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

Here,

(2) |

The three primary choices of polarization basis are a) horizontal and vertical linear polarizations (

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

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

the differential reflectivity

the differential phase upon backscatter,

and the parameter

The latter quantity measures the extent to which

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

In these expressions, the superscripts and denote the transmitted and received quantities, respectively. The net effect is that

Similarly, forward scattering from the aligned particles retards the phase of the horizontal component relative to the vertical, thereby reducing . Thus,

where 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

where 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,

or

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,
values are biased by the cumulative differential attenuation (*DA*)^{2},
and
values are affected by any differential phase upon backscatter,
.
The presence of
can be identified from non-monotonic
changes in
with range (e.g., Bringi et al., 1990; Tan et al., 1991;
Hubbert et al., 1993).