## 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 *Z*_{DR}, the differential phase Φ_{DP}, the specific differential phase *K*_{DP}, the depolarization ratio LDR, and the cross-correlation coefficient *ρ*_{hv}.

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 *Z*_{DR}, LDR, and *ρ*_{hv} biases caused by the antenna pattern mismatch are presented in the book by Bringi and Chandrasekar (2001). The errors in *Z*_{DR} 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 Φ_{DP} result in spurious values of its radial derivative *K*_{DP} and strong biases in the *K*_{DP}-based estimates of the rain rate. These adverse effects are commonly manifested as the appearance of negative *K*_{DP} 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 *ρ*_{hv} is closely related to the distribution of the differential phase within the radar resolution volume. Large cross-beam gradients of Φ_{DP} may cause noticeable decrease of *ρ*_{hv}, 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 *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

**V**

*) and received (*

^{t}**V**) waves are related as

*S*

_{hh},

*S*

_{vv}, and

*S*

_{hv}represent backscattering coefficients of hydrometeors in the radar resolution volume and

*T*

_{hh}and

*T*

_{vv}describe phase shifts and attenuations for H and V waves along the propagation path:

_{h,v}is the phase shift and Γ

_{h,v}is the attenuation. The differential phase Φ

_{DP}is defined as

*C*

_{1}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.,

**V**

*= (*

^{t}*V*,

^{t}*V*)], then

^{t}*S*

_{hv}in (5)–(6), which is reasonable assumption for rain and aggregated snow (Doviak et al. 2000).

*Z*

^{(e)}

_{h,v}at orthogonal polarizations as

*Z*

_{h,v}, the differential reflectivity

*Z*

_{dr}, and the cross-correlation coefficient

*ρ*

_{hv}are defined from the second moments of the scattering matrix 𝗦:

*C*

_{2}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.

*Z*

^{(m)}

_{h,v}and the covariance

*R*

^{(m)}

_{hv}are weighted by the radar antenna pattern

*I*(

**r**,

**r**

_{0}) as follows (see the appendix for details):

^{(m)}

_{DP}and cross-correlation coefficient

*ρ*

^{(m)}

_{hv}are

_{DP}

^{(m)}and

*ρ*

_{hv}

^{(m)}depend on the distributions of

*Z*

^{(e)}

_{h,v}and

*R*

_{hv}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):

*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).

*Z*

^{(e)}

_{h,v}and

*Z*

_{hv}expressed in logarithmic scale vary linearly in both cross-beam directions,

*θ*and

*ϕ*:

_{DP}:

*ρ*

_{hv}| ≈ 1, then

*ξ*

_{2}is usually very close to 1, hence we will ignore this term in our further considerations.

*σ*via the antenna beamwidth Ω, we finally arrive at the following approximate formulas for the biases of

*Z*

_{DR}, Φ

_{DP}, and

*ρ*

_{hv}that will be used in the subsequent analysis in the paper:

_{DP}, ΔΦ

_{DP}, Ω,

*θ*, and

*ϕ*are expressed in degrees, whereas

*Z*

_{H},

*Z*

_{HV}, Δ

*Z*

_{DR}, and

*Z*

_{DR}are in decibels.

*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 ΔZ

_{H}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 *Z*_{H}, *Z*_{HV}, *Z*_{DR}, and Φ_{DP} 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 *Z*_{H}, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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}, *Z*_{HV}, and *Z*^{(e)}_{DR}, which are affected by attenuation according to their definition in (7)–(11). High values of Φ_{DP} in the eastern sector are accompanied by negatively biased *Z* and *Z*_{DR} and a pronounced drop in the cross-correlation coefficient *ρ*_{hv}. While the drop in *Z*_{DR} well below −2 dB is caused by differential attenuation, the decrease in *ρ*_{hv} is a result of NBF.

This is confirmed by Fig. 2 where the fields of the parameters Δ*Z*_{DR}, ΔΦ_{DP}, and *ξ* computed from Eqs. (36)–(38) are displayed together with *Z*_{H}. The *ρ*_{hv} depression in Fig. 2d is very well correlated with the observed decrease of the measured *ρ*_{hv} in Fig. 1. The magnitude of the negative *ρ*_{hv} 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 *ρ*_{hv} are frequently observed in the KOUN polarimetric data. Their primary cause is large vertical gradient of Φ_{DP}. 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 Φ_{DP} stops increasing, whereas Φ_{DP} 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 Φ_{DP} are responsible for the decrease in *ρ*_{hv}. In contrast, perturbations of Φ_{DP} are determined by both the gradients of the differential phase and the reflectivity factor. As a result, ΔΦ_{DP} 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 Φ_{DP} dominate and apparent radial features are evident in the field of ΔΦ_{DP}.

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 *Z*_{DR} 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 *Z*_{DR} with height. The *Z*_{DR} biases, as well as the biases in Φ_{DP} and *ρ*_{hv}, 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 *ρ*_{hv} depression is well predicted from the analysis of gradients. The perturbations of the Φ_{DP} 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 Φ_{DP} (thin curves) are compared with radial profiles of ΔΦ_{DP} 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 ΔΦ_{DP}, the correlation between the Φ_{DP} and ΔΦ_{DP} profiles is surprisingly high. The most pronounced excursions of the Φ_{DP} curves, such as spikes and depressions, are well reproduced in the modeled ΔΦ_{DP}. Thus, they are primarily attributed to NBF rather than pure statistical errors in Φ_{DP} 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 *Z*_{DR} maximum and *ρ*_{hv} 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*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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 Φ_{DP} profile in the melting layer as was explained by Ryzhkov and Zrnic (1998). Below the melting layer, the mean value of Φ_{DP} is less biased but differential phase becomes more noisy due to lowering of *ρ*_{hv} 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 *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} according to Eqs. (36)–(38). The procedure for gradient estimation is simple and straightforward.

If the magnitudes of Δ*Z*_{DR}, ΔΦ_{DP}, and the difference 1 − *ξ* exceed certain thresholds, then the corresponding variables (*Z*_{DR}, *K*_{DP}, and *ρ*_{hv}) 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 *Z*_{DR} bias has to be less than 0.2 dB if *Z*_{DR} is utilized for rainfall estimation. The biases of Φ_{DP} within ±2° are acceptable because the statistical fluctuations of the Φ_{DP} estimate are between 1° and 2° for typical dwell times used for operational weather radars. The bias of 0.02 in *ρ*_{hv} may also be tolerable for classification purposes.

In addition to the negative impact on the quality of polarimetric classification, the decrease of *ρ*_{hv} is detrimental for statistical accuracy of the estimates of *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} itself. Indeed, the standard deviations of the estimates for all three variables are proportional to (1 − *ρ*^{2}_{hv})^{1/2} (Bringi and Chandrasekar 2001). This means that if *ρ*_{hv} drops from 0.99 to 0.90, the corresponding errors increase 3 times.

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

The magnitudes of Δ*Z*_{DR}, ΔΦ_{DP}, 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 Φ_{DP} and *ρ*_{hv} are wavelength dependent because the differential phase and its gradients are inversely proportional to the radar wavelength *λ*. The impact on ΔΦ_{DP} is proportional to *λ*^{−1}, whereas the *ρ*_{hv} 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 *Z*_{DR}. In some situations, these changes in the *Z* and *Z*_{DR} gradients may offset the increase in the gradient of Φ_{DP} and its greater impact on the NBF-related biases in Φ_{DP} and *ρ*_{hv}. 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 *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} because the bias estimates are very approximate due to many simplifying assumptions made in derivation of these equations. Instead, we recommend using Δ*Z*_{DR}, ΔΦ_{DP}, 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 Δ*Z*_{DR}, ΔΦ_{DP}, and *ξ* along with a signal-to-noise ratio, the total differential phase (which characterizes potential impact of attenuation/differential attenuation), the magnitude of *ρ*_{hv} (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 *Z*_{DR}, the differential phase Φ_{DP}, and the cross-correlation coefficient *ρ*_{hv} assuming linear gradients of radar reflectivity *Z*_{H}, *Z*_{DR}, and Φ_{DP} 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 *Z*_{DR} does not depend on the radar wavelength, whereas the biases of Φ_{DP} and *ρ*_{hv} increase at shorter wavelength (proportionally to *λ*^{−1} in the case of Φ_{DP} and to *λ*^{−2} in the case of *ρ*_{hv}). Thus, the NBF effects are stronger at C and X bands than at the S band.

Horizontal and vertical gradients of *Z*_{H}, *Z*_{DR}, and Φ_{DP} 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 *ρ*_{hv} depression and oscillatory behavior of the Φ_{DP} profiles. The latter usually manifests itself as the appearance of negative *K*_{DP}. It is also shown that polarimetric signatures of the melting layer rapidly degrade with distance as the antenna beam widens.

Although correcting *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

**V**

*) and received (*

^{t}**V**) waves in the case of individual scatterer are related as (Bringi and Chandrasekar 2001)

*s*

_{hh},

*s*

_{vv}, and

*s*

_{hv}represent backscattering coefficients of the scatterer, and

*T*

_{hh}and

*T*

_{vv}describe phase shifts and attenuations for H and V waves along propagation path:

_{h,v}is the phase shift, Γ

_{h,v}is the attenuation, and Φ

_{DP}= 2(Φ

_{h}− Φ

_{v}) is a differential phase. The coefficient

*C*

_{1}is defined as

*G*is the antenna gain,

*λ*is the radar wavelength,

*R*is the distance between the radar and scatterer, and

*f*

^{2}is the normalized one-way antenna power pattern. It is assumed that the antenna patterns for orthogonal polarizations are the same.

*P*= |

_{t}*V*|

^{t}^{2}[

**V**

*= (*

^{t}*V*,

^{t}*V*)] and

^{t}*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}.

*R*

^{(m)}

_{hv}is defined as

*R*

_{0}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,

*K*= (

_{w}*ɛ*− 1) / (

_{w}*ɛ*

_{w}+ 2), where

*ɛ*is the dielectric constant of water. Substituting (A4) into (A6), we obtain

_{w}*n*is the concentration of scatterers.

*ρ*

_{hv},

_{DP}= Φ

_{DP}+ arg(

*ρ*

_{hv}).

*Z*

_{h,v}, Γ

_{h,v},

*ρ*

_{hv}, and Φ

_{DP}are constant within the radar resolution volume, then the measured covariance

*R*

^{(m)}

_{hv}is equal to its intrinsic value

*R*

_{hv}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:

*σ*= Ω/4(ln2)

^{1/2}.

Composite plot of (a) Z, (b) ΔZ_{DR}, (c) ΔΦ_{DP}, and (d) *ξ* (multiplicative factor of *ρ*_{hv}) corresponding to PPI in Fig. 1. The biases of Z_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite plot of (a) Z, (b) ΔZ_{DR}, (c) ΔΦ_{DP}, and (d) *ξ* (multiplicative factor of *ρ*_{hv}) corresponding to PPI in Fig. 1. The biases of Z_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite plot of (a) Z, (b) ΔZ_{DR}, (c) ΔΦ_{DP}, and (d) *ξ* (multiplicative factor of *ρ*_{hv}) corresponding to PPI in Fig. 1. The biases of Z_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite plot of *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite plot of *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite plot of *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite plot of (a) *Z*, (b) Δ*Z*_{DR}, (c) ΔΦ_{DP}, and (d) *ξ* (multiplicative factor of *ρ*_{hv}) corresponding to PPI in Fig. 3. The biases of *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite plot of (a) *Z*, (b) Δ*Z*_{DR}, (c) ΔΦ_{DP}, and (d) *ξ* (multiplicative factor of *ρ*_{hv}) corresponding to PPI in Fig. 3. The biases of *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite plot of (a) *Z*, (b) Δ*Z*_{DR}, (c) ΔΦ_{DP}, and (d) *ξ* (multiplicative factor of *ρ*_{hv}) corresponding to PPI in Fig. 3. The biases of *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Radial profiles of Φ_{DP} (thin curves) and its bias (ΔΦ_{DP}, 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

Radial profiles of Φ_{DP} (thin curves) and its bias (ΔΦ_{DP}, 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

Radial profiles of Φ_{DP} (thin curves) and its bias (ΔΦ_{DP}, 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

Composite RHI plot of *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} measured with the KOUN WSR-88D radar on 7 Apr 2002.

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

Composite RHI plot of *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} measured with the KOUN WSR-88D radar on 7 Apr 2002.

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

Composite RHI plot of *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} measured with the KOUN WSR-88D radar on 7 Apr 2002.

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

Composite RHI plot of simulated *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite RHI plot of simulated *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

Composite RHI plot of simulated *Z*, *Z*_{DR}, Φ_{DP}, and *ρ*_{hv} 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

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

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

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

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

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

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