## 1. Introduction

The problem of a limited Nyquist velocity with Doppler radar sampling is a long standing issue. This is even more so for radars operating at higher frequencies. For example, a 5-cm wavelength (C band) radar with a maximum unambiguous range of 150 km has a Nyquist velocity of only about 13 m s^{−1}. With such a Nyquist, common “dealiasing” algorithms that may be satisfactory for 10-cm wavelength observations (e.g., Eilts and Smith 1990) have serious problems, and even manual unfolding may be impossible. A widely used approach to mitigate this problem is using a dual pulse repetition time (PRT). This may be on a pulse-pair to pulse-pair basis, where the time between pulses is varied on every pulse pair [referred to as a staggered sampling (Doviak et al. 1976; Sirmans et al. 1976)], or where alternate records of, for example 32 pulses, are sampled at a single, but different PRT (dual PRT). The former has intrinsic advantages for unambiguous unfolding, but the staggered sampling limits the performance of simple ground clutter filters (e.g., Banjanin and Zrnic 1991). A common approach to avoid this is the latter strategy, which is an approximation of the first. This approximation is employed on operational Doppler radars in Canada and Australia and is available on some commercial signal processors (SIGMET 1992). However, there are problems unfolding the data if the azimuthal gradient of the radial velocity is large. The most obvious example of this occurs within mesocyclones. The goal of this paper is to evaluate and demonstrate the limitations associated with the dual PRT method in the presence of wind fields with large horizontal shears. Data have been simulated to explore the limitations with some simple calculations and some numerical examples. Errors in the velocity estimates are also induced by the clutter filter response being aliased through the unfolded velocity interval, but this effect is not examined here. A modeling approach has been undertaken in order to examine how mesocyclone and microburst signatures may be distorted and if dual PRT unfolding error correction is possible.

## 2. Velocity unfolding with a dual PRT radar

*i,*is given by where

*θ*

_{i}and

*T*

_{i}are the phase of the autocorrelation function at the first lag and the time between pulses, respectively, and

*λ*is the radar wavelength. Velocity estimates at adjacent rays use different values of

*T*

_{i}, usually in a 4:3 or 3:2 ratio, allowing a tripling or doubling of the effective Nyquist velocity compared with the ray with the shorter

*T*

_{i}.

*V*

_{N2}) are added or subtracted to

*V*

_{2}so that it lies within the Nyquist interval of

*V*

_{c}. An example of this is shown in Fig. 1. The radial velocity measurements of the two rays (

*V*

_{1},

*V*

_{2}) are combined to produce an estimate

*V*

_{c}. Then, the dual PRT unfolded estimate for the radial velocity for ray 2 is given by

*V*

_{2}+ 2

*V*

_{N2}.

*V*

_{2}would be used to unfold the next (adjacent) radial velocity estimate, not

*V*

_{2}+ 2

*V*

_{N2}. However, there are clearly situations where large azimuthal gradients of the true radial velocity may cause problems. If the difference in the real (i.e., unfolded) radial velocities of the field exceeds a threshold, errors occur. This limit is reached when the velocity difference between the combined estimate and the true value of

*V*

_{2}exceeds the Nyquist velocity of ray 2 (

*V*

_{N2}), that is, Note that this condition can be exceeded in situations where the individual radial velocity estimates are

*not*aliased. That is, the dual PRT approach can have folding errors even where the original data is not folded. The possible impact of this on real observations is now explored using some simple model calculations.

## 3. The model and some results

There are a number of situations that can lead to extreme values of azimuthal shear. These include outflow boundaries where the wind is essentially linear, microburst outflows, and highly rotational flow, such as occur in mesocyclones. For the purposes of this study, mesocyclone and microburst signatures are examined.

For the mesocyclone case, a simple model of a Rankine vortex^{1} embedded within a mean flow is utilized. The vortex is placed at some distance and azimuth from the radar, the radial velocities from the model wind field are calculated, and individual radial velocity estimates are aliased in a manner consistent with the dual PRT sampling. This approach is very similar to the one taken by Wood and Brown (1997, 2000) in their studies of sampling effects on mesocyclone detection. To examine microburst signatures, the model uses a pure divergent flow from a “touchdown point” out to a preset microburst radius embedded with a mean flow. The outflow speed is constant from the touchdown point to the microburst radius. The radial wind components are then calculated as for the mesocyclone case.

An option where Gaussian distributed noise is added to the radial velocities is also included to allow for simulations with realistic uncertainties. For the simulations, a Nyquist for the shorter PRT is given as 13.75 m s^{−1}, which corresponds to a PRT of 1 ms and a wavelength of 5.5 cm. Simulations with varying PRT ratios have been performed with ratios of 4:3 and 3:2, giving a tripling and doubling of the effective Nyquist velocity with similar results.

Figure 2a shows the model radial velocity field for a 20 m s^{−1} vortex with a radius of maximum wind (RMW) of 2.5 km centered at a range of 40 km with zero mean flow. For perfect unfolding this field would be reproduced. Figure 2b shows the corresponding dual PRT measured field using a 3:2 unfolding and with no measurement noise, and Fig. 2c is the difference between the model and “measured” radial velocity fields. The target (heavy line) and measured (thin line) radial velocities at a range of 39.75 km are also plotted as a function of azimuth (Fig. 2d). This case represents a strong vortex and produces considerable errors in the retrieved velocity near the vortex center. The general outer signature and the high degree of structure are maintained. The errors are associated with azimuthal gradients, and for a mesocyclone a series of successive range bins have similar unfolding errors, producing a striped effect. The exact nature of the stripes is dependent on where the vortex center sits relative to the ray. Some errors occur for vortices with maximum tangential speeds greater than about 6 m s^{−1} for this size vortex. Larger vortices have smaller gradients for a given maximum tangential speed. However, the dependence on vortex size is relatively weak. For example, the maximum tangential speed before unfolding errors occur for a vortex with an RMW of 2.5 km and 3:2 unfolding is about 8 m s^{−1}, but for a much larger vortex, say 13 km in radius, the limit is only 14 m s^{−1}, less than twice. The addition of a mean background wind has no effect, as long as the radial wind speeds remain less than the dual PRT effective Nyquist.

The presence of the large excursions suggests that a dealiasing algorithm may correct the data. The distribution of these velocity errors is discrete, as the errors are multiples of twice *V*_{N1} or *V*_{N2}. In practice this most commonly results in errors of ±*V*_{N}.

A simple dealiasing algorithm based on the work of Eilts and Smith (1990) has been applied to the data in the dual PRT velocity fields. This traces the velocity along a radial and, if there is a jump with a magnitude greater than half the effective Nyquist velocity, then the data are folded back by an interval of twice the effective Nyquist. Note that for conventional aliasing the jumps are of twice the Nyquist velocity rather than the Nyquist as occurs here. This type of algorithm dealiasing is straightforward to implement here, because there are no data gaps in the velocity field or data with very large uncertainties in the radial velocity, but will be much more difficult for real data. Other approaches are being explored by May and Joe (2001).

Two examples of azimuthal cross-sections of the dual PRT radial velocity estimates through a vortex (as in the lower right-hand panel of Fig. 2) are shown in Fig. 3 for a strong vortex with no measurement error and a weaker vortex that includes random velocity errors with a standard deviation of 0.6 m s^{−1}. With the dealiasing errors, the weaker vortex has what appears to be a tornado-like signature embedded within it. The radial velocity fields after the dealiasing are mostly corrected. The essential mesocyclone velocity azimuthal profile is retrieved, although there remains some dealiasing errors. The magnitude of the remaining errors is always fixed at the magnitude of the Nyquist velocities of the individual PRTs. Thus, there may still be large errors induced that may significantly affect, for example, dual-Doppler retrievals. A mesocyclone detection algorithm would probably still obtain a detection signature despite the distortion.

For the assumed measurement errors of 0.6 m s^{−1}, there was only a small number of additional dealiasing errors induced in the measured field. The numbers increased significantly when the standard deviation of the errors was increased to 1.5 m s^{−1} with a large amount of speckle being introduced to the wind fields. This may be important in real data, particularly for measurements in the clear air, where the signal to noise ratio of the data can be quite low and the statistical measurement errors correspondingly large.

The microburst simulations show similar problems. Again, it is radial velocity variations as a function of azimuth that provide the problems, and errors tend to occur at the four locations around the microburst at angles of 45°, 135°, 225°, and 315° from the microburst center where the azimuthal shear is maximized (Fig. 4). There seem to be no problems at the center of the microburst. Again, the essential microburst signatures are maintained so that automated algorithms should be able to recognize the event.

## 4. Conclusions

The ability to resolve microburst and mesocyclone signatures with dual PRT radars is important, since this radar configuration is being used operationally in severe weather situations in Canada and Australia. There are several conclusions that can be drawn from these results. It is apparent that mean winds are unfolded easily, but that perturbations that are associated with mesocyclones and microbursts create distinctive patterns of errors in the unfolded velocity field. The general spatial pattern of the error characteristics is well defined, but is sensitive to small changes in vortex strength. These errors are large enough to contaminate automated detection algorithms, and the data must be quality controlled. Note that the perturbation velocities involved may be less than the individual Nyquist velocities. The dependence of vortex size on the results is quite weak.

Strong vortices produce a striped effect in the velocity data. The exact nature is dependent on where the vortex center sits relative to the ray. If the vortex is offset by about half of a beam width, the stripes are quite symmetrical. The bulk of these errors can be dealiased using fairly standard techniques, but it is worth noting that the magnitude of the errors revolved around the unfolded Nyquist velocity rather than twice the Nyquist. Some individual velocity values will retain errors with magnitudes of the individual Nyquist velocities of the rays.

Additive noise increases the problems in retrieving accurate velocity fields. This is exacerbated for smaller PRT ratios (giving larger effective Nyquist velocities), so it is possible that using a 3:2 unfolding ratio and living with a smaller effective Nyquist may be a better solution than, say, using a 4:3 ratio where statistical measurement errors have a greater effect.

These results show that the dual PRT analysis will generally work quite well if some postprocessing is applied. This is straightforward in these simulated data with continuous fields, but is potentially difficult in real data with echo-free regions. These issues will be explored further as real operational data with mesocyclones and microbursts become available.

## Acknowledgments

I would like to thank Paul Joe for many helpful discussions. I would also like to thank the anonymous reviewers for their constructive comments that helped to clarify the manuscript.

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^{1}

A Rankine vortex is defined by the tangential wind speed, *υ* = *C*_{1}/(*r*^{0.5}) for *r* > RMW and *υ* = *C*_{2}*r* for *r* < RMW, where *r* is the distance from the center of the vortex and RMW is the radius of maximum wind. Here *C*_{1} and *C*_{2} are constants defined by the RMW and the value of *V* at this radius.