Dataset Construction for Radiative Transfer Modeling: Accounting for Spherical Curvature Effect on the Simulation of Radiative Transfer Under Diverse Atmospheric Scenarios

Abstract

Conventional radiative transfer (RT) models often adopt the plane-parallel (PP) approximation, which neglects Earth’s curvature and leads to significant optical path errors under large solar or sensor zenith angles, particularly for high-latitude regions and twilight conditions. The spherical Monte Carlo method offers high accuracy but is computationally expensive, and the commonly used pseudo-spherical (PSS) approximation fails when the viewing zenith angle exceeds 80°. With the increasing application of machine learning in atmospheric science, the efficiency and angular limitations of spherical RT simulations may be overcome. This study provides a physical and quantitative foundation for developing a hybrid RT framework that integrates physical modeling with machine learning. By systematically quantifying the discrepancies between PP and spherical RT models under diverse atmospheric scenarios, key influencing factors—including wavelength, solar and viewing zenith angles, aerosol properties (e.g., single scattering albedo and asymmetry factor), and PP-derived radiance—were identified. These variables significantly affect spherical radiative transfer and serve as effective input features for data-driven models. Using the corresponding spherical radiance as the target variable, the proposed framework enables rapid and accurate inference of spherical radiative outputs based on computationally efficient PP simulations.

1. Introduction

The atmospheric radiative transfer model is crucial for satellite remote sensing inversion, directly influencing the quality and application value of data. Traditionally, it simplifies calculations by treating Earth’s atmosphere as an infinitely extended plane-parallel layer and ignoring its curvature. However, this assumption leads to significant light path errors at large zenith angles (>75°). In 1969, Plass and Kattawar [1] revealed that the plane-parallel approximation begins to break down at solar zenith angles (SZAs) exceeding 75–80°, which leads to modeling or simulation errors that degrade the remote-sensing accuracy of satellites. In contrast, the spherical atmospheric radiation transport model considers the spherical geometry of Earth, making light path calculations more accurate. The model better simulates radiation conduction, particularly in the morning and evening (Figure 1a) and at high latitudes (Figure 1b), thus improving satellite remote sensing accuracy.

The radiative transfer theory in a spherical atmosphere originated in the 1960s. Lenoble et al. and Smokty et al. first established the theoretical framework for the propagation of light in spherical atmospheres [2,3]. Meinel et al. subsequently developed a systematic method for calculating light transport paths in spherical atmospheres [4]. However, their model only considered single scattering and failed to fully capture the multiple scattering phenomena that are observed under real atmospheric conditions. With the development of computational technology, Marchuk pioneered the use of the Monte Carlo method to study radiation transport in a spherical atmosphere [5]. By simulating the scattering trajectories of a large number of photons (on the order of 10⁶ to 10⁸ [6]) under various atmospheric conditions—including absorption, aerosol scattering, and surface reflection—they were able to calculate the spatial distribution of the radiation field. This method, based on the statistical principle of Plass and Kattawar, models radiative transfer through stochastic photon interactions with the scattering phase function [1]. Its primary advantage is accurately handling multiple scattering in complex scenarios such as three-dimensional cloud systems and non-uniform atmospheres. Therefore, it has been widely used to develop benchmark models, such as the Monte Carlo Method (MYSTIC) and the Spherical Planetary Atmospheric Radiative Transfer Algorithm (SPARTA) [7]. However, Monte Carlo methods are computationally inefficient. As early as 1964, Hammersley and Handscomb found that the extensive computation required for reliable statistical results severely limits the use of these methods [8].

Caudill et al. introduced a pseudo-spherical approximation (PSS) method [9] to enhance computational efficiency by accounting for Earth’s curvature with an optical range length correction factor while maintaining the effectiveness of the plane-parallel geometric framework. However, this assumption, which ignores the actual curvature of the atmosphere, limits the PSS accuracy in high-precision remote sensing. Mei et al. addressed this by applying spherical corrections to the single-scattering term while retaining the plane-parallel approach for multiple scattering [10]. This hybrid algorithm design improves the computational efficiency by 2–3 orders of magnitude compared with traditional methods and maintains high simulation accuracy (error < 5%) for zenith angles < 84°.

Spurr et al. extended the PSS method to vector radiative transfer [11,12] and accurately simulated a polarized radiation field with a spherical correction factor. However, Korkin et al. identified limitations, including an upper limit constraint of 79.9° in the zenith angle range [13,14]. He et al. also claimed that the influence of Earth’s curvature increases rapidly with SZA, reaching 1%, 3%, and 12% for SZAs of 75°, 80°, and 85°, respectively [15].

Zhai and Hu proposed an improved pseudo-spherical approximation (IPSS) [16] that overcomes the bottleneck of the zenith angle accuracy of the traditional PSS method. This method uses the single/multiple scattering radiation ratio conservation property proposed by Adams et al. [17,18]. It maps radiation solutions between plane-parallel and spherical geometries by (1) solving single scattering in a spherical atmosphere, (2) calculating multiple/single scattering intensity ratios in a plane-parallel atmosphere, and (3) verifying whether the ratios remain constant in a spherical geometry to derive multiple-scattering solutions. This method significantly outperforms the conventional PSS method by maintaining the error within 3% over the zenith angle range of 75–86° in the Rayleigh scattering ideal experiment. However, this model is still confined to pure Rayleigh atmospheres, and its applicability under conditions involving aerosols and clouds has not been confirmed.

Combining physical mechanism models with data-driven methods is an important development direction. Korkin et al. polarized radiative transfer in a spherical atmosphere simulated using Monte Carlo and discrete ordinate methods and yielded benchmark results to assess and improve approximate models under strong curvature effects [14]. The ratio conservation assumption used in the IPSS method acts as a physically constrained machine-learning idea, thereby guiding new hybrid algorithm development. Future research should focus on three key issues: developing generalized algorithms for nonuniform atmospheres, creating a neural network architecture that accounts for Earth’s curvature, and improving accuracy degradation for large zenith angle observations. Specifically, new spherical radiative transfer models must be developed for high temporal and spatial resolution remote sensing with the new generation of geostationary meteorological satellites, thereby ensuring physical accuracy and computational efficiency.

Considering the effect of Earth’s spherical curvature, traditional physical models become increasingly complex, which in turn affects their computational efficiency. Meanwhile, pseudo-spherical approximation methods are subject to limitations at large solar zenith angles. In contrast, machine learning methods have been widely applied in recent years due to their ability to address complex nonlinear problems. For example, Hao Li et al. employed a neural network to perform atmospheric correction of ocean color remote sensing data under high solar zenith angle conditions, and it significantly improved the stability and usability of the retrieved reflectance [19]. Therefore, this study aims to construct an accurate and diverse dataset to facilitate the integration of traditional physical models (plane-parallel atmospheric models) with machine learning approaches in subsequent modeling efforts. This integration seeks to enhance computational efficiency and model simulation result accuracy.

To ensure the accuracy of the dataset and more realistically capture the propagation paths and scattering processes of photons in the spherical atmosphere, the Monte Carlo method was employed to consider photon trajectories and provide an accurate dataset of radiative intensity in the spherical atmosphere. This dataset will subsequently be used to identify strongly correlated factors and develop the spherical atmospheric radiation models that combine traditional physical methods with machine learning. Radiative intensities under plane-parallel and spherical geometries are simulated and compared across five representative atmospheric conditions: clear-sky, aerosol, water cloud, ice cloud, and aerosol–water cloud mixed atmospheres. Through feature selection, key variables are identified, including wavelength, solar and viewing geometry, surface albedo, aerosol optical properties, and plane-parallel radiance. These efforts provide a robust data foundation for the subsequent development of machine learning models.

2. Data and Methods

The open-source software library libRadtran (version 2.0.4) (www.libradtran.org) was used for the atmospheric radiation simulations. According to foundational studies [20,21], this software uses radiative transfer theory to simulate light propagation in the atmosphere by solving the radiative transfer equation (RTE). It encompasses molecular absorption (ozone and water vapor), aerosol scattering, cloud microphysical effects (liquid water clouds, ice clouds), and surface reflection. The software further integrates various solvers, including the Discrete Ordinates Radiative Transfer (DISORT) [22], Polarization Solver, and MYSTIC, to satisfy the accuracy and efficiency requirements for different scenarios, including irradiance, polarized radiation, and 3D complex cloud structure. However, in the current release of libRadtran (version 2.0.4), vector radiative transfer calculations are only available for clear-sky atmospheres, while simulations involving clouds and aerosols remain limited to one-dimensional scalar mode. Accordingly, this study incorporates Q-component analysis exclusively under clear-sky conditions. Extension to more complex atmospheric scenarios will be pursued as soon as vector solvers become available for cloudy and aerosol-laden cases in future software updates.

MYSTIC is a 3D vector solver used in libRadtran to solve the radiation transmission issue, employing photon random-path tracking [23,24]. In plane-parallel geometry, MYSTIC simulates the multiple scattering process of photons entering from the top of the atmosphere using forward tracking. The initial photon position is determined by the solar zenith angle (SZA) and azimuth (PHI). Scattering events involving molecules, aerosols, and clouds are considered until the photons are absorbed or escape the atmospheric boundary. For spherical geometry, MYSTIC can provide more accurate results in subsequent experiments compared to a pseudo-sphere model.

MYSTIC is particularly suitable for satellite remote sensing, cloud–aerosol interaction, and polar radiation balance research. Moreover, its algorithm supports full vector polarization calculations and analyzes changes in the polarization state with Stokes parameters. MYSTIC is superior to traditional methods, such as DISORT or second-rate approximation, when accounting for the spherical effect of large SZAs. In this study, the results for the parallel plane atmosphere were obtained using a model based on the MYSTIC method with the assumption of a plane-parallel approximation, where the atmosphere is considered as a horizontally uniform, infinitely extending plane layer. The spherical atmospheric model, on the other hand, fully considers Earth’s curvature effects and employs the unique vector radiative transfer algorithm of MYSTIC. This algorithm accurately simulates the atmospheric radiative transfer process under various angle conditions by tracing the random paths of photons in a spherical coordinate system. Both models are configured with identical atmospheric conditions and use the MYSTIC algorithm to eliminate any differences that could arise from varying computational approaches when calculating the I, Q, U, and V components at the top of the atmosphere. In the aerosol-laden atmosphere, aerosol optical thickness in libRadtran is represented by adjusting the Angstrom parameters. In this study, the Angstrom exponent value is held constant while the turbidity coefficient is varied from 0.2 to 1 to investigate the impact of different aerosol optical thicknesses on radiative intensity errors. In addition, for the water cloud atmosphere, the cloud optical thickness is set to 15.0 and the single scattering albedo to 0.9999. To ensure efficiency and accuracy, 1,000,000 photons were used in calculations for all data output. The parameter settings under various atmospheric conditions in this study are defined in

Table 1. Parameters list of experiments.

Atmospheric Type	Parameter	Values/Options
Clear atmosphere	Atmospheric Profile File	TRO, MLS, MLW, SAS, SAW, STA
	Wavelength	450 nm, 550 nm, 650 nm
	Solar Zenith Angle (SZA)	[0°, 89°]
	Sensor Azimuth Angle (PHI)	0°, 90°
	Viewing Zenith Angle (VZA)	[0°, 79°]
	Surface Albedo ($\alpha$)	0.08 (ocean), 0.15 (tree), 0.2 (grass), 0.3 (stone), 0.4 (ice), 0.8 (snow) [25]
Aerosol-laden atmosphere	Aerosol Type	Water-insoluble (INSO), Water-soluble (WASO), Black Carbon (SOOT), Sulfate Droplets (SUSO), Sea Salt Coarse Mode (SSCM), Transported Mineral (MITR)
	Single Scattering Albedo (SSA)	Based on the OPAC database: (450 nm): 0.705586, 0.968506, 0.245706, 0.999999, 0.999996, 0.755423 (550 nm): 0.730000, 0.961548, 0.208889, 0.999999, 0.999999, 0.837441 (650 nm): 0.750693, 0.953461, 0.175677, 0.999999, 0.999993, 0.876215
	Asymmetry Factor (gg)	Based on the OPAC database: (450 nm): 0.915526, 0.908406, 0.509439, 0.941813, 0.943903, 0.908330 (550 nm): 0.911533, 0.895515, 0.469539, 0.940180, 0.942755, 0.896900 (650 nm): 0.908112, 0.881702, 0.431983, 0.936384, 0.941450, 0.889530
	Optical depth	Angstrom parameters: $\alpha =$$1.1, \beta =0.2,$ 1
Water cloud atmosphere	Asymmetry Factor (gg)	Based on the OPAC database: 0.950884 (450 nm), 0.950601 (550 nm), 0.949917 (650 nm)
Ice cloud atmosphere	Asymmetry Factor (gg)	Based on OPAC database: Columnar ice cloud: 0.864696 (450 nm), 0.868618 (550 nm), 0.871774 (650 nm) Platelike ice cloud: 0.874529 (450 nm), 0.885377 (550 nm), 0.891587 (650 nm)

. The data primarily cover four types of atmospheric conditions: clear-sky atmosphere, aerosol-laden atmosphere, water cloud atmosphere, and a mixed atmosphere containing both clouds and aerosols. Six typical profiles—tropical (TRO), mid-latitude summer (MLS), mid-latitude winter (MLW), sub-polar summer (SAS), sub-polar winter (SAW), and U.S. standard (STA)—were also simulated to characterize the atmosphere at various latitudes and explore the influence of different temperature and pressure conditions on radiation intensity.

Based on the physical and chemical characteristics of various aerosols and their distinct impacts on the radiation transmission mechanism, six representative aerosol types were selected: water-insoluble (INSO), water-soluble (WASO), black carbon (SOOT), sulfate droplets (SUSO), sea salt coarse mode (SSCM), and mineral transport (MITR). INSO and MITR, as representative mineral aerosols, exhibit non-spherical particle characteristics that amplify scattering effects through extended optical paths within a spherical geometry. This phenomenon is especially pronounced at large SZAs, where the spherical model significantly enhances multiple scattering contributions. Due to their hygroscopic nature, WASO and SUSO dynamically adjust their particle sizes and refractive indices under varying humidity conditions, resulting in differential scattering effects in plane-parallel and spherical models [26]. SOOT, a highly absorbent aerosol, exhibits significant variations in shortwave absorption efficiency within the spherical model, as changes in optical path length lead to notable differences in radiative forcing [27].

SSCM represents coarse-mode sea salt particles typical of marine environments, with scattering characteristics that reveal radiative transfer effects in the marine boundary layer. Additionally, MITR’s long-range transport properties enable the assessment of how the vertical distribution of dust aerosol in a spherical atmosphere impacts radiative balance. This study covers scattering-dominated (INSO and SSCM), absorption-dominated (SOOT), and mixed-type (WASO and SUSO) aerosols, thus providing a multidimensional perspective for quantifying the uncertainties in aerosol radiative effect assessments introduced by geometric model assumptions (plane-parallel vs. spherical). Information related to the optical properties of the six aerosols and water clouds at different wavelengths was obtained from the OPAC database (https://opac.userweb.mwn.de/radaer/opac.html (accessed on 2 April 2025)) to ensure accurate simulation conditions [28].

Given that ice clouds have a relatively smaller impact on solar radiation compared to liquid water clouds, and their complex morphologies present significant challenges for quantification in subsequent modeling efforts, this study is limited to analyzing the radiative intensity differences of two representative ice cloud types—columnar and platelike—across different wavelengths.

Additionally, the Medium Resolution Spectral Imager-III (MERSI-III) onboard the FY-3F satellite (National Satellite Meteorological Centre, China) was used to select wavelengths based on their key scattering and absorption characteristics: 450, 550, and 650 nm. The 450 nm (blue light) wavelength in the Rayleigh scattering region is highly sensitive to atmospheric molecular scattering. The 550 nm (green light) wavelength is near the vegetation reflection peak and visible light radiation, characterizing the surface–atmosphere interaction. The 650 nm (red light) wavelength is greatly influenced by aerosol scattering and absorption and is related to vegetation’s “red edge” characteristics.

3. Results Analysis

3.1. Clear Atmosphere

By analyzing the radiation intensity vector under clear atmospheric conditions, this study preliminarily concludes that the I-component and Q-component exhibit certain patterns, while the distribution of the U-component appears more irregular. In addition, the V-component is close to 0 under the wavelength and azimuth angle settings used in this study.

The Rayleigh scattering effect was significant at 450 nm, with atmospheric path length affecting radiative transfer. The radiative intensity differences between plane-parallel and spherical models grew nonlinearly with increasing SZA (Figure 2). When the SZA was <60° and VZA = 0°, the error was <1%, indicating the high accuracy of the plane-parallel assumption (Figure 2a). At 60° < SZA < 75°, the error increased slightly. However, when SZA > 75°, errors increased rapidly, peaking near 59% at SZA = 89°. This aligns with He et al. [1], who found that the plane-parallel model overestimates optical path length by neglecting Earth’s curvature, thereby underestimating radiative intensity. Moreover, the radiative intensity deviation between the plane-parallel model and spherical models slightly increased when VZA $\in$ [0°, 60°] but rose rapidly when VZA > 60°. For instance, at SZA = 89°, the spherical model’s radiative intensity was markedly higher than that of the plane-parallel model at VZA = 79° due to a shorter optical path, resulting in up to a 90% difference (Figure 2a).

The azimuth angle had an insignificant effect on the error, with no apparent change in the overall trend between PHI = 90° and 0° (Figure 2a,b). In addition, the errors in cases I and Q were evaluated based on vector radiative transfer. The error in the Q-component was relatively small at SZA < 80° and VZA $\in$ [0°, 60°] but increased significantly at SZA > 80° or VZA > 60° (Figure 2c,d). Notably, at equal SZAs and VZAs (VZA = SZA = 0°), certain simulated results in Q became abnormal. Therefore, to ensure reliable simulations, cases where VZA and SZA were the same were avoided by adjusting their difference by a small degree.

At wavelengths of 550 and 650 nm (Figure 3 and Figure 4), the error trends for I and Q components generally resembled those at 450 nm but were larger. Specifically, the error in the I component increased sharply at VZA $\in$ [75°, 79°] with increased wavelength. For instance, when PHI = 0° and SZA = 75°, the error of the I component at VZA $\in$ [75°, 79°] and 650 nm was 106% higher than that at 450 nm (Figure 2a and Figure 4a). The error in the Q component was notably significant at SZA < 80°. For example, when PHI = 0° and VZA = 75°, the error for the Q component at SZA $\in$ [0°, 79°] increased from 35% at 450 nm (Figure 2c), to 55% at 550 nm (Figure 3c), and reaching 70% at 650 nm (Figure 4c).

To investigate whether surface albedo affects radiative intensity differences, this study compares the variations in the I and Q components under six different surface albedo conditions while keeping other parameters constant. Analysis of Figure 5 and Figure 6 reveals that the results varied across different surface albedos [25]. Higher surface albedo indicates stronger ground-scattered radiation, which leads to a more pronounced influence when considering the spherical curvature effect. For instance, a comparison between Figure 5a,f—which represent the surfaces with the greatest difference in albedo—shows that at SZA $\in$ [0°, 60°] and VZA $\in$ [75°, 79°], the difference between plane-parallel and spherical radiative intensities for ocean and snow surfaces differs substantially by approximately 35%. Similarly, Figure 6a,f show notable differences even at moderate SZA and VZA values, with maximum deviations reaching approximately 40%. Moreover, the differences remained significant at larger angles. These results suggest that surface albedo exerts a measurable influence on radiative intensity differences; therefore, it will be considered a relevant factor in future model development.

Figure 7 and Figure 8 illustrate the differences in radiation intensity at a wavelength of 450 nm under various atmospheric profiles when the azimuthal angle is 0°. Despite some fluctuations (i.e., for some atmospheric profiles where both SZA and VZA are either 0° or approximately 79°), the VZA was adjusted by a small degree to address partial abnormal data. Therefore, the fluctuations observed in some of the corresponding curves under these two conditions can be attributed to the inherent variability in Monte Carlo simulations when the solar zenith angle and viewing zenith angle are similar. Moreover, the variation patterns in radiation intensity percentage with SZA and VZA were relatively consistent across the six profiles. Given that the different profiles correspond to various temperature and pressure conditions, it was concluded that temperature and pressure exert minimal effects on the differences between spherical and plane-parallel models.

3.2. Aerosol-Laden Atmosphere

Among the different aerosol types, the radiative transfer differences between the spherical and plane-parallel models exhibited significant variations in wavelength and SZA. These differences were driven primarily by the combined effects of aerosol optical properties (absorption/scattering ratio and complex refractive index) and particle physical characteristics (size distribution and shape). It is worth noting that the images of the I and Q components under a clear atmosphere showed generally consistent trends. Therefore, under atmospheric conditions containing aerosols or water clouds, only the most representative I-component images are presented in this experiment.

The differences for the absorbing aerosol-type SOOT remained close to 0% within SZA $\in$ [0°, 80°] (Figure 9c, Figure 10c and Figure 11c). This was attributed to the high absorption properties of black carbon, which reduced the impact of multiple scattering and minimized the differences in scattering paths between the spherical and plane-parallel models with the absorption effect. For the scattering aerosol-type WASO (Figure 9f, Figure 10f and Figure 11f), which is characterized by smaller particle sizes, the scattering was strongly isotropic at 450 nm, resulting in a relatively flat variation in the difference curves. In contrast, the scattering phase function of the large-particle, non-spherical scattering aerosol type and mineral-transported aerosols (Figure 9b, Figure 10b and Figure 11b) was complex. In particular, under long-path conditions (SZA > 80° or VZA > 60°), the spherical model more accurately simulated the curvature effects of forward scattering. Mixed-type aerosols, which are insoluble, exhibited nonlinear growth in error with increasing SZA, except at SZA = 80° for 450 and 550 nm (Figure 10a and Figure 11a). At 450 nm, forward scattering overlapped with Rayleigh scattering, rendering the spherical model more sensitive to the absorption-enhancing effects under long optical paths. Initially, differences decreased, followed by a rapid increase. Conversely, at 650 nm, where scattering predominated, differences consistently accumulated. The large-particle SSCM particles, governed primarily by Mie scattering, exhibited higher sensitivity of the scattering phase function to model the differences at 550 nm and 650 nm. At 450 nm, forward scattering peaked significantly, stabilizing model differences at approximately −2% (Figure 9d). However, at 550 nm and 650 nm and within SZA $\in$ [60°, 80°], the spherical model exacerbated the negative bias in backscattering due to the curvature effect, leading to a pronounced negative increase in differences (Figure 10d and Figure 11d). Unlike SSCM, SUSO had smaller particles. At 450 and 550 nm, forward scattering partially offset the plane-parallel approximation errors, resulting in a difference of approximately 0% at SZA = 80° (Figure 9e and Figure 10e). However, backscattering increased at 650 nm (Figure 11e). The spherical model more accurately characterized the spatial distribution of backscattering, with a −15% error, indicating that the plane-parallel models underestimated backscattering.

In addition to examining the radiative intensity variations associated with different aerosol types, this study also investigates the influence of aerosol optical depth (AOD) using SOOT as a representative example. Specifically, the radiative intensity differences at wavelengths of 450 nm, 550 nm, and 650 nm under varying AOD conditions are analyzed, as illustrated in Figure 12. The results clearly show that although the overall trend of increasing error with larger solar zenith angles remains unchanged, AOD exerts a pronounced impact on radiative intensity. This effect is particularly evident at 650 nm, where increasing AOD leads to greater fluctuations in the differences between the plane-parallel and spherical models across solar zenith angles. These findings further underscore the significance of AOD as an essential optical property of aerosols in shaping the accuracy of radiative transfer simulations and highlight its importance as a critical feature for future machine learning model development.

Theoretically, this conclusion is also consistent with the principles of light scattering. In the presence of aerosols in the atmosphere, Mie scattering dominates. This parameter is distinct from Rayleigh scattering under clear-sky conditions. The scattering phase function, particularly the asymmetry factor, plays a critical role in the calculation of multiple scattering. Variations in the phase functions of different types of aerosols lead to differences in multiple scattering paths. Consequently, when Earth’s curvature is considered, the multiple scattering trajectories are also affected when the Earth’s curvature is taken into account and thereby affect the results of radiation calculations.

3.3. Liquid Water Cloud Atmosphere

To clarify the assessment of atmospheric conditions influenced by water clouds, this study focuses on the results at several specific angles rather than the radiance differences for each angle. The trends in radiance angle differences between the plane-parallel and spherical models were compared at 450, 550, and 650 nm with an azimuth angle of 0°. Significant nonlinear growth in radiative intensity differences occurred after an SZA of 80° at a viewing angle of 0°, with the values fluctuating around 0% within 0–80° (Figure 13a). In contrast, when the viewing angle was 75°, significant growth in the differences began at an SZA of 75°, with a slight upward fluctuation trend between 0° and 75°. Moreover, compared to 450 nm, the radiative intensity differences at 550 and 650 nm were slightly larger (Figure 13a–c). However, the overall variation trends were consistent across all wavelengths.

3.4. Ice Cloud Atmosphere

In Figure 14 and Figure 15, they illustrate two representative types of ice clouds—columnar ice clouds and platelike ice clouds—as examples and reveal that radiative intensity differences under ice cloud conditions are generally large across all angular configurations. This effect is particularly pronounced at longer wavelengths (650 nm), where the discrepancies between the plane-parallel and spherical models are greater than those observed under other atmospheric conditions, with less evident regularity. These characteristics are primarily attributed to the complex optical properties of ice clouds. That is, the radiative-intensity differences between the spherical and plane-parallel atmospheric models vary across cloud-particle conditions, a behavior consistent with cloud particles adhering to the same fundamental light-scattering principles as aerosols. Therefore, the contribution of ice cloud conditions to the performance of the forthcoming machine learning model remains uncertain because the underlying nonlinear relationships are considerably complex.

3.5. Liquid Water Cloud and Aerosol-Laden Atmosphere

To evaluate atmospheric conditions influenced by water clouds and aerosols, two aerosol types (SOOT and SSCM) with significantly different particle sizes and optical properties were compared (Figure 16). The radiative intensity differences between SOOT and SSCM were similar at 450 nm, particularly for large SZAs, suggesting a minor impact of aerosols on overall radiance under conditions with water clouds and aerosols. When water clouds and aerosols were included, the radiative intensity error between the plane-parallel and spherical models was similar in growth rate and magnitude to that for water clouds alone within SZA $\in$ [85°, 89°]. The minor discrepancies between the two aerosol types and the water cloud-only atmosphere at smaller angles (SZA $\in$ [0°, 80°]) may have resulted from slight fluctuations in Monte Carlo calculations.

4. Conclusions

This study systematically examined the theoretical advantages of the spherical atmospheric radiative transfer model versus the plane-parallel approximation model, the sources of associated errors, and their differences under varying atmospheric conditions.

The plane-parallel model neglects Earth’s curvature, leading to significant optical path errors, particularly at large SZAs or VZAs > 60°. When the SZA surpasses 75°, the error increases sharply. Even at small SZAs, the error exceeds 5% when the VZA is greater than 75°. Comparative analysis across various atmospheric profiles indicates that these error characteristics are only weakly influenced by vertical variations in temperature and pressure.

In an aerosol-laden atmosphere, aerosols alter the divergence between spherical and plane-parallel models by affecting the scattering phase function. Coarse-mode aerosols, in particular, amplify this divergence, necessitating more substantial curvature corrections. When water clouds are present alongside aerosols, the contribution of aerosol properties to the total error diminishes. However, the presence of clouds—especially under large zenith angles—introduces stronger nonlinearities and increases overall model discrepancies. Even more pronounced radiative intensity differences are observed in ice cloud atmospheres, which exhibit the largest deviations across all angular configurations. Although the ice cloud types analyzed in this study are not exhaustive, the inclusion of two representative forms is sufficient to demonstrate the significant impact of ice clouds. Due to their complex and irregular optical properties, the associated errors are more substantial and less predictable compared to those under other atmospheric conditions.

As shown in the flowchart (Figure 17), based on the multi-scenario radiative dataset constructed in this study, which encompasses clear-sky, aerosol, water cloud, and mixed atmospheric conditions, we conducted feature screening and identified key variables including wavelength, solar geometry, viewing geometry, surface albedo, aerosol optical properties, and plane-parallel radiance. These features form the basis of a framework designed to rapidly and accurately infer spherical radiative intensity using a computationally efficient plane-parallel model, thereby significantly enhancing the performance of vector radiative transfer simulations. This study also generates a diverse and high-quality training dataset, thereby providing a solid foundation for subsequent machine learning model development and laying the groundwork for building a radiative transfer model capable of achieving Monte Carlo-level accuracy with over a 100-fold improvement in computational efficiency. The resulting model is expected to substantially improve the accuracy and efficiency of remote sensing retrievals from dawn-dusk orbit satellites such as FY-3E, as well as from high-latitude satellites. Moreover, it will enhance global monitoring capabilities of solar radiation and the diurnal variations of geophysical parameters, thereby enabling more accurate characterization of global climate change. However, it is important to note that due to the complex optical properties of ice clouds, the radiative intensity differences under such conditions exhibit irregular and highly nonlinear behavior. Therefore, careful consideration of ice cloud optical characteristics is essential in the early stages of model development to avoid compromising model accuracy.