The Impact of Beam Broadening on the Quality of Radar Polarimetric Data

Alexander V. Ryzhkov Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, and NOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma

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Abstract

The impact of beam broadening on the quality of radar polarimetric data in the presence of nonuniform beam filling (NBF) is examined both theoretically and experimentally. Cross-beam gradients of radar reflectivity Z, differential reflectivity ZDR, and differential phase ΦDP within the radar resolution volume may produce significant biases of ZDR, ΦDP, and the cross-correlation coefficient ρhv. These biases increase with range as a result of progressive broadening of the radar beam. They are also larger at shorter radar wavelengths and wider antenna beams.

Simple analytical formulas are suggested for estimating the NBF-induced biases from the measured vertical and horizontal gradients of Z, ZDR, and ΦDP. Analysis of polarimetric data collected by the KOUN Weather Surveillance Radar-1988 Doppler (WSR-88D) demonstrates that frequently observed perturbations of the radial ΦDP profiles and radially oriented “valleys” of ρhv depression can be qualitatively and quantitatively explained using the suggested NBF model.

Corresponding author address: Alexander V. Ryzhkov, CIMMS/NSSL, 120 David L. Boren Blvd., Norman, OK 73072. Email: Alexander.Ryzhkov@noaa.gov

Abstract

The impact of beam broadening on the quality of radar polarimetric data in the presence of nonuniform beam filling (NBF) is examined both theoretically and experimentally. Cross-beam gradients of radar reflectivity Z, differential reflectivity ZDR, and differential phase ΦDP within the radar resolution volume may produce significant biases of ZDR, ΦDP, and the cross-correlation coefficient ρhv. These biases increase with range as a result of progressive broadening of the radar beam. They are also larger at shorter radar wavelengths and wider antenna beams.

Simple analytical formulas are suggested for estimating the NBF-induced biases from the measured vertical and horizontal gradients of Z, ZDR, and ΦDP. Analysis of polarimetric data collected by the KOUN Weather Surveillance Radar-1988 Doppler (WSR-88D) demonstrates that frequently observed perturbations of the radial ΦDP profiles and radially oriented “valleys” of ρhv depression can be qualitatively and quantitatively explained using the suggested NBF model.

Corresponding author address: Alexander V. Ryzhkov, CIMMS/NSSL, 120 David L. Boren Blvd., Norman, OK 73072. Email: Alexander.Ryzhkov@noaa.gov

1. Introduction

The Joint Polarization Experiment (JPOLE) and other validation studies demonstrate superior performance of dual-polarization radar for rainfall estimation and radar echo classification (e.g., Ryzhkov et al. 2005a). These polarimetric products, however, have been validated at relatively close distances from the radar. To our knowledge, the maximal distance at which polarimetric analysis and/or classification was ever verified using in situ measurements is 120 km (Loney et al. 2002). In most studies, the largest range to which polarimetric rainfall estimation was tested with rain gauges does not exceed 100 km (e.g., Brandes et al. 2001, 2002; May et al. 1999; Le Bouar et al. 2001; Ryzhkov et al. 2005b). On the other hand, Giangrande and Ryzhkov (2003) and Ryzhkov et al. (2005a) show that although the polarimetric method for rain measurements still outperforms the conventional one beyond 100 km from the radar, the degree of improvement decreases with distance.

Progressive beam broadening and stronger impact of nonuniform beam filling (NBF) is one of the reasons why the quality of polarimetric information deteriorates with range. Beam broadening is a common problem for both polarimetric and conventional (nonpolarimetric) radar. The issue of the vertical profile of reflectivity (VPR) correction for precipitation measurements with conventional radar is addressed in extended literature (see, e.g., the overview in Meischner 2004). Much less effort has been made to assess similar problems regarding polarimetic variables such as the differential reflectivity ZDR, the differential phase ΦDP, the specific differential phase KDP, the depolarization ratio LDR, and the cross-correlation coefficient ρhv.

Adverse effects of NBF on polarimetric measurements are further exacerbated if the antenna patterns for horizontal and vertical polarizations are not identical. Theoretical formulas for the ZDR, LDR, and ρhv biases caused by the antenna pattern mismatch are presented in the book by Bringi and Chandrasekar (2001). The errors in ZDR due to mismatched copolar patterns together with intrinsic reflectivity gradients across the beam can be quite high at the periphery of strong storm cores (Herzegh and Carbone 1984; Pointin et al. 1988).

NBF may also cause significant perturbations of the radial profile of the differential phase (Ryzhkov and Zrnic 1998; Gosset 2004). Such perturbations of ΦDP result in spurious values of its radial derivative KDP and strong biases in the KDP-based estimates of the rain rate. These adverse effects are commonly manifested as the appearance of negative KDP in the regions of strongly nonuniform precipitation and become more pronounced as the physical size of the radar resolution volume increases at longer distances.

The magnitude of the cross-correlation coefficient ρhv is closely related to the distribution of the differential phase within the radar resolution volume. Large cross-beam gradients of ΦDP may cause noticeable decrease of ρhv, which is, in its turn, accompanied by higher statistical errors in the measurements of all polarimetric variables (Ryzhkov 2005).

Strong vertical gradients of radar variables are commonly observed in the presence of the bright band in startiform rain. Beam broadening causes notable smearing of the brightband polarimetric signatures at the distances as close as 40–50 km from the radar (Giangrande et al. 2005). Such a smearing makes polarimetric classification of the melting layer more difficult, and estimation of rainfall becomes a challenge.

In this paper, we attempt to quantify the effects of beam broadening on polarimetric measurements using a simple model of NBF. We assume that the antenna patterns at the two orthogonal polarizations are perfectly matched and the biases of the measured ZDR, ΦDP, and ρhv are solely due to linear cross-beam gradients of different radar variables. In section 2, closed-form analytical solutions for the biases are obtained using this simplified model of gradients and the Gaussian antenna pattern. Section 3 contains analysis of the cross-beam gradients and the corresponding biases estimated from real data collected with the polarimetric prototype of the S-band Weather Surveillance Radar-1988 Doppler (WSR-88D) in Oklahoma. In section 4, we simulate the smearing effect of beam broadening on the polarimetric signatures of the melting layer for different antenna beamwidths and compare results of simulations with observational data. Finally, in section 5 we discuss practical implications of the observed effects.

2. Theoretical analysis

In the case of weather scatterers, the voltage vectors of the transmitted (Vt) and received (V) waves are related as
i1520-0426-24-5-729-e1
where matrix elements Shh, Svv, and Shv represent backscattering coefficients of hydrometeors in the radar resolution volume and Thh and Tvv describe phase shifts and attenuations for H and V waves along the propagation path:
i1520-0426-24-5-729-e2
i1520-0426-24-5-729-e3
where Φh,v is the phase shift and Γh,v is the attenuation. The differential phase ΦDP is defined as
i1520-0426-24-5-729-e4
The coefficient C1 is a constant depending on radar parameters and range from the scatterers (see the appendix). If both H and V waves are transmitted simultaneously [i.e., Vt = (Vt, Vt)], then
i1520-0426-24-5-729-e5
i1520-0426-24-5-729-e6
In our analysis we will neglect the cross-coupling terms proportional to Shv in (5)(6), which is reasonable assumption for rain and aggregated snow (Doviak et al. 2000).
Using (5) and (6), we introduce effective radar reflectivity factors Z(e)h,v at orthogonal polarizations as
i1520-0426-24-5-729-e7
and
i1520-0426-24-5-729-e8
the effective differential reflectivity
i1520-0426-24-5-729-e9
and the covariance
i1520-0426-24-5-729-e10
where
i1520-0426-24-5-729-e11
In (7)(11), intrinsic values of the radar reflectivities Zh,v, the differential reflectivity Zdr, and the cross-correlation coefficient ρhv are defined from the second moments of the scattering matrix 𝗦:
i1520-0426-24-5-729-e12
Overbars in (7), (8), and (10) mean expected values and brackets in (12) stand for ensemble averaging. The factors C2 and C′ are constants defined in the appendix. In the absence of propagation effects and cross coupling, the effective reflectivity factors are equal to their intrinsic values.
The radar-measured reflectivities Z(m)h,v and the covariance R(m)hv are weighted by the radar antenna pattern I(r, r0) as follows (see the appendix for details):
i1520-0426-24-5-729-e13
i1520-0426-24-5-729-e14
In (13) and (14), it is assumed that antenna patterns are identical at the two orthogonal polarizations. The measured differential phase Φ(m)DP and cross-correlation coefficient ρ(m)hv are
i1520-0426-24-5-729-e15
The values of ΦDP(m) and ρhv(m) depend on the distributions of Z(e)h,v and Rhv within the radar resolution volume and on the shape of antenna pattern. In this study, we assume that a two-way antenna power pattern is axisymmetric and Gaussian (Doviak and Zrnic 1993):
i1520-0426-24-5-729-e16
where θ and ϕ are elevation and azimuth, respectively, and σ = Ω/4(ln2)1/2 (Ω is a one-way 3-dB antenna pattern width).
Next we assume that reflectivity factors Z(e)h,v and Zhv expressed in logarithmic scale vary linearly in both cross-beam directions, θ and ϕ:
i1520-0426-24-5-729-e17
i1520-0426-24-5-729-e18
Similar assumption is made for differential phase Φ′DP:
i1520-0426-24-5-729-e19
Note that, throughout the paper, an uppercase subscript is attributed to radar reflectivity and differential reflectivity in logarithmic scale, whereas lowercase subscript signifies the corresponding variables expressed in the linear scale. Arguments (0, 0) in Eqs. (17)(19) correspond to the center of the antenna beam.
As a result,
i1520-0426-24-5-729-e20
i1520-0426-24-5-729-e21
where
i1520-0426-24-5-729-e22
i1520-0426-24-5-729-e23
i1520-0426-24-5-729-e24
i1520-0426-24-5-729-e25
The measured differential reflectivity is expressed as
i1520-0426-24-5-729-e26
Since
i1520-0426-24-5-729-e27
we can further simplify
i1520-0426-24-5-729-e28
Or, equivalently,
i1520-0426-24-5-729-e29
The measured differential phase can be written as
i1520-0426-24-5-729-e30
where
i1520-0426-24-5-729-e31
Similarly, the magnitude of the measured cross-correlation coefficient is expressed as
i1520-0426-24-5-729-e32
where
i1520-0426-24-5-729-e33
i1520-0426-24-5-729-e34
If |ρhv| ≈ 1, then
i1520-0426-24-5-729-e35
The coefficient ξ2 is usually very close to 1, hence we will ignore this term in our further considerations.
Expressing the parameter σ via the antenna beamwidth Ω, we finally arrive at the following approximate formulas for the biases of ZDR, ΦDP, and ρhv that will be used in the subsequent analysis in the paper:
i1520-0426-24-5-729-e36
i1520-0426-24-5-729-e37
i1520-0426-24-5-729-e38
In Eqs. (36)(38), Φ′DP, ΔΦDP, Ω, θ, and ϕ are expressed in degrees, whereas ZH, ZHV, ΔZDR, and ZDR are in decibels.
Similar formulas can be obtained for the NBF-related bias of the radar reflectivity factor at horizontal polarization:
i1520-0426-24-5-729-e39
As expected, the NBF-induced bias of Z is always positive if reflectivity varies linearly in both orthogonal directions within the radar resolution volume. It follows from Eq. (39) that for a 1° beam the corresponding Z bias exceeds 1 dB if the gradient of Z is higher than 10 dB deg−1 in any of the two transverse directions. We will not address ΔZH anymore in the paper since the focus of this study is on the impact of beam broadening on the quality of polarimetric variables.

3. NBF effects in the case of the mesoscale convective system

The gradients of ZH, ZHV, ZDR, and ΦDP in Eqs. (36)(38) can be approximately estimated from real data by comparing the corresponding variables at adjacent radials. We perform such estimation in the case of a mesoscale convective system (MCS) that was observed with the polarimetric prototype of the S-band WSR-88D radar (hereafter KOUN) in central Oklahoma on 2 June 2004. The analysis was conducted using the data from two lowest plan position indicators (PPIs) at elevations of 0.44° and 1.45°. Horizontal gradients were computed from the data collected at the lowest elevation, whereas vertical gradients were estimated using the data at both elevations.

Strictly speaking, such a procedure underestimates the magnitude of intrinsic gradients because the data are smeared with the antenna beam. Indeed, the transverse dimension of the radar resolution volume exceeds 3 km at 200 km from the radar if the antenna beamwidth is 1°. Hence, smaller-scale cross-beam nonuniformities of the precipitation field are not resolved. Nevertheless, as will be shown later, these approximate estimates of gradients prove to be very useful for evaluating the quality of polarimetric data.

A composite plot of ZH, ZDR, ΦDP, and ρhv at elevation 0.44° (Fig. 1) corresponds to the time when an extensive squall line passes over the radar and produces tremendous attenuation and differential attenuation that are clearly visible in the eastern sector. The radar reflectivity factor and differential reflectivity are deliberately not corrected for attenuation in order to estimate the gradients of Z(e)H, ZHV, and Z(e)DR, which are affected by attenuation according to their definition in (7)(11). High values of ΦDP in the eastern sector are accompanied by negatively biased Z and ZDR and a pronounced drop in the cross-correlation coefficient ρhv. While the drop in ZDR well below −2 dB is caused by differential attenuation, the decrease in ρhv is a result of NBF.

This is confirmed by Fig. 2 where the fields of the parameters ΔZDR, ΔΦDP, and ξ computed from Eqs. (36)(38) are displayed together with ZH. The ρhv depression in Fig. 2d is very well correlated with the observed decrease of the measured ρhv in Fig. 1. The magnitude of the negative ρhv bias exceeds 0.2. Such a strong bias adversely affects the quality of the polarimetric classification of radar echoes and induces large statistical errors in the estimates of all polarimetric variables. Similar radial features or “valleys” of lower ρhv are frequently observed in the KOUN polarimetric data. Their primary cause is large vertical gradient of ΦDP. The ray at a higher elevation overshoots precipitation at closer distances from the radar than the ray at lower tilt. Therefore, the differential phase at higher tilt stops increasing earlier (i.e., at closer slant ranges) than the one at lower tilt. While both higher and lower rays are still in rain, the differential phases at the two rays grow proportionally and the difference between them is not high. However, once the higher ray intercepts the freezing level, the corresponding ΦDP stops increasing, whereas ΦDP at the lower ray continues its growth. This explains a radial character of the observed artifacts and their severity, which progresses with range.

According to (38), large gradients of ΦDP are responsible for the decrease in ρhv. In contrast, perturbations of ΦDP are determined by both the gradients of the differential phase and the reflectivity factor. As a result, ΔΦDP exhibits more complex and nonmonotonic behavior along the radial than the factor ξ. If the reflectivity field is relatively uniform as in the stratiform region of the MCS north-northeast of the squall line, then the gradients of ΦDP dominate and apparent radial features are evident in the field of ΔΦDP.

The NBF-related bias in differential reflectivity can also be significant and may exceed several tenths of a decibel as Fig. 2b shows. Positive biases of ZDR are common in convective areas of the storm not far away from the radar, whereas negative biases are prevalent at longer distances in convective and stratiform parts of the MCS. The latter feature is explained by the general decrease of ZDR with height. The ZDR biases, as well as the biases in ΦDP and ρhv, tend to increase with range as a result of beam broadening.

A similar analysis was performed on the data collected for the same storm but 2 h after the squall line passed over the radar and was viewed at a different angle (Figs. 3 and 4). At that moment, attenuation effects were much weaker and the differential phase was significantly lower. Again, the area of ρhv depression is well predicted from the analysis of gradients. The perturbations of the ΦDP radial profiles are also in good agreement with their estimates from the gradients in accordance with Eq. (37).

In Fig. 5, measured range dependencies of ΦDP (thin curves) are compared with radial profiles of ΔΦDP calculated from (37) (thick curves) for six successive azimuths belonging to the sector indicated in Fig. 4d. Despite many simplified assumptions made in the evaluation of ΔΦDP, the correlation between the ΦDP and ΔΦDP profiles is surprisingly high. The most pronounced excursions of the ΦDP curves, such as spikes and depressions, are well reproduced in the modeled ΔΦDP. Thus, they are primarily attributed to NBF rather than pure statistical errors in ΦDP estimation or to the contribution from the backscatter differential phase.

4. Beam-broadening effects in the case of stratiform rain

The melting layer or bright band is a special case of strong vertical nonuniformity in stratiform precipitation. The bright band is associated with very well pronounced polarimetric signatures such as the sharp ZDR maximum and ρhv minimum. These signatures have very important prognostic value because the top of the melting layer corresponds to the freezing level and its bottom represents the boundary between pure liquid and mixed-phase hydrometeors. The latter one marks the onset of the brightband contamination in radar rainfall estimates. Accurate designation of the melting layer is a key for successful discrimination between liquid and frozen hydrometeors (Giangrande et al. 2005).

Because the thickness of the bright band is only few hundreds of meters, the corresponding polarimetric signatures degrade very rapidly with range even for the radar beam as narrow as 1°. This degradation is illustrated in the range–height indicator (RHI) plot of Z, ZDR, ΦDP, and ρhv measured with the KOUN radar on 7 April 2002 (Fig. 6). To quantify the degree of such deterioration at longer distances from the radar one has to use a more sophisticated model of NBF than is described in section 2.

For the case illustrated in Fig. 6, we obtained average vertical profiles of all radar variables at very close distances from the radar and modeled the RHI fields of Z, ZDR, ΦDP, and ρhv at the S band for different antenna beamwidths assuming the horizontal homogeneity of the intrinsic fields of these radar variables. The results of such modeling studies are presented in Figs. 7 and 8 for antenna beamwidths at 1° and 2°. Modeled fields in Fig. 7 are very consistent with what was actually observed with the same antenna beamwidth (Fig. 6). This means that the model adequately reproduces observational data.

A twofold increase of the radar beamwidth leads to the enhanced brightband contamination of the low-altitude echoes in rain (Fig. 8). At the lowest elevations, the differential reflectivity and cross-correlation coefficients quickly acquire the values typical for melting hydrometeors. As in the case of the MCS, vertical nonuniformity causes wavelike perturbation of the ΦDP profile in the melting layer as was explained by Ryzhkov and Zrnic (1998). Below the melting layer, the mean value of ΦDP is less biased but differential phase becomes more noisy due to lowering of ρhv at the altitudes below the physical (i.e., intrinsic) bottom of the bright band.

5. Discussion

The findings in this study may have important practical implications to all users of polarimetric radar data. This is significant in view of the forthcoming polarimetric upgrade of the U.S. National Weather Service network of the WSR-88D radars. One should avoid using polarimetric variables in a quantitative manner in the areas where these variables are significantly affected by NBF. Such areas can be identified by computing horizontal and vertical gradients of the radar reflectivity, the differential reflectivity, and the differential phase as well as estimating the biases of ZDR, ΦDP, and ρhv according to Eqs. (36)(38). The procedure for gradient estimation is simple and straightforward.

If the magnitudes of ΔZDR, ΔΦDP, and the difference 1 − ξ exceed certain thresholds, then the corresponding variables (ZDR, KDP, and ρhv) should not be used for estimating polarimetric products in these areas. The choice of such thresholds is dictated by tolerable errors that depend on particular applications. For example, the ZDR bias has to be less than 0.2 dB if ZDR is utilized for rainfall estimation. The biases of ΦDP within ±2° are acceptable because the statistical fluctuations of the ΦDP estimate are between 1° and 2° for typical dwell times used for operational weather radars. The bias of 0.02 in ρhv may also be tolerable for classification purposes.

In addition to the negative impact on the quality of polarimetric classification, the decrease of ρhv is detrimental for statistical accuracy of the estimates of ZDR, ΦDP, and ρhv itself. Indeed, the standard deviations of the estimates for all three variables are proportional to (1 − ρ2hv)1/2 (Bringi and Chandrasekar 2001). This means that if ρhv drops from 0.99 to 0.90, the corresponding errors increase 3 times.

Perturbations of the ΦDP radial profile produce erroneous estimates of KDP of both signs. Although negative KDPs are easily identified (and sometimes taken out as unphysical), positively biased KDPs usually go undetected. Since KDP is a slope of the ΦDP radial profile, the bias in KDP is not necessarily zero if ΔΦDP = 0. Thus, the data with ΔΦDP = 0 in the vicinity of large |ΔΦDP| should be also scrutinized.

The magnitudes of ΔZDR, ΔΦDP, and 1 − ξ depend on the square of antenna beamwidth. Such a strong dependence may preclude the use of wide-beam antennas for polarimetric measurements. A twofold increase of the beamwidth from 1° to 2° leads to 4-times-larger biases and significant deterioration of the melting layer designation as Figs. 7 and 8 show.

The biases of ΦDP and ρhv are wavelength dependent because the differential phase and its gradients are inversely proportional to the radar wavelength λ. The impact on ΔΦDP is proportional to λ−1, whereas the ρhv bias is approximately proportional to λ−2. Enhanced attenuation and differential attenuation at shorter wavelengths may either increase or decrease the gradients of Z and ZDR. In some situations, these changes in the Z and ZDR gradients may offset the increase in the gradient of ΦDP and its greater impact on the NBF-related biases in ΦDP and ρhv. However, cursory analysis of the C- and X-band-simulated and observed polarimetric data reveals stronger NBF effects compared to the S band (Ryzhkov and Zrnic 2005). Although range coverage of the shorter-wavelength radars is usually smaller than the one for S-band weather radars and the antenna beam is not as broad at closer distances, all mentioned problems should be taken seriously. In convective situations, both attenuation and beamwidth effects may restrict the use of polarimetric methods on short-wavelength radars (particularly with antenna beams wider than 1°).

We emphasize that Eqs. (36)(38) cannot be used for correction of ZDR, ΦDP, and ρhv because the bias estimates are very approximate due to many simplifying assumptions made in derivation of these equations. Instead, we recommend using ΔZDR, ΔΦDP, and ξ as quality indexes for the corresponding radar variables. Such an approach is used in the algorithms for hydrometeor classification and rainfall estimation developed at the National Severe Storms Laboratory (NSSL) for operational utilization with the polarimetric prototype of the WSR-88D radar. According to this approach, each radar variable is supplemented with its confidence factor that may depend on ΔZDR, ΔΦDP, and ξ along with a signal-to-noise ratio, the total differential phase (which characterizes potential impact of attenuation/differential attenuation), the magnitude of ρhv (which characterizes the noisiness of polarimetric data), etc.

6. Conclusions

In this study, we evaluate the impact of nonuniform beam filling (NBF) on the quality of polarimetric measurements. It is shown that such an impact can be quite significant, especially at longer distances from the radar due to progressive broadening of the antenna beam.

Relatively simple analytical formulas have been obtained for the NBF-induced biases of the differential reflectivity ZDR, the differential phase ΦDP, and the cross-correlation coefficient ρhv assuming linear gradients of radar reflectivity ZH, ZDR, and ΦDP in the cross-beam directions within the radar resolution volume. It is found that the biases are proportional to the square of the antenna beamwidth. The bias of ZDR does not depend on the radar wavelength, whereas the biases of ΦDP and ρhv increase at shorter wavelength (proportionally to λ−1 in the case of ΦDP and to λ−2 in the case of ρhv). Thus, the NBF effects are stronger at C and X bands than at the S band.

Horizontal and vertical gradients of ZH, ZDR, and ΦDP were estimated from polarimetric data collected by the S-band KOUN WSR-88D radar in a mesoscale convective system. Joint analysis of the measured fields of polarimetric variables and their NBF-induced biases computed from the cross-beam gradients proves that nonuniform beam filling combined with beam broadening is responsible for such commonly observed artifacts as radial “valleys” of ρhv depression and oscillatory behavior of the ΦDP profiles. The latter usually manifests itself as the appearance of negative KDP. It is also shown that polarimetric signatures of the melting layer rapidly degrade with distance as the antenna beam widens.

Although correcting ZDR, ΦDP, and ρhv for such biases is not practical because the biases cannot be estimated with sufficient accuracy, their approximate estimates are important as “quality indexes” of the corresponding polarimetric variables. One should abstain from any quantitative use of the variable if the respective NBF-caused bias exceeds the threshold of acceptability.

These considerations should be taken into account in using polarimetric data at different wavelengths and various angular resolutions and in developing robust algorithms for polarimetric hydrometeor classification and rainfall estimation.

Acknowledgments

Funding for this study was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA/University of Oklahoma Cooperative Agreement NA17RJ1227, the U.S. Department of Commerce, and from the U.S. National Weather Service, the Federal Aviation Administration (FAA), and the Air Force Weather Agency through the NEXRAD Products Improvement Program. I am grateful to Dr. D. Zrnic for reading this manuscript and for making useful comments. The support from the NSSL and the CIMMS, University of Oklahoma staff who maintain and operate the KOUN WSR-88D polarimetric radar is also acknowledged.

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APPENDIX

Equations for the Covariance Rhv

The voltage vectors of the transmitted (Vt) and received (V) waves in the case of individual scatterer are related as (Bringi and Chandrasekar 2001)
i1520-0426-24-5-729-ea1
where matrix elements shh, svv, and shv represent backscattering coefficients of the scatterer, and Thh and Tvv describe phase shifts and attenuations for H and V waves along propagation path:
i1520-0426-24-5-729-ea2
where Φh,v is the phase shift, Γh,v is the attenuation, and ΦDP = 2(Φh − Φv) is a differential phase. The coefficient C1 is defined as
i1520-0426-24-5-729-ea3
In (A3), G is the antenna gain, λ is the radar wavelength, R is the distance between the radar and scatterer, and f2 is the normalized one-way antenna power pattern. It is assumed that the antenna patterns for orthogonal polarizations are the same.
In the case of many scatterers filling the radar resolution volume, the basic Eq. (A1) can be rewritten as
i1520-0426-24-5-729-ea4
where Pt = |Vt|2 [Vt = (Vt, Vt)] and
i1520-0426-24-5-729-ea5
Index i in (A4) and (A5) stands for a number of scatterer. In our derivation we neglect the cross-coupling terms proportional to s(i)hv.
The measured covariance R(m)hv is defined as
i1520-0426-24-5-729-ea6
where
i1520-0426-24-5-729-ea7
The overbar in (A6) means averaging in time. In (A7), R0 is the distance to the center of the radar resolution volume; c is the speed of light; τ is the radar pulse duration; Ω is the one-way 3-dB antenna pattern width, Kw = (ɛw − 1) / (ɛw + 2), where ɛw is the dielectric constant of water. Substituting (A4) into (A6), we obtain
i1520-0426-24-5-729-ea8
i1520-0426-24-5-729-ea9
The summation and time averaging in Eq. (A8) can be replaced by integration over the radar resolution volume:
i1520-0426-24-5-729-ea10
where brackets stand for ensemble averaging and n is the concentration of scatterers.
According to the definition of the cross-correlation coefficient ρhv,
i1520-0426-24-5-729-ea11
and
i1520-0426-24-5-729-ea12
Hence,
i1520-0426-24-5-729-ea13
where Φ′DP = ΦDP + arg(ρhv).
If variables Zh,v, Γh,v, ρhv, and ΦDP are constant within the radar resolution volume, then the measured covariance R(m)hv is equal to its intrinsic value
i1520-0426-24-5-729-ea14
because
i1520-0426-24-5-729-ea15
in the case of the Gaussian axisymmetric antenna pattern (Doviak and Zrnic 1993).
If the covariance Rhv varies within the radar resolution volume but its variation along the radial direction is neglected due to much smaller radial dimension of the radar volume compared to its transverse dimensions at longer ranges from the radar, then the general expression (A13) can be simplified as follows:
i1520-0426-24-5-729-ea16
where
i1520-0426-24-5-729-ea17
and σ = Ω/4(ln2)1/2.

Fig. 1.
Fig. 1.

Composite plot of Z, ZDR, ΦDP, and ρhv measured by the KOUN WSR-88D radar at 2038 UTC 2 Jun 2004. Elevation is 0.44°. Overlaid are contours of Z. No correction for attenuation has been made. The data are displayed for SNR > 5 dB.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2003.1

Fig. 2.
Fig. 2.

Composite plot of (a) Z, (b) ΔZDR, (c) ΔΦDP, and (d) ξ (multiplicative factor of ρhv) corresponding to PPI in Fig. 1. The biases of ZDR, ΦDP, and ρhv are attributed to NBF and computed from Eqs. (36)(38). Overlaid are contours of Z. The data are displayed for SNR > 10 dB.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2003.1

Fig. 3.
Fig. 3.

Composite plot of Z, ZDR, ΦDP, and ρhv measured by the KOUN WSR-88D radar at 2231 UTC 2 Jun 2004. Elevation is 0.44°. Overlaid are contours of Z. No correction for attenuation has been made. The data are displayed for SNR > 5 dB.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2003.1

Fig. 4.
Fig. 4.

Composite plot of (a) Z, (b) ΔZDR, (c) ΔΦDP, and (d) ξ (multiplicative factor of ρhv) corresponding to PPI in Fig. 3. The biases of ZDR, ΦDP, and ρhv are attributed to NBF and computed from Eqs. (36)(38). Overlaid are contours of Z. The data are displayed for SNR > 10 dB.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2003.1

Fig. 5.
Fig. 5.

Radial profiles of ΦDP (thin curves) and its bias (ΔΦDP, thick curves) caused by NBF at six adjacent azimuths within the sector shown in Fig. 4c. The antenna elevation is 0.44°.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2003.1

Fig. 6.
Fig. 6.

Composite RHI plot of Z, ZDR, ΦDP, and ρhv measured with the KOUN WSR-88D radar on 7 Apr 2002.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2003.1

Fig. 7.
Fig. 7.

Composite RHI plot of simulated Z, ZDR, ΦDP, and ρhv for the beamwidth of 1°. Intrinsic vertical profiles of radar variables were obtained from averaging the measured data at close distances from the radar (Fig. 6). Horizontal uniformity is assumed.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2003.1

Fig. 8.
Fig. 8.

Same as in Fig. 7, but for the beamwidth of 2°.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2003.1

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  • Fig. 1.

    Composite plot of Z, ZDR, ΦDP, and ρhv measured by the KOUN WSR-88D radar at 2038 UTC 2 Jun 2004. Elevation is 0.44°. Overlaid are contours of Z. No correction for attenuation has been made. The data are displayed for SNR > 5 dB.

  • Fig. 2.

    Composite plot of (a) Z, (b) ΔZDR, (c) ΔΦDP, and (d) ξ (multiplicative factor of ρhv) corresponding to PPI in Fig. 1. The biases of ZDR, ΦDP, and ρhv are attributed to NBF and computed from Eqs. (36)(38). Overlaid are contours of Z. The data are displayed for SNR > 10 dB.

  • Fig. 3.

    Composite plot of Z, ZDR, ΦDP, and ρhv measured by the KOUN WSR-88D radar at 2231 UTC 2 Jun 2004. Elevation is 0.44°. Overlaid are contours of Z. No correction for attenuation has been made. The data are displayed for SNR > 5 dB.

  • Fig. 4.

    Composite plot of (a) Z, (b) ΔZDR, (c) ΔΦDP, and (d) ξ (multiplicative factor of ρhv) corresponding to PPI in Fig. 3. The biases of ZDR, ΦDP, and ρhv are attributed to NBF and computed from Eqs. (36)(38). Overlaid are contours of Z. The data are displayed for SNR > 10 dB.

  • Fig. 5.

    Radial profiles of ΦDP (thin curves) and its bias (ΔΦDP, thick curves) caused by NBF at six adjacent azimuths within the sector shown in Fig. 4c. The antenna elevation is 0.44°.

  • Fig. 6.

    Composite RHI plot of Z, ZDR, ΦDP, and ρhv measured with the KOUN WSR-88D radar on 7 Apr 2002.

  • Fig. 7.

    Composite RHI plot of simulated Z, ZDR, ΦDP, and ρhv for the beamwidth of 1°. Intrinsic vertical profiles of radar variables were obtained from averaging the measured data at close distances from the radar (Fig. 6). Horizontal uniformity is assumed.

  • Fig. 8.

    Same as in Fig. 7, but for the beamwidth of 2°.

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