Summary and Discussion.

The results presented in this paper show that linear polarization quantities such as $Z_{\rm DR}$, $\phi_{dp}$, and $\rho _{HV}$ are readily measured by transmitting simultaneous H and V signals and receiving the backscattered returns in parallel H and V channels. The measurements are improved over those obtained when H and V is transmitted on alternate pulses because the correlation between the two polarizations is measured simultaneously rather than from alternate pulses.

The basic advantages of transmitting H and V simultaneously instead of on alternate pulses are as follows:

1.

$\phi_{dp}$ and $\rho _{HV}$ have less uncertainty because they are measured from simultaneous signals and are therefore not contaminated by Doppler effects. The uncertainties of alternating pulse measurements include a possible bias in $\rho _{HV}$ when the Doppler spectrum is not gaussian.

2.

The measurements are speeded up by a factor of two because signal estimates are obtained from each transmitted pulse rather than from pairs of pulses, and by some additional factor because of not having to average out the Doppler effects.

3.

A high-power polarization switch is not needed; it is replaced by a power divider. Parallel receiving channels are required, but modern technology and receiving techniques make this a relatively simple matter with no loss of performance. Good signal-to-noise ratios are maintained in the parallel channels, as in the alternating pulse techique. (The s/n values are 3 dB lower than those of an alternating pulse system due to the power divider.)

4.

The relative phases of the H and V components can be adjusted to transmit circular polarization, which has the advantage of being a factor of two more sensitive to the presence of randomly oriented particles than linear polarization transmissions. In addition, alignment directions and overall depolarization rates can be determined if desired.

5.

Polarization diversity measurements can be obtained using an electronic phase shifter to alternate the relative phase of the H and Vcomponents. The time saved by making the measurements simultaneously could be used to alternate between circular and linear polarization, which would aid in identifying the scattering processes and particle types.

Many of the above advantages have been recognized by previous investigators, as noted in the introduction.

The insights gained by looking at the problem in a geometric way from the standpoint of the Poincaré sphere provide a valuable framework for understanding dual-polarization observations, and for determining the best ways to make polarization measurements. Circularly polarized transmissions optimally sense all types of hydrometeors, including spherical, horizontally aligned, randomly oriented, and non-horizontally aligned particles. ${45^\circ}$slant linear transmissions have the same response to horizontally aligned particles but are less affected by randomly oriented particles. Hor V transmissions by themselves are not depolarized by horizontally aligned particles, leaving only the unpolarizing effect of randomly oriented particles, and any effects due to canting of horizontally oriented particles. Elliptical polarizations are affected in intermediate ways depending on the orientation angle of the ellipse.

The geometric approach also shows how traditional linear and circular polarization measurements are related, and shows how radars which alternate between H and V polarizations effectively simulate the transmission of ${45^\circ}$ linear polarization. The geometric approach is also readily extended to provide an interpretation of linear depolarization measurements.

The linear depolarization ratio (LDR) is determined by transmitting, say, horizontal polarization and measuring the backscattered return in parallel H and V channels. It is defined as the ratio of the cross-polar and co-polar powers, W_V/W_H, when H is transmitted. The cross-polar power is produced in essentially two ways: by scattering from randomly oriented non-spherical particles and from horizontal particles that are canted at an angle from vertical. The canting can occur either randomly about zero or relative to some mean value. The Poincaré sphere approach provides a straightforward method of understanding and quantifying the different contributions. In addition, the measurements need not be limited to incoherent quantities, but can (and should) include the co-polar cross-polar correlation (i.e., W, or $\rho _{HV}$ and $\phi_{{HV}}$). The primary reason to measure the coherent quantities is that the co-polar return serves as a reference signal for retrieving the cross-polar return from below the receiver noise level. As shown by Krehbiel et al. (1996), the quantity W/W₂ represents a coherent version of the depolarization ratio and has the significant advantage of being totally independent of the cross-polar signal-to-noise ratio. It is well determined down to the noise level of the co-polar signal. By contrast, incoherent LDR values are well-determined only down to the noise level of the cross-polar return. The use of W/W₂ in an H-V basis can be exploited to improve LDRmeasurements, and is the subject of continued study.

The time saved by the simultaneous approach makes it practical to use additional polarizations for more complete observations, if so desired. The Poincaré sphere results help in understanding the advantages of transmitting different polarizations. Optimal choices of two transmitted polarizations would be circular and ${45^\circ}$ slant linear, or circular and H (or V). The first set of polarizations would be obtained by alternately increasing the relative phase of the H and V signals between 0 and $90^\circ$(see Figure 1). The second set (LHC and H, say) would be obtained in the same way except that the orthomode transducer of the antenna feed would be rotated by ${45^\circ}$ so that the two transmitted components are $\pm 45^\circ$ slant linear rather than H and V. Equal powers would then be transmitted in the slant linear basis and the signals would be received in the same basis. The measurements would be readily transformed into an H-Vbasis (Krehbiel and Scott, 1999). Such transformations are also useful in `fine tuning' the alignment of the antenna feed and in calibrating dual-polarization systems (Scott, 1999).

The LHC-H polarization pair may prove to be the most useful because the contribution of horizontally aligned particles is nulled out when His transmitted, leaving only the contribution of randomly oriented and other particles. (The nulling is not perfect because of canting of the horizontal particles.) The CSU-CHILL radar alternates between ${45^\circ}$ slant linear and H-V polarizations (Brunkow et al., 1997) but could just as easily transmit a circular-linear combination. Orthogonal polarization transmissions are not necessarily useful (one exception being that LHC and RHCmeasurements can be used to distinguish between backscatter and propagation effects by aligned particles whose alignment direction is not known).

The study by Torlaschi and Holt (1998) concluded that of three possible transmitted polarizations - circular, slant ${45^\circ}$ linear, or horizontal/vertical - circular polarization is optimal for meteorological observations. The returns were considered to be received in the same basis as the transmitted signals in their study. The results of this study show that, if only a single polarization is transmitted, the optimal choice would be circular, received in an H-V basis.

Dual-polarization measurements are of interest because they provide independent information that can be used to discriminate or identify hydrometeor types. The identification is often done in an ad-hoc manner by dividing up the polarization variable space, or empirically by assessing the contributions of rain and hail to the polarization variables (e.g., Vivekanandan et al., 1993, 1999). A fully objective approach is possible by dividing the scatterers into different classes based on the way in which they depolarize the radar signal. As described in this paper, the polarization responses group the particles into spherical, horizontally aligned, randomly oriented, and non-horizontally aligned classes. Spherical particles are characterized by a single quantity, their reflectivity factor Z. Horizontally aligned particles are characterized by four quantities (Z_V or Z_H, $Z_{\rm DR}$, $\delta_\ell$ the shape correlation factor f), and possibly by a mean and rms canting angle. Randomly oriented particles are characterized by two quantities, $Z_{\rm avg}$ and the sphericity parameter g. Non-horizontally aligned particles are characterized by some or all of the aligned particle parameters plus the alignment direction $\tau$. Different subsets of the various quantities would be likely at different locations in a storm. The pertinent quantities can in principle be determined from independent polarization measurements.

An analytical basis for determining the scattering parameters comes from the fact that the covariances are additive for different classes of particles (Krehbiel and Scott, 1999). Each covariance variable is the sum of the covariances for each type of particle. For example,

$$\langle \hat E_H \hat E_H^\ast \rangle$$ W_H = (19)

$$W_{H(\rm spherical)} + W_{H(\rm aligned)} + W_{H(\rm random)} \; .$$ =

Similar expressions apply for W_V and W. The covariances themselves are functions of the scattering parameters, as for example in Equations 3. Determination of the scattering parameters then becomes a straightforward problem of specifying the dependence of the covariances upon the scattering parameters for each class of scatterers and then (somewhat less straightforwardly) obtaining sufficient independent measurements to solve for the parameter values. This provides a fully objective approach to the measurement problem.

A difficulty in the above approach is that, for a given class of scatterer, each backscatter parameter has an associated propagation effect, which is cumulative with range. For example, differential propagation phase $\phi_{dp}$ is associated with the differential backscatter phase $\delta_\ell$, and differential propagation attenuation DA is associated with differential reflectivity $Z_{\rm DR}$ (Less well recognized have been the incoherent depolarization factors f for aligned particles and g for randomly oriented particles, and their associated propagation terms.) The backscatter and propagation quantities are inextricably linked to each other in that changes from one range gate to the next can be caused by increases and/or decreases in either quantity. A number of schemes have been devised for extracting this information based on expected physical correlations between the different parameters (e.g., Torlaschi and Holt, 1993), but the basic ambiguities remain and are difficult to overcome. An objective method for extracting the scattering parameters would have to take this into account. This constitutes a substantial complication but is one that needs to be addressed in any particle identification scheme.