The PPA neglects the Earth's curvature by defining the surface and all the atmospheric layer boundaries as infinite parallel horizontal planes. This approximation allows to simplify the equation of radiative transfer (Eq. ) by making it depend on three variables only, which are the altitude z, the zenith angle θ and the azimuth angle φ. Equation (1) can then be expressed as : 3cos⁡θdI(z,θ,φ)κextρdz=S(z,θ,φ)-I(z,θ,φ), and can be written as follows by merging it with Eq. ():

4cos⁡θdI(z,θ,φ)κextρdz=κscaκext∫ΩP(θ)I(z,θ,φ)4πdΩ-I(z,θ,φ).

The plane-parallel assumption enables the use of simple analytical resolutions of the radiative transfer equation, which drastically lower the computation time. PPA is therefore frequently made in radiative transfer solvers such as the discrete ordinate radiative transfer (DISORT) solver , included in the libRadtran software , the Doubling-Adding (DOAD) method , available in the ARTDECO software (https://www.icare.univ-lille.fr/artdeco/, last access: 27 April 2026) and the modified Sobolev approximation analytical solver , used for aerosol properties retrieval from Meteosat satellite data .

In Sect. , we show how PPA is found to be accurate in several studies for angular geometries close to the zenith/nadir configurations, but limited for high zenith angles. Figure  illustrates the geometric differences between the plane-parallel assumption and the true Earth's sphericity (represented by the spherical-shell assumption described in Sect. ). One can see that neglecting the Earth's sphericity leads to an overestimation of the thickness of the atmosphere, and therefore of the optical path (colored in red on the figure). This miscalculation can induce important inaccuracies in the simulated TOA reflectances, which in turn can negatively impact the estimation of surface and aerosol properties. It is worth noting that these errors may grow with view and solar zenith angles (due to the increase in the optical path error when using PPA), which may become especially important in the case of geostationary satellites allowing observations in a broad angular range.

The SSA assumes a spherical Earth's surface, surrounded by a multi-layer spherical atmosphere defined by their corresponding radii as shown in Fig. . This assumption is closer to reality than the PPA, and can be implemented in various ways depending on the level of precision required.

First, radiative transfer codes referred to as pseudo-spherical methods integrate SSA corrections in otherwise PPA-based solvers. Following the statement by that most of the multiple scattering occurs in a narrow cone around the zenith, many pseudo-spherical methods consist in computing single scattering in SSA for both incident and reflected beams, while calculating the contribution of multiple scattering in PPA. This approach was already applied by for ozone vertical distribution estimation purposes. More recently, a similar logic is found in the enhanced pseudo-spherical method implemented in LIDORT and VLIDORT, respectively the linearized and vector upgrades of the discrete-ordinates RT code DISORT , and in the radiative transfer solver RTSOS radiative transfer model based on the successive order of scattering;. Another well-known pseudo-spherical approach was developed by in the atmospheric-correction scheme developed for the SeaWiFS (Sea-Viewing Wide-Field-of-View Sensor) satellite mission. There, the contribution of Rayleigh scattering to the total radiance is calculated using a SSA geometry, while the remaining pieces of the radiative transfer algorithm are calculated in PPA.

Second, radiative transfer algorithms based on Monte Carlo methods allow simulations in true spherical geometry by modeling particle-based light propagation. These statistical approaches relying on random sampling of backward-scattered photons were first used for radiative transfer purposes by . More recently, one can cite the well-known Monte Carlo based MYSTIC (Monte Carlo code for the physically correct tracing of photons in cloudy atmospheres) solver from the libRadtran package and the radiative transfer code running on graphics processing units (GPU) named Speed-Up Monte Carlo Advanced Radiative Transfer code with GPU SMART-G;, which is described in Sect. . These full-spherical RT codes have proven their accuracy even for high zenith angles configurations . However, their main limitation is the computation requirements that are significantly heavier than for PPA-based algorithms.

It is worth noting that the spatio-temporal resolution of FCI full-disk images (available every 10 min at 1 km resolution) makes full-spherical and pseudo-spherical radiative transfer codes unsuitable if near real-time data processing constraints must be met. Hence, this study will focus on assessing the errors caused by the use of the PPA in the context of geostationary satellites, in order to help develop approximate but fast corrections methods suited for operational applications.

Rayleigh scattering is the main radiance-inducing atmospheric process in the short visible spectrum, corresponding to the first measuring channels of most geostationary satellite imagers. Since the main difference between the PPA and SSA geometries lies in the difference in the optical path traveled by the radiation, it will greatly affect Rayleigh scattering. First, we describe the distinct effects of Earth's sphericity in the occurrence of high SZA or high VZA, which results in different impacts on TOA radiances. While both geometries involve long optical path, they affect TOA radiance differently due to the fundamental differences between PPA and SSA .

At high SZA, the primary effect is due to the attenuation of the incoming solar beam. In PPA, the optical path of the direct solar beam scales as 1/cos⁡(θs) and tends toward infinity as SZA approaches 90°, leading in excessive extinction of the incident light. In contrast, the SSA geometry allows the solar beam to cross a finite atmosphere or illuminate layers tangentially (also known as the twilight effect), allowing more photons to interact and scatter. Consequently, PPA tends to underestimate the scattered radiance compared to SSA, inducing a negative bias in PPA-computed TOA radiances at the beginning and end of the day (i.e., when SZA is high). Although high VZA causes an analogue overestimation of the optical path, the difference is due to the integration path along the line of sight. In PPA, a high VZA implies an optical path through an infinite slab, artificially increasing the scattering optical depth and thus the Rayleigh signal. In SSA, in the contrary, a line of sight near the limb crosses a finite path before exiting. This finite path limits the amount of scattering contributing to the radiance compared to the infinite PPA path. As a result, PPA overestimates the observed radiance at high VZA, inducing a positive bias in PPA-computed TOA radiances near the edge of the geostationary field of view (i.e., where VZA is high) .

However at larger wavelengths, in the red and near-infrared spectrum, Rayleigh scattering becomes weak while molecular absorption grows important, becoming the main process to affect radiation propagation in the atmosphere. Although many earlier studies have studied the Earth's sphericity effects, and specifically the impacts of using PPA over SSA on TOA radiance simulations, depending on the scene's geometry , we find a lack of studies on sphericity effects in the red and near infrared, and on the assessment of such effects in weak Rayleigh diffusion and therefore more gas-absorbing dominated wavelengths. Furthermore, discrepancies exist in the literature regarding the angular thresholds beyond which the PPA should not be used, which can be explained by the differences in the requirements of each study. Indeed, the simulations configurations vary depending on the research application, and therefore the quantification of the PPA impact is given for application-specific parameters including the scene's geometry, the atmosphere description, the wavelength of observation, the consideration or neglect of polarization, etc. The following literature review aims to give an order of magnitude of the sphericity effects and the validity of PPA depending on the sun-sensor geometry.

Regarding view zenith angle, found that using PPA geometry instead of SSA with VZAs > 60° could cause errors up to 20 % in simulated TOA radiances in a Rayleigh scattering atmosphere of optical thickness 0.25 over a black surface. confirmed that VZAs > 60° can lead to important relative errors between PPA and SSA simulations in their case study, up to to 12 % on TOA Rayleigh scattering radiances at 412 nm over a flat sea surface depending on the value of SZA (both zenith angles are limited to a maximum value of 85° in this study). It is worth noting a general lack of VZA-dependent studies, which makes difficult to reach a consensus on the validity of PPA according to view geometry. As for solar zenith angle, several works tried to identify the SZA value above which SSA should be preferred over PPA. For example, found that SZA = 72° caused a relative error of 4 % between the radiance computed with PPA and SSA geometries. showed that the absolute relative error on TOA Rayleigh radiances exceeds 2 % for SZAs > 80° for all VZA values at 412 nm over a smooth ocean surface in the cross azimuthal plane and concluded their study on the comparison between PPA and SSA performances for ocean color purposes by stating that “the effects of the curvature are negligible for SZA values below 70°”. showed that SZA = 75° at 412 nm over a flat sea surface caused a 1 % relative error between PPA and SSA computed Rayleigh radiances, while SZA = 80° and SZA = 85° resulted in a 3 % and 12 % relative error, respectively. relies on and to state that SSA should be preferred for SZAs > 75°. features examples of comparisons between PPA and SSA in a pure Rayleigh atmosphere of optical thickness 0.3262 at 412 nm over a black surface, showing the relative error exceeds 1 % from SZA values around 80°.

As expected due to the variability of radiative configurations and applications, our literature overview shows no evidence of a consensus on angular thresholds above which the Earth's curvature should be accounted for. One can also notice the predominance of ocean-color remote sensing-oriented studies assessing the effects of the Earth's curvature, due to the high precision requirements of such inversion purposes, in therefore short visible wavelengths where Rayleigh scattering predominates. Most importantly, there appears to be a lack of studies on the assessment of the sensitivity of common radiative parameters to the Earth's curvature, especially for the estimation of land surface parameters in the full shortwave range where Rayleigh scattering can be surpassed by molecular absorption in some instrumental channels. For example, the spectral dependency of sphericity effects are poorly documented, as well as the impact of the atmospheric profile and surface properties. In conclusion, we believe that there is a need for comparative studies establishing the specific configurations where PPA should not be applied, specifically in the scope of geostationary observations.

The experiments conducted in our study aim to assess the impact of sphericity effects on FCI shortwave observations, while also determining the key parameters on which these effects depend. The rationale of this work consists in comparing simulations in which only the geometry assumption (PPA or SSA) varies. An accurate Monte Carlo code is chosen for this purpose, as explained in Sect. . We consider various simulation cases, which allow us to assess the sensitivity of different radiative transfer parameters to the plane-parallel approximation. We aim to consider the broad angular ranges allowed by geostationary imagers by investigating the impact of the sun-sensor geometry, through the variation of VZA, SZA and the relative azimuth angle (RAA) (Sect. ). We also carry out TOA reflectance simulations for several wavelengths corresponding to the selected shortwave FCI channels (i.e., important for aerosol and surface albedo retrieval) to find out if the effects of the Earth's sphericity vary spectrally or not (Sect. ). For this purpose, we use the REPTRAN (representative wavelengths absorption parameterization applied to satellite channels and spectral bands) parametrization which enables realistic modeling of the gas absorption happening in FCI spectral channels. REPTRAN allow us to make fast satellite simulations with less than 1 % error with respect to accurate but expensive line-by-line simulations. This is achieved by running a few monochromatic simulations at “representative” wavelengths, which are then linearly recombined with appropriate weights. Both the wavelengths and the weights are determined by REPTRAN for each channel. Such parametrization is essential in our study to ensure proper simulations of molecular absorption processes. It is worth noting that although earlier works could assume pure Rayleigh scattering atmospheres by focusing on wavelengths with very low molecular absorption (see Sect. ), we consider both Rayleigh scattering and molecular absorption in all our simulations since absorption cannot be neglected in the considered long visible and near-infrared channels. Realistic surface albedo values are tested to examine a possible dependency of the Earth's curvature effects on surface brightness (Sect. ). Several standard atmospheric profiles are also investigated, in order to find out if changes in atmospheric composition and vertical distribution modulate sphericity effects (Sect. ). Finally, we assess the impact of adding an aerosol layer (Sect. ), with varying particle type, vertical distribution and optical thickness. All these parameters intervening in the radiative transfer simulation of TOA radiance were chosen to account for the typical variability observed in geostationary observations.

The Speed-Up Monte Carlo Advanced Radiative Transfer code with GPU is a statistical radiative transfer solver based on Monte Carlo processes that simulates light propagation in the surface-atmosphere coupled system . It accounts for absorption and scattering by molecules and aerosols as well as light polarization and is valid over the entire solar spectrum. SMART-G enables top of atmosphere radiance simulations to run in either PPA or SSA configurations, which is a key feature to study the impacts of neglecting the Earth's curvature. A local estimate variance reduction feature allows radiance evaluation for each altitude and direction by considering the propagation of photons as a Markov Chain . The statistical error introduced by this process allows the solver to estimate the standard deviation of the radiance, which is provided as a simulation output, thereby enabling the calculation of an uncertainty.

SMART-G presents great consistency with well-validated Monte Carlo solvers such as MYSTIC and SASKTRAN-MC , while being approximately 30 times faster . also showed the intensities simulated by these three radiative transfer solvers differed by less than 0.2 % for single scattering only, and by less than 1 % when accounting for multiple scattering.

One should note the word “photons” is used in this study for “virtual particles of light” that spread backwards in the computation, i.e. from the sensor to the sun, according to Monte Carlo related constraints. We acknowledge this is a misuse of language that does not cover its full definition given by quantum physics.

Our default setup consists of strictly similar simulations of TOA reflectance run twice, in PPA and SSA configurations. The simulations consist of sending photons in the visible (VIS) and near-infrared (NIR) wavelengths corresponding to FCI spectral channels of interest (using the REPTRAN parametrization in SMART-G) for joint aerosol and surface properties estimation: VIS 0.4 , VIS 0.5, VIS 0.6, VIS 0.8, NIR 1.6 and NIR 2.2, whose spectral characteristics can be found in Table . These photons propagate in a cloud-free gaseous atmosphere accounting for both Rayleigh scattering and molecular absorption for O₃, H₂O, NO₂ and O₂, with or without aerosol particles. The atmospheric profiles are selected among the Air Force Geophysical Laboratory (AFGL) atmospheric constituent profiles (defined in the altitude range of 0–120 km) , while the aerosol properties are set using fine particle-dominated and coarse particle-dominated models defined in the Optical Properties of Aerosols and Clouds (OPAC) database . We also set a Lambertian surface, determined by an albedo value changing depending on the experiment. Simulations are computed for numerous solar zenith angle values to mimic a full day of geostationary observations and for several moderate and extreme sensor zenithal geometries, in the principal plane of relative azimuth (i.e., RAA = 0, 180°), which is convenient for examining the effects of aerosols. The configurations used to compute the results presented in Sect.  are summarized in Table . One should mention these same experiments were also conducted in the cross-principal plane of relative azimuth (i.e., RAA = 90, 270°). The results corresponding to the experiment exploring several spectral channels (see Sect. ) can be found in Appendix , but results from other experiments for the cross-principal plane are not presented here for the sake of brevity, and since they do not lead to different conclusions compared to the principal plane experiments.

In order to analyze and compare our simulation results, we calculate the relative error as well as mean percentage error (MPE) and mean absolute percentage error (MAPE) scores between PPA and SSA simulated TOA reflectances. To clarify the statistical significance of each result, the plots of reflectance and relative error in reflectance consistently feature error bars corresponding to the calculated uncertainty, which corresponds to a 95 % trust interval (see Appendix ).

We define two criteria to conclude that the bias induced by the use of PPA is “significant”. The first criterion is an abrupt rise of the relative error value. This is intuitive, as we consider that the bias induced by the use of PPA becomes of importance as soon as the relative error between PPA and SSA TOA reflectances increases sharply. This means that a future compensation for the sphericity effects should start from the identified cut-off value. The second criterion, meant to be more quantitative than the first, is a threshold on the relative error between PPA and SSA TOA simulated reflectances. The acceptable error in aerosol optical depth remote sensing (an essential first step before surface properties retrieval) cannot however be determined straightforwardly, as it depends on many parameters including geometry, wavelength and aerosol-surface properties themselves . However, in unfavorable conditions when aerosol sensitivity is low e.g., in the occurrence of bright surfaces, or low aerosol scattering due to high scattering angles;, small errors on TOA reflectance can result in significant AOD biases, which in turn can result in biased surface reflectance. Therefore in this study we have decided to consider a 1 % error as the threshold above which the Earth's sphericity effects should be corrected.

Figure a presents the simulated TOA reflectances both in PPA and SSA for VZA values of 0, 60, 70, 75 and 80°, as a function of SZA across the principal plane, i.e. mimicking the full diurnal cycle of an imaginary location observed from the geostationary orbit. We consider FCI channel VIS 0.4 centered at 444 nm, in a aerosol-free US standard AFGL atmosphere, with a Lambertian surface of albedo 0.1, in the principal azimuthal plane. For VZA = 0°, the reflectance appears constant for SZA values below 60° in both PPA and SSA. The other curves suggest that the reflectance increases with the VZA. Indeed, when VZA increases, the illuminated volume observed by the satellite sensor grows, and thus the TOA reflectance increases too, as explained in Sect. . We also notice an asymmetry in the reflectance values around noon (SZA = 0°) for VZA values different than 0°. Indeed, when VZA is non-zero, the intensity received by the satellite varies depending on whether it is in backscattering mode (RAA = 0°) or forward scattering mode (RAA = 180°). For all VZA values, the PPA and SSA reflectance curves split above SZA = 80°. The SSA reflectances continue increasing since fewer solar beams reflect on the surface before reaching the sensor. Meanwhile, the PPA reflectances start dropping since considering an infinite plane surface does not account for this reduction of the amount of reflections. Moreover, as explained in Sect. , PPA leads to an overestimation of the solar beam attenuation in Rayleigh-dominated wavelengths.

Figure b features the relative error computed between PPA and SSA simulated TOA reflectances for each angular configuration shown in Fig. a. Figure c and d are the same plot zoomed respectively on the plateau and on the drop of the relative error curves. In these plots, we can see how the absolute relative error generally increases with VZA and exceeds 1 % at all times for VZA ≥ 75°, as one can also see in Table  summarizing all the obtained statistics. To fully understand these results, we need to keep in mind the opposite effects high VZA and high SZA have on simulated intensities in Rayleigh-dominated wavelengths. As explained in Sect. 2.2, high SZA values result in an underestimation of TOA intensity by the PPA compared to SSA, causing a negative bias in the resulting relative error according to Eq. (). This explains the sudden drop in the relative error curves, which for example starts around SZA = 60° for the VZA = 0° curve and causes the relative error to reach a MPE of -18.55 ± 0.26 % for SZA ∈ [80°;90°]. In contrast, high VZA values result in an overestimation of TOA intensity by the PPA compared to SSA, resulting in a positive bias in the relative error according to Eq. (). This can be observed when VZA = 80° and SZA ∈ [0°;60°], where the MPE is 1.84 ± 0.08 %. The combination of these two phenomena explains the increase preceding the drop of the relative error curves, causing this drop to occur at higher SZA values for high VZA values. This also explains why the MPE between the PPA and SSA simulations in the [80°;90°] range reaches only -11.94 ± 0.11 % for VZA = 80° against -18.55 ± 0.26 % for VZA = 0°. It also enables the understanding of the near-zero MPE values in the [70°;80°] SZA range for moderate VZA values. Indeed, the underestimation induced by high SZA values starts prevailing, causing the relative error to drop and eventually become negative, creating a transition region where the error appears to be extremely small despite the multiple biases arising.

These results demonstrate that the impact of the PPA highly depends on both SZA and VZA. According to the first criterion on the sharp increase of the PPA-induced relative error (defined in Sect. ), we can affirm that not taking into account the Earth's curvature by using a PPA instead of a SSA geometry when simulating TOA intensities has a major impact for SZA values exceeding 70°. The errors induced by high VZA values are lower, but according to the second criterion (defined in Sect. ), we still consider their impact to be significant beyond 70° as the relative error exceeds 1 % in such configurations.

Figure a presents the simulated TOA reflectances both in PPA and SSA for the VIS 0.4, VIS 0.5, VIS 0.6, VIS 0.8, NIR 1.6 and NIR 2.2 FCI visible and near-infrared channels interesting for albedo and aerosol retrieval, as a function of SZA. We consider an aerosol-free US standard AFGL atmosphere using REPTRAN to account for molecular absorption, for a VZA of 45°, with a Lambertian surface of albedo 0.1, in the principal azimuthal plane. The simulated TOA reflectance is visibly higher in the blue than in the green and even higher than in the red and infrared channels, which can easily be explained by the contribution of the Rayleigh scattering in the simulated standard atmosphere that is added to the surface albedo of 0.1. The PPA and SSA curves start diverging for SZAs around 80° for the three channels in the visible spectrum, with the SSA reflectances still increasing and the PPA reflectances starting to diminish. This divergence is more difficult to discern for the three near-infrared curves.

Figure b features the relative error computed between PPA and SSA simulated TOA reflectances for all channels as a function of SZA. Figure c and d show the same plot focused respectively on the plateau and on the drop of the relative error curves. For moderate SZAs, the relative errors are very close between the simulations in VIS 0.5 and VIS 0.6, and between the simulations in VIS 0.6 and VIS 0.8, as shown by the merging error bars. The NIR 1.6 and NIR 2.2 simulations stand out with relative errors respectively below 1 % and 2 % for SZA values below 85°. One can notice in Table  that for SZA < 60°, the relative error between PPA and SSA simulated TOA decreases when the wavelength increases (MPE = 0.39 ± 0.14 % for VIS 0.4, MPE = -0.01 ± 0.01 % for NIR 2.2). However for extreme SZA values over 80°, the relative error is larger in the red (MPE = -23.64 ± 0.23 % for VIS 0.6) than in the blue (MPE = -17.88 ± 0.23 % for VIS 0.4). This suggests that shorter visible channels present a stronger Rayleigh scattering-induced overestimation impact in PPA than longer visible channels, noticeable for moderate zenithal geometries. However, one can notice this effect does not persist for near-infrared channels for SZA values over 80° (MPE = -6.09 ± 0.58 % for VIS 0.8; MPE = -8.43 ± 0.47 % for NIR 2.2), since absorption is predominant in such wavelengths (see Appendix  for evidence).

These results highlight a dependency of the magnitude of PPA's impact on the wavelength for all zenith angles. We start to observe overall significant biases around SZA = 70° for all FCI channels, although near-infrared channels VIS 0.8, NIR 1.6 and NIR 2.2 appear less affected by the use of PPA. This implies that the FCI images used for aerosol and surface properties retrieval will undergo different sphericity effects depending on the spectral channel considered.

Figure a presents the simulated TOA reflectances both in PPA and SSA for Lambertian surfaces of varying albedo and as a function of SZA. We consider FCI channel VIS 0.6 centered at 640 nm in a aerosol-free US standard AFGL atmosphere, for a VZA of 45°, in the principal azimuthal plane. The “red” FCI channel VIS 0.6 was chosen this time for assessing the dependency of the sphericity effects on surface albedo, since land surfaces show a greater albedo variability at 640 nm compared to 444 nm, where they are darker overall . Albedo values were set to 0.005, 0.1, 0.2 and 0.4, which in red wavelengths correspond to the following cover types: ocean, dense forest, grassland and sand desert.

Figure a shows that simulated TOA reflectances are very low for a surface of near-zero albedo since the only radiance contribution comes from Rayleigh scattering in the atmosphere, which is rather weak around in VIS 0.6. As expected, reflectances are greater for surfaces of albedo 0.1, 0.2 and 0.4. We notice a splitting of the PPA and SSA curves which occurs around SZA = 75° for the 0.4 albedo reflectances, against SZA = 80° for the 0.005 albedo reflectances.

Figure b features the relative error between PPA and SSA simulations for each surface setting. Figure c and d show the same plot focused respectively on the plateau and on the drop. The relative errors for surface of albedo 0.1, 0.2 and 0.4 are extremely close, with MPE values of respectively 0.24 ± 0.08 %, 0.17 ± 0.05 % and 0.16 ± 0.03 % for SZAs < 60° as one can see in Table . For a surface of near-zero albedo and in the same angular configurations, the relative error between SSA and PPA simulations is more important (MPE = 0.34 ± 0.38 %), because molecular scattering and absorption are the only contributions in the computed reflectances, making the optical path differences more significant. One can notice the error bars are very large for this dark surface, which can be explained by the very small TOA reflectance values. However, for SZAs > 80°, the MPE is larger for an albedo of 0.4 (-24.91 ± 0.15 %) than for a surface of near-zero albedo (-22.51 ± 0.38 %), suggesting that the Earth's sphericity effects slightly depend on surface albedo.

Figure  is the same as Fig. , but at VZA = 80° in order to explore more extreme geometries found in geostationary observations. This time, the curves are clearly separated from each other, even for moderate SZA values. One should note this experiment was also conducted for VZA = 70° (not shown), and a dependence of the relative error on surface albedo was also observed for all SZA values. This suggests that for high VZA (i.e., 70° and beyond), the value of surface albedo impacts the error caused by the use of the PPA.

Overall, we start to observe significant biases around SZA = 70°, and absolute relative errors reach more than 10 % for SZAs above 85° and more than 50 % for SZAs above 88°, regardless of the surface albedo. This leads to the conclusion that the Earth's curvature should be taken into account at least above SZA = 70°, regardless of the surface reflectivity. One should also note that the effects induced by the use of the PPA on simulated TOA intensities slightly vary depending on surface albedo, especially for SZA > 80° and VZA > 70°.

Figure a presents the simulated TOA reflectances both in PPA and SSA for some selected AFGL atmospheric profiles as a function of SZA. We consider FCI channel NIR 2.2, for a 45° VZA, with a Lambertian surface of albedo 0.1, in the principal azimuthal plane, without aerosols. The investigated profiles are “US standard”, “tropical” and “subarctic winter” are chosen for their different temperature and molecular concentration profiles, making them appropriate to account for the diversity of atmospheric characteristics on Earth. The near-infrared FCI NIR 2.2 channel was chosen this time to investigate the impact of gas absorption on the Earth's sphericity effects. Indeed, Rayleigh scattering is the main radiance-inducing atmospheric process in short visible low-absorbing wavelengths, while fluctuations in Rayleigh optical depth between atmospheric profiles are below 1 % , suggesting there will be no evidence of a possible dependency on the atmospheric profile in short wavelengths. However, the difference in water vapor concentration between the three selected profiles (see figures in Appendix 1 of for details) could have an impact on the sphericity effects in NIR 2.2, which is a channel specifically affected by water vapor absorption (see Table ).

The first thing one can see in Fig. a is PPA simulated reflectances are very similar between the different atmospheric profiles to the point where it is difficult to differentiate the US standard and subarctic winter curves. The same observation applies to the simulated SSA reflectances. Figure b features the relative error between PPA and SSA simulations for each atmospheric profile. Figure c and d show the same plot focused respectively on the plateau and on the drop of the curves. The relative errors are very similar between the three atmospheric profiles for all SZA values, to the point where the error bars of the curves merge. As one can observe in Table , there are no deviations that are beyond the uncertainties between the MPE values obtained with the different profiles for SZA values < 80°. Only for SZAs > 80°, one can notice discrepancies in MPE values between the atmospheric profiles, that seem to occur mostly beyond SZA = 85° according to Fig. d.

Figure  is the same as Fig. , but at VZA = 80° to explore more grazing geometries. This time, the curves are slightly separated from each other, even for moderate SZA values. This suggests that for VZA values exceeding 80°, the choice of the atmospheric profile impacts the error caused by the use of the PPA.

The error induced by the use of the PPA on simulated TOA reflectances seem to depend on the atmospheric profile only for very extreme SZA (over 85°) or VZA (over 80°) and in absorbing channels. This leads to the conclusion that the composition of the atmosphere has an existing but limited impact on the Earth's sphericity effects.