Numerical simulations were carried out using the Monte Carlo algorithm created by Jacek Piskozub and applied previously in Piskozub and Freda (2013). The algorithm solves the vector radiative transfer equation for the atmosphere–ocean system using the successive orders of scattering method and the Stokes formalism to track the polarization of photons. The algorithm collects information about virtual photons involved statistically in optical processes: reflection at the rough sea surface, refraction at the air–water interface, scattering and absorption within the water body and reaching the ToA. Moreover, the original Monte Carlo algorithm has been modified to track polarization changes of each photon during these processes. The unmodified version of the algorithm was successfully used, with results published in Piskozub et al. (2001), Stramski and Piskozub (2003), McKee et al. (2008, 2013) or Piskozub and McKee (2011).

Polarization information is described by four elements of the Stokes vector: S=[I,Q,U,V]T, where I is the total radiance of light, Q describes the radiance of linearly polarized light (vertical to horizontal), U describes the radiance of linearly polarized light (diagonal right skewed to left skewed), V describes the circular polarization (clockwise to counterclockwise), and T denotes the transposition. Three elements (Q, U and V) of the Stokes vector may be both positive or negative. The single quantity that characterizes these properties is the degree of polarization: 1DoP=Q2+U2+V2I. The defined degree of polarization is often replaced by the degree of linear polarization (DoLP). 2DoLP=Q2+U2I The latter hardly differs from DoP because circular polarization is relatively rare in nature (Cronin and Marshall, 2011). The circular polarization represented by the V element of the Stokes vector does not appear in seawater as a result of scattering; see off-diagonal Mueller matrix elements in Voss and Fry (1984). It is measurable in light internally reflected from the underside of the sea's surface (Ivanoff and Watermann, 1958).

Our polarized radiative transfer model involves a virtual light source to send randomly polarized photons and track their pathways in the means of the probability of occurrence of the processes mentioned above. Reflection and refraction processes are described by Fresnel equations, and the slopes of the sea surface are characterized by the wind-dependent distribution of Cox and Munk (1956). The algorithm does not take into account additional depolarization due to enhanced whitecap fraction, described by Hu et al. (2008) – that is likely for high wind speed. The probability of processes within the water body is determined by the corresponding coefficients of absorption and scattering (including multiple scattering). Angular distribution of scattered photons is described by phase functions that, for both atmosphere and sea depth, are characterized separately for molecular scattering and particle scattering. Polarization properties of particle scattering are described by Mueller matrices that for seawater are taken from Voss and Fry (1984) and for atmospheric aerosol particles from Volten et al. (2001). The model outputs the angular distribution of the upwelling radiance and its degree of polarization at any desired level, which are presented in form of polar plots. Some of the results are additionally specified in the principal plane.

This section reports the input parameters used in the computations. The dataset of the absorption and attenuation coefficients of seawater constituents come from in situ measurements in the southern Baltic contained in Sagan (2008). It is the largest dataset of ac-9 (WET Labs, Inc.) measurements in the southern Baltic that was published in a tabular form of average values, extreme values and standard deviations. The instrument was calibrated in ultrapure water and routinely checked for stability with air readings. The standard recommended data processing was performed (Zaneveld et al., 1994). Absolute precision of measurement is 0.005 m-1, while relative precision is estimated from 4 % in clear waters to 12 % in the areas of turbid waters. The data have been recorded during cyclical cruises aboard R/V Oceania in 1999 and 2003 to 2005. Measurements were made in different months of the year. The dataset was divided into two seasons, here called “winter”, for the months from November to March, and “summer”, for the months from April to October. The summer season is characterized by strong phytoplankton growth and the winter season by low biological activity. In addition, Sagan (2008) distinguished three regions: open Baltic, gulfs (Gulf of Gdańsk and Pomeranian Gulf) and coastal waters. The measuring stations, divided into these three types of water, are shown in Fig. 1. For defined regions – open Baltic and coastal areas – those two datasets are statistically significantly different for all optical parameters (t test for means) at the level p<0.05 by Sagan (2008).

Total absorption coefficient taken to the simulation is a sum at=apg+aw, where apg is an average absorption coefficient (particle and dissolved fraction) of the N number of measured depth profiles (see Table 1) made with the ac-9 after Sagan (2008), and aw is the pure water absorption coefficient and comes from Pope and Fry (1997). Similarly, the total attenuation coefficient is defined as ct=cpg+aw+bw, where the component cpg comes directly from Sagan (2008), but to get the total attenuation coefficient, it was enlarged by clean water components of absorption aw and scattering bw (Smith and Baker, 1981). According to Sagan (2008), the highest values of IOPs and their highest variability are observed for the water of gulfs and estuaries of rivers that are located nearby. The simulations are carried out at nine wavelengths, namely, 412, 440, 488, 510, 532, 555, 650, 676 and 715 nm, which correspond to the ac-9 and are commonly dedicated to ocean color analysis.

Solar zenith angles in the southern Baltic region depend strongly on the season. In months described by Sagan (2008) as the summer season, the highest Sun position over the horizon, which means the minimum of SZA during Sun culmination, varies between 31∘ in June (the longest day of the year) and 69∘ at the end of October. In the modeling, a single value of 45∘ was chosen as a summer SZA. For months of the winter season, the minimum of SZA varies between 50∘ in the end of March and 78∘ in December (the shortest day). Given values are reached at approximately noon and are higher in the rest of the days. That is why SZA of 75∘ is chosen as a representative for the winter season. Computations are performed for the direction of wind twisted by 45∘ from the Sun reflection plane, chosen arbitrarily. Two wind speeds of 5 and 15 m s-1 are considered. Aerosol optical thickness of 0.12, independent of the wavelength and the same for both seasons, was applied to all simulations. This allowed us to observe DoP variations and correlations that were not affected by aerosol changes.

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.

The full set of simulation results is available in the repository Freda (2019). It contains tables of Stokes vector elements for sectors marked with zenith–azimuth coordinates, which allows one to calculate the DoP. Examples of simulation results are presented in Fig. 2a and b in the form of polar plots of the degree of polarization of upwelling radiance just above the sea surface. Figure 2a shows the DoP for the average IOPs measured in the open waters of the Baltic Sea for a wavelength of 412 nm in the summer season, while Fig. 2b depicts an analogous case for the winter season. These two plots are characterized by one of the highest values of the peak of DoP of 0.88 for summer and 0.84 for winter. The azimuth position of the Sun is 0∘ in all cases. Corresponding values of computed upwelling radiance I (in units of Wm-2 sr-1 nm-1) are shown on the plots of Fig. 2c and d on the logarithmic scale (due to their high angular variability).

The small SZA of the summer season (45∘) resulted in low values of upwelling radiance that are stretched from the direct reflection point to the horizon, where it is extended both left and right from the azimuth of 180∘. The high SZA of the winter season (75∘) resulted in much higher values of reflected light, which are also stretched from reflection point to the horizon.

Examples of the lowest values of the maximum DoP, referred to as max(DoP), are shown in Fig. 3a and b. They were obtained for the regions of Gulf of Gdańsk and Pomeranian Gulf, simulated for the spectral band of 555 nm, for the summer season (Fig. 3a) and for the winter season (Fig. 3b).

The maximum values of DoP presented in Fig. 2a (summer season, open Baltic waters) are visible for azimuth angles close to 180∘ (direction of reflected Sun) and a zenith angle of approximately 55∘ although the solar zenith angle is 45∘, while the max(DoP) in Fig. 3a (summer season, waters of gulfs) is visible for a zenith angle of approximately 60∘. In contrast to the summer season case, the maximum DoPs in the winter season are close to the zenith angle of 48∘ (Fig. 2b) and 54∘ (Fig. 3b), while SZA is 75∘. The lower position of the Sun results in a higher position of the maximum DoP of upwelling radiation than its reflection angle, and a higher position of the Sun results in a lower position of the maximum DoP. Another interesting effect is the higher DoP observed for directions close to the incident rays of the Sun (azimuth of 0∘). In general, in the winter season, the values of DoP are higher than in summer, and zenith angles of this effect are lower in the winter than in the summer season.

The results of Monte Carlo simulations of angular characteristics of DoP of upwelling radiance are presented in Fig. 4. These results are obtained for average IOPs of open Baltic waters for wind speed of 5 m s-1 for three wavelengths (440, 555, 650 nm) and for both seasons. Vertical cross-sections of such polar plots for the same type of water (open Baltic Sea) but additionally for all examined wavelengths and for two speeds of wind, 5 and 15 m s-1, are presented in Fig. 5. Such cross-sections show the DoP in the principal plane, including the direction of incident Sun beam (on the left side of the plot), zenith and direction of Sun reflection beam for calm sea surface (on the right side). The azimuth direction of the Sun position, described as 0∘ in the polar plots, is marked by negative zenith angles in Fig. 5, while azimuth directions of 180∘ that include the Sun reflection beam are marked by positive zenith angles.

The analysis of individual spectral bands shows that high values of DoP correspond to the high absorption-to-attenuation ratio for the total of visible light domain (see Table 1). High values of absorption coefficient for 650–676 nm wavelengths (in the red spectral region) are caused by pure water (see Pope and Fry, 1997), while high absorption coefficients for wavelengths of the blue–green range are caused mainly by CDOM (Kowalczuk et al., 2005). The lowest values of max(DoP) for each type of water and for each season are observed for the 555 nm spectral band. The lowest values of absorption and weak spectral variability of the scattering coefficient imply that the wavelength of 555 nm is characterized by the lowest absorption-to-attenuation ratios due to the existence of a minimum of absorption for seawater containing phytoplankton. Algae cells, depending on the composition of their pigments, may have a minimum of absorption in a wide range of spectral bands from 550 nm to 660 nm (Bricaud et al., 2004). Considering the absorption of pure water that is increasing with wavelength (Pope and Fry, 1997), the minimum of the absorption in Baltic waters for the spectral band of 555 nm results.

The spectral shape of the DoP cross-sections contains two maxima, and their angular positions depend on the absorption-to-attenuation ratio, the season and the wind speed. The angular position of the higher maximum depends mostly on the season, varying from approximately 60∘ in the summer to 35–50∘ in the winter (see Fig. 5). The lower maximum is observed at the zenith angles between -70 and -90∘ in the summer as well as between -55 and -70∘ in the winter. Higher wind speed of 15 m s-1, in comparison to 5 m s-1, causes the irregular shape of peaks. Moreover, the higher wind speed causes an increase of the DoP value for a lower maximum in the 650 and 676 nm spectral bands and its shift to a higher position (toward the zenith). At the same time, the DoP values for shorter wavelength bands are decreased and shifted to a lower position (toward horizon).

Computations of DoP were carried out in three optically different regions of the southern Baltic. Such a division is justified in previous studies of optical and hydrological properties of the south Baltic waters (Olszewski et al., 1992). They showed a relationship between the measured values of IOPs and their location in relation to river estuaries, distance from the shore or bathymetry of the bottom.

Comparison of water type influence on the DoP is shown in Fig. 6 for two wavelengths: 440 (Fig. 6a) and 555 nm (Fig. 6b). The type of water has less influence on the DoP than the season and its representative SZA.

However, the highest values of DoP for most zenith angles and the highest values of its peak max(DoP) for each season are observed for open Baltic Sea water. Coastal waters and gulfs are characterized by similar values of DoP in each season. For the wavelength of 440 nm, in the summer, the differences of max(DoP) between the open Baltic and other regions reach 0.02–0.04, while in the winter those differences exceed 0.05. For the 555 nm band, in the summer, the differences of max(DoP) between the open Baltic and other regions reach 0.06–0.09 and in the winter these differences reach 0.07–0.09, respectively. We also observed another regional difference in the angular position of the maxima of DoP that is noticeable in the winter season only. Two maxima of the DoP cross-sections are closer for open Baltic waters than for gulfs and coastal waters.

In the following section, we explain that the degree of polarization depends on the absorption-to-attenuation ratio, and all its regional changes are the result of the absorption-to-attenuation ratio variability.

The results of our simulations are in qualitative agreement with the measurements of above-water DoLP of the total upwelling radiance presented by Freda et al. (2015). This agreement is the similarity of the peak of degree of polarization on the polar plots, which are stretched along the azimuth angles. Freda et al. (2015) obtained lower values of measured DoLP with a maximum of 30 %–40 % (see Figs. 1 and 2 in Freda et al., 2015), which is presumably caused by different weather conditions and unknown environmental parameters during measurements, such as a high absorption coefficient in the waters of the river mouth, different aerosol optical depth or other parameters. However, despite the differences in the maximum degree of polarization, the angular distribution patterns are similar, with the peak in the vicinity of the Sun reflection azimuth angle.

This section contains the comparison of the degree of polarization for summer and winter seasons as a function of the absorption-to-attenuation ratio.

The total a(λ)/c(λ) ratio (see Table 1) is higher in the open Baltic water than in other regions because of the low scattering coefficients (Sagan, 2008). The value of the latter is determined mainly by the concentration of suspended matter, which in open waters is significantly lower than in gulfs or coastal/nearshore waters. According to Sagan (2008), the average particle scattering coefficient does not depend strongly on wavelengths, and in open Baltic waters in the winter season it varies between 0.15 (for 676 nm) and 0.19 (for 412 nm). In the same season, but in the waters of gulfs, the average particle scattering coefficient varies between 0.77 (for 676 nm) and 1.00 (for 412 nm). For the influence of water type on DoP, regardless of the wavelength and wind speed, the lowest max(DoP) values in winter are observed in the waters of gulfs. These waters are characterized by the highest scattering coefficients because of the high inflow of particulate matter with river waters. However, in the summer season, the lowest peak of DoP is observed for spectral bands from 412 to 532 nm in coastal waters and in wavelengths from 555 to 676 nm in gulfs. The values of a(λ)/c(λ) for these types of water differ by less than 5 % except for the 650 and 676 nm spectral bands.

All the values of maximum DoP of above-water upwelling radiance obtained for each absorption-to-attenuation ratio are collected in Fig. 7. The summer season case is depicted in Fig. 7a, while the winter case is depicted in Fig. 7b. The water types are marked with different symbols, and two wind speeds are marked with different colors. The correlations of the max(DoP) to the ratio of a(λ)/c(λ) are approximately linear. However, correlation coefficient analysis has shown that the power functions are better matched. The reason for the nonlinearity of this correlation may be related to the potential obtaining or even exceeding one by the value of DoP for certain combinations of absorption and attenuation coefficients. The trend line for the plot depicted in Fig. 7a shows the relationship of the maximum of DoP to the ratio of a(λ)/c(λ) for the summer season and in Fig. 7b for the winter season, respectively. The trend lines presented may be described by the following power functions: 3max⁡(DoP)=Aa(λ)c(λ)B, whose parameters are collected in Table 2. These correlations are obtained for various spectral channels. Hence, they are wavelength-independent for the examined visible spectral range.

An analysis of all collected data shows that higher values of maximum DoP are observed for lower wind speed. Moreover, max(DoP) has a higher range of variability in the summer season than in the winter. Figure 7a (summer season) shows that values of max(DoP) are between 0.64 and 0.91 for the wind speed of 5 m s-1 and between 0.61 and 0.83 for the wind speed of 15 m s-1. However, in Fig. 7b (winter season), the values are between 0.73 and 0.9 for the wind speed of 5 m s-1 and between 0.66 and 0.87 for the wind speed of 15 m s-1. Power trend lines for the same wind speed for summer and winter seasons (in Fig. 7a and b) intersect. For a wind speed of 5 m s-1, by a(λ)/c(λ) equal to 0.26, both power functions reach the same max(DoP) of 0.77. For a wind speed of 15 m s-1, by a(λ)/c(λ) equal to 0.32, both power functions reach the same value of 0.75. For lower absorption-to-attenuation ratios, winter DoPs have higher values than summer and for higher a(λ)/c(λ) values, the summer DoPs are higher.

The reason for the correlation of the maximum DoP with the absorption-to-attenuation ratio is the occurrence of multiple scattering in water depth. The degree of polarization tends to decrease after multiple scattering events. A high absorption-to-attenuation ratio means simply low scattering-to-attenuation impact and hence shallow penetration of light in the water column and low participation in multiple scattering that decreases the DoP. Such conclusion is in accordance with Piskozub and Freda (2013), who examined the influence of single scattering albedo on the polarization of water-leaving radiance. Their results show that in the Sun reflection plane, the highest value of DoP is observed when the total scattering coefficient is the lowest (see Fig. 3 in Piskozub and Freda, 2013).

The influence of wind speed on the DoP values shown in Fig. 7a and b is very clear: sea-surface roughness depolarizes the reflected light. Zhou et al. (2013) demonstrated that wind speed and wind direction can change the polarization patterns of reflected skylight from a rough sea surface to a certain extent. Our study shows, in particular, that high wind speed results in lower values of max(DoP) of the total upwelling radiance. Such regularity is filled for all types of water and all spectral bands.

Our algorithm does not consider possible additional depolarization, which is likely especially for high wind speeds (15 m s-1). This can be caused by whitecaps (Hu et al., 2008), air bubble entrainment and possibly more sea spray generation. Moreover, additional factors that may decrease the DoP value for high SZA (winter season) are wave-shadowing effects of incident and multiple reflected rays (see Hieronymi, 2016), which are not considered in this paper.

The results of the correlation of the maximum DoP with the absorption-to-attenuation ratio seem to be coincident with the results of Ibrahim et al. (2012), who studied the degree of linear polarization just below the air–water interface. Their correlation of attenuation-to-absorption ratio with DoLP displays a hyperbolic shape (see Figs. 5 to 8 in Ibrahim et al., 2012). Therefore, for an inverted absorption-to-attenuation ratio, it would be near linear. The modeling results of Ibrahim et al. cannot be compared directly to the results presented in this paper because they received DoLP just below the sea surface, and we focused on DoP just above the surface. However, our choice of seawater absorption-to-attenuation ratio, which can be called the relative absorption value (to total attenuation), as a parameter correlated to degree of polarization seems to be more suitable.