In this study, the beam broadening effect on Doppler spectrum was investigated with theoretical derivation and numerical simulation. It is found that vertical wind may cause a beam broadening spectrum not in exact Gaussian shape. Moreover, the azimuth angle variation of the beam broadening spectral width in the presence of vertical wind is more significant than that in the absence of the vertical wind. The widths of the beam broadening spectra for oblique beams pointing in the opposite directions are different. Further, wind shear effect on the azimuth anisotropy of the beam broadening spectral width is also studied in this article. In the context of three-dimensional wind field with vertical shear of horizontal wind, an analytic expression of the beam broadening spectral width is proposed. A comparison between the analytic expression and simulation result suggests that the expression is applicable in general.
 Wind profilers have been used successfully for nearly four decades to study structure and dynamics of the atmosphere since the 1970s [Woodman and Guilen, 1974; Larsen and Röttger, 1982; Balsley and Gage, 1982]. A large variety of atmospheric parameters and phenomena in MST (meso-, strato-, tropo-sphere) region are measured and observed by using wind profilers at different frequencies, including gravity and planetary waves, tropopause, layer thickness, bright band structure, terminal velocity and drop size distribution of hydrometeor, meteor events, turbulence strengths, atmospheric stability, wind velocity, eddy dissipation rates, etc. [Ecklund et al., 1979; Hocking, 1997]. The atmospheric parameters are commonly estimated from three lowest moments of observed Doppler spectrum (e.g., power, radial velocity and spectral width) in accordance with theoretical relationships. However, a number of factors may broaden or narrow the Doppler spectra, including turbulence activity, wind field, aspect sensitivity of backscatter and integration effect of wave modulation. Therefore, the estimations of the true atmospheric parameters are complicated by such unwanted contributions.
 The beam broadening effect on the observed Doppler spectra have long been the topic of interest in the Doppler radar community. This effect is especially important for wind profilers which have large beamwidths. Even using polarimetric weather radar with relatively narrow beamwidth, the data quality of atmospheric information still deteriorates with range [Ryzhkov, 2007]. Therefore large measurement errors may be induced by finite range and angular resolution [e.g., Sato and Fukao, 1982; Hocking, 1983; Fukao et al., 1988; May et al., 1988; Meischner, 2004]. Further, Candusso and Crochet  have addressed the problem of overlapping and wind velocity profilers smearing due to beam broadening. Besides, the effect of non-uniform reflectivity cannot be ignored [Fukao et al., 1988; Chu and Wang, 2003].
 A number of theoretical expressions of the beam broadening spectral widths for different atmospheric targets have been developed. For example, Chu and Wang  estimated the velocity fluctuation and spatial distribution of ionospheric field-aligned irregularities in sporadic E region after removing the beam broadening contribution from observed spectral width. Hocking  established a method to estimate the root mean square velocity of clear-air, in which some spectral broadening effects are considered, such as beam broadening, shear broadening and finite pulse lengths. With this method, the turbulent eddy dissipation rate can be obtained from the corrected spectral width. The results are in good agreement with those obtained using backscattered power method for a common data set [Cohn, 1995]. Wakasugi et al.  proposed a deconvolution algorithm to remove clear-air turbulence and beam broadening effects from the observed precipitation Doppler spectrum to estimate the raindrop size distribution. In addition, Nastrom and Eaton  derived an expression to estimate quantitative contribution of gravity wave to spectral width such that the wave effect can be removed from the observation data.
 Beam broadening effect has a profound influence on the measured Doppler spectrum, mainly governed by mean background wind [Chu, 2002, 2005]. This broadening effects for various targets have been investigated for years, and there is no general agreement on the quantitative estimation of the resultant spectral moments. Nastrom  proposed an analytic formula of spectral width caused by finite beamwidth and vertical shear of the horizontal wind. Chu  derived theoretical beam broadening spectrum as a function of three-dimensional wind vector and antenna beamwidth. Moreover, Nastrom and Tsuda  reported that the observed spectral widths are smaller for the beams parallel to horizontal wind than those perpendicular to horizontal wind. This feature was attributed to non-uniform distribution of scatters in the azimuth direction. In this article, an attempt is made to investigate the beam broadening spectral width induced by not only three-dimensional wind vector but also vertical wind shear of horizontal wind. In section 2, a numerical simulation model of beam broadening spectrum is established. The study of the beam broadening effect due to wind vector is presented in section 3, in which a comparison between the analytic expression and numerical simulation is made. The wind shear effect on the beam broadening spectrum is investigated in section 4. Conclusions are drawn in section 5.
2. Numerical Simulation
 Numerical modeling is the most feasible way to study the beam broadening effects on Doppler spectrum. The beam broadening power spectrum can be simulated by giving wind field, reflectivity distribution, radar parameters and beam geometry. With the effective radar illumination and the wind field, the relative backscatter power as a function of the radial velocity is obtained by numerically integrating the backscatter along radial velocity contour in the illumination volume. Consequently, the theoretical beam broadening spectrum is achieved and its properties can be investigated accordingly.
2.1. Weighting in the Radar Volume
 The first step of numerical simulation is to give the spatial weighting caused by the antenna pattern and range weighting function, which are both assumed to be in Gaussian shape [Doviak and Zrnić, 1993]. The antenna sidelobes are not taken into consideration in the simulation. The normalized two-way antenna power pattern for a particular beam direction is given [Scipión et al., 2008]
where θx and θy represent the zonal and meridional angles, respectively, (θxp, θyp) is the pointing direction of the radar beam axis, θx3 and θy3 are the one-way 3-dB (half power) beam width in x- and y-directions, respectively. In this study, x, y, z are selected in the east, north and vertical in Cartesian coordinate system, and the antenna center is located at the origin. The relations between (θx, θy) and zenith angle α, azimuth angle ϕ and x, y, z can be given by
where r = . It is worth to point out that θx3 = θy3 is valid for a circularly symmetric antenna beam pattern. The geometry of the radar beam is given in Figure 1.
 We now need to determine the range weighting function. As mentioned before, for a Gaussian weighting function, we have [Scipión et al., 2008]
where R0 is the range from radar to the center of the illumination volume, σr2 is the second central moment of the range weighting function. For a rectangular transmitted pulse and a Gaussian receiver frequency response under matched filter conditions, we have [Doviak and Zrnić, 1993]
where c represents the speed of light, τ is the pulse width. As a result, the composite spatial weighting is formed by multiplying the two-way antenna pattern and the range weighting function. Therefore the magnitude of the spatial weighting at an arbitrary location can be obtained once radar pulse width, illuminating volume center and beamwidth are given.
2.2. Radial Velocity Distribution
 Another crucial issue in investigating the beam broadening spectrum is to know the radial velocity distribution in the illumination volume. In the numerical simulation, the wind field with linear wind shear is considered. The wind vector [u v w] at the location (x, y, z) in rectangular coordinate is given by
where u, v and w represent the zonal, meridional and vertical components of the atmospheric wind field, respectively, u′o, v′o and w′o represent the components of the wind vector. It should be noted that the wind vector is equal to [u′ov′ow′o] at (0, 0, R0). Thus the radial velocity distribution Vr(x, y, z) can be expressed as follows
Figure 2 shows the contour plots of the radial velocity distribution over a horizontal plane at a height of 5000 m for the wind fields without (Figure 2, top) and with (Figure 2, bottom) vertical wind, in which the uniform wind are considered. It is obvious from Figure 2 that the locus of the contours are different between these two cases. The vertical velocity not only shifts but also deforms the radial velocity contours, which may lead to a change in the Doppler spectral shape from Gaussian pattern. Note that, except for vertical wind, the curved contours of the radial velocity can also be caused by divergence and vorticity of the horizontal wind [Larsen and Palmer, 1997; Palmer et al., 1997]. Although inhomogeneous wind field can also be included in the simulation model, it cannot be obtained from a wind profiler without beam steering capability. In view of this, the effects of inhomogeneous wind field on the beam broadening spectrum are not discussed in this article.
2.3. Beam Broadening Spectrum
 Once the radar illumination volume and the radial velocity distributions are given, the corresponding beam broadening spectrum can be simulated with the Doppler sorting process [Palmer et al., 1997]. Figure 3 shows the simulated beam broadening spectrum for vertical and oblique beams, in which the zenith and azimuth angles of the oblique beam are 20° and 90° (pointed eastward), respectively, and a circularly symmetric antenna beam pattern with 5° half power beam width (HPBW) is employed. The parameters used for simulations are that: [u v] = [8 6] ms−1, θx3 = θy3 = 5°, R0 = 5000 m and τ = 1 μs. As shown, the vertical velocity may shift and slightly skew the spectrum for vertical pointing beam, causing a little asymmetry of the beam broadening spectrum. For the oblique beam, the vertical wind effect on the beam broadening of the Doppler spectrum is more significant than the vertical beam. As shown in Figure 3 (bottom right), the corresponding beam broadening spectral width is about 30% smaller than that in the absence of vertical wind. Furthermore, the resultant beam broadening spectrum is skewed more than that for vertical beam, leading to erroneous estimate of wind velocity. Although the asymmetry seems to be negligibly small for the present cases, intense vertical wind with the absence of horizontal wind may result in significant asymmetry of the beam broadening spectrum shape [Chu, 2005]. In addition, a asymmetric beam broadening spectrum can be induced by a asymmetric beam pattern which may be caused by beamforming techniques [Palmer et al., 1993]. Because of these, the beam broadening effect on wind field measurements should be taken into consideration. Some applications of adaptive beamforming technique to measure wind field have been documented by Chen et al.  and Cheong et al. .
 The same exercise can be carried out for various parameters to investigate the properties of the beam broadening spectrum. Nevertheless, the low-order spectral moments can be directly obtained in accordance with following expressions
where W(x, y, z) represents the composite weighting of antenna pattern and range weighting function. Therefore, it appears that the Doppler sorting process is not necessary for the quantitative estimations of mean radial velocity and spectral width of the beam broadening spectrum.
3. Spectral Width Broadening Due to Uniform Wind Field
 Conventionally, for a uniform wind field distribution, the beam broadening spectral width has been considered to be a function of antenna beamwidth and horizontal wind [Gage and Balsley, 1978; Hocking, 1985]. However, considering the three-dimensional uniform wind field and azimuth angle of the beam axis respect to horizontal wind direction, Chu  derived a general expression to calculate the spectral width of the beam broadening spectrum, that is given by
where ϑ is the half power half width, U is the horizontal wind velocity ( = ), w is the vertical wind, ξ is the angle between the horizontal wind direction and the azimuth of the radar beam, ha is the zenith angle of the radar beam and β represents the angle between the projection of the horizontal wind and that of the azimuth of the radar beam axis on the cross-beam plane. The schematic diagram of the configuration is also shown in Figure 1.
 It should be noted that expression (13) is obtained from the integration of the radar power along the contour of the radial velocity distributed on the cross section of the radar beam, which implicitly is a two-dimensional problem. The assumptions made to derive expression (13) are Gaussian antenna beam pattern, low ratio of vertical wind to horizontal wind (<0.35), narrow antenna beam width (<10°) and small zenith angle of antenna beam (<30°) [Chu, 2002]. With these assumptions, the beam broadening spectrum is very close to Gaussian form and the mean radial velocity is almost identical to that of the conventional ones. Although expression (13) makes advance in the estimate of the beam broadening spectral width, there is still a room for it to modify.
Figure 4 shows the beam broadening spectral width as a function of azimuth angle of the radar beam and vertical air velocity. The dashed lines are the realizations of the expression (13) for the horizontal wind velocities of u = 15 ms−1 and v = 0 ms−1. The zenith angle and HPBW of the oblique beam are set to be 20° and 5°, respectively. For comparison, the simulation results (marked with discrete symbols) obtained on the basis of algorithms described in section 2 are also presented in Figure 4, in which the same horizontal wind velocities and R0 = 5000 m and τ = 1 μs are employed. As expected, for the conditions of low ratio of w to U, the simulation results are almost coincident with the prediction of (13), and both of them show the dependence of the beam broadening spectral width on azimuth angle. In the absence of vertical wind, the beam broadening spectral width is minimum (maximum) as the radar beam is steered parallel (perpendicular) to the horizontal wind direction with a 5% variation in amplitude with the azimuth angle. However, the azimuth angle variation of the beam broadening spectral width is more significant in the presence of vertical wind. The difference in spectral width between eastward and westward beams can be as large as 50% for w = 15 m s−1. Figure 4 also shows that the analytical beam broadening spectral width computed by using (13) is underestimated in the presence of strong vertical velocity, suggesting that the application to severe weather condition is limited.
 Observations made by radars show that the vertical velocities of the atmosphere and precipitation particles associated with a thunderstorm can be as large as 20 ms−1 and 9 ms−1, respectively [Chilson et al., 1993], indicating large ratio of vertical velocity to horizontal velocity. As a result, the theoretical beam broadening spectral width estimated in accordance with (13) cannot be applied to the echoes from precipitation particles with large fall velocity and clear-air turbulence with strong vertical wind. An attempt is made in this paper to modify the expression (13) such that it can be applied to the situation with large ratio of vertical to horizontal wind velocities. Following the derivation made by Chu , we redefined β to be the angle between the projection of the three-dimensional wind vector and that of the radar beam axis on the cross-beam plane. With this new definition, we re-calculated the beam broadening spectral width (solid curves) and compared with the simulation result (discrete symbols). As presented in Figure 4, a good agreement between them is shown, suggesting better performance of the modified expression (13) with the new definition of β.
Figure 5 compares the zenith angle variations of the simulated (discrete symbols) and calculated (solid curves) beam broadening spectral widths with selected vertical wind velocities. The azimuth angles of the radar beam are selected at 0°, 90°, 180° and 270°. The parameters employed for computation are the same as those in Figure 4. Figure 5 (left) (Figure 5 (right)) shows the cases while the radar beam is perpendicular (parallel) to the horizontal wind direction. As shown, for the perpendicular cases, the beam broadening spectral width is a monotonically increased function of the zenith angle, with slope depending on the vertical velocity. If there is no vertical velocity, the beam broadening spectral width does not vary with the zenith angle. For the parallel cases, vertical velocity effect on the zenith angle dependence are more significant and complex than those for the perpendicular cases. As shown, the beam broadening spectral width may increase or decrease with the zenith angle of radar beam, depending on the direction of the vertical velocity. In the absence of vertical velocity, the beam broadening spectral width decreases with the zenith angle of radar beam. These results suggest that the directions of radar beam with respect to the three-dimensional wind vector should be taken into account for acquiring true beam broadening spectral width.
 Although the modified expression (13) with the re-defined β can perfectly predict the azimuth and zenith angle dependence of the beam broadening spectral width in the condition of strong vertical velocity, it can not be applied to the environment with wind shear. In the following section, an analytic expression of the beam broadening spectral width in the presence of wind shear will be derived and discussed.
4. Spectral Width Broadening Due to Wind Shear
4.1. Two-Dimensional Beam Model
 An examination of the beam broadening spectral width subjected to horizontal wind with vertical shear proposed by Nastrom , shows that Jacobian term seems to be not included in the derivation. Nevertheless, the error induced by the miss of the Jacobian term is insignificant for the condition of R0 ≫ ΔR, that is always valid for wind profiler. An attempt is made in this section to derive the broadening spectral width due to three-dimensional wind field and vertical shear of horizontal wind.
 Using the form analogous to expressions (10) and (11) in coordinate proposed by Nastrom , the first and second moments of the radial velocity are given by, respectively,
where r is the range, ϕ is the angular extent from the beam axis. In the presence of vertical shear (∂U/∂z) of horizontal wind (U), the radial velocity distribution can be expressed as follows [Nastrom, 1997]
where α is the zenith angle of the beam axis and ξ is the angle between the horizontal wind direction and the azimuth of the radar beam, as shown in Figure 1.
 Assume that the composite weighting W is a constant within the radar volume and zero elsewhere. The integration limits for r and ϕ in deriving and are set to be, respectively,
where ϑ is the half power half width of the radar beam, R0 is the range from radar to the center of the resolution volume, ΔR is the range resolution (= cτ/2). By substituting (16) into (14) and (15), and employing (12), we can obtain the beam broadening spectral width. Further, the term due to pure background wind is substituted by expression (13). Consequently, an expression is obtained as follows
where Γn = sin(nϑ)/nϑ and wo is the vertical velocity. Comparing (19) with the expressions proposed by Nastrom , they are in agreement with each other under the condition of no vertical wind. Since an analytic expression of the beam broadening spectral width is obtained, it is necessary to validate the applicability.
4.2. Comparison and Discussion
 The analytic expression is further examined in terms of numerical simulation. For comparison, both of two- and three-dimensional beam are implemented in the simulations. Moreover, uniform and Gaussian weighting of the radar illumination are both used to find the difference. The parameters employed for the computation are that HPBW = 5°, uo = 15 ms−1, vo = 0 ms−1, wo = 0 ms−1, R0 = 5000 m and ΔR = 150 m. The zenith and azimuth angles of the oblique beam are 20° and 90°, respectively (thus ξ = 0, U = uo).
Figure 6 shows the variations of beam broadening spectral width with vertical shear of horizontal wind. The dashed line is calculated from expression (19) and the discrete symbols represent the results from numerical simulation. Both results seem to have a parabolic shape, but the minimum value is not at zero wind shear. As expected, the analytic solution is very close to the simulation results obtained in the condition of uniform weighting. The difference in the beam broadening spectral widths between two- and three-dimensional radar beam with the same weighting function are negligibly small. However the discrepancy between the results from uniform and Gaussian weighting is apparent and increasing with wind shear. It implies that the beam broadening spectral width obtained from (19) is slightly underestimated owing to the use of unrealistic weighting function. Nevertheless, the analytic expression is usually an adequate approximation in the condition of light wind shear. More comparisons are presented in the following.
 In Figure 7, the range resolution variations of the beam broadening spectral width are shown, in which ∂u/∂z = ±0.01 s−1. As presented, the spectral width increases with ΔR. Moreover, the difference in spectral width between Gaussian and uniform weighting function increases with the increase of ΔR. Because the simulation results for two- and three-dimensional beams are very close, we present the simulation results with three-dimensional beam only.
Figure 8 shows the changes of beam broadening spectral width with range. As shown, for the negative wind shear case (Figure 8, left), the beam broadening spectral width increases with the range. However, it is not the case for positive wind shear (Figure 8, right). The minimum spectral width occurs at around R0 = 12 km as analytic prediction.
Figure 9 shows the variation of beam broadening spectral width with zenith angle in the presence of vertical shear of horizontal wind. We note that the pattern of zenith angle dependence is variable. Nevertheless, the analytic solution is in coincident with the simulation for uniform weighting, and slightly deviates from results for Gaussian weighting.
 In the absence of vertical wind, the azimuth angle variations of the beam broadening spectral width with selected wind shears are shown in Figure 10. The simulation results obtained by using three-dimensional beam with Gaussian weighting are marked by discrete symbols for comparison. As presented, a good agreement between the simulation result and analytic expression is shown, especially for low wind shear cases. Moreover, the spectral width has maximum (minimum) for negative (positive) wind shear in the context of radar beam direction parallel to the horizontal wind. When radar beam is perpendicular to the horizontal wind, the beam broadening spectral width is expected to be constant regardless of the wind shear. In the presence of vertical velocity (wo = 5 ms−1), the azimuth angle variations of beam broadening spectral width become more complex, as shown in Figure 11. Although the discrepancy between numerical simulation and analytic solution becomes larger than that in the absence of vertical velocity, the error is less than 15% for the presented case. Expression (19) is adequate to predict the corresponding beam broadening spectral width in the presence of both vertical velocity and wind shear. After surveying the results using various magnitudes of the parameters, we suggest a criteria for the applicability of the expression: the ratio of vertical velocity to horizontal velocity less than 1/3.
 The spectral broadening effect due to finite beam width has been investigated in this paper, in which isotropic turbulence scatter and uniform reflectivity are considered. First, a numerical model is established to simulate the actual beam broadening spectrum. The characteristic of the radial velocity distribution and the illumination within the radar volume are the crucial issues. In addition, the shape of the beam broadening spectrum is obtained using the Doppler sorting process. It is found that the resultant beam broadening spectral shape may not be in an exact Gaussian shape. The estimation of the wind velocity could be in error from the distorted spectrum. Furthermore, an analytic expression is suggested to predict the precise beam broadening spectral width. It can be applied to the cases of homogeneous wind field even under the condition of large vertical velocity compared to horizontal velocity. Moreover, large vertical velocity will aggravate the azimuth angle variation of the beam broadening spectral width. The spectral width may be different using opposite beam directions. These results show that not only horizontal wind but also vertical velocity should be taken into account in quantitative estimation of the beam broadening spectral width.
 Based on a two-dimensional uniform illumination beam model, an expression of beam broadening spectral width caused by three-dimensional wind vector and the vertical shear of horizontal wind has been proposed. The variations of beam broadening spectral width as a function of several parameters are presented. The comparisons with simulation results show that the beam broadening spectral width predicted from the analytic expression is underestimated due to the uniform weighting of the illumination. However the expression is usually an adequate approximation to the simulation results. Moreover, the azimuth angle variation of the beam broadening spectral width is more complex in the presence of both vertical wind and wind shear. Nevertheless, the analytic expression is appropriate to estimate the beam broadening spectral width for the cases of light wind shear and low ratio of vertical velocity to horizontal velocity, which is typical situation in normal atmospheric condition. The work for more complicated circumstances may be accomplished using numerical simulation.
 This work was supported by the National Science Council of the Republic of China in Taiwan under grant NSC 98-2811-M-008-040.