On the optimum radar beam angle to minimize statistical estimation error of momentum flux using conjugate beam technique



[1] Multi beam experiments were conducted using MST radar at Gadanki (13.5°N, 79.2°E) to examine the dependence of statistical estimation error of momentum flux of wind fluctuations on radar beam angle. We find that an optimum angle between 9°–12° with sufficient integration minimizes the error at a value nearly equal to the irreducible error in the troposphere. The balance between the angle dependence of horizontal velocity measurement precision and spatial stationarity seems to decide the optimal beam separation. We also show that the radial velocities obtained by Doppler beam-swinging method must be appropriately corrected for anisotropy of VHF radio wave scatterers, particularly for lower beam angles, which are found to get affected substantially. Momentum flux of wind fluctuations estimated using low radar beam angles need a longer time of integration to reduce statistical error to its minimum value.

1. Introduction

[2] It is well established that the conjugate beam technique developed by Vincent and Reid [1983] is an unbiased estimator of momentum flux of wind fluctuations under the assumptions that the second order statistics of the wind field are horizontally homogeneous. The technique has been used by various workers to estimate momentum flux of wind fluctuations [Fritts and Vincent, 1987; Worthington and Thomas, 1996; Chang et al., 1997; Fritts et al., 2006]. But it is only recently that attention has been paid to estimate the errors associated with the method. The effect of geophysical noise in the measurement of momentum and heat flux has been addressed by several workers [Tao and Gardner, 1995; Thorsen et al., 1997; Gardner and Yang, 1998; Riggin et al., 2004; Dutta et al., 2005]. Kudeki and Franke [1998] showed that the dominant contribution to the uncertainty of momentum flux estimates scales as the geometric mean of the horizontal and vertical wind fluctuation variances and to obtain statistically significant measurements of momentum flux, long integration times are necessary. Riggin et al. [2004] estimated the momentum flux of short period wind fluctuations together with the estimation uncertainties using tropospheric and stratospheric wind data obtained at Jicamarca. They observed day-to-day variability of the flux profiles and felt that stationarity assumption could be violated during integration periods approaching or exceeding a day. Dutta et al. [2005] used wind data collected by Gadanki MST radar to estimate momentum flux of high frequency wind perturbations. A careful error analysis showed that the statistical estimation error becomes almost irreducible after about 15–16 hours of integration and that the stationarity assumption was not violated during this period.

[3] Thorsen et al. [2000] extended the work of Kudeki and Franke [1998] by including the influence of measurement noise and finite spatial correlation of the wind fluctuation on the statistical error in the dual coplanar beam technique. They derived a generalized form for the variance of the Vincent and Reid momentum flux estimator. Assuming a functional form for the spatial correlation and using typical values for the temporal and spatial correlation length scales, they showed that there is an optimal beam separation angle that minimizes the statistical estimation error and for typical mesopause parameters this beam separation is approximately ±13°.

[4] It is observed that there has been no study to simultaneously examine the effect of temporal and spatial (beam separation) stationarity on the statistical estimation error. The present work is motivated by the theoretical framework of Thorsen et al. [2000] and aims at examining the dependence of estimation error of momentum flux of gravity waves on radar beam angles and time of integration in the troposphere. The paper also brings out the importance of incorporating correction factors in the radial winds measured by MST radars, which get affected due to the anisotropy of VHF radio wave scatterers in the atmosphere.

2. Data and Analysis

[5] Multi beam experiment was conducted using MST radar at Gadanki on 16–17th August, 2006. The radar was operated continuously for 22 hrs in three beam directions (E, W, Zenith) each for different angles. The experimental specifications (ESF) are given in Table 1. A detailed description of the radar is given by Rao et al. [1995]. Stringent data processing was carried out to obtain height profiles of radial velocities [Dutta et al., 2005; Sasi et al., 1998] and echo powers. Wind data obtained between 6 and 12 km were found to be of very high quality for all the beam angles. This tropospheric data with high temporal (6 minutes) and spatial (150 m) resolutions have been used for the present study.

Table 1. Experimental Specifications
Specifications16th–17th August 2006
No. of Range bins150
No. of FFT points256
No. of coherent integration64
No. of incoherent integration1
Inter pulse period1000 ms
Pulse width16 μs (complementary code with 1 μs baud)
Beam angles3°, 6°, 9°, 12°, 15°

[6] VHF radars are extensively used for routine wind measurements, but they do suffer from underestimation of radial velocities due to the anisotropic nature of VHF radio wave scatterers in the atmosphere [Hocking, 1997; Rottger, 1980]. Hocking [2001] has shown that the radial velocities measured by VHF radars underestimate the true velocities and hence appropriate correction factors must be employed. We have corrected radial velocities measured by the radar with different beam angles following Hocking [2001].

equation image

where vmeas is the radial velocity measured by the radar and vtrue is the value it would have measured if the scatterers were isotropic. w, “winds ratio factor”, is given by

equation image

where P(0) and P(θt) are the powers received on the vertical and off-vertical beams, the respective beam angles being 0 and θt. θ0 is a measure of Gaussian e-folding half width of the antenna beam as described by the polar diagram of the vertically directed beam which is 1.8° for Gadanki MST radar [Ghosh et al., 2004]. Corrected radial velocities measured by 3° beam angle showed a few high values which were removed and interpolated back. Time series of uncorrected and corrected radial velocities were then detrended and filtered for <2 hr (high-pass) and >2 hr (low-pass) to retain wind fluctuations with corresponding periods. Vertical flux of horizontal momentum was then obtained using symmetric beam method developed by Vincent and Reid [1983]. Momentum flux profiles were smoothed by taking a three point running mean with weighted average and were integrated over different lengths of time i.e., (1, 2, 3, −22) hrs and the corresponding standard deviations were calculated. The estimation error of each block (1, 2, 3, −22) hrs for each angle was obtained as equation image, where σ is the standard deviation of the population and N is the corresponding number of points. Errors, so calculated, account for both instrumental and geophysical noise.

3. Results and Discussion

[7] The radial velocity profiles obtained before and after employing the correction factor for beam angles 3°, 9° and 15° and the corresponding percentage of corrections are illustrated in Figure 1. Significant difference of radial velocities can be noticed after correction particularly for 3° beam angle and in the altitude region where aspect sensitivity is more pronounced. The correction becomes less and inappreciable for 9° and 15° tilt angles, since the scattering becomes more isotropic. The percentage of correction is obviously very high for the lowest beam zenith angle of 3° which is affected by aspect sensitivity most adversely. The momentum flux profiles calculated using corrected and uncorrected radial velocities show similar differences.

Figure 1.

(top) Height profiles of East beam radial velocities with error bars before (dot) and after correction (line) for 3°, 9°, and 15°. (bottom) Corresponding absolute percentage of correction.

[8] The dependence of statistical estimation errors, obtained by using radial velocities before and after corrections, with radar beam angles for <2 hr and >2 hr wind fluctuations are shown in Figures 2 and 3. Second order polynomials have been fitted to the data points. It can be noticed that the statistical estimation errors show a gradual decrease with beam angles initially and a slight increase after 12°, which is clearer for gravity waves with period >2 hr. It can also be seen that the statistical error becomes more after correction of radial velocities for 3° and 6°, whereas the differences of errors in momentum flux estimates using corrected and un-corrected sets become inappreciable for 9°, 12° and 15°. The variation of winds ratio factor (w) with beam angle is shown in Figure 4. The factor beautifully increases to ∼1 at 15° showing isotropic nature of the scattering for larger beam angles.

Figure 2.

Dependence of estimation error of mean momentum flux of short period waves (<2 hr) with beam angles before and after correction of radial velocities.

Figure 3.

Same as Figure 2 but for long period waves (>2 hr).

Figure 4.

Variation of winds ratio factor (east and west beams with error bars on east beam) with angle.

[9] It is established that the horizontal velocities measured by smaller radar beam angles are more erroneous since vertical wind components contribute fairly. So the estimate itself will be biased and underestimate of true momentum flux which will be translated in the statistical estimation error. Further the anisotropy of the scatterers introduce significant error at low beam angle thereby masking the true variation of estimation error as a function of beam angle, particularly if the quality of data is not extremely good. Estimation errors of both uncorrected and corrected data-sets are found to be minimum somewhere between 9°–12° in the troposphere. Thorsen et al. [2000] has shown that the location of the minimum is controlled by the ratio of vertical to horizontal wind variance and the ratio of measurement error to wind variance. As either ratio becomes larger, the minimum statistical error moves to larger zenith angles. For typical mesopause values of vertical to horizontal wind variance (W/U = 0.1), the optimal beam separation is at ≈±13°. They also point out that singularities are found at small and large zenith angles because the formulation of the dual coplanar momentum flux requires dividing by sinθ cosθ. In the light of these observations, our results appear to be highly justified and illuminating.

[10] The contour plot (Figure 5) depicts the variation of estimation error of zonal momentum flux of wind fluctuations (<2 hr and >2 hr) with beam angle and time of integration in the troposphere. The error is found to be minimum around a beam angle of 9°–10° after ∼(14–16) hrs of integration and between 8°–12° after longer integration periods <2 h and >2 h periodicities respectively. Momentum flux values are found to be of the same order of magnitude as reported earlier by Dutta et al. [2005] for the same station. The time series of momentum flux estimates have been tested for statistical stationarity following Dutta et al. [2005]. The mean values of momentum flux of six consecutive blocks into which the total data was divided are found to be confined within ±1σ for 15 hr and ±2σ for 22 hr, which are statistically significant. Figure 5 also illustrates that the result is valid for gravity waves of both period bands, i.e., of short (comparable with radar beam separation) and medium scale horizontal wavelengths. So it encourages us to conclude that the time of integration needed for estimation error to be irreducible depends on radar beam angle and a beam separation of 9° to 12° appears to be optimum for tropospheric observations.

Figure 5.

Contour plot of variation of estimation error of zonal momentum flux with beam angle and time of integration for (top) short period (<2 hr) waves and (bottom) long period (>2 hr) waves.

4. Conclusions

[11] The study shows that the Gadanki MST radar with steerable beams is suitable for making optimal momentum flux measurements. The statistical estimation error for the conjugate beam momentum flux estimator, integrated over a sufficient length of time, is found to be minimum for radar beam angle between 9°–12° (broadly) in the troposphere. We find that spatial stationarity (beam separation) has to be maintained along with temporal stationarity to minimize the error to its irreducible value. For smaller beam angles, spatial stationarity condition is satisfied but measurement error of horizontal velocity becomes appreciable. And for larger separation, precision of horizontal velocity measurement is better but spatial stationarity condition could be violated. The optimal angle seems to be controlled by a balance of these two factors. These observations are believed to be the first of its kind to experimentally verify the dependence of statistical estimation error with radar beam separation. The study also highlights the necessity of employing “winds ratio factor” to correct the radial velocities for anisotropy of VHF radio wave scatterers in the atmosphere, particularly for low radar beam angles. Future work will emphasize on seasonal and latitudinal dependence of estimation error.


[12] The National Atmosphere Research Laboratory (NARL) is an autonomous facility under DOS with partial support from CSIR. Authors are grateful to Indian Space Research Organization (RESPOND), Govt. of India, for providing financial assistance to run the project. The authors also wish to thank UGC-SVU centre for MST radar Applications, S. V. University, for providing necessary facilities to carry out the work. The suggestions provided by W. K. Hocking, University of Western Ontario, Canada, are gratefully acknowledged. The authors would like to thank the Chairman and Secretary of Anwarul-Uloom College for their active support and kind encouragement. The authors express their deep gratitude to the anonymous referees, whose comments have substantially improved the quality of the paper.