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  • View in gallery

    Illustrations showing the coordinate variables used in the text. Terms and θ are the solar and viewing zenith angles, respectively. Terms and are the solar and viewing azimuth angles, respectively. Relative azimuth, defined as , is shown in (bottom) Poincaré sphere. Also shown in the bottom drawing are the two polarization angles: χ, the angle relative to the principal plane, and ψ, the angle relative to the scattering plane.

  • View in gallery

    One-dimensional P distribution comparison for a typical PDM region , for the case where (left) all aerosols are allowed vs (right) AOD < 0.1.

  • View in gallery

    Mean degree of polarization and for the cloudless-sky-over-ocean scene for the 670-nm band (top left) with and (top right) without the AOD < 0.1 constraint (see Table 1 for the full list of constraints). The PDM differences (bottom left) between 670 and 490 nm and (bottom right) between 865 and 490 nm for the cloudless-sky-over-ocean scene without the AOD restriction, with the rest of the constraints as shown in Table 2. In all the plots, is restricted to values between 40° and 50°.

  • View in gallery

    (left) Systematic uncertainty vs P for the 670-nm band without the AOD constraint. The graph was fitted with the polynomial . (right) Uncertainty in the intercalibrated reflectance as a function of polarization [Eq. (5)] for the 670-nm band derived from the dependence shown in the left plot. The imager sensitivity to polarization was set to 0.03 (roughly MODIS and VIIRS sensitivity), and its relative uncertainty was set to 10% (third curve from the top, black), 20% (second curve from the top, green), and 100% (top curve, blue). Also shown is the uncertainty in reflectance if the polarization is assumed to be zero (bottom line, red).

  • View in gallery

    (top) Residuals after subtracting [Eq. (5)] for the 865-nm band from those for the 490-nm band. The residuals for (left) and (right) are shown. No AOD constraints were imposed. The subtraction was performed using the PDM data values and standard deviations, such as the ones seen in Fig. 4, right. (bottom) Residuals after subtracting [Eq. (5)] without the AOD constraint from those with AOD < 0.1 for the (left) 670- and (right) 865-nm bands. Term was set to 10% in both cases.

  • View in gallery

    Terms and for the cloudless-sky over ocean scene for the 670-nm band (left) without and (right) with the AOD < 0.1 constraint (see Tables 1 and 2 for the full list of constraints). (bottom left) The χ(670 nm) − χ(490 nm) residuals; (bottom right) the χ(865 nm) − χ(490 nm) residuals. No AOD constraints were imposed for this comparison.

  • View in gallery

    (left) Theoretical χ PDM obtained using single-scattering approximation [see Eq. (6)]. (right) Empirical PDM for the 670-nm channel without any AOD constraint (see Fig. 6)–single-scattering PDM residuals.

  • View in gallery

    Terms (top) and (bottom) obtained by directly applying the χ definition in Eq. (3). (bottom left) An example of a χ distribution for the sample region , showing the wraparound effects when χ is close to its range limits of [0°, 180°]. (bottom right) Corrected χ distribution with χ translation applied to the distribution on the left.

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Empirical Polarization Distribution Models for CLARREO-Imager Intercalibration

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  • 1 Science Systems and Applications, Inc., and NASA Langley Research Center, Hampton, Virginia
  • | 2 NASA Langley Research Center, Hampton, Virginia
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Abstract

Polarization effects bias the performance of various existing passive spaceborne instruments, such as MODIS and the Visible Infrared Imaging Radiometer Suite (VIIRS), as well as geostationary imagers. It is essential to evaluate and correct for these effects in order to achieve the required accuracy of the total reflectance at the top of the atmosphere.

In addition to performing highly accurate decadal climate change observations, one of the objectives of the Climate Absolute Radiance and Refractivity Observatory (CLARREO) mission recommended by the National Research Council for launch by NASA is to provide the on-orbit intercalibration with the imagers over a range of parameters, including polarization. Whenever the on-orbit coincident measurements are not possible, CLARREO will provide the polarization distributions constructed using the adding–doubling radiative transfer model (ADRTM), which will cover the entire reflected solar spectrum. These ADRTM results need to be validated using real data. To this end the empirical polarization distribution models (PDMs) based on the measurements taken by the Polarization and Anisotropy of Reflectances for Atmospheric Sciences Coupled with Observations from a Lidar (PARASOL) mission were developed. Examples of such PDMs for the degree of polarization and the angle of linear polarization for the cloudless ocean scenes are shown here. These PDMs are compared across the three available PARASOL polarization bands, and the effect of aerosols on them is examined. The PDM-derived dependence of the reflectance uncertainty on the degree of polarization for imagers, such as MODIS or VIIRS, after their intercalibration with the CLARREO instrument is evaluated. The influence of the aerosols on the reflectance uncertainty is examined. Finally, the PDMs for the angle of linear polarization is cross-checked against the single-scattering approximation.

Denotes Open Access content.

Corresponding author address: D. Goldin, NASA Langley Research Center, MS 420, Hampton, VA 23681. E-mail: daniel.goldin@nasa.gov

Abstract

Polarization effects bias the performance of various existing passive spaceborne instruments, such as MODIS and the Visible Infrared Imaging Radiometer Suite (VIIRS), as well as geostationary imagers. It is essential to evaluate and correct for these effects in order to achieve the required accuracy of the total reflectance at the top of the atmosphere.

In addition to performing highly accurate decadal climate change observations, one of the objectives of the Climate Absolute Radiance and Refractivity Observatory (CLARREO) mission recommended by the National Research Council for launch by NASA is to provide the on-orbit intercalibration with the imagers over a range of parameters, including polarization. Whenever the on-orbit coincident measurements are not possible, CLARREO will provide the polarization distributions constructed using the adding–doubling radiative transfer model (ADRTM), which will cover the entire reflected solar spectrum. These ADRTM results need to be validated using real data. To this end the empirical polarization distribution models (PDMs) based on the measurements taken by the Polarization and Anisotropy of Reflectances for Atmospheric Sciences Coupled with Observations from a Lidar (PARASOL) mission were developed. Examples of such PDMs for the degree of polarization and the angle of linear polarization for the cloudless ocean scenes are shown here. These PDMs are compared across the three available PARASOL polarization bands, and the effect of aerosols on them is examined. The PDM-derived dependence of the reflectance uncertainty on the degree of polarization for imagers, such as MODIS or VIIRS, after their intercalibration with the CLARREO instrument is evaluated. The influence of the aerosols on the reflectance uncertainty is examined. Finally, the PDMs for the angle of linear polarization is cross-checked against the single-scattering approximation.

Denotes Open Access content.

Corresponding author address: D. Goldin, NASA Langley Research Center, MS 420, Hampton, VA 23681. E-mail: daniel.goldin@nasa.gov

1. Introduction

The Climate Absolute Radiance and Refractivity Observatory (CLARREO) (Wielicki et al. 2013) is a NASA decadal survey mission recommended by the National Research Council. CLARREO’s objectives are to conduct highly accurate climate change observations and to serve as an on-orbit intercalibration reference for other operational instruments by measuring spectral reflectance and to monitor their response function parameters, including gain, offset, nonlinearity, spectral response of the optics, and sensitivity to polarization. In this article we focus on polarization.

The polarization and its effect on radiance measurement have been considered in the past by the CLARREO team. Lukashin et al. (2013) used 12 days of data collected by Polarization and Anisotropy of Reflectances for Atmospheric Sciences Coupled with Observations from a Lidar (PARASOL) (Fougnie et al. 2007) to construct a global set of the empirical polarization distribution models (PDMs). Using a few representative values of the sensitivity to polarization, its corresponding uncertainty, and the uncertainty on the degree of polarization, the authors estimated the uncertainty in the reflectance measurements by an imager after its intercalibration with CLARREO.

In this work we extend the PDM statistics from 12 days of 2006 level-1 and level-2 PARASOL data to the entire 2006 dataset using the three available polarization bands, at 490, 670, and 865 nm. We show the PDM means and standard deviations for the degree of polarization P and angle of polarization χ. We also take a closer look at the effects the aerosols have on the PDMs and quantify the dependence of the uncertainties in P on the values of P at all aerosol optical depths (AODs) and with the AOD restriction of AOD < 0.1. We then use these dependencies to improve on the reflectance uncertainty estimates in Lukashin et al. (2013) using the polarization sensitivity of Moderate Resolution Imaging Spectroradiometer (MODIS) (Xiong et al. 2013) and Visible Infrared Imaging Radiometer Suite (VIIRS) (Scalione et al. 2003). Finally, we compare the χ empirical PDMs with the theoretical ones constructed using the single-scattering approximation.

At 490, 670, and 865 nm, the three polarized bands from the PARASOL sample the blue, yellow, and near-infrared part of the reflected solar spectrum. The PDMs derived from these channels will be used to validate the polarization modeling over the entire reflected solar spectrum using the adding–doubling radiative transfer model (ADRTM) (Sun and Lukashin 2013). We note that such PDMs would provide the complete set of polarization parameters to the low-Earth-orbiting (LEO) imagers, such as MODIS and VIIRS, and geosynchronous (GEO) instruments, such as GOES, which can subsequently be used to correct for the polarization biases (see, e.g., Lyapustin et al. 2014; Kulkarny et al. 2010).

In this work we considered the cloudless-sky over ocean scene. This choice was motivated by several factors. First, as is well known, the ocean alone, the biggest water body on the planet, covers approximately two-thirds of the earth’s surface. Second, we wish to consider the worst-case scenario, where polarization affects the radiance measurements the most. The unpolarized sunlight becomes polarized through atmospheric scattering and surface reflection, with the water surface being the best polarizer among the surface types identified by the International Geosphere–Biosphere Programme (IGBP) (Bartholomé and Belward 2005).

This article is structured in the following way. In section 2 we define the viewing geometry and polarization parameters. In section 3 we briefly describe the PARASOL instrument and discuss the PARASOL data format, which is used in forming our datasets (discussed in section 4), which, in turn, are used to construct the PDMs. In section 5, first, for the three available PARASOL polarization bands, we examine the empirical P PDMs for the cloudless-sky over ocean scene and the effect of the aerosols on them. The P PDMs can then be used to derive the uncertainty dependence of the imagers, such as MODIS and VIIRS, on the degree of polarization after their intercalibration with CLARREO; this is illustrated in section 6. In section 7 we show examples of χ PDMs with and without aerosols and, as a cross-check, compare the χ PDM with the single-scattering approximation. Finally, in the appendix we discuss the correction that needs to be applied to χ PDMs in order to obtain well-defined means and standard deviations across the entire PDM.

2. Viewing geometry, polarization, and PDM definitions

In this section we define the geometrical and polarization variables used throughout this article. By convention, we use polar coordinates with the z axis coinciding with the local zenith (Fig. 1). The position of the PARASOL instrument is described in terms of the viewing zenith angle and viewing azimuth as . The current position of the sun in terms of the solar zenith angle and solar azimuth is denoted as , while the relative azimuth is defined as

Fig. 1.
Fig. 1.

Illustrations showing the coordinate variables used in the text. Terms and θ are the solar and viewing zenith angles, respectively. Terms and are the solar and viewing azimuth angles, respectively. Relative azimuth, defined as , is shown in (bottom) Poincaré sphere. Also shown in the bottom drawing are the two polarization angles: χ, the angle relative to the principal plane, and ψ, the angle relative to the scattering plane.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-15-0165.1

The polarization of light (Kattawar et al. 2016) may be described by the following four Stokes parameters: I, Q, U, and V, where I is the total intensity, Q and U contain information about the linear polarization, and V describes circular polarization. The reflected solar radiation polarization is linear to a high degree and V can be neglected (Coulson 1988). The remaining Stokes components may be expressed as (Hansen and Travis 1974)
eq1
where the quantities on the right-hand side are the radiation intensities along the measurement directions indicated by the indices.
Linear polarization is fully specified by the degree of polarization, the polarization angle, and the total intensity. The degree of polarization P may be expressed in terms of the ratio of polarized and total intensities:
e1
The angle of linear polarization relative to the scattering plane is commonly denoted by ψ (Fig. 1). Since the light scattered in the atmosphere is polarized perpendicularly to the scattering plane (s polarized), ψ values are found mainly around 90°. The angle of linear polarization may also be defined relative to the meridian plane, which is the plane of the detector, in our case, and it can be expressed in terms of the Stokes parameters as (Hansen and Travis 1974)
e2
Since the common definition of arctan is over the [−90°, 90°] range, following the convention that the χ range be between [0°, 180°] Eq. (2) is transformed as (Sun and Lukashin 2013)
e3
Following the scheme proposed in Lukashin et al. (2013), PDMs are defined as P or χ distributions across the viewing zenith angle θ and relative azimuth ϕ, with constraints on the scene type, such as surface type and wind speed, as well as the solar zenith angle. In this article the PDMs are represented by the two-dimensional histograms with ϕ on the x axis, θ on the y axis, and the degree of polarization P or angle of polarization χ indicated by the color bar. Each P or χ bin is averaged over the data-taking period. For this work we have considered the entire year of 2006 PARASOL data1 with the global coverage, save for the polar regions, which were not sampled.

3. PARASOL description and data format

The PARASOL mission was active between 2004 and 20132 and consisted of the microsatellite flying as a part of A-Train formation at 705-km altitude. The mission’s primary aim was to study aerosols and clouds. The measurements were carried out by the wide-field imaging radiometer/polarimeter called POLDER. The instrument’s spatial resolution was 5.3 km × 6.2 km at nadir. POLDER’s charge-coupled device (CCD) detector took measurements from nine spectral channels from blue (443 nm) to infrared (1020 nm), three of which—490, 670, and 865 nm—were polarized. This work relies on the three polarized bands.

The data collected by PARASOL are separated into level-1B (Bréon 2006) and level-2 (Bréon 2011) data products. Level-1 data contain the date and time of the first and last image acquisition, geolocation data, data quality flags, angular information (, θ, ), normalized radiances, and the Q and U components of the Stokes vector.

The level-2 data product is broken up into three processing lines: radiation budget (RB), ocean color (OC), and land surfaces (LS). The RB line includes quantities such as surface type indicator and cloud parameters (fraction, pressure, optical depth, spherical albedo, phase, and water vapor column). The OC line contains measurements and derived quantities over the water bodies, such as the integrated aerosol optical depth for the 670- and 865-nm channels, as well as optical depths for different aerosol types, such as coarse, fine, and spherical, and nonspherical. The LS line contains land surface–related quantities, such as the aerosol optical depth, the angstrom coefficient, the index of refraction, and the aerosol index.

An important feature of PARASOL’s measurements is the multiangular sampling, which allowed for the same footprint to be imaged up to 15 times at different viewing angles, providing fuller reflectance information about each footprint. To account for the motion of the satellite in imaging a single footprint, a correction to θ and ϕ is applied for the 490- and 865-nm bands (Bréon 2006).

4. The description of the datasets used in constructing the PDMs

We will use three datasets corresponding to the cloudless-sky over ocean scenes: one, approximating the idealized conditions of no wind, clouds, or aerosols; and the other two, with the aerosols3 included. For the scene approximating the idealized conditions (sections 5a and 7b), we require AOD < 0.1 and . The first aerosol sample has no AOD constraints4 (sections 5b and 7a) and the requirement, while the second aerosol sample has the AOD > 0.1 restriction and the requirement (section 6). It is well known that different aerosol types have different polarizing behaviors (see, e.g., Herman et al. 2005); however, because of the limitations in statistical sample size, the PDMs shown here are global aerosol averages. The size of uncertainties on the PDMs, however, implicitly takes into account those differences.

In all three datasets, the wind speed is required to be less than 3.5 m s−1, which maximizes the polarization while keeping the statistical sample reasonably large. To ensure that cloudless sky is selected, the cloud fraction is set to less than 0.01. The water body selection is ensured by setting the IGBP surface type index (Bartholomé and Belward 2005) to 17. The three sets of selection requirements are summarized in Tables 13, respectively.

Table 1.

Constraints applied to select the cloudless-sky over ocean subset of data approximating the pristine conditions.

Table 1.
Table 2.

Constraints applied to select the cloudless-sky over ocean subset, including the aerosols.

Table 2.
Table 3.

Constraints applied to select the cloudless-sky over ocean subset to select maximum polarization with aerosol contribution.

Table 3.

We note that the chosen data samples are large enough to be insensitive to statistical fluctuations while, as magnitudes of the standard deviations (in panels in Figs. 3 and 6) indicate, reflecting the natural variability in P and χ, and having sufficient sensitivity to the aerosol content in the all-AOD case. We also remark that the dataset restrictions chosen for this work are just a few representative values from the full suite of cloudless-sky over ocean PDMs, which will be binned within each solar zenith angle, AOD, and wind speed ranges.

5. Polarization distribution models for the degree of polarization P

a. P PDMs for the cloudless-sky over ocean scene for AOD < 0.1

In the idealized case of the cloudless-sky over ocean scene without aerosols and winds present, the temporal or spatial variations either in the degree or the angle of linear polarization are expected to be nonexistent; therefore, averaging over the entire year and the entire globe for such a dataset is beneficial, since it results in better statistical accuracy. Since this work is concerned with real data obtained by an instrument, we must choose an AOD cutoff value that approximates such conditions, without sacrificing the statistics. Such a cutoff condition is chosen to be AOD < 0.1. That this is a reasonable approximation is illustrated for the case of a typical PDM region in Fig. 2, where the AOD < 0.1 constraint results in the disappearance of the depolarizing secondary peak present at low values of P when the aerosols are included and in the symmetrization of the PDMs around the 180°, as required by the scattering matrix invariance (Hovenier 1969) (see section 5b for more details).

Fig. 2.
Fig. 2.

One-dimensional P distribution comparison for a typical PDM region , for the case where (left) all aerosols are allowed vs (right) AOD < 0.1.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-15-0165.1

Using the PDM definitions in section 2, the P PDMs and the corresponding relative error , defined as the ratio of the standard deviation of the P distribution in a given bin over its mean, may be constructed. An example of the P PDMs for the 670-nm channel is shown in the top-left panel of Fig. 3. For this plot the aerosol optical depth was required to be below 0.1 and , with the rest of the constraints as shown in Table 1. The distribution is roughly symmetric about the axis. As mentioned above, this is to be expected due to the invariance properties of the scattering matrix in the ϕ direction. The maximum is around 0.9 and occurs at approximately , which is the Brewster’s angle for the air–water surface interface, where the reflected light is maximally polarized. The area around the maximum is the glint region. In the nadir region (), the degree of polarization varies between 0.2 and 0.3. The relative uncertainty on the degree of polarization shown in the middle plot is seen to vary between 0.2 and 0.4. The size of uncertainties depends crucially on the aerosol content: our findings indicate that applying a more stringent constraint on the aerosol optical depth results in significant reduction of .5 The highest relative uncertainty in Fig. 3 is observed in the region around and and, symmetrically, around with , where P is close to 0. This is the backscatter region (scattering angle of 20° or less), where scattered light intensity is weak and multiple scattering dominates.

Fig. 3.
Fig. 3.

Mean degree of polarization and for the cloudless-sky-over-ocean scene for the 670-nm band (top left) with and (top right) without the AOD < 0.1 constraint (see Table 1 for the full list of constraints). The PDM differences (bottom left) between 670 and 490 nm and (bottom right) between 865 and 490 nm for the cloudless-sky-over-ocean scene without the AOD restriction, with the rest of the constraints as shown in Table 2. In all the plots, is restricted to values between 40° and 50°.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-15-0165.1

b. P PDMs with the aerosol contribution for the cloudless-sky over ocean scene for all AODs

Removing the AOD < 0.1 aerosol requirement while keeping the rest of the constraints the same as in section 5a results in the P PDMs as shown in the top-right panel in Fig. 3. The comparison of the PDMs with and without the AOD constraint shows that the inclusion of the aerosols results in lower values of polarization, while the standard deviations increase. For the right panel, the asymmetry around for is observed. This asymmetry is an artifact produced by depolarization due to the aerosols and the limited statistics in this region. For , , the Northern Hemisphere, where aerosol concentration is higher, was primarily sampled, whereas the symmetrical region of , corresponds to the Southern Hemisphere with lower aerosol concentration.6

We get a deeper insight into the aerosol contribution by taking a typical PDM region and comparing one-dimensional P distributions with and without the requirement. In Fig. 2 we make such a comparison for the 670-nm band for , . A prominent feature of the aerosol contribution is a peak around . This type of enhancement is uniformly distributed in latitude and longitude, and is likely to be due to ocean foam. It contributes to lower means and higher relative and root-mean-square (RMS) errors, as the comparison of the top two left-hand-side plots with those on the right-hand side in Fig. 3 indicates.

The PDMs for the 490- and 865-nm bands are analogous to the 670-nm band. To better highlight the differences, we can compare the residuals for the 670 and 865 nm, relative to the 490-nm band. The (670 nm) − (490 nm) and (865 nm) − (490 nm) are shown in the bottom of Fig. 3. We observe that, while the polarization decreases slightly with wavelength as expected in the Rayleigh regime, most of the values lie inside the range, demonstrating that the polarization varies slowly across the reflected solar spectrum. Greater differences are observed in the two aerosol regions above , as discussed above, where the 490-nm band is found to be the most sensitive to the presence of aerosols.

6. Estimating the reflectance uncertainty due to polarization after intercalibration with CLARREO

The reflectance of the imager that is being calibrated depends on the polarization state in the following way (Lukashin et al. 2013) (Sun and Xiong 2007):
e4
where is the imager reflectance before the polarization intercalibration is applied and m is the imager’s sensitivity to polarization. The latter depends crucially, among other parameters, on the polarization angle χ. For the MODIS instrument, the ground-based tests showed a periodic dependency of m on χ (Sun and Xiong 2007). In the production version of the reflectance calibration dataset, the m values will be quantified using CLARREO reflectance measurements as well as P and χ PDMs for each individual viewing geometry configuration. Since CLARREO is not yet operational, in the analysis in this section we take an averaged value of m, independent of χ, and the magnitude of oscillations around this mean as the uncertainty .
Since the variables are independent of each other and the covariance terms are zero, straightforward propagation of errors gives the following relative uncertainty on reflectance:
e5
where , , and are the relative uncertainties () on , m, and P, respectively. Term is composed of three components (Lukashin et al. 2013): CLARREO’s own instrument accuracy (0.15%),7 intercalibration sampling uncertainty after averaging (0.1%), and the target sensor stability uncertainty (0.1%). The combined value of the three uncertainties is 0.2%. Next, we choose the value of m to be 0.03, which is roughly the sensitivity to polarization for both MODIS (Sun and Xiong 2007) and VIIRS (Kulkarny et al. 2010).

To find the maximum uncertainty in the reflectance , we need to construct a dataset corresponding to the maximum polarization uncertainties that simultaneously have the highest degree of polarization [see Eq. (5)]. Since the maximum polarization occurs at Brewster’s angle (53° for the water surface), we constrain the solar zenith angle to be between 50° and 60°, while given our dataset binning, the uncertainties in polarization are the greatest for the AOD > 0.1 range (the rest of the restrictions are as shown in Table 3). Using the PDMs one can derive the dependence of the relative uncertainty on polarization on . In Fig. 4, left, we plot such dependence for the 670-nm band. We note that the relative uncertainties in this dataset are larger than for the dataset used in the previous section and are on the order of 50% and that decreases with P.

Fig. 4.
Fig. 4.

(left) Systematic uncertainty vs P for the 670-nm band without the AOD constraint. The graph was fitted with the polynomial . (right) Uncertainty in the intercalibrated reflectance as a function of polarization [Eq. (5)] for the 670-nm band derived from the dependence shown in the left plot. The imager sensitivity to polarization was set to 0.03 (roughly MODIS and VIIRS sensitivity), and its relative uncertainty was set to 10% (third curve from the top, black), 20% (second curve from the top, green), and 100% (top curve, blue). Also shown is the uncertainty in reflectance if the polarization is assumed to be zero (bottom line, red).

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-15-0165.1

From the fits to the versus P plots using Eq. (5), we obtain the dependence on P. In Fig. 4, right, we show the results for the 670-nm band for the PDMs without the AOD constraint. A MODIS/VIIRS imager sensitivity to polarization m of 0.03 (Sun and Xiong 2007) was taken as an example and the relative uncertainty on the sensitivity was set successively at 10%, 20%, and 100%, with the latter, extreme, value shown only for comparison. The values of 10% and 20% represent realistic uncertainties on the imager sensitivity to polarization.8 At the maximum polarization, , , and 20% the relative uncertainty in reflectance of the intercalibrated imager is about 1%. This value represents the highest uncertainty in reflectance for any surface type that the imagers will encounter. (Since water surface, the best polarizer, is chosen here, a 2.5% value is reached for the completely uncertain value of m.) Term appear to be virtually indistinguishable for and 20%, indicating a weak dependence on the precision of the polarization sensitivity constant m.

We compare the results for the 670-nm channel with the other two polarization bands by plotting the residuals. The top two panels in Fig. 5 show that, while the mean uncertainties on reflectance are generally within 0.1% of each other for the three bands, they are indistinguishable from each other if the standard deviations on are included. One also observes, comparing the top-left and top-right plots, that is insensitive to the 20% variation in the sensitivity parameter m. We note that the 490- and 865-nm bands were chosen as the two extremes of the values and that the 670–490-nm residuals are lower than the ones shown in Fig. 5.

Fig. 5.
Fig. 5.

(top) Residuals after subtracting [Eq. (5)] for the 865-nm band from those for the 490-nm band. The residuals for (left) and (right) are shown. No AOD constraints were imposed. The subtraction was performed using the PDM data values and standard deviations, such as the ones seen in Fig. 4, right. (bottom) Residuals after subtracting [Eq. (5)] without the AOD constraint from those with AOD < 0.1 for the (left) 670- and (right) 865-nm bands. Term was set to 10% in both cases.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-15-0165.1

In the bottom of Fig. 5, we examine the influence of the aerosols on the intercalibration uncertainty. We compare with and without the AOD < 0.1 constraints for the 670-nm band (left) and 865-nm band (right). Clearly, the inclusion of the aerosols affects the , with the differences between constrained and unconstrained values of around 0.3% (up to 0.5% if standard deviations are included) for both channels. For the 670-nm band, these differences are especially significant, with roughly 3σ differences for P > 0.7. The results for our second test value of 20% are not shown since, as seen in Fig. 5, they are virtually the same as for .

In summary, we have used the versus dependence to quantify the uncertainty in intercalibrated reflectance as a function of the degree of polarization. Setting the sensitivity to polarization to 0.03, which is roughly equivalent to the MODIS (Sun and Xiong 2007) and VIIRS (Kulkarny et al. 2010) values, and the error on the sensitivity to 10% and 20%. We found to vary between 0.2% and 1%, increasing with the degree of polarization P. Not taking the error on sensitivity into account () results in up to 2.5% error on . Examining the variation of across the PARASOL’s three polarized bands reveals no change in values if is increased from 10% to 20%. The 490–865-nm interband variations in are within the 0.2% envelope and are mostly within one standard deviation of each other. The differences between with the aerosols included and the aerosol influence reduced by the AOD < 0.1 constraint can be as high as 0.3%, or up to 3σ difference. Within the constraints on the CLARREO’s target reflectance uncertainty of 0.15% (Wielicki et al. 2013), the variations in reflectance uncertainty due to the interband differences and the uncertainty on the sensitivity of polarization are negligible; however, the intercalibrated reflectance is sensitive to the presence of the aerosols.

7. Polarization distribution models for the angle of linear polarization

a. χ PDMs for the cloudless-sky over ocean scene

We now turn to the angle of linear polarization χ. As mentioned in the beginning of section 6, for the precise evaluation of the uncertainty in reflectance due to polarization, its dependence on χ for each viewing geometry configuration needs to be included. Thus, the χ PDMs, analogous to P PDMs, need to be constructed. Using the definitions in Eqs. (2) and (3), in Fig. 6 we show an example of such χ PDM for the 670-nm band. In the left panel of Fig. 6, the PDM shown was subject to the constraints in Table 1, whereas the right panel shows the PDM for the same channel without the AOD constraint (Table 2). On both sides of Fig. 6, we observe that χ is symmetric around the principal plane. For the case when the meridian and the scattering planes coincide ( and 180°) , as expected. At nadir (), one degree in ϕ corresponds to one degree in χ and since at , the full range of corresponds to , which is indeed the case. The same argument applies to the ϕ values between [270°, 360°] wrapping around to [0°, 90°]. Away from the nadir, the one-to-one ϕχ correspondence no longer holds; thus, the broadening of the χ range is seen. At the relative azimuth values extending to (20°, 40°) and up to (330°, 40°), the values of χ wrap around the χ range of 0° and 180° and the PDM suffers a discontinuity. The means and standard deviations around these discontinuity regions become unphysical and need to be corrected. The PDMs in Fig. 6 makes use of the algorithm developed by us to correct for this effect. It is described in more detail in the appendix.9

Fig. 6.
Fig. 6.

Terms and for the cloudless-sky over ocean scene for the 670-nm band (left) without and (right) with the AOD < 0.1 constraint (see Tables 1 and 2 for the full list of constraints). (bottom left) The χ(670 nm) − χ(490 nm) residuals; (bottom right) the χ(865 nm) − χ(490 nm) residuals. No AOD constraints were imposed for this comparison.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-15-0165.1

The standard deviations in the bottom-left plots in Fig. 6 show values of standard deviations of . For the case, the highest is observed in the , and, symmetrically, at the , regions. As described in section 5a, these correspond to the backscatter region. The top-right and top-left panels in Fig. 6 are virtually identical, except for the broadening of these two regions due to higher multiple scattering by aerosols.

In Fig. 6 we compare the PDMs for the 490-nm band without the AOD restrictions to its analog for the 670- and 865-nm bands by plotting the residuals. Most of the residuals are contained in the ±10° range. As expected, the largest deviations are seen in the regions around , and , . One observes large deviations in the wraparound regions (see the appendix); however, those are limited to a narrow, ±2°, band.

b. Comparing empirical and theoretical χ PDMs

The angle of linear polarization χ within the single-scattering approximation may be found using the sine property of the spherical triangle (see Fig. 1):
e6
where is the scattering angle, which is defined as
e7
Using Eqs. (6) and (7), we can construct the as a function of two variables, ϕ and θ. To approximate the empirical PDM, we fix the solar zenith angle at 45° (the empirical χ PDM constraint is set at ). For better comparison we convert the functional form of into a histogram with the same binning as the empirical PDM. Since one-dimensional per-bin χ distributions were not available in this case, the wraparound regions were adjusted differently from the empirical PDMs. This was accomplished by setting in Eq. (6) to ±1 (corresponding to crossover) and expressing θ as a function of ϕ. The results are shown in Fig. 7 (left).
Fig. 7.
Fig. 7.

(left) Theoretical χ PDM obtained using single-scattering approximation [see Eq. (6)]. (right) Empirical PDM for the 670-nm channel without any AOD constraint (see Fig. 6)–single-scattering PDM residuals.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-15-0165.1

In Fig. 7 (right) we compare the empirical χ PDM for the 670-nm channel with the AOD < 0.1 constraint (Fig. 6) to the theoretical one by plotting the residuals. The differences between the empirical mean χ and values are contained within ±5°. With the uncertainties in our sample being roughly the same magnitude (see Fig. 6), single scattering provides a useful tool for cross-checking the empirical χ PDMs. Larger deviations found in the wraparound regions (discussed in the appendix) are explained by the different treatment of the solar zenith angle constraint in the theoretical and the empirical cases, as described above. Large deviations are also observed in the areas around . At , the scattering angle is 0 and becomes undefined. In the vicinity of this point, the single-scattering approximation breaks down and multiple scattering dominates. Empirically, the Stokes parameters Q and U recorded by PARASOL in this region are small and have large uncertainties. This results in the large uncertainties on χ in Fig. 6, as well as those for P in these regions (Fig. 3). To summarize, except for the regions around the solar zenith angle constraint and within the uncertainties of our PARASOL data sample, we find that the empirical χ PDMs can be adequately described by the single-scattering approximation.

8. Conclusions

We have discussed the empirical PDMs for the degree of polarization P and the angle of polarization χ for the cloudless-sky over ocean scene that we developed from the 2006 PARASOL level-1 and level-2 cloud and aerosol data products. We compared the PDMs with pristine approximation (AOD < 0.1) with those with the aerosols included (all AODs) across the three available polarized bands, at 490, 670, and 865 nm. We found little difference in P PDMs across the three channels for the AOD < 0.1 case. For the all-AOD case, the P PDM differences across the three channels were similarly small, except for the region where the viewing zenith angle is greater than 50° due to the asymmetrical aerosol contamination in the Northern and Southern Hemispheres.

Using the P PDMs, we estimated the uncertainty in the reflectance as measured by an imager, such as MODIS or VIIRS, after its intercalibration with the CLARREO spectrometer. Having assumed the imager’s sensitivity m to be 0.03, as is the case for both MODIS (Sun and Xiong 2007) and VIIRS (Kulkarny et al. 2010), the relative uncertainty on the imager reflectance and using the dependence of the uncertainty of the degree of polarization on P derived from the PARASOL data, we examined how the relative intercalibrated uncertainty on the reflectance depends on the wavelength, on the imager’s uncertainty in sensitivity to polarization , and on the presence of aerosols. We found that at the CLARREO’s target accuracy of 0.15%, is insensitive to the choice of the wavelength as well as the differences in when the values were confined to 0% and 20%. The inclusion of the aerosols (AOD > 0.1), however, results in the increase in intercalibration uncertainty of up to 0.3% exhibiting the sensitivity of the intercalibrated reflectance to aerosols. Since the worst-case scenario was considered here (the highest polarization of the best polarizer having the largest polarization uncertainties), 0.3% is the maximum uncertainty on the CLARREO-intercalibrated imager reflectance for any scene and surface type for our choice of CLARREO and imager parameters.

We have developed an algorithm to correct the empirical χ PDM distributions around the wraparound regions; that is, the regions where χ is discontinuous and where the 0°/180° transition occurs, making use of the per-bin χ distributions. We employed this algorithm to construct the corrected χ PDMs. The PDMs were found to exhibit little variation in χ across the three polarized bands, except for the backscatter regions. Apart from the backscatter regions and within the accuracy of the PARASOL measurements, the presence of the aerosols was not found to have any significant effect on the χ distributions. We have cross-checked the empirical χ PDM means against the single-scattering approximation and found a ±5σ agreement. Given that the standard deviations in our statistical samples for χ PDMs are also on the order of ±5, we find that the single-scattering approximation adequately describes our empirical results.

Finally, we note that we have examined only two special types of the cloudless-sky over ocean scenes, with the aerosols present and with their influence reduced, using only one set of constraints. The PDMs that we will soon make publicly available will be based on the broad range of AOD and the solar zenith angle constraints. In addition, work is underway to develop the cloudless-sky PDMs for other surface types, as well as for the cloudy scene types.

Acknowledgments

We gratefully acknowledge François-Marie Bréon for the helpful comments on the PARASOL data and Wenying Su for the discussion of the aerosol-related measurements. We would also like to thank the PARASOL data distribution centers at CNES and ICARE, France, for providing the data and guidance on its use. This study was funded by the NASA CLARREO project.

APPENDIX

Correcting χ PDMs for the Wraparound Effect

Directly applying the χ definition in Eq. (2) to construct the χ PDMs leads to distributions markedly different from the ones shown in Fig. 6. Instead, a PDM, such as the one for the 670-nm band in Fig. A1, top, is obtained. Large values of σ are seen for bins starting at and ending at (30°, 40°), and starting at (0°,270°) and ending at (320°, 40°). The nature of these large standard deviations becomes clear if one considers a typical subregion and plotting the χ distribution for it (Fig. A1, bottom left). From this plot it is clear that not only the standard deviations but also the means are ill defined due to the wraparound effects when χ values are close to their range limits of 0° and 180°.

Fig. A1.
Fig. A1.

Terms (top) and (bottom) obtained by directly applying the χ definition in Eq. (3). (bottom left) An example of a χ distribution for the sample region , showing the wraparound effects when χ is close to its range limits of [0°, 180°]. (bottom right) Corrected χ distribution with χ translation applied to the distribution on the left.

Citation: Journal of Atmospheric and Oceanic Technology 33, 3; 10.1175/JTECH-D-15-0165.1

To obtain sensible means and standard deviations for χ in the wraparound regions of the PDM, we implemented the following.10

  1. For each bin in Fig. A1, top, construct a one-dimensional distribution in χ, such as shown in Fig. A1.
  2. We find that outside the wraparound region, the most frequent value of is . Thus, if in a particular per-bin distribution , then we leave the mean and unchanged.
  3. For the case when , χ values less than 90° (half the range of χ) are translated as (see Fig. A1).
    • If of the translated distribution is found to be less than the original value, then the new is taken to be the true RMS value. And if the translated χ mean is found to be within the χ range limit of [0°, 180°], then this new mean is recorded as the correct mean. However, if the new mean is higher than the 180° range limit (see Fig. A1), then the recorded mean needs to be reverse translated as .
    • If of the translated distribution is found to be greater than that of the original one (typically for broad distributions with low statistics), then we revert to the original distribution and take the original and .
Applying this algorithm to the wraparound bins, we calculate the new and leading to the correct PDM shown in Fig. 6.

In the sample bin in Fig. A1 (bottom), the wraparound effect leads to the misleading values of and . Following the algorithm steps, since is found to be greater than 5°, then the translation is performed, as described in step 3. As the new mean value of 189 lies outside the range, the resulting distribution is reverse translated as . The final distribution with the corrected mean of 9.2 and the RMS error of 7.7 is shown in Fig. A1. We note that this algorithm is applied to all the bins in the χ PDM, not only to those inside the wraparound regions. However, there are a large number of bins whose is only slightly above 5°. In such cases the application of the forward and reverse translations result in slight shifts of mean relative to the original values. This explains the slightly more granular nature of PDMs (Fig. 6) in the nonwraparound regions as compared to its counterpart in the top panels of Fig. A1. However, we also note that these variations are well within the per-bin standard deviations and do not present any concern.

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1

The choice of 2006 data was motivated by the quality of data acquired by PARASOL.

2

In this article we focus on the third incarnation of the Polarization and Directionality of the Earth’s Reflectances (POLDER) instrument on board the PARASOL satellite. The previous two missions, POLDER-1 (launched in 1996) and POLDER-2 (launched in 2002), lasted less than a year each due to the loss of both satellites.

3

Throughout the text we use the term aerosol loosely, since in our data-driven sample we are unable to distinguish real aerosols from phenomena such as ultrathin clouds. The difference between these phenomena is immaterial in our case, as we are interested only in how they affect polarization.

4

We note that since our goal was to intercompare the PDMs for all three polarization bands and the AOD measurement was not performed at the 490-nm band, the constraint-free dataset was chosen as a stand-in for the AOD > 0.1 dataset. The comparison of the P and χ PDM residuals showed such a substitution to be acceptable.

5

An AOD < 0.05 constraint, for example, leads to values in the range between 0.05 and 0.15 in the region of . Such a constraint, however, also results in the significant increase in statistical uncertainties and gaps in data points and therefore was not used here.

6

A functional fit to the PDM, the feasibility of which is currently being explored by the authors, would eliminate this artifact.

7

The uncertainties presented in this section are taken at the 68.27% confidence level ( for the case of the Gaussian distribution). Using the nomenclature of metrology (BIPM 2008), this corresponds to the coverage factor.

8

These values are close to those reported by the MODIS team for band 8 in Sun and Xiong (2007).

9

This χ wraparound effect is also present in the simulations, such as ADRTM (Sun and Lukashin 2013), and needs to be corrected.

10

We note that the empirical PDMs in the wraparound region were not corrected in Lukashin et al. (2013).

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