• Augros, C., , Perier L. , , Bousquet O. , , Kergomard A. , , Dupuy P. , , and Tabary P. , 2010: Improvements of the Doppler measurements quality inside the French radar network and experimentation of a national low levels wind shear mosaic. Proc. Sixth European Conf. on Radar Meteorology and Hydrology, Sibiu, Romania, ERAD, 6 pp. [Available online at http://www.erad2010.org/pdf/oral/thursday/quality3/03_ERAD2010_0220.pdf.]

  • Conan Doyle, A., 1890: The Sign of Four. Pitman & Sons, Ltd., 171 pp.

  • Pirttilä, J., , Lehtinen M. S. , , Huuskonen A. , , and Markkanen M. , 2005: A proposed solution to the range–Doppler dilemma of weather radar measurements by using the SMPRF codes, practical results, and a comparison with operational measurements. J. Appl. Meteor., 44, 13751390.

    • Search Google Scholar
    • Export Citation
  • Ruzanski, E., , Hubbert J. C. , , and Chandrasekar V. , 2008: Evaluation of the simultaneous multiple pulse repetition frequency algorithm for weather radar. J. Atmos. Oceanic Technol., 25, 11661181.

    • Search Google Scholar
    • Export Citation
  • Sirmans, D., , Zrnić D. S. , , and Bumgamer B. , 1976: Extension of maximum unambiguous Doppler velocity by use of two sampling rates. Preprints, 17th Conf. on Radar Meteorology, Amer. Meteor. Soc., 23–28.

  • Tabary, P., , Guibert F. , , Perier L. , , and Parent-du-Châtelet J. , 2006: An operational triple-PRT Doppler scheme for the French radar network. J. Atmos. Oceanic Technol., 23, 16451656.

    • Search Google Scholar
    • Export Citation
  • Zrnić, D. S., 1977: Spectral moment estimates from correlated pulse pairs. IEEE Trans. Aerosp. Electron. Syst., 13, 344354.

  • View in gallery

    Doppler velocity plan position indicators (PPIs) observed at (left) 1.6° and (right) 2.5° elevation by the Cherves radar in France at 0100 UTC 21 Dec 2011. The range ring is at 100 km of the radar. The graphs below the maps show the Doppler velocity measured on the radials heading north (solid line) and northwest (dashed line). Jumps in velocity can be observed, and the altitude at which they occur is annotated in parenthesis. These jumps take the form of step functions that we are seeking to explain, and of a few excursions to negative or highly positive velocities associated with a faulty velocity dealiasing.

  • View in gallery

    Simulated observations of velocity measured with different Nyquist intervals as a Gaussian beam passes through two types of wind transitions. (a) Geometry of the first set of simulations where the beam (whose 3-dB edges are shown with thick lines) samples a wind pattern that undergoes a gradual transition from 0 m s−1 (bottom thin line) to 10 m s−1 (top thin line) as illustrated by arrows. (b) Geometry of the second set of simulations where the beam samples a wind pattern that undergoes a sharp transition from 0 to 10 m s−1 at the thin line. (c) Resulting Doppler measurements for the gradual wind transition case for radars having different Nyquist velocities from 6.67 m s−1 to “large” (1000 m s−1), with thin lines showing aliased velocities and thick lines showing velocities after dealiasing. (d) Resulting Doppler measurements for the sharp wind transition case. Contrasting the two scenarios, we see that when the beam crosses a gradual wind transition, the dealiased Doppler velocity patterns are all similar, the aliasing (seen as a sudden change in the velocity of thin lines) being well corrected. But when the beam crosses a sharp velocity transition, the dealiased Doppler velocity can differ significantly, as the aliasing becomes a gradual process instead of a sudden one.

  • View in gallery

    Computed Doppler velocity around the McGill radar using (top left) data at a single 1200-Hz PRF and three dual-PRF combinations with smaller Nyquist intervals [(top right) 400:300, (middle left) 300:240, and (middle right) 240:200 Hz], together with the difference between the velocities derived at 1200 Hz and those derived with (bottom) the two smaller Nyquist intervals. As the Nyquist velocity decreases, artificial transitions in Doppler velocity become increasingly apparent along arcs. The range ring is at 60 km. The radial velocity transition around 120° azimuth is due to a small mismatch in elevation angle between the beginning and the end of the PPI scan.

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The Case of Sharp Velocity Transitions in High Vertical Wind Shear When Measuring Doppler Velocities with Narrow Nyquist Intervals

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  • 1 McGill University, Montreal, Quebec, Canada
  • | 2 Météo-France, Toulouse, France
  • | 3 McGill University, Montreal, Quebec, Canada
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Abstract

An investigation was launched following unexpected observations of step-function transitions in Doppler velocities from scanning radars in regions of high vertical wind shear. It revealed that, if wind velocity transitions are sufficiently sharp and strong, then the measured velocity at each pulse repetition frequency undergoes an unusual smooth transition, from being in the right Nyquist interval to being in the wrong one. Hence, when these velocities are dealiased, a step function in velocity appears at the transition between where velocities were aliased and where they were not. The extent and the impacts of this data artifact are considered.

Corresponding author address: Frédéric Fabry, Department of Atmospheric and Oceanic Sciences, McGill University, 805 Sherbrooke Street West, Montreal QC H3A 0B9, Canada. E-mail: frederic.fabry@mcgill.ca

Abstract

An investigation was launched following unexpected observations of step-function transitions in Doppler velocities from scanning radars in regions of high vertical wind shear. It revealed that, if wind velocity transitions are sufficiently sharp and strong, then the measured velocity at each pulse repetition frequency undergoes an unusual smooth transition, from being in the right Nyquist interval to being in the wrong one. Hence, when these velocities are dealiased, a step function in velocity appears at the transition between where velocities were aliased and where they were not. The extent and the impacts of this data artifact are considered.

Corresponding author address: Frédéric Fabry, Department of Atmospheric and Oceanic Sciences, McGill University, 805 Sherbrooke Street West, Montreal QC H3A 0B9, Canada. E-mail: frederic.fabry@mcgill.ca

1. The mystery: Observations of sharp vertical wind shear

During a visit to Météo-France in January 2012, Frédéric Fabry was courteously shown some of the latest work from his hosts. One of these presentations by Clotilde Augros dealt with a new algorithm for aviation safety. The algorithm seeks to detect lines of low-level wind shear from C-band and S-band Doppler data (Augros et al. 2010). Most of the lines detected did come from low-level phenomena. But, occasionally, the algorithm also detected sudden wind shifts aloft. And the lines of wind shifts varied with elevation (Fig. 1). The French forecasters who were shown these images were skeptical that these transitions were true sudden wind shifts, as they often occurred in stratiform precipitation. And the mystery remained, one that demanded answers: Are those true lines of horizontal wind shifts or not? What could be their cause?

Fig. 1.
Fig. 1.

Doppler velocity plan position indicators (PPIs) observed at (left) 1.6° and (right) 2.5° elevation by the Cherves radar in France at 0100 UTC 21 Dec 2011. The range ring is at 100 km of the radar. The graphs below the maps show the Doppler velocity measured on the radials heading north (solid line) and northwest (dashed line). Jumps in velocity can be observed, and the altitude at which they occur is annotated in parenthesis. These jumps take the form of step functions that we are seeking to explain, and of a few excursions to negative or highly positive velocities associated with a faulty velocity dealiasing.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00151.1

With the position of the wind shift line varying with elevation, and the shape of the line being an arc at nearly constant range, it seemed likely that the signature was caused by a vertical, rather than by a horizontal, wind shift. But to observe a step-function change in velocity along the radial direction, as seen in Fig. 1, is unexpected. Instead, we expect a smooth transition if only because of the smoothing caused by the radar beam. So the puzzle remained, encouraging us to launch an investigation.

2. Following the trail

a. Narrowing down the suspects

How can one measure, over an arc of significant length, a sudden wind shift in height while pointing at an elevation of a couple of degrees? Meteorologically and instrumentally speaking, it is impossible: Even with a sudden wind shift in height, and given a relatively uniform reflectivity field, the smoothing caused by the radar beam pattern should lead to a gradual transition between one velocity regime and the next. With the list of plausible culprits diminishing rapidly, the focus then shifted toward signal generation and its processing, particularly velocity dealiasing.

At shorter radar wavelengths, it is common to use multiple pulse repetition times (PRTs) or frequencies (PRFs) in order to obtain data with acceptable unambiguous ranges and Nyquist velocities (Sirmans et al. 1976). In particular, the French weather radar network uses an aggressive triple-PRT scheme on its C-band radars (Tabary et al. 2006). But given sufficient signal power and dwell time, the dealiasing algorithm should work well, and these conditions were met in the example shown in Fig. 1. One should hence not expect the dealiasing algorithm to fail, except for a few pixels where the signal strength is low or the spectrum width is high.

Nonetheless, “when you have eliminated the impossible, whatever remains, however improbable, must be the truth” (Conan Doyle 1890, chapter 6). The problem therefore has to be either in the raw radar signal or in the velocity dealiasing. To catch the culprit in the act, a simulation test was devised.

b. Setting up the trap

The controlled environment where we could monitor what would happen was arranged as follows. A Gaussian-shaped radar beam pointing at a low grazing angle was assumed to cross a 10 m s−1 transition in Doppler velocity. Two scenarios are considered: In the first instance, the velocity transition was made to be gradual (Fig. 2a); in the second instance, the transition was made to be sudden (Fig. 2b). Using the beam pattern and assuming a constant reflectivity field, the Doppler spectrum was computed as a function of range. These spectra were then used to simulate the raw time series (I, Q) that would be measured by the radar. Different time series were generated assuming different Nyquist velocities. No noise was introduced in the time series. Once the time series were generated, pulse-pair processing (Zrnić 1977) was used on them to retrieve velocities. Finally, the velocities obtained were dealiased by bringing the velocities in the Nyquist velocity interval that was the closest to the truth; this basically assumes that the dealiasing algorithm does exactly and perfectly what it is asked to do; errors in dealiasing will not be considered in this simulation. Hence, if we fail to reproduce what we observed, then the dealiasing algorithm must be the culprit.

Fig. 2.
Fig. 2.

Simulated observations of velocity measured with different Nyquist intervals as a Gaussian beam passes through two types of wind transitions. (a) Geometry of the first set of simulations where the beam (whose 3-dB edges are shown with thick lines) samples a wind pattern that undergoes a gradual transition from 0 m s−1 (bottom thin line) to 10 m s−1 (top thin line) as illustrated by arrows. (b) Geometry of the second set of simulations where the beam samples a wind pattern that undergoes a sharp transition from 0 to 10 m s−1 at the thin line. (c) Resulting Doppler measurements for the gradual wind transition case for radars having different Nyquist velocities from 6.67 m s−1 to “large” (1000 m s−1), with thin lines showing aliased velocities and thick lines showing velocities after dealiasing. (d) Resulting Doppler measurements for the sharp wind transition case. Contrasting the two scenarios, we see that when the beam crosses a gradual wind transition, the dealiased Doppler velocity patterns are all similar, the aliasing (seen as a sudden change in the velocity of thin lines) being well corrected. But when the beam crosses a sharp velocity transition, the dealiased Doppler velocity can differ significantly, as the aliasing becomes a gradual process instead of a sudden one.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00151.1

c. The two faces of velocity aliasing revealed

The bottom of Fig. 2 illustrates the result of this experiment. When the beam crosses the gradual wind transition, the measured Doppler velocities for all Nyquist intervals behave as expected: velocities move gradually from one velocity to the next, the gradual nature of the transition being due to the convolution between the wind transition itself and the beam pattern. If the velocity exceeds the Nyquist velocity, then a clear aliasing is observed, and this aliasing causes a sudden ±2Vnyq transition in velocity. The dealiasing algorithm then adds another transition of opposite sign from this range onward, and the aliasing can be corrected easily and properly.

But when the wind transition is sudden, sudden being defined as much smaller than the radar beamwidth, the modus operandi changes. If the Nyquist velocity is larger than the wind speed transition, then we still get similar results as in the gradual transition. But when the Nyquist velocity Vnyq is smaller, the measured wind speed changes smoothly from the original velocity (0 m s−1) to the aliased one (2Vnyq − 10) m s−1 without passing by Vnyq, as is the case in the gradual transition example. Because aliasing is now a smooth process instead of being a sharp one, when velocity dealiasing is done by adding 2mVnyq to the measured velocity Vmeas, where m is an integer chosen so that (Vmeas + 2mVnyq) falls into the correct Nyquist interval, a transition of magnitude 2mVnyq is introduced. And this is what we observe in Fig. 1. In the end, the unexpected velocity transition occurs because the dealiasing algorithm does exactly what it is designed to do, not because it failed. It is the unexpected smooth transition between correct and aliased velocities that occurred in these special conditions that is the cause of the problem. And because this aliasing does not occur in the expected way, its correction by traditional dealiasing algorithms fails near the wind transition.

When velocities are computed using multiple PRT, like on Météo-France radars, this effect remains. On a triple-PRT system, depending on the magnitude of the wind velocity transition, one, two, or all three Doppler velocities may jump at the level of the wind transition, as shown in Fig. 2. This will lead to a sudden change in the final Doppler velocity, the magnitude of which depends on how the velocities from the three PRTs are combined.

3. Reenacting the crime: Signal processing simulation

To further examine and reproduce the phenomena observed, we collected some raw level 1 (I, Q) data from the McGill S-band scanning radar on 31 January 2012. Each sample k of the complex time series collected will be referred to as Ak = Ik + jQk, where j is the square root of −1. Thanks to a Nyquist velocity Vnyq of 31.25 m s−1, the McGill radar data are immune to the effect described. We then simulated what would have been observed at smaller Nyquist velocities and for several dual-PRF approaches. First, single-PRF velocities υdop(n) with Nyquist velocities of (31.25/n) were calculated by computing velocities with a pulse-pair algorithm combining samples separated by n pulse intervals using
e1
where Arg is the argument of a complex number and N is the number of pulses that falls within each 1° azimuth, in our case 32. As shown in (1), we chose to compute υdop(n) using all available sample pairs Nn, and not only the ones that would really be available if an actual dual-PRF or dual-PRT scheme were used. This is because we did not want poor estimates of velocity obtained with very few samples to contaminate our experiment, an experiment made with a McGill radar scanning at a much faster rate (6 rpm) than that of radars that would actually collect data using dual-PRT schemes. Then, from pairs of aliased velocities [υ(n), υ(n+1)], dealiasing is performed in an attempt to recover υ(1), assumed to be correct. Dealiasing is done on both υ(n) and υ(n+1), and the results are averaged to obtain the final velocity from the simulated dual-PRF approach. The Doppler velocities obtained are shown in Fig. 3.
Fig. 3.
Fig. 3.

Computed Doppler velocity around the McGill radar using (top left) data at a single 1200-Hz PRF and three dual-PRF combinations with smaller Nyquist intervals [(top right) 400:300, (middle left) 300:240, and (middle right) 240:200 Hz], together with the difference between the velocities derived at 1200 Hz and those derived with (bottom) the two smaller Nyquist intervals. As the Nyquist velocity decreases, artificial transitions in Doppler velocity become increasingly apparent along arcs. The range ring is at 60 km. The radial velocity transition around 120° azimuth is due to a small mismatch in elevation angle between the beginning and the end of the PPI scan.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00151.1

If we ignore a few dealiasing mistakes (e.g., the green pixels at the far range in the south), then the main difference between the velocities at 1200-Hz PRF and those using pairs of estimates at lower PRFs is the appearance of sharp transitions at some ranges. The sharp velocity transitions are similar to what was both observed on Météo-France radars (Fig. 1) as well as simulated in idealized conditions (Fig. 2). Velocity differences exceeding 3 m s−1 can be measured over large rings between the 1200-Hz data and the 240:200-Hz data (with 6.25 and 5.2 m s−1 Nyquist velocities).

Could we detect regions where this mismatch occurs? A few clues are available, though they are not all unique to this phenomenon. The most characteristic one is a growing dispersion between estimates coming from different PRFs, with the differences between the two velocities Vdop(n1) − Vdop(n2) moving away from the {2m[Vnyq(n1) − Vnyq(n2)]} values expected for different aliasings, where m is an integer. Also, as a result of the broadening spectrum width, the ratio of the lag-1 to lag-0 power will diminish, while the number of dealiasing mistakes will increase.

4. The victims: Quantitative uses of radar data

Biases such as these are not restricted to French radars. All systems that use a combination of Doppler velocities with small Nyquist intervals to obtain Doppler velocity are subject to this problem. Here we illustrated the problem using a traditional multiple-PRT approach. Velocities obtained using the simultaneous multiple pulse repetition frequency codes (Pirttilä et al. 2005; Ruzanski et al. 2008) will also experience jumps, though generally because of the nature of the fitting process.

It is hard to quantify the frequency of these events, as it depends on meteorological events, radar hardware, and processing choices. The key ingredient remains the presence of a wind shear that is both over a height range that is much smaller than the beamwidth and that exceeds in magnitude the Nyquist velocity of one of the PRFs used to compute the final velocity. Hence, it is more frequent at far range, where the beam is wider, than at near range. Table 1 shows the minimum wind shear required over specific beamwidth fractions required before a sharp wind transition is observed. For example, at 60-km range where a 1° beamwidth corresponds to 1 km, a radar with one PRF at 5 m s−1 Nyquist like those of Météo-France would observe sharp transitions for a 7.5 m s−1 vertical shear over 550 m, a gradient that can be observed with some regularity in widespread weather systems.

Table 1.

Critical values for the combinations of the magnitude of wind shear and its vertical extent in elevation required to cause step-function velocity transitions. In this table, Vnyq is the Nyquist velocity and ϕbeam is the 3-dB beamwidth. For example, sharp velocity transitions will be observed when the shear exceeds 120% of Vnyq over an elevation angle smaller or equal to 0.29 ϕbeam.

Table 1.

Given that, are these transitions a real problem? As long as the radar data are used qualitatively, the occasional jumps in velocities are a minor inconvenience compared to the success obtained in dealiasing velocities. In fact, as pointed out by a French colleague (P. Tabary 2012, personal communication), at small Nyquist velocities, the restituted radial wind speed is closer to the actual wind profile with its sharp transition than the one obtained with the large Nyquist velocity, which always smoothens the transition because of the effect of the beamwidth (Fig. 2). However, the sharpening of the gradients only occurs when the gradient is both sufficiently sharp and large. If the magnitude of the wind change is weak or if the transition is gradual, then the data from the radar with the low Nyquist velocity revert to the same smoothing as its higher-Nyquist-velocity cousin.

But perhaps more importantly, we enter into an era where we try to use radial velocity data quantitatively. And it is for these uses, such as data assimilation and multiple Doppler retrievals as well as for specialized algorithms, such as Augros’ horizontal wind shear detection, that the biased winds and especially the jumps in velocities will lead to erroneous results. In data assimilation—for example, where the observed wind may be simulated in the model by smoothing the model radial velocities, like a 1° beam radar with a large Nyquist velocity would do—the result will not match the observations; the assimilation system will then try to modify its wind to get a sharper simulated wind transition, leading to excessive shear and then less accurate forecasts. Unless one can simulate the radar observations at the signal processing level to take into account the narrow Nyquist interval, it will be extremely difficult to assimilate data near these transitions. Perhaps the best method in such circumstances would be to detect those transitions and flag the velocity data in these regions as doubtful for assimilation.

Therefore, while it is technically possible to dealias Doppler velocities by combining two or more velocities with narrow Nyquist intervals, the results may be markedly different from the velocities expected with a wide Nyquist interval. The interpretation of these velocities and their simulation in a numerical model for data assimilation purposes become as a result much more difficult to accomplish.

Acknowledgments

This project was undertaken with the financial support of the government of Canada, provided through the Department of the Environment, but the views expressed herein are solely those of McGill University.

REFERENCES

  • Augros, C., , Perier L. , , Bousquet O. , , Kergomard A. , , Dupuy P. , , and Tabary P. , 2010: Improvements of the Doppler measurements quality inside the French radar network and experimentation of a national low levels wind shear mosaic. Proc. Sixth European Conf. on Radar Meteorology and Hydrology, Sibiu, Romania, ERAD, 6 pp. [Available online at http://www.erad2010.org/pdf/oral/thursday/quality3/03_ERAD2010_0220.pdf.]

  • Conan Doyle, A., 1890: The Sign of Four. Pitman & Sons, Ltd., 171 pp.

  • Pirttilä, J., , Lehtinen M. S. , , Huuskonen A. , , and Markkanen M. , 2005: A proposed solution to the range–Doppler dilemma of weather radar measurements by using the SMPRF codes, practical results, and a comparison with operational measurements. J. Appl. Meteor., 44, 13751390.

    • Search Google Scholar
    • Export Citation
  • Ruzanski, E., , Hubbert J. C. , , and Chandrasekar V. , 2008: Evaluation of the simultaneous multiple pulse repetition frequency algorithm for weather radar. J. Atmos. Oceanic Technol., 25, 11661181.

    • Search Google Scholar
    • Export Citation
  • Sirmans, D., , Zrnić D. S. , , and Bumgamer B. , 1976: Extension of maximum unambiguous Doppler velocity by use of two sampling rates. Preprints, 17th Conf. on Radar Meteorology, Amer. Meteor. Soc., 23–28.

  • Tabary, P., , Guibert F. , , Perier L. , , and Parent-du-Châtelet J. , 2006: An operational triple-PRT Doppler scheme for the French radar network. J. Atmos. Oceanic Technol., 23, 16451656.

    • Search Google Scholar
    • Export Citation
  • Zrnić, D. S., 1977: Spectral moment estimates from correlated pulse pairs. IEEE Trans. Aerosp. Electron. Syst., 13, 344354.

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