1. Introduction

Solar radiation drives many important biogeochemical processes in the ocean, e.g., primary production and ocean heating. These processes can be studied on basis of radiative transfer models of the atmosphere-ocean system. The air-sea interface plays a distinct role by wave-characteristic change of the radiance distributions. A prominent example is the sun glint pattern, which can be used to retrieve wind speed information. In optical remote sensing, which is the background of this study, reflected sun and sky light are disturbing factors that have to be carefully considered in order to obtain the water-leaving radiance. The spectral shape of the water-leaving radiance enables estimates regarding water constituents, e.g., concentrations of chlorophyll, mineral particles, and colored dissolved organic matter.

Reflectance and transmittance properties of the sea surface depend on prevailing wind and wave conditions. Up to now, studies are focused on sea surface roughness using the Cox and Munk (CM) model, which is based on wind speed-dependent wave slope statistics [1]. Important basic research includes Preisendorfer and Mobley [2] and Mobley [3]. Effects of wind have been incorporated into many radiative transfer models [4–6] and the Cox-Munk model is still widely used [7,8]. In view of ocean color remote sensing, wind effects are mainly attributed to “atmospheric correction” [9,10] whereby the maximum permissible wind is 9 m s⁻¹ [11]. But we should not forget that wind affects the underwater light field too [12].

The use of wind-roughened sea surfaces in radiative transfer models can be satisfactory for many applications. However, the slope statistics method has limitations by neglecting the wave structure which causes special light-wave interactions. This is why the method is mostly limited to moderate wind speeds and sun zenith angles that are not near the horizon. In the last years, however, this issue has been tackled by different researchers using high resolution sea surfaces realizations [13–15]. Mobley [15], who used state of the art sea-surface modeling also added the aspect of polarization-dependency of sea surface reflectance and estimated an irradiance reflectance error of 10 to 18% when using slope statistics and unpolarized ray tracing and provided improved wind-dependent radiance reflectance factors.

But it must be remembered, that same prevailing wind speeds occur at different sea states and swell seas occur very frequently [see Fig. 1]. By definition, swell is wave motion not generated by local wind. This fact illustrates that wind speed can be misleading when used as unique feature for description of the wave system, and thereby being the sole link to surface albedo and irradiance reflectance. Hanley et al. [16] presented climatologies that emphasize the relative lack of correspondence between local wind and wave fields and highlighted the importance of swell in the global oceans. They showed that there are regions of the ocean that are usually in the wind-driven wave regime and others that are generally in the wave-driven wind regime. The wind-driven wave regime is found to occur most often in the mid-latitude storm tracks where wind speeds are generally high. The wave-driven wind regime is found to be prevalent in the tropics where wind speeds are generally light and swell can propagate from storms at higher latitudes. Furthermore, statistical analysis shows that the global ocean is strongly dominated by swell waves, even along the extratropical storm areas, where the relative weight of the wind-sea part of the wave spectra is highest [17].

 

Fig. 1 (a) ERA-Interim climatological wave classes with combinations of significant wave height H_S, peak wave period T_P, and average wind speed U₁₀. (b) Probability density function PDF of wave classes (white means no occurrence and thus no consideration for this study). The white frame marks the case used in the example.

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Seen over the course of a year, many sea areas often have flat irradiation angles; this in particular refers to areas in high latitudes, where high wind speeds with corresponding high wave heights must be expected [17–19]. Furthermore, Khon et al. [20] predicted that the significant wave height and its extremes will increase over different Arctic areas due to reduction of sea ice cover and regional wind intensification in the 21st century.

Near-polar and sun-synchronous-orbiting ocean color sensors like OLCI (on board ESA’s Sentinel-3) provide data from high latitudes several times a day. Geostationary satellites allow observations of the ocean in higher temporal resolution; nonetheless, their pixel size is growing considerably with latitude. One limiting factor to make effective use of the satellite data is the sun-viewing geometry (next to clouds, etc.). The main issue is, that wind speed-dependent atmospheric correction becomes increasingly challenging when the sun zenith angle increases, in particular above 70°. Modern satellites (e.g., ESA’s Sentinel mission) deliver information on local wind speed but also on the significant wave height of a sea area. Thus, we actually have more information on the state of the sea and do not necessarily have to make assumptions on wave age, possibly fetch limitations, or shallow water effects.

The underlying idea of this article is a revision of reflectance and transmittance properties of the wind-blown sea surface; emphasis is on influences of sea state. Nonlinear wave shapes and polarization of light are considered. Goal of the paper is to study the mechanisms and interactions of light with the air-sea interface and consequently to reduce uncertainties related to large zenith angles and high wind speeds in order to extend the usability of satellite data.

2. Methods

2.1 Sea surface generation

Following the approach of Hieronymi and Macke [12], wave effects on the light transfer are investigated for all relevant sea states and different local wind-wave regimes (e.g., for inland waters). The wave profiles under consideration are stationary and two-dimensionally (2-D) in wave-propagation and -elevation direction. A simplification is that the wind equals the wave direction. 3-D effects, foam, and whitecaps are neglected but discussed in Section 4.

2.1.1 Considered sea states

Naturally occurring sea states are under consideration for systematic analysis of wave effects on radiative transfer functions. The wave classes with corresponding average wind speed and occurrence frequency are extracted from the ECMWF Re-Analysis (ERA-Interim) data set [19]. The data refer to all sea areas globally with a grid resolution of 1 x 1 degree and every day at 12 UTC of the year 2013. The atmospheric model delivers the wind components 10 m above surface. From this the prevailing total wind speed U₁₀ is calculated with the assumption that the wind propagates into the same direction as the waves. The wave model provides the mean wave period, assumed to be the peak period T_P, and the significant wave height H_S of combined wind waves and swells. Statistically, one in one thousand waves from a wind see in deep water will have a height of H_max = 1.86 H_S [21]. The wave classes used cover H_s from 0.5 to 10 m ( ± 0.25 m each) and T_P from 3 to 14 s ( ± 0.5 s each), comprising about 98% of all cases in the ECMWF climatology. Figure 1 shows the wave classes with prevailing wind speed and their global probability density function PDF. The white fields in Fig. 1 signify no occurrence in the climatology and thus no consideration in this study. The most frequent sea state has a wave height H_S of 2 m, a period T_P of 9 s, and the prevailing wind speed is 6 m s⁻¹. Very rarely, extreme storm events with Beaufort number 10 and an average wind speed of 26 m s⁻¹ occur according the climatology. The corresponding wave class has a significant wave height of 10 m and a mean wave period of 11 s. In this case the sea is characterized by large patches of foam from wave crests that give the sea a white appearance and large amounts of sea spray reduce the visibility – this is certainly not a “clear sky event” with large use in the radiative transfer problematic (see Discussion). For the sake of completeness, however, such rare cases are investigated. As an aside, ESA’s Sentinel-3 is able to monitor wave heights from 0 to 20 m with a H_S goal accuracy of 4%.

2.1.2 Modelling of 2-D wave profiles

The use of averaged climatological wind and wave data implies that there is no information about possible fetch limitation, water depth effects, or about wave age (i.e., narrowband, broadband, or bimodal spectral shape). Hence, it is not expedient to use a purely wind-dependent long- and short-wave spectrum [22,23]. Instead, the unified spectrum requires a long-wave part, which depends on significant wave height and mean wave period, and a quasi-independent short-wave part, which is characterized by the prevailing wind speed. The spectral transition in the wavelength range of decimeters to a few meters is difficult to determine mainly for technical measurement reasons. This mirrors the enormous variance of amplitudes in the elevation and curvature spectra of different models [23–27]. Plant [28] emphasized the dependence of the short-wave spectrum on the amplitude of the long waves and points out the related kinematic short-wave-long-wave interactions. Hwang [25] observed the swell influence on ocean surface roughness, for example a spectral density shift from the intermediate-scale wave range towards the long-wave portion, and concludes that the spectral density in the decimeter length scale becomes less sensitive to wind speed variations as swell intensity increases. Due to these uncertainties and from a wave modelling point of view, the transition range is a regulating skew for adaptation of the required slope variance from observations [15,23].

The utilized omnidirectional elevation spectrum in this study follows the model of Bringer et al. [23], which yields consistency with multi-band microwave satellite observations (an example is shown in Fig. 2(a)). One difference is the applied long-wave spectrum S_lw, which in this work is based on Pierson and Moskowitz [29,30]:

 

Fig. 2 Workflow description of sea surface generation and ray tracing: (a) Wave elevation spectrum of the sea state with H_S = 2 m, T_p = 9 s, and U₁₀ = 6.1 m s⁻¹. (b) Example of a wave profile generated from the spectrum with color-marked wave slopes. (c) Corresponding distribution (probability density function) of wave slopes with their vertical allocation. (d) Definition of angle directions for wave slopes as well as incidence, reflection, and transmission angles with respect to global coordinates. (e) Determination of wave shadowing. (f) Example of the mapping of ray tracing with multiple (seven) reflections.

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Fast Fourier transform (FFT) is applied in order to extract single harmonics from the unified and omnidirectional elevation spectrum (S = S_lw + S_sw) [32]. The wave elevation in propagation direction x is:

If we look more closely at the sea surface, we have to realize that real waves show a small but easily noticeable departure from a Gaussian surface. Crests are higher and sharper than expected from summation of sinusoidal waves with random phase and troughs are shallower and flatter [21]; to be more precise, measured wave crests were about 10% higher than expected from linear theory [33]. In order to better reflect observations of wave crest distributions, the second-order correction scheme by Forristall [33] is applied to all sea states in this work. Every single wave from the spectrum interacts with others (where, for two waves, the wave frequencies are f₁ and f₂). The wave-wave interactions were truncated for f₁ + f₂ greater than four times the peak frequency of the long-wave spectrum. This is partly because the energy in the first-order spectrum is very small at higher frequencies, but also because the energy in the measured spectrum at these high frequencies appears to be mostly due to nonlinear interactions. Thus, short waves from the spectrum are not affected at all. In practice, positive and negative second-order terms are added to the first-order simulation. The positive term is approximately a harmonic with half peak period; therefore, peaks of wave crests and wave minima are elevated.

Actually modelled wave profiles are 10 km long with a horizontal resolution of 1 cm. The mean water line is at ζ = 0 m; the Earth’s curvature is not taken into account. The frequency domain covers 1000 logarithmically distributed single waves reaching from 3 mm (k_max = 2000 rad m⁻¹) to 140 to 3000 m length (one-tenth of the peak wave number for T_P = 3 to 14 s). One example of a wave profile is shown is Fig. 2(b) with its corresponding vertical allocation of wave slopes in Fig. 2(c). By taking a closer look at the 2-D slope distribution, the effect of the nonlinear correction can be seen, namely a slightly non-Gaussian slope distribution in conjunction with slightly modified slope variance in wave-elevation direction (this feature is more pronounced in steeper seas). With respect to reflectance properties, the average reflectance converges with increasing surface length. The used length is sufficient for all cases; no significant changes can be expected with longer wave profiles.

Inverse tests were carried out to verify, whether the simulated waves still match the given input, namely H_S, T_p, and U₁₀. Indeed, the chosen profile length does not represent 1000 cycles of the characteristic wave as it would be required in a strict statistical sense; it is a compromise between statistical needs and computing efforts. However, the length is adequate to nearly capture the wave crest distribution including very high waves. The wave height distribution was examined by locating the zero-up crossings and determination of the subsequent difference between crest maximum and trough minimum. Per definition, the significant wave height H_S represents the average of the highest third of the zero crossing waves. Alternatively, $H_S=4 \sqrt{m_0}$, where m₀ is the variance of the wave spectrum, i.e., the lowest moment of the input Fourier spectrum. Forristall [33], Longuet‐Higgins [21], and others have shown that the “one-third” definition usually gives values about 5% lower than the definition from the variance. Against this background, the deviation of effective H_S from the input gives on average 8.7% lower values; the range is between −18 and + 7% for all wave classes but −14 to −4% for the most frequent cases (covering 80% of the PDF). The retrieved peak period deviates on average by 6.8%. It is therefore concluded that the wave profiles are representative for the given classes of sea states and that the corresponding wave elevations, and thus the wave energy, are adequately represented.

Another approach of testing concerns the wave slope variance. Mobley [15] underlines the importance of fully resolving the slope variance mainly in view of comparability with the classical Cox-Munk [1] model. Under Mobley’s approach, surface slope variance from the short-wave spectral range (not fully resolved by computational considerations) is virtually shifted towards longer waves, resulting in enhanced amplitudes (and slopes) mainly in the decimeter wave scale. The wave profiles in the present study fully represent the long and short-wave elevation spectrum with a discretization of 1 cm. It should be mentioned that the used model [23] usually gives higher amplitudes in the short-wave spectrum than the Elfouhaily et al. [22] model which is basis for [14,15], and others. However, a resolution of 1 cm does not accurately display capillaries too; they are partly smoothed out. The resulting slope variance of all wave classes is shown in Section 4.1.

Some of the sea states wave profiles yield very small mean-square slopes mss, and although they are not completely unfounded, their variances are corrected for inter-comparison purposes. A lower limit is applied, where $mss=0.002+0.0012 U_{10}$. The slope (not the amplitude) of each wave segment is adopted in 10% steps up to the slope, where the total distribution yields greater mss than the introduced limit. The correction is applied for 34 low-wind cases out of the 155 sea states under consideration. The variance adjustment is conducted in order to account for the discretization and the disregarding of nonlinear small-scale wave interactions and shape features not captured by purely linear wave theory.

2.1.3 Waves from local wind

Fifteen additional wave profiles were generated in order to study small-scale wave effects in more detail. The waves represent a spatially very limited “local” wind situation. For convenience, the short-wave spectrum was taken directly from Bringer et al. [23] with a somehow artificial cut at waves with 1 m length. Thus, the profiles are composed of single waves with lengths between 3 mm and 1 m (without application of the second-order correction). Remember that the main contribution for mss arises from those short gravity and gravity-capillary waves. However, their mss are somewhat smaller in comparison to CM. The additional wave profiles are realized for wind speeds between 3 and 17 m s⁻¹, each with a length of 1 km and a horizontal resolution of 1 mm. The maximum wave heights are < 20 cm.

2.2 Ray tracing

Local reflectance and transmittance properties are determined for every point on the stationary wave by means of geometrical optics. Due to the high number of points, different statistical values can be reliably determined. Moreover, the approach allows retracing of wave-light interactions and investigation of the total effects due to the wave structure such as wave focusing under water [12,34] and its equivalent above.

2.2.1 Determination of the directional distribution

The ray tracing refers to a global coordinate system based on which local vectors are rotated. The directional definitions are illustrated in Fig. 2(d). Light incidence occurs in 1° steps beginning from 0° (zenith or normal light incidence) to 90° (sun at the horizon). Indeed, just one half of the upper hemisphere is treated; since the wave slope distribution in this study exhibits no (and in reality negligibly small) skewness, the statistics are identical on both sides.

The first step is the determination of wave shadowing, due to self-shading of a slope angle and due to neighboring waves. For this purpose a line with 1 km horizontal length is used (the length mirrors the mean free path of rays in air). The line starts plane-parallel at each wave segment. By stepwise changing the line’s inclination, we find the first angle with undisturbed light incidence [Fig. 2(e)]. There are comparatively few cases, namely at the top of an outstanding wave crest only, where no shadowing occurs.

The second step is the actual ray tracing, which required most computational time. Basis is Snell’s law of refraction written for the air-incident case:

2.2.2 Determination of reflectance and transmittance

The fraction of incident power that is reflected from the interface is given by the reflectance R and the fraction that is transmitted (directional refracted) is given by the transmittance T. Under the assumption that no absorption occurs at the interface, which is possibly not fully true in case of a dense sea surface microlayer (which damps the roughness anyway), it is $R+T=1$. The same applies if we differentiate between the state of (linear) polarization, namely perpendicular and parallel to the incident plane, i.e., $R_{\perp }+T_{\perp }=1$ and $R_{\text{II}}+T_{\text{II}}=1$ . The two polarization modes are independent of each other; in principle, they stay either perpendicular or parallel to the incident plane after a surface interaction. The so called Fresnel reflectances for the two polarization modes of incident light are:

3. Results

3.1 Ray tracing

Results are demonstrated using the example of the most frequently occurring sea state with 2 m significant wave height, 9 s mean wave period, and a wind speed of 6.1 m s⁻¹ [Figs. 1-2]. The shadowing with light incidence directions from 0 to 90° is illustrated in Fig. 3(a). It is obvious that wave segments located below the mean water line receive significantly less input than above and that the side of the wave facing the insolation direction has higher input than the far side [Fig. 3(b)]. If one regards the entire upper hemisphere (−90 to 90°), the maximum, most unshaded, insolation shifts towards flat slopes, only a few peaks of high wave crests gain undisturbed insolation from the entire upper half-space. The most shaded wave facets are strongly tilted and located on the backside of a high wave crest; they obtain insolation from roughly 30% less directions. Shadowing plays an increasingly important role for light incidence angles starting from 65° [Fig. 4(c)]. Only 15% of the horizontal sea surface is illuminated undisturbedly for light incidence from the horizon.

 

Fig. 3 Surface contacts from a ray tracing example: (a) Shadowed areas of the wave visualized by the number of free light incidence angles between 0 and 90°; below: sum and spatial allocation of all contact points after multiple reflections from all directions (values between 0 and 45). (b) Mean percentage free light incidence of tilted and vertically attributed wave facets (values between 70 and 100%). (c) Allocation of entry from all multiple reflections.

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Fig. 4 Shadowing and multiple reflections with incidence angle for the example sea state (solid lines) and the corresponding local wind case with U₁₀ = 6 m s⁻¹ (dashed). (a) Upper contour lines of histogram levels of the horizontal scatter path length. (b) Probability density functions for the occurrence of multiple reflections with number of contacts. (c) Percentage area of unshaded wave segments.

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After first interaction of light with the water interface the reflected part of the light beam can be directed further downward or slightly upwards towards the wave structure. In this case second and possibly further interactions occur. The frequency of multiple scattering of second to seventh order is shown in Fig. 4(b) as function of light incidence angle. Multiple reflections at the surface start with (global) light incidence angles of 36° and reach a maximum near the horizon with θ_ig = 80°. After that, more and more surface is shadowed towards 90°, decreasing again the total amount of multiple scattering.

Due to the continuously curved wave, the majority of reflected rays hit neighboring wave facets. For θ_ig = 85°, the free path length after reflection is less than 10 cm for one third of the cases and less than 1 m for approximately 70%. The greatest distance to the next surface contact point is approximately 900 m. Figure 4(a) marks the outer contour lines of the PDF of free path lengths or scatter distances respectively. It shows on which scales light-wave inter-actions occur, namely from millimeter to kilometer or capillary to swell waves respectively.

The exact location of all contact points from multiple scattering is marked in Fig. 3(a). The corresponding histogram clearly points out the accumulation on the backside of wave crests slightly below the mean water line [Fig. 3(c)]. Most multiple reflections take place with very low light incidence angles relative to the wave slope. The local incidence angle is between 60 and 90°, the maximum is at 85°. Thus, always a high percentage of the light beam is reflected. So, the wave’s far side is partly shadowed and therefore gains less insolation. But this is particularly the area where multiple reflections and therefore additional input into the water body occurs. The net effect is rather balanced in the slope-elevation histogram.

Differences due to wave elevations are illustrated in Fig. 4. In addition to the sea state with H_S = 2 m here the corresponding local wind situation with U₁₀ = 6 m s⁻¹ and H_S ≈0.05 m is included (note that the resolutions are different, namely 1cm and 1 mm). Self-shading of wave facets remains unchanged but large-scale shadows due to gravity waves are eliminated. The percentage of unshaded area expands, e.g., 12% more sea surface gains undisturbed insolation at 85° [Fig. 4(c)]. This also concerns the occurrence frequency of multiple surface reflections at large incidence angles, i.e., for rays coming from near the horizon more reflections can occur [Fig. 4(b)]. The PDF maxima are comparable, but higher values are reached for multiple scattering of higher orders for local wind. One reason is also the higher resolution. Free path lengths onto the subsequent scattering point can be shorter. With reference to θ_ig = 85°: in 25% of all cases the scatter distance is shorter than 1 cm (resolution of sea state), 70% are < 10 cm, but still 2% of path lengths are greater than 1 m. The main difference due to sea state elevations, however, is that multiple scattering can already occur at lower light incidence angles.

Regarding shadowing of the horizontal surface, the steeper the wave system is, the more areas are shaded. It is not necessarily the wave height that characterizes this behavior, it is more its ratio to the wavelength. The sea state steepness is H/L, where the dominant wave length is derived from the peak period $L= \left(g/2\pi \right)T_P^2$, g is the acceleration of gravity. Very steep seas exhibit some large wave slopes, e.g., α > 45° (up to 65°). Thus, facet self-shading and shadowing by the wave structure already starts with incident angles < 45°. Differences between sea states are especially visible at 85°, where the fraction of shaded area varies between 1 and 80%. At grazing angle of 90°, the shadowing is near 100% for all cases.

A similar relationship can be observed for second and more reflections. If a wave slope is greater than 45°, light from zenith direction (θ_ig = 0°) is reflected further downward and multiple reflections must occur. Thus, steep seas exhibit multiple reflections already at perpendicular light incidence; whereas they start to occur only from 80° for very flat seas. In total, the sea state with 1 m wave height and 3 s wave period is exposed to approximately ten times more multiple reflections than the sea state with 1 m and 14 s. Up to incidence angles of approximately 70°, more multiple reflections occur with increasing steepness. Beyond, the wave structure-dependent shadowing becomes increasingly influential and the total number of secondary reflections decreases again. This effect is additionally amplified because the remaining unshaded wave slopes that are facing the incident direction are generally steeper. Thus, relative incidence angles are steeper and resulting reflection angles are larger; more light is reflected away from the sea surface.

Multiple reflections where traced up to 10th order. For wind speeds below 15 ms⁻¹, this number of surface contacts is mostly not reached, neither for the sea state cases nor for local wind situations. With increasing wind speed and wave steepness the total number of tenfold reflection increases slightly. In contrast to this, there are significantly more tenfold-reflections for all 34 cases where the mean-square slope was artificially enhanced by steepening the slopes. Note that these cases have low wind speeds between 1.6 and 4.8 m s⁻¹. Up to 40,000 tenfold and 100,000 sevenfold-reflections where registered in case of the swell-dominated sea with 1.5 m wave height, 11 s period, 3.1 m s⁻¹ wind speed, and mss of 0.0057. In an equivalent unadjusted case with mss = 0.0028, seven surface reflections are reached only once. In general, we gain significantly more multiple surface reflections with the approach of wave slope variance adaptation. Although steeper wave slopes mean more self-shading of the wave segment, light from more directions (lower zenith angles) are further reflected into directions with wave contact. In the uncorrected case, larger wave slopes are more located in areas that are shadowed by the superposed wave structure. Thus, it makes a difference where slopes are located. Furthermore, steeper slopes generally cause shorter distances to the following interaction point, which in turn entails higher numbers of surface contacts. So, in spite of identical sea states, reflectance properties can vary for different mean-square slopes, which are mainly governed by small-scale waves, i.e., local wind.

3.2 Reflectance and transmittance distribution functions

Reflectance distribution functions RDF (θ_ig, θ_v) and transmittance distribution functions TDF (θ_ig, θ_v) are introduced for the air-incident case. The data comprise (global) light incidence zenith angles θ_ig from 0 to 90° in 1° steps and viewing zenith angles θ_v averaged in the boundaries [-90, −89: 2: 89, 90], thus, nominal θ_v of [-89.5, −88: 2: 88, 89.5]. The 2° averaging is necessary in order to remove computational artifacts from raster detection, i.e., jaggies. Figure 5 illustrates RDF and TDF results for the example. As expected from Fresnel’s equations, there is an angular range around Brewster’s angle, where the contribution of parallel polarized reflected light is nearly zero [Fig. 5(a)]. The only contribution in this range is by light perpendicular to the incident plane [Fig. 5(b)]. Beyond 70°, differences in amplitudes become clearly visible. Next to the expected higher reflection values for perpendicular light, another reason is the redistributed gain from multiple reflections, again each with large incidence angles [Fig. 5(d)]. Similar behavior can be observed for transmitted light [Fig. 5(e-g)]. The characteristic differences between RDF and TDF are the magnitudes of respective fractions and the width of angle distribution. The refraction angle (in water) is less than the incidence angle, which mirrors in narrower angular range of transmitted light.

 

Fig. 5 Reflectance (top) and transmittance (below) distribution functions for the example sea state: Parallel (a, e) and perpendicular (b, f) to the incident plane and total unpolarized (c, g) distribution functions. (d) and (h) show the angel-dependent redistribution of reflectance and transmittance due to multiple reflections in terms of differences to single scattering.

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The contributions and angular pattern-changes of RDF and TDF due to multiple reflections relative to single scattering, i.e., Fresnel reflection/transmission of 1st order, are illustrated in Figs. 5(d) and 5(h). The percentage difference for reflectance properties is:

Changes of RDF and TDF are visualized as contour plots in Fig. 6. All sea states are included in this plot (155 data sets). Indeed, the most harmonic evolving of RDF and TDF is observed for dependency on the mean-square slope of the sea. No other correlation parameter does develop such a continuous trend as (nominal) wind speed, wave height, wave period or sea steepness. The connection of mss from modelled sea states with the other parameters is charted in Fig. 7. There is a wide range of wind speeds and sea states with similar mean-square slope. However, the relationship between wind speed and sea surface roughness in terms of mss is generally known, e.g., from observations by Cox and Munk [1].

 

Fig. 6 Contour plots of RDF (θ_ig, θ_v) (top) and TDF (θ_ig, θ_v) (bottom) as functions of mean-square slopes of all sea states at different light incidence angles: (a, e) 0°, (b, f) 30°, (c, g) 60°, and (d, h) 85°. White color corresponds to values near zero or at zero and maximum values can exceed upper legend boundary.

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Fig. 7 (a) Up/down-wind component of mss from modelled wave profiles with wind speed. The color of the data points shows the sea state steepness H/L. The blue lines show the up- and cross-wind relationship of Cox and Munk [1]. (b) Sea states with corresponding ID number of mean-square slope cases from Table 1. Colored lines mark levels of mss [Fig. 1].

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Figure 6 shows the affected angular distribution (in viewing angles θ_v) for light incidence from individual directions. A perfectly flat sea surface reflects the incident light beam directly and solely into the direction with negative incidence angle. If the surface roughens, fractions of reflected light shift into directions alongside the main reflection axis. As soon as the roughness increases, the affected angular range expands and reflection/transmission rates redistribute, i.e., are getting smaller per angle. High mss, i.e., heavy seas, causes reflections of light into the entire upper hemisphere [Fig. 6(a)]. The directional distributions shift with increasing incidence angles. In case of θ_ig = 60°, the direction of highest reflection rates changes with increasing roughness towards larger reflection angles [Fig. 6(c)]. In case of θ_ig = 85°, the maximum moves from θ_v = −85° at low mss to −70° at highest mss [Fig. 6(d)]. The redistribution of reflected fractions due to multiple scattering at the surface impacts the RDF, also in terms of total reflectance. Furthermore, Fig. 6(d) illustrates the increasing shadowing of reflection directions very near the horizon (< −85°) with higher and steeper seas. Regarding TDF, the directional distribution is much narrower [Fig. 6(e-h)] and a small spread of the affected angle range can be observed for increasing incidence angles. Near the horizon, however, the right side of the distribution function (towards zero) is blocked due to wave shadowing [Fig. 6(h)].

In principle the cases with local wind fit very well into the distributions [Fig. 6]. However, a small shift of the reflectance maximum towards larger viewing (reflection) angles can be observed for large incidence angles (not displayed in Fig. 6). This is due to the lack of shadowing from higher waves. In contrast, cases with artificially enhanced slope variance may not necessarily fit into the shape of the distributions for all incidence angles (not included in Fig. 6, even though the effects are small and they would not be well visible). With identical mss and at θ_ig = 85°, mss-enhanced cases exhibit their maximum of RDF slightly closer to the edge (of −90°) with higher reflectance values. For example, with identical mss = 0.006 the sea state with H_S = 1 m and T_P = 7 s yields a RDF_max (θ_ig = 85°, θ_v = −84°) of 0.0724, but in a mss-corrected case with H_S = 0.5 m and T_P = 14 s, RDF_max (θ_ig = 85°, θ_v = −86°) is 0.0856. A reasonable explanation again is the missing blockage of reflection directions due to steeper and higher waves.

3.3 Usage of reflectance and transmittance distribution functions

A number of RDF (θ_ig, θ_v) and TDF (θ_ig, θ_v) are provided as supplementary material [Data File 1, Data File 2, Data File 3, and Data File 4]. The data comprise both components of polarization, parallel and perpendicular to the plane of incidence, for the air-incident case. Corresponding total unpolarized data can be calculated using Eq. (8). The data are sorted and averaged for sea state cases with similar mean-square slope; Fig. 7(b) tabulates the mss ID numbers with respect to Table 1 for all combinations of wave height and wave period. Furthermore, Table 1 provides details on the upper boundary of mss per case, the range of nominal wind speeds used for wave modelling, as well as ranges of significant wave height, mean wave period, and sea state steepness of the averaged cases. In addition, the mss-equivalent wind speed W from Cox and Munk [1] is given for the two components, up/down-wind and cross-wind (for clean surface):

Table 1. Specifications of provided mss-averaged RDF (θ_ig, θ_v) and TDF (θ_ig, θ_v) data (see Data File 1, Data File 2, Data File 3, and Data File 4, where acronyms “p” is for parallel and “s” for perpendicular to the plane).

View Table

 

Fig. 8 Reflectance distributions for 10 m s⁻¹ wind speed separated according polarization mode and wind direction components (ID = 12 and 9) (logarithmic color scale).

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The sea surface is now defined for various wave systems including wave propagation (or wind) direction. It can be implemented into (polarized) vector radiative transfer models of coupled atmosphere-ocean systems such as [8], replacing the (for low sun altitudes inaccurate)Cox-Munk model with application of Snell’s law and Fresnel’s equations. The orientation of the sea surface relative to global coordinates of the radiative transfer model determines the azimuth angle of the sea surface and the Stokes vector. By means of the provided data, irradiance and radiance reflectances can be computed using desired sky radiance distributions from clear sky, overcast or partly cloudy sky conditions. The incident angles of RDF and TDF refer to any radiance contribution from the corresponding direction, from direct sun light or diffuse sky light.

3.4 Total surface reflectance and transmittance

Summation over all viewing angles per hemisphere yields the total surface reflectance R_t (θ_ig) and transmittance T_t (θ_ig), respectively. Total surface reflectance is also known as (downward) albedo or irradiance reflectance of the sea surface. For the sake of energy conservation, the sum of reflected and transmitted light for each incident angle must be one. Multiple reflections strongly affect R_t (θ_ig) and T_t (θ_ig). In comparison to the single scattering case, the total transmittance increases between 60 and 90°, e.g., at θ_ig = 85°, T_{t, 1st} = 0.566 and T_{t, multiple} = 0.618 (D = 8.4%). This is clear since some of the first reflected light beams further interact with the surface and each time they lose portions of their power to the total transmittance.

Total reflectance as function of sea state is depicted for selected incidence angles in Fig. 9. The upper panels denote parallel fractions, middle panels perpendicular parts, and the lower panels show the total unpolarized reflectance [Eq. (8)]. Differences between parallel and perpendicular orientation can be observed for all angles of light incidence. At normal incidence, values range from 0.016 to 0.026. Strongest differences, and therefore possible effects in case of partial polarization, are observed for high mss and high sea state steepness. Differences are almost fully compensated for unpolarized light (equal weighting of both polarization modes), the values vary slightly around 2.11% for all seas, which is the same value as for a perfectly flat sea. As expected, largest differences in polarization are near the polarization angle at 53°. The parallel contribution is close to zero at 60°, independent of sea state [Fig. 9(c)]. The lower panels replicate familiar patterns in view of wind speed [2]. Up to approximately θ_ig = 50°, the albedo is nearly independent of wind speed or sea state; with light incidence towards the horizon, differences become significant and in fact wind-characteristic. Again, the pattern in Fig. 9(l) mirrors the wind speed-dependency [Fig. 1(a)], the related roughness-dependency [Fig. 7(b)], but also the steepness of the sea state.

 

Fig. 9 Total reflectance of all sea states under different light incidence angles: (top) parallel, (center) perpendicular to the incident plane, and (bottom) totally unpolarized. The incidence angles are: (a, e, i) 0°, (b, f, j) 30°, (c, g, k) 60°, and (d, h, l) 85°.

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4. Discussion

4.1 Slope variance and wind speed

RDF (θ_ig, θ_v) and TDF (θ_ig, θ_v) primarily are functions of the mean-square slope of the sea surface. Different research groups derived the probability distribution function and variance ofsurface wave slopes using sunglint images taken from an aircraft (Cox and Munk; CM [1]), from geocentric sun-synchronous satellite (Breon and Henriot [35]), and from geostationary satellite (Ebuchi and Kizu [36]), or using satellite-based lidar backscatter [37]. In general, the sea surface slope distribution is nearly Gaussian, the up-wind distribution is wider than the cross-wind one, the distribution is slightly peaked, and slightly skewed up-wind. Breon and Henriot [35] showed that the CM model permits an excellent fit to their observations. Whereas Ebuchi and Kizu [36], who obtained their data mainly from subtropical seas, observed narrower slope distributions and much less directivity relative to the wind direction than that reported by CM. All three [1,35,36] observed very similar cross-wind components of mss. For this reason, Eq. (11) can be used to determine the cross-wind roughness.

Hu et al. [37] analyzed lidar data that are not sensitive to wind direction related issues found in sunglint measurements. They show a very large scatter in the lidar backscatter histogram, which may explain some of the observational differences for different sea areas. On average, their observations agree well with CM for wind speeds between 7 and 13.3 m s⁻¹. But, according to their data, the Cox-Munk model overestimates the wind speed by 1-2 m s⁻¹ for wind speeds less than 5 m s⁻¹; here the slope variance is proportional to the square root of the wind speed. For wind speeds larger than 13.3 m s⁻¹ [37] observed a log-linear relationship.

A recent study by Zappa et al. [27] underlines the complex interplay of processes in shaping the roughness of the sea surface. Their small-scale slope observations by a polarimetric imaging camera highlight the possible over-simplification of the view that mss increases linearly with wind speed, too; they suggest the potential importance of upper ocean currents in addition to the wind. In fact, Zappa et al. [27] presented mss from field experiments that are widely spread and clearly beneath CM. Roughness-variability can also be explained by the influence of natural surfactants, which generally damps the roughness.

The differences in the wind relations – derived from different methods – raise the question, why wave slope variance imperatively has to be adapted to the CM wind speed parameterization [15,23]. In general, wave systems exhibit different levels of development and the slope distribution can differ from the results of Cox and Munk.

In this study, the modelled wave trains, with partly second-order shape modification, yield mean-square slopes as displayed in Fig. 7(a). It is found that the wide scattering of mss is connected to the wave height and period, and, thus, the steepness of the sea state, which is related to the wave age. The best agreement of mss from modelled wave profiles with CM (including their uncertainty range) is at sea state steepness between 0.03 and 0.05. Swell-dominated low-wind cases as well as stormy and extreme seas exhibit clear deviations to CM, but can be found in most instances at the edge of the data histogram by Hu et al. [37]. According the ERA-Interim climatology, 98% of all ocean wave systems have steepness classes between 0.01 and 0.04 [19]. Steeper seas are seldom and very likely not captured by the observations by CM and by others. It is clear, that the scale of waves contributing to mss is mainly on the order of 1 mm to approximately 1 m, as compared with 1 m to approximately 100 m for the wave energy spectrum [26]. However, the superposition of fully developed high gravity waves must lead to steeper wave slopes, and thus higher mss. Therefore, sea state characterizing parameters are useful in context of wind retrievals. The hypothesis of sea state steepness dependency of mss is supported by observations of Vandemark et al. [31], where inland, coastal, and open ocean regions were separated.

Figure 7(b) shows that different wave classes, mostly with similar steepness, can have similar mss. The attempt to artificially enhance mss of swell-dominated low-wind cases lose sight of the overall wave structure, i.e., gravity waves, which govern shielding of reflection directions near the horizon. Flat seas shield much less directions. In addition, little tilted wave slopes mostly reflect light into the sky, steepening of wave slopes leads to more reflection directions towards the wave structure and therefore significantly more multiple reflections at the sea surface. This is why the mss-corrected cases (slightly) fall out of the continuously run of RDF and TDF as a function of mss.

4.2 Effects of wave height and sea state

Some of the strong-wind sea states under consideration occur very rarely, e.g., U₁₀ ≥ 15 m s⁻¹ appears in less than 3.3% cases and U₁₀ ≥ 20 m s⁻¹ has a probability of less than 1%. It is probable that they coincide with inhomogeneous sunlight conditions due to clouds, sea spray, etc. In particular from a satellite remote sensing point of view, these cases are mostly not of interest. However, most light-wave interactions appear at very steep seas; for this reason, they are part of the analysis.

With regards to the applied long-wave spectrum, this study is based on the Pierson-Moskowitz (PM) spectrum. The use of other spectra, like the comparatively narrower JONSWAP spectrum or the double-peaked Ochi-Hubble spectrum, would possibly be more convenient in some cases [30]. In shelf sea regions for example, waves can be affected by the ocean bottom (shallow water waves) with strong wave shape effects, here the choice of a PM spectrum is suboptimal. At the open ocean, however, and due to the lack of additional directional and wave age information, PM is a reasonable choice. In the presence of swell, the applied spectra fit to the observations [25]; with increasing swell intensity, the spectral density increases in the long-wave portion and decreases in the short-wave portion of the intermediate-scale waves. Thus, the range of transition between long and short-waves, for all cases under consideration, is inside the variance of observations [23,25,26]. However, the choice of the long-wave spectrum rather marginally affects the total slope variance.

But, within this work it has been shown that gravity waves significantly shadow light incidence on large parts of the sea surface [Figs. 3 and 4], which affect the remaining mss of unshaded areas [Fig. 3(b)]. Furthermore, gravity waves have certain shielding effect on the reflectance/transmittance distribution functions, in particular for large viewing zenith angles near the horizon. Steep gravity waves have a larger influence than flat waves. Hence, swell waves, aside from their possible impact on roughness, are less important. Free path lengths after first surface reflection can be in the order of swells [Fig. 4(a)]. However, distances between multiple reflections are mainly associated with the length of small gravity waves.

The utilized second-order wave shape correction [33] is unimportant for flat seas, but significant for steep wave classes. As a guideline, the wave crest amplitude is enhanced by less than 1% in the case of H_S = 0.5 m and T_p = 9 s, in the frequent case with H_S = 2 m the enhancement is 3 to 6%, but 10 to 30% for very steep sea states (H_S = 8 m and T_p = 9 s). In view of reflectance, more interesting is that slopes are changing accordingly. In comparison with linear wave theory, however, mss increases only slightly by less than 1% in the example with H_S = 2 m and T_p = 9 s. Note that a minimum wave profile length is required in order to obtain a robust level of slope variance, e.g., up to 5 km for steep seas.

In conclusion, wave elevations play a role regarding slope variance as well as reflectance and transmittance at low light incidence; their neglect, as in case with pure slope statistics (CM), is no longer adequate nowadays. The application of a second-order wave theory is a step towards more realistic sea surfaces. The question is whether something comparable could be applied in future studies to short waves as well in order to account for micro-physical effects of air-sea interactions.

4.3 Non-regarded wave breaking, asymmetry, and 3-D effects

The modelled sea surfaces do not include wind speed-dependent wave breaking; an effect that would produce quite different reflectance properties of the affected wave section. In cases of wave breaking the water no longer satisfies the set of hydrodynamic equations that are used to describe the wave shape [24]. Air bubbles are mixed into the water column and make the breaker appear white (whitecaps). Whitecaps persist for some time after the mechanism that created it has ceased. The whitecap coverage is a function of air and water temperature and wind speed [38,39]. The fraction of whitecaps at the sea surface is relatively small; up to a wind speed of 14 m s⁻¹, the whitecap coverage is less than 1% and during a “gale” with 20 m s⁻¹, it is still less than 5% [38]. However, whitecaps are not unimportant in terms of the total reflectance of the sea surface. The spectral reflectance of an oceanic whitecap is expected to vary from a maximum of 60 to 0% as it passes from the newly formed state to extinction [40,41]. Some more aspects related to wave breaking exist, which are not yet described and incorporated into radiative transfer theory in a satisfactory manner. These include depolarization effects of whitecaps [37] or formation of sea spray aerosols [39,40]. Because of whitecaps it is likely that the presented results are accurate to within a few percent at low wind speeds, but may differ somewhat more from reality at high wind speeds.

Asymmetry effects of the wave profile haven’t been investigated as well; its influence on reflectance properties is assumed to be negligible. Microscale breaking of short gravity waves generates parasitic capillary waves that are created on the leeward side. The breaking process is more subtle than with large scale waves, no bubbles are created in the process [42]. Moreover, not all waves travel in the direction of the wind; weak structures propagate against the wind direction. The linear dispersion relation can be used for large-scale waves. Small-scale waves have turbulence-dependent dynamics with nonlinear interactions [28,42].

The sea surface is a three-dimensional (3-D) structure that moves continuously. Generally wind generates and leads to growing of waves; but in nature waves of different age superpose, cross seas may occur, ocean swell can interfere, and on occasion the wind direction cannot be determined clearly, e.g., near islands or in frontal activity areas [16]. For modelling purposes, the directional characteristics of waves are sometimes assumed to be uncoupled from their spectral properties, and then the spectrum of waves travelling within a given range of headings is taken to be some portion of that measured at one point [30]. On this basis, many researchers employ directional spectra with direction-dependent spreading functions in order to generate 3-D ocean surfaces [13–15,43,44]. The resulting 3-D wave spectrum exhibits a clear symmetrical peak in the relevant long-wave range. High resolution 3-D sea surface realizations, like the ones in [14] or [15], result in very symmetric reflectance pattern with partly very good agreement to the comparatively simple wave slope statistics model [1], in particular for the cross-wind case. However, both references document cases with large differences (3-D vs. CM). From a different point of view, the visualization of the underwater lensing effect of surface waves also shows a preferential direction of the wave propagation, and, hence, the relative unimportance of the cross-wind component [12,34,44–46]. At the open ocean, gravity waves are often nearly two-dimensional, in that the crests appear very long in comparison with the wavelength (long-crested waves). Similar considerations apply to extreme waves; they are strongly focused in the peak wave propagation direction. For those and for practical reasons this study considers 2-D (nonlinear) wave systems only; in view on the analysis of maximum wave height effects this is reasonable. For lateral wind conditions, the CM-based mss-selection seems to be a feasible starting point as explained (Table 1).

4.4 Polarization

The sun’s direct beam usually is unpolarized, whereas incident sky radiance usually is partially linearly polarized, i.e., there is more power in one polarization mode than another [15]. As supplement of this paper, linearly polarized reflectance and transmittance properties are provided. Hence, the potential users can choose sky radiance distributions according to their atmospheric specifications, i.e., clouds, water vapor, aerosol types, ozone concentration, stratification, etc. These specifications decide on the polarization pattern and degree of polarization and enables generation of very accurate surface reflectance factors. Mobley [15] stated that errors in irradiance surface reflectance due to ignoring polarization were typically greater than errors due to using Cox-Munk rather than FFT surfaces, or errors due to using single rather than multiple scattering. Since polarization information is required in many remote sensing applications, e.g., retrieval of aerosol properties, wind speed, or sensor calibration, it is advisable to use the revised reflectance properties as a function of surface roughness with information related to sea state.

Regarding unpolarized light, the question has been addressed if differences occur with initially polarized vs. unpolarized ray tracing. For both RDF and TDF, selected angle combinations, θ_ig and θ_v, exhibit deviations of more than 5%. This is mainly related to shallow light incidence, wave shadowing, and multiple reflections. On the whole, differences are insignificantly small. Differences in total surface reflectance and transmittance are within ± 2%; most affected are very steep sea states with strong roughness and high wind speed. As stated before, these cases are subject to a degree of uncertainties in view of depolarization and further unaccounted effects due to whitecaps.

4.5 Angular resolution

The provided data have relatively high resolution of 1° for incidence zenith angles and 2° for viewing zenith angles. Some marine radiative transfer models may utilize such high resolution, e.g., Monte Carlo models [34,44,47]. The widely used model Hydrolight computes the radiance averaged over quads with 10° zenith angle θ by 15° azimuthal angle φ [43]. This angular resolution is a balance between the conflicting needs of having sufficiently high angular resolution in the radiance distribution and keeping run times acceptably small. Furthermore, from a measuring technique point of view, at sea it can be sometimes very difficult to maintain optimal sensor viewing geometry (within a small range of angles) in order to derive remote sensing reflectance, the water color-containing signal. Thus, Mobley’s [15] resolution of the surface reflectance factor of 10 by 15° is often totally sufficient.

Figure 10 illustrates differences of angular resolution Δθ_v with respect to RDF (θ_ig, θ_v) and TDF (θ_ig, θ_v) of the example. The blue stair curves represent the original Δθ_v of 2°; the red curves correspond to 10° (like [15]). Within 10°, reflectance rates can vary quite strongly in particular at the flanks of the distribution; as expected, inter-variability is highest for shallowlight incidence at the edge of the upper hemisphere (near −90°). Note that here the reflectance rates are significantly higher as compared to normal incidence. TDF is typically much narrower than RDF. Inter-variability of transmittance rates with Δθ_v = 2° is very high, between 0 and > 200% relative to 100% of TDF (Δθ_v = 10°) [Fig. 10(e-f)]. In conclusion, the higher resolution more accurately displays the angular distributions. Thereby, it can be expected that uncertainties are reduced for example in connection with sun glint analysis.

 

Fig. 10 RDF (θ_ig, θ_v) and TDF (θ_ig, θ_v) of the example for two viewing angle resolutions, namely Δθ_v = 2° (blue) and 10° (red).

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4.6 Air-incident reflectance and transmittance

A direct comparison with previous works is not straightforward. For example, Preisendorfer and Mobley [2] applied CM slope statistics for their analysis of reflectance and transmittance properties. The above discussion showed that there may be some discrepancies of the nominal wind speed due to inter-variability of mss from the modelled sea states. Furthermore, it has been shown that reflectance to some extend depends on a sea state steepness. However, if we take an example with a wind speed of 10 m s⁻¹ ( ± 0.4 m s⁻¹; Beaufort wind force 5), the total surface reflectance at θ_ig = 90° varies between 0.355 and 0.405, the mean value is $\overline{R} =0.384$. In this range, the related steepness H/L decreases from 0.0356 to 0.0212; the corresponding mss varies between 0.035 and 0.019 (on average lower than CM; see Fig. 7(a)). Estimating comparable values from their Fig. 18, Preisendorfer and Mobley [2] deduced R ≈0.32. Thus, the derived values are relatively close to each other; the mean reflectance from this study is slightly higher, but considering the same mss (based on CM), values come a bit closer. Variability can be explained by different sea states. Looking at other wind speeds, similar relationships can be observed, i.e., on average this study yields somewhat higher reflectance values, i.e., lower net transmission. The investigations of Mobley [15] point into a similar direction, that, in the end, deviations due to surface realization (CM versus 3-D) are rather small if the slope variance is similar. Another publication by Haltrin et al. [13] yield much higher Fresnel reflection coefficients of wavy sea surfaces, for the corresponding example 0.4437 to 0.8389, but little is known about their applied methods. Besides critical questioning related to slope variance in comparison with CM, insufficient considerations of multiple reflections can lead to stronger deviations; thus, net effects must be evaluated carefully [34].

It is known that the influence of surface roughness on the light transfer is minimal for incidence angles < 60°. Beyond, wave interactions become increasingly important, i.e., shadowing of insolation directions, redistribution of reflection fractions into more upwardly directed angles, and gain of total transmittance due to multiple reflections. The present study delivers revised reflectance and transmittance properties, distinguished according polarization type, with comparable high angular resolution, and related to a wide range of sea state parameters. In comparison with the slope statistics approach, although differences are not very high, wave-related uncertainties have been reduced.

4.7 Water-incident case

In general, radiative transfer models of the atmosphere-ocean system require information on the total bi-directional reflectance distribution function BRDF. This implies reflections into both directions, in and above water. The present work treats one aspect of the BRDF namely the air-incident case only, i.e., reflections above the surface and transmission into the water body. The water-incident case has some complex aspects. These aspects include the dependency on inherent optical properties (IOPs) of the water, in the first line scattering properties, which determine the mean free path length, which is necessary to estimate the wave-underside shadowing. Furthermore, attenuation during multiple reflections cannot be ignored. If wind and wave-dependency of the BRDF are considered, air bubbles in the water must be taken into account; they form large-scale layers which significantly influence the water’s IOPs and resulting water-leaving radiance [48].

The water-incident part of BRDF affects the water-leaving radiance and the theoretical construct of the f/Q factor, both important parameters in ocean color remote sensing. The latter describes the angular distribution of the upwelling radiance field in ocean waters and depends on (downwardly directed) sun and sky radiance distribution and its transmission into water [49]. The factor f is a linear coefficient which relates the irradiance reflectance just below the sea surface to IOPs, namely the ratio total backscattering to total absorption coefficient, b_b/a. Q describes the angular distribution of the upwelling radiance just below the surface; it is the ratio of upwelling irradiance to upwelling radiance. Both f and Q depend on sea surface roughness, i.e., wind speed. So, beside uncertainties of the f/Q factor particularly related to optically more complex, i.e., Case 2, waters [50], there are unknown uncertainties related to water-incident RDF and TDF.

Water-incident ray tracing is challenging. Light transmission is possible within Snell’s window with a cone angle of ± 48.26° (critical angle for given refractive index). If the angle of incidence is larger, total internal reflection occurs. Light that is transmitted is refracted away from the normal line with a transmission angle of 90° in case of the critical angle. A flat surface is the only case, where no water-to-air-and-back-to-water light transmission occurs.

BRDF also depends on the refractive index of seawater n_w which is a function of temperature, salinity, and wavelength [51]. The refractive index is 1.33 for cool and fresh water and red light (700 nm), and 1.35 for warm and salty water and blue light (400 nm). Using Eq. (9), the corresponding reflectances are R(1.33) = 0.0200 and R(1.35) = 0.0221, a relative difference of 10%. The polarization angle changes by less than a half degree. The cone width of Snell’s window is 97.6° in case of n_w = 1.33, but 95.6° for 1.35. Thus, proportionally more red than blue light transmits into the water and the angular range from where light can leave the water body is larger.

Another aspect are cases of low sun elevation, where transmitted sun light, with some absorption and scattering losses, re-transmits into air on the backside of a wave crest. Such “trans-illumination” of wave crests can be seen by eye if the water is clear enough. This part of the “water-leaving radiance” is not what the actual term defines; it is rather associated with back-scattering of water constituents than with forward-scattering at the deflected surface.

In conclusion, future studies are necessary that focus on wave effects and water-incident aspects of the BRDF. The appropriate tools are modern Monte Carlo radiative transfer models [34,44,47]. The present study is a first step towards a reduction of uncertainties of in situ optical radiometry for the determination of water-leaving radiance from in-water profile data and in particular from above-water data [52–54].

5. Conclusions

Air-incident linear polarized reflectance and transmittance properties of the (whitecap-free) sea surface have been investigated. The focus was laid on discussing wave effects on light interactions at the air-sea interface. For this purpose, a number of nonlinear wave profiles were modelled, accounting for almost all occurring sea states including fetch-limited seas. It is found that reflectance and transmittance distribution functions primarily depend on the mean-square slope of modelled sea surfaces; a better and more distinct relationship as compared to the related wind speed, significant wave height, mean wave period, or steepness of the sea state can be provided. The principle correlation of mean-square slope and wind speed is known and already used for a long time. But using slope statistics loses sight of the large-scale wave structure, which governs shadowing of incident light as well as shielding and redistribution of reflection directions near the horizon. The sea state steepness increases with surface roughness too and is an important factor controlling multiple reflections at the surface. A high degree of accuracy for redistribution of reflectance fractions from multiple reflections can only be achieved by using fully shaped surfaces; it cannot be accomplished with random and unrelated dicing of wave slopes according slope distributions. Multiple scattering originates mainly at the backside of steep wave crests, whereby free path lengths to the next reflection points are mostly short within some centimeters. Indeed, it makes a difference where steep slopes are located. The net effect of multiple reflections is to redistribute reflectance and transmittance fractions in the respective hemispheres and to slightly increase the net transmission of light into the sea – this relates to large light incidence angles, mostly > 60°. However, an overall assessment of shadowing and multiple reflections yields lower net transmission in comparison to previous studies [2], i.e., on average this study yields somewhat higher reflectance values with implications to energy flux and heating rates, but there is high variability related to sea states. In many sea regions, significant wave height and mean wave period seem to be better indicators of surface roughness and therefore reflectance properties. Consequently, this work contributes to the reduction of wave-related uncertainties of reflectance and transmittance properties.

RDF (θ_ig, θ_v) and TDF (θ_ig, θ_v) parallel and perpendicular to the plane of incidence are provided as function of the mean-square slope of the sea surface and are related to other sea state parameters. mss is a measure for the roughness of the sea, but also a measure of the degree of the nonlinearity of transport processes, such as gas transport. The mss-related reflectance and transmittance functions can be used for a large variety of sea states and wind speeds as up/down-wind and cross-wind properties of the sea surface. By means of the data with high angular resolution, irradiance and radiance reflectances can be computed using desired environmental conditions, i.e., sky radiance distributions from clear or cloudy skies. In conclusion, the provided data should be implemented into vector radiative transfer models of the coupled atmosphere-ocean system.