## 1. Introduction

Weather radars have been used widely, not only for operationally monitoring and issuing warnings of severe and hazardous weather, but also for providing important measurements to advance our understanding of the atmosphere (e.g., Serafin and Wilson 2000; National Research Council 2002). Reflectivity, mean radial velocity, and spectrum width are the three fundamental radar measurements, which are defined from the zeroth, first, and second moments of a Doppler spectrum and can be estimated by either the autocovariance method or spectral method (Doviak and Zrnić 1993). In the spectral approach, a Doppler spectrum can be initially obtained by the periodogram method and, consequently, the three spectral moments can be estimated by the moment method without using a priori knowledge of spectral shape (e.g., Doviak and Zrnić 1993; Bringi and Chandrasekar 2001). Spectral processing was identified by Fabry and Keeler (2003) as one of the trends in meteorological radar signal processing to enhance accuracy and sensitivity of weather information. For example, interference and/or clutter with frequency components different from the signals of interest can be readily edited on a Doppler spectrum to improve the data quality. Ice et al. (2004) have shown that Gaussian model adaptive processing (GMAP) (Siggia and Passarelli 2004) for clutter filtering in the spectral domain can provide better spectral moment estimation. Bachmann and Zrnić (2007) have applied spectral processing of polarimetric data to recover the underlying wind field when the clear-air signals are contaminated by birds and insects. Moreover, Doppler spectra reveal weighted radial velocity distribution to provide an additional view of the dynamics within the radar volume when the three spectral moments are not sufficient. For example, a tornado vortex can produce bimodal or flattened signatures (Zrnić and Doviak 1975; Zrnić and Istok 1980; Bluestein et al. 1997; Yu et al. 2007) and have been recently used to improve tornado detections that rely primarily on the velocity signature (Wang et al. 2008).

For weather radar, the three spectral moments are often obtained using the autocovariance method because of its robustness and computational efficiency (Sirmans and Bumgarner 1975; Zrnić 1977). In the autocovariance method, although the computation of Doppler spectrum is not needed, a Gaussian spectral model is assumed to derive the estimators of mean velocity and spectrum width. In other words, a bias in velocity and spectrum width resulted if the Doppler spectrum deviates from symmetry and Gaussian, respectively. Janssen and Spek (1985) reported that approximately 25% of their examined spectra from precipitation at close ranges, collected by a phased array system operated at 5.56 GHz with 4° beamwidth, are not Gaussian. Thus, it is of interest to characterize non-Gaussian spectra and systematically investigate their impact on the autocovariance estimators. Sirmans and Bumgarner (1975) briefly discussed in their appendix that the velocity bias caused by an unrealistic spectrum with asymmetric pattern of sawtooth is negligible. In this work, it will be shown that weather Doppler spectra with features such as dual peak, wide flat top, or a single Gaussian with strong tail are observed from a supercell thunderstorm by the research Weather Surveillance Radar-1988 Doppler (WSR-88D), KOUN, in Norman, Oklahoma. Motivated by these observations, a spectral model based on a mixture of two Gaussian functions is introduced to characterize various degrees of deviations from a Gaussian shape. In Vito et al. (1992), a similar but less flexible model was introduced, where the two Gaussian patterns have equal widths and a fixed power ratio of 0.5. It was shown that the bias of velocity is also small but increases with the difference of the two mean velocities. Moreover, a dual-Gaussian function was used to model both weather and ground clutter components for the purpose of clutter filtering (Nguyen et al. 2008).

This paper is organized as follows: in section 2 the model of dual-Gaussian spectrum is introduced. The bias of mean velocity and spectrum width estimated by the autocovariance method will be derived as a function of spectral moments of both Gaussian components. In addition, a nonlinear fitting algorithm is proposed to estimate the six spectral moments. In section 3 the theoretical bias derived in the previous section will be demonstrated and validated using four experiments of numerical simulations. Moreover, in section 4 spectra from a supercell thunderstorm will be shown to demonstrate the feasibility of the dual-Gaussian model and the proposed fitting algorithm. A hypothesis of the mechanisms for producing these non-Gaussian spectra will be presented and qualitatively verified using simulations in section 5. Finally, a summary and conclusions will be provided in section 6.

## 2. Dual-Gaussian Doppler spectrum

### a. Model of Doppler spectrum

The Doppler spectrum observed by weather radar, denoted by *S*(*υ*), represents the power weighted radial velocity distribution, where the power includes backscattered power from all the scatterers within the radar resolution volume that have radial velocity between *υ* and *υ* + *dυ*, radar’s range weighting function, and a two-way radar beam pattern. A Gaussian-shaped Doppler spectrum is typically assumed for radar signals and can be fully characterized by the three spectral moments of signal power, mean velocity, and spectrum width. However, they may not be sufficient to describe the spectral shape if the spectrum deviates from the Gaussian such as bimodal spectra from tornado vortices (e.g., Zrnić and Doviak 1975; Zrnić et al. 1985) and spectra with multiple peaks from insects, birds, and clear air (Bachmann and Zrnić 2007).

*S*

_{1}(

*υ*) and

*S*

_{2}(

*υ*) are the two Gaussian spectral components; and

*S*,

_{i}*υ*, and

_{i}*σ*,

_{i}*i*= 1 and 2 are the signal power, mean velocity, and spectrum width of the

*i*th component, respectively. A Gaussian spectrum is obtained if the three spectral moments of both components are identical. Although a generalization of

*n*Gaussian components can be formulated as presented in Boyer et al. (2003, 2004) and Zhang and Doviak (2008), the number of variables needed to define such a spectrum increases as 3

*n*. Moreover, from spectra of interest in this work, two components are likely to be sufficient for most weather signals. Nguyen et al. (2008) used the dual-Gaussian to model weather and clutter spectra and to extract weather information using a parametric estimation similar to those in Boyer et al. (2003, 2004). The signal power of the dual-Gaussian spectrum is obtained by the zeroth spectral moment

*S*=

*S*

_{1}+

*S*

_{2}. The normalized Doppler spectrum

*S*(

_{n}*υ*) =

*S*(

*υ*)/

*S*can be thought of as a mixture of two Gaussian probability density functions (pdfs). Its moments have been derived in the literature (e.g., Everitt and Hand 1981) and are presented as a function of

*S*,

_{i}*υ*, and

_{i}*σ*in appendix A. Hereafter, spectral moments are referred to as those of the dual-Gaussian model rather than a single Gaussian’s components unless specified otherwise. Note that the mean radial velocity is defined by the first moment as shown in (A1), the spectrum width (

_{i}*σ*) is defined by the square root of the second central moment (standard deviation) (A2), the skewness is defined by the third central moment normalized by the third power of the standard deviation (sk =

_{υ}*μ*

_{3}/

*σ*

_{υ}^{3}), and the kurtosis is defined by the fourth central moments normalized by the fourth power of standard deviation (kt =

*μ*

_{4}/

*σ*

_{υ}^{4}).

### b. Bias of spectral moments estimated by the autocovariance method

*m*is the temporal lag and

*υ*=

_{a}*λ*/(4

*T*) is the maximum unambiguous velocity with radar wavelength of

_{s}*λ*and the pulse repetition time

*T*. In the autocovariance method, the mean velocity, denoted by

_{s}*υ*, is defined from the argument of the autocorrelation at the first temporal lag. In other words, where

_{r}^{a}*A*=

_{i}*S*exp[−

_{i}*πσ*/

_{i}*υ*)

_{a}^{2}] and

*ϕ*= −(

_{i}*πυ*/

_{i}*υ*),

_{a}*i*= 1 and 2. Furthermore, the spectrum width can be obtained from the autocorrelation function at lags zero and one using the following form:

The signal power estimated from the autocorrelation function at zero lag is an unbiased estimator. The theoretical biases of the velocity and spectrum width estimates in the autocovariance method are defined by *b*(*υ _{r}^{a}*) =

*υ*−

_{r}^{a}*υ*and

_{r}*b*(

*σ*) =

_{υ}^{a}*σ*−

_{υ}^{a}*σ*, respectively, with

_{υ}*υ*and

_{r}*σ*given in (A1) and (A2). The bias of velocity and spectrum width estimators depends on the spectral moments of both Gaussian components. In practice,

_{υ}*υ*and

_{r}^{a}*σ*are obtained from the estimated autocorrelation function and therefore the biases will also be a function of the signal-to-noise ratio (SNR) and the number of samples. Numerical simulations with consideration of these practical issues are used to verify these biases in section 3b. It is important to point out that if spectral moments of both Gaussian components are of interest, the autocovariance estimators only provide a weighted average as shown in (A1) and (A2) even though they are unbiased.

_{υ}^{a}### c. Dual-Gaussian moments estimation

*n*Gaussian components and the noise power are derived. In this work, the six spectral moments from both Gaussian components and the noise level are estimated by minimizing the mean-square errors (MSEs) between the model spectrum of (1) and observed spectrum, defined in the following equation: where

*Ŝ*(

*υ*) is the estimated Doppler spectrum using

*M*samples of raw time series data and

*σ*

_{n}^{2}/(2

*υ*) is the noise level. Note that the proposed dual-Gaussian fitting algorithm is implemented in decibel scale to suppress inherent statistical fluctuations from the spectral estimation in order to provide robust results. The minimization is solved using a Gaussian–Newton method with Levenberg–Marquardt modification (Seber and Wild 2003). Consequently, a dual-Gaussian spectrum can be reconstructed using the estimated spectral moments and noise level. The initial conditions for the optimization will be discussed in more detail in sections 3b and 4 for simulations and real data, respectively.

_{a}## 3. Experiments of numerical simulations

Four numerical simulation experiments were designed to study various degrees of non-Gaussianarity of spectra using the dual-Gaussian model with the following two goals. The first goal, in section 3a, is to demonstrate and quantify the theoretical biases in velocity and spectrum width estimates of the autocovariance method. The second goal is to verify these biases using simulations in which practical conditions such as the finite number of samples, statistical fluctuations, and limited SNR are considered, in section 3b. In each experiment *S*_{1}(*υ*) is fixed with signal power (*S*_{1}) of 10 dBm, mean velocity (*υ*_{1}) of 10 m s^{−1}, and spectrum width (*σ*_{1}) of 2 m s^{−1}. But the second Gaussian component has mean velocity (*υ*_{2}) that varies from −10 to 10 m s^{−1} with a 1 m s^{−1} interval. In addition, its signal power (*S*_{2}) and spectrum width (*σ*_{2}) for the four experiments are provided in Table 1.

### a. Theoretical biases

The theoretical mean velocity (*υ _{r}*) and spectrum width (

*σ*) of a dual-Gaussian Doppler spectrum, derived in (A1) and (A2), as a function of

_{υ}*δυ*=

_{r}*υ*

_{2}−

*υ*

_{1}for the four experiments are denoted by solid lines in Fig. 1 (top).

Theoretical results from experiments I to IV are depicted by red, blue, green, and black lines, respectively. In addition, the mean velocity (*υ _{r}^{a}*) and spectrum width (

*σ*) derived from the idealized autocorrelation function using (3) and (4) are denoted by dashed lines with the same color scheme as the theoretical results. The theoretical bias of mean velocity estimates as a function of skewness is presented in the lower left panel and the bias of spectrum width estimates as a function of kurtosis is shown in the lower right panel. Note that the theoretical values of skewness and kurtosis from a Gaussian spectrum are zero and three, respectively.

_{υ}^{a}In the upper left panel of Fig. 1, the theoretical mean velocity (*υ _{r}*) in experiments I and II varies linearly with

*δυ*due to the equal power of both Gaussian components. Additionally, the mean velocities estimated by the autocovariance method are all unbiased (i.e.,

_{r}*υ*=

_{r}^{a}*υ*) in experiment I, which are manifested by a single point of zero skewness and zero bias in the lower left panel. Thus, only the result of

_{r}*υ*in experiment II (blue solid line) is shown. In experiments II, III, and IV, the autocovariance velocities are biased due to the asymmetric spectra, except for

_{r}*δυ*= 0. It is also apparent that the velocity bias increases with the increasing separation between the two Gaussian components for both experiments II and III. For experiment IV, although the asymmetry of spectra can still be reflected by nonzero skewness, the theoretical and estimated mean velocities are determined by the first component with dominant power. It is important to note that although skewness is a measure of asymmetry of a spectrum, the amount of velocity bias cannot be directly inferred by the value of the skewness. For these experiments, the maximum velocity bias of approximately 2.5 m s

_{r}^{−1}is obtained from experiment II.

For spectrum width, it is apparent in experiments I, II, and III that spectrum widths and the biases of spectrum width estimates increase with the separation of the two Gaussian components. In experiment IV, the spectrum width is mainly determined by the first Gaussian that has much stronger power. It is shown in the lower right panel that the kurtosis increases with the decrease of |*δυ _{r}*| in both experiments I and II. In addition, the kurtosis reaches the reference value of 3 for

*δυ*= 0 m s

_{r}^{−1}in experiment I, when the spectrum becomes Gaussian. In experiment III, spectra with strong tails or wide skirt, contributed by the second Gaussian, are obtained, and produce kurtosis of larger than 3. Extremely large values of kurtosis can be caused by the secondary Gaussian in experiment IV, even though it has a negligible effect on velocity and spectrum width estimates. Moreover, the autocovariance method overestimates the spectrum width for cases when kurtosis is smaller than 3 (flat top) and tends to underestimate the width for kurtosis larger than 3. However, the degree of deviation of kurtosis from 3 cannot readily reflect the amount of bias in spectrum width estimates either.

### b. Numerical simulations

To further verify these theoretical results, time series of complex-value weather radar signals with realistic statistical fluctuations and noise contributions were generated using the scheme similar to Zrnić (1975). The SNR is set to 30 dB for all the experiments. The number of samples (*M*) and the maximum unambiguous velocity (*υ _{a}*) are 64 and 25 m s

^{−1}, respectively. For each

*δυ*in an experiment, the estimated autocorrelation function and spectrum are averaged over eight independent trials to reduce their variances. In practice, this can be achieved by range averaging with a trade-off of reducing range resolution. The means of the biases for the autocovariance-estimated velocity and spectrum width over 200 realizations for the four experiments are denoted by filled circles in the lower two panels of Fig. 1 with the same color scheme defined previously. It is evident that simulation results agree well with the theoretical bias derived in section 3b for SNR = 30 dB. Although it is not shown, the standard deviation of the velocity and spectrum width biases generally decreases with decreasing |

_{r}*δυ*| because the true spectrum width decreases.

_{r}Examples of estimated and reconstructed spectra from one realization are shown in Fig. 2.

The four columns from left to right represent results from the four experiments, and the three rows from top to bottom are for the cases of *δυ _{r}* = −18, −14, and −2 m s

^{−1}, respectively. As expected, the autocovariance method is not sufficient to characterize the spectra except when only a single Gaussian pattern is apparent, such as for

*δυ*= −2 m s

_{r}^{−1}in experiments I, II, and IV. On the other hand, spectra reconstructed from the dual-Gaussian fitting algorithm can better characterize the simulated non-Gaussian spectra. The fitting algorithm is initialized by a number of initial conditions, and the final estimates are obtained by the one that produces the minimal MSE. A similar procedure was used in Boyer et al. (2004) and Nguyen et al. (2008). The initial values of

*S*

_{1}and

*S*

_{2}are both fixed at

*R̂*(0)/2, where

*R̂*(0) is the total power estimated by the autocorrelation at zero lag. The initial value of noise level is the mean of the lower 30% of spectrum values. The first guess of

*υ*

_{1}is always the location of the peak spectrum. Moreover, the initial value of

*υ*

_{2}varies from −

*υ*to

_{a}*υ*with an interval of 2.5 m s

_{a}^{−1}, while the initial values of both

*σ*

_{1}and

*σ*

_{2}increase from 1 to 7 m s

^{−1}every 1 m s

^{−1}.

The root-mean-square error (RMSE; ɛ) derived from the simulated spectrum and spectra reconstructed using both autocovariance and dual-Gaussian fitting techniques as shown in (5) is calculated for each realization. The pdfs of RMSE for the four experiments are presented in Fig. 3.

The pdf is obtained by the number of occurrences from a histogram and normalized by the total number of cases. The RMSE of the autocovariance method in experiment I has the widest distribution with the maximum error of approximately 18 dB due to the overestimated spectrum width as demonstrated in Fig. 2. It is also shown that the RMSE from the dual-Gaussian fitting technique exhibits more concentrated distribution with smaller mean than those from the autocovariance method for the four experiments. The mean of the RMSE distribution for the dual-Gaussian fitting is approximately 1.48, 1.81, 1.69, and 1.49 dB for the four experiments, respectively. These statistical results indicate that the spectral moments estimated from the fitting algorithm can better characterize the spectra. However, it should be noted that the dual-Gaussian fitting algorithm will not provide any information in addition to that from the autocovariance method if the Doppler spectrum is Gaussian and is more computationally expensive due to the iterative procedure in the minimization. Note that in these experiments the estimated spectral moments from the dual-Gaussian fitting algorithm are robust, especially at large values of *δυ _{r}* (> −4 m s

^{−1}). When the separation of the two Gaussians becomes smaller, the estimated moments are susceptible to initial conditions and fluctuations in a spectrum and the results can be biased. Although finer partitions of the seed values can improve the estimation, the computational cost becomes expensive. In these cases, the maximum likelihood estimation (Boyer et al. 2003, 2004) might provide more reliable estimates.

## 4. Experimental results of radar observations

As discussed earlier, the dual-Gaussian fitting algorithm is most advantageous when the Doppler spectrum deviates from a Gaussian shape and can be characterized by a mixture of two Gaussian functions. The three spectral moments of each component can be obtained, while the conventional autocovariance method is limited and can produce biased estimates of mean velocity and spectrum width. In this section, many interesting spectra that deviate from Gaussian are shown from a tornadic supercell observed by the KOUN operated by the National Severe Storms Laboratory (NSSL). In addition, the dual-Gaussian fitting algorithm was applied to these spectra to retrieve the six moments. KOUN has a typical spatial resolution of approximately 1° in angle and 250 m in range and has the capability of continuously collecting time series of in-phase and quadrature signals for several hours. Time series data from a supercell thunderstorm in central Oklahoma on 10 May 2003 were collected. The autocovariance-processed fields of reflectivity, mean radial velocity, and spectrum width at approximately 0405 UTC are presented from the left to the right columns in Fig. 4, respectively. In addition, the top and bottom rows are those fields from elevation angles (*ϕ*) of 1.5° and 0.5°, respectively. The KOUN is located at the origin and the data associated with reflectivity less than 20 dB*Z* are not shown for clarity.

From the lowest elevation scan, evident signatures of hook-shape reflectivity and strong inbound and outbound velocities can be observed at 25 km east and 40 km north of KOUN, indicating the presence of a tornado. In addition, large spectrum widths can be observed in this small region, which has been incorporated into an artificial intelligent algorithm to improve tornado detection (Wang et al. 2008). Generally speaking, these fields from the two consecutive elevation scans exhibit fair continuity, except for a region of enhanced spectrum widths (>6 m s^{−1}) only at the lowest elevation angle of 0.5°, which appears as a fan shape on the south side of the storm and northeast of the hook echoes. It has been reported that the median spectrum width from isolated tornadic storms has values smaller than 2 m s^{−1} (Fang et al. 2004). Moreover, the fields of spectral variability, a measure of the non-Gaussianity of Doppler spectrum (Janssen and Spek 1985), and estimated skewness and kurtosis are shown in appendix B, where more detailed discussions of these variables are provided. The results indicate that the region of enhanced spectrum width is associated with non-Gaussian spectra that are manifested by high values of variability, deviation of skewness, and kurtosis from zero and three, respectively. Another interesting region is located at 38 km east and 45 km north of KOUN that is characterized by extremely negative skewness (<−2), large positive kurtosis (>7), but moderate spectrum width (approximately 3–4 m s^{−1}). It is of interest to investigate what can produce these non-Gaussian spectra.

The dual-Gaussian fitting algorithm was applied to all the regions with reflectivity larger than 20 dB*Z*. Spectra from the region of enhanced spectrum widths are exemplified using those from the azimuthal angle (*θ*) of 35° and range between 55 and 75 km (depicted by the line *AB*

Spectra were initially obtained using the periodogram method with von Hann window to increase the dynamic range (Doviak and Zrnić 1993). To reduce statistical fluctuations, spectra were subsequently averaged in range by a running window with a size of eight gates (i.e., 2 km). Averaged spectra from every other of the six gates are displayed in decibels in the third column of Fig. 5 with solid lines indicating those from the elevation angle of 0.5° and dashed–dotted lines denoting those from *ϕ* = 1.5°. The maximum unambiguous velocity (*υ _{a}*) is approximately 32 m s

^{−1}. Note that the autocorrelation functions were also smoothed using the same scheme of averaging. The fields of mean velocity and spectrum width shown in Fig. 4 were those obtained from the range-averaged autocorrelation functions. At each range gate, a Gaussian-shape spectrum can be reconstructed using the three spectral moments estimated by the autocovariance and is denoted by

*Ŝ*(

_{a}*υ*). Moreover, the dual-Gaussian fitting algorithm was applied to the averaged spectrum using two sets of initial conditions. The first set of initial values is obtained by repeating the three moments from the autocovariance-derived moments except the power is halved. The second set is the six estimated spectral moments from the previous gate whenever they are available. This initial condition can improve the estimation when these moments are continuous in range. Consequently, the reconstructed spectrum

*Ŝ*(

_{d}*υ*) is obtained by the resulting moments from the one with the lower MSE. Those reconstructed spectra are shown in the first and second columns for the two elevation scans, respectively. Averaged spectra shown in the third column are also included and denoted by the thick gray line. It is shown qualitatively that the observed spectra can be better characterized by the dual-Gaussian model for both scans. Note that spectra at

*ϕ*= 0.5° possess signatures of broad spectrum with relatively flat-top or dual-peak patterns becoming narrower with increasing range. These spectra can be reconstructed by two Gaussians with distinct mean velocities but comparable power, which is similar to the scenario of experiment I in section 3. It is also evident that the autocovariance method produces a wider spectrum with mean velocities approximately between the locations of the two peaks, as shown in the first column of Fig. 5. For the case of

*ϕ*= 1.5°, spectra exhibit signatures of a single dominant peak and sometimes with tails (wider bases) such as those from ranges between 60 and 68 km. In the dual-Gaussian model, the dominant peak is characterized by a strong and narrow Gaussian component, and the tail can be described by the second component with smaller power and larger spectrum width. This scenario is similar to the conditions in numerical experiments III and IV. As a result, the autocovariance method can estimate the mean velocity of the dominant component but has the tendency to overestimate its spectrum width as the tail becomes stronger, as demonstrated by those spectra between 60 and 70 km. As spectra become more Gaussian-like between 72 and 76 km, the difference of reconstructed spectra from the autocovariance and dual-Gaussian fitting methods becomes smaller. Another interesting result is that spectra from

*ϕ*= 1.5° overlapped with the right portion of the spectra at

*ϕ*= 0.5°, which can be exemplified by the spectra between 65 and 70 km as shown in the rightmost column of Fig. 5. A hypothesis of the underlying process that produces these non-Gaussian spectra from the two consecutive elevation scans will be discussed in section 5.

To further investigate spectral moments estimated by the autocovariance method and dual-Gaussian fitting method, we define the two Gaussian components: *Ŝ*_{1}(*υ*) has the larger mean velocity and *Ŝ*_{2}(*υ*) is the other component. In other words, *Ŝ*_{1}(*υ*) in Fig. 5 represents the right component of a spectrum. The estimated signal power, mean velocity, and spectrum width of both components as a function of range are shown in the top six panels in Fig. 6 for both elevation angles.

Spectral moments of *Ŝ*_{1}(*υ*) and *Ŝ*_{2}(*υ*) are denoted by green and red lines, respectively, and those from the autocovariance method are indicated by blue solid lines. The estimated signal power, mean radial velocity, and spectrum width from *Ŝ _{i}*(

*υ*) are denoted by

*Ŝ*,

_{i}*υ̂*, and

_{i}*σ̂*,

_{i}*i*= 1 and 2, respectively; and the three moments estimated by the autocovariance method are denoted by

*Ŝ*,

*υ̂*, and

_{r}*σ̂*. It is shown that

_{υ}*Ŝ*

_{1}and

*Ŝ*

_{2}are comparable with a maximum difference of 8.5 dB at approximately 63.4 km. In the top second column, the mean velocities estimated by the autocovariance method

*υ̂*have values between those from the two Gaussian components,

_{r}*υ̂*

_{1}, and

*υ̂*

_{2}. As discussed in appendix A, the mean velocity of a dual-Gaussian spectrum is a weighted average of the mean velocities from each component. The weights are determined by the signal powers from each component. The autocovariance method would provide such an estimate if the spectrum is symmetry (i.e., unbiased). The weighted mean velocity, defined by

_{r}= (

*Ŝ*

_{1}

*υ̂*

_{1}+

*Ŝ*

_{2}

*υ̂*

_{2})/(

*Ŝ*

_{1}+

*Ŝ*

_{2}), is also included and is depicted by a blue dashed line. Good agreement between

_{r}and

*υ̂*suggests that the bias in the velocity estimate from the autocovariance method is relatively small. In addition, spectrum widths from the two Gaussian components have comparable magnitudes of 2–3 m s

_{r}^{−1}between 60 and 73 km, while the spectrum widths from the autocovariance method have values that are approximately doubled due to flat-top spectra.

For *ϕ* = 1.5°, *Ŝ*_{1}(*υ*) usually represents the Gaussian component with dominant peak and *Ŝ*_{2}(*υ*) describes the tail. It appears clearly that *Ŝ*_{2}(*υ*) has signal power approximately 20–30 dB lower than the one of *Ŝ*_{1}(*υ*) and has relatively large spectrum width between 57 and 71 km. For the autocovariance-derived mean velocities from *ϕ* = 1.5°, they are determined by the stronger component *Ŝ*_{1}(*υ*), which is consistent with results from numerical experiment IV. The weighted mean velocity _{r} also agrees well with *υ̂ _{r}*, which is suggested by symmetric spectra in Fig. 5.

The same analysis is performed for spectra from the azimuth angle of 41°, as indicated by *CD*

For *ϕ* = 0.5°, the dual-peak pattern is still evident for spectra between 66 and 76 km and, similarly, the autocovariance method overestimates the spectrum widths. At closer ranges, note that the spectrum widths from the autocovariance method are relatively small (<5 m s^{−1}) compared to those from azimuth of 35°. However, spectra are still deviated from Gaussian and can be better described by the dual-Gaussian model. As the range decreases, the left Gaussian component [*Ŝ*_{2}(*υ*)] decreases in amplitude and *Ŝ*_{1}(*υ*) becomes dominant. Nevertheless, the mean velocities from the autocovariance method are generally consistent with those obtained by the weighted average (_{r}). Discrepancy larger than 1 m s^{−1} can be observed between approximately 66 and 68 km in the lower second panel of Fig. 6. This could be caused by the uncertainty introduced in the dual-Gaussian fitting algorithm. Most spectra from *ϕ* = 1.5° at this radial still show a single dominate peak that appeared to be related to the right portion of the spectra from the lower elevation scan, similar to the results from *θ* = 35°. It is also shown in the lower fifth panel that the mean velocities from *Ŝ*_{1}(*υ*) and *Ŝ*_{2}(*υ*) are similar as well as those from the autocovariance method. The abrupt change of signal power and spectrum width, such as those from azimuth of 35° and elevation angle of 1.5° at approximately 73.8 and 75 km and from azimuth of 41° and elevation angle of 1.5° at approximately 61, 63, and 74 km, occurs when the two components have crossed over in range. For example, at a given range the dual-Gaussian has the stronger component with larger mean velocity and at the next gate the stronger component becomes the one with smaller mean velocity. This can be caused by the variation of wind field and/or reflectivity in consecutive ranges. Although this problem can be avoided by assigning the two Gaussian components based on their estimated signal power, there are cases that the mean velocities from the two components, on the other hand, can exhibit these abrupt changes. Therefore, one should be aware of how the two Gaussians are assigned for interpreting the fields of these six moments.

## 5. Discussion

*ϕ*= 0.5° is characterized by the flat-top and dual-peak spectra. Let us now investigate what is responsible for such spectra at

*ϕ*= 0.5° and how they are related to spectra from a higher elevation scan. A dual-peak pattern may be obtained if two independent processes are present simultaneously within the radar resolution volume. For example, for a VHF or UHF profiler radar, dual-Gaussian spectra can result when both precipitation and clear-air echoes are presented and have different mean velocities (e.g., Boyer et al. 2003, 2004). For weather radar, spectra with a dual-peak signature were simulated and observed from tornadic vortices when the radar volume is sufficiently large to encompass both the inbound and outbound radial components and no significant velocity aliasing occurs (e.g., Zrnić and Doviak 1975; Bluestein et al. 1997; Yu et al. 2007). However, the region of enhanced spectrum width is fairly large and is not in the vicinity of the tornado. Nevertheless these studies provide the idea that strong shear has the potential to produce such a spectral signature. Since the dual-peak or sometimes the flat-top spectra are observed in a number of consecutive radials, it is not likely such a shear pattern can repeatedly occur in the azimuthal direction. Therefore, the shear would be likely in the vertical direction. Indeed, it has been shown that vertical shear plays an important role in convective storms (e.g., Weisman and Klemp 1982; Weisman and Rotundo 2000). It should be noted that different combinations of reflectivity and radial velocity distributions can produce a similar spectrum. Therefore, it is not our goal to retrieve the radial velocity and reflectivity distribution from spectra. Instead, we have attempted to qualitatively illustrate how a reasonable distribution of radial velocity and reflectivity could produce spectra similar to those from the observations. Since only the vertical direction is of interest, mathematically the spectrum can be simplified to a 1D case as shown in the following formula (Doviak and Zrnić 1993): where

*f*

_{b}^{4}(

*ϕ*) is the two-way beam pattern in the elevation direction;

*Z*(

*z*) and

*υ*(

*z*) are the reflectivity and radial velocity in the vertical direction, respectively; and

*C*is a parameter that is a function of radar wavelength, peak transmitted power, range, antenna gain, and range weighting function. From (6),

*S*(

*υ*)

*dυ*represents the return power from all scatterers with radial velocity between

*υ*and

*υ*+

*dυ*and is obtained by numerical simulations in this work. We use the simulation scheme similar to that of Bluestein et al. (1993) and Yu et al. (2007). Initially, the reflectivity and velocity are sampled at a finer scale. A Gaussian-shaped beam pattern from (11.117a) in Doviak and Zrnić (1993) is used with a half-power beamwidth of 1°. The height of each ray path is calculated using

*z*=

*r*

^{2}+

*R*′

^{2}+ 2

*r R*′ sin

*ϕ*

*R*′, where the range

*r*= 65 km,

*R*′ is 4/3 of the earth radius, and

*ϕ*is the elevation angle of each ray path (Rinehart 2004). The integration in (6) is subsequently performed by a summation of return power at heights with radial velocities between

*υ*and

*υ*+

*dυ*over the radar volume, which is defined by the angle between −2° and 2° from the center of radar beam.

To obtain some idea of a shear pattern, we first examine the relationship of spectra between *ϕ* = 0.5° and 1.5° from the third column of Figs. 5 and 7. It appears that *Ŝ*_{1}(*υ*) maintains fairly good consistency from the two consecutive elevation scans, which suggests that the shear pattern responsible for *Ŝ*_{1}(*υ*) at *ϕ* = 0.5° could occur in the upper portion of the beam and continuously extend to higher altitudes. Additionally, the radial velocities at the lower portion of the beam at *ϕ* = 0.5° are smaller in order to produce *Ŝ*_{2}(*υ*) with smaller mean velocity. The profile of radial velocity is modeled using *υ _{r}* =

*υ*tanh(

_{s}*z*/

*z*), which is the same formula as the one used in Weisman and Klemp (1982). We set

_{s}*υ*= 20 m s

_{s}^{−1}and

*z*= 800 m. The profile is shown in the upper left panel of Fig. 8.

_{s}Additional velocity fluctuations generated from a normal distribution with a standard deviation of 2 m s^{−1} were added. Two cases of reflectivity profiles presented in the second column were used in simulations and the resulted spectra are shown in the lower two panels. For each case, spectra at both *ϕ* = 0.5° and 1.5° are generated and spectra with dual-peak signature are obtained from 0.5° for both cases. For case I both components of the dual-peak pattern have comparable power, and for case II *S*_{1}(*υ*) (the one with larger mean velocity) has relatively large power, which is similar to spectra observed from approximately 66 km and azimuths of 35° and 41°, respectively. Moreover, spectra at the elevation angle of 1.5° exhibit tail signature and their strong components are similar to the right component of the spectra [*S*_{1}(*υ*)] at 0.5°. Note that a Doppler spectrum represents a weighted radial velocity distribution within the radar volume. In other words, the velocity profile determines the radial components in a spectrum. Additionally, the shape of spectrum is determined by the effective weighting function, which is defined by the product of reflectivity and beam pattern [*W* ≡ *f _{b}*

^{4}(

*z*)

*Z*(

*z*)] and is shown in the top third column for both cases. The ordinate on the right represents corresponding elevation angles, where the center of the two beam locations is denoted by two arrows. Their heights are approximately 0.86 and 2.05 km, respectively. It can be observed that the effective weighting function shows a dual-peak pattern for the beam centered at

*ϕ*= 0.5° for both cases. In addition, it is apparent that more retuned power comes from the lower portion of beam (where smaller velocities occur) in case I than in case II. Moreover, it is hypothesized that the spectral tail at 1.5° is partially due to scatterers from a lower altitude through the beam pattern. In practice, the spectral shape can vary slightly for different window functions used in the estimation of the spectrum. Note that the tail is not as symmetrical as that from observations. To produce such a result, a more complicated radial velocity profile is needed and probably cannot be modeled by a simple mathematical equation such as those used here. Nevertheless, our simple approach can produce spectra that are similar to those observed to some extent. Furthermore, the region of non-Gaussian spectra, defined by spectral variability larger than 15 (dB)

^{2}(see appendix B), is consistent with the region of the polarimetric signature termed

*Z*

_{DR}arc, which is a shallow signature and is typically observed on the south side of the storm where strong reflectivity gradient is present (Kumjian and Ryzhkov 2007, 2008). It has been shown using numerical simulations that

*Z*

_{DR}arc can result from the size sorting of hydrometeors in a strong vertical shear environment. The independent polarimetric measurements further support our hypothesis that those non-Gaussian spectra resulted from vertical shear.

## 6. Conclusions

It is typically assumed that weather Doppler spectra have a Gaussian shape and, therefore, can be fully defined by three spectral moments, namely, signal power, mean velocity, and spectrum width. These moments can be estimated by the robust and efficient autocovariance method, where the Doppler spectrum is assumed to be Gaussian-shaped. In other words, biases of mean velocity and spectrum width estimates may be obtained if a Doppler spectrum is asymmetrical and/or deviates from a Gaussian. In this work, a generalized dual-Gaussian model is introduced to characterize various non-Gaussian weather spectra. As a result, the mean power, Doppler velocity, and spectrum width of a dual-Gaussian spectrum can be derived analytically as a function of the three spectral moments of each component. Additionally, skewness and kurtosis, which have been used to measure asymmetry and peakness/tail of a distribution, respectively, are also provided. Theoretical bias of mean radial velocity and spectrum width estimated by the autocovariance method in relation to the six spectral moments is also derived.

Four experiments were designed to demonstrate the theoretical biases of the autocovariance method. Simulations of realistic weather radar signals with considerations of statistical fluctuations, noise contamination, and finite number of samples are performed to further verify these biases. The results have shown that the bias of velocity and spectrum width is neither linearly nor uniquely related to skewness and kurtosis, respectively. Even though the bias of autocovariance-estimated velocity and spectrum width is moderate for the four experiments, the three spectral moments only represent weighted averages of the moments from each Gaussian component. Therefore, a nonlinear fitting algorithm is introduced to estimate the six spectral moments as well as the noise level. It is shown via simulations that statistically, spectra reconstructed from the dual-Gaussian fitting algorithm can better describe these non-Gaussian spectra than those from the autocovariance method.

Examples of spectra from a supercell thunderstorm collected by the KOUN radar in Norman, Oklahoma, are presented. Interesting spectra with dual-peak or flat-top features are observed at the lower elevation angle of 0.5°, which causes the autocovariance method to overestimate the spectrum widths. At the consecutive higher elevation scan, spectra correlate well with the right components of the spectra from the lower elevation scan. In addition, tails are often observed for those spectra that seem to relate to the left component of the spectrum at the lower scan. A hypothesis of vertical shear pattern with enhanced reflectivity structure at lower altitudes is proposed. Spectra from hypothesized profiles of reflectivity and radial velocity are simulated in 1D. Similar features to those from observations can be reproduced to some extent using these idealized profiles. A comprehensive analysis of spectra in a supercell is planned using a more sophisticated radar simulator based on numerical weather models. In this work, it is shown that spectra can reveal the radial velocity distribution, which has the potential to provide insight into detailed dynamics within the radar volume. Moreover, it is important to study how these additional spectral moments from the dual-Gaussian fitting algorithm can be used to assist the analysis of storm dynamics.

This work was primarily supported by NOAA/NSSL under Cooperative Agreement NA17RJ1227. Part of this work was supported by the National Science Foundation through Grant ATM-0532107 and DOD EPSCoR Grant N00014-06-1-0590. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation. The authors would also like to acknowledge the technical support of NSSL in the collection of KOUN data. The authors would also like to thank Drs. Melnikov and Zrnić at NSSL for useful discussions of spectra.

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# APPENDIX A

## Spectral Moments of a Dual-Gaussian Spectrum

*S*,

_{i}*υ*, and

_{i}*σ*will now be presented. The signal power of the dual-Gaussian spectrum is obtained by the zeroth moment

_{i}*S*=

*S*

_{1}+

*S*

_{2}. Moreover, the normalized Doppler spectrum

*S*(

_{n}*υ*) =

*S*(

*υ*)/

*S*can be thought of as a mixture of two Gaussian probability density functions (pdfs) and its moments have been derived in the literature (e.g., Everitt and Hand 1981). The first moment and second central moments are obtained by the following equations of (A1) and (A2), respectively: where

*P*

_{1}=

*S*

_{1}/

*S*and

*P*

_{2}=

*S*

_{2}/

*S*. The mean velocity of the dual-Gaussian spectrum is defined by the first moment and is a weighted average of the mean velocities from each Gaussian component. For example,

*υ*is the arithmetic average of

_{r}*υ*

_{1}and

*υ*

_{2}if the two Gaussians have equal powers. In addition, the spectrum width (

*σ*) is defined by the square root of the second central moment (standard deviation), which depends on the weighted average of the spectrum widths from individual Gaussians and the separation of the their mean velocities. If the two Gaussians have the same mean velocities, the second moment is the weighted average of

_{υ}*σ*

_{1}

^{2}and

*σ*

_{2}

^{2}. Hereafter, spectral moments are referred to as those of the dual-Gaussian model rather than single-Gaussian components unless specified otherwise.

*μ*

_{3}/

*σ*

_{υ}^{3}). The third central moment can be obtained by the following formula (Everitt and Hand 1981): The skewness is zero for a symmetric spectrum while positive (negative) values indicate that the right (left) side of the spectrum is heavier than the left (right) side. It is apparent from (A3) that a symmetric pattern can be obtained if

*υ*

_{2}=

*υ*

_{1}or

*P*

_{2}=

*P*

_{1}and

*σ*

_{2}=

*σ*

_{1}.

*μ*

_{4}/

*σ*

_{υ}^{4}), is used to characterize the peakness or the strength of the tail in a pdf. The fourth central moment is provided in the following equation:

The theoretical value of kurtosis for a Gaussian distribution is 3, which can be verified by a Gaussian spectrum using *P*_{1} = *P*_{2} = 1/2, *υ*_{1} = *υ*_{2}, and *σ*_{1}^{2} = *σ*_{2}^{2} in (A4). Kurtosis with values smaller than 3 can result from a spectrum with variations around the mean velocity smoother than a Gaussian (i.e., flat-top spectrum). On the other hand, kurtosis larger than 3 can be obtained if a spectrum decreases faster than a Gaussian from its mean (i.e., sharper peak) or the spectrum has significant tails.

# APPENDIX B

## Identification of Non-Gaussian Spectra

*Ŝ*(

*υ*)] and Gaussian-fitted spectra [

*G*(

*υ*)] in decibel scale as shown in the following formula: Note that (B1) is similar to (5) except for

*G*(

*υ*), which can be obtained using the fitting algorithm presented in section 2c but with a model of single-Gaussian spectrum. In this work, the spectral moments from the autocovariance method are used for the initial condition for the optimization. Since it is known that

*Ŝ*(

*υ*) has a

*χ*

^{2}distribution, Janssen and Spek (1985) have shown that the expected value of the spectral variability can be derived from the variance of

*Ŝ*(

*υ*), which is a function of the known window function in the spectral estimation and the degrees of freedom of the

*χ*

^{2}distribution. A large value of spectral variability indicates that the observed spectrum significantly deviates from the fitted spectrum, which always has a Gaussian shape. A detailed derivation is provided in the appendix of Janssen and Spek (1985). In this work, the expected value of spectral variability is approximately 2.5 (dB)

^{2}for the case of 64 samples and 8-gate averaging.

A plan position indicator (PPI) plot of the spectral variability from the 0.5° elevation angle of the KOUN data is presented in the first panel of Fig. B1. The estimated skewness and kurtosis are also provided in the second and third panels, respectively. In Janssen and Spek (1985), spectral variability larger than 15 (dB)^{2} is identified as a non-Gaussian spectrum. The same criterion is applied and a contour of spectral variability of 15 (dB)^{2} is depicted by black lines. It is apparent that spectra from the south side of the storm (i.e., 25–60 km in the east–west direction and 45–55 km in the north–south direction) are deviated from Gaussian. The southeast edge of this non-Gaussian region also exhibits features of skewness with large negative values (≤−2) and kurtosis larger than 7, which is the second region of interest in section 4. These non-Gaussian spectra are exemplified in Fig. 7 from the line of *CD**AB*

Control parameters for the four experiments. The signal power and spectrum width for the second Gaussian component are denoted by *S*_{2} and *σ*_{2}, while the first Gaussian is centered at 10 m s^{−1} with fixed power of 10 dBm and width of 2 m s^{−1}. The mean velocity of the second Gaussian varies from −10 to 10 m s^{−1} with a 1 m s^{−1} interval.