The SMART-P Algorithm for Aerosol and Ocean Properties Part 1: Algorithm Theoretical Basis and Expected Accuracy of PolCube Aerosol Products

Highlights

What are the main findings?

• Simulated PolCube-polarized radiances are highly sensitive to fine-dominated aerosols (e.g., sulfate and smoke), primarily due to distinct polarized radiance shapes at backscattering angles.
• The theoretical error analysis based on the Bayesian theorem revealed that the SMART-P algorithm, using PolCube-polarized radiances, can retrieve τ_aer, n, and F_num with sufficient sensitivity compared with radiance-only measurements.
• Increased aerosol loading significantly enhances sensitivity to k, while reducing information content for τ_aer due to interference from enhanced absorption.

What are the implications of the main findings?

• PolCube aerosol products retrieved by SMART-P provide insights into aerosol microphysical properties related to particle composition.
• The SMART-P algorithm will support the operation of the PolCube onboard the BusanSat-B from 2026, providing aerosol optical properties for climate and air quality studies.

Abstract

The SMART-P (Spectral Measurements for Atmospheric Radiative Transfer–Polarimeter) algorithm was developed to retrieve aerosol and ocean parameters from PolCube measurements. The PolCube is a multi-angular polarimeter (MAP) aboard the BusanSat-B CubeSat scheduled for launch in 2026, which measures polarized radiances at 410, 555, 670, and 865 nm from four viewing angles. This study presents the theoretical basis of the algorithm and conducts a sensitivity analysis of aerosol inversions over the ocean processed by SMART-P under the expected measurement conditions for PolCube observations. The results indicate that the degree of linear polarization (DoLP) significantly increases the information content of the real part of the refractive index and of the fine-mode particle-size parameters relative to radiance-only measurements. Enhanced measurement sensitivity enables more accurate retrieval of fine-dominated aerosol properties, such as smoke and sulfate. The sensitivity analysis also shows that the ocean surface reflectivity is the most critical forward-model parameter affecting aerosol-property retrievals. The SMART-P algorithm will support the BusanSat-B mission to understand the role of aerosol particles in the climate system and air quality.

1. Introduction

As aerosol sources become more diverse and their lifetimes vary, their chemical and physical compositions have become increasingly heterogeneous, thereby complicating the understanding of the Earth system [1,2]. In particular, reliable information on the atmosphere (e.g., meteorology and composition) and the ocean (e.g., chlorophyll concentration and turbidity) globally is essential for understanding their interaction [3]. For example, when dust aerosols, composed of minerals and iron, are transported into the ocean, local biogeochemical processes (e.g., chlorophyll) change. Gao et al. (2001) reported that the Oceans in the Northern Hemisphere are seasonally affected by atmospheric particles (e.g., from the Saharan Desert, Africa) [4]. In contrast, the impact is smaller in the Southern Hemisphere due to lower aerosol concentrations [4]. In addition, aerosols themselves play a critical role in our understanding of climate systems, particularly through their influence on direct and indirect radiative forcing, which is quantified by their physicochemical properties, including particle-size distribution (PSD), morphology, and complex refractive indices [5,6].

Satellite data are essential, as they provide information on aerosols and the ocean over the globe [5,7]. Previous satellite-based aerosol remote sensing techniques utilized reflected solar radiances measured by various multi-channel sensors such as Moderate-Resolution Imaging Spectroradiometers (MODIS) and Visible Infrared Imaging Radiometer Suite (VIIRS). The aerosol-retrieval heritage from the Deep Blue [8] and Dark Target [9] algorithms has been applied over both land and ocean. They have successfully provided reliable long-term records on aerosol burden for climate and air quality research [10,11]. In addition, continuous efforts have focused on improving the algorithm not only to reduce uncertainties, but also to extend aerosol parameters (i.e., Aerosol Layer Height, ALH, and Single Scattering Albedo, ω₀) by combining multiple satellite measurements [12].

Despite considerable efforts, aerosol remote sensing has limitations in accurately retrieving microphysical aerosol properties from radiance measurements. Even with observations from multiple angles (e.g., Multi-angle Imaging Spectro Radiometer, MISR), measurement sensitivity remains insufficient to derive information on major aerosol optical properties [13,14]. As an additional source of information, linear polarization can provide unique insights into aerosol properties. Unlike radiance, polarization shows distinct scattering phase function characteristics and is more sensitive to aerosol particle size, shape, and complex refractive indices [15,16,17,18]. The benefits of multi-angular polarimeter (MAP) measurements in aerosol property retrievals have been reported in previous studies [14,19,20,21]. Hasekamp (2010) demonstrated that the reduction in retrieval errors of aerosol properties with the number of measurement types and viewing angles [19]. Knobelspiesse and Nag (2018) further showed that information content increases when polarization is used in aerosol retrieval over the ocean [20].

Over the years, various MAPs have been developed. The use of polarimeters began with the Polarization and Directionality of the Earth’s Reflectances (POLDER) generations, which operated from 1996 to 2013 [22]. This heritage continued with the Aerosol Polarimetry Sensor (APS) on the National Aeronautics and Space Administration (NASA)’s Glory mission [17]. After a gap in the launch of polarimetric instruments, the Directional Polarimetric Camera (DPC) series was launched by China in 2018 and 2021 [23]. Then, two passive multi-angular polarimeters were launched in February 2024 as part of the Plankton, Aerosol, Cloud, ocean Ecosystem (PACE) mission [21]. These include the Hyper-Angular Rainbow Polarimeter 2 (HARP2) from the University of Maryland, Baltimore County (UMBC), and the Spectro-polarimeter for Planetary Exploration one (SPEXone) from the Space Research Organization Netherlands (SRON) [24]. HARP2 is influenced by the AirHARP and CubeSat HARP instruments [25,26]. The Multi-Angle Imager for Aerosols (MAIA), which operates in targeting mode, is scheduled for launch in the near future [27].

Alongside these developments, the PolCube polarimeter [28], which is onboard the BusanSat-B, was developed through international collaboration and is scheduled for launch in 2026. PolCube’s unique feature is its four viewing angles, enabled by two cameras that support dual-angle observations and operate in two modes: nadir and forward-looking [29]. It measures radiance and linear polarization across four channels in the visible and near-infrared (NIR) spectrum (i.e., 410, 555, 670, and 865 nm). Before its space deployment, an airborne campaign was conducted with the airborne version of PolCube (hereafter Air–PolCube). Air–PolCube was flown over the two major target regions (i.e., Busan and Incheon, South Korea) and showed qualitative agreement with the VIIRS Deep Blue aerosol product [30]. We anticipate that these datasets will enhance synergy with current and future satellite Earth observation missions in polar and geostationary orbits, including VIIRS, MODIS, and the Geostationary Environment Monitoring Spectrometer (GEMS).

Several algorithms have been developed to retrieve aerosol properties from MAP observations, each tailored to specific instrument characteristics. The Generalized Retrieval for Aerosol and Surface Properties (GRASP), designed for multi-pixel retrieval, has been successfully applied to POLDER3 [31,32,33]. SRON developed the Remote sensing of Trace gas and Aerosol Products (RemoTAP) algorithm, adapted for SPEXone aerosol retrievals [34,35]. This algorithm employs a coupled-retrieval approach integrated with a deep neural network to replace the radiative transfer model in retrieval. It is designed to handle large volumes of data from its multispectral, multi-angle observations. Initial retrieval results demonstrated promising performance [36]. NASA’s Goddard Space Flight Center (GSFC) has developed the Fast Multi-Angular Polarimetric Ocean coLor (FastMAPOL) algorithm, based on a neural network for HARP2 observations [37,38]. The Microphysical Aerosol Properties from Polarimetry (MAPP) algorithm at NASA Langley Research Center (LaRC) employs a joint-retrieval approach, using data from both the HARP2 and SPEXone sensors [39].

This study aims to establish the theoretical basis for the SMART-P (Spectral Measurements for Atmospheric Radiative Transfer–Polarimeter) algorithm, developed to process PolCube measurements operated by Busan Metropolitan City, South Korea. The SMART algorithm, initially developed by NASA’s GSFC, is based on the optimal-estimation method (OEM) [6,40], which aims to extract maximum information from measurements with minimal assumptions [6]. In this study, we assessed the expected accuracy of PolCube aerosol products using theoretical experiments, in preparation for operations in 2026. Section 2 outlines the theoretical background and experimental methods. In Section 3, we demonstrate the expected accuracy of the PolCube aerosol products using the OEM. Section 4 summarizes the overall results.

2. Methods

2.1. Generations of PolCube Synthetic Polarized Radiances

The PolCube is a 12U CubeSat, multi-angle, multi-band push-broom imaging polarimeter. It is designed to measure radiance and linear polarization at four bands: 410, 555, 670, and 865 nm, each with a full width at half maximum (FWHM) of 20 nm. Polarization states are collected at I_0° and I_90° for the 410, 555, and 865 nm bands, and at I_0°, I_60°, and I_120° for the 670 nm band. By leveraging the spacecraft’s attitude control system, PolCube can capture data from four distinct viewing angles. PolCube’s required radiometric and degree of linear polarization (DoLP) uncertainties are 2% and 0.5%, respectively. Before its launch, an Engineering Qualification Model onboard an airplane (hereafter Air–PolCube), flew over two major port cities, Busan and Incheon, in South Korea from 8 to 10 May 2024. Figure 1 shows an RGB image and a sample of raw data captured during the flight over the New Port of Busan on 10 May 2024 (T 11:44 LST), with annotations on the right describing the measurement types and wavelengths. The right panel in Figure 1 illustrates how the instrument captures polarization across multiple spectral channels. The thin stripes in the raw data result from the spacing between filters for different wavelengths and polarization angles and will be removed during Level 1b data production.

Figure 1. An example of measurements from the Air–PolCube, which flew over the New Port of Busan on 10 May 2024 (T 11:44 LST). (Left panel) RGB image; (Right panel) raw data with annotations for wavelengths (410, 555, 670, and 865 nm) and polarization angles.

Stokes vectors (i.e., $I$, $Q$, and $U$) and their weighting functions are calculated using the linearized pseudo-spherical vector Discrete Ordinate Radiative Transfer (VLIDORT) at carefully selected observation geometries for the sensitivity analysis [41,42,43]. The viewing zenith angles (57.0°, 52.0°, 0.0°, and 5.0°) were carefully selected, given the CubeSat’s limited capabilities. 0.0° and 5.0° represent typical Nadir observations, whereas 57.0° and 52.0° were selected to measure the maximum distinct scattering angles from Nadir while avoiding extreme geometry [29]. The solar zenith angle (SZA) at the overpass time in a sun-synchronous orbit typically ranges from 10° to 60° over the Korean Peninsula, depending on the season. For the sensitivity analysis, we calculated the radiances at SZA of 30° and four relative azimuth angles (RAAs) to represent nominal observation geometries, as listed in Table 1. Note that the overall results showed no meaningful differences across SZAs (10–70°). We assumed an aerosol optical depth (τ_aer) of 0.5 at 555 nm and a wind speed of 5.0 m/s, typical values in East Asia. The selection of τ_aer enhances the sensitivity of simulated radiances to aerosol properties, enabling an evaluation of retrieval sensitivity under significant aerosol loading (e.g., polluted conditions). While the primary results of this study focus on τ_aer of 0.5, τ_aer values from 0.1 to 1.0 (i.e., 0.1, 0.3, and 1.0) were additionally evaluated to account for the diverse aerosol loadings over the ocean. Since the SMART-P algorithm primarily targets aerosol and ocean retrievals, we adopted the Cox–Munk bidirectional reflectance distribution function (BRDF) model for surface reflectance [44]. The simulation conditions are summarized in Table 1.

Table 1. Summary of PolCube Geometries and Input Parameters for Synthetic Polarized Radiance Calculations.

Specifications
Wavelengths	410, 555, 670, 865 nm (All bands polarized)
Radiometric & DoLP uncertainty	2.0% for radiometric 0.5% for DoLP
SZA (°)	30.0
VZA (°)	57.0, 52.0, 0.0, 5.0
RAA (°)	0.0, 45.0, 90.0, 135.0
τaer (555 nm)	0.5
Wind speed	5.0 m/s

To represent various aerosol types, we adopted PSD parameters for sulfate, smoke, and dust from the OMAERUV algorithm, which is based on AERONET climatology [6,45]. The complex refractive indices were determined based on the OPAC database and Dubovik et al. (2002) to ensure realistic values [46,47]. The aerosol optical properties, including ω₀, are summarized in Table 2. The lognormal number size distribution is given by:

$$n_i\left(r\right) = \sum _i{\displaystyle \frac{N_i}{\sqrt{2\mathsf{\pi }}\ln {\mathsf{\sigma }}_i}}{\displaystyle \frac{1}{r}}\exp \left[-{\displaystyle \frac{{\left(\ln r -\ln r_i\right)}^2}{2{\ln }^2\left({\mathsf{\sigma }}_i\right)}}\right]$$

(1)

Table 2. Summary of aerosol models used in this study, including the particle size distribution parameters (r_f, σ_f, r_c, σ_c, and F_num), real (n) and imaginary (k) parts of the complex refractive indices, and single-scattering albedo (ω₀) at 555 nm.

Aerosol Models	${\mathit{r}}_{\mathbf{f}}$	${\mathsf{\sigma }}_{\mathbf{f}}$	${\mathit{r}}_{\mathbf{c}}$	${\mathsf{\sigma }}_{\mathbf{c}}$	${\mathit{F}}_{\mathbf{num}}$	$\mathit{n}$	$\mathit{k}$	ω0 (555 nm)
Sulfate	0.088	1.499	0.509	2.160	0.999596	1.40	0.0001	0.99
Smoke	0.080	1.492	0.705	2.075	0.999795	1.50	0.0100	0.92
Dust	0.052	1.697	0.670	1.806	0.995650	1.55	0.0020	0.95

We assumed the aerosols are spherical particles with a bimodal PSD characterized by the geometric mean radius (r_i), standard deviation (${\mathsf{\sigma }}_i$), and the number fraction (N_i). To describe the bimodal distribution, we define i as either f (fine mode) or c (coarse mode). Therefore, the total distribution can be written as n(r) = n_f(r) + n_c(r). N_f is the number fine-mode fraction (F_num). The last two parameters in Table 2 are the real (n) and imaginary (k) parts of the complex refractive index.

The intensity phase function (P₁₁) and the polarized phase function (−P₁₂/P₁₁) show distinct patterns for different aerosol types, particularly in the backward-scattering direction. Figure 2 shows examples of P₁₁ and −P₁₂/P₁₁ for sulfate, smoke (both fine-dominated; green and red, respectively), and dust (coarse-dominated; blue) aerosols at PolCube wavelengths: (a) 410, (b) 555, (c) 670, and (d) 865 nm. The −P₁₂/P₁₁ exhibits different scattering shapes across aerosol types and wavelengths, particularly at scattering angles between 60° and 180°. Because sun-synchronous satellites typically measure reflected radiance within this scattering angle, the −P₁₂/P₁₁ provides additional information for satellite-based aerosol remote sensing.

Figure 2. Examples of the intensity phase function (P₁₁), and the polarized phase function (−P₁₂/P₁₁) for sulfate (green), smoke (red), and dust (blue) aerosol at PolCube wavelengths: (a) 410, (b) 555, (c) 670, and (d) 865 nm. τ_aer for three aerosol models is assumed to be 0.5 at 555 nm.

2.2. Error Characterization Based on the Optimal Estimation Method

The measurement types and the number of viewing angles are strongly related to the information content of aerosol retrieval parameters, thereby affecting their uncertainties. An OEM-based algorithm with a well-tested linearized radiative transfer model provides reliable error estimates and analyses, which are valuable for studies using the products [6,48]. In this study, the sensitivity of the state vectors utilizing the PolCube measurements is analyzed based on the OEM [40]. While NASA/LaRC has performed a preliminary analysis of PolCube’s capabilities to retrieve aerosol and ocean parameters simultaneously using the MAPP algorithm [29], this study focuses on assessing the expected accuracy of the aerosol products to be provided by Busan Metropolitan City using the SMART-P package. We defined the state vectors (x) as Equation (2), which include bimodal PSD (i.e., r_f, σ_f, r_c, σ_c, and F_num), ALH (i.e., z_p and h), and complex refractive indices for both modes (i.e., n_f, n_c, k_f, and k_c). The parameters z_p and h denote the peak height and vertical dispersion parameter of the Gaussian aerosol extinction profile, respectively [42,43].

$$\mathbf{x} =[ r_{\mathrm{f}} {\mathsf{\sigma }}_{\mathrm{f}} r_{\mathrm{c}} {\mathsf{\sigma }}_{\mathrm{c}} F_{\mathrm{num}} z_p h {\mathsf{\tau }}_{\mathrm{aer}}(\mathsf{\lambda }) n_{\mathrm{f}}(\mathsf{\lambda }) k_{\mathrm{f}}(\mathsf{\lambda }) n_{\mathrm{c}}(\mathsf{\lambda }) k_{\mathrm{c}}(\mathsf{\lambda }){]}^{\mathrm{T}}$$

(2)

Based on the Bayesian theorem, the optimal solution at each iteration step is determined by the Levenberg–Marquardt method [40] as expressed in Equation (3). The state vector (x_i) is updated to x_i+1 in each iteration step with respect to the weighting function (K). The covariance matrix of measurement errors (S_ϵ) is diagonal, indicating that there is no cross-correlation between the measurement errors. The damping factor (γ) controls the step size between gradient descent and Gauss–Newton methods during optimization. A priori mean state (x_a) and the error covariance (S_a) are calculated by AERONET climatology [49,50] at Yonsei University in Seoul from 2011 to 2023. An example of a priori error covariance for SMART-P is illustrated in Figure 3. Wind speed is the most important forward-model parameter (b) and will be adopted from the European Centre for Medium-Range Weather Forecasts reanalysis 5 (ERA5) [51]. We assumed the uncertainty of approximately 0.5 m/s in this study.

$${\mathbf{x}}_{\mathit{i}+1}={\mathbf{x}}_{\mathit{i}}+{ [{\mathbf{K}}^{\mathbf{T}}{\mathbf{S}}_{\mathit{ϵ}}^{-1}\mathbf{K}+{(1+\mathsf{\gamma }){\mathbf{S}}_{\mathit{a}}}^{-1}]}^{-1}{\{\mathbf{K}}^{\mathbf{T}}{\mathbf{S}}_{\mathit{ϵ}}^{-1}\left[\mathbf{y}-\mathbf{F}\left({\mathbf{x}}_{\mathit{i}},\mathit{b}\right)\right]-{\mathbf{S}}_{\mathit{a}}^{-1}[{\mathbf{x}}_{\mathit{i}}-{\mathbf{x}}_{\mathit{a}}]\}$$

(3)

Figure 3. An example of the a priori covariance matrix for PolCube measurement derived from AERONET climatology at Yonsei University from 2011 to 2023. The indices 1–7 are non-spectral parameters, including PSD (1–5) and aerosol layer height (ALH; 6–7) parameters. The larger indices present aerosol optical depth (τ_aer), real (n), and imaginary (k) parts of the complex refractive indices at PolCube wavelengths.

Equation (4) defines the normalized cost function (χ), which is minimized to find the optimal solution. The first term represents both the radiance and DoLP measurement constraints, whereas the second term is the a priori constraints. N_m and N_a are the number of measurements and a priori data, respectively.

$${\mathsf{\chi }}^2=\frac{1}{N_m}\Vert {\mathbf{S}}_{ϵ}^{-{\displaystyle \frac{1}{2}}}\left(\mathbf{y} -\mathbf{Kx}\right){\Vert }_2^2+{\displaystyle \frac{1}{N_a}}\Vert {\mathbf{S}}_a^{-{\displaystyle \frac{1}{2}}}\left(\mathbf{x} -{ \mathrm{x}}_a\right){\Vert }_2^2$$

(4)

In this study, we used two scenarios for measurements (y): (1) using only radiance, and (2) combining radiance and DoLP (hereafter L + DoLP; Equation (5)). The radiance data are normalized by solar irradiance. DoLP is defined as $\sqrt{Q^2+U^2}/I$, as the circular polarization is negligible in nature [52]. We aim to retrieve optimum aerosol optical properties as possible using the second scenario (i.e., L + DoLP). Figure 4 illustrates an example of the K matrix generated using Equation (5).

$$\mathbf{y}=[\mathrm{Normalized} \mathrm{Radiance}(\mathrm{\lambda })\hspace{1em}\mathrm{DoLP}(\mathrm{\lambda }){]}^{\mathrm{T}}$$

(5)

Figure 4. An example of the weighting function (K) matrix using both normalized radiance and degree of linear polarization (DoLP). This calculation uses the sulfate aerosol model with specifications provided in Table 1.

VLIDORT provides the Ks for the $I$, $Q$, and $U$ of the Stokes vector to x. The K for DoLP is calculated as follows:

$${\displaystyle \frac{\partial \mathrm{DoLP}}{\partial {\mathrm{x}}_i}}={\displaystyle \frac{Q\frac{\partial Q}{\partial {\mathrm{x}}_i} + U\frac{\partial U}{\partial {\mathrm{x}}_i}}{I \sqrt{Q^2+U^2}}}-{\displaystyle \frac{\mathrm{DoLP} \frac{\partial \mathrm{I}}{\partial x_i}}{I}}$$

(6)

Averaging kernel matrix (A; Equation (7)), which provides the sensitivity of retrievals to true states, is calculated with measurement gain matrix (G_y) defined as G_y $={\displaystyle \frac{\partial \widehat{\mathbf{x}}}{\partial \mathbf{y}}}$, and a weighting function.

$$\mathbf{A} ={\mathbf{G}}_{\mathrm{y}}\mathbf{K}={\displaystyle \frac{\partial \widehat{\mathbf{x}}}{\partial \mathbf{x}}}$$

(7)

In the retrieval process, random errors are included. We estimated retrieval errors by three main sources: smoothing, retrieval noise, and forward model parameter error. The smoothing error is given by the covariance of the smoothing error (S_s) defined as Equation (8) with the identity matrix of parameters (I_n). Since estimating the absolute smoothing error is difficult, we can use S_a as an appropriate replacement [6,40].

$${\mathbf{S}}_{\mathrm{s}}=\left(\mathbf{A}-{\mathbf{I}}_{\mathrm{n}}\right){\mathbf{S}}_a{(\mathbf{A}-{\mathbf{I}}_{\mathrm{n}})}^{\mathrm{T}}$$

(8)

Considering measurement uncertainties, the covariance of retrieval noise is calculated using the following equation:

$${\mathbf{S}}_m={ \mathbf{G}}_{\mathrm{y}}{\mathbf{S}}_{ϵ}{\mathbf{G}}_{\mathrm{y}}^{\mathrm{T}}$$

(9)

As another critical consideration, BRDF, which depends on satellite and solar geometries, is important for analyzing error propagation to state vectors through the covariance of forward model parameter errors (S_f), as defined in Equation (10). Over the ocean, whitecap and shadow effects are incorporated into the Cox–Munk BRDF model through wind speed as a key parameter [44]. As discussed, S_b represents the covariance matrix of the forward model error suggested by ERA5 data uncertainty. We defined the retrieval error (ε_ret) as the square root of the sum of the diagonal elements of S_s, S_m, and S_f.

$${\mathbf{S}}_f={\mathbf{G}}_{\mathrm{y}}{\mathbf{K}}_b{\mathbf{S}}_b{\mathbf{K}}_b^{\mathrm{T}}{\mathbf{G}}_{\mathrm{y}}^{\mathrm{T}}$$

(10)

3. Results

3.1. Information Contents

To assess the retrieval sensitivity of state vectors, A is analyzed for three aerosol models (i.e., sulfate, smoke, and dust aerosols). The diagonal elements of the A (D_A) represent the sensitivity of the retrievals to the true state. When the value of D_A is close to one, the parameter is expected to be well inverted. On the other hand, the off-diagonal elements in rows of A (R_A) indicate retrieval interferences with other state vectors. This means that a retrieval parameter can be strongly affected by other elements of the state vector due to the cross-correlated measurement sensitivity.

Figure 5 shows the As for three aerosol models using only radiance measurements (upper panels), and L + DoLP (lower panels). These As are averaged over various scenarios summarized in Section 2.1. The PSD parameters (i.e., r_f, σ_f, r_c, σ_c, and F_num) are presented in 1 to 5, and ALH parameters (i.e., z_p and h) are indicated in 6 and 7. The following larger indices illustrate the spectral parameters, including τ_aer, n, and k at the PolCube wavelengths.

Figure 5. (Top) Averaging kernel matrices of (a) sulfate, (b) smoke, and (c) dust aerosols using only radiance data. (Bottom) Averaging kernel matrix of these aerosols (panels d–f) using L + DoLP. For this calculation, we assume τ_aer at 555 nm is 0.5 with geometries provided in Table 1.

As shown in panels (a)–(c) in Figure 5, D_As of PSD parameters show partial sensitivity depending on the aerosol models. The D_As of PSD parameters for fine-dominated aerosols (i.e., sulfate and smoke) are sensitive to fine-mode parameters (i.e., r_f, σ_f, and F_num). For coarse-dominated aerosols (i.e., dust), sensitivity is observed in coarse-mode parameters (i.e., r_c and σ_c) in addition to fine-mode parameters. The high D_A value of F_num with low R_A at other state vectors indicates that this parameter is more reliable when retrieved for fine-dominant aerosol. D_As of τ_aer are consistently high across all aerosol models, as the radiance is highly sensitive to the aerosol burden. However, R_As of τ_aer show that τ_aer is notably influenced by F_num, k_f, and k_c. D_As of n_f and n_c show wavelength-dependent sensitivities, which also possess strong interferences with other parameters. Interestingly, for sulfate aerosols (Figure 5a), the D_As for n_c show slightly higher sensitivity than for n_f except for the 410 nm, despite the aerosols being fine-particle dominated. This can be attributed to its high ω₀: the radiance is also influenced by the coarse mode, due to its strong scattering. In contrast, smoke is moderately absorbing (ω₀ = 0.92), which can suppress the scattering signal from the coarse mode. For the dust model, moderate sensitivity is specifically observed at 865 nm, where the coarse-mode scattering is most prominent. Conversely, D_As of k_f and k_c generally show low sensitivity across all aerosols. Overall, when multi-angle radiances are utilized, the current observation geometries and information content are insufficient for inversion of various aerosol optical properties.

When using the case of L + DoLP (Figure 5d–f), the sensitivities of several state vectors increased relative to those obtained from radiances alone. In particular, the retrieval sensitivity of fine-mode PSD parameters (i.e., r_f and σ_f) is enhanced with DoLP. This improvement mainly stems from the sensitivity of polarization measurements to the angular information in the scattering of fine particles [53]. As demonstrated for the radiance measurements, D_As of τ_aer also exhibit sufficient sensitivity when DoLP data are used across all aerosol models and wavelengths. D_As for n_f in the sulfate and smoke models, and for n_c in the dust model, show a significant increase compared to those in Figure 5a–c. The values of D_As for k_f and k_c are also partially increased, depending on the dominant aerosol mode. For smoke aerosols, the sensitivity of k_f increases slightly more than k_c. On the contrary, for dust aerosols, D_As of $k_{\mathrm{c}}$ show a greater enhancement compared to those of k_f.

In the case of ALH, D_As remain consistently low in sensitivity across all aerosol models, with or without DoLP. In passive remote sensing at visible or infrared wavelengths, measurements have limited sensitivity to aerosol vertical profiles compared with active sensors such as the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP). To retrieve ALH parameters from MAP measurements, ALH information is derived from UV or blue-band measurements, in which Rayleigh scattering is strong. A previous study using Research Scanning Polarimeter (RSP) data, with 152 viewing angles, reported a strong correlation between retrievals and true ALH values [54]. Moreover, Wu et al. (2016) demonstrate that wavelengths shorter than 410 nm are essential to retrieve ALH parameters [54]. Although the PolCube includes the 410 nm wavelength, the results reveal that the DoLP does not meaningfully enhance sensitivity to ALH, primarily due to the limited number of angular observations.

As described previously, the diagonal elements of As provide quantitative information content of each state vector, which can be expressed in terms of the degrees of freedom (d_f). The following Figures in Section 3.1 compare the d_f results for cases with and without DoLP measurements. The d_f with blue triangles represent results obtained from radiance measurements only, whereas those with red squares indicate results obtained from both radiance and DoLP.

Figure 6 shows the d_fs of τ_aer for all aerosol models. For fine-dominated aerosols such as sulfate and smoke, the d_fs are sufficiently large across all wavelengths, regardless of whether DoLP is included. For dust aerosols, d_fs remain generally high, except at 410 nm. At 410 nm, absorption predominates, and scattering from coarse-dominant particles is less sensitive to τ_aer. Nonetheless, these results demonstrate that the PolCube measurements contain sufficient information on τ_aer for both measurement types.

Figure 6. Estimated degrees of freedom (d_f) for τ_aer of (a) sulfate, (b) smoke, and (c) dust. The blue triangles and red squares denote the d_f for radiance-only and for radiance and DoLP, respectively.

In Figure 7, the d_fs of the complex refractive indices are averaged across all aerosol models. When both radiance and polarization measurements are used, the average d_fs of n_f remains high in the visible bands, while that of n_c increases from the visible to the NIR band. Moreover, clear differences are observed between d_fs with and without DoLP for n. This enhancement occurs because linear polarization measurements play a significant role in the angular scattering phase function. Although the d_fs for k show noticeable improvements when polarization is used, the PolCube observations appear to be less sensitive to k than to n. However, despite these results, n is among the most difficult to invert because of interactions between n and PSD parameters (see Figure 5). Therefore, obtaining sufficient measurements across multiple angles and a broader wavelength range is essential for effectively characterizing the differences between them.

Figure 7. Estimated mean d_f for the real part of the refractive index (n) for (a) fine-mode and (b) coarse-mode. (c,d) Similar plots for the imaginary part of the refractive index (k) for fine- and coarse-mode, respectively.

Figure 8 shows the mean d_fs of the PSD parameters (i.e., r_f, σ_f, r_c, σ_c, and F_num). The results indicate that combining radiance and DoLP measurements yields more reliable information for most PSD parameters, except for σ_c. Although PolCube is a multi-wavelength, multi-angle polarimeter, its spectral bands (410, 555, 670, and 865 nm) are insufficient to characterize variations in σ_c, which typically require longer wavelengths in the Short-Wave Infrared (SWIR). Furthermore, as shown in Figure 2, −P₁₂/P₁₁ shows a more pronounced backscattering signal for fine-dominated aerosols. Given that PolCube primarily detects the backscattered light, the σ_c retrievals contain less information than the other parameters.

Figure 8. Estimated mean d_fs for PSD parameters (i.e., r_f, σ_f, r_c, σ_c, and F_num). The error bar indicates the standard deviation of different aerosol types.

3.2. Retrieval Errors of Measurement Types

In this section, ε_rets are evaluated under both nominal and perturbed calibration conditions to assess how radiometric and DoLP precision impact retrieval accuracy. For the perturbed conditions, perturbations of ±1.0% and ±0.1% were applied to the radiometric (2%) and DoLP (0.5%) calibration uncertainties. These results reflect the anticipated operational variations in radiometric and polarimetric precision. For this analysis, we assume the scene is over the ocean, using each aerosol model’s characteristics (see Table 2) with τ_aer of 0.5 at 555 nm.

In Figure 9, the blue series of triangles indicates ε_rets when using only radiance data, whereas the red series of squares represents those when both radiance and DoLP data are used. Using both radiance and DoLP measurements significantly reduces ε_rets across all aerosol models. In particular, panels (a) and (b) in Figure 9 show that including DoLP data markedly improves τ_aer retrieval accuracy for fine-mode aerosols, such as sulfate and smoke. For the dust case (panel c), retrieval accuracy is slightly higher than in the absence of DoLP. Overall, these results demonstrate that the accuracy of both radiometric and DoLP measurements is crucial to τ_aer retrieval performance. Notably, polarimetric measurements at the PolCube wavelengths are more effective for fine-mode aerosols such as sulfate and smoke.

Figure 9. Estimated retrieval errors (ε_ret) for τ_aer under different calibration accuracy in radiometric and DoLP for all aerosol models: (a) sulfate, (b) smoke, and (c) dust aerosol. The blue and red series lines represent ε_rets computed from radiance alone and from L + DoLP, respectively. For this calculation, we assume the scene is over the ocean, using each aerosol model’s characteristics (see Table 2), with τ_aer of 0.5 at 555 nm.

Figure 10 shows results similar to those in Figure 9, with the exception of the complex refractive indices. For this analysis, the ε_rets are averaged over all aerosol models. Panels (a) and (b) in Figure 10 show that retrieval accuracies for n_f and n_c are substantially improved when polarization measurements are used. For n_f, retrieval errors are lower in the visible bands, which are more sensitive to fine-mode aerosols, than in the NIR band. By contrast, ε_rets for n_c decrease from visible to NIR wavelengths, as NIR bands are sensitive to coarse-mode aerosols. In panels (c) and (d), including DoLP data also improves the retrieval accuracy of k in both aerosol modes. However, the improvement in k retrievals is smaller than that for n, as discussed in Figure 7. As shown in Figure 9 and Figure 10, the spectral parameters are notably influenced by the radiometric measurements and DoLP accuracies. This suggests that reliable radiometric calibrations are essential for accurate and consistent retrieval of spectral parameters.

Figure 10. Estimated ε_rets of the complex refractive indices under different radiometric and DoLP accuracies for both fine and coarse-mode aerosols. The results show when using radiance (blue-colored series triangles) and L + DoLP (red-colored series squares). Panels (a,b) display results of the real part (n) of refractive index for fine and coarse aerosols, while panels (c,d) show the results of the imaginary part (k) for the same aerosol modes.

Figure 11 shows the ε_rets for PSD parameters (i.e., r_f, σ_f, r_c, σ_c, and F_num), with standard deviations indicating variation across aerosol models. Unlike τ_aer, which is highly sensitive and effectively reflects measurement accuracy, PSD parameters are much less sensitive (compare Figure 9 and Figure 11). Due to the limited information content, PSD retrievals are more likely constrained by the a priori datasets. Consequently, the PSD retrieval results remain less variable regardless of improvements in measurement accuracy. As discussed for Figure 8, PolCube’s limited spectral coverage explains why coarse-mode parameters are significantly less sensitive to improvements in measurement accuracy. The retrieval accuracies of r_c and F_num vary significantly by aerosol type. In general, the trends in Figure 9, Figure 10 and Figure 11 are consistent with previous research on the retrieval sensitivity of aerosol optical properties derived from radiances or polarization data [14,18,19,20].

Figure 11. Estimated ε_rets, averaged over all aerosol models, for particle-size distribution parameters (r_f, σ_f, r_c, σ_c, and F_num) under different radiometric and DoLP accuracies. Error bars represent the standard deviation for different aerosol types.

One of the major forward model parameters is the wind speed over the ocean, which significantly influences the spectral radiance and polarization computed by the forward model. As wind speed determines ocean surface roughness, it substantially influences the ocean surface BRDF depending on observation geometry. Furthermore, the bidirectional polarization distribution function (BPDF) is also sensitive to the direction between the incident and reflected light, and the location of the glint region [20,55]. Therefore, the impact of wind speed on retrieval is evaluated separately.

To evaluate the contribution of wind-speed uncertainties, the forward-model parameter error (ε_f) is compared with ε_ret. As shown in Figure 12 and Figure 13, the hatched and gray bars indicate ε_f and ε_ret, respectively. Both errors are averaged across all aerosol models and estimated for the L + DoLP case with known calibration accuracy [28]. Error bars indicate their standard deviation. Figure 12 compares ε_f and ε_ret for the PSD parameters. The contribution of ε_f to ε_ret is generally small across all PSD parameters. Nevertheless, it has a non-negligible impact on the retrieval of σ_f and r_c.

Figure 12. Comparison of averaged ε_ret (filled gray) and forward model parameter errors (ε_f; hatched) for PSD parameters (i.e., r_f, σ_f, r_c, σ_c, and F_num). For this analysis, both radiance and DoLP measurements are assumed to be available with known calibration accuracies. The error bars indicate the standard deviation for different aerosol models.

Figure 13. Comparison of averaged ε_ret (gray filled) and ε_f (hatched) with error bars (standard deviation) for (a) τ_aer, (b) n, and (c) k.

In Figure 13, the results are similar to those in Figure 12, but for (a) τ_aer, (b) n, and (c) k. The ε_f values of all spectral τ_aer are considerably larger, contributing 33 ± 1% than half of the total uncertainties. This suggests that wind speed is a major contributor to the retrieval of τ_aer. Therefore, accurate wind-speed data are essential for its retrieval. On the other hand, the ε_f values of n and k vary with wavelength and aerosol mode. Overall, the ε_fs of n and k contribute about 34 ± 2% and 19 ± 1% of the ε_ret.

3.3. Retrieval Characterization Across Diverse Aerosol Loadings

To account for the diverse aerosol loading scenarios over the ocean, τ_aer values of 0.1, 0.3, 0.5, and 1.0 were evaluated, with results averaged across all aerosol models as shown in Table 3. The aerosol properties are typically retrieved for the τ_aer above 0.3, as sufficient aerosol loading is required to provide the signal for retrieval. For PSD and n, the d_fs generally improve with increasing τ_aer and remain stable for τ_aer above 0.3. Notably, the d_fs for k significantly increase as τ_aer increases. Conversely, the d_fs for τ_aer itself decreases at high aerosol loadings. This seems likely due to the absorption effect of k (see Figure 5); the enhanced d_fs for k at high τ_aer complicates, thereby reducing the d_fs for τ_aer retrieval. The ε_ret for PSD also improves alongside d_fs as τ_aer increases, showing stabilized values for τ_aer above 0.3. While the absolute ε_rets for τ_aer increases with higher loading, the relative error significantly decreases from 31% at τ_aer of 0.1 to 14.8% at τ_aer of 1.0. Regarding the complex refractive indices, ε_rets for n (both fine and coarse modes) are improved with increasing τ_aer, whereas no remarkable changes for k are observed in the ε_rets across both modes.

Table 3. Comparison of averaged information contents (d_fs) and retrieval errors (ε_ret) across all aerosol models, for PSD, τ_aer, and complex refractive indices across different τ_aer scenarios: 0.1, 0.3, 0.5, and 1.0.

dfs
Parameter		τaer (555 nm)
		0.1	0.3	0.5	1.0
PSD	rf	0.47	0.59	0.61	0.61
	σf	0.80	0.87	0.88	0.88
	rc	0.31	0.38	0.41	0.42
	σc	0.02	0.03	0.04	0.04
	Fnum	0.82	0.84	0.84	0.084
τaer	410	0.99	0.94	0.86	0.73
	555	1.00	0.99	0.95	0.82
	670	1.00	0.98	0.95	0.75
	865	1.00	0.98	0.95	0.75
nf/nc	410	0.57/0.34	0.77/0.50	0.80/0.53	0.82/0.52
	555	0.39/0.59	0.55/0.68	0.60/0.69	0.59/0.67
	670	0.67/0.67	0.75/0.74	0.76/0.75	0.77/0.75
	865	0.17/0.84	0.35/0.92	0.41/0.94	0.43/0.95
kf/kc	410	0.06/0.17	0.14/0.28	0.30/0.34	0.52/0.41
	555	0.05/0.26	0.09/0.41	0.13/0.43	0.29/0.48
	670	0.03/0.23	0.08/0.34	0.10/0.37	0.19/0.45
	865	0.03/0.36	0.09/0.48	0.12/0.51	0.18/0.56
εret
PSD	rf	0.015	0.012	0.012	0.012
	σf	0.150	0.106	0.097	0.090
	rc	0.14	0.115	0.110	0.107
	σc	0.05	0.049	0.049	0.049
	Fnum	0.0011	0.0009	0.0008	0.0008
τaer	410	0.044	0.092	0.137	0.197
	555	0.031	0.059	0.088	0.148
	670	0.025	0.047	0.071	0.134
	865	0.022	0.035	0.053	0.105
nf/nc	410	0.090/0.082	0.055/0.062	0.049/0.058	0.048/0.058
	555	0.108/0.061	0.066/0.043	0.056/0.040	0.054/0.040
	670	0.072/0.053	0.046/0.034	0.040/0.030	0.040/0.030
	865	0.091/0.045	0.073/0.026	0.066/0.021	0.065/0.018
kf/kc	410	0.009/0.008	0.008/0.006	0.007/0.006	0.005/0.005
	555	0.008/0.007	0.007/0.005	0.006/0.005	0.005/0.004
	670	0.007/0.006	0.006/0.005	0.005/0.004	0.005/0.004
	865	0.008/0.007	0.007/0.005	0.006/0.004	0.006/0.004

4. Summary and Discussion

This study evaluated the PolCube aerosol products processed by the SMART-P algorithm, including PSD, τ_aer, and the complex refractive indices, which are planned for launch in 2026. Radiance and linear polarization were simulated under nominal measurement conditions and using the instrument characteristics of PolCube. The results in Section 3.1 clearly demonstrate that including DoLP data enhances sensitivity to most aerosol properties relative to using only radiance measurements. Theoretically, this additional polarimetric information from PolCube provides valuable constraints on aerosol properties during inversion. Accordingly, these enhancements enable reasonable retrieval of key aerosol parameters, particularly spectral τ_aer, n, and fine-mode PSD parameters (i.e., r_f and F_num).

The retrieval accuracy of aerosol properties is assessed under various calibration accuracies, as expected from onboard calibration. Overall, incorporating polarization data improves retrieval accuracy relative to omitting it, except for σ_f. In particular, the retrieval accuracies of τ_aer for fine-dominated aerosols and n_f are improved by approximately a factor of two and are highly sensitive to calibration accuracy. Therefore, precise on-orbit calibration is essential to meet the required retrieval performance. Because wind speed is a major parameter determining ocean surface conditions, its uncertainty is assessed to evaluate its impact on aerosol properties. The ε_f of wind speed contributes about 33% of the ε_ret in τ_aer and approximately 34% and 19% of the ε_rets in n and k.

Analysis of d_fs and ε_ret indicates that τ_aer above 0.3 ensures stable d_fs and improved accuracy for PSD and n. While enhanced absorption at high loadings reduces the d_fs for τ_aer, overall retrieval fidelity strengthens as loading increases, with τ_aer relative error reducing from 31% to 14.8%. These results confirm that higher aerosol loading enhances the characterization of most physical properties.

As natural aerosols are generally composed of particles with variable sphericity, the impact of sphericity will also be analyzed in the next study. Since polarimetric measurements are sensitive to particle shape, the polarized phase function is expected to show greater variability for coarse-dominant aerosols, such as desert dust, than in the preliminary sensitivity tests (Section 2.1). Therefore, the impact of sphericity will be rigorously analyzed in subsequent studies. Future improvements will address the complexity of particle sphericity and ALH.