Relation of g and f.

The depolarization produced by randomly oriented scatterers can be shown to depend only on the sphericity parameter g (Scott, 1999; Krehbiel and Scott, 1999), where

$$g = \frac{4{\rm Re}\{\langle S_{xx}S_{yy}^*\rangle\}}{\langle \vert S_{xx}+ S_{yy}\vert^2 \rangle} \, .$$

g can be related to the parameters that the scatterers would have if they were all aligned. This is done by noting that the quantity ${\rm Re}\{\langle S_{xx}S_{yy}^*\rangle\}$in normalized form is equal to

 

$$\frac{{\rm Re}\{\langle S_{xx}S_{yy}^*\rangle\}}{\sqrt{\langle{\v... ...le{\vert S_{yy}\vert^2}\rangle}} = {\rm Re}\{ \hat f\} = (f\cos\delta_\ell)\, .$$ (33)

Here, $\hat f= fe^{j\delta_\ell}$ is the complex version of the shape correlation function (7), of which f is the magnitude. The fact that $\delta_\ell$ is the phase of $\hat f$ follows from (6). From this it can be shown that

$$\frac{\langle \vert S_{xx}+ S_{yy}\vert^2 \rangle}{{\rm Re}\{\lan... ...rac{1}{f\cos\delta_\ell}\left [ \sqrt{ZDR} + \frac{1}{\sqrt{ZDR}} \right ] \, ,$$

where ZDR is the value that the particles would have if they were aligned. Thus,

 

$$g= \frac{4}{2 + \frac{1}{f\cos\delta_\ell}\left [ \sqrt{ZDR} + \frac{1}{\sqrt{ZDR}} \right ]} \, .$$ (34)

Therefore, g is a function of the ZDR value that the randomly oriented particles would have if they were aligned, and of $(f\cos\delta_\ell)$. We recall that f has a value between unity and zero depending on the variety of particle shapes; therefore $-1 \le (f\cos\delta_\ell) \le 1$.

Figure 11 shows how g varies with ZDR and $f\cos\delta_\ell$. The top graph shows that g is a maximum when the particles are spherical (ZDR= 0 dB) and decreases as the particles depart from sphericity; hence the term sphericity parameter for g. The bottom graph shows how g is related to to $(f\cos\delta_\ell)$ for different values of ZDR. For $\cos\delta_\ell$ positive, 0 < g< 1. g is unity only in the limiting case when ZDR and $f\cos\delta_\ell$are unity. The dashed ${45^\circ}$ line in the upper right quadrant of the graph indicates when the two quantities are equal; g is greater or less than $(f\cos\delta_\ell)$ depending on ZDR and the particular value of $(f\cos\delta_\ell)$.

  

Figure 11: Variation of the sphericity parameter g for randomly oriented particles, versus the ZDR value that the particles would have if they were aligned (top), and versus $f\cos\delta_\ell$ (bottom), for $f\cos\delta_\ell=$ between 0.5 to 1.0.

An interesting situation occurs if $\cos\delta_\ell$ were to ever reverse sign. The terms in the denominator of (34) would then tend to cancel and g would become large and negative. It turns out that this would cause the polarization state to suddenly switch from the top to the bottom half of the Poincaré sphere (or vice versa) and be near the circular polarization pole (Krehbiel and Scott, 1999). Such a sign reversal would occur if $\vert\delta_\ell \vert > \pi /2$. Since $\delta_\ell$ is the phase difference of backscattered Hand V signals if the aligned particles were aligned, such a situation would arise only at higher order resonances in the Mie scattering regime or when the particles are highly elongated. Such an effect, if it occurs in nature, would provide a strong signature of randomly oriented particles.