Physical Parameterization of Hyperspectral Reﬂectance in the Oxygen A-Band for Single-Layer Water Clouds

: Previous studies have shown that it is feasible to retrieve multiple cloud properties simultaneously based on the space-borne hyperspectral observation in the oxygen A-band, such as cloud optical depth, cloud-top height, and cloud geometrical thickness. However, hyperspectral remote sensing is time-consuming if based on the precise radiative transfer solution that counts multiple scatterings of light. To speed up the radiation transfer solution in cloud scenarios for nadir space-borne observations, we developed a physical parameterization of hyperspectral reﬂectance in the oxygen A-band for single-layer water clouds. The parameterization takes into account the inﬂuences of cloud droplet forward-scattering and nonlinear oxygen absorption on hyperspectral reﬂectance, which are improvements over the previous studies. The performance of the parameterization is estimated through comparison with DISORT (Discrete Ordinates Radiative Transfer Program Multi-Layered Plane-Parallel Medium) on the cases with solar zenith angle θ , the cloud optical depth τ c , and the single-scattering albedo ω in the range of 0 ≤ θ ≤ 75, 5 ≤ τ c ≤ 50, 0.5 ≤ ω ≤ 1. The relative error of the cloud reﬂectance is within 5% for most cases, even for clouds with optical depths around ﬁve or at strong absorption wavelengths. We integrate the parameterization with a slit function and a simpliﬁed atmosphere to evaluate its performance in simulating the observed cloud reﬂection at the top of the atmosphere by OCO-2 (Orbiting Carbon Observatory-2). To better visualize the possible errors from the new parameterization, gas molecular scattering, aerosol scattering, and reﬂection from the underlying surface are ignored. The relative error of the out-of-band radiance is less than 4% and the relative error of the intra-band radiance ratio is less than 4%. The radiance ratio is the ratio of simulated observations with and without in-cloud absorption and is used to assess the accuracy of the parameterization in quantifying the in-cloud absorption. The parameterization is a preparation for rapid hyperspectral remote sensing in the oxygen A-band. It would help to improve retrieval e ﬃ ciency and provide cloud geometric thickness products.


Introduction
Inferring cloud geometrical properties from the space-borne observation in the oxygen A-band has been recognized for a long time [1]. The principle is that cloud scattering affects the photon path, and the photon path length determines the total oxygen absorption along the path. The cloud-top height (CTH) and the cloud geometrical thickness (CGT) are important factors affecting the photon transfer in the cloud. In the next section, we will explain the in-cloud scattering and absorption difference between weak and strong absorption situations. Then, we present the improved parameterization and a simplified atmosphere model used in accuracy evaluation. We verify the necessity of the improvements and estimate the accuracy of the parameterized cloud reflectance and the simulated hyperspectral radiance in section 3. Section 4 discusses the source of errors, deficiencies, and future research directions. Finally, in section 5, we summarize the improvements and the performance of the physical parameterization. The relative error between the precise radiance and the radiance calculated according to the previous study [32]. The observations of OCO-2 and GOSAT have sharp and deep absorption channels, while those of GOME-2 do not. When applying the previous formula [32] to hyperspectral simulation, the relative error is doubled. The sampling number per FWHM is 2.5. The cloud optical depth used in the simulation is 15, the cloud top height is 2 km, the cloud geometrical thickness is 1 km, and the solar zenith angle is 40°.

Methodology
The physical parameterization originates from the absorption-free semi-infinite cloud case. The absorption refers to the absorption of oxygen molecules, as the absorption of water vapor and cloud droplets is far less than that of oxygen in the A-band and ignored in cloud retrievals. Therefore, the out-of-band absorption optical depth is regarded as 0, and the single scattering albedo of the cloud layer is 1. In the asymptotic theory, the cloud-top reflectance can be parameterized by the viewing geometry [33,34]. The superscript (0) of means no absorption in the cloud layer, and the subscript (∞) means infinite cloud optical depth. Oxygen molecules absorb solar radiation in the Aband, reducing the cloud reflectance [35,36]. With the function S representing the attenuation, due to absorption by oxygen, the reflectance ∞ of the semi-infinite cloud could be expressed as: Since most real-world clouds are optically limited, part of the solar radiation penetrates clouds and further reduces the reflectance [30]. With the function H standing for the loss due to penetration, the cloud reflectance R can be represented as: Kokhanovsky et al. divide into a single-scattering part ( , ), and a multi-scattering part ( , ), depending on whether related to the cloud phase function directly [30,32,34,37]. They also The relative error between the precise radiance and the radiance calculated according to the previous study [32]. The observations of OCO-2 and GOSAT have sharp and deep absorption channels, while those of GOME-2 do not. When applying the previous formula [32] to hyperspectral simulation, the relative error is doubled. The sampling number per FWHM is 2.5. The cloud optical depth used in the simulation is 15, the cloud top height is 2 km, the cloud geometrical thickness is 1 km, and the solar zenith angle is 40 • .
For fast and accurate hyperspectral remote sensing, we established a parameterization of the cloud reflectance that applies to both weak and strong absorption wavelengths. The parameterization consists of several functions with physical meaning, which reproduce the process of radiation transfer in the cloud. In the next section, we will explain the in-cloud scattering and absorption difference between weak and strong absorption situations. Then, we present the improved parameterization and a simplified atmosphere model used in accuracy evaluation. We verify the necessity of the improvements and estimate the accuracy of the parameterized cloud reflectance and the simulated hyperspectral radiance in Section 3. Section 4 discusses the source of errors, deficiencies, and future research directions and summarize the improvements and the performance of the physical parameterization.

Methodology
The physical parameterization originates from the absorption-free semi-infinite cloud case. The absorption refers to the absorption of oxygen molecules, as the absorption of water vapor and cloud droplets is far less than that of oxygen in the A-band and ignored in cloud retrievals. Therefore, the out-of-band absorption optical depth is regarded as 0, and the single scattering albedo of the cloud layer is 1. In the asymptotic theory, the cloud-top reflectance R 0 ∞ can be parameterized by the viewing geometry [33,34]. The superscript (0) of R 0 ∞ means no absorption in the cloud layer, and the subscript (∞) means infinite cloud optical depth. Oxygen molecules absorb solar radiation in the A-band, reducing the cloud reflectance [35,36]. With the function S representing the attenuation, due to absorption by oxygen, the reflectance R ∞ of the semi-infinite cloud could be expressed as: Remote Sens. 2020, 12, 2252 4 of 18 Since most real-world clouds are optically limited, part of the solar radiation penetrates clouds and further reduces the reflectance [30]. With the function H standing for the loss due to penetration, the cloud reflectance R can be represented as: Kokhanovsky et al. divide R 0 ∞ into a single-scattering part (R 0 ∞,ss ), and a multi-scattering part (R 0 ∞,ms ), depending on whether related to the cloud phase function directly [30,32,34,37]. They also propose another formula for R 0 ∞ in SACURA [4,33], where the phase-function-dependent part would vanish for nadir observations. In this study, we focused on nadir observations but considered cloud phase function, so the cloud reflectance, based on [30,34,37], could be written as: Dividing the reflectance into single-and multi-scattering parts could better explain how the cloud reflectance is related to the viewing geometry. Still, the same functions S and H are used in both parts. For weak absorption, the single-scattering part accounts for a small proportion in cloud reflection, as shown in Figure 2a, and hence it is unnecessary to define exclusive functions for R 0 ∞,ss . However, as the absorption increases, it is inadequate to use one function S for the two parts with different in-cloud absorption attenuation, as shown in Figure 2b. Similar, different H functions are also needed for the two parts.
Remote Sens. 2020, 12, x FOR PEER REVIEW 4 of 18 propose another formula for in SACURA [4,33], where the phase-function-dependent part would vanish for nadir observations. In this study, we focused on nadir observations but considered cloud phase function, so the cloud reflectance, based on [30,34,37], could be written as: Dividing the reflectance into single-and multi-scattering parts could better explain how the cloud reflectance is related to the viewing geometry. Still, the same functions S and H are used in both parts. For weak absorption, the single-scattering part accounts for a small proportion in cloud reflection, as shown in Figure 2a, and hence it is unnecessary to define exclusive functions for , . However, as the absorption increases, it is inadequate to use one function S for the two parts with different in-cloud absorption attenuation, as shown in Figure 2b. Similar, different H functions are also needed for the two parts. Moreover, forward-scattering impacts should be well resolved, since it retains direct-light-like directivity, which will enhance the radiance in the incidence direction, particularly crucial for optically thin clouds because of less multi-scattering. Considering that the single-scattering part is closely related to the cloud phase function, the phase-function-dependent part, Rph, is the result of the scatterings of direct solar radiation and the forward-scattered radiance. The other part, Rms, is the reflectance that does not depend directly on the cloud phase function because of the randomizing effect of multi-scattering.
Our work consisted of two parts: quantifying the contribution of the forward-scattering to Rph, and modifying Rms to handle the enhanced absorption. The physical parameterization expressed the reflectance (R) in terms of the attenuation due to absorption (Sph, Sms) and loss due to penetration (Hph, Hms) as: The single-scattering part (R 0 ∞,ss , R ∞,ss ) was calculated according to the definition of single-scattering. The cloud reflectance (R 0 ∞ , R ∞ ) was calculated by DISORT. The multi-scattering part (R 0 ∞,ms , R ∞,ms ) is the difference between the cloud reflectance and the single-scattering part.
Moreover, forward-scattering impacts should be well resolved, since it retains direct-light-like directivity, which will enhance the radiance in the incidence direction, particularly crucial for optically thin clouds because of less multi-scattering. Considering that the single-scattering part is closely related to the cloud phase function, the phase-function-dependent part, R ph , is the result of the scatterings of direct solar radiation and the forward-scattered radiance. The other part, R ms , is the reflectance that does not depend directly on the cloud phase function because of the randomizing effect of multi-scattering.
Our work consisted of two parts: quantifying the contribution of the forward-scattering to R ph , and modifying R ms to handle the enhanced absorption. The physical parameterization expressed the reflectance (R) in terms of the attenuation due to absorption (S ph , S ms ) and loss due to penetration (H ph , H ms ) as: Remote Sens. 2020, 12, 2252 5 of 18 R ∞,ms = R 0 ∞,ms S ms .
Three types of clouds are involved in the above formulas and the reflectance of each type contains parts that are dependent and independent of the phase function. Except for the fact that the sum of R ph and R ms is the reflectance of the real-world cloud, R ∞,ph and R ∞,ms correspond to the reflectance of the semi-infinite cloud (R ∞ ), and R 0 ∞,ph and R 0 ∞,ms correspond to the reflectance of the absorption-free semi-infinite cloud (R 0 ∞ ).

Phase-Function-Dependent Reflectance
Based on the definition of reflectance, R ph for the nadir observation is: where F is the solar radiation at the cloud top, and µ is the cosine of solar zenith angle θ. τ and ω are the extinction optical depth and the single-scattering albedo of the cloud layer, respectively. The phase-function-dependent source function, J ph , consists of contributions from direct solar radiation (J 1 ) and forward-scattered radiance (J 2 to J n ). The subscript n of J n indicates that the radiation source has previously scattered forward (n − 1) times. Each forward-scattered radiance has the same direction as direct solar radiation. For the single-scattering of direct solar radiation, the source function J 1 can be written as: where Fexp(−τ /µ) is the attenuated solar radiation at the optical depth τ , p(ϑ) is the water cloud phase function, and ϑ is the scattering angle. For the nadir observation, ϑ = π − θ. For the forward-scattering, ϑ = 0 • . The source function J 2 stands for the single-scattering of the first forward-scattered radiance: The integral in Equation (14) is the sum of all the first forward-scattered radiance reaching the extinction optical depth τ . Simplifying the above formula, we have: Similarly, the source function J n formed by the (n − 1)-times-forward-scattered radiance is: Remote Sens. 2020, 12, 2252 6 of 18 Substituting Equation (16) into Equation (12), the source function that depends on the phase function is: Compared with Equation (13), the additional positive term τ ωp(0)/4π in Equation (17) means that, except for the attenuated sunlight, forward-scattered radiance also contributes to R ph . The factor exp[τ ωp(0)/4π] only counts the forward-scattered radiance with a scattering angle of 0 • , and the scattered radiance, which has small scattering angles, also contributes to J ph . So, we replaced p(0)/4π with an undetermined coefficient c to take in more sources that give rise to the phase-function-dependent reflectance. Substituting Equation (17) into Equation (11), each term of R ph for nadir observation can be expressed as: In the derivation, the nothingness beneath the optically finite cloud L can be thought of as a disappeared semi-infinite cloud L ∞ . It has the same single-scattering albedo with L. If L ∞ were still there, the reflection from the top of L would be brighter than if it was only L. The cloud-top reflection reduces with the disappearance of L ∞ . The three terms in H ph accurately describe the penetration loss, which penetrates L from top to bottom, reflected by L ∞ , and finally passes through L again from the bottom up. After L ∞ disappears, this radiation cannot return to the cloud top and should be subtracted.

Phase-Function-Independent Reflectance
The phase-function-independent reflectance is composed of R 0 ∞,ms related to the viewing geometry, S ms representing absorption, and H ms standing for the penetration loss. Multi-scattering makes R 0 ∞,ms less dependent on the phase function. A rational polynomial would well fit the reflectance changed with the viewing geometry: The coefficients d 0 , d 1 , and d 2 are determined in Section 2.3. SACURA uses a parameter y to play the role of absorption in the radiative transfer of diffused light [37]. With absorption enhancement, high-order terms are needed to ensure accuracy [35]. After adding the fourth-order polynomial with nine undetermined coefficients (e 00 , e 10 , etc.), the improved function S ms applicable to absorption lines could be expressed as: S ms (µ, ω) = exp[−y e 00 + e 10 µ + e 01 y + e 11 µy + e 02 y 2 + e 12 µy 2 + e 03 y 3 + e 13 µy 3 + e 04 y 4 ], The function H ms , similar to H ph , consists of the three processes: (a) penetrating through L, (b) reflected by L ∞ , and (c) penetrating L again. SACURA gives each process a parameterization when ω > 0.95:

1.
The global transmittance t defines the first penetration. Besides, a correction ∆t is proposed for optically relatively thin clouds [30].

2.
The diffuse reflection of L ∞ is quantified as a spherical albedo and approximately equals to exp(-y) [38].

3.
When the diffused light penetrates L from the bottom up, the attenuation is similar to the radiative transfer in the optically deep region, which is approximated by exp(−x) [37].
Since the parameterizations (t, ∆t, exp(−y), and exp(−x)) are integrals of the diffuse radiation over incidence and exit directions, the asymptotic theory uses the escape function K to deal with the viewing geometry. The function H ms in SACURA is [30]: For cases with strong absorptions (i.e., ω < 0.95), we made modifications so that each part of the function H ms was close to the numerical solution of the radiative transfer equation, such as adding exp x m 0 + m 1 y + m 2 y 2 , y p 2 + p 3 µ + p 4 µ 2 , and (1 + q 3 y) into the functions t, K, and ∆t, respectively. The improved formulas are: The modifications would not affect the applicability to weak absorption, because of the degeneration from the improved formulas to the previous when y→0. The coefficients m 0 , p 0 , q 0 , etc. are determined in Section 2.3.

Coefficient Fitting
The improved formulas make up the physical parameterization for single-layer water clouds: As the reflectance R ∞ is a part of R, the coefficients in Equations (29) and (30) should be fitted in two steps. In the search for the best coefficients, the relative errors of the parameterization were used as the fitting cost, whose mean square should descend in iterations. However, it is not appropriate to take the relative error of R ∞ as its own cost, because the infinite optical depth makes R ∞ much larger than R, the error of the first fitting would destroy the second one. To solve the problem, we used the reflectance of low-level stratocumulus clouds over the ocean as the denominator of the cost in the first step, which had a statistically average cloud optical depth of 7 [39]. The cost functions δ 1 and δ 2 for the two-step fitting are: where the truth reflectance R ∞ and R were derived from DISORT (Discrete Ordinates Radiative Transfer Program for a Multi-Layered Plane-Parallel Medium) [40] with the assumption of cloud C1 phase function. The coefficient fitting was conducted on the cases with solar zenith angle θ, the cloud optical depth τ c , and the single-scattering albedo ω in the range of 0 ≤ θ ≤ 75, 5 ≤ τ c ≤ 50, 0.5 ≤ ω ≤ 1.
The fitting results are shown in Table 1.

Hyperspectral Simulation
The hyperspectral radiance simulation reproduces the radiative transfer of solar radiation passing through the atmosphere, reflected by clouds to the top of the atmosphere (TOA), and finally captured by the instrument: λ is the central wavelength of a channel of the instrument, and F is the extraterrestrial solar radiation at TOA. The simulation is a convolution of the reflected radiance at TOA and the instrument response function f. The cloud reflectivity is µR/π, whose optical depth is τ c and geometrical thickness is l. The formula also contains the radiative transfer above clouds. The symbol h is the cloud-top height. T inc is the transmissivity of the solar radiation from TOA to the cloud top, and T ref is the transmissivity from the cloud top to the space-borne sensor at the nadir observation. The simulations used the same instrument response function as the OCO-2 L2 retrievals [41,42], which are recorded in the OCO2_L1B_Science product (https://oco2.jpl.nasa.gov/oco-2-data-center/). The solar radiation is from the AER solar irradiance model (Atmospheric and Environmental Research, http://rtweb.aer.com/solar_frame.html) [43].
For estimating the parameterization in the hyperspectral application, a simplified atmosphere model was introduced in simulations, ignoring the molecular scattering, aerosol scattering, and reflection from the underlying surface. The spectral oxygen absorption coefficients came from LBLRTM (Line-By-Line Radiative Transfer Model) (Atmospheric and Environmental Research, Lexington, MA, USA) [44] with the assumption of the US 1976 standard atmosphere. To further simplify the atmosphere model, the oxygen absorption coefficients were fixed as a constant k(λ) below h 0 = 5 km, which only changed with the wavelength. So, the total oxygen zenith optical depth (τ O2,total ) from TOA to a certain height was a linear function of height (h < 5 km), and the in-cloud oxygen absorption optical depth (τ O2 ) was a linear function of l, too. The single-scattering albedo at each wavelength was calculated according to τ c and τ O2 : Remote Sens. 2020, 12, 2252 9 of 18 The transmissivities of the plane-parallel's atmosphere T inc and T ref depend on the slant path and τ O2,total . For nadir observation: The simulation is a bridge between observations and cloud properties. The observed radiance outside the oxygen A-band (I 0 ) is suitable for retrieving the cloud optical depth because the observations are not disturbed by oxygen absorption (T inc = T ref = 1).
Inside the oxygen A-band, the in-cloud absorption varies with the cloud geometrical properties (e.g., the cloud geometrical thickness). For evaluating the in-cloud absorption embodied in the parameterization, we considered a hypothetical case as a control, in which the cloud was seen as an opaque slab with a reflectance equal to R (µ,τ c ,ω = 1). The radiance I was an imaginary observation that was not affected by in-cloud absorption: In Equation (40), cloud reflectance is the only difference from Equation (33). We used the radiance ratio I/I as the quantification of the in-cloud absorption.

Contribution of Forward-Scattering
We compared the new parameterization with the previous study [32] to verify the necessity of estimating forward-scattering impacts in R ph . Figure 3 shows the truth reflectance (from DISORT) with two local peaks: θ < 5 • and θ ≈ 37 • . The two local peaks correspond to the local peaks of phase function at the backward and the primary rainbow (ϑ ≈ 143 • ) which means that part of the reflectance is closely related to the phase function. This is why we kept the phase-function-dependent part in the parameterization for nadir observations. When we divide the reflection into single-and multi-scattering parts, the remaining local peaks in the multi-scattering curve (dashed line) indicate that, in addition to R ss , there are other parts related to the phase function. Instead, the improved formula cleanly extracts the phase-function-independent reflectance (dotted line), a curve with almost no peaks, fitting well with a simple rational polynomial (red cross). The vanishing peaks prove the necessity of quantifying the influence of forward-scattering on the reflectance.

Improvements for Nonlinear Absorption
Next, we tested the performance of higher-order terms in S ms , which is to approximate the nonlinear absorption attenuation in R ∞,ms . Figure 4a shows that the function S ms decreases with increasing absorption and the declining rate varies with the single-scattering albedo. The variable rate suggests that the exponent in S ms cannot be linear and the attenuation function S in SACURA is a second-order function of y, which is not enough at ω < 0.95. Therefore, the hyperspectral application needs a high-order modification. Without the modification, the relative errors of the function S are higher than 5% when ω < 0.95, and exceed 20% when ω < 0.8, as shown in Figure 4b. On the contrary, the relative errors of the modified function S ms are less than 5% for ω > 0.5, as shown in Figure 4a.
The decline in the relative error demonstrates that the modified function S ms is suitable for either weak or strong absorption cases.
with two local peaks: θ < 5° and θ ≈ 37°. The two local peaks correspond to the local peaks of phase function at the backward and the primary rainbow (ϑ ≈ 143°) which means that part of the reflectance is closely related to the phase function. This is why we kept the phase-function-dependent part in the parameterization for nadir observations. When we divide the reflection into single-and multiscattering parts, the remaining local peaks in the multi-scattering curve (dashed line) indicate that, in addition to Rss, there are other parts related to the phase function. Instead, the improved formula cleanly extracts the phase-function-independent reflectance (dotted line), a curve with almost no peaks, fitting well with a simple rational polynomial (red cross). The vanishing peaks prove the necessity of quantifying the influence of forward-scattering on the reflectance.

Improvements for Nonlinear Absorption
Next, we tested the performance of higher-order terms in Sms, which is to approximate the nonlinear absorption attenuation in ∞,ms . Figure 4a shows that the function Sms decreases with increasing absorption and the declining rate varies with the single-scattering albedo. The variable rate suggests that the exponent in Sms cannot be linear and the attenuation function S in SACURA is a second-order function of y, which is not enough at ω < 0.95. Therefore, the hyperspectral application needs a high-order modification. Without the modification, the relative errors of the function S are higher than 5% when ω < 0.95, and exceed 20% when ω < 0.8, as shown in Figure 4b. On the contrary, the relative errors of the modified function Sms are less than 5% for ω > 0.5, as shown in Figure 4a. The decline in the relative error demonstrates that the modified function Sms is suitable for either weak or strong absorption cases.   [32]. The accurate S is calculated as ⁄ . Tests were at three solar zenith angles (θ). Figure 5 is similar to Figure 4, but shows the performance of the modified function Hms. Without the modification, the relative errors of the function H are higher than 10% when ω < 0.95 for θ = 40° and θ = 60°, and exceed 25% when ω < 0.8 for all test cases, as shown in Figure 5b. With the modifications, the relative errors decrease to less than 5% when ω > 0.95 and less than 10% when ω > 0.85.
where R 0 ∞ is the truth reflectance of the absorption-free semi-infinite cloud. (b) The function S and the relative errors were calculated based on the previous study [32]. The accurate S is calculated as R ∞ / R 0 ∞ . Tests were at three solar zenith angles (θ). Figure 5 is similar to Figure 4, but shows the performance of the modified function H ms . Without the modification, the relative errors of the function H are higher than 10% when ω < 0.95 for θ = 40 • and θ = 60 • , and exceed 25% when ω < 0.8 for all test cases, as shown in Figure 5b. With the modifications, the relative errors decrease to less than 5% when ω > 0.95 and less than 10% when ω > 0.85. Figure 5 is similar to Figure 4, but shows the performance of the modified function Hms. Without the modification, the relative errors of the function H are higher than 10% when ω < 0.95 for θ = 40° and θ = 60°, and exceed 25% when ω < 0.8 for all test cases, as shown in Figure 5b. With the modifications, the relative errors decrease to less than 5% when ω > 0.95 and less than 10% when ω > 0.85.  [32]. The accurate H is calculated as − . Tests were at three solar zenith angles (θ). Figure 6 comprehensively shows the calculated cloud reflectance and relative errors. The parameterization performs better on optically thick clouds than thin clouds. The modifications in Rph make the approximations match the truth at 5° ≤ θ ≤ 75°, including the local peaks, even when applied to the optically relatively thin cloud. For cases with strong absorption, the modifications in Sms and

Errors in Parameterization
The function H and the relative errors were calculated based on the previous study [32]. The accurate H is calculated as R ∞ − R. Tests were at three solar zenith angles (θ). Figure 6 comprehensively shows the calculated cloud reflectance and relative errors. The parameterization performs better on optically thick clouds than thin clouds. The modifications in R ph make the approximations match the truth at 5 • ≤ θ ≤ 75 • , including the local peaks, even when applied to the optically relatively thin cloud. For cases with strong absorption, the modifications in S ms and H ms keep the relative error δ 2 within 5%. Figure 6 shows the usability of the improved physical parameterization in cases with different solar zenith angles, cloud optical depths, and single-scattering albedos.

Errors in Parameterization
Remote Sens. 2020, 12, x FOR PEER REVIEW 11 of 18 Hms keep the relative error δ2 within 5%. Figure 6 shows the usability of the improved physical parameterization in cases with different solar zenith angles, cloud optical depths, and singlescattering albedos.

Accuracy Analysis Outside the Oxygen A-Band
The parameterization is to simulate the hyperspectral nadir observation in the oxygen A-band rapidly, and Figure 7 shows the process. In Figure 7b, the improved parameterization gives a set of cloud reflectance, which is close to the precise value from DISORT. The simulated TOA radiance, as shown in Figure 7c is a combination of the solar radiation, as shown in Figure 7a, the transmittivity above the cloud, the cloud reflectance, and the instrument response function.

Accuracy Analysis Outside the Oxygen A-Band
The parameterization is to simulate the hyperspectral nadir observation in the oxygen A-band rapidly, and Figure 7 shows the process. In Figure 7b, the improved parameterization gives a set of cloud reflectance, which is close to the precise value from DISORT. The simulated TOA radiance, as shown in Figure 7c is a combination of the solar radiation, as shown in Figure 7a, the transmittivity above the cloud, the cloud reflectance, and the instrument response function. The radiation outside the band is absorption-free and suitable for retrievals of cloud optical depth. Thus, it is necessary to test the accuracy of the simulated radiance outside oxygen A band. Figure 8a shows a series of radiances calculated by the physical parameterization corresponding to different cloud optical depths. The radiance changes significantly in some channels, including channels located near 759.3 nm and 771 nm outside the oxygen A-band, and some non-absorption channels located inside the band that can only be observed by hyperspectral sensors (e.g., channels near 767.6 nm). Figure 8b shows the relative errors of the hyperspectral simulations, and at least half of the samples have a relative error of less than 0.5% on the channels mentioned above. The relative error increases to about 4% if all the samples are counted. Due to the modification for enhanced absorption, the simulated radiance also has a high accuracy of about 2% in the absorption channels, even in channels containing oxygen absorption lines. The radiation outside the band is absorption-free and suitable for retrievals of cloud optical depth. Thus, it is necessary to test the accuracy of the simulated radiance outside oxygen A band. Figure 8a shows a series of radiances calculated by the physical parameterization corresponding to different cloud optical depths. The radiance changes significantly in some channels, including channels located near 759.3 nm and 771 nm outside the oxygen A-band, and some non-absorption channels located inside the band that can only be observed by hyperspectral sensors (e.g., channels near 767.6 nm). Figure 8b shows the relative errors of the hyperspectral simulations, and at least half of the samples have a relative error of less than 0.5% on the channels mentioned above. The relative error increases to about 4% if all the samples are counted. Due to the modification for enhanced absorption, the simulated radiance also has a high accuracy of about 2% in the absorption channels, even in channels containing oxygen absorption lines.

Accuracy Analysis Inside the Oxygen A-Band
The absorption in the oxygen A-band is the basis of the cloud structure retrievals. The parameterization considers the influence of in-cloud absorption and is established for retrievals of cloud geometrical thickness. We used the radiance ratio to quantify the in-cloud absorption. The numerator radiance was simulated as the process in Figure 7, and the denominator radiance was simulated under the opaque slab assumption, as shown in Figure 9. The radiance ratio is less than 1 in the absorption channel-the stronger the absorption, the smaller the ratio.

Accuracy Analysis Inside the Oxygen A-Band
The absorption in the oxygen A-band is the basis of the cloud structure retrievals. The parameterization considers the influence of in-cloud absorption and is established for retrievals of cloud geometrical thickness. We used the radiance ratio to quantify the in-cloud absorption. The numerator radiance was simulated as the process in Figure 7, and the denominator radiance was simulated under the opaque slab assumption, as shown in Figure 9. The radiance ratio is less than 1 in the absorption channel-the stronger the absorption, the smaller the ratio.

Accuracy Analysis Inside the Oxygen A-Band
The absorption in the oxygen A-band is the basis of the cloud structure retrievals. The parameterization considers the influence of in-cloud absorption and is established for retrievals of cloud geometrical thickness. We used the radiance ratio to quantify the in-cloud absorption. The numerator radiance was simulated as the process in Figure 7, and the denominator radiance was simulated under the opaque slab assumption, as shown in Figure 9. The radiance ratio is less than 1 in the absorption channel-the stronger the absorption, the smaller the ratio.   Figure 10a shows a series of radiance ratios, which decrease gradually in the absorption channels as the cloud geometrically thickens. The change is most apparent near the oxygen absorption lines, such as 760.6 nm and 763.8 nm. Figure 10b shows the relative error of the simulated radiance ratio. At least half of the samples have a relative error of less than 1%, and the relative errors of all samples are about 4% on the absorption channels.
Remote Sens. 2020, 12, x FOR PEER REVIEW 14 of 18 Figure 10a shows a series of radiance ratios, which decrease gradually in the absorption channels as the cloud geometrically thickens. The change is most apparent near the oxygen absorption lines, such as 760.6 nm and 763.8 nm. Figure 10b shows the relative error of the simulated radiance ratio. At least half of the samples have a relative error of less than 1%, and the relative errors of all samples are about 4% on the absorption channels.

Discussion and Conclusions
The accurate parametric cloud reflectance is the premise of cloud retrieval. The parameterization developed for space-borne moderate spectral resolution observations should not be used to calculate the hyperspectral reflectance directly. The hyperspectral measurements in the oxygen A-band have several strong absorption channels where the in-cloud scattering and absorption are different from those in weak absorption channels. We show that the proportion of single-scattering in the reflection increases gradually with the absorption enhancement and the influence of cloud droplets forwardscattering is important for the optically thin cloud. The single-scattering represents the radiative transfer dependent on the phase function, while the multi-scattering represents the radiative transfer independent on the phase function. The two scatterings are different, and it is necessary to establish formulas for each of them to parameterize their contribution to the cloud-top reflectance.
In this study, we developed a fast physical parameterization method for cloud reflectance in the oxygen A-band for hyperspectral remote sensing. Firstly, we quantified the influences of forwardscattering on reflectance. Secondly, the nonlinear absorption was parameterized, and the modified formulas were applicable to both strong absorption lines and weak absorption bands. After integrating all the improvements, the relative error of the parameterization was less than 5% when 5° ≤ θ ≤ 75°compared with the precise value from DISORT (Discrete Ordinates Radiative Transfer Program Multi-Layered Plane-Parallel Medium), even for the optically relatively thin clouds (the cloud optical depth is higher than 5) or strong absorption lines (the single-scattering albedo is lower than 0.95). For optically thick clouds, the relative error reduced to 2%. The performance of the parameterization in the hyperspectral simulation was estimated, too. The simulation was carried out in a simplified atmospheric model with the focus on the accuracy in calculating the observed cloud reflection, which ignores the molecular scattering, aerosol scattering, and reflection from the underlying surface. The relative errors of all test cases were less than 5% for the radiance outside the

Discussion and Conclusions
The accurate parametric cloud reflectance is the premise of cloud retrieval. The parameterization developed for space-borne moderate spectral resolution observations should not be used to calculate the hyperspectral reflectance directly. The hyperspectral measurements in the oxygen A-band have several strong absorption channels where the in-cloud scattering and absorption are different from those in weak absorption channels. We show that the proportion of single-scattering in the reflection increases gradually with the absorption enhancement and the influence of cloud droplets forward-scattering is important for the optically thin cloud. The single-scattering represents the radiative transfer dependent on the phase function, while the multi-scattering represents the radiative transfer independent on the phase function. The two scatterings are different, and it is necessary to establish formulas for each of them to parameterize their contribution to the cloud-top reflectance.
In this study, we developed a fast physical parameterization method for cloud reflectance in the oxygen A-band for hyperspectral remote sensing. Firstly, we quantified the influences of forward-scattering on reflectance. Secondly, the nonlinear absorption was parameterized, and the modified formulas were applicable to both strong absorption lines and weak absorption bands. After integrating all the improvements, the relative error of the parameterization was less than 5% when 5 • ≤ θ ≤ 75 • compared with the precise value from DISORT (Discrete Ordinates Radiative Transfer Program Multi-Layered Plane-Parallel Medium), even for the optically relatively thin clouds (the cloud optical depth is higher than 5) or strong absorption lines (the single-scattering albedo is lower than 0.95). For optically thick clouds, the relative error reduced to 2%. The performance of the parameterization in the hyperspectral simulation was estimated, too. The simulation was carried out in a simplified atmospheric model with the focus on the accuracy in calculating the observed cloud reflection, which ignores the molecular scattering, aerosol scattering, and reflection from the underlying surface. The relative errors of all test cases were less than 5% for the radiance outside the oxygen A-band that can be used for cloud optical depth retrievals, and at least half of the test cases had a relative error of less than 0.5%. The relative errors in all test cases were less than 4% for the radiance ratio in the oxygen A-band related to the cloud geometrical thickness, and at least half of the test cases had a relative error of less than 1%.
Generally, the accuracy of the parameterized cloud reflectance has been improved, but there are still cases whose relative error is a little bit higher than others (e.g., the cases at θ < 5 • ), as shown in Figures 3 and 6. Compared with the slant incidence case, the near-vertical transfer would encounter fewer cloud droplets and increase the proportion of reflections that experience only a few scatterings. Optically thin clouds are in a similar situation; e.g., the larger relative error in Figure 6d than that in Figure 6b). Although we have introduced the correction ∆t for these reflections, their parameterization is still incomplete. There is another error at θ > 75 • in Figure 6, mainly from the error of the escape function K at large angles [37], but the shadows caused by large solar zenith angles are more severe than the error of the parameterization.
This paper is an extension of the previous work and can replace the part of SACURA that calculates the cloud reflectance without interfering with other parts that deal with aerosol scattering and underlying surface reflection. It helps improve retrieval efficiency based on hyperspectral observations and provides cloud geometric thickness products. On the other hand, more details, such as polarized radiative transfer, multidirectional measurements, and cloud droplet effective radius, need to be considered before applying the parameterization widely. For example, on OCO-2 (Orbiting Carbon Observatory-2), a linear polarizer is installed to reduce the stray light [45]. If the parameterization is to be applied to OCO-2 observations, it is necessary to modify the parameterization to adapt to the polarized radiative transfer. Several studies suggest that the variability in angular measurements is sensitive to the cloud geometrical thickness [46,47], and the parameterization can also be extended to these studies if the observation zenith angle is added to the input of the parameterization. Besides, the influence of the cloud effective radius on cloud scattering is also unresolved, which is related to both phase function and cloud droplet number density. At present, the C1 phase function is used in the parameterization, which can potentially be biased and not applicable to a wide variety of clouds. It may improve the parameterization's applicability in cloud retrievals that introduce the cloud droplet effective radius into the parameterization. It is also necessary to detect and quantify other clouds, such as the water cloud covered by a thin ice cloud [48] or the profile of an optical thin cloud [49].
The cloud reflectance parameterization lays the foundation for the future rapid retrievals of marine water clouds, especially for the cloud geometric thickness of clouds. In the future, we will use OCO-2 oxygen A-band observations, based on the rapid algorithm, to obtain the cloud optical depth, cloud top height, and cloud geometric thickness, taking into account the instrument characteristics, such as polarization and instrument characteristics, including the observation noise and the wavelength registration error [50], as well as the atmospheric model containing aerosol scattering and surface reflection. Although the cloud retrieval algorithm has been studied based on OCO-2 observations, the cloud products retrieved by the rapid algorithm are not yet available. The rapid retrieval usually comes at the cost of accuracy and the unknown quality of cloud products obtained by the rapid retrieval algorithm is the topic for our future research.