Combined Doppler and Polarimetric Radar Measurements: Correction for Spectrum Aliasing and Nonsimultaneous Polarimetric Measurements

Transcription

1 MARCH 2004 UNAL AND MOISSEEV 443 Combined Doppler and Polarimetri Radar Measurements: Corretion for Spetrum Aliasing and Nonsimultaneous Polarimetri Measurements C. M. H. UNAL IRCTR, Delft University of Tehnology, Delft, Netherlands D. N. MOISSEEV Faulty of Civil Engineering and Geosienes, Delft University of Tehnology, Delft, Netherlands (Manusript reeived 5 Marh 2003, in final form 24 September 2003) ABSTRACT Combining Doppler and polarimetri information is advantageous for atmospheri studies. On the one hand, Doppler information gives insight into the mirophysial and dynami properties of radar targets, that is, radial veloity and its variability. The polarization diversity, on the other hand, has a strong link to the mirophysial properties of targets suh as shape and orientation. Polarimetri measurements, however, have an adverse effet on Doppler proessing. Measurements of the omplete sattering matrix require at least two pulses and result in the redution of the maximum unambiguous Doppler veloity that an lead to Doppler spetrum aliasing. Moreover, dynami properties of targets, beause of the nonsimultaneity of the measurements performed with different polarizations, affet the auray of polarimetri radar measurements. A solution to these two problems is given in this paper. It is shown that by applying a relatively simple proessing tehnique the effet of nonsimultaneous polarimetri measurements an be redued even in the ase of strong Doppler spetrum aliasing. This leads to better estimates of the opolar ross-orrelation oeffiient. Furthermore, this proessing results in the maximum unambiguous Doppler veloity as if no polarimetri measurements were performed, with the added advantage of obtaining the atual Doppler veloities. To illustrate the proposed proessing tehnique, preipitation measurements taken with the Delft Atmospheri Researh Radar (DARR) are used. 1. Introdution Radar polarimetry is widely used in radar remote sensing. Probably the more popular appliations of radar polarimetry are radar target haraterization (Lüneburg et al. 1991; Boerner et al. 1998; Ryzhkov 2001) and target ontrast enhanement (Swartz et al. 1988; Tragl 1990; Ryzhkov and Zrnić 1998; Moisseev 2002). Even though these two appliations are different they often employ seond-order-moment analysis of the polarimetri data. The hoie of this formalism is aused by the fat that most of the natural targets have time-dependent harateristis. This analysis, however, has one problem that is aused by measurement limitations (Sahidananda and Zrnić 1989). With the exeption of radars that simultaneously transmit two orthogonally polarized signals by means of speial signal oding (Giuli and Faheris 1991; Giuli et al. 1995) or by using a hybrid polarization measurement mode (Sott et al. Corresponding author address: Christine Unal, IRCTR, Faulty of Eletrial Engineering, Mathematis, and Computer Siene, Delft University of Tehnology, Mekelweg 4, 2628 CD Delft, Netherlands. .m.h.unal@ewi.tudelft.nl 2001), most radars employ one of these polarization swithing shemes to measure a full target sattering matrix as disussed in Santalla and Antar (2002). In this ase the variability of the radar targets may ause hanges in the properties of the targets from one yle of measurements to another. It should be noted that even for swithing on a pulse-to-pulse basis a hange in sattering properties of natural targets may be observed. The problem of nonsimultaneous polarimetri measurements in ases of radar observations has been studied by several authors. In Balakrishnan and Zrnić (1990) the effet of nonsimultaneous polarimetri measurements in atmospheri studies was analyzed. The authors of this paper have proposed a method that allows for ompensation of this problem, if two assumptions are met. The first assumption is that the Doppler power spetrum of meteorologial ehoes has the Gaussian shape. The seond one is that the distributions of radial veloities, anting angles, and shapes are independent of one another. Even though this method allows for a relatively simple ompensation sheme, it relies on two assumptions that are not always satisfied. It was observed that in many ases the Doppler power spetra of atmospheri ehoes do not have a Gaussian shape. For 2004 Amerian Meteorologial Soiety

2 444 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 21 raindrops, the distributions of radial veloities and shapes are related when there is no signifiant turbulene (Unal et al. 2001; Verlinde et al. 2002). Another widely used approah is to interpolate the time series, for example hh and in order to obtain a oinident set of hh and time samples. This tehnique was demonstrated by Illingworth and Caylor (1991) and Zrnić et al. (1994). This is a better method beause there is no assumption on the shape of the power spetra; however, this approah does not work in ase of Doppler spetrum aliasing, as will be shown later. An alternative way to solve the problem of nonsimultaneous polarimetri observations is to perform polarimetri measurements with three different transmit polarizations. This approah is introdued by Santalla et al. (1999) and further disussed in Santalla and Antar (2002). The proposed tehnique is insensitive to Doppler properties of radar ehoes and therefore an be used in ases where the two previously disussed tehniques fail. But unfortunately, this method requires a more ompliated radar measurements sheme. Namely, it needs transmission of three alternately polarized radar signals instead of the usual two and the radar must have a dual-hannel reeiver. Moreover, due to the fat that it needs three alternately polarized signals instead of two, as is ommonly the ase, it has a strong adverse effet on Doppler measurements of radar objets as disussed later. For most radar systems the measurement of a target sattering matrix requires at least two pulses, or three pulses in the ase of a single-hannel reeiver system. As a result the polarimetri measurements redue the maximum unambiguous Doppler veloity and thus inrease the probability of having Doppler aliasing, whih in its turn ompliates interpretation of radar data (Sahidananda and Zrnić 1989). Therefore, the seond main topi that is addressed in this paper is the effet of polarimetri measurements on Doppler observations. An inrease in sampling time, for example due to polarization swithing, auses a derease in Nyquist frequeny and thus inreases the probability of spetral aliasing. This problem is often solved by algorithms using range ontinuity and/or Doppler veloities ontinuity properties. But refletivity-free areas and wind shears may ause disontinuities in the measured Doppler field. For these ases, the algorithms annot orret for aliased veloities. Another issue is that a lot of unfolding tehniques exist, but they do not provide the atual Doppler veloities and thus annot give the atual mean Doppler veloity. Another dealiasing proedure is desribed by Hennington (1981) and Doviak and Zrnić (1993); however, this method requires an additional piee of meteorologial information, whih is the environmental wind knowledge as a funtion of height. Also the radar observation sheme is often modified to inrease the maximum unambiguous Doppler veloity. One of these suh measurement shemes is staggering the pulse repetition time (PRT; see Sirmans et al. 1976; Doviak and Zrnić 1993). This tehnique gives improved results for the refletivity and Doppler veloity retrievals. But this type of measurement inreases the omplexity and the time of the measurement, espeially when the measurement of the sattering matrix has to be added. In this artile we analyze the problem of nonsimultaneous polarimetri measurements for high-resolution Doppler polarimetri atmospheri radars, suh as the Delft Atmospheri Researh Radar (DARR; Niemeijer 1996) and the Transportable Atmospheri Radar (TARA; Heijnen et al. 2000). These S-band radars are used to provide high-resolution Doppler and polarimetri data of atmospheri ehoes to failitate studies of preipitation, louds, and atmospheri turbulene. Unal et al. (2001) and Verlinde et al. (2002) have shown that the ombination of Doppler and polarimetri measurements as provided by suh radars gives us an opportunity to gain a more detailed study of atmospheri phenomena. In this artile we use a Doppler phase ompensation method to ompensate for effet of nonsimultaneous polarimetri measurements. It is shown that the phase ompensation is a good orretion in the ase of Doppler aliasing. Moreover, it leads to a new possibility for dealiasing preipitation profiles. The proposed dealiasing tehnique uses polarization measurements of atmospheri ehoes to unfold Doppler spetra. It is shown that this tehnique allows an inrease in maximum unambiguous Doppler veloity as if no polarimetri measurements were performed with the added advantage of obtaining the atual Doppler veloities. It is not only the estimation of the opolar ross-orrelation oeffiient that is obtained, whih is signifiantly improved when folded Doppler spetra are orretly unfolded, but also other polarimetri parameters, and the Doppler moments. The performane of the method is illustrated on a slant profile measurement of light preipitation. Furthermore, the domain of validity of the proposed dealiasing method is disussed in details. For this purpose, a statistial model of the differential phase spetrum is introdued and experimentally verified to obtain the standard deviation of this phase for different measurement onditions (number of averages, ross-orrelation oeffiient). The paper onsists of the following setions. Setion 2 defines the spetral target ovariane matrix, whih is used for Doppler polarimetry and introdues the orretion for the nonsimultaneity of polarimetri measurements hh and. In setion 3, we disuss the proposed phase orretion and its onsequenes for the ase of Doppler aliasing. In order to elaborate upon a dealiasing algorithm, a statistial model of the phase of the ross-spetrum hh, is presented in setion 4. After phase ompensation in the Doppler frequeny domain, we assume that the mean value of the ross-spetrum phase is about 0 at S band. This assumption is disussed in setion 5. In setion 6, the dealiasing proedure is

3 MARCH 2004 UNAL AND MOISSEEV 445 explained and illustrated on a slant profile of light rain measured during a windy day. Better estimates of the ross-orrelation oeffiient are obtained in setion 7 with the proposed phase orretion and dealiasing tehnique. The omparison is made with already existing orretions. We onlude the paper in setion Doppler polarimetri formalism The polarimetri ontent of the radar signal an be expressed by a target ovariane matrix, whih desribes the mean polarimetri properties of the hydrometeors. Well-known polarimetri meteorologial parameters, like the differential refletivity Z dr and the omplex ross-orrelation oeffiient o, are derived from this matrix. By ombining Doppler and polarimetri information, we aim to measure and interpret these polarimetri parameters per Doppler veloity bin. For this purpose, a new polarimetri matrix, the spetral target ovariane matrix, is defined in this setion. Before this, the lassial target ovariane matrix is briefly introdued and will be related to the spetral target ovariane matrix. The phase orretion for the nonsimultaneity of the polarimetri measurements is introdued at the end of the setion and will lead to the final form of the spetral target ovariane matrix. a. Target ovariane matrix To desribe polarimetrially time-dependent targets, the seond-order moments of the time series of sattering matries are usually alulated. They lead to the size-averaged polarimetri properties of the targets. The sattering matrix elements must be simultaneously measured. In pratie, the opolar elements of the sattering matries S hh and S are measured with a time differene t, whih must be smaller than the deorrelation time of the hydrometeors. If the sattering matrix elements an be desribed in terms of ergodi disrete random proesses, the orrelation funtions R x,x and the rossorrelation funtions R x,y, where x or y stands for hh, h, or, an be used for analysis of their statistial properties. For example, the orrelation funtion R hh,hh at time lag mt m is defined as R (mt ) S (nt )S*[(n m)t ] x,x m x m x m k il1 x 1 S (nt )S*[(n m)t ] (1) kl i1 n(i1)l m x m and the ross-orrelation funtion R x,y is R x,y(mtm t) S x(nt m)s * y [(n m)tm t] (2) k il1 1 x kl i1 n(i1)l S (nt ) S *[(n m)t t]. (3) y The time between two onseutive measurements (or m m samples) with the same polarization setting is denoted by T m. In the above equations, t, a fration of T m,is the time differene between two onseutive measurements x and y. The samples numbers are denoted by n and m. The symbol denotes the time average and onsists of a double summation where L is the number of samples for the Doppler proessing and where k is the number of averages of the Doppler spetra. The Doppler proessing will be defined in the next setion. Sine the random proesses are assumed stationary, the orrelation and ross-orrelation funtions do not depend on the time nt m (sample n) but on the time lag mt m. The elements of the target ovariane matrix [C], alulated for eah range bin, are thus the orrelation funtions and ross-orrelation funtions of the random proesses S hh, S h, and S, alulated at zero time lag (m 0): [C] (4) S S* 2S S* S S* hh hh hh h hh az 2ShS* hh 2ShS* h 2ShS*. SS* hh 2SS* h SS* The target ovariane matrix is expressed in mm 6 m 3 for atmospheri targets, the sattering matrix has units of meters (m), and the onstant a Z is a unit onversion fator from m 2 to mm 6 m 3. The omplex ross-orrelation oeffiient, whih is disussed in the paper, is defined as ShhS* o 2 2 S S hh jdp j(dp o) o e o e, (5) where dp is the differential phase, dp is the differential propagation phase, and o is the differential baksattering phase (Bringi and Chandrasekar 2001). The orrelation funtions and ross-orrelation funtions, alulated at different time lags (m 0), give a more omplete desription of the targets. Another equivalent approah to enhaning the polarimetri desription is Doppler polarimetry. b. Spetral target ovariane matrix The purpose of the spetral target ovariane matrix is to desribe both polarimetri and dynami properties of the radar targets. When a random proess is stationary, it an also be desribed in the frequeny domain by power spetra (x y) and ross spetra (x y) (Therrien 1992): L1 jl D mt F (l ) T R (mt )e m x,y D m x,y m, (6) m0 where l denotes the Doppler frequeny bin and D is the Doppler frequeny resolution. The number of Dopp-

4 446 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 21 ler frequeny bins is L. The orrelation funtions and ross-orrelation funtions are the disrete Fourier transforms inverse of the power spetra and ross spetra, respetively (Therrien 1992): L/21 D jl D mt R (mt ) F (l )e m x,y m x,y D. (7) 2 ll/2 A spetral target ovariane matrix an be then defined for eah range bin and eah Doppler bin l: F (l ) 2F (l ) F (l ) hh,hh D hh,h D hh, D D [C(l ˆ D)] az 2F h,hh(l D) 2F h,h(l D) 2F h,(l D). (8) 2 F,hh(l D) 2F,h(l D) F,(l D) The spetral target ovariane matrix is expressed in mm 6 m 3. For example, a Z ( D /2) F hh,hh (l D ) gives us a variane of hh signal for a frequeny range [(l 1/2) D,(l 1/2) D [, and is named the spetral refletivity. Moreover, the sum of all spetral refletivities leads to the ommonly used refletivity value of the onerned range bin. The spetral target ovariane matrix an be equivalently expressed (Doviak and Zrnić 1993) as S ˆ (i,l )S ˆ* (i,l ) 2S ˆ (i,l )S ˆ* (i,l ) S ˆ (i,l )S ˆ* (i,l ) k hh D hh D hh D h D hh D D a Z [C(l ˆ ˆ ˆ* ˆ ˆ* ˆ ˆ D)] 2S h(i,l D)S hh(i,l D) 2S h(i,l D)S h(i,l D) 2S h(i,l D)S * (i,l D). (9) k i1 ˆ ˆ ˆ ˆ ˆ ˆ S (i,l D)S * hh(i,l D) 2S (i,l D)S * h(i,l D) S (i,l D)S * (i,l D) The aret symbol ˆ indiates that the matrix or parameter is expressed in the frequeny domain, meaning that Ŝ x (l D ) is defined as a disrete time Fourier transform of S x (nt m ).. Effet of nonsimultaneous polarimetri measurements on the off-diagonal elements of the target ovariane matrix The problem of nonsimultaneous polarimetri measurements is enountered in the estimates of the offdiagonal elements of the target ovariane matrix. It should be noted that in the ase where t is muh smaller than the deorrelation time of S x (nt m ), the effet of nonsimultaneous polarimetri measurements is negligible, otherwise it should be taken into aount. Let us illustrate the effet of nonsimultaneous polarimetri measurements on the ross-spetrum hh,. The orresponding ross-orrelation funtion is R hh,(mtm t) S hh(nt m)s * [(n m)tm t] L/21 D jldmtm hh, D 2 ll/2 L/21 D jldmtm jldt hh, D F (l )e F (l )e e. 2 ll/2 (10) It an be seen that the orret ross spetrum, F hh,, and the estimated one, F hh,, are obtained from one another just by multiplying by a phasor F hh, jl e Dt. This phasor determines the effet of polarization swithing on ross-orrelation estimates. Therefore, the ross-orrelation funtions R an be obtained just by jl multiplying the estimated ross spetrum by e Dt. This is alled Doppler phase ompensation. This orretion is general and an be applied for all moving targets and/or moving lutter, if no strong Doppler aliasing, resulting in spetrum overlapping, is present. In partiular for m 0, R (0) S (nt )S (nt ) hh, hh m * m L/21 D hh, 2 ll/2 F (l ). (11) In the paper, we will use the omplex ross-orrelation oeffiient per Doppler frequeny bin (or spetral omplex ross-orrelation oeffiient), whih is defined as follows: ˆ o(l D) F hh,(l D) F (l )F (l ) hh,hh D, D j ˆ dp (l D) ˆ o(l D) e. (12) 3. Analysis of the Doppler phase ompensation a. Experimental verifiation on rain Lets onsider in detail the argument of the opolar ross-orrelation oeffiient, namely, the differential phase dp. The differential phase dp (5) is determined as the sum of the differential propagation phase dp and D

5 MARCH 2004 UNAL AND MOISSEEV 447 FIG. 1. (a) Example of a rain Doppler power spetrum hh (thik line) with the orresponding ross-spetrum phase (dotted line) without phase ompensation for nonsimultaneous measurements of hh and. The spetral refletivity is given in mm 6 m 3, and the sum over all of the frequenies gives the ommonly used refletivity value. It an be seen that the time differene between the hh and measurements reveals itself in the slope of the linear trend of the phase spetrum. (b) Phase of the ross spetrum in rain after the phase ompensation. The linear trend has been ompensated for. It should be noted, however, that aliased and nonaliased parts of the spetrum have different mean phase values. the differential baksattering phase o. For an S-band radar it is expeted that the differential baksattering phase is lose to zero (Zrnić and Ryzhkov 1999; Bringi and Chandrasekar 2001). Therefore, the spetral differential phase ˆ dp does not depend on Doppler veloity and therefore is equal to the differential phase dp. The dp in its turn is fully determined by the differential propagation phase dp. For the sake of simpliity we shall assume dp to be equal to zero in this setion. It should be noted that this assumption is valid for a wide range of radar sensing senarios, and will be onsidered in more detail in setion 5. Under the urrent assumptions, the argument of the ross-spetrum F hh, is then the differential propagation phase per Doppler veloity bin, where the measured Doppler veloity is defined as m V l l D D D, (13) 22 with D being the Doppler veloity resolution and the radar wavelength. For Doppler veloities V D V D,max, (14) where V D,max is the maximum unambiguous Doppler veloity and V D is assumed to orrespond to the middle m of a Doppler veloity bin for simplifiation (V D V D ), the phase spetrum, without Doppler phase ompensation, is ˆ 4 Arg[F hh,(l D)] dp(l D) l Dt ˆ dp(l D) D(l D). (15) After adding the phase D(l D) D(l D ) (16) per Doppler veloity bin, the Doppler phase ompensation is performed and the phase spetrum beomes Arg[F (l )] ˆ (l ) 0. hh, D dp D (17) From the time series of sattering matries, we arry out Fourier transforms and alulate the seond moments of the obtained spetral sattering matries, whih leads to the spetral target ovariane matrix. In our ase, one spetral target ovariane matrix is obtained eah 0.96 s (LT m ) with L and T m being 256 and 3.75 ms, respetively. Beause of the single-hannel reeiver of DARR, three pulses are used to obtain S hh, S h, and S. The time offset t between hh and measurement is 2.5 ms (2T m /3). The maximum unambiguous Doppler veloity is 6.03 m s 1 [(/2)(1/2T m )] and the Doppler veloity resolution is 4.71 m s 1 [(/2)(1/LT m )], the radar wavelength being 9.05 m. The negative Doppler veloities indiate that the hydrometeors are approahing the radar. The Doppler phase ompensation is illustrated in Fig. 1. The averaged Doppler power spetrum (hh) ofone range bin of rain is plotted (Fig. 1a, thik line) with the orresponding ross-spetrum phase, without the phase orretion (Fig. 1a) and with phase orretion (Fig. 1b). This Doppler power spetrum is aliased. There is a ground lutter peak entered at 0 m s 1. Considering the top of the Doppler spetrum where the spetral refletivity values are between 0 and 18 db, the related ross-spetrum phase after orretion has a stable value of 0 between 6.0 and 1.5 m s 1, whih is the expeted phase value when the Doppler veloities are smaller than the maximum unambiguous Doppler veloity; see (14) and (17). Between 4 and 6 m s 1, the ross-spetrum phase is also stable but its value is about 120, whih indiates the presene of Doppler aliasing. We will disuss this aspet in detail later in the next setion. In the same areas of Doppler veloities (suffi-

6 448 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 21 iently large spetral refletivity), we an learly see the slope of the ross-spetrum phase versus Doppler veloity, whih equals (4/)t (20 m 1 s) when the Doppler phase ompensation is not yet arried out, see (15). b. Doppler phase ompensation in the ase of Doppler spetrum aliasing In Fig. 1b it was observed that in the area where spetrum aliasing is observed the ompensated phase shows values different from 0. This is aused by the use of the wrong Doppler veloity values in the Doppler phase ompensation proedure. This drawbak of the method, however, an be used for our advantage, sine it allows us to perform a Doppler spetrum dealiasing, as will be disussed in this setion. 1) SINGLE-DOPPLER ALIASING First, let us analyze what exatly is happening in the Fig. 1. As we know in this example we have a spetrum that has been aliased one, meaning that the Doppler veloities, V D, on the right side of the spetrum are V D VD,max and V D 3V D,max (18) or, speaking more generally, the orresponding measured Doppler veloities are m VD VD 2VD,max l D, (19) with for V D belonging to the interval [V D,max,3V D,max [ and for V D belonging to the interval [3V D,max, V D,max [. Before Doppler phase ompensation, Eq. (15) holds, and after the Doppler phase ompensation, whih onsists of adding the phase 4 8 D l Dt D VD,maxt D D,max, (20) the ompensated phase spetrum beomes Arg[F (l )] ˆ (l ) hh, D dp D D,max t 2. (21) T m Meaning that in the ase of Doppler spetrum aliasing the Doppler phase ompensation results in the addition of a onstant phase to the argument of the ross spetrum. This onstant phase depends on whih side the Doppler aliasing ours (negative Doppler veloities or positive Doppler veloities) and on the time differene between hh and measurement, t, ompared to the sampling time T m. 2) MULTIPLE-DOPPLER ALIASING In a more general ase of multiple-doppler aliasing, Eq. (21) an be generalized to TABLE 1. Argument of the opolar ross spetrum after Doppler phase ompensation ase of multiple aliasing. Atual Doppler veloity interval n D Arg[F (l D )] [5V D, max, 3V D, max [ [3V D, max, V D, max [ [V D, max, V D, max [ [V D, max,3v D, max [ [3V D, max,5v D, max [ hh, 8 (120) 3 4 (120) (120) 3 8 (120) 3 Arg[F hh,(l D)] nd D,max, (22) where n D 0, 1, The integer n D orresponds to no aliasing (n D 0), one aliasing (n D 1), and so on. We use a yle of three pulses to measure the hh, h, and elements of the sattering matrix. Therefore, if t 2T m /3, as in the ase of DARR, Arg[ F hh, (l D )] beomes 4 Arg[F (l )] n. (23) hh, D D 3 The phase an only be unambiguously defined in the interval [, ]. In our partiular ase we obtain phase ambiguity from n D 3 and we an solve two different aliasing problems. In Table 1 opolar phase spetrum values after the Doppler phase ompensation are given in the ase of multiple-doppler spetrum aliasing. It an be seen that the Doppler veloity spetrum an unambiguously be reonstruted if the veloities belong to, for example, intervals [5V D,max, V D,max [, [3V D,max, 3V D,max [, et. For a lassial measurement of the sattering matrix with a dual-hannel reeiver, only two pulses are used to get four polarimetri measurements (hh, h) and (h, ). In this ase t T m /2, whih leads to Arg[F hh,(l D)] nd. (24) Phase ambiguity ours from n D 2 and a single aliasing an be orreted. 4. Statistial model of the ross-spetrum phase Sine the dealiasing algorithm uses the phase between hh and measurements, it is neessary to model the statistial behavior of this phase and its dependeny on the number of averages. For this purpose, a statistial model of the ross-spetrum phase is hereby introdued and experimentally verified. Many hydrometeors are ontained in the radar resolution volume (range bin), or in our ase there are many hydrometeors in a given radar volume having similar Doppler veloities, so they are onfined to the same Doppler range bin. Moreover, no single one of them

7 MARCH 2004 UNAL AND MOISSEEV 449 dominates the others. In this ase the spetral sattering matrix an be modeled as having a multivariate omplex Gaussian distribution (Goodman 1985). The real and imaginary parts of any two omplex elements of the spetral sattering matrix are assumed to have then a irular Gaussian distribution. Under these assumptions, a probability density funtion of the phase differene between hh and measurements is given by Lee et al. (1994). And the probability density funtion an be written as k p () for the phase k k (1 ˆ ) o 2 k1/2 2(k)(1 ) 2 k (1 ˆ o ) 1 2 F k, 1, ; (25) 2 2 [ ] k 1 hh k i1 (l) Arg S ˆ (i, l)s ˆ* (i, l). (26) This phase orresponds to the differential phase per Doppler veloity bin after k time averages. The probability density funtion depends on the spetral omplex ross-orrelation oeffiient (12) and on the number of time averages k. The parameter is defined as follows: ˆ os( ˆ o dp ) (27) and the funtions and F (more preisely 2 F 1 ) are the gamma funtion and the Gauss hypergeometri funtion, respetively. The spetral omplex ross-orrelation oeffiient depends on meteorologial onditions (for heavy preipitation, ˆ dp will be different from 0 and the modulus of ˆ o may derease). Sine for our dealiasing proedure we have to onsider the statistis of the phase spetrum, it is neessary to verify the above presented formulation, whih was originally introdued for a time series of observations. For the experimental verifiation of the above presented model we have used a light preipitation measurement. We have estimated the opolar ross-orrelation oeffiient, ˆ o, per Doppler veloity bin using a Doppler ross-spetrum average of 40 spetra. The measurement speifiations are the same as the ones given in setion 3. The rain and preipitating loud areas show high values of this parameter, typially 0.99, and FIG. 2. Comparison of the theoretial pdf (thik line) with the experimental pdf (thin line) of the ross-spetrum phase as measured in stratiform rain. Two ases are onsidered: no averaging and averaging over 10 onseutive spetra. The phase used for this study was extrated from all Doppler veloity resolution ells for all range gates in the preipitation region, where the magnitude of the spetral ross-orrelation oeffiient is It is expeted that sine the spetral analysis was performed on onseutive nonoverlapping time intervals, all obtained Doppler spetra are statistially independent. That is onfirmed by the presented figure, whih shows good agreement between theoretial and experimental pdf s. we have found that ˆ o is about 0.97 in the melting layer at the top of the Doppler spetrum. These values are harateristi of stratiform light rain. It should be noted that the values of the spetral ross-orrelation oeffiient are generally larger than the values of the onventional ross-orrelation oeffiient. This an be explained by the fat that the ommon ross-orrelation oeffiient is determined by the sum of the ross spetra over all Doppler veloity bins. Therefore, it is an integral over all hydrometeors with different sizes and shapes. This integration results in a derease of the oherene of the signal and thus in the redution of the ross-orrelation oeffiient. For the experimental validation of the proposed model we have seleted only that Doppler range bins where the spetral ross-orrelation oeffiient was omposed of values between and Therefore, in (25) and (27) we have used ˆ o being equal to For the seleted Doppler range bins we have alulated the phase spetrum without any averaging and by averaging the ross spetrum over 10 spetra. A omparison between the model (25) and the data (26) is shown in Fig. 2. From this figure we an onlude that there is a good fit between the theoretial and experimental probability density funtions, with or without average. Therefore, we an use Eqs. (25) (27) to desribe the statistial behavior of the phase spetrum. Sine the model of the ross-spetrum phase distribution is verified, the related standard deviation k 2 k p () d (k) (28) p () d an be omputed versus the magnitude of the spetral ross-orrelation oeffiient for several averages (1, 2, 4, 10, 20, 30, 40, 50). This is plotted in Fig. 3. For this omputation, the mean value of is 0. After Doppler

8 450 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 21 FIG. 3. Standard deviation of the phase of the ross spetrum hh, for different numbers of averages (1, 2, 4, 10, 20, 30, 40, 50). For most atmospheri ehoes the opolar spetral ross-orrelation oeffiient belongs to the interval [0.7, 1]. proessing, the magnitude of the ross-orrelation oeffiient per Doppler veloity bin is generally omposed of values between 0.9 and 1 for light preipitation. In that ase, the largest standard deviation, about 40, is obtained for ˆ o 0.9 when there is no average. When there is a single aliasing, the mean phase is 120 (with our measurement speifiations), whih indiates that the highest standard deviation possible is 60 before a phase ambiguity ours. Therefore, a dealiasing tehnique based on the ross-spetrum phase an be suessfully used when there is no average. Nevertheless, Fig. 3 shows that it is worthwhile to arry out an average on two Doppler spetra, whih leads to a strong redution of the standard deviation, whih beomes about Mean value of the ross-spetrum phase Until now, the mean value of the ross-spetrum phase, or differential phase dp, was onsidered to be about 0 for atmospheri targets at S band. For Rayleigh sattering, only the differential propagation phase dp ontributes to dp. With our measurement speifiations, the mean value of the ross-spetrum phase will be 120 when there is Doppler aliasing, depending on whih side of the Doppler veloity spetrum the Doppler aliasing ours. Using these three harateristi phase values, 0, 120, and 120, and the standard deviation disussed in the previous setion, a relatively easy dealiasing algorithm an be implemented to orretly obtain the Doppler spetra of polarimetri parameters. In ases of heavy preipitation and for horizontal profiles, the differential propagation phase is signifiantly different from 0. With the measurement speifiations as used, the largest standard deviation permitted is 60. The standard deviation depends on the magnitude of the FIG. 4. The dependene of the maximum differential phase of atmospheri targets, whih allows for unambiguous veloity reonstrution, on the number of spetra taken for estimation of the Doppler ross spetrum. The dependene is shown for different numbers of averages (1, 2, 4, 10, 20, 30, 40, 50) and different opolar spetral ross-orrelation oeffiients (0.7, 0.8, 0.85, 0.9, 0.95, 0.99). spetral ross-orrelation oeffiient, whih is generally high for atmospheri targets, thus reduing and the number of averages. The maximum value for dp (in ) an then be expressed as follows: dp,max dp,max 60. (29) When the standard deviation is large beause of a small average number, the possible values of dp are redued. 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 speifi differential phase K dp ( km 1 ) is built and the estimate of rain rate R (mm h 1 ), using the Pruppaher Beard equilibrium shape model, is omputed (Bringi and Chandrasekar 2001): 0.85 R 40.5(K dp ). (30) This relation is given for horizontal sounding of preipitation and thus provides the upper-limit estimate of the differential propagation phase. This is the worst ase for the dealiasing algorithm. In this ase the differential propagation phase is dp 2Kdp r, (31) where r is the range. Figure 5 gives the domain of validity of the dealiasing algorithm for horizontal profiles of preipitation. The differential propagation phase must not exeed 60. Magnitude values of the spetral rossorrelation oeffiient indiate the possible maximum values of the differential propagation phase. For example, ˆ o 0.95 shows that the maximum values of dp are inreasing from 30 to 60 with the number of averages. The radars TARA and DARR are intended for detailed studies of the atmosphere with high range

9 MARCH 2004 UNAL AND MOISSEEV 451 was used for the dealiasing algorithm. The large and known jump in the phase of the ross-spetrum hh, at a fixed range (see Fig. 1), due to aliasing, an be easily deteted. In that ase, the absolute values of the ross-spetrum phase must not be onsidered but the phase jump in the Doppler spetrum should be. The onsequene is that the unfolding of the Doppler spetra an be suessfully performed but the knowledge of the atual Doppler veloities may be lost. For a lassial measurement of the sattering matrix (dual-hannel reeiver), the same approah an be followed with a limitation of 90 instead of 60 for the differential propagation phase; see Eq. (24). FIG. 5. Dependene of observed differential phase on rain rate and propagation path. This dependene is shown by equiphase urves that orrespond to dp values of 10, 20, 30, 40, and 60. Sine this dependene is alulated for horizontal sounding, this figure gives the upper limit on the observed dp. resolution and high Doppler resolution. Their operational maximum range is about 40 km. The maximum range used is urrently 15 km, whih maximally allows for a rain rate of 73 mm h 1 for horizontal profiling. With these radars, vertial and slant profiles (elevation of 30 or 45) are generally measured. Therefore, the failure of the dealiasing algorithm due to values of the differential propagation phase signifiantly different from 0 has not yet ourred. It is important to note that for a horizontally profiling weather radar the same strategy an be onsidered as 6. Dealiasing proedure a. Dealiasing of a slant profile of preipitation To illustrate the dealiasing proedure, a slant profile (Fig. 6) is seleted. It shows light stratiform preipitation at hh polarization. The radar looks at an elevation angle of 30 and the averaging time is 48 s. The range resolution is 75 m. The melting layer is loated between 2 and 3.1 km. Under the melting layer, there is rain. The preipitating loud layer is loated between 3.1 and 9.25 km. The Doppler resolution is 4.71 m s 1. The negative Doppler veloities indiate that the hydrometeors are approahing the radar. The measurements are arried out with the maximum Doppler veloity being too low. For slant profiles, the Doppler veloities annot be related diretly to the fall speed of the hydrometeors and ontain the wind Doppler veloity. The seleted slant profile shows large variations in the Doppler wind veloity as a funtion of height in Fig. 6, espeially in the rain area. FIG. 6. Slant profile of preipitation: (left) range profile of refletivity and (right) orresponding Doppler power spetra.

10 452 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 21 FIG. 7. Slant profile of preipitation: ross-spetrum phase after orretion for the nonsimultaneity of the measurements hh and. To alulate the ross spetrum, averaging over 50 spetra was used. The phase of the ross-spetrum F hh, is plotted in Fig. 7. Three main phase values are seen. The gray area in Fig. 7 is the only part of the preipitation event, rain in the near range, that belongs to the Doppler veloity interval [6 ms 1,6ms 1 [. The phase of the ross spetrum, after Doppler phase ompensation, is about 0. This preipitation measurement shows two other harateristi phases: one is about 120 (dark gray) and the other one is about 120 (light gray). These values indiate the presene of Doppler aliasing and make it possible to determine to whih Doppler interval they belong; see Table 1. In this example of the slant profile, the Doppler veloities are expeted to be negative (the mean Doppler veloities per range bin alulated from the Doppler spetrum part related to the phase value 0 are near 6 ms 1 ). When the preipitation event orresponds to the Doppler veloities interval [18ms 1, 6 ms 1 ], the onstant phase 4/3 [see (23)] is added to the phase of the ross-spetrum F hh,. This onstant phase is 240 or equivalently 120 and orresponds to the dark gray area. The same ours for veloities belonging to [30 m s 1, 18ms 1 [, then the onstant phase 8/3 is added to the ross-spetrum phase and leads to the light gray area (480 or equivalently 120). Using the phase of the ross spetrum and the ontinuity of the mean Doppler veloity, the dealiased slant profile is given in Fig. 8. The number of averages is 50 for this event. Only the areas orresponding to a rossorrelation oeffiient per Doppler bin larger than 0.7 and having a ross-spetrum phase omprised in the interval (mean phase 2 ) have been onsidered. The largest standard deviation is 6 and three mean phases are found. For this example, dp may reah 6. The phase window an be hosen larger in order to allow for a larger dp than 6. FIG. 8. Doppler power spetra (hh) after dealiasing for all the measured range bins. It an be seen that parts of the signal that orrespond to lutter and spetral tails of the melting layer were suppressed. This is due to a rather narrow phase window that was used for the dealiasing proedure. b. Consequenes of the dealiasing proedure The maximum unambiguous Doppler veloity in the ase of nonpolarimetri measurements would be ms 1. Therefore, Doppler aliasing would still be present for this slant profile where Doppler veloities up to 23 ms 1 are measured. This example with two onseutive Doppler aliasings shows that the proposed polarimetri dealiasing proedure is not only unfolding the Doppler spetra but is able to obtain the atual Doppler veloities with the benefits of full polarimetri information. The phase of the ross-spetrum F hh, also indiates problemati zones beause of the presene of very different phase values than expeted. Two examples of these values, present in Fig. 7, are briefly disussed: ground lutter and the melting layer. The ross-spetrum phase is very useful for deteting the Doppler ells ontaminated by ground lutter. After removal of the stable ground lutter, the time-varying ground lutter shows a rather broad spread of phases (see Fig. 7 around the Doppler veloity 0 m s 1 ) (Ryzhkov and Zrnić 1998; Moisseev et al. 2002). Therefore, during the dealiasing proedure (also if the threshold on the magnitude of the ross-orrelation oeffiient per Doppler veloity bin is not applied), the Doppler ells signifiantly affeted by ground lutter are disarded, whih an be seen in Fig. 8, under and above 1 km in the rain area (missing data around Doppler veloity 12 ms 1 ). When is large beause there is no time average or a signifiant propagation phase an be expeted, the threshold has to be applied for lutter suppression. In the seond area, the proposed dealiasing tehnique may not perform optimally. This is the melting layer (between 2 and 3.1 km). Under the refletivity peak of the melting layer (2.75 km), the Doppler spetra beome wide (2.5 km). In the tail of these Doppler spetra, the spetral refletivity values are similar to those of a pre-

11 MARCH 2004 UNAL AND MOISSEEV 453 radar is near the horizontal wind diretion (dashed line), about Comparison of different orretions for nonsimultaneous polarimetri measurements on the ross-orrelation oeffiient o The aim of this setion is to ompare well-known proposed orretions (Illingworth and Caylor 1991; Balakrishnan and Zrnić 1990) for the nonsimultaneous measurements of the sattering matrix with the method disussed in this paper. This omparison is arried out on range profiles of the modulus of the ross-orrelation oeffiient. FIG. 9. Comparison of the radar estimates of the Doppler mean veloity attributed to the horizontal wind with the wind profile provided by a radiosonde. ipitating loud but the spetral ross-orrelation oeffiient magnitude is small and the ross-spetrum phase differs from expetations. At the present moment, the dealiasing proedure suppresses the tails of these Doppler spetra. This effet an be learly seen for a few range bins, from 2.2 to 2.5 km, in the dealiased profile of Fig. 7. These Doppler spetra in the lower part of the melting layer (under the refletivity peak) are investigated in Verlinde et al. (2002).. Comparison with a meteorologial balloon In order to ompare the Doppler veloities obtained after dealiasing the slant profile, one vertial profile of the wind onditions (balloon in Fig. 9) was supplied by the Netherlands Meteorologial Institute (KNMI). The two sensors are not olloated but a omparison an still be done. The mean veloity measured by the balloon (thik line) is mainly omposed of values between 22 and 23 m s 1 from 0.2 to 4.5 km (height). The estimated horizontal wind mean veloities from the radar data, alulated for a vertial profile, are also plotted in Fig. 9 (asterisk). They orrespond to the Doppler mean veloities of the slant profile divided by os(30), whih means that the ontribution of the hydrometeors to the Doppler mean veloity is here negleted. Apart from the first 0.5 km (1-km range in the slant profile) there is a good agreement for the estimated wind veloities measured by the radar if the azimuth diretion of the a. Computation of o without orretion If no orretion is applied, the ross-orrelation oeffiient is diretly alulated from the time series of the sattering matrix. For the slant profile, the number of averages (k) is 50 and the number of the Doppler veloity bins (L) is 256: S hh(nt m)s * (ntm t) o(t). (32) 2 2 S (nt ) S (nt t) hh m m As was disussed above, the smaller the deorrelation time of S hh (nt m ) [or S (nt m )], as ompared to t, the larger the bias in the o (t) estimate; this an be observed in Fig. 10. b. Computation of o with Doppler phase ompensation and dealiasing The ross-spetrum F hh, (11) and the power spetra F hh,hh and F, are integrated on the relevant Doppler spetrum veloity bins. When the modulus of the ross spetrum is smaller than 0.7 or the phase of the ross spetrum does not belong to the predited phase intervals, whih are used for dealiasing, the orresponding Doppler bins are disarded for the integration. The number of integrations is dependent on the range bin: l max hh, ll min F (l D) (0). (33) o lmax llmin lmax llmin F (l ) F (l ) hh,hh D, D In the ase of Doppler aliasing and before integration, the phase of the ross spetrum is ompensated with the appropriate onstant phase: d jn D F (l ) F (l )e D,max hh, D hh, D. (34). Computation of o with time interpolation The time series hh and are interpolated in order to obtain a oinident set of hh and time samples.

12 454 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 21 This is done by first omputing the Fourier transform of the data, padding the transform with an equal number of zeros to double the Nyquist frequeny, and then taking the inverse Fourier transform (Illingworth and Caylor 1991). Then the sattering matrix obtained with two pulses (dual-polarized reeiver) an be estimated. In our ase, where one hannel reeiver is used for sattering matrix measurements, three pulses are needed. Therefore, the hh and time samples have to be oversampled three times, whih leads to a less favorable interpolation. d. Computation of o with the Gaussian power spetrum assumption When the Doppler power spetra are assumed Gaussian, the temporal orrelation oeffiient an be expressed as 8( mt /) 2 s j(4vmt s/) (mt s) e e. (35) where is the Doppler veloity spetrum width, V is the mean Doppler veloity, and mt s the time lag. Furthermore the ross-orrelation oeffiient at time lag mt s is assumed to ontain independent ontributions from the distribution of Doppler veloities and the distribution of anting angles and shapes (Balakrishnan and Zrnić 1990), whih leads to (mt ) (mt ) (0) (mt ). (36) o s s o s o With our measurement speifiations, the ross-orrelation oeffiient at time lag t 2T s in (32) and the temporal orrelation oeffiient at time lag T m 3T s, S hh(nt m)s * hh(ntm 3T s) S (ntm 2T s)s * (ntm 5T s) (3T s), (37) 2 2 S (nt ) S (nt 2T ) hh m m s an be estimated. Using (35) and (36), the magnitude of the ross-orrelation oeffiient an be expressed as o(2t s) o. (38) 4/9 (3T ) s e. Results and disussion Four range profiles of the magnitude of the rossorrelation oeffiient are plotted in Fig. 10 for the slant profile. The best result (thik line) is obtained with the orretions proposed in this artile. When no orretion is applied (line with xs), o dereases when the Doppler spetrum width inreases. The largest Doppler spetrum width (between 1.4 and 1.8 m s 1 ) is found between 0 and 1 km (rain area). The magnitude of the rossorrelation oeffiient dereases when the baksattered signal of atmospheri targets is ontaminated by lutter. Another ause of the dereases of o is the derease of the signal-to-noise ratio. This an be learly seen from the 8.5-km range. The method of estimating the ross-orrelation oeffiient at time lag 0 proposed by Balakrishnan and Zrnić (1990) gives good results (thin line). The method is not affeted by Doppler aliasing but an provide values of o larger than 1 (see Fig. 10: first and last range ells, range ell before 6 km). The major drawbak of the time interpolation method is its ollapse when Doppler aliasing ours. A series of minima of o oinides with the ranges affeted by Doppler aliasing in Fig. 10. FIG. 10. Comparison of four methods to estimate the ross-orrelation oeffiient: 1) no orretion (line with xs), 2) orreted with Doppler phase ompensation and polarimetri dealiasing (thik line), 3) padding or time interpolation (dashed line), and 4) Zrnić or Gaussian assumption (thin line).

13 MARCH 2004 UNAL AND MOISSEEV Conlusions Simultaneously ombining polarimetry and Doppler analyses leads to two major measurement diffiulties. The first one is the nonsimultaneity of the measurements performed with different polarizations. The seond is the redution of the maximum unambiguous Doppler veloity beause of the transmission of different polarizations, whih leads to Doppler spetrum aliasing effets. A solution to these two problems is given in the paper. This solution is based on the phase of the rossspetrum hh, or equivalently the differential phase ˆ dp per Doppler veloity bin. The proposed Doppler phase ompensation also orrets for the nonsimultaneity of the polarimetri measurements hh and in ases of Doppler spetrum aliasing but adds a onstant phase, whih depends on the interval of the atual Doppler veloities. This seeming drawbak of the proessing is a very useful tool for Doppler spetrum dealiasing. This dealiasing tehnique uses phase jumps and absolute phase values to dealiase and to retrieve the atual Doppler veloities. This method is espeially useful for slant profiles, whih are affeted by the wind veloity. The proposed polarimetri dealiasing method is onsidered for atmospheri ehoes at S band. This tehnique is relatively easy to implement. Its domain of validity is disussed in terms of maximum standard deviation and maximum propagation phase. When large values of dp are enountered (heavy preipitation and horizontal profiling), the algorithm an be extended. In that ase, the absolute values of the ross-spetrum phase hh, must not be onsidered but the jump of this phase in the Doppler spetrum at a fixed range should be. Then the unfolding of the Doppler spetra an be performed but the knowledge of the atual Doppler veloities may be lost. Combining a orretion for the nonsimultaneity of the polarimetri measurements, whih is valid in ases of Doppler aliasing and a new polarimetri dealiasing tehnique, allows for full polarimetri measurements of the atmosphere to be obtained with the atual Doppler veloities of the hydrometeors and the maximum unambiguous Doppler veloity of the orresponding single polarized measurement sheme. A omparison between the proposed method and known orretions for the offdiagonal elements of the target ovariane matrix shows that the proposed method provides a better estimate of the-ross orrelation oeffiient. Aknowledgments. The authors would like to thank their olleague Dr. Silvester Heijnen for his help with validation of the results and the Netherlands Meteorologial Institute (KNMI) for providing the radiosonde measurements. REFERENCES Balakrishnan, N., and D. S. Zrnić, 1990: Use of polarization to haraterize preipitation and disriminate large hail. J. Atmos. Si., 47, Boerner, W.-M., H. Mott, E. Lüneburg, C. Livingstone, B. Briso, R. Brown, and J. S. Paterson, 1998: Polarimetry in radar remote sensing: Basi and applied onepts. Manual of Remote Sensing: Priniples and Appliations of Imaging Radar, F. M. Henderson and A. J. Lewis, Eds., John Wiley and Sons, Bringi, V. N., and V. Chandrasekar, 2001: Polarimetri Doppler Weather Radar: Priniples and Appliations. Cambridge University Press, 500 pp. Doviak, R. J., and D. S. Zrnić, 1993: Doppler Radar and Weather Observations. Aademi Press, 562 pp. Giuli, D., and L. Faheris, 1991: SAR appliation of a signal oding tehnique for single hit measurement of the target sattering matrix. Eur. Trans. Teleommun., 2, ,, L. Baldini, M. Gherardelli, and S. Moni, 1995: Fully oherent simultaneous measurements in weather radar. Pro. Third Int. Workshop on Radar Polarimetry (JIPR), Nantes, Frane, IRESTE, Goodman, J. W., 1985: Statistial Optis. John Wiley and Sons, 576 pp. Heijnen, S. H., L. P. Ligthart, and H. W. J. Russhenberg, 2000: First measurements with TARA: An S-band transportable atmospheri radar. Phys. Chem. Earth, 25B, Hennington, L., 1981: Reduing the effets of Doppler radar ambiguities. J. Appl. Meteor., 20, Illingworth, A. J., and I. J. Caylor, 1991: Co-polar orrelation measurements of preipitation. Preprints, 25th Conf. on Radar Meteorology, Paris, Frane, Amer. Meteor. So., Lee, J. S., K. W. Hoppel, S. A. Mango, and A. R. Miller, 1994: Intensity and phase statistis of multilook polarimetri and interferometri SAR imagery. IEEE Trans. Geosi. Remote Sens., 32, Lüneburg, E., V. Ziegler, A. Shroth, and K. Tragl, 1991: Polarimetri ovariane matrix analysis of random radar targets. Pro. AGARD EPP Symp. on Target and Clutter Sattering and Their Effets on Military Radar Performane, Ottawa, ON, Canada, AGARD-CP-501, Moisseev, D. N., 2002: Contrast enhanement for depolarizing radar targets. Ph.D. dissertation, Delft University of Tehnology, 144 pp. [Available from Delft University of Tehnology Library, 2600 WV Delft, Netherlands.], C. Unal, H. Russhenberg, and L. Ligthart, 2002: A new method to separate ground lutter and atmospheri refletions in the ase of similar Doppler veloities. IEEE Trans. Geosi. Remote Sens., 40, Niemeijer, R. J., 1996: Doppler-polarimetri radar signal proessing. Ph.D. dissertation, Delft University of Tehnology, 221 pp. [Available from Delft University of Tehnology Library, 2600 WV Delft, Netherlands.] Ryzhkov, A. V., 2001: Interpretation of polarimetri radar ovariane matrix for meteorologial satterers: Theoretial analysis. J. Atmos. Oeani Tehnol., 18, , and D. S. Zrnić, 1998: Polarimetri rainfall estimation in the presene of anomalous propagation. J. Atmos. Oeani Tehnol., 15, Sahidananda, M., and D. S. Zrnić, 1989: Effiient proessing of alternately polarized radar signals. J. Atmos. Oeani Tehnol., 6, Santalla, V., and Y. M. M. Antar, 2002: A omparison between different polarimetri measurement shemes. IEEE Trans. Geosi. Remote Sens., 40, ,, and A. G. Pino, 1999: Polarimetri radar ovariane matrix algorithms and appliations to meteorologial radar data. IEEE Trans. Geosi. Remote Sens., 37, Sott, R. D., P. R. Krehbiel, and W. Rison, 2001: The use of simultaneous horizontal and vertial transmissions for dual-polarization radar meteorologial observations. J. Atmos. Oeani Tehnol., 18, Sirmans, D., D. S. Zrnić, and W. Bumgarner, 1976: Extension of maximum unambiguous Doppler veloity by use of two sampling