Eigenvalue Signal Processing for Phased-Array Weather Radar Polarimetry: Removing the Bias Induced by Coherent Antenna Cross-Channel Coupling

Abstract— We present a novel digital signal processing procedure, named Eigenvalue Signal Processing (ESP), patented by the first author with Brookhaven Science Associates in 2012. The method enables the removal of antenna cross-channel coupling in weather radar measurements at LDR mode, ATSR mode and STSR orthogonal mode. ESP is however not applicable at STSR hybrid mode.

In this work we focus on the LDR mode and consider reflectivity at horizontal transmit (Z_H), linear depolarization ratio at horizontal transmit (LDR_H) and degree of polarization at horizontal transmit (DOP_H).

The eigenvalue signal processing method is substantiated by an experiment carried out in November 2012 with a parabolic reflector C-band weather radar located at the Selex Systems Integration facilities in Neuss, Germany. The experiment consists of the comparison of weather radar measurements taken 1.5 minutes apart in two hardware configurations, namely with cross-coupling on (cc_on) and cross-coupling off (cc_off).

It is experimentally demonstrated that eigenvalue-derived variables are invariant with respect to antenna cross-channel coupling. This property had to be expected, since the eigenvalues of the Coherency matrix are SU(2) invariant.

Keywords: Coherency matrix, Covariance matrix, polarimetric phased array weather radar, eigenvalues.

<>                                                                                                             I.                               Introduction

The present paper proposes eigenvalue signal processing as a solution for the problem of antenna cross-polarization isolation. It is potentially suitable for implementation in polarimetric phased array antennas, but also in conventional parabolic reflectors, whenever the antenna cross-polarization isolation is sub-optimal. LDR mode corresponds to horizontal polarization transmit, and simultaneous reception of H and V; ATSR mode corresponds to H transmit and simultaneous H and V receive, followed by V transmit and simultaneous H and V receive. STSR orthogonal corresponds to simultaneous transmission of phase-coded H and V pulses, in order to retrieve the full scattering matrix in one PRT. STSR hybrid mode corresponds to the conventional simultaneous transmission of H and V as implemented, for example, in the NEXRAD network. <>                                                                                         II.                              Eigenvalue-Derived variables

All eigenvalue-derived variables are very robust with respect to antenna cross-channel coupling.  An analytical proof of this fact is given in [1] for the degree of polarization at horizontal transmit, which is just one among other eigenvalue-derived variables. In this paper, we propose eigenvalue-derived variables that are exact replacements for Reflectivity (Z_H, Z_V), Differential Reflectivity (Z_DR) and Linear Depolarization Ratio (LDR_H, LDR_V).

Standard radar meteorological variables can be replaced by new eigenvalue-derived variables, indicated with a subscript ESP (and protected by Brookhaven patent), that exactly correspond to their unbiased counterparts.

ZH→ZH_ESP≡λ1H

ZV→ZV_ESP≡λ1V

ZDR→ZDR_ESP≡λ1Hλ1V

LDRH→LDRH_ESP≡λ2Hλ1H

LDRV→LDRV_ESP≡λ2Vλ1V

Here, the subscript H indicates the eigenvalues of the Coherency matrix at horizontal transmit, and the subscript V indicates the eigenvalues of the Coherency matrix at V transmit. The subscript 1 indicates the larger eigenvalue, the subscript 2 indicates the smaller eigenvalue. Other eigenvalue-derived variables, not of common use in weather radar polarimetry, are the trace and the degree of polarization.

DOPH=λ1H-λ2Hλ1H+λ2H

DOPV=λ1V-λ2Vλ1V+λ2V

TraceH=λ1H+λ2H

TraceV=λ1V+λ2V

Obviously, relationships exist among the above-mentioned variables:

DOPH=1-LDRH_ESP1+LDRH_ESP

DOPV=1-LDRV_ESP1+LDRV_ESP

TraceH=ZH_ESP1+LDRH_ESP=ZH1+LDRH

TraceV=ZV_ESP1+LDRV_ESP=ZV1+LDRV

<>                                                                          III.                             Eigenvalue Signal Processing Experiment

Eigenvalue Signal Processing was tested for Z_{H_ESP}, LDR_{H_ESP} and DOP_H in an experiment conducted on November 10^th 2012 at Selex Gematronik facilities in Neuss, Nordrhein-Westfalen, Germany at around 16:20 local time, when ground temperature was +11ûC. The parabolic reflector C-band radar acquired a PPI at 1.5û elevation in a weather event consisting of light stratiform rain, with a melting band visible as a low LDR ring around the radar at about 50 km distance.

The radar was operated at LDR mode, in two different configurations indicated with cc_on (red curve in the plots) and cc_off (blue curve in the plots). The cc_on acquisition was taken between 16:18:21.122 and 16:19:40.826 CET (Central European Time), whereas the cc_off acquisition was taken between 16:19:40.826 and 16:21:00.643 CET. The two acquisitions are spaced in time by about 1.5 minutes, and it can reasonably be assumed that the illuminated scatterers are the same.

In the cc_off acquisition the radar was operated in its standard configuration, whereas in the cc_on acquisition, the detrimental effects of a suboptimal antenna were simulated by disconnecting the V transmit waveguide and by injecting into the Tx port of the V circulator a signal sample extracted from the H transmit channel via a 20 dB coupler.

In the following we present plots corresponding to radials at 352û azimuth, where light rain is present from 10 to 40 km and where wet aggregates (melting band) are present from 45 to 60 km.

A          Reflectivity at horizontal transmit Z_H, in dBZ.

B          Reflectivity at vertical transmit Z_V, in dBZ (cross-polar reflectivity, i.e. received in the V channel).

C          Cross-polar correlation coefficient, r_xh.

D          Linear Depolarization Ratio LDR, in dB.

E          Linear Depolarization Ratio ESP corrected. Note how the coupled configuration yields results identical to the uncoupled configuration, that is, coherent cross-polar power is automatically removed.

F          Degree of Polarization at horizontal transmit. Since the Degree of Polarization can be expressed as a function of the eigenvalues of the Coherency matrix at horizontal transmit, it automatically enjoys the properties of SU(2) invariance. Specifically, it is not affected by antenna cross-channel coupling.

Fig.2 In the panels from A to F are polarimetric variables at LDR mode; red is the coupled configuration, blue is the standard configuration. It can be observed how the coupled configuration (in red) affects variables that are NOT derived from the eigenvalues, like copolar reflectivity Z_H (indeed, in this case the bias is very small and not directly visible from these plots), cross-polar reflectivity Z_V, linear depolarization ratio (LDR) and cross-polar correlation coefficient (r_xh). The variables that are eigenvalue-derived, i.e. the degree of polarization at horizontal transmit DOP_H and LDR_{H_ESP}, are unbiased by antenna cross-channel coupling.