## Abstract

Even though it is well known that both the magnitude and detailed angular shape of scattering (phase function, PF), particularly in the backward angles, affect the color of the ocean, the current remote-sensing reflectance (${R}_{\mathrm{rs}}$) models typically account for the effect of its magnitude only through the backscattering coefficient (${b}_{b}$). Using 116 volume scattering function (VSF) measurements previously collected in three coastal waters around the U.S. and in the water of the North Atlantic Ocean, we re-examined the effect of particle PF on ${R}_{\mathrm{rs}}$ in four scenarios. In each scenario, the magnitude of particle backscattering (i.e., ${b}_{bp}$) is known, but the knowledge on the angular shape of particle backscattering is assumed to increase from knowing nothing about the shape of particle PFs to partially knowing the particle backscattering ratio (${B}_{p}$), the exact backscattering shape as defined by ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$ (particle VSF normalized by the particle total scattering coefficient), and the exact backscattering shape as defined by the ${\chi}_{p}$ factor (particle VSF normalized by the particle backscattering coefficient). At sun zenith $\text{angle}=30\xb0$, the nadir-viewed ${R}_{\mathrm{rs}}$ would vary up to 65%, 35%, 20%, and 10%, respectively, as the constraints on the shape of particle backscattering become increasingly stringent from scenarios 1 to 4. In all four scenarios, the ${R}_{\mathrm{rs}}$ variations increase with both viewing and sun angles and are most prominent in the direction opposite the sun. Our results show a greater impact of the measured particle PFs on ${R}_{\mathrm{rs}}$ than previously found, mainly because our VSF data show a much greater variability in ${B}_{p}$, ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, and ${\chi}_{p}$ than previously known. Among the uncertainties in ${R}_{\mathrm{rs}}$ due to the particle PFs, about 97% can be explained by ${\chi}_{p}$, 90% by ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, and 27% by ${B}_{p}$. The results indicate that the uncertainty in ocean color remote sensing can be significantly constrained by accounting for ${\chi}_{p}$ of the VSFs.

© 2017 Optical Society of America

## 1. INTRODUCTION

The study of ocean color is carried out by measuring the spectral radiance (${L}_{w}$; $\mathrm{W}\text{\hspace{0.17em}}{\mathrm{m}}^{-2}\text{\hspace{0.17em}}{\mathrm{sr}}^{-1}$) leaving the ocean and normalizing it by the incident irradiance (${E}_{d}$; $\mathrm{W}\text{\hspace{0.17em}}{\mathrm{m}}^{-2}$) at the surface, forming the spectral remote-sensing reflectance (${R}_{\mathrm{rs}}$; ${\mathrm{sr}}^{-1}$):

The VSFs influence the remote sensing reflectance [9,10] through both the backscattering coefficient and the bidirectional reflectance distribution function. ${b}_{b}$ describes the overall magnitude of VSFs in the backward directions and is generally responsible for reflecting incident sunlight out of the water. Assuming azimuthal symmetry of scattering, ${b}_{b}$ can be calculated as

where $\gamma $ is the scattering angle. The shape of VSFs influences the distribution of the underwater light field, which, in turn, impacts the ${R}_{\mathrm{rs}}$. However, due to limited knowledge on the VSFs of natural particle assemblage, most of the current ocean color algorithms [6,7,11] have ignored the shape effect. In other words, for two waters with the same $a$ and sun-sensor geometry, ${R}_{\mathrm{rs}}$ is assumed to be the same if the backscattering magnitudes are also the same, regardless of how the actual angular scattering might differ from each other. Optical closure studies, however, have shown that ${R}_{\mathrm{rs}}$ simulated using measured $a$ and ${b}_{b}$ but with an assumed angular scattering shape could differ significantly from the measured ${R}_{\mathrm{rs}}$ [12–17].The shape of a VSF is typically described using the phase function (PF),

where $b=2\pi {\int}_{0}^{\pi}\beta (\gamma )\mathrm{sin}\gamma \mathrm{d}\gamma $, representing the total scattering coefficient. Because of the importance of backward scattering in ocean color [4,8], the shape of a VSF in the backward angles is often described using two additional parameters, the backscattering probability (fraction or ratio),The angular shape of $\tilde{\beta}$ is contributed by both molecules and particles. Since the scattering by pure seawater is relatively well known [22,23], our focus in this study is on the effect of angular scattering by the particles, which represents the contribution by everything other than pure water and sea salt molecules and will be denoted by a subscript $p$. While Gordon [24] found that the PFs at angles $<15\xb0$ have a $<5\%$ impact on ${R}_{\mathrm{rs}}$, other studies have shown that the VSF shapes at larger angles could have a significant impact.

Bulgarelli *et al.* [12] found $\sim 40\%$ variation among simulated upwelling
radiance using the Fournier–Forand (FF) [25–27] PFs with
${B}_{p}=0.011$ and 0.033 and the average Petzold PF,
which was estimated using the measurements taken by Petzold [28] in the San Diego harbor and has a
${B}_{p}=0.018$. Similarly, Tzortziou
*et al.* [17] found ${L}_{w}$ simulated using the average Petzold PF
can differ by 50% from ${L}_{w}$ measured in a water with considerably
smaller ${B}_{p}$ values, with an $\text{average}=0.0128\pm 0.0032$. Using FF and a few other theoretical PFs
that have the same ${B}_{p}$ value as the average Petzold PF, Mobley
*et al.* [25] found that simulated ${R}_{\mathrm{rs}}$ vary only by 10%. They further compared
${R}_{\mathrm{rs}}$ among measured, simulated with measured
VSFs, and simulated with ${B}_{p}$-equal FF PF, and concluded that
“… the exact shape of the phase function in backscatter
directions does not greatly affect the light field, so long as …
the phase function have [sic] the correct backscatter fraction.”
The conclusion has since been tested. Chami *et al.*
[29] found up to 20%
differences in ${R}_{\mathrm{rs}}$ simulated using their measured PFs and
the corresponding ${B}_{p}$-equal FF PFs. They also showed that an
increase of the VSF between 10° and 100° could lead to
increased reflectance due to multiple scattering. Tonizzo
*et al.* [16] also compared ${R}_{\mathrm{rs}}$ simulated using measured PFs and the
corresponding ${B}_{p}$-equal FF PFs over a range of case I and
II waters and found an average difference of 20%.

The phase functions used in these studies, some of which are based on
simplified theoretical models, are limited in their representation of
natural variability of the VSFs. In addition, these studies considered
only the general backscattering fraction, not the full VSF shapes, while
theoretical derivation [8,9] indicates that the influence on
${R}_{\mathrm{rs}}$ arises from both the magnitude and the
exact shape of the VSF in the backward directions. For example, recent
studies by Pitarch *et al.* [15] and Lefering *et al.*
[14] showed that accounting for
the detailed shape of PFs is important in achieving optical closure in
${R}_{\mathrm{rs}}$.

We have measured VSFs over various coastal waters of the U.S. and in clear
waters of the North Atlantic Ocean [30,31], with
chlorophyll concentration ranging from 0.07 to $40\text{\hspace{0.17em}}\mathrm{\mu g}/\mathrm{L}$. The data show that the natural
variability of particle VSFs is greater than what had been known or
assumed previously in terms of both the backscattering probability and the
detailed backward shape (Fig. 1). For example, ${B}_{p}$ values estimated from our measured VSFs
vary over a factor of 30 from 0.0015 to 0.0543, which is much greater than
the range of 0.0073–0.0194 estimated from Petzold’s
measurements and the range of 0.002–0.020 measured in Tonizzo
*et al.* [16], but comparable to the range of 0.0015–0.0454
reported in Mankovsky and Haltrin [34,35] over the oceans
and lakes worldwide.

Therefore, we believe that the impact of angular scattering by oceanic
particles on ${R}_{\mathrm{rs}}$ may have been underestimated. Using the
latest measurements of the VSFs in a wide range of natural environments,
we have re-examined the uncertainty that the particle PFs might have on
the remote sensing of the color of the ocean. Our study differs from
previous ones in two aspects. First, we are interested in the natural
variability of particle PFs and their impact on ${R}_{\mathrm{rs}}$. Therefore, we used only PFs derived from
the measurements. Second, we examined the impact on
${R}_{\mathrm{rs}}$ by both general backward shape as defined
by ${B}_{p}$ and the exact backward shape as well.
Mobley *et al.* [25] and Pitarch *et al.* [15] also looked into the exact shape
of PFs, but they used only a few analytical PFs. While they are easy to
generate, FF PFs with varying ${B}_{p}$ values [25] or other analytical PF models may not cover full
variability of VSF shapes of natural particles. And some of them may not
even be suitable to represent the angular scattering by oceanic particles
[15].

## 2. METHODOLOGY

#### A. Measured VSFs

The VSFs at or near the surface (within 1 optical depth) were measured
in the field using a prototype multispectral volume scattering meter
(MVSM) and a LISST-100X sensor (Sequoia Scientific Inc., Bellevue,
Washington DC). The details of these measurements have been reported
[30,31]. The near full angular VSFs were generated by
combining the LISST-100X data at 30 near-forward angles
(0.07°–9.45°) and the MVSM data at 0.25°
angular resolution from 9.5° to 179°, both at
532 nm [30]. Laboratory
experiments using polystyrene spheres showed the combined LISST and
MVSM has an overall uncertainty of 5% in resolving the VSFs over the
entire angular range [30] and
inter-instrument comparison shows differences
$<10\%$ between the combined VSFs and the
scattering measurements by an ac-9 or ac-s (WET Labs Inc., Philomath,
Oregon), an ECO-VSF (WET Labs Inc.), and a HydroScat-6 (HOBI Labs
Inc., Bellevue, Washington DC) [31,36]. For this
study, we used a total of 116 VSFs collected in three coastal waters
of the United States (Chesapeake Bay in 2009, Mobile Bay in 2009, and
Monterey Bay in 2010) and in the North Atlantic Ocean during
NASA’s Ship Aircraft Bio-Optical Research (SABOR) cruise in
2014. The VSFs due to seawater were calculated following Zhang
*et al.* [22] using concurrently measured temperature and salinity
and subtracted from the bulk VSFs, generating
${\beta}_{p}$’s, which were used in the
following analysis.

To be used in HydroLight [37], a measured ${\beta}_{p}$ needs to be interpolated logarithmically between 0.1° and 180° at every 0.1°. In addition, the ${\beta}_{p}$ at near-forward angles need to be adjusted to ensure that their values at angles $<0.1\xb0$ follow a power-law function:

where $A$ and $m$ are estimated using ${\beta}_{p}$ values at ${\gamma}_{1}=0.1\xb0$ and ${\gamma}_{2}=0.2\xb0$.The particle scattering coefficient is then computed as

A precondition, $m<2$, is necessary for Eq. (8) to be physically valid [37]. There are 16 (out of 116) measured VSFs with $m\ge 2$. For these VSFs, we modified ${\beta}_{p}({\gamma}_{1})$ to ensure $m=1.9$. The tests were performed with those VSFs with $m\ge 2$ by changing $m$ values between $-1$ to 1.9, and the results showed that the differences on HydroLight-simulated ${R}_{\mathrm{rs}}$ were $<1.8\%$. These modified ${\beta}_{p}$ were used in generating the PF shapes in Fig. 1 and in the rest of this study.

#### B. HydroLight Simulation

We used HydroLight 5.2 to simulate ${R}_{\mathrm{rs}}$ following the methods described in the IOCCG report No. 5 [38]. Briefly, the absorption coefficients and backscattering coefficients of particles (${a}_{p}$ and ${b}_{bp}$) were generated as a function of chlorophyll concentration in 20 discrete values between 0.03 and $30.0\text{\hspace{0.17em}}\mathrm{\mu g}/L$, and at each value, 25 sets of spectral particle absorption and backscattering coefficients (${a}_{p}$ and ${b}_{bp}$) at 532 nm were generated, providing a total of 500 sets of ${a}_{p}$ and ${b}_{bp}$. The values of ${b}_{b}/(a+{b}_{b})$ at 532 nm are between 0.03 and 0.34. For each generated synthetic data set of ${a}_{p}$ and ${b}_{bp}$, ${R}_{\mathrm{rs}}$ was simulated ingesting each of the 116 measured ${\tilde{\beta}}_{p}$. Also, the particle scattering coefficients (${b}_{p}$) were estimated from ${b}_{bp}$ using ${B}_{p}$ associated with each ${\tilde{\beta}}_{p}$. Water is assumed homogeneous and infinitely deep, illuminated by a semi-empirical sky model (based on RADTRAN-X) with 0 cloud coverage and annual average climatology condition. The simulations were performed for ${\theta}_{s}$ ranging from 0° to 75° with an increment of 15°, ${\theta}_{v}$ ranging from 0° to 70° with an increment of 10°, and $\varphi $ ranging from 0° to 180° with an increment of 15°. In total, we had 32,016,000 simulated ${R}_{\mathrm{rs}}$ over a range of optical and viewing conditions that we believe cover a sufficiently extensive variability that an ocean color sensor may encounter.

## 3. RESULTS

#### A. ${\mathsf{R}}_{\mathsf{rs}}$$({\mathsf{\theta}}_{\mathsf{s}}=\mathsf{30}\xb0\mathsf{,}{\mathsf{\theta}}_{\mathsf{v}}=\mathsf{0}\xb0)$

The ${R}_{\mathrm{rs}}$ simulated with different particle PFs at ${\theta}_{s}=30\xb0$ and ${\theta}_{v}=0\xb0$ are shown in Fig. 2. As expected, ${R}_{\mathrm{rs}}$ increases with ${b}_{b}/(a+{b}_{b})$. For a set of $a$ and ${b}_{b}$ values, the variability of ${R}_{\mathrm{rs}}$ also increases with ${b}_{b}/(a+{b}_{b})$. This confirms that the impact of the particle VSF shapes on ${R}_{\mathrm{rs}}$ increases with ${b}_{bp}$. In the following, we will examine the impact of the VSF shape in four scenarios with increasing constraint on the backscattering: (1) no knowledge of ${\tilde{\beta}}_{p}$, (2) some knowledge of backscattering ratio, i.e., ${B}_{p}$, (3) some knowledge of exact backward shape of ${\tilde{\beta}}_{p}$, i.e., ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, and (4) some knowledge of both ${B}_{p}$ and ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$. In all four scenarios, the particle backscattering coefficient ${b}_{bp}$ is assumed to be known. To quantify uncertainty for ${R}_{\mathrm{rs}}$ due to ${\tilde{\beta}}_{p}$, we use pairwise percentage difference, $d{R}_{\mathrm{rs}}$ estimated as

### 1. No Knowledge of ${\tilde{\mathsf{\beta}}}_{\mathsf{p}}$

For two waters with the same ${b}_{bp}$ (and ${a}_{p}$), the current ocean color models, e.g., Eq. (2), would predict the same ${R}_{\mathrm{rs}}$ for a given viewing geometry. Therefore, any variations shown in the simulated ${R}_{\mathrm{rs}}$ would represent the uncertainty caused by the difference in ${\tilde{\beta}}_{p}$ shapes between the two waters [Fig. 3(a)]. With the natural variability exhibited by the ${\tilde{\beta}}_{p}$ shown in Fig. 1, the median differences in ${R}_{\mathrm{rs}}$ are $<15\%$ but the maximum differences could reach $>60\%$.

### 2. Knowledge of ${\mathsf{B}}_{\mathsf{p}}$

Among the 116 measured ${\tilde{\beta}}_{p}$, we found 44 pairs that have the
same ${B}_{p}$ values (within 0.5%). Since
${b}_{bp}$ and $a$ are assumed to be the same, the
same ${B}_{p}$ effectively means the same
$b$ (and $c$). With this additional constraint
on the general backward shape, the median difference in
${R}_{\mathrm{rs}}$ decreased to
$<10\%$, and the maximum difference
decreased to 35% [Fig. 3(b)]. Under the same set of constraints, Mobley
*et al.* [13] found a difference of up to 10% [dotted
line in Fig. 3(b)]
using Petzold’s average ${\tilde{\beta}}_{p}$ and a few other theoretical PFs
with the same ${B}_{p}$ as the Petzold’s average
${\tilde{\beta}}_{p}$. A difference of up to 20%
[dashed line in Fig. 3(b)] was found by Chami
*et al.* [29] using several measured PFs and the
${B}_{p}$-matched FF functions, while a
similar study by Tonizzo *et al.* [16] found differences up to
40% [dot-dashed line in Fig. 3(b)]. Our results are consistent with these
earlier studies, showing that the knowledge of
${B}_{p}$ is not sufficient to explain all
the variations in ${R}_{\mathrm{rs}}$ caused by PF shapes.

### 3. Knowledge of ${\tilde{\mathsf{\beta}}}_{\mathsf{p}}(\mathsf{\gamma}\ge \mathsf{90}\xb0)$

Among the 116 measured ${\tilde{\beta}}_{p}$, 66 pairs of measured particle PFs [Fig. 3(c)] have similar backward shape, with $<1\%$ difference between the logarithmic values of ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$. With this knowledge, the median difference in ${R}_{\mathrm{rs}}$ was lowered to $<6\%$ and the maximum difference to 20%.

### 4. Knowledge of Both ${\mathsf{B}}_{\mathsf{p}}$ and ${\tilde{\mathsf{\beta}}}_{\mathsf{p}}(\mathsf{\gamma}\ge \mathsf{90}\xb0)$

From Eq. (6), the same ${B}_{p}$ and ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$ are equivalent to the same ${\chi}_{p}$. Therefore, this scenario also represents the constraint of “knowledge of ${\chi}_{p}^{\prime \prime}$. Among the 116 measured ${\tilde{\beta}}_{p}$, we found 25 pairs that have the ${\chi}_{p}$ factors similar to each other within 1% at all angles between 90° and 180°. The ${R}_{\mathrm{rs}}$ simulated with the same ${b}_{bp}$ and similar ${\chi}_{p}$ factors are very similar to each other, with a median difference $<2\%$ and a maximum difference of $\sim 10\%$ [Fig. 3(d)].

Common to all four scenarios is that $d{R}_{\mathrm{rs}}$ increases with ${b}_{b}/(a+{b}_{b}$), seemingly reaching a saturation at approximately ${b}_{b}/(a+{b}_{b})>0.2$. As ${b}_{b}/(a+{b}_{b})$ increases, the backscattering will shift from water molecule-dominated to particle-dominated scattering, and therefore the shape of particle PFs will have an increasing impact on ${R}_{\mathrm{rs}}$. However, when ${b}_{b}/(a+{b}_{b})$ is high enough, the multiple scattering will be increased so much that the impact on ${R}_{\mathrm{rs}}$ due to the shape differences between particle PFs will not increase anymore [29].

#### B. ${\mathsf{R}}_{\mathsf{rs}}$$({\mathsf{\theta}}_{\mathsf{s}}\mathsf{,}{\mathsf{\theta}}_{\mathsf{v}})$

To examine how the impact of the particle PFs on ${R}_{\mathrm{rs}}$ varies with sun-viewing geometry, we plot the maximum percentage difference in ${R}_{\mathrm{rs}}$ in Fig. 4 as a function of sun zenith angles, viewing zenith, and azimuth angles. Without any knowledge on the shape of particle PF [corresponding to Fig. 3(a)], the maximum differences in ${R}_{\mathrm{rs}}$ vary between 52% and 107% [Fig. 4(a)]. With a knowledge of ${B}_{p}$ [corresponding to Fig. 3(b)], the simulated ${R}_{\mathrm{rs}}$ show a maximum difference varying between 26% and 54% [Fig. 4(b)]. If the backward shape of ${\tilde{\beta}}_{p}$ is similar within 1% [corresponding to Fig. 3(c)], the maximum differences in ${R}_{\mathrm{rs}}$ vary between 17% and 30% [Fig. 4(c)]. A knowledge of both ${B}_{p}$ and ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$ [corresponding to Fig. 3(d)] will reduce the maximum variations in ${R}_{\mathrm{rs}}$ between 8% and 16% [Fig. 4(d)]. For all four cases, the difference in ${R}_{\mathrm{rs}}$ follows the same geometric pattern: increasing with both the sun and viewing zenith angles, and being more prominent when viewing direction is opposite to the sun.

## 4. DISCUSSION AND CONCLUSIONS

Even though the commonly used ${R}_{\mathrm{rs}}$ models do not explicitly account for the shape of the VSFs [e.g., Eq. (2)], its impact on the color of the ocean is well recognized [4,8,23,39,40]. In fact, Jerlov [40] pointed out that the observed variation of the reflectance with sun angles is a direct consequence of the shape of the VSF. In Eqs. (4)–(6), three commonly used parameters describing the backward shape of a VSF are introduced: $B$, $\tilde{\beta}(\gamma \ge 90\xb0)$ and the $\chi $ factor. So, a logical question to ask is which of these three shape factors is more important in affecting ${R}_{\mathrm{rs}}$.

Under quasi-single-scattering approximation (QSS), Gordon [41] showed the remote sensing
reflectance (the detailed derivation is given by Mobley
*et al.* [42]):

Inserting the second equality of Eq. (6) into Eq. (11), we have

As shown in Eq. (11), when either ${\theta}_{v}$ or ${\theta}_{s}$ increases, the value of $\frac{1}{\mathrm{cos}\text{\hspace{0.17em}}{\theta}_{v}+\mathrm{cos}\text{\hspace{0.17em}}{\theta}_{s}}$ increases, and hence the impact of backward VSF increases. Therefore, the uncertainty due to the shape of VSF increases with both viewing and sun angles (Fig. 4).

For ${B}_{p}$, ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, or ${\chi}_{p}$, the field measurements indicate that the
natural variability of the particle PFs is much greater than what has been
assumed (Fig. 1).
Consequently, the impact of the shape of the particle PFs on
${R}_{\mathrm{rs}}$ may have been underestimated. For
example, Fig. 5 compares the
${R}_{\mathrm{rs}}$ predicted using the Lee’s
*et al.* [11] model, which was developed based on a limited range of
phase functions (green lines in Fig. 1), with the ${R}_{\mathrm{rs}}$ simulated using one particular phase
function (red line in Fig. 1). Both Lee *et al.* [11] and our study followed the IOCCG Report
No. 5 in generating the IOPs to drive the HydroLight. Since the
final formula of the Lee *et al.* [11] model accounts for only a
statistic average of the PFs used in their study, the scatter of the
comparison shown in Fig. 5
is expected. However, the Lee *et al.* [11] model consistently over-predicted
the ${R}_{\mathrm{rs}}$ simulated for this particular PF by
approximately 30%–50%. While the comparison shown in
Fig. 5 represents an
extreme case with the differences close to the maximum shown in
Fig. 3(a), it does
illustrate the potential impact of the VSF shape on the color of the
ocean.

Our results have demonstrated the increasing importance of ${B}_{p}$, ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, and ${\chi}_{p}$ in regulating ${R}_{\mathrm{rs}}$, and it will be of interest to quantify their respective importance in terms of fraction of total uncertainty in ${R}_{\mathrm{rs}}$ due to the VSF shapes that they can explain. To do this, we started with ${\tilde{\beta}}_{p}$, which describes the general shape of a VSF, includes ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, and can be used to derive both ${B}_{p}$ and ${\chi}_{p}$ [Eqs. (5) and (6)]. Let ${\sigma}_{s}$ denote the total uncertainty in ${R}_{\mathrm{rs}}$ explained by ${\tilde{\beta}}_{p}$. In the first test with no constraint on the shape, ${R}_{\mathrm{rs}}$ were found to differ up to 65% for nadir-viewed geometries [Fig. 3(a)]. In other words, the maximum uncertainty due to VSF shapes is 65%, i.e., ${\sigma}_{s}=0.65$. A constraint on ${B}_{p}$ was introduced in the second test, and, removing the uncertainty due to ${B}_{p}$, the maximum uncertainty due to VSF shapes was reduced to 35% [Fig. 3(b)], i.e., ${\sigma}_{s}^{2}-{\sigma}_{Bp}^{2}={0.35}^{2}$, where ${\sigma}_{Bp}$ denotes the uncertainty in ${R}_{\mathrm{rs}}$ explained by ${B}_{p}$. A constraint on ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$ was introduced in the third test, and, removing the uncertainty due to ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, the maximum uncertainty due to VSF shapes was reduced to 20% [Fig. 3(c)], i.e., ${\sigma}_{s}^{2}-{\sigma}_{>90}^{2}={0.20}^{2}$, where ${\sigma}_{>90}$ denotes the uncertainty in ${R}_{\mathrm{rs}}$ explained by ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$. A constraint on ${\chi}_{p}$ was introduced in the fourth test, and, removing the uncertainties due to ${\chi}_{p}$, the maximum uncertainty due to VSF shapes was reduced to 10% [Fig. 3(d)], i.e., ${\sigma}_{s}^{2}-{\sigma}_{\chi}^{2}={0.10}^{2}$, where ${\sigma}_{\chi}$ denotes the uncertainty in ${R}_{\mathrm{rs}}$ explained by ${\chi}_{p}$. Eq. (13) summarizes the results of these four tests:

Solving Eq. (13), we have ${\sigma}_{Bp}=0.55$, ${\sigma}_{>90}=0.62$, and ${\sigma}_{\chi}=0.64$. Therefore, among the total uncertainties in ${R}_{\mathrm{rs}}$ due to the shape of a VSF, about 71% ($={\sigma}_{Bp}^{2}/{\sigma}_{s}^{2}$) can be explained by ${B}_{p}$, 90% ($={\sigma}_{>90}^{2}/{\sigma}_{s}^{2}$) by ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, and 97% ($={\sigma}_{\chi}^{2}/{\sigma}_{s}^{2}$) by the ${\chi}_{p}$ factor.

From the perspective of ocean color remote sensing, it is difficult to
measure ${B}_{p}$ or ${\tilde{\beta}}_{p}(\gamma \ge 90\xb0)$, which requires $b$, which unfortunately cannot be retrieved
from ocean color, at least for now. On the other hand, both the QSS
approximation [Eq. (11)] and the exact ${R}_{\mathrm{rs}}$ models, such as the one developed by
Zaneveld [8,9], indicate that ${R}_{\mathrm{rs}}$ is directly proportional to the
$\chi $ factor. Gordon [43] showed an example to retrieve the VSF shape,
which was represented using an analytic equation first applied by
Beardsley and Zaneveld [44].
Recently, Zhang *et al.* [45] found that the ${\chi}_{p}$ factor of a VSF can be represented by a
linear mixing of two end members, the scattering by particles of sizes
much smaller than the wavelength of light, and the scattering by particles
of sizes much greater than the wavelength of light. The
${\chi}_{p}$ factors of both end members can be
derived analytically. They also showed that the mixing ratio can be
related to ${b}_{bp}$, which can be retrieved from ocean color
observation. However, this relationship is still preliminary and needs
further validation. Looking forward, it remains to be tested whether the
uncertainty in ${R}_{\mathrm{rs}}$ can be significantly constrained by
directly accounting for the $\chi $ factor of a VSF.

Our analysis was conducted at 532 nm, at which the VSFs were
measured. Based on Eq. (11), how the particle PFs affect the spectral variation of
${R}_{\mathrm{rs}}$ mainly depends on if and how the
${\chi}_{p}$ factor varies spectrally. While some
studies found no spectral dependence in ${\chi}_{p}$ for oceanic particles [18] or phytoplankton cultures [46,47], Chang *et al.* [13] indicated that spectral
differences could contribute to the up to 20% difference between simulated
${L}_{w}$ and measured ${L}_{w}$, and Chami *et al.*
[19] found
${\chi}_{p}$ varied spectrally by
$\pm 6\%$ in a non-blooming coastal water but up to
$\pm 20\%$ in algal cultures. How VSF shapes affect
the spectral variation of ${R}_{\mathrm{rs}}$ remains to be studied.

## Funding

National Science Foundation (NSF) (1458962); Directorate for Geosciences (GEO); National Aeronautics and Space Administration (NASA) (NNX13AN72G, NNX15AC85G); U.S. Naval Research Laboratory (NRL) (72-1C01); Office of Naval Research (ONR).

## Acknowledgment

This study has benefited from discussion with Dr. Zhongping Lee. The comments by two anonymous reviewers and by Dr. Emmanuel Boss have greatly improved the paper.

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