### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Computer Simulations
- 3. Simulation Results
- 4. Doppler Beam Swinging/Spaced Antenna (DBS/SA) Experiment
- 5. Discussion and Concluding Remarks
- Acknowledgments
- References

[1] In this paper we estimate the random errors in the spaced antenna (SA) winds by two techniques and compare them with measurements made by the Doppler technique. With a one spatial dimension computer simulation, we apply typical parameters for the middle and upper atmosphere (MU) radar without noise. *Briggs* [1984] full correlation analysis (FCA) is commonly applied to the SA technique, while *Doviak et al.* [1996] and *Holloway et al.* [1997] used another simple analytic form based on scattering theory to derive generalized expressions. This simple analytic form should have equivalent results to the Briggs FCA for the horizontal wind estimation. With our simulation results, this form generally shows similar results to the FCA, and sometimes shows better performance than the FCA although it needs simpler calculation than the FCA. The estimation error of this simple analytic form is better than that of the Doppler for large wind, and better than that of the FCA for small antenna spacing. MU radar observations are compared with these two analyses for the SA technique, and the simple analytic form also shows similar performance to the FCA.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Computer Simulations
- 3. Simulation Results
- 4. Doppler Beam Swinging/Spaced Antenna (DBS/SA) Experiment
- 5. Discussion and Concluding Remarks
- Acknowledgments
- References

[2] The quality of the horizontal wind velocity estimates made with the spaced antenna (SA) technique (hereafter called the “SA wind”) has been investigated through many kinds of experiments. The SA wind velocities have been compared with velocities measured by in situ and Doppler radar techniques. For example, comparisons with rocketsonde and radiosonde data were performed by *Vincent et al.* [1977, 1987]. Comparisons with the Doppler technique [*Vincent et al.*, 1987; *Van Baelen et al.*, 1990] and the interferometer technique [*Sheppard et al.*, 1993; *Brown et al.*, 1995] were also carried out. *Hines et al.* [1993] compared the SA wind collected during the Arecibo Initiative in Dynamics of the Atmosphere (AIDA'89) campaign with those obtained from an incoherent scatter radar, a meteor-wind radar, and a Fabry-Perot interferometer. Averaged wind profiles were compared by the authors and possible sources of systematic errors in the SA technique were discussed. Theoretical studies performed by *Briggs* [1980] showed that the Doppler and SA techniques use the same information on the scattering irregularities that produce the radar echoes. *May* [1990] compared the techniques from theoretical and experimental points of view and showed that give similar accuracies. The SA technique determines velocities from diffraction patterns on the ground by calculating the cross correlation and autocorrelation between signals received by spatial domain antennas; the velocities are twice the actual horizontal wind velocity. This velocity of diffraction patterns is calculated by estimating a temporal lag at the maximum cross correlation coefficient, and generally biases because spatial and temporal variations of the diffraction pattern are not considered, which is called as “apparent” velocity. The full correlation analysis (FCA) was proposed to estimate the true horizontal wind velocities with the SA technique [e.g., *Briggs*, 1984]. Accuracies of the FCA were investigated by many authors, and sometimes the FCA is found underestimates for small antenna spacing. *Meek* [1990] studied this defect, knowing as “triangle size effect”, and showed that noise is the sole cause of the triangle size effect. Computer simulations without noise for the FCA were also performed to investigate the random errors in wind estimation with the Doppler and the SA techniques [*Tahara et al.*, 1997]. It was shown that, in some particular conditions, the performance of the FCA results can be better than that obtained with the Doppler technique when the horizontal wind velocity *u* ≥ 70 m s^{−1}. However, the FCA results are worse for small antenna spacing although this simulation model did not include noise. *Holdsworth* [1995] also investigated the triangle size effect by using model-generated and experimentally obtained data, and reported that there are four possible sources that cause the triangle size effect; receiver noise, receiver characteristics differences, coarse digitization, and receiver saturation. The source of the triangle size effect was discussed in detail by *Holdsworth* [1999] and showed that the triangle size effect is reduced by minimization of these four sources. *Doviak et al.* [1996] showed generalized expressions for reduction to a simple analytic form related to the cross correlation/spectrum of signals in the SA technique to the turbulent diffraction flow based on scattering theory. *Holloway et al.* [1995, 1997] applied the simple analytic form for development of time domain and frequency domain algorithms. This form is an another approach of the FCA for the horizontal wind estimation, which should provide equivalent results given by the FCA. It is expected that, however, the FCA will be better performance than this simple analytic form because the FCA involves much more correlation functions in estimation. In this paper, we apply this another form in computer simulations of one spatial dimension and compare with the FCA. Simple hypotheses are used for these simulations: 1) the correlation function of the refractive index fluctuations within the scattering layer is taken to be Gaussian, 2) noise effects on the estimations are not taken into account. These hypotheses were also employed by *Tahara et al.* [1997] for the FCA. First we compare computer simulation results using this simple analytic form with the FCA results [*Tahara et al.*, 1997], and then we show an experimental observation conducted with the middle and upper atmosphere (MU) radar. This paper is organized as follows. In Section 2, the FCA, this another form with the SA technique, and the method of the computer simulation used in this paper are presented. In Section 3, computer simulations of one spatial dimension are shown. Section 4 presents an experimental observation with the MU radar.

### 3. Simulation Results

- Top of page
- Abstract
- 1. Introduction
- 2. Computer Simulations
- 3. Simulation Results
- 4. Doppler Beam Swinging/Spaced Antenna (DBS/SA) Experiment
- 5. Discussion and Concluding Remarks
- Acknowledgments
- References

[8] Simulation parameters are selected to correspond to typical stratosphere observations with the MU radar. In (4), θ_{T} = 2.16° and θ_{R} = 6.3°, then ξ_{0.5} = 48 m, assuming isotropic scatter. In (2), *T*_{0.5} ≈ 2 s for typical observations with the MU radar. Correlation times *T*_{0.5} of 1.0, 2.0, and 4.0 s provide turbulent velocities ν_{v} of 0.6, 0.3, and 0.15 m s^{−1}, respectively, where *ν*_{v} is expressed as follows [*Briggs*, 1980],

where λ is the radar wavelength.

[9] Errors of horizontal wind estimation with the Doppler technique are derived from the results with computer simulation. *Yamamoto et al.* [1988] studied random errors in radial wind estimates without noise, proposed an equation for the estimation error ε_{v} of the radial velocity with the least square fitting as

where *c* is the speed of light, *f*_{0} is the radar frequency, σ_{v} is the standard deviation of the Doppler spectrum, and *T* is the observation period. The constant 0.63 is derived from their computer simulation results. With the Doppler technique, the horizontal wind is usually estimated from two oblique beams. When a uniform wind field is assumed over the radar, the horizontal wind velocity *u* parallel to the azimuth direction of the oblique beams is estimated as

where *v*_{1} and *v*_{2} are radial velocities measured in the oblique beams, and θ_{z} is a zenith angle of the oblique beams. Then estimation error of *u* is written by

Figure 1 shows variation of the estimation errors ε_{u} for the horizontal wind velocity. Because the record length of individual time series is fixed in our system, the observation period *T* changes with the number of incoherent integrations *n*. Coherent integration was performed over 38 samples, giving an effective sampling interval of 0.076 s. Time series of 256 complex data were then stored every 19.45 s, which corresponds to incoherent integration *n* = 1. In the radar observations, incoherent integrations of the correlation functions are performed. Figure 1 shows results for *n* = 3, 6, 9, 18, and 36, which correspond to the periods of 58.4, 116.7, 175.1, 350.1, and 700.2 s, respectively. Solid lines show ε_{u} estimated by the FCA, dashed lines show ε_{u} estimated by the PCA, and dotted lines show ε_{u} estimated by the Doppler technique (20). Each line is generally decreasing as observation period *T* is increasing. This effect is caused by the reduction in statistical fluctuation, which is proportional to , therefore each line varies as . This effect can be seen with the Doppler technique plotted by the dotted line [*Yamamoto et al.*, 1988]. For larger *T*_{0.5}, ε_{u} is smaller because the maximum of the cross correlation function is larger for larger *T*_{0.5}. For *T*_{0.5} = 1.0 s, ε_{u} from the PCA is similar to that from the FCA for *T* ≤ 116.7 s, but is worse for *T* ≤ 175.1 s. For *T*_{0.5} = 2.0 s, ε_{u} from the PCA is slightly better than that from the FCA for all *T*. For *T*_{0.5} = 2.0 s, ε_{u} from the PCA is better than that from the FCA for all *T*, and the difference between the ε_{u} estimates is larger than that obtained for smaller value of *T*_{0.5}.

[10] Figure 2 shows variation of ε_{u} due to the horizontal wind velocity *u*. For each value of *T*_{0.5}, ε_{u} generally decreases with increasing *u*. Figure 3 shows examples of cross correlation functions with small (top panel) and large (bottom panel) winds. For a small wind, the maximum cross correlation is smaller than that for a larger wind, and the lag of the maximum cross correlation is larger than that for a larger wind. These effects cause ε_{u} to decrease as the wind increases. ε_{u} estimated by the Doppler technique (vertical dashed line) is larger for the large wind because its spectrum is wide; this is called the beam broadening effect [e.g., *Hocking*, 1983]. For *T*_{0.5} = 1.0 s, ε_{u} from the PCA is slightly better than that from the FCA but worse than that from the Doppler. For *T*_{0.5} = 2.0 s, ε_{u} from the PCA is smaller than that from the FCA. ε_{u} from the PCA is also better than that from the Doppler for winds over 70 m s^{−1} while ε_{u} from the FCA is better for winds over 75 m s^{−1}. For *T*_{0.5} = 4.0 s, ε_{u} from the PCA is clearly smaller than that from the FCA, while ε_{u} from the PCA is better than that from the Doppler for winds over 42 m s^{−1}; ε_{u} from the FCA is better for winds over 51 m s^{−1}. These results support that the PCA estimation is better than the FCA estimation in most cases, but both are worse than the Doppler estimation for winds below 70 m s^{−1}. Figure 4 shows the variation of ε_{u} with antenna spacing ξ_{0}. In the top panel, for *T*_{0.5} = 1.0 s, ε_{u} of apparent velocity is always worse than ε_{u} from the PCA and the FCA because spatial and temporal variations of the diffraction pattern are not considered. ε_{u} from the FCA increases as ξ_{0} decreases at ξ_{0} < 20 m and again increases ξ_{0} increase at ξ_{0} > 100 m, while ε_{u} is stable in the region of 20 ≤ ξ_{0} ≤ 100 m. ε_{u} from the PCA is stable in the region of 20 ≤ ξ_{0} ≤ 100 m and increases as ξ_{0} increases at ξ_{0} > 100 m, which is quite similar to ε_{u} from the FCA, but still stable at ξ_{0} < 20 m. In the middle panel, for *T*_{0.5} = 2.0 s, ε_{u} from the FCA shows a stable region at 30 ≤ ξ_{0} ≤ 100 m, increases as ξ_{0} increases over 100 m and increases again as ξ_{0} decreases under ξ_{0} = 30 m. Note that ε_{u} is worse than that of the apparent velocity at ξ_{0} = 5 m. ε_{u} from the PCA is similar to that from the FCA at ξ_{0} > 30 m while it still remains stable at ξ_{0} < 30 m. In the bottom panel, for *T*_{0.5} = 2.0 s, ε_{u} from the FCA shows a stable region at 50 ≤ ξ_{0} ≤ 200 m, ε_{u} is quite worse than that of the apparent velocity at ξ_{0} < 20 m, while ε_{u} from the PCA shows a stable region for 5 ≤ ξ_{0} ≤ 70 m with values much better than the FCA values. Figure 5 shows variations of averaged velocity (top panel) and standard deviation (bottom panel) versus antenna spacing ξ_{0} for *T*_{0.5} = 2.0 s. The averaged velocity from the FCA is stable at 30 ≤ ξ_{0} ≤ 150 m but underestimated for larger antenna spacing. This defect for large antenna spacing is thought to be due to the small value of the maximum cross correlation for large ξ_{0}. On the other hand, for small antenna spacing, the averaged velocity from the FCA is underestimated for small antenna spacing, but the averaged velocity from the PCA is stable. This underestimate for small antenna spacing seems to be caused by the triangle size effect [*Meek*, 1990]; however, this simulation model had not included noise. *Holdsworth* [1999] investigated the triangle size effect in detail and showed that the FCA velocity is constant for small antenna spacing without noise. We consider this may be due to our simplified simulation model, nevertheless, the PCA is almost constant for small antenna spacing. In the bottom panel, standard deviation from the FCA is large at ξ_{0} < 10 m and ξ_{0} > 150 m, while that from the PCA is stable at ξ_{0} < 120 m. These results indicate that the PCA is more stable in small antenna spacing in our simulation.

[11] Figure 6 shows variations of ε_{u} versus correlation distance ξ_{0.5}. In (3), ξ_{0.5} depends on antenna size parameters, which are constants for a given array configuration and an aspect angle θ_{s}. In isotropic scatter, θ_{s} is ignored, ξ_{0.5} is 48 m, while for anisotropic scattering with the strong aspect sensitivity such as that observed in the lower stratosphere, θ_{s} = 2° [*Hocking et al.*, 1986], which corresponds to ξ_{0.5} = 70 m. ε_{u} decreases as ξ_{0.5} decreases in Figure 6, which indicates that large θ_{T} and θ_{R} give better ε_{u}, which is consistent to the result by *Briggs* [1980], but this causes that signal to noise ratio to be poor, which must be taken into account for real observations. Within the region of actual observation parameters, 48 ≤ ξ_{0.5} ≤ 70 m, ε_{u} from the PCA is generally similar to that from the FCA.

### 4. Doppler Beam Swinging/Spaced Antenna (DBS/SA) Experiment

- Top of page
- Abstract
- 1. Introduction
- 2. Computer Simulations
- 3. Simulation Results
- 4. Doppler Beam Swinging/Spaced Antenna (DBS/SA) Experiment
- 5. Discussion and Concluding Remarks
- Acknowledgments
- References

[12] This observation was conducted from 1830 local time (LT) on January 16 to 0300 LT on January 17, 1996 with the MU radar located at Shigaraki, Japan (34.51°N, 136.6°E, 375 m Mean Sea Level). The receiving antenna configuration for this experiment is shown in Figure 7, which has the triangle array configuration with a antenna spacing ξ_{0} = 52 m. The whole array of 475 Yagis was employed as transmitting antenna. With the active phased array system, the radar beam was steered every inter pulse period (IPP) through the vertical and four oblique directions (north, east, south, and west) at 10° off zenith. On the other hand, the receiving antenna array was divided into 4 subarrays forming 4 channels. Channel 1 was connected to the whole antenna array for the Doppler analysis in the 5 beam directions. Channels 2, 3, and 4 corresponded to three equally sized subarrays composed of 57 Yagi antennas for applying the SA technique on the signals received from the vertical beam. Signals received from the four channels were independently and simultaneously sampled at 64 heights every 150 m from the altitude range from 5.1 km to 14.55 km. Coherent integration was performed over 38 samples, giving an effective sampling interval of 0.076 s. Time series of 256 complex data were then stored every 19.45 s for Fast Fourier Transform. Incoherent integrations over 36 records were performed. Thus, we obtained the spectra and cross correlation and autocorrelation functions approximately every 12 minutes. the FCA and the PCA wind estimates were obtained by procedure already explained. On the other hand, the horizontal wind simultaneously estimated every 12 minutes with the Doppler technique. This observation is presented in detail by *Kawano et al.* [2001]. In this paper we calculate the horizontal wind only via FCA, PCA, and the Doppler techniques. Our simulation results shown in Figure 2 indicate that the estimation errors for the Doppler are smaller than that for the FCA/PCA, the estimation errors are then derived from differences between the FCA/PCA winds and Doppler winds. Figure 8 shows the difference ε_{u} between the FCA and Doppler winds and between the PCA and Doppler winds versus horizontal wind. The data are averaged into 10 m s^{−1} bins for this plot. ε_{u} from the FCA and the PCA in the observation show similar results and remain at constant level even in large wind. ε_{u} in the simulation show smaller for all winds than that in the observation. We consider some reasons; this is mainly caused by noise because our simulation does not take noise into account. In addition, ε_{u} in the observation are not smaller for large winds, which is not consistent to ε_{u} in the simulation. We take *T*_{0.5} = 2.0 and ξ_{0.5} = 52 m in this simulations shown in Figure 8 derived from corresponding observation results, but *T*_{0.5} and ξ_{0.5} are not constant during the observation. *T*_{0.5} is smaller for large winds in this observation, for examples, *T*_{0.5} ≃ 1.9 at *u* = 20 m s^{−1}, *T*_{0.5} ≃ 0.6 at *u* = 50 m s^{−1}, so that smaller *T*_{0.5} provides larger estimation errors. Therefore we consider that the meteorological conditions, such as a passage of a trough and gravity waves, could affect to the SA results more than stable conditions. Moreover, note that ε_{u} here is just the difference between the FCA and Doppler winds or between the PCA and Doppler winds, not an estimation error from a true velocity as shown in Figure 2. ε_{u} of this observation is derived from differences between the Doppler and the SA winds. The SA wind has estimation errors ε_{uSA} and the Doppler wind also has estimation error ε_{uDPL}. ε_{u} from the observation should be shown as , which means that ε_{u} of this observation is larger than real estimation error of the SA. ε_{uDPL} is larger for large wind, which contributes to ε_{u} in larger wind. We consider that these cause the differences between observation and simulation results; however, the PCA shows similar ε_{u} to the FCA in this observation.

### 5. Discussion and Concluding Remarks

- Top of page
- Abstract
- 1. Introduction
- 2. Computer Simulations
- 3. Simulation Results
- 4. Doppler Beam Swinging/Spaced Antenna (DBS/SA) Experiment
- 5. Discussion and Concluding Remarks
- Acknowledgments
- References

[13] In this paper we studied estimation errors of the horizontal wind with the spaced antenna technique. Full correlation analysis (FCA) is generally employed for the SA, while *Holloway et al.* [1997] applied the simple analytic form based on the scattering theory, which for convenience, we called “partial correlation analysis (PCA)”. This simple analytic form is the alternative approach to the *Briggs*' [1984] FCA; therefore this should have equivalent results to the FCA. We expected that, however, the FCA would be better because the FCA involves much more correlation functions. We compared with results from these two techniques for the SA and the Doppler technique. Typical observation parameters for the MU radar were applied for this simulation. The PCA generally shows similar performance to the FCA, and performed better for large winds, *u* > 70 m s^{−1} for *T*_{0.5} = 2.0 s, than the Doppler technique. For the variation of estimation error ε_{u} due to the antenna spacing ξ_{0}, ε_{u} for the FCA increases as ξ_{0} decreases, at ξ_{0} < 40 m for *T*_{0.5} = 2.0 s; on the other hand, ε_{u} for the PCA is stable and performs better than the FCA for small ξ. *Holdsworth* [1999] showed that the FCA is constant for all antenna spacing; we consider our result may be due to simplified simulation model. For the small antenna spacing, the lag of cross correlation maximum is close to zero. If this position of the cross correlation maximum causes large estimation errors, it must provide similar results to the larger wind although the FCA and PCA show better performance for the large wind in Figure 2. Then we conduct the simulation for far larger wind, and find that *u* > 150 m s^{−1}, which cannot be realized in the real observation in the troposphere and stratosphere; ε_{u} of the FCA increases as *u* increases, which is consistent to the results in the small antenna spacing. ε_{u} of the PCA also increase as *u* increases, but it is still better than that of the FCA as shown in Figure 9. This result indicates that the PCA is more stable when the cross correlation function is close to the autocorrelation function in our model. Finally we showed an observation result with the MU radar. The estimation differences of the FCA and the PCA were stable for all wind speeds. These results show that the PCA has similar performances to the FCA although the PCA needs simpler calculation than the FCA. We considered this simulation without noise; however, it is expected that noise would be one of the major sources of estimation errors for the SA technique as shown by *Meek* [1990], and this is a subject for our future works.