Impact of the Uncertainties of Polarized Water-Leaving Radiance on the Retrieval of Oceanic Constituents and Inherent Optical Properties in Global Oceans via Multiangle Polarimetric Observations

Abstract

Compared with traditional single-view and radiometric-only observations, multiangle polarimetric observations of water-leaving radiation play a crucial role in enhancing the retrieval of ocean constituents and aerosol microphysical properties. In this study, the impacts of uncertainties in the degree of polarization (DOP) of water-leaving radiance (L_w) on the retrieval of oceanic constituents and inherent optical properties (IOPs) were investigated via global radiative transfer (RT) simulations and the fully connected U-Net (FCUN) model. The uncertainties in the retrieval of oceanic constituents and IOPs were further investigated with various sensor azimuth angles. The results indicated that the global mean absolute percentage errors (MAPEs) for differing oceanic constituents and IOPs significantly decreased as the number of observation angles increased. Taking the retrieval of Chla as an example, the global MAPEs between the FCUN predictions and RT simulation inputs for Chla concentrations under differing observation angles were 7.41%, 3.76%, 2.70%, 2.44%, 2.62%, and 1.82%. Moreover, the MAPEs at sensor azimuth angles of 0° and 30° were significantly lower than those at other azimuth angles for the single-view observations. As the number of observation angles increased, the variation in MAPEs with the sensor azimuth angle gradually weakened. Furthermore, the impact of errors in the L_w DOP on the retrieval uncertainties decreased as the number of observation angles increased, and the global MAPEs of Chla after adding the various random instrument noises were 46.56% (46.91%), 6.59% (7.21%), 5.21% (5.79%), 4.72% (4.98%), 3.99% (4.52%), and 3.64% (4.03%). Overall, the multiangle polarimetric observations can suppress or balance the impact of uncertainties in the L_w DOP on the retrieval of oceanic constituents and IOPs.

1. Introduction

The polarization characteristics of water-leaving radiation are highly sensitive to the micro-optical properties of marine particles [1]. The spectral and angular polarization of water-leaving radiance (L_w), observed by multiangle polarimetry, contains extensive ocean color information and can be used for the quantitative retrieval of oceanic constituents, including phytoplankton (Chla), non-algal particles (NAP), colored dissolved organic matter (named yellow matter or gelbstoff, CDOM), and inherent optical properties (IOPs) [2,3]. The oceanic constituents play a critical role in determining the IOPs of seawater, and are key factors in underwater light propagation. And the IOPs include absorption coefficients, scattering coefficients, and backscattering coefficients, which are directly influenced by the concentration and composition of oceanic constituents. Multiangle polarimetric observations of water-leaving radiance (L_w) provide a distinctive ability to effectively distinguish and retrieve variations in oceanic and atmospheric properties, addressing the limitations inherent to single-view or radiometric-only methods [4,5,6]. Furthermore, the incorporation of polarized L_w information significantly enhances scalar passive ocean color remote sensing, which typically relies on traditional radiation intensity [7,8].

The incorporation of polarimetry and multiangle observations enables the retrieval of aerosol and hydrosol characteristics [9]. International space agencies have extensively supported the development of spaceborne polarimetric instruments, as evidenced by deployments such as POLDER (POLarization and Directionality of the Earth’s Reflectances) on ADEOS-I and II satellites, PARASOL satellites [10], the APS instrument on the Glory mission [11], and SGLI on GCOM-C [12], alongside the hyperangular rainbow polarimeter (HARP) CubeSat [13]. Additionally, numerous airborne spectropolarimeters, including the Spectropolarimeter for Planetary Exploration (SPEX) [14,15], Research Scanning Polarimeter (RSP) [16,17], Air-HARP, AirMSPI, the Versatile Imager for Coastal Ocean (VICO) [18] and the Observing System Including PolaRisation in the Solar Infrared Spectrum (OSIRIS) [19], have contributed valuable insights. Consequently, observations from the coupled atmosphere–ocean system (AOS) through multiangular polarimetry have opened avenues for global aerosol characterization, offering enhancements to atmospheric correction (AC) and inversion models and supplementing ocean color instrument (OCI) remote sensing reflectance (Rrs) retrievals [20,21,22,23,24]. Additionally, advanced efforts have focused on field measurements of polarized L_w values involving instruments such as the CCNY HyperSAS-POL [25], snapshot hyperspectral imager [26], POLRADS [27], NRL-DC Polarimeter [28], and the direct measurement of the polarization of L_w (POLWR) [29].

Field measurements, along with theoretical and radiative transfer (RT) simulations, have demonstrated the sensitivity of L_w polarization to various marine parameters [3,4,30], such as the attenuation-to-absorption ratio [31,32], chlorophyll-a (Chla) fluorescence [33], turbidity [34], and wind speed [35]. Thus, L_w polarization complements spectral and angular measurements in enhancing ocean color retrieval, especially in coastal waters with complex aerosol and hydrosol optical properties [5,36]. However, these multiangular polarimetry methods were designed primarily for atmospheric particle property retrieval to support climate studies, indirectly assisting ocean color analysis [37,38,39,40]. Moreover, the current atmospheric correction, which is based on joint inversions, retrieves only the L_w and aerosol parameters simultaneously [20]. The retrieval of full Stokes vectors of L_w from operational satellite multiangle polarimetry remains challenging on a global scale [21]. Owing to imperfectly polarized atmospheric correction algorithms and instrumental noise, it is anticipated that polarized L_w will have significant uncertainties, which may affect the retrieval of oceanic constituents and IOPs. Therefore, the impact of these uncertainties on the retrieval of oceanic constituents and IOPs in global oceans via multiangle polarimetric observations remains unclear. Similarly, the variations in the uncertainties of oceanic constituent and IOP retrieval with observation azimuth angles have not been discussed in detail.

In this study, we used a vector radiative transfer (RT) model, named PCOART [41,42,43], to simulate the Stokes vectors of L_w on the basis of global ocean color operational products. We examined the influence of L_w degree of polarization (DOP) uncertainties on the retrieval of oceanic constituents and IOPs via the fully connected U-Net (FCUN) model. Additionally, the global uncertainties of the inversion algorithm were analyzed with respect to various sensor azimuth angles. Finally, the impact of L_w DOP uncertainties on the retrieval of oceanic constituents and IOPs, with varying levels of L_w DOP uncertainties, was assessed.

2. Materials and Methods

2.1. Radiative Transfer Simulations

The PCOART model was used to simulate the global distributions of intensity and the polarization state of L_w within the AOS. When the radiative transfer model necessitates predetermined IOPs in the hydrosol and atmosphere, the Stokes vectors of the upward and downward radiation fields within any stratified AOS are simulated. As illustrated in Figure 1a, the topmost layer is defined by atmospheric Rayleigh scattering, featuring a single-scattering albedo ($\omega$) of 1 and a depolarization factor of 0.0279 [44]. The middle layer consists of tropospheric and maritime atmospheric aerosols, described by Shettle and Fenn, with varying humidities and aerosol optical thicknesses (AOTs). Note that the IOPs of hydrosols are determined by pure water, Chla, nonalgal particles (NAP), and CDOM. First, the scattering coefficients for pure water were obtained from Morel and Prieur [45]. Specifically, the spectral absorption coefficients of pure water were obtained from Lee et al. [46] and Pope and Fry [47]. Second, the absorption and scattering coefficients for Chla and NAP were computed using the bio-optical model proposed by Bricaud et al. [48] and Babin et al. [49], respectively. Third, the absorption coefficient of CDOM (a_g), which is derived from ocean color operational Generalized IOP (GIOP) model products, was obtained with an improved a_g satellite-derived algorithm [50]. Notably, Mie theory was employed to compute the scattering phase matrices of hydrosols via distinct complex refractive indices (1.05 for Chla and 1.165 for NAP) and a Junge size distribution (0.1–50 µm) with an exponent value of 4. Afterward, the scattering phase matrices of hydrosols could be calculated on the basis of their scattering coefficients. Note that the Moderate Resolution Imaging Spectroradiometer (MODIS) operational products from 2002 and 2021, including aerosol optical thicknesses (AOTs), Chla, IOPs, and R_rs, were adopted and processed to derive annually averaged products. Additionally, the global cross-calibrated multi-platform (CCMP) analysis climatology wind speed data at 10 m above the sea surface, with a spatial resolution of 0.25°, were obtained from Remote Sensing Systems (RSS). All of the annually averaged products were merged with a spatial resolution of 3° for the global RT simulations.

The incident solar irradiances at the TOA were sourced from Thuillier et al. [51]. The global RT simulations were conducted at eight wavelengths corresponding to the band settings of the Geostationary Ocean Color Imager (GOCI) satellite, including 412, 443, 490, 555, 660, 680, 745, and 865 nm. The solar zenith angles (${\theta }_{\mathrm{s}}$) varied from 0° to 75° at intervals of 3°. Note that ${\theta }_{\mathrm{s}}$ was set to 22° to generate the independent validation database. The Stokes vectors of L_w were then acquired with azimuth angles (${\theta }_{\mathrm{Az}}$) ranging from 0° to 180° at intervals of 5° and sensor zenith angles (${\theta }_{\mathrm{SZ}}$) ranging from 0° to 90° at intervals of 6°. Here, two PCOART simulations were carried out to quantify the contribution of the radiance emerging from the ocean: one for the background with the atmosphere and a black (totally absorbing) ocean and the other with the atmosphere and nonblack ocean. We subsequently calculated the DOP of L_w according to Equation (1).

$$\mathrm{DOP}={\displaystyle \frac{\sqrt{Q^2+U^2}}{I}}$$

(1)

where I is the total intensity or radiance; here, I is the L_w, and Q and U are used to describe the linearly polarized component of L_w.

2.2. Polarization-Based Retrieval Algorithm Using the Deep Learning Method

A fully connected U-Net (FCUN) model was selected to develop a polarization-based algorithm for inverting oceanic constituents and IOPs by integrating the spectral and polarization features of L_w while effectively addressing inversion challenges for hydrosols at the global scale. The L_w DOP is highly sensitive to changes in oceanic constituents and IOPs, making it more accessible and less susceptible to random instrument noise. U-Net is a lightweight backbone network with few parameters and consists of encoder–decoder paths that can extract multiscale features. Its concatenated structure effectively fuses these features [52,53].

Figure 2 shows a schematic diagram of the FCUN for polarization-based inversion. It employs four fully connected upsampling layers to extract detailed features from the input L_w DOP, thereby expanding the number of features, which can be interpreted as an enhancement of both spectral and polarization information. Seven fully connected downsampling blocks were subsequently used to capture high-level features at a coarser scale. These features were then copied and concatenated with the high-resolution features generated through upsampling, creating multiscale features for polarization-based inversion. The network consists of six levels, with the feature length at each level being half or double that of the previous level, depending on whether it is part of the downsampling or upsampling process.

The model estimates the K parameters of oceanic constituents and IOPs, with the predictions returned as a K×1 vector, where K equals 12. The parameters include Chla, NAP, a_g(443), a, b, a_ph, b_ph, a_NAP, b_NAP, b_b, b_bph, and b_bNAP.

Table 1. Fully connected block diagram of the base architecture.

Block	Operation	Input	Output
FC upsampling	Linear(m,2m), Relu(),	m × 1 vector	(2m) × 1 vector
FC downsampling	Linear(m, m/2), Relu(),	m × 1 vector	(m/2) × 1 vector
Individual	FC Linear(m, n), Relu()	m × 1 vector	n × 1 vector

shows the detailed structure of the FCUN upsampling/downsampling blocks. Linear(in_len, out_len) indicates a linear transform to the input vector. The in_len represents the size of the input sample, and the out_len represents the size of the output sample. In the FC upsampling/downsampling blocks, the linear transform was followed by a ReLU activation. We used five layers containing linear transform and ReLU since multiple layers can increase the nonlinearity to provide compelling quality without adding complexity to the model. To determine the optimal architecture for the FCUN model, the number of layers was varied (2, 4, 8, 16, and 32), resulting in inversion accuracies of 21.01%, 11.66%, 9.76%, 8.76%, and 7.33%, respectively. Balancing training time and accuracy, the FCUN model with 16 layers was ultimately selected. For a comprehensive description of the FCUN model, readers are referred to Chen et al. [52].

The estimation of oceanic constituents and IOPs for learning the end-to-end mapping function F in polarization-based inversion involves minimizing the loss or difference between the FCUN training predictions and the corresponding RT simulation inputs. In this context, the parameters were denoted by θ, the reconstructed FCUN training predictions were expressed as F(I;θ), and the RT simulation inputs were represented by A. Typically, the mean absolute percentage error (MAPE) is employed as an effective metric for quantitatively assessing the accuracy of the reconstructed oceanic constituents and IOPs, which can be calculated as follows:

$$MAPE=\sqrt{{{\displaystyle \sum _{k=1}^K\left(A_k^{(n)}-A_k^{(n)\prime }\right)}}^2}/{\displaystyle \sum _{k=1}^K{A}_k^{(n)}}$$

(2)

where $A$ represents the RT simulation inputs, $A^{\prime }$ represents the FCUN training predictions, and n represents the given number of samples used for the training process. The loss functions can be defined as follows:

$${\mathcal{l}}_S={\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=1}^{N}MAPE}$$

(3)

where N represents the total number of samples used for training, with 5323 × 26 × N_s (samples × ${\theta }_{\mathrm{s}}$ × $N_{\mathrm{s}}$). N_s is the number of observation angles. Finally, the ${\mathcal{l}}_S$ values are used to highlight the consistency of the absolute values between the reconstructed oceanic constitutions and the IOPs and RT simulation inputs.

The paired L_w DOP at four wavelengths (412 nm, 443 nm, 490 nm, and 555 nm) and the corresponding oceanic constitutions and IOPs were randomly selected and used for training and testing from the dataset. The input variables included the L_w DOP at the four specified wavelengths and the cosine values of the observation geometry, with differing solar zenith angles, viewing zenith angles, and azimuth angles. And the inputs and outputs were normalized based on the maximum values before training. The global oceanic constituents and IOPs were retrieved using the FCUN through the following training process. First, the paired L_w DOP data and their corresponding oceanic constituents and IOPs were randomly selected from the dataset for training and testing. Next, the predefined hyperparameters and randomly generated parameters for the FCUN were used to initialize the network model. The network was then trained, with the parameters updated iteratively until the loss function reached its minimum. It is important to note that the retrieval performance of the FCUN tends to decrease as the number of block layers increases. In this process, 70% of the RT simulation data were used for training, 15% for testing, and the remaining 15% for model validation. To assess the FCUN model’s retrieval performance in terms of uncertainty and suitability, we generated independent synthetic global RT simulations with a solar zenith angle of 22°. And then, we changed the input variables of the L_w DOP at different viewing zenith angles, azimuth angles, and polarimetric uncertainties.

3. Results

3.1. Global Uncertainty Distributions of the Retrieval of Oceanic Constitutions and IOPs

Given the uncertainty in the L_w DOP from in situ measurements by the POLWR, three independent synthetic global RT-simulated L_w DOPs were generated by adding random Gaussian noise. The recorded polarimetric uncertainties of the L_w DOP for POLWR from the in situ measurements were 6.09% (in situ) (412 nm), 5.38% (443 nm), 6.67% (490 nm), and 3.86% (555 nm), respectively [29]. Additionally, owing to limitations in the AC algorithm, the inversion errors of the L_w DOP derived from spaceborne polarization satellites are significantly larger than those from in situ measurements. Currently, there is still a lack of globally retrieved L_w DOP products, and the retrieval error distribution characteristics of global L_w DOP products remain unknown. Hence, the recorded polarimetric uncertainties of the L_w DOP were amplified by factors of five and ten and then added to the global RT-simulated L_w DOP database to examine the impact of the uncertainties of the L_w DOP from multiangle polarimetric observations on the global retrieval of oceanic constitutions and IOPs.

Taking Chla concentration retrieval as an example, Figure 3 illustrates the global uncertainty distributions as the number of observation angles of the L_w DOP increased from 1 to 11. The global MAPE values between the FCUN predictions and RT simulation inputs for Chla concentrations at different observation angles were 7.41%, 3.76%, 2.70%, 2.44%, 2.62%, and 1.82%. Additionally, the R² values increased from 0.876 to 0.999, whereas the RMSE decreased from 0.85 mg/m³ to 0.080 mg/m³. As the number of observation angles increased, the global MAPE decreased significantly. Notably, the lowest MAPE values were observed primarily in tropical and subtropical oceans, whereas the highest MAPE values occurred in oligotrophic subtropical gyres and continental shelves, aligning with regions with extreme Chla concentrations.

Figure 4 shows the global distributions of the mean relative error (MRE, %) between the FCUN predictions and RT simulation inputs for Chla at observation angles of 1, 3, 5, 7, 9, and 11. Overall, the globally retrieved Chla concentrations were generally overestimated in tropical and subtropical oligotrophic regions, whereas they were underestimated in nearshore areas with high concentrations. As the observation angle increased, the uncertainties in the polarization-based global oceanic constituents and IOPs retrieval algorithm decreased sharply. Encouragingly, the retrieval uncertainties for Chla in coastal waters also decrease with an increasing observation angle. Additionally, the MRE for oceanic constituents and IOPs between the FCUN predictions and RT simulation inputs approached zero. Specifically, the maximum (minimum) MRE values for Chla retrieval were 47.53% (−45.90%), 34.03% (−31.30%), 18.36% (−21.80%), 22.80% (−21.41%), 14.56% (−21.00%), and 13.01% (−19.73%). Furthermore, an average MRE greater than 0 indicated a global overestimation of 3.41%; in contrast, an average MRE less than 0 reflected a global underestimation of 3.42%.

Figure 5 demonstrates the high fitting accuracy of the FCUN model under varying solar zenith angles. The FCUN model-predicted values and RT simulation inputs closely aligned along the 1:1 line, indicating a small mean bias. A strong correlation existed between the FCUN model predictions and RT simulation inputs. Specifically, the R² values for Chla were 0.968, 0.994, 0.998, 0.999, 0.999, and 1.00. The density plots further confirmed that the majority of the data points closely aligned with the identity line, indicating a robust correspondence for the oceanic constituents and IOPs predicted by the FCUN model, except in cases of extremely high Chla concentrations. Additionally, the global retrieval errors followed a Gaussian distribution, with the mean values of the fitted Gaussian functions being 14.46, 7.85, 5.52, 5.23, 4.81, and 3.97. These results were obtained with the inclusion of recorded polarimetric uncertainties of the L_w DOP from in situ measurements, whereas the errors associated with atmospheric correction algorithms were not considered. Overall, multiangle polarimetric observations showed superior performance in retrieving oceanic constituents and IOPs.

3.2. Angular Distributions in the Uncertainties of the Retrieval of Oceanic Constituents and IOPs

Figure 6 and Figure 7 illustrate the performance of the FCUN model in retrieving global oceanic constituents and IOPs as the L_w DOP with observation angles ranging from 1 to 11 and various levels of random instrument noise. Specifically, the MAPEs for various constituents (Chla, a_g(443), NAP) and IOPs (a, b, a_ph, b_ph, a_NAP, b_NAP, b_b, b_bph, b_bNAP) at 443 nm for the single-view observations are significantly greater than those for the multiangle observations. The MAPEs increased as the sensor azimuth angles ranged from 0° to 180° (see

Table 2. Global mean MAPEs (%) of the FCUN-retrieved ocean constituents and IOPs under different sensor azimuth angles and instrument noises with an observation angle of 3.

	Chla	NAP	ag	a	b	aph	bph	aNAP	bNAP	bb	bbph	bbNAP
0°	3.76	1.38	6.94	0.62	1.06	2.39	2.76	1.36	1.37	1.05	2.76	1.36
30°	4.46	1.53	7.27	0.59	1.00	2.69	3.12	1.52	1.52	1.15	3.18	1.52
60°	4.65	1.78	8.58	0.73	1.09	2.90	3.58	1.77	1.77	1.34	3.48	1.79
90°	4.38	1.46	7.50	0.58	1.01	2.73	3.20	1.44	1.45	1.10	3.17	1.44
120°	4.32	1.37	7.26	0.66	0.87	2.46	2.86	1.37	1.36	1.04	2.89	1.37
150°	3.97	1.18	6.83	0.54	0.80	2.34	2.85	1.18	1.20	0.89	2.92	1.18
180°	4.23	1.14	7.12	0.89	0.87	2.64	3.27	1.15	1.15	0.86	3.20	1.15
0°	6.07	2.31	11.25	1.09	1.29	3.84	4.44	2.30	2.30	1.73	4.43	2.30
30°	6.76	2.13	12.09	1.57	1.27	4.15	4.90	2.13	2.11	1.58	4.92	2.11
60°	7.33	2.46	12.35	1.67	1.36	4.61	5.32	2.46	2.47	1.82	5.31	2.47
90°	6.92	2.08	11.70	1.57	1.28	4.46	5.16	2.09	2.11	1.54	5.06	2.09
120°	6.77	2.19	12.17	1.40	1.20	4.21	4.92	2.19	2.19	1.60	4.93	2.18
150°	6.38	1.96	12.83	1.43	1.14	3.96	4.93	1.97	1.94	1.42	4.78	1.95
180°	5.87	1.72	11.65	1.30	1.15	3.74	4.37	1.72	1.71	1.28	4.35	1.71
0°	6.61	2.30	9.42	1.14	1.62	4.08	4.88	2.32	2.32	1.74	4.69	2.30
30°	7.37	2.42	9.34	1.45	1.65	4.41	5.17	2.42	2.43	1.82	5.12	2.42
60°	7.93	2.61	10.09	1.77	1.81	4.71	5.55	2.61	2.62	1.97	5.47	2.60
90°	7.24	2.46	10.62	1.55	1.69	4.56	5.36	2.45	2.45	1.83	5.37	2.46
120°	7.49	2.30	9.88	1.45	1.59	4.46	5.28	2.30	2.29	1.71	5.18	2.30
150°	7.05	2.02	10.04	1.51	1.63	4.43	5.10	2.03	2.02	1.52	5.14	2.03
180°	6.81	1.84	10.10	1.36	1.48	4.26	4.96	1.84	1.82	1.37	4.92	1.84

). Notably, the MAPEs at sensor azimuth angles of 0° and 30° were significantly lower than those at other azimuth angles for single-view observations. The polarization-based retrieval algorithm performed well at 0° and 30° azimuth angles. In contrast, as the number of sensor viewing angles increased, the variation in MAPEs with the sensor azimuth angle changed, and the variation gradually diminished. The MAPEs reached their maximum at a sensor azimuth angle of 60° and then gradually decreased as the azimuth angle increased. Moreover, multiangle observations greatly improved the retrieval of oceanic constituents and IOPs, with the global mean MAPEs decreasing as the number of viewing angles increased. Specifically, the global mean MAPEs for oceanic constituents and IOPs at observation angles ranging from 1° to 11° were 27.56%, 2.39%, 1.79%, 1.51%, 1.32%, and 1.20%, respectively.

The MAPEs for Chla, a_g(443), a_ph, b_ph, and b_bph were approximately twice as high as those for the remaining parameters. The retrieval algorithm generally performed better in estimating inorganic suspended particles and their IOPs, such as NAP, a_NAP, b_NAP, b_b, and b_bNAP. This is primarily because the polarization of L_w is strongly correlated with inorganic particles, which are refractive in coastal waters. Importantly, the enhancement effects of multiangle observations on the retrieval of oceanic constituents and IOPs varied. The ratios of MAPEs for the retrieval of oceanic constituents (Chla, NAP, a_g(443)) and IOPs (a, b, b_b, a_ph, a_g, a_NAP, b_ph, b_NAP, b_bph, b_bNAP) for three observation angles relative to single-view observations were 0.093, 0.077, 0.174, 0.027, 0.043, 0.078, 0.084, 0.076, 0.077, 0.066, 0.084, and 0.077, respectively. Therefore, the enhancement effects of multiangle observations for inorganic suspended particles and their IOPs are promising. Special attention should be given to the retrieval of a_g(443), which exhibited a MAPE of 42.93%. The global mean MAPEs of a_g(443) for observation angles ranging from 1° to 11° were 42.93%, 9.93%, 8.70%, 7.22%, 6.44%, and 6.03%, respectively. In fact, the enhancement effects of multiangle observations for the retrieval of a_g(443) were weaker than those for other parameters. Compared with the other parameters, the retrieval of a_g(443) via multiangle observations yielded higher MAPEs as the observation angle increased. Additionally, the MAPEs of a_g(443) decreased as the sensor azimuth angles ranged from 90° to 180°. This is primarily due to the globally low a_g(443) values having a relatively weak influence on L_w polarization at shorter wavelengths.

3.3. Influence of Lw DOP Uncertainties on the Retrieval of Oceanic Constituents and IOPs

To quantify the influence of L_w DOP uncertainties on the retrieval of oceanic constituents and IOPs, the recorded polarimetric uncertainties of the L_w DOP were amplified by factors of five and ten, and subsequently incorporated into the global RT simulation databases. Figure 8 shows the MREs of Chla retrieval under various random instrument noise conditions. Notably, the MREs decreased as the number of observation angles increased. The errors in the retrieval of oceanic constituents and IOPs under single-view observation conditions drastically increased as the uncertainties in the L_w DOP increased. Additionally, the regions exhibiting the largest positive and negative MREs gradually expanded, and their values also increased. These results indicated that the uncertainties in the L_w DOP significantly impacted the retrieval of oceanic constituents and IOPs. Taking Chla retrieval as an example, the MAPEs for the single-view observations with a sensor azimuth angle of 0° were 7.41%, 38.84%, and 39.02%. Furthermore, the global MAPEs for the recorded polarimetric uncertainties of the L_w DOP, as the observation angle increased from 1 to 11, were 45.80%, 4.25%, 3.11%, 2.73%, 2.46%, and 2.23%, respectively. After various random instrument noises were added, the global MAPEs for Chla were 46.56% (46.91%), 6.59% (7.21%), 5.21% (5.79%), 4.72% (4.98%), 3.99% (4.52%), and 3.64% (4.03%), respectively.

The global MAPEs for the retrieval of oceanic constituents and IOPs significantly decreased as the number of observation angles increased. As shown in Figure 9, Figure 10 and Figure 11, the global MAPEs for Chla at all sensor azimuth angles with three observation angles were 4.46%, 6.76%, and 9.37%, respectively. The corresponding R² (RMSE) values were 0.993 (0.402 mg/m³), 0.986 (0.555 mg/m³), and 0.981 (0.644 mg/m³), respectively. Notably, in the equatorial and polar regions, where oceanic dynamic processes are most active, the polarization-based retrieval algorithm tended to overestimate Chla, whereas Chla retrieval was underestimated in the Southern Ocean. As the number of observation angles increased, the white areas in the global MRE distribution gradually expanded. However, the MAPEs did not significantly decrease once the observation angle exceeded 5° (Figure 9), and the enhancement capability of multiangle observations gradually weakened. The L_w DOP is highly sensitive to changes in ocean constituents and IOPs, is more achievable, and is less influenced by random instrument noise. Moreover, as a dimensionless ratio, it is unaffected by certain systematic errors, providing higher measurement precision. The uncertainties in the retrieval of oceanic constituents and IOPs decreased significantly as the observation angle increased. Under multiangle polarimetric observation conditions, the impact of errors in the L_w DOP on the retrieval uncertainties gradually decreases with increasing number of observation angles. Therefore, multiangle polarimetric observations can effectively suppress or balance the impact of L_w DOP uncertainties on the retrieval of oceanic constituents and IOPs.

4. Discussion

With the increasing availability of multiangle polarimetric observations from various platforms, such as satellites, aircraft, and ground-based systems, significant progress has been made in the development of multiangle quantitative remote sensing models [5,37]. These advancements have enhanced preprocessing capabilities for multiangle remote sensing data and improved the comprehensive application potential of multisource data. Unlike single-angle remote sensing, multiangle polarimetric remote sensing introduces an additional dimension of viewing angles, enabling the acquisition of the micro-optical properties of oceanic particles and addressing challenges that traditional remote sensing techniques cannot resolve [6]. This makes multiangle polarimetry a crucial tool in quantitative ocean color remote sensing research. Furthermore, it complements the spatial, temporal, and spectral dimensions of information, offering both strong theoretical foundations and practical applications in ocean color remote sensing [5].

In this study, the use of multiangle polarization of L_w to retrieve oceanic constituents and IOPs was examined for global-scale applications, incorporating sensor observation directional variations and the polarimetric uncertainties of the L_w DOP. The influence of these uncertainties on the retrieval of oceanic constituents and IOPs was also investigated. The global MAPE values between the FCUN predictions and RT simulation inputs for Chla concentrations under varying observation angles were 7.41%, 3.76%, 2.70%, 2.44%, 2.62%, and 1.82% (Figure 3). Additionally, the R² values increased from 0.876 to 0.999, whereas the RMSE decreased from 0.85 mg/m³ to 0.080 mg/m³. As the observation angles increased, the global MAPEs decreased significantly, demonstrating that the retrieval of oceanic constituents and IOPs is strongly dependent on satellite observation geometries and the number of observation angles. The broader the range of scattering angles, the more comprehensive the oceanic constituent information retrieved from multiangle observations.

Multiangle polarimetric remote sensing is widely applied in the retrieval of various environmental parameters, including albedo, vegetation characteristics, cloud properties, aerosol optical properties, and sea ice texture [9]. However, the bidirectional characteristics of L_w are often overlooked, and the multiangle polarization features of L_w are not typically incorporated into traditional ocean color remote sensing algorithms. The multiangle polarization of L_w is influenced by the microphysical properties of hydrosols, particularly the real part of the bulk refractive index. The uncertainties in the polarization-based global retrieval algorithm for oceanic constituents and IOPs decreased sharply as the number of observation angles increased. The global retrieval errors followed a Gaussian distribution, with the mean values of the fitted Gaussian functions approaching zero (Figure 5). Consequently, multiangle polarimetric observations of L_w significantly increase the accuracy of retrieving oceanic constituents and IOPs.

Additionally, the angular distributions of uncertainties in the retrieval of oceanic constituents and IOPs were examined with sensor azimuth angles ranging from 0° to 180° at 30° intervals. The polarization-based retrieval algorithm demonstrated strong performance for sensor azimuth angles of 0° and 30° (Figure 6). Generally, the MAPEs for oceanic constituents and IOP retrievals decreased as the sensor azimuth angles ranged from 90° to 180°. RT simulations indicated that the scattered light directions exhibiting the maximum DOP typically lie beyond the concentric viewing circles associated with backscattering directions for sensor zenith angles greater than 15°. For these geometries, the L_w DOP was highly sensitive to oceanic constituents and IOPs, particularly inorganic particles, which are more effective at inducing greater depolarization than organic particles are.

Monte Carlo simulations using the T-matrix method revealed that the M₁₂ element of the Mueller matrices for organic particles had a higher absolute value than that for inorganic particles in the backscattering direction [1]. As a result, the L_w DOP exhibited greater sensitivity to oceanic constituents and IOPs in the backscattering direction. Furthermore, the variation in MAPEs with sensor azimuth angle decreased in angular distributions as the sensor viewing angle increased (Figure 7 and Table 1). By covering a broader range of scattering angles, multiangle observations provide a richer set of bidirectional polarimetric characteristics, significantly enhancing the retrieval of oceanic constituents and IOPs. The angular distributions also offer more flexibility for angle selection in multiangle polarimetric observations, extending beyond the limitations of the solar principal plane.

Subsequently, the influence of L_w DOP uncertainties on the retrieval of oceanic constituents and IOPs was further analyzed. The global MAPEs for the retrieval of oceanic constituents and IOPs significantly decreased compared with the single-view observations. However, the MAPEs did not substantially decrease with increasing L_w DOP uncertainty for multiangle observations when the observation angles exceed 5. The uncertainties in the L_w DOP were found to be the primary factors influencing the retrieval uncertainties of oceanic constituents and IOPs. Importantly, multiangle polarimetric observations can mitigate or balance the impact of these uncertainties on the retrieval accuracy.

Single-sensor observations of the polarization characteristics of water-leaving radiation are often limited by the sampling angle, which restricts the accuracy of retrievals. By integrating multiangle observations, it becomes possible to capture the bidirectional polarization characteristics of water-leaving radiation. Integrating multiangle observations is important for retrieving the bidirectional polarization characteristics of water-leaving radiation, which can improve the distribution and quantity of angular sampling from multiangle observations. The combination of polarization and angle of L_w can enhance the precision of oceanic constituent and IOP retrieval (Figure 8 and Figure 9). However, the angular information of the L_w DOP is not fully utilized for the current inversion algorithm.

Those traditional inversion algorithms rely on certain a priori knowledge, and errors in prior knowledge can further affect the retrieval uncertainty of oceanic constituents and IOPs [54]. As a result, those algorithms tend to exhibit considerable uncertainties, especially in complex coastal waters. Taking the retrieval of aerosols as an example, the multiparameter optimization framework was typically employed to jointly estimate these parameters via both radiance intensity and polarization observations from multiangle observations [55,56]. The above RT simulations indicate that the number of viewing angles and their corresponding geometries can cause significant changes in the amount of observational information used in multiangle polarimetric observation mode and can effectively increase the retrieval accuracy under various levels of L_w DOP uncertainties.

5. Conclusions

In this study, we analyzed the global uncertainty distributions of the polarization-based inversion algorithm for retrieving oceanic constituents and IOPs on the basis of global RT simulations. First, we compared the global uncertainties in retrieval between single-angle and multiangle polarimetric observations. It was observed that globally retrieved Chla concentrations tended to be overestimated in tropical and subtropical oligotrophic regions, whereas they were underestimated in nearshore areas with high concentrations. Specifically, the global MAPE values between the FCUN predictions and RT simulation inputs for Chla concentrations under different observation angles were 7.41%, 3.76%, 2.70%, 2.44%, 2.62%, and 1.82%, respectively. As the observation angle increased, the global MAPEs for various oceanic constituents and IOPs decreased significantly. Second, we analyzed the global uncertainties of the inversion algorithm under different sensor azimuth angles. The MAPEs at sensor azimuth angles of 0° and 30° were significantly lower than those at other azimuth angles for single-view observations. As the sensor viewing angle increased, the variation in MAPEs with respect to the sensor azimuth angle changed, and the influence of the azimuth angle gradually diminished. Finally, we examined the impact of L_w DOP uncertainties on the retrieval of oceanic constituents and IOPs. The uncertainties in the L_w DOP had a significant effect on the retrieval of oceanic constituents and IOPs. However, the MAPEs did not decrease significantly with increasing observation angle once the number of observation angles exceeded 5.

Overall, the global uncertainty distributions revealed that multiangle observations can cover a broader range of scattering angles, which can possess a richer set of bidirectional polarimetric characteristics, and multiangle polarimetric observations greatly enhance the retrieval of oceanic constituents and IOPs. The angular distributions provided more options for angle selection in multiangle polarimetric observations, which are not limited solely to the solar principal plane. In summary, multiangle polarimetric observations can suppress or balance the impact of the uncertainties of the L_w DOP on the retrieval of oceanic constituents and IOPs. The impacts of imperfectly polarized atmospheric correction algorithms and instrumental noise on the retrieval of oceanic constituents and IOPs can be eliminated by employing multiangle polarimetric observations.