Polarization trajectory.

A useful way of visualizing the polarization measurements is in terms of their location on the Poincaré sphere. Figure 2 shows such a representation for the observations of the preceding section. In particular, the figure shows the trajectory of the polarization state as a function of range along the cursor through the rain region of the storm. The Poincaré sphere is described in more detail in the next section (Figure 3); in Figure 2 it is shown in projection view as it would be observed from above its `north' pole. To orient the reader, the north and south poles of the Poincaré sphere correspond to left- and right-hand circular polarizations, respectively, while the equator corresponds to linear polarizations of varying orientation angle. The equator corresponds to the outermost circle in the projection view, with horizontal (H) and vertical (V) linear polarizations being at the bottom and top of the equatorial circle, respectively, and $+{45^\circ}$(+) and $-45^\circ$ (-) linear polarizations on the right and left sides. Left-hand circular polarization (L), being at the top of the sphere, corresponds to the center of the projection view. The Stokes parameters Q, U, and V constitute the cartesian coordinates of the Poincaré sphere and have their origin at the sphere's center. In the projection view the Q axis is pointed downward and the U axis is pointed to the right. V points vertically upward out of the page. The Q value of a given polarization state corresponds to the difference of the H and V powers of the signal; thus ${{\rm Q}} = W_H - W_V$. Differential reflectivity due to horizontally aligned particles increases the H component of the signal relative to the V component and thus displaces the polarization state toward the H polarization point at the bottom of the projection view. The fact that the entire trajectory lies below the U axis (slightly positive Q values) is a result of positive differential reflectivity along the entire length of the cursor.

  

Figure 2: Trajectory of the polarization state on the surface of the Poincaré sphere, along the cursor of Figure 12 (see text). The meridional and latitudinal lines indicate angular displacements of $22.5^\circ$ in the ($\alpha$,$\phi$) spherical coordinate system of Figure 5.

The left-hand end of the trajectory corresponds to the polarization state immediately after entering the storm. The radar transmitted LHC polarization and at the first range gate of the trajectory the polarization state was displaced downward from LHC due to the positive $Z_{\rm DR}$ of the rain (as discussed above). With increasing range the polarization state was displaced to the right as a result of differential phase effects, which cause the polarization state to move toward $+{45^\circ}$ linear polarization. As described earlier, the differential phase had additive propagation ($\phi_{dp}$) and backscatter ( $\delta_\ell$) contributions. Upon leaving the $\delta_\ell$ region the phase change reversed sign and the polarization trajectory partially retraced itself. During the remainder of the trajectory, on the far side of the main precipitation region, the polarization state gradually moved back to equal H and V powers as a result of decreasing $Z_{\rm DR}$ and increasing differential attenuation. It did not return to the LHCstate however, because of the cumulative differential propagation phase.

As discussed later, changes in $\rho _{HV}$ along the trajectory cause radial motion toward and away from the center of the sphere. These changes are not depicted in the view of Figure 2. The three types of polarization changes (differential phase, differential reflectivity and attenuation, and decorrelation) are in orthogonal directions on the Poincaré sphere.