B. Suhai and G. Horva´th Vol. 21, No. 9/September 2004/J. Opt. Soc. Am. A 1669 How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study Bence Suhai and Ga´bor Horva´th Biooptics Laboratory, Department of Biological Physics, Lora´ nd Eo¨ tvo¨ s University, H-1117 Budapest, Pa´ zma´ ny se´ta´ ny 1, Hungary Received December 18, 2003; revised manuscript received April 5, 2004; accepted April 28, 2004 We present the first high-resolution maps of Rayleigh behavior in clear and cloudy sky conditions measured by full-sky imaging polarimetry at the wavelengths of 650 nm (red), 550 nm (green), and 450 nm (blue) versus the ␪ ⌬␣ ϭ ͉␣ solar elevation angle s . Our maps display those celestial areas at which the deviation meas Ϫ ␣ ͉ ␣ ϭ ␣ Rayleigh is below the threshold thres 5°, where meas is the angle of polarization of skylight measured by ␣ full-sky imaging polarimetry, and Rayleigh is the celestial angle of polarization calculated on the basis of the single-scattering Rayleigh model. From these maps we derived the proportion r of the full sky for which the single-scattering Rayleigh model describes well (with an accuracy of ⌬␣ ϭ 5°) the E-vector alignment of sky- ␪ Ͻ Ͻ ␪ light. Depending on s , r is high for clear skies, especially for low solar elevations (40% r 70% for s р 13°). Depending on the cloud cover and the solar illumination, r decreases more or less under cloudy con- ϭ ditions, but sometimes its value remains remarkably high, especially at low solar elevations (rmax 69% for ␪ ϭ s 0°). The proportion r of the sky that follows the Rayleigh model is usually higher for shorter wave- lengths under clear as well as cloudy sky conditions. This partly explains why the shorter wavelengths are generally preferred by animals navigating by means of the celestial polarization. We found that the celestial E-vector pattern generally follows the Rayleigh pattern well, which is a fundamental hypothesis in the studies of animal orientation and human navigation (e.g., in aircraft flying near the geomagnetic poles and using a polarization sky compass) with the use of the celestial ␣ pattern. © 2004 Optical Society of America OCIS codes: 010.3920, 100.0100, 120.5410, 260.5430, 280.0280, 290.1310. 1. INTRODUCTION birefringent crystal called ‘‘sunstone.’’ A similar celestial polarimetric method was used in aircraft flying in the vi- Many animals are sensitive to the linear polarization of cinity of the geomagnetic poles.25 Obviously, such a navi- light, and several species can orient by means of the ce- gation is practicable only if the single-scattering Rayleigh lestial polarization pattern.1 These animals use the dis- tribution of the electric field vector (E vector) of skylight predictions are correct for the sky. Even small E-vector in the ultraviolet, blue, or green part of the spectrum.2 deviations can produce large errors if used in a strictly In the models and theories explaining the orientation be- geometrical way to locate the sun, impairing the naviga- havior of these animals, it is always assumed that in any tor’s ability to reach his actual goal. Hence in consider- point of the celestial hemisphere the E vector of skylight ing how accurately a navigator could orient by this is perpendicular to the scattering plane, i.e., the plane method, we must look also at how the E-vector pattern of through the sun, the observer, and the point observed.3–22 real skies differs from the single-scattering Rayleigh pat- In other words, in the literature dealing with the sky- tern. compass orientation of these animals it is hypothesized Although there are some studies of how the celestial 28,29 that the celestial E-vector pattern follows the rules of the E-vector pattern follows the Rayleigh pattern, these primary Rayleigh scattering of sunlight in the atmo- investigations are restricted to relatively small numbers sphere. This hypothesis originates from Karl von of points in the sky, owing to the use of scanning point- Frisch,23 who supposed that this condition is realized for source polarimeters. Only the technique of full-sky im- 30–33 most areas of the clear sky when he tried to interpret his aging polarimetry made it possible to determine the pioneering observations on the celestial orientation of map of the Rayleigh behavior of the E-vector direction in honeybees. Hence for these studies it is important to real skies. Such a map displays those celestial areas at ⌬␣ ϭ ͉␣ Ϫ ␣ ͉ know how the celestial E-vector pattern follows the rules which the deviation meas Rayleigh is below an ␣ of primary Rayleigh scattering. arbitrary threshold, where meas is the angle of polariza- A widespread belief is that the Vikings were able to tion of skylight measured by full-sky-imaging polarime- ␣ navigate on the open sea by means of the E-vector direc- try, and Rayleigh is the angle of polarization of skylight tion of clear (blue) patches of the sky when the sun was calculated on the basis of the single-scattering Rayleigh occluded by clouds. It is hypothesized24–27 that under model. The aim of this work is to present the first high- partly cloudy conditions, a Viking navigator could locate resolution maps of Rayleigh behavior in clear and cloudy the sun if he knew that the solar direction is perpendicu- sky conditions measured by full-sky imaging polarimetry lar to the E vector of skylight determined by a mysterious in the red (650 nm), green (550 nm), and blue (450 nm) 1084-7529/2004/091669-08$15.00 © 2004 Optical Society of America 1670 J. Opt. Soc. Am. A/Vol. 21, No. 9/September 2004 B. Suhai and G. Horva´th ␪ spectral ranges versus the solar elevation angle s . measured for clear and cloudy skies and presented earlier From these maps we derived the proportion r of the sky by Pomozi et al.34 These patterns were measured by full- for which the single-scattering Rayleigh model describes sky imaging polarimetry in Tunisia at wavelengths of 650 well the E-vector alignment of skylight. nm (red), 550 nm (green), and 450 nm (blue). For the ␪ clear and cloudy sky series, the solar elevation angles s ␪ were approximately the same (see column s in Tables 2. MATERIALS AND METHODS 1–3 below). After an appropriate rotation of a given pat- In this work we used the ␣ patterns (the angle of polar- tern of the cloudy sky series around the zenith, the solar ization ␣ is measured clockwise from the local meridian) azimuth angle becomes the same as that for the corre- ␪ Fig. 1. A, Distribution of the total radiance of clear skies versus the solar elevation angle s from the horizon. The center of the ␾ ␾ ϭ ␾ circular pictures is the zenith, the perimeter is the horizon, and the zenith angle is proportional to the radius ( zenith 0°, horizon ϭ 90°). B, C, D, Maps of the proportion r of the sky that follows the Rayleigh model for clear skies at the wavelengths 650 nm (red), ␪ ⌬␣ ϭ ͉␣ Ϫ ␣ ͉ р 550 nm (green), and 450 nm (blue) versus s . ‘‘Rayleigh’’ points with meas Rayleigh 5° are shaded gray, ‘‘non-Rayleigh’’ points with ⌬␣ Ͼ 5° are white, and overexposed points are black. The approximately hourly positions of the sun are represented by dots or the disk of the sun occulter. The radial bar in the circular pictures is the wire of the sun occulter. The compass rose shows the geographic (true) compass directions. Note that east and west are transposed in the compass rose, because we are looking upward at the skydome rather than downward at a map. The numerical values of r, n, and o determined for these skies are shown in Table 1. B. Suhai and G. Horva´th Vol. 21, No. 9/September 2004/J. Opt. Soc. Am. A 1671 sponding pattern of the clear sky series. The clouds were film; the maxima and half-bandwidths of its spectral sen- ␭ ϭ Ϯ ␭ ϭ detected algorithmically in the color pictures of the skies sitivity curves were red 650 30 nm, green 550 34 Ϯ ␭ ϭ Ϯ as described by Pomozi et al. 30 nm, blue 450 50 nm) in a roll-film photo- The polarimeter and the evaluation procedure are de- graphic camera (Nikon F801). From a given sky, three scribed in detail by Ga´l et al.32 About the calibrated in- photographs were taken for the three different align- strumentation and evaluation procedures, we mention ments of the transmission axis of the polarizers on the here only the following: An angle of view of 180° was en- built-in rotating disk. The camera was set up on a tripod sured by using a fisheye lens (Nikon-Nikkor, f2.8, focal such that the optical axis of the fisheye lens was vertical. length 8 mm) including a built-in rotating disk mounted With a personal computer, after 8-bit digitization (by a with three broadband (275–750 nm) neutral density lin- Hewlett Packard ScanJet 6100C) and evaluation of the early polarizing filters (HNPЈB, Polaroid Corporation) three developed color images for a given sky, the patterns with three different polarization axes (0°, 45°, and 90° of radiance and the degree and angle of polarization of from the radius of the disk). The detector was a photo- skylight were determined as high-resolution, color-coded, emulsion (Fujichrome Sensia II ASA 100 color reversal two-dimensional circular maps. These patterns were ob- Fig. 1 (continued). 1672 J.