Polarimetric Decomposition of Near-Earth Asteroids Using Arecibo Radar Observations

The polarization state of radar echoes from planetary bodies contains information about the scattering mechanisms present on the surface and thus the near-surface physical properties. Polarimetric radar scatter from complex surfaces, such as those observed for spacecraft-visited near-Earth asteroids (NEAs), is not well understood in terms of relating observed polarimetry to surface properties. Here we present an improved methodology for polarimetric analyses of ground-based radar observations of NEAs, extending techniques derived for larger bodies. We calculate the Stokes vector for delay-Doppler images of NEAs and use this to perform the m-chi decomposition and derive polarimetric products such as the degree of polarization, circular polarization ratio, and degree of linear polarization. We apply this methodology to radar observations of NEAs (53319) 1999 JM8, (101955) Bennu, and (33342) 1998 WT24 obtained by the Arecibo Observatory. We also perform numerical simulations of the m-chi decomposition for irregular boulders to augment the interpretation of the results for NEAs. Our analyses show that significant components of radar echoes are depolarized (random polarization) and linearly polarized. The numerical simulations confirm that depolarization is increased by single scattering from nonspherical wavelength-scale particles. Our analysis suggests that 1999 JM8 is possibly covered in regolith and that surface scatterers dominate the scattering properties of Bennu. The NEA 1998 WT24 displays diverse polarimetric properties, which we reconcile with optical and thermal observations by assuming a fine-grained regolith mantling a rugged, dense interior. In this work, we demonstrate the usefulness of radar polarimetry in characterizing the physical properties of planetary surfaces.


Introduction
NASA's Origins, Spectral Interpretation, Resource Identification, Security, Regolith Explorer (OSIRIS-REx) mission to near-Earth asteroid (NEA) (101955) Bennu (Lauretta et al. 2019) and JAXA's Hayabusa2 mission to NEA (162173) Ryugu (Watanabe et al. 2019) have recently revealed that these similar C-complex objects have morphologically complex surfaces. Modeling of ground-based radar imagery resulted in a three-dimensional shape model for Bennu (Nolan et al. 2013) that has been shown to be consistent with optical imagery returned by OSIRIS-REx Nolan et al. 2019). Polarimetric analysis of this same radar imagery suggested a surface for Bennu that is smooth at decimeter scales, likely mantled by a fine-grained regolith (Nolan et al. 2013). In contrast to this interpretation, angular boulders over a wide size distribution are visible on the surface in OSIRIS-REx optical imagery with only sparse coverage by fine-grained regolith Walsh et al. 2019). The preencounter radar albedo (s =  0.12 0.04 OĈ ; Nolan et al. 2013) and disk-integrated thermal inertia (Γ = 310 ± 70 J m −2 K −1 s −1/2 ; Emery et al. 2014) for Bennu indicate high porosity, which, despite the numerous boulders on the surface, is compatible with thermal inertia measurements by OSIRIS-REx (DellaGiustina et al. 2019). This implies intraboulder porosity, which has also been observed for boulders on Ryugu (Grott et al. 2019;Sugita et al. 2019;Okada et al. 2020), and could be a contributing factor to the low circular polarization ratio (CPR) for Bennu measured by radar (μ C = 0.18 ± 0.03; Nolan et al. 2013;Hickson et al. 2020). Interpretations of radar scattering from NEAs often derive conclusions from comparison with lunar radar studies due to the low number of NEA surfaces imaged by spacecraft missions. OSIRIS-REx has shown that this lunar-like interpretation is not viable for NEA surfaces (Lauretta et al. 2019), and that we lack a full understanding of radar scattering from NEAs.
The CPR is arguably the most popular metric for polarimetric analyses of ground-based asteroid radar observations, derived almost exclusively from data obtained using the Goldstone Solar System Radar (GSSR) and Arecibo Observatory (AO). The CPR has been shown to be sensitive to wavelength-scale surface roughness ; Campbell 2012) and particle shape (Virkki & Muinonen 2016;Virkki & Bhiravarasu 2019), ice content (Black et al. 2001), boulder abundance (Fa & Cai 2013;Fa & Eke 2018), particle size distribution (Virkki & Bhiravarasu 2019), composition Virkki & Muinonen 2016), and viewing geometry (Thompson et al. 2011). The convolution of these factors often makes physical interpretation of measured CPR values ambiguous without additional constraints (Virkki & Bhiravarasu 2019). These constraints may be in the form of terrestrial analog radar data (e.g., Campbell 2002;Carter et al. 2011) or additional data for the observed surface at other wavelengths. The CPR is a relative measure of the circularly polarized components of the radar echo; by also measuring the randomly and linearly polarized components, we acquire more information on the scattering mechanisms present and thus the near-surface physical properties, as is explained in more detail below.
Improving the capabilities of radar polarimetry at the GSSR and AO will therefore significantly improve the science return.
Radio observations of extrasolar objects often take advantage of the full polarization state of received signals by measuring the Stokes vector (see Section 2.1; Robishaw & Heiles 2018). In addition to the power in orthogonal receiver channels, this analysis also considers the complex crosscorrelation between either channel or, more intuitively, the relative phase of the two signals. While this is common practice for quadrature-polarimetric (quad-pol) Earth-observing synthetic aperture radars (SARs; van Zyl et al. 1987;Cloude & Pottier 1996) and dual-or hybrid-polarimetric spaceborne SARs, such as Mini-RF on board the Lunar Reconnaissance Orbiter (Raney et al. 2011) and MiniSAR on board Chandrayaan-1 (Spudis et al. 2009), this has only been attempted with AO planetary radar data a handful of times (Hagfors et al. 1965;Hagfors & Campbell 1974;Stacy 1993;Carter et al. 2004Carter et al. , 2006Carter et al. , 2011Carter 2005;Campbell 2012).
Expanding on early polarimetric radar studies of the Moon (Hagfors et al. 1965) and Venus (Hagfors & Campbell 1974), Stacy (1993) performed bistatic polarimetric S-band radar measurements of the Moon using the AO to transmit and the (now nonexistent) Los Caños antenna, located roughly 11 km northeast of the main site, to receive. The Stokes vector was calculated from the received signal and used with a radar scattering model, accounting for quasi-specular and diffuse surface and subsurface single scattering, to estimate the real part of the complex relative permittivity of the radar-probed material. The significant linearly polarized component of the received signal was attributed to subsurface reflection, owing to the different Fresnel transmission coefficients for orthogonal linear polarizations at the vacuum-regolith interface at the lunar surface. This same effect results in the angle of the linearly polarized component being in the local plane of incidence and therefore correlated with changes in local topography, which was observed in these data. Carter (2005) explored Stokes analyses of and Carter et al. (2004Carter et al. ( , 2006Carter et al. ( , 2011 performed polarimetric S-band radar observations of Venus from the AO and derived the Stokes vector for all observations. A substantial amount of linear polarization was measured in these experiments, which, under similar assumptions as in Stacy (1993), was used to infer mantling of finegrained material and subsurface reflections. Campbell et al. (2010) and Carter et al. (2011) carried out bistatic polarimetric S-band radar measurements of the Moon using the AO to transmit and the Green Bank Telescope (GBT) to receive, revealing buried lava flows and pyroclastic deposits from variable radar backscatter and CPR values. They observed a somewhat uniform linearly polarized component in the received signal, suggesting uniform subsurface scattering at the S band across various lunar terrain. Carter (2005) explored a Stokes analysis of AO S-band radar observations of NEA (53319) 1999 JM 8 , identifying an appreciable linearly polarized component of the radar echo indicating a penetrable regolith covering. This work highlighted the potential for deeper physical interpretation of delay-Doppler imagery of NEAs through improved polarimetric analysis.
In this paper, we explore the potential for ground-based radar observations of NEAs to exploit the full polarization state of radar echoes in order to apply tighter constraints on radar scattering processes and near-surface physical properties. We present and discuss in Section 2 the methodology to extract the Stokes vector from radar observations at the AO and use it to derive the degree of polarization (DP), degree of linear polarization (DLP), CPR, and m-chi decomposition (Raney et al. 2012). While we focus on observations of NEAs from the AO, this method can be adapted to other radar systems and targets. In Section 3, we apply this methodology to AO S-band radar observations of NEAs (53319) 1999 JM 8 , (101955) Bennu, and (33342) 1998 WT 24 . We improve the interpretation of the m-chi decomposition for NEA surfaces through numerical simulations of rough particle scattering in Section 4. In Section 5, we discuss the interpretation of our analyses from Section 3 in the context of these numerical simulation results and ultimately in terms of the physical properties of each asteroid's surface. We conclude our findings and propose useful avenues for future research in Section 6.

Method
Typical AO S-band radar observations consist of transmitting a 2.38 GHz circularly polarized signal to a planetary target and receiving the radar echo in orthogonal circularly polarized receiver channels in the same-sense circular polarization (SC) and opposite-sense circular polarization (OC) than that transmitted. At the feed, the AO S-band radar system transmits a right-circularly polarized (RCP) signal, so that the traditional SC and OC labels in radar astronomy correspond to RCP and left-circularly polarized (LCP), respectively. The observational setup and basic data processing have remained relatively unchanged over the decades this system has been operating and are described in detail in Ostro (1993) and Magri et al. (2007). In this scheme, the received complex voltages from the OC and SC channels are converted to power for analysis. For targets with intrinsically high signal-to-noise ratios (S/Ns), the crosscorrelation between the complex voltages of the OC and SC channels can be calculated to compute the Stokes vector of the signal. The additional steps in data reduction to compute delay-Doppler images of each of the Stokes parameters are described in Carter (2005). Our work is based on and expands this procedure, as described in the next sections.

Stokes Parameters and Polarimetric Decomposition
The four-element Stokes vector, pletely describes the polarization state of electromagnetic radiation (Jackson 1999). In this paper, we adopt the notation for the Stokes vector most commonly used in radar astronomy (e.g., Carter et al. 2011;Raney et al. 2012;Campbell 2016): The elements of the Stokes vector are related to the amplitude and phase of the orthogonal components of the electric field of an electromagnetic wave. The feed of a radio telescope converts an incident electric field into a voltage, which is eventually digitized and recorded as a complex voltage wave. For the S-band-narrow (SBN) receiver at the AO used for S-band radar observations, the Stokes vector can be calculated from the complex voltages recorded in the RCP (SC) and LCP (OC) channels, V R and V L , respectively, in the frequency The angle brackets in Equation (2) refer to a time or spatial average, and δ is the relative phase between the RCP and LCP voltage waves. In radio astronomy, a correlator is used to measure average Stokes parameters over the integration time. In radar astronomy, we measure instantaneous voltages as a function of time, so the averaging in Equation (2) refers to an ensemble average of the signal from all scattering elements within the resolution of the radar (Tinbergen 1996). Here S 1 is the total power in the received signal, S 2 is the excess linearly polarized power in the horizontal compared with the vertical direction, S 3 is the excess linearly polarized power at +45°with respect to horizontal compared with −45°, and S 4 is the difference in power between the RCP and LCP components of the signal. In Equation (2), S 4 is defined according to the IEEE and IAU convention, and in terms of radar scattering, in the backscattering alignment (BSA) coordinate convention, which defines the wave polarization vectors relative to the radar antenna (Ulaby & Long 2014). Note that S 4 is defined in the BSA convention in Raney et al. (2012) with the LCP and RCP power terms exchanged and a negative sign; however, this is mathematically equivalent to Equation ( In practice, we use the third column of Equation (2) to calculate the Stokes parameters as auto-and cross-correlations of the received RCP and LCP complex voltage signals. The Stokes parameters are related to the variables of the Poincaré sphere, χ, ψ, and m, as follows. Here we adopt the notation of Raney et al. (2012), where χ is the ellipticity of the polarization ellipse, ψ is the linear polarization angle (or orientation of the polarization ellipse), and m is the DP: Note that often in the polarimetric radar literature, χ refers to the linear polarization angle (e.g., Stacy 1993;Carter et al. 2004Carter et al. , 2006, so we caution the reader to keep in mind our definitions as stated above. Similarly, our expression for S 4 in Equation (2) omits the negative sign in Equation 3(a) found in Raney et al. (2012). The DLP, m ℓ , describes the fraction of the total power that is linearly polarized: The CPR, μ C , can be calculated from the ratio of the radar cross sections measured in the SC and OC channels, ( ) The handedness of circular polarization flips for quasi-specular reflection from a smooth surface at wavelength scales, so that the OC signal is dominated by single quasi-specular reflections and the SC signal arises from geometric or compositional complexity at the wavelength (decimeter) scale. The CPR is often used as a measure of surface roughness by gauging the relative amount of quasi-specular single scattering compared with multiple and diffuse scattering. First introduced in Raney et al. (2012), the m-chi decomposition technique is a compact-polarimetric (compact-pol) decomposition that has been applied in analyzing SAR data of the Moon from Mini-RF and MiniSAR (Saran et al. 2012;Mukherjee & Singh 2015;Mitchell et al. 2018). Compact-pol refers to a system that transmits one polarization and receives two orthogonal polarizations, such as the AO S-band radar system. While this is not sufficient to derive the 4 × 4 scattering matrix, as is done in quad-pol systems, it is enough to calculate the Stokes vector and Poincaré variables using Equations (2) and (3). The m-chi decomposition utilizes two of the three principal components of the Poincaré variables: the DP, m, and the ellipticity parameter, χ (hence the name m-chi). The DP inherently encompasses the degree of depolarization (1 − m), which represents the fraction of total backscatter intensity that is randomly polarized, largely comprising the diffuse component present in both OC and SC channels. The sign of the ellipticity parameter denotes the dominant handedness of the circularly polarized component of the total backscatter intensity and is thus a robust measure for the number of scattering events for quasi-specular scattering. The m-chi decomposition is presented as a false-color RGB image derived from the following equations (in the BSA convention; Raney et al. 2012 Raney et al. (2012), blue indicates odd bounce scattering (dominated by single quasi-specular reflection), red indicates even bounce scattering (dominated by double-bounce quasi-specular reflection), and green indicates the amount of depolarized total backscatter intensity, often attributed to volume scattering mechanisms. Although Raney et al. (2012) defined blue as arising primarily from Bragg-like/quasispecular scattering, we consider constructive interference from periodic structures precisely half wavelengths apart unrealistic for asteroid surfaces covered with a wide particle size distribution (following the definition of Bragg scattering in Raney et al. 2012). As Virkki & Bhiravarasu (2019) demonstrated, single scattering by wavelength-scale particles can complement the traditionally used number of bounces as an interpretation; while a double bounce between two surface facets can increase the echo power in the SC polarization, there are other ways for the SC component to rise than an even number of bounces. A wealth of literature has demonstrated that the level of SC polarization can be used as an indicator for irregularity of scatterer shapes. If a surface is covered by rough, nonspherical wavelength-scale particles, the cumulative echo power from the single scattering from each particle will play a role in the polarization characteristics of the received echo. Therefore, we perform simulations of the m-chi decomposition of wavelength-scale particles rather than using the interpretation by Raney et al. (2012). Using the m-chi decomposition is more powerful than a single-parameter interpretation, such as the CPR, because the relative proportions of linear and random polarization in the received signal are accounted for in the DP. No single parameter can unambiguously account for both the linear and circular components simultaneously. If there is no linearly polarized component to the radar echo, S 2 = S 3 = 0, m = |S 4 |/S 1 and the m-chi decomposition reduces to In terms of the OC and SC portions of the echo, red is dominant if SC is greater than OC, blue is dominant if OC is greater than SC, and green increases with both increasing SC and OC.

Calibrations and Corrections
The measured Stokes parameters are sensitive to the relative gain and phase of either receiver channel. Separate electronic chains constituting the signal path for either receiver channel have different system temperatures owing to their different hardware components. In addition, slight variations in the electrical length of either chain induces a phase offset between the two channels. Gain calibration scales the relative power in either channel to its system temperature. Phase calibration corrects for the systematic phase bias by measuring a correlated noise signal simultaneously in either receiver channel. The linearly polarized power from the noise diode should be entirely within S 3 ; however, the phase offset converts some of this power to S 2 as a function of frequency. Computing the polarization angle, ψ, from the Stokes vector shows a linear dependence with frequency when in theory, it should be independent. A linear fit to this is used to rotate the phase of one receiver channel to account for the system phase offset. This is necessary to calculate the polarization angle of a radar echo; however, the DLP is unaffected by this process, since the total amount of power between S 2 and S 3 is preserved. The linear polarization angle is further rotated by the parallactic angle due to the orientation of the antenna feed relative to the target as the telescope tracks the object. This can be corrected postmeasurement using the tracking information within the ephemeris used for an observation. Gain and phase calibration by noise injection cannot account for imperfections in the antenna feed. These imperfections are significant in passive observations of extragalactic radio sources, which typically have polarizations of <10%, and can be characterized by Müller matrices derived from measurements of known polarized sources (Heiles et al. 2001;Robishaw & Heiles 2018). In active radar astronomy, the transmitted signal is 100% polarized, so that radar echoes are typically polarized ?10%, rendering these imperfections less significant to the measured polarization state.

Results for Select Asteroids
Here we apply the methodology described above to AO radar observations of NEA (53319) 1999 JM 8 , allowing direct comparison with previous polarimetric analyses in Benner et al. (2002) and Carter (2005) and observations of NEAs (101955) Bennu and (33342) 1998 WT 24 , demonstrating the polarimetric diversity among NEAs. The NEA 1998 WT 24 is a good example of E-type asteroids, for which the measured CPR is typically much greater than for S-and C-type asteroids . These data are gain calibrated using daily system temperature measurements. Correlated calibration data, required for phase calibration, do not exist for these data, so we do not calculate the polarization angle; however, we reiterate that the measured DLP is unaffected by phase calibration.
Our basic data processing follows the methodology described in Carter (2005). We refer to those data pertaining to one transmit/receive cycle as one "run" and to each independent delay-Doppler image contained within each run as one "look." For a given run, we first decode and Fourier transform the raw complex voltage time series to produce complex voltage delay-Doppler images for both OC and SC channels. We cannot incoherently sum complex delay-Doppler images, so these data contain one look per run. We then use these images with Equation (2) to create delay-Doppler images of each Stokes parameter for each run. The background (system) noise level is calculated from off-source areas and subsequently subtracted from images of S 1 and S 4 . There is no system noise component in the images of S 2 and S 3 as a result of the cross-correlation in Equation (2) (Campbell 2012;Raney 2019). Images of each Stokes parameter (resulting from individual runs) are gain calibrated and summed to increase the resulting S/N (increasing the number of looks), as is discussed in more detail for each asteroid. These summed Stokes parameter images are then used to derive the CPR, DLP, DP, and m-chi for each asteroid. All images presented in this paper are masked to only show pixels with an S/N higher than 4σ (4 standard deviations of the noise power), a threshold that was arrived at through trial and error.
The correction for changes in parallactic angle described in Section 2.2 redistributes power between S 2 and S 3 . Since we have not phase calibrated our observations, this added correction is not relevant for our analysis, and we do not apply parallactic angle corrections. As mentioned in Section 2.2, the DLP is invariant to rotation and so is unaffected by parallactic angle correction. We also omit Müller matrix corrections from the present analysis, since these are higher order than phase calibration; however, we note that these are important to consider for future planetary radar polarimetry efforts in measuring the linear polarization angle.
3.1. (53319) 1999 JM 8 (53319) 1999 JM 8 (alternate designation 1990HD 1 ) is a P-type (Binzel et al. 2004; SMASSII taxonomy) NEA and potentially hazardous asteroid (PHA) that was observed by the GSSR in 1999 July-August and the AO in 1999 August at a distance of ∼0.06 au (Benner et al. 2002). With an estimated diameter of ∼7 km and rotation period of ∼7 days, these observations had a high S/N with thousands of pixels in delay-Doppler images at a range resolution of 15 m pixel −1 . Due to the limited integration time for each run and slow rotation period, these data required significant zero-padding of the complex voltage time series for 1999 JM 8 to be resolved in Doppler with a corresponding resolution of 0.0047 Hz pixel −1 . This causes leakage of the signal into adjacent Doppler bins, resulting in horizontal smearing in delay-Doppler images. In our analysis, we consider data obtained at the AO on 1999 August 1-3 ( Figure 1). We sum all runs from each day to increase the S/N without significantly smearing the image owing to the slow rotation period of ∼7 days. Summing ∼60 minutes of data per day corresponds to 2.4°-2.8°of rotation. We spatially average the images to further increase the S/N by convolving the images with a 2 × 2 boxcar filter, resulting in an effective range resolution of 30 m pixel −1 and Doppler resolution of 0.0094 Hz pixel −1 .
Total power images (S 1 ) in Figure 1 highlight the numerous concavities, facets, and topographic variations visible on the surface. The CPR images are clipped to an upper value of 0.5 to highlight variations among the relatively low values across the surface. At low incidence angles near the subradar point, the CPR is near minimum in each image and generally increases with incidence angle as the dominant form of backscattering transitions from quasi-specular to diffuse. As noted in Benner et al. (2002), the CPR image for August 2 shows a region near the trailing edge with significantly lower CPR than its surroundings, which is likely a flat facet perpendicular to the radar line of sight. The DLP images are clipped to an upper value of 0.5 and follow a similar trend as CPR, with the largest values close to the trailing edge. Similar values of DLP for each day were reported in Carter (2005) that were interpreted as indicating a penetrable regolith layer on the surface. Images of CPR and DLP in Figure 1 show that our methodology for Stokes polarimetry is able to replicate previous results from Benner et al. (2002) and Carter (2005). The DP image shows increasing depolarization with incidence angle, with some concentrated regions near either Doppler extent. The low CPR region identified in the CPR image from 1999 August 2 is visible in the DP image as being highly polarized relative to its surroundings, supporting the interpretation that this region is a relatively flat facet perpendicular to the radar line of sight resulting in increased quasi-specular single scattering. The mchi decomposition shows that blue is the dominant color, with some concentrated regions of green and very little red. We discuss the implications of these colors further in Section 5.

(101955) Bennu
(101955) Bennu (1999RQ 36 , hereafter Bennu) is a B-type (Clark et al. 2011; Bus-DeMeo taxonomy) NEA and PHA and at the time of writing was being orbited by NASA's OSIRIS-REx spacecraft with plans for future sample return to Earth. The GSSR and AO observed Bennu in its discovery apparition  Carter (2005) noted that the north-south ambiguity of delay-Doppler images can be particularly prominent for morphologically complex small bodies, implying that the Stokes parameters for a given pixel can represent an average of several physical surface locations. They discuss that for spherical objects, summing runs over a full rotation increases this averaging effect, allowing for a view of the average surface properties. Given Bennu's surface, with thorough coverage of decimeter-scale and larger rubble and relatively spherical shape, we perform a polarimetric analysis of 229 runs with 0.075 Hz pixel −1 Doppler resolution (some minor zero-padding) summed from data obtained on 1999 September 24 to understand what radar scattering processes dominate across the entire surface ( Figure 2). Analysis of radar polarimetry projected onto the OSIRIS-REx shape model would likely produce interesting results for specific surface orientations but is outside the scope of our preliminary analysis. Summing 229 scans corresponds to averaging the surface properties over ∼173°of rotation. This spatial averaging also reduces the prominent "self-noise," or Rayleigh-fading, noise component that is particularly present in those 1999 data (Nolan et al. 2013;Ulaby & Long 2014).
In Figure 2, there is power leakage, or "ringing" (Magri et al. 2007), visible at the edges of the asteroid, most prominently the leading edge, which is typical of high-S/N delay-Doppler images and exaggerated by slight zero-padding in the Doppler dimension. For this reason, pixels at the horizontal extents of delay-Doppler images of Bennu, and all other image regions where this effect could be relevant, should be interpreted with caution. Some of these edge effects might also be due to the deviations of Bennu from a perfectly spherical shape. Values of CPR in Figure 2 are low, consistent with the disk-integrated CPR of 0.18 ± 0.03 derived from analysis of continuous wave data (Nolan et al. 2013). The DLP values are low across the image, with some higher values near the trailing edge at higher incidence angles (∼80°). Anomalously high (0.4-0.5) values of DLP near the leading edge and either Doppler extent are likely due to noise. The DP image shows that 37% of the radar echo is depolarized ( Table 1). As expected, at low incidence angles near the subradar point, the return is dominantly polarized; however, the rest of the surface is somewhat homogeneously depolarized. The m-chi decomposition image indicates that blue and green are the dominant colors, with virtually no red present, similar to 1999 JM 8 . The bistatic observation between the PO and CDSCC represents the second detection of an asteroid using planetary radar from the Southern Hemisphere (Benson et al. 2017). The NEA 1998 WT 24 has a diameter of ∼415 m (Busch et al. 2008) and relatively short rotation period of 3.697 hr (Krugly et al. 2002;Pravec et al. 2007;Busch et al. 2008). In this paper, we use AO radar data from 2015 December 15, when 1998 WT 24 was observed at ∼0.028 au at a range resolution of 7.5 m pixel −1 and Doppler resolution of ∼0.06 Hz pixel −1 (no zero-padding). We analyze sums of 10 and 100 runs to compare the average radar scattering properties with those for a specific surface orientation. Figure 3 shows our resulting images when summing 10 (∼19.5°of rotation) and 100 (∼196°o f rotation) runs of AO data from 2015 December 15. When summing 10 runs of 1998 WT 24 , the prominent ridge between large basins mentioned in Busch et al. (2008) is visible in the images. The CPR values are high across the entire surface, with the lowest found at the subradar point. The DLP similarly shows high values, although not in as many areas as the CPR. When summing 100 runs, the prominent ridge is no longer apparent, and the resulting image looks more like a semicircle, as expected. As with 10 runs, the CPR values for 100 runs are high, often higher than 1. However, the DLP summed over 100 runs shows lower values when compared with the DLP summed over 10 runs. The NEA 1998 WT 24 shows a much wider range of values for the DP and m-chi decomposition compared to 1999 JM 8 and Bennu. The m-chi decomposition is strikingly different when compared with those for 1999 JM 8 and Bennu, with green being the dominant color and roughly equal amounts of blue and red present. The  DP summed over 100 runs is statistically distinct from the sum over 10 runs, although the difference is small ( Table 1). The mchi decomposition summed over 100 runs shows similar properties to that for 10 runs, although with green more dominant over the entire image.

Quantitative Comparison
For each image given in Figures 1-3, we compute a histogram corresponding to all pixel values greater than 4σ in total power, S 1 (since these images only show pixels that meet this criterion, the histograms correspond to all visible pixels). Each histogram is given as a probability density, normalized so that the integral of each equals 1. The histograms for the CPR and DLP are given in Figure 4 and for the DP and m-chi decomposition in Figure 5. Data from each day of observations of 1999 JM 8 are similar, so we only report histograms of those data from 1999 August 2.
Visually, some of the histograms appear Gaussian-distributed, whereas others appear closer to a Rayleigh distribution. The echo power in a single look is Rayleigh-distributed due to self-noise; however, as the number of looks in an image increases, this distribution should tend toward Gaussian due to the central limit theorem. For the sake of quantitative comparison between the different data sets, we fit all histograms with either the Gaussian probability density function (PDF) or the offset-Rayleigh PDF described in Carter et al. (2017): In Equation (7), c refers to the offset in the Rayleigh distribution from zero, and a is the mode. The PDF with the lowest rms error fit is plotted for each histogram, with the fitted parameters in Table 1. We calculate the uncertainty in our DLP measurements following Carter (2005), assuming the same 4% error due to cross-coupled power as was measured for Venus. The uncertainties in the remaining parameters are propagated from the thermal and self-noise following standard error propagation, except for the CPR, for which an equivalent confidence interval is calculated using Fieller's theorem (Ostro et al. 1992). We separately fit PDFs to histograms derived from image pixel values ±1σ to estimate the uncertainty in the modes of the histograms. These exclude systematic uncertainties, such as the gain variations across the primary reflector, which we estimate as an additional 10% error (Ostro et al. 1992;Magri et al. 2007;Shepard et al. 2015). The variance of each distribution results from a combination of true random error and actual variability in the surface properties of the observed asteroids, whereas the 1σ uncertainty reported for each mode is only the random error. For each histogram, we note that the PDF standard deviation is greater than the 1σ uncertainty in the mode of the fitted PDF, indicating that there is likely a variation in surface properties of all three asteroids over the range of observed rotation phases. Images composed of many looks, such as for 1999 JM 8 and Bennu, that are best fit by the offset-Rayleigh distribution further indicates that there is surface variation, since the distribution has not reduced to a Gaussian. The polarimetric radar properties of 1999 JM 8 and Bennu are quantitatively very similar. In comparison, radar echoes from 1998 WT 24 have a higher CPR and DLP and are more depolarized. The standard deviation in the PDFs for CPR and DLP images of 1998 WT 24 is reduced from 10 to 100 runs, as is expected from the increased number of looks. The modes of PDFs for DP, m-chi green, and m-chi blue images of 1998 WT 24 are statistically distinct from 10 to 100 runs, indicating that the average surface properties are different than those sampled in a more specific geometry of the 10 runs. Raney et al. (2012) interpreted the m-chi decomposition colors as double bounce (red), single bounce including Bragg scattering (blue), and degree of depolarization (green). Raney et al. (2012) conducted the m-chi decomposition analysis for the lunar surface, which has to a great extent smoother plains and less decimeter-to-meter scale rubble than asteroid surfaces such as that of Bennu does. Therefore, the polarimetric properties of the rubble itself should also be analyzed when considering asteroid surfaces. For example, Virkki & Muinonen (2016) and Virkki & Bhiravarasu (2019) built on a wealth of literature demonstrating that single scattering by nonspheroidal particles can enhance the SC echo power. Here we use the same particle scattering matrices (Virkki 2019) to visualize the m-chi decomposition of both single and biparticle scattering (scattering involving two particles) by nonspherical particles. The four particle shapes considered are labeled A, B, C, and D and visualized in Figures 7 and 8 next to their corresponding simulation results. The particle morphologies are derived from scanning electron microscope images of atmospheric dust particles, which have similar shapes to rocks collected from the Moon. The particle properties are described briefly below; for further details, see Virkki & Bhiravarasu (2019).

Numerical Analysis
Following Virkki & Bhiravarasu (2019), we consider two refractive indices, 1.78+0.001i and 2.54+0.01i, which correspond to relative dielectric permittivities of 3.17+0.003i and 6.45 +0.052i, respectively. While the first refractive index is considered as a mix of mostly water ice with traces of silicates in Virkki & Bhiravarasu (2019), a permittivity of 3.17 could also correspond to an effective permittivity, ε eff , of silicate or carbonaceous rock with 46% microporosity using the Looyenga-Landau-Lifshitz mixing formula e e e = +p p 1 eff 0 1 3 1 3 3 ( ( ) ), where p is porosity and ε 0 is equal to the relative dielectric permittivity of a vacuum, 1 . A relative dielectric permittivity of ε = 6.45 +0.052i is a typical value for silicates with low or negligible microporosity or basalts with low-to-medium porosity according to Campbell & Ulrichs (1969). We find a negligible change in the mchi decomposition for changes in the imaginary part of the permittivity from zero to 0.05. The latter is a reasonable estimate for the upper limit of S-and C-type asteroids according to laboratory measurements of ordinary chondrite meteorites (Heggy et al. 2012) and a carbonaceous chondrite regolith simulant (Boivin et al. 2018).
Virkki & Bhiravarasu (2019) parameterized the scatterers in terms of a size parameter, x = 2πr/λ, where r is the radius and the wavelength λ = 12.6 cm, with an assumed size distribution that varies from x ve,min to x ve,max , where "ve" refers to a volumeequivalent sphere. We consider a range of size parameters from = x 0.5 ve,min with a step size of 0.5. Using λ = 12.6 cm, this size parameter range corresponds to physical volume-equivalent sphere radii from 1 to 16 cm. In order to save computation time but ensure a constant step size in the sizefrequency averaging, we interpolate the scattering matrix elements of size parameters x = 3.5, 4.5, 5.5, 6.5, and 7.5. We use a number-weighted average for the 4 × 4 scattering matrices (defined as the Müller matrix relating an incident Stokes vector to a scattered Stokes vector, averaged over all scatterer orientations), a r Z , ( ), where the number of particles as a function of size is assumed to be a power law with differential exponent −3: n(r) ∝ r −3 . The scattering matrices we use include the scattering cross section. If normalized phase matrices are used, the size-frequency weights, w(x), should include the scattering cross section explicitly: w(x) = n(x)C sca (x) (Virkki & Muinonen 2016).
The weights are normalized to sum up to 1. For single scattering, the phase angle, α, is the angle between the backscattering direction and the viewing angle (i.e., 0°in the backscattering direction; Figure 6). Figure 7 shows the m-chi decomposition for the four different particle morphologies for all phase angles from 0°to 90°without averaging over a size-frequency distribution when only single scattering is accounted for. Each line (one per size parameter) is normalized to the maximum color saturation while keeping the ratios of red, green, and blue as given for the m-chi decomposition. It is important to note that this neglects the relative backscattered power contribution between the different sizes but enhances what the m-chi decomposition color is if the given size were the maximum particle size. Using the scattering matrix elements, the m-chi decomposition equations are as follows (in BSA convention for consistency, although the scattering matrices were computed in forwardscattering reference, which was accounted for in the computations) and using an incident wave fully in RCP: Here we should note that Equation (8) is the equivalent scattering matrix representation of Equation (6).
We also consider biparticle scattering to investigate double bounces through the nonspheroidal particles. Figure 8 shows the m-chi decomposition for the four different particle morphologies when, in addition to the single scattering, the incident wave goes through another similar particle before scattering back. The biparticle angle (β) is defined here as the lesser angle between the line through the geometric centers of the particles and the direction of the incident wave ( Figure 6). Considering possible surface scatterer geometries, at an incidence angle of 0°, the incident wave scatters twice in a 90°angle, which is also the biparticle angle. When the incident angle increases from 0°to 90°, the minimum biparticle angle (β min ) decreases from 90°to 0°. At larger incidence angles, the biparticle angle can take any value up to 90°, depending on the azimuth angle between the particles, and the scattering angles to account for range from 90°− β to 90°+ β. The azimuth angle affects the linear polarization components but not the circular polarization components. Although the double-scattering transaction could happen in a variety of angles when θ i ? 0, averaging over all of the possible angles is not relevant in this work, as we do not distinguish between the different linear polarization orientations or a specific phase of the echo. Furthermore, as Figure 8 proves, the biparticle angle plays a minor role in the polarimetric properties of the echo compared to the roles of the particle size and shape.
Ground-based radar observations of solar system bodies measure the echo power at phase angles of zero or very close to zero in both monostatic and bistatic configurations, given the large distances to targets compared with the relatively short separation between the transmitter and receiver in bistatic observations. Figure 7 shows that single scattering at zero phase angle can significantly depolarize the signal, as evidenced in green. Particles below the Rayleigh size limit do not depolarize the signal effectively, resulting in blue. At larger particle sizes, yellow and cyan indicate a combination of red/blue and green, indicating partial depolarization of the scattered wave. Figure 8 demonstrates that there is less difference between the m-chi decomposition of the particles with different permittivities when using particle shapes B and C than when using particle shapes A and D. This supports, although does not fully rule out, that red would arise from differences in the rubble composition. Blue and green are dominant m-chi decomposition colors for most particle morphologies and the tested range of permittivities. These simulations reveal that irregular particles can depolarize radar echoes through both single and biparticle scattering. For surfaces with a rich diversity of particles in terms of sizes and shapes, it would be an oversimplification to account all SC echo power to double bounces, as is implied in Raney et al. (2012), at least for small particles. These results indicate that the lunar-like interpretation of the m-chi decomposition colors does not account for all of the different scattering mechanisms that can take place on asteroid surfaces.

Discussion
In this paper, we have defined the degree of depolarization, (1 − m), as the fraction of total backscatter intensity that is unpolarized. This is different than the common use of the term "depolarized" in radar astronomy to refer to the SC backscatter (Harmon & Ostro 1985;Campbell et al. 2010). This distinction is important when discussing radar polarimetry. A completely unpolarized or randomly polarized radar echo would result in equal intensity measured by LCP and RCP receiver channels. In other words, as the DP approaches zero, the CPR approaches 1. Campbell (2012) described this situation as the single scattering backscatter from randomly oriented dipoles, and that double-bounce scattering between dipole-like elements can result in an upper limit of 2.2 for the CPR. Our simulations of biparticle scattering show that for the considered particle morphologies, the m-chi decomposition results in almost exclusively blue and green, which corresponds to an upper limit of 1 for the CPR. Thus, to measure a CPR greater than 1 or for the SC backscatter to be greater than the OC backscatter requires a mechanism other than depolarization, such as a dihedral double bounce (Fa & Cai 2013) or coherent backscatter enhancement (CBE), as observed for planetary ice deposits (Black et al. 2001) and E-type asteroids in optical photometry (Muinonen et al. 2010), or another as-yetunidentified scattering mechanism.
Nonzero values of DLP in Figure 1 and green in the m-chi decomposition of 1999 JM 8 , indicating depolarization, both support the existence of a fine-grained regolith. However, multiple scattering can similarly result in nonzero DLP, since the Fresnel reflection coefficients for orthogonal linear polarizations are not equivalent (i.e., more power in one linear polarization will be reflected). The numerical simulation results in Section 4 show that significant depolarization can result from single scattering from nonspherical wavelength-scale particles. Thus, the green in the m-chi decomposition of 1999 JM 8 could arise from morphologically complex surface rubble as opposed to fine-grained regolith. This scenario is plausible given the quantitative similarities between the decomposition of 1999 JM 8 and Bennu and the abundance of boulders confirmed on the surface of Bennu from OSIRIS-REx. The lack of red in the m-chi decomposition of 1999 JM 8 indicates that little dihedral double-bounce scattering or CBE is occurring. If we consider boulders on the surface, this means that either the geometry of the boulder orientations is unfavorable for double-bounce scattering or the reflectivity of the material is low enough that the power in double-bounce scattering is low (Fa & Cai 2013). Although we can directly measure the radar backscatter coefficient, or radar albedo, relating this to the Fresnel Figure 6. Geometry for the numerical simulations. Here θ i is the incidence angle, α is the phase angle, and β is the biparticle angle. Red and blue refer to single and biparticle scattering geometries, respectively (no relation to m-chi decomposition colors). reflectivity of the surface is nontrivial. To further explore the effect of reflectivity on the m-chi decomposition would require additional modeling, which will be the focus of future work. The variation in DLP with range (vertical axis) in Figure 1 implies a dependence with incidence angle, which provides additional support for the presence of regolith (Carter 2005). The DLP image for 1999 August 1 shows concentrated regions of elevated values that correspond with the interiors of concavities visible in the total power image. If the DLP is connected with regolith, this could be showing fine-grained material pooling at low gravitational potentials, as is visible in the four potential landing sites imaged by OSIRIS-REx on Bennu. Furthermore, the concavities visible in total power images do not exhibit as significant double-bounce scattering as observed at some craters on the Moon (Raney et al. 2012;Mitchell et al. 2018), providing additional evidence that the surface material is made up of moderate-to low-reflectivity material. To summarize, our results suggest that 1999 JM 8 likely has an appreciable penetrable regolith layer, although a heterogeneous distribution of porous surface boulders cannot be completely ruled out.
Images of the total power and CPR (Figure 2) of Bennu hint at a smooth surface at wavelength scales due to the homogeneity in both total power and CPR and overall low CPR values. However, the nonzero DLP measured for Bennu reveals that there is some structural complexity that is not captured by circular polarization alone. Similarly, the DP image shows nonnegligible depolarization (37%) that is not captured by higher CPR values. The linear polarization angle has been shown to be parallel to the local plane of incidence for subsurface scattering on the Moon (Stacy 1993) and Venus (Carter et al. 2006). If the linearly polarized component of radar echoes from Bennu similarly originate from subsurface scattering, it is reasonable to assume the same mechanism is true in this case. As a result, the linear polarization angle varies across Figure 7. Single particle scattering m-chi decomposition simulation results. On the left, the relative dielectric permittivity of the particles is 3.17+0.003i, and on the right, it is 6.45+0.052i. The vertical axis gives the size parameter of the particle, x = 2πr/λ, where r is the radius of a volume-equivalent sphere, and λ = 12.6 cm; no size averaging is used here. The values at each row have been normalized to maximum color saturation. The particle shape models for particles A, B, C, and D are given on the right side of the plot for two orientations.
the surface with topography, and averaging over multiple points on the surface effectively cancels the linear polarization vectors. Our images represent radar returns averaged over 173°of rotation, so it is expected that linear polarization is underestimated due to the effects mentioned above. Since we still measure nonzero DLP for Bennu despite this averaging, it is possible that the linearly polarized component of the radar echoes is caused by something other than subsurface scattering. The interpretation that nonzero DLP and depolarization indicates fine-grained regolith assumed for 1999 JM 8 is therefore not readily applicable to Bennu. Indeed, our numerical simulations in Section 4 indicate that large-scale morphologically complex surface particles can cause depolarization, such as that observed for Bennu. The homogeneity of CPR, DLP, and DP in the images of Bennu show that the scattering properties of the surface are independent of incidence angle, suggesting that surface scatterers, such as boulders, play a significant role. OSIRIS-REx imagery has confirmed that the surface of Bennu is largely devoid of fine-grained regolith (Walsh et al. 2019), supporting the interpretation that the observed depolarization is due to boulders and possibly also the observed linearly polarized component of radar echoes. We note that the 1σ uncertainty in the mode of the fitted PDF to the histogram created from the DLP image of Bennu is consistent with zero (Table 1). However, the upper values of the PDF (even when accounting for outliers caused by the ringing effect mentioned earlier) are still nonzero. Nonzero DLP measured for 1998 WT 24 averaged over ∼196°of rotation (100 runs) also supports our theory that surface boulders can convert circular polarization to linear polarization, as is discussed more below. Busch et al. (2008) developed a shape model for 1998 WT 24 using GSSR and AO radar imagery from 2001 and fit the OC and SC images with the same empirical cosine radar scattering law. Based on the conclusion that the scattering processes are the same in either polarization (and across the entire surface), Figure 8. Biparticle m-chi decomposition simulation results. On the left, the relative dielectric permittivity of the particles is 3.17+0.003i, and on the right, it is 6.45 +0.052i. The particle size distribution ranges from x = 0.5 to the maximum size parameter, x = 2πr/λ, where r is the radius and λ = 12.6 cm (the vertical axis) with a power-law size-frequency distribution proportional to r −3 . The biparticles are of equal size; scattering between particles of different sizes is not accounted for. The color saturation has been normalized to maximum for each row. The particle shape models for particles A, B, C, and D are given on the right side of the plot for two orientations. the authors concluded that the radar echoes are completely depolarized and characterized by diffuse scattering. The authors constrained the bulk density of the near surface to be from 3 to 5 g cm −3 based on the total (OC + SC) radar albedo, as compared with radar albedos of (433) Eros and (25143) Itokawa, both of which have densities measured by spacecraft. This estimate, coupled with their high disk-integrated CPR measured for 1998 WT 24 (μ C = 0.97 ± 0.10), led them to conclude that the surface is likely covered in centimeter-tometer-sized complex angular boulders. This high-density, boulder-strewn surface is contradictory to thermophysical modeling by Harris et al. (2007), who estimated a thermal inertia of 200 ± 100 J m −2 s −0.5 K −1 , a value lower than expected for bare rock, implying significant areas of thermally insulating regolith. Additionally, E-type asteroids exhibit a strong opposition surge in optical photometry near zero phase angles (Muinonen et al. 2010), which hints at CBE occurring in a particulate regolith (Hapke et al. 1993).
The m-chi decompositions for sums of 10 and 100 runs of 1998 WT 24 in Figure 3 indicate that red is as prominent as blue. If red and SC power arise primarily from dihedral doublebounce scattering, this implies that the surface roughness or topography must be complex in order to increase the favorable geometry for double-bounce scattering, and also that the material must have high reflectivity in order for double-bounce scattering to have equal power to single-bounce scattering. Our numerical simulations do not show significant red in m-chi decompositions of biparticle scattering scenarios; however, red is dominant in some single scattering at high phase angles, independent of permittivity. The observed red could also arise from CBE, as is discussed in more detail below. The lower DLP values in the image, averaged over ∼196°of rotation (100 runs) as compared with that for ∼19°.5 (10 runs), could be showing a cancellation of linear polarization vectors as discussed above, which corresponds with a regolith. Regardless, there is still appreciable nonzero DLP when summing 100 runs (note that the image scale is from 0.0 to 1.0), so another mechanism for producing linear polarization, such as that postulated for Bennu, could be present on 1998 WT 24 as well. The DP is lower when averaged over ∼196°of rotation (100 runs) as compared with that for ∼19°.5 (10 runs). Contributions to a given delay-Doppler pixel from flat facets will be weighted less when averaged over multiple runs, as the asteroid rotates to expose various topographies. The orientation-averaging fades out all possible orientation-dependent effects and enforces the overall polarimetric properties of the rubble and the underlying near-surface volume. This is likely a factor in the increase in green seen in the m-chi decomposition of 100 runs as compared with that for 10 runs. To explain the observations at optical, thermal, and radar wavelengths, we propose 1998 WT 24 to have a thin, fine-grained, low-loss regolith covering a rough, dense material, presumably solid rock, making up the interior. Fine-grained regolith is necessary to explain the low thermal inertia (Γ = 200 ± 100 J m −2 s −0.5 K −1 ; Harris et al. 2007), opposition surge in optical photometry (Muinonen et al. 2010), and reduction in our measured DLP when averaging over more rotation phases (summing more runs). Radar wavelengths penetrate through this layer, revealing the subsurface structure. The high CPR can be explained by either a subsurface layer of rugged, solid rock  or rocky inclusions embedded in the low-loss regolith resulting in CBE. Since 1998 WT 24 , and E-types in general, exhibit high radar albedos, this subsurface scattering must have a very high albedo to overcome attenuation through the regolith layer, however low-loss that layer may be. The CBE in ice is compatible with this scenario, although the retention of significant ice in NEAs is questionable. Recent thermal modeling of ice retention in asteroids shows that <1 km diameter NEAs could retain ice under favorable conditions, such as a low thermal inertia mantle and small spin axis tilt (Schörghofer & Hsieh 2018). We calculate an axial tilt of ∼105°for 1998 WT 24 using the pole direction in Busch et al. (2008) and method in Lhotka et al. (2013). Even with a finegrained regolith surface layer, considering the axial tilt, diameter of ∼415 m, and semimajor axis of 0.718 au, it seems physically unlikely that 1998 WT 24 contains enough ice to be responsible for CBE (Schörghofer & Hsieh 2018). A rocky interior that increases the CPR through double-bounce and multiple scattering can also explain the observed CPR without the need for CBE and matches well the observed depolarized scatter. It is possible that CBE occurs on 1998 WT 24 with some other homogeneous low-loss medium facilitating the multipath enhancement; future work is necessary to uncover the true nature of the mysterious E-type asteroids.

Conclusions and Future Work
We have described the methodology to analyze the Stokes vector of radar echoes from NEAs observed at the AO, a process that is transferable to all planetary radar observations, including those obtained using the GSSR. We demonstrated the usefulness of this approach by applying it to archival radar data of NEAs (53319) 1999 JM 8 , (101955) Bennu, and (33342) 1998 WT 24 . We compared our results with polarimetric analyses of 1999 JM 8 in Benner et al. (2002) and Carter (2005) and found them to be consistent. By expanding on this earlier work with the use of the m-chi polarimetric decomposition technique, we give further confirmation that 1999 JM 8 is plausibly covered in a fine-grained regolith, although the possibility of porous boulders on the surface cannot be ruled out. We improved the interpretation of m-chi decompositions of asteroid radar observations through numerical simulations of single and biparticle scattering from irregular particles. As demonstrated by Virkki & Bhiravarasu (2019) and Campbell (2001), the depolarization of a radar echo, typically attributed to multiple scattering, can be caused by single scattering from irregular particles. Here we added an m-chi decomposition analysis of the particles to illustrate their polarimetric properties through the tricolorization. Our analysis of Bennu revealed more complex polarimetric signatures than revealed by analyses of only the circularly polarized component. We demonstrate that single scattering by boulders and cobbles of diverse shapes and sizes plays a role in the observed depolarization in addition to multiple scattering contributions that are commonly attributed to playing the dominant role. We reconcile multiwavelength observations and diverse radar polarimetry of 1998 WT 24 by assuming a finegrained regolith mantling a dense rocky interior. Nonzero DLP in images of Bennu and 1998 WT 24 averaged over ∼173°and ∼196°of rotation implies that the transmitted circular polarization was converted to the received linear polarization through some other mechanism than transmission through and subsequent subsurface scattering within a regolith. This unknown mechanism could be related to surface boulders and needs to be explored through further polarimetric analysis of NEA radar observations. Intentionally smearing images could be useful in this endeavor; however, the effects of a nonspherical shape on the resulting images is not well understood.
The polarimetric analysis presented in this paper could be further improved by better calibration data. Continuing the application and analysis of Stokes polarimetry applied to archival data and ongoing observations is essential for accurate interpretations of radar imagery. Furthermore, the interpretation of the polarimetry would be significantly improved with additional laboratory experimentation and numerical simulations. In particular, the role of single and double-bounce scattering in SC polarization for different particle morphologies needs to be better constrained. Comparisons of radar polarimetry with optical polarimetry for the same object could potentially be used to investigate the changes in surface properties at these different length scales. Applying a Stokes analysis for radar-observed NEAs with shape models would allow polarimetry to be mapped to physical coordinates on the surface, breaking the delay-Doppler north-south ambiguity and providing a means to compare measured linear polarization angles with surface topography.