PULSED Doppler weather radars are capable of measuring

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1 1240 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 4, APRIL 2012 Detection and Mitigation of Second-Trip Echo in Polarimetric Weather Radar Employing Random Phase Coding Qing Cao, Guifu Zhang, Robert D. Palmer, and Lei Lei Abstract This study presents a new identification and mitigation scheme of second trip contamination for pulsed Doppler polarimetric weather radars with the ability of random phase coding. This scheme can be easily implemented in a magnetron radar without any hardware changes. For relatively weak contamination, identification and mitigation are based on a multilag processing method, which uses multiple lags of both the autoand cross-correlation functions to estimate radar moments. For relatively strong contamination, instantaneous phase variations of horizontal and vertical polarization channels are combined into a simple fuzzy-logic scheme to complete the identification. Data from the C-band OU-PRIME radar are used to demonstrate the effectiveness of the proposed scheme for identification and mitigation of second-trip echoes. Index Terms Phase coding, polarimetric measurements, weather radar. I. INTRODUCTION PULSED Doppler weather radars are capable of measuring reflectivity and velocity fields and have great potential in meteorological applications such as storm detection and tracking, quantitative precipitation estimation, and forecasting [1], [2]. Of course, radar algorithms for those applications depend greatly on high-quality measurements. However, the so-called Doppler dilemma [1], [2] is a fundamental problem which always exists for a pulsed Doppler radar. Given a pulse repetition time (PRT) of T s, the maximum unambiguous range of radar is r a = ct s /2, where c is the speed of light. Furthermore, the maximum unambiguous velocity is v a = λ/(4t s ), where λ is the radar wavelength. As a result, the product of r a and v a is limited by r a v a = cλ/8. (1) This relation gives a theoretical limitation of radar range velocity ambiguity. Given a specific radar wavelength, Manuscript received November 17, 2010; revised April 1, 2011; accepted July 31, Date of publication October 3, 2011; date of current version March 28, This work was supported in part by the National Science Foundation under Grant AGS Q. Cao is with the Atmospheric Radar Research Center, University of Oklahoma, Norman, OK USA ( qingcao@ou.edu). G. Zhang and R. D. Palmer are with the Atmospheric Radar Research Center and the School of Meteorology, University of Oklahoma, Norman, OK USA ( guzhang1@ouo.edu; rpalmer@ou.edu). L. Lei is with the Atmospheric Radar Research Center and the School of Electrical and Computer Engineering, University of Oklahoma, Norman, OK USA ( leilei@ou.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TGRS if v a is increased to measure velocity with lesser ambiguity, r a will decrease yielding more range aliasing. Taking the S-band Weather Surveillance Radar-1988 Doppler (WSR-88D) as an example, the long-prt mode can obtain an r a of 460 km and a v a of 9 m/s while the short-prt mode has a shorter r a of 148 km but a larger v a of 28 m/s [3]. Equation (1) also implies that the radar with a higher frequency may have more serious range velocity ambiguity. For example, the C-band Polarimetric Radar for Innovations in Meteorology and Engineering (OU-PRIME) [4] normally uses a mode with an r a of 125 km and a v a of 16 m/s, both of which are smaller than those of WSR-88D s short-prt mode. Many methods have been applied to mitigate the range velocity ambiguity of weather radar. Generally, those methods can be classified into two different approaches. The first approach depends on the optimal usage of dual/multiple PRTs or pulse repetition frequencies (PRFs). Simply put, the echo power can be de-aliased by a long PRT T 1 and the velocity can be de-aliased by a short PRT T 2, e.g., the batch mode of WSR-88D [3]. Cho [5] extended the dual-prt blocks to multiple-prt blocks and proposed the multiplepulse-repetition-interval processing for the C-band Terminal Doppler Weather Radar. This technique can effectively separate different trip echoes if out-of-trip precipitation does not spread over a long distance in the radial direction. Compared with transmitting pulse block with a uniform PRT, the range velocity ambiguity can also be effectively reduced by a staggered-prt technique which transmits pulses by alternating the use of two PRTs [3], [6] [8]. Under this scheme, the restriction of unambiguous range and velocity becomes r a v a = T 1 cλ (T 1 T 2 ) 8. (2) If the difference of two PRTs is small, the maximum unambiguous range and velocity can be greatly increased. However, its moment estimation error increases as the difference of two PRTs decreases. Recently Tabary et al. [9] extended the staggered dual-prt to the staggered triple-prt for the French radar network. Torres et al. [10] introduced an alternating dual-pulse dual-frequency (ADPDF) technique, which extended the staggered dual-prt to dual-frequency channels. Pirttila et al. [11] proposed a more complicated multi-staggered-prt method called simultaneous multiple PRF. This technique has been evaluated by Ruzanski et al. [12]. Theoretically, it is able /$ IEEE

2 CAO et al.: DETECTION AND MITIGATION OF SECOND-TRIP ECHO 1241 to achieve an unlimited maximum unambiguous range and velocity under the perfect statistical conditions. However, its performance could be severely limited by factors such as finite sampling number, the intensity of overlaid signal, and the weather s coherence time. Another approach of ambiguity mitigation is based on phase coding of the transmitted pulses [7], [13] [17]. Considering the fact that the radar receiver is only synchronous for the first-trip pulse, if transmitted pulses are coded with a random phase, the received second-trip echo (also called out-of-trip echo) will be purely random. As a result, the out-of-trip echoes contribute to the total signal as noise causing overlaid echoes in reflectivity field when the conventional power estimator is used. However, moment estimation, e.g., using the first lag of autocorrelation function (ACF) or the spectral method, can greatly alleviate the effect of out-of-trip echoes. Phase coding has also been introduced to mitigate the out-of-trip echoes. Sachidananda and Zrnic [15] proposed a systematic phase code (SZ code), which has better performance than random phase coding in the separation of overlaid echoes. An enhanced phase coding algorithm, which uses the SZ(8/64) code, has recently been recommended for the initial deployment of range velocity ambiguity mitigation on the new signal processors of the WSR- 88D network [16]. Bharadwaj and Chandrasekar [17] evaluated the performance of the systematic phase coding technique on the polarimetric moment estimation using simulations based on observations from the S-band CSU-CHILL radar. Compared with traditional radar signal-processing schemes, the aforementioned PRT-based and phase-coding-based techniques generally require the installation of additional hardware/control components to fulfill changing PRT or coding phase for pulses to be transmitted. The modification of hardware to a specific processing scheme makes the system less flexible for adapting to a subsequent new processing scheme. Moreover, it would be cheaper to keep the hardware unchanged. It is worth noting that a magnetron transmitter generates pulses with a randomly varying phase. That is, the realization of random phase coding is automatic for a magnetron radar (e.g., OU-PRIME). This characteristic of the magnetron radar provides convenience to develop advanced signal-processing algorithms for the purpose of second-trip-echo mitigation. It is noted that random phase coding makes the second-trip echoes essentially like noise. However, some signal processing methods, which can reduce the noise effect for moment estimation [18], [19], are not effective for the mitigation of second-trip contamination. It is not so for a multilag processing method, which has been proposed and implemented recently [20] [22]. This method was originally developed for crossbeam wind estimation using spaced antenna interferometry [20] and extended to polarimetric radar moment estimation in the presence of noise [21]. The implementation on the OU-PRIME radar shows a satisfactory mitigation for noise and second-trip contamination [22]. For a magnetron-based polarimetric radar (i.e., with random phase coding), the second-trip weather echoes simultaneously received on both polarimetric channels are correlated while noise signals are not, making it possible to identify and estimate both the first and second-trip weather echoes. The current study presents such characteristics of second-trip echoes. Based on the multilag method, a new algorithm is proposed to identify and mitigate the secondtrip contamination for a polarimetric weather radar employing random phase coding. This algorithm is tested with OU-PRIME data in this study. II. CHARACTERISTICS OF SECOND-TRIP ECHOES Due to the Doppler dilemma, the received radar signal is generally the combination of first-trip echo, measurement noise, and second-trip echo (here, the radar echo beyond the first trip, which might be from the third or fourth trip, is considered as the second-trip echo). If a significant storm is located at a distance exceeding the unambiguous range of radar, the second-trip echo can have serious contamination to the first-trip echo. To have high-quality weather radar data, the contamination due to the second-trip echo needs to be mitigated. Before introducing the mitigation algorithm, the signal statistics for the second-trip echo are analyzed and compared with that of the noise and weather signals. A. Instantaneous Phase Supposing that only the second-trip weather signal is measured by the radar, the measured voltage V can be expressed using a complex form such as V = Ae j(φ t+φ r +Φ s ) where A is the signal magnitude, the symbol Φ denotes the phase terms, and the subscripts t, r, and s represent transmitter phase, range phase, and scattering phase, respectively. As for the second-trip echo, the transmitter phase randomly changes pulse by pulse. The range phase is attributed with the range of target. Scattering phase reflects the backscattering phase change of target scatters and the forward scattering effect of scatterers along the propagation path. Phase terms Φ r and Φ s are time dependent. Measurements of horizontal and vertical polarization channels, V h and V v, are then expressed as V h = A h e j(φ th+φ rh +Φ sh ) V v = A v e j(φ tv+φ rv +Φ sv ). (3) (4a) (4b) Because Φ th (or Φ tv ) randomly changes for different pulses, the instantaneous phase of V h (or V v ) has a uniform distribution. However, these instantaneous phases are correlated. First, the range phases of the two channels are identical. Second, the scattering phase difference only depends on the scattering of hydrometeors. Third, H and V channels have a constant shift of transmitter phase because the simultaneous transmission system usually applies one transmitter for the two channels by splitting its power. Therefore, the randomness among different pulses stays the same for both channels and can be canceled by comparing the signals of the two channels. The instantaneous differential phase φ hv is then obtained by φ hv =arg[φ h /φ v ]=Φ sh Φ sv + constant. (5)

3 1242 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 4, APRIL 2012 Fig. 1. Instantaneous differential phase histograms of two weather signals simulated with different correlations. According to the random scattering theory [23, Sec. III.2], [24], the phase difference of two signals has a probability density function (PDF) that depends on the correlation of these two signals. Fig. 1 shows a simulation of instantaneous differential phase for two Gaussian random signals that have different correlations. If two signals are highly correlated (e.g., H and V channel measurements of the precipitation), the phase difference will have a narrowly distributed PDF and vice versa. In particular, if the two signals are uncorrelated (e.g., noise or noiselike measurements), the phase difference will be uniformly distributed. The phase distribution information can be utilized to assist the identification of second-trip echo and will be discussed in the next section. B. ACF and CCF This section describes the characteristics of ACF and crosscorrelation function (CCF) for second-trip echoes. The nth sample from H and V channels, i.e., h(n) and v(n), can be written as h(n) =h 1 (n)+h 2 (n)+ε h (n) v(n) =v 1 (n)+v 2 (n)+ε v (n) (6a) (6b) where subscript 1 (or 2) denotes the first (or second)-trip echo, ε is the measurement noise, and subscripts h and v represent H and V channels, respectively. Because the secondtrip echo h 2 (or v 2 ) contains a random phase, it is completely independent of h 1 (or v 1 ). However, unlike the white noise in the two channels, h 2 (n) and v 2 (n) are correlated for the same pulse. The ACF C hh (or C vv ) and CCF C hv are usually estimated by Ĉ xy (m) = 1 N m x (n)y(n + m) (7) where the notation ˆ denotes the estimation, subscripts x and y represent h and v, respectively, superscript means the complex conjugate, m is the lag number, and N indicates the total sampling number. The appendix gives details on the estimation of ACF and CCF with the second-trip-echo contamination. Since the three components in (6) are independent, the expectation of the correlation estimate is Ĉxy (m) =C x1 y 1 (m)+c x2 y 2 (m)δ(m)+c εx ε y (m)δ(m) (8) where the notation represents the expected value and δ(m) is the Kronecker delta function. This equation suggests that the second-trip echo would only bias the ACF/CCF estimations at lag 0, which are traditionally used to estimate radar moments [1], [2]. Moreover, estimated ACF/CCFs at other lags are unbiased although there may be an increase of variance due to the second-trip echo (see Appendix). Fig. 2 shows an example of the estimated ACF/CCF magnitudes of a weather signal, which is contaminated by the secondtrip echo. The data points indicate the estimates from (7). The solid lines are Gaussian shapes fitted with several nonzero lags of ACF/CCFs (lags ±1, ±2, ±3, and ±4). It is shown that the peaks (mainly attributed to the second-trip echo) exist at lag 0 for both ACF and CCF. As for the ACF, it looks as if there was a large noise power added to the weather signal (first trip). The CCF is similar to the ACF because second-trip weather echoes

4 CAO et al.: DETECTION AND MITIGATION OF SECOND-TRIP ECHO 1243 the CCF, respectively, integer m indicates the lag number, and parameter w is the decorrelation length of ACF/CCF in terms of lag spacing. The estimates of radar moments S h,v, σ, and ρ hv for the first-trip echo are obtained by Ŝ h,v = C f h,v (0) v a ˆσ p = π w p (10a) (10b) ˆρ hv = C f hv (0). (10c) C f h (0)Cf v (0) Fig. 2. ACF and CCF example of weather signal contaminated with the second-trip echo (OU-PRIME observations on May 10, 2010). Legacy results: S h = 263.3, S v = 557.6, σ v =4.01 m/s, and ρ hv = MLAG results: S h = 204.4, S v = 413.1, σ v =1.92 m/s, and ρ hv = are correlated for H and V channels. The noise has a different effect on the CCF. The H and V channel noise sequences are independent so they do not bias the CCF estimation at lag 0. Since it is widely accepted that the weather signal ACF/CCF can be well modeled by a Gaussian function [1], [2], [20], [21], [25] [27] and the ACF/CCF at other lags is not biased by the second-trip contamination, the Gaussian model can be used to fit the ACF/CCF for first-trip weather. Consequently, the estimation of radar moments such as power (S h,v ), spectrum width (σ), correlation coefficient (ρ hv ), and differential phase (Φ DP ) can be effectively improved. III. IDENTIFICATION AND MITIGATION OF SECOND-TRIP CONTAMINATION According to the power ratio of the second-trip echo to the first-trip echo, the contamination can be considered as two types: weak and strong contamination. For weak contamination, the contribution of first-trip echo to nonzero lags of ACF/CCF is dominant. Using the multilag processing, its contribution to lag 0 can be estimated. Since the second-trip contamination only biases the lag 0 ACF/CCF, it can therefore be identified and estimated with the estimation of first-trip echo. For strong contamination, however, the phase information would be more useful for identification because the multilag processing is not feasible when the estimation of ACF/CCF is greatly degraded by the strong contamination. The processing for these two types is described in detail in the following. A. Multilag Processing Multilag processing is applied to identify the first type of contamination. Based on the magnitude of estimated ACF/CCF Ĉ h,v,hv (m), the Gaussian ACF/CCF for the first-trip weather echo is fitted through multiple lags (excluding zero lag) as ) Cp f (m) =Cp f (0) exp ( m2 (9) 2w 2 p where superscript f stands for the fitting model, subscript p = h, v, and hv represent two ACFs (H and V channels) and The estimation of Φ DP is based on the phase of ACF/CCF at multiple lags (excluding zero lag). To accomplish Φ DP estimation, the Doppler phase shift should be first removed from the CCF phase at different lags. It is known that the ACF phase only contains the Doppler shift. Here, ACF phases of H and V channels are averaged and removed from the CCF phase, as shown in (11a). As a result, an estimate of differential phase can be obtained at every lag. Next, the Φ DP is estimated by the weighted average of multiple estimates of the differential phase, as described in (11b) ] ˆφ(m) =arg[ĉhv (m) 0.5 ( ] ]) arg [Ĉh (m) +arg[ĉv (m) (11a) ˆΦ DP = M m 0 C f hv (m) ˆφ(m) M m 0 C f hv (m). (11b) Fig. 3 shows the superiority of multilag processing over the conventional method using simulated data. The simulated first-trip echo has σ =0.8 m/s, Φ DP =0, and ρ hv =0.98. The simulated second-trip echo has σ =1.6 m/s, Φ DP = 22.5, and ρ hv =0.96. TheX-axis denotes the power ratio of the first-trip echo (P 1 ) to the second-trip echo (P 2 ).The Y -axis denotes the estimated error, which is calculated by (estimation truth)2 /M. Dashed lines show the estimation from lag 0 (or 0 and 1) of ACF/CCF. Solid lines represent the estimation from multiple lags. The simulation repeats N = 400 times for each power ratio P 1 /P 2.Thesampling number is 128. The noise effect has been ignored in this simulation. As shown in Fig. 3, conventional estimates are susceptible to the second-trip echo, which contributes to the total magnitude and phase of ACF/CCF at lag 0. The multilag results, however, are less affected by the second-trip echo and hence have much less error. The identification of second-trip contamination depends on whether the second-trip echo causes peaks on both ACF and CCF at lag 0. The co- and cross-polar powers of second-trip echo are calculated by Ŝh,v s = Ĉh,v(0) Ŝh,v ˆN h,v (12a) Ŝhv s = Ĉhv(0) Ŝhv (12b)

5 1244 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 4, APRIL 2012 Fig. 3. Moment estimation comparison based on simulated weather signals: (a) Echo power S h (or S v). (b) Differential phase Φ DP. (c) Correlation coefficient ρ hv. (d) Spectrum width σ. Fig. 4. ACF and CCF examples of real radar data from OU-PRIME: (a) Tornado signal (first trip). (b) Weak precipitation signal with a strong contamination of second-trip echo. where the superscript s stands for the second trip and N h and N v are noise powers. If both Ŝs h,v and Ŝs hv appear to be large, the contamination of second-trip echo will be identified. The identification ignores the second-trip power if it is less than the noise level and, moreover, 14 db ( 14 db =0.04) below the first-trip power. That is, the criterion for identifying the contamination is Ŝs h,v,hv > 0.04 Ŝh,v,hv and Ŝs h,v > ˆN h,v. This is mainly due to the following considerations: 1) A 4% threshold would reduce the false alarm of the identification because the estimation error of power (S h, S v,or S hv ) by multilag processing is less than 4% for higher SNR (> 15 db) [22, Fig. 2], and 2) according to (A10), much weaker contamination (e.g., 14 db) introduces a small portion of error and therefore results in little degradation for the estimation of ACF/CCF at different lags. The property of the first-trip echo can be well kept through the multilag processing. B. Phase-Aided Identification Phase-aided processing identifies the second type contamination. When the second-trip echo is much stronger than the first, the total ACF or CCF will have a prominent peak. The signal with this peak could be regarded as a first-trip echo that decorrelates very quickly. Fig. 4 shows two examples for such ACFs and CCFs. The radar measurements of a tornado [e.g., Fig. 4(a)] may have a wide spectrum and hence a very narrow ACF/CCF. If the first-trip echo reflects the clear air (or weak cloud) environment, and the second-trip echo comes from a heavy precipitation [e.g., Fig. 4(b)]; the shape of total

6 CAO et al.: DETECTION AND MITIGATION OF SECOND-TRIP ECHO 1245 Fig. 5. (a) Dependence of PDE on the correlation between two Gaussian random signals. (b) (d) Dependence of PDE on the SNR: (b) PDE of single channel. (c) PDE of cross-polarization channels with a correlation of 0.98 for two channels. (d) PDE of cross-polarization channels with a fixed spectrum width. ACF/CCF might be very close to that of a tornadic signal. Consequently, the ACF/CCF alone is insufficient, and other information is needed to aid the identification. Consider the following three instantaneous phases: [ ] h(n) φ h,v (n) =arg φ hv (n) =arg h(n +1) [ h(n) v(n) or arg [ v(n) ] v(n +1) (13a) ]. (13b) As described in Section II-A, the phase difference PDF depends on the correlation of two random signals. When the spectrum width of weather signal is less than 5 m/s, the samplings h(n) and h(n +1), v(n) and v(n +1),orh(n) and v(n) are quite correlated and the distribution of φ h, φ v,orφ hv is generally narrow. It is a little different for the second-trip weather echo. Due to the phase randomness from pulse to pulse, its φ h and φ v are uniformly distributed. The wide distributions of φ h and φ v as well as the much narrower distribution of φ hv make it easy to distinguish the second-trip echo from the first. In practice, the degree of phase distribution broadness can be represented by the phase distribution evaluation (PDE) defined as 1 N x(n) PDE = (14) N x(n) n=1 where x(n) =h(n)/h(n +1), v(n)/v(n +1), and h(n)/v(n) correspond to PDE h,pde v,pde hv, respectively, and N is the sampling number. Fig. 5(a) shows the PDE dependence on the signal correlation. The PDE approaches one when two signals seem to be completely correlated. When the phase tends to be uniformly distributed, the PDE value decreases toward zero. There are several factors which may reduce the signal correlation and therefore decrease the PDE value. Fig. 5(b) (d) shows the dependence of three PDEs on the signal-to-noise ratio (SNR) and other factors. Generally, the PDE value decreases with decreasing SNR. This is because the increase of noise reduces the signal correlation between h(n) and h(n +1), v(n) and v(n +1),orh(n) and v(n). When the SNR is below 10 db, the PDE decreases rapidly. Weather spectrum width is another factor which influences PDE h and PDE v.pde hv is almost independent of σ but dependent on ϱ hv. The randomness from pulse to pulse would cause a very wide spectrum for h(n) or v(n). Therefore, the PDE h or PDE v of second-trip echo should be very small, as shown by the PDE curve for a wide spectrum width in Fig. 5(b). Second-trip echo has the same PDE hv characteristics as those of first-trip echo [as shown in Fig. 5(c) and (d)] because the randomness can be canceled for each pulse. In summary, if the intrinsic (without the effects of noise and random phase) weather spectrum is narrow and intrinsic ϱ hv is high, the PDE h and PDE v of a strong second-trip echo should tend to be zero and PDE hv should be close to one. This property distinguishes the strong second-trip echo from a first-trip echo, which contains strong turbulence (i.e., causes a wide spectrum) with a relatively low ϱ hv.to identify the second type of contamination, a simple fuzzy logic scheme is applied in this study. The aggregation value F is defined as F =[f h (PDE h )+f v (PDE v )+f hv (PDE hv )] /3 (15)

7 1246 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 4, APRIL 2012 is the first-trip echo with a relatively wide spectrum (σ/v a = ) and a low ϱ hv ( ). The other one is the secondtrip echo with a relatively narrow spectrum (σ/v a = ) and a high ϱ hv ( ). The simulation also takes noise into account, with the SNR varying from 3 to 40 db. The membership functions in Fig. 6(c) are apparently capable of discriminating between these two categories. With a threshold F =0.7, 99.76% of first-trip echo in Fig. 6(a) and 98.25% of second-trip echo in Fig. 6(b) are successfully identified. As a result of the aforementioned analysis, thresholds F>0.7 and SNR > 3 db are chosen for the identification of second-type contamination in this study. Fig. 6. Construction of membership functions for fuzzy logic scheme: (a) PDE distribution of first-trip echo which has a relatively wide spectrum and low ρ hv. (b) PDE distribution of second-trip echo which has a relatively narrow spectrum and high ρ hv. (c) Membership functions for three types of PDE. where f h, f v, and f hv are membership functions for PDE h, PDE v, and PDE hv, respectively. The membership functions, whose parameters are decided from Figs. 5 and 6(a) and (b), are given in Fig. 6(c). As for the first-trip echo, most PDE h and PDE v values are larger than 0.4 when SNR > 3 and are not likely to be smaller than 0.2, even for a wide spectrum width (e.g., σ v =0.4v a ). As for the second-trip echo, PDE h and PDE v can be reasonably assumed within this range (< 0.4). Given an SNR larger than 10 db, most PDE hv values are larger than 0.8 for a ϱ hv around 0.98 and less than 0.4 for a low ϱ hv (< 0.5). Even for a lower SNR ( 3 db),pde hv is still larger than 0.4 if ϱ hv is high. Fig. 6(a) and (b) shows the PDE distributions of simulated weather signals of two categories. One C. Identification and Mitigation Procedure Fig. 7 shows the identification and mitigation flowchart to mitigate the second-trip echo. First, the I/Q data of two polarimetric channels are inputted to estimate the signal ACF and CCF using (7). Then, in step A.3, the spectrum width is estimated from ACF and CCF with fixed multiple lags, i.e., using lags 0-1, 1-2, 1-3, 1-4, etc., in [21] and [22]. In this way, many spectrum width estimates are obtained. The reliability of these estimates is assessed with a couple of rules: 1) Spectrum widths estimated from ACF and CCF should be consistent (e.g., within 10% difference), and 2) the lags used for the estimation should be usable lags predicted by the estimated spectrum width. Through the comparison, the best matched spectrum width is found and then used to determine the usable lag number M. If there are at least two usable lags, the Gaussianshaped ACF and CCF of the first-trip echo can be obtained using the multilag fitting (step B.1). With (12) and criterion Ŝ s h,v,hv > 0.04 Ŝh,v,hv, the ACF and CCF peaks at lag zero can be easily discerned. These peaks indicate the existence of second-trip contamination. Accordingly, the multilag estimator is recommended to estimate radar moments (step D.0) for the first-trip echo. The estimation of S h,v, σ, ρ hv, and Φ DP using (10) and (11) can be effectively improved. In step B.0, a usable lag number less than two means that the signal decorrelates quickly. In such a case, the phase-aid identification mentioned in Section III-B should be applied (step B.2). If the criterion of the fuzzy logic scheme is satisfied (i.e., F>0.7), a relatively strong contamination is identified. For this type of contamination, the lag-1 estimator [25] is recommended for the moment estimation (step D.0) and the power of the secondtrip echo can be greatly reduced in the estimation of S h,v. If no second-trip contamination is identified, the legacy lag-0 estimator [1], [2] will be applied for the moment estimation (step D.0). There are some additional remarks on the moment estimation at step D.0. For the estimation of radial velocity, both lag-0 and lag-1 estimators utilize the ACF at lag 1. For the multilag estimator, the radial velocity is estimated using a weighted-average formula, which is similar to (11b). Its result, however, is very close to the velocity estimated from the ACF at only lag 1, which is unbiased by the second-trip contamination. Considering this fact, the velocity estimation using a multilag estimator is discussed less in the current study than the other moments S h,v, σ, ϱ hv, and Φ DP.

8 CAO et al.: DETECTION AND MITIGATION OF SECOND-TRIP ECHO 1247 Fig. 7. Identification and mitigation scheme flowchart of second-trip contamination. Fig. 8. NEXRAD MOSAIC reflectivity image, central plains, May 10, 2010, 2257 UTC ( The final identification of the second-trip contamination is classified into four categories. The first class has no contamination. The second one refers to a weak contamination, i.e., the power of second-trip echo is less than 10 db over the noise level. The other two classes indicate that the absolute power of the second-trip echo is not weak. Compared with the power of the first-trip echo, a relatively weak contamination is regarded as the third class. The fourth class means a relatively strong contamination, i.e., the power ratio of the first-trip echo to the second-trip echo is less than 0 db. IV. APPLICATION FOR OU-PRIME DATA ANALYSIS A. Case Description On May 10, 2010, there was a mesoscale convective system (MCS) moving across the central plains of the U.S. from west to east. The MCS was organized as a line-type system from north to south and spread over several hundred miles. Fig. 8 shows one reflectivity image of this event (from NEXRAD MOSAIC results, available online: ). It is noted that Oklahoma is on the lower portion of the map and

9 1248 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 4, APRIL 2012 Fig. 10. Identification of second-trip-echo contamination: No No contamination, C-WK weak contamination of the second-trip echo, C-RW relatively weak contamination when compared with the first-trip echo, and C-RS relatively strong contamination when compared with the first-trip echo. with A and B roughly indicate the regions with secondtrip contamination. According to KTLX s measurements, the second type of contamination is mainly within region A because the first-trip echo of OU-PRIME is from clear air or weak clouds. With strong weather echoes from the first trip, region B includes several first-type contaminations. Fig. 9. PPI images of radar reflectivity on May 10, 2010: (a) S-band KTLX radar, El =0.9, UTC. (b) Zoom-in of figure (a). (c) C-band OU- PRIME radar, El =0.95, UTC. The distance between the two circle rings in figure (a) indicates the maximum unambiguous range of OU-PRIME, which is about 125 km. OU-PRIME applies the sector scan, the coverage of which is denoted by a circular sector in figure (a). the MCS encompasses most of the region. Fig. 9 compares the reflectivity measurements between the S-band KTLX radar and the C-band OU-PRIME radar, both of which are located in central Oklahoma. The maximum unambiguous range r a of OU-PRIME for this data set is about 125 km, much shorter than that of KTLX (about 450 km). In Fig. 9(a), three rings spaced over the distance of OU-PRIME s r a are overlaid on KTLX s measurements. The out-of-trip echoes in Fig. 9(c) are clearly shown coming from two directions. Two elliptic circles marked B. Results Following the procedure in Fig. 7, the second-trip contamination in Fig. 9(c) is identified and shown in Fig. 10. The identification results are generally consistent with the measurements of KTLX and OU-PRIME in Fig. 9. For example, the second-trip echo identified at km north corresponds to the KTLX measured storm cells at km north. Within region A, the contamination area is mainly dominated by the second-trip echo and denoted by class C-RS. Within region B, some second-trip echoes are overlaid with relatively stronger first-trip weather signals. Therefore, the contamination is ascribed to class C-RW and can be mitigated using the multilag processing method. Fig. 11 shows images of three moments (Z H, σ, and ρ hv ) using the same OU-PRIME data set as shown in Fig. 9(c). The left-column moments are estimated using the conventional lag-0 (or lags 0 and 1) method. The right column moments are estimated using the identification and mitigation method described in Section III and Fig. 7. To make it visually clearer, all the images are filtered using a threshold of SNR > 0 db. It is worth noting that the strong contamination would cause a rapid decorrelation of ACF and CCF, which might make the multilag process inapplicable. In such a case, although the power of second-trip echo could be effectively reduced using the lag-1 estimator, the residue would still exist and affect the estimation of σ, ρ hv, and Φ DP. This could be a problem if the first-trip echo came from the clear air because the residue would be considered as the first-trip echo. Therefore, it is more meaningful to remove the residue. Of course, this will not be a problem if the first-trip echo dominates the total echo. The images on the

10 CAO et al.: DETECTION AND MITIGATION OF SECOND-TRIP ECHO 1249 Fig. 11. Improved moment estimation of OU-PRIME data set using the signal processing described in Fig. 7: (left column) Original moments and (right column) improved moments. Three rows are for images of reflectivity Z H, spectrum width σ, and correlation coefficient ρ hv, respectively. All the images are filtered with a threshold of SNR > 0. The images in the right column are additionally filtered by discarding the signals with a relatively strong contamination. right column discard some echoes, which have been identified as strong contamination. As shown in the images of improved moments, although there are some very weak residues that can be seen from the Z H and ρ hv images, most of the secondtrip echoes have been removed and the storm feature looks clearer. The relatively weak contaminations on weather echoes have also been mitigated through multilag processing. The improvement can be visually perceived in the region denoted by C where the spectrum width has been reduced, and it becomes more consistent in the region without contamination. Fig. 12 shows a quantitative analysis of the improvement showninfig.11.thex-axis denotes the difference between improved moments and original moments for contaminated echoes identified as C-RW. With the second-trip echo being mitigated, the signal power and the spectrum width should decrease. This trend is clearly shown in Fig. 12. Reflectivity Z H has generally been reduced by db, depending on the power of the second-trip echo. Spectrum width σ decreases about m s 1. These results are reasonable, regarding the second-trip echo as an effective noise. The improvement

11 1250 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 4, APRIL 2012 Fig. 12. Histogram of moment difference (shown in Fig. 11) for contaminated echoes identified as C-RW. The x-axis denotes the difference between improved moments and original moments. The y-axis indicates the occurrence frequency. of ϱ hv is not as evident as Z H and σ because the original ϱ hv is the combination of ϱ hv for the first-trip echo and the secondtrip echo [17]. The first trip ϱ hv might be smaller than the combined ϱ hv if the second trip ϱ hv is larger. However, if both echoes have the same ϱ hv, the combined ϱ hv should be less than that value. That is, the combination of signals tends to reduce the signal correlation. It is consistent with Fig. 12(c) that the improved ϱ hv tends to be slightly larger than the original one (with positive difference). V. C ONCLUSION The random phase coding technique has been applied for range velocity ambiguity mitigation in many previous studies. The current study enhances the application of random phase coding and presents a new identification and mitigation scheme for second-trip echoes. This scheme is particularly useful for a weather radar with a magnetron transmitter because of its random phase characteristic. This new scheme is designed for polarimetric measurements, as well as the usage of instantaneous phases on both polarimetric channels. The study originates from the recently proposed multilag processing method [21], [22]. Based on the power of the second-trip echo, it identifies the contamination as three classes: weak, relatively weak, and relatively strong. The latter two classes indicate the degree of relative intensity to the first-trip echo. The estimation of radar moments such as power/reflectivity, spectrum width, ρ hv, and differential phase can be improved by mitigating the contamination. The first-trip echo can be estimated well by the multilag processing method when it dominates the total echo. When the second-trip echo is much stronger than the first, the increase of estimation variance for the total ACF/CCF will deteriorate the performance of multilag processing. In such a case, the lag-1 estimator is recommended for moment estimation. When the first-trip echo comes from clear air or a very weak cloud, the lag-1 estimator may have some residue from the second-trip echo, although it is unbiased. This is attributed with the finite sampling effect. The stronger the second-trip echo, the more evident the residue. Because the first-trip echo is very weak and can be ignored, it is more reasonable to remove the residue. The new processing scheme is relatively simple to implement on a magnetron polarimetric weather radar and has been tested with OU-PRIME data. The data processing shows that the secondtrip-echo identification and mitigation are effective using this scheme. APPENDIX ESTIMATION OF ACF AND CCF WITH SECOND-TRIP-ECHO CONTAMINATION A. Effect of Second-Trip Echo on ACF and CCF Assume that the nth samples of H/V channel signals, i.e., h(n) and v(n), are h(n) =h 1 (n)+h 2 (n)+ε h (n) v(n) =v 1 (n)+v 2 (n)+ε v (n) (A1) (A2) where subscripts 1 and 2 denote the first- and second-trip echoes, respectively, ε is the measurement noise, and subscripts h and v represent the H and V channels, respectively. First, the measurement noise is independent of the first- and second-trip echoes, i.e., ε h,v (n)x 1,2 (n + m) = ε h,v (n)x 1,2(n + m) = ε h,v(n)x 1,2(n + m) = ε h,v (n)x 1,2 (n + m) =0 (A3) where the notation represents the expectation value, x means h or v, and m denotes the shifted lag number. For pulses generated by a magnetron, their phases are randomly changed pulse by pulse. Because received radar signals are only coherent within the first-trip echo, the second-trip echo can be regarded as being modulated with a random phase pulse by pulse. Therefore, the second-trip echoes, similar to the measurement noise, are also independent of the first-trip echoes, i.e., x 1(n)y 2 (n + m) = x 1(n)y2(n + m) = x 1 (n)y 2 (n + m) =0 (A4) where y also represents h or v. Although the phase of the second-trip echo changes pulse by pulse, in a single transmitter system using simultaneous transmission mode, the H and V echoes are highly correlated for the same pulse cycle. Therefore x 1(n)y 2 (n) 0

12 CAO et al.: DETECTION AND MITIGATION OF SECOND-TRIP ECHO 1251 while x 2(n)y 2 (n + m) = x 2(n)y 2(n + m) Thus (Ĉxy(m)) 2 = x 2 (n)y 2 (n + m) =0for m 0. (A5) According to these results, the ACF C hh or C vv and the cross-correlation function C hv can be obtained from C xy (m) = x (n)y(n + m) = C x1 y 1 (m)+c x2 y 2 (m)δ(m)+c εx ε x (m)δ(m) (A6) where δ(m) is the Kronecker delta function. This result indicates that the second-trip echo only contaminates ACF and CCF at zero lag. ACF/CCF at other lags can be used to recover the first-trip echo. B. Estimation of ACF and CCF The ACF/CCF C xy is reasonably estimated using Ĉxy (m) Ĉ xy (m) = 1 x (n)y(n + m). N m n=1 n=1 The expectation of the estimation is 1 = x (n)y(n + m) = C xy (m) N m (A7) = C x1 y 1 (m)+c x2 y 2 (m)δ(m)+c εx ε y (m)δ(m). (A8) It is obviously an unbiased estimator. To evaluate the variance of this estimator, we first investigate the expectation value of Ĉ2 xy(m) (Ĉxy(m)) ( 2 = = 1 N m 1 (N m) 2 n=1 i=1 x (n)y(n + m) j=1 x (j)y(j + m). ) 2 x (i)y(i + m) Considering weather signals have uniformly distributed random phases x 1(i)y 1(j) = x 2(i)y 2(j) = x 1 (i)y 1 (j) = x 2 (i)y 2 (j) =0for any i and j. (A9) = 1 (N m) 2 i=1 j=1 { [x 1(i)y 1 (i + m)+x 2(i)y 2 (i + m) + ε x(i)ε y (i + m)] [x 1(j)y 1 (j + m)+x 2(j)y 2 (j + m) + ε x(j)ε y (j + m)] + [x 1(i)y 1 (j + m)+x 2(i)y 2 (j + m) + ε x(i)ε y (j + m)] [x 1(j)y 1 (i + m)+x 2(j)y 2 (i + m) + ε x(j)ε y (i + m)] }. The estimation variance of ACF/CCF Ĉxy(m) is then obtained as follows: ] VAR [Ĉxy (m) ) 2 2 = (Ĉxy (m) Ĉxy (m) = = 1 (N m) 2 () 1 t=1 () 1 (N m) {[(N m) t ] [C x1 y 1 (m+t)+c x2 y 2 (m+t)δ(m+t) + C εx ε y (m+t)δ(m+t) ] [C x1 y 1 (m t)+c x2 y 2 (m t)δ(m t) + C εx ε y (m t)δ(m t) ]} () 1 t=1 () {[ 1 t ] N m } [C x1 y 1 (m+t)c x1 y 1 (m t)] + 2 [ 1 m ] N m N m [C x1 y 1 (2m)C x2 y 2 (0)+C x1 y 1 (2m)C εx ε y (0) ] + δ(m) [ Cx2 y N m 2 (m)+c εx ε y (m) ] 2. (A10) The first term in (A10) is determined only by the first-trip echo. The second and third terms are dependent on the secondtrip echo, the measurement error, and more specifically, their ACF/CCF at the zero lag, i.e., C x2 y 2 (0) and C εx ε y (0). Moreover, the third term only has an effect on the total ACF/CCF estimation at the zero lag.

13 1252 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 4, APRIL 2012 ACKNOWLEDGMENT The authors would like to thank the engineers of the Atmospheric Radar Research Center for maintaining the OU-PRIME. REFERENCES [1] R.J.DoviakandD.S.Zrnić, Doppler Radar and Weather Observations. New York: Academic, 1993, 562 pages. [2] V. Bringi and V. Chandrasekar, Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge, U.K.: Cambridge Univ. Press, [3] S. M. Torres, Y. F. Dubel, and D. S. Zrnić, Design, implementation, and demonstration of a staggered PRT algorithm for the WSR-88D, J. Atmos. Ocean. Technol., vol. 21, no. 9, pp , Sep [4] R. D. Palmer, D. Bodine, M. Kumjian, B. Cheong, G. Zhang, Q. Cao, H. B. Bluestein, A. Ryzhkov, T. Yu, and Y. Wang, Observations of the 10 May 2010 tornado outbreak using OU-PRIME: Potential for new science with high-resolution polarimetric radar, Bull. Amer. Meteorol. Soc., vol. 92, pp , Jul [5] J. Y. N. Cho, Multi-PRI signal processing for the terminal Doppler weather radar. Part II: Range velocity ambiguity mitigation, J. Atmos. Ocean. Technol., vol. 22, no. 10, pp , [6] D. Sirmans, D. S. Zrnić, and B. Bumgamer, Extension of maximum unambiguous Doppler velocity by use of two sampling rates, in Proc. 17th Conf. Radar Meteor., 1976, pp , Preprints. [7] D. S. Zrnić and P. Mahapatra, Two methods of ambiguity resolution in pulse Doppler weather radars, IEEE Trans. Aerosp. Electron. Syst., vol. AES-21, no. 4, pp , Jul [8] M. Sachidananda and D. S. Zrnić, Unambiguous range extension by overlay resolution in staggered PRT technique, J. Atmos. Ocean. Technol., vol. 20, no. 5, pp , May [9] P. Tabary, F. Guibert, L. Perier, and J. Parent-du-Chatelet, An operational triple-prt Doppler scheme for the French radar network, J. Atmos. Ocean. Technol., vol. 23, no. 12, pp , Dec [10] S. Torres, R. Passarelli, A. Siggia, and P. 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Technol., vol. 24, no. 5, pp , May Qing Cao (M 07) received the B.S. and M.S. degrees in electrical engineering from Wuhan University, Wuhan, China, in 1997 and 2005, respectively, and the Ph.D. degree in electrical engineering from the University of Oklahoma (OU), Norman, in He was a Telecom Engineer in China from 1997 to 2002, working on wireless communication systems. From 2002 to 2005, he was with the Radio Wave Propagation Laboratory, Wuhan University, performing research on high-frequency ground wave radar systems. Since 2006, he has been with the Atmospheric Radar Research Center, OU, where he was first a Graduate Assistant, then a Postdoctoral Researcher, and currently a Research Associate. His research interests include remote sensing techniques, radar signal processing, and radar polarimetry for precipitation estimation and forecast. Dr. Cao has been a member of the American Meteorological Society since Guifu Zhang (S 97 M 98 SM 02) received the B.S. degree in physics from Anhui University, Hefei, China, in 1982, the M.S. degree in radio physics from Wuhan University, Wuhan, China, in 1985, and the Ph.D. degree in electrical engineering from the University of Washington, Seattle, in From 1985 to 1993, he was an Assistant and Associate Professor with the Space Physics Department, Wuhan University. In 1989, he was a Visiting Scholar at the Communications Research Laboratory, Japan. From 1993 to 1998, he studied and worked in the Department of Electrical Engineering, University of Washington, where he was first a Visiting Scientist and later a Ph.D. student. He was a Scientist with the National Center for Atmospheric Research. He joined the School of Meteorology, University of Oklahoma, Norman, in His research interests include modeling of wave propagation and scattering in geophysical media and the development of remote sensing techniques for monitoring the Earth environment and understanding physical processes. He is currently interested in radar polarimetry and interferometry for weather quantification and forecasting.

14 CAO et al.: DETECTION AND MITIGATION OF SECOND-TRIP ECHO 1253 Robert D. Palmer was born in Fort Benning, GA, on June 3, He received the Ph.D. degree in electrical engineering from the University of Oklahoma (OU), Norman, in From 1989 to 1991, he was a Japan Society for the Promotion of Science Postdoctoral Fellow with the Radio Atmospheric Science Center, Kyoto University, Kyoto, Japan, where his major accomplishments were the development of novel interferometric radar techniques for studies of the lower and middle atmospheres. From 1993 to 2004, he was a member of the faculty of the Department of Electrical Engineering, University of Nebraska, Lincoln, where his interests broadened into areas including wireless communications, remote sensing, and pedagogy. He is currently the Tommy C. Craighead Chair of the School of Meteorology, OU, where he is also an Adjunct Professor with the School of Electrical and Computer Engineering. He serves as Director of OU s interdisciplinary Atmospheric Radar Research Center, which is the focal point for the weather radar research and educational activities on the Norman campus. Since he came to OU, his research interests have been focused primarily on the application of advanced radar signal processing techniques to observations of severe weather, particularly related to phased-array radars and other innovative system designs. He has published widely in the area of radar remote sensing of the atmosphere, with emphasis on generalized imaging problems, spatial filter design, and clutter mitigation using advanced array/signal processing techniques. Dr. Palmer is a member of URSI Commission F, the American Geophysical Union, and the American Meteorological Society. He has been the recipient of several awards for both his teaching and research accomplishments. Lei Lei received the B.S. degree in electrical engineering from Wuhan University, Wuhan, China, in 2006 and the M.S. degree in electrical and computer engineering from the University of Oklahoma, Norman, in 2009, where she is currently working toward the Ph.D. degree in electrical and computer engineering. She is also a member of Atmospheric Radar Research Center, University of Oklahoma, and has been a Graduate Research Assistant since Her research interests include weather radar signal processing, weather radar polarimetry, phased array technique, and antenna design.