Resolution Enhancement Technique Using Range ... - AMS Journals

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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY

VOLUME 23

Resolution Enhancement Technique Using Range Oversampling TIAN-YOU YU School of Electrical and Computer Engineering, University of Oklahoma, Norman, Oklahoma

GUIFU ZHANG School of Meteorology, University of Oklahoma, Norman, Oklahoma

ANIL B. CHALAMALASETTI School of Electrical and Computer Engineering, University of Oklahoma, Norman, Oklahoma

RICHARD J. DOVIAK

AND

DUSAN ZRNIC´

NOAA/National Severe Storms Laboratory, Norman, Oklahoma (Manuscript received 19 January 2005, in final form 12 July 2005) ABSTRACT A novel resolution enhancement technique using range oversampling (RETRO) is presented. Oversampled signals are radar returns from shifted and overlapped resolution volumes in range. It has been recently shown that these signals can be whitened and averaged to optimally reduce the statistical error of weather spectral moment estimations for the case of uniform reflectivity and velocity. Using the same oversampled data, when the resolution is of interest, RETRO can reveal the variation of reflectivity and velocity in range at finescale. The idea is to utilize the redundant information contained in oversampled signals, which come from common regions, to improve the resolution defined by the range weighting function. As a result, oversampled data are optimally combined to produce high-resolution signals for spectral moment estimations. RETRO is demonstrated and verified using numerical simulations for two cases. In the first case, range variation of a tornadic vortex with a diameter of 120 m can be reconstructed by RETRO at a scale of 25 m when a 250-m pulse and an oversampling factor of 10 are used. Application of RETRO to mitigate ground clutter contamination is demonstrated in the second case.

1. Introduction Spatial resolution of a pulse radar is determined by the size of radar resolution volume, which is defined by radar beamwidth (in angle) and pulse width with receiver bandwidth (in range). Improving resolution allows us to observe atmospheric structure at finescale, to advance the fundamental knowledge of atmospheric dynamics, and to improve detection and prediction. For example, multiple-vortex structure at subtornado scale has been observed and studied using high-resolution mobile weather radars with spatial resolution on the order of tens of meters (Wurman 2002; Bluestein et al.

Corresponding author address: T.-Y. Yu, School of Electrical and Computer Engineering, University of Oklahoma, Norman, OK 73019. E-mail: [email protected]

© 2006 American Meteorological Society

2003). However, for the operational Weather Surveillance Radar-1988 Doppler (WSR-88D) with approximately 1° beamwidth and pulse length of 250 m (Crum and Alberty, 1993), such a fine resolution cannot be achieved easily. In this work, an advanced technique to improve range resolution is developed and demonstrated. The standard approach of increasing the range resolution is to decrease the width of transmitted pulse. As a result, however, the sensitivity is degraded due to the decrease of transmitted energy contained in one pulse. Many efforts have been made to mitigate resolution limitation, such as pulse compression (Mudukutore et al. 1998; Schmidt et al. 1979) and a deconvolution scheme (Röttger and Schmidt 1979). Kudeki and Stitt (1987) have developed frequency domain interferometry (FDI) for the wind profiler radars to observe a single and narrow turbulent layer within one range gate

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by transmitting a pair of close-spaced frequencies. To alleviate the restrictive assumption of a single Gaussian layer in FDI and to further improve the resolution, range imaging (RIM) was developed by Palmer et al. (1999). Signals from multiple frequencies are optimally combined such that the finescale atmospheric structure can be reconstructed within the resolution volume in range. Yu and Brown (2004) have further exploited the idea of RIM to obtain high-resolution vertical profiles of three-dimensional wind with the use of additional receivers. Furthermore, Zhang et al. (2005) have proposed range interferometry (RI) to improve range resolution using the cross-correlation function of signals from a pair of spaced resolution volumes. Range resolution is gained at the expense of increasing the relative statistical error in the estimation. In this work, a resolution enhancement technique using range oversampling (RETRO) is developed based on a deconvolution method used in RIM. All available cross-correlation functions are combined optimally such that the statistical error can be reduced significantly. Radar signals from adjacent range gates are independent when they are digitally sampled at a rate of the reciprocal of pulse width (␶) and an ideal receiver of infinite bandwidth is used (Doviak and Zrnic´ 1993, section 4.6). If a higher sampling rate is employed, the range weighting function [defined as a convolution of the transmitted pulse shape and the receiver impulse response (Bringi and Chandrasekar 2001, section 5.3)] is overlapped in range. Therefore, signals from adjacent gates are correlated due to radar returns from a common region. This is termed range oversampling. Range oversampled signals can be whitened and averaged to reduce the statistical error of the three spectral moment estimates: reflectivity, mean radial velocity, and spectrum width (Ivic´ et al. 2003; Torres and Zrnic´ 2003). However, the range resolution is degraded by approximately a factor of 2 in the whitening processing. RETRO is developed using the same range oversampled data for the case that small-scale variations of reflectivity and velocity in range are of primary interest. The paper is organized as follows. In section 2, a model of range oversampled signals is established and is followed by the theory of RETRO. Oversampled signals are optimally combined to generate high-resolution signals and, consequently, signal power and velocity at fine resolution will be obtained. Theoretical RETRO bias and the noise effect will be discussed. Applications of RETRO to weather radar observations are demonstrated using numerical simulations in section 3. A comprehensive statistical analysis of RETRO is also presented. Finally, the conclusions are given in section 4.

FIG. 1. A schematic diagram of L (⫽3) range oversampling. Each oversampled signal Vi(t) consists of independent signals from L subvolumes (Si to SL⫹i⫺1).

2. Theory a. Model of range-oversampled signals For a pulse radar with infinite bandwidth, the range resolution is defined as ⌬R ⫽ c␶/2, where ␶ and c are the pulse width and the speed of light, respectively. In general, radar returns are digitally sampled at a rate of 1/␶ such that the centers of the resolution volumes are spaced by ⌬R in range, and therefore spectral moment estimates from each volume are independent (Doviak and Zrnic´ 1993, section 4.6). For the case of oversampling with a factor of L (sampling rate is L/␶), adjacent resolution volumes will be overlapped and shifted by ⌬R/L. Therefore, signals from successive volumes are correlated in range due to the radar returns from the region shared by adjacent resolution volumes. This is exemplified in Fig. 1 for L ⫽ 3. A resolution volume with a dimension of ⌬R in range consists of L “subvolumes” or “slabs.” Each subvolume has a size of ⌬R/L, chosen to be much larger than the radar wavelength. Thus, complex signals (in-phase and quadrature-phase components) from each subvolume are independent (Doviak and Zrnic´ 1993). In other words, 具Si(t ⫹ ␶)S*j (t)典 ⫽ 0 if i ⫽ j, where Si(t) is the signal from the ith subvolume, ␶ is the time lag, the asterisk is the complex conjugate, and angle brackets denote the ensemble average. Note that Si(t) is assumed to be a zero-mean Gaussian random variable. One radar sample Vi(t) from ⌬R is a weighted sum of those L independent signals (Si(t) to SL⫹i⫺1(t)), where the weights are determined by the range weighting function. This can be represented in the following matrix notation: V共t兲 ⫽ QS共t兲,

共1兲

where V(t) ⫽ [V1(t) V2(t) . . . VL(t)]T is a column vector of oversampled signals from L successively spaced resolution volumes, S(t) ⫽ [S1(t) S2(t) . . . S2L⫺1(t)]T is a vector of independent signals at 2L ⫺ 1 subvolumes

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within the range covered by L consecutive resolution volumes, and the superscript T is the transpose, Note that both Vi(t) and Si(t) are complex signals. Moreover, Q ⫽ [q1 q2 . . . q2L⫺1] is a matrix of range weighting functions and has a size of L ⫻ (2L ⫺ 1), where qi is a column vector of size L specifying the weights at the ith subvolume to produce oversampled signals in V(t). In this work, the value of each element in Q is assumed to be known. In practice, Q can be measured by injecting attenuated transmitted pulses into the receiver, as suggested by Torres and Zrnic´ (2003). Equation (1) can be exemplified by an ideal case, as demonstrated in Fig. 1, in which a rectangular pulse, a receiver of infinite bandwidth, and an oversampling factor L ⫽ 3 are considered. Thus, the range weighting function has a rectangular shape, and oversampled signals can be represented in the following form:

冋册冋册 V1共t兲 V2共t兲 V3共t兲

⫽

1 1 1 0 0 0 1 1 1 0 关S1共t兲S2共t兲S3共t兲S4共t兲S5共t兲兴T. 0 0 1 1 1 共2兲

Each oversampled signal is simply a summation of equally weighted independent signals at those subvolumes contained in one resolution volume. For example, V1(t) ⫽ S1(t) ⫹ S2(t) ⫹ S3(t). When the reduction of errors in spectral moment estimation is of interest, oversampled signals V(t) can be first whitening transformed to yield L uncorrelated signals, and subsequently these L signals are incoherently averaged (Torres and Zrnic´ 2003). It is termed the whitening-transforming-based (WTB) technique. The resulting range resolution in WTB is ⌬R(2L ⫺ 1)/L, which is approximately twice the resolution ⌬R for large oversampling factor L. Note that each of the whitened signals will consist of information from all 2L ⫺ 1 subvolumes after the whitening transformation. Therefore, no range resolution will be gained even though those signals are not averaged. When a range resolution higher than ⌬R is needed, the same oversampled signals can be processed differently to improve resolution. Zhang et al. (2005) have shown that the resolution can be gained by examining a cross-correlation function of signals from overlapped volumes. Let us examine the ideal scenario in Fig. 1 again. The cross correlation of oversampled signal V1 and V3 is RV13(␶) ⫽ 具V1(t ⫹ ␶)V*3 (t)典 ⫽ 具S3(t ⫹ ␶)S*3 (t)典 because 具Si(t ⫹ ␶)S*j (t)典 ⫽ 0 if i ⫽ j. As a result, power and mean Doppler velocity can be estimated at resolution of ⌬R/L using the crosscorrelation function at the zeroth and first temporal lag [Eqs. (4) and (5) in Zhang et al. (2005)]. Various range resolutions can be obtained depending on which pair of

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signals is used. To maximize the resolution improvement for a given oversampling factor, signals from the farthest spaced and overlapped resolution volumes are processed. The cost of the resolution refinement is the increasing statistical error because the magnitude of the cross-correlation function becomes smaller.

b. RETRO In this work, an innovative technique using all available cross-correlation functions is developed to estimate spectral moments at a fine resolution equivalent to the size of subvolume (⌬R/L). Given L oversampled signals, it is assumed that the high-resolution signal Yj(t) at the jth subvolume is a linear combination of oversampled signals, as shown in the following vector notation: Yj共t兲 ⫽ wjHV共t兲,

共3兲

where the weighting vector wj is a column vector of size L and j ⫽ 1, 2, . . . 2L ⫺ 1. The Hermitian (transpose conjugate) is denoted by superscript H. Furthermore, the autocorrelation function of Yj(t) has the following form: H RYj共␶兲 ⫽ 具Yj共t ⫹ ␶兲Y* j 共t 兲典 ⫽ wj RV共␶ 兲wj,

共4兲

where RV(␶) ⫽ 具V(t ⫹ ␶)V (t)典 is the autocorrelation matrix of range-oversampled signals with a size of L ⫻ L. The element of RV(␶) in the ith row and the jth column is defined as RVij(␶) ⫽ 具Vi(t ⫹ ␶)V* j (t)典, which is the cross-correlation function of the ith and jth oversampled signals. Substituting (1) into (4), it can be shown that H

RYj共␶兲 ⫽ wjHQRS共␶兲QHwj,

共5兲

where RS(␶) ⫽ 具S(t ⫹ ␶)SH(t)典 is the autocorrelation matrix of the true high-resolution signals at temporal lag ␶ and is a diagonal matrix with size of (2L ⫺ 1) ⫻ (2L ⫺ 1). Note that the vector wH j Q has a size of 2L ⫺ 1 and is defined as the RETRO response of the weighting vector at 2L ⫺ 1 subvolumes. One approach to obtain the weights in (3) for realvalued signals is to minimize a cost function with objective criterion (Farrar and Smith 1992; Stogryn 1978). This method provides a solution in which the noise amplification and resolution enhancement can be balanced by adjusting a subjective parameter. In this work, a relatively simple and robust method developed by Capon (1969) is used. The Capon method, also referred to as the minimum variance method or linearly constrained minimum variance (Naidu 2001), has been successfully implemented to improve angular resolution of radars using signals from spatially spaced receivers (Palmer et al. 1998; Yu et al. 2000) and to obtain high-

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resolution vertical profiles of reflectivity and threedimensional wind fields using shifted frequencies (Palmer et al. 1999; Yu and Brown 2004). Using Capon’s approach, the weighting vector in RETRO can be obtained by solving a constrained optimization: min RYj共0兲, wj

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subject to qjHwj ⫽ 1.

共6兲

It will be shown in section 2c that the power to be estimated at subvolume j, RYj(0) is a weighted sum of power from all 2L ⫺ 1 subvolumes. Signals from subvolumes other than the jth subvolume can be thought of as interference. The optimization ensures that the contamination of interference power is minimized while the response of the weighting vector at the subvolume in which the RETRO estimation is performed is unity. A detailed discussion of Capon interference suppression will be presented in section 2c. Note that only the power RYj(0) is minimized in (6), and no constraint about the velocity is considered in this work. As a result, the velocity estimate depends on the estimate of high-resolution power and will be discussed in section 2d. A similar approach has been used in radar imaging (Cheong et al. 2004) and high-resolution wind field mapping (Yu and Brown 2004). The Capon weighting vector can be estimated using a Lagrange method (Capon 1969; Palmer et al. 1998) and is also provided in the appendix (A5). Subsequently, time series of high-resolution signals at a subvolume Yj can be generated by substituting (A5) into (3). Signal power and mean radial velocity at subvolumes can be estimated by processing Yj(mTs), m ⫽ 0, 1, . . . , M ⫺ 1, using the autocovariance method (Doviak and Zrnic´ 1993, chapter 6). The number of samples in sample time is denoted by M, and Ts is the pulse repetition time. Note that both the whitened signals in the WTB technique and the high-resolution signals in RETRO are generated by a linear combination of L oversampled signals but with distinctly different goals. In WTB, the transformation matrix is designed to produce uncorrelated signals and is predefined by assuming uniform reflectivity (Torres and Zrnic´ 2003). On the other hand, the Capon weighting vector is adaptive to the measurements to improve the range resolution, and no assumption on the range dependence of reflectivity is needed. Moreover, to achieve the resolution of ⌬R/L given L oversampled signals, RETRO uses the cross-correlation functions estimated from all the pairs of oversampled signals, while RI only uses a single crosscorrelation function of signals V1 and VL.

c. RETRO power estimate The signal power at subvolume j, Pˆj, can be estimated by calculating the autocorrelation function of high-

ˆ (0)] in (4), resolution signals at temporal zero lag [R Yj which is equivalent to Pˆj ⫽

1 ˆ ⫺1q qjHR V j

共7兲

,

obtained by substituting (A19) into (4). Note that the ˆ ⫺1(0) is neglected. zero temporal lag in R V To understand the performance of RETRO power estimation, (5) is rewritten in the following form, given that RS is a diagonal matrix and ␶ ⫽ 0: 2L⫺1

Pˆj ⫽ EjTdiag{RS共0兲} ⫽

兺 E␴, 2 jl l

共8兲

l⫽1

where Ej ⫽ | QHwˆj| 2 is defined as the effective range weighting pattern (ERWP), which is the square of the RETRO response of the weighting vector. The weight Ejl is the lth element of Ej, and j ⫽ 1, . . . , 2L ⫺ 1. In addition, diag{RS(0)} ⫽ [␴21 ␴21 . . . ␴22L⫺1], where ␴2j is the true signal power at subvolume j that we would like to estimate. It is apparent that the RETRO power estimate is a weighted sum of the true signal power at 2L ⫺ 1 subvolumes. The weights are determined by the range weighting function and oversampled signals. An ideal ERWP would be a Kronecker delta function located at the jth subvolume, such that Pˆj ⫽ ␴2j . In the problem of range oversampling, it is not feasible to find appropriate weights to produce the deltalike ERWP. Thus, the performance of a high-resolution power estimate is determined by ERWP. The Capon ERWP will automatically adjust its values to minimize the contamination of interference to produce an optimal estimate with a resolution of the size of subvolume. An ideal example to demonstrate the Capon ERWP is given in Fig. 2 for ⌬R ⫽ 250 m and L ⫽ 10. The location of the subvolumes is illustrated in the top portion of the figure. Each oversampled data Vi(t) contains 10 independent signals at subvolumes [S i (t) to Si⫹L⫺1(t), i ⫽ 1, 2, . . . , L]. Model power at a resolution of subvolume (25 m) is denoted by a dashed line with circles, where three strong targets located at subvolumes 4, 10, and 16 are shown. Let us examine the case when signal power is estimated by RETRO at the 10th subvolume, which is the weighted sum of the model power over 19 subvolumes. The weights are ERWP at subvolume 10 (Ejl, j ⫽ 10 and l ⫽ 1, . . . , 2L ⫺ 1), denoted by a dotted line with diamonds. In this case, any signals at subvolumes other than 10 are regarded as interference and should be minimized. It is evident that the Capon ERWP adaptively produces minimum values at the locations of the two most significant interference subvolumes 4 and 16. Although the Capon ERWP has relative large values at other subvolumes, the model power is extremely low at these ranges. As a

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FIG. 2. Illustration of Capon RETRO for L ⫽ 10. Signals at subvolume (Sj) in relation to the first oversampled data V1 are shown on the top. RETRO power is estimated at the subvolume 10, indicated by an arrow, which is an ERWP weighted sum of model power (dashed line with circles) at all 19 subvolumes. Capon ERWP (Ejl, j ⫽ 10), denoted by a dotted line with diamonds, automatically suppresses interference at subvolumes 4 and 16. As a result, Capon RETRO can obtain a reliable power estimate at subvolume 10 using 19 oversampled data.

result, the signal power at the 10th subvolume is estimated by Pˆ 10 with a small bias. In addition, the value of ERWP at subvolume 10 (Ejl, j ⫽ l ⫽ 10) is unity (0 dB) owing to the constraint in (6). Applying the same procedure to all subvolumes, the Capon RETRO can grossly reconstruct the variation of power at 25-m resolution, as indicated by a solid line with downward triangles. Note that ERWP adaptively changes its shape with RETRO estimation at different subvolumes. In regions where the model power is low, small contamination from the interference can bias the RETRO power significantly. The bias in RETRO power estimator can be derived ˆ (0) ⫽ R (0). In other words, statisfor an ideal case R V V tical error caused by a finite number of samples (M) is not considered. As a result, RV(0) can be derived in terms of combinations of true signal power (␴2l ) at subvolumes. For example, L ⫽ 3,

RV共0兲 ⫽

冤

␴21 ⫹ ␴22 ⫹ ␴23

␴22 ⫹ ␴23

␴23

␴22 ⫹ ␴23

␴22 ⫹ ␴23 ⫹ ␴24

␴23 ⫹ ␴24

␴23

␴23 ⫹ ␴24

␴23 ⫹ ␴24 ⫹ ␴25

冥

.

共9兲 Consequently, the power estimate at the center of the 2L ⫺ 1 subvolumes can be evaluated using j ⫽ L in (7) and has the following form: L⫺1 2 ⫹ Pˆ L ⫽ ␴L

兺␴ l⫽1

2 ␴l2␴l⫹L

2 l

2 ⫹ ␴l⫹L

.

共10兲

The second term on the right-hand side of (10) is the theoretical bias of the Capon power estimator. When the reflectivity has a linear gradient on decibel scale, the signal power at subvolumes can be represented as

␴2l ⫽ ␴2s 10G(l⫺1)⌬R/(10L), 1 ⱕ l ⱕ 2L ⫺ 1, where G is a constant gradient in decibels per kilometer, ⌬R is also in kilometers, and ␴2s is a constant. The error of power estimates (in dB units) is defined by e(Pˆ L) ⫽ 10 log10Pˆ L ⫺ 10 log10␴2L. The error of the RETRO Capon power estimate as a function of reflectivity gradient is shown in Fig. 3 for L ⫽ 5, 10, and 15. The error of RETRO power is a constant for a given reflectivity gradient and oversampling factor. It is apparent that the error decreases with increasing gradient and decreasing L. However, resolution improvement will be limited when a small oversampling factor is used. Note that a more complex power profile can result in a smaller bias term in (10), for example, a Gaussian-shape power profile or the profiles shown in section 3.

d. RETRO mean velocity estimate The mean radial velocity at the jth subvolume can be estimated using the following equation:

␭ ˆ 共T 兲}, arg{R 共11兲 Yj s 4␲Ts ˆ (T ) is the eswhere ␭ is the radar wavelength and R s Yj timated autocorrelation function of Yj at the first temporal lag. To investigate the performance of RETRO velocity ˆ (T ) is derived in terms of the estimate, the phase of R Yj s phase in diag{RS(Ts)} assuming the spectrum width at each subvolume is constant: ␷ˆ j ⫽ ⫺

2L⫺1

兺 E␴

2 jl l

⫺1

ˆ 共T 兲} ⫽ tan arg{R Yj s

sin␾l

l⫽1

,

2L⫺1

兺 l⫽1

Ejl␴l2

cos␾l

共12兲

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FIG. 3. Bias of Capon RETRO (a) power and (b) mean velocity as a function of reflectivity gradient G (dB km⫺1). Small error in RETRO power is observed for strong gradient and small oversampling factor. The bias of RETRO velocity is calculated for a constant shear of 50 m s⫺1 km⫺1.

where ␾l ⫽ ⫺4␲␷lTs/␭ is the phase of the lth element in diag{RS(Ts)}, and ␷l is the true mean radial velocity at subvolume l. It is clear that ␷ˆ j is also a function of reflectivity structure through ERWP in (11) and (12). The RETRO velocity estimate would equal to the true radial velocity if ERWP were a delta function centered at the subvolume of interest. In practice, performance of RETRO velocity estimation depends on the range structure of both reflectivity and velocity. A special case is the uniform velocity profile (␾l is constant over ˆ (T )} ⫽ ␾ the 2L ⫺ 1 subvolumes). Therefore, arg{R Yj s l in (12) and RETRO velocity is unbiased and independent of ERWP. Moreover, the bias of RETRO velocity ˆ ) ⫽ ␷ˆ ⫺ ␷ , can at the Lth subvolume, defined as b(V L L be derived for the same condition in Fig. 3a. The resulting bias for a constant shear of 50 m s⫺1 km⫺1 in range is illustrated in Fig. 3b. Note that the velocity estimate is unbiased for the case of uniform reflectivity (G ⫽ 0) and constant linear shear, even though the power estimate is biased.

e. Noise effect In the presence of system noise, the oversampled signals in (1) are modified to V ⴔ QS ⫹ VN, where VN is a column vector of zero-mean white Gaussian noise. It is assumed that the system noises and signals are not correlated. Following the same steps from (3) to (8), the total power estimated in RETRO is derived: 2L⫺1

Pˆ j ⫽

兺 l⫽1

L

Ejl␴l2 ⫹

兺 |w | ␴ jl

2 2 Nl,

共13兲

l⫽1

where Pˆ j is the estimated total power, wjl is the lth element in the weighting vector wˆ j, and ␴2Nl is the average noise power in the lth oversampled signals. The second term on the right-hand side of (13) is the result-

ing noise power in the RETRO estimation and depends on both wˆ j and the noise powers in all L oversampled signals. The RETRO noise effect is demonstrated and discussed in section 3. Note that wˆ j is estimated using ˆ has both signal and noise compo(A5), in which R V nents. As a result, the RETRO signal power can be estimated using the following form: L

Pˆ sj ⫽ Pˆ j ⫺

兺 |w | ␴ jl

2 2 Nl.

共14兲

l⫽1

For most weather radars the averaged noise power can be measured and is a constant (␴2N) in range.

3. Numerical simulation RETRO is tested and verified using numerical simulations. Oversampled data were generated based on a simulation scheme similar to the one used in Torres and Zrnic´ (2003). Ideal time series were generated independently for each subvolume according to designed spectral moments (Zrnic´ 1975). Subsequently, range oversampled signals were obtained using (1) for a given range weighting vector and oversampling factor L. In this work, a rectangular pulse of 250 m is simulated to demonstrate the resolution improvement on WSR88D. Moreover, it is assumed that the receiver has a bandwidth much larger than 1/␶, the same condition used in Torres and Zrnic´ (2003) and Ivic´ et al. (2003). A sequence of white Gaussian noise is added to each set of oversampled data according to a designed signal-tonoise ratio (SNR).

a. Case I: High-resolution tornado observations A tornado vortex is simulated using a combined Rankine model (Lewis and Perkins 1953) in which the

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FIG. 4. (a) Mean profile of signal power using 250-m pulses without range oversampling (defined as CS), RI technique, and RETRO with L ⫽ 10 and SNR of 40 dB. Model signal power from the tornado at every 25 m, denoted by a dashed line with circles, can be grossly reconstructed using RETRO and RI. (b) As in (a) but for velocity profile. (c) Normalized SD of RETRO and RI power estimates. (d) Normalized SD of RETRO and RI velocity estimates.

diameter of the maximum wind is approximately 120 m and the maximum tangential wind is 50 m s⫺1. A doughnut-shaped reflectivity with a width of approximately 45 m is simulated, and the maximum reflectivity is located at the radius of 65 m from the center of the tornado. The tornado is located 10 km due north of a virtual weather radar. The radar has a 1° beamwidth and various range sampling schemes. The radar beam is pointing slightly east to the center of the tornado. Modeled range profile of signal power and mean radial velocity observed by the radar using a 25-m pulse is denoted by a dashed line with circles in Figs. 4a and 4b, respectively Radar measurement using a 250-m pulse without range oversampling (i.e., L ⫽ 1) was simulated over a 1-km range and termed conventional sampling (CS). Moreover, range-oversampled signals were generated for a 250-m pulse and an oversampling factor of 10. Each time series of oversampled and CS signals consists of 64 samples (M ⫽ 64). The aliasing velocity (Va) is 25 m s⫺1. All oversampled signals were generated at a constant SNR of 40 dB. Note that in practice the noise power is approximately a constant in range and might be obtained through the self-calibrating process of the radar. To enhance the resolution to ⌬R/L, L oversampled signals were RETRO-processed to obtain one power and velocity estimate at the range of the center of 2L ⫺

1 subvolumes. The next estimate is obtained from the next ten signals, which are shifted one subvolume in range. The RETRO signal power was estimated using ˆ has both signal and noise (14) and (7) in which R V components. The RETRO velocity estimate was obtained using (11). For RI only a pair of signals, V1 and VL, from L oversampled signals were used to estimate one power and velocity. The mean profile of the signal power and velocity estimated by CS, RI, and RETRO over 1000 realizations is shown in Figs. 4a and 4b. The standard deviations (SDs) of signal power and velocity estimators in Figs. 4c and 4d are normalized by the mean of the power estimates and 2Va, respectively. It is clear in Figs. 4a and 4b that conventional sampling with 250-m resolution (dashed–dotted line with asterisks) is not sufficient to resolve the variation of reflectivity and velocity of the tornado. On the other hand, both RETRO and RI can grossly reconstruct the modeled power profile and velocity variation at a resolution of 25 m, as shown by a solid line with downward triangles and a dashed line with diamonds, respectively. The erˆ ) for ror of signal power [e(Pˆ s)] and velocity bias b(V RETRO and RI are listed in Table 1 for range between 9.8875 and 10.1125 km. It is evident that RI can provide more accurate power estimates than RETRO at the expense of increasing the variance significantly. Moreover, RETRO velocity estimate is more reliable than

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YU ET AL. TABLE 1. Bias of power and velocity for RETRO and RI. Range (km)

RETRO e(Pˆ s) RI e(Pˆ s) ˆ) RETRO b(V ˆ) RI b(V

9.89

9.91

9.94

9.96

9.99

10.01

10.04

10.06

10.09

10.11

⫺1.70 0.12 0.01 0.13

⫺1.97 0.09 0.03 ⫺0.28

⫺2.19 0.06 ⫺0.01 ⫺0.05

⫺2.58 ⫺0.02 0.02 ⫺0.19

⫺2.14 0.12 ⫺0.35 ⫺1.15

⫺1.93 0.15 ⫺0.70 ⫺3.11

⫺1.93 0.00 ⫺0.09 ⫺0.42

⫺1.70 0.03 0.03 ⫺0.07

⫺2.49 0.04 ⫺0.07 0.09

⫺2.62 ⫺0.05 ⫺0.04 ⫺0.85

the one obtained by RI. It is interesting to point out that in the region where all L oversampled signals are obtained from uniform reflectivity and velocity, such as at ranges larger than 10.41 km or smaller than 9.59 km. It is expected in this case that the CS power estimate is 10 dB (i.e., 10log10L) larger than the model power. For uniform reflectivity, RETRO power estimation will be biased because ERWP cannot suppress interference effectively. The theoretical error in this case is approximately 7.4 dB for L ⫽ 10, as shown in Fig. 3. The discrepancy between the theoretical bias and the RETRO results in Fig. 4a for the region of uniform reflectivity will be reduced if a larger number of samples were used and is shown in Fig. 7a for M ⫽ 256. Given the same tornado model and pulse width used in Fig. 4, the performance of RETRO at three levels of SNR (0, 20, and 40 dB) is presented in Fig. 5. For low

SNR (i.e., 0 dB), the variation of signal power between 9.85 and 10.15 km becomes difficult to resolve, and the velocity bias in the region slightly increases, as shown in Figs. 5a and 5b. Note that the use of large receiver bandwidth in range oversampling will increase the amount of noise power. Now let us examine the noise effect at each subvolume after RETRO processing. The highresolution RETRO total power and noise power are estimated using (7) and the second term in (13), respectively. Therefore, the resulting RETRO SNR (R-SNR) at each subvolume can be obtained. The mean profile of R-SNR for Fig. 5 is shown in Fig. 6. Although all oversampled data have a constant SNR, the R-SNR varies with ranges after the RETRO processing and will affect the quality of power and velocity estimates. It is clear that higher values of velocity SD in Figs. 5c and 5d

FIG. 5. Comparisons of RETRO estimates for SNR ⫽ 0, 20, and 40 dB. Mean of (a) signal power and (b) velocity estimates over 1000 realizations are presented for L ⫽ 10 and M ⫽ 64. Modeled signal power and velocity are denoted by a dashed line with circles. The normalized SD of (c) power and (d) velocity estimates is also shown. The degradation of RETRO with decreasing SNR is evident with increased bias and SD, especially when SNR decreases from 20 to 0 dB.

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FIG. 6. Mean profiles of resulting SNR in RETRO given constant values of SNR of oversampled signals at 0, 20, and 40 dB.

correlate well with small values of R-SNR, especially for SNR at 0 dB. Generally speaking, when the SNR is medium and high (20 and 40 dB), the resulting R-SNR degrades but the resolution can be enhanced. On the other hand, for a low SNR resolution improvement is limited. However, the R-SNR is slightly improved (larger than zero) at 9.94 km, 10.06 km, and in regions in which the range is larger than 10.25 km or smaller than 9.75 km. As a result, relatively small statistical variances are observed in these regions, as shown in Figs. 5c and 5d. It is interesting that RETRO estimates obtained at 20 and 40 dB are extremely similar and

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they, in turn, are significantly different from the estimates at 0 dB. This result indicates that RETRO can improve the resolution without introducing significant estimation uncertainty at a medium SNR of approximately 20 dB. The performance of RETRO at a given SNR can be further improved using a larger number of samples. However, the dwell time for each estimate will increase. The mean and SD of RETRO power and velocity estimates for M ⫽ 64, 128, and 256 samples is shown in Fig. 7. A smaller error of RETRO power estimates is obtained for larger M at ranges between 9.85 and 10.15 km. The average error of power estimates [e(Pˆ s)] over the region is 2.11, 0.77, and 0.22 dB for M ⫽ 64, 128, and 256, respectively. However, the average bias in RETRO velocity is approximately a constant of 0.1 m s⫺1 for all three values of M. In the region of uniform reflectivity, the error in power for M ⫽ 256 is approximately 6.2 dB, which is closer to the theoretical error of 7.4 dB than the case of M ⫽ 64 as indicated in Fig. 4. It is also evident that the SD of RETRO power and velocity estimates decreases with the increasing number of samples. Note that the SD of RETRO also depends on the spectrum width of the true high-resolution signal Sj(t). The SD of RETRO power decreases with increasing spectrum width because the equivalent number of independent samples increases. In contrast, smaller SD

FIG. 7. Mean of (a) RETRO signal power, (b) RETRO velocity, and normalized SD of (c) signal power and (d) velocity estimates for 64, 128, and 256 samples. Less bias and smaller SD of RETRO estimates can be obtained for a larger number of samples.

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FIG. 8. (a) Mean of the RETRO power. (b) Mean of RETRO velocity. (c) Normalized SD pf RETRO power. (d) Normalized SD pf RETRO velocity for L ⫽ 5, 10, and 15. The SNR and the number of samples are 40 and 64 dB, respectively.

of RETRO velocity is observed for narrower spectrum width. This behavior is consistent with the results obtained in the autocovariance processing of nonoversampling signals (Doviak and Zrnic´ 1993, chapter 6). RETRO has shown to improve the range resolution to the size of subvolume ⌬R/L. Intuitively, a larger value of L would be desirable to obtain finer resolution. It is of interest to investigate the trade-off of oversampling factor in RETRO. The mean and SD of RETRO power and velocity estimates using L ⫽ 5, 10, and 15 is shown in Fig. 8. Although the model profiles are not shown, it should be noted that it has corresponding range resolution of 50 m, 25 m, and 16.67 m for the three values of L. In the region between 9.85 and 10.15 km, using a larger value of L can better resolve the reflectivity variation with a higher ratio of the power at peak (10.08 km) to the power at valley (10.0 km). However, more bias in RETRO power is produced for larger L. The average power error over the region is 0.41, 2.12, and 4.34 dB for L ⫽ 5, 10, and 15, respectively. In the region of uniform reflectivity, it is also expected that higher L will produce more bias, as illustrated in Fig. 3. For RETRO velocity estimation, the tip in velocity profile at approximately 10.05 km can be better resolved for L ⫽ 10 and 15. Although a larger value of L has the potential to provide finer resolution, the statistical error increases for both signal power and velocity estimation, as shown in Figs. 8c and 8d. To

reduce the error at high value of L more samples are needed, as shown in Fig. 7.

b. Case II: Ground clutter mitigation Ground clutter is often observed by weather radar such that weather signals are masked. The fine resolution afforded by RETRO can be used to mitigate the clutter contamination and retrieve some of the weather information within a conventional range bin. A ground scatterer located at approximately 9.9 km and a weather echo with uniform velocity of 10 m s⫺1 and Gaussian-shaped reflectivity centered at 10.01 km were simulated. The ground scatterer has zero mean Doppler velocity and a spectrum width of 0.5 m s⫺1. The clutter-to-signal ratio (CSR) at the subvolume where the clutter lies is 30 dB. Mean and normalized SD of the 1000 realizations of RETRO and RI estimates are shown in Fig. 9 for L ⫽ 5, M ⫽ 64, and SNR ⫽ 40 dB. In Figs. 9a and 9b, the conventional sampling with 250-m resolution (CS) cannot resolve clutter and weather components. In contrast, both RETRO and RI can reconstruct the range profile of return power at finescale such that the clutter can be separated from the weather. The error in signal power estimates for RETRO and RI is listed in Table 2 for a range between 9.8 and 10.2 km. Although the averaged error in RETRO power estimates is approximately 0.8 dB more than the error in RI, the SD of RETRO can be 10–20

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FIG. 9. A ground clutter is located within an atmospheric structure with Gaussian shape and a constant velocity of 10 m s⫺1. Clutter signal is suppressed to one subvolume using Capon RETRO, while clutter contamination is still significant over a wider range for conventional sampling due to the resolution limitation.

dB lower than the SD of RI, as demonstrated in Figs. 9a and 9c. Moreover, RETRO can limit the clutter contamination in velocity to the subvolume centered at 9.875 km in which ground clutter exists. The normalized SD of RETRO velocity is approximately 0.1. Therefore, the reflectivity and velocity from weather can still be retrieved at adjacent subvolumes using RETRO. Note that the bias of RETRO velocity is extremely small even when the power estimate is biased, such as at ranges larger than 10.25 km and smaller than 9.85 km. This is because the RETRO velocity estimator is theoretically unbiased for a constant velocity profile, as indicated in (12). However, the RI velocity estimate is significantly biased and the normalized SD is high except at a few ranges (9.875, 9.975, and 10.025 km). This limited performance is observed when one of the oversampled signals used in RI has much stronger power than the other one. For example, the ratio of the power for the two oversampled signals at 9.875, 9.975, and

10.025 km is approximately 3, while the power ratio at other ranges can be as high as a few thousands. This result can be caused by the large uncertainty in the estimation of cross-correlation function when the power ratio from the two signals is high. It can be improved using more samples. Furthermore, Doppler spectra of high-resolution signals (Yj) from one realization at and adjacent to the subvolume that clutter exists are denoted by dashed lines with circles in Fig. 10. Modeled signal spectra with estimated noise level using (13) are denoted by solid lines. It demonstrates that the modeled spectra at subvolumes can be grossly reconstructed using RETRO and the clutter signal is suppressed significantly at the subvolumes adjacent to the clutter subvolume. The resulting CSR at each subvolume is labeled in Fig. 10. The weather information in Fig. 10b can still be retrieved by filtering the clutter component around zero frequency. If the clutter extends to several subvolumes, RETRO

TABLE 2. Error of RETRO and RI power estimates. Range (km) RETRO e(Pˆ s) (dB) RI e(Pˆ s) (dB)

9.82

9.88

9.93

9.97

10.03

10.07

10.12

10.18

4.85 1.13

⫺0.07 ⫺0.07

⫺0.81 0.27

⫺1.45 0.05

⫺1.53 0.06

⫺1.18 0.10

⫺1.99 ⫺0.33

1.84 ⫺5.71

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FIG. 10. Doppler spectra of RETRO-processed signals at the subvolumes centered at (a) 9.825, (b) 9.875, and (c) 9.925 km. It is evident that the modeled spectra at subvolumes can be reconstructed using RETRO, and the clutter only occurs at one subvolume.

can still reconstruct the spectrum at subvolume successfully as long as the return power has sufficient range variation. Reflectivity and velocity form weather at these clutter subvolumes can be obtained through clutter filtering. In contrast, RI estimates the spectral moments using the cross–correlation and no highresolution spectra are available. Therefore, it may be difficult to separate clutter component from weather component at these clutter subvolumes.

4. Conclusions It has been shown by Torres and Zrnic´ (2003) that range oversampled signals can be whitened and averaged to optimally reduce the variance of spectral moment estimation for uniform reflectivity and velocity. When the resolution is of interest, Zhang et al. (2005) have shown that, using range interferometry (RI), the range resolution can be enhanced using a pair of the same oversampled signals. The resulting resolution of RI is determined by the size of common region from two overlapped volumes. In this work, the application of range oversampled signals is further exploited. Signals from all overlapped volumes are combined optimally to improve the resolution to the size of subvolume (⌬R/L). It is termed RETRO. RETRO can be postulated as an inversion problem, and many approaches can be used to solve it. In this work, the Capon method was used because of its simplicity and robustness. In addition, the Capon method has been successfully implemented in other approaches to improve range and angular resolution of radar measurement (Palmer et al. 1998, 1999). In RETRO it is assumed that high-resolution signals are a linear combination of oversampled signals. The weighting vector was obtained by solving a constrained optimization. As a result, RETRO power and velocity estimates are adaptive to the oversampled data through ERWP. RETRO is demonstrated using numerical simulations for two cases. In case I, the variation of tornado reflectivity and velocity in range was observed

at resolution of subvolume using Capon RETRO and RI, while the conventional processing techniques cannot resolve them. In case II, ground clutter contamination can be confined at one subvolume using RETRO. Both cases suggest that fine resolution can be achieved by transmitting a longer pulse and oversampling returns in range. Another approach to improve the range resolution is the pulse compression in which a longer pulse consisting of coded subpulses is transmitted. However, pulse compression requires a more complex system and a larger transmission bandwidth is required than for range oversampling. On the other hand, RETRO requires more receiving bandwidth and the resolution enhancement is limited when the SNR is low. Nevertheless, given medium SNR and sufficient variation in reflectivity, RETRO can provide robust power and velocity estimates at fine resolution. To consider a nonrectangular pulse and finite receiver bandwidth, RETRO needs to be modified and the matrix Q has to be properly defined. In addition, other approaches, such as the one used in Farrar and Smith (1992) and Stogryn (1978), can ensure the valid power estimates for uniform reflectivity and might be applied to RETRO. Acknowledgments. This work was partially supported by the National Oceanic and Atmospheric Administration (NOAA) CSTAR program under Grant NA17RJ1227. Thanks to Dr. J. Vivekanandan at the National Center for Atmospheric Research (NCAR) for preliminary discussions about radar resolution. Thanks Dr. S. Torres and C. Curtis at National Severe Storms Laboratory (NSSL) for their comments to improve the manuscript.

APPENDIX Derivation of the RETRO Capon Method The linearly constrained optimization in Capon RETRO is postulated in (6). This problem can be solved using the following Lagrange cost function:

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L共wj, ␭兲 ⫽ wjHRV共0兲wj ⫹ ␭共qjHwj ⫺ 1兲,

共A1兲

where ␭ is the Lagrange multiplier. The estimated weighting vector wˆ j is derived by differentiating (15) with respect to wj and setting it to zero:

␭ ⫺1 ˆ q, wˆ j ⫽ ⫺ R 2 V j

共A2兲

ˆ is the estimated autocorrelation matrix of L where R V oversampled signals at temporal zero lag. Note that the ˆ (0) is omitted for simplicity. The element time lag in R V ˆ is the crossin the ith row and the jth column of R V correlation function of Vi and Vj at zero lag and is estimated using the following equation: M⫺1

ˆ ⫽ R Vij

兺

1 V 共mTs兲V* j 共mTs 兲, M m⫽0 i

共A3兲

where Ts is the pulse repetition time, and M is the number of samples in sample time. The Lagrange multiplier can be solved by substituting (16) into the equation of constraint,

␭⫽

⫺2 ˆ ⫺1q qjHR V j

.

共A4兲

The Capon weighting vector is obtained by substituting (A4) into (A2) and is shown in the following equation: ˆ ⫺1q R V j wˆ j ⫽ H ⫺1 . ˆ q R q j

V

共A5兲

j

Note that the weighting vector is a solution to an ill-posed inversion problem due to the presence of noise. As a result, the solution is not unique (Haykin 2002, section 8.10). In practice, unreliable estimates can ˆ becomes closer to a singular matrix. occur when R V REFERENCES Bluestein, H. B., W.-C. Lee, M. Bell, C. C. Weiss, and A. L. Pazmany, 2003: Mobile Doppler radar observations of a tornado in a supercell near Bassett, Nebraska, on 5 June 1999. Part II: Tornado-vortex structure. Mon. Wea. Rev., 131, 2968–2984. Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar Principles and Applications. Cambridge University Press, 636 pp. Capon, J., 1969: High-resolution frequency–wavenumber spectrum analysis. Proc. IEEE, 57, 1408–1419. Cheong, B. L., M. W. Hoffman, R. D. Palmer, S. J. Frasier, and F. J. López-Dekker, 2004: Pulse pair beamforming and the

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