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Abstract
Regarding aerosol particle-laded turbid atmospheres, full-sky background radiation polarization patterns can be adversely affected, an important factor limiting their effective near-ground observation and acquisition. We established a multiple-scattering polarization computational model and measurement system and conducted the following three tasks. (a) We thoroughly analyzed the impact of aerosol scattering characteristics on polarization distributions, calculating the degree of polarization (DOP) and angle of polarization (AOP) patterns for a more comprehensive set of atmospheric aerosol compositions and aerosol optical depth (AOD) values than calculated in previous studies. (b) We assessed the uniqueness of the DOP and AOP patterns as a function of AOD. (c) By employing a new polarized radiation acquisition system for measurements, we demonstrated that our computational models are more representative of the DOP and AOP patterns under actual atmospheric conditions. We found that under a clear sky without clouds, the impact of the AOD on the DOP was detectable. With increasing AOD, the DOP decreased, and the decreasing trend became increasingly obvious. When the AOD was above 0.3, the maximum DOP did not exceed 0.5. The AOP pattern did not change notably and remained stable, except for the contraction point at the sun position under an AOD of 2.
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1. Introduction
The sky background radiation parameter is one of the important parameters of atmospheric optical properties due to the presence of atmospheric molecules and aerosol particles in the sky, which makes sunlight form polarized light from the nonpolarized state by scattering, showing a certain distribution pattern that contains DOP and AOP information [1,2]. The spatial distribution characteristics of sky background radiation polarization include symmetry, neutral points, zeniths, and solar positions [3,4], and such information can provide new research ideas and directions for many pioneering fields, such as space target detection, atmospheric environment monitoring, celestial polarization observation, and sky polarization navigation [5–17].
How different atmospheric environmental parameters affect the variation in sky background radiation polarization patterns is still a complex issue, which makes it difficult to stably apply sky background radiation polarization patterns for detection and navigation [18]. Aerosols, as an important component of the atmosphere, are an important part of the sky polarization properties. Nevertheless, the effect of aerosols on the atmospheric state is difficult to determine quantitatively, and this problem is mainly due to the high variability in the number, chemical composition, and size of aerosols in time and space, which is difficult to determine on a large scale [19,20]. Therefore, to accurately simulate the process of radiative transfer in aerosol media, it is necessary to consider the aerosol physical and optical properties.
The actual atmospheric scattering polarization can no longer be characterized simply by using Rayleigh or Mie single scattering models, and some previous empirical models can only input the effect of AOD, surface albedo, and other factor queries on DOP. Currently, the common methods used to calculate radiation transport in aerosol media are various numerical solutions to the vector radiative transfer equation, which mainly include the discrete-ordinates method, the adding-doubling method, the successive order of scattering (SOS) method, and the Monte Carlo method [21–23]. Pust et al. [24,25] studied the visible and near-infrared DOP in cloud-free weather and then compared the simulation results with the full-day polarimetric measurements, showing that accurate modeling of sky DOP depends heavily on the correct representation of aerosols in the model. Emde et al. [26,27] presented the results of the IPRT project which includes ten test cases, followed by use of Monte Carlo radiative transfer model to simulate the transmission properties of polarized radiation in the Earth's atmosphere, examining the effects of surface reflections and multiple scattering caused by molecules, aerosol particles, cloud droplets, and ice crystals on polarized radiation. Gao et al. [28] provided a flexible Monte Carlo error propagation method to compare the retrieval uncertainties from error propagation with errors from synthetic retrievals and discussed the effective uncertainty quantification for multi-angle polarimetric aerosol remote sensing over ocean. Compared with other radiative transfer equation solvers, the Monte Carlo algorithm is easier to model and implement without complicated mathematical derivation. Its main advantage is that it simulates each atmospheric process individually and exactly, and each quantity of the radiation field is easily accessible. This includes not only irradiance, actinic flux, and radiance, but also for example photon pathlength statistics.
Researchers have also investigated sky polarization patterns in real atmospheric environments by building various polarimetric instruments. Kreuter et al. [29] studied the effects of large-scale aerosols and surface albedo on DOP and sky radiance. Liu et al. [30] studied the effect of typical aerosol types on sky polarization sensitivity and analyzed the correction between aerosol and insect polarization sensitive spectra. Chen et al. [20] systematically analyzed the different effects of scattering aerosol patterns and absorbing aerosol patterns on sky polarization patterns. Wu et al. [31] studied sky polarization patterns in urban areas using a developed full Stokes imaging polarimeter. Li et al. [32] proposed a new division of focal plane (DoFP) polarimetric camera polarization visualization method that can visualize the distribution of sky polarization states and the polarization characteristics of target buildings and can be applied to navigate polarization generated in urban shaded environments. In previous studies, the acquisition of sky polarization patterns using polarimetric instruments has often considered only the variation brought by the AOD, without an in-depth analysis of the effect of aerosol particle scattering properties and the type of constituent aerosols on the polarization patterns, and the polarization information obtained by the traditional means of rotating the polarizer does not have the same temporal characteristics [33].
In this paper, we addressed the radiation-related multiple scattering phenomenon and established a polarization calculation model via the Monte Carlo algorithm based on the vector radiation atmospheric transport equation. The new capability of the developed model lies in classifying different aerosol types and AODs for calculation and analysis based on the optical properties of aerosols. This model can handle atmospheres in which parameters vary in three dimensions, not just altitude, and is easier to model and implement. Subsequently, starting from the aerosol particle scattering characteristics, we investigated the effects of aerosol type and AOD on the DOP and AOP distributions in the full-sky background radiation polarization patterns through calculations. We also assessed the uniqueness of the DOP and AOP patterns as a function of AOD. Finally, a DoFP polarization acquisition system with the same spatio-temporal characteristics was independently built for actual measurement and verification. This paper advances our knowledge of the factors determining the impact of aerosols on the polarization patterns, which better agrees with the actual atmospheric conditions, and this research is of great significance to practical applications such as sky light polarization measurement, polarization navigation, atmospheric remote sensing, and space target detection.
2. Methods
2.1 Vector radiation atmospheric transport equation
Radiation is a manifestation of electromagnetic waves, and the complete description of radiation and its polarization state can be expressed by the Stokes parameter, that is, ${\boldsymbol I} = {[{I,Q,U,V} ]^T}$, where I is the total light intensity, Q and U indicate two orthogonal directions of linearly polarized light, and V indicates circular polarized light. For a plane-parallel atmosphere, the radiative transfer equation [34] can be expressed as:
The above equation is a highly nonlinear equation, which is very complicated to solve, and some specific algorithms are usually utilized to statistically solve the numerical solution in the boundary range (as I mentioned in Introduction), instead of directly solving its analytical solution. In this paper, a Monte Carlo algorithm-based multiple-scattering model is used to count, analyze and construct the polarization states of each photon in the mode when it is scattered during transmission.
2.2 Multiple-scattering model based on the Monte Carlo algorithm
The Monte Carlo algorithm treats the scattering process as a collision process between photons and particles in the atmospheric medium. The path between two collisions of photons in the medium is related to the aerosol characteristic parameters, the photons will change their forward direction after the collision, the scattering angle is determined by the phase function, and the results of specific problems can be obtained by tracking and counting the behavior of a large number of photons [36]. In contrast to general numerical methods, the number of points selected by Monte Carlo method is not limited by dimensionality (e.g., spatial coordinates, wavelength of photons, and size, shape, and optical properties of scatterers), and the computation time is only proportional to the increasing dimensionality of the problem, which makes Monte Carlo method more applicable when dealing with high-dimensional problems.
The atmospheric radiative transfer model takes photon multiple scattering as the basic process, which reflects randomness and a certain regular statistical distribution, and the Monte Carlo process is more suitable as a classical sampling method to simulate the collision process of building photons and particles in the atmospheric medium than other methods mentioned in section 1. At the same time, since the scattering of light waves with atmospheric particles is only related to the state of the most recent light wave but not to the previous ones, this property satisfies the nature of the static Markov chain, so the whole multiple-scattering process reflects the Markov chain Monte Carlo properties. Specifically, based on the Mueller matrix, the scattered Stokes vector ${S_n}$ can be obtained as follows:
Based on the Markov chain, the important features of the past can be quickly forgotten, and the photon transport is transformed into a problem of solving the impulse response, essentially by solving the rotation matrix between the input and output and thus updating the Stokes vector, completing the transfer model between the single scattering elements and realizing the DOP P(θ) statistics of the measured point P:
The Monte Carlo method is not constrained by the plane-parallel atmosphere. It can solve the problem of both horizontal nonuniform atmospheres and spherical atmospheres, and an approximate numerical solution of the equation can be found using the mathematical statistics of the random variables. When the sample size is insufficient, the Monte Carlo algorithm suffers from the disadvantage of a large probability error. Therefore, we adopted the method of increasing the number of simulated photons to reduce the Monte Carlo simulation error, and after several statistical experiments, the quality of the polarization mode distribution could no longer be significantly improved when the photon number was increased beyond 50E6. We used the Monte Carlo method based on the libRadtran program to conveniently resolve the radiation transmission problem.
2.3 AOD measurement based on sun photometer
According to the Beer-Lamber law, the solar radiation voltage value $V(\lambda ,t)$ with wavelength $\lambda $ received by the sun photometer at time t can be written as:
where ${V_0}(\lambda )$ is the voltage that would be due to the intensity of the solar irradiance at the top of the atmosphere, ${d_0}/d$ is the solar-terrestrial correction factor, m is the optical air mass, which describes the increase in the direct optical path length from the sun to the detector, $\tau (\lambda ,t)$ is the atmospheric optical depth. After the calibration of the instrument, the value ${V_0}(\lambda )$ is obtained, and the total AOD can be obtained from Eq. (7) as follows:The total AOD consists of the following three parts: (I) Rayleigh scattering optical depth ${\tau _R}(\lambda )$. (II) Aerosol optical depth ${\tau _{AOD}}(\lambda )$; (III) Gas absorption optical depth ${\tau _G}(\lambda )$. The ${\tau _{AOD}}(\lambda )$ can be obtained from Eq. (9) using the total atmospheric optical depth to deduct the Rayleigh scattering optical depth and the gas absorption optical depth:
The main absorbing gas to consider in the visible-near infrared spectral interval is O3. Gas absorption optical depth is typically calculated as the product of the absorption coefficient and the concentration of the absorbing gas. For example, the optical depth of O3 can be expressed as:
where D is the content of ozone, ${k_{{O_3}}}(\lambda )$ is the ozone absorption coefficient of each wave band.The Rayleigh scattering optical depth can be obtained by integrating the scattering coefficient ${k_R}(h )$ with the height in the vertical direction [37]:
Generally speaking, the DOP decreases with increasing wavelength in the visible band for AOD greater than 0.2. Combining the atmospheric aerosol scattering effect with the specific test environment, the band used in this paper is 450 nm.
2.4 Skylight background radiation polarization measurement system
The skylight background radiation polarization measurement system uses a focal plane polarization camera to collect the sky polarization image in real time. The measurement system is shown in Fig. 1 and consists of a polarization acquisition module, an industrial control computer and a power supply. The polarization acquisition module consists of a fisheye lens and a CMOS polarization image detector with an FOV of 185° for capturing the whole sky image, and a CMOS chip consisting of 2448 × 2048 pixel units, each with a size of 3.45 µm × 3.45 µm, and a C-mount compatible with 2/3” optical format.
The DoFP polarization camera integrates a complete array of linear micropolarizers to detect linear polarization states, and each pixel of the acquired image is covered with one of the four linear polarizers, which are oriented at -45°, 0°, 45°, or 90°. Given that the polarimetric system can detect the full-sky polarization mode by a single shot, it eliminates the asynchronous errors caused by rotating polarizers and is more compact.
3. Results and analysis of simulation
To determine the impact of aerosols on the polarization patterns of full-sky background radiation, we first analyzed the results of the impact of the aerosol particle scattering characteristics on the polarization distribution, after which we analyzed the differences in the distribution maps of the DOP and AOP of full-sky background radiation by performing simulations involving different aerosol types and optical depth environments.
3.1 Impact of the aerosol particle scattering characteristics on the polarization distribution
The decay of light transmission in the atmosphere is mainly due to the scattering and absorption effects of aerosol particles in the atmosphere. Because of the variety of aerosol particles in the atmosphere and their complex composition, they exhibit various physical properties, making individual aerosol particles and aerosol particle groups exhibit different scattering polarization characteristics. Aerosol microphysical parameters, such as refractive index, size distribution and shape, are necessary input parameters to calculate optical properties, but to date, the influence of these parameters on the radiation polarization distribution has not been sufficiently analyzed.
The light scattering from aerosol particles can be described using Mie scattering theory, which can deal with light scattering from aerosol particles of arbitrary scale size and uniform complex refractive index. The Jünge spectrum is a special aerosol distribution mode, and measurements in the actual atmosphere show that the spectral distribution of near-surface aerosols in most cases also conforms to the Jünge spectral distribution to some extent. The Jünge spectral distribution [38] is expressed as follows:
where K is a constant related to the concentration of particles, υ is the Jünge shape factor, indicating the spectral shape characteristics, reflecting the proportion of aerosol particles of different particle sizes in the atmosphere, that is, a large υ value indicates a larger proportion of small particles, and a small υ value indicates a larger proportion of large particles. The υ index generally varies between 2 and 4.To demonstrate the sensitivity of the DOP to $\nu$, we specify the distribution of the relationship between DOP and scattering angle for different Jünge shape factor υ conditions, as shown in Fig. 2, with complex refractive index m = 1.45-i0.004, wavelength λ=450 nm, and particle radius range of $0.01 - 1\mu m$.
Fig. 2. Distribution diagram of the DOP and scattering angle under different Jünge shape factor (υ) values.
From Fig. 2, we see that for different particle scale spectra under the above simulation conditions, DOP and scattering angle show different variation relationships. When the Jünge shape factor is large (e.g., υ=4), i.e., the proportion of small particles is large, DOP is at the maximum near the scattering angle of 90°. As the Jünge shape factor decreases, that is, the proportion of large particles gradually increases, the maximum value of DOP gradually moves backward, and the maximum value does not exceed 0.5.
To demonstrate the sensitivity of the DOP to the refractive index of the aerosols, the relationship between DOP and the scattering angle under the conditions of variable complex refractive index real and imaginary parts is given, as shown in Fig. 3, where Fig. 3(a) and (b) represent the DOP distribution under different refractive index real and imaginary parts, respectively; the Jünge shape factor is taken as 3, wavelength λ=450 nm, and the radius range is $0.01 - 1\mu m$.
Fig. 3. Distribution diagram of the relationship between the DOP and scattering angle under the conditions of the real part (a) and imaginary part (b) for different refractive index values.
From Fig. 3, we see that the DOP and scattering angle show a large variation under different refractive index real part conditions, which indicates that the effect of scattering of different particles on the DOP is more obvious. The influence of the imaginary part of the refractive index is much smaller than that of the real part, and the influence of the light absorption of the particles on the DOP can be ignored to a certain extent.
3.2 Impact of the aerosol type on the polarization pattern
We investigate various aerosol types as tabulated in the OPAC (Optical Properties of Aerosols and Clouds) library by Hess et al. [39] The single scattering properties, including the full phase matrices, are taken directly from the cases of spheroidal aerosol particles and Sahara dust aerosols from the IPRT database [26]. We selected continental average, urban, maritime clean, and desert aerosols for our study, and their compositions are shown in Table 1, which can be further classified into nuclear mode (nuc.), accretion mode (acc.), and coarse mode (coa.) according to the aerosol particle size, with Ni as the number density and Mi as the mass concentration in Table 1.
Table 1. Compositions of the four aerosol types
In Table 1, the continental average-type aerosol has the lowest mass concentration and consists mainly of water-soluble aerosols and soot, representing continental areas under anthropogenic influence. The urban-type aerosol has the highest number density and again consists mainly of water-soluble aerosols and soot, representing strongly polluted urban areas. The maritime clean-type aerosol has the lowest mass density, contains a certain amount of accumulated modal particles, consists mainly of sea salt and sulfate, and represents a vast undisturbed ocean without soot, where the change in the vertical profile of aerosol is not very large [41]. The desert-type aerosol has the highest mass concentration, contains mineral and water-soluble aerosols, has more nuclear modal particles, and represents the desert area containing nonspherical particles. These different types of aerosols also have different effects on the sky polarization patterns.
Figure 4 shows the distribution of background radiance DOP and AOP under full sky for different aerosol types with a solar zenith angle of 30°, a solar azimuth angle of 30°, an observed wavelength of 450 nm, and a surface albedo of 0.
Fig. 4. Distribution map of the degree of polarization (DOP) and angle of polarization (AOP) of full-sky background radiation under the different aerosol types.
The full-sky background radiation DOP is symmetrical about the solar meridian and the antisolar meridian, and the regional line near the sun center has the smallest DOP, while the back-to-sun regional line has a larger DOP. Combined with Table 1, the overall mass concentrations of continental average-type aerosols and maritime clean-type aerosols are lower than urban-type and nonspherical desert-type aerosols. Furthermore, the DOP distribution for continental average-type and maritime clean-type aerosols are obvious, which is similar to the Rayleigh scattering effect. Urban type and nonspherical desert-type aerosols contain higher amounts of soot and mineral particles, which have a stronger weakening effect on the DOP. The maximum DOP values of the continental average, maritime clean, urban and desert spheroid aerosol types in the whole sky are 0.678, 0.704, 0.472 and 0.616, respectively.
We see the distribution of the background radiation AOP under full sky, which also shows a symmetric distribution of solar meridian and antisolar meridian, and its angle value gradually shrinks toward the solar meridian in an “$\infty $” shape. The larger the absolute value of AOP near the solar meridian is, the smaller the influence of different aerosol types on it, and the change law is basically the same and more stable. The AOPs of maritime aerosols and desert aerosols are partly distorted near the solar center, which may be caused by water clouds, ice clouds and a large number of nonspherical aerosols.
For comparison, in Fig. 5, we show the distribution map of the DOP and AOP for a clear atmosphere with no aerosols, as also computed in [1]. By comparing Fig. 4 against Fig. 5, we can see that the major influence of the aerosols is to weaken the DOP values of sky background radiation polarization to varying extents.
Fig. 5. Distribution map of the degree of polarization (DOP) and angle of polarization (AOP) of full-sky background radiation for a clear atmosphere with no aerosols.
3.3 Impact of the AOD on the polarization pattern
Angström [42] described the mathematical relationship between the optical depth of an aerosol and its wavelength, which is axiomatically shown as follows:
where β is the turbidity coefficient, which is related to the number of aerosol particles in the vertical column above the observation point, and α is the wavelength index, which indicates the size of aerosol particles in the vertical column. The turbidity factor β generally ranges from 0 to 0.5; when β ≤ 0.1, it indicates a clean atmosphere; when β ≥ 0.2, it indicates a relatively turbid atmosphere. The wavelength index α generally ranges from 0 to 2, and the average value of model input is approximately 1.3. The larger the particle is, the smaller the α value is. For example, when α is close to 0, the main particles of aerosol are large diameter dust particles; when α is close to 2, the main particles of aerosol are small diameter smoke particles.Under the same aerosol type, we can use a sun photometer to measure the AOD in the design wavelength band, then take the logarithm of both sides of Eq. (13) and fit it to obtain the values of α and β, and substitute the target wavelength and the fitted values of α and β into Eq. (13) to find the AOD at the target wavelength. Taking the general continental aerosol as an example, a wavelength of 450 nm, a solar zenith angle of 30°, a solar azimuth angle of 30°, and a surface albedo of 0.2 are used to simulate and analyze the effect of different AODs on the polarization pattern of background radiation under full sky.
Based on Fig. 6, as the AOD increases, the particle scattering characteristics are enhanced and the full-sky DOP decreases, and when the AOD increases from 0.2 to 2, the corresponding full-sky maximum DOP values are 0.50, 0.39, 0.27, and 0.17, with a maximum decrease by 66%. This result shows that the AOD has a large effect on the DOP. However, for the AOP distribution, the overall distribution trend remains stable, except for the contraction point at the sun position, which is shifted slightly at an AOD of 2.
Fig. 6. Degree of polarization (DOP) and angle of polarization (AOP) of full-sky background radiation with AODs of 0.2, 0.5, 1, and 2.
4. Comparison of the measurement results
To further clarify the impact of aerosols on the polarization patterns of the full-sky background radiation under real scenarios, we built a skylight background radiation polarization measurement system for testing at Hefei Science Island (31°54’N, 117°10’E), China.
The test was conducted on October 11, 2022, October 18, 2022, and October 22, 2022, under cloud-free conditions, all between 10 and 11 a.m. The AOD and aerosol visibility at 450 nm in this interval were calculated using a laboratory-based spectroscopic sun photometer [43], as shown in Fig. 7.
Fig. 7. Aerosol optical depth (a) and aerosol visibility (b) at 450 nm on October 11, 2022, October 18, 2022, and October 22, 2022.
Between 10:00 and 11:00 a.m., the average AOD values were 0.35, 0.20, and 0.66 on October 11, October 18, and October 22, respectively, and the average aerosol visibility values were 24.47 km, 34.00 km, and 8.72 km, respectively. The solute optical thickness is negatively correlated with the aerosol visibility. Knowing the above aerosol parameters under the actual measurement conditions, we next analyze the full-sky background radiation polarization images acquired using a DoFP polarization camera.
In Fig. 8, the distribution of the background DOP of the full-sky for Oct. 11, Oct. 18 and Oct. 22 changes, and the maximum DOPs for each day are 0.46, 0.53 and 0.35, respectively, according to the Stokes method, which is consistent with the vectorial radiative transfer polarization calculation model of the Monte Carlo method and allows us to use the Monte Carlo algorithm to solve the atmospheric radiation multiple-scattering transmission model. The feasibility of using the Monte Carlo algorithm to solve the atmospheric radiation multiple-scattering transmission model is verified. Therefore, with the increase in AOD, the DOP decreases continuously, and the decreasing trend will be increasingly obvious. When the AOD is above 0.3, the maximum DOP will not exceed 0.5.
Fig. 8. Measured distribution of the background DOP of full-sky radiation on October 11, 2022 (a), October 18, 2022 (b) and October 22, 2022 (c).
5. Conclusions
A calculation model of multiple-scattering polarization was established via the vector radiation atmospheric transport equation and Monte Carlo method. Through simulations, we analyzed the impact of aerosol scattering characteristics on the polarization distribution, and four aerosol types were modeled. The characteristics of continental average, urban, maritime clean, and desert aerosols and their impacts on the distributions of the DOP and AOP of the full-sky background radiation were investigated. Then, the impact of the AOD on the polarization mode was assessed via both the simulation and measurement methods, and a comparison revealed that under a clear sky without clouds, the impact of the AOD on the DOP was still noticeable. With increasing AOD, the DOP decreased, and the decreasing trend became increasingly obvious. When the AOD was above 0.3, the maximum DOP did not exceed 0.5. For the AOP distribution, the overall distribution trend did not change notably and remained stable, except for the contraction point at the sun position under an AOD of 2, which was offset to a certain extent.
Although our work provides a more comprehensive analysis of the impact of aerosols on the polarization patterns of full-sky background radiation, there remain some limitations. In subsequent research, the inclusion of water and ice cloud parameters in the simulation of polarization patterns in complex weather environments, such as clouds and haze, will be a focus. In addition, the DoFP polarimetry technique proposed in this paper may provide greater research value in practical applications.
Funding
National Key Research and Development Program of China (2019YFA0706004).
Acknowledgments
The authors would like to express their gratitude to Xiaoxiao Tang and Cao Yao for their assistance in data treatment, as well as Jiuming Cheng and Prof. Heli Wei for their valuable insights and discussions.
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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