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    The std dev of the phase of the co-to-cross covariance (ϕxH or ϕxV) as a function of the co-to-cross correlation coef (ρxH or ρxV) with the normalized spectral with συn as a parameter; συn = σV/λ/(2Ts)

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    Co-to-cross correlation coef as a function of mean backscatter canting angle (α) with std dev (σα) as a parameter.

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    LDRH as a function of mean backscatter canting angle (α) with the std dev of (σα) as a parameter

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    LDRH as functioin of principal plane ϕdp with mean propagation canting angle θ as a parameter. The mean backscatter canting angle α is 0°

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    Same as in Fig. 4, but for the co-to-cross correlation coefficient ρxH

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    The phases Ψ̂xV and Ψ̂xH as a function of ϕPdp with the ratio α/θ as a parameter. The left-hand axis corresponds to positive values of θ, while the right-hand vertical axis corresponds to negative values of θ

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    LDRH as a function of tilt angle error with σα as a parameter and ϕPdp = 0°

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    The ρxH as a function of tilt angle error with σα as a parameter and ϕPdp = 0°

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    The LDRH as a function of ellipticity angle error with σα as a parameter and ϕPdp = 0°

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    The ρxH as a function of ellipticity angle error with σα as a parameter and ϕPdp = 0°

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    Range profile of Ψdp and ΨxV − ΨxH

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    Range profile of ΨxV − Ψdp/2

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    Range profile of filtered ρxH and polarization error corrected ρxH

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    Range profile of filtered LDR and polarization error corrected LDR

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    Using the estimated polarization errors, (a) ρxH (solid), LDRH (dashed), and (b) ΨxH − Ψdp/2 as functions of principal plane ϕdp from the theoretical model

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Studies of the Polarimetric Covariance Matrix. Part II: Modeling and Polarization Errors

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  • 1 Colorado State University, Fort Collins, Colorado
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Abstract

A polarimetric radar covariance matrix model is described to study the behavior of the co-to-cross covariances in precipitation. The 2 × 2 propagation matrix with attenuation, differential attenuation, and differential phase is coupled to the backscatter matrix leading to a propagation-modified covariance matrix model. System polarization errors are included in this model as well. This model is used to study the behavior of the magnitude and phase of the co-to-cross covariances and the linear depolarization ratio (LDR) in rainfall. It is shown that the model predictions are consistent with data collected with the Colorado State University (CSU)–University of Chicago–Illinois State Water Survey (CHILL) radar in intense rainfall. A method is also given for estimating the system polarization errors from covariance matrix data collected in intense rainfall.

Corresponding author address: Dr. J. C. Hubbert, UCAR/ATD, Foothills Laboratory, Building 1, Room 2085, Boulder, CO 80303. Email: hubbert@ucar.edu

Abstract

A polarimetric radar covariance matrix model is described to study the behavior of the co-to-cross covariances in precipitation. The 2 × 2 propagation matrix with attenuation, differential attenuation, and differential phase is coupled to the backscatter matrix leading to a propagation-modified covariance matrix model. System polarization errors are included in this model as well. This model is used to study the behavior of the magnitude and phase of the co-to-cross covariances and the linear depolarization ratio (LDR) in rainfall. It is shown that the model predictions are consistent with data collected with the Colorado State University (CSU)–University of Chicago–Illinois State Water Survey (CHILL) radar in intense rainfall. A method is also given for estimating the system polarization errors from covariance matrix data collected in intense rainfall.

Corresponding author address: Dr. J. C. Hubbert, UCAR/ATD, Foothills Laboratory, Building 1, Room 2085, Boulder, CO 80303. Email: hubbert@ucar.edu

1. Introduction

The 3 × 3 radar polarimetric covariance matrix, consisting of three real powers and three complex covariance elements, provides a complete set of measurements from distributed particles such as precipitation. It is only recently that research radars have been configured to measure all of the covariance matrix elements in real time (e.g., Brunkow et al. 2000). While the theory of covariance matrices and associated optimum polarizations for random media have been known for some time (e.g., Mott 1992, and extensive references therein), the application to retrieval of microphysical information of precipitation is much more recent (Huang et al. 2001; Ryzhkov et al. 2002). The conventional polarimetric observables such as the differential reflectivity (Zdr), linear depolarization ratio (LDR), specific differential propagation phase (Kdp); and copolar correlation coefficient (ρHV) have been well studied in the past two decades with respect to their microphysical information content and are now, more or less, routinely applied to rain-rate estimation and hydrometeor-type classification (e.g., see Bringi and Chandrasekar 2001, and extensive references therein). The two co-to-cross covariance terms associated with 〈SHHS*VH〉 and 〈SVVS*HV〉 have only recently been receiving attention for the detection of oriented hydrometeors (Ryzhkov 2001; Ryzhkov et al. 2002), while the use of optimum polarizations has also only recently been investigated (Hubbert and Bringi 1996, 2001). There is promise that full covariance matrix analysis may lead to improved rain-rate estimation and to reducing ambiguities in hydrometeor classification based on the conventional five polarimetric measurands (ZH, Zdr, LDR, Kdp, ρHV). An important related application is experimental evaluation of the so-called “hybrid” mode proposed for polarimetric upgrade for the Weather Surveillance Radar-1988 Doppler (WSR-88D) radars (Doviak et al. 2000). Clearly, more understanding of the behavior of the magnitude and phase of the co-to-cross covariances is needed, especially in the presence of system polarization errors. Also, techniques need to be developed for constructing accurate covariance matrices from data for which the system differential gain and phase offsets need to be evaluated. This paper addresses these issues using a covariance matrix modeling methodology and analysis of covariance matrix data from the Colorado State University (CSU)–University of Chicago–Illinois State Water Survey (CHILL) radar. More specifically, the impact of the nonzero mean canting angle, spread of canting angles, and system polarization errors on the co-to-cross covariances and LDR are studied.

2. Background

As mentioned earlier, the members of the radar covariance matrix that result from the products of the copolar time series have been well studied (Doviak and Zrnić 1993; Bringi and Chandrasekar 2001). Because the horizontal (H) and vertical (V) copolar signals are typically well correlated [if the signal-to-noise ratio (SNR) is at least 10 dB], the data quality is high and the standard errors are low when at least 64 sample pairs are used per resolution volume (Liu et al. 1994). Recently, the calibration and interpretation of the co-to-cross covariances has been a topic of interest (Hubbert et al. 1999; Hubbert and Bringi 2001; Ryzhkov 2001; Ryzhkov et al. 2002; Hubbert et al. 2003, hereafter Part I). The co-to-cross covariances are difficult to interpret not only because they result from the weaker cross polar time series but more importantly, because the cross polar time series is typically weakly correlated with the copolar time series, which results in high standard errors in the estimates, or “noisy” estimates. Figure 1 shows σx, the standard deviation of ΨxH (argument of the co-to-cross covariance), as a function of ρxH (co-to-cross correlation coefficient for the H transmitted wave) with the normalized spectral width, συn, as a parameter where συn = συ/(λ/2Ts), where συ is the spectral width, λ is the wavelength, and Ts is the pulse repetition time. The Doppler spectra are assumed to be Gaussian shaped. The simulation is described by Chandrasekar et al. (1986) and Bringi and Chandrasekar (2001). As can be seen, for typical values of ρxH between 0.1 and 0.3 (e.g., in rain), the σx varies from about 18° to 140° for spectral widths of 1–6 m s–1 (for λ = 10 cm and Ts = 1 m s). Compare this to the standard deviation of ϕdp (copolar differential propagation phase) at S band of about 2°–3° (Sachidananda and Zrnić 1986).

It is known that the co-to-cross covariances are strong functions of mean canting angle and standard deviation of the canting angle of the precipitation medium. Ryzhkov (2001) has used an analytical model to study the effects of canting angle distribution on the co-to-cross covariances and has formulas for estimating the mean canting angle and standard deviation of canting angles of a rain medium. The interpretation of the co-to-cross covariances is perhaps clearer in the circular polarization basis. In terms of the circular polarization observables, the 〈SHHS*VH〉 and 〈SVVS*HV〉 elements can be expressed as
i1520-0426-20-7-1011-e1
where β0 is the mean canting angle, ρ2,4 are orientation parameters (ρ2 = 〈cos2β〉, ρ4 = 〈cos4β〉) where angle brackets are expectations over a symmetric canting angle distribution about the mean β0, ν is the circular backscatter amplitude ratio, |ν|2 is related to the circular depolarization ratio (CDR), and ηc is the circular reflectivity (see Bringi and Chandrasekar 2001, section 3.5.3). For narrow Gaussian canting angle distributions, ρ4 = ρ42 = exp(−8σ2β), where σ2β is the variance of β (Hendry et al. 1987). Rayleigh scattering is assumed. It follows that if β0 ≈ 0°, then the magnitude of the co-to-cross covariances is nearly zero and the phase fluctuates between 0° and 180°, depending on the sign of β0. In practice, however, the mean canting angles are never identically zero and, in addition, small antenna polarization errors exist, both of which prevent the co-to-cross covariances from being zero. In order to investigate these effects, a complete scattering model is developed here for the simulation of radar observables. Both 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 1971). The modeled parameters can be independently varied so that the sensitivity of the cross polar and co-to-cross covariances can be studied. Antenna polarization errors are modeled similar to McCormick (1981).

3. The scattering model

The scattering geometry used here is the backscatter alignment (BSA) convention. The model strictly addresses the case of monostatic backscatter for precipitation particles that are rotationally symmetric. Canting angles are measured counterclockwise from the horizontal in the plane of polarization (i.e., plane containing H and V axes perpendicular to the propagation direction).

Because forward scatter is coherent (van de Hulst 1957), the propagation medium can be completely described via a 2 × 2 scattering matrix 𝗣 as
θ0θ
where 𝗥 is the Cartesian rotation matrix and 𝗣0 is the principal plane propagation matrix
i1520-0426-20-7-1011-e4
where λ1,2 are the propagation constants along the principal planes of the propagation medium and z is the distance along the direction of propagation. It follows 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. In rain these two parameters can be related by APdp = κKPdp (Holt 1988; Bringi et al. 1990). The superscript P signifies principal plane values. The characteristics of the propagation medium are then controlled via the variables θ (the mean canting angle) and κ. The output of the model gives radar measurands as a function of ϕPdp = 2 r2r1 KPdp dr.
The propagation medium is coupled to the backscatter medium via the radar-scattering matrix (Kennaugh 1949–1954; Sinclair 1950)
i1520-0426-20-7-1011-e5
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)
i1520-0426-20-7-1011-e6
where 〈*〉 denote ensemble average. Note that ensemble averaging only applies to the particles in the resolution volume (i.e., backscatter medium). Each matrix member in (6) consists of propagation terms and backscatter covariances of the form 〈Sx1y1S*x2y2〉, (x1,2, y1,2 = H, V). The H/V basis backscatter covariance matrix ΣS is simulated via the T-matrix method for precipitation particles with specified size distribution (gamma), and orientation distribution (Fisher, Gaussian, etc.) with a mean canting angle of zero and some specified standard deviation of canting angle σα. 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)
ΣHVχ–1χΣ𝗦χ
where
i1520-0426-20-7-1011-e8
where χ is the polarization ratio EV/EH, in which EV and EH are the electric field phasors, “*” signifies complex conjugation, and Γ = (1 + χχ*)0.5. As shown in Hubbert (1994), the phase term ρ = ej tan–1(tanα tanε) 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)
i1520-0426-20-7-1011-e9
To our knowledge, there are no known precipitation particles that yield a nonzero ellipticity angle in the H/V basis and therefore ε is always taken as zero in this paper. This then completes the model. Summarizing, the backscatter medium is modeled via a covariance matrix (ΣS), which is calculated for a particular hydrometeor size distribution (gamma) and orientation distribution (Fisher, Gaussian, random, etc.) of precipitation particles with a zero mean canting angle (α) and specified spread of canting angles (σα). This ensemble of particles can be given an arbitrary mean canting angle via (7). The backscatter covariances are then substituted into (6); recall that each member of (6) consists of propagation terms and backscatter covariances. The parameters of the propagation matrix are controlled in (6) via (3). The mean canting angle of the propagation matrix θ is set in (3) while the Adp, AH, and ϕPdp are specified in (4) via λ1,2. The propagation medium is linked to the backscatter medium in (5). The resulting covariances in (6) are used to calculate the following radar measurands of interest:
i1520-0426-20-7-1011-eq1
where ϕdp, ϕxH, and ϕxV are due to propagation and δ, δxH, and δxH are due to the backscatter medium.1 These phases have typically been defined when the propagation medium has a 0° mean canting angle (i.e., diagonal propagation matrix), and under this assumption the incident polarization on the backscatter medium is either H or V, assuming H of V transmit polarization. Thus, the backscatter and propagation phases are easily separable in the arguments of the cross covariances. When the propagation mean canting angle is nonzero, there is a much more complex relationship between the propagation and backscatter medium. The cross covariances, 〈SbbS*aa〉, 〈SaaS*ab〉, and 〈SbbS*ab〉, become complicated functions the covariances of the backscatter medium and propagation matrix terms. Nevertheless, we retain the established nomenclature. There is no confusion, for example, by defining ΨxH = arg{〈SaaS*ab〉}. The δxH always is defined as the phase shift upon backscatter for incident pure H polarization and backscattered V polarization. The associated propagation phase is then unambiguously defined as ϕxH = ΨxHδxH. With these definitions in mind, we proceed, but the reader should remember that the cross covariances are complex functions of the backscatter covariances and the propagation parameters when the propagation matrix is nondiagonal. The above-defined radar measurands are plotted as functions of ϕPdp. Unless specified, the frequency is at S band (near 3 GHz).

4. Model results

a. Diagonal propagation matrix

If the propagation matrix 𝗣 is diagonal, the situation is greatly simplified. A diagonal propagation matrix will result when the mean canting angle of the precipitation particles is zero and the distribution of canting angles about the mean is symmetric (Bringi and Chandrasekar 2001). The overall covariance matrix becomes
i1520-0426-20-7-1011-e10
where A and B are the propagation parameters eλ1z and eλ2z, respectively. For this case, the backscatter components and the propagation components of the covariances are easily separable. The copolar differential phase [the phase of the (3,1) term in (10)] is routinely estimated (Mueller 1984; Sachidananda and Zrnić 1986) and the separation of propagation phase from backscatter phase can be accomplished using a filtering technique (Hubbert and Bringi 1995). The specific attenuation constants may be estimated in several ways that will allow for the correction of the magnitudes of the covariances though this continues to be an area of research (see Bringi and Chandrasekar 2001). The precise values for the specific attenuation constants are not relevant for the following discussion.
The behavior and the possible interpretation of the phases of the co-to-cross covariances (ΨxH and ΨxV) are not well documented. If the backscatter medium also has a zero mean canting angle α, then the co-to-cross covariances will be zero and the phase of the co-to-cross covariances are arbitrary. For nonzero α with δ = 0 (e.g., Rayleigh scattering; rain at S band), the ΨxH and ΨxV start at zero (positive α) or 180° (negative α) and increase (ΨxV) or decrease (ΨxH) by 0.5ϕdp as a function of ϕPdp. This is true because there is no depolarization due to the propagation media. But because the α will likely change sign with range (i.e., from resolution volume to resolution volume), both ΨxH and ΨxV will contain 180° discontinuities corresponding to the sign changes in α and, thus, plots of ΨxH and ΨxV versus range will appear noisy. Furthermore, because the magnitudes of ρxH and ρxV are typically low, the standard measurement error of ΨxH and ΨxV will be high (see Fig. 1). To eliminate the phase discontinuities caused by the sign change in α, the phases ΨxH and ΨxV are differenced. The model shows that for practically all values of α and θ,
ϕSXxVxHdp
The ϕSX then serves as a good check as to the quality of the ΨxH and ΨxV measurements. If ϕSX does not follow Ψdp, then there are likely problems with the radar system. Again, for this class of scatterers (i.e., Rayleigh, δ = 0°), δxH and δxV are either 0 or π, depending on the sign of α.
When δ is nonzero, the δxH and δxV are difficult to estimate accurately from range profiles of ΨxH or ΨxV. The ΨxHxV) will contain a propagation term of 0.5ϕdp and an additional backscatter phase term. However, it can be shown beginning with (7) that for ϕPdp = 0 (i.e., no propagation effects) and with α < 20° (i.e., 〈SHHS*VH〉, 〈SVVS*HV〉, and 〈|SHV|2〉 are small compared to other covariances) that
i1520-0426-20-7-1011-e12
where ;vR and ;im denote the real and imaginary parts, respectively. This is a good way to estimate δxV, (δxH) because 1) the copolar signals will have a higher SNR than the cross polar signals and 2) are typically well correlated, and 3) the copolar covariances are weak functions of θ. That is, if 〈SbbS*aa〉, 〈|Sbb|2〉, and 〈|Sbb|2〉 can be accurately estimated for the backscatter medium, then the δxH and δxV can be estimated accurately. This method works well in rain where ρHV is high (>0.95) and the LDR is low (<−25 dB), but may not be accurate in mixed precipitation type regions (i.e., ρHV is low) or when Zdr is close to 0 dB. Also, (12) and (13) are not functions of the sign of the mean canting angle α and thus only the magnitude of the canting angle can be determined from covariance matrices constructed using (12) and (13). The sign of the co-to-cross covariance will change when the sign on the α switches. To illustrate the size of the co-to-cross phases, when ϕPdp = 0° at S band (λ = 10 cm), consider an 8-mm raindrop at 0°C with a Pruppacher–Pitter axis ratio. The resulting phases are δ = −6.2°, δxH = −7.2°, and δxV = −13.5°. For raindrops with Deq ≤ 7 mm, where Deq is the diameter of an equivolumetric sphere, these phase are all nearly zero. At C band (λ = 5.5 cm, 0°C, Pruppacher–Pitter axis ratio) these phases are considerably larger. Table 1 gives these phases for Deq of 5, 6, 7, and 8 mm. For such modeled raindrops, the δxH is on the order of δ, while δxV is about twice δ. In general, the relationship among δ, δxH, and δxV is a function of the shape and dielectric constant of the medium (Ryzhkov 2001). However, it can be shown beginning with (12) and (13) that in general δxVδxH = δ if ρHV is high. This relationship is consistent with (11). Note the sign difference between the angles of Table 1 and those in Ryzhkov's (2001) Fig. 7, which are incorrect (Ryzhkov 2002, personal communication).

Next, consider the effect of nonzero α on the magnitude of the co-to-cross covariances. The backscatter medium is modeled as rain with an exponential size distribution with parameters of Dmax = 8 mm, D0 = 2.56 mm, and N0 = 8000 mm–1 m–1. The orientation distribution is Fisher (Mardia 1972; Hubbert and Bringi 1996). Figure 2 shows ρxH (solid lines) and ρxV (dashed lines) as a function of the mean backscatter canting angle (α) with the standard deviation (σα) as a parameter. As the σα increases, ρxH and ρxV become less sensitive to α. In light stratiform rain with low turbulence, where one would expect a small σα, ρxH, and ρxV could be large with just a small α [1.5° gives ρxH (ρxV) ≈ 0.63]. In a convective rain environment with more turbulence, one might expect larger σα of about 10° (Hubbert and Bringi 2001). In this case α = 1.5° gives ρxH (ρxV) ≈ 0.2 and α = 5° gives ρxH (ρxV) ≈ 0.42. Figure 3 shows the effect of α on LDRH with σα as a parameter. As expected, LDRH increases with increasing σα. Presently, the CSU–CHILL radar's LDR system limit is from around −33 to −34 dB; thus, detecting raindrop distributions with σα < 5° and α < 3° is impossible either with LDR or with ρxH (ρxV) for the case of the zero propagation phase.

b. Nonzero mean propagation canting angle

The separation of propagation and backscatter terms in the covariances becomes quite difficult when both α and θ are nonzero (recall that α refers to the backscatter medium and θ refers to the propagation medium) and thus the numerical model is necessary for the most accurate modeling. The backscatter medium is modeled as rain as above and with σα = 10°. The propagation matrix is assumed to have absolute and specific attenuation of AH/Kdp = 0.0165 dB deg–1 and Adp/Kdp = 0.004 dB deg–1, which are typical values for S band (Bringi et al. 1990). The model demonstrates that the conventional radar variables, ZH, Zdr, ϕdp, and ρHV are insensitive to mean propagation canting angles (θ), less than about 10° for the parameters selected here. For θ = 10° and ϕPdp of 180°, the ZH and Zdr deviate less than 0.5 dBZ and 0.2 dB, respectively, from their nominal values (i.e., when ϕPdp = 0°). At ϕPdp = 90° with θ = 10°, ϕdp is about 86°. The ρHV changes less than 0.002 from its nominal value. Nonzero θ does, however, dramatically change the cross polar power and the co-to-cross covariances. Shown in Figs. 4 and 5 are LDRH and ρxH as functions of the principal plane copolar differential phase ϕPdp with the mean canting angle of the propagation medium θ as a parameter. For simplicity, the mean backscatter canting angle is 0°. For θ < 0.5°, the effect on LDRH is minimal with LDRH increasing by less than 1 dB from its nominal value (nominal value closely follows the curve marked 0.2°) after about 150° of ϕPdp. The increasing trend of LDRH seen in the curve labeled 0.2° is caused by the assumed differential attenuation. The effect of θ on ρxH is more accentuated with ρxH increasing to more than 0.3 from the nominal value of zero after 100° of ϕPdp for θ ≥ 0.5°. If σα is less than 10°, the effect of θ on LDRH and ρxH is greater. For example, the case where σα = 5° was shown in Hubbert et al. (1999, see their Fig. 1, 2). This then suggests a way to check for the presence of nonzero θ in experimental data. If obvious increasing trends in ρxH and LDRH range profiles are detected along paths where there is large accumulation of ϕdp, then nonzero θ is likely present. Polarization errors can also cause such effects and are discussed later.

The effect of nonzero θ on the co-to-cross phases ΨxH and ΨxV is particularly complicated, especially in conjunction with nonzero α. However, the analysis is simplified by making a few observations. First, because ΨxV − ΨxH = Ψdp, this suggests creating the following two new phase variables:
i1520-0426-20-7-1011-e14
It appears that ΨxH and ΨxH are corrected for propagation phase; however, the resulting phases, Ψ̂xH and Ψ̂xH, are not to be interpreted as backscatter phases, that is, Ψ̂xHδxH and Ψ̂xVδxV. Interestingly, the model shows that Ψ̂xH and Ψ̂xV are nearly identical functions of ϕPdp with the restrictions α < 25° and θ < 10°. In addition, the model shows that the behavior of Ψ̂xH and Ψ̂xV is determined by the ratio of the backscatter and propa-gation mean canting angles α/θ. Figure 6 shows Ψ̂xV and Ψ̂xV as functions of ϕPdp with the ratio of α/θ as a parameter. The left-hand axis in Fig. 6 is for positive θ and for α/θ values shown on the left-hand part of the graph. All curves lie between 0° and 180°. The right-hand axis is for negative θ and for α/θ values shown on the right-hand part of the graph. All curves lie between 0° and −180°. Thus, if experimental curves of Ψ̂xH (Ψ̂xV) are strictly positive (negative) this would either indicate a mean positive (negative) θ is present or again could indicate the presence of polarization errors. As Fig. 6 shows, if the θ and α are small (as expected in rain) with a ratio of about 1, then the Ψ̂xH and Ψ̂xV will cluster around ±90° after a few tens of degrees of ϕdp are accumulated. In the experimental section this is shown to be the case for the CSU–CHILL radar data collected during the Severe Thunderstorm Electrification and Precipitation Study (STEPS).

5. Polarization errors

a. Theory

The ideal two-channel H/V system would radiate alternately pure H and V polarized electromagnetic waves and then also receive pure H and V polarized components of the scattered wave. In practice, polarization errors are introduced by cross coupling in the radar microwave circuitry and in imperfections in the polarization purity of the antenna pattern, which can vary throughout the beam, especially where cross-polarized lobes exist (Bringi and Chandrasekar 2001). For distributed precipitation media, the resulting error is an integrated effect. Polarization errors have been covered in detail by McCormick (1981) whose treatment was analytical and was applied primarily to circular polarization basis. Here, the polarization errors are integrated and represented by a single complex error term for each channel. The polarization errors are easily included in the model by pre- and postmultiplying 𝗦 of (5) by the error matrix ϒ,
eϒTϒ
where
i1520-0426-20-7-1011-e17
with constraints i2H + |ɛH|2 = i2V + |ɛV|2 = 1 where iH and iV are real. The polarization errors of the H and V channels are represented by the complex numbers ɛH and ɛV, respectively. The polarization errors can also be equivalently represented with the geometric ellipse parameters of tilt angle τ and ellipticity angle ε. These variables are related by (Azzam and Bashara 1989)
i1520-0426-20-7-1011-e18
where χ is the polarization ratio given in (9) and ℜ and ℑ denote real and imaginary parts, respectively. For H errors, χ = ɛH/iH and for V errors, χ = iVV. As can be seen from the equations, if the ɛHV) is real then ε is zero, and if ɛHV) is imaginary then τ is zero. If the errors are orthogonal, that is, ɛV = −ɛ*H, then ϒ is unitary and (16) represents an orthogonal change of polarization basis. Separating the polarization errors into their geometric components gives a convenient and intuitive way to analyze polarization errors. Because the copolar covariances are negligibly affected by small polarization errors, we next investigate the effect of polarization errors on just the covariances involving the cross polar signals. Figures 7 and 8 show the effect of tilt angle error on LDRH and ρxH, with σα as a parameter for ϕPdp = 0. For |τ| < 0.5°, ρxH reaches a maximum of 0.15 when σα = 5° and is always below 0.1 when σα > 10°. The LDRH is much less affected by tilt angle errors remaining nearly constant for σα = 10°, 20° and varying only a little more than 1 dB for σα = 5°. Obviously, LDRH is much more sensitive to σα than to τ for the range of values selected. The curves in Figs. 7 and 8 are identical to portions of the curves shown in Figs. 2 and 3. This occurs because tilt errors, when defined as above, produce the same effect as α; that is, τ = 2° is identical to α = −2° in terms of the resulting radar signatures. Figures 9 and 10 show the effect of ellipticity angle error ε on LDRH and ρxH with σα as a parameter. Both LDRH and ρxH are much more sensitive to ellipticity errors than tilt errors and the sensitivity increases for decreasing σα. As can be seen, for a constant error, as σα decreases LDRH decreases, but ρxH increases. For example, in light stratiform rain where raindrops are likely well aligned with a small σα, if there is a small tilt angle or ellipticity angle error, LDR can be very low while ρxV and ρxH can be quite high.

The co-to-cross phases ΨxH and ΨxV are very sensitive to polarization errors and understanding these effects is essential in determining the calibration offset phases (see Part I). If there are no polarization errors and the propagation matrix is diagonal, then ΨxHxV) start at 0° and decrease (increase) as ϕdp/2 is a function of ϕPdp. If the polarization errors are real only (i.e., tilt angle error with no ellipticity angle error), then the phases ΨxH and ΨxV are zero when ϕdp is zero. In other words, tilt angle errors do not affect the offset phase of ΨxH and ΨxV. If the polarization errors are imaginary only (i.e., ellipticity angle error with no tilt angle error), then the phases ΨxH and ΨxV are offset by ±90° when ϕdp is zero. When the polarization errors are complex (i.e., both tilt and ellipticity angle errors are present), the phase offsets of ΨxH and ΨxV can be any angle but they will be equal so that the condition ΨxV − ΨxHϕdp (11) still generally holds true.There are some discrepancies depending on the polarization errors, but these deviations are very small and (11) remains a very good approximation.

6. Estimating polarization errors from data

To construct covariance matrices that are accurate representations of the scattering medium, first the magnitude and phase biases due to the radar system need to be assessed as was done in Part I. That model, however, did not account for the possible presence of polarization errors that can affect the ΨxH and ΨxV greatly. Inclusion of polarization errors affects the phase offsets of the various interchannel covariances and further adjustments to the phase offset estimates, as given in Part I, may be required.

Evidence of phase offset due to polarization errors is found in the ice phase of storms where the mean canting angle of the precipitation in the backscatter resolution volume should be on average zero. There are circumstances where nonzero mean canting can be found in the ice phase, that is, due to electric fields (Caylor and Chandrasekar 1996), and these areas need to be avoided for the following technique to be valid. It is possible to determine the mean canting angle of the particles in the backscatter resolution volume from optimum polarizations determined via matrix methods (Tragl et al. 1991). Using this method, range profiles of mean canting angle in the ice phase of a convective storm can be constructed where very little or no propagation effects are present (e.g., no differential propagation phase or attenuation is present). If the average mean canting angle of all of the resolution volumes is not approximately zero, then the phases ΨxH and ΨxV can be adjusted by ϕerr as
i1520-0426-20-7-1011-e20
until the average of the mean canting angles is zero. The ϕx_off is estimated as discussed in Part I of this paper. This then completes the process of calibrating covariance matrices for the purpose of estimating polarization errors.
The polarization error estimation method is based on the premise that the sum of ρxV or ρxH along a range profile in rain will be minimum when the polarization basis is H/V. It is assumed that the mean canting angle of the propagation medium will be, on average, zero. If the polarization basis is H/V, then no depolarization occurs along the propagation path and ρxV and ρxH are determined solely by the backscatter medium. The model shows that if there is a mean canting angle of the propagation medium, then there will be depolarization along the propagation path and this will in general cause ρxV and ρxH to increase and be a function of ϕdp. Polarization errors effectively cause the same phenomena. With polarization errors present, depolarization occurs along the propagation path and in general this causes ρxV and ρxH to be higher than if the polarization basis were pure H/V. Thus, in general, polarization errors will increase the average value of the ρxV or ρxH over a range profile where significant (>90°) ϕdp accumulation is present. In order to observe any systematic shifts in ρxV or ρxH along range profiles, it is necessary to have significant accumulation of ϕdp. Depending upon the polarization errors, the range profiles of ρxV or ρxH can increase or decrease as a function of ϕdp achieving maximums or minimums for various values of ϕdp. But in any event, the average value of ρxV or ρxH over a significant range of accumulated ϕdp will be minimized in the H/V polarization basis. For each individual resolution volume, the covariance matrix will yield a nonzero tilt and ellipticity angle that minimizes ρxV or ρxH, but the mean value of ρxV or ρxH over range will be minimized when the polarization basis is H/V. This is the basis of the polarization error determination method. Thus, if ρxV and ρxH are high in rain data, then it is reasonable to assume that polarization errors are the cause barring other known anomalies (e.g., interference, three-body scattering, second trip echos, radar malfunction). Finding error terms that minimize ρxV or ρxH should yield unique polarization errors because eigenpolarizations (or optimum polarizations) of the propagation medium are unique (i.e., the wave polarizations that pass through the medium without depolarization; Azzam and Bashara 1989). Instead of minimizing the cross covariance, there is a more general condition for optimum polarizations. From the general covariance matrix theory (Tragl 1990; Tragl et al. 1991; Lüneburg et al. 1991), optimum polarizations for distributed scatterers are solutions to
SaaS*abχS*bbSabχ
where χ is polarization ratio variable as given in (7). The tilt angle of the characteristic optimum polarization basis (corresponding do the smallest eigenvalue) gives the mean canting angle of the precipitation medium (if no polarization errors are present) and can be found via analytical matrix methods (Tragl et al. 1991; Lüneburg et al. 1991), which are based on orthogonal basis transformations. However, polarization errors can be nonorthogonal, so that finding the tilt and ellipticity angles via the above-mentioned analytical method may not reveal the true error terms. Thus, the errors here are determined by finding values of the error terms, τH, εH, τV, εV, that minimize 〈SaaS*ab〉 − 〈S*bbSab〉, and this is done via a simple search method.
The mechanics of the polarization error estimation method are as follows. A range profile(s) of data is selected where there is significant increase in Ψdp, which will accentuate the errors thus making it possible to detect small polarization errors. For example, if significant tilt error is present (a few tenths of a degree is sufficient), then in general ρxH and ρxV will increase significantly with increasing Ψdp. Calibrated covariance matrices are constructed at each sample point (spaced at 0.15 km in the following case). For all the selected covariance matrices, error terms τH, τV, εH, and εV, are varied and the minimum of the sum
i1520-0426-20-7-1011-e23
is found where N is the total number of covariance matrices. The resulting tilt and ellipticity angles are considered the integrated polarization errors.

a. Data analysis

Shown in Figs. 1114 is a ray of data gathered with the CSU–CHILL radar (Brunkow et al. 2000) during STEPS on 21 July 2000 through a heavy rain cell with reflectivities of 40–60 dBZ. Figure 11 shows Ψdp increasing 100° over 50 km and a filtered version of ΨxV − ΨxH which mimics Ψdp very well, as expected. Figure 12 shows a range profile of Ψ̂xV [see (15)] with 150-m range sampling. According to the model, if the mean propagation canting angle is zero, then this phase should be 0° or 180° (assuming δ = 0°), depending on the mean canting angle of the backscatter resolution volume. Because this phase is nearly always positive, we surmise from Fig. 6 that θ must be positive or, more likely, that polarization errors are present. Thus, if the mean canting angle of the propagation medium is zero, then there is likely negative polarization tilt error present (tilt error τ is opposite in sign to θ). Figure 12 is not a single ray anomaly but rather is seen in nearly all range profiles of CSU–CHILL data gathered during STEPS that possess large amounts of increasing Ψdp. There are 333 covariance matrices available through the region of large phase accumulation that are used in the minimization procedure (23). The resulting polarization errors are τH = −0.5°, εH = −0.1°, τV = 89.5°, and εV = 0.4°. Using these error terms, a 3 × 3 transformation matrix can be constructed and the polarization errors can be removed from the data by pre- and post multiplying the measured covariance matrices by this matrix (Huang et al. 2001). Range profiles of the corrected data can then be made. Figure 13 shows measured ρxH and corrected ρxH and, as expected, corrected ρxH is significantly reduced to an average level that is consistent with rain. Similarly, Fig. 14 shows filtered and corrected LDR. Again, the the corrected LDR is reduced as it should be with the polarization errors removed. As an internal consistency check as to the validity of the estimated polarization errors, the error terms are next put into the scattering model. The canting angles θ and α are set to zero and the canting angle distribution is Fisher where σα is 10°. Figure 15a shows the resulting ρxH (solid line) and LDRH (dashed line) while Fig. 15b shows Ψ̂xH, all as functions of principal plane ϕdp. When comparing these figures to the experimental data shown in Figs. 1214, it can be seen that the trends and general behavior predicted by the model are also observed in the data.

7. Summary and conclusions

A complete numerical covariance matrix model is described where the parameters of the propagation and backscatter medium can be independently varied. Polarization errors are also included. The propagation medium was modeled via a 2 × 2 matrix (i.e., coherent scattering) and the backscatter medium was modeled by a covariance matrix constructed by the T-matrix method for an ensemble of raindrops.The model was used to examine the effect of a mean canting angle of both the propagation and backscatter medium, the standard deviation of the canting angles of the backscatter resolution volume, and polarization errors on the co-to-cross covariances and LDR. It was found that ρxH (ρxV) is more sensitive to the mean backscatter canting angle than LDR. Furthermore, as the standard deviation of canting angle increases, LDR increases but ρxH (ρxV) decreases. The ρxH (ρxV) is also more sensitive to the mean propagation canting angle, θ. As a practical application, examination of range profiles of ρxH (ρxV) data along propagation paths with large ϕdp may be a useful indicator of how well the “hybrid” method proposed for polarimetric upgrade of the WSR-88D would work (Doviak et al. 2000). The phases ΨxHxV)are also very sensitive to the mean canting angles, α and θ, as demonstrated in Fig. 6, especially after the accumulation of 30° or 40° of ϕdp. For these reasons, the co-to-cross covariances are particularly good for detecting small polarization errors. For CSU–CHILL data gathered during STEPS, it was noticed that whenever large accumulation of Ψdp was present, Ψ̂xH and Ψ̂xV would “saturate” around 90° after the accumulation of 30° to 40° of ϕdp. If θ = 0° and no polarization errors were present, then the phases Ψ̂xH and Ψ̂xV would appear almost random in range profiles. Thus, because raindrops should not possess any preferred mean canting angle (other than zero), we believe this observed systematic phase behavior indicates the presence of small system polarization errors. Using the developed algorithm, the integrated errors were determined to be τH = −0.5°, εH = −0.1°, τV = 89.5°, and εV = 0.4°. These polarization errors were then used in the scattering model (with α = θ = 0°) and the resulting ρxH, LDRH, and Ψ̂xHmatched the behavior of the experimental range profiles very well, offering an explanation for the observed behavior of the data and lending credence to the overall methodology.

Thus, the co-to-cross covariances of weather radars signals potentially contain valuable information about not only precipitation microphysics, but also the calibration of the radar. By direct inspection of the co-to-cross covariances in regions of large ϕdp accumulation it can be seen if significant polarization errors are present or if a nonzero mean propagation canting angle exists. If polarization errors are present and can be well modeled by the 2 × 2 matrix representation given in this paper, then it is possible to remove these errors from the data. The application of the full covariance matrix data to microphysical retrieval will be given in a future paper.

Acknowledgments

The CSU–CHILL radar is operated by Colorado State University via a cooperative agreement with the National Science Foundation (NSF) ATM-9500108. The authors were supported by this grant and via NSF Grant ATM-9982030. The authors acknowledge the outstanding effort of Dave Brunkow, Pat Kennedy, and Robert Bowie in deploying and operating the radar for STEPS.

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  • Waterman, P. C., 1971: Symmetry, unitarity and geometry in electromagnetic scattering. Phys. Rev., D3. 825839.

Fig. 1.
Fig. 1.

The std dev of the phase of the co-to-cross covariance (ϕxH or ϕxV) as a function of the co-to-cross correlation coef (ρxH or ρxV) with the normalized spectral with συn as a parameter; συn = σV/λ/(2Ts)

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 2.
Fig. 2.

Co-to-cross correlation coef as a function of mean backscatter canting angle (α) with std dev (σα) as a parameter.

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 3.
Fig. 3.

LDRH as a function of mean backscatter canting angle (α) with the std dev of (σα) as a parameter

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 4.
Fig. 4.

LDRH as functioin of principal plane ϕdp with mean propagation canting angle θ as a parameter. The mean backscatter canting angle α is 0°

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 5.
Fig. 5.

Same as in Fig. 4, but for the co-to-cross correlation coefficient ρxH

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 6.
Fig. 6.

The phases Ψ̂xV and Ψ̂xH as a function of ϕPdp with the ratio α/θ as a parameter. The left-hand axis corresponds to positive values of θ, while the right-hand vertical axis corresponds to negative values of θ

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 7.
Fig. 7.

LDRH as a function of tilt angle error with σα as a parameter and ϕPdp = 0°

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 8.
Fig. 8.

The ρxH as a function of tilt angle error with σα as a parameter and ϕPdp = 0°

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 9.
Fig. 9.

The LDRH as a function of ellipticity angle error with σα as a parameter and ϕPdp = 0°

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 10.
Fig. 10.

The ρxH as a function of ellipticity angle error with σα as a parameter and ϕPdp = 0°

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 11.
Fig. 11.

Range profile of Ψdp and ΨxV − ΨxH

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 12.
Fig. 12.

Range profile of ΨxV − Ψdp/2

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 13.
Fig. 13.

Range profile of filtered ρxH and polarization error corrected ρxH

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 14.
Fig. 14.

Range profile of filtered LDR and polarization error corrected LDR

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Fig. 15.
Fig. 15.

Using the estimated polarization errors, (a) ρxH (solid), LDRH (dashed), and (b) ΨxH − Ψdp/2 as functions of principal plane ϕdp from the theoretical model

Citation: Journal of Atmospheric and Oceanic Technology 20, 7; 10.1175/1456.1

Table 1.

The phases δ, δxH , and δxV as a function of Deq valid for 0° < α < 20° with ϕ Pdp = 0° at C band (no propagation effects)

Table 1.
*

Current affiliation: National Center for Atmospheric Research, Boulder, Colorado.

1

Note the difference of complex conjugation in the definition here of ρxH and ρxV as compared to Ryzhkov [2001, his Eqs. (3) and (4)].

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