Abstract

In this paper we propose a model to understand the polarization patterns of skylight when reflected off the surface of waves. The semi-empirical Rayleigh model is used to analyze the polarization of scattered skylight; the Harrison and Coombes model is used to analyze light radiance distribution; and the Cox-Munk model and Mueller matrix are used to analyze reflections from wave surface. First, we calculate the polarization patterns and intensity distribution of light reflected off wave surface. Then we investigate their relationship with incident radiation, solar zenith angle, wind speed and wind direction. Our results show that the polarization patterns of reflected skylight from waves and flat water are different, while skylight reflected on both kinds of water is generally highly polarized at the Brewster angle and the polarization direction is approximately parallel to the water's surface. The backward-reflecting Brewster zone has a relatively low reflectance and a high DOP in all observing directions. This can be used to optimally diminish the reflected skylight and avoid sunglint in ocean optics measurements.

©2013 Optical Society of America

1. Introduction

During atmospheric propagation, sunlight is scattered by molecules and aerosol particles. Thus light is usually partially polarized. The exception is on an overcast cloudy day or at the certain neutral points. Water surface is a typical polarizer in nature. The reflection process can change the polarization patterns of incident light. In addition to the atmospheric scattering process, most light which is polarized naturally rises from water surface reflection. The research on water reflection-polarization patterns shows special values in hydrophytic bionomics [1,2]. A lot of animal species have the ability of sensing polarized light. They can use polarization rather than the intensity of skylight and light reflected from water surface for navigation, hunting and escaping [3]. It is important to understand the distribution and changing rules of light when reflected off water surface in research on aquatic insects and bionic navigation theories. Most existing research operates under the assumption that incident light is unpolarized or the water surface is flat. This, however, is not consistent with the reality of nature [1].

Furthermore, in ocean remote sensing, the roughness of water surface can be deduced from the nature of the reflected light [4]. According to the polarization differences in field of view (FOV), water surface can be easily distinguished [5]. Nevertheless, surface re〉ected glint is a curse for ocean color remote sensing from above-water platforms [6]. The specular reflection of sunlight (sunglint) can create a saturation area in the sensor's FOV, and can produce a circular bright band which may obscure certain underlying information [7]. Skylight reflection is widespread but much weaker. It can change the spectrum components of water body. This kind of impact is very difficult to predict or compensate for [6]. To eliminate the negative effects of light reflected from water surface the most common method is to set reasonable observation geometry. But this method has many limitations and generally will not get ideal results. However, using the polarization approach to eliminate interventions caused by reflection from water surface is a new method of solving this problem [8]. Since light reflected from water surface is partially polarized—especially when a beam of unpolarizaed incident is reflected off a flat water surface at a Brewster zenith angle—it will become linearly polarized. Thus, one can use a linear polarizer to eliminate the reflected component completely. However, most of the natural water surfaces are not calm but rough. The light incident on water surface includes both direct solar radiation and atmosphere scattering components. Where sunlight can be treated as natural light (unpolarized), the scattered skylight is partially polarized and its polarization patterns change with incident directions. So the polarization patterns of reflections from wave water become much more complicated than flat water—which depend on the polarization of incident light, observation geometry, water surface condition, etc. At present, this method of eliminating light reflected off water surface by using a polarization filter is controversial [911]. This is mainly due to lacking of clearly understanding of the polarization patterns of light when reflected off waves.

Kinsell L. Coulson systematically described the atmosphere polarization effects in his monograph [12]. The “Coulson Table”, created early on, has been regarded as a benchmark of polarized skylight ever since. Gábor Horváth used a polarization imaging system and proved the rationality of Rayleigh model in describing the distribution of polarized skylight under clear conditions [13]. Nevertheless research on the polarization patterns of light reflected off water is rare. Cunningham made use of the polarization properties of reflected light to eliminate specular reflected sunlight in measuring the reflectivity of sea surface [8]. Horváth and his team had carried out a series of remarkable research projects on the polarization of light when reflected off a flat water surface. They used the semi-empirical Rayleigh scattering model and Fresnel law of reflection to simulate the polarization pattern of skylight reflected off a flat water surface when under clear sky conditions [1,5]. A few years later, they used a polarization imaging system to measure skylight reflected off a calm water surface at dusk. This verified the correctness of their early simulation results [14]. However, the polarization patterns of light reflected off wave water has been seldom investigated, even though most of natural water surfaces are in fact wavy.

For direct solar incidence, the reflection polarization patterns of calm water can be easily described by the Fresnel law. However with scattered skylight, the incident radiances come from the whole celestial hemisphere rather than an individual direction, thus the reflection calculation becomes more complicated. Furthermore, scattered skylight is partially polarized and affected by atmosphere molecules, aerosol, clouds, dust, etc. The effects of these other factors are hard to quantitatively describe. Natural water surface is usually wavy with different slopes and orientations for separate incident directions, so the modulation of polarization is an extraordinary complicated question. At present most research rests on the polarization patterns of skylight reflected off flat water and little research has been done on wave water. In order to effectively eliminate the light reflected off wave water by using the polarization method, we must first determine the polarization patterns and the distribution of reflected light. Only then can we investigate the proper stripping method and corresponding observation geometry. This is the major subject of our research.

Using basic scattering and reflecting theory, this paper focuses on the polarization patterns and the intensity of distribution of primary Rayleigh scattered skylight after being reflected off wave water. We also discuss how certain patterns (the degree of polarization, the direction of polarization and the intensity of reflected light) change depending on such factors as solar zenith angle (SZA), wind speed and wind direction. The polarization states are calculated theoretically and the results may perhaps provide valuable information regarding reflected light in ocean color remote sensing, water surface target detection and hydrophilic animal navigation study.

2. Methodology

2.1 The polarization patterns of skylight

To investigate the polarization of natural light reflected off wave water, we must first determine the properties of incident light. Direct sunlight is unpolarized, part of which will be scattered by atmosphere molecules and aerosol particles, thus becoming partially polarized skylight. The primary Rayleigh scattering usually dominates, especially under clear skies, while multiple scattering can depolarize skylight [15]. Since aerosol and cloud conditions change with time and space, it is hard to predict their multiple scattering effects on skylight. This paper is mainly concerned with primary Rayleigh scattering light under clear skies.

In order to understand the intensity of skylight in different directions, this paper refers to the clear skylight radiance distribution model proposed by Harrison and Coombes [16]. According to their theory, skylight intensity in a specific direction can be expressed as Eq. (1).

N(γ,θs,θv)=(A+Bemγ+Ccos2γcosθs)(1e1.90secθv)(1e0.53secθs).
Where the N(γ, θs, θv) stands for the relative intensity of skylight, θs is SZA, θv is the zenith angle of the incidence direction, m is the air mass and γ is the scattering angle. Having analyzed approximately 3000 measurements, Harrison and Coombes obtained the following fitting data: A = 1.63, B = 53.7, C = 2.04, m = 5.49.

During our calculation of light reflected from wave water, we have taken both direct sunlight and scattered skylight into consideration. The total incident light is set to 1 (dimensionless). The ratio of diffuse to direct solar irradiance is dependent on initially available solar irradiance, atmospheric and ground conditions, and topography [17]. In most of the following simulations, we set the solar irradiance to 0.9 (perpendicular to the solar rays). The skylight is separated into a series of incident directions which are almost evenly distributed in celestial hemisphere according to the above mentioned intensity model. The total irradiance of skylight is set to 0.1 which is the sum of the radiation perpendicular to each incident directions (a possible situation under clear sky, low aerosol depth and regardless of ground reflection) [18]. We also make a group of contrast in which the solar part is set to 6/7, and scattering part to 1/7 (the ratio of direct normal irradiance to skylight become 6:1, which may be caused by atmosphere variation).

The degree of polarization (DOP) of skylight changes with the scattering angle. Except for the four neutral points (Argo, Babinet, Brewster and the fourth neutral point) in celestial hemisphere [15]. The primary Rayleigh scattering model should adequately describe the skylight polarization patterns [19]. In Rayleigh atmosphere, the DOP of skylight can be expressed by Eq. (2) [12].

DOP=DOPmaxsin2γ1+cos2γ.
In the Rayleigh model, the maximum degree of polarization, DOPmax, is 100%. However, some of atmospheric phenomenon (e.g. multiple scattering, ground reflections, the presence of dust and molecular anisotropies) cause the deviations from the ideal Rayleigh model. One can partially take these effects into consideration by using an empirical relationship between DOPmax and the SZA. This approximation is the so-called semi-empirical Rayleigh model. The empirical values of DOPmax are 56%, 63%, 70% and 77% when θs is respectively 0° (sun appears at zenith), 30°, 60° and 90° under clear atmosphere [12].

The cosine of the scattering angle γ can be expressed as Eq. (3), with SZA (θs), view zenith angle (VZA, θv), and the azimuth angle (φ) between them.

cosγ=cosθscosθv+sinθssinθvcosφ.
In any incident direction of celestial hemisphere, the skylight polarization direction is perpendicular to the scattering plane decided by solar incidence and scattering light. The angle of polarization (AOP) is defined as the angle between the polarization direction and the scattering particle’s meridian plane, and the AOP can be calculated by Eq. (4).
AOP=arccos(sinφsinγsinθs)
So the AOP pattern of skylight is regularly distributed and changes with solar position.

2.2 Reflection of calm water surface

The Fresnel law can be used to describe the reflection and refraction process at the interface of the two media. When dealing with the polarization changes in reflection or refraction, the electric vector (E-vector) of light is usually divided into two oscillation directions, i.e. one parallel and one perpendicular to the incident plane. Different E-vectors have different reflectance which can be described as Eq. (5).

{rs=n1cosθ1n2cosθ2n1cosθ1+n2cosθ2rp=n2cosθ1n1cosθ2n2cosθ1+n1cosθ2.
Where the rs stands for reflectivity in a perpendicular direction and rp stands for reflectivity in a parallel direction, n1 is the refractive index of incidence medium (in this paper it refers to air, n1 = 1), n2 is the refractive index of reflective medium (in this paper it refers to water, n2 = 1.34), θ1 is the incidence angle, θ2 is the refractive angle.

When the incident light is unpolarized, the amplitude of perpendicular E-vector and parallel E-vector are equal. The DOP of reflected light can be expressed as Eq. (6).

DOPr=rs2rp2rs2+rp2.
When the incident angle reaches about 53° (near the Brewster angle), the DOP will reach a maximum value 1, which means the reflected light will become linearly polarized. The polarization direction is perpendicular to the incident plane.

Considering the partially polarized incident light, the polarization pattern of reflected light depends on the incident polarization states and reflection process.

To get the exact polarization information of reflected light, we use the Stokes parameters to express incident and outgoing light. The reflection process is represented by a Mueller matrix which is shown in Eq. (7).

[IrQrUrVr]=M[IiQiUiVi]=12(rs2+rp2rp2rs200rp2rs2rs2+rp200002rsrp00002rsrp)[IiQiUiVi].
Where I, Q, U, and V are the four Stokes components, the subscript r represents reflected light and subscript i represents incident light. The reference direction of the Mueller matrix is parallel to the incidence plane. The DOP and angle of polarization (AOP) can be expressed with Stokes components using Eqs. (8) and (9).
DOP=Q2+U2+V2I.
AOP=12tan1(UQ).
The scattering process in atmosphere or reflection above water surface seldom produce circular polarized light, so component V is set to 0 [12].

2.3 The Wave Surface Reflectance Model

Cox and Munk got the statistics of wave slope distribution from sunglint images of the sea surface which can be used to calculate the bidirectional reflectance property of wave surface [4]. Cox-Munk model considers wave surface as a collection of facets. The direction of each facet is represented by its slope. Probability distribution of facet slopes depends on wind speed and direction.

In a local right-handed coordinate system with the z axis pointing straight upward, a is the unit vector of a given direction. θ and φ are the zenith and azimuth angle of vector a, respectively where φ is taken clockwise from y axis. So that a can be represented as (θ, φ) or (sinθsinφ, sinθcosφ, cosθ). The incident direction s, reflected direction o and the outward normal of a wave facet n can be represented by (θs, φs), (θo, φo) and (β, α). It will not change the results if φs = 0, i.e. y axis aligned with the solar azimuth angle. The coordinate system and the directions mentioned above are shown in Fig. 1(a). Then the components of facet slope in x and y axis can be computed as Eq. (10).

{zx=z/x=sinαtanβ=sinθssinφs+sinθosinφocosθs+cosθozy=z/y=cosαtanβ=sinθscosφs+sinθocosφocosθs+cosθo.
The slopes of wave facet are affected by wind speed and direction. So the original coordinate O-xyz is rotated through χ in the horizontal plane with y axis aligned with the wind direction. In the new coordinate O-xyz’, the facet slope components in x' and y' axis, i.e. the crosswind and upwind direction, can be expressed as Eq. (11).
{zx=cosχzx+sinχzyzy=sinχzx+cosχzy.
The root mean square (RMS) of wave facet slopes in crosswind and upwind directions are given by σc and σu, respectively. With ξ = zx’/σc and η = zy’/σu, the probability of wave facet with certain slope components is given by Eq. (12) [4].
p(zx,zy)=12πσuσceξ2+η22[112C21η(ξ21)16C03(η33η)+124C40(ξ46ξ2+3)+14C22(ξ21)(η21)+124C04(η46η2+3)].
Parts of the parameters are determined by wind speed W (in m/s). The RMS components of wave slopes and other parameters in Eq. (12) can be found in [4]. Another group of RMS components derived from sunglint images taken from geostationary meteorological satellite are given by Eq. (13) [20].
{σu=0.0053+6.71×104Wσc=0.0048+1.52×104W.
We adopt the above formula which is recommended by the MERIS team [21]. Then we can calculate sunglint reflectance as long as the probability of wave slope is determined. The angle between incident direction s and observing direction o is calculated using Eq. (14).
cosΘ=cosθscosθo+sinθssinθocosΔφ.
Where Δφ = φo-φs. So the incident angle on the facet is ω = 0.5Θ. Wave shadowing can be a result of high wave elevation or large incident or observation zenith angle. So the shadowing factor S is taken into consideration which is given by Eq. (15) [22, 23].
S(θs,θo,σ2)=11+Λ(cot(θs))+Λ(cot(θo))Λ(x)=12[2πσxexp(x22σ2)erfc(x2σ)].
Where σ 2 = σu2 + σc2, is the sum of squares of facet slope components. And erfc is the complementary error function. Thus the sunglint reflectance can be written as Eq. (16) [4].
ρg(θs,θo,Δφ)=πr(ω)4cosθscosθocos4βp(zx,zy)S(θs,θo,σ2).
Where r(ω) is Fresnel reflectance with the incident angle ω. Then the polarization property of incident radiation is taken into consideration. The reflectance Mueller matrix of a single wave facet Fr(ω) is the same with matrix M in Eq. (7).

 figure: Fig. 1

Fig. 1 (a) Geometry of wave facet reflectance; (b) Sketch map of reference plane rotation.

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Considering that the reference (meridian) planes of incident and reflected radiation vectors are not in the same plane for wave surface, a rotated reference plane is needed. As shown in Fig. 1(b), the rotation angles i1 and i2 are determined according to the spherical trigonometry properties [24]. So the reflectance matrix of wave facet after reference plane rotation is written as Eq. (17).

Rf(θs,θo,Δφ)=C(πi2)Fr(ω)C(i1).
Where the rotation matrix C(i) is given by Eq. (18).
C(i)=[10000cos2isin2i00sin2icos2i00001].
Finally the reflectance Mueller matrix of the wave surface can be expressed as Eq. (19) [24].
R(θs,θo,Δφ)=π4cosθscosθocos4βp(zx,zy)S(θs,θo,σ2)Rf(θs,θo,Δφ).
The sunglint reflection is calculated directly by the above-mentioned model. When studying the reflected skylight, the incident radiation from the celestial hemisphere is dispersed to a series of evenly distributed incident directions (about 1460 incident directions are used in calculation). The Stokes vector of every individual incident direction is computed by the semi-empirical Rayleigh scattering model. Then the vector will be used by the wave surface reflectance model to calculate the patterns of the reflected Stokes vector within the celestial hemisphere. Afterwards the reflected energy in a given direction can be calculated by summing up contributions from all the incident radiation. In the same way, we can get the patterns of reflected Stokes vectors in the celestial hemisphere.

3. Results and Discussions

Using the methods mentioned above, this section simulates the polarization patterns of clear skylight. It also simulates the polarization and intensity patterns of light reflected off both calm water and wave water.

3.1 The polarization patterns of clear skylight

The DOP of clear skylight was calculated using the semi-empirical Rayleigh scattering model and the results are shown in Fig. 2. Equation (2) shows that when the scattering angle reaches 90°, the DOP of scattered light will reach the maximum, DOPmax. So the DOP is mainly decided by scattering angle and SZA. The highest DOP area appears in the direction which is perpendicular to the solar incident light. The DOP patterns of clear skylight under different SZAs (0°, 30°, 60°, 90°) are shown in Fig. 2. The corresponding DOPmax is 56%, 63%, 70% and 77%, respectively. These patterns change according to solar position and follow the Rayleigh law.

 figure: Fig. 2

Fig. 2 DOP patterns of clear sky at solar zenith angle θs = 0°, 30°, 60°, 90°. Here and in the following figures, the celestial hemisphere and its reflection patterns are represented in a two dimensional coordinate system. The zenith and the nadir are at the origin and the horizon is represented by the outermost circle. The zenith angle and azimuth angle are measured radially and tangentially, respectively. The solar azimuth angle is always set to 0.

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The corresponding AOP patterns of clear skylight at different solar zenith angles are shown in Fig. 3. The AOP ranges from 0° to 90° and is denoted by different colors. When the sun is at the zenith of the celestial hemisphere (θs = 0°), the polarization orientation of skylight in all observing directions is in the horizontal plane. For each incident ray, the E-vector is perpendicular to its own meridian, so the color of the full-sky area is red which means AOP = 90°. When θs = 30°, there are two singular points (0°,0°) and (0°,30°). When θs = 60°, there are two singular points (0°,0°) and (0°,60°). When θs = 90°, there are three singular points (0°,0°), (0°,90°) and (180°,90°). The couple of angles in parenthesis represents azimuth angle and zenith angle, respectively. The results show that singular points appear at solar position, anti-solar position and the zenith. Solar and anti-solar positions are singular points because they are unpolarized. Zenith is a singular point because the reference plane for polarization angle is the meridian plane of each observing direction.

 figure: Fig. 3

Fig. 3 AOP patterns of clear sky at solar zenith angle θs = 0°, 30°, 60°, 90°, where the reference plane is the meridian of each observing direction.

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We validate the semi-empirical Rayleigh model by comparing the DOP and AOP patterns with results of 6SV model under a specific situation. 6SV is one of the widely used vector radiative transfer model which is rigorously validated and publicly available [25]. The DOP and AOP patterns calculated by the two models are shown in Fig. 4.

 figure: Fig. 4

Fig. 4 Polarization patterns calculated by semi-empirical Rayleigh model and that calculated by 6SV. (a1) and (b1) are DOP and AOP patterns calculated by Rayleigh model, (a2) and (b2) are DOP and AOP patterns calculated by 6SV. Other parameters: SZA is 30°, aerosol optical depth of 550nm is 0.2, wind speed is 5 m/s, wind direction is 0°, the Midlatitude Summer atmosphere model and Maritime aerosol model are used.

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Figure 4 shows that the DOP and AOP patterns of the two different radiative transfer models are highly similar as a whole. The DOPs of 6SV results are a little higher than that of Rayleigh model. Considering that the maximum DOP is empirically restricted to 63% in Rayleigh model for average atmosphere conditions when SZA is 30°, while the used 6SV run aims at certain atmosphere conditions, this result is acceptable. The AOP patterns of the two models look the same except for the zenith zone and the location of the sun. This is mainly due to the larger zenith and azimuth angle interval used in our 6SV run. However, these tiny differences will not significantly influence the water reflection patterns. So the semi-empirical Rayleigh model is valid to describe the polarization characteristic of clear sky.

3.2 The polarization patterns of skylight reflected off a flat water surface and the corresponding reflectivity distribution

Before exploring the skylight polarization patterns reflected off wave water, we first consider a flat water surface situation. Figures 5 and 6 show the DOP and AOP patterns of reflected skylight. Figure 7 shows the reflectivity distribution.

 figure: Fig. 5

Fig. 5 DOP patterns of reflected clear skylight off a flat water surface at solar zenith angle θs = 0°, 30°, 60°, 90°, the DOPs range from 0 to 1.

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 figure: Fig. 6

Fig. 6 AOP patterns reflected clear skylight off flat water surface at solar zenith angle θs = 0°, 30°, 60°, 90°, the AOP ranges from 0° to 90°, the reference plane is the meridian of each observing direction.

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 figure: Fig. 7

Fig. 7 Reflectivity patterns reflected clear skylight off flat water surface at solar zenith angle θs = 0°, 30°, 60°, 90°.

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The polarization patterns of skylight reflected from a flat water surface regularly change with solar position. Comparing Fig. 2 with Fig. 5, we can find that the DOP of reflected skylight is generally higher than that of the clear skylight. The DOP of clear skylight has different upper limits in different solar positions but never exceeds 77%, which is decided by the semi-empirical Rayleigh model. However, part of the reflected light’s DOP exceeds 90% and even reaches 100%. The highly polarized parts often appear at the “Brewster zone” [13] (zenith angle between 40° to 60°). So it can be concluded that light will become highly polarized after reflected off a flat water surface near the Brewster angle under different solar positions.

Reflection and refraction happen when light travels across water surface. The intensity and polarization patterns of light will change during this process which can be described by the Fresnel law. The perpendicular E-vector is usually stronger than the parallel part in reflection; while it is reverse during refraction. When the incident angle reaches the Brewster zone, reflected light will become linearly polarized and transmission light will still be partially polarized.

The comparison of AOP between skylight [Fig. 3] and reflected light [Fig. 6] shows that reflection makes polarization orientation lean toward a perpendicular direction. As shown in Fig. 6, most of the polarization angle is greater than 85° especially near the Brewster angle. Thus we can set the observing zenith angle in Brewster zone so that the reflected skylight from water surface can be eliminated by a linear polarizer with its polarization direction perpendicular to the E-vector of reflected light. When the solar position is at the zenith, the incident and reflected light have the same angular pattern, i.e. the AOP is 90° at all observing angles. However, when the solar position approaches the horizon, the AOP pattern changes from “8” shape to “∞” shape after the reflection.

As shown in Fig. 7, the skylight patterns have nearly the same reflectivity distribution under different SZAs. Smaller reflection angle indicates lower reflectivity. Most of the detail differences appear in the zenith area where the zenith angle is less than 50°. We can find that the reflectivity distributions change regularly with SZA in the zenith area. The low reflectance area also becomes a “∞” shape at large SZA.

The patterns of skylight reflected from flat water surface are similar to the numerical results of Horváth [1]. The results agree well with measured data of reflected skylight from a flat water surface under clear sky at sunset [13]. The figures in this paper use color maps with minor steps so that most of the details can be found definitely. It indicates that the semi-empirical Rayleigh scattering model can properly describe the polarization pattern of clear sky and the Fresnel surface can properly model the reflection of skylight at a flat water surface.

3.3 The polarization patterns of skylight reflected by wave water surface and the corresponding intensity distribution

The Cox-Munk model was used to describe the probability of wave slope, and the reflection of a single facet follows the Fresnel law. The incidences from the celestial hemisphere in all directions are separated into equally spaced sampling directions (equal solid angles). The polarization and intensity of light from the sampling directions are calculated with semi-empirical Rayleigh scattering model and radiation intensity distribution model in section 2.1. The wave water surface reflect skylight in all directions in the celestial hemisphere, and this reflected light is represented by a series of observing directions. To calculate the reflected light in one observing direction, contribution from all the sampling incident beams need to be considered. The Stokes vector of each incident beam needs to be calculated before being reflected to the observing direction. By summing up all the Stokes vectors, we can get the final reflected light’s Stokes vector in one observing direction. Needless to say, it is a time consuming process to calculate the full reflection pattern.

By changing the incident light and wave conditions, we can get various reflection figures and make a series of contrasts. This includes the unpolarized skylight incidence, the effects of different ratios of direct solar irradiance and skylight, solar zenith angle, wind speed and wind direction.

3.3.1 Wave water surface reflection under an overcast sky

The skylight polarization patterns are seriously affected by atmospheric conditions. The DOP of skylight usually decreases with the increasing of aerosol optical depth. We mainly discuss the clear sky situation which can be described by the semi-empirical Rayleigh scattering model. In order to show the depolarization effects on reflected light from wave water, we make a typical contrast between clear and overcast skylight reflected by wave water. The overcast sky is an extreme condition that skylight is homogeneously distributed and unpolarized. The results are shown in Fig. 8.

 figure: Fig. 8

Fig. 8 Polarization and intensity patterns of reflected light from wave water under clear sky (a1,b1,c1) and overcast skylight (a2,b2,c2). (a1) and (a2) refer to AOP patterns. (b1) and (b2) refer to DOP patterns. (c1) and (c2) refer to intensity patterns. Other parameters: SZA is 30°, wind direction is 0°, wind speed is 5m/s, intensity ratio of sunlight to scattered skylight is 9:1.

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It is shown that the reflection patterns of overcast sky are quite different from those of clear sky. For overcast sky, most of the reflection directions have large AOPs (larger than 80°). The highly polarized area forms a circular ring whose zenith angle ranges from 40° to 60°. Generally, the DOP of reflected light under overcast sky is lower than that under clear sky. This means that aerosols and clouds can also depolarize the reflected light from wave water. We also compare our results under overcast sky with Horváth’s model for reflected light from flat water surface [1]. The DOP and intensity patterns show great agreement.

3.3.2 The impact of direct sunlight and scattered skylight

This comparison is meant to investigate the impact of different intensity ratios of sunlight (direct normal irradiance) to skylight on reflected polarization and intensity patterns. We compare the results when sunlight and skylight are of different percentages. This can represent different atmosphere conditions to a certain degree, and the results are shown in Fig. 9.

 figure: Fig. 9

Fig. 9 Polarization and intensity patterns of reflected light from wave water when the ratio of sunlight to skylight is 9:1 (a1,b1,c1) and 6:1 (a2,b2,c2). (a1) and (a2) refer to AOP patterns. (b1) and (b2) refer to DOP patterns. (c1) and (c2) refer to intensity patterns. Other parameters: SZA is 30°, wind direction is 0°, wind speed is 5m/s.

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It shows that reflected DOP and AOP patterns change little with the ratio of incident source while the intensity pattern seems more dispersed at a lower incident ratio (sunlight to skylight). By subtracting the two DOP patterns, we find that the average DOP ratio of 6:1 is about 0.2% higher than that of the ratio 9:1. We also calculate some other examples whose incident ratios vary from 1:1 to 10:1. This all indicates that the variation of incident ratio seldom changes wave water reflection patterns.

When sunlight incident reflects off wave water, the highest reflection area appear around solar specular reflecting point, shown as gray in Figs. 9(c1) and 9(c2). The sunglint often appears in this direction and there is little sunlight reflected to other directions especially in the reverse direction.

3.3.3 Wave water reflection patterns at different solar zenith angle

In this section, we investigate how the patterns of reflected light change according to different solar zenith angles. We also compare the results with flat water surface reflection figures, and try to find the optimal solution to eliminate light reflected from water surface in remote sensing. The results show that reflection patterns change regularly with solar zenith angle. This is consistent with the variation of skylight polarization patterns.

Comparing the DOP patterns of wave water [Fig. 10] with flat water [Fig. 5], we find that the distribution and variation trends have similar characteristics. However the DOP of reflection from flat water is somewhat higher. After subtracting the two DOP patterns, we find that the average DOP of wave water reflection is about 9.03% (θs = 0°), 5.00% (θs = 30°), 4.89% (θs = 60°) and 13.56% (θs = 90°) lower than reflection from flat water. This phenomenon indicates the water surface facets depolarize the reflected light, because the water surface modulates the reflection E-vectors into different directions. We analyze the DOP in solar meridian [Fig. 11(A)] and the vertical meridian [Fig. 11(B)]. In the solar meridian (0°-180°), the highly polarized reflections are between 40° and 60° zenith angles (Brewster zone), irrespective of SZA. However, in the vertical meridian (90°-270°), the highly polarized reflections depend on SZA. When SZA is small, highly polarized reflections are between the 40° and 60° zenith angles. When SZA is large, highly polarized reflections shift directions to being between 30° and 50° zenith angles.

 figure: Fig. 10

Fig. 10 DOP patterns of reflected skylight under different SZAs. Other parameters: wind direction is 0°, wind speed is 5m/s, intensity ratio of sunlight to skylight is 9:1.

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 figure: Fig. 11

Fig. 11 DOP patterns of reflected skylight in (A) solar meridian (positive and negative zenith angle refers to solar and anti-solar meridian, respectively) and (B) vertical meridian (positive and negative zenith angle refers to 90° direction and 270° direction, respectively.).

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Similar to flat water [Fig. 6], the AOP of light reflected from waves [Fig. 12] is large in most observing directions. The polarization angles often exceed 80° especially near the Brewster angle. This indicates that the reflection eliminating method mentioned above (in section 3.2) which is valid for flat water, is also available for wavy water. The major differences appear at the zenith zone and solar specular reflection zone.

 figure: Fig. 12

Fig. 12 AOP patterns of reflected full skylight under different SZAs. Other parameters: wind direction is 0°, wind speed is 5m/s, intensity ratio of sunlight to scattered skylight is 9:1.

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We use the relative intensity figures to describe the reflectance. The intensity figures [Fig. 13] indicate that highly reflected areas are near the solar specular reflecting position. The intensity of other areas is however relatively low and evenly distributed. Low reflection zone is denoted by dark blue which expands with the increasing of SZA.

 figure: Fig. 13

Fig. 13 Intensity patterns of reflected full skylight under different SZAs. Other parameters: wind direction is 0°, wind speed is 5m/s, intensity ratio of sunlight to scattered skylight is 9:1.

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3.3.4 The effects of wind speed & direction on light reflection from waves

In this study, we assume that water surface fluctuation is driven by wind. Wind speed and direction are two main factors affecting the surface shape of wave water, which decides the probability of wave slope. We analyze the polarization and intensity patterns of skylight reflected off waves under different conditions of wind speed and direction.

With the increasing velocity of wind speed, the eigen area of reflected light expands while the main characteristics remain the same [Fig. 14]. In particular, the zenith area (looks like a butterfly) in which the polarization angles are relative low, expands in the direction perpendicular to the wind. The areas with low DOP also expand in the same way. The DOP as a whole decreases with the increasing wind speed. When the wind speed increases from 1m/s to 5m/s, the average DOP decreases by 2.01%. When the wind speed increases from 5m/s to 10m/s, the average DOP decreases by 1.83%. It means that rough water surface will decrease the DOP of reflection. From the intensity reflection figures, we can see that the areas with high reflectance expand with wind speed which indicates that sunglint will cover wider areas. On the contrary, the reverse reflecting zone with relatively low reflectance (dark blue area) shrinks with the increasing wind speed.

 figure: Fig. 14

Fig. 14 Patterns of reflected light from wave water under different wind speed. Other parameters: SZA = 30°, wind direction is 0°, intensity ratio of sunlight to skylight is 9:1.

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The wind direction is defined relative to the solar meridian which is the 0° direction in Fig. 15. It can change wave water shape and the probability of wave slope. Figure 15 indicates that wind direction can change the eigen area shape of reflected light. However the distribution as a whole is not distinctly changed. The shifting of intensity reflection figures mainly appears at the sunglint area.

 figure: Fig. 15

Fig. 15 Patterns of reflected light from wave water under different wind directions. Other parameters: SZA = 30°, wind speed is 5m/s, intensity ratio of sunlight to skylight is 9:1.

Download Full Size | PPT Slide | PDF

Plass and Kattawar had done research on directly reflected sunlight (sunglint) and considered that wind speed is irrelevant for sunglint polarization [26]. We further demonstrate that wind speed and wind direction can change the polarization patterns of reflected skylight from wave water to a certain extent. Wind speed affects the size of eigen area, while wind direction affects the symmetry of reflected skylight. When using the polarization filter to eliminate light reflected off water surface, the optimal observing zenith angle is still near the Brewster angle. When the wind speed above water surface is high, the solar meridian direction is recommended to be the view direction and the wind direction effects can be neglected.

4. Conclusion

The DOP of flat water surface reflected light increases significantly compared with incident skylight. It can make the water surface more easily detectable for some insects or sensors with certain polarization sensitivities. This may be a useful method for these animals to find aquatic habitats and also useful as an effective solution in object detection. Skylight reflected off flat water surface usually has a large AOP, especially in the area near the Brewster zone. In addition, the DOP in this area is relatively high. This means it is feasible to eliminate surface reflected light with a polarization filter.

This paper further calculates the skylight reflected off wave surface with the Cox-Munk model. Taking the polarization of incident light (skylight) into consideration, a series of contrasts and analyses under different conditions have been carried out. The reflection of overcast sky has relative low DOP compared with clear sky, most of the reflection directions get large AOPs except for zenith area. We also find that the polarization patterns change little with the intensity ratio of sunlight and scattered skylight. The reflection patterns of skylight reflected off flat water and those reflected off wave water have similar patterns and shift trends. Similar to flat water, the DOP of skylight reflected off wave water is higher than that of scattered skylight as a whole. This means that wave water can be recognized by polarization method. Skylight reflected off both flat and wave water are highly polarized around the Brewster zone, and the polarization direction in this area is almost perpendicular to the local meridian. This means that we can use the similar elimination method to reduce the wave water reflection as we can with flat water. The investigation of wind speed and wind direction shows that the polarization pattern of skylight reflected off wave water can change slightly according to different wind conditions.

We use the semi-empirical Rayleigh scattering model to describe skylight polarization patterns. The effects of scattering factors such as aerosol, cloud and dust are partially important. The coupling effects during the reflection by underlying surfaces (water surface) are not yet considered. The reflection off the water’s bottom, refraction at water surface and scattering in water may modify the polarization patterns of water. So, further investigation is necessary for a deep understanding of the properties of reflection.

Acknowledgments

The authors would like to thank Dr. Jackson Turner, Jian Xing, Wenna Yang and Jiwen Wang for discussions. This work was supported by National Natural Science Foundation of China (Grant No. 40901168). We are grateful to the anonymous reviewers for their valuable comments and suggestions.

References and links

1. G. Horváth, “Reflection-polarization patterns at flat water surfaces and their relevance for insect polarization vision,” J. Theor. Biol. 175(1), 27–37 (1995). [CrossRef]   [PubMed]  

2. A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011). [CrossRef]   [PubMed]  

3. D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000). [CrossRef]  

4. C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun's glitter,” J. Opt. Soc. Am. 44(11), 838–850 (1954). [CrossRef]  

5. G. Horváth and D. Varjú, “Polarization pattern of freshwater habitats recorded by video polarimetry in red, green and blue spectral ranges and its relevance for water detection by aquatic insects,” J. Exp. Biol. 200, 1155–1163 (1997). [PubMed]  

6. S. P. Garaba and O. Zielinski, “Methods in reducing surface reflected glint for shipborne above-water remote sensing,” J. Europ. Opt. Soc. Rap. Public. 8, 13058 (2013). [CrossRef]  

7. S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009). [CrossRef]  

8. A. Cunningham, P. Wood, and D. McKee, “Brewster-angle measurements of sea-surface reflectance using a high resolution spectroradiometer,” J. Opt. A, Pure Appl. Opt. 4(4), S29–S33 (2002). [CrossRef]  

9. J. F. R. Gower, “Observations of in situ fluorescence of chlorophyll-a in Saanich Inlet,” Bound.-Lay. Meteorol. 18(3), 235–245 (1980). [CrossRef]  

10. K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993). [CrossRef]  

11. Z. P. Lee, K. L. Carder, T. G. Peacock, and R. G. Steward, “Remote sensing reflectance measured with and without a vertical polarizer,” Proc. SPIE 2963, 483–488 (1997). [CrossRef]  

12. K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (A. Deepak Publishing, 1988).

13. B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21(9), 1669–1676 (2004). [CrossRef]   [PubMed]  

14. J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76(1), 103–111 (2001). [CrossRef]  

15. G. Horváth, B. Bernáth, B. Suhai, A. Barta, and R. Wehner, “First observation of the fourth neutral polarization point in the atmosphere,” J. Opt. Soc. Am. A 19(10), 2085–2099 (2002). [CrossRef]   [PubMed]  

16. A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40(1), 57–63 (1988). [CrossRef]  

17. W. A. Peterson and I. Dirmhirn, “The ratio of diffuse to direct solar irradiance (perpendicular to the sun’s rays) with clear skies-a conserved quantity throughout the day,” J. Appl. Meteorol. 20(7), 826–828 (1981). [CrossRef]  

18. L. T. Wong and W. K. Chow, “Solar radiation model,” Appl. Energy 69(3), 191–224 (2001). [CrossRef]  

19. G. P. Konnen, Polarized light in nature (Cambridge University, 1985).

20. N. Ebuchi and S. Kizu, “Probability distribution of surface wave slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58(3), 477–486 (2002). [CrossRef]  

21. L. Bourg, F. Montagner and V. Billat. “MERIS ATBD 2.13 sun glint flag algorithm,” MERIS PO-TN-MEL-GS-0005(2011).

22. P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72(18), 4643–4649 (1967). [CrossRef]  

23. M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008). [CrossRef]  

24. X. Q. He, Y. Bai, Q. K. Zhu, and F. Gong, “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectr. Rad. Tr. 111(10), 1426–1448 (2010). [CrossRef]  

25. S. Y. Kotchenova, E. F. Vermote, R. Matarrese, and F. J. Klemm Jr., “Validation of a vector version of the 6S radiative transfer code for atmospheric correction of satellite data. Part I: Path radiance,” Appl. Opt. 45(26), 6762–6774 (2006). [CrossRef]   [PubMed]  

26. G. N. Plass, G. W. Kattawar, and J. A. Guinn Jr., “Isophotes of sunlight glitter on a wind-ruffled sea,” Appl. Opt. 16(3), 643–653 (1977). [CrossRef]   [PubMed]  

References

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  1. G. Horváth, “Reflection-polarization patterns at flat water surfaces and their relevance for insect polarization vision,” J. Theor. Biol. 175(1), 27–37 (1995).
    [Crossref] [PubMed]
  2. A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
    [Crossref] [PubMed]
  3. D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000).
    [Crossref]
  4. C. Cox and W. Munk, “Measurement of the roughness of the sea surface from photographs of the sun's glitter,” J. Opt. Soc. Am. 44(11), 838–850 (1954).
    [Crossref]
  5. G. Horváth and D. Varjú, “Polarization pattern of freshwater habitats recorded by video polarimetry in red, green and blue spectral ranges and its relevance for water detection by aquatic insects,” J. Exp. Biol. 200, 1155–1163 (1997).
    [PubMed]
  6. S. P. Garaba and O. Zielinski, “Methods in reducing surface reflected glint for shipborne above-water remote sensing,” J. Europ. Opt. Soc. Rap. Public. 8, 13058 (2013).
    [Crossref]
  7. S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
    [Crossref]
  8. A. Cunningham, P. Wood, and D. McKee, “Brewster-angle measurements of sea-surface reflectance using a high resolution spectroradiometer,” J. Opt. A, Pure Appl. Opt. 4(4), S29–S33 (2002).
    [Crossref]
  9. J. F. R. Gower, “Observations of in situ fluorescence of chlorophyll-a in Saanich Inlet,” Bound.-Lay. Meteorol. 18(3), 235–245 (1980).
    [Crossref]
  10. K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
    [Crossref]
  11. Z. P. Lee, K. L. Carder, T. G. Peacock, and R. G. Steward, “Remote sensing reflectance measured with and without a vertical polarizer,” Proc. SPIE 2963, 483–488 (1997).
    [Crossref]
  12. K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (A. Deepak Publishing, 1988).
  13. B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21(9), 1669–1676 (2004).
    [Crossref] [PubMed]
  14. J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76(1), 103–111 (2001).
    [Crossref]
  15. G. Horváth, B. Bernáth, B. Suhai, A. Barta, and R. Wehner, “First observation of the fourth neutral polarization point in the atmosphere,” J. Opt. Soc. Am. A 19(10), 2085–2099 (2002).
    [Crossref] [PubMed]
  16. A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40(1), 57–63 (1988).
    [Crossref]
  17. W. A. Peterson and I. Dirmhirn, “The ratio of diffuse to direct solar irradiance (perpendicular to the sun’s rays) with clear skies-a conserved quantity throughout the day,” J. Appl. Meteorol. 20(7), 826–828 (1981).
    [Crossref]
  18. L. T. Wong and W. K. Chow, “Solar radiation model,” Appl. Energy 69(3), 191–224 (2001).
    [Crossref]
  19. G. P. Konnen, Polarized light in nature (Cambridge University, 1985).
  20. N. Ebuchi and S. Kizu, “Probability distribution of surface wave slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58(3), 477–486 (2002).
    [Crossref]
  21. L. Bourg, F. Montagner and V. Billat. “MERIS ATBD 2.13 sun glint flag algorithm,” MERIS PO-TN-MEL-GS-0005(2011).
  22. P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72(18), 4643–4649 (1967).
    [Crossref]
  23. M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
    [Crossref]
  24. X. Q. He, Y. Bai, Q. K. Zhu, and F. Gong, “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectr. Rad. Tr. 111(10), 1426–1448 (2010).
    [Crossref]
  25. S. Y. Kotchenova, E. F. Vermote, R. Matarrese, and F. J. Klemm., “Validation of a vector version of the 6S radiative transfer code for atmospheric correction of satellite data. Part I: Path radiance,” Appl. Opt. 45(26), 6762–6774 (2006).
    [Crossref] [PubMed]
  26. G. N. Plass, G. W. Kattawar, and J. A. Guinn., “Isophotes of sunlight glitter on a wind-ruffled sea,” Appl. Opt. 16(3), 643–653 (1977).
    [Crossref] [PubMed]

2013 (1)

S. P. Garaba and O. Zielinski, “Methods in reducing surface reflected glint for shipborne above-water remote sensing,” J. Europ. Opt. Soc. Rap. Public. 8, 13058 (2013).
[Crossref]

2011 (1)

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

2010 (1)

X. Q. He, Y. Bai, Q. K. Zhu, and F. Gong, “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectr. Rad. Tr. 111(10), 1426–1448 (2010).
[Crossref]

2009 (1)

S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
[Crossref]

2008 (1)

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
[Crossref]

2006 (1)

2004 (1)

2002 (3)

G. Horváth, B. Bernáth, B. Suhai, A. Barta, and R. Wehner, “First observation of the fourth neutral polarization point in the atmosphere,” J. Opt. Soc. Am. A 19(10), 2085–2099 (2002).
[Crossref] [PubMed]

N. Ebuchi and S. Kizu, “Probability distribution of surface wave slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58(3), 477–486 (2002).
[Crossref]

A. Cunningham, P. Wood, and D. McKee, “Brewster-angle measurements of sea-surface reflectance using a high resolution spectroradiometer,” J. Opt. A, Pure Appl. Opt. 4(4), S29–S33 (2002).
[Crossref]

2001 (2)

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76(1), 103–111 (2001).
[Crossref]

L. T. Wong and W. K. Chow, “Solar radiation model,” Appl. Energy 69(3), 191–224 (2001).
[Crossref]

2000 (1)

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000).
[Crossref]

1997 (2)

G. Horváth and D. Varjú, “Polarization pattern of freshwater habitats recorded by video polarimetry in red, green and blue spectral ranges and its relevance for water detection by aquatic insects,” J. Exp. Biol. 200, 1155–1163 (1997).
[PubMed]

Z. P. Lee, K. L. Carder, T. G. Peacock, and R. G. Steward, “Remote sensing reflectance measured with and without a vertical polarizer,” Proc. SPIE 2963, 483–488 (1997).
[Crossref]

1995 (1)

G. Horváth, “Reflection-polarization patterns at flat water surfaces and their relevance for insect polarization vision,” J. Theor. Biol. 175(1), 27–37 (1995).
[Crossref] [PubMed]

1993 (1)

K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
[Crossref]

1988 (1)

A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40(1), 57–63 (1988).
[Crossref]

1981 (1)

W. A. Peterson and I. Dirmhirn, “The ratio of diffuse to direct solar irradiance (perpendicular to the sun’s rays) with clear skies-a conserved quantity throughout the day,” J. Appl. Meteorol. 20(7), 826–828 (1981).
[Crossref]

1980 (1)

J. F. R. Gower, “Observations of in situ fluorescence of chlorophyll-a in Saanich Inlet,” Bound.-Lay. Meteorol. 18(3), 235–245 (1980).
[Crossref]

1977 (1)

1967 (1)

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72(18), 4643–4649 (1967).
[Crossref]

1954 (1)

Bai, Y.

X. Q. He, Y. Bai, Q. K. Zhu, and F. Gong, “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectr. Rad. Tr. 111(10), 1426–1448 (2010).
[Crossref]

Barta, A.

Bernáth, B.

Carder, K. L.

Z. P. Lee, K. L. Carder, T. G. Peacock, and R. G. Steward, “Remote sensing reflectance measured with and without a vertical polarizer,” Proc. SPIE 2963, 483–488 (1997).
[Crossref]

K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
[Crossref]

Chen, R. F.

K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
[Crossref]

Chow, W. K.

L. T. Wong and W. K. Chow, “Solar radiation model,” Appl. Energy 69(3), 191–224 (2001).
[Crossref]

Coombes, C. A.

A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40(1), 57–63 (1988).
[Crossref]

Cox, C.

Cunningham, A.

A. Cunningham, P. Wood, and D. McKee, “Brewster-angle measurements of sea-surface reflectance using a high resolution spectroradiometer,” J. Opt. A, Pure Appl. Opt. 4(4), S29–S33 (2002).
[Crossref]

Davis, C. O.

K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
[Crossref]

Dirmhirn, I.

W. A. Peterson and I. Dirmhirn, “The ratio of diffuse to direct solar irradiance (perpendicular to the sun’s rays) with clear skies-a conserved quantity throughout the day,” J. Appl. Meteorol. 20(7), 826–828 (1981).
[Crossref]

Ebuchi, N.

N. Ebuchi and S. Kizu, “Probability distribution of surface wave slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58(3), 477–486 (2002).
[Crossref]

Erlick, C.

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Gál, J.

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76(1), 103–111 (2001).
[Crossref]

Garaba, S. P.

S. P. Garaba and O. Zielinski, “Methods in reducing surface reflected glint for shipborne above-water remote sensing,” J. Europ. Opt. Soc. Rap. Public. 8, 13058 (2013).
[Crossref]

Gong, F.

X. Q. He, Y. Bai, Q. K. Zhu, and F. Gong, “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectr. Rad. Tr. 111(10), 1426–1448 (2010).
[Crossref]

Gower, J. F. R.

J. F. R. Gower, “Observations of in situ fluorescence of chlorophyll-a in Saanich Inlet,” Bound.-Lay. Meteorol. 18(3), 235–245 (1980).
[Crossref]

Guinn, J. A.

Hamilton, M.

K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
[Crossref]

Harrison, A. W.

A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40(1), 57–63 (1988).
[Crossref]

He, X. Q.

X. Q. He, Y. Bai, Q. K. Zhu, and F. Gong, “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectr. Rad. Tr. 111(10), 1426–1448 (2010).
[Crossref]

Hedley, J. D.

S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
[Crossref]

Horváth, G.

B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21(9), 1669–1676 (2004).
[Crossref] [PubMed]

G. Horváth, B. Bernáth, B. Suhai, A. Barta, and R. Wehner, “First observation of the fourth neutral polarization point in the atmosphere,” J. Opt. Soc. Am. A 19(10), 2085–2099 (2002).
[Crossref] [PubMed]

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76(1), 103–111 (2001).
[Crossref]

G. Horváth and D. Varjú, “Polarization pattern of freshwater habitats recorded by video polarimetry in red, green and blue spectral ranges and its relevance for water detection by aquatic insects,” J. Exp. Biol. 200, 1155–1163 (1997).
[PubMed]

G. Horváth, “Reflection-polarization patterns at flat water surfaces and their relevance for insect polarization vision,” J. Theor. Biol. 175(1), 27–37 (1995).
[Crossref] [PubMed]

Kattawar, G. W.

Kay, S.

S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
[Crossref]

Kizu, S.

N. Ebuchi and S. Kizu, “Probability distribution of surface wave slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58(3), 477–486 (2002).
[Crossref]

Klemm, F. J.

Kotchenova, S. Y.

Labhart, T.

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000).
[Crossref]

Lambrinos, D.

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000).
[Crossref]

Lavender, S.

S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
[Crossref]

Lee, Z. P.

Z. P. Lee, K. L. Carder, T. G. Peacock, and R. G. Steward, “Remote sensing reflectance measured with and without a vertical polarizer,” Proc. SPIE 2963, 483–488 (1997).
[Crossref]

Lerner, A.

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Li, W.

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
[Crossref]

Matarrese, R.

McKee, D.

A. Cunningham, P. Wood, and D. McKee, “Brewster-angle measurements of sea-surface reflectance using a high resolution spectroradiometer,” J. Opt. A, Pure Appl. Opt. 4(4), S29–S33 (2002).
[Crossref]

Meyer-Rochow, V. B.

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76(1), 103–111 (2001).
[Crossref]

Möller, R.

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000).
[Crossref]

Muller-Karger, F.

K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
[Crossref]

Munk, W.

Ottaviani, M.

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
[Crossref]

Peacock, T. G.

Z. P. Lee, K. L. Carder, T. G. Peacock, and R. G. Steward, “Remote sensing reflectance measured with and without a vertical polarizer,” Proc. SPIE 2963, 483–488 (1997).
[Crossref]

Peterson, W. A.

W. A. Peterson and I. Dirmhirn, “The ratio of diffuse to direct solar irradiance (perpendicular to the sun’s rays) with clear skies-a conserved quantity throughout the day,” J. Appl. Meteorol. 20(7), 826–828 (1981).
[Crossref]

Pfeifer, R.

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000).
[Crossref]

Plass, G. N.

Reinersrnan, P.

K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
[Crossref]

Sabbah, S.

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Saunders, P. M.

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72(18), 4643–4649 (1967).
[Crossref]

Shashar, N.

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Spurr, R.

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
[Crossref]

Stamnes, K.

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
[Crossref]

Steward, R. G.

Z. P. Lee, K. L. Carder, T. G. Peacock, and R. G. Steward, “Remote sensing reflectance measured with and without a vertical polarizer,” Proc. SPIE 2963, 483–488 (1997).
[Crossref]

Su, W.

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
[Crossref]

Suhai, B.

Varjú, D.

G. Horváth and D. Varjú, “Polarization pattern of freshwater habitats recorded by video polarimetry in red, green and blue spectral ranges and its relevance for water detection by aquatic insects,” J. Exp. Biol. 200, 1155–1163 (1997).
[PubMed]

Vermote, E. F.

Wehner, R.

G. Horváth, B. Bernáth, B. Suhai, A. Barta, and R. Wehner, “First observation of the fourth neutral polarization point in the atmosphere,” J. Opt. Soc. Am. A 19(10), 2085–2099 (2002).
[Crossref] [PubMed]

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000).
[Crossref]

Wiscombe, W.

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
[Crossref]

Wong, L. T.

L. T. Wong and W. K. Chow, “Solar radiation model,” Appl. Energy 69(3), 191–224 (2001).
[Crossref]

Wood, P.

A. Cunningham, P. Wood, and D. McKee, “Brewster-angle measurements of sea-surface reflectance using a high resolution spectroradiometer,” J. Opt. A, Pure Appl. Opt. 4(4), S29–S33 (2002).
[Crossref]

Zhu, Q. K.

X. Q. He, Y. Bai, Q. K. Zhu, and F. Gong, “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectr. Rad. Tr. 111(10), 1426–1448 (2010).
[Crossref]

Zielinski, O.

S. P. Garaba and O. Zielinski, “Methods in reducing surface reflected glint for shipborne above-water remote sensing,” J. Europ. Opt. Soc. Rap. Public. 8, 13058 (2013).
[Crossref]

Appl. Energy (1)

L. T. Wong and W. K. Chow, “Solar radiation model,” Appl. Energy 69(3), 191–224 (2001).
[Crossref]

Appl. Opt. (2)

Bound.-Lay. Meteorol. (1)

J. F. R. Gower, “Observations of in situ fluorescence of chlorophyll-a in Saanich Inlet,” Bound.-Lay. Meteorol. 18(3), 235–245 (1980).
[Crossref]

J. Appl. Meteorol. (1)

W. A. Peterson and I. Dirmhirn, “The ratio of diffuse to direct solar irradiance (perpendicular to the sun’s rays) with clear skies-a conserved quantity throughout the day,” J. Appl. Meteorol. 20(7), 826–828 (1981).
[Crossref]

J. Europ. Opt. Soc. Rap. Public. (1)

S. P. Garaba and O. Zielinski, “Methods in reducing surface reflected glint for shipborne above-water remote sensing,” J. Europ. Opt. Soc. Rap. Public. 8, 13058 (2013).
[Crossref]

J. Exp. Biol. (1)

G. Horváth and D. Varjú, “Polarization pattern of freshwater habitats recorded by video polarimetry in red, green and blue spectral ranges and its relevance for water detection by aquatic insects,” J. Exp. Biol. 200, 1155–1163 (1997).
[PubMed]

J. Geophys. Res. (1)

P. M. Saunders, “Shadowing on the ocean and the existence of the horizon,” J. Geophys. Res. 72(18), 4643–4649 (1967).
[Crossref]

J. Oceanogr. (1)

N. Ebuchi and S. Kizu, “Probability distribution of surface wave slope derived using sun glitter images from geostationary meteorological satellite and surface vector winds from scatterometers,” J. Oceanogr. 58(3), 477–486 (2002).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

A. Cunningham, P. Wood, and D. McKee, “Brewster-angle measurements of sea-surface reflectance using a high resolution spectroradiometer,” J. Opt. A, Pure Appl. Opt. 4(4), S29–S33 (2002).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Quant. Spectr. Rad. Tr. (2)

M. Ottaviani, R. Spurr, K. Stamnes, W. Li, W. Su, and W. Wiscombe, “Improving the description of sunglint for accurate prediction of remotely sensed radiances,” J. Quant. Spectr. Rad. Tr. 109(14), 2364–2375 (2008).
[Crossref]

X. Q. He, Y. Bai, Q. K. Zhu, and F. Gong, “A vector radiative transfer model of coupled ocean–atmosphere system using matrix-operator method for rough sea-surface,” J. Quant. Spectr. Rad. Tr. 111(10), 1426–1448 (2010).
[Crossref]

J. Theor. Biol. (1)

G. Horváth, “Reflection-polarization patterns at flat water surfaces and their relevance for insect polarization vision,” J. Theor. Biol. 175(1), 27–37 (1995).
[Crossref] [PubMed]

Philos. Trans. R. Soc. Lond. B Biol. Sci. (1)

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Proc. SPIE (1)

Z. P. Lee, K. L. Carder, T. G. Peacock, and R. G. Steward, “Remote sensing reflectance measured with and without a vertical polarizer,” Proc. SPIE 2963, 483–488 (1997).
[Crossref]

Remote Sens. (1)

S. Kay, J. D. Hedley, and S. Lavender, “Sun glint correction of high and low spatial resolution images of aquatic scenes: a review of methods for visible and near-infrared wavelengths,” Remote Sens. 1(4), 697–730 (2009).
[Crossref]

Remote Sens. Environ. (2)

K. L. Carder, P. Reinersrnan, R. F. Chen, F. Muller-Karger, C. O. Davis, and M. Hamilton, “AVIRIS calibration and application in coastal oceanic environments,” Remote Sens. Environ. 44(2-3), 205–216 (1993).
[Crossref]

J. Gál, G. Horváth, and V. B. Meyer-Rochow, “Measurement of the reflection-polarization pattern of the flat water surface under a clear sky at sunset,” Remote Sens. Environ. 76(1), 103–111 (2001).
[Crossref]

Robot. Auton. Syst. (1)

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1-2), 39–64 (2000).
[Crossref]

Sol. Energy (1)

A. W. Harrison and C. A. Coombes, “Angular distribution of clear sky short wavelength radiance,” Sol. Energy 40(1), 57–63 (1988).
[Crossref]

Other (3)

K. L. Coulson, Polarization and Intensity of Light in the Atmosphere (A. Deepak Publishing, 1988).

L. Bourg, F. Montagner and V. Billat. “MERIS ATBD 2.13 sun glint flag algorithm,” MERIS PO-TN-MEL-GS-0005(2011).

G. P. Konnen, Polarized light in nature (Cambridge University, 1985).

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Figures (15)

Fig. 1
Fig. 1 (a) Geometry of wave facet reflectance; (b) Sketch map of reference plane rotation.
Fig. 2
Fig. 2 DOP patterns of clear sky at solar zenith angle θs = 0°, 30°, 60°, 90°. Here and in the following figures, the celestial hemisphere and its reflection patterns are represented in a two dimensional coordinate system. The zenith and the nadir are at the origin and the horizon is represented by the outermost circle. The zenith angle and azimuth angle are measured radially and tangentially, respectively. The solar azimuth angle is always set to 0.
Fig. 3
Fig. 3 AOP patterns of clear sky at solar zenith angle θs = 0°, 30°, 60°, 90°, where the reference plane is the meridian of each observing direction.
Fig. 4
Fig. 4 Polarization patterns calculated by semi-empirical Rayleigh model and that calculated by 6SV. (a1) and (b1) are DOP and AOP patterns calculated by Rayleigh model, (a2) and (b2) are DOP and AOP patterns calculated by 6SV. Other parameters: SZA is 30°, aerosol optical depth of 550nm is 0.2, wind speed is 5 m/s, wind direction is 0°, the Midlatitude Summer atmosphere model and Maritime aerosol model are used.
Fig. 5
Fig. 5 DOP patterns of reflected clear skylight off a flat water surface at solar zenith angle θs = 0°, 30°, 60°, 90°, the DOPs range from 0 to 1.
Fig. 6
Fig. 6 AOP patterns reflected clear skylight off flat water surface at solar zenith angle θs = 0°, 30°, 60°, 90°, the AOP ranges from 0° to 90°, the reference plane is the meridian of each observing direction.
Fig. 7
Fig. 7 Reflectivity patterns reflected clear skylight off flat water surface at solar zenith angle θs = 0°, 30°, 60°, 90°.
Fig. 8
Fig. 8 Polarization and intensity patterns of reflected light from wave water under clear sky (a1,b1,c1) and overcast skylight (a2,b2,c2). (a1) and (a2) refer to AOP patterns. (b1) and (b2) refer to DOP patterns. (c1) and (c2) refer to intensity patterns. Other parameters: SZA is 30°, wind direction is 0°, wind speed is 5m/s, intensity ratio of sunlight to scattered skylight is 9:1.
Fig. 9
Fig. 9 Polarization and intensity patterns of reflected light from wave water when the ratio of sunlight to skylight is 9:1 (a1,b1,c1) and 6:1 (a2,b2,c2). (a1) and (a2) refer to AOP patterns. (b1) and (b2) refer to DOP patterns. (c1) and (c2) refer to intensity patterns. Other parameters: SZA is 30°, wind direction is 0°, wind speed is 5m/s.
Fig. 10
Fig. 10 DOP patterns of reflected skylight under different SZAs. Other parameters: wind direction is 0°, wind speed is 5m/s, intensity ratio of sunlight to skylight is 9:1.
Fig. 11
Fig. 11 DOP patterns of reflected skylight in (A) solar meridian (positive and negative zenith angle refers to solar and anti-solar meridian, respectively) and (B) vertical meridian (positive and negative zenith angle refers to 90° direction and 270° direction, respectively.).
Fig. 12
Fig. 12 AOP patterns of reflected full skylight under different SZAs. Other parameters: wind direction is 0°, wind speed is 5m/s, intensity ratio of sunlight to scattered skylight is 9:1.
Fig. 13
Fig. 13 Intensity patterns of reflected full skylight under different SZAs. Other parameters: wind direction is 0°, wind speed is 5m/s, intensity ratio of sunlight to scattered skylight is 9:1.
Fig. 14
Fig. 14 Patterns of reflected light from wave water under different wind speed. Other parameters: SZA = 30°, wind direction is 0°, intensity ratio of sunlight to skylight is 9:1.
Fig. 15
Fig. 15 Patterns of reflected light from wave water under different wind directions. Other parameters: SZA = 30°, wind speed is 5m/s, intensity ratio of sunlight to skylight is 9:1.

Equations (19)

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N(γ, θ s , θ v )=(A+B e mγ +C cos 2 γcos θ s )(1 e 1.90sec θ v )(1 e 0.53sec θ s ).
DOP=DO P max sin 2 γ 1+ cos 2 γ .
cosγ=cos θ s cos θ v +sin θ s sin θ v cosφ.
AOP=arccos( sinφ sinγ sin θ s )
{ r s = n 1 cos θ 1 n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 r p = n 2 cos θ 1 n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ 2 .
DO P r = r s 2 r p 2 r s 2 + r p 2 .
[ I r Q r U r V r ]=M[ I i Q i U i V i ]= 1 2 ( r s 2 + r p 2 r p 2 r s 2 0 0 r p 2 r s 2 r s 2 + r p 2 0 0 0 0 2 r s r p 0 0 0 0 2 r s r p )[ I i Q i U i V i ].
DOP= Q 2 + U 2 + V 2 I .
AOP= 1 2 tan 1 ( U Q ).
{ z x = z / x = sin α tan β = sin θ s sin φ s + sin θ o sin φ o cos θ s + cos θ o z y = z / y = cos α tan β = sin θ s cos φ s + sin θ o cos φ o cos θ s + cos θ o .
{ z x = cos χ z x + sin χ z y z y = sin χ z x + cos χ z y .
p ( z x , z y ) = 1 2 π σ u σ c e ξ 2 + η 2 2 [ 1 1 2 C 21 η ( ξ 2 1 ) 1 6 C 03 ( η 3 3 η ) + 1 24 C 40 ( ξ 4 6 ξ 2 + 3 ) + 1 4 C 22 ( ξ 2 1 ) ( η 2 1 ) + 1 24 C 04 ( η 4 6 η 2 + 3 ) ] .
{ σ u = 0.0053 + 6.71 × 10 4 W σ c = 0.0048 + 1.52 × 10 4 W .
cos Θ = cos θ s cos θ o + sin θ s sin θ o cos Δ φ .
S ( θ s , θ o , σ 2 ) = 1 1 + Λ ( cot ( θ s ) ) + Λ ( cot ( θ o ) ) Λ ( x ) = 1 2 [ 2 π σ x exp ( x 2 2 σ 2 ) erfc ( x 2 σ ) ] .
ρ g ( θ s , θ o , Δ φ ) = π r ( ω ) 4 cos θ s cos θ o cos 4 β p ( z x , z y ) S ( θ s , θ o , σ 2 ) .
R f ( θ s , θ o ,Δφ)=C(π i 2 ) F r (ω)C( i 1 ).
C(i)=[ 1 0 0 0 0 cos2i sin2i 0 0 sin2i cos2i 0 0 0 0 1 ].
R( θ s , θ o ,Δφ)= π 4cos θ s cos θ o cos 4 β p( z x , z y )S( θ s , θ o , σ 2 ) R f ( θ s , θ o ,Δφ).

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