1. Introduction

The range velocity ambiguity is one of main challenges in weather radar observations. In cases where a uniform pulsing scheme is used, the maximum range, r_a, is related to the maximum unambiguous velocity, υ_a, as υ_ar_a = cλ/8, where c is the speed of light and λ is the radar wavelength. An increase in the pulse repetition time (PRT) results in an increase of the maximum unambiguous range, but decreases the maximum unambiguous velocity. Therefore, the uniform sampling of radar signals always implies a trade-off between an unambiguous Doppler velocity and a maximum range.

One of the solutions to this problem is to choose a pulsing scheme that alternates between two different pulse intervals, T₁ and T₂ (Skolnik 2001). If the pulse pair processing is applied to a staggered PRT observation the unambiguous Doppler velocity is determined by the PRT difference (Zrnic and Mahapatra 1985) and the maximum unambiguous range is related to the sum of the pulse repetition times.

Because of a nonuniform sampling, caused by the staggered PRT pulsing scheme, standard clutter suppression techniques cannot be directly applied to staggered PRT sequences. To overcome this problem, Cho and Chornoboy (2005) have defined a finite impulse response ground clutter filter with time-varying coefficients that can be applied to staggered PRT sequences. Sachidananda and Zrnic (2000) have introduced a spectral domain clutter filtering to solve this problem. However, those techniques suffer from velocity errors in certain Doppler frequency bands.

Sato and Woodman (1982) have shown that by applying a nonlinear least squares fit to calculated Doppler spectra, one can separate clutter and precipitation echoes. The procedure relies on the assumption that clutter and precipitation signal spectra follow a Gaussian functional form. Using the same assumptions, Boyer et al. (2003) have proposed a method, based on the maximum-likelihood approach, to separate precipitation echoes and ground clutter in the time domain. Nguyen et al. (2008) have examined the performance of this parametric time domain method for a variety of conditions and have shown that the method performs well even in cases of strong clutter contamination. Furthermore, Nguyen et al. (2008) have introduced a goodness of fit criteria that can be used to detect cases where precipitation Doppler spectra shapes deviate from Gaussian. It was proposed that in these cases a sum of two Gaussians can be used to approximate precipitation spectra.

In this study we extend Nguyen et al.’s (2008) method to the case of staggered PRT observations. The advantage of this approach is that both clutter filtering and spectral moments estimation are based on a parametric time domain model and therefore can be applied to a variety of waveforms that result in nonuniformly sampled radar signals (i.e., staggered PRT observations).

Based on radar signal simulations, error analysis of the proposed method is carried out for different observation scenarios. To demonstrate the performance of the method, radar data for two plan position indicator (PPI) observations using staggered PRT and uniform PRT pulsing schemes were collected by the Colorado State University–University of Chicago–Illinois State Water Survey (CSU–CHILL) radar. The parametric time domain method (PTDM) was applied to the staggered PRT observations and the spectral clutter filter (Siggia and Passarelli 2004) was applied to the data with the uniform PRT. It is shown that the results are comparable.

2. Staggered PRT observations

If a staggered PRT observation scheme is employed, the pulse repetition time alternates between two pulse intervals, T₁ and T₂, and the maximum unambiguous velocity, υ_a, is defined as

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where T₁ < T_2.

Zrnic and Mahapatra (1985) have suggested that since second trip echoes for two PRT arrive from different ranges and thus are uncorrelated, the maximum achievable range for the staggered PRT observations is r_a = c(T₁ + T₂)/2. However, the second trip echoes would result in a reduction of the quality of estimates, because their effect can be considered as an increase in the noise floor. Therefore, we will consider that the maximum unambiguous range is determined by T₁ and there are no echoes beyond this range.

Generally T₁ and T₂ are selected as multiples of a certain unit time, T_u. Zrnic and Mahapatra (1985) have shown that for the pulse pair estimation technique, the optimum T₁/T₂ ratio is 2/3. Therefore we have limited our study to this case only.

Despite obvious advantages of the staggered PRT pulsing scheme, its application is limited because of difficulties with clutter suppression. Sachidananda and Zrnic (2000) have proposed a spectral procedure for the ground clutter suppression. A calculated Doppler spectrum is a convolution of the true spectrum with a pulse transmission code. Because of the nonuniform sampling of radar signals, the clutter peaks appear at several frequency bands, defined by the waveform, and spectral moment estimates are compromised if a precipitation radial velocity lies close to those bands. The PTDM, on the other hand, uses a parametric time domain model that can directly be applied to a nonuniformly sampled sequence and should not suffer from such limitations.

3. PTDM methodology

Radar signals can be represented as a sum of individual signals backscattered from scatterers in a given radar resolution volume. It is shown that (Bringi and Chandrasekar 2001) the real and imaginary parts of a received signal are zero mean normal variables. Therefore, the vector of a complex sampled signal has a normal multivariate distribution. By rewriting the multivariate probability density function of the received complex voltage we get,

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where R̂_V = 𝗩𝗩^H is the sample covariance matrix. For staggered PRT observations the sample covariance matrix is given as

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Using an assumption that Doppler spectra of clutter and precipitation have a Gaussian shape, one can write an observed Doppler spectrum as follows:

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where P_p is the precipitation signal power, σ_p is the precipitation spectrum width, is the mean velocity of precipitation, P_c is the clutter power, σ_c is the clutter spectrum width, and N is the noise power. Here we have assumed that there could be more than one precipitation signal present, where K denotes a total number of precipitation echoes. Nguyen et al. (2008) have shown that deviations from the assumption of a single Gaussian-shaped precipitation spectrum can be detected. It was also proposed that in these cases it is advantageous to use the precipitation signal model consisting of two Gaussian spectra. This will be discussed in more detail in section 6 where an analysis of CSU–CHILL observations is presented.

Given the spectral representation (4), one can write the covariance function of the measured signal as (Bringi and Chandrasekar 2001)

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where λ denotes the radar wavelength.

And the negative log-likelihood function, L, can be written as follows (Wooding 1956):

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where θ = [σ_c, P_c, , σ_p, P_p, N] is the vector of unknown parameters, |·| is the determinant operator and tr(·) is the trace operator, R̂_V is given by expression (3), and elements of the matrix 𝗥(θ) are given by the expression (5). The parameter vector θ can be estimated by minimizing L(θ):

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4. PTDM implementation

b. Non-Gaussian precipitation spectrum

Janssen and van der Spek (1985) have observed that, at least in 25% of precipitation observations, weather echo spectrum shapes deviate from Gaussian. Nguyen et al. (2008) have shown that in those cases mean velocity estimates using the pulse pair and PTDM can be significantly different. To minimize the difference one can introduce a second precipitation echo into the model.

To detect cases where the assumption of a Gaussian-shaped precipitation spectrum fails, one can use a combination of a normalized trace:

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and the R² goodness of fit parameter based on the imaginary part of the autocovariance function (5) (Nguyen et al. 2008):

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where 〈·〉 denotes a mean value. The goodness of fit parameters are evaluated at the estimated parameters following the procedure given in section 4a. Both parameters are expected to deviate from unity in cases where the fit is bad and the assumption of a Gaussian-shaped precipitation spectrum fails. We can define four typical cases; see Nguyen et al. (2008) for more detailed explanations:

• R² > 0.9 and |tr_norm − 1| ≤ 0.2—Gaussian-shaped spectrum assumption is valid;

• R² < 0.9 and |tr_norm − 1| ≤ 0.2—the assumption fails. The PTDM fitted spectrum extends over the complete precipitation spectrum;

• R² > 0.9 and |tr_norm − 1| > 0.2—the assumption fails. The PTDM fits the stronger part of the precipitation spectrum; and

• R² < 0.9 and |tr_norm − 1| > 0.2—no convergence.

In all cases, except for the first one, a second precipitation echo can be introduced to minimize velocity errors. The seed values in these cases are found by using the following procedure: the spectrum widths for both precipitation echoes are set to be half of the estimated spectrum width, but not less than 2 m s⁻¹. The velocity seed values are found as υ̂ ± σ_p/4. One of the echoes is assumed to be a stronger one with the power being equal to the estimated power; the other signal power is assumed to be 10 dB or less. Then for the given seed values the likelihood function is calculated. Two cases are considered here where the velocities of stronger and weaker echoes are interchanged. The appropriate combination is the one that has the smallest likelihood function. Finally, using these seed values, the Nelder–Mead simplex method is used to find spectral moments for two precipitation signals. The combined spectral moments, that describe the non-Gaussian precipitation spectrum, are found by applying the pulse pair estimation procedure to the combined autocorrelation function:

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5. Performance of the PTDM

To evaluate the performance of the PTDM, the radar signals were simulated for a variety of observation scenarios. The radar signal simulation procedure is based on the work of Chandrasekar et al. (1986). Radar signal sequences were generated for different values of signal-to-noise ratio (SNR) and clutter-to-signal ratio (CSR). Also, different values of mean velocity and spectrum width were used to generate precipitation spectra (see Table 2).

Since the main advantage of the staggered PRT waveforms is an increased maximum unambiguous velocity for a given maximum range, as compared to the uniform spaced pulse transmission, we focus our attention on the accuracy of the velocity estimates for a variety of conditions. The simulation-based study of the sensitivity of the method to the non-Gaussian precipitation spectra was carried out by Nguyen et al. (2008) and is not included in this study. It will be discussed in more details in section 6 where an analysis of the radar observations is presented.

6. Validation using CSU–CHILL observations

To illustrate the performance of the PTDM on radar measurements, the staggered PRT observations were carried out on 20 December 2006 by the CSU–CHILL radar. Data for two PPI plots were collected. First, measurements using the staggered PRT scheme with T₁ = 1 and T₂ = 1.5 ms were taken. Then, 1 min later, measurements using the uniform pulse transmitting scheme with a PRT of 1 ms were carried out. It is expected that the staggered PRT observations would yield an unambiguous velocity range from −52 to 52 m s⁻¹ and a half of that for the uniform PRT waveform.

The measurements were collected in a snowstorm event where reflectivity values did not exceed 35 dBZ and spectrum widths in excess of 7 m s⁻¹ were observed. An application of the staggered PRT waveform to large spectrum width events is traditionally more limiting (see Figs. 1, 2). One would also expect that deviations of precipitation Doppler spectra from Gaussian shape are more common in such cases. Therefore, it is believed that these observations are appropriate for the PTDM validation.

To gauge the PTDM performance, the spectral clutter filtering method similar to one described by Siggia and Passarelli (2004) was applied to the uniformly sampled observations. In Fig. 7 the resulting PPI plots are shown. One can observe that both PTDM and spectral filtering result in comparable clutter suppression rates. The uniform PRT observations exhibit velocity aliasing in the northeast quadrant that is resolved by applying the PTDM to the staggered PRT observations. The color scale from −32 to 32 m s⁻¹ is selected here to show the radial velocity fields in full detail while avoiding scale ambiguities.

Even though the velocity fields, obtained using the PTDM and pulse pair processing, appear to be similar, there are differences that can be observed for the ranges from 50 to 80 km and the azimuth angles from 180° to 210°. For these measurements it appears that PTDM is underestimating the radial velocities. It can be more clearly observed in Fig. 8 where estimated radial velocities are plotted for the azimuth angle of 210.5°. In Fig. 8 one can see that the PTDM underestimates the radial velocities, by as much as 2 m s⁻¹, for the ranges larger than 60 km.

By applying the goodness of fit criteria, described in section 4, we have observed that the assumption of a Gaussian-shaped precipitation spectrum frequently fails. It is more evident for the ranges larger than 60 km. In Fig. 9 the map of observations that violate the spectrum shape assumption is shown. To mitigate this effect, the precipitation model is changed to include two weather echoes, as described in section 4. Using the modified PTDM, new spectral moments are estimated. In the right column of Fig. 7, the retrieved reflectivity, velocity, and spectrum width fields are shown. It can be observed that the differences between the pulse pair and PTDM estimated velocities are minimized. This can also be observed in Fig. 8. In Fig. 10 histograms of the velocity differences are shown. For this figure both PTDM and pulse pair methods were applied to the data collected using the uniform PRT waveform. For both PTDM methods, the mean velocity difference is zero. The shapes of the histograms are different, however. The application of one echo PTDM to the data results in the bimodal histogram. By introducing a second echo into the retrieval, the bimodality is removed and the resulting velocity difference histogram became narrower. By comparing estimated reflectivity values shown in Fig. 7, one can see that the introduction of the second precipitation signal in the PTDM does not affect the clutter suppression capabilities of the method.

7. Discussion

In this paper it is shown that the parametric time domain method can be applied to the staggered PRT sequences. Based on the simulation study, we have observed that the performance of the proposed method is good even in cases of strong clutter contamination and that the velocity estimates are unbiased for all values of radial velocities. Even though the simulation study results are only valid for the cases where the assumption of the Gaussian-shaped precipitation spectrum holds, they are indicative of the clutter suppression capabilities for a variety of conditions.

It was also observed that the measured precipitation spectra deviate from the assumed Gaussian shape. By comparing the goodness of fit criteria to the spectrum width observations, we can conclude that the Gaussian spectrum assumption fails more often in cases where the spectrum width is large. This can be explained by wind shear effects that are more pronounced at larger distances.

If the deviations of the true spectrum shape from the modeled one are not taken into account, the estimated velocity errors can be as large as 2 m s⁻¹. In these cases a second precipitation echo, with a Gaussian-shaped spectrum, is introduced into the parametric model. It was shown that by using the modified PTDM, the errors in the velocity estimates are reduced. It should be noted that this problem does not influence the retrieved reflectivity values. Moreover, it appears that a precipitation Doppler spectrum can be described as a combination of two Gaussian curves. The observations presented here suggest that this model can be used to describe most observations.

It was observed that the assumption of the clutter spectrum shape is not very important for the performance of the method. By subtracting the mean signal, the statistical model (2) implies the use of zero mean signals; any possible DC components of the clutter are removed. The system instabilities, such as phase noise, are mitigated by including the noise power as a free parameter into the estimation procedure.

The main drawback of the method is its computation cost. It was not a focus of this study to investigate how computationally intensive the procedure is. Our preliminary analysis shows that the PTDM is about 10 times slower than the spectral filtering method. The addition of the second echo into the model increases the computation time by a factor of 2–3.