1. Introduction

Clouds envelop 60% of the Earth's surface, exerting their influence on weather patterns and climate change [1]. Cloud PSD reflects a cloud's lifecycle and enhancing its retrieval is crucial for precise cloudy atmospheric remote sensing. Currently, a substantial portion of cloud PSD retrieval research depends on single-angle intensity data (dual reflectance method [2–4]) or active radar data (lidar pulses method [5–7]). The incorporation of multi-directional and polarization observations for atmospheric parameter retrieval still remains relatively limited [8]. Mishchenko et al highlighted that utilizing multi-directional polarized observations significantly outperforms algorithms based on single-angle measurements for enhancing cloud retrieval capabilities [9]. Therefore, examining the sensitivity of multi-angle polarized observations to PSD retrieval is important. In-orbit satellite multi-directional polarized sensors mainly come in two types: rotating elements and co-boresighted [10]. The former includes POLDER (POLarization and Directionality of the Earth's Reflectances) and 3MI (Multi-viewing Multi-channel Multi-polarisation Imager), as well as DPC (Directional Polarization Camera). POLDER can provide observations at 16 angles, covering a range from 443 nm to 1020 nm, with three polarization channels [11]. 3MI, the next generation of POLDER, offers 10 to 14 views of a scene and 12 bands from 410 to 2130 nm. It includes 9 polarized channels [12]. The DPC's observation wavelength range spans from 443 nm to 910 nm at 17 angles, featuring the same polarization bands as POLDER, but with higher resolution [13]. In co-boresighted instruments, APS (Aerosol Polarimetry Sensor) covers wavelengths from 410 nm to 2200 nm across 250 viewing angles. Despite launch failure, its airborne version, RSP (Research Scanning Polarimeter), significantly contributes to cloud property retrieval research [14]. Additionally, POSP (Particulate Observing Scanning Polarimeter) features design like APS. POSP and the previously mentioned DPC payload are co-deployed as PCF (Polarization Cross Fire) on the GF-5B(GaoFen-5B) satellite, enabling polarized crossfire observations [15].

Measurements at different wavelengths, directions, and polarization states vary in their sensitivity to different atmospheric parameters and the content of information they provide. Schuessler et al evaluated the information content of oxygen absorption bands for cloud height retrieval and provided insights into the sensitivity of cloud macro-physical parameters to cloud geometric thickness [16]. Similar assessments have also been made regarding this applied to aerosol altitude retrieval, such as Chen et al [17] and Wang et al [18]. King and Vaughan used Bayesian OE (Optimal Estimation) to derive the profile of liquid droplet effective radius profile as a function of COT and evaluate the information content across diverse wavelengths [19]. Zhang et al employed information content assessment methods to determine the optimal combinations of bands and viewing angles to retrieve ice cloud properties [20]. Similar studies have also conducted information content and posterior error assessments for cloud water vapor profiles, liquid water paths [21], and cloud top heights [16]. Besides, Dong et al assessed that the additional shortwave infrared bands of 3MI provide more information about coarse particles based on Bayesian theory and information content analysis. Its retrievals, compared to POLDER-3, exhibit a 50% decrease in uncertainty [22]. Similarly, performance assessments have been conducted for instruments such as CAPI (Cloud and Aerosol Polarimetric Imager) [23], PSAC (Polarized Scanning Atmospheric Corrector) [24], MAPI (Multi-Angle Polarization Imager) [25], TEMPO (Tropospheric Emissions: Monitoring of POllution) and GOES-R (Geostationary Operational Environmental Satellite R-series) [26–29]. Information content's independence from retrieval techniques allows it to pre-evaluate the sensitivity of a given observational mode to different observational targets. Hou et al evaluated the extent to which multi-spectral polarimetric measurements improve the retrieval of various aerosol parameters near the Earth's surface [30]. Xu et al model spectral fingerprints between 330 nm and 4000 nm, suggesting that hyperspectral measurements can retrieve heating rate estimates of absorbing aerosols above clouds [31]. Then, they employed information content analysis to assess the retrieval errors of 22 aerosol microphysical parameters related to a bimodal particle distribution function retrieved from actual AERONET (AErosol RObotic NETwork) measurements under fine and coarse models [32]. In conclusion, the evaluations of instruments or target state vectors aid in our quantitative assessment of the inversion potential of instruments or methods. A comprehensive evaluation method that covers both the optical device configuration and its influencing factors is essential to assess the potential of retrieving PSD across various cloud scenarios. Based on this framework, we utilize the DFS of cloud PSD as an evaluation metric and analyze the variations in information content within spectral signals. Changes in different factors, from incident to scattering, and ultimately to reception by the sensor throughout the process are involved. These adaptations allow us to develop a more comprehensive understanding of the mechanisms responsible for fluctuations in the information content of cloud PSD across various observational situations.

In this study, we used RT simulations and Bayesian-based OE to explore factors impacting cloud PSD retrieval. The second section covers information content theory, forward simulation assumptions, and prior knowledge configuration. The third section focuses on instrument design's impact on cloud PSD retrieval information, examining observational geometry, spectral bands, and the number of viewing angles. The fourth section discusses cloud characteristics’ influence, particularly COT and cloud PSD. The fifth section explores other elements along the observation path affecting cloud PSD retrieval. Finally, we summarize our study's conclusions.

2. Theory and method

2.1 Information theory

We utilize Bayesian-based OE method to assess the information content of retrieved properties. The premise of retrieval is to construct a forward model ($F$). We assume that $x$ represents the state vector, comprising $n$ elements awaiting inversion, while $\textrm{y}$ denotes the observation vector, consisting of $m$ observation elements. Assuming a linear relationship between observation and state vector, satellite observations can be expressed by Eq. (1):

Here $b$ signifies a non-state vector that will also affect the result. It refers to parameters in the forward model of satellite observation systems that affect the accuracy of inversion targets, such as surface, aerosol, or other cloud parameters. whereas $\varepsilon $ conveys the aggregate error originating from both the forward model and satellite observations. Under the assumption of a Gaussian distribution, Eq. (2) can be utilized to articulate the maximum likelihood solution for the state vector:

Here ${x_a}$ and ${S_a}$ denote the prior estimate and the error covariance matrix of the state vector $x$, respectively. ${S_\varepsilon }$ signifies the measurement error covariance matrix, while $K$ represents the Jacobian matrix, consisting of partial derivatives of each measurement with respect to each state vector. Consequently, the square root of the diagonal elements of $\widehat S$ pertains to the posterior error of the respective parameters in Eq. (3):

As shown in Eq. (4), the error in the inversion of $\hat{x}$ can be assessed by the root mean square of the diagonal elements of $\hat{S}$:

Therefore, the average kernel matrix $A$ could be defined in Eq. (5):

As shown in Eq. (6), DFS represents the independent or total information content retained by the satellite measurements. The diagonal elements of A, known as partial $DF{S_{\textrm{i,j}}}$, with values ranging from 0 to 1, indicate the partial capability to retrieve independent properties. When it exceeds 0.5, the corresponding state vector can be retrieved. The trace of a diagonal matrix $A$ is referred to as the total information content:

2.2 UNL-VRTM forward model

We used the UNL-VRTM (UNified Linearized Vector Radiative Transfer Model) (https://unl-vrtm.org) model to simulate information content for water cloud PSD retrieval under different observation situations [33]. The UNL-VRTM model incorporates the VLIDORT (Volume LInearized Discrete Ordinates Radiative Transfer) code, linear Mie scattering, T-matrix scattering algorithms, an OE-based information assessment module, as well as various surface reflectance models. These modules enable the model to support simulations of various observational geometries and particle mixtures among different cloud modes. Furthermore, it can calculate Stokes parameters under different conditions, as well as Jacobian matrices for describing cloud droplet distribution parameters under specified observation conditions. This allows it to be used for assessing the richness of information content in cloud PSD provided by the observational data acquired by the optical instrument before retrieval.

In our simulation, we utilized the US (United States) standard atmospheric conditions (1976), which encompassed 33 atmospheric layers. To achieve more realistic atmospheric observation simulations, the HITRAN (HIgh-resolution TRANsmission molecular absorption) gas absorption database is utilized here. The instrument's FWHM (Full-Width at Half Maximum) was configured to 0.01 nm. Table 1 presents the observation geometry range defined within the single-angle observation vector in our studies. The observation geometry is based on the actual observation capability of the actual satellite instrument.

Table 1. Configuration of observation geometry in UNL-VRTM model.

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2.3 Observation vector and error covariance

Representative spaceborne polarimetric observation payloads currently include CNES(Centre National d'Études Spatiales)'s POLDER and 3MI series, NASA(National Aeronautics and Space Administration)'s APS series, and CNSA(China National Space Administration)'s DPC and POSP series. They all have the capability to measure directional reflectance within the specified wavelengths, offering both intensity and partial band polarization capabilities. The settings of wavelength range are introduced in Fig. 1. The IPA (Independent Pixel Assumption) has been applied within our studies due to avoid the interference arising from pixel size and the heterogeneity among cloud pixels.

 

Fig. 1. Spectral configurations for mainstream spaceborne polarimetric instruments.

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In this paper, we employ both intensity and polarization to assess performance variations in the scalar and vector observation data within multi-directional polarization observation in cloud PSD retrieval capabilities. The definition of polarization is specified as DOLP (Degree Of Linear Polarization) in Eq. (7):

Based on the current spectrum configurations of mainstream payloads in Fig. 1, we explore the primary observational spectral range where they overlap. While, the ultraviolet and water vapor channel bands are typically not used for cloud parameter retrieval. Our study does not specifically delve into the impact of cloud layer height on retrieval information. Consequently, overall observation vector is defined as in Eq. (8):

Here $\textrm{y}$ represents the observation vector in the single-angle observation mode. ${I_{410}} - {I_{2250}}$ indicate intensity information, while $\textrm{DOL}{\textrm{P}_{410}} - \textrm{DOL}{\textrm{P}_{2250}}$ represent polarization information, all within the 410 nm to 2250 nm wavelength range. About instrument uncertainties, we referred to the instrument parameters of the DPC onboard GF-5B in Ref. [13,34]. The maximum relative error in radiance measurements is 0.05, while the absolute error in polarized radiance measurements is 0.02. Therefore, observation errors definitions can be seen in Eq. (9), where ${\varepsilon _\textrm{I}}$ and ${\varepsilon _{\textrm{DOLP}}}$ represent the absolute errors for radiance observations and polarized radiance observations, respectively:

In multi-views observation vectors, we incorporated additional directions by referencing the geometric information from DPC's multi-directional observation mode. The multi-views observation vector could be defined as in Eq. (10):

Here $\varepsilon _{\textrm{I}{\lambda _{1 \ldots \textrm{N}}}}^2$ and $\varepsilon _{\textrm{DOLP}{\lambda _{1 \ldots \textrm{N}}}}^2$ represent the standard deviations of radiance and polarized radiance observations in the $\lambda $ band, respectively. They can be calculated using Eq. (9).

2.4 Cloud model and state vector

Low clouds are primarily composed of spherical water droplets, and their particle size distribution conforms to a modified Gamma distribution [35]. It can be represented by the function in Eq. (12):

The $B$ in Eq. (12) can be represented by Eq. (13):

Here $N$ denotes the total particle density per cubic centimeter, measured in $\mu {m^3} \cdot \mu {m^{ - 2}}$ units. $a$ is a normalization constant. ${r_{\bmod }}$ corresponds to the mode radius of this model, while $\alpha $ and $\gamma $ characterize the slope of the droplet distribution. According to the theory of independent scattering of radiation by cloud droplets, the effective particle radius (${r_{\textrm{eff}}}$) and variance (${v_{\textrm{eff}}}$) can characterize the PSD of cloud [36] in Eqs. (14-15):

 

Fig. 2. Modified gamma distribution for five cloud model of OPAC. Abscissa is the radius of cloud particles, ordinate is the number density of cloud droplets.

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Table 2. Microphysical properties of the water cloud models from OPAC.

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The phase function of cloud droplets undergoes discernible variations with scattering angle and maintains a pronounced correlation with wavelength. As illustrated in Fig. 3, utilizing Lorenz-Mie scattering theory, we conducted computations for the scattering phase function (F11) and the linear polarization degree (-F12/F11) of water cloud droplets with an effective particle radius of 10 µm and an effective particle variance of 0.1. As shown in Fig. 3 (a), cloud droplets demonstrate significant forward diffraction and minor backward scattering. The scattering phase function depends relatively little on wavelength in the visible and near-infrared regions. Figure 3 (b) shows that the linear polarization degree of water cloud droplets exhibits pronounced oscillations with increasing scattering angles. Meanwhile, in the wavelength from 410 nm to 2250 nm, the peak positions and magnitudes of the oscillations vary with wavelength.

 

Fig. 3. F11 phase function of radiance component (a) and -F12/F11 linear polarization degree component (b) as a function of scattering angle from 410 nm to 2250 nm.

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As depicted in Fig. 4, we have opted for the visible (490 nm) and near-infrared (865 nm) bands to scrutinize the changes in F11 and -F12/F11 concerning scattering angle across various combinations of particle radius and particle variance. When the effective particle radius of water droplets is fixed at ${r_{\textrm{eff}}}$=10$\mu m$ in Fig. 4(a-b), fluctuations in the effective particle variance of the cloud have a negligible impact on F11. However, when the scattering angle approaches 140 degrees, variations in the effective particle variance of water droplets within cloud have resulted in noticeable fluctuations in -F12/F11. This indicates that in the visible to near-infrared spectral range, intensity is insensitive to variations in the effective particle variance of cloud droplets. However, polarization exhibits sensitivity at specific scattering angles. As shown in Fig. 4(c-d), when the effective particle variance of water droplets was fixed at ${v_{\textrm{eff}}}$= 0.1 while the effective particle radius of the cloud is adjusted, both F11 and -F11/F12 of water cloud particles exhibit significant fluctuations. This suggests that the effective radius of cloud particles was sensitive to both intensity and polarization information and remains largely unaffected by the observation angle.

 

Fig. 4. F11 phase function of radiance component and -F12/F11 linear polarization degree component across different cloud droplet distributions (a, b: effective radius = 10 µm; c, d: effective variance = 0.1) in visible and near-infrared bands.

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In this condition, neglecting surface contributions, we assume the presence of a uniform single layer of water cloud at an altitude of 1-2 km. In forward RT simulations, COT of 10 is employed to ensure the cloud radiative information at the TOA(Top Of Atmosphere) reaches saturation. Unless specified, subsequent simulations utilize identical settings for cloud layers.

3. Instrument design

The scattering properties of cloud droplet are intricately tied to the observation direction and wavelength. The configuration of wavelength and view settings of satellite instrument can significantly impact the richness of information content gathered through remote sensing. Here, we investigate how the instrument's observational geometry, combinations of multi-spectrum, and the number of observation angles within the hemisphere affect the information content of cloud droplet distribution in imaging results. Then, we present the optimal combination of polarized and non-polarized spectrums, along with the number of observation angles, to enhance the retrieval of cloud droplet size distributions utilizing the multi-directional polarized measurements. The analytical results assist in evaluating the instrument ‘s capability to perform cloud droplet distribution retrieval.

3.1 Geometry of observation

As shown in Fig. 5, the results of the TOA reflectance and DOLP with respect to variety in observation geometry are presented for the simulation described above. With the exception of water vapor absorption at the 1380 nm band, cloud droplet reflectance demonstrates similar characteristics throughout the 410 nm to 2250 nm range. By comparing the phase functions of cloud droplets in Fig. 3, we observe that the scattering phase function values of the cloud droplets are very similar across all wavelengths, indicating that the cloud's reflectance remains consistent from visible to near-infrared bands. Similarly, the polarization of water clouds exhibits a pronounced variation with the scattering angle. The values of the linear polarization degree exhibit systematic oscillations as the scattering angle changes. The DOLP of water clouds significantly intensifies for scattering angles greater than 135°, resulting in the formation of a rainbow effect. The magnitudes of these oscillation peaks and their scattering angle positions encapsulate the information of cloud particles distribution. In other words, the observation geometry of instruments can influence the information content contained in the observation data. Adding polarized observations at specific angles contributes to obtaining more accurate cloud parameter retrieval results in the hemispherical space.

 

Fig. 5. TOA reflectance (a-i) and DOLP (j-r) in the pure water cloud case when COT = 10 and solar zenith angle of 40 degree from 410 nm to 2250 nm, respectively. The polar radius is view zenith angles from 0 to 75 degree and the polar angles is relative azimuth from 0 to 180 degree, as specified in Table 1.

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Based on the observation vector in Eq. (8), we have established the observation vector S1, which includes the intensity from 410 nm to 2250 nm in vector $\textrm{y}$. S2 comprises the polarization observation vector in same spectral observation. S1 + S2 encompasses all observation vectors in $\textrm{y}$. According to above observation vectors, the distribution and statistical DFS for the effective particle radius, effective particle variance, and total information content within the half-sphere space are presented in Fig. 6 and Table 3. The DFS analysis of the effective radius for both S1 and S2 indicates that the observation information from multi-band observation vectors has reached saturation for the effective particle radius, with mean values of 0.987 and 0.848, respectively. The values of DFS range from 0 to 1, and when they exceed 0.5, it is considered feasible for inversion. Therefore, we believe that increasing observation bands at this point no longer contributes to improving the inversion capability of effective particle radius. For effective particle variance, both S1 and S2 exhibit distinct angular characteristics. Based on the sun's position, we can infer that the cyclic variations in its information content are, in fact, related to the cloud's rainbow effect. S1 and S2 have peak values of 0.126 and 0.190, respectively, for effective particle variance. The combined S1 + S2 observation vector can reach a maximum of 0.230. The high values in different observation modes all originate from the cloud's rainbow region. Furthermore, polarization contributes more information content regarding effective variance compared to scalar observations. In terms of the total information content, the total DFS obtained from the S1 + S2 observations ranges from 0.978 to 1.214, with a mean of 1.074. Almost all the total information content is contributed by the effective cloud particle radius. This result suggests that if we aim to individually retrieve the effective cloud particle radius, it can be achieved solely based on intensity information. However, to simultaneously retrieve both the effective radius and variance, incorporating polarized observation information at specific scattering angles significantly enhances the capability for PSD retrieval. For the observation vectors mentioned above, despite the enhanced information content of polarization contribution at certain observe angles with the addition of multi-band observation, effective variance’s DFS still remains below 0.5. Therefore, retrieving it using single-angle observations alone is difficult.

 

Fig. 6. DFS of polar coordinates plotted for effective radius, variance and total in S1, S2 and S1 + S2 observation modes. Each row corresponds to different state vectors, and each column corresponds to different observation modes. The coordinates are defined similarly to Fig. 5.

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Table 3. DFS statistical result of different state vector in different observation modes.

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The above statistics demonstrate the correlation between information content and observation geometry, specifically scattering angle. In Fig. 7, the box plots illustrate the information content of corresponding state vectors as a function of the scattering angle block under the observation mode. The dependence of cloud droplet size distribution information content on the scattering angle is stronger for polarization (S2) than for intensity (S1). Compared to S1, S2 increases the information content for cloud droplet size distribution in the rainbow region, specifically in the scattering angle range of 135-165 degrees. Additionally, the mean information content of effective particle radius contrast for S2 is almost twice that of S1 between 155-165 degrees(from 0.067 to 0.131). Referring to Fig. 3, we find that this region corresponds to the oscillation region of water cloud droplets -F12/F11. This suggests that adding polarization observation bands in the rainbow region of water cloud droplets can be beneficial for cloud droplet size distribution retrieval. At the same time, the DFS sensitivity of cloud effective particle variance to the scattering angle is significantly higher than that of cloud effective particle radius. The inclusion of polarization information significantly enhances the information content of effective particle variance for S1 + S2. the cumulative mean DFS refers to the aggregation of 12 statistical interval means derived from box plots within all scattering angle range of 50-180 degrees. It indicates the cumulative potential for retrieving cloud droplet size distribution across all observation angles within the hemisphere. Compared to S1, the cumulative mean DFS of S1 + S2 has increased by 0.44 across all intervals in the simulated scattering angle range from 50 to 180 degrees. Above all, we believe that adding polarization observation in the rainbow region of clouds can increase the information content of cloud effective particle variance, thus enhancing the overall information content for cloud droplet size distribution retrieval.

 

Fig. 7. Box plot of DFS with respect to scattering angle range for effective radius, variance, and total in S1, S2, and S1 + S2 observation modes. Each row corresponds to different state vectors, and each column corresponds to different observation modes. The orange horizontal line represents the mean, while the blue ten-character symbol indicates the outliers.

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3.2 Spectral configuration

Reflecting on the above findings, we contend that the incorporation of more bands may not necessarily enhance the information accessible for cloud droplet size distribution retrieval. As such, it becomes crucial to evaluate the individual contributions of both intensity and polarization data within the 410nm-2250 nm range towards the information content available for retrieval. Figures 8 and 9 illustrates the DFS calculation results for both single-angle and single-wavelength observations across the 9 spectral bands covered by the observation vector $\textrm{y}$. The cloud model and atmospheric observation conditions remain consistent with those described in Section 3.1. For intensity data, the DFS values for effective radius are consistently high across all spectral bands in the context of known COT. Conversely, there is minimal information content regarding the effective variance. The limited information content of cloud effective variance in scalar observations is the reason why current mainstream reflectance-based methods can only retrieve cloud particle radius and face challenges in estimating particle variance. Polarization observation data demonstrate notably high information content for effective radius at specific angles across various spectral bands. While the information content for effective variance is lower than that of effective cloud radius, it is significantly higher than the information content of intensity observations. This showcases the potential of utilizing polarization observations for cloud droplet size distribution retrieval. Nevertheless, single-angle polarization observations at a single wavelength still fall short of meeting the retrieval requirements.

 

Fig. 8. Subfigures (a-i) and (j-r) respectively illustrate the DFS contained in intensity of a single-band when considering the effective radius and effective particle variance as state vectors. The polar coordinates are defined similarly to Fig. 5.

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Fig. 9. Similarly, to Fig. 8, subfigures (a-i) and (j-r) correspond to the scenario with observation vectors using DOLP.

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Because different wavelengths contribute varying amounts of information to the retrieval of cloud droplet size distribution, in a single-angle observation mode, the information content for retrieving the cloud droplet size distribution may saturate in specific spectral bands. Hence, we evaluate the information content for various spectral band combinations to ascertain the optimal choices for different numbers of observation bands. We utilize the maximum value from multi-band observations to represent the highest potential for cloud droplet size distribution retrieval for the given number of observation bands. Presented in Fig. 10 and Table 4, we compare the total information content values for varying numbers of observation bands, leading to the following insights. Firstly, once the combination reaches four bands, the increase in total DFS becomes relatively marginal. Secondly, upon reaching six bands, the value exceeds 1.2, approaching the information content within the complete observation vector $\textrm{y}$. Lastly, incorporating longer wavelengths, particularly in the near-infrared spectrum, significantly boosts the information content available for cloud PSD retrieval. This aligns with the investigations conducted by Xu et al [31]. In summary, in a single-angle observation mode, multi-bands and single-angle observation vectors consistently fail to achieve a total DFS exceeding 1.5. Single-angle within multi-bands observation vectors alone is insufficient to meet the retrieval requirements for cloud droplet size distribution.

 

Fig. 10. Total cloud droplet distribution DFS of different observation numbers by adding intensity or DOLP observations in single-angle mode from 410 nm to 2250 nm. The lowermost curve (black) shows the DFS when only a single observation element is used. The second curve (red) shows the DFS for two observation elements, when the first element is the best-fit element for the single-element case (2250I). Similarly, the third curve represents the three-element case, when the first two elements are fixed to those providing the best DFS for the two-element case (2250I and 2250P), and so forth.

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Table 4. Statistical combinations that yield the highest DFS value under various observation band combinations.

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3.3 Combination of multiple viewing angles

Utilizing multi-angle observation information is a crucial approach to augment the quantity of observational data. The majority of multi-angle polarimetric satellites are situated in polar orbits, characterized by periodic observation orbits. In this context, leveraging the observation geometry of the GF-5B DPC sensor during its orbit on October 9, 2021 (Orbit 3896), we perform an analysis of the variations in multi-angle observation information content. Considering the movement of the orbital center during the DPC revisit period, we chose 8 sets of observation geometries along the cross-track direction above the equator. The coordinates of the sampling points are shown in Table 5. The solar geometry follows the settings in section 3.1. Detailed multi-angle observation geometry information can be found in Fig. 11 (a).

 

Fig. 11. Adoption of multi-view observation geometries (a) and illustration of observations with different view counts (b). Location of eight solid circular symbols depict the view zenith and azimuth corresponding to multi-view observation data obtained through sampling. The position of the pentagram indicates the zenith and azimuth angle of the solar direction.

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Table 5. The sampling positions and scattering angle range of geometry 1-8 of observational geometries.

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To avoid potential band combination influences, the observation vector includes intensity and polarization observation from above 9 bands spanning 410 to 2250 nm. We sequentially increased the number of observation angles, starting from small to large zenith angles, as shown in Fig. 11 (b), to form observation vectors with varying numbers of observation angles. Table 5 and Fig. 12 provide the scattering angle range and distribution for these 8 sets of observation geometries. From the perspective of the scattering angle distribution under the current solar incidence conditions, the targets closer to the center of the satellite's orbital path exhibit larger scattering angle extremes, covering a broader range of scattering angles.

 

Fig. 12. Distribution of the range of scattering angle corresponding to the geometry 1-8. Based on the sampling plan in Fig. 11(a), distinct observation angles with varying sampling quantities (N) are color-represented to represent the scattering angle distribution characteristics for each number of angles.

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As shown in Fig. 13, compared to the information obtained from single-angle observations, it becomes evident that the addition of observation angles markedly enhances the information content. The degree of enhancement significantly surpasses the gains achieved by adding extra bands. In those geometries, the total DFS average for multi-angle configurations is enhanced by 0.49 compared to single-angle, representing a growth of 48%. When geometry 5 reaches 12 observation angles, the information content of effective variance exceeds 0.5, and the total DFS consistently exceeds 1.5. This implies the simultaneous fulfillment of the requirements for cloud effective particle radius and variance retrieval. Considering its scattering angle distribution, it can be inferred that a broader scattering angle range may have contributed to the increase in DFS. Meanwhile, geometries 4-8 all achieved effective variance information content surpassing 0.5. This implies that multi-angle observations in the rainbow region can support cloud PSD retrieval when enough observation angles are available.

 

Fig. 13. DFS variation of effective particle radius and variance for 8 sets of sampled observation geometries as the number of observation angles increases from 1 to 17. The red triangles represent the DFS of effective radius, while the blue rectangles represent the DFS of effective variance.

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4. Cloud properties

Water droplets with distinct concentrations and properties exhibit different scattering properties, resulting in variations in the information content detected by instruments. To assess the extent to which the inherent characteristics of cloud particles are responsive to the information contained in multi-angle polarized observations, an analysis was undertaken to investigate the impact of different COT, effective particle radii, and variances on the information content. The observation vectors employed in this context are derived from geometry 6 in section 3.3.

4.1 Cloud optical thickness

In the context of the five classic water cloud models provided in the OPAC dataset, we conducted an analysis to examine how the information content related to the distribution of cloud droplet sizes changes with varying COT in multi-angle polarized observations. COT is based on the COT at 565 nm wavelength, and the spectral COT in other wavelength bands is controlled through a consistent loading columnar volume. In Fig. 14, the variation of information content for cloud particle effective radius and effective particle variance is presented as a function of COT ranging from 1 to 100 in these five cloud models. Considering the potential impact of prior COT values on retrieval, we analyze scenarios with both no COT uncertainty and another of 50% COT uncertainty.

 

Fig. 14. The DFS variations in cloud effective radius (a: 0 COT uncertainty,c:50% COT uncertainty) and variance (b: 0 COT uncertainty,d:50% COT uncertainty) for five cloud models from the OPAC database as a function of COT, using an observation vector consisting of 17 viewing angles in geometry 6.

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Firstly, DFS for cloud particle effective radius and particle effective variance both exhibit a trend of decreasing information content as COT increases, when the prior COT is assumed to be accurate. This implies that an increase in the loading columnar volume of cloud particles is not conducive to retrieving cloud droplet size distribution. But, once DFS decreases to a certain extent, it tends to stabilize. We believe that when COT is sufficiently large, its reflectance information tends to saturate. The saturation of the observation vectors leads to the stabilization of the DFS. Secondly, the uncertainty in COT adversely affects the DFS of cloud particle radius and variance, and this impact becomes more pronounced at lower COT values. As COT surpasses 10, the diminishing effect of COT uncertainty on information reduction gradually weakens. Ultimately, as DFS stabilizes with COT variations, the DFS values with COT errors end up slightly lower than those without COT uncertainty.

Lastly, the trend of DFS decreasing with increasing COT varies among different cloud models. It can be observed that variations in cloud models have a minimal impact on the information content related to cloud particle radius, with differences among different cloud models being less than 1%. Referring to Table 2, we have observed that the rate at which DFS decreases for effective particle variance is influenced by the cloud particles’ effective variance values. The smaller the effective particle variance in the model, the faster the DFS of effective particle variance decreases with COT. CUCC and CUMA exhibit comparable effective particle variances, resulting in closely overlapping descent curves. Nonetheless, for COT less than 20, variations in their variance information content become evident. This discrepancy arises from differences in their effective particle radii. Therefore, we contend that effective particle variance is not the exclusive determining factor of DFS. Both the effective particle radius and effective particle variance collectively influence the observational information content.

4.2 Particles distribution of cloud droplets

The results of OPAC cloud model information content as a function of COT show that the cloud particle size distribution itself can impact the inversion information content. Furthermore, the influence of cloud effective particle radius and effective particle variance on the information content is a combined effect. As shown in Fig. 15, we assessed the impact on cloud droplet size distribution information content by varying the effective particle radius (1-20$\mu m$) and effective particle variance (0.01-0.2) of water droplets in both the high information fluctuation range (COT = 2) and the stable information range (COT = 40).

 

Fig. 15. The DFS variations in cloud effective radius (left) and variance (right) for COT =2 (upper) and 40 (bottom). The horizontal axis represents the effective particle variance, the vertical axis represents the effective particle radius, and the color bands represent the DFS values. The positions of the five cloud models in the graph are indicated by red circles as those in Fig. 14.

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For effective particle radius DFS, the dependence on cloud droplet size distribution is minimal, DFS consistently approaching saturation. Hence, at COT = 2, the DFS of the effective particle radius in OPAC's five cloud models exhibits similar values. However, as section 4.1 demonstrated, the DFS of cloud PSD is related to COT. At COT = 40, the sensitivity of DFS to the PSD is heightened. This results in a scenario where, with increasing effective particle variance and radius, there is a decrease in effective radius DFS. Therefore, the DFS of cloud particle radius only experiences a slight decrease under conditions of large COT, large particles, and large variance. STMA achieved the highest value of ${\nu _{\textrm{eff}}}$= 0.13 among the five models due to its relatively large effective particle variance. However, all five cloud models of OPAC are in a relatively stable region of high-DFS. Therefore, variations have a minimal impact on the DFS of effective radius. Regarding DFS of effective variance, it is more strongly influenced by COT. At COT = 2, particle variance DFS for five OPAC models is STCO(0.94), STMA(0.93), CUCC (0.88), CUCP (0.83) and CUMA (0.90), all capable of retrieval. But, those all experienced a significant decline at COT = 40, such as STCO(0.73), STMA(0.75), CUCC (0.52), CUCP (0.41) and CUMA (0.66). Specifically, the information content for CUCP is smaller than 0.5, reaching an un-retrievable level. Under high COT, effective particle variance DFS decreases as variance increases, also related to effective particle radius.

Due to complex cloud structure, the actual multi-directional signals do not often come from a single location within the cloud but rather from different layers. Multi-directional reflectance may primarily be influenced by different heights within the cloud. Consequently, PSD variations across vertical levels result in a blending of signals from various heights. The blending of different cloud droplets within the field of view could potentially disrupt the DFS of cloud PSD. Therefore, we analyze how the DFS of effective particle variance changes in various scenarios of mixing PSD for OPAC's five cloud models in Fig. 16. This experiment is achieved by introducing cloud droplets with effective particle radii ranging from 1 to 20 µm and effective particle variance varying between 0.01 and 0.2, all mixed in a 1:1 ratio (COT = 10) within the same vertical height. In other words, the extra cloud droplets mixed with the OPAC cloud make a 50% contribution to the particle number concentration, or to the COT, or to cloud liquid water content. The results suggest that partially blending PSD leads to the effective particle variance DFS falling below 0.5, in proportions of 45.28% for STCO, 82.15% for STMA, 93.63% for CUCC, 100% for CUCP, and 100% for CUMA. Due to the greater sensitivity of polarization to small particles, small particles exhibit stronger interference capability in cloud PSD retrieval. Therefore, when more small particles are introduced, the decrease in DFS becomes more pronounced.

 

Fig. 16. The variations of cloud effective variance DFS in mixed cloud droplet situations for five OPAC cloud models. The horizontal axis represents the mix particle effective variance, the vertical axis represents the mix effective radius, and the color bands represent the DFS values of effective variance.

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5. Other components in the observation path

5.1 Absorptive aerosols above cloud

The influence of aerosols on cloud observations primarily revolves around the disruption of scattering information from cloud droplets detected by the sensor. The presence of aerosols at the cloud top or within the cloud affects the retrieval of cloud PSD. Hence, building upon the previous cloud layer settings, we introduce smoke aerosols with Gaussian distribution, featuring varying peak heights between 2-3.5 km. It occupies a vertical range of ±1km around the peak height. Additionally, the AOD varies between 0 and 1 to investigate the impact of aerosols on cloud droplet information content. The smoke-absorbing aerosol model is referenced from Xu et al [31]. This aerosol model follows a logarithmic distribution, characterized by an effective radius of 0.12 $\mu m$ and an effective variance of 0.18. Considering the minimal impact of aerosol presence on the effective particle radius of cloud droplets, we would solely concern about its influence on the effective particle variance of clouds. In Fig. 17, we can observe the variation in information content related to the effective particle variance concerning the peak height and AOD. When viewed as a function of the peak height of the aerosol, the substantial decrease in information content directly results from the interplay between aerosol and cloud within a vertical interval. This affirms that the interaction zone between clouds and aerosols presents a challenge in retrieving cloud parameters. The information content of the five OPAC cloud models in the mixed region with aerosols, compared to the non-mixed region, has decreased by averages of 0.28 (STCO), 0.27 (STMA), 0.39 (CUCC), 0.59 (CUCP) and 0.58 (CUMA), respectively. In most situations, the decrease in information content resulting from vertical mixing of cloud and aerosol is smaller for smaller aerosol loading. However, when the AOD exceeds 0.2, the mixing of CUCP and aerosols results in a decline in information content of cloud effective particle variance to an impractical retrieval level of DFS < 0.5. In the vertical mixed domain, CUCC, CUCP, and CUMA demonstrate a diminished resilience against interference from aerosols in comparison to STCO and STMA. We attribute this to the smaller effective particle variance in these cloud models. Due to the AOD typically not exceeding 0.2, we believe that aerosols have relatively minor interference in the inversion process.

 

Fig. 17. The variations of cloud effective variance DFS in mixed aerosol particles situations for five OPAC cloud models. The horizontal axis represents the AOD, the vertical axis represents the peak height of aerosol, and the color bands represent the DFS values of effective variance.

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5.2 Contribution from surface

In cloudy scenes, due to the strong light extinction capability of clouds, the contribution from the underlying surface is generally minimal. However, when the cloud cover is thin, the surface contribution may potentially penetrate through the cloud layer and be detected by the sensor. Here, we set the COT to be between 1 and 20. The surface is assumed to be Lambertian, with surface albedo ranging from 0 to 1. The remaining configurations align with those used in the geometry 6 experiment described in Section 3.3. Figure 18 illustrates how the cloud PSD information varies with COT and surface albedo for view numbers of 4, 10, and 17.

 

Fig. 18. The DFS variations in cloud effective radius (left) and variance (right) for number of views =4 (upper), 10 (middle) and 17 (bottom). The horizontal axis represents the COT, the vertical axis represents the surface albedo, and the color bands represent the DFS values.

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Analysis from Section 3.3 reveals that the information content of cloud effective particle radius is insensitive to the number of observation angles. Consequently, as the number of observation angles increases from 4 to 17, the variation in effective particle radius information content does not exceed 3%. However, with a lower number of observation angles, the contribution from the surface slightly reduces the information content of cloud effective particle radius. This reduction can be negligible when observation angles are abundant. Effective particle variance information is significantly more affected by surface interference than effective particle radius. At 17 observation angles, when COT is less than 5, and the surface albedo is below 0.25, the high information content of cloud effective particle variance aligns with the conclusions in section 4.1. When the surface albedo exceeds 0.25, an increase in surface reflectance interferes with the information content of the PSD, resulting in an information content of cloud effective particle variance retrieval less than 0.5. This phenomenon occurs due to the penetration of radiative information from the surface through the cloud, which is subsequently received by the sensor, thereby introducing interference with the cloud particle variance information. On the other hand, when COT exceeds 8, the influence of the surface contribution tends to stabilize. Even with a bright surface, cloud entirely shields the surface, preventing any interference with the PSD retrieval. Due to the strong extinction capacity of liquid water clouds, we believe surface contribution has a relatively minor impact on PSD retrieval.

6. Conclusion

Our study initiates an analysis of the potential to employ multi-angle polarization data for inferring cloud PSD within the unified framework of the RT process, utilizing principles of information theory. We delve into the scrutiny of cloud properties and other factors that might affect the retrieval. Our studies considered the influences of the receiving instrument, target characteristics, and along the radiation transmission path. We offer a quantization of the principles that underpin the improvements achieved through the utilization of multi-directional polarized information in PSD. For mainly satellite polarized instruments, we have identified the optimized combinations of observation wavelengths, directions, and views. Meanwhile, we analyze the information content variations within cloud cluster observations. This considered different COT, effective particle radii and variances, utilizing the OPAC cloud model. Finally, we carried out simulations that encompassed contributions from cloud-top aerosols and underlying surface, all while examining the mechanisms of their interference in PSD retrieval.

The cloud effective particles radius can be inverted by using intensity alone. But, if retrieval of variance is desired, polarization observations are required to be incorporated. The inversion of PSD is not sensitive to the number of bands, but when using longer-wavelength near-infrared bands the information content increases. Meanwhile, cloud particle variance is sensitive to the scattering angle range of observations and the number of observed angles. To simultaneously retrieve cloud particle radius and variance, spaceborne sensors need to enhance multi-angle polarimetric observation capabilities, particularly in the rainbow region. Therefore, our research findings affirm the necessity of integrating multi-angle polarimetric observation data at 1380 nm, 1610 nm, and 2250 nm wavelengths using the latest multi-angle polarimetric observation payloads, specifically PCF and 3MI. In the analysis concerning OPAC cloud models, we observed that a reduction in the DFS of cloud PSD correlates with an escalation in the COT value. The decrease in DFS for cloud effective variance contrast is greater than that for cloud particle effective radius. Additionally, for cloud with smaller effective particle variances, the decline is faster. However, when the COT exceeds 40 to complete reflectance saturation, the DFS tends to stabilize. In other words, during actual inversion, the accuracy fluctuations in PSD retrieval will be larger in regions with lower COT. Lower DFS necessitates a greater number of known factors to constrain the inversion of unknown properties. The effective radius and variance of the cloud itself can also interfere with the satellite's ability to retrieve its PSD. The combined effect of both on inversion information content is nonlinear, and it is correlated with the COT. Absorbing aerosols above clouds can also interfere with the DFS, with the aerosol loading playing a crucial role. When AOD is high and mixed with clouds, especially when AOD > 0.7, it renders the inversion of cloud variance from retrievable to un-retrievable. In contrast, the effect of cloud and aerosol mixture on effective variance DFS is minor, and higher mixing only slightly weakens the inversion capability. Surface contributions depend on the surface albedo and the COT itself. When the COT is greater than 5, surface contributions have little impact on inversion accuracy. Otherwise, when the surface albedo exceeds 0.25, an increase in surface albedo gradually reduces the DFS of cloud PSD.

In summary, this study is based on the variation of information entropy during the RT process, aiming to enhance the accuracy of cloud PSD retrieval. This lays the groundwork for the collaborative development of future instrument optical designs and inversion algorithms. Naturally, the shortcomings of this study are unavoidable. We have not yet considered complex cloud scenarios involving mixed-phase states and multi-modal cloud particle distributions. Furthermore, the influence of the 3-D(3-Dimensional) cloud structure on information quantity remains unexplored. This matter pertains to the accurate 3-D positioning of cloud radiative data within cloudy atmosphere. Despite the limitations presented above, we still believe that the analysis we have discussed is significant and deserves further exploration. Cloud pixel heterogeneity depends on sensor's imaging pixel size and particle mixing condition. In the future, we will further investigate its impact on changes in information content. We believe that conducting a more comprehensive examination of information content for optical observation will facilitate the optimization of satellite multi-angle polarized payload configurations and the enhancement of retrieval algorithm development efficiency.