Combined Doppler and Polarimetric Radar Measurements: Correction

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443

Combined Doppler and Polarimetric Radar Measurements: Correction for Spectrum Aliasing and Nonsimultaneous Polarimetric Measurements C. M. H. UNAL IRCTR, Delft University of Technology, Delft, Netherlands

D. N. MOISSEEV Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, Netherlands (Manuscript received 5 March 2003, in final form 24 September 2003) ABSTRACT Combining Doppler and polarimetric information is advantageous for atmospheric studies. On the one hand, Doppler information gives insight into the microphysical and dynamic properties of radar targets, that is, radial velocity and its variability. The polarization diversity, on the other hand, has a strong link to the microphysical properties of targets such as shape and orientation. Polarimetric measurements, however, have an adverse effect on Doppler processing. Measurements of the complete scattering matrix require at least two pulses and result in the reduction of the maximum unambiguous Doppler velocity that can lead to Doppler spectrum aliasing. Moreover, dynamic properties of targets, because of the nonsimultaneity of the measurements performed with different polarizations, affect the accuracy of polarimetric radar measurements. A solution to these two problems is given in this paper. It is shown that by applying a relatively simple processing technique the effect of nonsimultaneous polarimetric measurements can be reduced even in the case of strong Doppler spectrum aliasing. This leads to better estimates of the copolar cross-correlation coefficient. Furthermore, this processing results in the maximum unambiguous Doppler velocity as if no polarimetric measurements were performed, with the added advantage of obtaining the actual Doppler velocities. To illustrate the proposed processing technique, precipitation measurements taken with the Delft Atmospheric Research Radar (DARR) are used.

1. Introduction Radar polarimetry is widely used in radar remote sensing. Probably the more popular applications of radar polarimetry are radar target characterization (Lu¨neburg et al. 1991; Boerner et al. 1998; Ryzhkov 2001) and target contrast enhancement (Swartz et al. 1988; Tragl 1990; Ryzhkov and Zrnic´ 1998; Moisseev 2002). Even though these two applications are different they often employ second-order-moment analysis of the polarimetric data. The choice of this formalism is caused by the fact that most of the natural targets have time-dependent characteristics. This analysis, however, has one problem that is caused by measurement limitations (Sachidananda and Zrnic´ 1989). With the exception of radars that simultaneously transmit two orthogonally polarized signals by means of special signal coding (Giuli and Facheris 1991; Giuli et al. 1995) or by using a hybrid polarization measurement mode (Scott et al. Corresponding author address: Christine Unal, IRCTR, Faculty of Electrical Engineering, Mathematics, and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, Netherlands. E-mail: [email protected]

q 2004 American Meteorological Society

2001), most radars employ one of these polarization switching schemes to measure a full target scattering matrix as discussed in Santalla and Antar (2002). In this case the variability of the radar targets may cause changes in the properties of the targets from one cycle of measurements to another. It should be noted that even for switching on a pulse-to-pulse basis a change in scattering properties of natural targets may be observed. The problem of nonsimultaneous polarimetric measurements in cases of radar observations has been studied by several authors. In Balakrishnan and Zrnic´ (1990) the effect of nonsimultaneous polarimetric measurements in atmospheric studies was analyzed. The authors of this paper have proposed a method that allows for compensation of this problem, if two assumptions are met. The first assumption is that the Doppler power spectrum of meteorological echoes has the Gaussian shape. The second one is that the distributions of radial velocities, canting angles, and shapes are independent of one another. Even though this method allows for a relatively simple compensation scheme, it relies on two assumptions that are not always satisfied. It was observed that in many cases the Doppler power spectra of atmospheric echoes do not have a Gaussian shape. For

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raindrops, the distributions of radial velocities and shapes are related when there is no significant turbulence (Unal et al. 2001; Verlinde et al. 2002). Another widely used approach is to interpolate the time series, for example hh and yy in order to obtain a coincident set of hh and yy time samples. This technique was demonstrated by Illingworth and Caylor (1991) and Zrnic´ et al. (1994). This is a better method because there is no assumption on the shape of the power spectra; however, this approach does not work in case of Doppler spectrum aliasing, as will be shown later. An alternative way to solve the problem of nonsimultaneous polarimetric observations is to perform polarimetric measurements with three different transmit polarizations. This approach is introduced by Santalla et al. (1999) and further discussed in Santalla and Antar (2002). The proposed technique is insensitive to Doppler properties of radar echoes and therefore can be used in cases where the two previously discussed techniques fail. But unfortunately, this method requires a more complicated radar measurements scheme. Namely, it needs transmission of three alternately polarized radar signals instead of the usual two and the radar must have a dual-channel receiver. Moreover, due to the fact that it needs three alternately polarized signals instead of two, as is commonly the case, it has a strong adverse effect on Doppler measurements of radar objects as discussed later. For most radar systems the measurement of a target scattering matrix requires at least two pulses, or three pulses in the case of a single-channel receiver system. As a result the polarimetric measurements reduce the maximum unambiguous Doppler velocity and thus increase the probability of having Doppler aliasing, which in its turn complicates interpretation of radar data (Sachidananda and Zrnic´ 1989). Therefore, the second main topic that is addressed in this paper is the effect of polarimetric measurements on Doppler observations. An increase in sampling time, for example due to polarization switching, causes a decrease in Nyquist frequency and thus increases the probability of spectral aliasing. This problem is often solved by algorithms using range continuity and/or Doppler velocities continuity properties. But reflectivity-free areas and wind shears may cause discontinuities in the measured Doppler field. For these cases, the algorithms cannot correct for aliased velocities. Another issue is that a lot of unfolding techniques exist, but they do not provide the actual Doppler velocities and thus cannot give the actual mean Doppler velocity. Another dealiasing procedure is described by Hennington (1981) and Doviak and Zrnic´ (1993); however, this method requires an additional piece of meteorological information, which is the environmental wind knowledge as a function of height. Also the radar observation scheme is often modified to increase the maximum unambiguous Doppler velocity. One of these such measurement schemes is staggering the pulse repetition time (PRT; see Sirmans et al. 1976;

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Doviak and Zrnic´ 1993). This technique gives improved results for the reflectivity and Doppler velocity retrievals. But this type of measurement increases the complexity and the time of the measurement, especially when the measurement of the scattering matrix has to be added. In this article we analyze the problem of nonsimultaneous polarimetric measurements for high-resolution Doppler polarimetric atmospheric radars, such as the Delft Atmospheric Research Radar (DARR; Niemeijer 1996) and the Transportable Atmospheric Radar (TARA; Heijnen et al. 2000). These S-band radars are used to provide high-resolution Doppler and polarimetric data of atmospheric echoes to facilitate studies of precipitation, clouds, and atmospheric turbulence. Unal et al. (2001) and Verlinde et al. (2002) have shown that the combination of Doppler and polarimetric measurements as provided by such radars gives us an opportunity to gain a more detailed study of atmospheric phenomena. In this article we use a Doppler phase compensation method to compensate for effect of nonsimultaneous polarimetric measurements. It is shown that the phase compensation is a good correction in the case of Doppler aliasing. Moreover, it leads to a new possibility for dealiasing precipitation profiles. The proposed dealiasing technique uses polarization measurements of atmospheric echoes to unfold Doppler spectra. It is shown that this technique allows an increase in maximum unambiguous Doppler velocity as if no polarimetric measurements were performed with the added advantage of obtaining the actual Doppler velocities. It is not only the estimation of the copolar cross-correlation coefficient that is obtained, which is significantly improved when folded Doppler spectra are correctly unfolded, but also other polarimetric parameters, and the Doppler moments. The performance of the method is illustrated on a slant profile measurement of light precipitation. Furthermore, the domain of validity of the proposed dealiasing method is discussed in details. For this purpose, a statistical model of the differential phase spectrum is introduced and experimentally verified to obtain the standard deviation of this phase for different measurement conditions (number of averages, cross-correlation coefficient). The paper consists of the following sections. Section 2 defines the spectral target covariance matrix, which is used for Doppler polarimetry and introduces the correction for the nonsimultaneity of polarimetric measurements hh and yy. In section 3, we discuss the proposed phase correction and its consequences for the case of Doppler aliasing. In order to elaborate upon a dealiasing algorithm, a statistical model of the phase of the cross-spectrum hh, yy is presented in section 4. After phase compensation in the Doppler frequency domain, we assume that the mean value of the cross-spectrum phase is about 0 at S band. This assumption is discussed in section 5. In section 6, the dealiasing procedure is

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explained and illustrated on a slant profile of light rain measured during a windy day. Better estimates of the cross-correlation coefficient are obtained in section 7 with the proposed phase correction and dealiasing technique. The comparison is made with already existing corrections. We conclude the paper in section 8. 2. Doppler polarimetric formalism The polarimetric content of the radar signal can be expressed by a target covariance matrix, which describes the mean polarimetric properties of the hydrometeors. Well-known polarimetric meteorological parameters, like the differential reflectivity Zdr and the complex cross-correlation coefficient rco, are derived from this matrix. By combining Doppler and polarimetric information, we aim to measure and interpret these polarimetric parameters per Doppler velocity bin. For this purpose, a new polarimetric matrix, the spectral target covariance matrix, is defined in this section. Before this, the classical target covariance matrix is briefly introduced and will be related to the spectral target covariance matrix. The phase correction for the nonsimultaneity of the polarimetric measurements is introduced at the end of the section and will lead to the final form of the spectral target covariance matrix. a. Target covariance matrix To describe polarimetrically time-dependent targets, the second-order moments of the time series of scattering matrices are usually calculated. They lead to the size-averaged polarimetric properties of the targets. The scattering matrix elements must be simultaneously measured. In practice, the copolar elements of the scattering matrices Shh and Sy y are measured with a time difference Dt, which must be smaller than the decorrelation time of the hydrometeors. If the scattering matrix elements can be described in terms of ergodic discrete random processes, the correlation functions Rx,x and the crosscorrelation functions Rx,y, where x or y stands for hh, hy, or yy, can be used for analysis of their statistical properties. For example, the correlation function Rhh,hh at time lag mTm is defined as R x,x (mTm ) 5 ^S x (nTm )S*x [(n 1 m)Tm ]& 5

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1 kL

O O k

(1)

i51 n5(i21)L

and the cross-correlation function Rx,y is R x,y (mTm 1 Dt) 5 ^S x (nTm )S*y [(n 1 m)Tm 1 Dt]& 5

1 kL

O O k

[C]

(4) 



* Ï2^S hhS*hy & ^S hhS*y y &  ^S hhS hh &  * *  5 a Z Ï2^S hy S hh & 2^S hy S hy & Ï2^S hy Sy*y & .   Ï2^Sy y S*hy & ^Sy y S*y y &  hh &  ^Sy y S* The target covariance matrix is expressed in mm6 m23 for atmospheric targets, the scattering matrix has units of meters (m), and the constant aZ is a unit conversion factor from m2 to mm6 m23. The complex cross-correlation coefficient, which is discussed in the paper, is defined as

rco 5

^S hhS*y y & Ï^|S hh | 2 &Ï^|Sy y | 2 &

5 | rco |e j Cdp 5 | rco |e j(Fdp1d co ) ,

(5)

where Cdp is the differential phase, Fdp is the differential propagation phase, and dco is the differential backscattering phase (Bringi and Chandrasekar 2001). The correlation functions and cross-correlation functions, calculated at different time lags (m ± 0), give a more complete description of the targets. Another equivalent approach to enhancing the polarimetric description is Doppler polarimetry. b. Spectral target covariance matrix

iL21

S x (nTm )S*x [(n 1 m)Tm ]

samples) with the same polarization setting is denoted by Tm. In the above equations, Dt, a fraction of Tm, is the time difference between two consecutive measurements x and y. The samples numbers are denoted by n and m. The symbol ^ & denotes the time average and consists of a double summation where L is the number of samples for the Doppler processing and where k is the number of averages of the Doppler spectra. The Doppler processing will be defined in the next section. Since the random processes are assumed stationary, the correlation and cross-correlation functions do not depend on the time nTm (sample n) but on the time lag mTm. The elements of the target covariance matrix [C], calculated for each range bin, are thus the correlation functions and cross-correlation functions of the random processes Shh, Shy, and Sy y, calculated at zero time lag (m 5 0):

(2)

The purpose of the spectral target covariance matrix is to describe both polarimetric and dynamic properties of the radar targets. When a random process is stationary, it can also be described in the frequency domain by power spectra (x 5 y) and cross spectra (x ± y) (Therrien 1992):

iL21

OR

L21

S x (nTm )

Fx,y (lv D ) 5 Tm

i51 n5(i21)L

x,y

(mTm )e2jlv D mTm ,

(6)

m50

3 S*y [(n 1 m)Tm 1 Dt].

(3)

The time between two consecutive measurements (or

where l denotes the Doppler frequency bin and v D is the Doppler frequency resolution. The number of Dopp-

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ler frequency bins is L. The correlation functions and cross-correlation functions are the discrete Fourier transforms inverse of the power spectra and cross spectra, respectively (Therrien 1992):



 Fhh,hh (lv D ) 2p 

ˆ (lv D )] 5 a Z v D Ï2Fhy ,hh (lv D ) [C 

Fy y ,hh (lv D )

The spectral target covariance matrix is expressed in mm6 m23. For example, aZ(vD /2p) Fhh,hh(lvD) gives us a variance of hh signal for a frequency range [(l 2 1/2)vD, (l 1 1/2)vD[, and is named the spectral reflectivity.

ˆ (lvD)] 5 a Z [C k

O k

i51



Sy y (i,lv D )S hh (i,lv D )

R hh,y y (mTm 1 Dt) 5 ^S hh (nTm )Sy*y [(n 1 m)Tm 1 Dt]&

5

vD 2p

(8)

Moreover, the sum of all spectral reflectivities leads to the commonly used reflectivity value of the concerned range bin. The spectral target covariance matrix can be equivalently expressed (Doviak and Zrnic´ 1993) as

Ï2Sy y (i,lv D )S hy (i,lv D )

The problem of nonsimultaneous polarimetric measurements is encountered in the estimates of the offdiagonal elements of the target covariance matrix. It should be noted that in the case where Dt is much smaller than the decorrelation time of Sx(nTm), the effect of nonsimultaneous polarimetric measurements is negligible, otherwise it should be taken into account. Let us illustrate the effect of nonsimultaneous polarimetric measurements on the cross-spectrum hh, yy. The corresponding cross-correlation function is

O O

Ï2Fhh,hy (lv D ) Fhh,y y (lv D )   2Fhy ,hy (lv D ) Ï2Fhy ,y y (lv D ) .  Ï2Fy y ,hy (lv D ) Fy y ,y y (lv D ) 

ˆ ˆ* Ï2Sˆ hh (i,lv D )Sˆ *hy (i,lv D ) Sˆ hh (i,lv D )Sˆ *y y (i,lv D )   S hh (i,lv D )S hh (i,lv D )  ˆ hy (i,lv D )Sˆ *hh (i,lv D ) Ï2S 2Sˆ hy (i,lv D )Sˆ *hy (i,lv D ) Ï2Sˆ hy (i,lv D )Sˆ y*y (i,lv D ) .  ˆ  ˆ* ˆ ˆ* ˆ ˆ*

c. Effect of nonsimultaneous polarimetric measurements on the off-diagonal elements of the target covariance matrix

vD 2p

O

v D L / 221 F (lv )e jlv D mTm . (7) 2p l52L / 2 x,y D A spectral target covariance matrix can be then defined for each range bin and each Doppler bin l: R x,y (mTm ) 5



The caret symbol  indicates that the matrix or parameter is expressed in the frequency domain, meaning that Sˆx(lv D) is defined as a discrete time Fourier transform of Sx(nTm).

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Sy y (i,lv D )Sy y (i,lv D )



e jlv DDt. This phasor determines the effect of polarization switching on cross-correlation estimates. Therefore, the cross-correlation functions Rc can be obtained just by multiplying the estimated cross spectrum by e2jlv DDt. This is called Doppler phase compensation. This correction is general and can be applied for all moving targets and/or moving clutter, if no strong Doppler aliasing, resulting in spectrum overlapping, is present. In particular for m 5 0, c R hh, y y (0) 5 ^S hh (nTm )S* y y (nTm )&

5

vD 2p

O

L / 221

F chh,y y (lv D ).

Fhh,y y (lv D )e jlv D mTm

(11)

l52L / 2

In the paper, we will use the complex cross-correlation coefficient per Doppler frequency bin (or spectral complex cross-correlation coefficient), which is defined as follows:

rˆ co (lv D ) 5

F chh,y y (lv D ) ÏFhh,hh (lv D )ÏFy y ,y y (lv D )

5 | rˆ co (lv D )|e j Cˆ dp(lv D) .

L / 221

(9)

(12)

l52L / 2 L / 221

F chh,y y (lv D )e jlv D mTm e jlv DDt .

l52L / 2

(10) It can be seen that the ‘‘correct’’ cross spectrum, F chh,y y , and the estimated one, Fhh,y y , are obtained from one another just by multiplying F chh,y y by a phasor

3. Analysis of the Doppler phase compensation a. Experimental verification on rain Lets consider in detail the argument of the copolar cross-correlation coefficient, namely, the differential phase Cdp. The differential phase Cdp (5) is determined as the sum of the differential propagation phase Fdp and

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FIG. 1. (a) Example of a rain Doppler power spectrum hh (thick line) with the corresponding cross-spectrum phase (dotted line) without phase compensation for nonsimultaneous measurements of hh and yy. The spectral reflectivity is given in mm6 m23, and the sum over all of the frequencies gives the commonly used reflectivity value. It can be seen that the time difference between the hh and yy measurements reveals itself in the slope of the linear trend of the phase spectrum. (b) Phase of the cross spectrum in rain after the phase compensation. The linear trend has been compensated for. It should be noted, however, that aliased and nonaliased parts of the spectrum have different mean phase values.

the differential backscattering phase dco. For an S-band radar it is expected that the differential backscattering phase is close to zero (Zrnic´ and Ryzhkov 1999; Bringi and Chandrasekar 2001). Therefore, the spectral differˆ dp does not depend on Doppler velocity ential phase C and therefore is equal to the differential phase Cdp. The Cdp in its turn is fully determined by the differential propagation phase Fdp. For the sake of simplicity we shall assume Fdp to be equal to zero in this section. It should be noted that this assumption is valid for a wide range of radar sensing scenarios, and will be considered in more detail in section 5. Under the current assumptions, the argument of the cross-spectrum F chh,y y is then the differential propagation phase per Doppler velocity bin, where the measured Doppler velocity is defined as V Dm 5 ly D 5 l

l vD , 2 2p

(13)

with yD being the Doppler velocity resolution and l the radar wavelength. For Doppler velocities | VD | # VD,max,

(14)

where VD,max is the maximum unambiguous Doppler velocity and VD is assumed to correspond to the middle of a Doppler velocity bin for simplification (VD 5 VmD), the phase spectrum, without Doppler phase compensation, is ˆ dp (ly D ) 1 4p ly D Dt Arg[Fhh,y y (ly D )] 5 C l ˆ 5 Cdp (ly D ) 1 Df D (ly D ).

(15)

After adding the phase DfDc(lyD) 5 2DfD(lyD)

(16)

per Doppler velocity bin, the Doppler phase compensation is performed and the phase spectrum becomes ˆ dp(lyD) ø 08. Arg[F chh,y y(lyD)] 5 C (17) From the time series of scattering matrices, we carry out Fourier transforms and calculate the second moments of the obtained spectral scattering matrices, which leads to the spectral target covariance matrix. In our case, one spectral target covariance matrix is obtained each 0.96 s (LTm) with L and Tm being 256 and 3.75 ms, respectively. Because of the single-channel receiver of DARR, three pulses are used to obtain Shh, Shy, and Sy y. The time offset Dt between hh and yy measurement is 2.5 ms (2Tm/3). The maximum unambiguous Doppler velocity is 6.03 m s21 [(l/2)(1/2Tm)] and the Doppler velocity resolution is 4.71 cm s21 [(l/2)(1/LTm)], the radar wavelength l being 9.05 cm. The negative Doppler velocities indicate that the hydrometeors are approaching the radar. The Doppler phase compensation is illustrated in Fig. 1. The averaged Doppler power spectrum (hh) of one range bin of rain is plotted (Fig. 1a, thick line) with the corresponding cross-spectrum phase, without the phase correction (Fig. 1a) and with phase correction (Fig. 1b). This Doppler power spectrum is aliased. There is a ground clutter peak centered at 0 m s21. Considering the top of the Doppler spectrum where the spectral reflectivity values are between 0 and 218 dB, the related cross-spectrum phase after correction has a stable value of 08 between 26.0 and 21.5 m s21, which is the expected phase value when the Doppler velocities are smaller than the maximum unambiguous Doppler velocity; see (14) and (17). Between 4 and 6 m s21, the cross-spectrum phase is also stable but its value is about 1208, which indicates the presence of Doppler aliasing. We will discuss this aspect in detail later in the next section. In the same areas of Doppler velocities (suffi-

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ciently large spectral reflectivity), we can clearly see the slope of the cross-spectrum phase versus Doppler velocity, which equals (4p/l)Dt (208 m21 s) when the Doppler phase compensation is not yet carried out, see (15). b. Doppler phase compensation in the case of Doppler spectrum aliasing In Fig. 1b it was observed that in the area where spectrum aliasing is observed the compensated phase shows values different from 0. This is caused by the use of the wrong Doppler velocity values in the Doppler phase compensation procedure. This drawback of the method, however, can be used for our advantage, since it allows us to perform a Doppler spectrum dealiasing, as will be discussed in this section. 1) SINGLE-DOPPLER

ALIASING

First, let us analyze what exactly is happening in the Fig. 1. As we know in this example we have a spectrum that has been aliased once, meaning that the Doppler velocities, VD, on the right side of the spectrum are |V D | . V D,max

and

|V D | # 3V D,max

(18)

or, speaking more generally, the corresponding measured Doppler velocities are VmD 5 VD 7 2VD,max 5 lyD,

(19)

with 2 for VD belonging to the interval [VD,max, 3VD,max[ and 1 for VD belonging to the interval [23VD,max, 2VD,max[. Before Doppler phase compensation, Eq. (15) holds, and after the Doppler phase compensation, which consists of adding the phase Df cD 5 2

4p 8p ly D Dt 5 2Df D 6 V Dt l l D,max

5 2Df D 6 Df D,max ,

(20)

the compensated phase spectrum becomes ˆ dp (ly D ) 6 Df D,max Arg[F chh,y y (ly D )] 5 C ø 62p

Dt . Tm

TABLE 1. Argument of the copolar cross spectrum after Doppler phase compensation—case of multiple aliasing. Actual Doppler velocity interval

nD

[25V D, max, 23V D, max [

2


1

[2V D, max, V D, max [

0

[V D, max, 3V D, max [

1


2

Arg[Fchh,yy (ly D )] 8p (21208) 3 4p 2 (1208) 3 0 4p (21208) 3 8p (1208) 3 2

Arg[F chh,y y(lyD)] ø 6nDDfD,max,

(21)

ALIASING

In a more general case of multiple-Doppler aliasing, Eq. (21) can be generalized to

(22)

where nD 5 0, 1, 2 . . . . The integer nD corresponds to no aliasing (nD 5 0), one aliasing (nD 5 1), and so on. We use a cycle of three pulses to measure the hh, hy, and yy elements of the scattering matrix. Therefore, if Dt 5 2Tm /3, as in the case of DARR, Arg[F chh,y y(lyD)] becomes Arg[F chh,y y (ly D )] ø 6n D

4p . 3

(23)

The phase can only be unambiguously defined in the interval [2p, p]. In our particular case we obtain phase ambiguity from nD 5 3 and we can solve two different aliasing problems. In Table 1 copolar phase spectrum values after the Doppler phase compensation are given in the case of multiple-Doppler spectrum aliasing. It can be seen that the Doppler velocity spectrum can unambiguously be reconstructed if the velocities belong to, for example, intervals [25VD,max, VD,max[, [23VD,max, 3VD,max[, etc. For a classical measurement of the scattering matrix with a dual-channel receiver, only two pulses are used to get four polarimetric measurements (hh, yh) and (hy, yy). In this case Dt 5 Tm /2, which leads to Arg[F chh,y y(lyD)] ø 6nDp.

Meaning that in the case of Doppler spectrum aliasing the Doppler phase compensation results in the addition of a constant phase to the argument of the cross spectrum. This constant phase depends on which side the Doppler aliasing occurs (negative Doppler velocities or positive Doppler velocities) and on the time difference between hh and yy measurement, Dt, compared to the sampling time Tm. 2) MULTIPLE-DOPPLER

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(24)

Phase ambiguity occurs from nD 5 2 and a single aliasing can be corrected. 4. Statistical model of the cross-spectrum phase Since the dealiasing algorithm uses the phase between hh and yy measurements, it is necessary to model the statistical behavior of this phase and its dependency on the number of averages. For this purpose, a statistical model of the cross-spectrum phase is hereby introduced and experimentally verified. Many hydrometeors are contained in the radar resolution volume (range bin), or in our case there are many hydrometeors in a given radar volume having similar Doppler velocities, so they are confined to the same Doppler range bin. Moreover, no single one of them

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dominates the others. In this case the spectral scattering matrix can be modeled as having a multivariate complex Gaussian distribution (Goodman 1985). The real and imaginary parts of any two complex elements of the spectral scattering matrix are assumed to have then a circular Gaussian distribution. Under these assumptions, a probability density function of the phase difference between hh and yy measurements is given by Lee et al. (1994). And the probability density function can be written as

1

G k1 p k (c) 5

2

1 (1 2 | rˆ co | 2 ) kb 2

2Ïp G(k)(1 2 b 2 ) k11/ 2 1

1

2

(1 2 | rˆ co | 2 ) k 1 F k, 1, ; b 2 2p 2

(25)

for the phase

c (l) 5 Arg

[

1 k

O Sˆ (i, l)Sˆ * (i, l)] . k

hh

yy

(26)

i51

This phase corresponds to the differential phase per Doppler velocity bin after k time averages. The probability density function depends on the spectral complex cross-correlation coefficient (12) and on the number of time averages k. The parameter b is defined as follows: ˆ dp) b 5 | rˆ co | cos(c 2 C

(27)

and the functions G and F (more precisely 2F1) are the gamma function and the Gauss hypergeometric function, respectively. The spectral complex cross-correlation coefficient depends on meteorological conditions ˆ dp will be different from 0 (for heavy precipitation, C and the modulus of rˆ co may decrease). Since for our dealiasing procedure we have to consider the statistics of the phase spectrum, it is necessary to verify the above presented formulation, which was originally introduced for a time series of observations. For the experimental verification of the above presented model we have used a light precipitation measurement. We have estimated the copolar cross-correlation coefficient, | rˆ co | , per Doppler velocity bin using a Doppler cross-spectrum average of 40 spectra. The measurement specifications are the same as the ones given in section 3. The rain and precipitating cloud areas show high values of this parameter, typically 0.99, and we have found that | rˆ co | is about 0.97 in the melting layer at the top of the Doppler spectrum. These values are characteristic of stratiform light rain. It should be noted that the values of the spectral cross-correlation coefficient are generally larger than the values of the conventional cross-correlation coefficient. This can be explained by the fact that the common cross-correlation coefficient is determined by the sum of the cross spectra over all Doppler velocity bins. Therefore, it is an in-

FIG. 2. Comparison of the theoretical pdf (thick line) with the experimental pdf (thin line) of the cross-spectrum phase as measured in stratiform rain. Two cases are considered: no averaging and averaging over 10 consecutive spectra. The phase used for this study was extracted from all Doppler velocity resolution cells for all range gates in the precipitation region, where the magnitude of the spectral cross-correlation coefficient is 0.989. It is expected that since the spectral analysis was performed on consecutive nonoverlapping time intervals, all obtained Doppler spectra are statistically independent. That is confirmed by the presented figure, which shows good agreement between theoretical and experimental pdf’s.

tegral over all hydrometeors with different sizes and shapes. This integration results in a decrease of the coherence of the signal and thus in the reduction of the cross-correlation coefficient. For the experimental validation of the proposed model we have selected only that Doppler range bins where the spectral cross-correlation coefficient was composed of values between 0.9885 and 0.9895. Therefore, in (25) and (27) we have used rˆ co being equal to 0.989. For the selected Doppler range bins we have calculated the phase spectrum without any averaging and by averaging the cross spectrum over 10 spectra. A comparison between the model (25) and the data (26) is shown in Fig. 2. From this figure we can conclude that there is a good fit between the theoretical and experimental probability density functions, with or without average. Therefore, we can use Eqs. (25)–(27) to describe the statistical behavior of the phase spectrum. Since the model of the cross-spectrum phase distribution is verified, the related standard deviation

sc (k) 5

Î

E E p

c 2 p k (c) dc

2p

(28)

p

p k (c) dc

2p

can be computed versus the magnitude of the spectral cross-correlation coefficient for several averages (1, 2, 4, 10, 20, 30, 40, 50). This is plotted in Fig. 3. For this computation, the mean value of c is 0. After Doppler

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FIG. 3. Standard deviation of the phase of the cross spectrum hh, yy for different numbers of averages (1, 2, 4, 10, 20, 30, 40, 50). For most atmospheric echoes the copolar spectral cross-correlation coefficient belongs to the interval [0.7, 1].

processing, the magnitude of the cross-correlation coefficient per Doppler velocity bin is generally composed of values between 0.9 and 1 for light precipitation. In that case, the largest standard deviation, about 408, is obtained for | rˆ co | 5 0.9 when there is no average. When there is a single aliasing, the mean phase is 61208 (with our measurement specifications), which indicates that the highest standard deviation possible is 608 before a phase ambiguity occurs. Therefore, a dealiasing technique based on the cross-spectrum phase can be successfully used when there is no average. Nevertheless, Fig. 3 shows that it is worthwhile to carry out an average on two Doppler spectra, which leads to a strong reduction of the standard deviation, which becomes about 208. 5. Mean value of the cross-spectrum phase Until now, the mean value of the cross-spectrum phase, or differential phase Cdp, was considered to be about 08 for atmospheric targets at S band. For Rayleigh scattering, only the differential propagation phase Fdp contributes to Cdp. With our measurement specifications, the mean value of the cross-spectrum phase will be 61208 when there is Doppler aliasing, depending on which side of the Doppler velocity spectrum the Doppler aliasing occurs. Using these three characteristic phase values, 08, 11208, and 21208, and the standard deviation discussed in the previous section, a relatively easy dealiasing algorithm can be implemented to correctly obtain the Doppler spectra of polarimetric parameters. In cases of heavy precipitation and for horizontal profiles, the differential propagation phase is significantly different from 08. With the measurement specifications as used, the largest standard deviation permitted is 608. The standard deviation depends on the magnitude of the

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FIG. 4. The dependence of the maximum differential phase of atmospheric targets, which allows for unambiguous velocity reconstruction, on the number of spectra taken for estimation of the Doppler cross spectrum. The dependence is shown for different numbers of averages (1, 2, 4, 10, 20, 30, 40, 50) and different copolar spectral cross-correlation coefficients (0.7, 0.8, 0.85, 0.9, 0.95, 0.99).

spectral cross-correlation coefficient, which is generally high for atmospheric targets, thus reducing sc and the number of averages. The maximum value for Fdp (in 8) can then be expressed as follows: Cdp,max 5 Fdp,max 5 60 2 sc.

(29)

When the standard deviation is large because of a small average number, the possible values of Fdp are reduced. This is illustrated in Fig. 4. These values of the maximum differential phase are related in the following to rain rates. For this purpose, a set of values of the specific differential phase Kdp (8 km21) is built and the estimate of rain rate R (mm h21), using the Pruppacher–Beard equilibrium shape model, is computed (Bringi and Chandrasekar 2001): R 5 40.5(Kdp)0.85.

(30)

This relation is given for horizontal sounding of precipitation and thus provides the upper-limit estimate of the differential propagation phase. This is the worst case for the dealiasing algorithm. In this case the differential propagation phase is Fdp 5 2Kdpr,

(31)

where r is the range. Figure 5 gives the domain of validity of the dealiasing algorithm for horizontal profiles of precipitation. The differential propagation phase must not exceed 608. Magnitude values of the spectral crosscorrelation coefficient indicate the possible maximum values of the differential propagation phase. For example, | rˆ co | . 0.95 shows that the maximum values of Fdp are increasing from 308 to 608 with the number of averages. The radars TARA and DARR are intended for detailed studies of the atmosphere with high range

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451

was used for the dealiasing algorithm. The large and known jump in the phase of the cross-spectrum hh, yy at a fixed range (see Fig. 1), due to aliasing, can be easily detected. In that case, the absolute values of the cross-spectrum phase must not be considered but the phase jump in the Doppler spectrum should be. The consequence is that the unfolding of the Doppler spectra can be successfully performed but the knowledge of the actual Doppler velocities may be lost. For a classical measurement of the scattering matrix (dual-channel receiver), the same approach can be followed with a limitation of 908 instead of 608 for the differential propagation phase; see Eq. (24). 6. Dealiasing procedure a. Dealiasing of a slant profile of precipitation FIG. 5. Dependence of observed differential phase on rain rate and propagation path. This dependence is shown by equiphase curves that correspond to Fdp values of 108, 208, 308, 408, and 608. Since this dependence is calculated for horizontal sounding, this figure gives the upper limit on the observed Fdp.

resolution and high Doppler resolution. Their operational maximum range is about 40 km. The maximum range used is currently 15 km, which maximally allows for a rain rate of 73 mm h21 for horizontal profiling. With these radars, vertical and slant profiles (elevation of 308 or 458) are generally measured. Therefore, the failure of the dealiasing algorithm due to values of the differential propagation phase significantly different from 08 has not yet occurred. It is important to note that for a horizontally profiling weather radar the same strategy can be considered as

To illustrate the dealiasing procedure, a slant profile (Fig. 6) is selected. It shows light stratiform precipitation at hh polarization. The radar looks at an elevation angle of 308 and the averaging time is 48 s. The range resolution is 75 m. The melting layer is located between 2 and 3.1 km. Under the melting layer, there is rain. The precipitating cloud layer is located between 3.1 and 9.25 km. The Doppler resolution is 4.71 cm s21. The negative Doppler velocities indicate that the hydrometeors are approaching the radar. The measurements are carried out with the maximum Doppler velocity being too low. For slant profiles, the Doppler velocities cannot be related directly to the fall speed of the hydrometeors and contain the wind Doppler velocity. The selected slant profile shows large variations in the Doppler wind velocity as a function of height in Fig. 6, especially in the rain area.

FIG. 6. Slant profile of precipitation: (left) range profile of reflectivity and (right) corresponding Doppler power spectra.

452


FIG. 7. Slant profile of precipitation: cross-spectrum phase after correction for the nonsimultaneity of the measurements hh and yy. To calculate the cross spectrum, averaging over 50 spectra was used.

The phase of the cross-spectrum F chh,y y is plotted in Fig. 7. Three main phase values are seen. The gray area in Fig. 7 is the only part of the precipitation event, rain in the near range, that belongs to the Doppler velocity interval [26 m s21, 6 m s21[. The phase of the cross spectrum, after Doppler phase compensation, is about 08. This precipitation measurement shows two other characteristic phases: one is about 1208 (dark gray) and the other one is about 21208 (light gray). These values indicate the presence of Doppler aliasing and make it possible to determine to which Doppler interval they belong; see Table 1. In this example of the slant profile, the Doppler velocities are expected to be negative (the mean Doppler velocities per range bin calculated from the Doppler spectrum part related to the phase value 08 are near 26 m s21). When the precipitation event corresponds to the Doppler velocities interval [218 m s21, 26 m s21], the constant phase 24p/3 [see (23)] is added to the phase of the cross-spectrum F chh,y y . This constant phase is 22408 or equivalently 1208 and corresponds to the dark gray area. The same occurs for velocities belonging to [230 m s21, 218 m s21[, then the constant phase 28p/3 is added to the cross-spectrum phase and leads to the light gray area (24808 or equivalently 21208). Using the phase of the cross spectrum and the continuity of the mean Doppler velocity, the dealiased slant profile is given in Fig. 8. The number of averages is 50 for this event. Only the areas corresponding to a crosscorrelation coefficient per Doppler bin larger than 0.7 and having a cross-spectrum phase comprised in the interval (mean phase 62sc) have been considered. The largest standard deviation sc is 68 and three mean phases are found. For this example, Cdp may reach 68. The phase window can be chosen larger in order to allow for a larger Cdp than 68.

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FIG. 8. Doppler power spectra (hh) after dealiasing for all the measured range bins. It can be seen that parts of the signal that correspond to clutter and spectral tails of the melting layer were suppressed. This is due to a rather narrow phase window that was used for the dealiasing procedure.

b. Consequences of the dealiasing procedure The maximum unambiguous Doppler velocity in the case of nonpolarimetric measurements would be 18.09 m s21. Therefore, Doppler aliasing would still be present for this slant profile where Doppler velocities up to 223 m s21 are measured. This example with two consecutive Doppler aliasings shows that the proposed polarimetric dealiasing procedure is not only unfolding the Doppler spectra but is able to obtain the actual Doppler velocities with the benefits of full polarimetric information. The phase of the cross-spectrum F chh,y y also indicates problematic zones because of the presence of very different phase values than expected. Two examples of these values, present in Fig. 7, are briefly discussed: ground clutter and the melting layer. The cross-spectrum phase is very useful for detecting the Doppler cells contaminated by ground clutter. After removal of the stable ground clutter, the time-varying ground clutter shows a rather broad spread of phases (see Fig. 7 around the Doppler velocity 0 m s21) (Ryzhkov and Zrnic´ 1998; Moisseev et al. 2002). Therefore, during the dealiasing procedure (also if the threshold on the magnitude of the cross-correlation coefficient per Doppler velocity bin is not applied), the Doppler cells significantly affected by ground clutter are discarded, which can be seen in Fig. 8, under and above 1 km in the rain area (missing data around Doppler velocity 212 m s21). When sc is large because there is no time average or a significant propagation phase can be expected, the threshold has to be applied for clutter suppression. In the second area, the proposed dealiasing technique may not perform optimally. This is the melting layer (between 2 and 3.1 km). Under the reflectivity peak of the melting layer (2.75 km), the Doppler spectra become wide (2.5 km). In the tail of these Doppler spectra, the spectral reflectivity values are similar to those of a pre-

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radar is near the horizontal wind direction (dashed line), about 2308. 7. Comparison of different corrections for nonsimultaneous polarimetric measurements on the cross-correlation coefficient rco The aim of this section is to compare well-known proposed corrections (Illingworth and Caylor 1991; Balakrishnan and Zrnic´ 1990) for the nonsimultaneous measurements of the scattering matrix with the method discussed in this paper. This comparison is carried out on range profiles of the modulus of the cross-correlation coefficient. a. Computation of rco without correction If no correction is applied, the cross-correlation coefficient is directly calculated from the time series of the scattering matrix. For the slant profile, the number of averages (k) is 50 and the number of the Doppler velocity bins (L) is 256: FIG. 9. Comparison of the radar estimates of the Doppler mean velocity attributed to the horizontal wind with the wind profile provided by a radiosonde.

cipitating cloud but the spectral cross-correlation coefficient magnitude is small and the cross-spectrum phase differs from expectations. At the present moment, the dealiasing procedure suppresses the tails of these Doppler spectra. This effect can be clearly seen for a few range bins, from 2.2 to 2.5 km, in the dealiased profile of Fig. 7. These Doppler spectra in the lower part of the melting layer (under the reflectivity peak) are investigated in Verlinde et al. (2002). c. Comparison with a meteorological balloon In order to compare the Doppler velocities obtained after dealiasing the slant profile, one vertical profile of the wind conditions (balloon in Fig. 9) was supplied by the Netherlands Meteorological Institute (KNMI). The two sensors are not collocated but a comparison can still be done. The mean velocity measured by the balloon (thick line) is mainly composed of values between 22 and 23 m s21 from 0.2 to 4.5 km (height). The estimated horizontal wind mean velocities from the radar data, calculated for a vertical profile, are also plotted in Fig. 9 (asterisk). They correspond to the Doppler mean velocities of the slant profile divided by cos(308), which means that the contribution of the hydrometeors to the Doppler mean velocity is here neglected. Apart from the first 0.5 km (1-km range in the slant profile) there is a good agreement for the estimated wind velocities measured by the radar if the azimuth direction of the

rco (Dt) 5

^S hh (nTm )Sy*y (nTm 1 Dt)& Ï^|S hh (nTm )| 2 &Ï^|Sy y (nTm 1 Dt)| 2 &

.

(32)

As was discussed above, the smaller the decorrelation time of Shh(nTm) [or Sy y(nTm)], as compared to Dt, the larger the bias in the rco(Dt) estimate; this can be observed in Fig. 10. b. Computation of rco with Doppler phase compensation and dealiasing The cross-spectrum F chh,y y (11) and the power spectra Fhh,hh and Fy y,y y are integrated on the relevant Doppler spectrum velocity bins. When the modulus of the cross spectrum is smaller than 0.7 or the phase of the cross spectrum does not belong to the predicted phase intervals, which are used for dealiasing, the corresponding Doppler bins are discarded for the integration. The number of integrations is dependent on the range bin:

OF lmax

rco (0) 5

!O lmax

l5lmin

c hh,y y

l5lmin

Fhh,hh (ly D )

(ly D )

!O lmax

.

(33)

Fy y ,y y (ly D )

l5lmin

In the case of Doppler aliasing and before integration, the phase of the cross spectrum is compensated with the appropriate constant phase: c 7jn D Df D,max F cd . hh,y y (ly D ) 5 F hh,y y (ly D )e

(34)

c. Computation of rco with time interpolation The time series hh and yy are interpolated in order to obtain a coincident set of hh and yy time samples.

454


This is done by first computing the Fourier transform of the data, padding the transform with an equal number of zeros to double the Nyquist frequency, and then taking the inverse Fourier transform (Illingworth and Caylor 1991). Then the scattering matrix obtained with two pulses (dual-polarized receiver) can be estimated. In our case, where one channel receiver is used for scattering matrix measurements, three pulses are needed. Therefore, the hh and yy time samples have to be oversampled three times, which leads to a less favorable interpolation. d. Computation of rco with the Gaussian power spectrum assumption When the Doppler power spectra are assumed Gaussian, the temporal correlation coefficient can be expressed as

r(3Ts ) 5

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r(mTs ) 5 e28(p s y mTs /l) 2 e2j(4p V mTs /l) .

(35)

where sy is the Doppler velocity spectrum width, V is the mean Doppler velocity, and mTs the time lag. Furthermore the cross-correlation coefficient at time lag mTs is assumed to contain independent contributions from the distribution of Doppler velocities and the distribution of canting angles and shapes (Balakrishnan and Zrnic´ 1990), which leads to

rco (mTs ) 5 r(mTs )rco (0) 5 r(mTs )rco .

(36)

With our measurement specifications, the cross-correlation coefficient at time lag Dt 5 2Ts in (32) and the temporal correlation coefficient at time lag Tm 5 3Ts,

^S hh (nTm )S*hh (nTm 1 3Ts ) 1 Sy y (nTm 1 2Ts )S*y y (nTm 1 5Ts )& Ï^|S hh (nTm )| 2 &Ï^|Sy y (nTm 1 2Ts )| 2 &

,

(37)

can be estimated. Using (35) and (36), the magnitude of the cross-correlation coefficient can be expressed as | rco | 5

| rco (2Ts )| . | r(3Ts )| 4/9

(38)

e. Results and discussion Four range profiles of the magnitude of the crosscorrelation coefficient are plotted in Fig. 10 for the slant profile. The best result (thick line) is obtained with the corrections proposed in this article. When no correction is applied (line with xs), | rco | decreases when the Doppler spectrum width increases. The largest Doppler spectrum width (between 1.4 and 1.8 m s21) is found between 0 and 1 km (rain area). The magnitude of the crosscorrelation coefficient decreases when the backscattered signal of atmospheric targets is contaminated by clutter. Another cause of the decreases of | rco | is the decrease of the signal-to-noise ratio. This can be clearly seen from the 8.5-km range. The method of estimating the cross-correlation coefficient at time lag 0 proposed by Balakrishnan and Zrnic´ (1990) gives good results (thin line). The method is not affected by Doppler aliasing but can provide values of | rco | larger than 1 (see Fig. 10: first and last range cells, range cell before 6 km). The major drawback of the time interpolation method is its collapse when Doppler aliasing occurs. A series of minima of | rco | coincides with the ranges affected by Doppler aliasing in Fig. 10.

FIG. 10. Comparison of four methods to estimate the cross-correlation coefficient: 1) no correction (line with xs), 2) corrected with Doppler phase compensation and polarimetric dealiasing (thick line), 3) padding or time interpolation (dashed line), and 4) Zrnic´ or Gaussian assumption (thin line).

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8. Conclusions Simultaneously combining polarimetry and Doppler analyses leads to two major measurement difficulties. The first one is the nonsimultaneity of the measurements performed with different polarizations. The second is the reduction of the maximum unambiguous Doppler velocity because of the transmission of different polarizations, which leads to Doppler spectrum aliasing effects. A solution to these two problems is given in the paper. This solution is based on the phase of the crossspectrum hh, yy or equivalently the differential phase ˆ dp per Doppler velocity bin. C The proposed Doppler phase compensation also corrects for the nonsimultaneity of the polarimetric measurements hh and yy in cases of Doppler spectrum aliasing but adds a constant phase, which depends on the interval of the actual Doppler velocities. This seeming drawback of the processing is a very useful tool for Doppler spectrum dealiasing. This dealiasing technique uses phase jumps and absolute phase values to dealiase and to retrieve the actual Doppler velocities. This method is especially useful for slant profiles, which are affected by the wind velocity. The proposed polarimetric dealiasing method is considered for atmospheric echoes at S band. This technique is relatively easy to implement. Its domain of validity is discussed in terms of maximum standard deviation and maximum propagation phase. When large values of Cdp are encountered (heavy precipitation and horizontal profiling), the algorithm can be extended. In that case, the absolute values of the cross-spectrum phase hh, yy must not be considered but the jump of this phase in the Doppler spectrum at a fixed range should be. Then the unfolding of the Doppler spectra can be performed but the knowledge of the actual Doppler velocities may be lost. Combining a correction for the nonsimultaneity of the polarimetric measurements, which is valid in cases of Doppler aliasing and a new polarimetric dealiasing technique, allows for full polarimetric measurements of the atmosphere to be obtained with the actual Doppler velocities of the hydrometeors and the maximum unambiguous Doppler velocity of the corresponding single polarized measurement scheme. A comparison between the proposed method and known corrections for the offdiagonal elements of the target covariance matrix shows that the proposed method provides a better estimate of the-cross correlation coefficient. Acknowledgments. The authors would like to thank their colleague Dr. Silvester Heijnen for his help with validation of the results and the Netherlands Meteorological Institute (KNMI) for providing the radiosonde measurements. REFERENCES Balakrishnan, N., and D. S. Zrnic´, 1990: Use of polarization to characterize precipitation and discriminate large hail. J. Atmos. Sci., 47, 1525–1540.

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itation measurements: Introduction of the spectral differential reflectivity. Preprints, 30th Int. Conf. on Radar Meteorology, Munich, Germany, Amer. Meteor. Soc., 316–318. Verlinde, J., D. Moisseev, N. Skaropoulos, S. Heijnen, F. V. Zwan, and H. Russchenberg, 2002: Spectral polarimetric measurements in the radar bright band. Proc. URSI-F Open Symp. on Propagation and Remote Sensing, Garmisch-Partenkirchen, Germany, URSI and DLR, CD-ROM, P113. Zrnic´, D. S., and A. V. Ryzhkov, 1999: Polarimetry for weather surveillance radars. Bull. Amer. Meteor. Soc., 80, 389–406. ——, N. Balakrishnan, A. V. Ryzhkov, and S. L. Durden, 1994: Use of copolar correlation coefficient for probing precipitation at nearly vertical incidence. IEEE Trans. Geosci. Remote Sens., 32, 740–748.