Theoretical Formulations.

A dual-polarization radar measures the complex amplitudes of the backscattered electric field in two orthogonal polarizations and estimates the covariances

 

$$\langle \hat E_1 \hat E_1^\ast \rangle$$ W₁ =

$$\langle \hat E_2 \hat E_2^\ast \rangle$$ W₂ = (1)

$$\langle \hat E_1 \hat E_2^\ast \rangle = \vert W\vert e^{j\phi} \ .$$ W =

Here, E₁ and E₂ denote the orthogonal components of the electric field vector and $\phi$ is the phase difference between the components. The four quantities, W₁, W₂, |W|, and $\phi = \angle W$ characterize the polarization state of the radar signal. An alternative way of representing the covariances is in terms of the polarization ratio W₁/W₂ and the normalized cross-covariance $W/\sqrt{W_1W_2}$. The magnitude of the latter is the correlation coefficient of the orthogonal signals,

$$\rho = \frac{\vert W\vert}{\sqrt{W_1W_2}}\; .$$ (2)

The three primary choices of polarization basis are a) horizontal and vertical linear polarizations (H,V), b) left- and right-hand circular (L,R), and c) $\pm 45^\circ$ or slant linear polarization (+,-). The power differences in each of these bases are the Stokes parameters Q, U, and V, as discussed later. When the receivers operate in an H-V polarization basis, W₁ = W_H, W₂ = W_V, and $\rho = \rho_{HV}$. The polarization state is therefore be characterized by the quantities W_V (or W_H), W_H/W_V, $\rho _{HV}$, and $\phi_{{HV}}$. We refer to these as the rationalized covariances.

The reason that an H-V polarization basis is useful in meteorological polarimetry is that horizontally aligned particles, most notably liquid drops, transform the rationalized covariances in a simple way. In particular, backscatter from horizontally aligned particles changes the covariances from the values incident upon a scattering volume to

 

$$W_V\vert^{\rm s} Z_V\cdot W_V\vert^{\rm i}$$ =

$$\left . \frac{W_H}{W_V} \right \vert^{\rm s} ZDR \cdot \left . \frac{W_H}{W_V} \right \vert^{\rm i}$$ = (3)

$$\rho_{HV}\vert^{\rm s} f\cdot \rho_{HV}\vert^{\rm i}$$ =

$$\phi_{{HV}}\vert^{\rm s} \delta_\ell+ \phi_{{HV}}\vert^{\rm i} \, .$$ =

Here, the superscripts i and s denote the incident and scattered values, respectively. The backscattering is characterized by the reflectivity values Z_H or Z_V,

 

$$N\langle{\vert S_{hh}\vert^2}\rangle$$ Z_H =

$$N\langle{\vert S_{vv}\vert^2}\rangle\; ,$$ Z_V = (4)

the differential reflectivity

 

$$ZDR = \frac{Z_H}{Z_V}\, ,$$ (5)

the differential phase upon backscatter,

 

$$\delta_\ell= \angle\langle{S_{hh}S_{vv}^*}\rangle\,$$ (6)

and the parameter

 

$$f= \frac{\vert\langle{S_{hh}S_{vv}^*}\rangle\vert}{\sqrt{\langle{\vert S_{hh}\vert^2}\rangle\langle{\vert S_{vv}\vert^2}\rangle} } \; .$$ (7)

The latter quantity measures the extent to which S_hh and S_vv are correlated with each other; it is unity when all the particles have the same relative shape and is less than unity when the particles have a variety of shapes (Balakrishnan and Zrnic, 1990a). It is sometimes referred to simply as $\rho _{HV}$ but this implicitly assumes that $\rho_{HV}\vert^{\rm i}$ is unity. We term this variable the shape correlation coefficient and denote it by fto identify it as a parameter of the scatterers, as distinguished from the radar measurable $\rho _{HV}$. f has been calculated to have a value of 0.99 in rain having equilibrium drop shapes (Sachidananda and Zrnic, 1985).

In propagating from the radar to the scattering volume and back the signal undergoes additional depolarization due to the effects of the propagation medium. If the medium also consists of horizontally aligned particles, several effects occur that affect the polarization state. Differential attenuation causes the polarization ratio W_H/W_V to be reduced by attenuating the H component relative to the V component. This causes W_H/W_V incident upon the scatterers to be different from the transmitted value according to

$$\left . \frac{W_H}{W_V}\right \vert^{\rm i}= \frac{1}{DA} \cdot \left . \frac{W_H}{W_V}\right \vert^{\rm t}\, .$$ (8)

DA is the differential attenuation, defined here as $DA= {A_V}/{A_H}\ge 1$, where A_V and A_H are the attenuation factors for vertical and horizontal signals, respectively. An equal amount of differential attenuation occurs in propagating back to the radar,

$$\left . \frac{W_H}{W_V}\right \vert^{\rm r}= \frac{1}{DA} \cdot \left . \frac{W_H}{W_V}\right \vert^{\rm s} \, .$$ (9)

In these expressions, the superscripts ${\rm t}$ and ${\rm r}$ denote the transmitted and received quantities, respectively. The net effect is that

$$\left . \frac{W_H}{W_V}\right \vert^{\rm r}= \frac{ZDR}{(DA)^2} \cdot \left . \frac{W_H}{W_V}\right \vert^{\rm t}\, ,$$ (10)

Similarly, forward scattering from the aligned particles retards the phase of the horizontal component relative to the vertical, thereby reducing $\phi_{{HV}}$. Thus,

$$\phi_{{HV}}\vert^{\rm r}= -2\phi_{dp}+ \delta_\ell+ \phi_{{HV}}\vert^{\rm t}\, ,$$ (11)

where $\phi_{dp}$ is the one-way propagation differential phase shift. Finally, the forward scattering during propagation introduces an unpolarized component when the particles have a variety of shapes, which causes the HVcorrelation coefficient to be reduced. By analogy with the above,

$$\rho_{HV}\vert^{\rm r}= f_{\rm prop}^{\; 2}\cdot f\cdot \rho_{HV}\vert^{\rm t}\, ,$$ (12)

where $f_{\rm prop}$ is the one-way effect of shape variability on the propagation. The above set of equations is completed by adding the corresponding expression for one of the reflectivity values,

$$W_V\vert^{\rm r}= {A_V}^2 Z_V\cdot W_V\vert^{\rm t}$$ (13)

or

$$W_H\vert^{\rm r}= {A_H}^2 Z_H\cdot W_H\vert^{\rm t}\; .$$ (14)

The propagation effects are cumulative with range and can significantly affect or even dominate the backscatter terms. In addition, it is generally not possible to distinguish between the backscattering effect and its corresponding propagation term (e.g., Torlaschi and Holt, 1993). Thus, $Z_{\rm DR}$ values are biased by the cumulative differential attenuation (DA)², and $\phi_{dp}$ values are affected by any differential phase upon backscatter, $\delta_\ell$. The presence of $\delta_\ell$ can be identified from non-monotonic changes in $\phi_{{HV}}$ with range (e.g., Bringi et al., 1990; Tan et al., 1991; Hubbert et al., 1993).