## Abstract

The recently installed S-band phased-array radar (PAR) at the National Weather Radar Testbed (NWRT) offers fast and flexible beam steering through electronic beam forming. This capability allows the implementation of a novel scanning strategy termed beam multiplexing (BMX), with the goal of providing fast updates of weather information with high statistical accuracy. For conventional weather radar the data acquisition time for a sector scan or a volume coverage pattern (VCP) can be reduced by increasing the antenna’s rotation rate to the extent that the pedestal allows. However, statistical errors of the spectral moment estimates will increase due to the fewer samples that are available for the estimation. BMX is developed to exploit the idea of collecting independent samples and maximizing the usage of radar resources. An improvement factor is introduced to quantify the BMX performance, which is defined by the reduction in data acquisition time using BMX when the same data accuracy obtained by a conventional scanning strategy is maintained. It is shown theoretically that a fast update without compromising data quality can be achieved using BMX at small spectrum widths and a high signal-to-noise ratio (SNR). Applications of BMX to weather observations are demonstrated using the PAR, and the results indicate that an average improvement factor of 2–4 can be obtained for SNR higher than 10 dB.

## 1. Introduction

The Weather Surveillance Radar-1988 Doppler (WSR-88D) has proven to be a vital instrument for meteorological and climatological applications. It has been shown that the warning lead time for severe weather has been significantly improved after the installation of the national network of WSR-88D (Polger et al. 1994). The current WSR-88D system completes a volume coverage pattern (VCP) in a minimum of approximately 4 min for 14 elevation steps (Lee and Steadham 2004). However, faster updates are often required for fast-evolving convective storms (e.g., Carbone et al. 1985). In addition, Shapiro et al. (2003) have shown that single-Doppler wind retrieval can be improved using data with rapid updates on the order of 1 min. Recent results from mobile radars have shown that the dynamics and structure of tornadoes can vary significantly over a few minutes (e.g., Alexander and Wurman 2005; Bluestein et al. 2003). Therefore, rapid scanning is needed not only to increase the warning lead time but also to advance understanding of fast-evolving weather systems. An intuitive way to reduce the data acquisition time for a given VCP or a sector scan is to increase the antenna rotation rate. However, the statistical error of the three spectral moments (reflectivity, mean Doppler velocity, and spectrum width) will increase due to a decrease in the number of samples for processing. Moreover, the rotational rate is constrained by the mechanical limitation of a pedestal. A novel approach to address these fundamental limitations is presented using a phased-array weather radar (PAR), which can achieve the goals of rapid scanning and high data quality.

In estimation theory, the statistical error (i.e., the standard deviation) of an estimator can be reduced by averaging a number of independent measurements (Papoulis and Pillai 2002). Radar measurements from adjacent ranges and radials can be averaged to improve the quality of the estimate at the expense of resolution degradation. For example, reflectivity measurements from four range gates are averaged to achieve the required accuracy of 1 dB in the WSR-88D (Doviak and Zrnić 1993). Recently, Torres and Zrnić (2003) have developed the whitening transform technique for range-oversampled data. These highly correlated signals are whitened to become independent and, consequently, are averaged to reduce the statistical error. In this work, the idea of the collection of independent samples is exploited in sample time to optimize the acquisition time and data quality. Because neither range nor azimuthal averaging is performed, there is no additional degradation in spatial resolution.

For radar with a mechanically rotating antenna, long dwell times are often needed to acquire a sufficient number of independent samples, because successive signals in sample time are highly correlated. Koivunen and Kostinski (1999) developed a technique to whiten these signals to improve the power estimates through the increase in the equivalent number of independent samples. However, the problem becomes ill conditioned and sensitive to the noise when the autocorrelation is a Gaussian-shaped function. Further, the width of the Gaussian function is not available but is needed a priori. In this work, a novel scanning strategy is developed to directly collect independent samples (Zrnić 1977). During the wait time for signals to become uncorrelated, the radar is tasked with probing other regions to maximize the usage of the radar resources, as suggested in Smith et al. (1974). Hence, the radar beam is multiplexed over a designated region to provide measurements with low statistical errors in a relatively short period of time. This approach is termed beam multiplexing (BMX). To implement BMX, the radar beam needs to be steered rapidly from one location to another, and for such a task an electronically steered PAR is an ideal candidate.

Phased-array systems were developed in the mid-1960s primarily for military use (Skolnik 2001). A radar with a single agile beam in elevation (mechanically rotating in azimuth) was proposed for weather applications (e.g., Smith et al. 1974; Keeler and Frush 1983; Josefsson 1991; Joss and Collier 1991). A recent report from the National Research Council also identified phased-array systems as one of the promising technologies for the next-generation weather radar (National Research Council 2002). A mobile radar both frequency scanning in elevation and mechanically rotating in azimuth was developed by Wurman (2003). Moreover, a mobile X-band phased-array system with a spaced-antenna configuration was developed to measure transverse winds in severe convective storms (Pazmany et al. 2004; Hardwich et al. 2005). A phased-array system also has the potential for multiple functions, such as performing simultaneous meteorological observations and en route air traffic control (Weber et al. 2005). An S-band PAR was recently installed at the National Weather Radar Testbed (NWRT) in Norman, Oklahoma. The NWRT initiative was developed through a unique research partnership including the U.S. Navy, federal agencies, private sectors, and a university (Forsyth et al. 2001, 2002). The PAR at NWRT is a national asset and has become available to research communities since September 2003 (Forsyth et al. 2005). The PAR is equipped with a military SPY-1A antenna from the Navy, which allows electronic scanning in both azimuth and elevation on a pulse-to-pulse basis. Preliminary results have shown that the PAR can provide meteorological information that is consistent with those obtained by a nearby WSR-88D (Forsyth et al. 2005).

The paper is organized as follows. An overview of the statistical errors for two sampling schemes, contiguous-pair sampling (CPS) and independent-pair sampling (IPS), is given in section 2. BMX is developed based on independent-pair sampling, while the conventional scanning strategy in weather radar uses contiguous-pair sampling. Design criteria and implementation of BMX are presented and discussed. Moreover, an improvement factor of data acquisition time is defined to quantify the performance of BMX based on the statistical properties of the moment estimates. In section 3, results from an experiment using the PAR are presented to demonstrate the application of BMX to weather observations. Furthermore, results from BMX are compared with those obtained from step scanning, which is similar to the conventional sampling. Finally, conclusions are given in section 4.

## 2. Beam-multiplexing scanning strategy

### a. Contiguous-pair sampling and independent-pair sampling

It is beneficial to review two radar sampling schemes, CPS and IPS (Zrnić 1977), which are fundamental for the conventional and BMX scanning strategies, respectively. In CPS, a number of pulses is transmitted every pulse repetition time (PRT), denoted by *T _{s}*. The three spectral moments are estimated from the

*M*uniformly spaced samples (see the upper panel of Fig. 1). As a result, the dwell time is

*MT*. CPS is currently employed in most weather radars unless the staggered PRT techniques (e.g., Torres et al. 2004; Cho and Chornoboy 2005) are used. Note, in this work the spectral moments are estimated using the autocovariance method (commonly called the pulse-pair processor) (Doviak and Zrnić 1993). Signal power is estimated as the mean of

_{s}*M*power estimates from each sample. In addition, the mean radial velocity is derived as the mean of the autocorrelation function at lag one from

*M*− 1 contiguous pairs. If these

*M*samples are independent, the variance of the mean power estimate is reduced optimally by

*M*. However, if signals are correlated, a larger number of samples is needed to achieve the same statistical variance. This problem can be alleviated by IPS, in which

*L*spaced pairs of pulses are transmitted. The time separation between pairs, defined as the revisit time (

*T*), is sufficiently large such that signals from adjacent pairs are uncorrelated. This is depicted in the lower panel of Fig. 1. A pair of correlated samples is required for the estimation of the mean radial velocity and spectrum width in the autocovariance method. Because of the increase in the equivalent number of independent samples using IPS, the statistical accuracy can be achieved with fewer samples than are required for the CPS. However, the dwell time of IPS (

*LT*) would be longer than

*MT*. In BMX independent pairs of samples are collected to minimize the statistical variance of the estimates. Moreover, during the revisit time the radar is tasked to scan other radials to minimize the data acquisition of a sector scan.

_{s}The goal of BMX scanning strategies is to reduce the scan time without compromising the accuracy of radar measurements. Now let us examine the statistical errors of moment estimates using data from both CPS and IPS. The variance of the signal power and mean velocity estimators using CPS is shown in the following equations (Zrnić 1977; Doviak and Zrnić 1993):

where *Ŝ _{C}* and

*υ̂*are estimators of signal power and mean velocity, respectively;

_{C}*S*is the true signal power;

*M*is the number of samples; and the signal-to-noise ratio is defined as SNR =

*S*/

*N*, where

*N*is the noise power. Moreover,

*ρ*(

*lT*) is the correlation coefficient of weather signals, and

_{s}*K*=

*λ*

^{2}/[32

*π*

^{2}

*T*

^{2}

_{s}ρ^{2}(

*T*)], where

_{s}*λ*is the radar wavelength. The standard deviation (SD) of reflectivity (

*Z*) estimate can be obtained from SD{

*Z*} = 10 log

_{C}_{10}(1 + SD{

*Ŝ*}/

_{C}*S*) (Doviak and Zrnić 1993). For a given SNR,

*M*, and

*T*, the variance of signal power increases with decreasing spectrum width, as shown in Fig. 5.7 in Bringi and Chandrasekar (2001). This is because signals are more correlated at smaller spectrum widths and, therefore, the equivalent number of independent samples is smaller. In contrast, the variance of mean velocity estimate decreases with decreasing spectrum width for large SNR because correlated signals are favorable for the pulse-pair processor (Zrnić 1977). Nevertheless, both variances decrease with increasing

_{s}*M*and SNR. The data acquisition time of a sector scan that consists of

*n*radials using CPS is defined as

_{b}*T*=

_{AC}*n*. Thus, for a given PRT the acquisition time can be reduced using fewer samples. For a mechanically rotating radar this can be achieved by increasing antenna rotation rate. However, the variance of the estimates will increase due to a smaller number of samples, as shown in (1) and (2).

_{b}MT_{S}In IPS, mean power is estimated by the arithmetic mean of 2*L* power estimates from each sample. The autocorrelation function at lag one is estimated from *L* independent pairs because correlation exists only within each independent pair. The variance of signal power and mean velocity estimates can be approximated by using *M* = 2 in (1) and (2) and dividing by the number of pairs *L*. The resulting variances are shown in the following equations (Zrnić 1977; Doviak and Zrnić 1993):

where *Ŝ _{I}* and

*υ̂*are the signal power and mean velocity estimators using IPS. It is apparent that both variances depend on the correlation coefficient only up to lag one because only a pair of correlated samples is available for the estimation. Note the dwell time of IPS is

_{I}*LT*. If the radar is idle during the revisit time, the data acquisition time for

*n*radials is

_{b}*T*=

_{AI}*n*, which could be much longer than

_{b}LT*T*.

_{AC}### b. Beam multiplexing

BMX is developed to exploit the idea of IPS and to make full use of the radar. During the revisit time, the radar beam will be rapidly steered within the region of interest to collect independent pairs of samples at many beam locations. As a result, the data quality and acquisition time can be optimized. In designing the BMX scanning strategy, two fundamental issues need to be addressed. First of all, the revisit time *T* should be large enough to ensure the collection of independent pairs. At the same time, *T* should be small enough such that the weather of interest does not change significantly over the dwell time of *LT*. In other words, the 2*L* weather samples are collected from a stationary process with the same statistical properties. The decorrelation time of the weather signal is defined by the time when the correlation coefficient has decreased by two orders of magnitude. For example, for an S-band radar the revisit time should be larger than 24.2 ms for spectrum widths equal to and larger than 1 m s^{−1}. The second issue is that the angular separation between any two consecutive beam locations should be large enough to suppress the second-trip echoes from the previous beam location, if they are present. If two locations are too close, the data collected in the current location will be contaminated by the second-trip echo from the previous beam location through large sidelobes or even the main lobe. Therefore, BMX requires a radar to steer the beam from one location to another one, which should be a few degrees apart, every couple of PRTs. This is different from the conventional scanning strategy, in which the beam location varies continuously because of the mechanical rotation of the antenna. Thus, a phased-array radar is an ideal instrument for implementing BMX because of its fast and flexible beam steering. The suppression of second-trip echoes can be determined by multiplying the antenna’s receiving pattern at the current location by the transmitting pattern centered at the previous beam location.

Note that the antenna pattern for a phased-array radar is a function of steering angle. The suppression factor as a function of steering angle and angular separation between the two consecutive beam locations is obtained using the simulated PAR antenna pattern and is shown in Fig. 2. It is assumed that the PAR has the same receiving and transmitting patterns. White contour lines of −30 dB are superimposed. The result indicates that an angular separation of 6°–13° between two successive beam locations can suppress the second-trip echoes from the previous beam location by approximately 30 dB for steering angles less than 20°. Note that signals from the second pulse of each pulse pair can still contain second-trip echoes. However, these should not bias the velocity estimates. But, an increase of variance would occur because the cross product of the first- and second-trip return acts as noise. For WSR-88D, two scans with one long and one short PRT are used to recover range-folded velocity estimates. A similar algorithm can be used to mitigate the second-trip echoes within the pulse pair.

To maximize the use of the radar, the number of beam locations that can be visited during the revisit time is determined by *n _{b}* =

*T*/(2

*T*), where

_{s}*n*is an integer. In designing the BMX scanning strategy, it is important to select

_{b}*n*to address the two issues discussed earlier. An example of a possible BMX strategy to survey a sector with 1° angular sampling is illustrated in Fig. 3.

_{b}In practice, no a priori knowledge of the decorrelation time is available. Nevertheless, Fang et al. (2004) have shown that the median spectrum width ranges approximately from 1.5 to 5.4 m s^{−1} for various types of weather. To ensure signals are sufficiently decorrelated for most weather conditions, a spectrum width of 1 m s^{−1} was used in the design of BMX. As a result, the revisit time is 28 ms for *n _{b}* = 14 and

*T*= 1 ms. A smaller

_{S}*n*will make the angular separation of two consecutive beams less than 6°, which will be demonstrated shortly. Therefore, the sector to be scanned by BMX consists of 14 angular locations, which are denoted by

_{b}*a*

_{1},

*a*

_{2}, . . . ,

*a*

_{14}with 1° spacing and are presented on the top panel of Fig. 3. The BMX scanning strategy is demonstrated in the lower portion of Fig. 3. The first pair of pulses is transmitted at

*a*

_{1}and the next pair is transmitted at

*a*

_{8}, which is 7° from

*a*

_{1}. Subsequently, two pulses are transmitted at

*a*

_{2}, which is 6° from the previous location

*a*

_{8}, and are followed by another pair of pulses at

*a*

_{9}and so on. As a result, the radar beam is multiplexed over the 14 beam locations. After

*T*seconds, a pair of samples would have been received at each location. This pattern will be repeated until a desired number of independent pairs are collected at each location. Therefore, the data acquisition time for BMX to scan the

*n*radials is defined as

_{b}*T*=

_{AI}*LT*= 2

*n*. Although the second-trip echo will be suppressed by approximately 30 dB in this case, third- or higher-trip echoes, if present, can still contaminate the data. Note that the BMX strategy can be used in either azimuth, elevation, or both directions. Surveying a large sector using the described BMX strategy with a large

_{b}LT_{s}*n*can make the dwell time (2

_{b}*Ln*) too long, and thus violate the stationarity of the weather signals. To avoid this problem a number of small sectors with BMX can be scanned sequentially. For example, six sectors each 14° wide can be used in sequence to survey an 84° sector. It should be emphasized that in conventional scanning strategies spectral moments are estimated using samples from different time intervals (i.e., radial by radial). In contrast, in BMX spectral moments at different radials are estimated by the samples from the same time interval because the dwell time and data acquisition time are the same. In other words, the spectral moments are averaged over a longer period of time than those obtained from conventional strategies given the same number of samples.

_{b}T_{s}It is of interest to quantify the gain in data acquisition time using BMX while the estimation accuracy obtained by a CPS-based scanning strategy is maintained. Therefore, the improvement factor of data acquisition time, which is defined by the ratio *T _{AC}*/

*T*, can be derived by equating the variance between IPS and CPS and then solving for

_{AI}*M*/(2

*L*). As a result, the improvement factor derived from the signal power (

*I*) and mean velocity (

_{S}*I*) estimates for a large number of samples is shown in the following equations:

_{υ}where Σ_{S} and Σ_{υ} are the summation terms in (1) and (2), respectively. The correlation coefficient is typically assumed to have a Gaussian form of *ρ*(*lT _{s}*) = exp[−2(

*πσ*)

_{υn}l^{2}] (e.g., Doviak and Zrnić 1993). The normalized spectrum width is

*σ*=

_{υn}*σ*/2

_{υ}*υ*, where

_{a}*σ*is the spectrum width and

_{υ}*υ*is the maximum unambiguous velocity.

_{a}The improvement factors of acquisition time as a function of normalized spectrum width are shown in Fig. 4 for SNR = 20, 5, and 0 dB. If *I _{S}* and

*I*are smaller than one, no gain in acquisition time will be obtained by BMX. For the case of signal power, the improvement is more significant at smaller spectrum widths for all SNRs. This is because signals are more correlated and, consequently, more samples are needed in CPS to achieve the required accuracy of power estimates. Moreover,

_{υ}*I*is larger than unity when spectrum width is small and the SNR is relatively high. The reason is that the noise from the common pulse of adjacent pairs in CPS will be canceled in the averaging process (Zrnić 1977). Thus, the required performance of the velocity estimator can be obtained by CPS with fewer samples at low SNR. It is evident that the most favorable conditions for BMX are a small spectrum width and large SNR for both cases. For weather application, the SNR is often reasonably high and the median spectrum width ranges approximately from 1 to 5 m s

_{υ}^{−1}for various types of weather phenomena (Fang et al. 2004).

## 3. Experiment results

In this work, the feasibility and application of BMX to weather observations are demonstrated using the PAR at NWRT. The PAR is equipped with a SPY-1A passive phased-array antenna that is comprised of 4352 elements. The antenna is mounted on a pedestal to provide a full 360° view of the atmosphere. The PAR has a beamwidth of 1.5° at the center and can be electronically steered ±45°, within a 90° volume, on a pulse-to-pulse basis without moving the pedestal. In addition, the PAR has a WSR-88D transmitter that was modified to transmit at 3.2 GHz. A customized environmental processor (EP) was used for receiving and data processing. Some of the PAR’s specifications are summarized in Table 1. More detailed information can be found in Forsyth et al. (2001, 2002, 2003, 2004, 2005).

An experiment was conducted to demonstrate and verify BMX on 2 May 2005. A 28° sector from azimuth 183° to 156° was scanned using two scanning strategies. The 28° sector was first scanned using BMX with two 14° sectors. In each sector, the PAR was beam multiplexed over 14 beam positions as demonstrated in Fig. 3, where 32 pairs of pulses were collected at each beam position. As a result, the acquisition time of BMX is 1.792 s, with a PRT of 1 ms. The other scanning strategy is a step scan (SS), which was devised to probe the same 28° sector with 28 discrete beam positions. Data from 64 pulses were collected before the beam was steered to the next azimuth location. These azimuthal locations are 1° apart. It is similar to the conventional scanning strategy used in weather radar with a mechanically rotating antenna. But, the SS will not produce a spectral broadening effect (Doviak and Zrnić 1993) because the antenna is stationary. Note that a direct demonstration of the improvement factor is not possible without a priori knowledge of the spectrum width and SNR. Therefore, to make a comparison, the data acquisition time of BMX is set to be same as SS (i.e., *M* = 2*L*). If the standard deviations of power and velocity estimates from BMX are smaller than those from CPS, it indicates that the improvement in acquisition time could be obtained using BMX when the estimation accuracy is the same as that of CPS. Moreover, to perform statistical analysis, the two scanning strategies were alternated 50 times. The reflectivity and velocity fields obtained by the two strategies from the 17th scan are shown in Fig. 5.

The BMX reflectivity and velocity are shown in the upper panels, while the SS results are in the lower panels. It is evident that the gross structures of the reflectivity and velocity from both scanning strategies are consistent. Note that the fields from BMX appear to have less spatial fluctuation. One possible reason is that the BMX moments at 14 radials are obtained by samples from the same period, which is longer than the SS. Another reason is that the variance of BMX estimates is smaller. Additionally, the dashed lines indicate the radial of 159° from which the statistics will be discussed later.

Because the weather continuously varies with time, it is important to estimate the time interval for which the underlying processing can be considered stationary. Thus, the statistics of spectral moments can be calculated from various scans within that period. In this work, the length of time for statistical analysis is defined as the time for scatterers within the radar volume to be totally replaced under the assumption that the fields of interest are homogeneous. An intuitive approach is used in an attempt to estimate the period of time. Considering a radar volume centered at 50 km and an azimuth of 159°, its dimension is approximately 250 m in the radial and 1.3 km in the transverse directions, given a 1.5° beamwidth. The mean radial velocity at 50 km is approximately 2 m s^{−1} and, therefore, it takes 125 s for scatterers to totally replenish in the radial direction. The wind component in the azimuthal direction can be roughly estimated by tracking the motion of a reflectivity structure with a maximum of 45 dB*Z* from the images. It was estimated to be approximately 57.3 s for scatterers to move across the radar volume in the azimuthal direction. Therefore, the length of time for statistical analysis is determined by the transverse motion and is 57.3 s. This is approximately the time for 16 scans in each scanning strategy. For both BMX and SS, statistics of the reflectivity and velocity estimates were estimated independently in three time periods (*T*_{1}, *T*_{2}, and *T*_{3}), each with 16 scans.

Statistical results at an azimuth of 159° are selected to exemplify the performance of BMX and SS because of a large region of high SNR. The mean and SD of the three spectral moments were estimated from *T*_{2} (i.e., from the 17th to the 32d scan in each strategy). Statistical results from other time periods are similar. Note that a linear trend in the moment estimates was removed, if it existed, before the calculation of the SDs. The procedure is performed in an attempt to remove the nonhomogeneous portion of the estimates. The mean profile of the reflectivity, radial velocity, and spectrum width is shown in Fig. 6.

The results of BMX are denoted by solid lines while the results of SS are indicated by dashed lines. It is evident that the means of three spectral moments estimated from both scanning strategies are extremely similar. The difference in reflectivity, radial velocity, and spectrum width between BMX and SS and averaged over entire ranges is 0.05 dB*Z*, 0.003 m s^{−1}, and 0.08 m s^{−1}, respectively. The SD of reflectivity and mean velocity from both scanning strategies is shown in Fig. 7. The mean SNR from BMX over the same 16 realizations is superimposed and is denoted by black lines. The value of SNR is referred to the vertical axis on the right. The SD of BMX reflectivity and velocity is denoted by red lines, while the SD of reflectivity and velocity from SS is denoted by blue lines. It is evident that BMX can provide more accurate reflectivity and velocity estimates up to approximately 90 km, where the SNR is larger than 10 dB and the mean spectrum width is 1–2 m s^{−1} (as shown in Fig. 6c). In other words, the results indicate that the data acquisition time can be reduced by BMX while maintaining the same accuracy as in the SS. To further verify the theoretical relations discussed in section 2a, the mean SNR and spectrum width were substituted into (3) and (4) to obtain theoretically fitted SDs for BMX. The resultant SD of reflectivity is denoted by the green line. Similarly, the theoretically fitted SD of reflectivity for SS can be obtained from (1) and (2), and is denoted by the cyan line. It is evident that the fitted SD of reflectivity agrees reasonably well with the observational results for both BMX and SS if the SNR is higher than approximately 5 dB. The discrepancy between fitted and observational SDs beyond 120 km may be caused by inaccurate estimation of the spectrum width in the autocovariance method at low SNR. Furthermore, the fitted SDs of velocity for both BMX and SS are also consistent with the SDs measured directly from observations between 20 and 95 km. However, a discrepancy in SS velocity estimates between 40 and 60 km is observed. A possible reason for the discrepancy can be that the high-reflectivity structure is not homogeneous within the radar volume. The reflectivity in this region decreases with a constant rate of approximately 0.08 dB s^{−1}.

The improvement factor of acquisition time is defined such that the same SD is achieved for both BMX and SS. In this experiment with the same acquisition time, the reduction in SD using BMX can be translated into the improvement factor. In other words, the improvement factors *I _{s}* and

*I*can be obtained by the ratio of var{

_{υ}*Ŝ*}/var{

_{C}*Ŝ*} and var{

_{I}*υ̂*}/var{

_{C}*υ̂*}, respectively. The minimum of

_{I}*I*and

_{s}*I*at each range gate is used to represent the overall improvement factor because both signal power and velocity estimates are of interest. Moreover, the mean improvement factor at each radial is obtained by averaging data whose SNR is larger than 10 dB over the entire ranges. The resultant measure of performance is shown in Fig. 8 for the three time periods.

_{υ}The improvement factors from the three time periods are consistent. It is clear that the data acquisition time can be improved by an average factor of 2–3.8 in this case. Note that the first few radials are associated with high-reflectivity regions and, hence, estimates therein can be improved more significantly than elsewhere.

## 4. Conclusions

A fundamental limitation of rapid scans using a mechanically rotating antenna is the degradation of data quality. In this work, a novel scanning strategy that exploits the collection of independent radar samples and maximizes the radar resources is developed. This is termed beam multiplexing (BMX), whereby the radar beam is multiplexed to provide optimal data quality and acquisition time. Two criteria for designing BMX are provided and discussed. The improvement factor of acquisition time is theoretically derived for the cases of signal power and mean velocity estimates. It has been demonstrated that the acquisition time can be reduced significantly, especially at small spectrum widths and high SNR.

The phased-array radar (PAR) was recently installed at the National Weather Radar Testbed (NWRT) and is a unique instrument for phased-array weather radar research. In this work, BMX has been demonstrated using the PAR for weather observations. The BMX results were compared with those obtained from a step scan (SS), which is similar to the conventional scanning strategy used in most weather radars. Statistical analysis has shown that BMX can provide fields of the three spectral moments that are consistent with those from SS. Moreover, the variance of the spectral moment estimates can be smaller than those from SS for the same data acquisition time. The results indicate that rapid scans can be achieved by BMX, and the data quality required by conventional radar is still maintained. The improvements in data acquisition time, with an average factor of 2–4, were obtained from the data. Note that additional decrease in scan time can be achieved if more antenna faces are used. Although BMX is not favorable when the spectrum width is large and/or SNR is relatively low, especially for the case of mean velocity estimates, the PAR has the potential to adaptively switch to a mode of SS to provide optimal scanning strategy. Moreover, it should be noted that the clutter filtering (e.g., Torres and Zrnić 1999), as well as spectral analysis for moment estimation (Doviak and Zrnić 1993), is limited in BMX because only a pair of correlated samples is available for the processing. In a volume scan strategy, it is possible to implement SS at the lowest elevation angle and BMX at higher elevation angles to mitigate clutter contamination because ground clutters are likely more severe at lower elevation angles. Another approach to alleviate this limitation is to transmit multiple pulses with variable interpulse periods rather than only a pair of pulses. A thorough study is needed to investigate the trade-off between the effectiveness of clutter filtering and the improvement factor for acquisition time. An advantage of the variable interpulse periods, which is the staggered PRT, is to mitigate range–velocity ambiguities.

## Acknowledgments

This work was primarily supported by NOAA/NSSL under Cooperative Agreement NA17RJ1227. Part of this work was supported by DOD, EPSCoR Grant N00014-06-1-0590. The authors would also like to thank the technical support of NSSL in the collection of PAR data.

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## Footnotes

*Corresponding author address:* T.-Y. Yu, School of Electrical and Computer Engineering, University of Oklahoma, Norman, OK 73019. Email: tyu@ou.edu