(a) Stability of the per cent polarization phase function
The PPPFs (per cent polarization versus scattering angle) measured at 2 m depth over a 28 m deep sea floor and at all solar zenith angles (SZAs) in clear and semi-turbid water conditions are shown in . In clear waters (a–c), the PPPF is symmetric around the 90° scattering angle where the maximum polarization is achieved, for all Sun elevations, as described by equation (1.1). However, in semi-turbid waters (d–f), the symmetry around the 90° scattering angle is preserved only at low SZAs (d, SZA < 30°); at other angles, the symmetry breaks and the maximum per cent polarization shifts to other scattering angles. Therefore, the PPPF is consistent with the Sun's elevation in clear waters only.
Figure 2. Per cent polarization phase function (PPPF; per cent polarization versus scattering angle) measured at a depth of 2 m, (a–c) in clear and (d–f) semi-turbid waters at three solar zenith angles (SZA). Wavelength = 450 nm. SZA; (a,d) 0°–30°; (more ...)
(b) Stability of the e-vector orientation pattern
In , the e-vector orientation, ψ, calculated using equation (1.2), which accounts for refraction of the direct light from the Sun into the water and single Rayleigh scattering within the water only, is presented for different SZA (SZA = θs) related to the different angles of refraction into the water, θr. Note that this theoretical calculation is considered to be more potentially relevant outside of Snell's window than inside of Snell's window, since it does not account for refraction of radiation that was already scattered in the atmosphere into the water. Nevertheless, for completeness, the solution is presented for all viewing (sensor) zenith angles, θp, and for all differences between the azimuthal angle of the Sun and the sensor, Δϕ. Owing to the symmetry of the solution, Δϕ is presented in the 0–180° range only. The colour bar represents the values of e-vector orientation calculated with respect to the horizon. In general, the (Δϕ, θp) points at which the e-vector orientation equals the angle of refraction vary with the Sun's elevation. However, at horizontal viewing directions, in a plane perpendicular to the Sun's bearing (Δϕ = 90°, θp = 90°), ψ equals θr at all Sun elevations. As the Sun's elevation increases, the e-vector field becomes more and more horizontal (a–c).
Figure 3. Numerical solutions of equation (1.2) for the e-vector orientation, ψ, presented in the viewing (sensor) zenith angle (VZA = θp) and the sun-viewing azimuthal difference (Δϕ = ϕs − ϕp) plane. The (more ...)
Note that one must consider the angular resolution capability of the observer. In , pixels in which θr − δ < ψ <θr + δ, where δ represents the detection resolution, are coloured white. Increasing δ (i.e. allowing lower detection resolution) would decrease the precision of the prediction of the Sun's position and with it the accuracy of the compass.
A linear regression model was applied to fit the measured e-vector orientations to the e-vector orientations calculated according to equation (1.2). The slope and intercept of the regression, and the correlation coefficient r, are presented in as a function of the VZA (see statistical details in ), where maximum correlation is achieved for slope = 1, intercept = 0 and r = 1. (Note that all three parameters, slope, intercept and correlation coefficient, are necessary, as a deviation of any one of them from its maximum correlation value will suggest involvement of a different scattering mechanism than the one assumed.) Although maximum correlation was not achieved at any VZA, as the parameter values did not attain the 95% confidence level (95 CI; ), the highest prediction was achieved under the clear conditions at 5 m depth (95 CI of slope = 1.01–1.03, intercept = 5.81–5.93, r = 0.85 and p < 0.001). In general, in both water types at both depths, the correlation was high in the horizontal viewing direction (VZA = 90°; mean ± 95 CI of the slope = 1.0 ± 0.2, intercept = 0.0 ± 5.93 and r > 0.8, p < 0.001), and decreased towards the viewing zenith/nadir. In the horizontal viewing direction (θp = 90°; ), the fitted slope in clear waters at 2 m depth over the shallow sea floor (mean ± 95 CI of the slope = 0.67 ± 0.01, p < 0.001) is significantly lower than all other slopes at both the 2 m depth and the 5 m depth over the deeper sea floor. In the deeper sea-floor sets, all slopes are significantly different from each other, except that in clear waters at 2 m depth and that in semi-turbid water at 5 m depth (mean ± 95 CI of the slope = 0.89 ± 0.01 and 0.91 ± 0.02, p < 0.001, respectively). The slope in semi-turbid water at 5 m depth over the deeper sea floor is significantly lower than the other slopes over that sea-floor depth (mean ± 95 CI of the slope = 0.81 ± 0.02, p < 0.001).
Figure 4. Linear regression (a) slope, (b) intercept and (c) coefficient r between the measured e-vector orientation and the e-vector orientation calculated using equation (1.2) for each sensor viewing zenith angle (VZA, θp), in clear (open symbols) and (more ...)
Table 1. Major axis type-II linear regression line estimates of the correlation between the measured e-vector orientation and the e-vector orientation calculated using equation (1.2) for different water types and viewing directions. n = number of measurements (more ...)
(c) Using the e-vector orientation in the horizontal viewing plane to detect the angle of refraction
Using the e-vector orientation (ψ) to detect the angle of refraction requires that the absolute maximum value of the orientation angle perceived does not exceed the absolute angle of refraction into the water θr, and that the absolute maximum value of the orientation angle equals θr at at least one azimuthal direction at the moment of detection. e-vector orientations measured in clear and semi-turbid waters, respectively, at different VZAs are plotted in and against the SZA and against θr (denoted by the green line, calculated for refractive index in water = 1.337 and wavelength = 450 nm). In clear waters (), the above requirements are met completely at all SZAs in the horizontal viewing direction (VZA = 90°, c) and at VZA = 70° for SZA < 75° (b). At VZA = 30° (a), ψ exceeds the absolute value of θr. This is to be expected, since this VZA is within Snell's window, and therefore the predictions of equation (1.2) are considered less relevant, even in clear waters (see above). In semi-turbid waters (), the above requirements are not met at any VZA, not even near the horizon. The absolute maximum of ψ differs from θr, and the difference between the two increases with increasing absolute VZA (looking away from the horizon).
Figure 5. e-vector orientation versus absolute solar zenith angle (SZA = θs) measured in clear waters at eight azimuthal angles of 8°, 53°, 98°, 143°, 188°, 233°, 278° and 323° defined from (more ...)
Figure 6. Same as for e-vector orientations measured in semi-turbid waters at 5 m depth with sensor viewing zenith angles (VZA, θp) directed from zenith to nadir at 30° intervals and at azimuthal angles of 8°, 53°, (more ...)