Horizontally aligned particles.

Scattering by horizontally aligned particles is characterized by four quantities: Z_V (or Z_H), $Z_{\rm DR}$, $\delta_\ell$, and f. Each of the quantities has an associated propagation effect, namely A_V (or A_H), DA = (A_H/A_V), $\phi_{dp}$, and $f_{\rm prop}$. In each case it is the latter three quantities that affect the polarization state, as in expressions (10) through (12). The manner in which the different quantities change the polarization state on the Poincaré sphere is shown in Figure 5. The changes are best described in terms of the spherical angles $2\alpha$ and $\phi$. $2\alpha$ is the angle that the polarization state makes with the +Q or H axis and represents the polar angle of a spherical coordinate system for which ${\rm Q}$ or H is the z-axis. Positive $Z_{\rm DR}$ values cause the polarization state to move toward the H polarization point by enhancing the H component of the signal relative to the V component. Positive $Z_{\rm DR}$ therefore decreases $\alpha$. Differential attenuation (DA) by horizontally oriented particles does the opposite and causes the polarization state to move toward the Vpolarization point, increasing $\alpha$.

  

Figure 4: A two-sphere representation of the degree of polarization. The outer sphere represents the total power (polarized and unpolarized) and is of radius I = I_p + I_u. The inner sphere is the Poincaré sphere and has radius I_p = p I. The ratio of the radii is the degree of polarization p.

  

Figure 5: The polarization changes produced by horizontally aligned particles, and the ($\alpha$,$\phi$) spherical coordinate system in which the changes are best described.

The differential phase effects of horizontally aligned particles ($\phi_{dp}$and $\delta_\ell$) change the relative phase of the H and V components but not their amplitudes. Q therefore remains constant and the polarization state changes along a circle parallel to the U-V plane. This is called the $\phi$ direction; $\phi$ is the azimuthal angle of the spherical coordinate system. It is also the same as the phase of the cross-covariance W(Equation 1). $\phi_{dp}$ due to horizontally aligned particles causes $\phi$ to decrease (Equation 11). A positive $\delta_\ell$would cause $\phi$ to increase, but $\delta_\ell$ for horizontally oriented particles is negative and therefore changes the polarization state in the same direction as $\phi_{dp}$.

Finally, changes in the unpolarized component of the radar signal cause the degree of polarization to change, and thereby changes the radius of the Poincaré sphere. Aligned particles produce an unpolarized component when the particles have a variety of shapes. The unpolarized component can be generated during the backscattering itself or during propagation by means of forward scattering, and directly affects the correlation coefficient $\rho _{HV}$ (Equation 12). $\rho _{HV}$ and pare therefore related; the relationship is presented in the appendix. When the H and V powers are nearly equal, $\rho_{HV}\simeq p$; exact equality occurs when the powers are equal. The condition for this being true is that the polarization state remains in the vicinity of the ${{\rm U}}$-${{\rm V}}$plane, which will generally be the case if the radar transmits nearly equal H and V powers and is not depolarized by an unusual amount. (It also holds both for simultaneous or alternating H and V transmissions.)

In the vicinity of the ${{\rm U}}$-${{\rm V}}$ plane, therefore, shape variability changes the polarization state in the radial direction. The degree of polarization is reduced by the same amount as $\rho _{HV}$, namely by the fractional shape correlation coefficient f. The different types of changes are therefore orthogonal or nearly orthogonal to each other in the Poincaré sphere space.

Independent of the polarization state, the polarization changes due to horizontally aligned particles are azimuthally symmetric about the ${\rm Q}$ or H axis of the Poincaré sphere. In other words, they do not depend on the particular value of $\phi$. Thus, circular, slant linear, or elliptical polarizations may be transmitted with the same effect, as long as they contain the same relative amounts of H and V power. H and Vsignals lie on the axis of symmetry and are therefore do not have their polarization state changed by horizontally aligned particles. These are termed the the characteristic polarizations of the particles (Oguchi, 1983).

The above describes the polarization changes qualitatively. The different effects have been seen in the observations of the previous section. A quantitative description of the changes is presented by Scott (1999) and by Krehbiel and Scott (1999). The changes are determined by transforming from the rationalized covariance variables (W_V, W_H/W_V, $\rho _{HV}$, $\phi_{{HV}}$) of Equations (10) through (13) to the spherical Stokes coordinates (I, $\alpha$, $\phi$, p).