The polarization states of a skylight can be used for navigation by a lot of insects as well as by human beings. However, the polarization patterns of skylight are greatly influenced by the atmospheric conditions. This paper studied the polarization patterns of skylight under different sky conditions by polarized imaging measurements, in which the AOT (Aerosol Optical Thickness) and clouds were taken into account. The results showed that both the aerosol and cloud disturbed the polarization patterns of the skylight, but the patterns of AOP (Angle of Polarization) showed great robustness. The symmetry of the AOP images was found to be quite steady under most of the sky conditions. We proposed a navigation method by finding out the solar meridian according to the symmetry of an AOP map. The results showed that the solar meridian can be identified accurately under different situations. The calculation errors slightly fluctuated along with the aerosol and cloud.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The skylight is usually partially polarized, due to the sunlight scattered by the atmospheric particles . According to the Rayleigh scattering theory, the polarization pattern of skylight is mainly determined by the solar position. Therefore the solar position can be retrieved by the polarization pattern of skylight. Based on this mechanism, the partially polarized skylight is utilized by kinds of creatures for navigation, such as ants, crickets, beetles, etc [2–8]. These creatures usually sense the scattering light of the whole sky dome or patches of the sky through their unique compound eyes and optic nerve systems, and then orientate their bodies according to the polarization patterns of skylight. It was reported that the Vikings (between AD 900 and AD 1200) used to orient during their sailings with the help of the “sun-stone”, which was applied to search the solar position by observing the skylight through the “sun-stone” [9,10]. The sun is usually blocked by the clouds and fogs, but the polarization map of the partly polarized skylight remains the same as in the clear sky. Inspired by these creatures and the Vikings, scientists have developed kinds of polarization navigation sensors, which can be divided into two main groups: the “point-source” polarization sensor and the polarized imaging navigation sensor. The former one orientates itself by detecting the polarized skylight in a particular direction [11–16]. This is a kind of non-imaging detecting system which obtains the composited polarization characteristics of the sky light within the scope of the field of view. The “point-source” detector usually got a small field of view and aimed at the zenith of sky dome. The latter one detects the sky light by polarized imaging, and generally has a large field of view, which can obtain the polarization maps of the skylight from a patch of sky or the hemispherical sky [17–22]. The direction information can be obtained by analyzing the polarization images. Compared with the “point-source” polarization navigation sensor, the imaging polarization navigation system can obtain more polarization information from variety of scattering directions of the skylight. Therefore it has a higher data redundancy, and the imaging polarization navigation method has got a stronger ability of anti-interference, such as the cloudy sky which DOP (Degree of Polarization) is relatively low. The “point-source” sensor will more easily be affected by clouds. It is supposed that the polarized imaging is a better solution for the polarization navigation compared with the “point-source” way. This paper focuses on the polarization properties of skylight and proposes a new navigation method based on the polarized image of the hemispheric skylight. Since there is almost no circular polarized component in the skylight, this paper ignores the circular polarization and the DOP refers to the degree of linear polarization for convenience.
The polarization patterns of skylight in clear weather can generally be described by the Rayleigh scattering theory, because the radius of atmospheric molecules is much smaller than the wavelength of the incident radiation in the visible band. A horizontal coordinate system needs to be established when describing the polarization patterns, as shown in Fig. 1.
In the coordinate system above, S stands for the solar position and Z stands for the zenith. P stands for the direction of scattered light. and stands for the elevation angle of sun and scattered skylight, respectively. and stands for the azimuth angle of sun and scattered skylight, respectively. stands for the scattering angle. stands for AOP, which is the angle between the polarization direction ( vector) and the reference plane (OPZ).
As described by the Rayleigh theory, the polarization vector is always perpendicular to the scattering plane (the plane decided by OPS in Fig. 1). The scattered light remains completely unpolarized in the forward and backward directions, whereas at the 90◦ scattering angle, the scattered light becomes completely polarized. In other directions, the scattered light is partially polarized . The polarization pattern of the hemispheric sky can be described graphically as the pattern shown below (Fig. 2).
However, beside the gas molecules, the scattering particles in the atmosphere include all kinds of aerosol particles, water droplets, ice crystals and other scattering particles with large size. The radius of these kinds of particles is usually larger than the wavelength of the incident radiation, which will introduce Mie scattering and significant multiple scattering. The scattering effects are different for different particle types, different sizes, and the thickness of the scattering media layers. The polarization states of skylight will be affected by the large scattering particles, and this effect can hardly be accurately estimated. Although some polarized radiative transfer software (such as MODTRAN-P, 6SV, RT3) takes into account the influence of some important factors such as aerosol and ground reflection, it is still unable to accurately calculate the polarization patterns of the sky dome [24–26].
Through the polarization measurements under variety of weather, Horváth and his associates demonstrated that the polarization patterns of the skylight were “stable”, especially for the AOP images [27–29]. Nathan and Joseph proved that the AOP of light scattered by a cloud in a partly cloudy sky remains the same as in the clear sky for most cases . Though there are obvious differences between the AOP images in different sky conditions, a lot of identifiable characteristics remain the same, among which the symmetry of the AOP maps is similar to that of the Rayleigh model. Their researches enlighten us to orientate during the polarization navigation with the AOP map by finding out the symmetrical axis, which is the solar meridian. In this paper, we measured the polarization patterns of the skylight under the condition of different AOT (Aerosol Optical Thickness) and different cloud coverage. We found that most of the AOP patterns of skylight are axisymmetric. This kind of symmetry can hardly be destroyed by clouds or aerosol. Then, a new method of searching out the solar meridian was proposed based on the AOP map. The feasibility and accuracy of this method were verified under different cloudy skies and AOT situations. Our research could provide a new idea and supporting data for the polarization navigation study.
2. Methods and experiments
We used a ground-based full-sky imaging polarimeter based on liquid crystal variable retarders (LCVRs) to photograph the polarization images of the hemispherical sky. The polarization imaging system was developed by ourselves, and the specific working principles and parameters can be found in . The effective field of view of the system is about 173°. The standard deviations of DOP and AOP measurements are 1.52% and 4.91° for linear polarized light in the whole area of the CCD. Each group of polarized imaging needs to collect 4 intensity maps under different phase delay states. The acquisition time of each intensity map can be adjusted according to the intensity of incident light, about 5ms to 30ms. Therefore, it will take a very short period of time to complete a polarimetric imaging of the sky dome. The polarization pattern of skylight can be considered as unchanged in such a short period of time. The four intensity images obtained by each group are calculated combined with the instrument Muller matrix, which is calibrated accurately before, so as to obtain the Stokes parameters of each pixel of the sky dome image. For each pixel, there are four Stokes parameters which can be used to calculate the AOP and DOP data of that pixel. The LCVR camera has 5 alternative shooting bands. The central wavelength of each band is 476nm, 514nm, 530nm, 676nm and 750nm, respectively. Before the measurements, the LCVR camera was adjusted to aim at the zenith point of sky dome, so the center of the polarized image is the zenith point. The following picture (Fig. 3) shows the scene of experiment.
We measured the skylight under the conditions of different AOT and different cloud coverage. A typical gray scale image obtained from a single shot is shown in Fig. 4.
After acquiring a series of polarized images, we analyzed the polarization characteristics of sky under different conditions.
2.1 Polarization patterns of the sky with different AOT
All the measurements in this paper were collected at Beihang University, which latitude and longitude is 39°58′38″N,116°20′34″E, respectively. The aerosol is urban type. The device for measuring the AOT is the hand&held multi&band sunphotometer (MICROTOPS II), which was produced by the Solar Light Company in US. This portable sunphotometer can simultaneously measure the AOT of 5 bands, which includes 440nm, 500nm, 675nm, 870nm and 936nm. The choice of different AOT conditions excluded the influence of clouds, and all the measurements were carried out under cloudless sky. The variance of AOT was mainly caused by air pollution and sand storm. We contrasted the polarization patterns of three typical AOTs, which included the clear sky, the slightly polluted sky and the heavily polluted sky. In the clear sky, the air was clean with high horizontal visibility, and the AOT measured in 500nm is less than 0.3. In the slightly polluted sky, the air was mainly polluted by vehicle exhaust, and the AOT measured in 500nm was around 0.65 to 0.85. In the heavily polluted sky, there was a dust storm which produced a very high AOT which reached above 2.5. Table 1 shows the AOT data during the three measurements. The polarized imaging data of the skylight obtained in the above conditions are shown in Fig. 5.
From the above three sets of measurements we can easily find that the polarization states of the skylight are highly influenced by the aerosol, especially for the DOP. With the increase of the AOT, the DOP of the skylight is decreased, but the distribution of the DOP is consistent, that is, the DOP is relatively low near the sun. With the increase of the scattering angle, the DOP of the skylight increases gradually. When the scattering angle is near 90°, the DOP of the scattered light approaches the maximum. It is consistent with the DOP patterns decided by the Rayleigh scattering theory. From the AOP map, it is found that the AOP distribution of the skylight stay the same in different wavelengths. Although affected by the aerosol, the mode of the AOP pattern of the skylight is not significantly changed. However, when the AOT was extremely high, such as in group C, the AOP patterns of skylight are seriously affected and becoming “noisy”, especially for the longer wavelength. The AOP patterns of 676nm and 750nm seems to be weakened to a certain extent, which makes the AOP maps become fuzzy. The AOP patterns of the skylight at shorter wavelength, such as 476nm and 514nm, seem to be steady even under the highest AOT situation. The AOP maps are similar to each other either in the clear sky or in the heavily polluted situation.
Therefore, we think that the polarization patterns of the shorter wavelength are more stable, that is to say, the scattered skylight with short wavelength is more beneficial for the polarization navigation. This result coincided well with the former research carried out by Chu and his associates . We made a comparison between the measured AOP map at 476nm under clear weather and the simulated AOP map which was drawn according to the Rayleigh scattering theory. The measured AOP map and the simulated map look quite similar to each other, but not exactly the same, especially in the regions with lower DOP (regions near the solar position). It means that the Rayleigh scattering theory cannot accurately describe the polarization patterns of the sky dome. It may lead to errors by measuring only one direction of the scattered skylight during the polarization navigation.
2.2 Polarization patterns of skylight under different cloudy skies
In order to study the influence of clouds on the polarization patterns of skylight, we measured the polarization maps of the sky dome under three different cloudy skies, which include the less cloudy sky, the partly cloudy sky and the overcast sky. Then we compared the results with the polarization maps of clear sky. Figure 6 shows the gray images of the clear sky and the three cloudy skies.
From the measurements, we can find that the clouds significantly affected the polarization pattern of skylight. First of all, the overall DOP of the skylight in the entire wavelength is significantly reduced by the clouds. However, the distribution rule of the DOP is similar to that of the clear sky, which accords with the distribution characteristics described by Rayleigh scattering theory. We also find that the DOP of the cloud areas is generally lower than that of the clear sky. For the overcast sky, the DOP of the whole sky is quite low, and there is no significant regularity of distribution. Secondly, the clouds also change the AOP patterns of the skylight. When the sky is cloudy, there seems to be some turbulence in the AOP maps, especially near the clouds. With the increase of the wavelength, the influences on the AOP maps become more significant, that make the AOP images fuzzy. But generally speaking, the cloudy AOP pattern still maintains the characteristics of that of the clear sky. The results consistent with the measurements carried out by Pomozi et al.  under different cloudy skies. In this situation, the orientation of the sun can be easily found through the AOP map of the sky. For the completely overcast sky, the AOP images become chaotic; no specific rule can be found in the image. The image is too nosy to figure out the solar position from the AOP maps.
By analyzing the measured results, we believe that clouds can bring disturbances to polarization navigation. With the increase of cloud thickness and coverage, the polarization navigation will be affected more. For the overcast condition, the DOP of the skylight may be too low that the polarization navigation can hardly work. Although Hegedüs and his colleagues demonstrated that even in the overcast sky the skylight still maintain the same AOP patterns with that of the clear sky, but their AOP map had also been distorted by thick clouds . The deviation may come from the differences in cloud type, thickness and height, ground reflection and the instrumental error of the camera. In the cases of partially cloudy sky, the short wavelength is more likely to retain the AOP pattern of the clear skylight. It also proves that the shorter wavelength is more suitable for the application of polarization navigation in cloudy weather. This conclusion agreed well with the simulation researches by Liu et al. about the atmospheric polarization properties under water cloud condition .The researches of Barta and his associates showed that a variety of species are more sensitive to the polarized light in the short wavelength (range from 330nm to 540nm) . Our research may offer an explanation to this phenomenon. It is known that the polarized skylight can be affected by clouds and the real situation can be extremely complex due to the cloud type, the coverage, the thickness, the height of cloud and even the solar elevation angle. Our research in this paper only concerns the effects of cloud coverage. In the future, some more systematic researches need to be carried out in both theoretical research and experimental study to solve these problems caused by clouds in polarization navigation and other fields.
2.3 A method for identifying the solar meridian
It has been proved in our previous measurements that the polarization patterns of the skylight can be affected by the aerosol and clouds. The polarization maps of the actual skylight are somewhat different from the polarization patterns of the skylight described by the Rayleigh model, especially under the large AOT of atmosphere or in the cloudy weather. However, it is easy to find that the AOP maps generally keep symmetry even in the cloudy weather or in the atmosphere with large AOT. The symmetry axis is always the solar meridian plane for the AOP map of sky dome. This rule enlightens us to find the solar meridian plane through the symmetry of the AOP map, and then to orientate according to the solar azimuth. It is a new strategy for polarization navigation, which has already been proposed by many scholars [35–39]. In this paper, we took advantage of the polarimetric images obtained by the LCVR camera and further studied the navigation method based on the DOP and AOP maps. A new polarization navigation method based on symmetry axis scanning and curve fitting is proposed. Our method does not need to carry out preprocessing for DOP and AOP images and it uses the DOP map to exclude the invalid data. Two kinds of symmetry scanning are applied to improve efficiency. Almost the full data sets of an AOP map can make contribution to the orientation. The curve fitting can help us getting the optimal estimation of the solar meridian. This section will introduce this method in detail, and make a simple assessment of the results.
First, by analyzing the degree of polarization (DOP) map, the valid areas can be determined, and the image data of over exposed or under exposed areas should be excluded. The lower limit of the valid DOP in our work is set to 5%, since several researches showed that insects can navigate with the polarized skylight above 5% of DOP . Too low DOP will lead to large errors in the AOP calculation. The higher limit of the valid DOP is set to 90%, because the scattered skylight can hardly be fully polarized, mostly partially polarized . The pixels of a DOP image beyond the limits will be excluded and the valid pixels can form a mask, which is used to determine the validity of the AOP map. Taking a group of our measurements for example (476nm under clear weather), the DOP image, the mask image and the AOP image are shown in Fig. 8.
After getting the mask, the symmetry scanning method is used to analyze the solar meridian. The zenith point in the AOP image will be taken as the center, and the symmetry axis is certainly across the central point. For an arbitrary axis across the center, the symmetry level of this axis will be evaluated. For an arbitrary pixel in the AOP image, there will be one and only one symmetric pixel which can be decided by the axis to be evaluated. The symmetry factor of the pair of pixels is calculated by subtracting the two AOP values of the pair of pixels. We can find out numerous pairs of symmetry pixels and calculate all of the symmetry factors. Then we sum up the absolute values of all the symmetry factors. The summed value is defined as the symmetry value. The larger the sum value, the worse the symmetry is, vice versa. The calculation process for the symmetry verification is as follows.
A coordinate system need to be established for the AOP image. The origin of the coordinate system is defined as the pixel corresponding to the zenith point. The coordinate system is shown in Fig. 9. The angle of the symmetry axis to be evaluated is , and the axis is . An arbitrary random pixel is indicated by , its symmetric pixel about is . In order to figure out the coordinates , the vector method is used to solve the intermediate point at first.
The function of can be expressed as Eq. (1). (x,y) represents any point in the line.
is perpendicular to , that can be expressed as Eq. (2).
Therefore, the coordinates of M can be determined by solving Eq. (3).
It is easy to obtain the coordinates of by Eq. (4)
The symmetry factor of and is defined as which is calculated in Eq. (5).
is the AOP value of the pixel . By randomly collecting N pair of pixels, we can get the total symmetry value at the angle as shown in Eq. (6). We randomly selected the pixels when calculating the symmetry direction to ensure that the pixels are distributed uniformly in the valid areas of an AOP image. There are about 3 × 105 valid pixels in an AOP image, which makes it time consuming to calculate the solar meridian angle with all of these pixels. But too few pixels will lead to the increase of the error. By balancing the computational efficiency and accuracy, we randomly chose 1000 pair of pixels in one calculation. The calculation of the symmetry angle did not change remarkably with much more pixels.
The optimal solar azimuth estimation can be obtained by searching the symmetry value at different. In order to improve the calculation efficiency, we designed two scanning methods: sparse scanning and dense scanning. The angle of the sparse scanning ranges from 0° to 180° with the step of 1°. The initial angle of the solar meridian can be determined by sparse scanning. The dense scanning will be carried out around the preliminarily determined solar meridian angle with the step of 0.05°. During the calculation, it is found that the obtained by dense scanning has random errors in a certain range, which makes it difficult to accurately determine the optimal value. In order to find out the with lowest , the curve fitting method was applied to the data generated by the dense scanning. We found that a quartic polynomial curve fitting could satisfy the requirement. The minimum value of the fitted curve can be figured out along with the optimal angle estimation of the solar meridian. The results of sparse scanning, the dense scanning and curve fitting are shown in Fig. 10.
The final result of the solar meridian plane determined by the symmetry scanning method is shown in Fig. 11.
We carried out several group of tests to verify the performance of our scanning method in different sky conditions. The clear sky with different AOT and the partly cloudy sky are tested. Since the polarization patterns of shorter wavelength are more stable in high AOT and cloudy weather, so we chose the data of 476nm for solar meridian scanning. The results are shown in Fig. 12. To evaluate the accuracy of this algorithm, we calculated the solar azimuth angle hundreds of times, and 1000 pairs of random pixels were chosen in each calculation. The random error was indicated by the standard deviation of the calculated solar azimuth angles.
The results show that the solar meridian can be correctly recognized in the five sky conditions. The aerosol and clouds can affect the random error in the calculation of solar meridian angle. The random error increases with the increase of AOT. The highest error appears in the sandy weather with the largest AOT. In partly cloudy sky, the random error increases with the coverage of cloud. When the sky was covered by thick stratus, the polarization of skylight became quite weak that made the AOP pattern hard to recognize. Consequently, the polarization navigation method by finding the solar meridian would possibly be unavailable in the overcast situation.
The polarization pattern of skylight is mainly decided by the solar position. Theoretically, the variance of the solar zenith angle (SZA) will not change the symmetry of the polarization pattern of skylight. But in the real situation, the DOP of the full sky will decrease along with the SZA. When the sun moves near to the zenith point, the AOP of different directions will tend to be consistent, and it will lead to the AOP symmetry becoming fuzzy. To validate our algorithm, we chose the data measured at the SZA around 50°. The algorithm is still effective at other SZA situations, especially when the SZA is large, there will be more pixels representing higher DOP values thus more reliable AOP values, which results in more accurate solar azimuth determination. But the calculation error will be much larger when the SZA becomes too small. The solar meridian cannot be accurately recognized with our method at partly cloudy sky and the high AOT situation when the SZA is lower than 15°. In addition, it is worth noting that when the SZA becomes too small, the solar azimuth angle can hardly be calculated, but the solar elevation angle might be derived with AOP and DOP images in other ways. For the overcast skies, it may be promising to get the solar meridian direction with the AOP and DOP data at low solar elevations according to the researches of Hegedüs. It is believed that the high accuracy equipment might be a solution to the overcast skies with extremely low DOP. A more in-depth study needs to be carried out in the future study to figure out how to deal with the situations with clouds and low SZA.
We measured the polarization patterns of the skylight with a LCVR camera under different aerosol optical thickness and different cloud coverage. After analyzing the polarization images, we found that the overall DOP of the skylight can be seriously weakened by aerosol and clouds. However the AOP patterns are more stable compared with the DOP pattern, especially for the shorter wavelength. The AOP patterns of all five wavelengths are similar to each other while in the clear sky. But when affected by thick aerosol or clouds, the AOP patterns of the longer wavelength such as 676nm and 750nm seem to be fuzzier than that of the shorter wavelength. Thus, we believed that it is better to use the shorter wavelength for polarization navigation. The actual AOP pattern of skylight is not exactly the same with that of the Rayleigh model even in the clear sky. The polarization patterns of skylight are affected by Mie scattering and multiple scattering. Yet the symmetry of the AOP map is constant, which can hardly be destroyed by aerosol and clouds. It will be a better strategy for polarization navigation by finding out the solar meridian from the AOP image of the sky dome, rather than measuring the polarization angle in one or several directions. Thus we believe that the polarized imaging method is more reliable than the “point-source” method in polarization navigation.
Based on the symmetry of the AOP map, we proposed a polarization navigation method by symmetric axis scanning and curve fitting. This method was applied to the polarization images acquired by our LCVR camera at different situations. The results are good, that the solar meridians are correctly figured out with acceptable random errors. The aerosol and cloud could affect the errors in searching for the solar meridian. Generally speaking, the larger AOT or higher cloud coverage bring in larger random errors.
Although the polarization patterns of the skylight may become unrecognizable for the overcast sky, the polarization navigation can be carried out in most of the sky conditions. Only if the AOP pattern of the skylight is not totally destroyed by clouds or other factors, the solar meridian could be figured out for orientation.
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