## 1. Introduction

It is only recently that research weather radars have been configured for measuring the full polarimetric covariance matrix (Bringi and Chandrasekar 2001). Two such radars are the Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) radar operated by CSU and supported by the National Science Foundation (Brunkow 1999; Brunkow et al. 2000) and the S-Band Dual-Polarization Doppler Radar (S-Pol) operated by the National Center for Atmospheric Research (NCAR) (Keeler et al. 2000), both operating at S-band frequency near 3 GHz. The construction and interpretation of radar covariance matrices has been a recent topic of research, with modeling being the primary focus (Ryzhkov 2001; Hubbert et al. 1999). This paper focuses on a procedure for construction of calibrated covariance matrices. Calibrated covariance matrices have recently been used by Huang et al. (2001) to estimate both mean canting angle and the spread of canting angles of rain drops, and by Hubbert and Bringi (2001) to estimate beam-averaged antenna polarization errors. Even though both radars record all the necessary data for constructing the full covariance matrix, calibration of all the covariances has not been explicitly addressed so far, especially the co-to-cross covariance phases. In this paper only the relative covariance matrix is addressed so that absolute power calibration is not discussed. Our primary focus here to is construct well-calibrated relative covariance matrices to which interesting matrix techniques can be applied (to be addressed in future papers). Absolute power is irrelevant in these techniques. It follows that two relative power terms remain to be calibrated: 1) differential reflectivity (*Z*_{dr}), and 2) linear depolarization ratio (LDR). Three phase offsets also need to be determined for 1) the copolar differential phase (Ψ_{dp}), 2) the co-to-cross phase (arg{〈*S*_{HH}*S*^{*}_{VH}*S*_{VV}*S*^{*}_{HV}

A novel method to calibrate *Z*_{dr} is given that depends only on a sun calibration measurement and the difference of the two crosspolar powers obtained from precipitation. This solution is analytical and requires no assumptions other than reciprocity. A more established method to determine the *Z*_{dr} system offset uses vertical pointing data in precipitation where it is assumed that *Z*_{dr} is 0 dB when data are averaged over a full 360° rotation of the antenna (Bringi and Chandrasekar 2001; Gorgucci et al. 1999). No such assumptions are necessary with the method presented here.

Finally, a novel technique for the estimation of LDR is given that utilizes the cross-to-cross covariance. This technique is immune to background noise and thus no knowledge of the noise temperature is needed for noise correction, which makes the technique attractive for low signal-to-noise ratio (SNR) regions. Radar data from the Severe Thunderstorms Electrification and Precipitation Study (STEPS) field campaign are used to illustrate the LDR estimation technique and *Z*_{dr} calibration technique.

## 2. Theory

### a. CSU–CHILL radar

*P*

_{H,V}are the input transmitter powers,

*C*

^{T}

_{H,V}

*C*

^{R}

_{H,V}

*W*

_{H,V}are waveguide losses,

*G*

^{A}

_{H,V}

*G*

^{R}

_{H,V}

*R*

_{H,V}are the received powers. The dotted line represents the measurement plane where both the transmit power is monitored and test signals are injected. From Fig. 1 it follows that where

*Z*

^{m}

_{dr}

^{m}

_{H}

*S*

_{HH}|

^{2}〉/〈 |

*S*

_{VV}|

^{2}〉 =

*Z*

_{dr}and 〈 |

*S*

_{VH}|

^{2}〉/〈 |

*S*

_{HH}|

^{2}〉 = LDR

_{H}are the intrinsic values we wish to isolate, with 〈*〉 denoting time average. Available for calibration are sun calibration measurements (

_{V}=

^{m}

_{V}

_{V}= 〈 |

*S*

_{HV}|

^{2}〉/〈 |

*S*

_{VV}|

^{2}〉. If

*Z*

_{dr}and LDR can be determined quite accurately. However, in practice, there are errors associated with these measurements that can bias

*Z*

_{dr}. These errors can be eliminated if the crosspolar powers are utilized. The ratio of the two received crosspolar power is Because of reciprocity (Saxon 1955), 〈 |

*S*

_{VH}|

^{2}〉 = 〈 |

*S*

_{HV}|

^{2}〉, and it follows that This equation is important in that

*Z*

^{m}

_{dr}

*Z*

_{dr}is calibrated via vertical pointing data in precipitation, which relies upon the assumption that the average

*Z*

_{dr}is intrinsically zero for hydrometeors at vertical incidence when the polarization basis is rotated 360° in azimuth. This indeed should be true; however, it is evident from examining vertical pointing data from several radars around the world that other scattering mechanisms can be present other than direct scatter from the precipitation in the main beam of the radar. It is known that vertically pointing

*Z*

_{dr}can be a function of azimuth angle (Gorgucci et al. 1999; Bringi and Chandrasekar 2001), having sinusoidal-like variations with peak-to-peak excursions of several tenths of a decibel. This behavior has been surmised to be caused by antenna pattern sidelobes interacting with ground clutter. For the CSU–CHILL radar, these sinusoidal-like variations are observed but are not always present. Furthermore, these variations are not observed in clear air. If ground clutter were responsible, then these sinusoidal-like variations would be seen even when vertical pointing data are taken in clear air. We speculate that back lobes of the antenna pattern interact with the precipitation medium via multipath scattering to cause the azimuthal variation. If this is true, then

*Z*

_{dr}from this multipath scatter will not necessarily give 0 dB when

*Z*

_{dr}is integrated over 360° since the ground scatter cross sections will not be the same for H and V polarizations. It is also possible that the azimuthal variation of

*Z*

_{dr}is an artifact of a wet radome or due to imperfections in the isolation of the dual-channel rotary joint with azimuth angle (though this proved not to be the case for the CSU–CHILL radar). Other practical problems with vertical pointing data are infrequent occurrence of precipitation over the radar and the minimum range needed to be in the far field (2

*D*

^{2}/

*λ,*where

*D*is the antenna diameter and

*λ*is the wavelength). The procedure based on (9) offers significant advantages since crosspolar power will routinely be available from precipitation. Furthermore, ground clutter targets should also obey reciprocity so that ground clutter data can also be used. Since the ratio of

*R*

_{HVHV}and

*R*

_{VHVH}represents a system bias that should be constant over 5–10-min intervals, this ratio can be more accurately estimated by averaging over an entire volume of precipitation. Examples are given in section 3, “Data analysis.”

#### 1) Phase calibration

_{i}, VV

_{i}, VH

_{i}, and HV

_{i}, with the V-polarized pulse being transmitted first. It is easy to show that (Sachidananda and Zrnić 1986; Mueller 1984) Combining these equations gives Equations (15) and (16) are interesting because they show that not only the receive phase difference,

*ϕ*

_{Hr}−

*ϕ*

_{Vr}, but also the transmit phase difference,

*ϕ*

_{Ht}−

*ϕ*

_{Vt}, can be determined solely from precipitation measurements. Unfortunately, neither the CSU–CHILL radar nor S-Pol routinely record the cross-to-cross covariances, 〈

^{*}

_{i}

_{i}〉 and 〈HV

_{i}

^{*}

_{i}

_{dp}from (15) and (16) either data can be selected from ranges where insignificant Ψ

_{dp}has accumulated, or, alternately, Ψ

_{dp}can be be estimated and removed. The receive phase difference can also be determined from sphere calibration or remote horn measurements, but the convenience of determining these phase offsets, which may vary on a day-to-day basis, from precipitation data is obvious. To find the Ψ

_{dp}offset, Ψ

_{dp_off},

_{dp}is typically calculated (Mueller 1984). Since the intrinsic range profiles of Ψ

_{dp}should begin at 0°, Ψ

_{dp_off}= −(

*ϕ*

_{Ht}−

*ϕ*

_{Vt}) − (

*ϕ*

_{Hr}−

*ϕ*

_{Vr}) can be easily found from range profiles of (

*θ*

_{H,V}represent the intrinsic co-to-cross covariance phases (Hubbert et al. 1999). If the cross-to-cross covariances are available, (16) can be used to determine the offset phase,

*ϕ*

_{Hr}−

*ϕ*

_{Vr}, but this offset can also be found as follows. It is known that

*θ*

_{H}+

*θ*

_{V}= Ψ

_{dp}(Jameson 1985; Hubbert et al. 1999), so that

_{dp}

*ϕ*

_{Hr}

*ϕ*

_{Vr}

*ϕ*

_{Hr}−

*ϕ*

_{Vr}. A constant phase can be added to both

_{dp}range profile. Since range profiles of

_{dp}for calculating

*K*

_{dp}, specific differential phase; Hubbert and Bringi 1995). This technique works best when there is a significant amount of Ψ

_{dp}accumulation (e.g., propagation path through intense multiple rain cells), and it is attractive since it is based on data alone. The reason for this is that in regions of large Ψ

_{dp}accumulation, the SNR is typically large and data quality is high.

*ϕ*

_{x_off}=

*ϕ*

_{Hr}−

*ϕ*

_{Vr}and Ψ

_{dp_off}= −(

*ϕ*

_{Ht}−

*ϕ*

_{Vt}) −

*ϕ*

_{x_off}. [See Tragl (1990) for covariance matrix background.]

### b. S-Pol

*Z*

_{dr}that can be caused by using two separate receivers that may not be perfectly matched in gain over the entire dynamic range. Because of the transfer switch, the analysis is slightly more complicated. Since there are four possible paths through the receiver chain, four sun calibration numbers are required (see the appendix). It can be shown that for the S-Pol configuration, where Γ

_{co,V}, Γ

_{co,H}, and Γ

_{x,V}represent the insertion losses associated with various paths through the switch. For example, Γ

_{co,V}represents the path from the V switch input through the switch to the copolar receiver. The ratio of the crosspolar powers becomes The difference between (27) and (8) is that (8) contains the ratio of the two receiver gains, whereas (27) contains a ratio of insertion loss due to receive paths through the switch. For the S-Pol configuration,

*Z*

^{m}

_{dr}

^{m}

_{H}

_{1},

_{2}, and

_{3}are as defined in the appendix.

#### 1) S-Pol phase calibration

_{dp_off}= −(

*ϕ*

_{Ht}−

*ϕ*

_{Vt}) − (

*β*

_{Hr}−

*β*

_{Vr}) − (

*α*

_{co,H}−

*α*

_{co,V}) and again can be determined by forcing range profiles of (

*β*

_{Hr}−

*β*

_{Vr}+

*α*

_{co,V}−

*α*

_{co,H}+

*α*

_{x,V}−

*α*

_{x,H}, can be determined by finding the phase constant that will make range profiles of

_{dp}). However, this phase offset is not the phase offset required in (37) and (38). As can be seen from (37) and (38), the phase difference of the two receivers,

*ϕ*

_{GRco}

*ϕ*

_{GRx}

*Z*

_{dr}calibration. The mean canting angles of ice particles in the ice phase of convective storms should on average be zero due to tumbling (the same assumption is used in practice to calibrate

*Z*

_{dr}). The mean canting angle can be determined from eigenpolarization analysis of the covariance matrix (Tragl 1990). Since the mean canting is quite sensitive to the co-to-cross phases arg{

*ρ*

_{xH}} and arg{

*ρ*

_{xV}}, the unknown phase offset term can be adjusted until the mean canting in the ice phase is zero on average. The topics of determining system polarization errors and mean and standard deviation of canting angles from analysis of covariance matrix data will appear in Part II (Hubbert and Bringi 2003) and Part III of this paper.

### c. An alternate LDR estimator

_{i}

^{*}

_{i}

_{i}

^{*}

_{i}

_{i}

^{*}

_{i}

_{i}

^{*}

_{i}

_{s}

*σ*

^{2}

_{n}

_{i}

^{*}

_{i}

_{s}is the signal portion and

*σ*

^{2}

_{n}

_{i}and VH

_{i}is uncorrelated because of the time lag between the two time series, so that

_{i}

^{*}

_{i}

_{i}

^{*}

_{i}

_{s}

_{H}estimator: where

*ρ*

_{HH}(2)

^{0.25}=

*ρ*

_{HH}(1) for Gaussian-shaped spectra, and

*ρ*

_{HH}(2),

*ρ*

_{HH}(1) are the first and second lag autocorrelations of the HH

_{i}sequence. The correction term in the denominator is required because the HV

_{i}and VH

_{i}time series are not simultaneously sampled. This is the same correction factor used in calculating the copolar correlation coefficient [see (21)]. If the

*ρ*

_{HH}(2) > 0.75, the correction factor will only increase LDR by 0.31 dB. Thus, the correction factor represents a very small change in LDR, and even if this correction factor is poorly estimated, LDR should not be greatly affected. The LDR estimator of (42) also needs to be calibrated. Using similar analysis as was done for the CSU–CHILL system, it can be shown that

## 3. Data analysis

*R*

_{VHVH}and

*R*

_{HVHV}, from precipitation. The scales are in decibels and the lower sensitivity limit of the radar is about −113 dBm. Figure 5 shows a plot of the statistics obtained by binning the horizontal axis into 3-dB steps and then averaging the data points that fall into these bins. The vertical lines represent a standard deviation on both sides of the mean (i.e., the vertical lines are 2

*σ*long). The two small horizontal lines about each mean (barely visible) show the 95% confidence interval. As can be seen, the confidence intervals are quite small except perhaps for the last three bins (

*R*

_{VHVH}> −75 dBm), and thus the mean values should be quite accurate estimates of the actual receiver gain difference (including receive circulators) as a function of received power. To reveal the receiver gain difference as a function of input power, the receive power difference is plotted as a function of the mean value of

*R*

_{VHVH}denoted by

*Z*

^{m}

_{dr}

*Z*

_{dr}that should be corrected if possible. Examining (8), since the term (

*P*

_{H}

*C*

^{T}

_{H}

*P*

_{V}

*C*

^{T}

_{V}

*C*

^{R}

_{V}

*G*

^{R}

_{V}

*C*

^{R}

_{H}

*G*

^{R}

_{H}

*Z*

^{m}

_{dr}

*C*

^{R}

_{V}

*G*

^{R}

_{V}

*C*

^{R}

_{H}

*G*

^{R}

_{H}

*W*

_{V}

*G*

^{A}

_{V}

*W*

_{H}

*G*

^{A}

_{H}

*C*

^{R}

_{V}

*G*

^{R}

_{V}

*C*

^{R}

_{H}

*G*

^{R}

_{H}

*p*can estimated from

*p*

_{ref}

*p*

_{ref}

*p*) is the sun calibration figure as a function of measured received power,

_{ref}is the actual measured sun calibration figure,

*p*) is the average crosspolar power difference as a function of measured input power (Fig. 6), and

_{ref}is the value of

*p*) at the same measured power level of

_{ref}. Then

*Z*

_{dr}can be calibrated as a function of measured power using

*Z*

_{dr}

*Z*

^{m}

_{dr}

^{2}

_{ref}

*p*

^{2}

_{ref}

*Z*

_{dr},

*p*can be estimated by averaging

*R*

_{HHHH}and

*R*

_{VVVV}, the two copolar input powers.

This *Z*_{dr} correction method is now applied to an RHI volume scan from 11 June 2000 collected at 2327 UTC. Around this time a sun calibration was performed (at 2344 UTC) and vertical pointing data was also gathered (at 2328 UTC); thus, it is possible to compare the *Z*_{dr} correction method given by (45) and the vertical pointing data method. From the vertical pointing data the *Z*_{dr} bias is determined to be 1.3 dB. The sun calibration figure is 0.559 dB, and the sun power is about −100 dBm. The data in Fig. 6a (labeled 2327:13) are used with 3-dB bin intervals. Differential reflectivity *Z*_{dr} is corrected as a function of the received copolar horizontal power (*R*_{HHHH}). Figure 7 shows a vertical profile of *Z*_{H}, *Z*^{vp}_{dr}*Z*^{m}_{dr}*Z*^{xp}_{dr}*Z*^{m}_{dr}*ρ*_{HV} | . The data come from a trailing stratiform region of a convective cell and are averaged over a 5-km range (26–31 km). The *Z*^{vp}_{dr}*Z*^{xp}_{dr}*Z*_{dr} = 0 dB. However, it is also frequently seen that *Z*_{dr} is positive (or negative) in stratiform regions or in storm anvils due to oriented pristine ice crystals (Caylor and Chandrasekar 1996; Bader et al. 1987). Thus it may be inaccurate to use the assumption of *Z*_{dr} = 0 dB in the ice phase to calibrate *Z*_{dr}. Furthermore, long periods of time frequently elapse with no precipitation over the radar, and thus vertically pointing data may not be available for *Z*_{dr} calibration. Using the difference of the crosspolar powers is an attractive, viable, and accurate way to calibrate *Z*_{dr}.

Figure 8 shows another vertical profile of data from 23 June 2001 through an intense convective core. No vertical pointing data was available that day. Data were averaged over a 5-km range (52–57 km). As can be seen, very high reflectivities (56 dB*Z*) are found through the upper level of the core. The *Z*^{xp}_{dr}*Z*_{dr} should be 0 (dB). This example again shows that the crosspolar power method for calibrating *Z*_{dr} works well. For this case the received power levels were very high in the −55 to −65 dBm range. Figure 6b shows that the crosspolar power difference is only available up to −65 dBm, and thus correcting *Z*_{dr} at power levels greater than −65 dBm cannot be directly accomplished using crosspolar powers. However, by examining the calibration curves made on that day, the characteristics of the receivers and circulators at high power input levels can be observed. The receiver calibration curves for 23 June indicate that there was about a 0.05-dB decrease in receiver gain difference from about −75 dB to −55 dBm input power levels. This gain difference was incorporated into the *Z*_{dr} correction in Fig. 8.

### a. The alternate LDR estimator

CSU–CHILL time series data from 24 June 2000 are used to illustrate the alternate LDR estimator in (42). The data are from from elevation angles greater than 4° at ranges 64–69 km and are therefore likely from a region of ice. Only light precipitation was present in the region between the radar and about 60 km, and thus the actual background noise present in the data will be close to the background noise of “clear air” [i.e., the noise temperature of ice and light rain is low; Gordon and Morgan (1993)]. Sun calibration data and “clear air” data were gathered immediately following the time series volume scan, so the calibrations for the time series dataset should be very accurate. The data are divided into low- and high-SNR categories. The threshold for the low-SNR data is −113 dBm < *R*_{VHVH} < −110 dBm and the threshold for the high-SNR data is *R*_{VHVH} > −103 dBm. A scatterplot of LDR_{x,H} versus LDR_{H} for low-SNR data is shown in Fig. 9a. The LDR_{H} has been corrected for noise by subtracting the measured background noise (obtained from the clear air scan) from the measured crosspolar power. The two LDR estimators agree quite well considering the very low SNR. The statistics of the scatterplot are given in Fig. 9b, and it can be seen that the LDR_{x,H} estimator gives a slightly lower value on average by about a half a decibel. Similar plots are shown in Fig. 10 for the high-SNR case. The spread of the scatter is reduced considerably and the two LDR estimators are nearly identical on average, as expected, as seen in Fig. 10b. The LDR_{x,H} estimate should actually be slightly lower than the standard LDR_{H} estimate since unpolarized energy will not be included in the LDR_{x,H} estimate whereas it will be present in the standard LDR_{H} estimate.

Since accurate background noise estimates may not always be available, LDR_{x,H} is an attractive alternate estimator. Furthermore, the background noise power is a function of the type and density of the precipitation medium intersected by the radar beam (Seminario et al. 2001; Gordon and Morgan 1993); that is, the noise power obtained from clear air measurements may not be indicative of the actual background noise for precipitation data. For example, a 50-dB*Z* echo region at 15 km can raise the noise floor by more than 2 dB when looking through the rain region (Seminario et al. 2001)! The LDR_{x,H} estimator is immune to this problem.

## 4. Summary and conclusions

This paper gives a methodology to construct calibrated covariance matrices for both the CSU–CHILL radar and S-Pol configurations. The two radars are configured differently; that is, the CSU–CHILL uses two separate transmitters and two separate receivers (H receiver and V receiver), whereas S-Pol uses a single transmitter with a mechanical switch and two receivers (copolar receiver and crosspolar receiver via a transfer switch). A procedure for determining the two relative power calibration terms and three relative phase offsets was described. For S-Pol it was shown that the phase offsets for the co-to-cross covariances cannot be determined from precipitation data whereas they can be for the CSU–CHILL system. In order to determine the S-Pol phase offsets, external test horn measurements are required. Alternately, the co-to-cross covariance phase offsets can be determined from meteorological data by using the assumption that the mean canting angle of precipitation particles in the ice phase of convective storms is zero. It was also shown that for CSU–CHILL, the differential receiver power gain is a function of the input power level as shown by Fig. 6. Even though the amount of differential gain variation as a function of the input power is less than about 0.15 dB, this is significant when calculating *Z*_{dr}. To overcome this, a method was given to correct measured *Z*_{dr} for these differential receiver gain variations that utilize the crosspolar powers and sun calibration measurements. The differential receiver gain variation for the S-Pol configuration should be less than the CSU–CHILL receiver variation since all copolar signals are sent to the same receiver. To date, both CSU–CHILL and S-Pol *Z*_{dr} data have been calibrated either with vertical pointing data or with measurements in the ice phase of storms. Vertical pointing data are not always available, and indeed a *Z*_{dr} calibration method that relies upon rain over the radar site may not be reliable. The assumption that *Z*_{dr} is 0 (dB) in the ice phase of storms is also not always reliable, due to, for example, alignment of ice crystals or other anomalies in the radar data, such as reflectivity gradients, sidelobes, or three body scattering. Thus, the proposed *Z*_{dr} calibration technique is very attractive. It relies only on accurate sun calibrations and reciprocity of the crosspolar power measurements.

An alternate LDR estimator was also given that used the cross-to-cross covariances in contrast to using just the autocovariance of the crosspolar time series to calculate crosspolar power. The magnitude of the cross-to-cross covariance contains no background noise power since the white noise portion of the two crosspolar time series is uncorrelated. However, the magnitude of the cross-to-cross covariance is not a direct estimate of the crosspolar power, due to the time lag between the two time series. The amount of decorrelation due to this time lag can be estimated from the second lag correlation of the HH_{i} time series, *ρ*_{HH}(2) (assuming Gaussian spectra), as is routinely done when estimating the copolar correlation coefficient. The LDR is typically corrected for noise by using background noise estimates obtained by pointing the radar at a precipitation-free region. However, it is known that this noise figure is also dependent on the precipitation that intersects the radar beam. Additionally, these noise measurements may not always be available. The new estimator, LDR_{x,H}, is immune to these effects. This new LDR estimator was applied to CSU–CHILL radar data and was shown to be in good agreement with the standard LDR_{H} estimator.

The CSU–CHILL radar is operated by Colorado State University via a cooperative agreement with the National Science Foundation (ATM-9500108). Two of the authors (JCH and VNB) were supported by this grant and NSF Grant ATM-9982030. The authors acknowledge the outstanding effort of Pat Kennedy and Robert Bowie in deploying and operating the radar for STEPS. The authors also acknowledge the technical discussions with Jon Lutz and Bob Rilling of the National Center for Atmospheric Research concerning S-Pol.

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# APPENDIX

## S-Pol Sun Calibration Measurements

_{co,H}is the attenuation along the path beginning from the H input port of the switch to the input of the copolar receiver

*G*

^{R}

_{co}

^{m}

_{H}

*Z*

^{m}

_{dr}