, and D. J. Gray, “Interpretation of scattering by oceanic particles around 120 degrees and its implication in ocean color studies,” Opt. Express 25, A191–A199 (2017).
[Crossref]

J. Churnside, R. Marchbanks, C. Lembke, and J. Beckler, “Optical backscattering measured by airborne lidar and underwater glider,” Remote Sens. 9, 379 (2017).
[Crossref]

X. Zhang, R. H. Stavn, A. U. Falster, J. J. Rick, D. Gray, and R. W. Gould, “Size distributions of coastal ocean suspended particulate inorganic matter: amorphous silica and clay minerals and their dynamics,” Estuarine Coast. Shelf Sci. 189, 243–251 (2017).
[Crossref]

G. Xu, B. Sun, S. D. Brooks, P. Yang, G. W. Kattawar, and X. Zhang, “Modeling the inherent optical properties of aquatic particles using an irregular hexahedral ensemble,” J. Quantum Spectrosc. Radiat. Transfer 191, 30–39 (2017).
[Crossref]

B. Sun, P. Yang, G. W. Kattawar, and X. Zhang, “Physical-geometric optics method for large size faceted particles,” Opt. Express 25, 24044–24060 (2017).
[Crossref]

2016 (2)

2015 (5)

2014 (5)

2013 (7)

H. Tan, R. Doerffer, T. Oishi, and A. Tanaka, “A new approach to measure the volume scattering function,” Opt. Express 21, 18697–18711 (2013).
[Crossref]

M. J. Behrenfeld, Y. Hu, C. A. Hostetler, G. Dall’Olmo, S. D. Rodier, J. W. Hair, and C. R. Trepte, “Space-based lidar measurements of global ocean carbon stocks,” Geophys. Res. Lett. 40, 4355–4360 (2013).
[Crossref]

J. H. Lee, J. H. Churnside, R. D. Marchbanks, P. L. Donaghay, and J. M. Sullivan, “Oceanographic lidar profiles compared with estimates from in situ optical measurements,” Appl. Opt. 52, 786–794 (2013).
[Crossref]

J. H. Churnside, “Review of profiling oceanographic lidar,” Opt. Eng. 53, 051405 (2013).
[Crossref]

X. Zhang, Y. Huot, D. J. Gray, A. Weidemann, and W. J. Rhea, “Biogeochemical origins of particles obtained from the inversion of the volume scattering function and spectral absorption in coastal waters,” Biogeosciences 10, 6029–6043 (2013).
[Crossref]

L. Bi, P. Yang, G. W. Kattawar, and M. I. Mishchenko, “Efficient implementation of the invariant imbedding T-matrix method and the separation of variables method applied to large nonspherical inhomogeneous particles,” J. Quantum Spectrosc. Radiat. Transfer 116, 169–183 (2013).
[Crossref]

A. Borovoi, A. Konoshonkin, and N. Kustova, “Backscattering by hexagonal ice crystals of cirrus clouds,” Opt. Lett. 38, 2881–2884 (2013).
[Crossref]

2012 (4)

M. Twardowski, X. Zhang, S. Vagle, J. Sullivan, S. Freeman, H. Czerski, Y. You, L. Bi, and G. Kattawar, “The optical volume scattering function in a surf zone inverted to derive sediment and bubble particle subpopulations,” J. Geophys. Res. Oceans 117, C00H17 (2012).
[Crossref]

X. Zhang, D. J. Gray, Y. Huot, Y. You, and L. Bi, “Comparison of optically derived particle size distributions: scattering over the full angular range versus diffraction at near forward angles,” Appl. Opt. 51, 5085–5099 (2012).
[Crossref]

M. Babin, D. Stramski, R. A. Reynolds, V. M. Wright, and E. Leymarie, “Determination of the volume scattering function of aqueous particle suspensions with a laboratory multi-angle light scattering instrument,” Appl. Opt. 51, 3853–3873 (2012).
[Crossref]

C. Li, W. Cao, J. Yu, T. Ke, G. Lu, Y. Yang, and C. Guo, “An instrument for in situ measuring the volume scattering function of water: design, calibration and primary experiments,” Sensors 12, 4514–4533 (2012).
[Crossref]

2011 (2)

X. Zhang, M. Twardowski, and M. Lewis, “Retrieving composition and sizes of oceanic particle subpopulations from the volume scattering function,” Appl. Opt. 50, 1240–1259 (2011).
[Crossref]

H. Czerski, M. Twardowski, X. Zhang, and S. Vagle, “Resolving size distributions of bubbles with radii less than 30 µm with optical and acoustical methods,” J. Geophys. Res. Oceans 116, C00H11 (2011).
[Crossref]

2010 (2)

2009 (4)

2008 (1)

2007 (1)

2006 (1)

H. Loisel, J.-M. Nicolas, A. Sciandra, D. Stramski, and A. Poteau, “Spectral dependency of optical backscattering by marine particles from satellite remote sensing of the global ocean,” J. Geophys. Res. Oceans 111, C09024 (2006).
[Crossref]

2005 (1)

2003 (1)

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571 (2003).
[Crossref]

2002 (1)

2001 (2)

1998 (1)

H. Volten, J. F. de Haan, J. W. Hovenier, R. Schreurs, W. Vassen, A. G. Dekker, H. J. Hoogenboom, F. Charlton, and R. Wouts, “Laboratory measurements of angular distributions of light scattered by phytoplankton and silt,” Limnol. Oceanogr. 43, 1180–1197 (1998).
[Crossref]

1997 (1)

1996 (2)

R. A. Maffione and D. R. Dana, “In-situ characterization of optical backscattering and attenuation for lidar applications,” Proc. SPIE 2964, 152–162 (1996).
[Crossref]

E. Aas, “Refractive index of phytoplankton derived from its metabolite composition,” J. Plankton Res. 18, 2223–2249 (1996).
[Crossref]

1992 (3)

R. A. Maffione and R. C. Honey, “Instrument for measuring the volume scattering function in the backward direction,” Proc. SPIE 1750, 15–26 (1992).
[Crossref]

A. N. Bogaturov, A. A. D. Canas, J. C. Dainty, A. S. Gurvich, V. A. Myakinin, C. J. Solomon, and N. J. Wooder, “Observation of the enhancement of coherence by backscattering through turbulence,” Opt. Commun. 87, 1–4 (1992).
[Crossref]

M. I. Mishchenko, “Enhanced backscattering of polarized light from discrete random media: calculations in exactly the backscattering direction,” J. Opt. Soc. Am. A 9, 978–982 (1992).
[Crossref]

1991 (2)

C. F. Bohren and S. B. Singham, “Backscattering by nonspherical particles: A review of methods and suggested new approaches,” J. Geophys. Res. Atmos. 96, 5269–5277 (1991).
[Crossref]

G. D. Hickman, J. M. Harding, M. Carnes, A. Pressman, G. W. Kattawar, and E. S. Fry, “Aircraft laser sensing of sound velocity in water: Brillouin scattering,” Remote Sens. Environ. 36, 165–178 (1991).
[Crossref]

1989 (1)

M. S. Quinby-Hunt, A. J. Hunt, K. Lofftus, and D. Shapiro, “Polarized-light scattering studies of marine Chlorella,” Limnol. Oceanogr. 34, 1587–1600 (1989).
[Crossref]

1988 (2)

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

A. Ishimaru and L. Tsang, “Backscattering enhancement of random discrete scatters of moderate sizes,” J. Opt. Soc. Am. A 5, 228–236 (1988).
[Crossref]

1986 (1)

1985 (1)

1984 (2)

1982 (2)

1979 (1)

1977 (3)

P. W. Holland and R. E. Welsch, “Robust regression using iteratively reweighted least-squares,” Comm. Statist. Theory Methods 6, 813–827 (1977).
[Crossref]

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[Crossref]

V. Khare and H. M. Nussenzveig, “Theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[Crossref]

Aas, E.

E. Aas, “Refractive index of phytoplankton derived from its metabolite composition,” J. Plankton Res. 18, 2223–2249 (1996).
[Crossref]

Agrawal, Y. C.

W. H. Slade, Y. C. Agrawal, and O. A. Mikkelsen, “Comparison of measured and theoretical scattering and polarization properties of narrow size range irregular sediment particles,” in OCEANS, San Diego, California, USA, 2013, pp. 1–6.

Audouy, C.

Babin, M.

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semianalytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924 (1988).
[Crossref]

Beckler, J.

J. Churnside, R. Marchbanks, C. Lembke, and J. Beckler, “Optical backscattering measured by airborne lidar and underwater glider,” Remote Sens. 9, 379 (2017).
[Crossref]

Behrenfeld, M.

J. Hair, C. Hostetler, Y. Hu, M. Behrenfeld, C. Butler, D. Harper, R. Hare, T. Berkoff, A. Cook, J. Collins, N. Stockley, M. Twardowski, I. Cetinić, R. Ferrare, and T. Mack, “Combined atmospheric and ocean profiling from an airborne high spectral resolution lidar,” in EPJ Web of Conferences (2016), Vol. 119, p. 22001.

Behrenfeld, M. J.

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Figures (8)

Fig. 1.
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).
Fig. 2.
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 $ .
Fig. 3.
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)].
Fig. 4.
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.
Fig. 5.
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.
Fig. 6.
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.
Fig. 7.
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.
Fig. 8.
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.

Tables (1)

Tables Icon

Table 1. Summary of Field VSF Measurements Used in This Study and Ranges (Median Values) of the Particulate Backscattering Coefficient ( b b p ) and χ p ( π ) Derived Using the Inverse-Forward Modeling Method with Three Inversion Kernels: Homogenous Sphere (HS), Coated Sphere (CS), and Homogenous Hexahedra (HH)

Equations (4)

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β ( θ ) = lim Δ v 0 I ( θ ) E Δ v .
χ p ( θ ) = b b p / 2 π β p ( θ ) ,
b b p = 2 π π / 2 π β p ( θ ) sin θ d θ .
β p ( θ ) = i = 1 M b p , i β ~ p , i ( θ ) ,

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