Dealiasing of Doppler Radar Velocities Using a Torus Mapping

The radial velocity of scattering particles is determined from their observed phase difference between successive radar pulses. There is a maximum velocity that can be determined unambiguously. This maximum velocity is called the Nyquist velocity,

where PRF is the pulse repetition frequency and λ is the wavelength of the radar. The time between two successive radar pulses, and thus the PRF, also determines the maximum range R_a that can be resolved unambiguously. This leads to the fundamental equation for the maximum Nyquist range and radial velocity measurable by a Doppler radar.

where c is the speed of light. Therefore, a trade-off has to be made between the maximum velocity and the maximum range. For a typical C-band radar with a maximum range of 250 km, a maximum velocity of only 7.55 m s⁻¹ is obtained. Velocities higher than the unambiguous velocity will be folded back into the fundamental velocity interval. This process is called “aliasing.” The observed radial wind velocity V_o is therefore related to the unaliased velocity through

where n is an unknown integer called the Nyquist number. Velocity aliasing can usually be identified in radar images by detecting abrupt velocity changes of about 2V_a between neighboring measurements. Obviously, the specific radar parameters λ and PRF increase or reduce the impact of velocity folding.

There exist two main approaches to tackling the aliasing problem: one is based on improving the measurement technique of the radar system. For example, velocity aliasing can be reduced significantly by measuring radial winds with alternating sets of pulses using high and low PRFs (Doviak and Zrnić 1993). The dual-PRF method has the disadvantage that measurements performed at slightly different times or locations are combined, which can lead to representativeness errors.

The other approach focuses on postprocessing methods. The basic assumption for most of them is that the true wind field is sufficiently smooth and regular; this is true for the greater part of the weather situations, with the exception of mesocyclones, tornado vortices, and highly sheared environments. The elementary dealiasing techniques are based on local statistics (Ray and Ziegler 1977; Bargen and Brown 1980; Leise 1981; Mohr and Miller 1983; Miller et al. 1986) or on local continuity (Eilts and Smith 1990; Liang et al. 1997). Both methods need a starting point; therefore, they are not capable of dealiasing isolated areas of radar data without additional information on the environmental wind. This information could be provided as a profile from a nearby sounding (e.g., radiosonde or wind profiler) or from an NWP model. Defolding of radar winds can also be accomplished in a straightforward manner by always taking the Nyquist number that results in the smallest deviation from a given wind profile (Doviak and Zrnić 1993). More sophisticated dealiasing techniques based on, for instance, two- or more-dimensional variational methods have been developed during the last two decades (Merritt 1984; Boren et al. 1986; Bergen and Albers 1988; Desrochers 1989; Jing and Wiener 1993; Wüest et al. 2000). They try to identify regions with the same Nyquist number. Then adjacent regions are compared to detect and correct folded data.

Siggia and Holmes (1991) proposed a variation of the volume velocity processing (VVP) algorithm (Waldteufel and Corbin 1979), in which dealiasing is built into the procedure itself, rather than being done as a separate pass. Unlike the methods mentioned above, they do not need independent wind information. Their technique achieves a defolding factor of 3 by running 22 VVPs simultaneously on the same input dataset. Each VVP assumes a different trial wind field, and the one that yields the best fit (lowest variance) is adopted as the correct solution.

Based on the idea of Siggia and Holmes (1991) to combine VVP and defolding, a new dealiasing algorithm has been developed at SMHI recently. The innovation of the new method is that it maps the measurements onto the surface of a torus (section 2). Section 3 describes its validation and a comparison with the defolding approach of Siggia and Holmes (1991). Examples for dealiased radar products are presented in section 4, followed by concluding remarks in section 5.

The new method is based on a linear wind model V_m, in which the radial wind velocity in a specified height interval can be expressed as a function of azimuth (Φ) and elevation angle (Θ):

It should be mentioned that the zonal (u) and meridional wind speeds (υ) are assumed to be proportional to the horizontal transport of precipitation. For the sake of simplification, the vertical velocity of hydrometeors is neglected.

Assuming that the elevation angle and the distance to the radar are constant, each observation at a given azimuth angle is assigned a radial velocity. Unfortunately, the resulting curve could have discontinuities due to aliasing difficulties (Fig. 1a). To avoid this problem, we map the measurements onto the surface of a torus and yield a continuous parametric curve (Fig. 1b):

The unit vectors e₁, e₂, and e₃ describe the geometry of the new system. The choice of the torus radius R is arbitrary as long as R > V_a/π. Note that the possibly aliased velocities V_o and the unaliased velocities V now map onto the same points because

In the following, any of the parametric curve components (F₁, F₂, or F₃) can be used to derive the proposed algorithm. No extra information is gained by considering more than one component, since all three originate from Eq. (4). However, we chose F₃ because it simplifies the calculations.

The component F₃ can be approximated using Eq. (7) in conjunction with the linear wind assumption (V_m ≈ V):

Using Eq. (4), the third component of the tangent vector along all azimuth angles at constant range and elevation angle can be formulated as

Applying Eq. (6) and the linear wind assumption (V_m ≈ V), Eq. (8) yields

The derivative D can be estimated using a filter and a method that handles sparse data (e.g., Knutsson et al. 1993).

By dividing the set of observations into a number of subsets (e.g., corresponding to different height intervals), different solutions can be obtained for each subset. Suppose we have a subset with N observations. The model velocities u and υ within this subset can then be solved, using Eq. (9) and a least squares approach:

Afterward, the wind speeds u and υ are inserted into Eq. (4) and used to dealias the observed radial wind velocities:

Here, m is the assumed maximum Nyquist number. Finally, the dealiased radial wind velocity is obtained by inserting n into Eq. (3).

In order to make a validation of the new dealiasing algorithm (section 2) as realistic as possible, we decided to apply it on Doppler measurements from an existing radar network. The distribution of such measurements is a priori more natural than for a synthetically generated wind field.

In the validation process, Swedish Doppler radar observations are artificially aliased to a wind speed lower than the Nyquist velocity. Afterward, the method's capability to reconstruct the original wind field is examined.

Each C-band radar of the Swedish radar network generates volume data every 15 min. In the Doppler mode, two PRFs (900 and 1200 Hz) with corresponding Nyquist velocities of 12 and 16 m s⁻¹, respectively, are being used alternately during the time to scan a lobe width. The extended unambiguous velocity interval of the dual-PRF data thus becomes 48 m s⁻¹ (corresponding to a defolding factor of 3 in terms of the high PRF). Velocities close to zero are suppressed with a filter intended to cancel ground clutter. Selected characteristics of the weather radars in Sweden are compiled in Table 1.

Unfortunately, the presence of discrete dealiasing errors (so-called velocity outliers) in large areas of high quality data is characteristic for data obtained using the dual-PRF technique (e.g., Holleman and Beekhuis 2003). Since these velocity outliers will influence the results of the realiasing and re-dealiasing validation procedure negatively (in particular if the Nyquist velocity differs from the original high-PRF unambiguous velocity of the chosen radars), we decided to blacklist them. For this purpose, an algorithm proposed by Joe and May (2003) is used to detect dealiasing errors. In their analysis, the measurement is assumed to have zero variance. The detection is done by applying a Laplacian operator on the dual-PRF data to compute a discrimination parameter χ for each pixel. If the value of |χ| is less than the Nyquist velocity, then a pixel is assumed to be correct. If the value is greater, then an error is assumed to be detected and the pixel is blacklisted. In order to keep data manipulation to a minimum, we have not tried to corrected marked pixels.

The validation study considers volume Doppler velocity measurements for two typical weather situations: case “s” represents a stratiform wind field observed by the radar in Hemse, Sweden (57.24°N, 18.38°E), at 1047 UTC 2 July 2003, whereas case “c” typifies a convective wind field observed by the radar in Norrköping, Sweden (58.58°N, 16.15°E), at 0932 UTC 16 August 2001. In both cases, the maximum Nyquist number in Eq. (11) was m = 4. The filtered observations are shown in Fig. 2. In case “s,” 37 out of 443 805 valid pixels have been discarded, whereas in case “c,” 999 out of 147 883 valid pixels were blacklisted.

The results of the validation study are summarized in Table 2. The maximum unambiguous velocity has been assigned to 7.55 m s⁻¹, as it is for the lowest elevation angles of the Finnish radar network (Table 1). Velocity outliers have been filtered using the detection algorithm mentioned above. The columns of Table 2 distinguish between cases where the realiasing procedure was activated and those when it was not. The rows provide information whether the re-dealiasing was successful or not. Generally, the new dealiasing algorithm works better for the stratiform case than for the convective. For case “s,” 0.2% of all valid pixels have been reconstructed falsely, and for case “c” this figure is 5.5%. This might be caused by erroneous observations or a strong violation of the linear wind assumption. Another error source is the spatial and temporal inhomogeneity of the radial wind observations within each height interval (see section 2). There is a risk that data coming from different elevation scans with different time stamps may not match. Additionally, insufficient azimuthal radar data coverage reduces the performance of the new dealiasing technique. Furthermore, the estimation of D in Eq. (9) becomes more unstable for small Nyquist velocities.

Moreover, we compared the new method with the defolding technique developed by Siggia and Holmes (1991), which is part of SIGMET's radar software package IRIS. The comparison was motivated by two reasons: (i) neither algorithm needs independent wind information and (ii) IRIS is operated at many locations all over the world (at SMHI since 2004); therefore, it is assumed that the results of its implemented dealiasing technique are verified by numerous users under various weather situations.

Compared to the new algorithm, the percentage of falsely reconstructed pixels is significantly higher for both cases (Table 2), especially convective pixels that have been aliased artificially are processed incorrectly (16.3%).

Since data from the Finnish radar network are strongly affected by aliasing problems (currently, only single-PRF data are available for the lowest four elevation angles), we applied the new technique to defold Doppler winds measured by radar in Vantaa, Finland (60.27°N, 24.87°E), at 1200 UTC 4 December 1999. At this time a storm passed the Gulf of Finland eastward.

Figure 3a shows a plan position indicator (PPI) of the observed wind velocities for this particular case. Aliasing effects are clearly visible. After applying the new defolding algorithm, velocity jumps are almost eliminated and the wind field looks more reasonable (Fig. 3b). However, there are still some suspicious areas (e.g., in the southwestern corner). For possible explanations, see section 3b.

An important goal of the new dealiasing algorithm is to improve the assimilation of radial winds in NWP models via vertical azimuth display (VAD; Andersson 1998) and VVP profiles, respectively, or superobservations (see section 4c).

We generated wind profiles based on the VVP method, which is typically applied to thin layers of data at successive heights. Wind speed and direction can be extracted via a multidimensional and multiparameter linear fit of all observations in a certain height interval.

Figure 4 shows the azimuthal radar data coverage versus height for Vantaa radar at 1200 UTC 4 December 1999. Between ground level and 2900-m height, measurements from all directions are used in the VVP technique. Above 2900 m, the number of unique azimuth angles decreases drastically, which results in a more erroneous wind retrieval at these altitudes. This should be considered when interpreting the radar-derived wind profiles. If the azimuthal data coverage is less than 1/3 the wind retrieval is rejected at this height level.

A comparison of vertical profiles of wind speed and direction generated by the commercial radar software package IRIS and the new algorithm is presented in Fig. 5 (same date and radar location as in Fig. 3). Note that both methods use identical radial wind observations as input. The gray-shaded area indicates the standard deviation of the dealiased velocity measurements from the VVP solution for the novel technique. It is clearly visible that the two curves almost coincide. Long-term comparisons (30 h) between both processing algorithms for five Finnish radars reveal mostly good agreement.

Unfortunately, there is no radiosonde sounding available for the radar location in Vantaa. Instead, the radiosonde observation for Tallinn, Estonia (59.38°N, 24.58°E), is shown in Fig. 5 (approximately 100-km distance from Vantaa). Although the vertical resolution is much lower than for the radar measurements, structures in the wind speed and direction profiles are similar. The High Resolution Limited Area Model (HIRLAM; Undén et al. 2002) forecast (22-km gridpoint spacing, no assimilation of radar winds in 1999) reveals the same trend as the radar observations but not as detailed. Therefore, forecasts would probably benefit from an assimilation of dealiased radar radial winds. Currently, VAD profiles are assimilated operationally into HIRLAM in order to complement the radiosonde soundings in time and space. A defolding routine is not applied.

The dealiased volume radar data can be used in variational assimilation schemes for NWP models through the generation of so-called superobservations (Lindskog et al. 2000). A superobservation is an intelligently generalized observation created through smoothing in space based on high-resolution data. It includes also a number of derived variables that collectively serve to describe the characteristics of a given observation (Michelson 2003). At the Swedish Meteorological and Hydrological Institute (SMHI), a method for generation of radial wind superobservations through horizontal averaging in polar space of the raw polar volume data is already implemented.

An example how radial wind superobservations benefit from the new dealiasing technique is illustrated in Fig. 6. The generalized treatment of the original high-resolution polar data can be clearly discerned, as can the preservation of the polar nature of the derived superobservation product (4° azimuthal and 5-km radial resolution). In the output bins with only few high-resolution input bins, the superobservation generator has the effect of filling in gaps, which leads to more complete and smoothed results.

The radial wind superobservations have been used for the development of a Doppler data assimilation system for HIRLAM (Lindskog et al. 2000). The use of superobservations instead of high-resolution data has three major benefits: (i) higher representativeness in respect of the NWP model resolution, (ii) lower correlation of the observation errors, and (iii) lower computational costs during the assimilation cycle. Additionally, a radial wind superobservation provides more spatial information compared to a corresponding VAD and VVP profile, respectively.

A novel dealiasing algorithm for Doppler radar velocity data has been developed. The validation and comparison study revealed that the new method is an accurate and robust tool. Elimination of multiple folding is possible. However, the validation should be extended with more weather scenarios (e.g., squall lines, clear-air echoes) in order to study the sensitivity of the technique. Currently, we are preparing a dealiasing experiment for the Swedish and Finnish radar network, including representative time series for a winter and a summer period.

Unlike most other concepts, the method presented here does not depend on wind information from a nearby sounding (e.g., radiosonde or wind profiler) or from an NWP model. This results in independent data making computations efficient and real-time applications possible (no time-consuming iterations). Moreover, additional wind observations can be used for further validation purposes.

In the near future, the dealiased volume radar data will be applied to the HIRLAM variational assimilation scheme through the generation of wind profiles and superobservations. Their use is expected to improve substantially with the introduction of the proposed dealiasing method.

This work is financed in part by the European Commission under Contract EVG1-CT-2001-00045, “CARPE DIEM.” It contributes also to the COST-717 action entitled “Use of Radar Observations in Hydrological and NWP Models.” The authors gratefully acknowledge Magnus Lindskog and Daniel Michelson at SMHI for providing the HIRLAM forecasts and the Swedish radar observations, respectively. Special thanks also to Elena Saltikoff, Harri Hohti, and Thomas Skogberg (Finnish Meteorological Institute), who made the IRIS data available to us.