Detailed scattering contributions of different ocean water types to multiangle, polarimetric spaceborne observations in the UV-NIR regime

Global retrievals of ocean parameters greatly benefit from spaceborne missions equipped with passive multispectral, multiangle and polarimetric capabilities. Here we present a compendium of advanced radiative transfer simulations of  such observations at the top of the atmosphere, where the total reflectance and its polarization counterpart are partitioned among the scattering contributions from the atmosphere, a rough ocean surface, and the ocean body. The focus is on the spectral contributions of different water types in an extensive wavelength range, which impact the retrievability of their descriptive parameters. Beside accurately quantifying such contributions for observations in heritage ocean-color bands, the results highlight the detectability of the ocean signal in the total reflectance measured in the ultraviolet range of the spectrum. Also, polarization signatures emerge for highly-complex waters in the green region and at larger wavelengths. This textbook exercise in radiative transfer provides the basis for a correct interpretation of spaceborne measurements, and can be exploited in studies related to instrument design and spectral information content.

1 Introduction

Airborne and spaceborne remote sensing in the VISible (VIS) portion of the electromagnetic spectrum has revolutionized our ability to monitor the Earth’s system. For the oceans, it provides crucial insights into their biogeochemical and ecological processes, including phytoplankton dynamics, carbon cycling and oceanic productivity. With the coming of age of accurate multiangle polarimetric sensors, satellite observations have access to a considerably higher information content than previous instruments capable of measuring only the total reflectance (Dubovik et al., 2021; 2019; Hasekamp et al., 2019; Xu et al., 2019; Chowdhary et al., 2012; Stamnes et al., 2018; Gao et al., 2018; 2021). This improvement is even more significant as the range of observational wavelengths is extended to the UltraViolet (UV) and the Near InfraRed (NIR), which contain unique signatures of particulate matter present both in the atmosphere and the ocean (NASA, 2018a; 2018b). At longer (ShortWave InfraRed, SWIR) wavelengths, multiangle polarimetry also provides unique information on the air-water interface itself, such as sensitivity to contaminants afloat (Ottaviani et al., 2019; 2012).

A prime example of these technological advances is the successful launch of the NASA Plankton, Aerosol, Cloud and Ecosystem (PACE) satellite mission (Werdell et al., 2019), which offers the highest quality known to date of freely accessible, passive optical observations collected from a single platform, including those of two state-of-the-art polarimeters: the Hyper-Angular Rainbow Polarimeter, HARP2 (Martins et al., 2018), and the Spectropolarimeter for Planetary EXploration one, SPEXone (Hasekamp et al., 2019). The availability of such additional measurements will boost both the number and the retrieval accuracy of ocean remote sensing products, provided the modeling is informed with a thorough understanding of the Radiative Transfer (RT) processes involving the multitude of constituents of the ocean-atmosphere system: how does each scattering contribution vary with the pixel-specific viewing geometry, the observational wavelength and the water type? In each case, to what degree are the oceanic and atmospheric scattering contributions radiatively coupled? Only a few studies have analyzed selected aspects of such intricacies (Chowdhary et al., 2019; 2012; 2006; Ottaviani et al., 2018; 2008), mostly because an accurate decomposition of the signal simulated at any level in the ocean-atmosphere system requires specialized, advanced radiative RT computer codes.

Here we address these needs by exploiting an advanced vector code (Chowdhary et al., 2020) which can accurately quantify each of the contributions to the total Stokes vector (Born and Wolf, 1999) measured at the Top of The Atmosphere (TOA). We consider viewing geometries corresponding to a high and a low solar position in the sky, and relative azimuths for which both off-sunglint and sunglint-contaminated views can be defined in viewing-zenith-angle space. The extent of these regions depends on wavelength (here, six selected bands spanning the UV-NIR portion of the spectrum) and water type (oligotrophic Case I waters, global-average Case I waters, and coastal Case II waters containing minerals). Off-sunglint observations are of interest to remote sensing retrievals of atmospheric and/or oceanic properties, and may still contain small residual sunglint contributions which need to be corrected for. On the other hand, sunglint observations are of interest to remote sensing retrievals of ocean surface roughness and/or refractive index (Ottaviani et al., 2019) and aerosol absorption (Kaufman Y. et al., 2002; Ottaviani et al., 2013), provided that corrections are applied for any measurable atmospheric or oceanic contribution. For maritime atmospheres that contain low amounts of coarse-mode aerosol particles, as chosen here, the variations of the atmospheric and oceanic scattering contributions with scattering angle for each wavelength are small, and for most practical purposes can be ignored. Hence, the atmospheric and oceanic scattering contributions relative to the off-sunglint geometries identified here are indicative also of the scattering contributions at azimuth angles further away from the solar principal plane.

The paper is structured as follows. Section 2.1 introduces the atmosphere-ocean system model used for the RT computations. Section 2.2 describes the RT code itself and define the individual scattering contributions which add up to the total signal. We then discuss in detail the spectral behavior (between $385\text{\hspace{0.17em}nm}$ and $865\text{\hspace{0.17em}nm}$) of each inherent optical property (IOP) chosen as input for the RT calculations, for three different water types: oligotrophic Case I waters, global-average Case I waters, and coastal Case II waters containing minerals (Section 2.3). The constant, maritime atmosphere (Section 2.4) contains a low amount of salt particles and background fine-mode aerosols.

The results of the TOA simulations are presented in Section 3.1. Each wavelength is discussed separately in a dedicated subsection, with the visual aid of intuitive, cumulative diagrams showing the relative contribution of each scattering process as a function of water type and viewing geometry (the Supplementary Material contains the complete spectral collection). The implications for multiangle, polarimetric remote sensing is the subject of Section 3.2, where an alternative view of the cumulative plots in the form of bar diagrams emphasizes the spectral trend of each component. The conclusions of the study are drawn in Section 4. The Appendix contains tables compiled with the numerical values for the IOPs and other quantities of interest, together with a complete glossary for the many symbols and acronyms used in the paper.

2 Methodology and optical properties

2.1 Atmosphere-ocean system model

The atmosphere-ocean system model used for the RT calculations (Section 2.2) is depicted in Figure 1. For the (homogeneous) ocean layer we consider three different cases that cover a wide range of Earth’s water types (see Figure 2), with realistic properties taken from literature:

$•$ Scene A: typical Case I (i.e., open-ocean) waters with a Chlorophyll-a concentration [Chl] = 0.1 mg m⁻³ (Chowdhary et al., 2012).

$•$ Scene B: extremely oligotrophic, Case I waters with [Chl] = 0.03 mg m⁻³ (Claustre et al., 2008), and anomalously low amounts of colored dissolved organic matter (CDOM), as in Morel et al. (2007).

$•$ Scene C: highly turbid, Case II waters common for off-shore and coastal areas around the globe (Wei et al., 2021). We consider [Chl] = 3 mg m⁻³ and large amounts of suspended particulate matter ([SPM] = 10 mg m⁻³) as retrieved, for example, in the outer region of the Rio de la Plata estuary (Matano and Palma, 2010; Dogliotti et al., 2016; Moreira and Simionato, 2019).

Figure 1

Figure 1. Vertical structure of the atmosphere-ocean system model used in the radiative transfer simulations.

Figure 2

Figure 2. Water types modeled in the simulations. The image in the background is a non-deseasonalized MODIS composite of all daily Chl-a maps for the period 03/21-06/20 in 2014. Scene A represents typical open-ocean waters with low CDOM content, Scene B highly oligotrophic waters near the South Pacific gyre (Claustre et al., 2008) and Scene C highly turbid waters from the Rio de la Plata estuary (Matano and Palma, 2010; the quasi-true-color image in the inset is from Dogliotti et al., 2016).

The atmosphere is the same for all scenes and consists of three homogeneous layers, each hosting a unique mixture of molecules and spherical aerosol particles. The aerosol scenario is chosen to represent typical conditions over open oceans, therefore targeting the majority of pixels in a global image. For coastal regions, the fine-mode aerosols can be more abundant and more absorbing. Conversely, the coarse-mode aerosols can be larger in size and at higher loads in regions affected by dust outflows. While the numbers discussed in Section 3 can change depending on the specific type of aerosols, the main conclusions drawn for the contributions of different water types to TOA remote sensing observations and the spectral trends remain valid. The derivation of the IOPs used as input for the RT computations is discussed in detail in Section 2.3 (ocean) and Section 2.4 (atmosphere).

2.2 Radiative transfer (RT) simulations

For the forward RT computations, we employ the extended General Adding Program (eGAP) used to generate benchmark results in Chowdhary et al. (2020) and routinely used at Goddard Institute for Space Studies (GISS) to model measurements by the Research Scanning Polarimeter (RSP) sensor (Cairns et al., 1999) over ocean for the retrieval of aerosol, cloud, surface, and ocean properties (Chowdhary et al., 2001; 2002; 2005; 2012; Stamnes et al., 2018; Ottaviani et al., 2019; 2012). The eGAP code simulates the Stokes vector of the upward and downward light field in the atmosphere-ocean system at any desired set of illumination and viewing angles, altitude and user-specified accuracy.

Of interest to this work is the ability of eGAP to decompose the Stokes vector $\mathbf{S}$ into the scattering contributions schematically depicted in Figure 3, corresponding to light that has interacted with:

$•$ only the atmosphere $({\mathbf{S}}_1)$;

$•$ only the ocean surface, i.e., the direct sunglint contribution $({\mathbf{S}}_2)$;

$•$ only the ocean body (including transmission through the surface), i.e., the direct ocean contribution $({\mathit{S}}_3)$;

$•$ both the atmosphere and the ocean surface, i.e., the diffuse skyglint contribution $({\mathit{S}}_4)$;

$•$ all the atmosphere and ocean surface and the ocean body components, i.e., the diffuse ocean contribution $({\mathit{S}}_5)$.

Figure 3

Figure 3. Contributions of the different scattering processes to the light field at the top of the atmosphere. Note that each sparkly symbol can actually represent more than a single scattering event.

Hence, ${\mathit{S}}_4$ describes the radiative coupling of light scattered by the atmosphere and ocean surface, and ${\mathit{S}}_5$ describes the radiative coupling of light scattered by the atmosphere and ocean body.

The total reflectance $(R_I)$, the polarized reflectance $(R_{\mathrm{P}})$ and the Degree of Linear Polarization (DoLP) are defined in terms of the I, Q, and U components of $\mathit{S}$, as in Equations 1–5 below:

$R_I=\frac{\pi r_0^2}{F_0\cos {\theta }_0}I(1)$

$R_Q=\frac{\pi r_0^2}{F_0\cos {\theta }_0}Q(2)$

$R_U=\frac{\pi r_0^2}{F_0\cos {\theta }_0}U(3)$

$R_{\text{P}}=\sqrt{{R_Q}^2+{R_U}^2}(4)$

$DoLP=\frac{R_{\text{P}}}{R_I}\times 100(5)$

where the annual average of the extraterrestrial solar irradiance, ${\mathit{\text{F}}}_0$, is corrected for the Sun-Earth distance $({\mathit{\text{r}}}_0)$ and for the cosine of the solar zenith angle $({\theta }_0)$. Effects of circular polarization, described by the fourth Stokes parameter $(\mathit{\text{V}})$, are ignored because they are largely irrelevant to remote sensing (Kawata, 1978).

The waviness of the ocean surface is described by the slope distribution parameterized by Cox and Munk (1954) as a function of the wind speed, $W$, specified at 10 m above the ocean surface. For all scenes, we use W = 7 ms⁻¹ representative of the global average (Kent et al., 2013). The corresponding reflection and transmission matrices are computed using the geometrical optics approach (Chowdhary, 1999), and include shadowing effects of the wave facets (Sancer, 1969; Saunders, 1967). Renormalization of the scattering matrix is applied to ensure conservation of energy (Chowdhary et al., 2006). Note that the treatment for shadowing effects was developed for the total reflectance component, and at extreme geometries the renormalization can occasionally cause the sum of all contributions to the total polarized reflectance at the TOA to exceed $100\%$ (although by just a few percent). Since the process involves multiple reflections at the surface, these spurious cases are corrected by subtracting the excess $R_{\text{P}}$ from the diffuse component ($p_4$, see Equation 22). For the spectral refractive index of seawater, $m_w(\lambda )$ where $\lambda$ is the wavelength, we use the data tabulated by Segelstein (1981). The viewing geometries are discussed in Section 3.1.

The simulation wavelengths were chosen for their relevance in the remote sensing of atmospheric and oceanic parameters, and include a list of channels typically used by heritage satellite instruments for aerosol and ocean color retrievals: 410 nm (including a small range of adjacent wavelengths, also referred to as “Violet” in the text that follows), 470 nm (“Blue”), 550 nm (“Green”), 670 nm (“Red”) and 865 nm (“NIR”). We add simulations at 385 nm (“UV”) to address observations of ocean color in the UV, but also other aspects such as the corrections in retrievals of absorbing aerosols from Total Ozone Mapping Spectrometer (TOMS) and Ozone Monitoring Instrument (OMI) data (e.g., Torres et al. 2007).

2.3 Oceanic inherent optical properties

From a RT perspective, the differences among the three scenes are quantified by the following IOPs: the particulate scattering coefficient $b_p$, the particulate backscattering coefficient $b_{b,p}$, and the total absorption coefficient a (Chowdhary et al., 2019). In each case, the optical depth of the (homogenous) ocean layer is set to 20. For the RT properties of particles that co-vary with [Chl] (i.e., all particles except minerals) we use the detritus-plankton (DP) mixtures described in Chowdhary et al. (2012) for phytoplankton-based particles in Case I oceans. The associated IOPs are the DP particulate scattering coefficient $b_{DP}$, the backscattering efficiency $q_{DP}$, the backscattering coefficient $b_{b,DP}$ = $q_{DP}\times b_{DP}$ and the total absorption coefficient $a_{DP}$. The scattering IOPs are obtained from the bio-optical model described in Loisel and Morel (1998), Morel and Maritorena (2001), and Huot et al. (2008). The total absorption coefficient $a_{DP}$ can also be obtained from these models except for Scene B, for which the amount of CDOM is anomalously low. In this case we follow Chowdhary et al. (2012) and obtain $a_{DP}$ from the diffuse attenuation coefficient $K_d$ (Gordon, 1989) measured by Morel et al. (2007) for the Pacific Ocean gyre. Note that $K_d$ measurements include absorption by pure seawater, and therefore so does $a_{DP}$. To compute the DP scattering matrices, we assume both types of particles are homogeneous spheres and assign to each of them a unique size distribution and refractive index. These particles are then mixed as a function of [Chl] (Chowdhary et al., 2012). Figure 4 and Appendix provide a summary of the scattering and absorption spectral properties of these mixtures for [Chl] = 0.1 (Scene A), 0.03 (Scene B), and 3.0 (Scene C) mg m⁻³. To study the individual impacts of anomalies in CDOM or mineral content, Figure 4 also shows the results for Scene B′, which is the same as Scene B except for containing typical amounts of CDOM, and Scene C′, which is the same as Scene C without mineral particles.

Figure 4

Figure 4. Spectral IOPs for DP particulates in each scene: (a) scattering coefficient; (b) backscattering efficiency; (c) backscattering coefficient; (d) absorption coefficient.

The determination of the scattering properties of mineral particles in Scene C follows Woźniak and Stramski (2004). Assuming homogeneous spherical particles, they retrieve a complex refractive index m($\lambda$)$=1.15-0.0079935\exp (-0.007186\lambda )$i and a Junge-type differential size distribution between 0.05 and $50\mathrm{\mu }\text{m}$ with a Junge exponent $\gamma =-4$ (Loisel et al., 2008). The corresponding mineral density is 2.77 $\times$ 10⁶ mg m⁻³ (Woźniak and Stramski, 2004), which leads to a particle number density of 1.95 $\times$ 10⁸ m⁻³ for [SPM] = 10 g m⁻³. The resulting IOPs are labeled with the subscript ‘M’ and are plotted in Figure 5. The explicit values are listed in Table A2.

Figure 5

Figure 5. Spectral IOPs for mineral particles in Scene C ([SPM] = 10 g/m³). The total scattering coefficient $b_M$ is divided by a factor of 100.

In the notation that follows we drop for simplicity the dependence of the IOPs of DP particles on [Chl] and of mineral particles on [SPM]. Equations 6–9 describe the scattering IOPs and scattering matrices for particulates (denoted by the subscript ‘p’), which includes DP particles for all scenes and mineral particles for Scene C:

$b_p\left(\lambda \right)=b_{DP}\left(\lambda \right)+b_M\left(\lambda \right)(6)$

$q_p\left(\lambda \right)=\frac{b_{DP}\left(\lambda \right)\times q_{DP}+b_M\left(\lambda \right)\times q_M}{b_{DP}\left(\lambda \right)+b_M\left(\lambda \right)}(7)$

$b_{b,p}\left(\lambda \right)=b_p\left(\lambda \right)+q_p\left(\lambda \right)(8)$

${\mathit{F}}_p\left(\lambda \right)=\frac{b_{DP}\left(\lambda \right)\times {\mathit{F}}_{DP}+b_M\left(\lambda \right)\times {\mathit{F}}_M\left(\lambda \right)}{b_{DP}\left(\lambda \right)+b_M\left(\lambda \right)}(9)$

In all scenes, a volume element of the bulk ocean (subscript ‘blk’ for the associated quantities) includes DP particles and pure seawater. In addition, bio-optical models for Case I waters (Chowdhary et al., 2012) apply to CDOM amounts higher than in Scene B and mineral amounts lower than in Scene C. The IOPs and scattering matrices in these cases are described by Equations 10–13:

$b_{blk}\left(\lambda \right)=b_w\left(\lambda \right)+b_p\left(\lambda \right)(10)$

$b_{b,blk}\left(\lambda \right)=b_{b,w}\left(\lambda \right)+b_{b,p}\left(\lambda \right)=b_w\left(\lambda \right)\times {\mathit{\text{q}}}_w+b_p\left(\lambda \right)\times {\mathit{\text{q}}}_p\left(\lambda \right)(11)$

${\mathit{F}}_{blk}\left(\lambda \right)=\frac{b_w\left(\lambda \right)\times {\mathit{F}}_w\left(\lambda \right)+b_{DP}\left(\lambda \right)\times {\mathit{F}}_{DP}+b_M\left(\lambda \right)\times {\mathit{F}}_M\left(\lambda \right)}{b_w\left(\lambda \right)+b_{DP}\left(\lambda \right)+b_M\left(\lambda \right)}(12)$

$a_{blk}\left(\lambda \right)=a_{DP}\left(\lambda \right)+a_M\left(\lambda \right)(13)$

where $b_w$ and $b_{b,w}$ are the scattering and backscattering coefficients, and ${\mathit{F}}_w$ and $q_w$ the scattering matrix and the backscattering efficiency, for pure seawater. As mentioned above, absorption by pure seawater is implicitly included in $a_{DP}$. We use the values of $b_w$ from Smith and Baker (1981), assume Rayleigh scattering with a depolarization factor 0.039 to compute ${\mathit{F}}_w$, and set $q_w$ to 0.5 (cf. Chowdhary et al., 2012; Chowdhary et al., 2019).

Figures 6a,b show $b_{blk}(\lambda )$ and $b_{b,blk}(\lambda )$ for all scenes and the corresponding values for pure seawater. The values at 865 nm are omitted because virtually no light emerges from the ocean at this wavelength except for Scene C, where scattering by large amounts of highly-reflective minerals compensates for water absorption. Scenes C and C′ exhibit in fact the largest particulate scattering at UV-VIS wavelengths, even when no minerals are present as in Scene C′, because of the large values of $b_{DP}$. Note further that $b_{b,w}>b_{b,DP}$ for Scenes A, B and B′ when $\lambda \le 600$ nm (Figure 4c). Therefore, for these scenes $b_{b,blk}$ is comparable to $b_{b,w}$ in the UV-Violet, consistent with the remarks in Chowdhary et al. (2019) for pure seawater properties in clear oceans.

Figure 6

Figure 6. Spectral behavior, for the different scenes, of the bulk (a) scattering coefficient; (b) backscattering coefficient; (c) absorption; (d) ocean single-scattering albedo. The curves in dark blue are for pure seawater.

Figures 6c,d show the variations in $a_{blk}(\lambda )$ and ${\omega }_{blk}(\lambda )$, the latter computed from Equation 14:

${\omega }_{blk}\left(\lambda \right)=\frac{b_{blk}\left(\lambda \right)}{b_{blk}\left(\lambda \right)+a_{blk}\left(\lambda \right)}(14)$

The strong absorption of pure seawater drives the rapid increase in $a_{blk}$ and the corresponding decrease in ${\omega }_{blk}$ at $\lambda >550$ nm. For Scene C and C′, $a_{blk}$ is much larger than for all other scenes because of the strong absorption of DP particulates when [Chl] = 3.0 mg m⁻³, although ${\omega }_{blk}\gg 0.8$ in the mid-VIS because of the large values of $b_{DP}$ and, for Scene C, also because of the large values of $b_M$. For the other scenes, ${\omega }_{blk}\le 0.8$ at all wavelengths.

The IOPs for bulk ocean waters can be used to obtain a first-order approximation for the spectral variation in ocean brightness. Let $A_{blk}(\lambda )$ define the ratio of upwelling to downwelling irradiance just below the ocean surface:

$A_{blk}\left(\lambda \right)=\mathit{\text{f}}\times \frac{b_{b,blk}\left(\lambda \right)}{a_{blk}\left(\lambda \right)}(15)$

with $\mathit{\text{f}}\approxeq 0.3$ (Morel, 1988, and references therein). Using multiple scattering computations, Morel (1991) provides a more detailed expression for f as a function of ${\theta }_0$ and of the ratio ${\eta }_b(\lambda )$, which quantifies the dominance of backscattering in seawater as shown in Equation 16:

${\eta }_b\left(\lambda \right)=\frac{b_w\left(\lambda \right)\times q_w}{b_{blk}\left(\lambda \right)}(16)$

Figure 7a shows the spectral variations of f for all scenes and for ${\theta }_0=20\text{°}$. Note that f varies slightly with wavelength except for Scene C, where it remains constant at 0.335 (although it still varies with ${\theta }_0$) because the large scattering coefficients of the mineral particles (see Table A2) cause ${\eta }_b$ to be one order of magnitude smaller than for the other scenes. Morel (1991) further remarks that f increases also with ${\omega }_{blk}$ if ${\omega }_{blk}>0.8$. For Scene C, ${\omega }_{blk}>0.90$ for UV-VIS wavelengths (see Figure 6d); however, no parameterizations are available for f$({\omega }_{blk})$. Finally, Morel (1991) derives f using the particulate scattering function measured by Petzold (1972), whose ${\mathit{\text{q}}}_p=0.019$ is comparable to $0.018\le {\mathit{\text{q}}}_p\le 0.021$ obtained from Equation 7 for Scene C. However, his ${\mathit{\text{b}}}_p$ values are at least one order of magnitude smaller than those obtained from Equation 6. Hence, the constant value of f for Scene C will likely lead to an underestimation of ${\mathit{\text{A}}}_{blk}$ if rigorous multiple scattering computations are replaced with Equation 15.

Figure 7

Figure 7. Spectral variations of (a) f and (b) ${\text{A}}_{\mathit{blk}},$ for ${\theta }_0=20\text{°}$.

Figure 7b shows $A_{blk}$ obtained from Equation 15. From Scene A to Scene B′ to Scene C′, the ocean brightness increases in the UV-Violet and decreases for $\lambda >510$ nm. The resulting “hinge” point in response to variations in [Chl] is a known feature of open-ocean waters (Nobileau and Antoine, 2005). The very low CDOM amount in Scene B causes this hinge point to shift towards larger wavelengths, whereas the surplus of minerals in Scene C shifts it towards smaller wavelengths. Also, Scene C is brighter than Scene C′ by almost one order of magnitude due to the presence of minerals, and $\ge 15$ times brighter than Scenes A and B in the Green-Red spectral range. Only in the UV-Violet range does Scene B become brighter than Scene C, although as pointed out earlier $A_{blk}$ is likely to be underestimated by Equation 15 for Scene C (a statement that can be true also in the Green-Red range). Finally, we remark that Equation 15 does not account for the bidirectionality and the polarization of upwelling underwater light, whose impacts will be discussed in Section 3.

2.4 Atmospheric inherent optical properties

We simulate clean, maritime atmospheric conditions by dividing the atmosphere in three homogeneous layers. The first layer resides just above the ocean surface, is 1-km thick and contains background air molecules as well as non-absorbing, coarse-mode particles representative of sea salt. Their effective radius is $r_{eff}=1.5\mathrm{\mu }m$, the effective variance $v_{eff}=0.3$, the optical thickness ${\tau }_{a,c}=0.02$ at 550 nm and the (spectrally constant) refractive index ${\mathit{\text{m}}}_c=1.41+0$i. The second layer, situated on top of the first, is 2-km thick and contains molecules as well as non-absorbing fine-mode, sulfate-like particles with $r_{eff}=0.2\mathrm{\mu }m$, $v_{eff}=0.15$, ${\tau }_{a,f}=0.1$ and ${\mathit{\text{m}}}_f=1.45+0i$. The top layer contains only background air molecules and 300 DU of ozone (O₃). The spectral variations of the aerosol (${\tau }_{a,f}$, ${\tau }_{a,c}$) and of the molecular (${\tau }_m$, from Hansen and Travis (1974)) optical depths are given in Table A3 for each layer, together with the depolarization factor ${\delta }_m$ used to compute the molecular scattering matrix (Bodhaine et al., 1999).

The scattering and absorption properties of aerosols and molecules affect the transmission of light through the atmosphere. Let $t^{\downarrow }({\theta }_0,\lambda )$ denote the total transmittance of the atmosphere for downwelling sunlight reaching the ocean surface when the solar zenith angle is ${\theta }_0$ (red curve in Figure 8a):

${\mathit{\text{t}}}^{\downarrow }\left({\theta }_0,\lambda \right)={\mathit{\text{t}}}_{dir}^{\downarrow }\left({\theta }_0,\lambda \right)+{\mathit{\text{t}}}_{dif}^{\downarrow }\left({\theta }_0,\lambda \right)(17)$

where $t_{dir}^{\downarrow }$ (orange curve) and $t_{dif}^{\downarrow }$ (blue curve) are the transmittances for direct (i.e., unscattered) and diffuse light, respectively, with $t_{dir}^{\downarrow }=1$ and $t_{dif}^{\downarrow }=0$ if the atmosphere is completely transparent. For ${\theta }_0=20\text{°}$ (solid lines), $t_{dir}^{\downarrow }$ increases with wavelength, whereas $t_{dif}^{\downarrow }$ decreases due to the spectral decrease in ${\tau }_a$ and ${\tau }_m$. Also, $t_{dir}^{\downarrow }>t_{dif}^{\downarrow }$ especially at longer wavelengths. For ${\theta }_0=50\text{°}$ (dashed lines), $t_{dif}^{\downarrow }$ is larger than for ${\theta }_0=20\text{°}$ because of the longer path traveled by the downwelling light to the ocean surface, causing a corresponding decrease in $t_{dir}^{\downarrow }$ and in $t^{\downarrow }$.

Figure 8

Figure 8. (a) Total (red), direct (orange) and diffuse (blue) spectral transmittances for upwelling and downwelling light. Solid lines are for ${\theta }_0=20\text{°}$ and dashed lines for ${\theta }_0=50\text{°}$. (b) Reflectance for the three scenes and for geometry 1 (solid) and 2 (dashed); (c) Products of transmittances; (d) Direct (dashed) and diffuse (solid) two-pass transmittances, for ${\theta }_0=\theta =20\text{°}$ (green), ${\theta }_0=\theta =50\text{°}$ (red), and for ${\theta }_0=50\text{°}$ and $\theta =20\text{°}$ (blue).

Note also that the upwelling counterparts ($t^{\uparrow }$, $t_{dir}^{\uparrow }$ and $t_{dif}^{\uparrow }$) of the downwelling transmittances, for light reaching a satellite at a viewing angle $\theta$, are equal to those in Equation 17 when $\theta ={\theta }_0$ because symmetry relationships ensure reciprocity (Hovenier, 1969).

The two-pass transmittance $\chi (\theta ,{\theta }_0,\lambda )$ for light traveling from the TOA to the ocean and from the ocean back to a sensor at the TOA at a viewing angle $\theta$ can be defined as:

$\chi \left({\theta }_0,\theta ,\lambda \right)=t^{\uparrow }\left(\theta ,\lambda \right)\times t^{\downarrow }\left({\theta }_0,\lambda \right).(18)$

It should be stressed that $t^{\downarrow }$ is a scalar flux quantity, obtained by integrating the Bidirectional Transmission Distribution Function (BTDF) for downward light over all transmission angles, and it ignores the polarization of incident light as well as the polarization of light scattered in the atmosphere (as opposed to the exact computations in eGAP). The same arguments are true for $t^{\uparrow }$, obtained instead by integrating the upward BTDF over all incident angles. Hence, $\chi$ in Equation 18 represents also the two-pass transmission for light traveling through the atmosphere if the lower boundary behaves as a Lambertian reflector (De Haan et al., 1991; Qin et al., 2001).

Combining Equations 17 and 18, and omitting the dependence on wavelength and angles, we obtain:

$\chi =t_{dir}^{\uparrow }\times t_{dir}^{\downarrow }+\left(t_{dir}^{\uparrow }\times t_{dif}^{\downarrow }+t_{dif}^{\uparrow }\times t_{dir}^{\downarrow }+t_{dif}^{\uparrow }\times t_{dif}^{\downarrow }\right)\equiv {\chi }_{dir}+{\chi }_{dif}.(19)$

The product term defining ${\chi }_{dir}$ contains only direct transmittance components and corresponds to the two-pass transmittance for light that is attenuated but not scattered in the atmosphere, i.e., it describes the atmospheric impact on components ${\mathit{S}}_2$ and ${\mathit{S}}_3$. Figure 8d shows that ${\chi }_{dir}$ increases with wavelength just like the individual direct-transmittance components in Figure 8a. On the other hand, the three product terms adding up to ${\chi }_{dif}$ in Equation 19 contain at least one diffuse transmittance component and therefore at least one scattering event in the atmosphere. Hence, ${\chi }_{dif}$ describes the atmospheric impact on contributions ${\mathit{S}}_4$ and ${\mathit{S}}_5$. Two out of the three product terms in ${\chi }_{dif}$ contain a direct transmittance component, and the spectra of direct and diffuse transmittances are anti-correlated (cf. blue and orange curves in Figure 8a). To understand the non-trivial spectral behavior of ${\chi }_{dif}$, Figure 8c shows the spectral variation of each product term for ${\theta }_0,\theta \in \{20\text{°},50\text{°}\}$.

All terms involving the product of one diffuse and one direct transmittance component decrease smoothly with wavelength for $\lambda ⪆500$ nm, but they exhibit a maximum in the UV-Blue resulting from the trade-off between the increase in direct transmittance and the decrease in diffuse transmittance when the atmospheric optical depth decreases. These transmittances vary with both ${\theta }_0$ and $\theta$ but become comparable in the UV-Blue (cf. again blue and orange curves in Figure 8a), and the maximum shifts towards larger wavelengths for larger ${\theta }_0,\theta$. The gray curves in Figure 8c show the results for the product term $t_{dir}^{\uparrow }\times t_{dir}^{\downarrow }$ in ${\chi }_{dif}$, which decreases rapidly with increasing wavelength across the entire UV-VIS spectrum just like the individual diffuse transmittance components in Figure 8a. These curves become comparable to the other product terms depending on the specific $({\theta }_0,\theta )$ pair. The overall atmospheric impact on TOA contributions ${\mathit{S}}_4$ and ${\mathit{S}}_5$ in the UV, shown in Figures 8d, is described by the sum of all product terms in ${\chi }_{dif}$. It therefore depends on the balance between diffuse and direct transmittances, which by themselves vary with ${\theta }_0$ and $\theta$. The cyan, green, and red curves are for $({\theta }_0,\theta )=(20\text{°},50\text{°})$ or $(50\text{°},20\text{°}),(20\text{°},20\text{°})$ and $(50\text{°},50\text{°})$, respectively. The maxima in Figure 8c manifest again in Figure 8d, but they are less pronounced and at slightly smaller wavelengths.

2.5 Spaceborne ocean color observations: semi-analytical model

Assuming an isotropic water-leaving radiance, its TOA contribution can be predicted to a first approximation for any given ${\theta }_0$, $\theta$, $\lambda$, [Chl], and [SPM]. The associated reflectance ${\rho }_w$ (i.e., the ratio between radiance and the extraterrestrial solar flux) can be approximated following Gao et al. (2000) and Gao et al. (2009):

$\begin{array}{ccc}\hfill {\rho }_w\left({\theta }_0,\theta ,\lambda ,\left[\text{Chl}\right],\left[\text{SPM}\right]\right)\propto & \chi \left({\theta }_0,\theta ,\lambda \right)\times A_{blk}\left({\theta }_0,\lambda ,\left[\text{Chl}\right],\left[\text{SPM}\right]\right)=\hfill & \left[{\chi }_{dir}\left({\theta }_0,\theta ,\lambda \right)+{\chi }_{dif}\left({\theta }_0,\theta ,\lambda \right)\right]\times A_{blk}\left({\theta }_0,\lambda ,\left[\text{Chl}\right],\left[\text{SPM}\right]\right)\propto \hfill \\ \hfill & \hfill & {\mathit{\text{S}}}_3\left({\theta }_0,\lambda ,\left[\text{Chl}\right],\left[\text{SPM}\right]\right)+{\mathit{\text{S}}}_5\left({\theta }_0,\lambda ,\left[\text{Chl}\right],\left[\text{SPM}\right]\right)\hfill \end{array}(20)$

Note that we have dropped the bold-font notation for ${\mathit{S}}_3$ and ${\mathit{S}}_5$, since the relationship applies to the radiance and can therefore be considered scalar. Morel and Gentili (1996) discuss in detail the proportionality factor in Equation 20, which is equal to $\mathcal{R}/\mathcal{Q}$, where $\mathcal{R}$ describes the transmission and internal reflection properties of the ocean surface and $\mathcal{Q}$ accounts for the angular variation of the upwelling underwater light. Note that $\mathcal{R}$ depends on the Fresnel coefficients for the ocean surface, which vary with $({\theta }_0,\theta )$ and also with $\lambda$ by means of $m_w(\lambda )$. However, $m_w(\lambda )$ is fairly constant at visible wavelengths (Zhang and Hu, 2009) so that the spectral dependence of $\mathcal{R}$ can be ignored. This is not necessarily true for $\mathcal{Q}$, which depends on the IOPs of the bulk ocean and therefore also on $\lambda$ (Equations 10–13). Nevertheless, Morel et al. (2002) found that $\mathcal{Q}$ remains fairly constant for open-ocean waters (scenes A and B) in the $412-670$ nm spectral range if [Chl] remains the same. Hence, Equation 20 can be used to approximate the relative changes in the spectral variations of ${\rho }_w$ for scenes A and B. In Figure 8b the solid curves are the spectral values of ${\rho }_w$ for ${\theta }_0=20\text{°}$ and $\theta =0\text{°}$ (geometry 1), and the dashed curves for ${\theta }_0=50\text{°}$ and $\theta =60\text{°}$ (geometry 2). While the results for Scene A and B are more accurate, the results for Scene C (a lower bound for the actual ${\rho }_w$) are included to show that Scene C is still by far the brightest except for in the UV-Blue. The values at 865 nm are excluded because they are very small (${\rho }_w<3\times 10^{-6}$ for scenes A and B, and ${\rho }_w=0.002$ for Scene C).

A comparison with Figure 7b shows that factoring in the two-pass atmospheric transmittance does not impact the location of the “hinge point” seen for $A_{blk}$. We recall that Equation 20 ignores light that has bounced back and forth between the atmosphere and ocean, which can become substantial in the UV (Chowdhary et al., 2019). It also does not provide information on the polarized component of ${\rho }_w$, can underestimate the actual contribution for Scene C obtained from rigorous RT computations, and assumes that the water-leaving radiance is isotropic (Frouin et al., 2019) which to a first approximation is true in the UV-VIS (Chowdhary et al., 2006). However, the approximated curves in Figure 8b are useful to interpret the relative contributions of the water-leaving radiance to TOA observations obtained from eGAP, as discussed in the next section.

3 Results and discussion

3.1 TOA simulations of total and polarized reflectance

We first present the simulations of the total signals computed at the TOA. The angular and spectral variations of $R_I$ and $R_{\text{P}}$ for all scenes are given in Figure 9 for ${\theta }_0=20\text{°}$, and in Figure 10 for ${\theta }_0=50\text{°}$. The solid and dashed lines correspond to azimuth angles relative to the Sun of $\phi =0\text{°}$ and $\phi =50\text{°}$, respectively. Sunglint observations along the principal-plane $(\phi =0\text{°})$ are less typical than other remote sensing geometries, but are included because they provide useful insights on the maximum range of variation of some scattering contributions. The x-axis is the viewing zenith angle, $\theta$, spanning the $[0\text{°},60\text{°}]$ range with a fine grid.

Figure 9

Figure 9. Spectral and angular variations of total (top row) and polarized (bottom row) reflectance for Scene A (left column), B (middle column) and C (right column), and for ${\theta }_0=20\text{°}$.

Figure 10

Figure 10. Same as 9, but for ${\theta }_0=50\text{°}$.

Note that ${\theta }_0=50\text{°}$ is very close to the Brewster angle (Born and Wolf, 1999) for a flat surface, which in the case of pure water is $\approxeq 53°$. At this angle of incidence, the component of the electric field vibrating parallel to the principal plane vanishes upon specular reflection, and water therefore acts as a perfect polarizer (DoLP = $100\%$ for the specularly-reflected beam). The Brewster observational geometry can be obtained also if $\phi \ne 0\text{°}$, because real oceans are never perfectly flat and the wave facets can easily direct the specular reflection off the principal plane. However, Brewster geometries achieved away from the principal plane have a lesser impact on the TOA polarized radiance.

Sunglint is a prominent feature for any ${\theta }_0$ when $\phi =0\text{°}$. At high illumination angles $({\theta }_0=50\text{°})$, $R_I$ and $R_{\text{P}}$ within the sunglint region increase with wavelength following the spectral behavior of the dominating contribution, ${\mathit{S}}_2$. When $\phi =50\text{°}$, residual sunglint is present for ${\theta }_0=20\text{°}$ and also for ${\theta }_0=50\text{°}$ but only for $\lambda ⪆550$ nm.

The TOA reflectances correspond to the sum of the individual contributions $r_k$ (for $R_I$) and $p_k$ (for $R_{\text{P}}$), each associated with the respective scattering process in Figure 1, as per Equations 21, 22:

$R_I={\displaystyle \sum _{k=1,\dots ,5}}{r}_k(21)$

$R_{\text{P}}={\displaystyle \sum _{k=1,\dots ,5}}{p}_k(22)$

For the relative contributions we use overhead hat symbols, viz.:

${\widehat{r}}_k=\frac{r_k}{R_I}\times 100(23)$

${\widehat{p}}_k=\frac{p_k}{R_{\text{P}}}\times 100(24)$

Note that Equations 23, 24 also imply that any individual relative DoLP contribution to the total DoLP:

${\widehat{DoLP}}_k=\frac{{DoLP}_k}{DoLP}\times 100=\frac{100\times p_k/R_I}{100\times R_{\text{P}}/R_I}\times 100={\widehat{p}}_k(25)$

is identically equal to the respective ${\widehat{p}}_k$, so Equation 25 states that the relative contributions to the polarized reflectance and to the DoLP are the same.

To describe the dominant contribution to the surface reflectance, $r_{24}=r_2+r_4$, Equation 26 defines the surface reflectance quotient $Q_{24}$ (%) as:

$Q_{24}\equiv \frac{r_2}{r_2+r_4}\times 100=\frac{{\widehat{r}}_2}{{\widehat{r}}_2+{\widehat{r}}_4}\times 100(26)$

where the right-hand side follows after substituting Equation 23. If the ocean surface reflection is dominated by direct sunglint, then $Q_{24}>50\%$. The quantification of the $r_3$ component is relevant to instrument calibration procedures using the sunglint, and for the retrieval of parameters descriptive of the water surface (Ottaviani et al., 2019).

Similarly to $Q_{24}$, to describe the dominant contribution to the ocean reflectance, $r_{35}=r_3+r_5$, Equation 27 defines the ocean reflectance quotient $Q_{35}$ as:

$Q_{35}\equiv \frac{r_3}{r_3+r_5}\times 100=\frac{{\widehat{r}}_3}{{\widehat{r}}_3+{\widehat{r}}_5}\times 100(27)$

Finally, Equation 28 defines the degree of linear polarization of the ocean reflectance, ${DoLP}_{35}$:

${DoLP}_{35}\equiv \frac{p_3+p_5}{r_3+r_5}\times 100=\frac{{\widehat{p}}_3+{\widehat{p}}_5}{{\widehat{r}}_3+{\widehat{r}}_5}\times 100(28)$

The results can be conveniently collected in cumulative plots where each relative contribution, ${\widehat{r}}_k$ and ${\widehat{p}}_k={\widehat{DoLP}}_k$, is depicted as a layer building up to the total $R_I$ or $R_{\text{P}}$ (or DoLP) at the TOA. The sensitivity of remote sensing observations to contribution k can then be directly assessed by comparing ${\widehat{r}}_k$ or ${\widehat{p}}_k$ to the instrument-specific accuracy for $R_I$ and $R_{\text{P}}$. The radiometric uncertainty is typically $3\%$ for modern-age satellite sensors and can be reduced to $1\%$ post-launch by means of vicarious calibration protocols (National Research Council, 2011). The polarimetric uncertainty depends, in addition, on the accuracy of the DoLP measurements which is typically $0.3\%-1\%$ for modern-age polarimeters (the smaller the value, the closer the uncertainties of $R_I$ and $R_{\text{P}}$).

The spectral trends of the scattering contributions $r_3$ and $r_5$ vary with those of the atmospheric transmission and $A_{blk}$. For this reason, in the following subsections each wavelength is discussed separately. For the sake of brevity, the complete set of the “cumulative” plots is available in the Supplementary Material. Here we include the cases for 385 nm (rarely utilized insofar for ocean color remote sensing purposes) and 550 nm (where the differences between the three scenes can be dramatic due to effects of the Chlorophyll-a main peak and the presence of minerals). In each figure, the upper (lower) four panels are for ${\theta }_0=20\text{°}(50\text{°})$, and the top (bottom) row in each block of panels is for $\phi =0\text{°}(50\text{°})$. The columns are for Scene A, B and C.

3.1.1 385 nm (“UV”)

The cumulative plots for the wavelength 385 nm are given in Figure 11 for the total reflectance and in Figure 12 for the polarized reflectance. Let the superscript “*” identify the angular location of the sunglint maximum, ${\widehat{r}}_2^{*}=\max ({\widehat{r}}_2)$. (Note that, normally, the global maximum of the curves across all viewing zenith angles coincides with the peak of the sunglint). For sunglint views at ${\theta }_0=20\text{°}$, ${\widehat{r}}_2^{*}$ varies in the range $17\%\le {\widehat{r}}_2^{*}\le 19\%\hspace{0.17em}(\phi =0\text{°})$ and $11\%\le {\widehat{r}}_2^{*}\le 12\%\hspace{0.17em}(\phi =50\text{°})$ across the different scenes. In correspondence of the angular location of ${\widehat{r}}_2^{*}$, the atmospheric contribution ${\widehat{r}}_1$ is $60\%-67\%\hspace{0.17em}(\phi =0\text{°})$ and $66\%-73\%\hspace{0.17em}(\phi =50\text{°})$ from scene to scene. For off-sunglint views, $75\%\le \max ({\widehat{r}}_1)\le 86\%$. The diffuse skylight reflection contribution ${\widehat{r}}_4$ amounts to $6\%-9\%$ for all viewing geometries and across all scenes. Hence, the signal at the TOA contributed by the surface reflection, $r_{24}$, is dominated by direct sunglint for sunglint views, where the surface reflectance quotient $Q_{24}$ becomes as large as $73\%(64\%)$ for $\phi =0\text{°}\hspace{0.17em}(\phi =50\text{°})$. The diffuse skylight illumination dominates instead $r_{24}$ for off-sunglint observations, where $Q_{24}$ is as low as $10\%\hspace{0.17em}(1\%)$ for $\phi =0\text{°}\hspace{0.17em}(\phi =50\text{°})$. The ocean contribution ${\widehat{r}}_{35}$ amounts to $8\%$ (Scene A) and $18\%$ (Scene B and C) at nadir. Most of this contribution originates from diffuse skylight illumination ($40\%\le Q_{35}\le 42\%$ across all scenes for near-nadir viewing directions), which implies a strong radiative coupling between the atmosphere and the ocean body.

Figure 11

Figure 11. Cumulative contributions of different scattering processes to the total reflectance observed at the TOA for ${\theta }_0=20\text{°}$ (top six panels) and ${\theta }_0=50\text{°}$ (bottom six panels), for the wavelength of 385 nm. In each subfigure, the first row is for principal plane observations $(\phi =0\text{°})$ and the bottom row for $\phi =50\text{°}$. The three columns pertain to Scene A, B and C.

Figure 12

Figure 12. Same as Figure 11, but for the polarized reflectance components.

As ${\theta }_0$ increases, direct sunlight must travel a larger airmass before reaching the ocean surface and the atmospheric contribution slightly increases for off-sunglint views along the principal plane ($77\%\le {\widehat{r}}_1^{*}\le 88\%$ for ${\theta }_0=50\text{°}$ and $\phi =0\text{°}$). The subsequent decrease in ${\chi }_{dir}$ (see Figure 8b) is partially countered by the rapid increase in surface reflectivity governed by the Fresnel laws for specular reflection past the Brewster angle. However, normalization by $R_I(\theta )$ causes ${\widehat{r}}_2^{*}$ to increase only to 21%−22% as ${\theta }_0$ increases from $20\text{°}$ to $50\text{°}$ for $\phi =0\text{°}$. For observations far off the principal plane $(\phi =50\text{°})$, fewer wave facets are oriented to specularly reflect the incoming direct sunlight into a sensor’s field of view at the TOA (or at any altitude). If the wind speed increases, the incident direct beam is reflected over a larger angular span and the sunglint region broadens. For any given viewing geometry, the number of facets properly oriented for specular reflection varies with wind speed, but at $\phi =50\text{°}$ it always decreases with increasing solar zenith angle. Indeed, at this azimuth $r_2$ is one order of magnitude smaller for ${\theta }_0=50\text{°}$ than for ${\theta }_0=20\text{°}$, and ${\widehat{r}}_2^{*}$ rapidly fades below $1\%$ (Figure 11). Hence, there is no sunglint (as defined in Section 3.1) for ${\theta }_0=\phi =50\text{°}$, and the total signal is still dominated by the atmospheric contribution $(74\%\le {\widehat{r}}_1^{*}\le 88\%)$. The diffuse skylight reflection remains non-negligible because ${\chi }_{dif}$ increases with ${\theta }_0$ (see green and red curves in Figure 8d), and because $r_4$ includes reflection by more surface facets than just those reflecting specularly into the field of view. Indeed, the maximum $Q_{24}$ decreases with ${\theta }_0$ to $69\%$ for $\phi =0\text{°}$ and $9\%$ for $\phi =50\text{°}$. The minimum $Q_{24}$ value also decreases to $<2\%$ and $<0.1\%$ for the same geometries. All in all, $r_4$ increases slightly (to $6\%-10\%$ for ${\theta }_0=50\text{°}$) despite of $r_4$ being normalized by larger values of $R_I$.

The larger airmass and surface reflectivity at large solar zenith angles also imply less direct sunlight being transmitted into the ocean. As a result, $r_3$ decreases (Figure 12) across all relative azimuths, whereas $r_5$ grows larger because of the increase in ${\chi }_{dif}$ at ${\theta }_0=50\text{°}$. At nadir, where the dependence on $\phi$ is minimized, $Q_{35}$ decreases to $32\%-33\%$ and ${\widehat{r}}_{35}$ decreases across all scenes by less than $1\%$ because of the opposite trends in $r_3({\theta }_0)$ and $r_5({\theta }_0)$. For any given ${\theta }_0$, ${\widehat{r}}_{35}$ remains much larger for Scene B than for Scene A, because $A_{blk}$ increases with decreasing [Chl] in the UV-Blue for Case I waters (Figure 8b) and with decreasing CDOM concentrations at all wavelengths (Morel et al., 2007). For Scene C, ${\widehat{r}}_{35}$ remains larger than Scene A but equal to Scene B at nadir. These arguments are again consistent with Figure 8b, and result from the balance between a decrease in $A_{blk}$ caused by the large [Chl] and a (larger) increase in $A_{blk}$ caused by SPM scattering (Figure 4a). The results for the polarized reflectance in Figure 12 are similar to those for the total reflectance except for the following:

$•$ ${\widehat{p}}_2^{*}$ increases to $34\%$ for $\phi =0\text{°}$ and ${\theta }_0=50\text{°}$, which is close to the Brewster geometry, while it is smaller $(12-19\%)$ for lower solar zenith angles;

$•$ ${\widehat{p}}_1$ is everywhere larger than ${\widehat{r}}_1$ except when ${\theta }_0$ approaches the Brewster geometry (i.e., when $p_{24}$ dominates $R_{\text{P}}$). For off-sunglint observations, $86\%\le \max ({\widehat{p}}_1)\le 93\%$;

$•$ ${\widehat{p}}_{35}$ is small for all scenes ($3\%-5\%$ for Scene A and C and $7\%-9\%$ for scene B at nadir);

$•$ The fraction of polarized light emerging from the ocean is very small for Scene C ($22\%\le {DoLP}_{35}\le 33\%$ at nadir, compared to $43\%-53\%$ for Scene A and $40\%-49\%$ for B). The reason is that mineral particles have higher refractive indices and are therefore less polarizing than plankton particles. The bulk ocean single-scattering albedo ${\omega }_{blk}$ and the number of underwater light scattering events $n\propto 1/(1-{\omega }_{blk})$ (Morel, 1991) are then also much higher for Scene C than for Scene A and B (see Figure 3d), leading to further depolarization (Hansen and Travis, 1974).

3.1.2 410 nm (“Violet”)

The molecular and fine-mode-aerosol optical depths at 410 nm are smaller than at 385 nm (Table A3), and the implications for TOA observations (Supplementary Figures S2 and S8) are several-fold. Firstly, the absolute atmospheric scattering contribution $r_1$ is also smaller. The total reflectance $R_I$ then also decreases outside the sunglint, and all relative contributions ${\widehat{r}}_k$ increase except for ${\widehat{r}}_1$, which is $48\%-66\%$ at the peak of sunglint and $68\%\le \max ({\widehat{r}}_1)\le 85\%$ for off-sunglint views. Secondly, the direct two-pass transmittance ${\chi }_{dir}$ increases spectrally by a factor ${\xi }_{\chi ,dir}=1.3$ for $\theta ={\theta }_0=20\text{°}$ and ${\xi }_{\chi ,dir}=1.4$ for $\theta ={\theta }_0=50\text{°}$ (cf. Figure 8b). In turn, the direct sunglint contribution increases ($15\%\le {\widehat{r}}_2^{*}\le 26\%$ for ${\theta }_0=20\text{°}$ and $29\%\le {\widehat{r}}_2^{*}\le 32\%$ for ${\theta }_0=50\text{°}$ for $\phi =0\text{°}$), but remains negligibly small ($1\%$, hence not classified as sunglint) for ${\theta }_0=50\text{°}$ and $\phi =50\text{°}$ because there are fewer contributing facets. However, the spectral change in ${\chi }_{dif}$ is non-trivial: it decreases if $\theta$ and ${\theta }_0$ are both small (${\xi }_{\chi ,dif}=0.97$ for $\theta ={\theta }_0=20\text{°}$, see Figure 8d), but it can increase if ${\theta }_0$ and/or $\theta$ are large (${\xi }_{\chi ,dif}>1$ for $\theta ,{\theta }_0\ge 50\text{°}$). The small spectral increase in ${\widehat{r}}_4$ seen at all geometries ($6\%\le {\widehat{r}}_4\le 11\%$ for ${\theta }_0=20\text{°}$ and $7\%\le {\widehat{r}}_4\le 11\%$ for ${\theta }_0=50\text{°}$) is mostly caused by the decrease in $R_I$. The absolute contribution $r_4$ actually decreases for most geometries, except (depending on $\phi$) for $\theta \ge 55\text{°}$ if ${\theta }_0=20\text{°}$, and for $\theta \ge$25° if ${\theta }_0=50\text{°}$, all in accordance with the behavior of ${\xi }_{\chi ,dif}$. Regardless of $r_4$, $r_2$ dominates the surface reflection in sunglint views. The surface reflectance quotient $Q_{24}$ is $\approx 5\%$ higher than at 385 nm at the glint peak both when ${\theta }_0=20\text{°}$ ($78\%$ for $\phi =0\text{°}$ and $71\%$ for $\phi =50\text{°}$) and when ${\theta }_0=50\text{°}$ ($74\%$ for $\phi =0\text{°}$). At ${\theta }_0=50\text{°}$ and $\phi =50\text{°}$, where the sunglint is negligibly small, the maximum $Q_{24}$ value still increases to $12\%$. For all other off-sunglint observations, $r_4$ and ${\widehat{r}}_4$ can still be much larger than $r_2$ and ${\widehat{r}}_2$, but the diffuse skylight reflection is less dominant: the minimum $Q_{24}$ values are $13\%(\phi =0\text{°})$ and $1\%(\phi =50\text{°})$ when ${\theta }_0=20\text{°}$, and remain about $2\%$ (regardless of $\phi$) when ${\theta }_0=50\text{°}$.

From Figure 7b, $A_{blk}(\lambda )$ decreases by a factor ${\xi }_R=0.89$ (0.84) for Scene A (B), and increases by a factor ${\xi }_R=1.15$ for Scene C. These changes are much smaller than the increase in ${\xi }_R$ (i.e., ${\xi }_R\times {\xi }_{\chi ,dir}\ge 1$ regardless of ${\xi }_R$) seen in Figure 8d. As a result, trends in $r_3$ and ${\widehat{r}}_3$ are dominated by ${\chi }_{dir}$ and they both increase across all scenes and angles. Conversely, the non-trivial spectral changes of ${\chi }_{dif}(\theta ,{\theta }_0)$ do not manifest in those of $r_5$ and ${\widehat{r}}_5$, which retain the same sign for any $(\theta ,{\theta }_0)$ pair as those of ${\xi }_R$ because $r_5\propto {\xi }_R\times {\xi }_{\chi ,dif}$ is driven by the small spectral changes in $A_{blk}$. As a result, $r_5$ decreases for Scene A and B and increases for Scene C, while ${\widehat{r}}_5$ remains about the same for Scene A and B, and increases slightly for Scene C regardless of $\theta$ and ${\theta }_0$. Summing the two relative contributions, ${\widehat{r}}_{35}$ increases slightly for Scene A and B, and substantially for Scene C, to $10\%,20\%$, and $24\%$ in the nadir viewing direction for ${\theta }_0=20\text{°}$. The corresponding quotient is $46\%\le Q_{35}\le 49\%$ and decreases to $39\%\le Q_{35}\le 41\%$ when ${\theta }_0=50\text{°}$ because the decrease in ${\chi }_{dir}$ (which determines ${\widehat{r}}_3$) is partially compensated by the increase in ${\chi }_{dif}$. As a result, the variation with ${\theta }_0$ is ${\widehat{r}}_{35}\le 1\%$ in the nadir direction. The smaller difference in $r_{35}$ between Scene B and C is again in agreement with Figures 7b and 8b.

The spectral changes between 385 and 410 nm for the polarized reflectance are similar to those for the total reflectance described above: ${\widehat{p}}_1$ decreases for all scattering geometries ($41\%\le {\widehat{p}}_1^{*}\le 72\%$; $82\%\le \max ({\widehat{p}}_1)\le 90\%$ for off-sunglint views), along with the polarized reflectance $R_{\text{P}}$ outside the sunglint. The spectral increase in ${\chi }_{dir}$ leads to higher maxima $(44\%-45\%)$ for ${\widehat{p}}_2^{*}$ at ${\theta }_0=50\text{°}$. The ocean contribution ${\widehat{p}}_{35}$ increases slightly to $4\%-5\%$ (Scene A), $6\%-8\%$ (Scene C) and $8\%-10\%$ (Scene B) at nadir, but the corresponding ${\widehat{DoLP}}_{35}$ values remain the same: $43\%-53\%$ (Scene A), $41\%-51\%$ (Scene B) and $22\%-33\%$ (Scene C).

3.1.3 470 nm (“Blue”)

Increasing the wavelength to 470 nm, the fine-mode-aerosol and molecular optical depths (Table A3), and therefore $r_1$, continue to decrease and all the normalized contributions ${\widehat{r}}_k$ (Supplementary Figure S3) to increase except for ${\widehat{r}}_1$ ($30\%\le {\widehat{r}}_1^{*}\le 52\%$; $51\%\le \max ({\widehat{r}}_1)\le 81\%$ for off-sunglint views). The associated increase in ${\chi }_{dir}$ (see Figure 8d) by a factor ${\xi }_{\chi ,dir}=1.44(>1.7)$ for $\theta ={\theta }_0=20\text{°}(\theta ={\theta }_0=50\text{°})$ leads to an increase in $r_2$ ($23\%\le {\widehat{r}}_2^{*}\le 43\%$ for ${\theta }_0=20\text{°}$ and $43\%\le {\widehat{r}}_2^{*}\le 51\%$ for ${\theta }_0=50\text{°}$). However, the diffuse skylight reflection continues to spectrally change in a non-trivial manner. While the range in ${\widehat{r}}_4$ increases slightly to $5\%-15\%({\theta }_0=20\text{°})$ and $6\%-14\%({\theta }_0=50\text{°})$, $r_4$ decreases for most geometries except for ${\theta }_0=50\text{°}$, where it can actually increase if $\theta \ge 45\text{°}$ depending on $\phi$. Consistent with the complex spectral trends discussed in Section 2.4, ${\chi }_{dif}$ changes by a factor ${\xi }_{\chi ,dif}=0.87$ for $\theta ={\theta }_0=20\text{°}$. When ${\theta }_0=50\text{°}$ and $\theta =60\text{°}$, ${\xi }_{\chi ,dif}$ is slightly above 1. Within the sunglint region, the spectral increase in $r_2$ remains always larger than that of $r_4$ and the maximum $Q_{24}$ value increases by more than $7\%$ to become $85\%(\phi =0\text{°})$ and $80\%(\phi =50\text{°})$ when ${\theta }_0=20\text{°}$, and $82\%$ for $\phi =0\text{°}$ when ${\theta }_0=50\text{°}$. Even for ${\theta }_0=50\text{°}$ and $\phi =50\text{°}$, when sunglint is negligible as for the previous wavelengths, the maximum $Q_{24}$ value increases to $19\%$. The minimum value of $Q_{24}$ also slightly increases to $21\%(4\%)$ for $\phi =0\text{°}$ and $3\%(<1\%)$ for $\phi =50\text{°}$ when ${\theta }_0=20\text{°}(50\text{°})$.

The large spectral increase in ${\chi }_{dir}$ leads also to the direct ocean reflection to dominate the ocean contribution $r_{35}$, as inferred from $Q_{35}$ growing to $59\%(61\%)$ for Scene C (Scene A and B) at nadir when ${\theta }_0=20\text{°}$. The actual changes in $r_3(r_5)$ depend not only on the spectral changes in ${\chi }_{dir}({\chi }_{dif})$, but also on $A_{blk}$, which decreases in the Violet-Green for Scene A (B) by a factor ${\xi }_R=0.64$ (0.41), and increases for Scene C by a factor ${\xi }_R=1.19$ (cf. Figure 4). The spectral impacts on $r_3$ and $r_5$ are determined by the products ${\xi }_R\times {\xi }_{\chi ,dir}$ and ${\xi }_R\times {\xi }_{\chi ,dif}$, respectively. For Scene A, $r_3$ moderately decreases (increases) for small (large) values of $\theta$ and ${\theta }_0$, whereas $r_5$ decreases substantially for all geometries and so does $r_{35}$. For Scene B, $r_3$ and $r_5$ decrease substantially for all geometries. For Scene C, $r_3$ and $r_5$ moderately increases for all geometries. In summary, $r_{35}$ decreases for Scene A and even more so for Scene B but it remains the smallest for Scene A. For Scene C, it increases and becomes largest among all scenes.

At nadir, these trends are generally preserved in ${\widehat{r}}_{35}$ (Supplementary Figure S3), which becomes $10\%$ (Scene A), $15\%$ (Scene B) and $36\%$ (Scene C) when ${\theta }_0=20\text{°}$. Increasing ${\theta }_0$ from $20\text{°}$ to $50\text{°}$, the quotient $Q_{35}$ lowers to $53\%(54\%)$ for Scene C (Scene A and B) because changes in $r_3$ with ${\theta }_0$ caused by ${\chi }_{dir}$ are again partially compensated by changes in $r_5$ caused by ${\chi }_{dif}$. The overall result is that ${\widehat{r}}_{35}$ increases slightly to $11\%$ (Scene A), $16\%$ (Scene B) and $41\%$ (Scene C) when ${\theta }_0=50\text{°}$. Compared to 410 nm, ${\widehat{r}}_{35}$ remains about the same (Scene A), decreases (Scene B), and increases (Scene C), consistent with the trends for ${\rho }_w$ shown in Figure 7.

The spectral changes in the polarized reflectance follow those for the total reflectance described above. The atmospheric scattering contribution $p_1$ (Supplementary Figure S9) decreases for all scattering geometries, and $R_{\text{P}}$ decreases outside the sunglint because of the spectral decrease in atmospheric optical depth. The percent contribution is now $24\%\le {\widehat{p}}_1^{*}\le 56\%$, and $74\%\le {\widehat{p}}_1\le 85\%$ for off-sunglint views. The direct sunglint component $p_2$ also increases spectrally following ${\chi }_{dir}$, and $62\%\le {\widehat{p}}_2\le 63\%$ when ${\theta }_0=50\text{°}$ for observations close to the Brewster geometry. Finally, $p_3$ and $p_5$ follow the spectral trends discussed above for $r_3$ and $r_5$ with the minor exception that $p_3$ for Scene A increases at any $\theta$ when ${\theta }_0=50\text{°}$. At nadir, ${\widehat{p}}_{35}=5\%$ (Scene A), $7\%-8\%$ (Scene B) and $10\%-13\%$ (Scene C), and ${DoLP}_{35}$ increases slightly for Scene A $(45\%-56\%)$, B $(45\%-54\%)$ and C $(25\%-35\%)$.

3.1.4 550 nm (“Green”)

Geometry-dependent maxima like those observed for the transmittances in the UV-Violet-Blue (Figure 8c) and in the scattering contribution $r_4$ do not appear in the Green and beyond. The monotonic spectral decrease in the fine-mode-aerosol and molecular optical depths drive the decrease in $r_1$ and $R_I$ outside the sunglint region, leading to $17\%\le {\widehat{r}}_1^{*}\le 37\%$ and $34\%\le \max ({\widehat{r}}_1)\le 81\%$ for off-sunglint views at 550 nm (Figure 13). Consequently, also increasing are ${\chi }_{dir}$ (${\xi }_{\chi ,dir}=1.30$ for $\theta ={\theta }_0=20\text{°}$ and $>1.46$ for $\theta \ge 50\text{°}$ when ${\theta }_0=50\text{°}$) and ${\widehat{r}}_2^{*}$ ($20\%-52\%$ for $\phi =50\text{°}$ and $40\%-64\%$ for $\phi =0\text{°}$ at ${\theta }_0=20\text{°}$, and $55\%-69\%$ for $\phi =0\text{°}$ at ${\theta }_0=50\text{°}$). The ${\widehat{r}}_2$ contribution is now also detectable when ${\theta }_0=50\text{°}$ and $\phi =50\text{°}$ in Scene A and B $(5\%\le {\widehat{r}}_2^{*}\le 7\%)$, but not large enough yet to be classified as sunglint. However, unlike in the UV-Violet-Blue, ${\chi }_{dif}$ decreases for all geometries (${\xi }_{\chi ,dif}=0.79$ for $\theta ={\theta }_0=20\text{°}$ and 0.84 for $\theta ={\theta }_0=50\text{°}$) as does $r_4$, and the range in ${\widehat{r}}_4$ increases to $4\%-19\%$ for ${\theta }_0=20\text{°}$ and to $5\%-19\%$ for ${\theta }_0=50\text{°}$. When $\phi =0\text{°}(50\text{°})$, the changes in $r_2$ and $r_4$ lead to $31\%\le Q_{24}\le 90\%\hspace{0.17em}(4\%\le Q_{24}\le 87\%)$ for ${\theta }_0=20\text{°}$, and $8\%\le Q_{24}\le 87\%\hspace{0.17em}(0.2\%\le Q_{24}\le 29\%)$ for ${\theta }_0=50\text{°}$.

Figure 13

Figure 13. Same as Figure 11, but for the 550 nm wavelength.

Similarly, $Q_{35}$ is larger for all scenes and geometries. At nadir, $71\%\le Q_{35}\le 73\%$ when ${\theta }_0=20\text{°}$ and $66\%\le Q_{35}\le 67\%$ when ${\theta }_0=50\text{°}$. However, compared to 470 nm, $A_{blk}$ is significantly lower for Scene A $({\xi }_R=0.26)$ and even more so for Scene B $({\xi }_R=0.15)$, whereas it increases for Scene C $({\xi }_R=1.24)$. As a result, $r_{35}\propto {\xi }_R\times ({\xi }_{\chi ,dir}+{\xi }_{\chi ,dif})$ becomes comparable for Scene A and B. For Scene C, the increase in $r_3$ is significantly larger than the small decrease in $r_5$ for all geometries and $r_3+r_5$ is over an order of magnitude larger than for scenes A and B. These results are consistent with the spectral trends for ${\rho }_w$ (Figure 8b) and, as shown in Supplementary Figure S4, lead to ${\widehat{r}}_{35}\sim 4\%(\sim 5\%)$ for Scene A and B, and $49\%(61\%)$ for Scene C at nadir when ${\theta }_0=20\text{°}(50\text{°})$.

The behavior of the polarization components $p_1$, $p_2$ and $R_{\text{P}}$ follows that of the total reflectance counterparts: $12\%\le {\widehat{p}}_1^{*}\le 40\%$ and $61\%\le \max ({\widehat{p}}_1)\le 82\%$ for off-sunglint views, and $75\%\le {\widehat{p}}_2\le 77\%$ near the Brewster angle (Figure 14). Finally, also $p_3$ and $p_5$ follow the spectral trends of $r_3$ and $r_5$. At nadir, the ocean contribution to polarization is $2\%\le {\widehat{p}}_{35}\le 3\%$ (Scene A and B) and $18\%\le {\widehat{p}}_{35}\le 19\%$ (Scene C), with corresponding ${DoLP}_{35}$ values of $46\%-59\%$ and $31\%-36\%$.

Figure 14

Figure 14. Same as Figure 12, but for the 550 nm wavelength.

3.1.5 670 nm (“Red”)

The spectral trends moving towards 670 nm follow those noted for 550 nm. The cumulative plots for this wavelength are given in Supplementary Figures S5, S11. The direct transmittance increases for all geometries (${\xi }_{\chi ,dir}=1.21$ for $\theta ={\theta }_0=20\text{°}$ and ${\xi }_{\chi ,dir}>1.31$ for $\theta ={\theta }_0\ge 50\text{°}$), while a decrease is observed for (i) $R_I$ outside the sunglint; (ii) ${\widehat{r}}_1$ ($11\%\le {\widehat{r}}_1^{*}\le 23\%$ regardless of φ when ${\theta }_0=20\text{°}$ and at $\phi =0\text{°}$ when ${\theta }_0=50\text{°}$; $42\%\le \max ({\widehat{r}}_1)\le 81\%$ for all off-sunglint views); and (iii) ${\chi }_{dif}$ (${\xi }_{\chi ,dif}=0.70$ for $\theta ={\theta }_0=20\text{°}$ and ${\xi }_{\chi ,dif}\ge 0.73$ for $\theta ={\theta }_0\ge 50\text{°}$). The larger ${\chi }_{dir}$ also increases ${\widehat{r}}_2$ in the sunglint views identified in the UV-Green ($59\%\le {\widehat{r}}_2^{*}\le 80\%$ for $\phi \le 50\text{°}$ when ${\theta }_0=20\text{°}$, and $79\%\le {\widehat{r}}_2^{*}\le 81\%$ for $\phi =0\text{°}$ when ${\theta }_0=50\text{°}$), but sunglint emerges now also at ${\theta }_0=50\text{°}$ and $\phi =50\text{°}(9\%\le {\widehat{r}}_2^{*}\le 14\%)$. On the other hand, the change in ${\chi }_{dif}$ decreases $r_4$ for all geometries but normalization by $R_I$ still leads to slight spectral increases in ${\widehat{r}}_4$ for Scene A and B. At the peak of sunglint, $Q_{24}$ then also increases to $92\%-94\%$ regardless of the solar zenith angle if observing along the principal plane, but is only $43\%$ for ${\theta }_0=50\text{°}$ when $\phi =50\text{°}$, implying again strong radiative coupling between the atmosphere and the surface.

The spectral changes in $r_3$ and $r_5$ are not only affected by those in ${\chi }_{dir}$ and ${\chi }_{dif}$ but also by those in $A_{blk}$, which decreases (Figure 6b) most for Scene B $({\xi }_R=0.065)$, less for Scene A $({\xi }_R=0.089)$ and least for Scene C $({\xi }_R=0.23)$. In turn, $r_3\propto {\xi }_R\times {\xi }_{\chi ,dir}$ and $r_5\propto {\xi }_R\times {\xi }_{\chi ,dif}$ decrease for all scenes and geometries. For Scenes A and B, $r_{35}$ becomes very small at 670 nm and ${\widehat{r}}_{35}<1\%$, but remains non-negligible for Scene C and is increasingly dominated by $r_3$: at nadir, ${\widehat{r}}_{35}=21\%(40\%)$ and $Q_{35}=81\%(77\%)$ for ${\theta }_0=20\text{°}(50\text{°})$.

The polarization components vary similarly to those of the total reflectance: $R_{\text{P}}$ decreases outside the sunglint and ${\widehat{p}}_1$ at all geometries ($7\%\le {\widehat{p}}_1^{*}\le 24\%$ regardless of $\phi$ when ${\theta }_0=20\text{°}$ and along the principal plane when ${\theta }_0=50\text{°}$, $55\%\le {\widehat{p}}_1^{*}\le 68\%$ at $\phi =50\text{°}$; $51\%\le \max ({\widehat{p}}_1)\le 79\%$ for off-sunglint views). Increases are observed for ${\widehat{p}}_2$ in the sunglint ($62\%\le {\widehat{p}}_2^{*}\le 78\%$ regardless of $\phi$ when ${\theta }_0=20\text{°}$, and $84\%\le {\widehat{p}}_2^{*}\le 86\%$ for $\phi =0\text{°}$ when ${\theta }_0=50\text{°}$) and for ${\widehat{p}}_4$ at all viewing angles ($4\%-23\%$ for ${\theta }_0=20\text{°}$ and $7\%-27\%$ for ${\theta }_0=50\text{°}$). Finally, ${\widehat{p}}_{35}<1\%$ for Scene A and B for all geometries but remains significant for Scene C: at nadir, ${\widehat{p}}_{35}=13\%$ ($18\%$) for ${\theta }_0=20\text{°}$ ($50\text{°}$), and the corresponding ${DoLP}_{35}$ is $61\%$ ($46\%$).

3.1.6 865 nm (“NIR”)

At 865 nm, the coarse-mode aerosol optical depth becomes comparable to that of the fine mode (Table A3). Around this wavelength, the total aerosol optical thickness therefore transitions through a minimum from a fine-mode to a coarse-mode aerosol scattering regime. Compared to shorter wavelengths, the spectral change is only moderate and ${\xi }_{\chi ,dir}=1.13$ (1.19) for $\theta ={\theta }_0=20\text{°}(50\text{°})$. In turn, $r_2$ and ${\widehat{r}}_2$ (Supplementary Figure S6) increase in the sunglint region: $83\%\le {\widehat{r}}_2^{*}\le 90\%$ regardless of $\phi$ when ${\theta }_0=20\text{°}$ and $26\%\le {\widehat{r}}_2^{*}\le 89\%$ along the principal plane when ${\theta }_0=50\text{°}$. On the other hand, ${\chi }_{dif}$ depends on the optical depth and on the scattering function of each aerosol mode. Coarse-mode aerosols have a larger size parameter, $2\pi r_{eff}/\lambda$ (see Figure 12 in Hansen and Travis (1974)), than fine-mode aerosols and scatter therefore less diffusely. Hence, from 670 nm to 865 nm, ${\chi }_{dif}$ still experiences a robust decrease at least for the adopted maritime-atmosphere model: ${\xi }_{\chi ,dif}=0.62(\ge 0.65)$ for $\theta ,{\theta }_0=20\text{°}(\ge 50\text{°})$. Also decreasing are $R_I$ (outside the sunglint), and $r_1$ and ${\widehat{r}}_1$ ($7\%\le {\widehat{r}}_1^{*}\le 11\%$ when ${\theta }_0=20\text{°}$ and $6\%\le {\widehat{r}}_1^{*}\le 54\%$ when ${\theta }_0=50\text{°}$; $45\%\le \max ({\widehat{r}}_1)\le 76\%$ for off-sunglint views).

Finally, $r_4$ decreases for all geometries but normalization by $R_I$ still leads to slight increases in the range of ${\widehat{r}}_4$ ($4\%-24\%$ for ${\theta }_0=20\text{°}$ and $5\%-27\%$ for ${\theta }_0=50\text{°}$). Altogether, $Q_{24}$ still increases in the sunglint to $95\%-96\%$ for ${\theta }_0=20\text{°}$, and to $95\%(59\%)$ when $\phi =0\text{°}(50\text{°})$ for ${\theta }_0=50\text{°}$. Hence, at $865\text{\hspace{0.17em}nm}$ the radiative coupling between atmosphere and ocean surface does no longer dominate observations in the sunglint.

The ocean becomes essentially black ($A_{blk}\ll 0.001$, see Figure 7b) for Scene A and B: ${\xi }_R<0.01$ and ${\widehat{r}}_{35}\ll 1\%$ regardless of $(\theta ,{\theta }_0)$. However, $A_{blk}=0.003$ and ${\xi }_R=0.07$ for Scene C. The resulting ocean contributions $r_3\propto {\xi }_R\times {\xi }_{\chi ,dir}$ and $r_5\propto {\xi }_R\times {\xi }_{\chi ,dif}$ still add up to non-negligible values when ${\theta }_0=50\text{°}$, and ${\widehat{r}}_{35}=8\%(Q_{35}=86\%)$ in the nadir-viewing direction.

The polarized reflectance $R_{\text{P}}$ (Supplementary Figure S12) follows $R_I$ and decreases (increases) outside (within) the sunglint compared to 670 nm. The decrease is caused by smaller values of $p_1$ (leading to $44\%\le \max ({\widehat{p}}_1)\le 75\%$ for off-sunglint views) and of $p_4$ (leading to $4\%\le {\widehat{p}}_4\le 26\%$ when ${\theta }_0=20\text{°}$, and $5\%-32\%$ when ${\theta }_0=50\text{°}$, for all views). Similarly, within the sunglint $R_{\text{P}}$ increases because of larger values of $p_2$, and $83\%\le {\widehat{p}}_2^{*}\le 89\%$ when ${\theta }_0=20\text{°}$ and $25\%\le {\widehat{p}}_2^{*}\le 92\%$ when ${\theta }_0=50\text{°}$. For the same sunglint views, $7\%\le {\widehat{p}}_1^{*}\le 12\%$ and $3\%\le {\widehat{p}}_1^{*}\le 55\%$, respectively. Finally, ${\widehat{p}}_{35}\ll 1\%$ for Scene A and B regardless of the geometry but remains non-negligible for Scene C when ${\theta }_0=50\text{°}$: here ${\widehat{p}}_{35}=4\%$ and ${DoLP}_{35}=22\%$ in the nadir viewing direction.

3.2 Implications for remote sensing observations

We conclude the analysis with some recommendations for the remote sensing of the various scenes. To this end, we present additional viewgraphs illustrating the spectral behavior of each scattering contribution to the total reflectance (Figure 15) and the polarized reflectance (Figure 16). We also define the following classes for a relative TOA scattering contribution, according to the intervals given in parentheses: “dominant” $(\ge 50\%)$, “major” (between $20\%$ and $50\%$), “significant” (between $10\%$ and $20\%$), “detectable” (between $3\%$ and $10\%$) and “undetectable” $(<3\%)$. Isolating the signal of a particular atmosphere-ocean system component is, for a particular geometry and spectral band, as successful as (i) the associated contribution is substantial compared to the others; and/or (ii) the other scattering contributions can be constrained to a suitable low absolute uncertainty.

Figure 15

Figure 15. Spectral behavior of the [$\mathit{\text{min}}$, $\mathit{\text{max}}$] range for each scattering contribution to the total reflectance. The three scenes are in the three columns, and each row is for the indicated geometry. The color coding is the same as for Figures 11–14, and the gray horizontal lines within the bars represent the value at nadir.

Figure 16

Figure 16. Same as Figure 15, but for the polarized reflectance.

In this context, we summarize the optimal spectral ranges for radiometric (Section 3.2.1) and polarimetric (Section 3.2.2) observations. In each case, we differentiate between “off-sunglint observation” defined for the range of viewing zenith angles at which sunglint is at most just detectable $({\widehat{r}}_2<10\%)$, and “bright-sunglint observations” defined for the angular regions where sunglint is dominant $({\widehat{r}}_2\ge 50\%)$. Table 1 highlights the cases where these conditions are met at least for one viewing angle. Off-sunglint observations are of interest to remote sensing retrievals of both atmospheric and oceanic properties. For this reason, the left side of Table 2 quantifies the total ocean contribution by reporting the classification of the maximum values of ${\widehat{r}}_{35}$ within these regions: since the sunglint contribution can simultaneously be detectable, it should be corrected for when attempting a retrieval. On the other hand, bright-sunglint observations are of interest to remote sensing retrievals of ocean surface roughness and/or refractive index (Ottaviani et al., 2019), but also of aerosol absorption (Kaufman Y. J. et al., 2002; Ottaviani et al., 2013), provided corrections are applied for any measurable atmospheric and/or oceanic contribution.

Table 1

Table 1. Spectral behavior, for all scenes and solar geometries, of the class for the minimum and the maximum $({\widehat{r}}_2^{*})$ of the direct sunglint contribution across all viewing zenith angles, according to the classification given in Section 3.2: UND = undetectable; DET = detectable; SIG = significant; MAJ = major; DOM = dominant. Cases highlighted in bold correspond to either where $\min ({\widehat{r}}_2)$ is detectable or undetectable (at least at one viewing angle), discussed in the main text under “Off-sunglint observations”, or where ${\widehat{r}}_2^{*}$ reaches the dominant status (at least at one viewing angle), discussed in the main text under “Bright-sunglint observations”.

Table 2

Table 2. Spectral behavior (for all scenes and solar geometries) of the class for the maximum total ocean contribution, $\max ({\widehat{r}}_{35})$, in the subset of viewing angles corresponding to off-sunglint $({\widehat{r}}_2<10\%)$ and bright sunglint $({\widehat{r}}_2\ge 50\%)$ geometries: UND = undetectable; DET = detectable; SIG = significant; MAJ = major; DOM = dominant. Cells marked with “-” indicate that the subset does not exist for that case (see Table 1).

3.2.1 Radiometric observations

3.2.1.1 Off-sunglint observations

Subsets of viewing angles where ${\widehat{r}}_2<10\%$ are always present for any given (${\theta }_0$, $\phi$) pair, except in the Red-NIR when ${\theta }_0=20\text{°}$ and $\phi =0\text{°}$ regardless of the Scene (see Table 1). Neglecting these particular cases, the atmospheric contribution ${\widehat{r}}_1$ is dominant at all wavelengths and for all scenes, except in the Green-NIR for Scene C where it can be major. The diffuse contribution ${\widehat{r}}_4$ is between detectable and significant in all cases, except for significant-major in the Red (Scenes A and B) and NIR (all scenes).

For Scenes A and B, the maximum total ocean contribution $\max ({\widehat{r}}_{35})$ is undetectable in the Red-NIR, detectable in the Green and significant in the Blue. In the UV-Violet it continues to be significant for scene B, but drops to detectable for Scene A. Marked differences appear for Scene C, where it is detectable in the NIR, and significant-major in the UV-Red.

Therefore, ${\widehat{r}}_1$ can be successfully isolated provided that one corrects for the significant-major contribution of ${\widehat{r}}_4$ especially in the Red-NIR, and the significant (significant-major) contribution of ${\widehat{r}}_{35}$ in the UV-Green (UV-Red) for Scenes A and B (Scene C) as seen in Table 2. To instead extract ${\widehat{r}}_{35}$, nadir views are better suited since ${\widehat{r}}_1$ increases with ${\theta }_0$ (although the simultaneous decrease in ${\widehat{r}}_3$ is in part balanced by an increase in ${\widehat{r}}_5$). The most favorable observations are in the Violet-Blue for Scene A and UV-Blue for Scene B (detectable-significant contribution), and in the Blue-Red for Scene C (significant-dominant contribution). For all geometries and scenes, $Q_{35}<50\%$ in the UV-Violet, $\max (Q_{35})<75\%$ in the Blue-Green and $\max (Q_{35})>75\%$ in the Red. In the NIR, $\max (Q_{35})<85\%$ for Scene C. The implication is that ignoring the atmosphere-ocean coupling term $r_5$ leads to a $>50\%(<25\%)$ error in ${\rho }_w$ in the UV-Violet (Red-NIR). To avoid such errors, retrievals of atmospheric and oceanic parameters must be performed simultaneously as opposed to sequentially (Stamnes et al., 2018; Gao et al., 2019; Frouin et al., 2019, and references therein).

3.2.1.2 Bright-sunglint observations

Observations along the principal plane and at $\phi =50\text{°}$ if ${\theta }_0=20\text{°}$ are strongly contaminated by sunglint. As shown by the cumulative plots and Table 1, ${\widehat{r}}_2$ indeed dominates viewing geometries within a cone centered around the direction of specular reflection in the Green-NIR for Scene A and B, and in the Red-NIR for Scene C. If ${\theta }_0=50\text{°}$ and $\phi =0\text{°}$, the dominant status extends to the Blue for Scene A and to the Green for Scene C. For all these cases, the associated diffuse contribution ${\widehat{r}}_4$ is detectable-significant, and the atmospheric contribution ${\widehat{r}}_1$ is significant-major in the Red and detectable-major in the NIR. In the Green, ${\widehat{r}}_1$ is major for Scene A and B, and significant for Scene C. For Scene A and B the ocean contribution ${\widehat{r}}_{35}$ is still detectable in the UV-Violet. For Scene C, it is significant-major in the Blue-Green, and detectable in the Red (Table 2, right side). Extracting ${\widehat{r}}_2$ from sunglint observations is most effective in the Red-NIR for Scene A and B, and in the NIR for Scene C, where ${\widehat{r}}_1$ is minimized and the degree to which light scattered in the atmosphere and reflected by the ocean surface are decoupled is maximized $(Q_{24}\ge 85\%)$. Still, one must account for the significant contribution of ${\widehat{r}}_4$ (all scenes) and also of ${\widehat{r}}_{35}$ for Scene C in the Red.

3.2.2 Polarimetric observations

3.2.2.1 Off-sunglint observations

For all scenes, ${\widehat{p}}_1$ is dominant for all geometries and wavelengths. Furthermore, ${\widehat{p}}_1>{\widehat{r}}_1$ and ${\widehat{p}}_{35}<{\widehat{r}}_{35}$ in the UV-Blue, indicating that in this spectral range polarimetry is more effective than radiometry for aerosol retrievals. The sunglint contribution ${\widehat{p}}_2$ is at most detectable except when ${\theta }_0=20\text{°}$ and $\theta =60\text{°}$ along the principal plane, when it becomes significant in the Red and major in the NIR. The maximum sunglint contribution $\max ({\widehat{p}}_2)$ is detectable in the NIR for all scenes and all geometries, and becomes significant in the Red only when ${\theta }_0=50\text{°}$ and $\phi =50\text{°}$. At shorter wavelengths, it varies between detectable and significant depending on the scene but for ${\theta }_0=50\text{°}$ and $\phi =50\text{°}$ it reduces to detectable in the Green and undetectable in the UV-Blue regardless of the scene. Similarly to $\max ({\widehat{r}}_4)$, the maximum diffuse surface reflection $\max ({\widehat{p}}_4)$ increases with wavelength and ranges from detectable in the UV, to detectable-significant in the Violet, to significant in the Blue, to significant-major in the Green-Red, and to major in the NIR. The maximum ocean contribution $\max ({\widehat{p}}_{35})$ is detectable for Scene A and B in the UV-Blue (with $40\%\le {DoLP}_{35}\le 45\%$ at nadir when ${\theta }_0=50\text{°}$), and undetectable in the Green-NIR. For Scene C, $\max ({\widehat{p}}_{35})$ is also detectable in the UV-Violet (${DoLP}_{35}=22\%$ at nadir when ${\theta }_0=50\text{°}$) but it increases to significant in the Blue-Red $(22\%\le {DoLP}_{35}\le 46\%)$, after which it decreases to detectable in the NIR $({DoLP}_{35}\approxeq 67\%)$.

In summary, ${\widehat{p}}_1$ can be extracted from off-sunglint observations of all scenes and at any wavelength (except for ${\theta }_0=20\text{°}$ and $\phi =0\text{°}$ in the NIR) after accurately accounting for ${\widehat{p}}_2$ and for ${\widehat{p}}_4$. For off-sunglint views of Scene C, ${\widehat{p}}_{35}$ also becomes significant in the Green-Red but can only be isolated if one accurately corrects for especially ${\widehat{p}}_1$ and ${\widehat{p}}_4$.

3.2.2.2 Bright-sunglint observations

The maximum direct sunglint contribution $\max ({\widehat{p}}_2)$ is major at all wavelengths and for all scenes. The maximum atmospheric contribution $\max ({\widehat{p}}_1)$ for these cases is also major except for Scene C in the Green-Red, where it is significant for principal plane observations when ${\theta }_0=50\text{°}$. Furthermore, $\max ({\widehat{p}}_4)$ is significant for all bright-sunglint observations with the exception of the detectable values for Scene A and B in the Green when ${\theta }_0=20\text{°}$, and Scene C in the Red. Finally, $\max ({\widehat{p}}_{35})$ is undetectable for Scene A and B regardless of wavelength. For Scene C it remains detectable (occasionally significant in the Red) except in the NIR when ${\theta }_0=20\text{°}$, where it becomes undetectable.

In summary, ${\widehat{p}}_2$ is best isolated from strong sunglint views, pending accurate accounting of ${\widehat{p}}_1$ and ${\widehat{p}}_4$ for all scenes, and of ${\widehat{p}}_{35}$ for Scene C in the Red.

3.2.3 Further remarks

It is evident that the diffuse sunglint contribution (${\widehat{r}}_4$ or ${\widehat{p}}_4$) often interferes with the extraction of the other contributions from TOA observations. This component strongly depends on the scattering properties of the atmosphere and of the ocean surface. While an exhaustive sensitivity study lies beyond the scope of this work, we can provide a few general comments. Firstly, the total AOD affects the diffuse illumination ${\chi }_{dif}$ of the ocean surface, and therefore ${\widehat{r}}_4$ and ${\widehat{p}}_4$, in a non-trivial manner (Section 2.4 and Section 3.1): the magnitude and spectral behavior of ${\widehat{r}}_4$ in the UV-Blue consequently depend on the AOD of the fine mode and, in the Red-NIR, on that of the coarse mode. Secondly, the ocean surface reflection properties vary with the wind speed and the surface refractive index (${\mathit{\text{m}}}_s$). Jin et al. (2011) show that the ocean surface albedo for isotropic skylight illumination, ${\alpha }_{dif}^s$, increases with decreasing $W$ by $\approxeq 1\%$ for $\mathrm{\Delta }W=1m{s}^{-1}$ when $5m{s}^{-1}\le W\le 15m{s}^{-1}$ and ${\mathit{\text{m}}}_s=1.34$. They also show that ${\alpha }_{dif}^s$ increases by $\approxeq 28\%$ if ${\mathit{\text{m}}}_s$ changes from 1.34 to 1.45 for a given $W$. Hence, assuming that the diffuse skylight illumination is approximately isotropic and that $r_4$ varies linearly with ${\alpha }_{dif}^s$, $r_4$ will increase by $5\%$ if $W$ decreases by 5 ms⁻¹ or if ${\mathit{\text{m}}}_s$ increases by 0.02. Of relevance to the latter scenario, ${\mathit{\text{m}}}_s>1.4$ in the UV-NIR for oil slicks (Otremba, 2000; Ottaviani et al., 2019). In addition to the intensity properties, the magnitude, spectrum and angular behavior of $p_4$ also depend on the polarization properties of the diffuse skylight illumination and of the surface reflection.

4 Conclusions

This paper presents a comprehensive set of vector radiative transfer computations of the total and polarized reflectance (and/or the degree of linear polarization) measured over three different types of ocean waters by a hypothetical satellite instrument with multispectral, multiangle and polarization capabilities. The added information offered by this measurement strategy, together with the higher accuracies warranted by recent technological advances, can be exploited to infer ocean parameters within the uncertainties that meet the demands of present Earth system models.

For every case, we analyze in detail the angular and spectral properties of each individual contribution (atmospheric and underwater scattering, and ocean surface reflection) to the total signal. The complete set of simulations, contained in the Supplementary Material, is visualized in the form of intuitive cumulative plots where each contribution as a function of the viewing geometry is easily quantifiable for each scene. In addition, the angular information is collapsed into bar diagrams representing the full ranges of variability for each component as a function of wavelength (Figures 15 and 16), emphasizing therefore their spectral variations. The discussion addresses why:

$•$ the ocean can be relatively bright in the UV (Torres et al., 2007) or in the Red (Frouin et al., 2019);

$•$ the radiative coupling between the atmosphere and the ocean exceeds $50\%$ in the UV-Violet, indicating that aerosol and hydrosol properties should be simultaneously retrieved when including wavelengths in this region (Stamnes et al., 2018; Gao et al., 2019; Frouin et al., 2019);

$•$ the ocean polarization cannot be ignored in the UV-Blue, but its angular features do not vary significantly with water type and can potentially be scaled among different scenes (see Chowdhary et al., 2012). It becomes insignificant in the Green-NIR for Scenes A and B, but large in the Green-Red for Scene C (Loisel et al., 2008);

$•$ the sunglint contribution dominates the signal only in the Green and wavelengths beyond (Ottaviani et al., 2019);

$•$ the atmosphere scattering is not always the dominant signal in the Green-NIR, an aspect that can affect applications like, e.g., the black-pixel approximation (Siegel et al., 2000).

The three analyzed water types span a large range of conditions, so that the results can serve as a valid reference for the interpretation of many typical satellite measurements over ocean and coastal areas. However, we considered a single aerosol scenario (vertical distribution, amount, and particle properties such as size, shape, and refractive index) to represent the average clean, maritime atmosphere over open ocean, and some modeling assumptions are involved in description of the optical behavior of hydrosols (bio-optical equations, spectral properties for DP and mineral particles). Also, the radiative transfer treatment assumes a plane-parallel geometry and does not include inelastic scattering. These assumptions are reasonable for the majority of open-ocean environments; differences can occur in coastal environments with excessive amounts of absorbing fine-mode and/or dust aerosols, and excessive amounts of SPM in the ocean (e.g., see Bi and Hieronymi, 2024). Still, our method provides a rigorous framework to compute the exact value of each individual contribution for any scene of interest, which can then be compared to the reference cases given here.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

MO: Writing – original draft, Writing – review and editing, Funding acquisition, Conceptualization, Methodology, Visualization. JC: Writing – original draft, Writing – review and editing, Funding acquisition, Conceptualization, Formal analysis, Methodology.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. This work was funded through the NASA ROSES grant 80NSSC20M0207 and, in part, by grant 80NSSC20M0205 (PACE Science and Applications Team).

Acknowledgments

This work was funded through the NASA ROSES grant 80NSSC20M0207 and, in part, by grant 80NSSC20M0205 (both belonging to the PACE Science and Applications Team): support from the program leadership is gratefully acknowledged. We are truly indebted to the reviewers for their thoughtful questions and comments, which helped clarify many of the details of this intricate paper.

Conflict of interest

Author MO was employed by Terra Research Inc.

The remaining author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The reviewer WRE declared a shared parent affiliation with the authors to the handling editor at the time of review.

Correction note

This article has been corrected with minor changes. These changes do not impact the scientific content of the article.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frsen.2025.1550120/full#supplementary-material

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Appendix

In this Appendix we report the tables of values used to generate Figure 4 (Table A1) and Figure 5 (Table A2). Table A3 reports the spectral molecular and aerosol optical depths of the chosen atmospheric model. A glossary for all acronyms used in the paper is also included.

TABLE A1

TABLE A1. Spectral IOPs for DP particulates in Scene A ([Chl]=0.1 $\mathrm{m}\mathrm{g}{\mathrm{m}}^{-\mathrm{3}}$), B and B’ ([Chl]=0.03 $\mathrm{m}\mathrm{g}{\mathrm{m}}^{-\mathrm{3}}$) and C and C’ ([Chl]=3 $\mathrm{m}\mathrm{g}{\mathrm{m}}^{-\mathrm{3}}$), corresponding to the plots in Figure 4. The contributions from mineral particles are not included, but $a_{DP}$ includes absorption by pure seawater and by CDOM.

TABLE A2

TABLE A2. Spectral IOPs for mineral particles in Scene C ([SPM] = $10\mathrm{g}{\mathrm{m}}^{-\mathrm{3}}$). See Figure 5.

TABLE A3

TABLE A3. Layer-resolved model values of the optical depths of the fine- (${\tau }_{a,f}$) and coarse-mode (${\tau }_{a,c}$) aerosol particles, as well as the optical depths for molecular scattering (${\tau }_m$) and for absorption by ozone (${\tau }_{O_3}$). Also reported is the molecular depolarization factor (${\delta }_m$).

Nomenclature

$A_{blk}$ Upwelling-to-downwelling irradiance ratio just below the ocean surface

$a_{DP},a_M$ Absorption coefficient of DP and mineral particles

a Total absorption coefficient

BTDF Bidirectional transmission distribution function

$b_p,b_{DP},b_M$ Scattering coefficient for particulate, DP particles and mineral particles

$b_{b,p},b_{b,DP},b_{b,M}$ Back-scattering coefficient for particulate, DP particles and mineral particles

$b_{blk},b_w$ Scattering coefficient for bulk ocean and pure seawater

$b_{b,blk},b_{b,w}$ Back-scattering coefficient for bulk ocean and pure seawater

CDOM Colored Dissolved Organic Matter

DoLP Degree of linear Polarization

${\mathit{D}\mathit{o}\mathit{L}\mathit{P}}_{\mathrm{35}}$ DoLP of the ocean reflectance

DP Detritus-Plankton

eGAP Extended Generalized Adding Program

${\mathbf{F}}_w$, ${\mathbf{F}}_p$, ${\mathbf{F}}_{DP}$, ${\mathbf{F}}_M$ Scattering matrix for pure seawater, particulate, DP and mineral particles

$F_0$ Annual average of the extraterrestrial solar irradiance

f Scaling factor for the upwelling-to-downwelling ratio

GISS Goddard Institute for Space Studies

HARP2 Hyper-Angular Rainbow Polarimeter 2

IOP Inherent Optical Property

I, Q, U First, second and third component of the Stokes vector

$K_d$ Diffuse attenuation coefficient

$m_w$, $m_s$ Refractive index of pure seawater and of the ocean surface

m Complex refractive index of aerosols

NIR Near InfraRed

OMI Ozone Monitoring Instrument

$p_{\mathit{1}},\dots ,p_{\mathit{5}}$ Absolute scattering contributions to the TOA polarized reflectance

${\widehat{p}}_{\mathit{1}},\dots ,{\widehat{p}}_{\mathit{5}}$ Relative scattering contributions to the TOA polarized reflectance

${{\widehat{p}}_k}^{*}$ Value of contribution ${\widehat{p}}_k$ at the sunglint maximum total reflectance

PACE Plankton, Aerosol, Cloud, ocean Ecosystem (satellite mission)

${\mathit{Q}}_{\mathrm{24}}\mathrm{,}{\mathit{Q}}_{\mathrm{35}}$ Surface and ocean reflection quotients

$\mathcal{Q}$ Bidirectional function of upwelling radiance just below the surface

$q_p,q_{DP},q_M,q_w$ Backscattering efficiency for particulate, DP particles, mineral particles, pure seawater

$\mathcal{R}$ Impact factor of water surface on water-leaving radiance

$r_{eff}$ Effective radius of aerosol particles

$R_{\text{P}}$ Polarized reflectance

$R_I$, $R_Q$, $R_U$ Reflectance associated with Stokes vector components I, Q, U

RSP Research Scanning Polarimeter

RT Radiative Transfer

${\widehat{r}}_{\mathit{35}}$ Relative ocean reflectance contribution

$r_{\mathrm{0}}$ Sun-Earth distance

${\mathit{r}}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{r}_{\mathrm{5}}$ Absolute scattering contributions to the TOA total reflectance

${\widehat{\mathit{r}}}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{\widehat{\mathit{r}}}_{\mathrm{5}}$ Relative scattering contributions to the TOA total reflectance

${{\widehat{r}}_k}^{*}$ Value of contribution ${\widehat{r}}_k$ at the sunglint maximum total reflectance

${\mathit{r}}_{\mathrm{24}}\mathrm{,}{\mathit{r}}_{\mathrm{35}}$ Absolute surface and ocean reflectance contributions

$\mathbf{S}$, ${\mathbf{S}}_1$, …, ${\mathbf{S}}_5$ Stokes vector and its individual TOA contributions

SPEXone Spectro-polarimeter for Planetary space EXploration

SPM Suspended Particulate Matter

SWIR Short-Wave InfraRed

[SPM] SPM concentration

$t^{\downarrow }$, $t_{dir}^{\downarrow }$, $t_{dif}^{\downarrow }$ Downward direct transmittance, and its direct and diffuse components

$t^{\uparrow },t_{dir}^{\uparrow },t_{dif}^{\uparrow }$ Upwelling direct transmittance, and its direct and diffuse components

TOA Top Of the Atmosphere

TOMS Total Ozone Mapping Spectrometer

$W$ Wind speed at 10 m above the surface

UV UltraViolet

$v_{eff}$ Effective variance of aerosol particles

VIS Visible

${\alpha }_{dif}^s$ Ocean surface albedo for isotropic skylight illumination

$\gamma$ Junge exponent

${\eta }_b$ Scattering-to-backscattering ratio for pure seawater

${\theta }_{\mathit{0}},\theta$ Solar and viewing zenith angle

$\lambda$ Wavelength

${\xi }_{\chi ,dir},{\xi }_{\chi ,dif}$ Spectral increase factors for ${\chi }_{dir}$ and ${\chi }_{dif}$

${\rho }_w$ Water-leaving reflectance at the TOA

${\xi }_R$ Spectral increase factor for $A_{blk}$

${\tau }_a,{\tau }_m,{\tau }_{O_{\mathit{3}}}$ Aerosol, molecular and ozone optical depth

${\tau }_{a,f},{\tau }_{a,c}$ Fine- and coarse-mode aerosol optical depth

$\phi$ Viewing azimuth angle (relative to the Sun)

${\omega }_{blk}$ Single-scattering albedo for the bulk ocean

$\chi ,{\chi }_{dir},{\chi }_{dif}$ Two-pass transmittance, and its direct and diffused components