Multi-PRI Signal Processing for the Terminal Doppler Weather Radar. Part II: Range–Velocity Ambiguity Mitigation (2024)

Suppose we have a PRI sequence T₀, T₁, T₂, etc. Then the sum of all the out-of-trip signals aliasing into the first trip at range gate R_i and time index k is computed from the following equations:

where S is signal power. Here, L is set to the maximum integer that keeps r_j within maximum range of the long-PRI scan. This received pulse is marked clean if S_Oik < LP_n or S_ik > αS_Oik, where P_n is noise power and S_ik is first-trip signal at range gate R_i and time index k. The first condition corresponds to negligible range aliasing while the second condition allows some overlay power as long as the ratio of first-trip to out-of-trip powers exceeds α. The value of α is determined as follows.

a. Clutter filter problem

An additional complication to range-overlay protection is the ground clutter filter (GCF). Applying a GCF coherently across all PRI pulse sets convolves information from different pulses and destroys the independence of range aliasing between PRI sets. In other words, even if only one PRI set is contaminated by an overlaid signal, application of the GCF will mix some of this unwanted signal into the time series of all the other PRI sets. Therefore, the GCF should only be applied when absolutely necessary. The following is the procedure we use to minimize interference from the GCF for each range gate. First, apply the adaptive multi-PRI GCF (Part I) if the clutter signal is present in the long-PRI scan data and all pulses are clean from range aliasing or the estimated clutter power exceeds the average range-aliased power. Then if the power removed by the GCF is nonnegligible (i.e., greater than the noise power), use the filtered time series for further processing; otherwise, use the unfiltered data. Second, if there is nonnegligible clutter power present, but the multi-PRI GCF was not applied because of range-overlay contamination, then the means are subtracted from the in-phase and quadrature (I&Q) components (clean pulses only) of each PRI set separately. In this way, the zero-Doppler power is removed from each PRI set without cross-contamination. Obviously, the clutter suppression is limited in this case, but it is an improvement over no suppression.

b. PRI set selection

What PRIs should be used for optimal RV ambiguity mitigation? Initially, we examined combinations of simple integral ratios for velocity dealiasing using the Chinese remainder theorem (CRT). However, it turned out that a clustering algorithm applied to the dealiased velocities (Trunk and Brockett 1993) performed as well as the CRT-based method (see section 3). This result then freed us to choose any combination of PRIs.

The range of usable PRIs for velocity estimation on the TDWR is limited at the upper end by the coherence criterion, T < λ/(4πσ_υ), where λ ∼5.3 cm for TDWRs. For σ_v = 4 m s⁻¹ [corresponding to median values observed in squall lines; Fang et al. (2004)] this yields T < 1050 μs. (The current operational RDA has a maximum PRI for velocity estimation of 938 μs.) For high-elevation scans, the shortest PRI is determined by the transmitter capability (518 μs), but for low-elevation scans the shortest PRI is limited by the FAA operational coverage requirement of 48 n mi in range (594 μs).

For maximum range-overlay protection and velocity dealiasing accuracy, the spread of PRIs should be as big as possible. Furthermore, without the need to form simple integral ratios, and with no preferred locations from which range aliasing takes place, the PRIs should be spaced evenly to avoid gaping “holes” in protection. The question then is how many PRIs to place within that spread.

Suppose there is a strong point target at true-range r_t. For T₁ with unambiguous range r_a1 < r_t, there will be range-overlay contamination at r_f1 = r_t − r_a1. If we add T₂ with unambiguous range r_a2 < r_t to the transmission sequence, then r_f2 = r_t − r_a2 will also have range-overlay contamination. However, because r_f1 and r_f2 also have “clean” pulses from which the moments can be estimated, we have reduced the number of contaminated range gates from 1 to 0 by adding a PRI. In this way, increasing the number of PRIs can provide increased range-overlay protection. On the other hand, if there was strong ground clutter at r_f1 and r_f2, then the interaction with the GCF will destroy the range-overlay protection capability and data from both range gates will be corrupted, thus increasing the number of “bad” range gates from 1 to 2. Furthermore, by increasing the number of PRIs, the number of pulses per PRI is necessarily decreased for a constant dwell time. This in turn increases the estimate variance coming out of each PRI set, which is detrimental to velocity dealiasing. Therefore, there is no single number of PRIs that is optimal for all situations. In the eventual adaptive processing system, we will have a number of different PRI sets available, from which the algorithm will select an optimum set for each radial based on the range-aliasing and ground-clutter distributions provided by the initial long-PRI scan. It will also preferentially protect areas that are most important to the TDWR’s mission, such as the airport and approach/departure corridors.

a. CRT versus clustering

Velocity dealiasing for Doppler weather radars utilizing PRI diversity is usually limited to two PRIs, and the technique employed is the Chinese reminder theorem (e.g., Ding et al. 1996). For a pair of PRIs with a ratio reducible to relatively prime integers, CRT can be used to prove that the difference of the respective velocity estimates maps uniquely onto an unambiguous velocity interval larger than either of their associated Nyquist intervals. In this way the effective unambiguous velocity can be extended beyond the Nyquist limit. For example, the weather channel on the operational Airport Surveillance Radar-9 (ASR-9) processes a block-staggered dual-PRI sequence (Weber 2002), while the commercial SIGMET RVP8 radar signal processing system employs an alternating-dwell dual-PRI waveform to dealias velocity across radials (SIGMET 2005). Both systems use the difference in velocity estimates from the two PRIs to compute the dealiased velocity based on rules generated by the CRT.

For our case, however, velocity estimates from more than two PRIs may be produced. In the beginning we tried to extend the traditional dual-PRI approach by choosing PRI sets that included simple integral ratios that were amenable to the CRT approach on a pair-by-pair basis. However, it became clear that a more general approach was desirable, since censoring of range-overlay contaminated PRI sets can break up those special pairs. Direct application of the CRT to multiple PRIs with a large common divisor also did not produce good results. We decided, therefore, to use the clustering algorithm (Trunk and Brockett 1993), which does not have a preference for any particular PRI relationship and can be applied in the same manner to any number of PRIs.

Suppose there are velocity estimates from m PRIs. In the clustering algorithm, for each PRI velocity estimate, all possible unfolded velocities are computed up to ±υ_MAX, which is set by the user. Then all the velocity values are sorted from smallest to largest and the average squared error is computed in a sliding window of length m that is incremented across the entire list. The median value in the window with the smallest error (the “best cluster”) is the dealiased velocity. One of the advantages of this algorithm is that υ_MAX can be set to any value. In other words, the trade-off between the maximum speed that can be dealiased and dealiasing error can be adjusted in a smooth, continuous fashion. Decreasing υ_MAX increases the dealiasing success rate as long as most of the velocity distribution lies within ±υ_MAX. With the rule-based CRT technique, such a trade-off can only be realized in large, discontinuous jumps and is dependent on the particular PRI ratio used (Torres et al. 2004).

Figure 2 shows the velocity dealiasing success rate versus υ_MAX for dual-PRI using the clustering algorithm. Here, T₁ was set to 600 μs and T₂ was set to 1.3 (solid), 1.4 (dashed), and 1.5 (dashed–dotted) times T₁. The input velocity had a zero-mean Gaussian distribution with a standard deviation of 10 m s⁻¹. On top of this, an independent Gaussian random error with standard deviation of 2 m s⁻¹ was added to each PRI input velocity. The resulting velocities were aliased into the corresponding Nyquist intervals. These velocity pairs were then fed into the clustering algorithm. Dealiasing was deemed to be successful if the absolute difference between the input and dealiased velocities was smaller than the unambiguous velocity υ_a2 corresponding to T₂. Ten thousand Monte Carlo runs were averaged for each data point. Note that at small υ_MAX the success rate increases with υ_MAX, because there are enough velocities in the tail of the Gaussian distribution greater than υ_MAX. At large υ_MAX, the success rate decreases with υ_MAX, because there are virtually no velocities in this region and increasing υ_MAX only serves to raise the chances of false dealiasing. For reference, the percentage of input velocities for PRI T₁ that lies within ±υ_a1 (±22 m s⁻¹) is 97%. This means that velocity dealiasing success rates above this figure are an improvement over what could be achieved using only the shorter PRI for this particular velocity distribution and error.

Let us now compare the velocity dealiasing performances of the clustering algorithm and the rule-based CRT technique (Torres et al. 2004). Figure 3 shows the velocity dealiasing success rates for dual-PRI using the clustering (solid) and CRT (dashed) algorithms versus the standard deviation of the input zero-mean Gaussian velocity distribution: T₁ = 600 μs and T₂ = 900 μs. As with Fig. 2, a Gaussian random error with standard deviation of 2 m s⁻¹ was added for each PRI input velocity and aliased into the corresponding Nyquist intervals. These velocity pairs were then fed into the respective algorithms, and the outputs were the means of the dealiased pairs. Ten thousand Monte Carlo runs were averaged for each data point. Dealiasing success was measured in the same way as for Fig. 2. For the CRT method, a PRI ratio of 2:3 yields a maximum extended unambiguous velocity of 3υ_a2 = 44.5 m s⁻¹, so υ_MAX was also set to this value. The small difference in the performance is due to the difference in the output velocity range of the two techniques. With the CRT approach, υ₁ has a maximum dealiased range of ±3υ_a1 and υ₂ has a maximum dealiased range of ±3υ_a2, even though the dealiasing rules are only unambiguous up to ±3υ_a2, which is narrower than ±3υ_a1. With the clustering algorithm, ±υ_MAX is the dealiasing range for both υ₁ and υ₂. Thus, the clustering algorithm performs slightly better if the input velocity distribution rarely exceeds ±3υ_a2, and the CRT algorithm works slightly better as the velocity distribution tail extends beyond ±3υ_a2.

For number of PRIs greater than 2, we tried various simple integral ratio combinations to use with the CRT technique, but none was able to outperform the clustering algorithm. Note that Trunk and Brockett (1993) also showed the clustering algorithm to perform better than a CRT approach for range dealiasing using three PRIs. Therefore, also considering the flexibility of PRI selection and maximum dealiasing range that the clustering algorithm allows, we conclude that the clustering algorithm should generally be chosen over the CRT method, unless computational speed is the number one priority.

b. Estimation performance dependence on PRI loss

Let us now examine the effect that PRI-set censorship has on velocity dealiasing performance. For example, consider a four-PRI sequence with T = 600, 700, 800, and 900 μs. Velocity dealiasing can be performed on the resulting velocity estimates as long as at least two PRI sets are clean from range-overlay contamination. However, we would expect the dealiasing performance to be different for different combinations. This expectation is supported by simulation results. Figure 4 shows dealiased velocity estimation error versus different available PRI combinations. The input velocity generation procedure was the same as that used for Figs. 2 and 3, with a zero-mean Gaussian distribution of standard deviation 15 m s⁻¹ and an additive Gaussian error with standard deviation of 2 m s⁻¹. Here, υ_MAX was set to 40 m s⁻¹ and 10 000 Monte Carlo runs were made for each data point. The four-digit binary number over each point indicates which PRI velocity estimates were used; for example, 1001 means that velocity estimates from T = 600 and 900 μs were used to compute the dealiased velocity. The results were sorted for the plot in order of increasing velocity estimation error.

As seen in Fig. 4, the primary PRI-combination factor in determining velocity dealiasing performance is the maximum difference in available unambiguous velocities. The unambiguous velocities corresponding to the four PRIs used here are υ_a = 22.3, 19.1, 16.7, and 14.8 m s⁻¹. So the maximum difference for a pair is υ_a1 − υ_a4 followed by υ_a1 − υ_a3, υ_a2 − υ_a4, υ_a1 − υ_a2, υ_a2 − υ_a3, and υ_a3 − υ_a4. A secondary factor is the total number of PRIs available.

Not included in this analysis is the dependence of the velocity error on the PRI itself, because that is, in turn, dependent on the characteristic of the weather spectrum [e.g., Zrnić (1977), for pulse-pair velocity estimation]. In most cases smaller PRIs would yield more accurate velocity estimates, so the ordering in Fig. 4 should not be affected by this factor.

c. False dealias correction

Because of the inherent variance in real weather radar data, false velocity dealiasing is unavoidable for some fraction of cases no matter what technique is used. The human eye can usually detect such errors, because of the available contextual information in space and/or time. Similarly, automated algorithms can also detect and correct such errors based on continuity. We developed the following algorithm for a two-dimensional (2D) range–azimuth field of dealiased velocity data.

First, for every range cell, three values of velocity are stored: the undealiased (raw) velocity, the dealiased velocity, and the second-choice dealiased velocity. The second-choice dealiased velocity corresponds to the dealiased velocity cluster with the second-smallest error given by the clustering algorithm. Of these three velocities, none or two or three could be the same, depending on the value of υ_MAX relative to the υ_as. Second, for each cell a weighted median (Arce 1998) of the dealiased velocity is computed over a 2D range− azimuth grid (e.g., 3 × 3) centered on that cell. The weights are provided by the magnitude of the correlation coefficients of the time series at lag one [denoted as signal quality index (SQI) in the SIGMET manuals], a quantity that varies between zero and one. Because of the discontinuous nature of dealiased velocity data, the median is a more accurate measure of the background value than the mean. The weighting by the SQI diminishes the contribution to the median by less reliable data points. Third, of the three possible velocity values, the one closest to the weighted median is chosen. In this way, falsely dealiased velocity values are restored to the correct value, as long as the bad values are not clustered too densely compared to the grid dimensions used. Results from this algorithm will be presented in the following section.

a. Range-overlay protection

The first set of scans was taken on 17 March 2003 starting at 2040 UTC, while convective storm cells were active in the vicinity. The scan elevation angle was 0.3° with an antenna rotation rate of 21.6° s⁻¹. Figure 9 shows the reflectivity fields computed from the long-PRI (3.06 ms) scan for the full 460-km radius (top panel) and zoomed in to a radius of 77 km (bottom panel). Very strong scattering targets exist at many different ranges and at azimuths, thus making this a challenging case for range-overlay protection. In some azimuths there are multiple trips aliasing into the first trip. The resulting near-range reflectivity suffers much contamination as can be seen in the single-PRI (667 μs) scan (Fig. 10, top panel). The MBS (518, 578, 638, 698, 758, 818, 878, 938 μs × 16 pulses each) scan reflectivity (Fig. 10, bottom panel), however, looks very similar to the “truth” provided by the long-PRI scan (Fig. 9, bottom panel). There is still some unfiltered overlay power in the south-southeast and west-northwest sectors, but otherwise the differences are small. These problem areas are associated with out-of-trip patches that have relatively long continuous radial extent (Fig. 9, top panel). As expected these areas are protected better in a constant-PRI phase-code processed scan (not shown), which supports our plan for an adaptive signal transmission and processing scheme.

Here we insert a note about the clutter filtering. As with the simulated data processing, we used the GMAP GCF for the constant-PRI scans. For the multi-PRI scans, however, the full, multilevel adaptive GCF as presented in Part I was not applied. In the initial prototype version, the PRI transmission sequence was not synched to the beginning of each 1° azimuthal dwell boundary. Consequently, for each dwell, different filter coefficient matrices had to be defined for each possible sequence permutation, in this case 64. Therefore, we only generated one set of coefficient matrices at a 60-dB suppression. We used the estimated clutter power from the long-PRI scan to turn this GCF on or off for a given cell. Furthermore, the initial SIGMET receiver had limited dynamic range compared to the legacy operational receiver, which led to saturation on very strong clutter targets. The current version has a dynamic range slightly better than the legacy system. The present RDA prototype is also capable of initializing the PRI sequence at every 1° azimuthal dwell boundary, allowing for easy implementation of a multilevel adaptive GCF for multi-PRI signals. Therefore, the overall clutter filtering performance with the present prototype is better than what is shown here. We used the older data, because they contained the best examples of range aliasing.

The crucial parameter that must be protected from range-overlay contamination is not reflectivity, which is essentially available from the long-PRI scan, but velocity. Figure 11 shows the comparison for velocity between the single-PRI scan (top panel) and the multi-PRI scan (bottom panel). Although there is no truth available for comparison, it is clear that the multi-PRI scan has protected many of the areas that were corrupted by range aliasing in the single-PRI case.

The single-PRI scan shown here is representative of estimates produced by the current operational TDWR. Even though the TDWR utilizes an adaptive PRI selection algorithm at the lowest-elevation scan in an attempt to minimize range-overlay obscuration in certain key areas, such as the airport and approach/departure corridors, it is still a single PRI for the entire scan (Crocker 1988). Therefore, the leverage to protect all azimuth and range cells is very limited.

b. Velocity dealiasing

Since the velocities in the previous example were not strong enough to test the velocity dealiasing algorithm, we present another case. The following scans were taken on 3 April 2003 starting at 1820 UTC at an elevation angle of 2.6°. No significant range aliasing was present. The velocity plots are shown in Fig. 12 for the single-PRI (598 μs) scan (top panel) and the multi-PRI (same sequence as before) scan (bottom panel). The unambiguous velocity of the single-PRI scan was 22 m s⁻¹, which was clearly exceeded over significant areas. The multi-PRI dealiasing algorithm does an excellent job of restoring the velocity field. The problem regions (for both scans) to the northwest and to the southwest at close range are areas of very low SNR.

The current operational TDWR also has a velocity dealiasing procedure, but it uses two consecutive scans at the same elevation with two different PRIs (Wieler and Hu 1993). Being able to perform the dealiasing within a single scan will save time and allow a faster volume scan or reduced estimate variance via a slower scan rate. Also, of course, the multi-PRI processing simultaneously provides vastly improved range-overlay protection.