The core of the PACE simulator is a monochromatic radiative transfer model based on the successive order of scattering method (RTSOS) (Zhai et al., 2009; Zhai et al., 2010). This model was recently validated in a comprehensive comparison and testbed study (Chowdhary et al., 2020). The atmospheric and ocean optical properties as a function of wavelength is provided as input to RTSOS to calculate the polarized radiance field at user specified locations. The total radiance field is denoted by L t = ( I ,   Q ,   U ,   V ) T , where I , Q , U , and V are the Stokes parameters and the superscript T stands for matrix transpose. In RTSOS, L t is decomposed into contributions from different orders of scattering L n : L t ( λ , θ s , θ v , ϕ v ) = ∑ n = 1 N L n ( λ , θ s , θ v , ϕ v ) (9) where N is the maximum order of scattering included in the series, θ s is the solar zenith angle, and θ v and ϕ v are the viewing zenith and azimuth angles, respectively. Note that we define ϕ v = 0 as the half plane where the Sun glint is located in, so that the azimuth angle of the solar ray is always zero. The first order of scattering solution L 1 can be solved analytically in the atmospheric and ocean system (Zhai et al., 2012), and each higher-order scattering solution can be obtained by performing optical depth and solid angle integrations of the previous order solution (Zhai et al., 2010).

For the scattering functions with a large forward peak, we implemented several truncation methods to increase the efficiency and maintain the accuracy, including the Delta-M method (Wiscombe, 1977), δ -fit method (Hu et al., 2000), and Delta-M+ method (Lin et al., 2018). The default option is the δ -fit method, which gives the best solution in most of the situations. RTSOS solves the vector radiative transfer equation at viewing angles corresponding to discrete Gaussian quadrature points. An advanced interpolation scheme based on the integration of source function is used to obtain the radiance field at arbitrary viewing angles (Zhai et al., 2013).

To simulate inelastic scattering processes in ocean waters, the excitation radiation field is obtained by looping elastic RTSOS over the excitation wavelengths to evaluate the inelastic source function at the emission wavelength (Zhai et al., 2015; Zhai et al., 2017b). The chlorophyll fluorescence quenching processes can be modeled by allowing the quantum yield, and the fraction of photons reemitted in the total number of photons absorbed by chlorophyll molecules, to vary with the instantaneous photosynthetically available radiation (IPAR) (Morrison and Goodwin, 2010). For more details on how inelastic scattering is implemented in the radiative transfer model, readers are referred to Zhai et al. (2015), Zhai et al. (2017a).

The PACE simulator calls RTSOS in each instrument channel to simulate the radiance field at the center wavelength at a specified location and viewing direction. For channels with little or weak gas absorption, an averaged gas absorption optical depth τ ¯ g can be calculated for each atmospheric layer: T ( λ i ) = ∫ − ∞ ∞ exp [ − τ g ( λ ) ] I L S ( λ i , λ ) d λ / ∫ − ∞ ∞ I L S ( λ i , λ ) d λ (10) τ ¯ g ( λ i ) = − ln [ T ( λ i ) ] (11) where λ i is the center wavelength of i th channel; τ g ( λ ) is the monochromatic optical depth at λ ; and I L S ( λ i , λ ) is the instrument line shape function for i th channel. Using τ ¯ g , we only need to call RTSOS once for each channel to simulate the band averaged polarized radiance.

In some strong absorption bands of gases, such as the Oxygen-A band centered at 765 nm, the absorption cross-section can vary by several orders of magnitude within the full width at half maximum (FWHM) of the considered channel. The approximation made in Eqs 10, 11 will introduce a significant error that cannot be tolerated. In this case, we adopt the philosophy of the double-k method (Duan et al., 2005) to model the monochromatic radiance in a channel by: L t ( λ ) = { L c / [ τ g ( λ ) − d ] e τ g ( λ ) ∈ [ 0 ,   1 ] L c ′ ⁡ exp { − [ f ⋅ τ g ( λ ) ] } τ g ( λ ) ∈ ( 1,5 ] L c ′ ′ ⁡ exp { − [ h ⋅ τ g ( λ ) ] } τ g ( λ ) ∈ [ 5 , ∞ ) (12) where L c , L c ′ , L c ′ ′ ,   d , e , f , and h are fitting parameters. To find these fitting parameters, the PACE simulator calls RTSOS m times within [ λ i − FWHM 2 , λ i + FWHM 2 ] for a channel centered at λ i . The number of calls m depends on the maximum value of the gas absorption optical depth τ g , m a x , i.e., m = 3 if τ g , m a x < 1 ; m = 5 if 1 < τ g , m a x < 5 ; m = 7 if τ g , m a x > 5 . The resultant radiance vectors are denoted as L t ( λ i ,   λ j ) , where j = 1,2 , … , m . The first three wavelengths λ j = 1,2,3 are sampled in τ g ( λ j ) ∈ [ 0 ,   1 ] ; λ j = 4,5 are sampled in τ g ( λ j ) ∈ [ 1 ,   5 ] , and λ j = 6,7 are sampled in τ g ( λ j ) ∈ [ 5 ,   10 ] . In all three intervals ln ⁡ τ g ( λ j ) is uniformly distributed in [ ln ⁡ τ g , m i n ′ ln ⁡ τ g , m a x ′ ] where τ g , m a x ′ and τ g , m i n ′ are the maximum and minimum values in the gas absorption optical depth interval. If τ g , m a x ′ >10, we set τ g , m a x ′ = 10 to avoid unnecessary simulations of small reflectance. After the fitting parameters are found in Eq. 12, the band averaged radiance can be found with: L t ( λ i ) = ∫ λ m i n λ m a x L t ( λ )   I L S ( λ i , λ ) d λ (13a) where the integration limit λ m i n and λ m a x are determined by the negligible values of I L S ( λ i , λ ) . The integration can be numerically approximated by the following summation: L t ( λ i ) = ∑ k L t ( λ k )   I L S ( λ i , λ k ) Δ λ k (13b) where λ k ∈ [ λ m i n ,   λ m a x ] is the discretized integration wavelength and the wavelength step Δ λ k is small enough to resolve the spectral features of the gas absorbing bands. In the weakly absorbing bands, Δ λ k is mostly 0.1 nm. In strongly absorbing bands, for instance, the oxygen A (∼765 nm) and B (∼686 nm) bands, Δ λ k is as small as 0.005 nm. A sensitivity test (not shown) indicated that the fitting scheme Eqss 12, 13 provides an accuracy level better than 0.5% for most OCI channels, which meets the calibration goal of OCI (Werdell et al., 2019). For a few strongly absorbing bands, including oxygen A and B bands, and the error could be as large as 1%.

The PACE simulator is used to simulate the spectral TOA reflectance, ρ t ( λ ) = π L t ( λ ) / E d ( λ ) , where L t ( λ ) and E d ( λ ) are the upwelling radiance and the downwelling irradiance at TOA, respectively. The molecular density is from the 1976 US standard atmosphere. The aerosol model is the Urban aerosol model from Shettle and Fenn (1979) with the relative humidity of 0.90. The aerosol optical depth at 550 nm is 0.1. In the ocean, three sources of the absorption coefficient of the pure ocean water are used: IOC (Twardowski et al., 2018), LEE (Lee et al., 2015), and MCF (Mason et al., 2016). Two [Chla] values are used: 0.03 mg/m³ and 0.1 mg/m³. For each [Chla] value, two CDOM absorption bio-optical models are used to see the impacts of different waters: MG2009 and MAG 2007. Other inherent optical properties are the same as those outlined in Section 2.1.2. No sediment is included in the simulation. Ocean water depth is set as 200 m so that the bottom effect is minimal at TOA. The solar zenith angle is 30°.

Figures 3A,B show the TOA reflectance at nadir as a function of wavelength for [Chla] = 0.03 mg/m³ and 0.1 mg/m³, respectively. FWHM of 5 nm is used in the simulation. Figure 3C shows the percentage differences of the reflectances calculated with LEE and IOC with respect to those with MCF for [Chla] = 0.03 mg/m³. Figure 3D is the same as Figure 3C except for [Chla] = 0.1 mg/m³. It can be seen that the impact of the different a w values is larger when [Chla] is small (0.03 mg/m³), which is expected because the contribution of a w in the total absorption coefficient becomes smaller when [Chla] becomes larger, as both a p and a C D O M increase with [Chla] in the UV. Figures (C–D) show that adopting MAG2007 leads to a larger percentage difference between different a w data, as a C D O M based on MAG2007 is smaller than those of MG2009 so that the relative importance of a w is larger. Overall, the impacts of different a w source are in the range of −6–2% for [Chla] = 0.03 mg/m³ and −2%–1% for [Chla] = 0.1 mg/m³, respectively. This is small albeit detectable by OCI, whose calibration accuracy is 0.5%. In ocean color remote sensing the water leaving contribution is mostly smaller than 10% of the total signal at TOA at 440 nm. The uncertainty of the TOA reflectance would be amplified some 10 times in terms of the accuracy of the water leaving signals, which becomes worse in the UV due to large atmospheric signal contribution.

We used the PACE simulator to simulate polarized reflectance at the TOA for nine cases with different amounts of brown aerosol and CDOM in the coupled atmosphere and ocean system. The wind speed is 5 m/s. The ocean water is assumed to be a mixture of pure seawater, phytoplankton particles, and CDOM. The absorption coefficient of the phytoplankton particles follows the bio-optical model Eq. 4 with [Chla] = 1 mg/m³. The backscattering coefficient at 660 nm b b p ( 660 ) is 0.00347 m⁻¹; the spectral exponent S b p is 0.15 nm⁻¹; and the backscattering fraction B b p is 0.01 and has no spectral dependence (Whitmire et al., 2007). This leads to b p ( 660 ) = 0.347 m⁻¹ which is consistent with Morel and Maritorena (2001) when [Chla] = 1 mg/m³. For the absorption coefficient of CDOM, we use reference wavelength of λ r e f = 440 nm, and a C D O M ( 440 ) = 0.0316 m⁻¹ and S c d o m = 0.018 nm⁻¹, which are calculated by MG2009 with [Chla] = 1 mg/m³. In addition, we perturb the CDOM absorption by setting Φ = 0.5 and 2 in Eq. 8 so that two more values of a C D O M ( 440 ) are used: 0.0158 and 0.0632 m⁻¹, which creates a variability of a C D O M so that it can be compared with the effect of variable amounts of absorbing aerosols.

The aerosol optical depth is 0.1 at 550 nm. The aerosol model is assumed to be a bio-modal lognormal distribution, with the fine and coarse modes to be 90 and 10% of the total volume, respectively. The coarse mode is assumed to be sea salt with the effective radius and variance of 2.0194  μ  m and 0.672, respectively. The fine mode is assumed to be an internal mixture of dust-like, water-soluble, and brown carbon, and soot carbon aerosols, with the effective radius and variance of 0.15  μ  m and 0.437, respectively. Three brown carbon volume fractions are used in the fine mode: 0.02, 0.04, and 0.06 (Schuster et al., 2016; Shi et al., 2021). The volume ratio of dust-like to water-soluble aerosols is fixed at 1:3, while the volume ratio of brown carbon to soot carbon aerosols is assumed to be 2, which is consistent with Schuster et al. (2016). If brown carbon aerosol fraction in the fine mode is 0.02, the corresponding soot, dust-like, and water soluble aerosol volume fraction are: 0.01 (1–0.02-0.01)/4 = 0.2425 (1–0.02-0.01) 3/4 = 0.7275, respectively. These numbers will need to be multiplied by 90% to get their total fraction in aerosols including both fine and coarse modes. The fractions of the different components of the other two cases can be calculated similarly, with the fractions showing in Table 1.

The refractive indices of soot carbon, dust-like, water-soluble, and sea-salt are from Shettle and Fenn (1979). The real refractive index n of brown carbon is set to 1.55 for all wavelengths and the imaginary part k between 375 and 532 nm is based on the Kramers–Kronig fitting of the measurement data from Sumlin et al. (2018). For λ < 375  nm, we use the exponent fitting (Mok et al., 2016): k ( λ ) = k ( 375 ) ( 375 λ ) 5.7 (14) where k ( 375 ) = 0.01235 is from Sumlin et al. (2018) to ensure continuity.

Figure 4 shows the singe scattering albedo of the aerosol models with three different brown carbon volume fractions: 0.02, 0.04, and 0.06 in the fine mode. It shows that the single scattering albedo ranges between 0.915 and 0.965. It is smaller for smaller brown carbon fraction, primarily due to soot carbon which is ½ of brown carbon volume fraction. The separation between the three lines in the UV bands are slightly larger than other spectral region, indicating the influence of brown carbon. In addition, the DoLP signal has different sensitivity to single scattering albedo, which may be used to further differentiate the influence of CDOM and absorbing aerosols.

We calculate the percentage difference of the TOA signals for each case in comparison to a reference case, which is a C D O M ( 440 ) = 0.0316 m⁻¹ and the brown carbon aerosol fraction of 0.04. The left and right diagrams of Figure 5 show the variation of the TOA reflectance and degree of linear polarization (DoLP), respectively. The solar zenith angle is 30° and the viewing angle is set to be 60° with the relative azimuth angle of 0°, which is in the same half-plane as the glint. The influence of different a C D O M ( 440 ) values is mainly for wavelength shorter than 600 nm, while the different values of brown carbon fractions move the reflectance up and down in the whole spectral range. The DoLP plot shows different trends from those of reflectance, which can be used to enhance the accuracy of the retrieval algorithms. The calibration requirement of OCI is 0.5%, which is sufficient to discriminate the variation shown in Figure 5. The polarization signal variation is between -0.008 and 0.008, which can be detected by SPEXone, which aims to achieve 0.003 of DoLP accuracy (Werdell et al., 2019).

The MAP instruments will measure the polarized radiance at different viewing angles, which can provide extra information on the aerosol and hydrosol microphysics. Figure 6 shows the dependence of DoLP in the principal plane as a function of viewing zenith angle. The principal plane is defined as the plane containing the direct solar ray and local vertical. Positive viewing zenith angles indicate the half plane with an azimuth angle of 0°, which contains the Sun glint, while negative viewing zenith angles are the half plane of an azimuth angle of 180°. Two HARP wavelengths, 441 and 669 nm, are chosen to show the angular dependence. At 441 nm, both the brown carbon fractions and the CDOM absorption coefficients have visible influences on the DoLP at TOA (see Figure 6A, the difference of DoLP with those of a reference case). At 669 nm, the influence due to CDOM absorption diminishes and only brown carbon aerosols would have influences on DoLP (See Figure 6B). The DoLP change at 441 nm is as large as 0.008. It is smaller at 669 nm for which the DoLP change is mostly smaller than 0.00015, which is hard to be differentiated by SPEXone. The angular dependence of DoLP at TOA at different wavelength can be used to differentiate the brown carbon fraction and CDOM absorption if a proper data fitting algorithm is implemented.