Empirical Polarization Distribution Models for Use in CLARREO Pathfinder-VIIRS Intercalibration

Highlights

What are the main findings?

• Polarization can introduce bias in radiometric intercomparisons over polarized Earth scenes, especially for polarization-sensitive instruments like VIIRS. To mitigate this, the CLARREO Pathfinder team developed empirical Polarization Distribution Models (ePDMs) using POLDER data that characterize scene polarization using DOP and AOP as functions of scene type, geometry, and wavelength.

What are the implications of the main findings?

• The development and application of ePDMs enable more accurate intercalibration of polarization-sensitive VIIRS RSBs against CPF’s high-accuracy benchmark measurements. By selecting low-polarization scenes for comparison, the CPF team reduces systematic bias introduced by differing polarization diattenuation coefficients, improving accuracy in the radiometric intercalibration.

Abstract

In this work, we discuss the impact of polarized scene radiances on the intercalibration of CPF and VIIRS reflective solar bands and the mitigation of these effects using empirical Polarization Distribution Models (ePDMs). The ePDMs, derived from multidirectional polarized reflectance measurements taken by the POLDER instrument, can provide the polarization state of the reflected solar radiation in terms of the Degree and Angle of Polarization, DOP and AOP, for each spatially, temporally, and angularly matched intercalibration footprint between CPF and VIIRS. The CPF science team will leverage these ePDMs to identify scenes with low polarization to reduce intercalibration uncertainties for specific VIIRS channels that are polarization-sensitive. The study also demonstrates that, in the absence of ePDM-based filtering of intercalibration samples, polarization-induced biases in VIIRS reflectance measurements for shortwave bands (e.g., M3 0.49 μm) can be as high as 2.4% for clear-sky over ocean scenes.

1. Introduction

NASA’s Calibration Absolute Radiance and Refractivity Observatory (CLARREO) Pathfinder (CPF) [1] is an Earth-viewing reflected solar spectrometer to be mounted onboard the International Space Station. The instrument will measure the Earth-reflected solar radiation with a radiometric uncertainty of 0.3–0.6% (1σ), traceable to the standards of the International System of Units (known as “SI-traceability”). The high-accuracy reflectance measurements from CPF will serve as an SI-traceable reference for intercalibrating other reflective solar (RS) instruments in orbit. The CPF team will showcase an innovative on-orbit intercalibration approach, wherein two other RS sensors—the shortwave (SW) channel of the Clouds and the Earth’s Radiant Energy System (CERES) and the Reflective Solar Bands (RSB) of the Visible Infrared Imager Radiometer Suite (VIIRS) [2]—are intercalibrated against CPF benchmark measurements. This approach aims for an intercalibration methodology uncertainty of 0.3% (1σ). By leveraging the CPF payload’s two-axis pointing capability, moderate spatial sampling resolution (0.5 km), and wide spectral coverage, the CPF instrument will enable near-simultaneous temporally, spatially, angularly, and spectrally matched observations with intercalibration targets. The CPF science team has developed novel methods to account for spatial, spectral, polarization, and angular differences between CPF and the target instruments’ intercalibration footprints to achieve the stringent 0.3% intercalibration methodology uncertainty. The CPF instrument is designed to have a low sensitivity to polarized radiances. In the range of wavelengths below 1400 nm, CPF has a polarization diattenuation coefficient of less than 1%. On the other hand, the NOAA-20 VIIRS reflective solar bands (RSBs) are more polarization-sensitive, with diattenuation coefficients up to 6% [3] for some channels. This has the effect of biasing the radiometric measurements over polarized Earth scenes. To mitigate this impact, the CPF team has developed Polarization Distribution Models (PDMs) to characterize the polarization state in terms of the degree and angle of polarization, DOP and AOP, respectively, classified by the scene type, solar and view geometry, and wavelength [4,5]. By employing these PDMs, the CPF science team will identify low-polarized scene radiances to intercalibrate the polarization-sensitive VIIRS RSBs against the CPF benchmark measurements.

Two types of PDMs were developed by the CPF team: empirical (ePDMs) and theoretical PDMs (tPDMs). The former are derived using the multidirectional measurements from visible and near-infrared channels of the POLarisation and Directionality of the Earth’s Reflectances (POLDER) instrument [6,7] mounted onboard the PARASOL microsatellite flying as a part of A-Train formation at an altitude of 705 km. The mission’s primary aim was to study aerosols and clouds. POLDER, whose spatial resolution was 5.3 km × 6.2 km at nadir, took measurements from nine spectral channels from the visible (443 nm) to infrared (1020 nm), of which three—490, 670, and 865 nm—were polarized. The ePDMs are constructed using the Stokes parameters measured by these three polarized bands from top-of-atmosphere. For the polarized bands at 490, 670, and 865 nm, the Stokes parameters are retrieved from the three radiance measurements taken at polarizer orientations of −60°, 0°, and 60°. The tPDMs, on the other hand, were produced using simulations based on the Adding-Doubling Radiative Transfer Model (ADRTM). CPF makes use of three simulated scene types: clear-sky [8] and overcast ocean, and clear-sky deserts [9]. Whereas ePDMs primarily cover the 488, 555, 672, 746, and 865 nm VIIRS channels, the tPDMs extends this coverage further: to 412 and 445 nm on the lower end, and to 1240, 1378, 1610, and 2250 nm on the higher end of the VIIRS spectral channels. For scene types where both ePDMs and tPDMs are available between 488 and 865 nm, the ePDM and tPDM values are useful for evaluating consistency and validation of the two approaches for deriving polarization parameters for intercalibration footprints. However, tPDMs are beyond the scope of this paper and are not discussed further; the focus here is exclusively on ePDMs.

After briefly introducing the relevant polarization-related mathematical definitions in Section 2, in Section 3.1 we describe the procedure for constructing the DOP ePDMs, and in Section 3.2 we describe the derivation of DOP and AOP ePDMs, the AOP PDMs. In each case, we will detail how PDMs are stratified by scene type, i.e., clear scenes, with or without the presence of aerosols and cloudy ones, as well as by surface type and surface conditions. Finally, in Section 4 we will discuss the sensitivity of radiometric measurements to the two parameters, DOP and AOP, which make use of ePDMs corresponding to the highly polarized scene types.

2. Polarization: Mathematical Definitions

A polarization state at the top of the atmosphere (TOA) is specified by three Stokes parameters I, Q, U where I is radiance, and Q and U describe linear polarization. (In the context of most of the TOA remote sensing applications, the fourth Stokes parameter, V, is negligibly small and can be ignored.) Alternatively, a polarization state may also be described by the degree and angle of polarization, DOP and AOP, expressed in terms of the Stokes components. The degree of polarization, DOP, which we will denote as P, may be expressed in terms of the ratio of the Stokes parameters as:

$$P= {\displaystyle \frac{\sqrt{Q^2+U^2}}{I}}.$$

(1)

From this definition it follows that the polarization state can range from completely unpolarized (Q = 0, U = 0) to fully polarized $\sqrt{Q^2+U^2}=I.$

We denote the angle of linear polarization relative to the scattering plane as ψ. Light scattered in the atmosphere is predominantly s-polarized, so that ψ values are found mainly around 90°. The angle of linear polarization may also be defined relative to the meridian, or detector, plane and can be expressed in terms of the Stokes parameters as:

$$\chi ={\displaystyle \frac{1}{2}}{\tan }^{-1}\left({\displaystyle \frac{U}{Q}}\right).$$

(2)

The two polarization angles, χ and ψ are related as

$$\chi =\psi -\alpha$$

(3)

where α may be expressed in terms of the three view geometry variables, relative azimuth (RAZ), solar-zenith, and view-zenith angles (SZA and VZA, respectively), as [10]

$$\tan \alpha ={\displaystyle \frac{\sin \left(SZA\right)}{\frac{\sin \left(VZA\right)}{\tan \left(SZA\right)}-\cos \left(VZA\right)\cos \left(RAZ\right)}}.$$

(4)

By default, it is the angle χ that is being referred to as the angle of polarization (AOP) throughout the text. We note, since the light scattering in the atmosphere is dominated by single scattering, AOP may be represented by a simple formula [5]:

$$\cos \chi ={\displaystyle \frac{\sin \left(SZA\right)\sin \left(RAZ\right)}{\sin \mathsf{\Theta }}},$$

(5)

where Θ is the single scattering angle. Figure 1 illustrates these polarization and geometry angles.

When the light incident upon a detector is polarized and the detector itself is polarization-sensitive the relationship between the uncorrected and corrected reflectances is given by [3,11]:

$$\rho =c_{\rho }{\rho }_0={\displaystyle \frac{{\rho }_0}{1+aP\cos 2\left(\chi +\varphi \right)}}$$

(6)

where ρ₀ is the uncorrected reflectance and ρ is the reflectance corrected for detector polarization effects, a is the detector diattenuation coefficient and ϕ is the phase angle caused by the detector optics. Both a and ϕ are measured during pre-launch characterization.

3. ePDMs from POLDER/PARASOL Data

POLDER-derived data demonstrate that polarization phenomena are driven primarily by three physical phenomena: atmospheric Rayleigh scattering, Mie scattering by aerosols and clouds, and, in the case of DOP, polarization due to surface reflection.

The entire 2006 POLDER dataset from the three polarized channels was used in constructing the ePDMs, with the multi-angular sampling of each POLDER footprint (up to 15 times per footprint) augmenting the statistics. In analogy with the CERES Angular Distribution Models (ADMs) [12] and POLDER bidirectional polarization distribution function (BPDF) [13,14], each PDM may be represented by a histogram binned in terms of viewing geometry variables: relative azimuth (RAZ) and view zenith angle (VZA). The RAZ range spans 0° to 180°, with a 2° binning, while the VZA range is 0° to 70°, with 1° bins. Each ePDM is constrained within 10° sub- ranges in solar zenith angle (SZA), with SZA spanning 10° to 80° (

Table 1. The SZA ranges used to construct all ePDMs.

Constraint	Ranges
SZA (deg.)	[10, 20], [20, 30], [30, 40], [40, 50], [50, 60], [60, 70], [70, 80]

). DOP or AOP values are computed from the Stokes parameters obtained by the POLDER instrument for Earth observations from the top of atmosphere and are averaged for each RAZ–VZA bin. In addition to the mean DOP or AOP, the corresponding bin-wise standard deviations, σ_P and σ_χ, are recorded in the same RAZ–VZA format as the means [4,5]. This data format allows us to have a statistically robust set of polarization parameters. The scene type (clear or overcast), surface type (land or water surface) and the solar-view geometry uniquely determine ePDM DOP and AOP (mean) values and their corresponding standard deviations.

3.1. DOP ePDMs

We categorize PDM surface types by their International Geosphere-Biosphere Programme (IGBP) indices [15]. The ratios of the CPF-VIIRS intercalibration opportunities over each IGBP type, relative to all CPF-VIIRS intercalibration opportunities simulated within one year of CPF operations (i.e., the sampling frequency for each IGBP type) are shown in

Table 2. Surface types expected to be most frequently (sampling frequency > 0.01) encountered by the CLARREO Pathfinder during its intercalibration with the VIIRS instrument arranged by their corresponding International Geosphere-Biosphere Programme (IGBP) indices [15]. Since the Deciduous Needleleaf Forest (IGBP = 3), Permanent Wetlands (IGBP = 11), Urban and Built-up Lands (IGBP = 13), Snow and Ice (IGBP = 15), and Tundra (IGBP = 18) were found to be observed less than 0.1% during the one year of operations, they were excluded from consideration.

IGBP Index	Surface Type	Sampling Frequency (%)
1	Evergreen needleleaf forest	1.5
2	Evergreen broadleaf forest	1.4
4	Deciduous broadleaf forest	0.7
5	Mixed forest	1.5
6	Closed shrublands	0.2
7	Open shrublands	2.8
8	Woody savannas	1.5
9	Savannas	1.6
10	Grasslands	2.7
12	Croplands	4.0
14	Cropland Mosaics	4.0
16	Bare soil and rocks	2.9
17	Water bodies	75

. We note that although all IGBP indices were considered in our sampling study, in the interests of simplicity only the surface types with sampling frequencies greater than 0.1% were retained, with the combination of all rejected surface types constituting only 0.2% of the total CPF-VIIRS sampling coverage. The POLDER-derived PDMs have demonstrated that, for a given scene type, only DOP, but not AOP, is sensitive to the surface type for the reflected solar radiation. For clear sky over water surfaces (IGBP = 17) DOP displays sensitivity to aerosol optical depth and wind speed, requiring additional stratifications by both parameters. Due to its greater sensitivity, the ePDM stratifications are, therefore, primarily driven by DOP, rather than AOP. Quantitatively, the DOP ePDMs stratification is based on the requirement that the mean of the differences (residuals) in DOP between the adjacent ePDMs be about 0.05, the threshold, which, as we will show in Section 4, is generally below the sensitivity of the CPF-VIIRS intercalibration. The same, ∆P ⪅ 0.05, criterion was also used to group land surface ePDMs to reduce their overall number and improve sampling. In the sections that follow we describe the ePDM classifications in more detail.

3.1.1. DOP ePDMs for Clear-Sky over Water Bodies

The clear sky scene is identified by constraining the value of the cloud fraction at CF < 0.01. As Figure 2 demonstrates this constraint does, indeed, select mean cloud optical thickness (COT) close to 0. The water bodies are selected using the geolocated IGBP map by choosing footprints corresponding to IGBP = 17.

Table 3. Summary of the stratifications for clear sky over water bodies PDMs by values or ranges of values.

Constraint	Values/Ranges
IGBP Index	17
Wavelengths (nm)	490, 670, 865
SZA (°)	see Table 1
Cloud Fraction	[0, 0.01]
Wind Speed (m/s)	[0, 2], [2, 10], [10, 14], [14, 17], [17, 20]
AOD (670, 865 nm)	[0, 0.05], [0.05, 0.1], [0.1, 0.2], [0.2, 0.3],
	[0.3, 0.4], [0.4, 0.5], [0.5, 0.6]

summarizes the constraints applied in constructing the ePDMs for clear sky over water surfaces.

The variations in DOP for the clear sky over water surface scene types are primarily driven by the solar and view geometry, and, to a lesser degree, by the aerosol optical depth [16]. In the forward scattering region, especially at higher values of SZA, surface polarization becomes dominant, culminating in a sharp maximum around 53° surface incident angle, known as the Brewster angle. Under the idealized conditions the reflected light is fully polarized (P = 1) at VZA = SZA = 53°, RAZ = 0°. For the ePDM with the SZA ∈ [50°, 60°] constraint shown in Figure 3, the Brewster angle is indeed observed at (RAZ, VZA) ≈ (0°, 53°), although the maximum P from POLDER data is around 0.8, due in part to depolarization by coarse-mode aerosols. The broad enhancement around the Brewster’s angle is due to wind causing the inclination of the wave facets to act in concert to “smear” the reflected polarized radiation. This phenomenon has been described mathematically by Cox and Munk in 1954 [17]. For SZA values above and below SZA ∈ [50°, 60°] range, the DOP maximum decreases rapidly. Three examples from other SZA ranges, SZA ∈ [10°, 20°], [30°, 40°], and [70°, 80°] are shown in Figure 4. Standard deviations associated with the mean DOPs are also shown in Figure 3 and Figure 4. The standard deviation plots show that, in regions with high DOP, where radiance measurements from a polarization-sensitive detector, such as VIIRS, are expected to exhibit significant biases, the standard deviations are a fraction of mean DOP values. The low standard deviations suggest a high precision in the DOP estimates. The smooth transitions in DOP and the low standard deviations are supported by sufficiently large POLDER sample sizes across most angular bins (Figure 3), with the exception of edge bins (near VZA = 60° or higher), where the sample size is limited. For bins with VZA < 55°, the number of POLDER samples per angular bin ranges from 373 to 145,373, with an average of 26,453, and 94% of the bins contain more than 1000 samples. The sample distribution for SZA < 50° is significantly larger than this, whereas for SZA > 70° the sample size is somewhat smaller, with an average of 11,458 samples per bin. The corresponding sample distributions for Figure 4 are found similar to that of Figure 3. A notable feature of the PDM distribution in Figure 3 is the “valley” of the DOP means around RAZ ∈ [20°, 120°], VZA > 50° region. These lower DOP values are not caused by the lack of POLDER’s sampling, but by sampling of the dust-affected regions of the ocean [5]. We note that this bias may be eliminated using statistical fitting.

All three polarized POLDER channels were used in constructing the clear sky ePDMs; however, due to the absence of aerosol retrievals from the 490 nm POLDER band, only the ePDMs from the 670 and 865 nm bands were stratified by AOD. The bulk of aerosols over ocean, the primary water surface, are coarse-mode aerosols, such as sea foam and sand dust, which are depolarizing. As shown in Figure 5, this depolarizing dependence is approximately linear until AOD ≈ 0.6, after which it plateaus. As in the case of the wind speed, DOP PDMs are stratified by AOD, as indicated by the dashed lines in Figure 5. As noted above, for the case of no restrictions on aerosol optical depth, as in Figure 3, the DOP maximum is found to be about 0.8. For the PDMs where the aerosols’ influence is minimized by constraining AOD to <0.05, the polarization of the light reflected from the water surface rises to about 0.95, close to full polarization. As a function of wind speed, DOP shows a slight rise (Figure 6) between 0 and 5 m/s as the maximum broadens, followed by a drop-off when the water surface becomes rougher. This dependence was used to stratify PDMs, with PDM boundaries indicated by dashed lines in the figure.

Finally, we note that there is little variation in DOP between the three POLDER wave- lengths of 490, 670, and 865 nm for the clear sky over water scenes. This is demonstrated in Figure 7, where we plot mean DOP for each POLDER band without SZA restrictions and the corresponding standard deviations. Within the POLDER spectral range, the DOP varies smoothly with wavelength. Accordingly, DOP values for the VIIRS 555 and 746 nm bands, which are not directly sampled by POLDER, are estimated through linear interpolation between adjacent POLDER spectral channels. This assumption is further supported by comparisons with tPDMs, which exhibit consistent behavior at these intermediate VIIRS wavelengths.

3.1.2. DOP ePDMs for Clear Sky Over Land Surfaces

As noted above, in one year of operations, CLARREO Pathfinder will have statistically significant intercalibration opportunities over 12 land surface types (Table 2). Each clear-sky land surface PDM has fewer restrictions than the comparable clear-sky water surfaces. The selection requirements are summarized in

Table 4. Summary of clear-sky land surface PDM constraints.

Constraint	Values/Ranges
IGBP Index	Land surface indices from Table 2
Wavelengths (nm)	490, 670, 865
SZA (°)	See Table 1
Cloud Fraction	[0, 0.01]

. Given the three POLDER bands and seven SZA ranges, there are 252 land-surface DOP PDMs, together with their associated uncertainties. We would like to find a way, wherever possible, of grouping ePDMs based on their similarities, thereby reducing their number and improving data coverage by filling in gaps due to insufficient data sampling. Additionally, combining these PDMs provides insights into the similarity of various surface types based on their polarization state and their dependence on wavelength. Intuitively, one major benefit of combining PDMs would be the reduction of variability in P. This is, indeed, the case for the upper ranges of VZA and SZA, where the sampling is lacking, however, due to the fact that the overall statistical sample in each ePDM is large, the overall improvement to the PDM accuracy is relatively minor.

Table 5. Mean degree of polarization $\overline{P}$ and mean standard deviations σ_P for the PDMs from the three POLDER bands averaged over the RAZ–VZA–SZA domain for the statistically significant land-surface types encountered by CPF. For completeness, polarization for clear sky over water surfaces is shown in the last row.

IGBP Index	Surface Type	490 $\overline{\mathit{P}}$	nm σP	670 $\overline{\mathit{P}}$	nm σP	865 $\overline{\mathit{P}}$	nm σP
1	Evergreen needleleaf forest	0.18	0.09	0.12	0.08	0.04	0.04
2	Evergreen broadleaf forest	0.25	0.06	0.18	0.08	0.06	0.08
4	Deciduous broadleaf forest	0.18	0.10	0.11	0.07	0.04	0.04
5	Mixed forest	0.17	0.09	0.11	0.07	0.04	0.04
6	Closed shrublands	0.17	0.10	0.10	0.08	0.05	0.05
7	Open shrublands	0.14	0.09	0.07	0.05	0.04	0.04
8	Woody savannas	0.15	0.08	0.08	0.07	0.04	0.04
9	Savannas	0.22	0.07	0.12	0.06	0.05	0.05
10	Grasslands	0.16	0.08	0.08	0.05	0.04	0.03
12	Croplands	0.18	0.08	0.11	0.06	0.05	0.05
14	Cropland mosaics	0.19	0.09	0.11	0.07	0.05	0.05
16	Bare soil and rocks	0.15	0.07	0.07	0.05	0.05	0.05
17	Water bodies	0.28	0.10	0.27	0.05	0.24	0.06

shows DOP means and standard deviations across the three available POLDER bands for all relevant CPF-VIIRS surface types averaged over the RAZ–VZA–SZA domain. The similarity of some of the mean values suggests the feasibility of potential combinations of the land surface types. In the rest of this section, we will discuss in more detail how these combinations were obtained.

As the first step of our land-surface grouping scheme, for each SZA range (Table 1) and for each POLDER band we construct all pairwise PDM combinations for the surface types encountered by CPF (Table 2). For the 12 PDMs in Table 2, there are 66 pairwise combinations per SZA range for each POLDER band, resulting in a total of 1386 pairings across the seven SZA ranges and three POLDER bands. We consider a pair of values of the degree of polarization to be distinct if the difference between their mean ∆P values is greater than 0.05. In Figure 8, we plot the percentages of residuals with ∆P < 0.05 calculated for each PDM over the entire RAZ–VZA domain. Based on the sharp peak above 90% corresponding to correlated PDM pairs, our first requirement, then, is that a pair of PDMs be combined only if the corresponding percentage of the residual bins with ∆P < 0.05 is 90% or more.

We apply this threshold requirement to all 1386 PDM pairings, selecting only those pairs where 90% of the corresponding residuals are within ∆P = ±0.05 envelope. Pairs that contain one PDM in common may be further combined into larger groupings. For each grouping, we aim to find the maximum number of PDM pairings that satisfy the 90% requirement. For example, for a simple grouping composed of PDMs A, B, and C (A ∪ B ∪ C), all the pairings, A ∪ B, A ∪ C, and B ∪ C, must pass the 90% residual threshold. Here, we note that the 90% threshold has been carefully chosen, as the lower threshold would admit the uncorrelated PDMs, while a higher one would result in ambiguous grouping candidates, such as A ∪ B and A ∪ C passing, while A ∪ C failing the threshold constraint, using the previous three-PDM example. In

Table 6. Land-surface PDM groupings by IGBP number. The IGBPs are grouped according to the combination of the results from the residual differences between the P PDMs. IGPBs within a group are enclosed in parentheses, while separate groups are divided by commas in the right column.

SZA Range	PDM Groupings by IGBP Number
	490 nm
[10°, 20°]	(1,2,4,5,6,8,12,14), 9, (7,10,16)
[20°, 30°]	(1,2,4,5,7,12,14), (6,8,9), (7,10,16)
[30°, 40°]	(1,2,4,5,6,8,9,12,14), (7,10,16)
[40°, 50°]	(1,4,12,14), 2, (5,6,7,8,10,16), 9
[50°, 60°]	(1,4,6,10,16), 2, (5,8), 7, 9, (12,14)
[60°, 70°]	(1,5,6,10,16), 2, (4,12,14), (7,8), 9
[70°, 80°]	(1,4,5,12,14), 2, (6,10,16), (7,8), 9
	670 nm
[10°, 20°]	(1,2,4,5), (6,8,12,14), (7,9,10,16)
[20°, 30°]	(1,2,4,5), (6,8,9,12,14), (7,10,16)
[30°, 40°]	(1,4,5,6,8,9,12,14), 2, (7,10,16)
[40°, 50°]	(1,4,9,12,14), 2, (5,6,8), (7,10,16)
[50°, 60°]	(1,4,6,12,14), 2, (5,7,8,10,16), 9
[60°, 70°]	(1,4,5,6,12,14) 2, (7,8,10,16), 9
[70°, 80°]	(1,4,5,12,14,16), 2, (6,7,8,10), 9
	865 nm
[10°, 20°]	(1,2,4,5,6,7,8,9,10,12,14,16)
[20°, 30°]	(1,2,4,5,6,7,8,9,10,12,14,16)
[30°, 40°]	(1,2,4,5,6,7,8,9,10,12,14,16)
[40°, 50°]	(1,2,4,5,6,7,8,9,10,12,14,16)
[50°, 60°]	(1,2,4,5,6,7,8,9,10,12,14,16)
[60°, 70°]	(1,5,7,8,10), 2, (4,6,9,12,14,16)
[70°, 80°]	(1,4,5,6,7,8,10,12,14,16), 2, 9

, we show the final PDM groupings for all three POLDER bands and the seven SZA ranges. Grouping land surfaces, as we have done, results in reduction in the number of land-surface PDMs from 252 to 64. Tracking PDM combinations across the wavelengths and SZA ranges reveals that certain PDM pairs tend to cluster together, such as, perhaps unsurprisingly, croplands (IGBP = 12) with cropland mosaics (IGBP = 14), as well as bare soil and rocks (IGBP = 16) together with grasslands (IGBP = 10). On the other hand, two surface types with relatively high polarizabilities were the evergreen broadleaf forests (IGBP = 2) and savannas (IGBP = 9).

In Figure 9, we show examples of constituent PDMs in the second grouping in the SZA ∈ [30°, 40°] range at 490 nm. The residuals for each PDM pair in the group, shown at the bottom of Figure 9, are clearly within our prescribed ± 0.05 limits, with DOP values for IGBP = 7 and 10 matching most closely. For comparison, Figure 10 shows the residuals between three sample PDMs from group 1 and the PDM corresponding to IGBP = 16 from group 2 (Figure 9) for the same SZA range and wavelength. The fact that all three sets of residuals exceed the ∆P = ±0.05 constraint may be easily visualized in the plots indicating that the two combinations of surface types are indeed distinct. In Figure 11 the ePDMs corresponding to the two groupings in Table 6, IGBP = 1, 2, 5, 6, 8, 9, 12, 14, and IGBP = 7, 10, 16, are shown. Noticeable separation in mean DOP between the two groupings, which exceeds the size of their respective standard deviation, demonstrates the validity of our method.

During CPF-VIIRS intercalibration over a particular land surface type, the mean and standard deviation of the group in Table 6 to which that IGBP input belongs will be chosen. For example, for a desert site (IGBP = 16) at SZA = 35° and the given (RAZ, VZA) angle, the mean and the standard deviation of the PDM corresponding to the IGBP = 7, 10, 16 group (cf. Figure 11) will be taken. With this in mind, we plot the means and standard deviations for each land surface type, taking the mean and σ from the PDM combination to which the surface type belongs. In Figure 12 we show RAZ–VZA-averaged degree of polarization as it varies across the SZA ranges for the three POLDER bands. It should be noted that the evergreen broadleaf forests (IGBP = 2) and, to a lesser extent, savannas (IGBP = 9), both exhibit a high degree of polarization at 490 nm, with the mean P corresponding to IGBP = 2 even exceeding the polarization of the water surface at SZA > 60°. It is important to note, however, that the averages shown in these graphs mask the nature of the PDM distributions. The DOP maxima for water-surface ePDMs are higher than that of the equivalent ePDMs for the broadleaf forest; however, the DOP peak in the latter is broader than that of the water-surface ePDMs resulting in a comparable or, at some SZA ranges even higher, overall mean. At 490 nm, this broad maximum raises the average P for IGBP = 2 to levels higher than those of water for SZA > 50°. Both average polarizations for IGBP = 2 and IGBP = 9, as well as the rest of the land surface, decrease rapidly at higher wavelengths, whereas the mean P for water surfaces is close to constant throughout the POLDER range. In Figure 13, in addition to RAZ and VZA, averaging was also performed over SZA, with the degree of polarization plotted for each IGBP. By design of the grouping algorithm, we do not expect to see significant differences between the means in Figure 13 and the ones shown the Table 5. A significant drop in DOP with the increasing wavelength for all surface types can be observed in the figure, and at 865 nm the polarization is close to 0. This near extinction of DOP explains the results in Table 5, where, except for two cases, all the surface types in each SZA range are merged into single groups.

3.1.3. DOP ePDMs for Overcast Water Bodies and Land

In addition to the global selection criteria defined in Section 3, cloud parameters, such as the cloud thermodynamic phase, cloud fraction (CF), and cloud optical thickness (COT), were used to construct the ePDMs corresponding to the overcast scenes. These parameters are retrieved from POLDER Level-2 [18] datasets. The selection criteria for the PDMs for overcast scenes are listed in

Table 7. Summary of constraints on the PDMs corresponding to overcast scene types.

Constraint	Values/Ranges
IGBP Index	IGBP = 17 and combinations from Table 6
POLDER Bands (nm)	490, 670, 865
SZA	See Table 1
Cloud Phase	1 (water), 2 (ice), 3 (mixed)
Cloud Fraction (CF)	[0.9, 1.0]
COT	Variable COT binning depending on the cloud phase, band and SZA range

.

Due to its relatively low spatial resolution, the cloud fraction determined by POLDER is biased toward higher values. This is seen most clearly in the partly cloudy scenes: specifically, small broken clouds with a higher COT tend to trigger a cloudy pixel flag more often with POLDER than with a higher resolution instrument, such as MODIS or VIIRS [18]. This bias manifests itself in the slope of the COT vs. CF for values of CF ≲ 0.99 plotted in Figure 14. For values above 0.99, essentially corresponding to the overcast scene, the dependence of the CF on the COT and cloud size disappears. A looser selection of CF > 0.9, which allowed for a bigger sample size, without significantly impacting the ePDM distributions, was chosen as a cloud fraction cutoff for the overcast scene types.

We note, for simplicity of the ePDM development, the partly cloudy scenes were omitted. To stratify the ePDMs corresponding to overcast scenes by cloud phase, Level-2 flags provided by POLDER were used [19,20,21,22].

The PDMs for overcast scene types were stratified by cloud optical thickness using $\overline{∆P}$ ≲ 0.05 as a criterion to separate the individual PDMs. We note that, due to the approximate independence of COT of the wavelength (see, e.g., [23]), the COT values originally retrieved by POLDER at the 670 nm band over land surfaces were applied in constructing the land-surface ePDMs for the 490 nm and 865 nm bands. Similarly, the COT values obtained by POLDER from the 865 nm band over water surfaces were extended to 490 and 670 nm for water-surface ePDMs. DOP PDMs were stratified in batch as a function of COT using variable COT binning, depending on the cloud phase, POLDER band and the given SZA range. In Figure 15 we compare such stratification for water, ice, and clouds over water surfaces for the POLDER 670 nm band and SZA ∈ [30°, 40°].

Generally, as the cloud optical thickness increases, degree of polarization decreases due to multiple scattering through the volume of the cloud. This is evidenced in Figure 15 for all three cloud phases. However, the drop-off for water and mixed clouds is more gradual than for the ice clouds. This is due to two reasons: first, at similar COT ranges, water and mixed clouds transmit more polarized radiation from the surface than ice clouds. This is manifested as a larger maximum around water surface Brewster angle (Figure 16). Secondly, in PDMs corresponding to thinner water clouds, cloudbows appear in the form of a crest in the RAZ range between 80° and 140°. At higher optical depths, the height of this crest gradually decreases, although even for thicker water clouds and, to a lesser extent, mixed phase ones, it may still be observed (Figure 17). We note that the two sets of PDMs shown in Figure 16 and Figure 17 are taken at the lowest and highest COT ranges illustrated in Figure 15.

3.2. AOP PDMs

While in Section 4 we focused on the degree of polarization, in the following two sections we discuss the AOP PDMs corresponding to the clear-sky and overcast scenes.

3.2.1. Clear-Sky AOP PDMs

As noted in Section 2 and Ref. [5], angle of polarization χ depends to a much greater extent on scattering in the atmosphere than on surface reflection. This was demonstrated by the agreement between the atmospheric single scattering AOP function and empirical AOP throughout most of the solar-view geometry domain. Given the same SZA range, most of the differences between AOP PDMs in Figure 18, Figure 19 and Figure 20 occur in the backscatter region (RAZ > 140°). In this region the scattering angle Θ approaches 180° and single scattering becomes undefined (Equation (5)), while multiple scattering effects take over. The latter manifest themselves in terms of large standard deviations (σ_χ > 20°) seen in these figures. Whether aerosols in the atmosphere are present or absent has little influence on AOP, as demonstrated in Figure 18. While AOP is dependent on the given SZA, it is independent of the surface type. This is illustrated in Figure 19, where we show examples of AOP PDMs for six different surface types and two SZA ranges. Finally, AOP is also independent of the wavelength as shown in Figure 20.

Despite the reduced dependence on surface types, the clear-sky water body AOP PDMs are stratified in the same manner as the corresponding DOP PDMs for implementation convenience and consistency (Table 3). There were sufficient clear-sky water body samples available to support this categorization. The clear sky land AOP PDMs have been stratified separately from their water surface counterparts; however, to improve sampling statistics, the land surface IGBP index was not considered (see

Table 8. Summary of the stratifications for clear-sky land AOP PDMs.

Constraint	Values/Ranges
Surface type	IGBP ≠ 17
Wavelengths (nm)	490
SZA (°)	See Table 1
Cloud Fraction (CF)	[0.9, 1.0]
Cloud Fraction (CF)	[0, 0.01]

).

3.2.2. Overcast AOP PDMs

The same selection criteria used for DOP PDMs for overcast scene types (Table 7) were applied to classify the corresponding AOP PDMs. Like the clear-sky AOP PDMs, most of the SZA–RAZ–VZA space is dominated by single scattering, save for the RAZ > 140° region. In Figure 21, we show the comparison between water-cloud AOP PDMs for the 490, 670, and 865 nm POLDER bands. While the differences between these distributions are small, primarily manifesting themselves in terms of standard deviations, for consistency with the clear-sky AOP PDMs, the AOP PDMs were classified by POLDER bands. The three different thermodynamic cloud phases, water, ice, and mixed, result in three different “fingerprints” at the backscatter region, as illustrated in Figure 22. The AOP backscatter pattern shifts with increased cloud thickness for all three cloud phases, as may be seen by comparing Figure 21 with Figure 23, both plotted for the same SZA range. In analogy with the AOP PDMs for the clear-sky scenes, despite their independence of the surface type, overcast AOP PDMs were processed separately and classified into land and water surface types. The selection criteria are summarized in

Table 9. Summary of the stratifications for overcast land and water surface AOP PDMs.

Constraint	Values/Ranges
Surface type	IGBP = 17, IGBP ≠ 17
Wavelengths (nm)	490, 670, 865
SZA ranges	See Table 1
Cloud Phase	[0.9, 1.0]
Cloud Fraction (CF)	[0.9, 1.00]
COT	Same COT binning as overcast DOP PDMs (Table 7)

.

4. Sensitivity of Intercalibration to Variations in DOP and AOP

Finally, in this section we investigate the sensitivity of the intercalibration results to variations in the two polarization parameters, DOP and AOP, especially the polarization-induced bias on radiance or reflectance measurements of a polarization-sensitive instrument, such as VIIRS. This will help us understand the sensitivity of the CPF-VIIRS intercalibration to variations in DOP and AOP over the intercalibration footprints. The collection of ePDMs developed by the CPF team enables us to easily identify and select several “worst-case” scenarios with high DOP. Here, we consider three such scenarios: clear-sky and overcast water surfaces, and clear-sky over broadleaf forest scene types. We will rely on the following expression for the relative error (${\delta }_{\rho }$) in reflectance (or radiance) as a function of DOP and AOP derived from Equation (6):

$${\delta }_{\rho }={\displaystyle \frac{{\rho }_0-\rho }{\rho }}=aP\cos 2\left(\chi +\varphi \right)$$

(7)

For this study, we choose a phase angle ϕ = 0 to yield the highest value of δ_ρ and take the NOAA-20 VIIRS channel that falls within the ePDM wavelength range and has the largest diattenuation coefficient. This corresponds to the M3 (488 nm) channel for which a = 0.03 as reported by the VIIRS ground characterization team [3]. The CPF’s own diattenuation coefficient may be neglected here for simplicity, as it is expected to be several times smaller in magnitude than the VIIRS’ value. Taking P and χ values from the clear-sky water surface ePDM at SZA ∈ [50°, 60°] for Equation (7), we obtain the reflectance relative error dependence on DOP and AOP, as shown in Figure 24. (For each value of DOP from the ePDM, the value of the AOP from the corresponding AOP PDM was taken, leading to the final value of ${\delta }_{\rho }$. Conversely, for each value of AOP, the corresponding DOP from DOP ePDM is taken to compute ${\delta }_{\rho }.$) As it is clear from the figure, if polarization state is not considered, the reflectance is likely to be either over- or underestimated, depending on the value of DOP. The highest difference between the “true” and uncorrected reflectance occurs at the highest polarization for the clear sky over water surface at P ≈ 0.8 and is approximately 2.4%. As another example of a highly polarized scene type, we consider water clouds over water surface with SZA ∈ [30°, 40°], COT ∈ [1, 1.8] (Figure 25). The highest statistically robust DOP that may be obtained from POLDER data is around 0.4, which yields δ_ρ = 1.5%. The final scene type is the clear-sky evergreen broadleaf forest (IGBP = 2) at SZA ∈ [60°, 70°] (Figure 26). As can be seen in Figure 12, at 490 nm at larger values of SZA the mean DOP for this ePDM is even higher than that of the ocean, although, due to the difference in the corresponding AOP domains, the observed DOP maximum appears to be lower, at about P ≈ 0.7. The maximum and minimum DOP deviations are about 0.5% and −0.5%, respectively, with the minimum deviation being smaller than that observed for the clear sky over water. We also note the cyclical nature of the δ_ρ vs. χ plots due to the cosine term in Equation (7) and the difference in the gradients in the region of the crest-and-trough feature for 90° < RAZ < 180° (see, e.g., Figure 23) giving rise to step-like δ_ρ vs. χ transitions of around χ = 90°. To mitigate polarization effects, the CPF science team plans to use PDMs to identify intercalibration samples with low polarization and use these to intercalibrate the polarization-sensitive VIIRS bands against CPF benchmark measurements. The team intends to limit the DOP of the selected sample radiances to below 0.1 to meet the stringent intercalibration uncertainty requirements. Roithmayr et al. [24] showed that the total number of intercalibration opportunities between CPF and NOAA-20 VIIRS is approximately 800 per year. Additional simulation analyses conducted by the CPF science team indicate that this would yield roughly 70,000 intercalibration footprints of 15 × 15 km between VIIRS and CPF, out of which about 10,000 samples would pass the data-filtering criteria, including scene homogeneity and degree of polarization below 0.1. This sample size is sufficient to meet the uncertainty requirements for spatial and temporal matching, as well as to satisfy the 0.3% target intercalibration uncertainty budget.

5. Conclusions

In this article, we have described the empirical Polarization Distribution Models (ePDMs) developed by the CLARREO Pathfinder (CPF) team based on the multidirectional polarized reflectance measurements from the visible and near-infrared channels of the POLDER instrument. The CPF team will leverage these ePDMs to identify low-polarized intercalibration footprints required to reduce the uncertainty in the intercalibration of polarization-sensitive VIIRS RSB against the CPF benchmark measurements. Knowing the polarization state of an intercalibration footprint, the ePDMs may also be used to account for the polarization effects in the radiometric measured VIIRS reflectance. Two ePDM types corresponding to two polarization parameters, the degree and angle of polarization, DOP and AOP, were discussed. Using primarily DOP for classification, we described how ePDMs are stratified by atmospheric scene and surface types, which will be encountered during CPF-VIIRS intercalibration. We also showed several examples of radiometric biases due to polarization effect and discussed the sensitivity of the intercalibration results to variations in DOP and AOP.