Figure 2 shows time series of rainfall rate observed on 25 December 1994 at three locations, Chung-Li (C1C52), Hsin-Wu (C0C45) and Yang-Mei (C1C50) about 1 km east, 8 km south and 7.8 km west of the Chung-Li VHF radar site, respectively. The rain gauge data is collected with 1-min resolution. According to C1C52, C0C45 and C1C50 rain gauges the total rainfall of 15, 14 and 14-mm is recorded, respectively, on 25 December 1994. The time series of the rain rate shows less similar variations for the three stations. From the rain intensities observations we can anticipate a widespread showers over this region. As we know that vertical air motions largely control the aerial extent, intensity, and lifetime of a precipitation cloud system.
3.1. Estimation of Advection Speed and Direction of the Precipitating Cloud System Observed on 25 December 1994 From Surface Rain Gauges
 In general, the subtropical region's heavy rain was found in widespread as well as showery precipitation situations occur in compact groups rather than to be randomly scattered. The groups are often in the form of rainbands associated with frontal surfaces. Topographic effects, as well as fronts, can affect the structure and development of rain cells. Although the effects have been documented to some extent but it is not yet fully understood the patterns of the observed precipitation. The rainfall rate is a function of position on the surface and time. Following the practice in the theory of random processes, it is possible to define the autocorrelation function of rainfall rate in time or space. Earlier, Zawadzki  utilized space autocorrelation function to describe the fine-scale structure of widespread rain. To understand the structure of the showery precipitation, we have utilized the radar and three rain gauges. Chu et al.  and other groups applied the spaced antenna drift (SAD) method to ionospheric experiments for measurement of horizontal drifting velocity of electron density irregularities. We adapted similar procedure to measure the horizontal drift velocity of the rain cells during showery precipitation.
 We applied SAD method to the rainfall data collected from the three stations to measure the horizontal drift velocity of the rain cells. Figure 3 shows the schematic diagram of a rain pattern moving through rain gauges and Chung-Li VHF radar. The closed curves represent the contours of the rain cells, V is the drift velocity of the pattern, a, b and c are the distance between rain gauge pairs 2-3, 3-1 and 1-2, respectively. The dashed line DD′ is a reference line and is parallel to the drift direction of the rain cells on the ground and connects the rain gauges at the apex point 2. The dotted lines are drawn such that they are normal to the dashed line and line with the rain gauge array at the corresponding apex points. With the help of these complementary lines, the distances L and M can be estimated from the geometry of the figure in order to deduce the drift velocity of the rain cells. On the basis of this schematic plot, it is clear that the following equations can be obtained readily:
where t1 and t2 are time delays between the time series pairs of the rain intensities recorded by the rain gauge pairs 1-2 and 1-3, respectively. From equations (1), (2) and (3), we have
By substituting equation (4) into equation (1) and rearranging the formula, the angle α can obtained by the following equation:
As a result, the drift velocity, V of the rain pattern can be estimated by using the following equation:
Equations (5) and (6) indicate that the estimation of t1 and t2 from the observed rainfall rate plays a vital role in determining α and V. The t1 and t2 are estimated through the calculation of the cross-correlation function of the corresponding time series of the rainfall rate.
Figure 3. Schematic plot showing the geometry of the three rain gauges with a moving rain cell, where the curves represent the contours of the pattern. V is the drift velocity of the pattern, and the reference line DD′ is chosen arbitrarily.
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 Estimation of time delays between the time series of the rain intensity recorded by three tipping bucket rain gauges plays a key role in deriving drift velocity of the rain cells. The correlation analysis is performed for the rain intensities recorded by each pair rain gauges. The cross-correlation function, ρ(τ), of the time series x(t) and y(t) is defined as follows:
where τ is the time lag. Apparently, the maximum cross-correlation coefficient, ρm (τ), occurs at a specific time lag, τm, which represents the time shift between the time series x(t) and y(t). The sign of τm depends on whether the time series x(t) leads or lags behind the time series y(t). By definition, if τm is positive, this indicates that y(t) leads x(t), and vice versa. Figure 4a shows the cross-correlation functions of the rain intensities shown in Figure 2. As indicated in Figure 4a, the time delays between ρ12, ρ23 and ρ31 pair of rain intensities obtained from rain gauge 1-2, 2-3 and 3-1 are −7.74, 5.77 and −0.29 min, respectively. By substituting these time delays into equations (5) and (6), the horizontal drift velocity of the rain cells can thus be obtained, as presented in Figure 4b. It shows that observed drift velocity of the rain cell is 13.4 m/s from the direction of 44.6° in southwest to northeast. These results are fairly good agreement with the horizontal wind velocity observed with the VHF radar. Horizontal wind velocity observed over Pan-Chiao rawinsonde station around 0800 local time (LT) also confirms the orientation of the wind direction and the magnitude of the wind velocity (figures not shown here).
Figure 4. (a) The cross-correlation functions of the pairs of rain rate as shown in Figure 2. (b) The observed drift velocity and direction of the rain cells on 25 December 1994.
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3.2. Chung-Li VHF Radar Observation of Precipitating Cloud System
 Chung-Li VHF radar has been used in the recent past to study the adverse weather/meteorological phenomena, such as typhoon passage, Mei-Yu cold front due to the fact that precipitation measurements can be made at the same time as wind observations [Chu et al., 1999]. Due to these capabilities of Chung-Li VHF radar, we utilized this radar to investigate the useful information on the precipitating cloud systems and their vertical structure during (tropical depression) warm front over subtropical region of Taiwan.
 The echoes from refractive index fluctuations and precipitation particles can be identified when the precipitation is within the radar volume. Figure 5 shows an example of the Doppler spectra observed by vertical beam from 1.8 to 7.8 km with 300 m of height resolution on 25 December 1994 around 2218:44 LT. The first peak located near the position of the zero Doppler frequency is attributed to the Bragg scattering from refractive index fluctuations, and the second peak is due to Rayleigh scattering from the precipitation particles. These Doppler spectra, and all those presented in detail in this paper, were taken at vertical incidence. It is obvious from Figure 5 that the stronger echoes intensities and narrow Doppler velocities (centered near 0 ms−1) correspond to the radar returns of refractivity fluctuations. The fall velocity of the particles up to −7 ms−1 is caused due to the hydrometeors. We can separate the two echoes and find the reflectivity, mean velocity, and spectral width associated with each echo.
where S(ω) is the observed precipitation Doppler spectrum, St(ω) represents the Doppler spectrum of refractivity fluctuations, and Sp(ω − ωo) is the size distribution of precipitation particles in the Doppler spectral domain at Doppler frequency ωo, and ∗ represents the convolution operator. Note that the shape of St(ω) is usually assumed to be Gaussian because of the beam-broadening and turbulent broadening effects [Woodman and Chu, 1989]. Superficially, the shape of Sp(ω − ωo) is not Gaussian due to the exponential form or Gamma pattern of the drop-size distribution [Marshall and Palmer, 1948; Ulbrich, 1983]. However, because the radar echo power from precipitation is proportional to the 6th power of the diameter of the precipitation particle, the pattern of Sp(ω − ωo) will be quasi-Gaussian in the Doppler spectral domain [Atlas et al., 1973]. Consequently, for the sake of mathematical simplicity, the shape of Sp(ω − ωo) can be treated with Gaussian form, causing the reasonable approximation of Gaussian pattern S(ω). The spectral components for turbulent refractivity and precipitation can be separated unambiguously in the Doppler spectral domain as long as the radar beam is steered in the right direction and the radar parameters are set appropriately. The echo power, mean Doppler frequency shift, and spectral width for these two components are estimated separately with the least squares method, in which the Gaussian curve, is employed to best fit the corresponding Doppler spectral component [Chu et al., 1991, 1999; Cohn et al., 1995; Rao et al., 1999]. The data also contain spectra with only a single peak. In that case we must decide if the echo is from clear air or hydrometeors. This is done based on the mean fall speed and power of the echo.
 Figures 6a and 6b present time-height distributions of the VHF echo power from the refractivity fluctuations and precipitation observed by using the vertical pointed radar beam. Roughly speaking, the variation in echo power is the indicative of change in atmospheric stability. Stratified stable layers up to 3 km height are observed from 0007 to 1015 LT and a second, more-or-less stable layer, at ∼4 km from 0007 to 0200 LT. Because of the different radiation balances between night and daytime conditions stable lower atmospheres tend to dominate at night, and unstable ones at day. Figure 6b shows several intermittent precipitation events (circled characters indicated in Figure 6b) are caused due to the passage of precipitating cloud systems. Around 0110 and 0255 LT, precipitation is detected by the VHF radar, but no precipitation is observed at the surface (as shown in Figure 2). The lifetime of the individual precipitation showers ranges from 20 to 168 min. In this study, we mainly focused on predominant three precipitation echoes (F, G and H) observed around 1705, 1807, and 2055 LT. In the case “F” enhanced precipitation echo power, which is almost continuously observed in Figure 6b at an altitude of 4.2 km, is considered to be a bright band where falling particles pass through the melting (0°C) layer. According to Szoke and Zipser , one of the necessary conditions for the formation of bright band is the relatively small vertical air velocity (less than 2 ms−1). Examining the distribution of vertical velocity (as shown later in Figure 7a) reveals that for the present precipitation event the vertical air speed is weak that leads to the formation of the bright band. It is noteworthy to observe that inside the bright band the precipitation echoes are enhanced due to change in precipitation state from ice to liquid water and by the aggregation of ice crystals [Fletcher, 1972; Battan, 1973; Fukao et al., 1985b]. In Figure 6a it is also noticed (F, G and H) that the turbulence echo intensity variations during the occurrence of precipitation. The radar refractivity is a function of atmospheric humidity, temperature and pressure. The interaction between precipitation particles and the ambient atmosphere leads to the following effects. One is the cooling effect, arising from the absorption of latent heat associated with the evaporation of the raindrops and melting of the ice particles. This effect will lower the atmospheric temperature and changes the refractive index. The other one is the vaporizing effect, due to the release of water vapor through the processes of ice sublimation and raindrop evaporation. This effect will increase the atmospheric water vapor content and also influences the atmospheric refractive index. These two effects occur simultaneously in the precipitating atmosphere. However, it is believed that the latter one is more important than that of the former one. This because the fractional change of humidity, due to the turbulent-mixing process, is greatly larger than that of temperature below the middle troposphere (for example, below 8 km) over the Taiwan area [Chu et al., 1990]. Moreover, from Figures 6a and 6b it is evident that precipitation plays a vital role in atmospheric turbulence and turbulence mixing.
Figure 6. Time and height cross section of (a) turbulence echo power and (b) precipitation echo power observed on 25 December 1994.
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Figure 7. Time and height cross section of (a) vertical velocity and (b) the hydrometeor terminal velocity deduced from the vertical beam on 25 December 1994.
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 Radar measurements of vertical velocities are important in improving our understanding of the turbulence and precipitation growth in the showery precipitation. Figure 7a shows the height-time distribution of the vertical air velocity observed on 25 December 1994. The most striking feature of the vertical wind structures is random variations in the updrafts/downdraft before and during the showery precipitation. We believe that (in this study) the weak updraft/downdraft plays a major role in the formation and disappearance of turbulence layers and also in precipitation echoes during the rain “showers.”
 Figure 7b is the height-time variation of the terminal velocity (i.e., updrafts and downdrafts are removed from the fall velocity of the hydrometeor) deduced from the precipitation Doppler spectra observed by the vertically pointed radar beam. During the showery precipitation, above 4.2-km the frozen hydrometeors (snow/snowflakes) show terminal velocity of 1–2 ms−1. At the melting level, near 4.2, the melting droplets accelerate and a relative maximum (bright band) reflectivity can be observed (Figure 6b). In the rain region, i.e., from 4 to 1.8 km, the terminal velocities up to 8 ms−1 are observed. Our results are consistent with the earlier investigations with VHF radar [Fukao et al., 1985a, 1985b; Wakasugi et al., 1985; Chu et al., 1991] and also with conventional meteorological Doppler radar [Atlas, 1964; Battan, 1973].
 One of the most critical parameters deduced from radar measurements, and one of central importance in precipitation microphysics, is the distribution of the water particles as a function of diameter. To find the raindrop diameter accurately from Doppler radar spectra it is necessary to have accurate measurements of the particle terminal speeds, which in turn requires accurate estimation, made by the motion of the air through which the drops are falling. It is well known that the mathematical relation between raindrop terminal velocity v and its diameter D can be formulated empirically as follows [Gunn and Kinzer, 1949]:
where v is in meters per second and negative downward, D is in millimeters, ρo is the air density at the ground level, and ρ is the air density at the height of observation. This equation is, however, valid only if D is larger than Dmin that is the diameter to make the velocity, v is zero.
 We assume that there is no raindrop that has the drop-size smaller than Dmin for estimation of raindrop size distribution (DSD) during the passage of showery precipitating cloud systems on 25 December 1994. Figure 8 shows the estimated DSD using equation (9). The figure shows that the radar returns scattered from the raindrops with diameters from 1.75 to 3 mm contribute to the major portion of the observed precipitation echo power.
 The Doppler spectral width is an extremely important VHF radar echo parameter. Most of the atmospheric information can be evaluated from this radar parameter. However, there are quite a few physical mechanisms that can contaminate the width of the Doppler spectrum. For example, the beam-broadening effect, wind shear effect, drop-size-distribution-broadening effect, and gravity wave oscillation effect will broaden the Doppler spectral width. This problem is present for all VHF Doppler radars [Nastrom and Eaton, 1997] and especially more severe for the off-vertical beams [Nastrom and Tsuda, 2001]. The Doppler spectral width will also be narrowed by the aspect sensitivity for vertical or close to vertical pointing radar beam. Because the broadening and the narrowing effects are coexisting in the observed Doppler spectrum, the estimation of true atmospheric information from spectral width will be impossible if the contaminating factors are not thoroughly removed from the spectrum [Woodman and Chu, 1989]. Beam broadening is caused by the horizontal wind drifting of the atmospheric refractive index fluctuations across the radar beam. For the conventional microwave meteorological radar the beam width is always narrow (<1°); hence the beam-broadening effect can be ignored. Whereas, the VHF radar has the broad antenna beam width, and the beam-broadening effect cannot be neglected. Figure 9 shows the spectral width of the turbulence echo, which is separated from precipitation echo on 25 December 1994 from 0007 to 0000 LT. From this figure we anticipate that beam broadening of turbulence Doppler spectrum is significant and should be taken into consideration in the analysis of Doppler spectral width for further applications.
 As discussed above, turbulence medium has influence on the Doppler velocities associated with the hydrometeors. The precipitation particles are carried along with the turbulence, which have the effect of smearing their backscattered signal across the frequency bins in the Doppler spectrum. This in turn broadens the spectrum, which leads to an erroneous assessment of the actual diameters of the particles and their distribution function. It is possible to remove the contribution of the clear-air spectral width caused by beam-broadening and turbulent broadening effects from the spectral width of precipitation to estimate the hydrometeor size distribution provided the precipitation particles are frozen in the background wind [Wakasugi et al., 1986; Gossard et al., 1990]. Under this assumption, the mathematical relation of the Doppler spectra between precipitation particles and refractivity fluctuations for vertically pointed radar can be formulated as in equation (8). The information of precipitation particles can be separated from Sr(ω), which is contaminated by St(ω), through the following relation:
where σp2, σ2, and σt2 are the variances of Sp(ω − ωo), S(ω) and St(ω), respectively. Because σp can be treated as an indicator of the breadth of hydrometeor size distribution, the larger σp is, the broader the size distribution will be.
 The main factors influencing the magnitude of spectral width from precipitation are the reflectivity per unit area of the particles, their backscattering cross-sectional area and the distribution of the drop sizes. The reflectivity of each particle will depend on the complex index of refraction of its constituents. In conditions of vertical mixing in precipitation, the different hydrometeors become separated into layers according to the thermal structure of the atmosphere. The various dependencies of radar return then become important [Smith, 1986]. Figure 10 shows the distribution of precipitation particle spectral width observed on 25 December 1994. Above the melting layer, where the temperature is well below 0°C, are small ice crystals of low reflectivity that give a narrow spectral width. As indicated in the figure, in the height ranges at the bright band a broader Doppler spectral width is observed due to rapid increase in the dielectric constant of hydrometeors at the top of the melting layer followed by an increase in the terminal velocity of melting snowflakes toward the end of the melting process. Below the melting layer large drops tend to increase with rainfall rate and contribute to the radar reflectivity. Comparison of Figure 10 with Figures 6b and 7b shows that the relatively larger σp around bright band region coincides very well with the intense radar reflectivity, consistent with the theoretical anticipation.