Multi-angle polarimetric observations are obtained from HARP2, which is onboard the PACE satellite (Martins et al., 2018; Xu et al., 2024). HARP2 provides ultra-multi-angle (up to over 60 viewing angles) and multispectral polarimetric measurements, delivering high-precision Stokes parameters in the visible and near-infrared bands (Werdell et al., 2019). Among these, the 441 nm channel exhibits high sensitivity to variations in ALH. Multi-angle polarization information enhances constraints on aerosol vertical distribution and microphysical properties, and significantly improves retrieval stability and information content compared with single-angle radiance observations. Based on this capability, this study develops a joint retrieval method using multi-angle radiance intensity (I) and linear polarization measurements (Q and U) from HARP2 to improve the sensitivity and accuracy of ALH retrieval.

This study uses the PACE_HARP2.L1C standard product with a spatial resolution of 5.2 km. The product has undergone radiometric calibration, geometric correction, and resampling, and provides multi-angle observations on a unified geographic grid (Gao et al., 2023). At the 441 nm wavelength used in this study, HARP2 provides 10 discrete viewing directions, with nominal viewing angles ranging from approximately -55.9 to 53.8°. The datasets used in this study include: (1) solar zenith angle (SZA), viewing zenith angle (VZA), and relative azimuth angle (RAA), which characterize the observation geometry; (2) Stokes parameters I, Q, and U at 441 nm across 10 viewing angles, together with their corresponding uncertainty metrics (ISD, QSD, and USD).

This study incorporates aerosol product data from VIIRS onboard the SNPP satellite, specifically the AERDB_L2_VIIRS_SNPP dataset from the Level-2 Deep Blue aerosol product (NASA VIIRS Atmosphere Science Team, 2023). The following parameters were selected: (1) AOD at 550 nm as an indicator of column-integrated aerosol loading, with data pre-screened by an internal high-precision cloud masking algorithm (Hsu et al., 2013; Sayer et al., 2012); (2) surface reflectance at 488 nm to characterize surface reflection properties, providing auxiliary information for analyzing land–atmosphere coupling effects; and (3) the Ångström exponent (AE) to facilitate AOD conversion across different wavelengths.

This study incorporates surface bidirectional reflectance information from the MODIS MCD43A1 BRDF/Albedo product (Schaaf and Wang, 2015). Specifically, the BRDF Albedo Parameters dataset was used to characterize the directional surface reflection properties at the corresponding wavelength. The MCD43A1 product provides the three RossThick–LiSparseReciprocal BRDF model parameters, which describe the anisotropic scattering behavior of the land surface (Schaaf et al., 2002). These parameters were used as auxiliary inputs for the radiative transfer simulations to account for surface bidirectional reflectance effects in the aerosol layer height retrieval.

The Level-2 ALH product (version 02.08.00) from the TROPOMI instrument onboard Sentinel-5P is used for validation (European Space Agency, 2021). In this version, surface albedo is incorporated into the optimal estimation state vector as a fitting parameter and is simultaneously retrieved together with ALH and AOD. This improvement enhances the consistency between radiative transfer simulations and observed spectra, and significantly reduces the impact of land–atmosphere coupling errors on the fitting of O₂A band absorption features, thereby effectively mitigating the underestimation bias of ALH under high-reflectance surface conditions. Previous studies have shown that, compared with earlier versions, version 02.08.00 exhibits smaller systematic differences between land and ocean regions and achieves better agreement with active lidar observations (e.g., CALIOP and ATLID) (de Graaf et al., 2025; Nanda et al., 2019, 2020).

In addition, the TROPOMI ALH product provides a continuous Quality Assurance (QA) metric ranging from 0 to 1, where 0 indicates retrieval failure or unreliable results, and 1 indicates fully successful retrieval. To ensure data quality, only pixels with QA ≥ 0.5 are retained for validation in this study, while those with QA < 0.5 are considered unreliable and excluded. Furthermore, the ALH product has been cloud-screened using the VIIRS/SNPP Enterprise Cloud Mask (ECM).

The reference data used for ALH validation are obtained from the ATLID active remote sensing instrument onboard the EarthCARE satellite, which was launched on 28 May 2024 (Illingworth et al., 2015). ATLID is an active sensor based on High-Spectral-Resolution Lidar (HSRL) technology, operating at a wavelength of 355 nm in the ultraviolet band. By separating molecular (Rayleigh) scattering from particulate (Mie) scattering signals, ATLID can independently retrieve aerosol and cloud backscatter coefficients, extinction coefficients, and polarization properties without requiring a priori assumptions of the lidar ratio (Nishizawa et al., 2026). Its native vertical resolution is approximately 103 m within the 0–20 km altitude range and about 500 m above 20 km. The along-track horizontal sampling interval is about 140 m, which is typically aggregated to an effective resolution of approximately 280–300 m in standard products (van Zadelhoff et al., 2023). This high vertical resolution makes ATLID particularly well suited for resolving the fine structure of aerosol layers and accurately determining layer top heights.

The specific datasets used in this study include the Level-2A products ECA_EXAF_ATL_AER_2A and ECA_EXAD_ATL_FM__2A. The former provides aerosol optical profile parameters, specifically the extinction coefficient at 355 nm, which can be used to derive aerosol layer centroid height or layer top height and serves as a reference for validating retrieval results. The latter includes the FeatureMask dataset, which contains a feature detection index ranging from 0 to 10, where 0 represents clear-sky conditions and 10 indicates a very high probability of optically thick clouds. This index is constructed based on probabilistic analysis of ATLID backscatter signals. It should be noted that this mask does not distinguish between different particle types; instead, it is designed to differentiate strong signal regions, weak signal regions, and clear-sky areas, thereby facilitating the construction of vertical profiles for aerosol classification.

To ensure the reliability of the ATLID validation dataset, quality control and target-type screening were applied to the EarthCARE Level-2 products. First, observations were filtered based on the Quality Status flag (QALD) provided in the Quality Status dataset, retaining only samples with QALD values of 0, 1, and 2. Here, QALD = 0 indicates high-quality observations; QALD = 1 denotes valid data but with the number of aerosol layers exceeding a configurable threshold; and QALD = 2 indicates that the retrieved layer height shows some inconsistency with the A-TC (Target Classification) product, although the data remain valid. In addition, aerosol types were further screened using the A-TC product. Since this study focuses on the vertical structure of absorbing aerosols, only observations classified as Dust, Smoke, Dusty_smoke, and Dusty_aerosol_mix were retained for subsequent statistical analysis. This procedure helps reduce the influence of non-target aerosol types on the ALH validation results.

Owing to its high-precision vertical sensing capability and its physically based discrimination between aerosols and clouds, ATLID provides products with significant reference value for ALH validation. It offers an independent and reliable benchmark for evaluating ALH retrievals from passive remote sensing.

In the framework of optimal estimation theory (Hou et al., 2018), the relationship between the state vector x (containing n parameters) and the observation vector y (with m measurements) can be expressed as: 1y=Fx+∈ where F is the forward model that describes the physical relationship between satellite observations and atmospheric parameters (such as ALH), x is the state vector to be retrieved, and ∈ represents the errors arising from forward model simulations and satellite measurements. For the DFS analysis, the nonlinear forward model in Eq. (1) is linearized around a reference state x using the Jacobian matrix K=∂F∂x, which represents the sensitivity of the simulated observations to the retrieved parameters. Under the assumption that the measurement and forward model errors follow Gaussian distributions, the posterior probability density function of the retrieved state can be obtained through Bayesian estimation by combining the prior information with the measurement information. The corresponding posterior error covariance matrix S is then expressed as: 2S-1=KTSϵ-1K+Sa-1 Where, Sϵ denotes the measurement error covariance matrix, and Sa represents the a priori error covariance matrix. In this study, the measurement error covariance matrix Sϵ is assumed to be diagonal, indicating that the measurement errors among different observations are independent. It includes uncertainties from both instrument measurements and forward-model simulations. Specifically, Sϵ is defined as a diagonal matrix: 3Sϵ=diagσI,v12,…,σI,vM2,σDoLP,v12,…,σDoLP,vM2 where the radiance uncertainty is typically assumed to be 5 % of the measured value, accounting for calibration errors), while the absolute uncertainty of the Degree of Linear Polarization (DoLP) is usually set to 0.005 for high-precision polarimeters such as HARP2 (Sienkiewicz et al., 2025). This weighting scheme ensures that polarization signals have a higher constraint weight under conditions of high aerosol loading.

The superscripts v1–vM are used to represent individual viewing angles in multi-angle observations. Accordingly, the measurement vector y can be expressed as: 4y=Iλv1,…,IλvM,DoLPλv1,…,DoLPλvMT

To establish the relationship between the retrieved state and the true state, the averaging kernel matrix A is introduced: 5A=∂x^∂x=KTSϵ-1K+Sa-1-1KTSϵ-1K The trace of the averaging kernel matrix A (i.e., the sum of its diagonal elements) is defined as the Degrees of Freedom for Signal (DFS), which quantifies how much information about the retrieved parameters can be obtained from the satellite measurements. For a single state variable, the DFS value ranges between 0 and 1. If the DFS of an aerosol parameter is close to 0, it indicates that the measurements contain little to no information for retrieving that parameter. In contrast, a DFS value approaching 1 indicates that the observations provide substantial information and make a strong contribution to the retrieval process.

Using the Unified Linearized Vector Radiative Transfer Model (UNL-VRTM), satellite observations and their weighting functions (Jacobian matrices) with respect to retrieval parameters are simulated under various viewing geometries. UNL-VRTM is based on the VLIDORT model (Spurr, 2006) and computes the Stokes vector [I,Q,U,V]T, where I represents radiance, and Q and U denote the linear polarization components (Wang et al., 2014; Xu and Wang, 2019). Circular polarization (Stokes parameter V) is generally negligible in atmospheric studies. In this study, the DoLP is defined as: 6DoLP=Q2+U2I

Furthermore, the Jacobian of DoLP with respect to the state vector can be expressed as: 7∂DoLP∂x=-DoLP∂I∂xiI+Q∂Q∂xi+U∂U∂xiIQ2+U2

In radiative transfer calculations, aerosol scattering and absorption processes involve AOD (related to the extinction coefficient), single scattering albedo (SSA), and the scattering phase matrix. For aerosol particles, these optical properties are computed using a linearized Mie scattering code. Although dust particles are known to exhibit non-spherical shapes, spherical particle approximation is commonly adopted in LUT-based radiative transfer simulations to maintain computational efficiency. The common input parameters shared by the Mie scattering code and the radiative transfer model include those defining the bimodal log-normal volume size distribution (see Eq. 8), namely the median radius (r) and variance (σ) for both fine and coarse modes, as well as the real and imaginary parts of the complex refractive index. These microphysical parameters are specified based on previous studies. The aerosol size distribution and fine-mode fraction (FMF) follow Lee et al. (2015), while the complex refractive indices are adopted from Chen et al. (2021b) for smoke and Zeng et al. (2008) for dust, as summarized in Table 1. 8dVrdln⁡r=∑i=12Cv,i2πσie-12ln⁡r-ln⁡rv,iσi2 Here, rv,i and σ represent the median radius and standard deviation (SD) of the bimodal log-normal volume size distribution, respectively.

The aerosol vertical distribution is assumed to follow a Gaussian-like profile (Spurr and Christi, 2014), defined as: 9τz=Kexp⁡-σz-zpeak1+exp⁡(-σ|z-zpeak|)2 where K is related to the column aerosol loading; zpeak represents the height of the AOD peak, i.e., the ALH; and σt denotes the aerosol layer thickness, which is set to 1.76 km.

For quantitative comparison, the ATLID-based ALH (ALHATLID) is derived from the Level-2 355 nm extinction coefficient profiles following the method of Koffi et al. (2012), using an extinction-weighted height definition: 10ALHATLID=∑i=1nβext,iZi∑i=1nβext,i Where βext,i represents the aerosol extinction coefficient (km⁻¹) at altitude Zi for the ith vertical layer. The derived ALHATLID represents the extinction-weighted mean altitude of the aerosol layer. As discussed by Kylling et al. (2018), ALH can be defined in different ways, including geometric-mean, cumulative-extinction, and extinction-weighted heights, and these definitions may yield systematic differences when used for intercomparison. In this study, the extinction-weighted height is adopted to characterize the vertically integrated center of the aerosol extinction profile and to provide a physically consistent reference for comparison with the retrieved ALH (Chen et al., 2021b).

The surface reflectance is modeled using the kernel-driven BRDF model coupled within UNL-VRTM (Litvinov et al., 2011; Wang et al., 2014), expressed as: 11ρRμ,μ0,Δϕ,λ=fisoλ+fvolλKvolμ,μ0,Δϕ+fgeoλKgeoμ,μ0,Δϕ Where μ, μ0 and Δϕ denote the solar zenith angle, viewing zenith angle, and relative azimuth angle, respectively; Kvol(μ,μ0,Δϕ) is the volumetric scattering kernel derived from the Ross-Thick radiative transfer model; Kgeo(μ,μ0,Δϕ) is the geometric kernel derived from the Li-Sparse geometric-optical model (Wanner et al., 1995). The coefficients fiso, fvol and fgeo are BRDF kernel weights, obtained as the average values of the corresponding parameters for pixels with AOD > 0.3 from the MCD43A1 product over the study region. For surface polarized reflectance, the Bidirectional Polarization Distribution Function (BPDF) kernel for land surfaces developed by Maignan et al.(2009), based on long-term PARASOL observations, is adopted. The kernel function is given as follows:

12fmaignanθ,θ0,Δϕ=Ce-(ν+tan⁡γ)4μ0+μFrγ,m Where C is a constant characterizing the surface type; ν denotes the normalized difference vegetation index (NDVI), and the values of C and ν are given in Table 2; m is the refractive index of the vegetation canopy, typically set to 1.5. Fr(γ,m) represents the Fresnel reflection matrix with eight non-zero elements (Kokhanovsky et al., 2015).

To systematically evaluate the retrievability of ALH under different observational conditions, this section employs an information content analysis approach to investigate how ALH information varies with AOD, observation wavelength, and surface reflectance under smoke aerosol conditions.

Figure 1 presents the two-dimensional distributions of ALH–AOD₄₄₁–DFS for four spectral bands (441, 549, 669, and 873 nm) at a scattering angle of θ=120°, under different surface reflectance conditions As=0.00, 0.05, 0.20, 0.50. To ensure a physically consistent comparison among different spectral bands, AOD was specified at 441 nm as the reference aerosol loading condition. The corresponding AOD values at other wavelengths were calculated using the AE and used in the radiative transfer simulations (Eck et al., 1999). From the spectral perspective (comparison by columns), the information content of ALH shows a clear decreasing trend with increasing wavelength, i.e., 441 nm > 549 nm > 669 nm > 873 nm. Under typical aerosol loading conditions (AOD = 0.3–1), the DFS at 441 nm generally reaches 0.6–1.0, whereas at 873 nm it is typically only 0.2–0.5, corresponding to an overall reduction of approximately 40 %–70 % in information content. In comparison, the 549 and 669 nm bands fall within an intermediate range, with DFS values around 0.4–0.7. These results indicate that shorter wavelengths have a clear advantage for ALH retrieval. This behavior is primarily attributed to stronger Rayleigh scattering at shorter wavelengths, which enhances the sensitivity of the radiative field to changes in atmospheric vertical structure. In contrast, Rayleigh scattering weakens at longer wavelengths, leading to a significant reduction in sensitivity to layer height. This finding is highly consistent with previous studies (Wu et al., 2016), which identified near-ultraviolet and blue bands as the optimal spectral regions for ALH retrieval using multi-angle polarization measurements.

From the perspective of surface reflectance, it exerts a significant suppressing effect on ALH information content, with clear spectral dependence. The shortwave bands (441 and 549 nm) exhibit remarkable stability: even as surface reflectance As increases from 0 to 0.50, the DFS distributions for these bands remain almost entirely saturated (DFS > 0.9). This indicates that, at short wavelengths, atmospheric scattering dominates over surface reflection, and the extraction of ALH information is minimally affected by surface background. High sensitivity can thus be maintained even under low AOD conditions. In contrast, the longer wavelengths (669 and 873 nm) show pronounced degradation. As As increases, the DFS values for these bands decrease significantly, and the effective sensitivity region under low to moderate aerosol loading (AOD < 1.0) rapidly shrinks. Under dark surface conditions (As ≤ 0.05), regions with DFS > 0.5 can cover most of the range where AOD ≈ 0.5. However, under high-reflectance surfaces (As ≥ 0.2), a significant information response appears only when AOD > 1.0. This behavior arises from the competition between atmospheric scattering and surface reflection. At short wavelengths such as 441 nm, both Rayleigh scattering and aerosol scattering cross-sections are large, meaning that most of the radiance received by the sensor originates from atmospheric scattering rather than surface reflection. As a result, variations in layer height strongly modulate the signal, and the surface background is effectively “screened out.” As wavelength increases, atmospheric scattering weakens; when surface reflectance becomes stronger, its contribution to the total radiance increases, causing the ALH signal to be increasingly masked by the bright background. Therefore, shortwave bands are critical for achieving reliable ALH retrievals over high-reflectance land surfaces, whereas longer wavelengths exhibit a much stronger dependence on aerosol loading.

From the AOD perspective, the DFS exhibits a clear stage-wise behavior along the AOD dimension. When AOD is low (< 0.3), the DFS is close to zero, indicating that layer height information is difficult to detect. As AOD increases to the range of 0.5–1.5, the DFS rises rapidly, and under high aerosol loading conditions (> 2), the DFS tends to saturate. This pattern suggests the existence of a distinct aerosol loading threshold for ALH retrieval. When aerosol concentrations are insufficient, atmospheric scattering signals are weak and surface reflection dominates the total radiance. As a result, variations in layer height have only a limited impact on top-of-atmosphere radiation, making it difficult to extract ALH information from the observations.

Along the ALH dimension, the DFS generally shows a slight increase with increasing layer height, indicating that elevated aerosol layers are easier to detect than near-surface aerosols. This is because higher-altitude aerosols exert a stronger modulation on the solar radiation path and exhibit stronger coupling with molecular scattering layers, making the radiance more sensitive to height variations. In contrast, low-level aerosols are more easily masked by surface reflection contributions, leading to weaker sensitivity to layer height.

Overall, the information content for ALH retrieval exhibits significant variability across different observation conditions. Under scenarios characterized by high aerosol loading (AOD > 1), short wavelengths (≤ 550 nm), and dark surfaces (As ≤ 0.05), the DFS consistently exceeds 0.6, indicating robust retrieval capability. Conversely, under conditions of low aerosol loading (AOD < 0.3), long wavelengths (> 550 nm), and bright surfaces (As ≥ 0.20), the DFS typically falls below 0.3, leading to a substantial increase in retrieval uncertainty.

Under fixed conditions of solar zenith angle (SZA = 40°) and aerosol optical depth (AOD = 0.3), Figs. 2–6 present the ALH–DFS distributions for different viewing zenith angles (VZA = 0, 20, 40, 60, 80°) and surface reflectance conditions. These figures are used to compare two observation configurations: radiance-only (I) and combined radiance and polarization (I+P). In each figure, the upper panels show DFS derived using only radiance, while the lower panels show DFS derived using both radiance and polarization (I+P). The columns correspond to different surface reflectance values (As=0.00, 0.05, 0.20, 0.500). Overall, under moderate aerosol loading conditions (AOD = 0.3), the ALH information content is relatively low when using radiance-only observations, and it is strongly affected by surface reflectance. Over dark surfaces (As ≤ 0.05), the DFS still maintains a certain level; however, as surface reflectance increases, the DFS decreases significantly. In particular, under high-reflectance conditions (As = 0.50), the sensitivity of radiance to layer height is greatly reduced. This indicates that under moderate aerosol loading, surface reflection exerts a strong masking effect on the scalar radiation field, thereby weakening the detectability of ALH.

After incorporating polarization information (I+P), the information content of ALH is improved across all viewing geometries, with particularly pronounced enhancements over high-reflectance surfaces. For example, at VZA = 40° and As = 0.50 (comparing Fig. 4d and h), the inclusion of polarization increases the DFS from approximately 0.2 to the range of 0.5–0.7, corresponding to an information gain of about 0.5. This improvement arises from the strong sensitivity of polarization signals to atmospheric scattering components, which effectively suppresses the interference from surface reflection. As a result, even under complex background conditions with As=0.20–0.50, the ALH DFS can still be maintained above the retrievability threshold of 0.5, demonstrating the substantial advantage of incorporating polarization information in ALH retrieval.

In addition, DFS exhibits a strong dependence on VZA. As VZA increases from 0 to 80°, the longer atmospheric scattering path enhances the sensitivity of radiance measurements to aerosol vertical distribution, resulting in increased ALH information content. Under large viewing angles, intensity-only observations already provide considerable information due to the enhanced atmospheric path length. However, the additional contribution from polarization measurements is more pronounced at small viewing angles, where intensity-only observations provide relatively limited constraints on ALH. By incorporating polarization information, the DFS is substantially enhanced under these geometries, demonstrating that polarization measurements provide complementary information beyond radiance observations. Furthermore, under high surface reflectance conditions (As=0.50), polarization measurements help maintain higher DFS values by reducing the impact of surface background effects on ALH retrieval.

Under the same retrieval framework and identical aerosol layer conditions, the difference in information content between multi-angle observations and multi-angle polarimetric observations primarily arises from their different sensitivities to various scattering components involved in the radiative transfer process. Multi-angle radiance measurements primarily characterize the integrated radiative response of the coupled atmosphere–surface system, where the observed signal generally consists of contributions from surface reflection, aerosol single scattering, and multiple scattering processes. Under conditions of low AOD or high surface albedo, the surface-reflected component can become a dominant contributor to the measured radiance, while multiple scattering can obscure the signatures associated with single-scattering processes. Consequently, when only multi-angle radiance observations are used, the impact of ALH on the observed signal is mainly manifested through indirect modulation of radiative path length and scattering geometry. The resulting ALH sensitivity is relatively weak and can be strongly coupled with other parameters, such as AOD and surface reflectance, leading to a more ill-posed inverse problem (Dubovik et al., 2011).

In contrast, incorporating polarization information extends the measurement vector from radiance-only observations to a combined set of radiance and polarization states, thereby fundamentally altering the sensitivity structure of the observations to aerosol scattering processes. A key feature of polarized radiance is its enhanced sensitivity to single-scattering processes, whereas multiple scattering tends to reduce polarization through depolarization effects. Meanwhile, surface reflection generally contributes much less to polarized signals than to total radiance, particularly for rough or approximately isotropic surfaces. Therefore, polarization measurements can substantially reduce the influence of surface background and multiple-scattering contributions, providing a more direct constraint on aerosol scattering geometry and particle microphysical properties.

Under these conditions, the impact of ALH on polarization signals is primarily associated with changes in the scattering-angle distribution and the corresponding photon path geometry of single-scattering processes, resulting in enhanced physical sensitivity to aerosol vertical distribution. In addition, polarization measurements are more sensitive to the scattering characteristics of small-sized aerosol particles, such as fine-mode smoke aerosols, because particle-scale scattering processes produce stronger polarization responses. This further improves the detectability of variations in aerosol vertical structure. Within the optimal estimation framework, the inclusion of polarization information effectively expands the dimension of the observation vector, transforming the Jacobian matrix from one based solely on radiance sensitivity into a combined sensitivity matrix containing both radiance and polarization information. Since radiance and polarization observations exhibit different response patterns to aerosol and surface parameters, particularly with respect to the sensitivity relationships among ALH, AOD, and surface reflectance, the combined Jacobian matrix becomes less correlated, leading to improved parameter separability and increased DFS (Mishchenko et al., 2007). This enhancement is particularly pronounced under conditions where parameter coupling is strongest, such as high surface albedo (0.5) and low AOD (0.3). Under these conditions, radiance measurements are strongly influenced by surface contributions, whereas polarization measurements retain stronger sensitivity to atmospheric scattering, resulting in enhanced complementarity between the two observation types.

Overall, the incorporation of polarization information into multi-angle observations is not simply an increase in the number of measurements, but rather the introduction of an additional independent information dimension that is more sensitive to single-scattering processes and less sensitive by surface reflection. By reducing the coupling among aerosol and surface parameters, polarization observations improve the physical constraint on ALH retrieval. Consequently, the overall sensitivity to ALH is enhanced, while the dependence of retrieval performance on specific viewing geometries and surface conditions is reduced, enabling more robust ALH retrievals over a broader range of surface types and observation geometries.

This subsection compares the effects of the number of viewing angles and observation modes (I and I+P) on the DFS under different surface reflectance conditions (As=0.00, 0.05, 0.20, 0.50).

As shown in Fig. 7, polarization observations provide a significant enhancement in system information content. In all simulated scenarios, the DFS values corresponding to the radiance-plus-polarization configuration (red curves, I+P) are consistently higher than those derived from radiance-only observations (blue curves, I). Quantitatively, for typical configurations with 3–6 viewing angles, the inclusion of polarization information leads to an increase in DFS of approximately 5 %–30 %. This enhancement becomes more pronounced over high-reflectance surfaces (As=0.50), indicating a higher information contribution efficiency of polarization measurements under bright surface conditions. These results demonstrate that polarization observations can effectively enhance the sensitivity of atmospheric parameters and, to a certain extent, suppress the interference caused by surface reflection.

In addition, DFS exhibits a clear saturation tendency with increasing number of viewing angles. As the number of angles increases from 1 to approximately 5, the DFS under all scenarios undergoes a rapid growth phase, during which more than 70 % of the total information gain is typically achieved. However, when the number of viewing angles exceeds 6, the slope of the curve decreases markedly and the growth of information content becomes gradual. This result is consistent with the findings of Chen et al. (2021b), highlighting the nonlinear nature of information acquisition in multi-angle observations. These results suggests that multi-angle measurements provide the most significant information gain at the early stage, whereas beyond a certain number of viewing geometries, the system approaches information saturation and the marginal benefit of additional angles gradually diminishes (Gu et al., 2022).

Finally, the complexity of the surface environment imposes a direct constraint on the information content of the signal. As As increases from 0 to 0.5, the overall DFS decreases by approximately 10 %–40 %, while the shadowed region associated with SZA variations expands significantly. This indicates that under high-reflectance surfaces (e.g., desert regions), atmospheric signals are more easily overwhelmed by surface contributions, making the retrieval more sensitive to observation geometry. Therefore, in such complex surface environments, the incorporation of combined multi-angle and polarization observations can substantially improve the stability and robustness of retrieval results.

In recent years, frequent large-scale wildfires in North America have become a major factor affecting regional air quality and climate. Large amounts of smoke aerosols generated by fires can be uplifted into the middle and upper troposphere under strong convection and thermal buoyancy, and subsequently transported across regions or even intercontinentally by the westerly circulation (Peterson et al., 2018). The vertical distribution of smoke aerosols directly affects their radiative forcing and aerosol–cloud interactions; therefore, accurate retrieval of smoke layer height is essential for understanding their climatic impacts.

The selected smoke cases capture different physical stages and vertical evolution characteristics of smoke transport. The selected days cover a range of scenarios, including multilayer smoke plumes transported in the upper troposphere in late May, rapid expansion of smoke coverage at the end of May, and lower-tropospheric advection processes in early June. Such diverse smoke distributions, spanning from upper to lower troposphere and from local to regional scales, provide an ideal dataset for systematically evaluating the stability, spatial continuity, and altitude-dependent performance of the retrieval algorithm under complex atmospheric conditions.

The Sahara Desert is one of the largest sources of mineral dust in the world, emitting vast amounts of aerosols to the Atlantic Ocean and West Africa each year. During spring and summer, under the influence of the SAL and atmospheric systems such as African easterly waves, dust can be uplifted to altitudes of several kilometers and transported over long distances (Doherty et al., 2008; Prospero et al., 2014). The height of the dust layer not only affects its radiative forcing but also plays an important role in cloud formation processes. To evaluate the applicability of the retrieval algorithm under mineral dust conditions, this study selects three representative dust transport events over West Africa on 18 March, 30 May, and 17 September 2025.

To further quantitatively evaluate the accuracy of ALH retrieved from HARP2 multi-angle polarimetric observations, this study performs point-by-point collocation comparisons with ATLID lidar measurements. ATLID provides aerosol vertical profiles with high vertical resolution and is therefore widely regarded as a reliable reference for validating passive remote sensing retrievals.

During the collocation process, HARP2 and TROPOMI pixels are first spatially matched with the ATLID ground track, and the data are then aggregated within a 10 km × 10 km window to reduce the impact of spatial scale mismatches. Subsequently, a set of statistical metrics – including Root Mean Square Error (RMSE), Mean Bias (MB), Mean Absolute Error (MAE), correlation coefficient (R), and the proportion of errors within 1 km – is calculated to comprehensively evaluate the retrieval performance. Tables 5 and 6 summarize the statistical error characteristics for each case and under different aerosol types, respectively. Figure 13 presents the scatterplot comparison based on pixel-level collocation, while Fig. 14 illustrates the error distribution of the matched data pairs.

First, Table 5 presents the statistical comparison between HARP2 and TROPOMI retrievals and ATLID observations for the six selected cases. For the three smoke cases, the RMSE of HARP2 retrievals is 1.25, 1.06, and 1.10 km, respectively, which is consistently lower than that of TROPOMI (1.65, 1.29, and 1.55 km). Meanwhile, the mean bias of HARP2 remains close to zero across all cases, whereas TROPOMI exhibits a systematic negative bias of approximately -1 km in all three cases, indicating a general underestimation of smoke layer height. In terms of the proportion of retrievals with errors smaller than 1 km, HARP2 achieves values of 48.33 %, 58.68 %, and 50.00 % for the three smoke events, respectively, which are significantly higher than those of TROPOMI (24.24 %, 40.74 %, and 15.71 %). These results further demonstrate that HARP2 provides higher retrieval accuracy under smoke conditions. For the three dust cases, the RMSE of HARP2 is 1.09, 0.96, and 0.75 km, respectively, which is generally lower than the corresponding TROPOMI values of 1.32, 1.55, and 0.94 km. In addition, HARP2 shows relatively small biases, with slight positive bias observed in some cases, while TROPOMI consistently exhibits a pronounced negative bias across all dust events. In particular, for the dust event on 17 September 2025, HARP2 achieves an RMSE of only 0.75 km, with the proportion of errors within 1 km reaching 84.51 %, demonstrating superior retrieval performance.

On the basis of individual case statistics, Table 6 further summarizes the overall retrieval performance across all cases, as well as for smoke and dust events separately. For all cases combined, the HARP2 retrieval achieves an RMSE of 1.03 km, which is notably lower than that of TROPOMI (1.40 km), with a near-zero mean bias (-0.07 km). The obtained RMSE of 1.03 km is comparable to the theoretical HARP2 ALH retrieval uncertainty (∼ 0.94 km) reported by Gao et al. (2023), indicating good consistency between the validation results of this study and the expected retrieval performance of HARP2-based ALH observations. In contrast, TROPOMI exhibits an overall systematic underestimation of approximately 1 km. For the smoke cases, the RMSE of HARP2 is 1.12 km, compared with 1.51 km for TROPOMI. Meanwhile, TROPOMI shows a pronounced negative bias (-1.26 km) under smoke conditions, further indicating its tendency to underestimate ALH in elevated smoke plumes. For the dust cases, HARP2 achieves an RMSE of 0.92 km, significantly lower than the 1.31 km obtained from TROPOMI. In addition, the mean bias of HARP2 is 0.22 km, close to zero, whereas TROPOMI still exhibits a systematic negative bias of approximately -1 km.

To further place these results in the context of previous passive ALH retrievals, we compare the HARP2 results with the dust-layer height retrievals reported by Kylling et al. (2018). Using CALIOP observations as the reference, systematic negative biases were reported for the O₂ A-band retrievals, with mean differences of approximately -1.39 km for GOME-2 and -1.10 km for SCIAMACHY. In comparison, the HARP2 retrieval in the present study exhibits a small positive bias of +0.22 km for the three dust events, indicating a substantially smaller systematic offset relative to the ATLID reference. It should be noted that the two studies adopt different definitions of aerosol layer height: the present study uses an extinction-weighted aerosol layer height derived from ATLID extinction profiles, whereas Kylling et al. (2018) considered a cumulative-extinction-based layer-height definition. Nevertheless, the comparison provides useful context and suggests that the additional multi-angle polarization information exploited by HARP2 may provide a complementary constraint on aerosol vertical structure and help reduce some of the systematic biases associated with passive ALH retrievals based primarily on a single spectral absorption feature.

The consistency between the retrievals and ATLID observations is further illustrated in Fig. 13, which presents scatterplots of the HARP2 and TROPOMI retrieved ALH against ATLID measurements. Overall, both passive remote sensing retrievals show a positive correlation with ATLID observations; however, they exhibit different characteristics in terms of statistical performance. In general, the HARP2 retrievals are closer to the 1 : 1 consistency line and exhibit lower dispersion, resulting in a smaller RMSE, which indicates better agreement with ATLID in terms of absolute layer height values. In contrast, TROPOMI retrievals show higher correlation coefficients, suggesting stronger capability in capturing the relative variability and spatial–temporal patterns of ALH. When analyzed by aerosol type, the smoke cases show that HARP2 retrievals achieve a good linear correlation with ATLID (R=0.75), while TROPOMI exhibits an even higher correlation (R=0.93). However, TROPOMI also shows a pronounced negative bias (bias = -1.26 km), indicating systematic underestimation of smoke layer height in most conditions, which leads to a relatively larger RMSE. For dust cases, the correlation of both methods decreases to some extent, which may be associated with the non-spherical nature and more complex scattering properties of mineral dust particles.

Additionally, to analyze the differences between the two retrieval methods and ATLID observations from the perspective of error statistics, Fig. 14 presents histograms of ALH differences relative to ATLID, together with their statistical characteristics. As shown in Fig. 14a, for all cases, the error distribution of ALH-HARP2 is more concentrated around zero (mean = -0.07 km, median = -0.08 km), indicating good overall consistency with the reference. In contrast, ALH-TROPOMI exhibits a clear negative bias, with the distribution shifted toward negative values and a median significantly below zero. For the smoke cases (Fig. 14b), ALH-HARP2 shows a relatively larger dispersion (SD = 1.07 km), which may be attributed to the increased complexity of viewing geometry dependence and the amplification of random errors introduced during multi-source data coupling. Nevertheless, the proposed method does not exhibit a systematic underestimation for high-altitude smoke layers, demonstrating better physical consistency. In comparison, ALH-TROPOMI shows a highly concentrated negative deviation, indicating relatively stable retrieval behavior under strong absorption constraints, but with a persistent systematic low bias in elevated aerosol conditions. For the dust cases (Fig. 14c), both methods perform better than in smoke scenarios, which may be related to differences in the physical and optical properties of the two aerosol types. In general, smoke particles are typically smaller and more nearly spherical, exhibiting stronger polarization scattering signatures in the visible range, which is theoretically more favorable for polarimetric retrieval. In contrast, coarse-mode mineral dust particles are highly non-spherical and exhibit weaker polarization signals, reducing the sensitivity of polarization measurements to coarse dust properties (Kalashnikova et al., 2011). However, dust events often present relatively well-defined layered structures, such as the SAL, where aerosol concentration exhibits a clear vertical gradient. This structural feature facilitates the retrieval of ALH or centroid height. In contrast, smoke aerosols are often associated with multilayer structures or mixing with cloud layers, and their strong absorption further complicates radiative transfer processes. These factors increase retrieval uncertainty and lead to a relatively larger dispersion in smoke-related retrieval results.

Overall, in the validation against ATLID lidar observations, the ALH retrieved from HARP2 multi-angle polarimetric measurements shows generally good agreement with ATLID observations. For both smoke and dust aerosol cases, the HARP2 retrievals are able to capture the spatial variability of ALH and exhibit relatively small errors and stable statistical characteristics in most cases. This indicates that multi-angle polarimetric observations contain substantial information content for resolving aerosol vertical structure and have strong potential for ALH retrieval. In contrast, the intercomparison results show that the TROPOMI ALH product exhibits a certain degree of systematic underestimation in some cases, although it remains well correlated with ATLID observations overall. This suggests that the product is robust in capturing the variability of ALH. As an operational global aerosol height product, TROPOMI benefits from its wide swath coverage and stable observational capability, making it valuable for global aerosol monitoring and long-term variability studies. It should be noted that although the HARP2 multi-angle polarimetric approach demonstrates relatively high accuracy in ALH retrieval, the results are still affected by uncertainties in aerosol optical model assumptions, surface reflectance parameterization, and viewing geometry. In complex aerosol environments, these factors may introduce additional uncertainties. Future work could further improve the stability and applicability of ALH retrievals by introducing more refined aerosol optical models, improving surface reflectance parameterizations, and incorporating multi-source satellite observations for joint constraints.

Overall, the two passive remote sensing approaches have complementary strengths. HARP2 multi-angle polarimetric observations provide higher information content and show potential advantages in detailed characterization of aerosol vertical structure, while the TROPOMI product plays an important role in long-term, stable global aerosol monitoring. Together, they can provide more reliable data support for studies of aerosol three-dimensional structure and its climatic impacts.