We introduce a new technique derived from the classical Stokes parameters for analysis of polarimetric radar astronomical data. This decomposition is based on m (the degree of polarization) and chi (the Poincaré ellipticity parameter). Analysis of the crater Byrgius A demonstrates how m-chi can more easily differentiate materials within ejecta deposits and their relative thicknesses. We use Goldschmidt crater to demonstrate how m-chi can differentiate coherent deposits of water ice. Goldschmidt crater floor is found to be consistent with single bounce Bragg scattering suggesting the absence of water ice and further corroborating adsorbed H to mineral grains or an H2O frost as plausible explanations for a H2O/OH detection by near-infrared instruments.
 Impact cratering is the dominant weathering process on the surface of the Moon and is largely the determining factor of material distribution on the lunar surface [Melosh, 1989]. Characterization of the physical properties of these materials as well as their spatial distribution is crucial for a chronological account of events and determination of resources (e.g., water) potentially trapped in the near subsurface at the poles [Arnold, 1979]. Although much can be determined from optical observation of impact craters, ultimately optically bright, or immature, materials will mature in the upper microns of the surface due to space weathering and information regarding the distribution of materials can be masked [Lucey et al., 2000; Hapke, 2001]. Radar can observe the properties of the lunar surface to depths that are proportional to several radar wavelengths, thus on the order of 1 m for the 12.6 cm wavelength of the LRO S-band channel [Raney et al., 2011]. The polarimetric decomposition analysis technique introduced in this paper can provide additional and unique information regarding the vertical distribution of materials, and the presence (or absence) of coherent deposits of water ice.
 There is substantial heritage of both theoretical and observational studies that address the question of water on the surface of the Moon [Watson et al., 1961; Arnold, 1979; Feldman et al., 1998, 2000, 2001; Bussey et al., 2010; Mazarico et al., 2010]. However, recent ultraviolet and near-infrared spectral analyses of Chandrayaan-1's Moon Mineralogy Mapper (M3), the Visual and Infrared Mapping Spectrometer (VIMS), the Deep Impact High-Resolution Instrument-IR spectrometer (HRI-IR), and the Lyman Alpha Mapping Project (LAMP) as well as S-band radar measurements by the Chandrayaan-1 MiniSAR have presented corroborative and compelling evidence to confirm the presence of H2O/OH on the surface of the Moon [Clark, 2009; Pieters et al., 2009; Sunshine et al., 2009; Spudis et al., 2010; Gladstone et al., 2012]. Even more intriguing are results suggesting H2O/OH is not confined to permanently shadowed regions, but is spread widely across the lunar surface. This puts into question the physical form, or forms, that H2O/OH may be found within different geologic contexts. Interpretation of near-infrared data suggest H is adsorbed by mineral grain surfaces at microns to millimeter depths [Clark, 2009]. Interpretations of ultraviolet observations by LAMP suggest H2O is present in the form of a frost at 1–2 vol.% [Gladstone et al., 2012]. Still other radar interpretations suggest water is present at the poles in the form of water ice [Nozette et al., 1994; Spudis et al., 2010] although this conclusion remains to be independently verified. Here we evaluate the Anaxagoras-Goldschmidt crater region, an area recently noted for a high H2O/OH spectral enhancement [Pieters et al., 2009; Cheek et al., 2011], using an approach to analyzing polarimetric radar data not previously explored in the context of the Moon.
 The Miniature Radio Frequency (MiniRF) instrument flown on the Lunar Reconnaissance Orbiter (LRO) is a Synthetic Aperture Radar (SAR) with an innovative hybrid dual-polarimetric architecture [Raney, 2007; Raney et al., 2011], a form of compact polarimetry [Nord et al., 2009]. Examples used in this paper are from the S-band zoom mode, having a wavelength of 12.6 cm, 7.5-m pixels, and 8 looks [Raney et al., 2011]. The resulting data support calculation of the 2 × 2 coherency matrix of the backscattered field, from which follow the four Stokes parameters. These parameters are the basis of science products from the observations, including traditional thematic images, such as the circular polarization ratio, and polarimetric decompositions, which are relatively unknown in traditional radar astronomy. In particular, we examine the utility of the m (degree of polarization) – chi (ellipticity) decomposition of the MiniRF data.
 Conventional dual-polarized remote sensing radars transmit on one linear polarization (e.g., H) and receive on two polarizations, one matched to the transmitted polarization (H), and the other its orthogonal counterpart (V). Standard practice for Earth-based radar astronomy such as for the Arecibo Observatory is to use circular polarization on both transmitted and received signals [e.g., Green, 1968; Campbell, 2002; Ostro, 2002]. In contrast, a hybrid dual-polarimetric architecture receives orthogonal linear polarizations, while transmitting circular polarization. The precedent for this architecture may be found in radars used for meteorological purposes [Bringi and Hendry, 1990; Torlaschi and Holt, 1998], and in the early days of radar astronomy. In the 1960s the Moon was illuminated from the Arecibo Observatory by circular polarization, and the lunar backscatter was transformed into coherent linear dual-polarized components. The data were processed to align the observed orthogonal linear basis coordinates with the axis of lunar libration so that the orientation of the Ex and Ey components relative to the Moon's surface would be known. The results provided the first estimate of the mean thickness of the lunar dust layer [Hagfors et al., 1965].
 An orbital dual-polarized radar that is expected to make radar astronomy-class measurements of the lunar surface should transmit circular polarization [Raney, 2007]. However, the polarization basis of the receiver does not have to be circular, if the data product of the radar is stipulated to be the 2 × 2 coherency matrix of the backscattered field. This is true, because the values of the Stokes parameters obtained from the matrix elements are independent of the particular polarization basis in which the data are observed [Stokes, 1852; Wolf, 1954; Green, 1968]. The polarization plan of the MiniRF receivers was optimized with respect to engineering principles, without impacting science [Raney et al., 2011].
 The hybrid-polarimetric MiniRF and MiniSAR radars offer the same suite of polarimetric information (expressed through four Stokes parameters and their child products) from lunar orbit as Earth-based radar astronomy. Classical child parameters—such as the degree of depolarization, the degree of circularity, and the circular polarization ratio—may be determined from the Stokes parameters. The values of these parameters provide objective indications of geophysical properties of the surface [Stacy and Campbell, 1993; Carter et al., 2004, 2006].
Cloude and Pottier  have developed an effective decomposition technique (entropy-alpha) for quad-polarization data (earth-observing synthetic aperture radar jargon), where both polarizations are separately transmitted and both polarizations and their relative phase (i.e., full Stokes data) are received from each of the transmitted polarizations. We have developed a similar methodology (m-chi) for single-transmitted dual-receive polarization data, as obtained by MiniRF, or the terrestrial planetary radar systems (such as Arecibo or Goldstone). We use the m-chi decomposition to investigate the thickness of impact crater ejecta and to investigate the plausibility of water ice within craters (e.g., Anaxagoras and Goldschmidt) near the Moon's North Pole.
where τ represents the EM oscillation, θ0 is an arbitrary reference phase, and δ represents the relative phase between the two linearly polarized components. Stokes proved that such a field could be represented by four real numbers, known as the Stokes parameters (S1, S2, S3, S4) [Stokes, 1852]. (In the literature the Stokes parameters appear in a variety of notations, such as [S0, S1, S2, S3], [S1, S2, S3, S4], or [I, Q, U, V], where the order is always preserved, total power being the first parameter in each case.)
 Starting from focused, single-look complex SAR image data, in response to a left-circularly polarized transmitted field, the four Stokes parameters of the backscattered field are
In these expressions, the chevrons (< >) indicate spatial averaging, which is an essential attribute of this method, and Re and Im select the real or the imaginary value (respectively) of the complex cross-product amplitude. The negative signs on S4 are consistent with the backscattering alignment (BSA) coordinate convention which has become the standard in Earth-observing radar polarimetry [Boerner et al., 1998], in contrast to the forward scattering alignment (FSA) that is traditional in optics, and radio and radar astronomy.
 The first column in equation (2) follows directly from calculation of the Stokes parameters from the coherency matrix resulting from the arbitrary EM field of equation (1). The customary model for dual-polarization is to assume that the received basis is the same as the transmitted basis. In the traditional radar astronomy context that would be “transmit circular and receive circular”; the third column corresponds to this familiar case. The MiniRF radar, however, represents the unconventional alternative—the “transmit circular and receive linear” combination. The Stokes parameters in the second column correspond to this hybrid-polarity architecture. The fourth column represents the parameters in terms of the classical Poincaré variables [Poincare, 1892], where χ indicates the sign of rotation of the ellipse and its ellipticity, ψ is the orientation of the long axis of the polarization ellipse, and m is the degree of polarization
Note that the Poincaré sphere [Poincare, 1892; Boerner et al., 1998], which applies only to the polarized portion of a partially polarized quasi-monochromatic EM field, should be annotated to include explicitly the degree of polarization m as a factor on its radius dimension, as shown in Figure 2. This visualization is a helpful aid for interpreting the effectiveness of the m-chi decomposition method.
2.2. Signal Decomposition
 Traditional polarimetric radar astronomy is based on analysis techniques that use Stokes or child parameters individually, such as total power (S1) or the circular polarization ratio (CPR). In contrast, polarimetric analysis in quantum physics and related fields has developed a mature alternative technique known as decomposition, in which two or more suitably selected parameters are used jointly to classify fundamental characteristics of the observed field.
 The initial expectation on polarimetric decomposition of imaging radar data is that it should provide unambiguous differentiation of single bounce, double bounce, or randomly polarized backscatter. Although the radar transmits a fully polarized EM field, the resulting backscatter includes fully polarized and randomly polarized constituents. The randomly polarized part arises primarily from volumetric materials that give rise to multiple internal reflections which obliterate the polarization of the illumination. The polarized portion of the backscatter falls into two classes, single (odd) bounce and double (even) bounce. Single bounce includes Bragg scattering, as well as specular reflection from a quasi-planar surface at right angles to the incoming illumination, and reflections from a three-sided “corner,” either fabricated or naturally formed. Bragg scattering as used in this paper (and by the terrestrial imaging radar community) refers to the net return from that subset of backscattering features which are spaced apart in radar range by (multiples of) half-wavelengths. These reflections combine coherently, thus increasing their effective radar cross section in proportion to area2, in contrast to randomly phased backscatter whose radar cross section is proportional only to area. Double-bounce includes backscatter from two sided structures (dihedrals or di-planes), either fabricated or naturally formed, and returns from the internal reflections of frozen volatiles having thickness of many wavelengths, due to the coherent opposition backscatter effect (COBE) [Peters, 1992; Ostro, 1993].
 We choose as the first decomposition variable the degree of polarization m, which has long been recognized as the single most important parameter characteristic of a partially polarized EM field [Wolf, 1959]. The close relationship between entropy and degree of depolarization (1−m) has been verified experimentally [Aiello and Woerdman, 2005]. The degree of depolarization (1−m) is indicative of randomly polarized backscatter, typically arising from radar-quasi-transparent volumetric materials, such as lunar regolith. The mean value of m for lunar radar data (at S-band) is about 0.6, so that on average about 40% of the radar backscatter from the Moon is randomly polarized.
 The Stokes parameters offer several candidates for a second decomposition parameter. Of these, the Poincaré ellipticity parameter χ (Figures 1 and 2) is the most robust choice. It is one of the three principal components (m, χ, ψ) that are necessary and sufficient to describe the polarized portion of a partially polarized quasi-monochromatic EM field. Further, the sign of χ is an unambiguous indicator of even versus odd bounce backscatter, even if the radiated EM field is not perfectly circularly polarized.
 The m-chi decomposition and color-coding described here is a useful analysis tool for a variety of lunar investigations. Sin2χ (which is known formally as the degree of circularity) may be found from the Stokes parameters (equation (2)) by
Note that the negative sign in this expression is consistent with transmitted left-circular polarization, and the use of the BSA sign convention. Then the m-chi decomposition may be expressed through a color-coded image, where
In this scheme, Blue indicates single-bounce (and Bragg) backscattering, red corresponds to double-bounce, and green represents the randomly polarized constituent.
Aside: Decomposition strategies developed originally for Earth-observing radar remote sensing are designed for data that are quadrature-polarimetric, and hence represented by a 3 × 3 covariance (or coherence) matrix. For radar astronomical data such as produced by the MiniRF radar, the data are characterized by a 2 × 2 matrix. It has been shown that the entropy-alpha decomposition derived from the 3 × 3 matrix typical of the “random volume over ground” forestry model when adapted to 2 × 2 covariance matrix data reduces to an expression that is equivalent to the m-chi decomposition [Cloude et al., 2012].
 One situation in which there often is clear double-bounce geometry at the lunar surface is the combination of the floor and far wall (from the radar's perspective) of an impact crater, which together form a large natural dihedral. The surrounding backscatter arises from features that are typical of the surface, and these reflections are mapped at their appropriate distance (range) from the radar. In contrast, backscatter that corresponds to forward scatter from the floor of the crater to the far wall and then back to the radar travels an extra distance. These double-bounce reflections will appear at greater range in the radar image, hence appearing as if they come from an area beyond the far crater rim. Such double-bounce signatures will be strongest when the crater walls are terraced or relatively steep, where in this context terracing or steepness depends on the age, materials, and perhaps layering exposed by the generating impact. Figure 3a shows an example of a double-bounce signature highlighted by the m-chi decomposition. The double-bounce return is the red “halo” at the far range side of Kies C.
 It is instructive to look at these same data through an m-delta decomposition, as originally proposed by Raney . In this case (Figure 3b), the “halo” appears red on one side, but blue on the other. This indicates that the sign of δ (equation (1)) has been reversed from one side to the other, which is caused by the orientation ψ of the axis of the floor-wall dihedral feature relative to the linear component of the radar's incoming illumination (see the first column of equation (2)). The field transmitted by the MiniRF radar, although nominally circularly polarized, in practice is elliptically polarized, having a dominant linearly polarized component [Raney et al., 2011]. In response to such illumination, the m-delta decomposition leads to ambiguous interpretation, as evidenced by the red/blue halo shift, and also by the rather confused coloration of the surface backscatter. On the other hand, the effect suggests that a three-component decomposition (m, χ, ψ) could offer further insight into linear trends in the structure of the reflecting surfaces, thus taking advantage of the known ellipticity of the transmitted field.
3. Analysis and Discussion
3.1. Byrgius A—Physical Properties of Ejecta
 Byrgius A is a 19 km diameter Copernican crater with an estimated age of 48 Ma that is located in the lunar highlands east of the Orientale Basin and west of Mare Humorum [Wilshire, 1973; Morota et al., 2009]. Visible image data of the region obtained by the LROC wide-angle camera (WAC) at a resolution of 100 m/pixel show optically bright ejecta deposits associated with the crater that extend to radial distances of 100s of km, with near continuous deposits observed to an average radial distance of 70 km (Figure 4a).
 MiniRF Circular Polarization Ratio (CPR) information derived from S-band (12.6 cm) data of the region at a post-averaging resolution of 100 m show an increased roughness for Byrgius A and its ejecta deposits relative to the surrounding terrain (Figure 4b). This is a commonly observed characteristic of young, fresh craters, and indicates that the crater and its ejecta have a relatively high fraction of cm- to m-scale scatterers. These may be at the surface but also may be buried to depths of up to ∼1 m, thus visible to the radar since the radar's illumination and subsequent backscatter can penetrate up to ten wavelengths into the surface) [Thompson et al., 1974]. There is a markedly higher CPR associated with Byrgius A and ejecta deposited within approximately a crater radii than with ejecta deposited at greater distances. We presume this indicates the transition from the continuous to the discontinuous portions of the ejecta. However, as observed with visible image data, the increased roughness associated with the discontinuous ejecta of Byrgius A appears nearly continuous to a radial distance of ∼70 km.
 An m-chi decomposition of MiniRF S-band data for the region indicates that, for the top meter of the surface, the background lunar terrain can be characterized as predominately Bragg scattering (blue) while the impact crater interiors and ejecta blankets are characterized by a combination of double bounce and volumetric scattering (Figure 4c). In the case of Byrgius A, the utility of this product is twofold. First, by decomposing the scattering properties of the surface we observe that the portion of ejecta that extends radially from ∼10 to 70 km appears far less continuous than is suggested in either the optical data or the CPR information. The implication is that we are observing properties of the ejecta and lunar background terrain in the top meter of the surface. In other words the thickness of the ejecta in this distance range is on the order of a meter or less. Second, the scattering properties of the interior of Byrgius A and its nearby ejecta clearly include volumetric and randomly polarized components. This property is not evident in the CPR mapping. The implications of this difference are described in the following section.
3.2. Goldschmidt—Looking for Water-Ice
Figure 5 shows mosaics of Anaxagoras and Goldschmidt craters in optical and radar (S1, CPR, and m-chi decomposition scheme) data sets. In these images the width of each of the swaths is about 15 km, which provides a reference for the size of the crater and its associated ejecta blanket.
 Lunar soils at the North Pole, such as within the impact crater Goldschmidt, have significant 3000 nm absorption features consistent with enhanced H2O/OH spectral signatures [Pieters et al., 2009; Cheek et al., 2011] (Figure 5a). Goldschmidt crater in particular has enhanced near-infrared spectral absorptions suggestive of H2O/OH, but the physical form and vertical distribution of H2O/OH is still being debated. Initial thermal and epithermal neutron enhancements observed by Lunar Prospector suggested contradictory low hydrogen abundances in the upper meter of the lunar surface using single layer (dry-over-wet) models [Lawrence et al., 2006; Pieters et al., 2009]. However, Lawrence et al.  have recently reexamined this study by evaluating a two-layer model (wet-over-dry) that is more consistent with the near-infrared results although still inconclusively attributed to hydrogen. If relatively clean water ice is present within a many-wavelength-thick layer (>1 m at S-band), then anomalously high CPR ratios (approaching unity or higher) should be expected. Comparatively thin (less than one wavelength) water-ice populations will not cause elevated CPR. Several studies have detected high CPR values in the permanently shadowed regions of the lunar poles, however the interpretation of these radar measurements are the subject of much debate within the planetary radar community [Nozette et al., 1994; Campbell, 2003; Campbell et al., 2006; Ghent et al., 2008; Spudis et al., 2010]. However, in the context of evaluating the plausibility of water ice, high CPR alone can be ambiguous, since high CPR can result from double-bounce backscatter usually indicative of relatively large scale surface roughness or blocky boulder populations. However, relatively clean water-ice that has thickness of many wavelengths usually exhibits high CPR values. Ice-regolith mixtures generate lower CPR values [Peters, 1992]. Generally, areas that are interpreted to have water-ice using radar have also been in permanent shadow [Spudis et al., 2010]. The caveat for Goldschmidt is that it is not in permanent shadow [Bussey et al., 2010; Mazarico et al., 2010], and so it has been considered an unlikely candidate for water-ice, despite the recent discovery of H2O/OH in the near-infrared suggests the possibility of very thin layer of water to be present on surface deposits.
 The bulk of Goldschmidt's crater floor appears as blue in the m-chi decomposition image (Figure 5d), corresponding to Bragg scattering, suggesting moderately rough surface soils. This is inconsistent with volumetric water ice (which would show up red, indicative of double bounce scattering). This example is particularly interesting because, although the lunar surface is covered in regolith which gives rise to volume scattering, the decomposed image indicates that Bragg (polarized) characteristics dominate the observed backscatter.
 The blue-red-green coloration of the odd-even-random backscatter constituents seen through an m-chi decomposition is not always pure, but may have a mixture of scattering properties at work. The impact crater Anaxagoras is an example of this with an ejecta field appearing yellowish, which from the color wheel (inset, Figure 5d) indicates a mixture of volume and double-bounce backscattering.
 The MiniRF instrument on LRO (and its simpler sibling on India's Chandrayan-1) is the first polarimetric imaging radar outside of Earth orbit. Its architecture is hybrid-polarimetric, transmitting (quasi-) circular polarization, and receiving orthogonal linear polarizations and their relative phase. The four Stokes parameters that are necessary and sufficient to fully characterize the observed backscattered EM field are calculated from the received data. These Stokes parameters may be used to formulate an m-chi decomposition of the scene, which for radar astronomy is a new technique. This method facilitates unambiguous interpretation of lunar features according to single (odd) or double (even) bounce signatures in the polarized portion of the reflections, and characterization of the randomly polarized constituents. It is not a substitute for CPR, but rather a complementary product that is a more rigorous qualifier of surface scattering regimes for which CPR can be ambiguous in some contexts. The m-chi decomposition parameter also does not need to be combined or overlaid on S1 maps for morphological information as CPR, OC, and SC maps often do.
 Using the m-chi decomposition parameter as a tool to examine lunar craters we have demonstrated its utility to examine crater floors and walls, crater ejecta, and the presence (or lack thereof) of coherent deposits of water ice in the top meter of the lunar surface. With this tool we have demonstrated the ability to differentiate materials, more clearly within ejecta deposits (dominantly a combination of double bounce and volumetric scattering regimes) and their relative thicknesses. We have characterized the floor of Goldschmidt crater to be consistent with single bounce Bragg scattering suggesting the absence of water ice and further confirming adsorbed H to mineral grains as a plausible explanation for a H2O/OH detection by near-infrared instruments.
 This work was supported by the Lunar Reconnaissance Orbiter Project through contract NNN06AA01C with NASA, and was made possible by the larger MiniRF team, to whom we express our deepest appreciation. We are thankful for the constructive comments of two anonymous reviewers.