Raindrop Size Distribution Retrievals from a VHF ... - AMS Journals

JANUARY 2004

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LUCAS ET AL.

Raindrop Size Distribution Retrievals from a VHF Boundary Layer Profiler CHRISTOPHER LUCAS, ANDREW D. MACKINNON,

AND

ROBERT A. VINCENT

Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, South Australia, Australia

PETER T. MAY Bureau of Meteorology Research Centre, Melbourne, Victoria, Australia (Manuscript received 15 January 2003, in final form 15 July 2003) ABSTRACT The retrieval of raindrop size distributions (DSDs) in precipitation using boundary layer wind profiler operating at VHF is described. To make the retrievals, a Fourier transform–based deconvolution technique, optimized to run with little human input, is used. The sensitivities of the technique and its overall accuracy are investigated using simulated spectra. The retrievals have an error that depends on the drop diameter, with relative errors varying between ;10% and 35%. An overall average negative bias of about ;20% is also found. The magnitude and direction of this bias depend on the spectral width of the input spectrum. The radar and methodology are applied to a case study of a convective cell. Retrievals are made with ;300 m resolution between 800 and 4600 m. The temporal resolution is 2 min. Comparisons with a rain gauge show that both the magnitude and timing of the precipitation are well captured by the radar. The relationship between the observed rain rate and exponential fits applied to the DSDs agrees very well with previously published studies. A careful analysis of the characteristics of the DSDs within the descending rainshafts provides direct observations of drop size sorting within the precipitation and the formation of hybrid DSDs formed by the overlapping of consecutive rainshafts. This study highlights the potential of the boundary layer profiler in precipitation studies. Some drawbacks exist, such as the wide beam of the radar, which increases the spectral width of the radar and limits its use in windy conditions. However, when observations are available, they appear to be of high quality and fill a gap in observations unavailable to more conventional wind profilers. In the future, it is hoped that refinements in the technique will allow the temporal resolution of the radar to be increased and the quality of the retrievals to be improved.

1. Introduction A useful application of wind profiler technology lies in the retrieval of the drop size distributions (DSDs) from both precipitating and nonprecipitating clouds (e.g., Gossard 1988; Wakasugi et al. 1986; May and Rajopadhyaya 1996; Cifelli et al. 2000). The technique requires measuring the Doppler spectrum in the vertical and converting it via a known drop diameter–fall speed relationship into a drop size distribution. Using this information, the liquid water content and rain rate of the cloud can be estimated through a significant portion of its depth at a high time resolution. In principle, this can be accomplished by any vertically pointing Doppler radar that is sensitive to scattering from precipitation. However, as noted by Atlas et al. (1973), the accuracy of the vertical air motion measurement by standard meCorresponding author address: Dr. Christopher Lucas, Dept. of Physics and Mathematical Physics, University of Adelaide, Adelaide, South Australia 5005, Australia. E-mail: [email protected]

q 2004 American Meteorological Society

teorological radars is not sufficient to adequately correct for the effects of turbulent vertical motions typically found in clouds. The lower frequencies used in wind profilers, typically UHF and VHF, overcome this limitation as they can determine the clear-air vertical motion to an acceptable accuracy. Drop size distribution retrievals from vertically pointing radars have been made at a variety of frequencies. The particular frequency used imparts advantages and disadvantages to the retrieval process. A VHF profiler, operating around 50 MHz, can detect Bragg scattering from clear-air turbulence and Rayleigh scattering from precipitation (nominally) more intense than about 5 mm h 21 . However, drops smaller than about 1 mm are unable to be resolved at these wavelengths (Rajopadhyaya et al. 1993), as the relevant part of the precipitation spectrum is obscured by the clear-air spectral peak. Additionally, VHF profilers typically have their first usable range gate at about 1.5 km above the ground. Examples of DSD retrievals at 50 MHz can be found in Wakasugi et al. (1986, 1987), who examined the seasonal rain front in Japan, and in May and Rajopadhyaya (1996), who

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studied a tropical continental squall line near Darwin, Australia. A rain retrieval at VHF using the algorithm described here is presented in May et al. (2003). A UHF profiler, operating around 915 MHz, is primarily responsive to precipitation echoes and can resolve drops smaller than 1 mm. It can also detect the Bragg scatter from clear-air turbulence at low altitudes and low rain rates. However, the precipitation echo completely masks the clear-air echo in moderate to heavy rain (Rajopadhyaya et al. 1998). Gossard (1988) used a UHF profiler in Colorado to retrieve the DSDs in clouds and light precipitation. Recently, Williams (2002) has developed a technique to determine the ambient vertical air motion, the spectral broadening, and the DSD characteristics using only a UHF radar. To limit the shortcomings of a given single frequency, investigators have more recently performed dual-frequency retrievals, where the vertical motion parameters are estimated from the VHF system and used to correct the precipitation spectra observed with the UHF profiler. Examples of this methodology can be found in Currier et al. (1992), Maguire and Avery (1994), and Schafer et al. (2002). Rajopadhyaya et al. (1999) discussed the relative accuracy of the two methods, finding that the dual-frequency methods are more accurate under a wider range of meteorological circumstances, although again the VHF systems limits retrievals to heights above ;1.5 km. Techniques have also been developed to retrieve DSDs from Doppler radars operating at 94 GHz. These methods use the predictions of Mie scattering theory to estimate the vertical motion and perform the retrieval. Despite heavy attenuation, Kollias et al. (2002) successfully retrieved vertical air motions, rain rates, and drop size spectra from low levels in a convective storm in Florida. Besides differing in the frequencies used the retrieval techniques also vary in their approach to estimating the ‘‘pure’’ precipitation spectrum—that unaffected by spectral broadening, turbulence, and the effects of clearair vertical motion. Following Wakasugi et al. (1986), many studies use a parametric method. These studies assume a form of the DSD, usually an exponential or Gamma distribution (Marshall and Palmer 1948; Ulbrich 1983). This assumed form with variable parameters is convolved with the vertical motion statistics and nonlinear least squares curve fitting is applied to determine which set of parameters results in the best fit to the observed spectrum. Alternatively, the effects of clear-air broadening and vertical velocity can be directly deconvolved from the precipitation spectrum. In this method, the resulting form of the DSD is not assumed. This was first proposed by Gossard (1988) and expanded upon in Rajopadhyaya et al. (1993). Babb et al. (1999) presented a deconvolution methodology for use with 94-GHz Doppler radars to resolve cloud drop size spectra. Recently, Schafer et al. (2002) compared deconvolution and parametric tech-

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niques for dual-frequency retrievals, finding that deconvolution methods result in better retrievals of median diameter over a wider range of conditions. The ability of wind profilers to simultaneously (and directly) measure the characteristics of precipitation and the vertical motion provides a powerful tool in the study of precipitating clouds and mesoscale convective systems (MCSs). Additionally, if several receivers are used to measure the backscattered signal from the radar, a high time resolution measurement of the horizontal wind can be made using the spaced-antenna technique where the clear-air signal is strong enough (Briggs 1984). Recently, a new boundary layer version of VHF wind profilers has been developed at the University of Adelaide (Vincent et al. 1998). This radar allows wind measurements to heights as low as ;500 m, but has the disadvantage of using much wider beams than the profilers previously discussed. In this study, the potential of this VHF boundary layer wind profiler to study the dynamics and microphysics of MCSs is described. The characteristics of the VHF boundary layer radar are briefly described in section 2. Section 3 discusses the methodology used to retrieve the DSD. Direct deconvolution is chosen to make the retrievals. We choose deconvolution over the more-oft-used parametric methods for a couple of reasons. The first is based on the results of Schafer et al. (2002), which, as noted earlier, show that deconvolution performs as well as or better than parametric techniques, particularly at higher spectral widths, which are commonly encountered with the radar used here. Second, no assumptions about the shape of the DSD are required with the deconvolution procedures, so the solutions are not limited to a predetermined functional form of the DSD. While the basic methodology to do this deconvolution has been the subject of several other investigations (e.g., Rajopadhyaya et al. 1993; Schafer et al. 2002), some practical aspects of the procedure have been left unreported in previous publications. These are discussed here, along with procedures to automate the retrieval process such that a minimal amount of human input is required. Also in section 3, the sensitivity of the retrieval method to common radar variables such as spectral width and the noise level is discussed. Also addressed is the overall accuracy of the technique. This is done through the use of simulated spectra and by examining the effect of retaining different numbers of frequencies in the solution. In section 4, a brief analysis of a convective cell that passed over the profiler is presented. This is done to highlight the capabilities and limitations of both the methodology and the radar. Drop size sorting and the formation of hybrid or synthetic DSDs are directly observed within these observations. A summary of the findings is given in section 5, along with a discussion of the direction of future research. 2. The VHF boundary layer radar The data used in this study come from the VHF boundary layer radar located at Buckland Park, South

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TABLE 1. Typical boundary layer radar wind profiler operating parameters. Parameter

Value

Frequency Peak power 3-dB beamwidth, transmitter 3-dB beamwidth, receiver Nyquist velocity Vertical resolution Sample spacing Spectral points Dwell time Coherent integrations

54.1 MHz ;1 kW 178 318 27.7 m s21 150 m 100 m 1024 51.2 s 1024

Australia. This location is approximately 40 km north of Adelaide and roughly 5 km from the coast, with no significant topographic features. Some operating characteristics of the profiler are shown in Table 1. Complete details of the radar can be found in Vincent et al. (1998). Further details describing the calibration procedure for this radar are given in appendix A. Briefly, the radar is run in spaced-antenna mode with three receivers. Only a vertically incident beam is used. In this study, the atmosphere above the radar was sampled every 2 min. The complex time series (in-phase and quadrature terms) from the three receivers are summed to create the final raw signal for a single vertical ‘‘beam.’’ A Welch window is applied to the time series to reduce spectral leakage from nearby frequencies and the power spectrum of the time series is computed (Press et al. 1992). To reduce the noise and variability, averages are performed using the spectra one height above and below the level in question. Thus, a total of three spectra are averaged to get the final spectrum, corresponding to a height resolution of 300 m and a time resolution of 2 min. The precipitation retrieval described in the next section is applied to this average power spectrum. 3. Drop size distribution retrieval a. Methodology As noted in the introduction, the DSDs are retrieved by direct deconvolution. In particular, the methodology described by Rajopadhyaya et al. (1993), in which the deconvolution is performed using Fourier transforms, is followed. During implementation of this algorithm and its optimization for automation, several practical issues arose that have been left unreported in the literature to date. The first is the identification and separation of the ‘‘clear air’’ and precipitation portions of the VHF spectrum. This procedure is described in detail in appendix B. The second involves the truncation of the retrieved precipitation spectrum, namely what is the best method for doing this in an objective manner. This is described in the main text. Figure 1 shows a typical spectrum from the VHF boundary layer wind profiler processed by the method described in section 2. Two peaks are clearly visible.

FIG. 1. Doppler spectrum (dB) from the BLR at 0712 UTC 20 Feb 2000 between 1.4 and 1.6 km. Gaussian curve fit used to estimate clear-air spectral parameters shown as heavy dashed line. The dotted line indicates the precipitation spectrum, the solid horizontal line is the noise level, and the solid vertical line is the location of the divide between the clear-air and precipitation portions of the spectrum. The vertical velocity and spectral width from the clear-air fit, along with the SNR of the precipitation spectrum and the quality score are noted in the upper-left corner.

One is centered just below 0 m s 21 ; this is the clear-air peak. The width of this peak is the sum of the contributions from the width of the radar beam and turbulent motions within the sampled volume of the atmosphere. The offset from 0 m s 21 is due to the mean vertical motion of the air. The second, smaller peak centered near 210 m s 21 is the precipitation peak, which has been convolved with the clear-air peak and shifted due to the mean vertical air motion. The goal of the deconvolution is to remove this ‘‘smearing’’ of the ‘‘pure’’ precipitation signal and correct for the mean vertical motion of the air to obtain an accurate estimate of the DSD at this time and height. A necessary step to accomplish the deconvolution is the separation of the precipitation and clear-air signals. This allows for an estimate of the clear-air parameters unbiased by precipitation. These parameters, the clearair vertical velocity and the spectral width (the standard deviation), are estimated using a least squares Gaussian fit. The noise level is determined following Hildebrandt and Sekhon (1974). The separation also isolates the precipitation spectrum for further processing. The ‘‘divide’’ between the two should be near the local minimum where the two curves merge. This is easy to pick out by eye, particularly for the spectrum in Fig. 1, but is often difficult to identify objectively. An empirical routine to do this accurately has been developed, but it should be noted that it is next to impossible to accurately find the divide in all cases. This routine is detailed in appendix B. Once the precipitation spectrum is isolated and the clear-air parameters computed, the deconvolution can begin. This deconvolution procedure both removes the

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FIG. 2. Variation of ‘‘pure’’ precipitation spectrum with different numbers of frequencies retained for spectrum depicted in Fig. 1. Heavy solid line is the model precipitation spectrum. Solutions for 6 (solid), 11 (dotted), 16 (dashed), and 21 (dashed–dotted) frequencies retained are shown.

beam broadening and corrects for the mean vertical air motion. No separate step is required for the vertical velocity correction. The basic methodology of deconvolution by fast Fourier transform (FFT) used here follows that of Rajopadhyaya et al. (1993). In short, this method performs the deconvolution by dividing the FFT of the convolved precipitation signal by the FFT of the normalized clear-air spectrum, which returns the FFT of the pure precipitation spectrum. Taking the inverse transform, and solving P5

1 6 dD D N(D) Z dW

for N(D), gives the number of drops at a given diameter, or the DSD. Here P is the pure precipitation spectrum, D is the drop diameter, Z 5 # D 6N(D) dD is the reflectivity factor, and dD/dw is the variation of drop diameter with respect to the velocity. This last term is derived from the fall speed relations presented by Atlas et al. (1973) and Foote and duToit (1969). While the above equation is relatively straightforward, a complicating factor is the need to truncate P to retain only the low-frequency components. The need for truncation arises because of the small amplitude of the higher-frequency components in the clear-air spectrum. As a result of these small numbers, the equivalent highfrequency components of the returned, pure precipitation spectrum are greatly magnified, resulting in a noisy, meaningless solution. With the exception of Schafer et al. (2002), the method of truncation has been left unreported in the current literature. Figure 2 shows the variations of the solutions for the deconvolution with the 6, 11, 16, and 23 lowest frequencies retained (out of a possible 512) in units of spectral power for the spectrum shown in Fig. 1. For illustrative purposes, the corresponding precipitation spectrum for a gamma distribution N(D) 5 N 0G D m

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exp(2L G D), with N 0G 5 21 000 m 23 mm 212m , L G 5 8.08, and m 5 14, is also shown. These numbers are obtained by applying a least squares, gamma distribution curve fit on the DSD retrieved from this spectrum via the deconvolution procedure. The purpose of this curve is not to ‘‘verify’’ the retrieval, but rather to illustrate the general behavior of the solutions with different numbers of frequencies retained by providing a model spectrum that accurately reflects the likely true shape of the retrieved spectrum. Simulated spectra, where the true answer is known, show similar behavior to that described below. With six frequencies, the solution is much broader than the model spectrum, and of much lower amplitude. Where a small signal is expected, the solution oscillates about zero. With 21 frequencies, the artificially enhanced high-frequency portion of the spectrum predominates. This solution contains many ‘‘sidelobes’’ and this tendency continues as the number of frequencies is further increased. Where no signal is expected, these sidelobes persist, reflecting the contamination by noise noted earlier. When 11 and 16 frequencies are retained, the solutions resemble the model solution to some extent and are reasonably similar in shape, although the amplitudes do differ. In the regions where no signal is expected, relatively low-amplitude oscillations about zero are observed. Estimated rain rates for these solutions also vary systematically. With lower numbers of frequencies retained, the rain rate tends to be larger. A minimum value of rain rate is reached at some intermediate number of frequencies, after which the rain rate rapidly increases. Which of the solutions is most representative of the true rain rate? How can this solution be chosen automatically? We want to select a solution that accurately portrays the shape of the true precipitation spectrum while limiting the number and amplitude of the sidelobes. For most VHF spectra, the ‘‘best’’ solutions lie in a relatively narrow range, with only subtle differences between them. To differentiate between the solutions, a proxy for the strength of the sidelobes and the goodness of the solution must be determined. Such a metric is computed here by integrating the absolute value of the spectral power of the solutions between 0 and 15 m s 21 , a region where we otherwise expect no precipitation signal. This ‘‘noise parameter’’ for the example spectrum is shown in Fig. 3, where it is seen that this proxy varies little, showing only a slight increase up to 12 frequencies. After this, the metric begins to decrease slightly, reaching a minimum at 16 frequencies. With more frequencies, the noise parameter increases rapidly, reflecting the dominance of the artificially enhanced high-frequency components. This general pattern is seen in most cases and reflects the D 6 dependence for reflectivity. With too few frequencies retained the precipitation spectrum is too broad and inadequately resolved with broad sidelobes. Large numbers of small drops are required to match the spec-

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FIG. 3. Variation of ‘‘noise parameter’’ for different numbers of frequencies retained. Noise parameter is defined as the integral of the absolute value of solutions between 0 and 15 m s 21 . Examples of solutions shown in Fig. 2.

tral power at low fall speeds. With too many frequencies, noise dominates, resulting in an overestimate of the magnitude of the high-frequency components of the spectrum, and strong, but narrow sidelobes, again requiring a large number of small drops to match the spectral power. Both of these extremes result in a higher value of the noise parameter and an overestimate of the rain rate. The best answer lies between these two extremes, where the rain rate is lower, the noise parameter is minimized, and the retrieved pure precipitation spectrum best reflects the shape of the model curve. To choose an appropriate solution automatically, the following procedure is used. The pure precipitation spectra with between 5 and 45 frequencies retained (41 possible solutions) are calculated. Following the previous discussion, the solution with the lowest value of the noise parameter is identified. Letting n equal the number of frequencies retained, spectra with n 6 2 frequencies retained are averaged to get the final precipitation spectrum. This averaging is done to smooth the final profile and to help provide assurance that the correct solution is chosen. Alternatively, the rain rates for each solution can be substituted for the solution noise and a similar methodology followed. Testing of both methods shows that they give similar answers, varying by only one or two frequencies on average. However, sensitivity tests using simulated spectra (described in section 3b) suggest that the answers from the noise method are slightly superior, as they generally have a smaller negative bias on average. The minimum noise parameter–rain rate is the limit where the effects of residual broadening (too few spectral points) and/or noise amplification (too many spectral points) are minimized. A view of this process in spectral space is shown in Fig. 4. Only the lowest 21 frequencies are shown for clarity. Taking the FFT of any power spectrum returns the autocovariance (or autocorrelation) of that function.

FIG. 4. Autocovariance, the FFT of the power spectrum, for (a) the normalized clear-air spectrum, (b) the convolved precipitation spectrum, and (c) the pure precipitation spectrum for the case in Fig. 1. Vertical dotted line represents the number of frequencies–time lags chosen in the final solution.

The autocovariance of the normalized clear-air spectrum (Fig. 4a) is itself a Gaussian curve. For wider clear-air spectral widths this curve becomes narrower, limiting the number of time lags in the main peak. The peak at zero lag is due to the noise in the original spectrum. The autocovariance of the convolved precipitation spectrum is also quasi-Gaussian near the center (Fig. 4b). However, with larger numbers of time lags (not shown), a considerable number of nonzero spectral bins and smaller peaks are also observed. Taking the ratio of these two autocovariances yields the inverse FFT of the pure precipitation spectrum (Fig. 4c), which is quite flat (in this case) out to about 16 time lags, coinciding with the minimum in the noise parameter. This flatness is not observed in all cases, but is found relatively often. In some retrievals, one or more peaks are observed. Beyond the 16 shortest time lags, the clear-air autocovariance, the divisor in the deconvolution, is quite small. This results in the blowup of the pure autocovariance seen in the figure. In some instances, the values of individual spectral bins in the pure signal are frequently eight orders of magnitude larger than the main peak. In general, the truncation point occurs at or very near the minimum spectral bin strength of the retrieved spectrum, which generally coincides with the edges of the main peak of the clear-air autocovariance. Schafer et al. (2002) presented a simpler method for choosing the deconvolution solution for joint UHF– VHF wind profiler precipitation retrievals. Their method

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FIG. 5. Base-10 logarithm of number of drops N (D) vs diameter for the spectrum shown in Fig. 1. Fits using exponential (dashes) and gamma (dots) distributions also shown. Dashed vertical line represents the ‘‘minimum observable diameter.’’ Computed rain rates (mm h 21 ) for the complete distribution, drops .1 mm, exponential and gamma fits (in that order from left to right) in upper right. Also at upper right in same order are liquid water content (second line; g m 23 ) and median diameter (third line; mm). Fourth and fifth lines are parameters of exponential and gamma fits, respectively.

applies a so-called optimal filter to the convolved precipitation spectrum and truncates the spectrum at the bottom of the main peak. This method of solution was considered as well. The only difference between the method used here and the optimal filter is the selection criterion for the number of frequencies to retain. However, our testing suggested that the rigid selection criteria of the optimal filter method can cause, on occasion, a solution with too many frequencies to be chosen. This can happen in cases when the (convolved) precipitation autocovariance develops a ‘‘flare,’’ or a broadening of the main peak, which allows the high-frequency components to be magnified. It should be noted that this has only a minor effect on the measurement of parameters such as the median diameter, which depend only on the slope of the retrieved DSD, and not its final amplitude. The final retrieved DSD from the example spectrum is shown in Fig. 5. As discussed previously, the radar is unable to measure drops smaller than about 1 mm in diameter, equivalent to a fall speed of 4.0–4.5 m s 21 between the surface and 3 km. The exact minimum observable diameter is a function of the clear-air–precipitation divide, whose velocity can be converted to a diameter. This diameter for this spectrum is indicated by the vertical dashed line just above 0.8 mm. This ‘‘nonobservability’’ is reflected in the retrieved DSD by the large increase in the number of drops at sizes smaller than this diameter. The increase is typically of about four orders of magnitude and is a result of contamination by noise in the solution. Such an increase is also noted in the DSD retrievals of Wakasugi et al. (1987), indicating that this is not just an artifact of the technique

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used here. This deviation in the DSD has an undesirable effect on the computation of parameters such as liquid water content, rain rate, and median diameter, artificially increasing them in the case of the integral rain parameters and drastically lowering the estimate of the median diameter. To provide a better estimate, these final quantities are estimated using only that part of the DSD greater than 1 mm. Nonlinear least squares fits in the form of exponential [N(D) 5 N 0E exp(2L E D)] and gamma [N(D) 5 N 0G D m exp(2L G D)] distributions are also performed, parameterizing the characteristics of the DSD for further study. Characteristics of the DSD, including liquid water content, median diameter, and rain rate and the least squares fits are indicated in Fig. 5. See the caption for details. By comparing the various numbers in Fig. 5, the effects caused by the neglect of the smaller drops can be estimated. These effects are relatively small, given that the truncated rain rates and liquid water contents and those from the curve fits are in reasonably close agreement. The truncation has the effect of slightly increasing the median diameter. Also in Fig. 5, the sharp kink in the DSD located between 1.4 and 1.5 mm is also likely artificial, reflecting the location of a sidelobe in the solution. The kinks generally have an adverse affect on the gamma distribution fit, making likely unrealistic estimates of the curvature of the DSD. See the gamma fit in Fig. 5. An indicator of the quality of the retrieval has also been developed to assist in the interpretation of the retrievals. After performing the retrievals on literally thousands of spectra, certain common factors that negatively impact the quality of the retrievals have been identified. These factors include too wide (narrow) spectral widths, too many (few) frequencies retained, high noise levels (often caused by aircraft flying over the radar), errors in identifying the clear-air–precipitation divide, and a low clear-air signal-to-noise ratio (SNR), among others. An empirical (and somewhat arbitrary) algorithm has been developed to determine the severity of these potential errors and highlight points that may be of lower quality. A ‘‘quality score’’ is assigned based on the above criteria, with lower scores representing better retrievals. Scores of 6 and above are considered less reliable. This score is used only as an indicator—good retrievals can still occasionally have high scores. b. Sensitivity of solutions to clear-air and spectral parameters The sensitivity to various spectral parameters and overall accuracy of the method of DSD retrieval is considered here. Simulated VHF spectra, whose spectral parameters and DSDs can be varied, are created. The method used to create these simulated spectra is identical to that described by Rajopadhyaya et al. (1993), except that in this study the model uses N 5 16 384 points instead of 4096. In particular, the simulations are used to help understand the sensitivity of the solutions

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TABLE 2. Values of simulation parameters used in sensitivity experiments. Parameter

Value

Drop size distribution Rain rate Intercept Spectral width Vertical velocity No. of spectral points Noise level No. of spectra averaged

Exponential 20.0 mm h21 8000.0 m23 mm21 1.0 m s21 0.0 m s21 1024.0 5.0 (;6 dB) 8.0

to the spectral width, the number of spectral points, the noise level of the spectrum, and the effects of averaging several spectra together. To perform these experiments, the following methodology is used. Simulated VHF precipitation spectra are created using a DSD based on the parameters described by Marshall and Palmer (1948) with a rain rate of 20 mm h 21 . For each sensitivity test, 200 retrievals

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are performed. While the DSD is the same in all these tests, the random noise in the spectral bins differs. The spectral parameters (except for the one being tested) are also kept constant throughout. The parameters used are summarized in Table 2. The relative difference of each DSD retrieval from that specified is computed. These relative differences are then averaged to determine the mean characteristics of the solutions. From this, we can glean information on the general behavior of the solutions and use this knowledge to assist in the interpretation of data collected with the [boundary layer (BL)] radar. Figure 6 shows the sensitivities of the solutions to variations in the above-listed parameters. Regardless of the characteristics of the clear-air spectrum, the solutions have many common features. Most notable is the rapid increase in the number of drops retrieved at diameters smaller than ;1 mm. As noted in the previous section, this arises because the radar is unable to sense drops at these diameters.

FIG. 6. Relative error of simulated deconvolution retrieval solutions with variations in the (a) spectral width, (b) noise level, (c) number of spectra included in average, and (d) number of points in the spectra. Lines represent the different values of the tested parameter. See legend in corner of each plot. Each solution is the average of 200 individual retrievals.

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The spectral width of the clear-air spectrum (Fig. 6a) produces the largest variability in the mean solutions, suggesting that the goodness of the retrieval is quite dependent on the spectral width. Such a dependence was also noted by Rajopadhyaya et al. (1993). The location of the rapid increase noted above moves toward larger diameters with bigger spectral widths. When the spectral width is less than 1.75 m s 21 , the retrievals seem reasonable, with a typical negative bias of ;20%. With larger widths, the solutions have a larger negative bias at diameters greater than 2 mm and a large positive bias at lower diameters and generally appear far worse. Changing the spectral width is similar to choosing a different number of frequencies, as fewer frequencies are retained with larger spectral widths, as the main peak of the clear-air autocovariance is narrower in this situation (see Fig. 4). The signal-to-noise ratio of the precipitation spectrum (Fig. 6b) primarily affects the region between 1 and 2 mm. In these simulations, the precipitation signal is constant, while the noise level changes. At noise levels greater than ;5 dB, a low precipitation SNR, the retrievals are quite negatively biased in this diameter range. This effect lessens as the noise is reduced. However, further reducing the noise below about 210 dB apparently has little effect on the solutions. Altering the number of individual spectra that are incoherently averaged to produce to final spectrum (Fig. 6c) tests the sensitivity to the variance within individual spectral bins. Averaging more spectra together acts to reduce this variance. Relatively little benefit is seen by averaging more than four spectra. With higher spectral variance, the spread of solutions is larger. Changing the number of spectral points (Fig. 6d) has some effect on the solutions. With 128 points, the retrieved DSD is generally positively biased. Adding more points makes the bias mostly negative, and this amount of negative bias increases as more points are added. The results here suggest that using 256 or 512 points produces the best solutions, but this comes at the expense of diameter resolution, which increases with the number of points. c. Accuracy of solutions Overall, the results of the sensitivity analysis (Fig. 6) show that the solutions are oscillatory and generally have a negative bias of 10%–20% on average across the retrieved DSD at diameters greater than 1 mm. Larger negative relative errors are generally noted between 1and 2-mm diameters, followed by an increase between 2 and 3 mm. The magnitude of this structure depends significantly on the parameters chosen in the retrieval, particularly the spectral width. By examining the spread of the solutions in any given sensitivity experiment (not shown), we can estimate the relative uncertainties in the typical case due to random errors. This is shown in Table 3. Near diameters of 1 mm, the spread of the solutions

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TABLE 3. Relative uncertainty in solutions as a function of diameter. Diameter (mm)

Relative uncertainly (%)

1.0 1.5 2.0 2.5 3.0 3.5 4.0

35 23 19 12 11 9 8

is typically about 35%, lowering to about 8% at diameters of 4 mm. The spread in the DSD solutions arises as a result of the different realizations of the randomly generated noise in the different spectra, which make up a given sensitivity test. These different spectra lead to different numbers of frequencies being chosen for the final solution. Different frequencies are also emphasized between the solutions, resulting in different sidelobes appearing. In general, there remains some ambiguity in choosing the appropriate number of frequencies, which leads to the above uncertainty. A different estimate of this uncertainty can be obtained by averaging the solutions with different numbers of frequencies over a reasonable range, such as those along the ‘‘flat’’ portion of the pure precipitation spectrum in Fig. 4, and measuring the variance within those solutions. This gives an extreme range of the uncertainty. In general, the behavior mimics that of the sensitivity tests examined above. Estimates of the relative uncertainty are similar to the above in the typical case, as well. The numbers reported in Table 3 do not take into account the biases in the solution. These biases appear to be related to the number of frequencies chosen in the final solution. With only a few frequencies retained in the solution, the solutions are similar to the large spectral width solutions, suggesting a positive bias at small diameters and a large negative bias at large diameters. As more frequencies are retained, a ‘‘dip’’ suggestive of the sidelobes is found between l- and 2-mm diameters, suggesting a negative bias at these diameters. These sidelobes also affect the solutions at diameters greater than 4 mm, resulting in a larger uncertainty and a likely negative bias there as well. Between 2 and 4 mm, the solutions are apparently uninfluenced by the sidelobes, as the spread of the solutions is similarly small in cases with and without sidelobes. However, despite the uncertainty in interpreting these small peaks and troughs, the overall characteristics of the DSD, such as the slope from portions unaffected by sidelobes, remain reasonably constant. This suggests that the gross characteristics of the DSDs are being captured correctly. As noted in the introduction, the deconvolution method is not limited to a predefined form for the DSD. Simulations also show that even DSDs using a gamma distribution with a high m value, resulting in a sharply

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curved DSD, can be retrieved with the degree of accuracy noted above. These results suggest that the boundary layer radar (BLR) should be able to accurately retrieve most forms of the DSD over the full gamut of rain rates. However, the detection of multiple peaked DSDs, such as the equilibrium DSDs modeled by Hu and Srivastava (1995), cannot be performed with confidence at this time, as some uncertainty remains in distinguishing between artifacts caused by the sidelobes and real variations in the DSD. To fully estimate the amount of uncertainty in the actual solutions, the spectral width, the number of frequencies, and the presence of sidelobes in the data must be considered. 4. Example of DSD retrieval To illustrate the abilities and usefulness of the BLR in the study of precipitation systems, a small case study of a convective cell is presented in this section. The cell under scrutiny here passed over the BLR on 20 February 2000 around 0700 UTC. The period from 19 to 22 February 2000 was a time of active convection in the vicinity of Adelaide. A cyclone and associated cold front developed in the Great Australian Bight, moved ashore, and interacted with an in-feed of moist tropical air from the northeast, resulting in thunderstorms [the Australian Bureau of Meteorology (BoM 2000)]. This moist, tropical air brought dewpoints in excess of 208C to the vicinity, unusually high for Adelaide. During this 3-day period, 50–100 mm of rain were observed over the greater Adelaide region. Just under 75 mm were recorded by a BoM rain gauge located at the radar field site during the same period. For this analysis, a convective cell that occurred around 0700 UTC [1730 local daylight time (LDT)] on 20 February is examined. It was located over the radar for approximately 30 min, during which time ;6 mm of rain was recorded by the BoM gauge at the surface. Figure 7 shows the rainfall trace from the gauge. From the gauge, rainfall totals are reported every 6 min. These data are multiplied by 10 so as to convert them into an average rain rate during that 6-min period. A peak rain rate of 46 mm h 21 is recorded by the gauge between 0712 and 0718 UTC. Nonzero rain rates are recorded during four 6-min intervals during the time of study. Figure 7 also shows the rain-rate retrieval from 800 m, the lowest usable height of the BLR during this time, for the same time period. These data have been processed and calibrated as described earlier in the text. As previously noted, only drops with D . 1 mm are considered in the computation of the rain rate. The agreement between the two measurements is excellent. The peak values of the rain rate agree well, with a peak value of just over 42 mm h 21 seen by the radar. More importantly, the timing between the two curves agrees well; the start and stop times of the rainfall are similar, and the peak rain rates occur near the same time in both measurements. This gives confidence that the BLR is

FIG. 7. Surface rain gauge data (solid) and rain rate from precipitation retrieval at 800 m (dashed) from 20 Feb 2000 between 0648 and 0736 UTC. Only drops with D . 1 mm are included in rain-rate calculation from the profiler.

accurately retrieving the DSDs of the precipitation, and that the calibration procedure is robust. Figure 8 shows a time–height cross section of the equivalent reflectivity factor for the heights 800 to 4600 m and between 0648 and 0742 UTC. The divide between the clear-air and precipitation spectra was accurately chosen by the automatic algorithm in 90%–95% of the spectra in this analysis. The remainder required some manual editing to correctly separate the clear-air and precipitation portions of the spectrum. The strong reflectivity maximum associated with the high rain rates is clearly visible between 0708 and 0714 UTC as it descends. Peak reflectivities are in excess of 45 dBZ, and the main rain shaft is visible by the BLR prior to the precipitation reaching the surface. A smaller, weaker reflectivity maximum, presumably associated with the lighter rain observed just prior to the main rain shaft, is also visible during its descent between 0700 and 0704 UTC. At 0716 UTC, a third maximum in reflectivity, with a peak of over 50 dBZ, is also noted. The interpretation of this last peak is uncertain, as significant manual editing was required to fully extract the details presented here. The raw spectra could be interpreted either as having very wide spectral widths or as a signal merged by the combination of a strong updraft and a strong precipitation signal. The latter interpretation is presented here. Upward motion in excess of 3 m s 21 near 4-km altitude is also observed at this time. The further evolution of this cell is not visible as it moves out of the view of the radar. A few other isolated reflectivity peaks are also noted, although much of the apparent reflectivity is due to noise. This noise is more visible at higher altitudes due to the r 2 dependence of the echoes. This determination has been made by examining the raw spectra at different times/heights to see if the characteristic VHF precipitation signal described earlier (Fig. 1) is apparent. By

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FIG. 8. Time–height cross section of equivalent radar reflectivity factor (dBZ ) retrieved from the wind profiler from 0648 and 0736 UTC and between 800- and 4600-m altitude. Only drops with D . 1 mm are included in the calculation. The box circumscribes the subset of data used in the later analyses.

examining this and other cases, it has been established that the minimum detectable signal for the BLR is a rain rate of ;1 mm h 21 . This is lower than the minimum detectable rain rate of 5 mm h 21 stated in Rajopadhyaya et al. (1993). Simulations also suggest this is true. With model parameters set to mimic the BLR, DSDs with rain rates as low as 1 mm h 21 were easily detected and resolved with the same degree of accuracy described earlier using the deconvolution method. As long as some drops with D $ 1 mm are present, the radar should be able to perform a retrieval. Figure 9 shows the variation of the parameters of an exponential fit to the retrieved DSDs against rain rate for a subset of the data shown in Fig. 8. The extracted data come from times between 0706 and 0716 UTC and from 1000- to 2000-m altitudes; this subset of data is indicated in Fig. 8 by the box. These limits are chosen to include periods of significant rainfall and to limit any potential sampling problems caused by large beamwidths. Additionally, the quality scores suggest that a significant fraction of the data above ;2500 m is of marginal to poor quality. The quality of the retrievals is indicated in Fig. 9 via the symbol size. Larger symbols indicate high quality observations; the symbols are proportionally smaller for poorer quality returns. An exponential fit is appropriate here as both the observations of Sauvageot and Lacaux (1995) and the modeling results of Hu and Srivastava (1995) show that at diameters greater than about 1.0–1.5 mm the DSDs tend to follow an exponential form (although the value of the slope in the modeling and observations are very different). Some

errors are present in these fits due to the biases and forms of the solutions described earlier. Simulations show that, in wide spectral width cases, L E is overestimated (i.e., too many small drops) by up to 20%. In cases with apparent sidelobes, L E is underestimated (i.e., too many large drops) by about 5%–10%. Estimates of N 0E are similarly biased. Figure 9a shows the variation of L E with rain rate. The results of the similar curves from the long-term average DSDs of Sauvageot and Lacaux (1995) are shown for comparison. In general the two sets of data compare favorably, although some scatter exists in the BLR data. The scatter is particularly prevalent at low rain rates, say ,20 mm h 21 . At higher rain rates, the slopes from the BLR and the average data agree very well. Despite this general agreement, the slopes from rain rates above 50 mm h 21 appear to increase slightly. A few points, generally of lower quality, are well off the curve. In general, the data presented here support the idea that L E is independent of rain rate above a certain threshold. Figure 9b shows the variation of the exponential intercept parameter (N 0E ) with rain rate for the subset. Above R 5 10 mm h 21 , the agreement between the BLR data and the curves of Sauvageot and Lacaux (1995) are generally good, although the BLR values suggest a slight negative bias. At low rain rates, considerable differences are observed. Many of the BLR points are about one order of magnitude larger than the average data, particularly the cluster of points around

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FIG. 9. Variations of the (a) exponential fit slope (L E ) and (b) exponential fit intercept (N 0E ) with rain rate for the subset of data defined above. Smaller symbols indicate retrievals with lower quality scores. Overplotted in each panel are the same curves from the average data tabulated by Sauvageot and Lacaux (1995) for their E1 (diamonds), E2 (triangles), and E3 (crosses) sites. The Marshall–Palmer rain rate–slope relation (asterisks) is also noted in (a).

N 0E ; 20 000 m 23 mm 21 . However, considerable scatter in N 0E is a feature at low rain rates. Plotting the exponential fit parameters of the subset data against one another, as is done in Fig. 10, reveals that the shapes of the DSDs are varied both in time and with height. In the figure, clusters of points are easily identifiable. Nine clusters have been subjectively identified and their mean characteristics tabulated in Table 4. Table 5 indicates the time and height when the points in each cluster were observed. The separation into clusters is based on their location and degree of separation

FIG. 10. Scatterplot of exponential fit intercept (N 0E ) against exponential fit slope (L E ) for the subset of data. Circles bound the clusters of points discussed in the text. The number identifies each cluster. Smaller symbols indicate retrievals of lower quality.

from other groups. Two points in the subset are not identified as belonging to any cluster, as they are quite separate from the other groups and do not fit ‘‘naturally’’ into any. Each cluster of points is assigned an arbitrary number to identify it. This number is indicated in Fig. 10. The light rain initially observed belongs to groups 7 and 8. Both of these DSDs have a large L E and large N 0E , indicating the predominance of small drops over large. Group 7 is observed at the first time (0706 UTC); group 8, with the higher exponential parameters, is seen at the lowest two heights at 0708 UTC. The smaller median diameter suggests that these drops represent the smallest and slowest falling from the drops from the initial shower. Above this region of small drops at 0708 UTC, DSDs with very different characteristics are observed. From the top height down to 1700 m, group 5 is seen. The characteristics of this DSD fall near the middle of the range of characteristics shown in Table 4, and are the beginning of the significant precipitation. Between these ranges, groups 1 and 6 are observed. Group 1 is characterized by low values of both N 0E and L E , although reflectivity and rain rate are moderately high and the median diameter is quite large. These observations and their positions suggest that the DSDs primarily compose large drops, with very few small drops, and arise from size sorting in the main rain shaft. These DSDs are the vanguard of the approaching rain shaft—the largest, fastest falling drops that separate from the main group. Immediately below cluster 1 and above the tail end of

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TABLE 4. Mean characteristics of drop size clusters defined in Fig. 10. Shown are the group number, the number of points in the group, the intercept parameter (N 0E ), the slope (L E ), the rain rate (R), the equivalent reflectivity factor (ZE), the liquid water content (LWC), the vertical velocity (W), and the median diameter (D 0). All integral rain parameters and median of the diameter are computed directly from the retrieved DSD using only drops with D . 1 mm. Group

No. of points

N 0E (m23 mm21 )

L E (mm21 )

R (mm h21 )

Z E (dBZ )

LWC (g m23 )

W (m s21 )

D 0 (mm)

1 1a 2 3 4 5 6 7 8

5 2 6 18 7 7 5 12 2

740 2209 8369 28 860 68 308 20 946 4743 22 848 97 872

1.77 1.92 2.05 2.42 3.17 3.02 2.79 4.05 5.11

7.2 12.3 33.0 51.6 59.0 13.9 4.6 2.7 3.5

39.9 41.7 45.8 46.4 44.8 38.9 35.3 29.8 29.8

0.28 0.50 1.31 2.13 2.82 0.65 0.21 0.14 0.19

20.05 20.40 21.05 21.36 21.12 20.81 0.12 0.25 0.10

2.21 2.17 2.15 1.92 1.47 1.47 1.57 1.30 1.32

the previous shower (groups 7 and 8) is an observation of group 6 and the unclassifiable point with N 0E ; 1600 m 23 mm 21 . These observations are interpreted here as hybrid or ‘‘synthetic’’ DSDs, an intermediate combination of groups 1 and 7, formed by the overlapping of two DSDs from separate rain shafts. This phenomenon was described for disdrometer data at the surface by Sauvageot and Koffi (2000). At 0710 and 0712 UTC, structures similar to the previous observation are noted. A descending, high-reflectivity rain shaft (clusters 3 and 4) is noted at the highest levels. Below the base, both L E and N 0E gradually decrease, indicating fewer, larger drops falling more quickly and separating themselves from the main rain shaft. At 0714 UTC, clusters 3 and 4 are observed exclusively, with little variation with height seen. At 0716 UTC, the observations are a bit less clear due to the uncertainty in interpreting the heavy rain shaft noted earlier, although at the top heights of the subset, the clear-air– precipitation divide was determined automatically. In this rain shaft at this time, L E changes little. Primarily variations in N 0E are seen, suggesting mainly changes in the liquid water content. 5. Summary and concluding remarks In this study, the ability of a VHF boundary layer wind profiler to make retrievals of rainfall and drop size TABLE 5. Cluster number observed at each time and height for the subset of data. Refer to Table 4 and Fig. 10 for the identification and description of the clusters. Time (UTC)

Height (m)

0706

0708

0710

0712

0714

0716

2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000

7 7 7 7 7 7 7 7 7 7 7

5 5 5 5 1 1 6 ? 7 8 8

3 4 3 2 2 1a 1 1 1 1a 2

3 3 3 3 3 3 3 3 2 2 2

3 3 3 3 3 3 4 4 4 3 3

X 4 4 5 5 6 6 6 6 5 4

distribution has been demonstrated. These retrievals have been made at heights as low as 800 m. This is much lower than the 1.5–2.0 km obtained with more typical VHF wind profilers. With properly tuned, wellmaintained antennas, it is possible (and has been done) to obtain precipitation retrievals from heights as low as 400 m. The retrievals are made using a deconvolution technique similar to that outlined by Rajopadhyaya et al. (1993). The technique presented herein has been redesigned to work automatically, with minimal human intervention, although some manual editing is still required. The ‘‘clear air’’ and precipitation portions of the spectrum are separated using an objective algorithm based on derivatives of the spectrum. Estimates of the clear-air vertical velocity and spectral width are made. The spectral broadening is removed from the precipitation spectrum through the use of an FFT-based deconvolution algorithm. The methodology is described in some detail within the paper. This method has the advantage over the commonly used parametric methods in that a functional form of the DSD is not specified beforehand. Hence, multipeaked and truncated distributions can theoretically be retrieved. The concerns expressed by Wakasugi et al. (1987) regarding the longtime averages required to perform this operation have been overcome by truncation of the high frequencies in the FFTs, which reduces the noise in the final solution. The solutions retrieved are examined in some detail, by using both simulation and careful examination of the actual retrievals. The simulations show that the solutions are adversely affected by sidelobes. These sidelobes lead to negative biases and uncertainties in the solution. The sidelobes particularly manifest themselves between 1and 2-mm diameters, where simulations suggest the solutions may be ;50% too low. The magnitudes of these errors are also affected by the spectral width and the noise level of the spectrum. Sensitivities are also shown to the number of points in the spectrum and the variance within individual spectral bins. These numerical simulations and an examination of the solutions with different numbers of frequencies are used to estimate the magnitude of the uncertainty of the solutions.

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A brief case study is presented to highlight the potential use of the VHF BLR in precipitation studies. The radar has been calibrated to fix the absolute number of drops detected by the radar. In general, we find that the rain rates retrieved by the profiler generally agree well with those obtained from a collocated rain gauge. The agreement is quite good, identifying both the start and end times of the precipitation at the surface, as well as matching the peak rain rates. The BLR also allows for a high-resolution view through the lower portion of the cloud. The characteristics of the retrieved DSDs against rain rate also agree well with the average characteristics presented by Sauvageot and Lacaux (1995), suggesting that the BLR and the deconvolution technique are accurately retrieving the large drop portion of the rainfall. Finally, a detailed examination of the characteristics provides direct observations of drop size sorting within clouds and the formation of hybrid or synthetic DSDs. Serious limitations exist for the use of the VHF boundary layer radar in precipitation studies. The effects of sidelobes on the solutions limit the ability to interpret the DSD and identify more exotic multipeaked or truncated distributions. The wide beamwidth of the radar is also a limitation, as it results in an increased spectral width of the returned spectra. As shown earlier, the solutions are quite sensitive to the spectral width of the solution. Only a few frequencies can be obtained in the solution in these cases, limiting the resolution of the solutions. The mathematical relationship of spectral width and wind speed presented by Nastrom (1997) suggests that the spectral widths become too wide for accurate, reliable retrievals when the wind speed exceeds 10–15 m s 21 . At higher altitudes, the large beamwidth of the BLR leads to a large volume of the atmosphere being sampled. This large sampling volume raises issues of the representativeness of the sample and of beam filling as the horizontal area of the beam is approaching the scale of individual convective elements at higher altitudes. Despite these limitations, there is a strong potential upside for the use of this radar in studies of precipitation systems. Few other observing platforms are even able to obtain these observations at the altitudes presented here. Wind profilers operating at UHF can sample this low, but the Rayleigh scattering from precipitation dominates the Bragg scatter from clear air at these wavelengths and hence these radars are unable to accurately measurement the clear-air parameters required for the deconvolution. The high time resolution of the VHF radar is also an advantage. Here, we used a ‘‘1 min on, 1 min off’’ sampling scheme. The radar can easily be set to sample continuously. There is also the further possibility of subdividing the raw time series, to obtain an even higher time resolution, possibly as low as ;15 s. This would also reduce the number of spectral points, which may provide better accuracy based on the sensitivity tests shown in Fig. 6d. The radar also uses the spaced-antenna method of computing the wind. Without

the need to change beam directions, these measurements can be updated rapidly and, hence, provide high temporal resolution details of the kinematic structure of the precipitation system. Finally, there is also the possibility of estimating the thermodynamic properties of the systems through the use of the Radio Acoustic Sounding System (RASS), as was demonstrated by May et al. (2003). By combining all these operations, a complete picture of a precipitation system currently unavailable with other observational platforms could potentially be obtained. Further work remains to be done. Most importantly, an independent verification of the DSD retrievals needs to be performed. Such an experiment is currently under way using a disdrometer at the surface collocated with the radar. This will help solidify the nature and size of the errors in the DSD and of the calibration procedure. Ideally, a systematic experiment to verify the DSD using in situ sampling from aircraft, similar to the flights of opportunity presented by Rogers et al. (1993), would be performed to provide a more direct comparison. Acknowledgments. This research is supported by Australian Research Council Grant A69802414. Surface rain gauge data provided by the BoM Regional Forecast Office in Adelaide. APPENDIX A Radar Calibration To calibrate the radar and fix the absolute value of the number of drops, a radar equation for the equivalent radar reflectivity factor Z e of the precipitation is derived for the VHF boundary layer radar. The equation takes the form Ze 5

1024 ln(2)kl 2 BTN r02 (SNR)p , (P t e x )l 2 cp 3 g r g t u r u t l r t |K 2 |10218

where the variables measured by the radar are T N , the noise temperature; r 20 , the range to the scatterers; and (SNR) P , the signal-to-noise ratio of the precipitation signal only. The remainder of the terms are assumed to be constant for a given radar. The definitions, values, and uncertainties for each of these terms are shown in Table A1. The radar equation developed above follows a similar derivation made by P. Johnson of the NOAA/ Aeronomy Laboratory. Because of the bistatic nature of the BLR, some modifications must be made. In particular, the terms for the antenna beamwidth and the gain, usually written as u 2 and g 2 in traditional formulations, are separated into individual components for the transmitting and receiving antennas, subscripted with t and r, respectively. To obtain the noise temperature T N , a receiver calibration was performed with a noise generator. The response of each receiver to different input noise levels was linear. The results of this calibration from the three

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TABLE A1. Description of radar parameters and constants, their values, and the estimated uncertainties. Dash indicates negligible uncertainty. Varible

Description

Value

Estimated relative uncertainty (%)

B «TX gr gt ur ut lr PT t l c Kw l2 k

Bandwidth Transmission line loss Receiver antenna gain Transmitter antenna gain Receiving antenna half beamwidth Transmitting antenna half beamwidth Receiver loss term Transmitted power Pulse width Wavelength Speed of light Dielectric constant of water Atmospheric attenuation term Boltzmann’s constant

1 MHz 1.75 dB 15.22 dB 19.74 dB 318 178 2.3 dB 1000 W 1 ms 5.54 m 2.997 3 10 8 m s21 0.93 1 1.38 3 10223 J K 21

5 5 3 3 3 6 4 5 — — — — — —

different receivers were combined and a linear regression of these data is used to calculate the noise temperature from the observed noise level of an individual spectrum. The equation returns a value of reflectivity that should be expected given the observed strength of the precipitation signal, the range, and the noise level. This is compared to the reflectivity calculated (from the unmodified spectrum) from the DSD retrieval described previously. The ratio of the expected to the observed is taken as a ‘‘calibration factor,’’ which is applied to the DSD at the end of the calculation. For the case examined in this study, the calibration factor is ;1.8. Based on the uncertainties in the individual terms (Table A1), a final uncertainty of 30%–40% in this number is estimated. A second, cruder calibration is also performed by comparing the accumulated rainfall estimated from the DSD retrievals from the lowest usable range gate, typically about 800 m, to that measured at the surface with a collocated tipping bucket rain gauge operated by the Australian Bureau of Meteorology. It requires on the order of 2–3 min for raindrops observed at the lowest range gate of the profiler to reach the surfaces, and during that time many processes that affect the raindrops and their size distributions, such as advection and evaporation to name two, can occur. Hence, the two measurements should not necessarily be expected to exactly agree, although some degree of consistency is expected and observed. Following the methodology above, calibration factors from the gauge in the range 3.0–4.0 are computed. The relative agreement between the calibration factors computed from independent measurements suggests that our calibration procedure is robust and the numbers reasonably accurate. A calibration factor of 2.6, the upper end of the range of uncertainty, is used here. This gives a close agreement with the rain gauge and does not require large amounts of precipitation formation in the lower troposphere.

APPENDIX B Determination of the Precipitation–Clear-Air Spectral Divide A crucial element of VHF wind profiler precipitation retrievals is identifying the clear-air and precipitation portions of the full spectrum. An empirical routine, designed to be run automatically, is described in this appendix. The divide-picking routine uses a spectrum that has been twice smoothed over a 1 m s 21 velocity interval (19 points for this case). An example showing this smoothing for the spectrum in Fig. 1 is given in Fig. B1a. This double smoothing is only used in estimating the central peak and the divide and acts to eliminate many changes due to the fluctuations between individual spectral bins. The first derivative of the smoothed spectrum with respect to velocity is computed, and this in turn is smoothed twice (Fig. B1b). A second derivative (w.r.t. velocity) is also computed on the singly smoothed first derivative (Fig. B1c), which is used as a secondary check. Zero crossings and their direction (i.e., negative to positive or vice versa) are identified in the derivative fields. To begin, the central or ‘‘clear air’’ peak (see Fig. 1) must be identified. The first derivative is scanned in the negative velocity direction, looking for a negativeto-positive zero crossing and a spectral value that is at least 80% of the maximum spectral amplitude. These conditions are met near the central peak, although in the (relatively rare) cases where the precipitation echo is significantly greater than the clear-air signal, this algorithm will fail. These conditions are often met a second time near the spectral peak of the precipitation. Hence, we scan in the negative velocity direction and take the first occurrence of these conditions. This point is set as the starting point in the search for the divide. The divide is found by searching for the first positiveto-negative zero crossing after the central peak. In cases with a relatively weak precipitation spectrum, the actual

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results in the precipitation retrieval being performed partially on the clear-air spectrum. This leads to excessively high numbers of drops in the DSD and unrealistic rain rates and liquid water contents. This happens relatively rarely, and measures are taken within the routine to flag such occurrences for manual editing. The dividepicking algorithm can also perform poorly in light rain conditions when the precipitation spectrum is weak and the differentiation of the two signals is not as clear. In these cases, the precipitation spectrum is missed, and no precipitation is retrieved. REFERENCES

FIG. B1. Values of the (a) raw and smoothed spectrum (dB), (b) smoothed first derivative, and (c) smoothed second derivative used in determining the precipitation divide. Time and height are the same as depicted in Fig. 1.

divide may not be identified, as the heavy smoothing used prevents the first derivative from crossing zero. When precipitation is present, it is expected that the divide will be between the clear-air peak and ;10 m s 21 below that peak. If the divide is outside of the lower limit, a second check is performed that looks for the peak in the second derivative after the ‘‘dip’’ in that field associated with the central peak. These two estimates of the divide do not necessarily give the same answer, although both are in the same region. These different estimates of the divide will lead to different realizations of the precipitation spectrum and add to the uncertainty. However, even under ideal conditions some question remains as to where the divide should be placed. Ideally, the point at which the precipitation spectrum is completely isolated should be picked. But at the juncture of these two spectra, the total signal is some unknown (and unknowable) combination of the two signals and the exact point where the precipitation signal becomes dominant is not clear. Experiments suggest that getting the general location (to within ;1 m s 21 ) is sufficient to consistently retrieve the precipitation signal; choosing slightly different divide locations does not dramatically affect the outcome of the precipitation retrievals. Some serious errors in the location of the divide can occur when the variability within the spectral bins is arranged in such a way that the smoothed spectra has a small dip that mimics the precipitation–clear-air junction. This usually occurs near the clear-air peak and

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2002: Estimation of rainfall drop size distributions from dualfrequency wind profiler spectra using deconvolution and a nonlinear least squares fitting technique. J. Atmos. Oceanic Technol., 19, 864–874. Ulbrich, C. W., 1983: Natural variations in the analytical form of the raindrop size distribution. J. Climate Appl. Meteor., 22, 1764– 1755. Vincent, R. A., S. Dullaway, A. MacKinnon, I. M. Reid, F. Zink, P. T. May, and B. H. Johnson, 1998: A VHF boundary layer radar: First results. Radio Sci., 33, 845–860. Wakasugi, K., A. Mizutani, M. Matsuo, S. Fukao, and S. Kato, 1986: A direct method for deriving drop size distribution and vertical air velocities from VHF Doppler radar spectra. J. Atmos. Oceanic Technol., 3, 623–629. ——, ——, ——, ——, and ——, 1987: Further discussion on deriving drop-size distribution and vertical air velocities directly from VHF Doppler radar spectra. J. Atmos. Oceanic Technol., 4, 170–179. Williams, C. R., 2002: Simultaneous ambient air motion and raindrop size distributions retrieved from UHF vertical incident profiler observations. Radio Sci., 37, 1024, doi:10.1029/2000RS002603.