Handout (2.5 MB)

_{1}and W

_{2}of the return signals in two orthogonal polarizations and the magnitude and phase of the cross-covariance W between the two signals, from which the Stokes parameters are readily determined.

For meteorological radars, the return from a given set of scatterers in range has an average power that is the sum (or superposition) of the powers from each individual scatterer. This results from the scatterers being randomly distributed in range and constantly rearranging, so that the scattering is uncorrelated from one particle to the next. The important implication for dual-polarization measurements is that the various polarization effects, such as differential reflectivity, differential phase, and correlation coefficient (as well as reflectivity) are also additive, because each is the manifestation of power measurements. The overall polarization state is therefore the superposition of the polarization effects of different types or classes of particles.

Another effect of the random nature of the scattering is that the reflected radar signal has both a polarized and an unpolarized component. The unpolarized component results from any variation or randomness in the scatterers, such as their size, shape, and/or orientation, and is important in remotely sensing the presence of randomly shaped or oriented particles such as hail. It is important to note that the instantaneous return from a given arrangement of scatterers is completely polarized, in that the signals each consist of a sinusoid at the radar frequency f_{0} having some amplitude A and phase φ. If the particles were to be frozen in place relative to each other, A and φ would not change with time. The polarization state would be incompletely determined and would have no unpolarized component. The fact that the scatterers rearrange from one transmitted pulse to the next causes A and φ to vary with time in a fluctuating manner, thereby allowing the average power to be determined and also revealing the unpolarized component.

For interpreting dual-polarization observations, it is useful conceptually to categorize the particles into several basic types based on their polarization effects: a) spherical particles which do not depolarize, b) oriented or aligned particles which have differential reflectivity, differential phase, and correlation effects, and c) randomly shaped and/or oriented particles that primarily introduce an unpolarized component that affects the correlation. The different effects combine additively to give overall polarization 'trajectories' along radial beams through a storm.

By categorizing the particles according to their polarization effects, each type can be characterized in the Poincaré coordinate system that is natural to its class, and then added together using transformations between the different coordinate systems. For horizontally oriented particles such as liquid drops, the polarization changes are rotationally symmetric about the Q or H,V axis of the Poincaré sphere. They are best described in an H,V polarization basis and in terms of the rationalized covariances W_{v}, W_{h}/W_{v}, ρ_{hv} = |W|/(W_{h}W_{v})^{1/2}, and φ = arg(W). This well-known set of four parameters gives the backscatter quantities Z_{v} (or Z_{h}), ZDR, ρ_{hv}, and phase shift δ of the precipitation, as well as the associated propagation effects of attenuation, differential attenuation, and differential propagation phase.

The polarization effects of randomly oriented or shaped particles, such as hail, are symmetric about the vertical, circular polarization axis of the Poincaré sphere. The effects are best described in an LHC, RHC basis, and primarily alter the degree of polarization p = I_{polarized}/I_{total} of the polarization state. The resulting polarization changes can be used to evaluate the sphericity parameter g = 4Re{S_{xx}S_{yy}*}/|S_{xx}+S_{yy}|^{2} of the scatterers (Scott et. al., 2001).

For particles that are non-horizontally oriented, such as electrically aligned ice crystals, the polarization changes are symmetric about a rotated Q' axis. For particles oriented at +/-45 degrees, the rotation angle is 90 degrees, and the differential propagation phase changes characteristic of electrical alignment show up as +/-ZDR values for processing that assumes horizontal orientation. This highlights the fact that processing dual polarization data in one basis can give incorrect results when the depolarization is in a different basis. This is not the case for the depolarization caused by randomly oriented or shaped particles, as the degree of polarization is independent of the basis in which it is calculated.

In summary, the Poincaré formulations have the advantage that the different polarization effects can be conceptualized and visualized geometrically, which substantially aids in their understanding and interpretation. The results readily show the advantages and disadvantages of transmitting different polarization types, and provide the analytical framework for correctly interpreting incremental polarization changes from any given polarization state. Also, because the observed polarization changes are the linear superposition of those due to different classes of particles, in principle it should be possible to separate out the contributions of different particle types - for example in the mixed phase regions of storms. Finally, the geometrical formulations simplify understanding how polarization diverse measurements can be used to distinguish between propagation and backscatter effects.

Reference:

Scott, R.D., P.R. Krehbiel, and W. Rison, The use of simultaneous horizontal and vertical transmissions for dual-polarization radar meteorological observations, J. Atmos. and Oceanic Tech., 18, 629-648, 2001.