Vertical structure of raindrop size distribution in lower atmospheric boundary layer



[1] We examine the vertical structure of the raindrop size distribution (DSD) in the lower atmospheric boundary layer (ABL) below altitudes of 300 m, where conventional radars typically do not observe. The DSD in the lower ABL is retrieved using Ku-band broadband radar (BBR) having an observational range of 50 m to 15 km, and a high range resolution of several meters and a 3-dB beam width of 3 deg. The radar-retrieved DSD are in excellent agreement with the DSD measured with a co-located, 2 dimensional video disdrometer with correlation coefficients over 0.96 in both stratiform and convective rain events. While the DSD reveals no significant change in the stratiform event, the growth process increases about 2 times in the number of raindrops larger than 0.5 mm in diameter in the convective event. This growth process in the ABL is important when we discuss the rainfall rate from radar reflectivity factor.

1. Introduction

[2] A small-baseline weather radar network can detect small-scale weather phenomena such as thunderstorms, tornadoes, and downbursts in the atmospheric boundary layer (ABL). Collaborative Adaptive Sensing of the Atmosphere (CASA) [Junyent and Chandrasekar, 2009], and X-band radar network (X-net) [Maki et al., 2008], have proposed a new radar network system using short range X-band radars to observe lower troposphere. Most parameters that comprise precipitation profiles, in the ABL are influenced by interaction with ground. Understanding these interactions plays an important role in the accurate estimation of precipitation rates. We have been developing a Ku-band broadband radar (BBR) that observes precipitation profiles from a low altitude of 50 m up to 15 km with a high range resolution of several meters and a temporal resolution of about 1 min per 1 volume scan [Yoshikawa et al., 2010]. The spatial and temporal resolution of our BBR will improve with additional short-baseline installations.

[3] Previous studies [Williams et al., 2007; Nikolopoulos et al., 2008] used co-located wind profilers and disdrometers to study the localized features of vertical precipitation profile and uncertainties in radar observations of precipitation. Such studies indicate that changes in precipitation profiles, such as water vapor, can be significant in the ABL. Such changes in precipitation profiles results in observational errors in radar observations near ground. Precipitation profiles in the ABL are important as ground validation of Precipitation Radar products of the Tropical Rainfall Measuring Mission (TRMM) satellite, particularly when studying the global water cycle [Gage et al., 2002]. However, since conventional radars cannot observe below altitudes of several hundred meters (about 300 m at least) and operational radars focus on even higher altitudes, precipitation profiles in the lower ABL (under 300 m) are typically not observed by radar.

[4] In this study, DSD are measured with two co-located instruments, the BBR and a 2 dimensional video disdrometer (2DVD) in Osaka, Japan. The DSD profiles that are retrieved from the vertical Doppler spectrum (VDS) of the BBR in the lower ABL (50 m through 300 m) are compared to the DSD estimated by the 2DVD. Using these instruments, we discuss the differences of vertical structures of the DSD in the lower ABL between stratiform and convective rain events.

2. Equipments and Observation Characteristics

[5] We observed vertical precipitation profiles using a prototype BBR observing a fixed direction using a bistatic antenna system [Mega et al., 2007]. The antennas were upgraded to Luneburg lenses which achieve a 3-dB beam width of 3 deg. The observational characteristics of the BBR are shown in Table 1. The BBR transmits and receives wide band signals at Ku-band (15.75 GHz) using pulse compression to give us sufficient energy on a target for detection with high range resolution and good signal to noise ratio. The transmitting waveform of the BBR is a linear up-chirped and raised-cosine windowed (a = 0.3) pulse of 128 μs duration and 80 MHz bandwidth, which ideally achieves 3-dB range resolution of about 6 m. Since the digital sampling frequencies (250 or 400 MHz) have smaller wavelengths than the range resolution in this observation, the BBR products are averaged every 5 m. The VDS is measured every 3 seconds by the BBR outputs from a sequence of sixty-four pulses (there is a data-offload time of about 2.8 seconds). The 2DVD that is used for validation is an optical sensor that detects each rain drop [Kruger and Krajewski, 2002], and it was collocated a distance 15 m horizontally and 25 m below the BBR. The outputs of the 2DVD are accumulated and averaged over 1 min intervals to ensure that a statistically significant number of rain drops are sampled. Correspondingly, the BBR output is also averaged over 1 min intervals (20 values are averaged because the sequence interval is 3 sec).

Table 1. Observation Characteristics
 Observation IObservation II
Center Frequency15.75 GHz15.75 GHz
Band width80 MHz80 MHz
Operation modeVertical pointingVertical pointing
Power2 W (peak value)2 W (peak value)
Pulse length128 usec128 usec
ModulationLinear up chirpLinear up chirp
Weighting windowRaised cosine window (a = 0.3)Raised cosine window (a = 0.3)
Sampling frequency400 MHz250 MHz
Pulse repetition time200 usec200 usec
Pulse Number/Sequence64 pulses64 pulses
Sequence interval3 sec3 sec
Nyquist velocity23.79 m s−123.79 m s−1
Doppler velocity resolution0.37 m s−10.37 m s−1
Day & Time22:00, May 28, 2008 through 6:00, May 29, 200816:50 – 17:50, Sept. 5, 2008 and 8:50 – 9:30, Sept. 5, 2008
PlaceOsaka University, Suita, Osaka, JapanOsaka University, Suita, Osaka, Japan
RemarksStratiform rain eventConvective rain event

[6] Two separate events were observed, one stratiform and one convective. The observational periods for the stratiform and convective events were 480 and 100 minutes, respectively. The two measurements are time synchronization through cross-correlation analysis. The reflectivity factors calculated from the BBR at an altitude of 50 m are compared with the 2DVD measurement, and the 2DVD data are time-shifted to the BBR to give the highest cross-correlation coefficient. The reflectivity factor of the BBR is then calibrated by the mean difference between reflectivity factors of the BBR at the 50 m altitude and 2DVD.

3. DSD Retrieval

[7] In this section we describe the methodology of DSD retrieval employed in this study. The DSD retrieval consists of 3 steps; 1) the conversion of VDS to DSD, 2) the fitting to a Gamma DSD, and 3) the correction for precipitation attenuation. The VDS near velocities of 0 m s−1, and hence measurement of small-diameter raindrops, is contaminated by direct current (DC) components in the instrumentation electronics and by motionless targets, so only number densities over 0.9 mm are estimated in Step 1. In Step 2, the DSD estimated in Step 1 is fitted to a Gamma DSD to estimate number densities of precipitation under 0.9 mm in diameter. Step 3 corrects precipitation attenuation for a VDS at a given range bin using the DSDs of all the bins behind. Steps 2 and 3 are performed iteratively and deterministically beginning with the closest bin (50 m) in which no precipitation attenuation is assumed. The other assumptions contained in this study are described below.

3.1. Step 1: Conversion of VDS to DSD

[8] Assuming that background winds are neglected and that all the raindrops in the radar resolution volumes fall at terminal velocity, the relation between a VDS and DSD is approximated as

equation image

where vt is the terminal fall velocity (m s−1), S(vt) is the VDS (mm6 m−3 (m s−1)−1), λ is the wavelength (mm) of a transmitting wave, σb is the Mie back scattering cross section (mm2) [Ulaby et al., 1981], and N(D) is the DSD (m−3 mm−1). Also, the diameter of a raindrop D (mm) is empirically related to vt (m s−1) as reported by [Atlas et al., 1973]

equation image

Equation (2) precludes terminal velocities over 9.65 m s−1. In the previous works using wind profilers which receives both back scattering from raindrops and Bragg scattering from air (containing vertical air motions and turbulences) in a VDS [Williams et al., 2007], several superior methods to estimate DSD from a VDS that eliminate background winds have been proposed [Schafer et al., 2002; Lucas et al., 2004]. On the other hand, air motions are not detected in the BBR. In this study, turbulences are simply neglected due to the remarkable spatial resolution of the BBR (the beam width at 300 m is about 15 m). Vertical air motions are neglected because low altitudes (under 300 m) are being studied, although this may introduce errors into the calculations. However, the boundary condition of vertical air speed is 0 m s−1 at ground and the mean vertical velocity of air motions over about 1.5 km is 0.5 m/s [May and Rajopadhyaya, 1996], so the vertical air motion in the lower ABL is estimated to be about 0.1 m s−1, which yields a maximum error in number density of 1.5 dB. Strong downdrafts, such as microbursts, might break this assumption, so it is necessary to eliminate events in which these occur. Strong downdrafts occur more often in convective than stratiform events. We have sufficiently confirmed that the data analyzed in this study do not contain strong downbursts by examining the time-height cross sections of reflectivity and mean vertical velocity (not presented here).

3.2. Step 2: Fitting to Gamma DSD

[9] A 3 degree beam width will have some spectral broadening due to turbulence and horizontal wind across the beam, depending on Doppler spectrum resolution. A power of DC component leaks into adjacent bins of low velocities, which correspond to diameters under 0.9 mm in DSD. In Step 2, the DSD that were estimated in Step 1 are fitted to the Gamma DSD to estimate the number densities of precipitation under 0.9 mm diameter. The Gamma DSD [Ulbrich, 1983] is expressed as

equation image

where the exponent μ can be any positive or negative value, N0 and Λ are the coefficients of the units of (m−3 mm−1−μ) and (mm−1), respectively, and Dmax is the maximum drop diameter (mm). Since the DSD from a VDS have an insufficient number of samples for a good fit, μ is sensitive to fluctuations in VDS. Therefore, we employ an empirical and statistical relationship between N0 and μ [Ulbrich, 1983; Testud et al., 2001] shown in equation (4)

equation image

[10] Considering these characteristics, the optimal parameters of the Gamma DSD, N0, μ, and Λ, are determined by least squares error minimization on a logarithmic scale, that is, by minimizing the cost function

equation image

where NBBR(D) and NGAMMA(D) are the DSDs in equations (1) and (3), respectively. In this fit, Di is above 0.9 mm. Figure 1 shows the DSD that are estimated from the BBR along with the 2DVD, and that there is good agreement between the two for raindrops of diameters greater than 1 mm. The 2DVD underestimates the number density of raindrops smaller diameters, especially less than 0.75 mm because the smaller raindrop is frequently obscured by larger raindrops at the optical wavelengths used by the 2DVD measurement.

Figure 1.

An example of DSD retrieval from the BBR. Diamond marks indicate DSD retrieved from the VDS of the BBR. Solid line is the result of fitting BBR DSD to Gamma distribution whose parameters in this case are also shown on the bottom-left corner in the panel. Cross marks indicate DSD estimated by 2DVD.

3.3. Step 3: Correction for Precipitation Attenuation

[11] Although precipitation attenuation does not significantly affect the measurement of precipitation profiles with VDS at the close ranges of focus in this paper, an effort is made to apply a precipitation attenuation correction for accurate validation. The attenuation coefficient k (dB km−1) can be calculated from the DSD as,

equation image

where σe is the Mie extinction cross section (mm2) [Ulaby et al., 1981]. The reflectivity factor at a given range bin is corrected deterministically using the total attenuation of all preceding range bins. Though the deterministic approaches for precipitation attenuation correction are unstable over long ranges, it is known that they are more stable at the closer ranges of focus here [Iguchi and Meneghini, 1994].

4. Observation Results

4.1. Comparison of the BBR in the Minimum Altitude and 2DVD

[12] Figure 2 shows the comparison of DSDs from the BBR and 2DVD during a period of 60 minutes. Figures 2 (left) and 2 (right) are for the stratiform and convective events, respectively. Figures 2 (top) and 2 (middle) are, respectively, the results from the BBR and 2DVD, and the colors represent number densities in m−3 mm−1 from −20 to 40 dB. Figures 2 (bottom) indicate the correlation coefficient of both instruments in each minute. The time averages of the correlation coefficients are 0.74 and 0.78 in the stratiform and convective events, respectively. These correlation coefficients are calculated on a logarithmic scale for all diameters from 0 to 10 mm in diameter every 0.25 mm. If raindrops of diameter less than 0.75 mm are excluded, the correlation coefficients increase to 0.98 and 0.96 for stratiform and convective events, respectively. Thus, the two instruments perform well and are in good agreement for raindrops of diameter greater than 0.75 mm for the stratiform and convective events.

Figure 2.

Comparison of DSDs from the BBR at a 50 m altitude and 2DVD. (top) Time series of DSD retrieved from the BBR. (middle) That from 2DVD. The colors represent number density (m−3 mm−1) in dB. (bottom) Correlation coefficient between them in a logarithmic scale. The black lines are the correlation coefficient calculated with the DSD from BBR between 0 and 10 mm in diameter and the red lines are between 0.75 and 10 mm in diameter. (left and right) Results of the stratiform and convective event, respectively.

4.2. Vertical Profiles of DSD

[13] Figure 3 shows the vertical structures of the time-averaged (over 60 min) DSD for the stratiform and convective events in Figures 3 (right) and 3 (left), respectively. In the stratiform event (a-1), the DSD do not show significant change below an altitude of 300 m, indicating that the growth and breakup processes of raindrops are balanced to the equilibrium condition for the stratiform event in the lower ABL. In the convective event (b-1) the raindrops over a 0.9 mm diameter (which are directly retrieved) grow in size as altitude decreases. Since larger raindrops generally fall at faster terminal velocities, the lower region becomes denser even below 300 m altitude in the convective event. Figure 2 (bottom) shows the differences in number densities (in dB) from 300 m to lower altitudes (200, 100 and 50 m) for each event. The number density differences are probably not significant for precipitation of diameter less than 0.2 mm (because the density estimation may be inaccurate, see Section 3.1), or greater than 4 mm in the stratiform event (due to the relatively low number density). For the stratiform event (a-2), the most considerable shift is about −1.4 dB (a factor of 0.77) at 2.2 mm in the 50 m and 300 m difference. In the convective event, growth processes generate larger raindrops over 0.9 mm in diameter as stated above, especially for precipitation over 2 mm in diameter. In the convective event, the differences of the number densities between 100 m and 300 m, and between 50 and 300 m, respectively, have the maximum values of 2.9 dB at 4.5 mm and 3.2 dB at 4.1 mm. While the number densities from 3 to 5 mm in diameter approximately double from 300 m to 100 m in altitude, there are no significant changes in DSD under 100 m altitude.

Figure 3.

Vertical structures of DSD. (top left and top right) Results of stratiform and convective events, respectively. The colors represent number density (m−3 mm−1) in dB. Contour lines are depicted with a 2 dB interval. (bottom) Differences of DSDs at different altitudes in each event. Black, blue, and red lines correspond to DSD differences of 200 m, 100 m, and 50 m, respectively, from 300 m in altitude.

5. Discussions and Conclusions

[14] We have shown that the BBR can estimate DSDs accurately for precipitation of diameter greater than 0.75 mm. The measurement of DSD from the BBR is in excellent agreement with that of the collocated 2DVD, indicating that the BBR can effectively measure precipitation profiles in the lower ABL. We used the BBR measurements of DSD to study the precipitation profiles in the lower ABL under 300 m in a stratiform and a convective event. The vertical precipitation profile in the ABL indicates that there is no significant change in the DSD in the stratiform event, but in the convective event the growth of DSD from 300 m to 100 m altitude is clearly shown. Table 2 indicates that in the Z-R relation, the coefficient increases with altitude while the exponent almost never changes in the convective event. The changes in the Z-R relation amount to a maximum bias error of 2 percent to the estimated rainfall rate (although higher errors can occur at higher altitude). In the lower ABL, Z-R relation does not change, but Z and R increase as raindrops approach the ground. Thus, conventional radars may get the right Z-R relationship, but they cannot measure the increase in rainfall rate.

Table 2. Z-R Relations (the Convective Event)
50 m100 m200 m300 m
Z = 378R1.42Z = 378R1.43Z = 385R1.44Z = 405R1.43Z = 319R1.42


[15] This work was supported by a grant from Strategic Information and Communication R&D Promotion Programme (SCOPE) of Ministry of Internal Affairs, Japan, Communications and Ministry of Education, Science, and Sports and Culture, Japan, Japan Society for the Promotion of Science (JSPS). We acknowledge Y. Fujiyoshi at Hokkaido University, Japan for providing with the data of 2DVD, and Christopher J. Biagi in University of Florida for discussions and English correction.