1. Introduction

For most rain-rate studies the size distribution of droplets is enough to describe the population behavior of rainfall and it is adequate to measure drop size distributions at the ground. In particular, drop size distribution measurements are useful in understanding the rainfall intensity differences between surface point measurements of rain gauges and volume scan measurements of radar. However, conventional methods of obtaining drop distributions are compromised by local wind effects, splashing, and contamination by material and debris from the surface, also common to rain gauges (Nystuen 1998; Löffler-Mang 1998). Bradley (1996) suggested that a high-frequency Doppler acoustic precipitation radar may overcome some of these unwanted problems. Coulter and Martin (1986) verified that minisodars have the capacity to determine vertical wind profiles simultaneously with rain spectra. Bradley (1997) has successfully used sodars to measure vertical profiles of the size distribution of rain drops. In all these cases the instruments operated at audible frequencies of a few kilohertz. Bradley's sodar had a transmit frequency of 4500 Hz and a beamwidth of 8°.

We now consider optimization of the acoustic operating frequency to enhance the drop echoes and to limit the range so that rapid sampling can be used without pulse-to-pulse aliasing. The resulting ultrasonic instrument is described, together with field results indicating its potential.

2. Scattering by rain

The acoustic radar equation

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describes reception of scattered power from both turbulence and rain. Here P_R and P_T are, respectively, the received and transmitted acoustic powers, G_T and A_e are the transmit antenna gain and receiving effective area, r is the range, α is the atmospheric absorption coefficient, and σ is the scattering cross section. The differential acoustic scattering cross section for a spherical water drop of diameter D scattering angle θ is

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where S(θ) is a Legendre polynomial series with coefficients computed from spherical Bessel functions (Bowman et al. 1969), and the size parameter is

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for wavelength λ. In the Rayleigh limit, when x ≪ 1,

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For rain drops and at 40 kHz, the size parameter is in the range 0 ≤ x ≤ 4.5, but detailed calculations show that the Rayleigh assumption is valid to within a few percent even for the largest drops, which allows for simplification. The analysis method described later is also dominated by smaller drops: the drop-number-weighted mean size parameter is less than about 0.3 for rainfall rates less than 10 mm h⁻¹. For Rayleigh dependence and a scattering volume V,

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where there are n(D)dD drops of diameter D to D + dD per unit volume. For a monostatic sodar with beam half-width angle θ and pulse duration τ,

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where c is the speed of sound. Rossing and Fletcher (1995) showed that

αβf²_T(7)

where β = 10⁻¹⁰ m⁻¹ Hz⁻² for transmit frequency range 2 kHz < f_T < 100 kHz. The received power in frequency interval f_T + f to f_T + f + Δf is

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where f is the Doppler shift due to drops of diameter D falling at their terminal speed, and Δf is the Doppler frequency interval corresponding to drop diameter interval ΔD. The drop diameter D is related to Doppler shift f through drop terminal fall speed W via

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where W can be approximated by a simple function of diameter (e.g., Uplinger 1981). Equations (8) and (9) together allow n(D) to be found for a number of discrete frequency ranges or drop diameter ranges.

For a given range r the maximum total received power is obtained for a transmitted frequency determined by

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This optimum frequency is shown as a function of range in Fig. 1. A 4500-Hz sodar is predicted to give results to around 200 m, as confirmed by Bradley (1997). However, if the objective is to simply retrieve drop distributions above the immediate surface layer, at a few meters, then the combination of absorption and scattering gives an optimum frequency in the ultrasonic range.

3. Sodar design

Transducer pairs were readily available as matched pairs of 40-kHz burglar alarms or proximity sensors. A 3 × 3 array was used to provide a beam of half-width 7°. Other parameters in Eq. (1) are found by anechoic chamber measurements. Frequency dependence and atmospheric absorption were also measured. Some complexity in the Doppler spectrum due to finite beamwidth must also be allowed for (Webb 2000). In particular, drops falling in the periphery of the beam will have different Doppler shifts from drops of the same size falling in the center of the beam. Also, the range variation is significant over the sampling volume if the range gate is, say, from 8 to 12 m. In particular, if the range gate is too large, then the frequency dependence of the absorption term in the frequency spectrum needs to be included. All these effects lead to convolutions and smoothing of the retrieved drop size distribution, but since fine detail of size distributions is seldom warranted, these effects are not a major limitation to design.

The physical design consists of the array and a reflector so that the array remains dry. A thin plywood reflector was chosen after extensive tests of reflectivity and absorption on a range of materials. Experiments showed that water on this reflector did not compromise performance significantly. Separate transmitter and receiver units were made with absorbing foam encasing the baffles, as shown in Fig. 2. The centers of these two units were separated by about 300 mm, which is small compared to the range, so that the instrument is essentially monostatic. Conventional pulse, demodulation, and FFT methods were used. Table 1 summarizes system design.

4. Processing

Note that the effect of the D⁶ mapping of power to drop numbers further enhances the low-frequency disparity due to power leakage into this region. Although turbulence and wind broadening will also occur, this finite pulse broadening is much more significant.

As a consequence of this spreading, we concentrate on the higher-frequency part of the frequency spectrum that corresponds to the larger drops. As rainfall intensity increases, numbers of larger drops increase, as predicted by Eq. (12), and the spectrum peak moves toward higher frequencies. Figure 4 shows spectra for a range of idealized rain intensities. The small-drop errors are not significant for radar calibration purposes, but could be troublesome for optical extinction calculations or if the instrument is used as a fast rain gauge by integrating across the drop size distribution.

The power spectra have a nearly quadratic shape, so the basic peak position and height can be estimated by fitting the measured spectra with quadratics. Figure 5 shows a number of raw spectra and the associated quadratic fits. From these fits, the best-fitting Marshall–Palmer drop size distribution can be found. This gives an estimate of rainfall intensity through parameter Λ. If further drop size distribution structure information is required, it can then be found from the residual between the fit and the raw spectrum. This method is useful because it gives a robust estimate of rainfall intensity. Note that the spread of the spectrum through convolution with the pulse envelope spectrum really mitigates against more sophisticated analysis at this stage. But an indication of the potential of the instrument can be readily obtained by comparison with output from a conventional rain gauge. This method of validation consists of more testing than a theoretical error analysis.

We give the results of two tests of this instrument and the analysis technique is described. Figure 6 shows a comparison between fast “hydra” rain gauge intensities and the rain intensity derived from the ultrasonic sodar. The rain gauge was placed within a few meters of the ultrasonic disdrometer. The rainfall intensities shown in Fig. 6 are quite low. The agreement is good for the second rain period but not very good during the first period. However, it should be noted that agreement between two conventional rain gauges in such circumstances is not generally good for such short periods. Figure 7 shows a similar plot for a high-intensity short-duration downpour. Overall, the agreement here is very good.

In Fig. 8 we also show the correlation between the fast rain gauge and the sodar-derived rain intensity for the second period of Fig. 6. There appears to be a systematic calibration error (the slope is less than 1), due no doubt to uncertainties in antenna efficiency, but this can be readily corrected with a simple empirical scale factor. The correlation coefficient is rather low, suggesting that some integration beyond the 15 s employed is desirable. However, this largely defeats one of the design advantages of this instrument: its rapid sampling capability of 30 pulses per second.

5. Conclusions

A simple ultrasonic sodar has been developed for use as a rain drop size instrument. It has a useful range below 20 m, but this is sufficient to remove the surface wind and contamination problems that plague normal gauges.

There are problems caused by short transmit pulse length contaminating the drop part of the spectrum. These can be reduced partially by the Hanning shaping of the pulse, but spectral spreading into the low-frequency part of the spectrum still occurs. This biases the normal curve fitting so as to overestimate the rainfall intensity. An alternative fitting procedure is proposed, which appears to work reasonably well in the limited cases considered. This procedure should also be useful for minisodar estimates of rain intensity.

A more practical solution is to increase pulse length. This must then be allowed for with the short range used, since the range-dependent factors on the right-hand side of Eq. (6) must be integrated over the pulse length. Increasing pulse length also has the advantage of increasing signal power into the atmosphere, with consequent improvement in signal-to-noise ratio.

The correlation between the rain gauge and ultrasonic sensor is not particularly good. Since the acoustic sensor is volume-averaging, and the sampling volume is of order 20 m³, it would be expected to give a lower-variance estimate of rainfall intensity. Statistical variations in rain gauge output may explain some of the differences, but the relatively low signal-to-noise ratio of the acoustic device is also a problem. We are exploring use of both larger arrays and pulse-coding techniques, so that better-resolution results can be obtained and the high sampling rate capability utilized.