## Abstract

The lower atmospheric wind profiles are obtained by the postset beam steering (PBS) technique on middle and upper (MU) atmosphere radar data. The Capon beamformer is used to improve the beam synthesizing in the desired directions within the radar transmit beamwidth. From a synthesized beam, the power spectrum is obtained by various spectral estimators, such as Fourier, multiple signal classification (MUSIC), and eigenvector (EV). The wind vector components are derived from radial velocities estimated from the power spectra of synthesized beams. As the reliability of the PBS wind estimate depends on the choice of spectral estimators, a detailed analysis is carried out to compare the performance of estimators in deriving wind profiles on the radar data. The results suggest that EV shows a better performance in deriving possible spectrum parameters and is useful for reliable wind profiling up to the lower stratosphere. The wind profiles derived by PBS with EV are more consistent with near-time observations using GPS sonde and Doppler beam swinging (DBS) methods. The study also suggests that MUSIC cannot be used to reliably estimate atmospheric spectrum parameters.

## 1. Introduction

The wind profilers are Doppler radars capable of measuring winds over much of the troposphere and lower stratosphere. The profilers observe Doppler shifts and then measure radial wind velocities along antenna beam directions. By assuming the uniform wind field over beam separation, these radial wind velocities are combined to derive three orthogonal wind vector components, that is, zonal (east–west), meridional (north–south), and zenith (vertical) winds. To achieve a greater signal-to-noise ratio (SNR), these radars generally employ long dwell times, of the order of one to a few minutes, in a fixed direction such that the radar signal from several successive pulses are coherently averaged to filter out the noise on the required signal. The same process is repeated along other beam-pointing directions required for wind profiling. Today, numerous techniques such as Doppler beam swinging (DBS), spaced antenna (SA), and radar interferometry (RI) are available for deriving 3D atmospheric winds by radar remote sensing. DBS is most popular because of its simplicity of using a single receiver and tilting the beam to the minimum of three pointing directions (noncoplanar) one after the other using the electronic phase shifters. DBS has proven to be a reliable means of obtaining the 3D winds. The performance of other wind observations, such as SA, has been analyzed by considering DBS-observed winds as a reference (Praskovsky et al. 2004).

RI has been extensively studied by many authors (Röttger and Ierkic 1985; Adams et al. 1986; Woodman 1997; Palmer et al. 1995, 1998; Hélal et al. 2001; Hassenpug et al. 2008; Cheong et al. 2008). Several multireceiver techniques have been developed in the field of RI: spatial Doppler interferometry (SDI) (Palmer et al. 1995), imaging Doppler interferometry (IDI) (Adams et al. 1986), postset beam steering (PBS) (Röttger and Ierkic 1985), poststatistic steering (PSS) (Kudeki and Woodman 1990), radar interferometric imaging (RII) (Larsen et al. 1992), time domain interferometry (TDI), frequency domain interferometry (FDI) (Kudeki and Stitt 1987; Palmer et al. 1990), and others. Most of methods are mainly based on the concept introduced by Woodman and Guillen (1974), specifically that the radar power spectra and cross spectra are the representative of the Doppler distribution of a large number of randomly distributed volume scatterers. Radar imaging was introduced to make use of maximum available information about the scatterer distribution statistics from the received signals. Two-dimensional angular imaging using coherent radar imaging (CRI) and a variety of PBS methods were first implemented on mesosphere–stratosphere–troposphere (MST) radars (Palmer et al. 1998; Hélal et al. 2001), followed by multifrequency one-dimensional FDI experiments on MST radars and ultrahigh-frequency (UHF) wind profilers (Luce et al. 2001, 2006; Chilson et al. 2003; Chen 2004).

PBS is one of the multireceiver techniques. Basically, the technique involves multiplying the time series data of each antenna by an appropriate phase such that the combination of all time series effectively steers the effective beam in a specific direction (Röttger and Ierkic 1985). Recently, PBS (Sureshbabu et al. 2013) applied on the profiler data showed a flexibility to synthesize the time series data in optimum directions. The PBS improvements presented here include a number of aspects in signal processing procedures like beamforming, spectral estimation, and moments estimation (Sureshbabu et al. 2013). Since the reliability of the PBS estimates mainly depends on the performance of the spectral estimator, several statistical and simulation studies are carried out to show the performance of spectral estimators for reliable wind profiling. The beam synthesis in PBS is equivalent to the hardware steering employed by DBS and thus the study includes a number of analyses in PBS similar to standard DBS. The analysis includes the use of a long dataset obtained only in a quiet meteorological condition. This paper mainly focuses the advantages of signal processing works on PBS for obtaining wind profiles possibly with the temporal resolution of ~26 s and maximum height coverage. The present study includes the wind estimation up to the height of ~20 km due to the availability of the data. The obtained results are compared with the standard GPS sonde and DBS observations in contiguous time.

## 2. Simulation results

Several approaches (Boyer et al. 2001; Cheong et al. 2004) were presented to overcome the lack of spectral resolution and to identify overlapped echoes in the power spectrum. An earlier study (Boyer et al. 2001) showed that subspace methods (applied to the autocorrelation function) appear to be an alternative to Fourier-like techniques when the echoes are buried in the noisy environment. The study also revealed the limitation of Fourier in identifying spectrum parameters in the case of weak SNR and the advantage of Fourier in retrieving the spectral width better than the model-based spectral estimators. Meanwhile, subspace methods (Krim and Viberg 1996; Godara 1997; Boyer et al. 2001) have shown greater spectral resolution and SNR improvements. This section includes the signal processing aspect of Fourier, multiple signal classification (MUSIC), and EV in deriving spectrum parameters on the simulated data. Both MUSIC and EV assume a statistical model on the input signal through eigendecomposition (Krim and Viberg 1996). With the subspace method, the power spectrum is sharply peaked (line/pseudospectrum) at frequencies of sinusoidal components of the signal. Generally, the retrieval of spectral width is not observed by the subspace method (Boyer et al. 2001). As spectral width is an indication for the atmospheric turbulence, a simulation study has been carried out to show the possibility of retrieving spectrum parameters.

The goal of the simulation is to determine the matrix order of the signal subspace, which helps to produce an equivalent Fourier spectrum. A sample simulation is provided in Fig. 1. The simulation follows a methodology that includes four steps to determine the matrix order of the signal subspace. First, an autocovariance matrix (Krim and Viberg 1996) is obtained from the simulated time series data. Let the number of sample points in the time series data be *n*. So, the order of the matrix is equal to the number of sample points in the time series data. It is known that the matrix has *n* number of eigenvalues. The eigenvalues of are sequenced in descending order and indexed from 1 to *n*. The normalized eigenvalue distribution is obtained and shown in Fig. 1a. Second, a response curve is obtained by taking the gradient on normalized eigenvalues and is shown in Fig. 1b. Third, a very first valley point is identified such that the magnitude of the gradient increases abruptly more than ~10 dB. The value of *n* corresponding to the valley point is taken as *p* (dotted arrow line in Fig. 1b). Fourth, the signal subspace includes eigenvalues from 1 to *p* and corresponding eigenvector sets (sinusoids), whereas the noise subspace includes eigenvalues from *p* + 1 to *n* and corresponding eigenvector sets.

To obtain the simulated time series data, a normalized Gaussian distribution is generated with the input values, that is, mean Doppler frequency (), spectral width (σ), and unambiguous frequency range (*f*). The frequency range is set to be about ±4.8 Hz with the spectral resolution of ~0.04 Hz. The time series data are obtained by performing the inverse fast Fourier transform (IFFT) on the constructed Gaussian distribution. The white Gaussian noise of suitable strength is added with the obtained time series data to fix the required SNR, say, −20 dB. Let us consider the time series data obtained in this fashion as . The Fourier spectrum is obtained by taking FFT on and is shown in Fig. 1c. Meanwhile, MUSIC and EV spectra are obtained by following the methodology on as described in the previous paragraph. The MUSIC and EV spectra are shown in Figs. 1d and 1e, respectively. In Figs. 1c–e, MUSIC and EV have the tendency to denoise the signal compared to Fourier. The subspace methods show more spectral resolution compared to Fourier. Though it is an advantage, MUSIC shows a greater uncertainty in the spectral width retrieval and also exhibits a number of unwanted peaks within the spectral width. Further, the statistics on percentage of success (POS) within the error limit of ±5% in retrieving the spectral width in each method is shown in Fig. 2. The analysis includes statistics by performing the above simulation about 1000 times for all estimators. The performance of all estimators is better in retrieving small values of spectral width and that degrades gradually the width of large values of spectral width and SNR. However, EV shows a better performance in retrieving large values of spectral width even at the SNR of −20 dB. The statistics infer that EV can be an alternative to standard Fourier and overcome demerits of Fourier and MUSIC in deriving possible spectrum parameters. Moreover, the advantage gained in EV for better spectral resolution can be very useful to distinguish the target echo, which cannot be separated from other overlaid echoes by Fourier. MUSIC can be useful to denoise the signal but may not be useful to estimate spectrum parameters reliably. Though EV differs from MUSIC by the contribution of eigenvalues, its performance in retrieving possible parameters is observed to be better compared to Fourier and MUSIC. Thus, the analysis performed in this way is also tested to show the performance of estimators in deriving wind profiles on the use of profiler data. The performance analysis is presented in the section 4. It is noted that the methodology described in this section is applicable only for atmospheric radar data. The idea is common for atmospheric radar data and does not depend on techniques and radar sites.

## 3. System description and data processing

Middle and upper (MU) atmosphere radar located in Shigaraki, Japan (), is a monostatic pulsed phased array radar and has a large circular antenna array of 110 m in diameter. It operates at 46.5 MHz with a peak power of 1 MW. The array is configured with 475 crossed Yagi elements, which are grouped to form 25 separate receiver channels. The array is also capable of steering the beam electronically using phase shifters in transmit and receive paths. The transmission bandwidth of 3.5 MHz has been divided into five overlapping subbands with the interval of 0.25 MHz (i.e., 46, 46.25, 46.5, 46.75, and 47 MHz) and the bandwidth of 1.65 MHz. These subbands are alternatively switched on a pulse-to-pulse manner during the transmission. For receiving, the antenna array is separated into 25 subarrays (channels) that have independent signal processing and storage units. The observation was conducted with a full array of transmission (beamwidth of 3.6°) in the vertical direction for the PBS estimate. A 1- transmitted pulse was used for a 150-m range resolution. The sampling time, including all coherent integrations, was set to be 0.1024 s. Each record included 128 range gates, each of which consists of 256 data points. It is important to note that the datasets were collected for FDI-based observations. The datasets have been used for PBS in the present work. In fact, a single frequency of operation (say, 46.5 MHz) is sufficient for the PBS wind estimates.

For the PBS estimate, the systematic phase shifts are introduced on the time series data of 25 channels to synthesize new beams (within the transmit beamwidth) at a fixed tilt angle of 1.7° and equally separated 16 different azimuth positions. Beam synthesizing is done by the Capon beamforming method. From synthesized beams, Doppler power spectra are obtained by Fourier, MUSIC, and EV. Likewise, the power spectra are independently obtained for all overlapping subbands. The average spectra are obtained by integrating five spectra incoherently to further improve the SNR. The total power, mean Doppler frequency, and spectral width are estimated from the average spectrum by an adaptive moments estimation method (Anandan et al. 2005). The radial velocities are readily obtained on negatively multiplying derived moments by half of the wavelength of the center frequency (46.5 MHz). As a result, the 3D wind vector components are derived binwise by the least squares sense with the temporal resolution of ~26 s.

## 4. Results and discussion

To reveal the performance of Fourier, MUSIC, and EV in deriving winds, a detailed analysis was carried out on the vertical data collected by MU radar on 16 and 18 July 2008 at various time intervals. The analysis includes mainly the performance of spectral estimators in obtaining the high temporal winds up to the height coverage of the lower stratosphere. The procedure in determining the SNR and spectral width is the same for all the spectral estimators. For example, the mean noise level and three lower-order moments are estimated using the standard formulas (Anandan et al. 2005). MUSIC and EV produce the power spectral distribution (PSD) through eigendecomposition. The PSD is arbitrary and not a true power of digitizer counts, as the spectrum is obtained from the noise autocovariance matrix. Hence, the power correction (Madisetti and Williams 1998) to the obtained distribution leads to a reasonable spectrum similar to Fourier. Though the methodology followed in the simulation study helps to obtain the Gaussian-shaped PSD for EV, the demerits have been observed for MUSIC. As EV is derived from MUSIC, the analysis includes MUSIC for comparisons with Fourier and EV. The study indirectly relates an intercomparison between a parent and child with Fourier.

The contour maps of temporal variation of radar parameters (total power, spectral width, and vertical wind velocity) are given in Fig. 3 to show the nature of the atmospheric condition during times of observation. Since the study concludes the best estimator at the end, the plots in Fig. 3 show the contour maps of Fourier-derived spectrum parameters over the height and time. The contour map includes 1083 vertical profiles obtained every 26 s during observations. In Fig. 3, the signal power (zeroth moment), zenith wind velocity, and spectral width are observed to be ~22–68 dB, ~(−1)–(0.8) m s^{−1}, and ~(0.2)–(1.3) m s^{−1}, respectively. The zenith velocity and spectral width are obtained by statistical approximation and therefore the estimation of very small values in both quantities can be possible. In Fig. 3, a small range of temporal variations in the vertical wind velocity and spectral width indicates a quiet meteorological condition (Balsley et al. 1988; Chu and Lin 1994). The dataset analyzed on another date and time resembles the same. Hence, the performance of estimators in estimating winds is reported in a quiet meteorological condition. The reason for enhancement in the spectral width at a height interval of ~10–14 km (compared to other height regions) is explained later.

### a. Wind estimation

The horizontal winds derived by PBS in comparison with DBS-observed winds are shown in Figs. 4a and 4b. The profiles of DBS correspond to the observation at 0338 local time (LT) 16 July 2008 in Fig. 4a and at 0413 LT 18 July 2008 in Fig. 4b. The DBS observation uses Fourier for wind estimation, since the data collection by the system was on the power spectral mode. The profiles of PBS correspond to the observation at 0336 LT 16 July 2008 in Fig. 4a and at 0410 LT 18 July 2008 in Fig. 4b. PBS-derived winds (26 s) are comparable with standard DBS-derived winds up to the height of about 20 km (Figs. 4a,c). It shows a possibility of deriving 3D wind vector components even at the tilt angle of 1.7°. In the DBS wind estimate, the tilt angle is maintained at 10°, which makes a spatial separation (at north–south or east–west beam positions) of about 0.36 and 6.77 km at the height of 1.05 and 19.5 km, respectively. In the PBS wind estimate, the tilt angle is 1.7°, which makes the spatial separation (at north–south or east–west beam positions) of about 0.06 and 1.19 km at the height of 1.05 and 19.5 km, respectively. In a quiet meteorological condition, the horizontal wind velocity is expected to be uniform over the extent of several tens of kilometers. Therefore, the PBS-derived winds are expected to be consistent with DBS-observed winds. Nevertheless, there are deviations in the wind estimate of both observations in some height regions. Fourier and MUSIC show large deviations of more than 6 m s^{−1} in the height regions of 10–14 km compared to DBS. If both measurements are considered to be fully confident, then the deviation between them can be mainly attributed to the small-scale wind variability due to observations at the 10° and 1.7° tilt angles, the different signal processing involved, and the temporal resolution. Such small-scale variability in winds can be useful to derive the wind divergence and also to study the mass energy transport mechanism. For example, in the quiet meteorological condition, the magnitude of wind along the north–south or east–west direction is expected to be uniform over several tens of kilometers at a constant height. When there is inhomogeneity in wind during the disturbed atmospheric condition, the simultaneous wind estimations along the north–south and east–west directions can be useful to obtain the amount of variation in the horizontal wind components. This variation and the horizontal extent of beam positions can be used to derive the wind divergence.

Now the statistical quantities, such as slope (*s*), bias (*e*), and correlation coefficient (*c*), between both observations are provided in Table 1. The above-mentioned quantities are obtained by the polynomial fit of the order one. The above-mentioned statistics are also provided at height regions of ~11–20 km, where the power is observed to be low (Fig. 3a). For profiles shown in Fig. 4, the linearity (slope) and correlation are seen to be high (Table 1) in EV compared to Fourier and MUSIC. The analysis performed for a different date and time also shows that EV has greater linearity and correlation with DBS. It qualitatively infers the better performance of EV in deriving wind profiles reliably. An interesting point from Table 1 is that there is relatively little degradation in the correlation in the lower SNR region compared to the full profile in the case of the EV approach. In addition, the profile comparison between the PBS and GPS sonde observations is shown in Figs. 4b and 4d. The GPS sonde observations correspond to 0750–0910 (Fig. 4c) and 0750–0850 LT (Fig. 4d) on 16 and 18 July 2008, respectively. The PBS observations correspond to datasets in contiguous time to GPS sonde observations. A similar kind of statistical analysis performed with GPS sonde observation is tabulated. The statistical quantities of *s*, *e*, and *c* are given in Table 2. The small difference of ~1–2 m s^{−1} in the wind velocities of both observations can be attributed to the spatial separation usually with reference to the work of Jasperson (1982) on the spatial differences of simultaneous sonde ascents. Large differences in winds of both PBS and GPS sonde observations are noticed in few height regions. Nevertheless, the trends in wind profiles of both observations follow the same. The statistical quantities given in Table 2 also reveal the better performance of EV in deriving wind profiles reliably.

To justify a better estimator, a quantitative analysis is carried out using long datasets collected at 0000–0758 LT 16 July 2008. Figure 5 shows comparisons of the vertical profiles of the standard deviation (SD) in winds derived by estimators. The SD for the duration of every hour is computed relative to the mean wind of the corresponding estimator. The number of profiles obtained during an hour is about 130. In the height regions of ~10–14 km, the SD in zonal wind varied up to 3.6, 4.4, and 2 m s^{−1}; and in meridional wind up to 4, 3.5, and 2 m s^{−1} in Fourier, MUSIC, and EV, respectively. SDs in winds observed by estimators are consistent with each other in most of the height regions except in weak regions of ~10–14 km, where the return power is very low. Though the quiet meteorological condition is ensured, Fourier and MUSIC show inconsistent wind variations over time in height regions of ~10–14 km due to uncertainty in estimating the Doppler frequency within the spectral width in the case of Fourier and in estimating both the Doppler frequency and the spectral width in the case of MUSIC. However, the SD is noticed within the limit in the case of EV.

### b. Spectral width estimation

Figure 6 shows the vertical profile of spectral width and SNR derived in the case of Fourier and EV. Since MUSIC is inappropriate for the comparison of spectral width, it is excluded from Fig. 6. The spectral width derived by standard Fourier ranges from ~0.1 to ~1.2 m s^{−1} in various atmospheric layers for different time steps. In Fig. 6b, Fourier shows weak SNRs at heights of ~10–14 km, whereas the SNR improvement is achieved in the case of EV. By comparing Fig. 6a and Figs. 4a and 4b, it is clear that the rise of the spectral width has a close correlation with horizontal wind variations. The maximum horizontal wind velocity and shear are observed in between height regions of ~10 and 14 km. The maximum spectral width is ~1.2 and ~0.75 m s^{−1} in the case of Fourier and EV, respectively. It is noted that Fourier and EV show the maximum spectral width in height regions of ~10–14 km. This infers that the enhancement in the spectral width can be possibly due to the large horizontal wind motion and wind shear as described by Hocking et al. (1986). Here the large horizontal wind motion is termed as the high wind. From Figs. 4a and 4b and Fig. 6 in connection with the simulation results (Figs. 1c,e), EV reveals a little enhancement in the spectral width at heights of ~10–14 km due to the high wind and wind shear. Conversely, Fourier overestimates the spectral width in height regions of ~10–14 km, such that the major contribution to the spectral width is imparted by the statistical uncertainties rather than the wind shear and high wind. This can be the reason for the larger spectral widths noticed in Fig. 3b at a height interval of ~10–14 km. Meanwhile, the above-mentioned result reveals the better performance of EV in investigating the spectral width.

As a result the study reveals that EV has advantages in (i) identifying atmospheric signal buried in noisy environments, (ii) obtaining spectrum parameters (moments) with greater accuracy, (iii) cleaning the spectrum through the denoising process, (iv) improving SNR, and (v) having a high temporal (26 s) wind profiling with maximum height coverage (~20 km) on profiler radar data. The study also reveals that PBS with EV can reach a higher level and have a higher temporal resolution in wind profiling compared to earlier studies (Palmer et al. 1995, 1998).

## 5. Conclusions

Various statistical studies are carried out to show the performance of Fourier, MUSIC, and EV in the reliability of PBS wind estimates on the profiler radar. This study reveals that MUSIC cannot be used to estimate atmospheric spectrum parameters and can be very applicable to analyze signals exhibiting line spectra. The large uncertainty in horizontal winds is observed in the case of Fourier, particularly in weak regions where the return power is very low. The wind profiles derived by EV are more reliable up to the maximum height coverage (i.e., ~20 km in the present discussion) with a temporal resolution of ~26 s. The several qualitative and quantitative analyses carried out on the profiler datasets also justify that EV-derived winds are more consistent with DBS and GPS sonde observations in contiguous time. The study also reveals that a methodology incorporating EV helps to produce a Fourier-equivalent spectrum with greater spectral resolution. Results suggest that the EV can be a suitable estimator in high temporal wind profiling using the profiler data collected at small and large tilt angles. Such high temporal estimation can be reliably used to study the fast-changing nonhomogeneous wind fields during disturbed atmospheric conditions. This possible application will be addressed in a future study.

## Acknowledgments

This study was conducted using the MU atmosphere radar under support of the Asia-Africa Science Platform (AA-SP) program of JSPS, Japan, and the Indian Space Research Organization (ISRO), India.

## REFERENCES

*Digital Signal Processing Handbook.*CRC Press, 1776 pp.