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Doppler Weather Radar Velocity Measurements Using a Single Pulse 804 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 15 Doppler Weather Radar Velocity Measurements Using a Single Pulse R. G. STRAUCH AND R. FREHLICH Cooperative Institute for Research in the Environmental Sciences, University of Colorado, Boulder, Colorado 8 April 1997 and 10 September 1997 ABSTRACT Simulations of weather radar echoes were used to determine whether single-pulse estimates of velocity could be made with suf®cient accuracy to measure the velocity aliasing that is inherent with pulse Doppler weather radars. The results indicated that this type of velocity measurement can be made, but such a large number of single-pulse estimates need to be averaged to determine the velocity aliasing that the concept is not compatible with the spatial and temporal resolution requirements of the WSR-88D (NEXRAD). 1. Introduction weather signals is evaluated numerically by Frehlich Unlike pulse laser or pulse acoustic Doppler systems (1993), but a sense of single-pulse measurement accu- (lidar or sodar) that measure the radial velocity of at- racy is obtained from the simple expression for the Cra- mospheric scatterers by measuring the Doppler shift of mer±Rao lower bound for the rms frequency error for the backscattered radiation, pulse Doppler radars esti- a sinusoidal pulse in white noise (corresponding to a 1/2 mate the radial velocity by measuring the pulse-to-pulse point target) given by 1/[2pt0(2E/N0) ] (Swerling 1/2 change in the phase and amplitude of the backscattered 1970), where t0 5 t/(12) , t is the pulse duration, E radiation. Because pulse Doppler radars are sampling is the pulse energy, and N0 is the noise spectral density. systems, the pulse-to-pulse measurements have a max- For a radar with a matched ®lter receiver 2E/N0 is the imum radial velocity that can be measured without am- peak point target signal-to-noise power ratio (SNR) (Na- biguity given by l/(4T ), where l is the radar wave- thanson 1969, 279). For a 1-ms pulse duration and a 40- I dB SNR, the estimate uncertainty exceeds 5 kHz or 250 length and TI is the interpulse period. The unambiguous velocity is often less than the maximum velocity of the ms21 for a 10-cm-wavelength radar. For a weather sig- scatterers, so ambiguities arise that are sometimes dif- nal in white noise the Cramer±Rao lower bound is even ®cult to resolve, particularly in real time. Lidar and greater because the correlation time of the single-pulse sodar Doppler systems do not have this ambiguity prob- weather signal is less. Unlike usual Doppler weather lem; they derive the velocity estimate from the Doppler radar, where signal sampling occurs after an approxi- shift of the signal backscattered from a single pulse. mate matched ®lter and the signal for a particular range The coherence time of the scattered atmospheric sig- cell is sampled once after each transmitted pulse (with nal is given by l/[2s(2p)1/2] (Nathanson 1969, 90), the same time delay following each transmitted pulse), where s is the width of the Doppler velocity spectrum single-pulse estimation uses a broadband receiver (as in (assumed Gaussian shaped). This time exceeds the time lidar), and typically eight or more signal samples spaced between pulses for the typical Doppler radar but not for much less than t are acquired for each range cell. Each lidar or sodar. This is the fundamental reason that the successive sample of single-pulse data is therefore from latter must obtain velocity estimates from a single pulse, a collection of scatterers centered at a longer range; while radar uses pulse-to-pulse information. signal samples separated by t are from entirely different One can attempt to measure radial velocity with a scatterers, so they will be independent. The correlation single radar pulse, as proposed by Fetter (1975); how- time of a single-pulse weather signal (approximately ever, the uncertainty of the measurement is so large that t/2) will therefore be less than the correlation time of the result is meaningless. This can be seen from the a single-pulse point target signal (t), leading to larger Cramer±Rao lower bound for the rms error of the fre- errors in the estimate of velocity. quency estimate. The Cramer±Rao lower bound for In this note we use simulations to answer the question of whether single-pulse velocity estimates with a weath- er radar (such as the WSR-88D) can be used to deter- Corresponding author address: Dr. Richard G. Strauch, NOAA/ mine the velocity with suf®cient accuracy to resolve the ERL/ETL, R/E/ET4, 325 Broadway, Boulder, CO 80303. measurement ambiguity inherent with the more accurate E-mail: [email protected] pulse-to-pulse measurements. (Single-pulse estimation q 1998 American Meteorological Society Unauthenticated | Downloaded 09/30/21 12:08 PM UTC JUNE 1998 NOTES AND CORRESPONDENCE 805 would require modi®cations to the radar receiver and an unknown parameter. The spectrum width is known signal processor.) This requires a measurement uncer- (determined by the transmitted pulse because all other tainty of order 25 m s21, so that averaging independent spectral broadening effects are negligible), and the SNR estimates of velocity over space and time will be re- is assumed known for the simulations. (For actual data quired. If this were possible, the weather radar could with high SNR, the SNR can be estimated with suf®cient operate in a surveillance mode and simultaneously mea- accuracy from the data prior to velocity estimation.) The sure re¯ectivity and velocity aliasing and then use the ML estimate is the velocity that produces a maximum usual pulse Doppler mode to measure velocity. of the likelihood function (Zrnic 1979; Chornoboy 1993; Frehlich and Yadlowsky 1994, 1995). When the velocity is the only unknown, the ML estimate can be 2. Simulation of the single-pulse radar signal ef®ciently produced from estimates of the covariance Simulation of weather radar signals usually follows matrix and fast Fourier transforms. the procedure described by Zrnic (1975); however, be- When the SNR is low, the random fades in the signal cause the radar resolution volume increases in range as produce poor estimates that are characterized as random the signal is sampled, we use a modi®cation of this outliers. The probability density function of the velocity method that is used for lidar (Frehlich and Yadlowsky estimates is then described as a localized Gaussian dis- 1994, 1995). Here, we start with the correlation function tribution of good estimates centered on the true mean (which is solely determined by the transmitted pulse velocity and a uniform distribution of random outliers. because, as with lidar, a wideband receiver is used, and The performance of the velocity estimator is described we assume uniform re¯ectivity and velocity), calculate by the velocity error or standard deviation of the good the power spectrum, add noise, randomize the power estimates, the bias of the good estimates, and the fraction spectral density as given by Zrnic, generate complex of random outliers. These statistical parameters are ex- Fourier coef®cients with random phase, and inverse tracted from the velocity estimates using robust algo- Fourier transform to obtain complex time series data rithms (Frehlich 1997). Data from multiple pulses are that have the desired correlation function. The pulse used to improve the velocity estimates by accumulating shape, pulse duration, SNR, record length, and number the statistics from N statistically independent pulses and of samples are selectable. With single-pulse radar the then estimating the velocity (Rye and Hardesty 1993a,b; signal spectrum is centered near zero Doppler, and the Frehlich and Yadlowsky 1994, 1995; Frehlich 1996). velocity estimates are independent of s because the For the ML estimator, the average covariance matrix of width of the signal spectrum is determined by the trans- all the data is produced and then the maximum of the mitted spectrum, which is very much broader than the likelihood function is determined. For all the results width of the velocity distribution of the scatterers. The presented here, 105 ML velocity estimates were gen- receiver bandwidth for single-pulse radar would be of erated for each case. The bias of the estimates was neg- order 3/t so that the signal spectrum near zero velocity ligible for all cases. has a white noise background. Note that if the single- pulse radar used a matched ®lter, the noise spectrum 4. Single-pulse simulations would be almost the same as a signal spectrum, which would bias the velocity estimates. As has been shown The radar pulse shape was assumed Gaussian (pulse for lidar and is veri®ed in our simulations, at high SNR duration is de®ned as the full width at half power) to the accuracy of the velocity estimates is not degraded avoid high-frequency sidelobes associated with sharp if the receiver bandwidth is greater than the signal spec- pulses. These sidelobes degrade estimate accuracy for trum width. single-pulse estimates by broadening the signal spec- Because the SNR depends on the bandwidth, we ref- trum, a consideration not associated with usual weather erence the SNR to that of a weather radar with a ®lter Doppler radars where the receiver approximates a bandwidth of 200 kHz (similar to the long-pulse mode matched ®lter. We simulated data for three cases: the of the WSR-88D). The SNR therefore corresponds to WSR-88D transmitting ``short'' (1.57 ms) and ``long'' familiar weather radar measurements, and all simula- (4.71 ms) pulses and a hypothetical radar transmitting tions with the same SNR have the same signal statistics. 8-ms pulses. For each case we varied the number of samples (M) from 8 to 64 in a total record length (T) of8or16ms.
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