1. Introduction

Spectral water-leaving radiance (L_w, W m⁻² nm⁻¹ sr⁻¹) or its equivalent, remote sensing reflectance (R_rs, sr⁻¹), is a core quantity in interpreting ocean color observation and the subsequent retrieval of water constituents. Among the three observational schemes typically deployed in the field to measure L_w or R_rs [1], i.e., in-water, above-water, or a combination of in and above-water, the above-water method has been widely used [2] although each has its own advantages and disadvantages [3].

It has long been recognized [4] that the measured radiance (L_t) from an above-water sensor unavoidably includes surface-reflected skylight or glint, which has to be removed to derive L_w, i.e.,

 

Fig. 1 Schematic diagram showing the typical experimental setup for above-water water-leaving radiance measurements. As a convention, the sun is always placed in the X-Z plane.

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To aid correction of surface-reflected skylight, skylight coming from the direction (θ’, φ’) that would be specular-reflected into the sensor by a flat surface is often measured (Fig. 1). With the measured L_s(λ; θ’, φ’), the skylight correction is performed as:

Mobley [5] assumed the angular pattern of sky radiance (L_s in Eq. (3)) and water refractive index (r in Eq. (3)) are independent of wavelength, and hence so is ρ_s. Spectrally invariant ρ_s is also assumed in some other studies [e.g., 10,11]. Lee et al. [7] pointed out that ρ_s should vary spectrally because diffused sky radiance and direct sunlight both have spectral shapes different from L_s(λ; θ’, φ’). By comparing measured L_t(λ; θ_v, φ_v) and L_s(λ; θ’, φ’) with simulated L_w(λ; θ_v, φ_v), Lee et al. [7] also showed that ρ_s could increase from 400 to 800 nm by a factor of up to eight. However, it is still unclear how the spectral behavior of ρ_s varies with sensor setup, sky condition, sun angles, or sea states.

Effect of skylight polarization on ρ_s was recently investigated [12,13] and found to be significant. For example, for the viewing geometry shown in Fig. 1, both Harmel et al. [12] and Mobley [13] found that ρ_s at 550 nm would be underestimated by up to 25 – 30% depending on solar zenith angles and aerosol concentration if polarization of skylight was not accounted for. However, if and how polarization affects the spectral variation of ρ_s has not been studied yet.

The main objective of this study is to numerically investigate the spectral variation of ρ_s as a function of solar zenith angle, wind speeds and aerosol concentrations. Our approach has improved over the earlier studies (e.g., [5,7]) in three areas: First, the spectral variation of L_s distribution was explicitly considered. Second, we separated sun glint from sky glint. Since it is impossible to completely eliminate sun glint in above-water radiometry regardless of the setup, it is of interest to know how much the sun glint contributes to the sky glint [14]. Third, we accounted for the spectral variation of the refractive index of water, which increases toward shorter wavelength through normal dispersion and results in ~10% increase of Fresnel reflectance from 700 nm to 400 nm. Also, we explored the possible effect of skylight polarization on spectral variation of ρ_s.

2. Methodology

2.1 Theoretical basis

We assume that for skylight or sun light coming from an arbitrary direction, capillary wave facets within the surface area subtended by the sensor’s FOV have an orientation that reflect this incident light into the sensor. The probability of the facets having such orientation is determined by the slope distribution of facets which, in turn, is a function of wind [15]. The probability-weighted sum of reflected light coming from every direction within the sky dome represents the glint component that a sensor measures. Also, we explicitly separate direct sun beam (Lsun), considered a mathematical singularity, from the rest of skylight (L_sky), i.e., L_s = L_sky + L_sun. With this, Eq. (3) becomes:

2.2 Simulation

Following Mobley [5], the sky dome hemisphere surrounding the sensor FOV is partitioned into quads, except for the polar cap (i.e., θ = 0°) which is represented by a cone defined by its half angle. Radiance is assumed to be constant within each quad. Instead of partitioning the sky dome into quads of equal angular spacing (i.e., each quad is defined by the same same Δφ and Δθ), the quads in our partition are of equal area; each quad is defined by the same Δφ and Δu, where u = cosθ. In other words, each quad subtends the same solid angle, which, in our study, is set to be equal to Ωs. With an average sun-earth distance of 1.496 × 10⁸ km and the sun’s diameter of 1.393 × 10⁶ km, the half angle of the sun disk is α_0.5 = 0.2668°, which gives Ωs = 6.8096 × 10⁻⁵ sr. Therefore, in partitioning the sky dome, the half angle is α_0.5 for the polar cap; and Δφ = 9.3084 × 10⁻³ rad and Δu = 7.3155 × 10⁻³ rad for non-polar quads. With this equal solid-angle partition, the weighting of reflected skylight from each quad depends only on the orientation probability of capillary wave facets. If using the equal-angle partition, the weighting factor for each quad depends on both orientation probability and the solid angle that this quad subtends. Also, with the solid angle of each quad being the same as the sun’s, the contribution of the direct solar beam can be easily assessed by simply replacing skylight from the sun’s direction with solar radiation.

We used the coordinate system defined in Fig. 1 with the sun in the x-z plane. Let (θ_i, φ_i) be the polar coordinate (zenith and azimuth angles, respectively) of an arbitrary point on the ith quad and s the unit Cartesian vector representing incoming skylight from this point so that $s_i=(-\cos {\phi }_i\sin {\theta }_i,-\sin {\phi }_i\sin {\theta }_i,-\cos {\theta }_i)$. Let r be the unit Cartesian vector representing the reflected light entering the sensor’s field of view, the unit vector (n_i) normal to the wave facets that would reflect s_i into r is $n_i=\frac{r-s_i}{\Vert r-s_i\Vert }$. The angle of reflection $={\cos }^{-1}\left|s_i\cdot n_i\right|$. The polar coordinate for n_i, (θ_ni, φ_ni) = ($\arccos z_{ni}$, $\arctan y_{ni}/x_{ni}$), where x_ni, y_ni, and z_ni are n_i’s Cartesian x-, y-, and z- coordinates, and the slope of these wave facets is $\tan ({\theta }_{ni})$. As (θ_i, φ_i) moves around on this quad, the corresponding n_i also changes, defining ranges of θ_ni,min ≤ θ_ni ≤ θ_ni,max and φ_ni,min ≤ φ_ni ≤ φ_ni,max. Assuming independence of wind directions, the probability density (p_i) of wave facets having the orientations that would reflect skylight coming from the i^th quad into the sensor is [16]:

The same procedure was followed to find the probability of reflecting direct sunlight (or skylight coming from the polar cap) into the sensor except in this case the shape of the grid is not a quad but a cone. The Cartesian unit vector representing incoming sun beam from the peripheral of the sun is

In computing the Fresnel reflectance, we used the refractive index of seawater following [17]. Skylight radiance distributions L_sky(λ;θ,φ) and direct solar radiance L_sun(λ;θ_sun) for various solar zenith angles are simulated using MODTRAN [18] with a 1976 US standard atmosphere and standard maritime aerosol model with an aerosol optical depth at 550 nm (τ_a) of 0.05 (~23 km visibility).

3. Results

We will present the results computed for the sensor-sun geometry that is currently recommended using the coordinate system defined in Fig. 1 (φ_sun = 0): for the sensor, (θ_v, φ_v) = (40°, 45°); and for skylight to be measured, (θ’, φ’) = (40°, 225°).

3.1 Spatial distribution of p, r and L_sky

The estimated probability density function p are shown in Fig. 2 for wind speeds U = 0, 5, 10 and 15 m s⁻¹ and for sun zenith angle θ_sun = 30°. For low wind (U = 0 m s⁻¹), the skylight reflected into the sensor (‘□’ in Fig. 2) is mainly from directions immediately surrounding the sensor’s specular point (θ’, φ’) (‘ + ’ in Fig. 2). As wind speeds increase, the chances of the existence of wave facets with an orientation that would reflect skylight coming from directions other than (θ’, φ’) increase [15]. With a roughened surface, it is possible for the sensor to see reflected skylight coming from almost any direction. However, the probability remains the highest in the directions around (θ’, φ’) and decreases in other directions with the lowest probability in the directions behind the sensor and near the horizon. The probability of the sensor seeing the sun(the star symbol in Fig. 2) with a zenith angle of 30° increases by three orders of magnitude from ~2 × 10⁻⁶ (sr⁻¹) for U = 5 m s⁻¹ to ~8 × 10⁻³ (sr⁻¹) for U = 15 m s⁻¹. For measurements taken with near-nadir sun (zenith < 10°), the probability of seeing sun glint remains > 0.1 sr⁻¹ as long as U > 5 m s⁻¹.

 

Fig. 2 The contours of logarithmic probability density p (sr⁻¹) as a function of skylight direction (θ, φ) at wind speeds of 0, 5, 10 and 15 m s⁻¹. In each polar plot, the center represents the sensor’s FOV, zenith angles (θ) vary from 0 to 90° in radial direction, and azimuth angles (φ) are defined relative to the sun. The star, □ and + symbols denote the directions of the sun (θ_sun, φ_sun) = (30°, 0), the sensor (θ_v, φ_v) = (40°, 45°) and specular point of the sensor (θ’, φ’) = (40°, 225°), respectively.

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The mean reflectance angles and the corresponding Fresnel reflectance (r) at 532 nm for each quad are shown in Fig. 3. No dependence on wind speeds was found for either of the two variables, which is expected because the slope distribution for wave facets has a zero mean [15] regardless of wind speeds. For the sensor setup examined here, the reflectance angles for incoming skylight are limited to < 65°, increasing towards the directions opposite to the sensor and towards the horizon. The corresponding Fresnel reflectance at 532 nm varies between 0.021 to 0.089, relatively invariant for reflectance angles < 40° and increasing rapidly following the trend observed in Fig. 3(a) for reflectance angles > 40°.

 

Fig. 3 The reflectance angle (a) and Fresnel reflectance at 532 nm (b) estimated for each quad. The polar axes and symbols are the same as in Fig. 2.

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The ratio of L_sky(θ,φ) to L_sky(θ’,φ’) simulated using MODTRAN are shown in Fig. 4 for θ_sun = 10, 30, and 50° and for wavelength = 400 and 700 nm. The region with relatively low sky radiance is in the directions opposite to the sun (location with lowest sky radiance is indicated by the star symbol), and this relatively dim region expands with decreasing θ_sun. Also, the sky radiance at (θ’, φ’) (‘ + ’ in Fig. 4) is generally low, only < 20% brighter than the lowest sky radiance. There is a clear spectral variability in the skylight distribution; sky radiance at 400 nm is distributed relatively more uniform at non-solar directions than that at 700 nm, which tends to increase with a stronger gradient towards the sun.

 

Fig. 4 The normalized sky light distribution, L_sky(θ,φ) / L_sky(θ’,φ’), as a function of θ_sun (10, 30 and 50°) and wavelengths (400 and 700 nm). The yellow contour line, if shown, has a value of 16. The polar axes and symbols are the same as in Fig. 2. The star indicates the direction in the sky with the lowest radiance.

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Having examined the spatial variations of ρ, r, and L_sky individually, we will now examine their combined effect on ρ_s under various environmental conditions, particularly by accounting for the spectral variations of Fresnel reflectance and skylight distribution.

3.2 Spectral r_sky, R_sky and ρ_sky

As expected, r_sky depends only on winds (Fig. 5(a)). Under calm conditions, r_sky(λ) is close to (but always greater than) the Fresnel reflectance by a flat surface reflecting skylight coming from (θ’, φ’) (dotted line in Fig. 5(a)). The probability for the sensor to see skylight coming from directions other than (θ’, φ’) that have higher reflectance (Fig. 3) increases significantly with wind speeds (Fig. 2). As a result, r_sky increases by about 8-10% as wind speeds increase from 0 to 15 m s⁻¹ (Fig. 5(a)).

 

Fig. 5 Variations of r_sky, R_sky and ρ_sky with wind speeds (U = 0, 5, 10 and 15 m s⁻¹) and solar zenith angles (θ_sun = 0 and 30°). Note that the blue solid line (U = 0 m s⁻¹, θ_sun = 0°) and blue dotted line (U = 0 m s⁻¹, θ_sun = 30°) overlap each other in (b) and (c). The dotted line in (a) represents Fresnel reflectance for reflectance angle of 40° and the dashed line in (b) and (c) represents the values computed if the skylight distribution is assumed to be spectral-invariant and simulated using the Harrison and Coombes [19] model U = 10 m s⁻¹ and θ_sun = 30°.

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Because L_sky(λ; θ’, φ’) is generally dimmer than sky radiance in other directions and only < 20% brighter than the lowest sky radiance (Fig. 4), the ratio of average sky radiance L_sky(λ; θ_m, φ_m) to L_sky(λ; θ’, φ’), i.e., R_sky(λ), are mostly > 1 (Fig. 5(b)). R_sky(λ) is nearly spectrally flat under calm conditions with values ≈1. With low winds, reflected skylight come from a very limited direction around (θ’, φ’) (Fig. 2(a)), where skylight is relatively uniform (Fig. 4). With winds, R_sky(λ) typically increases towards longer wavelengths (Fig. 5(b)) because skylight in other directions are normally richer in longer wavelengths than L_sky(λ; θ’, φ’) (Fig. 4). Generally, the spectral slope is greater the stronger the wind and closer the sun is to the zenith.

ρ_sky increases with wind speeds (Fig. 5(c)), which affect both r_sky and R_sky; ρ_sky increases with decreasing solar zenith angles, which only affect R_sky. It can be easily derived from Figs. 5(a) & (b) that ρ_sky is always greater than the flat surface Fresnel reflectance. The spectral shapes of ρ_sky also change with wind speeds, from slightly blueish to relatively neutral to increasingly reddish as wind speeds increase (Fig. 5(c)). For example, the values of ρ_sky for θ_sun = 30° (dotted lines in Fig. 5(c)) are 0.0274 and 0.0246 at 350 nm and 1000 nm for U = 0 m s⁻¹, 0.0293 and 0.0302 for U = 5 m s⁻¹, and 0.0315 and 0.0377 for U = 10 m s⁻¹.

3.3 Spectral r_sun, R_sun and ρ_sun

The probability of the sensor seeing the sun glint increases rapidly with increasing wind speeds and decreasing solar zenith angles; so does r_sun (Fig. 6(a)). The values of r_sun vary over a wide range with maximums on the order of 10⁻⁷. As can be expected, direct sun beam is typically 5 – 6 orders of magnitude greater than L_sky(θ’, φ’), and due to Rayleigh scattering, R_sun always exhibit a strong reddish hue (Fig. 6(b)). As a result, ρ_sun always increases towards longer wavelengths. However, unlike ρ_sky which is always greater than the flat surface Fresnel reflectance expected for the given observation geometry, ρ_sun varies widely from negligible to significant as compared to ρ_sky (Fig. 6(c)). Its significance is mainly decided by r_sun. Under most conditions (open circles in Fig. 6(d)) ρ_sun is negligible, with values < 1% of ρ_sky. For sun glint to be significant (average ${\rho }_{\text{sun}}(\lambda )/{\rho }_{\text{sky}}(\lambda )>10\%$), r_sun should be greater than ~0.2 × 10⁻⁷, which corresponds to the conditions where the sun is at the zenith with moderate winds or the sea is much roughened (U > 10 m s⁻¹) with θ_sun < 20° (black circles in Fig. 6(d)).

 

Fig. 6 Variations of r_sun (a), R_sun (b) and ρ_sun (c) with wind speeds (U = 0, 5, 10 and 15 m s⁻¹) and solar zenith angles (θ_sun = 0 and 30°). Open, grey and black circles in (d) indicate those conditions under which average ρ_sun(λ)/ρ_sky(λ) < 1%, 1-10%, and > 10%.

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Spectral ρ_s (ρ_sky + ρ_sun) is examined in Fig. 7 for various wind speeds and solar zenith angles. We roughly partitioned ρ_s(λ) into three groups depending on the significance of sun glint. The first group of ρ_s(λ) (Fig. 7(a)) represents the cases when ρ_sun is negligible as compared to ρ_sky and corresponds to the conditions denoted by open circles in Fig. 6(d). In this group, ρ_s(λ) are typically < 0.04 for 350 < λ < 1000 nm and exhibit negligible spectral variation (the spectral slopes are between −0.1 and 0.1). The second group represents the intermediate cases (grey circles in Fig. 6(d)), where ρ_sun account for between 1 and 10% of ρ_sky(λ). In this group, ρ_s(λ) remain < 0.05 but show increasing spectral signature with slopes range between 0.1 and 0.3 (Fig. 7(b)). The third group represents the cases when sun glint amounts to > 10% of ρ_sky(λ), corresponding to the conditions denoted as black circles in Fig. 6(d). In this group, ρ_s(λ) (Fig. 7(c)) are generally greater in magnitude and steeper in spectral slope towards longer wavelengths than are the other two groups. The spectral slopes in the 3rd group vary between 0.4 and 0.8. In general, ρ_s increases with increasing wind speeds and decreasing solar zenith angles.

 

Fig. 7 Simulated ρ_s(λ) for various wind speeds and solar zenith angles are grouped depending on the importance of the sun glint. (a), (b) and (c) correspond to cases denoted by open, grey and black circles in Fig. 6(d). For the same wind speed (i.e., curves of same color), ρ_s(λ) have higher values for smaller solar zenith angle. The y-scale in (c) is different from (a) and (b).

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4. Discussion and conclusions

Results shown in Figs. 2–4 explain the logic behind the experiment setup currently recommended for above-water radiometry for ocean color:: (i) φ_v ≈45° reduces the probability of a sensor seeing sun glint while still avoiding seeing the sun-cast shadow (Fig. 2); (ii) θ_v ≈40° minimizes Fresnel reflectance (Fig. 3); and (iii) skylight coming from (θ’, φ’) ≈(40°, 225°) has relatively low radiance (Fig. 4). A combination of these factors ultimately reduces the sky- and sun-glint though it cannot be completely eliminated and their correction requires the knowledge of ρ_s (Eq. (3)). The variables in Eq. (3) that vary spectrally include incoming sky and direct sun light L_s(λ) and Fresnel reflectance r(λ). Due to normal dispersion of the refractive index of water, Fresnel reflectance at a given angle increases with decreasing wavelength within the visible and near infrared wavelengths. Although this spectral change small and probably won’t be detected by an above-water radiometer, it is opposite to the spectral change of ρ_s that has been observed to increase towards the longer wavelengths [7,20]. Therefore, the observed spectral variation of ρ_s must arise from L_s(λ).

To better understand this causal relationship, we separated direct sun beam L_sun(λ; θ_sun) from the rest of skylight L_sky(λ; θ, φ). Figure 4 shows that the distribution of skylight does vary with wavelengths. To examine the possible effect of ignoring the spectral component of skylight distribution, we followed [5] using the Harrison and Coombes [19] model to compute skylight distribution, which is assumed to be spectrally invariant, for U = 10 m s⁻¹ and θ_sun = 30°. The estimated R_sky(λ) is a constant (dashed line in Fig. 5(b)) and ρ_sky(λ) (dashed line in Fig. 5(c)), which has the same spectral shape as r_sky(λ) but of higher values, differ significantly from the result (yellow solid line in Fig. 5(c)) computed under the same wind speed and solar zenith angle but with a skylight distribution that varies spectrally. This confirms that the observed spectral behavior of ρ_s increasing towards longer wavelengths is largely due to the spectral variation of skylight.

Our simulation has been focused on the recommended above-water radiometry setup, i.e., (θ_v, φ_v) = (40°, 45°). The effects on ρ_s of changes in the sensor’s orientation and azimuthal positions were examined for U = 10 m s⁻¹ and θ_sun = 30° and the results are shown in Fig. 8. For θ_v changing from 40° to 30°, ρ_sky decreases (blue vs. yellow dotted lines in Fig. 8), but ρ_sun increases (blue vs. yellow dashed lines). For φ_v changing from 45° to 90°, ρ_sky decreases (blue vs. red dotted lines in Fig. 8), but ρ_sun increases (blue vs. red dashed lines). This interesting contrast between ρ_sky and ρ_sun results in less dramatic but still observable changes in ρ_s because the changes in ρ_sun are greater. For example, ρ_s varies by ~20% from 0.031 at 350 nm to 0.038 at 900 nm for (θ_v, φ_v) = (40°, 45°) (blue solid line), but varies by ~60% from 0.030 at 350 nm to 0.048 at 900 nm for (θ_v, φ_v) = (30°, 90°) (purple solid line). The recommended setup, i.e., (θ_v, φ_v) = (40°, 45°) not only lowers the value of ρ_s but reduces its spectral variation as well. For a station near Hawaii with U = 8 m s⁻¹, θ_sun ≈31°, (θ_v, φ_v) = (30°, 90°), Lee et al. [7] estimated ρ_s by comparing measured reflectance with simulated water-leaving reflectance. They found ρ_s increasing from about 0.03 – 0.04 at 400 nm to 0.08 – 0.25 at 800 nm. The spectral increases of ρ_s they derived are much greater than our simulated results (purple solid line in Fig. 8), probably due to noises in the simulated water-leaving reflectance in the longer wavelengths or due to instantaneous and random variabilities above water (e.g., clouds), in water (e.g., bubble or phytoplankton patches), or on the surface (e.g., foam) that the models could not account for. Regardless, part of their estimated spectral variation of ρ_s is due to stronger sun-glint contamination with smaller θ_v and larger φ_v (purple dashed line in Fig. 8) than the recommended observation geometry.

 

Fig. 8 ρ_s(λ) simulated for U = 10 m s⁻¹ and θ_sun = 30° and for different sensor viewing geometry as indicated in the legend. The dashed lines are ρ_sun(λ), dotted lines ρ_sky(λ) and solid lines ρ_s(λ).

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So far, the results are based on maritime aerosol with an optical depth τ_a = 0.05 at 550 nm. Spectral variation of skylight distribution is expected to vary with aerosol loads, and this effect is examined in Fig. 9, showing the changes in ρ_sky(λ), estimated as a ratio to ρ_sky(λ; U = 10 m s⁻¹; τ_a = 0.05), for various wind speeds and aerosol optical depths. ρ_sky becomes slightly more reddish with increasing aerosol concentrations (red dotted lines in Fig. 9). However, for τ_a varying between 0 – 0.3, which is greater than typical seasonal variability of aerosols (~0.2) [21], its effects (< 5%,) is very limited, particularly in comparison with the effect due to winds (about 5-10% for ΔU = 2 m s⁻¹; blue lines in Fig. 9).

 

Fig. 9 Ratios of ρ_sky(λ) to ρ_sky(λ; U = 10; τ_a = 0.05) simulated for θ_sun = 30° and various wind speeds and aerosol optical depths. From bottom to top, the blue lines are for U = 0, 5, 8, 10, 12 and 15 m s⁻¹ and τ_a = 0.05; the red dotted lines are for U = 10 m s⁻¹ and τ_a = 0, 0.05, 0.1 and 0.3.

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Polarization of skylight has been found to exert a considerable impact on ρ_sky [12,13]. However, it is still unknown how this impact varies spectrally. As an exploratory investigation, we computed ρ_sky(λ) using a vector radiative transfer model [22] with a roughened surface based on [15] and a maritime aerosol. Figure 10 shows the comparison of ρ_sky(λ) estimated with and without polarization for τ_a = 0 and 0.2 and the viewing geometry shown in Fig. 1. Clearly, ignoring polarization would lead to an underestimation of ρ_sky(λ), consistent with the prior results [12,13]. For a hypothetical molecules-only atmosphere with wind speeds = 10 m s⁻¹, Mobley [13; Fig. 19] estimated the ratios of ρ_sky with polarization to ρ_sky without polarization at 550 nm are highest (~1.3-1.4) when θ_sun = 30 - 40°, agreeing in general with our results (yellow solid lines in Fig. 10). The impact of polarization on ρ_sky decreases with increasing aerosol loads and wind speeds, because increased scattering by aerosols and roughened surface would further depolarize the skylight. Spectrally, the impact of polarization generally decreases towards longer wavelengths. On average, accounting for polarization would increase ρ_sky(λ) by about 8-20% at 400 nm to 0-10% at 1000 nm (dashed black lines in Fig. 10), which implies that our results shown in Fig. 5(c) have underestimated ρ_sky(λ) by the similar amounts. Apparently, the spectral effect due to polarization is opposite to the spectral effect due to skylight distribution. Comparison of Figs. 10 and 5(c) indicates that under situations where ρ_sky(λ) exhibits strong spectral variation towards longer wavelengths, e.g., the yellow solid line in Fig. 5(c) for U = 10 m s⁻¹ and θ_sun = 0°, the polarization effect is the weakest, e.g., the yellow lines in Fig. 10(a). On the other hand, the polarization effect is relatively strong (Fig. 10(b) and 10(c)) when ρ_sky(λ) exhibits weak spectral variation (see dashed lines in Fig. 5(c)). While results of ρ_sky(λ) shown in Fig. 5 are still valid in general, polarization needs to be accounted for conditions with low wind speeds or larger solar zenith angles. Also, the effect of aerosol becomes much more significant through polarization than what is shown in Fig. 9 when polarization is ignored.

 

Fig. 10 Ratios of ρ_s(λ) simulated with and without polarization for various solar zenith angles (θ_sun), wind speeds (U) and aerosol optical depth (τ_a). The mean of the ratios is shown as black dotted line in each panel.

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While clouds’ impact was not simulated in this study, it is not difficult to postulate that with increased scattering by clouds, skylight would become more uniform spatially and spectrally, which in turn would diminish the spectral variation of R_sky(λ) and ρ_sky(λ). Take an extreme case as an example. If skylight is isotropic over the entire sky dome, which applies more or less for overcast conditions, then R_sky ≈1 and ρ_sky(λ) is effectively the same as r_sky(λ) (Fig. 5(a)), which would probably appear spectrally flat for field observations. This is consistent with the results of Cui et al. [20] who estimated ρ_sky almost spectrally flat between 400 and 800 nm for overcast or mostly cloudy atmosphere conditions.

Sun glint could make significant contributions to the surface-reflected light in windy and high-sun conditions that are often encountered in field experiments (Fig. 6(d)) because the probability of a sensor seeing sun glint increases considerably under these conditions (Fig. 2). Sun glint affects skylight correction by increasing the overall value of ρ_s and further shifting its spectra favoring longer wavelengths (Fig. 6(c)). Since direct sun light is unpolarized, we recommend taking a two-step approach by first removing sun glint using Fig. 6(c) followed by skylight glint correction using Fig. 5(c) and Fig. 10.

In this study, the glint probability is computed directly from the distribution of capillary wave slopes and in the study by Mobley [5], it is estimated by the Monte Carlo method. For either method, one underlying assumption is that the entire hemispheric sky dome is sampled (e.g., see Eq. (10)). For field measurements, this means that the integration time of the sensor is long enough, the FOV is large enough, or both such that the capillary wave facets inside the FOV would cover enough variability during the integration time that light reflected from every direction in the sky (including the sun) are sampled. For sensors with short integration time (10⁻³ – 10⁻² s) or small FOV (< 100 cm²), skylight is under-sampled (lower ρ_sky) and sun glint, if by any chances is recorded, may appear as spikes. In this case, measurements with outlier L_t values may need to be filtered out [1,2]. For measurements with presumably fully sampled sky dome, ρ_sun is typically less than ρ_sky (Figs. 5(c) and 6(c)), meaning sun-glint contaminated measurements do not necessarily appear as outliers and need to be removed separately as we outlined here.

A recently suggested sky-blocking method allows L_w to be measured directly from above the sea surface by blocking surface-reflected light with a screen that extends into the water [23,24]. This would eliminate the need for sky- and sun-glint correction, but requires self-shading correction and sensors to be placed very close to the surface. Also, a majority of above-water measurements are taken from a fixed platform, for which reflected skylight and/or direct sun light is an unavoidable part and needs to be corrected.