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  • View in gallery

    Depiction of (top) continuous scan and (bottom) step scan. (top) The horizontal wide double-arrow bands depict the span of data used in computation of spectral moments. The number of returns (pulses) within the beamwidth θ1 is 2L. On the abscissa is time or angle for a continuously scanning beam. The curves above the ribbons represent window functions. (bottom) The single-arrow bands represent locations where the beam is pointing while L pulses are transmitted by the PAR. The size of the arrows insinuates the relative weight given to the samples from the three consecutive beam positions (spaced at θ1/2) for synthesizing a better estimate.

  • View in gallery

    Effective main lobe of the antenna pattern. The relative position in azimuth is normalized by the one-way 3-dB width of the antenna pattern. Step Tay100 and Step Tay30 are applied to the three positions (−0.5, 0, and 0.5) of the stepped scan. The Scan Tay54 is applied to the continuous scan between −0.5 and 0.5 of the normalized position.

  • View in gallery

    Equivalent number of independent samples for power estimation of pure noise. The numbers in the legend are the sidelobe levels (spectral) of the Taylor window. Dashed lines represent results for continuous window, and full lines are for step-equivalent window.

  • View in gallery

    Standard deviation of reflectivity estimates vs the total number of samples available from the three-beam position M = 3L. The curves are for the following conditions. The Rec M/3 means a single stationary beam collecting M/3 power samples. The Scan Tay54 2M/3 is for a scanning beam over a 1° interval and collecting 2M/3 power samples. The beam displacement has been accounted according to Eqs. (A6), (A7), and (A8). Step Tay100 M is for a combined estimate from the three pointing directions (L = M/3 estimates from each) and the decorrelation due to the three beam positions is accounted for via Eqs. (A10), (A11), and (A8). The Scan Rec 2M/3 is for a scanning beam over a 1° interval collecting 2M/3 power estimates, and the beam displacement is accounted for via Eqs. (A5) and (A6). The assumed SNR = 10 dB and the spectrum width συ = 1 m s−1.

  • View in gallery

    SD of velocity estimates vs the spectrum width and for the same conditions as in Fig. 4. The beam displacement in the Scan Tay54 and Scan Rec has been accounted for by inserting Eqs. (A6) and (A7) into Eq. (A13). For the Step Tay100 M curve, Eqs. (A10) and (A4) are combined into Eq. (A6), and then Eqs. (A6), (A7), and (A11) are inserted into Eq. (A13). The SNR = 8 dB.

  • View in gallery

    SD of spectrum width estimates obtained from R(0)/R(1) vs the spectrum width. The parameters are as in Fig. 5, but the SNR = 10 dB. The curves depict the results from simulations.

  • View in gallery

    SD of spectrum width estimates obtained from R(1)/R(2) vs the spectrum width. The parameters are as in Fig. 6. The curves depict the results from simulations.

  • View in gallery

    (a) Reflectivity field of the storm at 2346:46 UTC 31 May 2013. Azimuthal spacing of radials is half a beamwidth, and elevation angle is 0.51°. The number of samples for reflectivity estimates is 12, and all samples are uniformly weighted (i.e., Fixed Rec window). Data with SNR ≥ 10 dB is displayed. (b) As in (a), but the value Zw in each radial is obtained by combining powers from three adjacent radials and weighting these with the Step Tay100 coefficients. Thus, the total number of samples per estimate is 36.

  • View in gallery

    (a) Histogram of the difference field, Z (Fixed Rec window) in Fig. 8a minus Zw (Step Taylor100) in Fig. 8b. (b) Histogram of the texture (SD) of Zw from Fig. 8b.

  • View in gallery

    (a) Doppler velocity field of the storm at 2346:46 UTC 31 May 2013 (as in Fig. 8). The number of autocovariance samples for velocity estimates is 19 (one less than the sequence length of 20), and all samples are uniformly weighted (i.e., Fixed Rec window). Data with SNR ≥ 10 dB is displayed. (b) As in (a), but the value υw in each radial is obtained by combining autocovariance samples from three adjacent radials and weighting these with the Step Tay100 power coefficients. Thus, the total number of autocovariance samples per velocity estimate is 57.

  • View in gallery

    Histogram of the difference field, υ (Fixed Rec window) in Fig. 10a minus υw (Step Tay100) in Fig. 10b.

  • View in gallery

    (a) Doppler spectrum width field of the 31 May 2013 storm corresponding to the velocity field in Fig. 10a. The field is computed by applying the autocovariance estimator R(2)/R(1) to the same sequences as in Fig. 10a (i.e., Fixed Rec window is used). (b) As in (a), except the value συw in each radial is obtained by combining autocovariance samples as done in Fig. 10b (Step Tay100 window is applied).

  • View in gallery

    Histogram of the difference field, συ (Fixed Rec window) in Fig. 12a minus συw (Step Tay100) in Fig. 12b.

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Considerations for Oversampling in Azimuth on the Phased Array Weather Radar

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  • 1 Advanced Radar Research Center, University of Oklahoma, Norman, Oklahoma
  • 2 School of Meteorology, and School of Electrical and Computer Engineering, and Advanced Radar Research Center, University of Oklahoma, Norman, Oklahoma
  • 3 National Severe Storms Laboratory, Norman, Oklahoma
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Abstract

When spectral moments in the azimuth are spaced by less than a beamwidth, it is called oversampling. Superresolution is a type of oversampling that refers to sampling at half a beamwidth on the national network of Doppler weather radars [Weather Surveillance Radar-1988 Doppler (WSR-88D)]. Such close spacing is desirable because it extends the range at which small severe weather features, such as tornadoes or microbursts, can be resolved. This study examines oversampling for phased array radars. The goal of the study is to preserve the same effective beamwidth as on the WSR-88D while obtaining smaller spectral moment estimate errors at the same or faster volume update times. To that effect, a weighted average of autocorrelations of radar signals from three consecutive radials is proposed. Errors in three spectral moments obtained from these autocorrelations are evaluated theoretically. Methodologies on how to choose weights that preserve the desirable effective beamwidth are presented. The results are demonstrated on the fields of spectral moments obtained with the National Weather Radar Testbed (NWRT), a phased array weather radar at NOAA’s National Severe Storms Laboratory (NSSL).

Corresponding author address: Dr. L. Borowska, Advanced Radar Research Center, University of Oklahoma, 120 David L. Boren Blvd., Norman, OK 73072. E-mail: lesya.borowska-1@ou.edu

Abstract

When spectral moments in the azimuth are spaced by less than a beamwidth, it is called oversampling. Superresolution is a type of oversampling that refers to sampling at half a beamwidth on the national network of Doppler weather radars [Weather Surveillance Radar-1988 Doppler (WSR-88D)]. Such close spacing is desirable because it extends the range at which small severe weather features, such as tornadoes or microbursts, can be resolved. This study examines oversampling for phased array radars. The goal of the study is to preserve the same effective beamwidth as on the WSR-88D while obtaining smaller spectral moment estimate errors at the same or faster volume update times. To that effect, a weighted average of autocorrelations of radar signals from three consecutive radials is proposed. Errors in three spectral moments obtained from these autocorrelations are evaluated theoretically. Methodologies on how to choose weights that preserve the desirable effective beamwidth are presented. The results are demonstrated on the fields of spectral moments obtained with the National Weather Radar Testbed (NWRT), a phased array weather radar at NOAA’s National Severe Storms Laboratory (NSSL).

Corresponding author address: Dr. L. Borowska, Advanced Radar Research Center, University of Oklahoma, 120 David L. Boren Blvd., Norman, OK 73072. E-mail: lesya.borowska-1@ou.edu

1. Introduction

The WSR-88D network is undergoing a life extension program to improve its capability and performance until at least 2030. Parallel to this effort, the National Oceanic and Atmospheric Administration is looking ahead for candidate replacements. One potent nascent technology is the agile-beam phased array radar (PAR), which could augment the weather-observing mission (Zrnić at al. 2007) and serve other purposes, such as air traffic control and monitoring, protection of terminals, and homeland security (Weber et al. 2007).

To be a viable replacement, the PAR should meet all WSR-88D performance standards. One of these is the superresolution mode. The term means that spectral moments are produced at an azimuthal spacing of half a beamwidth while the antenna is rotating at a rate at which statistical errors produced a beamwidth apart are acceptable. In the superresolution mode, the effective beamwidth of the antenna pattern is about 25% narrower than in the standard mode (Torres and Curtis 2007). This increase in azimuthal resolution comes at a price of larger estimate errors.

Simulations indicate (Brown et al. 2002) the superresolution (at 0.5° on the WSR-88D) extends by at least 50% the detection range of mesocyclones and tornadoes. Moreover, the estimates of the Doppler velocity difference between the extreme positive and negative values also increase, better quantifying the strengths of the circulation. Thus, the superresolution data enable forecasters to more easily recognize small hazardous features and to forewarn the public of eminent dangers. Because of increased statistical errors in estimates, the superresolution data are only directly processed by the mesocyclone detection algorithm. For other algorithms, the superresolution data are recombined to form radials of spectral moments spaced 1° apart.

In principle, the PAR could scan as a conventional radar. However, that would degrade the inherent advantages of no beam smearing, such as better filtering of ground clutter (its spectrum width is not broadened) and smaller statistical errors in Doppler variables. This paper introduces a way to obtain superresolution data on the PAR at the same volume coverage speeds as on the WSR-88D with the same azimuthal resolution but lower statistical errors.

The next section discusses the proposed superresolution method, including step scans, window functions, and effective beamwidth. Predicted errors in spectral moment estimates from the PAR in the superresolution mode are quantified in section 3. Fields of spectral moments obtained with the superresolution PAR mode are presented in section 4. Section 5 summarizes the paper.

2. Superresolution mode on PAR

We start with a brief depiction of the superresolution on the WSR-88D and then contrast it to how data are collected on the PAR. Figure 1 is a schematic of a continuous scan with overlapping samples. Here 2L is the number of samples within the continuous scan. The two convex curves represent the weighting functions applied to these samples. The horizontal ribbon with two arrows indicates the beginning and end positions of the beam center. The horizontal axis corresponds to angles that are related to time via the antenna rotation rate. Because the beamwidth of the WSR-88D antenna is 1°, it follows that the superresolution spectral moments are spaced half a degree apart.

Fig. 1.
Fig. 1.

Depiction of (top) continuous scan and (bottom) step scan. (top) The horizontal wide double-arrow bands depict the span of data used in computation of spectral moments. The number of returns (pulses) within the beamwidth θ1 is 2L. On the abscissa is time or angle for a continuously scanning beam. The curves above the ribbons represent window functions. (bottom) The single-arrow bands represent locations where the beam is pointing while L pulses are transmitted by the PAR. The size of the arrows insinuates the relative weight given to the samples from the three consecutive beam positions (spaced at θ1/2) for synthesizing a better estimate.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

The PARs scan by stepping the beam from position to position and pausing long enough to obtain a sufficient number of independent samples (bottom of Fig. 1), and thus constrain the estimate errors. The vertically oriented ribbons with arrows indicate the positions of the beam (Fig. 1). To maintain the same speed as in a continuous scan, the number of samples per position must be L. This raises the issue of how to combine the data from adjacent azimuths and preserve similar or better angular resolution compared to the conventional radar while maintaining lower estimate errors. Combining data reduces estimate errors. Moreover, the Doppler spectral resolution of the combined sequence (3L) is 3 times finer than the resolution of the L sample sequence. Consequently, filtering ground clutter from the longer sequence in the spectral domain (as on the WSR-88D; Siggia and Passarelli 2004) is more efficient than that from the shorter one. Spectral processing is also crucial for separating overlaid echoes on the WSR-88D using systematic phase codes (Sachidananda and Zrnić 1999); hence, finer spectral resolution improves this procedure as well. The spectral aspects of the PAR processing and challenges due to discontinuities in position and/or time are not considered herein.

a. Combining data and window functions

Two related ways of combining data from three overlapping azimuths are contrasted. In one, the power or autocovariance samples are weighted by a power data window. This is dubbed a continuous window combination. In the other, all data at a single azimuth have the same weight, but the composite estimate is a weighted sum of estimates from the adjacent azimuths. We call this “step window” and choose the window coefficients, so that the relative contributions from the three adjacent radials are the same as in the continuous window case. The ribbons with vertical arrows in Fig. 1 (bottom) depict this concept. The length of the ribbon is proportional to the weight and the position indicates azimuthal location. As illustrated in Fig. 1, the total number of samples in the “step scan” mode is 3L (50% larger than in the continuous scan mode), while the angular span from beam center to beam center over which data are collected is the same. This is possible because the side beam of one central position and the main beam of the adjacent central position share the same samples. What matters is the effective beamwidth (Doviak and Zrnić 2006), which in the stepped scan can be equal to or smaller than in the continuous scan.

Let Pli, Pci, and Pri be the single-power samples from the left, center, and right radials. Then, the application of a power window produces the following estimate:
e1
where the total number of samples M= 3L. The is the data window power coefficient (dm is the amplitude window coefficient).
We define a step window with three weighting coefficients for powers or autocorrelations from three adjacent radials to construct the value for the center radial. The averaged power estimate is obtained at a single range location from the three consecutive radials as
e2
where the power estimates from the left l, central c, and right r radials are averages,
e3
The weights a2 and b2 are computed as
e4
This is to match the power from the three radial positions with the power that would be obtained if a standard window were applied to the combined sequence. This is because the effective beamwidth for the two cases would be equal; hence, the resolution of the data field would not be affected by the clutter filter. The side coefficient a2 is considerably smaller than the center coefficient b2, so that the returns in the center beam are weighted most to maximize power efficiency.
The velocity is estimated from the weighted averaged autocorrelation,
e5
The spectrum width can be estimated from Eqs. (1) and (5) by the lag zero/one autocorrelation estimator, R(0)/R(Ts), or lag one/two estimator R(Ts)/R(2Ts), [Eqs. (A14) and (A15)].

b. Effective beam pattern

Window functions determine the effective beam pattern as well as the variance reduction achievable by averaging. Herein we choose the Taylor window (Tay) for comparative studies because its spectral sidelobes are uniform. The ratio of the main lobe to sidelobe levels can be easily changed, making it convenient for analysis and drawing conclusions. Sidelobe suppressions of 30, 54, and 100 dB are considered, and the corresponding window functions are designated with Tay30, Tay54, and Tay100, respectively. The reasons for these choices are explained in this section.

Assume a Gaussian-shaped main beam such that the two-way power pattern in the principal (azimuth) plane is
e6
where θ1 is the one-way half-power beamwidth and ϕ is the azimuth with respect to the beam center. For a continuously scanning antenna, the effective two-way power pattern is defined in Doviak and Zrnić [2006, Eq. (7.31) and Fig. 7.24] and Zrnić and Doviak (1976). In the step scan case, the effective two-way antenna pattern is the sum of weighted by the window function patterns from the three directions. Assuming the pattern as in Eq. (6) and defining the normalized angle x = ϕ/θ1, we present the effective two-way power patterns f4(x) versus ϕ/θ1 in Fig. 2. The equation for these patterns is
e7
where
eq1
It is valid for either a gradually changing window (i.e., dk is the standard window) or a step window (dk equals a or b). The intrinsic pattern has a two-way 6-dB normalized width of 1. This facilitates comparison with values on the WSR-88D because the WSR-88D’s beamwidth is 1°.
Fig. 2.
Fig. 2.

Effective main lobe of the antenna pattern. The relative position in azimuth is normalized by the one-way 3-dB width of the antenna pattern. Step Tay100 and Step Tay30 are applied to the three positions (−0.5, 0, and 0.5) of the stepped scan. The Scan Tay54 is applied to the continuous scan between −0.5 and 0.5 of the normalized position.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

The window types we consider are listed in Table 1, where the number of samples is as on the National Weather Radar Testbed (NWRT); the values are very close to the ones on the WSR-88D. Of all the Taylor windows with length 40, the Tay54 is the best fit in the least squares sense to the von Hann window of length 40. The von Hann is applied in the superresolution mode to the WSR-88D data. The Step Tay100 applied to 60 samples (20 from each adjacent azimuth) produces the same width (defined by the 6-dB points) as the Scan (continuous) Tay54. It is about 20% broader than the intrinsic one (i.e., stationary antenna). Thus, as far as effective beamwidth is concerned, the Step Tay100 applied to 60 samples and the Scan Tay54 applied to 40 samples can be fairly compared in terms of estimate errors.

Table 1.

Summary of the window applications that are quantified in section 3.

Table 1.

3. Errors in estimates

Comparisons of errors in reflectivity, velocity, and spectrum width are made between the stepped scan (such as on the PAR) and the continuous scan (such as on the WSR-88D) assuming the times to cover one scan in azimuth are equal. We consider a scenario applicable to the NWRT because data from that radar are presented in section 4.

The NWRT has a phased array antenna mounted on a rotating pedestal. For data collection, the antenna is positioned in the azimuth to cover the weather of interest, and electronic scanning is applied to capture the event. In any single direction, two consecutive sequences of pulses are transmitted. The first contains 12 pulses at the pulse repetition time (PRT) of 3 ms for reflectivity estimation, whereas the second contains 20 pulses at the PRT of 0.8 ms for velocity estimation (Table 2). Thus, the reflectivity estimates are free of range ambiguities and are used to determine potential overlaid and/or range ambiguous returns in the short PRT sequence. This long–short sequence of pulses on the NWRT serves the same purpose as the long PRT in the surveillance scan followed by the short PRT in the Doppler scan on the WSD-88D (Zrnić 2011). Moreover, on the two radars the transmitters are the same and the PRTs are very similar (the other parameters are in Table 2). Pertinent to our analysis are the accuracy requirements of estimates for the WSR-88D (Table 3), which the future PAR should satisfy. Thus, these requirements determine the parameters used in the comparisons.

Table 2.

Parameters of the NWRT and WSR-88D.

Table 2.
Table 3.

Accuracy requirements for spectral moments on the WSR-88D.

Table 3.

a. Reflectivity

The effects of Taylor window functions with 30, 54, and 100 dB of sidelobe suppression on the estimate errors are examined. The 30-dB window is included for reference to show the tendencies in effective beamwidth and errors. Furthermore, this window type would be an efficient choice for some radar systems in which spectral spurious components are up to 30 dB below signal components; hence, a more aggressive window is not beneficial. The effect of these windows on the equivalent number of independent samples in the case of white noise power is presented in Fig. 3 [from Eq. (A9)]. It is evident (Fig. 3) that given the same window type, the step window produces about 8%–10% more independent samples and a proportional decrease in variance. The step window has a similar slight advantage if only the signal is present. The window with less sidelobe suppression produces a larger number of independent samples and is preferred. However, applications to spectral processing may favor windows with lower sidelobes.

Fig. 3.
Fig. 3.

Equivalent number of independent samples for power estimation of pure noise. The numbers in the legend are the sidelobe levels (spectral) of the Taylor window. Dashed lines represent results for continuous window, and full lines are for step-equivalent window.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

To quantify the advantage of the step scan we present the standard error of reflectivity estimates in Fig. 4. The results are obtained using theoretical expressions for the variance of estimates [Eq. (A8)]. On the abscissa is the total number of available samples M = 3L over the three beam positions. Here and throughout the paper M indicates the sum of the number of samples from three adjacent positions. Thus, for reflectivity M = 3 × 12 = 36 and for velocity it is 60 (see Table 1). The total M is actually used only in the Step Tay100 case; in all other cases, the appropriate numbers of available (and used) samples is smaller, as indicated in the legend. For example, the top curve, labeled Fixed Rec M/3, is for a single-beam position on a PAR with M/3 power samples. This is the worst among the examined possibilities, as the dwell time is shortest. The Scan Tay54 with 2M/3 samples is for the continuous window applied to power samples obtained from contiguous azimuthal locations within one beamwidth while the beam is scanning. It illustrates what is achievable on the WSR-88D in the superresolution mode and in the Step Tay100 scan of the NWRT, which has the same “effective beamwidth” (see Fig. 2). The latter achieves significantly lower estimate errors (dashed red curve in Fig. 4). The Scan Rec is included for reference. It produces the smallest errors but applies to radial spacing of one beamwidth in regular scans if clutter is not filtered and there is no phase coding for mitigation of range ambiguities.

Fig. 4.
Fig. 4.

Standard deviation of reflectivity estimates vs the total number of samples available from the three-beam position M = 3L. The curves are for the following conditions. The Rec M/3 means a single stationary beam collecting M/3 power samples. The Scan Tay54 2M/3 is for a scanning beam over a 1° interval and collecting 2M/3 power samples. The beam displacement has been accounted according to Eqs. (A6), (A7), and (A8). Step Tay100 M is for a combined estimate from the three pointing directions (L = M/3 estimates from each) and the decorrelation due to the three beam positions is accounted for via Eqs. (A10), (A11), and (A8). The Scan Rec 2M/3 is for a scanning beam over a 1° interval collecting 2M/3 power estimates, and the beam displacement is accounted for via Eqs. (A5) and (A6). The assumed SNR = 10 dB and the spectrum width συ = 1 m s−1.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

The values in Fig. 4 are for the SNR = 10 dB and συ of 1 m s−1. At this συ, the errors are larger than those at the συ = 4 m s−1 specified by the WSR-88D requirement (Table 2). At συ = 4 m s−1, any one of the four windows produces estimates that exceed the requirement. The requirement applies to the average over four samples in range and because our plot is valid for one sample, the curves should be divided by two for comparison with the value applicable to the WSR-88D. Thus, with 24 samples (two-thirds of 36) the four-sample average of the Scan Rec sequence produces a standard deviation SD(Z) smaller than 1 dB; whereas, for the other modes, the errors are larger. But in the superresolution processing on the WSR-88D, the von Hann equivalent to Scan Tay54 mode is used (full curve in Fig. 4) and it has noticeably poorer performance than the Step Tay100 at all spectrum widths. The reasons for using the von Hann equivalent are better clutter filtering and finer lateral resolution than possible with the rectangular window.

In conclusion the PAR with a step window can have the same effective beamwidth as the WSR-88D and can produce better estimates of reflectivity in the same volume update time. This is because it utilizes more independent samples.

b. Velocity

We next compare the standard deviations of velocity estimates as functions of the number of samples and radar parameters corresponding to the NWRT (Table 2). The fixed total number of samples M = 60 implies 20 returns from a discrete position of the PAR beam and 40 returns in the case of a scanning beam (like WSR-88D) over a beamwidth interval. We make relative comparisons between the proposed superresolution mode for the PAR and the superresolution mode as on the WSR-88D, but we assume the PAR parameters for both. The signal-to-noise ratio is set to 8 dB to match the WSR-88D requirement. The results are obtained using theoretical expressions for the variance of estimates [Eq. (A13)].

The estimates of velocity in the case of Fixed Rec and 20 returns have the largest errors (Fig. 5). Continuously sampled data weighted with the Tay54 window over 40 returns yield almost the same errors. Continuous sampling over one beamwidth and uniform weight (Scan Rec 2M/3) results in the smallest SD, whereas processing on the PAR (Step Tay100 M) produces errors that are between those of the two continuously scanning cases.

Fig. 5.
Fig. 5.

SD of velocity estimates vs the spectrum width and for the same conditions as in Fig. 4. The beam displacement in the Scan Tay54 and Scan Rec has been accounted for by inserting Eqs. (A6) and (A7) into Eq. (A13). For the Step Tay100 M curve, Eqs. (A10) and (A4) are combined into Eq. (A6), and then Eqs. (A6), (A7), and (A11) are inserted into Eq. (A13). The SNR = 8 dB.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

Considering the requirement for velocity estimate errors (Table 3), the implications of the result in Fig. 5 are as follows. The requirement is almost satisfied (it is off by 5%) for the scanning radar and processing of uniformly weighted 40 samples spread over a beamwidth (Scan Rec 2M/3 = 40, in Fig. 5); the scan has been designed to satisfy the requirement for the actual dwell time on the WSR-88D. The Step Tay100 is off by 23% (error of about 1.23 m s−1). This compares to the 45% larger error (1.45 m s−1) of Scan Tay54 applicable to the WSR-88D.

The conclusion from this analysis is that, the superresolution on the PAR achieved by application of the Step Tay100 window on data from three consecutive positions spaced half a beamwidth apart produces about 20% smaller errors than in the superresolution mode on a conventional radar. This is a definite advantage of the agile beam with step scan over the mechanically steered antenna.

c. Spectrum width

The recently accepted “hybrid” spectrum width συ estimator on the WSR-88D uses one of the autocorrelation ratios: R(0)/R(1), R(1)/R(2), or R(1)/R(3) (Meymaris 2012). At low to moderate spectrum widths, the R(1)/R(2) is chosen [Eq. (A15)], while at larger widths the choice is the R(0)/R(1) estimator [Eq. (A14)]. At very narrow spectrum widths, the R(1)/R(3) estimator is used. At the συ and SNR specified in the accuracy requirement (Table 3), theory predicts that the R(1)/R(2) estimator is better than the R(0)/R(1) estimator (see Doviak and Zrnić 2006, Figs. 6.6 and 6.7). At somewhat larger widths starting at the crossover point, συx = 5 m s−1 [after Eq. (A15)], the R(0)/R(1) estimator produces smaller errors. Because συx is close to 4 m s−1 for which the requirement is stipulated, we have evaluated both estimators for the same window types, as in the cases of the reflectivity and velocity estimators. The errors (Figs. 6, 7) are obtained from simulations (appendix, section c).

Fig. 6.
Fig. 6.

SD of spectrum width estimates obtained from R(0)/R(1) vs the spectrum width. The parameters are as in Fig. 5, but the SNR = 10 dB. The curves depict the results from simulations.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

Fig. 7.
Fig. 7.

SD of spectrum width estimates obtained from R(1)/R(2) vs the spectrum width. The parameters are as in Fig. 6. The curves depict the results from simulations.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

Examination of Fig. 6 reveals that the Step Tay100, R(0)/R(1) estimator outperforms the Scan Tay54 at συ larger than about 6 m s−1. None of the estimators except the Scan Rec meets the requirement (at the συ = 4 m s−1). This changes if the R(1)/R(2) estimator is applied (Fig. 7). Then the Step Tay100 (applicable only to PAR) also meets, albeit barely, the requirement, demonstrating again the advantage of the agile beam. Note, that at συ less than about 3 m s−1, the Step Tay100 performs a bit better than the Scan Tay54.

The fact stands that in the superresolution mode, the hybrid spectrum width estimator satisfies the requirement if the Step Tay100 window is applied to the PAR data. If the Scan Tay54 window is applied to the WSR-88D data, then the hybrid spectrum width estimator does not satisfy the requirement.

4. Examples of data from a supercell

Select procedures described in section 3 are applied to the data collected with the NWRT. We compare the application of the Fixed Rec window on a single radial (M/3 samples) with applications of the Step Tay30, Step Tay54, Step Tay100, and continuous Tay100 each on three consecutive radials and for reflectivity estimation. The continuous Tay100 is a sequence of power coefficients applied to the sequence of power samples from the three consecutive beam positions. Comparison among the three step windows is meant to show the influence on the spectral moments as the windows’ width changes (windows with lower spectral sidelobes are narrower in the time domain). The Step Tay54 is included because it is similar to Scan Tay54, whose proxy (von Hann) is applied in the WSR-88D scanning mode, but on two-thirds of the number of samples used here. Inclusion of the continuous Taylor is to show its marginal inferiority to the Step Taylor window. Comparison of the Fixed Rec on the single radial with the Step Tay100 shows PAR’s improvement if data from its three radials are combined.

The effects of three window types (Fixed Rec, Step Tay54, and Step Tay100) on the velocity and spectrum width fields are also examined. The relative outcomes of the Step Tay30 and Tay100 windows on the Z field are similar to the outcomes on the υ and συ fields and are hence not considered in sections 4b and 4c.

The mode of operation summarized in Table 2 was used to collect data from a supercell storm that occurred on 31 May 2013 and produced a devastating tornado in El Reno, Oklahoma (Atkins et al. 2014; Burgess et al. 2014; Wurman et al. 2014). The spacing of the radials in the azimuth was at half a beamwidth. The large leading storm (maximum Z of close to 70 dBZ) is depicted in Fig. 8a. It produced the tornado, but it dissipated 3 min before this field was recorded. The reflectivity (Fig. 8a) exhibits a double hook echo (x = −60 km, y = 20 km; and x = −70 km, y = 20 km) where trailing storms formed. Remnants of circulation and strong convergence are centered at x = −22 km, y = 22 km, as can be discerned in the velocity field (Figs. 10a,b). At ranges within 20 km of the radar, ground clutter is evident as the clutter filter is not applied.

Fig. 8.
Fig. 8.

(a) Reflectivity field of the storm at 2346:46 UTC 31 May 2013. Azimuthal spacing of radials is half a beamwidth, and elevation angle is 0.51°. The number of samples for reflectivity estimates is 12, and all samples are uniformly weighted (i.e., Fixed Rec window). Data with SNR ≥ 10 dB is displayed. (b) As in (a), but the value Zw in each radial is obtained by combining powers from three adjacent radials and weighting these with the Step Tay100 coefficients. Thus, the total number of samples per estimate is 36.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

a. Effects on the reflectivity field

We use Eq. (5) to compute the power in the central direction and compute the coefficients α and b in Eq. (4) to be equivalent to Taylor windows. We compare the averaged with the original Z field (from single radials with no window), quantify the difference between the two, and examine the texture of these fields. In Fig. 8b is the Zw field (w indicates windowed) obtained after application of the Step Tay100 window. It has a visibly smoother appearance, yet the fine structures (the encircled reflectivity core and few pockets of 60-dBZ values) are well preserved. The histogram of the difference between the original Z field (Fig. 8a) and the Zw field (Fig. 8b) is shown in Fig. 9a. It has a relatively narrow but skewed distribution. This asymmetry is primarily caused by the nonlinear transformation of powers into logarithmic units (dBZ) and conforms to theoretical prediction. The median is −0.08 dB and about 90% of points are within ±1 dB. Similar comparisons can be made if the other two step windows are applied.

Fig. 9.
Fig. 9.

(a) Histogram of the difference field, Z (Fixed Rec window) in Fig. 8a minus Zw (Step Taylor100) in Fig. 8b. (b) Histogram of the texture (SD) of Zw from Fig. 8b.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

We define texture as a 21-point running SD along range of the reflectivity field. This corresponds to a range interval of 1.26 km because the spacing of samples is 60 m (Table 2). The median textures after application of the four windows are listed in Table 4. SNR thresholds of 10 and 20 dB illustrate the SNR effects. Fixed Rec is applied to the 12 available samples from a single-beam position (for Z; Table 2). The other windows are applied to power samples from three consecutive positions in the azimuth.

Table 4.

Median texture (SD) of the Z field.

Table 4.

The results in Table 4 depict a progressive increase in SDs starting from the Step Tay30 case and moving to the aggressive Step Tay100 case. Note that the ratio of median SD(Z)s between the Fixed Rec and Step Tay100 case in Table 4 (SNR ≥ 10 dB) is 2.02/1.74 = 1.16. The ratio of SD(Z) in Fig. 4 (M = 36 points total) is 2.57/2.17 = 1.17. This agreement confirms the relation between these two processing options, suggesting that the other two window-processing applications are correctly quantified in Fig. 4. The graph is valid at SNR = 10 dB and the spectrum width of 1 m s−1, whereas the Z field is for SNR ≥ 10 dB with no restrictions on the spectrum width. The median spectrum width for these data is 2.7 m s−1 (section 4c), which is considerably larger than 1 m s−1 for which the graph in Fig. 4 is valid. At the 2.7 m s−1 spectrum width, the statistical fluctuation of estimates for the Fixed Rec and Step Tay100 windows [from Eq. (A8)] are 1.67 and 1.42 dB. These are smaller than the 2.02 and 1.74 dB estimated from the data (Table 4 at SNR ≥ 10 dB) for these two windows. The excess in measured values is attributed to small-scale fluctuations in the Z field. The theoretical ratio of SD(Z) at the spectrum width of 2.7 m s−1 is 1.67/1.42 = 1.18, again very close to the measured value and the value predicted at the smaller spectrum width of 1 m s−1. Thus, the use of the Step Tay100 window on the reflectivity data produces the expected decrease in the standard deviation of estimates but also smooths small-scale reflectivity features. The median values of SD(Z) at the SNR ≥ 20 dB in Table 4 are smaller than at the SNR ≥ 10 dB because the effect of noise is smaller. The trend of the SDs versus the window type is the same for the two SNR thresholds.

The histogram of the texture (Fig. 9b) is bimodal. The well-pronounced peak at small values of texture is primarily due to the weather signal, whereas the secondary wide mode is caused by ground clutter and possibly other nonmeteorological scatterers. This is because the clutter returns along range are almost uncorrelated. The spread is similar to the one in the running difference squared of reflectivity, which has been one criterion for identification of ground clutter (Hubbert et al. 2009).

b. Effects on the Doppler velocity field

We next examine the Doppler velocity field (Fig. 10a) estimated after application of the Fixed Rec (no window, single radial with 20 samples; Table 2) and the Step Tay100 window (Fig. 10b). It is evident that the latter field is smoother (see textures, Table 5). The S-like zero Doppler curve (within the ellipse in Fig. 10) indicates strong convergence close to the radar (bottom of the S curve), then some rotation (middle of the curve), and again tight convergence with a miniscule convergence couplet at the top of the S curve. This S curve is delineated equally well in both images (Figs. 10a,b), confirming that the reduction in azimuthal resolution by the step averaging is not detrimental to the recognition of the feature.

Fig. 10.
Fig. 10.

(a) Doppler velocity field of the storm at 2346:46 UTC 31 May 2013 (as in Fig. 8). The number of autocovariance samples for velocity estimates is 19 (one less than the sequence length of 20), and all samples are uniformly weighted (i.e., Fixed Rec window). Data with SNR ≥ 10 dB is displayed. (b) As in (a), but the value υw in each radial is obtained by combining autocovariance samples from three adjacent radials and weighting these with the Step Tay100 power coefficients. Thus, the total number of autocovariance samples per velocity estimate is 57.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

Table 5.

Median texture (SD) of the Doppler velocity field.

Table 5.

The histogram of υυw (the subscript w stands for the field after application of the Step Tay100) in Fig. 11 indicates very good agreement between the two fields. The median difference is smaller than 1 m s−1 and most values are within −1 to 1 m s−1.

Fig. 11.
Fig. 11.

Histogram of the difference field, υ (Fixed Rec window) in Fig. 10a minus υw (Step Tay100) in Fig. 10b.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

Next we compare the error of estimates at the spectrum width of 2.7 m s−1 (median of the field; see section 4c) and SNR ≥ 10 dB with the textures (Table 5) produced after application of the Fixed Rec and Step Tay100 windows. The values of errors in velocity estimates [obtained from Eq. (A13)] are 1.07 m s−1 (Fixed Rec) and 0.91 m s−1 (Step Tay100). These values are about 50% smaller than the corresponding textures (1.51 and 1.24 m s−1, respectively; Table 5). The excess texture over statistical errors is likely due to the spatial variations of the velocity field, which include effects of clutter and occasional aliasing. These variations are comparable contributors (in the rms sense) to the texture. Hence, the reduction of fluctuation (1.51/1.24 = 1.22) is larger than what it would be (1.07/0.91 = 1.18) if an intrinsically flat field were subjected to the same procedures. Clearly, the reduction in errors is appreciable, while the gradients in the azimuth (such as across the rotation signature) are faithfully preserved.

c. Effects on the spectrum width field

We present the συ field obtained from the R1/R2 estimator because it outperforms the other lag estimators in the range from about 2 to 5 m s−1 (Meymaris et al. 2009) and the median here is 2.7 m s−1. The median is larger than reported in three isolated severe storms (1.4–1.9 m s−1) by Fang et al. (2004), but is in line with the 2.9 m s−1 measured by these authors in a severe storm cluster.

Notable in Figs. 12a,b (encircled) is the S-like feature of increased συ coincident with the same feature of near-zero Doppler velocities in Figs. 10a,b. Because the feature delineates the maximum in radial and azimuthal shears of Doppler velocities, we conclude that the increased συ is mostly due to shear. Close examination of Figs. 12a,b suggests that the Step Tay100 better depicts this shear zone as well as the rest of the image. This attests to the effectiveness of the PAR with the proposed processing (Step Tay100) to achieve superresolution.

Fig. 12.
Fig. 12.

(a) Doppler spectrum width field of the 31 May 2013 storm corresponding to the velocity field in Fig. 10a. The field is computed by applying the autocovariance estimator R(2)/R(1) to the same sequences as in Fig. 10a (i.e., Fixed Rec window is used). (b) As in (a), except the value συw in each radial is obtained by combining autocovariance samples as done in Fig. 10b (Step Tay100 window is applied).

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

The median difference of the fields συσυw is 0.05 m s−1 and the histogram is in Fig. 13. The conclusion is consistent with the differences in the Z and υ fields (Figs. 9a, 11). The smoothed fields have no perceptible bias, and variations with respect to the original (i.e., Fixed Rec) fields are insignificant.

Fig. 13.
Fig. 13.

Histogram of the difference field, συ (Fixed Rec window) in Fig. 12a minus συw (Step Tay100) in Fig. 12b.

Citation: Journal of Atmospheric and Oceanic Technology 32, 9; 10.1175/JTECH-D-15-0018.1

The textures of the spectrum width fields are listed in Table 6. We compare the values for SNR ≥ 10 dB with statistical errors of estimates plotted in Fig. 7, which are 1.25 m s−1 (Fixed Rec) and 0.93 m s−1 (Step Tay100). These are very close to the corresponding textures (Table 6) of 1.18 and 0.97 m s−1, respectively, suggesting the statistical errors are the principal contributors to the texture.

Table 6.

Median texture (SD) of the spectrum width field.

Table 6.

5. Summary and conclusions

This paper presents a way to obtain superresolution data with a PAR at volume coverage speeds and the same azimuthal resolution as on the WSR-88D but with lower statistical errors. On the WSR-88D the superresolution is achieved by overlapping 50% of time series data and applying a window function to these data before computing spectral moments. Therefore, radial arrays of moments are produced at half-beamwidth spacing from data collected over beamwidth-wide azimuthal sections. The window function serves to narrow the effective beamwidth and where needed to be part of ground clutter filter.

For the PAR with step scans, we have proposed a weighted average of powers and autocorrelations from three consecutive radials as follows. We have defined the Step Taylor weighting by uniform weights at each radial. The middle radial has the strongest weights and the two lateral radials have equal weights. The weights are chosen so that the average power in a specific radial equals the average power that would be obtained with the part of the true Taylor window sequence applied to that radial.

We have assumed that the total number of samples M from the three consecutive radials is constant. Then we made comparisons of estimate errors that result if 1) PAR estimates are from a radial with M/3 uniformly weighted samples; 2) PAR estimates are from three radials, and the weighting of powers and autocorrelations is with the three Steps Tay100 coefficients, one per radial; 3) WSR-88D estimates are from 2M/3 samples uniformly weighted over one beamwidth-wide azimuthal sector; and 4) as in point 3 but the powers and autocorrelations are weighted with the Scan Tay54 window. This window is very close to the von Hann window used on the WSR-88D. The Step Tay100 window applied to the three radials of the PAR (M samples total) produces the same effective beamwidth as the Scan Tay54 applied to the WSR-88D (2M/3 samples total).

The errors of the three spectral moments are smaller on the PAR with the Step Tay100 than the corresponding errors on a conventional radar with the Scan Tay54 window. For example, the superresolution on the PAR has about 20% smaller errors of velocity estimates than the superresolution mode on conventional radar. Furthermore, in the superresolution mode on the PAR with the Tay100 window, the spectrum width estimator satisfies the WSR-88D requirements. At the same effective azimuthal resolution as on the PAR, the WSR-88D does not satisfy this requirement. This is a definite advantage of the agile beam with step scan over the mechanically steerable beam.

Our conclusion comes from theoretical computations whereby the pertinent radar parameters are those of the NWRT. Except for the beamwidth, these parameters are very similar to the parameters on the WSR-88D.

We have demonstrated the proposed superresolution processing on the spectral moment fields obtained by the NWRT. Visual inspection of the three spectral moment fields before and after application of the superresolution processing revealed no significant degradation of meteorological features. The histograms of the differences between the original and processed fields were mostly contained within ~1 dB and ~1 m s−1 for reflectivity and velocity/spectrum width, respectively. The textures of the superresolution fields indicate the decrease with respect to the texture of original fields as expected from theoretical formulations.

In summary the proposed superresolution processing on the PAR is superior to similar processing on conventional radar because it produces smaller errors of estimates and avoids decorrelation due to changes in beam position. The step scan is very flexible and adaptable to adjust weights for optimal performance (smallest errors) as a function of SNR, the number of samples, and/or spectrum width. This adjustment remains to be explored.

The challenging issue of spectral processing and clutter filtering is postponed to the future. As the phased array steps its beam to different positions, there will likely be time gaps between contiguous chunks of samples from each position. Thus, the spectral processing of the time series data across multiple beam positions would be done over an effectively uneven PRT sequence. We are testing a few procedures to do it.

Acknowledgments

The authors acknowledge discussions with Drs. C. Curtis, R. Doviak, I. Ivic, S. Torres, and R. Palmer; Mr. D. Prignitz; Mr. E. Forren; and Ms. H. Guifu. The NSSL’s engineering team and the NWRT Software Upgrades Project (MPARSUP) implemented controls and processing on the NWRT. Anonymous reviewers provided extremely useful suggestions. The two lead authors were supported by NOAA Grant NA11OAR4320072.

APPENDIX

Variances of Spectral Moments

a. Reflectivity estimates

Derivations for uniformly spaced samples are in Melnikov (2004, p. 64) and in Doviak and Zrnić (2015). Assume the reflectivity factor is computed from the power estimate by subtracting the known noise power N; that is, the signal power estimate . Then the standard deviation of becomes , where S is the mean signal power [Doviak and Zrnić 2006, Eq. (6.13b)]. The variances are related by . The power is estimated from the weighted average of M samples,
ea1
where is the power weighting coefficient. The vector of power coefficients D is
ea2
For the weather signal with noise, the expected value of the single-power measurement Pm is E(Pm) = and its variance is .
Assume scatterers are locally homogeneous over lateral regions corresponding to at least two beamwidths, and the mean Doppler velocity and the intrinsic spectrum width do not change. Thus, the parameters that determine the accuracy are decorrelation (due to change in beam position), the number of samples, and the window type. Consequently, the results do not depend on the range or the pointing direction of the PAR beam. The returned in-phase and quadrature phase components (I, Q) are uncorrelated Gaussian variables, but the complex time samples form a correlated process. Herein, the correlation coefficient of power samples is separated into the part due to the change in beam position during the dwell time ρb and the part ρs due to weather signal fluctuation. These two parts are expressed as [see Doviak and Zrnić 2006, Eq. (6.11)]
eaa3a
eab3b
where Δ is the spacing in azimuth between two beam positions, tm is time separation, συ is Doppler spectrum width, and θ1 is the 3-dB one-way antenna power pattern width.
Assuming that the beam position changes during dwell time, the correlation coefficient of powers becomes ρ(tm) = ρs(tm) ρb(tm). Consider M consecutive power samples taken at the same range delay so that tm = 0, Ts, 2Ts,…, (M−1)Ts. In absence of beam motion the correlation coefficient matrix is
ea4
and it is implicit that the integers in the arguments of the correlation coefficients indicate time increments in steps of Ts. Similarly, the matrix of correlation coefficients associated with change in beam position is
ea5
Then the matrix of correlation coefficients that accounts for both change in beam position and reshuffling of scatterers is expressed as
ea6
where means element-by-element multiplication of the two matrices; that is, the i, j element is cij = bijrij.
Define the window matrix of the weighting function T as
ea7
The elements of multiply the variances of power and autocorrelations.
The variance of estimate Eq. (A1) can be expressed as
ea8
In absence of signal, Eq. (A8) reduces to the third term, which defines the equivalent number of independent samples MI,
ea9
used to generate Fig. 3.
Next we evaluate variance for the three stepped beam positions. The adjacent positions are separated by Δ degrees in azimuth. It is assumed that L returns are processed from each position so that M = 3L and that the PRT is uniform across the three beam positions. To compact notation, we define the all-ones L × L square matrix L so that the beam displacement matrix is expressible as
ea10
For this three-beam position case, two window types were considered. In one, the amplitude coefficients dk have the Taylor tapering and is given by Eq. (A2). The other has the power weight a2/L for the left and right beam positions and b2/L for middle position. Therefore,
ea11
The a2 and b2 are given by Eq. (4) in which dm is the Taylor window coefficient. For the 60-term Tay100 window, a2 = 0.08 and b2 = 0.84. The variance of power estimates can be obtained using Eqs. (A10) and (A11) in Eq. (A8); this was done to generate the Step Tay100 M, PAR curve (Fig. 4).

If the L returns at one beam position are uncorrelated with the L returns at the adjacent position (as on the NWRT with two sequential PRTs; Table 1), the variance can be computed by inserting M/3 into Eq. (A8) and multiplying the result with (2a4 + b4). This, at small M (<20), produced negligibly lower variance than in the correlated case (uniform long PRT). These two cases are indistinguishable at M > 20.

b. Velocity estimates

We consider the pulse-pair estimator. Its variance is proportional to the variance of the autocorrelation function (Zrnić 1977). Hence, it suffices to compute this variance and inject it into the formula for the velocity variance.

A natural extension of the method applied to power samples is to use the power window coefficients on the samples of the autocorrelation at lag one R(1). Let the sequence of contiguous autocorrelation samples at lag one be represented as (R1, R2, …, RM-1), and the window power coefficients be as in Eq. (A7) but with one less term. Then the estimate of the autocorrelation is
ea12
All the Rs in Eq. (A12) are samples at lag one; the first L−1 of these are from the first radial, the ones in the second sum are from the central radial, and the ones in the third sum are from the third radial. The variance of velocity estimates is derived as in Zrnić (1977), thus
ea13

For comparison we considered the following.

1) PAR averaging from three beams spaced half a beamwidth apart

  1. The Taylor window sequence dm is applied to the M = 3L (60) time samples of (I, Q) voltages and then M−3 (57) autocorrelation samples at lag one are computed (omitted are the terms straddling beam transitions). The velocity is computed from the argument of the samples’ sum. The decrease in the terms of the autocorrelation matrix (because of the three beam positions) is accounted in Eq. (A10).
  2. The Taylor power window sequence is applied to the M−1 autocorrelation samples. Other aspects are as in (i). There was no perceptible difference between SDs obtained by the two approaches.
  3. The Step Taylor power equivalent window is applied to the autocorrelation sample sequence. The number of coefficients per left, central, and right radials is as in (i)—that is, 20 for power and 19 for autocorrelation at each beam position. Comparison between this approach (Step Tay100 with M = 60 samples) and the continuous Tay100 sequence applied across the three beam positions [as in (i)] revealed that the standard error of velocity estimates is smaller in the Step Tay100 case. The improvement is between 0.05 m s−1 (at συ = 1 m s−1) and 0.1 m s−1 (at συ= 10 m s−1), and supports the preference for the Step Taylor on the combined sequence.

In Fig. 5, the graph labeled Step Tay100 M, PAR has been obtained following the outlined procedure and Eq. (A12) in which is replaced with a2 in the first sum, b2 in the second sum, and a2 in the third sum. Then Eq. (A12) is substituted into Eq. (A13). The theoretical result has been verified by simulations.

2) Continuously scanning beam as on the WSR-88D

  1. Rectangular window is applied to 2M/3 autocorrelation samples assuming these are obtained in a continuous scan over one beamwidth. Accounting for beam smearing is accomplished by multiplying each term of with a corresponding term of as in Eq. (A6).
  2. Tay54 window over the 2M/3 samples [other details as in (i)].
The assumed (intrinsic) spectrum widths do not include displacement of the beam either over the three-beam position as on the PAR or over the one beamwidth swath as on the WSR-88D. These effects are added to the correlations via the outlined computation [Eq. (A6)] and appear in the variances as element-by-element products of the matrices and . The SDs are plotted versus the intrinsic spectrum widths. Thus, for the PAR pointing in a fixed direction, the measured and assumed spectrum widths are the same. But in the case of a continuously scanning beam (WSR-88D) or step scanning (PAR), the measured spectrum widths would be slightly larger than the assumed ones.

c. Spectrum width estimates

The R(0)/R(1) estimate of the spectrum width is [Doviak and Zrnić 2006, Eq. (6.27)]
ea14
where and are estimates of signal power and autocorrelation at lag one, respectively.
The R(1)/R(2) estimate of the spectrum width is [Doviak and Zrnić 2006, Eq. (6.32)]
ea15
There is a crossover point at συx dependent on the SNR and M, such that for συ larger than συx Eq. (A14) performs better and for smaller συ Eq. (A15) is better. For the NWRT parameters the συx is about 5 m s−1.
The variances of the estimates Eqs. (A14) and (A15) have been obtained by adapting the simulations of Zrnić (1975). It is assumed that the intrinsic spectrum width is constant in the region of the three beam positions. The increase (bias) in the spectrum width due to continuously scanning beam is (from Doviak and Zrnić 2006)
ea16
At the wavelength λ = 0.0937 m, Ts = 800 μs, and M = 40, the σac = 0.32 m s−1. In our figures we use the intrinsic spectrum width on the abscissa, but to the simulation routine we supplied the biased spectrum width because that is the one that affects the errors—that way a comparison of the results between the stationary beam case and scanning is realistic.
To simulate Step Tay100 we constructed three long (100 Mc samples) time series—x(n), y(n), and z(n). These have the same mean velocity and spectrum width but are uncorrelated. Then we computed s1 = (x + y)/√2, s2 = y, and s3 = (y + z)/√2. The 1/√2 is the correlation coefficient of signals coming from two beam positions separated by half a beamwidth. Therefore, this is the correlation coefficient between s1 and s2 as well as between s2 and s3. Then the following sequence (vector V) from the three positions is constructed,
ea17
where s1(1:L) = [s1(1), s1(2), s1(3), …, s1(L)] and the same explicit form holds for s2(L+1: 2L) and s3(2L+1: 3 L). The power weighting (Step Tay100) is applied to the powers or autocorrelations of the signal sequence Eq. (A17). The result for the estimator Eq. (A14) is in Fig. 6 (Step Tay100 PAR). We have verified this simulation procedure on velocity estimates (Step Tay100 PAR), and these compare very well with the theoretical result in Fig. 5, confirming that the procedure is correct. The results for the estimator Eq. (A15) are in Fig. 7. We have also computed variances in case the returns from the adjacent stepped beams are uncorrelated as on the NWRT because a 12-pulse-long PRT sequence precedes the 20-pulses short PRT sequence at each azimuth (Tables 1 and 2). The procedure to obtain these is essentially the same as for the contiguous trains of pulses (PAR) except the s1, s2, s3 in Eq. (A17) are replaced with uncorrelated sequences x, y, z. No perceptible difference was found between the variances obtained by these two procedures.

REFERENCES

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