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.
2.1. Stokes Parameters
 The underlying Stokes parameter theory is well known [Born and Wolf, 1959; Boerner et al., 1998; Touzi et al., 2004]. A monochromatic EM field is represented by the ellipse swept out by its electric potential vector E = [Ex Ey]T (Figure 1). In general analytic form, the orthogonal components of E are
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.
Figure 2. The Poincaré sphere for a partially polarized quasi-monochromatic EM field. Note the factor m (degree of polarization) on the radius vector.
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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.
Figure 3. (a) An m-chi decomposition and (b) an m-delta decomposition of floor-wall double bounce from the crater Kies C (26°S, 26°W), and surface backscatter from surrounding areas.
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 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.