# Hybrid Dual-Polarization Synthetic Aperture Radar
## 3 Hybrid Dual-Polarization
### HCP Lunar Data Analysis  
The values of the Stokes and their derived child parameters provide objective characterizations of geophysical properties of a lunar or planetary surface [19; 20; 29]. Along with the child parameters, a two-parameter set of classification variables (see Appendix C.3) was selected by the Mini-RF science team. The degree of polarization \(m\) was the first, which has long been recognized as the single most important parameter characteristic of a partially polarized EM field [30]. 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 close relationship between entropy and degree of depolarization has been verified experimentally [31].) The mean value of \(m\) for lunar radar data (at S-band) is about 0.6; on average about 40% of the radar backscatter from the Moon is randomly polarized.  
The Stokes parameters offer several candidates for a second classification variable. Of these, the Poincare ellipticity parameter _chi_ (\(\chi\)) [32] is the most robust choice. It is one of the four principal components (_I_, \(m\), \(\chi\), \(\psi\)) that are necessary and sufficient to describe the polarized portion of a partially polarized quasi-monochromatic EM field of total intensity \(I\). Further, the sign of sin\(\chi\) is an unambiguous indicator of even vs. odd bounce backscatter, thus directly responsive to the driving science requirements. The sign of \(\chi\) indicates rotation sense even when the radiated EM field is not perfectly circularly polarized [33], as was the case with the Mini-RF and Mini-SAR radars. The resulting _m-chi_ classification method has proven to be effective and reliable.  
Discrimination between single (in general, odd) or double (in general, even) bounces (radar reflections from within an illuminated pixel within the scene) is known as a key to discovering lunar ice water deposits [34]. The required single-bounce vs. double-bounce characterization is expressed through the sign of the fourth Stokes parameter. Sin2\(\chi\) (which is known formally as the _degree of circularity_) may be found from the Stokes parameters (see Appendix C.2) by sin2\(\chi\) = -S4/mS1. (_Alert_: The sign, negative in this expression, is dependent upon the L or R handedness of the transmitted polarization and the data's coordinate convention as well single vs. double bounce discrimination.)  
Three-color assignments for polarimetric imagery portrayal found in the literature usually use a combination of primary colors. The assignment of those colors to the three principal backscatter classes--random, single bounce, and double bounce--is at the discretion of the user.  
An example of an LRO Mini-RF imaging product of the lunar surface classified through the _m-chi_ method is shown in Figure 1. The Mini-RF science team adopted a color-coding backscatter classification based on three primary colors: green, blue, and red [24]. Green, indicating the randomly-polarized backscattered constituent, is proportional to (1-_m_). Blue usually was assigned to the single-bounce constituent, a class that includes Bragg scattering [35], a major constituent of lunar radar returns. In the coordinate system used for analysis, and respecting that the transmitted field was left-circularly polarized, this component is proportional to the square-root of (1 - sin2\(\chi\)). Red, indicating double-bounce, is proportional to the square-root of (1 + sin2\(\chi\)).  
Figure 1: Examples of lunar impact crater polarimetric images as seen from the Arecibo Observatory (**a**) and an _m-chi_ classification of hybrid dual-polarized data (**b**) from the Mini-RF radar aboard LRO (adapted from Figure 5 of [24]). The color wheel helps to retrieve meaning from the transition colors between primaries. In this example, yellow indicates dominant contributions from both random and double-bounce backscatter.