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  • Waterman, P., 1969: Scattering by dielectric particles. Alta Freq., (Speciale), 348–352.

  • View in gallery

    A block diagram showing the elements of the radar model. TX: transmitter block, PD: power division network, RH vertical receiver chain, and RV horizontal receiver chain.

  • View in gallery

    The polarization ellipse. Propagation is into the paper.

  • View in gallery

    S-Pol pseudo–antenna patterns obtained from passive solar scan measurements from data gathered on 19 May 2008 during TiMREX. (top) H and V antenna patterns and (bottom) magnitude and phase of the H to V correlation. Power is uncalibrated.

  • View in gallery

    SHV Zdr as a function of principal plane ϕdp with unbalanced transmit power as a parameter. For SHV mode ideally |Eht| = |Eυt|, and this nominal curve is shown (dashed line). The errors are independent of the phase difference between |Eht| and |Eυt|.

  • View in gallery

    As in Fig. 4, but with the mean canting angle of the propagation medium as a parameter. The transmission errors are zero, that is |Eht| = |Eυt|. The two curves between ±1° are for mean canting angles of ±0.5°.

  • View in gallery

    As in Fig. 4, but with the mean canting angle of the propagation medium as a parameter. Here, |Eht| = |Eυt| but there is a 90° phase difference; that is circular polarization is transmitted.

  • View in gallery

    As in Fig. 4, but with the mean canting angle of the propagation medium as a parameter. Here, |Eht| = |Eυt|, but there is a 90° phase difference; that is circular polarization is transmitted. (a) C band and (b) X band. There is very little difference between these two error plots and the analogous one for S band in Fig. 6.

  • View in gallery

    SHV mode Zdr for 1° antenna polarization errors. (top) ±1° tilt errors and (bottom) ±1° ellipticity errors.

  • View in gallery

    SHV mode Zdr bias for mixed tilt and ellipticity antenna error angles, which are given in Table 1. The antenna errors are orthogonal and the H and V transmitting signals are equal, that is Eh = Eυ. These antenna errors correspond to a system LDR limit of −31 dB.

  • View in gallery

    SHV mode Zdr bias for mixed tilt and ellipticity antenna error angles which are given in Table 1; however, the transmission polarization state is circular. The antenna errors are orthogonal. These antenna errors correspond to a system LDR limit of −31 dB.

  • View in gallery

    Normalized SHV mode Kdp as a function of principal plane ϕdp for the antenna error angles given in Table 3.

  • View in gallery

    SHV mode Zdr bias as a function of principal plane ϕdp with LDR system limit as a parameter. The antenna polarization errors are assumed to be orthogonal ellipticity angles. The sign of the H ellipticity angle is given in each quadrant. (a) The transmitting polarization is 45° linear, that is Eht = Eυt. The curves all mimic a sine wave shape. (b) The transmitting polarization is circular. The curves are symmetric about the vertical line through 180°. The corresponding antenna errors are given in Table 2.

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Modeling, Error Analysis, and Evaluation of Dual-Polarization Variables Obtained from Simultaneous Horizontal and Vertical Polarization Transmit Radar. Part I: Modeling and Antenna Errors

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Abstract

In this two-part paper the biases of polarimetric variables from simultaneous horizontally and vertically transmitted (SHV) data are investigated. Here, in Part I, a radar-scattering model is developed and antenna polarization errors are investigated and estimated. In , experimental data from the National Center for Atmospheric Research S-band dual-polarization Doppler radar (S-Pol) and the National Severe Storms Laboratory polarimetric Weather Surveillance Radar-1988 Doppler (WSR-88D) radar, KOUN, are used to illustrate biases in differential reflectivity (Zdr). The biases in the SHV polarimetric variables are caused by cross coupling of the horizontally (H) and vertically (V) polarized signals. The cross coupling is caused by the following two primary sources: 1) the nonzero mean canting angle of the propagation medium and 2) antenna polarization errors. The biases are strong functions of the differential propagation phase (ϕdp) and the phase difference between the H and V transmitted field components. The radar-scattering model developed here allows for the evaluation of biases caused by cross coupling as a function of ϕdp, with the transmission phase difference as a parameter. Also, antenna polarization errors are estimated using solar scan measurements in combination with estimates of the radar system’s linear depolarization ratio (LDR) measurement limit. Plots are given that show expected biases in SHV Zdr for various values of the LDR system’s limit.

Corresponding author address: J. C. Hubbert, 3450 Mitchell Lane, NCAR, Boulder, CO 80307. Email: hubbert@ucar.edu

Abstract

In this two-part paper the biases of polarimetric variables from simultaneous horizontally and vertically transmitted (SHV) data are investigated. Here, in Part I, a radar-scattering model is developed and antenna polarization errors are investigated and estimated. In , experimental data from the National Center for Atmospheric Research S-band dual-polarization Doppler radar (S-Pol) and the National Severe Storms Laboratory polarimetric Weather Surveillance Radar-1988 Doppler (WSR-88D) radar, KOUN, are used to illustrate biases in differential reflectivity (Zdr). The biases in the SHV polarimetric variables are caused by cross coupling of the horizontally (H) and vertically (V) polarized signals. The cross coupling is caused by the following two primary sources: 1) the nonzero mean canting angle of the propagation medium and 2) antenna polarization errors. The biases are strong functions of the differential propagation phase (ϕdp) and the phase difference between the H and V transmitted field components. The radar-scattering model developed here allows for the evaluation of biases caused by cross coupling as a function of ϕdp, with the transmission phase difference as a parameter. Also, antenna polarization errors are estimated using solar scan measurements in combination with estimates of the radar system’s linear depolarization ratio (LDR) measurement limit. Plots are given that show expected biases in SHV Zdr for various values of the LDR system’s limit.

Corresponding author address: J. C. Hubbert, 3450 Mitchell Lane, NCAR, Boulder, CO 80307. Email: hubbert@ucar.edu

1. Introduction

The simultaneous transmission and reception of horizontally (H) and vertically (V) polarized waves (SHV) has become a very popular way to achieve dual-polarization measurements for weather radar. Previously, radars achieved dual-polarization measurements by either employing a fast, high-powered waveguide switch or by using two separate transmitters (Brunkow et al. 2000; Keeler et al. 2000). Both technologies incur significant costs to the operation and maintenance of the radar. The premise of the SHV technique is that as the transmitted wave propagates, no cross coupling occurs between the H and V electric field components. Mathematically, this requires that the propagation matrix be diagonal (Vivekanandan et al. 1991). An advantage of the SHV technique is that the expense of either a fast waveguide switch or two transmitters is avoided. There are other advantages of SHV mode (see Doviak et al. 2000 for details). Disadvantages of SHV mode are that 1) the linear depolarization ratio (LDR) is not measured, and, 2) if there is cross coupling of the H and V waves, there can be measurement biases. Thus, the viability of the SHV dual-polarization technique is based on having 1) a zero mean canting angle of the propagation medium and 2) negligible antenna polarization errors. If either condition is not met, cross coupling occurs between the H and V channels, which can cause measurement biases.

Measurement biases in the SHV mode have been investigated (Sachidananda and Zrnić 1985). Doviak et al. (2000) evaluated cross-coupling errors of SHV mode and concluded that since the mean canting angle of rain is close to zero, the biases were acceptable. More recently, Wang and Chandrasekar (2006) and Wang et al. (2006) investigated the biases in SHV polarization measurements resulting from cross-coupling errors caused by the radar system polarization errors as a function of ϕdp (differential propagation phase). They concluded that system isolation between the H and V channels must be greater than 44 dB in order to ensure that the Zdr bias is within 0.2 dB for worst-case errors.

Ryzhkov and Zrnić (2007) examined the effects of nonzero mean canting angle of the propagation medium on SHV mode measurements. Data gathered in SHV mode with KOUN, the National Severe Storms Laboratory’s (NSSL’s) S-band dual-polarization Doppler radar, displayed Zdr radial bias “stripes” after the radar waves passed through the ice phase of either convective cells or stratiform precipitation. They propose that the nonzero mean canting angle of the propagation medium produces coupling between the H and V polarized waves that causes the anomalous Zdr signatures.

All reflector-type antennas will introduce polarization errors to the desired H and V transmitted polarization states, causing cross coupling between the H and V channels. This will bias polarization measurements of precipitation unless the antenna interchannel isolation is extremely low. This paper investigates and quantifies the expected magnitude and phase of antenna polarization errors via an analysis of solar measurements and a known typical LDR system limit. Importantly, the antenna polarization errors are interpreted in terms of their tilt and ellipticity angles, which has not been done before. Such an analysis provides insight as to the nature of the antenna polarization errors. Specifically, we show that the antenna polarization errors are dominated by the ellipticity angle errors, at least for the National Center for Atmospheric Research (NCAR) dual-polarization Doppler radar (S-Pol). The radar model that is introduced by Hubbert and Bringi (2003) is modified to accommodate arbitrary transmitting polarization states, which also has not been shown before. The transmit differential phase, as defined in this paper, is distinctly different and independent from the phases of the antenna polarization errors. Thus, the model allows for the separate parameterization of both transmit differential phase and antenna-induced polarization errors, and for an analysis of their impact on SHV mode Zdr and ϕdp biases. For example, the effect of cross coupling caused by the propagation medium is demonstrated with the model, and it is shown that small changes in ϕdp with range can cause significant bias in Zdr. It is shown that the Zdr bias is a strong function of differential transmit phase.

The paper is organized as follows. Section 2 describes the radar model of Hubbert and Bringi (2003) and how antenna polarization errors are accounted for. The model is then modified to permit arbitrary transmit polarizations, and model results are given. Section 3 shows how antenna polarization errors can be estimated from experimental data. The summary and conclusions are given in section 4. In Hubbert et al. (2009, hereafter Part II), experimental data from the NCAR S-Pol and from NSSL’s S-band research radar, KOUN, illustrate the theory established herein, in Part I.

2. The model

The radar scattering model is described in Hubbert and Bringi (2003), but is briefly reviewed here for convenience and clarity. The model is then expanded to accommodate arbitrary transmit polarization states (refer to Fig. 1). The particles in the backscatter volume and the coherent propagation medium are independently modeled. The “steady” propagation medium is modeled via a 2 × 2 matrix that includes absolute attenuation (Ah), differential attenuation (Adp), differential propagation phase (ϕdp), and mean canting angle (θ) as parameters. The resolution volume (or backscatter medium) is modeled as an ensemble of precipitation particles with gamma drop size distribution (DSD) and various spatial orientation distributions via the T-matrix method (Vivekanandan et al. 1991; Waterman 1969). Antenna polarization errors are modeled similar to McCormick (1981) and Bringi and Chandrasekar (2001) and are beam integrated so that a pair of complex numbers characterizes the H and V antenna polarization errors. Our modeling of antenna errors is very similar to Wang and Chandrasekar (2006). The transmit polarization state is defined at the reference plane as shown in Fig. 1. The antenna system refers to the microwave path from the reference plane through the antenna dish.

The scattering geometry used is the backscatter alignment (BSA) convention (Bringi and Chandrasekar 2001). Canting angles are measured counterclockwise from the positive horizontal axis in the plane of polarization (i.e., the plane containing the H and V axis perpendicular to the propagation direction). Further details of the radar model are given in appendix A.

a. Modeling antenna polarization errors

The radar antenna and surrounding microwave circuitry introduce microwave cross coupling that gives rise to polarization errors so that pure H or V polarization are not transmitted. Polarization errors have been covered in detail (McCormick 1981; Metcalf and Ussails 1984; Bringi and Chandrasekar 2001; Hubbert and Bringi 2003; Wang and Chandrasekar 2006). Some of the sources of polarization error are nonideal feedhorn, nonideal parabolic reflector, antenna support struts, and edge effects. These polarization errors are distributed across the radar antenna patterns and thus can vary across the beam, especially where the cross-polarized lobes exist (Ussailis and Metcalf 1983; Bringi and Chandrasekar 2001). For distributed precipitation media, the resulting error is an integrated effect and we model these distributed errors with a 2 × 2 polarization error matrix. The assumption is that the scattering medium is homogenous across the antenna pattern.

The polarization errors are included in the model by pre- and postmultiplication of 𝗦, where 𝗦 is the scattering matrix in Eq. (A4), by the error matrix ϒ,
i1520-0426-27-10-1583-e1
where
i1520-0426-27-10-1583-e2
with constraints ih2 + |ξh|2 = iυ2 + |ξυ|2 = 1, where ih and iυ are real. The polarization errors of the H and V channels are represented by the complex numbers ξh and ξυ, respectively. The polarization errors can also be equivalently represented in the following several ways: 1) the polarization ratio χ [see Eq. (A6)], 2) the geometric parameters of tilt angle α and ellipticity angle ε, and 3) the phasor descriptors γ and ζ. For H errors
i1520-0426-27-10-1583-e3
and for V errors
i1520-0426-27-10-1583-e4
because the definition of χ used here is the ratio of the V to H electric field components. Referring to the polarization ellipse of Fig. 2, the polarization error terms ξh and ξυ can also be related to the geometric polarization ellipse parameters α and ε, and the phasor descriptors, where γ = tan−1(|Eυ|/|Eh|) and arg(Eυ/Eh) = ζ = ζυζh. The tilt angle α is measured from the positive horizontal axis to the major axis of the polarization ellipse and ε is defined as tanε = (minor axis)/(major axis).
Mathematically, these variables are related by (Azzam and Bashara 1989)
i1520-0426-27-10-1583-e5
i1520-0426-27-10-1583-e6
and
i1520-0426-27-10-1583-e7
As can be seen from Eqs. (3) through (7), if the ξh(ξυ) is real then ε is zero and if ξh(ξυ) is imaginary then α is zero. If the errors are orthogonal,
i1520-0426-27-10-1583-e8
then ϒ is unitary and (1) represents an orthogonal change of polarization basis. Expressing the polarization errors using the geometric descriptors gives a convenient and intuitive way to analyze polarization errors.
Given the tilt and ellipticity angles of the polarization state (or ellipse), the corresponding phasor parameters are easily found using the relations in Eqs. (7). Then, the antenna error matrix terms ξh and ih are found as
i1520-0426-27-10-1583-e9
i1520-0426-27-10-1583-e10
and finally
i1520-0426-27-10-1583-e11
Similarly, the vertical polarization error ξυ is found from αυ and ευ (remembering that the polarization ratio is defined as χ = iυ/ξυ) as
i1520-0426-27-10-1583-e12
i1520-0426-27-10-1583-e13
and finally
i1520-0426-27-10-1583-e14
It is also useful to express the polarization ratio in terms of its real and imaginary parts as (Tragl 1990)
i1520-0426-27-10-1583-e15

b. Modeling simultaneous H and V transmission

The model thus far was constructed so that the transmit and receiving polarization states are the same according to radar polarimetry theory. The covariance matrix of Eq. (A5) is a convenient form for covariance analysis and for polarization basis transformations; however, it is not a transmission matrix. It does not express a transfer relationship between an arbitrary input polarization and the resultant output covariances, as does the Mueller matrix (Azzam and Bashara 1989). To model the H and V receiving covariances that result from arbitrary transmitting polarizations, a 4 × 4 covariance matrix is formed using the feature vector
i1520-0426-27-10-1583-e16
Taking the outer product of the feature vector yields the 4 × 4 covariance matrix in the H–V basis as
i1520-0426-27-10-1583-e17
where 〈·〉 denotes ensemble (spatial) or temporal averages (which are equivalent because of the assumption of ergodicity). Note that the covariance matrix is Hermitian. It can be shown that the matrix of Eq. (17) may be transformed to the Mueller matrix (Azzam and Bashara 1989), and thus the covariance matrix of Eq. (17) can also be used as a transfer function matrix
i1520-0426-27-10-1583-e18
where 𝗝i and 𝗝o are 4 × 1 input and output coherency matrices. In terms of the desired polarization characteristics of the incident polarization, namely, tilt angle (α) and ellipticity angle (ε), 𝗝i becomes
i1520-0426-27-10-1583-e19
If linear, a slant of 45° transmitting polarization is desired (i.e., SHV mode), and then α = 45° and ε = 0°. SHV variables of interest can then be calculated, for example, as
i1520-0426-27-10-1583-e20
i1520-0426-27-10-1583-e21
i1520-0426-27-10-1583-e22
where the superscript “shv” denotes SHV variables. In this way, the radar model is modified to allow for arbitrary transmitting polarizations (but still receiving H and V) by putting the covariances of Eq. (A5) into the matrix form of Eq. (17). The input vector is then controlled by Eq. (19) via the tilt and ellipticity angles of the desired transmitting polarization state.

The advantages of the presented radar model are that 1) no approximations are made in terms of the relative significance of the various error terms and 2) transmit, propagation, backscatter, and antenna model parameters are all independently set so that the effect of varying each parameter on the polarization variables of Zdr, ϕdp, etc., can be investigated.

3. Estimating antenna polarization errors

The estimation of the complex error terms ξh and ξυ for antennas is difficult, and they are not typically supplied by antenna manufacturers. There are ways, however, to estimate the magnitude of the error terms and to generally qualify their character. Two generally available and measurable quantities are LDR and passive solar scan measurements.

For well-designed radars with parabolic, center-fed antennas, the dominant cross-coupling factor is the antenna (Bringi and Chandrasekar 2001). The radar system lower limit of LDR can be estimated by measurement in drizzle where raindrops are considered circular so that the backscatter (and propagation) medium cause no cross coupling, and thus intrinsic LDR is −inf dB. Theoretically, LDR can be expressed as a function of the polarization errors ξh and ξυ (Bringi and Chandrasekar 2001). It can be shown (see appendix C) that LDR measurements in drizzle are approximately
i1520-0426-27-10-1583-e23

The LDR system limit values for S-band radars with well-designed, center-fed parabolic reflector antennas are typically reported to be in the −30- to −35-dB range (Keeler et al. 2000; Brunkow et al. 2000). To gauge the expected magnitude of the antenna polarization errors, we assume that ξh = ξυ and that ξh and ξυ are either completely real or imaginary. Then, let the system limit LDR = 10 log10(2ξh)2 = −30 dB [using Eq. (23)]. Solving gives |ξh| = |ξυ| = 0.0158. If the LDR limit is −35 dB, then |ξh| = |ξυ| = 0.00889. Equivalently, these errors correspond to tilt and ellipticity angles of the polarization state (polarization ellipse) of the antenna. The angles are 0.91° and 0.51° for LDR limits of −30 and −35 dB, respectively. If the ξh and ξυ are real, the angles are tilt angles, and if the ξh and ξυ are imaginary, the angles are ellipticity angles. Of course, in general the antenna errors will be complex. This calculation indicates the magnitude of the antenna errors that can be expected in terms of the geometric polarization ellipse parameters, which are used later in the model.

a. Solar scan measurements

Solar data can also be used to quantify antenna polarization errors. S-Pol collects solar data by performing a “box scan” of the sun in the passive mode. The sun here is considered to be an unpolarized radio frequency (RF) source subtending a solid angle of about 0.53° (Jursa 1985; Tapping 2001). The dimension of the box scan is approximately 3° high (elevation angle) by 7° wide (in azimuth), and the radar scan elevation steps are 0.2°. The scanning rate is 1° s−1 so that one complete solar box scan requires about 2 min. Noise samples are collected while the radar is pointing away from the sun so that the thermal background noise can be estimated and used to correct the measured sun data. S-Pol’s sensitivity is −113 dBm and the sun’s measured power is about −100 dBm when the main antenna beam is centered on the sun. The data are interpolated to a square 2° × 2° grid in 0.1° intervals. The data are first corrected for sun movement and for distortion caused by scanning in elevation and azimuth angle rather than in a rectangular grid.

Shown in Fig. 3 in the top panels are the H and V “pseudo”-antenna patterns, respectively, obtained from S-Pol data gathered on 19 May 2008 during the Terrain-Influenced Monsoon Rainfall Experiment (TiMREX) in southern Taiwan. The powers are uncalibrated. These are termed pseudo–antenna patterns because the sun is not a point source and thus the given antenna patterns are a convolution of the antenna beam pattern of S-Pol with the solar disk. The complex H and V time series data can be used to create a cross-channel correlation antenna pattern. The simultaneously received voltage time series from a single dwell angle, Vh(i) and Vυ(i) for the horizontal and vertical channels, respectively, are correlated in usual fashion as
i1520-0426-27-10-1583-e24
Thus, ω gives the pointwise (spatial) correlation from temporal averages. This correlation data from all dwell angles is interpolated to a grid. The resulting magnitude and phase of the correlation product of Eq. (24) are given in Fig. 3 in the bottom two panels. If solar radiation is unpolarized, then the correlation of data between any two orthogonal receiving polarization channels is zero by definition. The correlation magnitude in Fig. 3 (bottom left) shows two principal “lobes” in the lower two quadrants where the correlation increases to about 0.07. These large areas of increased correlation coefficient are manifestations of the antenna polarization errors. The antenna errors are obviously a function of azimuth and elevation angle and are not constant across the 2° × 2° antenna patterns shown. The areas of maximum correlation do, however, fall outside the 3-dB beamwidth of the antenna [which is about 0.9° (Keeler et al. 2000)], which helps reduce the magnitude of the cross coupling. Figure 3d shows the complex behavior of the phase of the correlation product, with the phases being fairly constant in the regions of the highest correlation. For the lower-left quadrant, this phase is −100°, while the lower right quadrant phase is about +60°. These antenna pattern correlations can be integrated to obtain a single complex correlation coefficient and this is discussed later in the text.

The radar model presented above represents the antenna polarization errors as a single complex number for the H and V polarizations, that is, the polarization errors are integrated. Even though antenna errors are distributed, this is a useful approximation that simplifies analysis and permits a realistic numerical evaluation and simulation of polarization errors. The assumption is that there is a homogeneous distribution of scatterers across the antenna beam.

It can be shown that for small polarization errors (see appendix B)
i1520-0426-27-10-1583-e25
where Ω is the pattern-integrated correlation coefficient.

b. General observations

Solving Eqs. (23) and (25) simultaneously yields
i1520-0426-27-10-1583-e26
i1520-0426-27-10-1583-e27
where LDR is LDR in linear units. The real parts are not solvable, but obviously ℜ(ξh + ξυ) = ℜ(ξ*h + ξυ).
Starting with Eqs. (23) and (25) several interesting observations are possible. The theoretical conditions for making LDR minus infinity and Ω zero are ξυ = −ξh, and ξυ = −ξ*h, respectively. In terms of the geometric polarization quantities, the system LDR limit minima condition is
i1520-0426-27-10-1583-e28
i1520-0426-27-10-1583-e29
and the minimum condition for Ω is
i1520-0426-27-10-1583-e30
i1520-0426-27-10-1583-e31
Again, for low solar cross correlation, the errors must be near orthogonal. This is consistent with the observation that the cross correlation of passive solar measurements from two orthogonal polarization states is, by definition, zero if the solar radiation is unpolarized, as is assumed. It is interesting to note that for minimum LDR, the H and V tilt angles are orthogonal while the ellipticity angles are not. This is a direct artifact of radar polarimetry theory and the radar voltage equation (Kennaugh 1949–1954; Hubbert 1994; Hubbert and Bringi 1996). Because the polarization basis of the radar is transformed from linear H and V polarizations to the circular polarization basis by increasing ε from 0° to 45°, the depolarization ratio from a circular scatterer (e.g., a circular raindrop) goes from −inf to +inf dB [e.g., see Fig. 7 from Hubbert (1994) or Fig. 3.10 of Bringi and Chandrasekar (2001)]. Thus, for increasing orthogonal ellipticity errors, the LDR system limit will increase while Ω is unaffected. Also, note that the orthogonal tilt angle errors will not increase the LDR system limit (as determined from measurements in drizzle) nor will they increase Ω. Therefore, neither LDR drizzle measurements nor solar correlation measurements will detect orthogonal tilt antenna errors. For example, these measurements would not reveal whether a radar was unleveled, that is, whether the desired H polarization state of the antenna was not parallel to the earth. Equivalently, it can be shown that if the H and V errors are orthogonal, this implies the phase relationship ζυ = π + ζh, where ζh = arg(ξh/ih) and ζυ = arg(iυ/ξυ). If the antenna errors meet the LDR minimum criteria, that is, ξυ = −ξh, then ζυ = πζh.

Thus, the cross-correlation Ω can be either zero or very low, indicating that the receive polarization states are orthogonal or nearly orthogonal, but this does not necessarily mean that the LDR system limit is low. Conversely, the LDR system limit can be low, indicating that the H and V tilt errors are nearly orthogonal and that the ellipticity errors are nearly equal, but Ω could be relatively high.

From the S-Pol solar data of Fig. 3, the integrated solar correlation coefficient is calculated to be Ω = 0.0038 + j0.00088. There is very likely system phase offset included in this number, such as the differential phase from the reference plane to the I and Q time series samples (see Fig. 1), so that the phase of Ω is not an accurate estimate of the phase of ξh + ξ*υ. However, the magnitude of Ω should be an accurate estimate of the magnitude of ξh + ξ*υ. The magnitude of Ω is about 0.0039, and from the measurements in drizzle S-Pol’s LDR limit is about −31 dB. Using Eqs. (23) and (25), we can write
i1520-0426-27-10-1583-e32
i1520-0426-27-10-1583-e33
where 0.028 = 10(−31dB/20). The real parts of the arguments of the absolute values in Eqs. (32) and (33) are equal, and therefore there must be significant canceling of the imaginary parts of ξh and ξ*υ in Eq. (32) to account for the large magnitude difference between 10(LDR/20) and |Ω|. Interpreting these measurements in conjunction with Eqs. (3)(7) gives some insight as to the nature of the S-Pol antenna errors. Orthogonal tilt angle errors (with no ellipticity angle errors) require that αυ = αh + 90°, ℜ(ξh) = −ℜ(ξυ), and ℑ(ξh) = ℑ(ξυ) = 0. Under this condition both LDR and Ω are both zero, so obviously orthogonal tilt angle errors cannot produce the observed measurement. Additionally, nonorthogonal tilt angle errors will increase both LDR and Ω and therefore they cannot account for LDR ≫ |Ω|2. For orthogonal ellipticity errors (with no tilt angle errors), ℜ(ξh) = ℜ(ξυ) = 0 and ℑ(ξh) = ℑ(ξυ). Therefore, for increasing orthogonal ellipticity errors, LDR increases while Ω is not affected. It follows that S-Pol antenna errors are likely dominated by ellipticity errors and the H and V ellipticity errors are such that
i1520-0426-27-10-1583-e34
Orthogonal tilt errors are detected by neither the system limit LDR nor the Ω measurements, and they could be significant. We assume, however, that physically the feed horn is well aligned (leveled) and this then would make these errors negligible. Under these conditions, ℜ(ξh) = ℜ(ξυ) ≈ 0, and then
i1520-0426-27-10-1583-e35
i1520-0426-27-10-1583-e36

Using Eqs. (3), (4), and (6), the H and V ellipticity angles are found to be εh = −0.91° and ευ = 0.69°. These antenna error values are used in the model in Part II. Summarizing, if LDR is significantly >10 log10(|Ω|2), then the H and V ellipticity errors must be closer to orthogonal, that is, ευ ≈ −εh, rather than close to equal, that is, ευεh.

The functional relationship between the antenna errors and SHV Zdr is found by calculating SHV mode Zdr in drizzle (see appendix C) as
i1520-0426-27-10-1583-e37
where β is the differential transmit phase arg(Eυt/Eht). Equation (37) shows that the first-order H and V antenna error terms appear in the form ξh + ξυ, as they do in the LDR system limit in Eq. (23). This shows that the sum of the antenna errors appears both in SHV Zdr and LDR, and antenna errors that increase LDR will also, in general, increase SHV Zdr bias. However, as shown in appendix C, Zdr is a strong function of differential transmit phase and differential propagation phase. The conclusion is that the LDR system limit of a radar is a good indicator of the expected SHV Zdr bias. Later, we present curves that relate the LDR system limit to the SHV Zdr bias.

4. Model results

Next, the model is used to examine biases in the SHV mode for the 1) transmit errors, 2) nonzero mean propagation canting angle, and 3) antenna polarization errors. Again, the radar variables are plotted as a function of principal plane . Because is the independent variable and Kdp is of more meteorological interest than ϕdp, the normalized SHV mode Kdp is expressed as
i1520-0426-27-10-1583-e38
where is normalized SHV Kdp, is SHV Kdp, and is the principal plane Kdp [see Eq. (A2) and surrounding text for clarification of principal plane]. For the relatively small antenna polarization errors examined here, and therefore the errors in are small.

The backscatter medium is modeled as rain with a zero mean canting angle, a standard deviation of canting angles of 5°, Zdr = 2.8 dB, and LDR = −35 dB. For this study, the characteristics of the backscatter medium are relatively unimportant in terms of characterizing antenna polarization errors and the effect of nonzero mean canting angle of the propagation medium on Zdr. Thus, the mean canting angle of the backscatter medium is always 0°. For the propagation medium, typical S-band values of absolute attenuation Ah = 0.0165 dB °−1 (of ) and differential attenuation Adp = 0.0035 dB °−1 (Bringi and Chandrasekar 2001) are used unless otherwise stated.

a. Transmit errors

Here we examine SHV Zdr bias caused by errors in the transmit polarization state. The transmit polarization state is defined by the transmit H and V electric fields Eht and Eυt at the reference plane shown in Fig. 1. For the SHV mode, ideally Eht = Eυt. Figure 4 shows for several values of |Eυt|/|Eht| versus . Note that the tilt angle of the transmit polarization state is defined as tan−1(|Eυt|/|Eht|) (V-to-H, angles are given in the plot for each curve), whereas Zdr is H-to-V power. The mean canting angle of the particles in the propagation medium is zero (i.e., diagonal propagation matrix) and the antenna polarization errors are zero, that is, ξh = ξυ = 0. The bias is independent of the phase difference between Eht and Eυt. The slope of the curves is caused by Adp = 0.0035 dB °−1. The dashed nominal line is considered ideal. As can be seen, the biases are constant as compared to the nominal curve, and such biases could be corrected via radar calibration. Even though the phase difference between Eht and Eυt does not affect , this phase difference is very important in the sections below. We note that Wang and Chandrasekar (2006) show the effect of differential antenna error phase on the bias. Here we show how the differential H and V transmit phase (as well as antenna phase error) affects the bias. The differential transmit phase and differential antenna phase error are distinctly different.

b. Nonzero mean propagation canting angle

The model is now used to illustrate bias caused by nonzero mean canting angle of the propagation medium θ, which causes cross coupling between the H and V components of the electric field. Antenna errors are zero and the transmit errors are zero, that is, Eht = Eυt. Figure 5 shows bias as a function of with θ as a parameter. The Zdr bias is defined as the difference between the modeled and the error-free, nominal Zdr shown in Fig. 4. As |θ| increases, the bias increases. If , the biases are kept to within about 0.1 dB. Figure 6 is similar to Fig. 5, except that the phase difference between Eht and Eυt is 90°, that is, the transmit polarization is circular. The bias now increases much more rapidly as a function of as compared to Fig. 5. The value must be less than about 1° in order for Zdr bias to be less than about 0.1 dB. This then shows that if θ is not near zero, very little needs to accumulate in order to cause significant bias. The phase difference of the H and V transmitted waves controls the amount of the constructive–destructive interference between the main copolar wave and the biasing cross-coupled component. This illustrates the importance of the phase difference between the transmit electric field components Eυt and Eht, and corroborates and explains the SHV Zdr bias in the ice regions of storms reported by Ryzhkov and Zrnić (2007).

When the backscatter medium is changed to ideal drizzle, that is, small spherical drops, the bias curves of Figs. 5 and 6 change very little, and thus are not given. Also, the propagation medium parameters can be changed to simulate C band with Adp = 0.018 dB °−1 and Ah = 0.075 dB °−1, and to simulate X band with Adp = 0.03 dB °−1 and Ah = 0.28 dB °−1. The corresponding bias curves are given in Figs. 7a,b, respectively. These bias curves are very similar to the S-band errors curves in Fig. 6. Thus, the primary difference between the S-, C-, and X-band biases here is that for the same propagation medium, will accumulate faster at C band and still faster at X band, as compared to S band. This means that the Zdr biases discussed here will be more problematic at C band and even more so at X band. We give one note of observational interest: If ice particles have a mean canting angle of around 45°, SHV ϕdp will be a constant as a function of range because it propagates through such a medium even though can be significant so that becomes biased.

c. Antenna errors

Antenna errors are quantified in the model by the complex numbers ξh and ξυ. These errors can be equivalently defined by the tilt and ellipticity angles, αh,υ and εh,υ, respectively, of the polarization ellipse as described above. First, the model is used to illustrate the different effects that tilt and ellipticity angle errors individually have on Zdr bias. Figure 8 shows for 1° orthogonal polarization tilt errors (top panel) and 1° orthogonal polarization ellipticity errors (bottom panel), both versus . The θ is zero and Eυt = Eht. The magnitude of these errors, that is |ξh + ξυ| corresponds to an LDR system limit of about −30 dB. The solid straight lines represent nonbiased Zdr that would be measured in fast alternating H and V transmit mode. The figure shows that Zdr bias is significant with a maximum error of about 0.6 dB when .

It is likely that true antenna errors will be some combination of tilt and ellipticity angle errors, and thus we present the following again as an illustrative example of how antenna errors can affect . Figure 9 shows bias for the H and V tilt and ellipticity error angles given in Table 1. The antenna errors are orthogonal [Eq. (8)]. The figure shows that the character of the Zdr bias is quite different for each curve, with a maximum bias of about 0.4 dB. These antenna errors all correspond to about a −31-dB LDR system limit.

These same antenna errors from Table 1 are used again in Fig. 10, but for circular transmit polarization. The Zdr biases curves have changed dramatically and demonstrate the importance of the phase difference between the H and V components of the transmit wave. Again the transmit wave is defined here at the reference plane given in Fig. 1.

As shown above, S-Pol’s antenna polarization errors are fairly well characterized by orthogonal ellipticity angles and by H and V tilt angles of 0° and 90°, respectively (i.e., no tilt angle errors). Using this restriction and given an LDR system limit value, the ellipticity error angle can be calculated. Table 2 gives the ellipticity error angles corresponding to several LDR system limit values. The corresponding values for the Im(ξh) (or equivalently εh in radians) are also given. The information in Table 2 is used below.

We next examine SHV Kdp biases caused by polarization errors given in Table 3. These antenna polarization errors correspond to an LDR system limit of −25 dB. Shown in Fig. 11 is as a function of principal plane ϕdp. The Kdp bias is fairly small, always being less than 3%. If the LDR system limit is less than −30 dB, the Kdp error is within 2%.

The biases of SHV ρhv for the LDR system limits as high −25 dB are less than 1% and are not given.

d. SHV Zdr as a function of LDR system limit

As shown in appendix C, the antenna error terms appear as ξh + ξυ in the expression for the LDR system limit and SHV Zdr in drizzle. Thus, the LDR system limit for a radar can be related to the SHV Zdr bias as a function of , with differential transmit phase as a parameter. Based on the antenna errors for S-Pol, the antenna errors are modeled as orthogonal ellipticity angles with no tilt angle errors. This is shown in Figs. 12a,b for slant 45° linear transmit polarization (i.e., Eht = Eυt) and circular transmit polarization, respectively. The shown ε denotes the sign of the H polarization ellipticity angle. The values of the ellipticity angle corresponding to each curve are given in Table 2. Note how not only the shape of bias curves changes, but also the maximum Zdr bias increases significantly for circular transmit polarization. The model shows that the most stringent cross-polar isolation criteria results from the circular polarization transmit condition. As can be seen, if the SHV Zdr bias is to be kept under 0.2 dB, the LDR system limit needs to be about −40 dB. Practically, if one of the circular transmit bias curves characterized a radar, the Zdr bias at ϕdp = 0 would likely be detected by the user and a Zdr offset correction factor would be used. Then, the maximum Zdr bias would occur for .

5. Summary and conclusions

Biases in simultaneous H and V transmit (SHV) radar data were investigated via a radar-scattering model. The model includes the transmit errors, antenna polarization errors, propagation medium, and backscatter medium. The validity of the SHV technique is based on the following two assumptions: 1) a zero mean canting angle of the propagation precipitation medium and 2) negligible antenna polarization errors. Violation of either condition will cause cross coupling of the H and V electric field components, which in turn biases polarimetric variables. Differential reflectivity (Zdr) is particularly sensitive to the cross coupling. How the cross-coupled signal combines (constructively versus destructively) with the primary copolar field component depends upon their relative phase, and thus the biases are strong functions of differential propagation phase ϕdp and differential transmit phase.

The zero mean canting angle is a good approximation for rain but not for ice particles. The Zdr bias resulting from oriented ice crystals has been reported before (Ryzhkov and Zrnić 2007), and the results reported here support that work. Demonstrated here for the first time was the impact of differential transmit phase on SHV Zdr. It was also shown that the phase difference between the H and V transmit signals at the plane of reference (see Fig. 1) will significantly shape the Zdr errors as a function of principal plane ϕdp. For example, consider the ice phase of storms where ice crystals can be aligned by electric fields so that their mean canting can be significantly away from 0°. If the transmit polarization state is circular, only 2° or 3° of need accumulate, which in turn will cause several tenths of a decibel bias in SHV Zdr. This bias will be difficult to correct and can have deleterious effects on particle classification schemes that rely on accurate Zdr estimates.

Antenna error–induced cross coupling was also investigated. The model indicates that antenna errors of the magnitude to be expected from well-designed, center-fed parabolic reflectors will cause significant SHV Zdr bias in rain after sufficient ϕdp accumulation. The model shows that if Zdr bias is to be kept below 0.2 dB, the LDR system limit must be reduced to about −40 dB. This is largely in agreement with Wang and Chandrasekar (2006) who quote a similar requirement of a −44-dB LDR system limit for worst-case antenna errors.

SHV Zdr bias in rain will also affect Zdr measurements in the ice phase of storms. Any SHV Zdr bias acquired in the rain will be carried forward into the subsequent ice phase along that radar radial; that is, if 0.3 dB of Zdr bias were accumulated in rain before the radar signal reached the ice phase, then the Zdr measurements in the ice phase would also manifest the 0.3-dB bias.

Estimates of the magnitude and type (tilt versus ellipticity angle) of antenna polarization errors for well-designed, center-fed parabolic reflector antennas, such as NCAR’s S-Pol, were estimated. Describing antenna errors with the geometric polarization ellipse parameters of tilt and ellipticity angles help provide insight as to the nature the antenna errors. If a radar has an LDR measurement limit of −30 dB (determined from measurements in drizzle), the corresponding magnitude of either tilt or ellipticity angle polarization error is about 1°. If the cross correlation of H and V solar measurements are also considered, then the relative values of the tilt and ellipticity angle error can be better estimated. For S-Pol it was shown that the H and V antenna errors are characterized by nearly orthogonal ellipticity angles. The ellipticity angles were estimated based on knowledge of the S-Pol LDR system limit and solar scan measurements. These antenna error estimates were used in the model and were found to predict the experimentally observed S-Pol SHV Zdr bias and the behavior of LDR measurements through a rain medium. This is further addressed in Part II.

Achieving an LDR limit figure of −40 dB is very difficult and the authors are not aware of any S-band weather radars that are capable of this, except for the new Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) dual-offset-fed antenna. It may not be cost effective to design a radar with a center-feed parabolic antenna that achieves such a level of cross-polar isolation, and this cost must be considered against the above-mentioned benefits of implementing the SHV mode dual-polarization radar systems.

Acknowledgments

This research was supported in part by the Radar Operations Center (ROC) of Norman, Oklahoma. The authors thank Terry Schuur of the of the Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma for supplying the KOUN data and software assistance, and for technical discussions. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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APPENDIX A

Radar Model Details

The details of the radar model can also be found in Hubbert and Bringi (2003). Because forward scatter is coherent (van de Hulst 1957), the propagation medium can be completely described via a 2 × 2 scattering matrix 𝗣 as
i1520-0426-27-10-1583-ea1
where 𝗥 is the Cartesian rotation matrix and 𝗣0 is the principal plane propagation matrix
i1520-0426-27-10-1583-ea2
where λ1 and λ2 are the propagation constants along the principal planes of the propagation medium and z is the distance along the direction of propagation. It can be shown that specific differential attenuation is Adp = −(8.686 × 103)ℜ(λ1λ2) dB km−1, and specific differential phase is Kdp = −103ℑ(λ1λ2) rad km−1, where ℜ and ℑ denote real and imaginary parts, respectively (Bringi and Chandrasekar 2001). In rain, these two parameters can be related by (Holt 1988; Bringi et al. 1990), where the superscript P signifies principal plane value and c is a constant. The output of the model gives radar measurements as a function of
i1520-0426-27-10-1583-ea3
where is principal plane ϕdp. The propagation medium is coupled to the backscatter medium via the radar-scattering matrix (Kennaugh 1949–1954; Sinclair 1950)
i1520-0426-27-10-1583-ea4
where Shh, Svv, and Shv are backscatter amplitudes in the H–V polarization basis. The propagation-modified covariance matrix is then formed as (Tragl 1990)
i1520-0426-27-10-1583-ea5
where 〈·〉 denotes ensemble average and * signifies complex conjugation. Note that ensemble averaging only applies to the particles in the resolution volume (i.e., backscatter medium). Each matrix member in (A5) consists of propagation terms and backscatter covariances of the form 〈Sx1,y1S*x2,y2〉, (x1,2, y1,2 = h, υ). The H–V basis backscatter covariance matrix Σs is simulated via the T-matrix method. An arbitrary mean canting angle (α) and mean ellipticity angle can be given to the backscatter covariance matrix Σs by rotating this simulated covariance matrix in the plane of polarization with (Tragl 1990)
i1520-0426-27-10-1583-ea6
where
i1520-0426-27-10-1583-ea7
and χ is the polarization ratio defined as the ratio of the V-to-H electric field components (or electric field phasors) as Eυ/Eh and Γ = (1 + χχ*)0.5. As shown in Hubbert (1994), the phase term ϱ = ejtan−1(tanαtanε) is necessary to maintain a constant phase difference between the elliptic basis polarization vectors. The χ can be expressed in terms of the mean tilt (or canting) angle α and mean ellipticity angle ε as (Azzam and Bashara 1989)
i1520-0426-27-10-1583-ea8
In this paper, cross coupling caused by antenna polarization errors and by the nonzero mean canting angle of the propagation medium are the focus and the backscatter medium is of secondary importance. Therefore, the mean tilt angle of the backscatter medium is assumed to be zero and this implies that the co- to cross-covariance terms in the backscatter 3 × 3 covariance matrix are zero.

APPENDIX B

Cross Correlation of Horizontal and Vertical Sun Measurements

Antenna beam-integrated passive solar measurements can be represented as
i1520-0426-27-10-1583-eb1
where ξh and ξυ are the complex antenna polarization errors of the H and V channels, respectively, and and Vh,υ are the incident electric fields and received voltages. Radar gain and proportionality constants are nonessential and are omitted. We wish to find the correlation
i1520-0426-27-10-1583-eb2
where 〈·〉 indicates time average. Expanding the numerator yields
i1520-0426-27-10-1583-eb3
Because solar radiation is unpolarized, the Ehi to Eυi cross correlations are zero so that
i1520-0426-27-10-1583-eb4
Similarly,
i1520-0426-27-10-1583-eb5
i1520-0426-27-10-1583-eb6
If the errors are small so that ξhih and ξυiυ, then
i1520-0426-27-10-1583-eb7
i1520-0426-27-10-1583-eb8
Using Eqs. (B4), (B7), and (B8), the correlation can be written as
i1520-0426-27-10-1583-eb9
The H and V solar radiation power should be equal, that is, |Ehi|2 = |Eυi|2. Assuming that |ξh| and |ξυ| are small, then ihiυ = 1. Then, Eq. (B9) can be approximated with
i1520-0426-27-10-1583-eb10
Finally, Ω = ξ*h + ξυ because by definition ξh,υ are beam-integrated, constant antenna errors.

APPENDIX C

An Estimate of System Limit LDR and SHV Zdr

LDR in drizzle

To derive radar covariances in drizzle we begin with equations for scattering from a single particle, taking into account antenna errors (Bringi and Chandrasekar 2001). Using Eqs. (1) and (2), the received voltages can be modeled in drizzle as
i1520-0426-27-10-1583-ec1
where shh and svv are the backscatter scatter amplitudes, ξh,υ are the H and V antenna polarization errors, and are the transmitted H and V electric fields. Radar gain and proportionality constants are nonessential and are omitted. Executing the matrix multiplication Eq. (C1) yields
i1520-0426-27-10-1583-ec2
Next, the appropriate covariance products are then taken and the products are integrated over the particle distribution and over the antenna pattern. Here we assume beam-integrated antenna errors. The LDR for the transmit state Eht = 1, Eυt = 0 is
i1520-0426-27-10-1583-ec3
where 〈·〉 stands for ensemble average.
Because we are interested in isolating the effects of antenna errors, several simplifying observations and assumptions are made. First, we assume that the antenna errors are small so that ξh,υih,υ and ihiυ ≈ 1. For example, if we assume that |ξh| ≈ |ξυ|, then for a LDR limit of −30 dB, ih = iυ = 0.99975. Eq. (C3) reduces to
i1520-0426-27-10-1583-ec4
Because, for drizzle, shh = svv, and because antenna errors are constants they can be brought outside the angle brackets
i1520-0426-27-10-1583-ec5

SHV Zdr bias and antenna errors

We begin with Eq. (C1). Propagation effects can be included in the model by attaching the coefficient e to the shh term, where ν is a complex constant. The voltages can then be expressed as
i1520-0426-27-10-1583-ec6
Without loss of generality, divide through the scattering matrix by svv and let . For simultaneous H and V transmissions, let Ehi = 1 and Eυi = e, where β represents the phase difference between the H and V transmission signals. Then,
i1520-0426-27-10-1583-ec7
The magnitude-squared operation could be executed, ensemble averages taken, and antenna errors separated from particle ensemble averages. Here we retain the argument of the magnitude-squared operation in order to examine the effects of antenna errors. Notice that the cross-polar term e(ihξυ3e + iυξh) appears in the numerator of LDR and also appears in both the numerator and denominator of SHV Zdr. Thus, in general, as the LDR system limit increases because of an increase in ξh + ξυ, the maximum bias in SHV Zdr also will increase. Where the maximum SHV Zdr bias occurs depends upon the phase of e and the magnitude and phase of e. Thus, because the LDR system limit of a radar is typically a know performance characteristic of a radar, it is useful to plot SHV Zdr bias using the LDR system limit as a parameter as is done in this paper. Again, to simplify, for small antenna polarization errors, ih = iυ ≈ 1 and the second-order ξ are dropped. Note that the model used in the paper retains all of these terms. Equation (C7) simplifies to
i1520-0426-27-10-1583-ec8
This equation shows how differential propagation phase (embedded in ν) and differential transmission phase (β) affect SHV Zdr. Importantly, it shows that for small z and a phase of e = 0, the antenna errors approximately appear as ξh + ξυ, which is the same form as found in the expression for the system LDR limit (as apposed to ξh + ξ*υ). Finally, if the precipitation medium is drizzle, then z = 1 and υ = 0 and Eq. (37) results.
Fig. 1.
Fig. 1.

A block diagram showing the elements of the radar model. TX: transmitter block, PD: power division network, RH vertical receiver chain, and RV horizontal receiver chain.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 2.
Fig. 2.

The polarization ellipse. Propagation is into the paper.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 3.
Fig. 3.

S-Pol pseudo–antenna patterns obtained from passive solar scan measurements from data gathered on 19 May 2008 during TiMREX. (top) H and V antenna patterns and (bottom) magnitude and phase of the H to V correlation. Power is uncalibrated.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 4.
Fig. 4.

SHV Zdr as a function of principal plane ϕdp with unbalanced transmit power as a parameter. For SHV mode ideally |Eht| = |Eυt|, and this nominal curve is shown (dashed line). The errors are independent of the phase difference between |Eht| and |Eυt|.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 5.
Fig. 5.

As in Fig. 4, but with the mean canting angle of the propagation medium as a parameter. The transmission errors are zero, that is |Eht| = |Eυt|. The two curves between ±1° are for mean canting angles of ±0.5°.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 6.
Fig. 6.

As in Fig. 4, but with the mean canting angle of the propagation medium as a parameter. Here, |Eht| = |Eυt| but there is a 90° phase difference; that is circular polarization is transmitted.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 7.
Fig. 7.

As in Fig. 4, but with the mean canting angle of the propagation medium as a parameter. Here, |Eht| = |Eυt|, but there is a 90° phase difference; that is circular polarization is transmitted. (a) C band and (b) X band. There is very little difference between these two error plots and the analogous one for S band in Fig. 6.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 8.
Fig. 8.

SHV mode Zdr for 1° antenna polarization errors. (top) ±1° tilt errors and (bottom) ±1° ellipticity errors.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 9.
Fig. 9.

SHV mode Zdr bias for mixed tilt and ellipticity antenna error angles, which are given in Table 1. The antenna errors are orthogonal and the H and V transmitting signals are equal, that is Eh = Eυ. These antenna errors correspond to a system LDR limit of −31 dB.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 10.
Fig. 10.

SHV mode Zdr bias for mixed tilt and ellipticity antenna error angles which are given in Table 1; however, the transmission polarization state is circular. The antenna errors are orthogonal. These antenna errors correspond to a system LDR limit of −31 dB.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 11.
Fig. 11.

Normalized SHV mode Kdp as a function of principal plane ϕdp for the antenna error angles given in Table 3.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Fig. 12.
Fig. 12.

SHV mode Zdr bias as a function of principal plane ϕdp with LDR system limit as a parameter. The antenna polarization errors are assumed to be orthogonal ellipticity angles. The sign of the H ellipticity angle is given in each quadrant. (a) The transmitting polarization is 45° linear, that is Eht = Eυt. The curves all mimic a sine wave shape. (b) The transmitting polarization is circular. The curves are symmetric about the vertical line through 180°. The corresponding antenna errors are given in Table 2.

Citation: Journal of Atmospheric and Oceanic Technology 27, 10; 10.1175/2010JTECHA1336.1

Table 1.

The H and V tilt and ellipticity error angles corresponding to Fig. 9.

Table 1.
Table 2.

Antenna polarization errors as a function of system LDR limit. The antenna errors are assumed to be orthogonal and elliptical.

Table 2.
Table 3.

The H and V tilt and ellipticity error angles corresponding to Fig. 11. The corresponding LDR system limit is 25 dB.

Table 3.

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

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