Polarization Performance Simulation for the GeoXO Atmospheric Composition Instrument: NO 2 Retrieval Impacts

. NOAA’s Geostationary Extended Observations (GeoXO) constellation will continue and expand on the capabilities of the current generation of geostationary satellite systems to support US weather, ocean, atmosphere, and climate operations. It is planned to consist of a dedicated atmospheric composition instrument (ACX) to support air quality forecasting and monitoring by providing similar capabilities to missions such as TEMPO (Tropospheric Emission: Monitoring Pollution), currently planned to launch in 2023, and Ozone Monitoring Instrument (OMI), TROPOMI (TROPOspheric Monitoring Instrument), 5 and GEMS (Geostationary Environment Monitoring Spectrometer) currently in operation. As the early phases of ACX development are progressing, design trade-offs are being considered to understand the relationship between instrument design choices and trace gas retrieval impacts. Some of these choices will affect the instrument polarization sensitivity (PS), which can have radiometric impacts on environmental satellite observations. We conducted a study to investigate how such radiometric impacts can affect NO 2 retrievals by exploring their sensitivities to time of day, location, and scene type with an ACX instru- 10 ment model that incorporates PS. The study addresses the basic steps of operational NO 2 retrievals: the spectral fitting step and the conversion of slant column to vertical column via the air mass factor (AMF). The spectral fitting step was performed by generating at-sensor radiance from a clear sky scene with a known NO 2 amount, the application of an instrument model including both instrument PS and noise, and a physical retrieval. The spectral fitting step was found to mitigate the impacts of instrument PS. The AMF-related step was considered for clear sky and partially cloudy scenes, of in the development of a composition instrument that will our to


Introduction
NOAA's Geostationary Extended Observations (GeoXO) constellation will continue and expand on the capabilities of the current generation of geostationary satellite systems to support US weather, ocean, atmosphere, and climate operations. It is planned to consist of a dedicated atmospheric composition instrument (ACX) to support air quality monitoring and forecasting. The mission will build on knowledge obtained from low earth orbit (LEO) and geostationary (GEO) satellite air quality 25 monitoring instruments such as TROPOspheric Monitoring Instrument ( TROPOMI) (Veefkind et al. (2012)), OMI (Ozone Monitoring Instrument) (Levelt et al. (2006(Levelt et al. ( , 2018), Geostationary Environment Monitoring Spectrometer (GEMS) (Kim et al. (2020)), and Sentinel 4 (Kolm et al. (2017)). Retrievals of trace gases like NO 2 derived from satellite platform observations have been used to relate top-down emissions estimates, air quality monitoring and forecasting, pollution events, trends, and health studies (Bovensmann et al. (2011); Levelt et al. (2018); Burrows et al. (1999); Bovensmann et al. (1999); Levelt et al. 30 (2006); Munro et al. (2016); Bak et al. (2017); Veefkind et al. (2012); Cooper et al. (2022); Hollingsworth et al. (2008)). The World Health Organization has designated NO 2 as a pollutant, since it has detrimental effects on human health (WHO (2021); Huangfu and Atkinson (2020)). It also impacts climate by contributing to the formation of aerosols in the upper troposphere that reflect incoming solar radiation, and, thus, cool the planet (Shindell et al. (2009). Over non-polluted regions, the stratospheric NO 2 participates in photochemical reactions that can affect the ozone layer (Crutzen (1979)). 35 In the near future, these phenomena will be monitored from geostationary (GEO) orbit over the greater North America as part of the TEMPO (Troposphere Emission: Monitoring Pollution) mission (Zoogman et al. (2017)), at an increased temporal frequency than available from its LEO counterparts. Like other atmospheric composition monitoring instruments, TEMPO is and ACX will be a hyperspectral imager with fine spectral sampling and resolution from the ultraviolet to the near-infrared allowing trace gas absorption features to be discriminated using the well-known differential optical absorption spectroscopy 40 (DOAS) technique. For total vertical NO 2 amount retrievals, the DOAS technique is applied around the 420 to 455 nm range (Bucsela et al. (2006); Lamsal et al. (2021); Marchenko et al. (2015); Boersma et al. (2007); Richter and Burrows (2002) ;Valks et al. (2011); Martin (2002)).
ACX is in its early stages of development with its initial performance requirements being formulated with respect to parameters like sampling and resolution to enable this DOAS approach. Other parameters such as pixel size, noise, and polarization 45 sensitivity (PS) are also being defined. These requirements may be updated as the instrument design choices are better understood. This study focuses on the requirements for instrument PS, which, for instance, may inform whether a polarization scrambler is needed. Air quality monitoring instruments such as OMI and TROPOMI were designed with polarization scramblers to reduce their PS (Bézy et al. (2017);Voors et al. (2017)).
Without PS suppression, the polarization state of incoming radiation will impact the at-sensor radiance for satellites sensors 50 in both GEO (Pearlman et al. (2015)) and LEO, though these impacts have been more extensively analyzed for LEO satellites (Meister and Franz (2011);Wu et al. (2017); Goldin et al. (2019)). GEO orbit presents unique challenges due to the highly variable solar angles throughout the day. This results in a variation in the degree of linear polarization of the at-sensor radiance throughout the day due to Rayleigh scattering in the Earth's atmosphere; for instance, light scattered in the normal direction to the incident light generates highly polarized radiation but not in the forward or backward direction. If the instrument is 55 sensitive to light with a certain polarization, this variation in degree of linear polarization translates to a variation in measured radiance throughout the day. Thus, limiting the PS of the satellite sensor can limit the radiometric uncertainty. These impacts can be derived by employing radiative transfer simulations to predict the at-sensor polarization state or Stokes parameters (S) and applying the instrument polarization impacts via its Mueller matrix (M).
The Stokes formulation expresses the polarization state consisting of its un-polarized (or randomly polarized) component, S 0 ; two terms describe its linear polarization state: the excess in horizontal linear polarization relative to the vertical direction, S 1 , and excess in linear polarization at 45 • relative to 135 • , S 2 ; one term describing its circular polarization through its excess of right circular relative to left circular polarization, S 3 . The Mueller matrix is a 4 x 4 matrix used to apply the optical effects of an element to generate an output Stokes vector. We model ACX as a Mueller matrix with a transmission of one and non-zero 65 linear polarization extinction elements (m 01 , m 02 , m 10 , and m 20 ). Since the system only detects total energy or radiance, not polarization state, only the first row is relevant. So the output term corresponding to the detected normalized Stokes parameter is: This detected radiance can differ from the true at-sensor radiance if ACX has linear PS, defined as (m 2 01 + m 2 02 ), which 70 can propagate to higher level satellite products. For instance, the retrieval of surface reflectance can suffer large uncertainties, especially when the signal from the surface is small compared to the atmospheric component. In this work, we discuss our study of NO 2 retrievals, and investigate the parts of the process that may be affected. To our knowledge, NO 2 retrievals dependence on instrument PS have not yet been fully documented. We describe an initial study to show the ways that these retrievals can be impacted and make initial estimates of those impacts associated with the current PS requirements, <5% PS for wavelengths 75 <500 nm.
Our NO 2 retrieval simulation approach discussed here follows a simplified version of the DOAS technique used for operational NO 2 retrievals and consists of two basic steps: One involves the DOAS spectral fitting step for the at-sensor radiance.
This fit is normally used to retrieve the NO 2 slant column amount -the total number of molecules along the atmospheric photon path to the satellite sensor. The second step converts this slant column amount to the vertical column amount through 80 the air mass factor (AMF), which depends on the geometrical path as well as the differences in scattering and absorption within the atmosphere between the slant and vertical paths. Our first approach for analyzing polarization effects deals with the DOAS spectral fitting step with clear sky scenes by simulating at-sensor Stokes parameters and applying an instrument model that includes a range of PS values in several orientations (defined by m 01 and m 02 ), as well as the instrument noise and spectral properties consistent with our current knowledge of ACX. The fits of these spectra are used to retrieve NO 2 vertical column 85 amount directly, not slant column, in our case; since these are simulations with the vertical profiles used as inputs, we do not need to use the AMF for converting slant column to vertical column amount. The second approach deals exclusively with the AMF derivation step. For this analysis, the AMF, required for operational retrievals, is affected by instrument PS when considering the potential for partially cloudy scenes. Retrievals in such situations are commonly performed for atmospheric monitoring instruments, since their large instantaneous fields of view make completely clear scenes rare. We will discuss the 90 formalism in detail for both approaches in the methods section. With these two approaches, referred to as the method for "clear scenes" and "partially cloudy scenes", we demonstrate the capability to investigate PS requirements.

Methods
As mentioned, the approach for clear scenes exploits the spectral features in the radiance spectra to retrieve the total vertical amount of NO 2 , and the approach for cloudy scenes relies on the AMF calculation.

Clear scenes
The overall method for clear scenes is illustrated in Fig. 1. In this process, simulated radiance spectra are propagated through an instrument model and the total vertical column NO 2 is retrieved using a look-up table (LUT) approach with the aid of a constrained energy minimization algorithm (CEM) algorithm (Farrand (1997)). Further details are discussed below. The at-sensor radiances from clear scenes are simulated using a vector radiative transfer code, the Unified Linearized Vector Radiative Transfer Model, UNL-VRTM, which integrates the linearized vector radiative transfer (VLIDORT) into a broader framework (Xu and Wang (2019)). The code can generate Stokes vectors from any scene defined by its view and solar geometry, surface reflectance, wavelength range, and atmospheric composition. Note that rotational Raman scattering is not included in the model. The ACX was assumed to be at 105 • West longitude viewing several locations across the continental US (CONUS).

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The time of day was chosen to generate solar zenith angles of 60 to 70 • , where PS is expected to be highest but still within the range where NO 2 retrievals are typically performed. The US Standard Model default profiles were used for 21 trace gases for all scenes (excluding NO 2 ). The default NO 2 profiles were modified by injecting a known amount uniformly into the troposphere below 2 km ( Fig. 1). Three basic surface spectra generated from spectral libraries were used. The water spectrum  We ran radiative transfer simulations for several US locations, with the three scene types, with varying amounts of tropospheric NO 2 . This produced a look up table (LUT) of scene type, NO 2 vertical amount, and at-sensor radiance spectra. This LUT was used in the retrieval discussed below. The reference radiance spectra corresponding to the NO 2 reference amounts over water, rural and urban scenes were modified by applying the instrument model (for several US locations). The instrument response model was based on the TEMPO design, which consists of of a reflective f/3 Schmidt-form telescope and a spectrometer assembly that utilizes a diffraction grating to form an image on CCD detector arrays (Zoogman et al. (2017)). The simulated radiance was modified by this instrument 125 response model, which sampled the radiance at 0.2 nm wavelength steps with a resolution of 0.6 nm, and applied a PS response. The PS response model was not specific to TEMPO as our goal was to understand the range of impacts associated with the ACX polarization requirements. The noise was also applied as defined by the ACX signal-to-noise (SNR) specification. Our instrument parameters from TEMPO were modified by assuming a sampling strategy or integration time modification that brought the noise in line with that specified by ACX. Table 3 shows the parameters included in this model. The noise was applied by generating 1000 spectra with different amounts of noise following a Gaussian distribution that are added to the at-sensor radiance (after being modified by the polarization response). All spectra were normalized by subtracting a second order polynomial fit to remove the sensitivity to absolute radiance as is done in the DOAS retrieval technique. The NO 2 vertical amount was retrieved using the look-up-table and the CEM algorithm:

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where C −1 and m are the inverse covariance matrix and mean over the noise spectra, respectively. The CEM was calculated for all (target) spectra in the LUT, t, with the noise spectra, C −1 and m. The spectrum, x that generated a CEM value closest to one was chosen, and its associated NO 2 vertical amount was retrieved.

Partially cloudy scenes
The process for "partially cloudy scenes" involves an AMF derivation process that includes the consideration of subpixel-scale 140 clouds. The typical instantaneous field of view for an atmospheric composition instruments means that most scenes contain some clouds. Operational trace gas retrievals are routinely done in partially cloudy scenes, so we derive PS impacts for such scenes primarily through their impact on the AMF.

Theoretical background
This approach assumes a simple cloudy scene model where each scene is assumed to be a combination of a fully cloud covered 145 subpixel and a clear sky subpixel weighted with an effective cloud fraction, f , consistent with previous approaches (Stammes et al. (2008)): where L obs is the observed radiance, L clr is the calculated radiance in a clear sky, and L cld is the cloudy radiance. To produce observed amounts of Rayleigh scattering and absorption, it was found that for this equation to work across most conditions, 150 we model L cld as a Lambertian surface (opaque) with surface reflectivity 0.80 at the effective cloud pressure, assumed here to be equivalent to a cloud at 2 km. Aerosols are not considered for the cloudy scenes, since they would have a negligible impact; the clouds would lie above the tropospheric NO2 and aerosol layer. This simple model has been demonstrated to represent the complex radiative transfer in clouds accurately (Stammes et al. (2008); Joiner (2004); Vasilkov et al. (2008)). So, we typically derive f at a wavelength with little absorption and use a surface climatology for L clr . Then, we simply invert the above equation 155 to give: For the trace gas retrievals, another quantity defines the fraction of scene radiance from the cloud versus the clear parts of the scene called the cloud radiance fraction, f r , which has wavelength dependence:

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A cloudy air mass factor (AMF) is computed along with the clear sky AMF. The total AMF is then computed with the clear and cloudy AMFs weighted by the cloud radiance fraction To compute the error in the NO 2 vertical column due to an error in f , we started with the calculation of the error in f due to an error from PS: and this would then propagate into the error in NO 2 vertical column density (N O 2,V CD ) through Equations 6,7 above along with: This process is shown graphically in Fig 3, where a clear and cloudy version of a scene are simulated. The clear version is 170 propagated through the instrument polarization response model, and, using the radiance generated from the cloudy scene, the impacts are propagated through the cloud fraction, cloud radiance fraction, AMF, and finally the NO 2 amount. Following the process by Kuhlmann et al. (2015), the AMFs for each atmospheric layer (also called box AMFs) were computed using a pre-calculated LUT with input parameters of altitude, z, solar zenith angle, view zenith angle, relative azimuth angle, surface reflectance, and surface altitude. The total AMF was calculated by linearly interpolating over all variables for each altitude and 175 summing over all layers to the top of atmosphere (TOA), where each layer dz has a vertical column amount V N O2 : where the integration assumes an exponential dependence within each layer. A correction term, α, is normally included in the AMF calculation to account for the temperature dependence of the NO 2 cross sections, though was neglected here by setting it to one. The NO 2 error derived through the conversion of slant to vertical amount is then computed. This error can be 180 considered as the effect of a change in detected radiance due to PS, which, in turn, leads to an error in the interpretation of the amount of clouds in the scene. This leads to an impact on the NO 2 retrieval over the total vertical column. Note that assuming a constant PS over the wavelength range, this error will also change negligibly as a function of wavelength. We perform this analysis at one wavelength (425.8 nm) in this study. By differentiating Equation 9, the NO 2 error in the total vertical column amount (∂(NO 2,total )) is then calculated in terms of the total vertical NO 2 amount (V NO2,total ), the AMF, and the AMF error 185 (∂AMF total ) as:

Radiative transfer modeling
We conducted the radiative transfer simulations as summarized in Table 3. Simulation A will be shown to define an upper bound for the retrieval error with a PS of 5% by using a NO 2 profile (similar to those defined in the clear scene simulations) 190 with a large NO 2 amount, the lowest reflectance scene, and high constant solar zenith angle over all of CONUS over a one degree latitude/longitude grid. Simulation B quantifies the retrieval impact of scene type -water, rural, and urban scene -over CONUS for a constant reference NO 2 profile. The scene types are the same as defined in Table 1 and are assigned to all pixels in CONUS for each run. Simulation C explores the retrieval impacts on the solar zenith angle and NO 2 amount for selected   in AMF (∂(AM F total )) and total vertical NO2 amount (∂(NO 2,total )) 3 Results

Clear scenes
As part of the method for clear scenes, the ACX instrument model was applied to the at-sensor radiance including sampling with a Gaussian slit function at the interval and resolution of 0.2 and 0.6 nm, respectively, and its noise as depicted in Fig.   4. The differences between the normalized solar irradiance (multiplied by a factor of 5 for visibility) and radiance spectra shows the atmospheric contribution and the effects of this resampling. The 1000 radiance spectra shown cannot be discerned 210 clearly given the high SNR (explicitly shown by the blue line). The noise was applied after modifying with the PS response.
The PS model parameters applied via Equation 2 using m 01 = ±PS and m 02 = 0, so that the PS was applied in the vertical or horizontal orientation. These orientations were chosen for most simulations for simplicity but other orientations will be discussed in the cloudy scene analysis section. Figure 4. Example of at-sensor radiance spectra simulated with an applied instrument model including resampling effects and added noise set by ACX instrument parameters. 1000 spectra are plotted (black lines), which appears as a slightly thicker line than the mean SNR (blue line and right axis). The normalized solar irradiance multiplied by a factor of 5 is shown for comparison to the resampled spectra.
The retrieval process effectively matches the spectral shape of the simulated detected spectra -affected by spectral sampling, 215 noise, and PS -to the most similar spectra in the LUT that contains a large range of tropospheric NO 2 amounts for the three surfaces. Figure 5(a) shows an example of a the adjusted sample spectrum with the the spectra in the LUT. Note that all spectra were adjusted using quadratic fits in the spectral fitting process. The CEM algorithm finds the spectrum from the spectra that is most similar. Figure 5(b) shows a summary of the NO 2 retrieval errors, average biases and standard deviations as a function of PS for several scene types for a particular location (Norman, Oklahoma). The errors are driven by a combination of the SNR, 220 view/solar geometry, surface reflectance spectrum, and aerosol model and are similar for all scene types. The flat dependence indicates that the PS does not affect the retrieval error in the DOAS spectral fitting retrieval step. The reason is that the PS is a smooth function of wavelength, and the radiometric error introduced are compensated through the spectral fitting process.
These results were similar for all locations (not shown). We note that other retrieval techniques that do not use a polynomial correction term in the spectral fitting approach may exhibit larger PS impacts.
225 Figure 5. Clear-sky scene retrieval results: (a) An example of an adjusted ACX simulated spectrum (cyan) with all spectra from the look-uptable (LUT) with varying amounts of tropospheric NO2 (b) The average error (or bias) and standard deviation for 1000 total vertical NO2 retrievals of the "high" amount (8.44 × 10 15 molecules/cm 2 ) for the three scene types (water, vegetation, and urban) at Norman, Oklahoma, assuming a vertical PS orientation.

Partially cloudy scenes
In contrast to the previous results, the AMF-related processing step showed more significant polarization impacts, where an error is induced when a clear scene scene is interpreted as a partially cloudy scene due to the instrument response model that includes PS (but not noise). Figure 6 shows the results as they are propagated through each step in the process (Fig. 3) for an example with an extremely high total vertical NO 2 amount, 20×10 15 molecules/cm 2 , over all of CONUS (Table 3, Simulation Stokes parameters, retrieved cloud fraction, and NO 2 error are particularly apparent. This example shows that the PS orientation can generate vastly different spatial dependence in NO 2 retrieval errors. The maximum NO 2 error of 1.4×10 15 molecules/cm 2 235 is above the specified TEMPO NO 2 precision (Zoogman et al. (2017)). Note that this is likely an upper bound, since NO 2 amounts like these are mostly found in industrialized areas in other regions of the world.
Similar simulations for more realistic NO 2 amounts using constant profiles across CONUS show how these retrieval errors change as a function of surface type (Table 3, Simulation B). Figure 7 shows a lower, more realistic, NO 2 amounts of 8.4 × 10 15 molecules/cm 2 corresponding to the "high" NO 2 case shown in Fig. 1 Table 3, Simulation A for more details).  Table 3, Simulation B for more details).   Table 3, Simulation C for more details) In contrast to the previous results with constant profiles across CONUS, Fig. 9 shows the results using GEOS-5 profiles, which appear qualitatively consistent with the results using the artificial profiles used above (Table 3, Simulation D). The NO 2 255 amounts for this day varied between 2.5 to 6.5 ×10 15 molecules/cm 2 are displayed . The figure shows the polarization impacts with 5% PS in the vertical orientation. The impacts are more apparent as the solar zenith angle increases and resemble the previous results in Fig. 7, where the solar zenith angle is fixed at 70 • . For instance, the NO 2 errors are larger at 20 UTC in the eastern regions where the solar angles are relatively large, and the NO 2 errors are larger in the western regions at UTC 16, where the solar zenith angles are larger. The higher cloud fraction decrease the retrieval errors, which can be seen in the 260 western regions at 16 UTC; although the southwest and southeast have similar solar zenith angles, the southwest has lower retrieval errors due to the increased cloud fraction. As a result of the cloud fraction and lower NO 2 amount, the maximum NO 2 errors found were 0.03 ×10 15 molecules/cm 2 for this day -a negligible value when compared to the TEMPO precision requirement. Figure 9. Solar zenith angles, total NO2 column amount, GOES-5 cloud fraction, and resulting NO2 errors at 20 UTC (top) and 16 UTC (bottom). GEOS-5 NO2 profiles were used assuming 5% PS with vertical orientation, all water scenes, and clouds at a 2 km altitude. (See Table 3, Simulation D for more details)

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We demonstrated a simulation and modeling capability to assess polarization effects for ACX predicted performance studies.
Our results show that the DOAS spectral fitting step mitigates PS effects in the NO 2 retrieval process. The AMF calculation step, however, can cause retrieval errors from instrument PS when considering partially cloudy scenes. The PS magnitude and orientation (Mueller matrix elements) impacts can cause different NO 2 retrieval errors depending on location, time of day, cloud fraction, and NO 2 amount. For a PS of 5 % with vertical orientation, the maximum NO 2 retrievals errors were 270 0.25 ×10 15 molecules/cm 2 for high pollution cases. In extreme cases, if NO 2 pollution significantly increases to levels on the order of the world's most polluted regions, these errors can reach 1.4 ×10 15 molecules/cm 2 . A more typical maximum error found through analyzing the GEOS-5 profiles was 0.03 ×10 15 molecules/cm 2 . This study shows that in most cases, the 5% PS requirement introduces retrieval uncertainties significantly lower than the TEMPO precision requirement except in the most extreme cases. Note that these estimates assume a particular set of instrument Mueller matrix elements. We emphasized 275 a vertical orientation based on an assumed vertical grating orientation where its polarization axis would likely be in this direction. In this configuration, the instrument effectively sweeps wavelengths over locations in the west-east direction. The Mueller matrix will be updated with the appropriate values as the instrument design matures to refine the estimates of NO 2 retrieval impacts. Our simplified retrieval approach may have neglected factors used in operational retrievals that could be affected by instrument PS and contribute to additional retrieval errors related to estimates of aerosols, surface reflectance, and 280 cloud parameters. Rotational Raman scattering, which has been used in cloud height retrievals (e.g., Vasilkov et al. (2008)), for instance, can be particularly sensitive to polarization. Other approaches for cloud height retrievals such as oxygen dimer absorption (Acarreta et al. (2004)) should be much less sensitive. We do not account for the PS to cloud height retrievals. The PS to cloud optical thickness is implicitly accounted for within the effective cloud fraction estimation. In addition, the limited set of surface reflectance types that were used and the directional and polarization surface effects that were neglected, can be 285 included in future work to improve the accuracy of the results. This capability can be utilized to support the development of ACX to continue and build on the legacy of atmospheric composition measurements to forecast and monitor air quality.

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Competing interests. We declare that the authors have no conflicts of interest. Joiner, J.: Retrieval of cloud pressure and oceanic chlorophyll content using Raman scattering in GOME ultraviolet spectra, Journal of