a. Data used

The data we used for this study are from collocated profilers near Darwin, Australia. The UHF profiler collected data in two different modes (high and low height coverage) alternately switching between each mode. The low height mode has a vertical resolution of 100 m, whereas the high height mode has a vertical resolution of 500 m that is sampled at 300-m intervals and 128 spectral points. For the purpose of this study, only the high mode UHF data were used. The time resolution of the UHF high mode data is about 100 s because the profiler was alternating between low height coverage and high modes and cycling between beams, so that coverage was not continuous. The vertical resolution of the VHF data is also 500 m sampled at 300-m intervals. Time resolution is 166 s and each spectrum has 256 spectral points. Description of radar characteristics is given in Table 1.

A well-developed mesoscale convective system passed over Darwin on 28 December 1993 during an active monsoon period. This storm consisted of alternating periods of high and low rain rates. A distinct precipitation echo was visible in the UHF Doppler spectra for almost an entire day (>16 h) with intense rainfall observed from 1000 to 1200 UTC. Figure 1 shows a time–height profile of the vertical velocity for the period 0500–1730 UTC. Well-developed convective cells with upward velocities exceeding 3 m s⁻¹ (actual values with 4–5 m s⁻¹ that do not show in the shading in the figure) were observed between 0700–0800 UTC and 0900–1130 UTC above the melting layer (∼5 km). Relatively homogeneous vertical velocities of about ±0.5 m s⁻¹ are visible after 1200 UTC, which is consistent with widespread stratiform conditions during this time.

Figure 2 shows a time–height profile of reflectivity (dBZ) estimated from the signal-to-noise ratio of the returned signals for the period 0500–1700 UTC. The height coverage is from 0.75 to 7.0 km. Relatively large reflectivities (above 35 dBZ) are seen up to about 5 km between 0930 and 1200 UTC. High reflectivities (above 35 dBZ) from 1330 to 1700 UTC between about 4 and 5 km indicate the presence of a bright band. However, the bright bands are not continuous, mainly due to the occasional occurrences of mixed convective regions in the low rain-rate regions (Williams et al. 1995). Occasionally, very weak reflectivities are observed in the lower height level from 0500 to 0900 UTC.

b. Retrieval technique

The radar reflectivity factor Z is proportional to the sixth power of the diameter of the hydrometeors and is expressed as (Doviak and Zrnic 1984)

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where N(D) is the number of particles in the diameter range between D and D + dD.

Maguire and Avery (1994) have also applied these techniques assuming a Gaussian-shaped DSD. While the gamma distribution is the more realistic, the Gaussian distribution has some computational advantages and may be a reasonable approximation when only bulk parameters, such as rain rate and median volume diameter, are needed. Although all the analysis in the study has been performed using a gamma distribution, the rainfall rate comparisons between retrieval and gauge differ only slightly when a fit to a Gaussian DSD is performed.

The terminal velocity is related to the diameter through the empirical relation of Beard (1976, 1977):

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and

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where w₀ is the terminal fall velocity at sea level in meters per second, drop diameter D is in millimeters, and C_i are polynomial-fit coefficients listed in Beard (1977). The variables ρ₀ and ρ are the densities of air at sea level and at the height of observation.

The mean vertical velocity and spectral width are estimated using the VHF (50 MHz) data that is interpolated in time and height to match the UHF (920 MHz) sampling. Here,

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and

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where A₀ and σ are the respective amplitude and spectral width of the vertical air motion spectrum that has been normalized to unit area. Note that this has implicitly assumed that the spectral width of the vertical air motion spectrum measured by the profilers will be the same at both frequencies. In general this will not be the case, which is an issue that is discussed in section 3e.

The amplitude of the vertical air motion echo detected by two radars operating at two different frequencies (50 and 920 MHz) is not the same due to the wavelength dependence of Bragg scatter. Additionally, the clear-air portion of the UHF spectrum is often very difficult to observe, as noted in the introduction. For this reason, the mean vertical velocity and spectral width information obtained from the VHF spectrum is used to locate the position of the vertical air motion echo in the UHF spectrum and assign a minimal signal strength (noise level) to this portion of the spectra. Thus, by incorporating the VHF data, the precipitation component of the UHF spectra can be isolated.

The vertical air motion information is extracted by fitting a Gaussian curve to the VHF spectra. However, the observed height and time of the VHF spectra often does not match with the corresponding height and time of the UHF spectra. As an example, Fig. 3 shows the four VHF spectra just before, after, above, and below the UHF observation time and range gate. A Gaussian fit to the spectrum is shown by the dotted lines. Even though the time resolution of the VHF spectra is just 166 s, there is a change of mean vertical motion from around −1.9 m s⁻¹ to about 1.3 m s⁻¹. The change in spectral width is also considerable (1.3 to 2.2 m s⁻¹). Therefore, a two-dimensional interpolation technique is applied to obtain an average estimate of the mean vertical velocity and spectral width.

Finally, a model precipitation drop size distribution is converted to a reflectivity-weighted fall speed spectrum [Eqs. (4) and (6)] and then convolved with G(w) [Eq. (5)]. A least squares technique is used to find a best fit to the observed spectrum by varying the parameters N₀, μ, and γ using Eq. (3). This process was first described by Currier et al. (1992) and is similar to that described by Wakasugi et al. (1986) except that a more general form of the DSD is used as well as the two different frequency radars.

The precipitation rate R (mm h⁻¹) and water content M (g m⁻³) are related to the size distribution of raindrops N(D) through

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and

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where ρ_w is the density of the drop. These parameters can be easily determined once N(D) is established through the fitting procedure just described.

a. Preliminary comments

Next we will discuss the four experiments described in Table 2. The first experiment will examine the accuracy of the full retrievals against rain gauge data. The next two experiments use the full retrievals as “truth” and examine the effect of the vertical air motion and spectral broadening. The last experiment examines the effect of the beam mismatch on the retrievals by introducing a modified spectral broadening.

Before discussing the intercomparisons it is useful to describe some of the observed features of the 28 December case. These features are shown in the time–height cross sections of the water content in the high and low rain-rate conditions (Fig. 4). In each case there is considerable structure and high-frequency variations. However, it is clear that the fluctuations have spatial and temporal coherence. This provides confidence that the fluctuations we observe are real manifestations of the precipitating systems and are not associated with noise in the retrieval process. The primarily vertically oriented structure also validates the idea of comparing the retrieved data with surface (gauge) measurements, although some differences due to the evolving precipitation field structure over the profiler will be evident. Section 3b will examine whether these data can be used quantitatively as well.

b. Experiment 1: Profiler versus rain gauge intercomparison

For the intercomparison of profiler retrieved rain rates and rain gauge measurements, UHF data from 1.9 km is used because it is just above the lowest VHF range gate unaffected by ground clutter. The air temperature at this level is typically about 33°C for a tropical wind environment.

Figure 5 is a time series of rainfall rates (solid line) estimated using a simple Z–R relation (Z = 200R^1.6) where the reflectivity has been estimated using the signal-to-noise ratio. The gauge rainfall rate is shown by the dotted line. The time resolution of the gauge data is about 35 s. The gauge data are smoothed by a six-point running mean, and the profiler data are smoothed by a three-point running mean. The different averaging is to account for differences in the sampling rate between the model retrieval and the rain gauge. Additionally, there is a time lag between the precipitation event at the ground and the profiler precipitation retrieval at 1.9 km. For this reason, the retrieved rain-rate series has been shifted to the right by about 5 min (the time required for a “typical” drop to fall from 1.9 km to the ground). The two time series are highly correlated (ρ ∼ 0.78), demonstrating that the temporal variations of the rainfall rates match in both estimates. However, the total accumulated rain using the Z–R values is just 53 mm, which is very low compared to the rain gauge total of 145 mm. The Z–R relation that we used here serves as an illustration of the technique. It is valid only for the Marshall–Palmer distribution with rainfall rate greater than 10 mm h⁻¹. Because there are so many Z–R relations for various precipitating conditions, all determined empirically but as a parameterizing in mean DSD, one must be careful in choosing a “correct” relationship. With profiler data we actually measure the DSD. In general we have found that the Z–R rainfall estimates using various relationships is low compared to the rain gauge estimates.

Figure 6 is a time series plot of the retrieved rain rate (solid line) and the gauge rain rate (dotted line) that shows much better agreement, although there are some discrepancies during the time of high rainfall (0930–1200 UTC). There are numerous reasons to expect differences. The profiler measurement is a volume sample 1.9 km above the earth’s surface compared with the essentially point measurement of the rain gauge. Therefore, time and spatial variability can cause these discrepancies. There are a large number of gaps in the data due to the sampling sequence between the vertical and oblique beams and only the high height mode of the 920-MHz profiler data is used, so further differences can be expected, particularly in highly variable convective rain. The difference between the two measurements could also depend on atmospheric conditions, such as background humidity and temperature, which determine the rate of evaporation of the drops that fall from 1.9 km (retrieved height) to the ground (gauge). The estimated total rainfall is 151 mm, which is very close to the gauge total (145 mm) for the same period.

c. Experiment 2: The effect of mean vertical velocity corrections

Atlas et al. (1973) have shown theoretically that the error in retrieving DSDs from radar data can be very large if the vertical air motions are not properly taken into account. It is easily seen that if there is an uncorrected upward velocity, then the apparent fall speed is reduced, leading to an estimate where there is a large number of small drops (since Z remains constant and is proportional to D⁶). Atlas et al. (1973) showed that errors in assigning diameters and number of drops exceeded 100% for situations where the uncorrected vertical air velocities exceeded 1 m s⁻¹. However, no detailed analysis has been done to discriminate the effects of the mean vertical velocity and spectral width in the retrieval process on real data. Nor there has been an analysis of the integrated error effect on realistic DSDs. We will consider the full retrievals as truth for examining the effect of vertical motion and spectral width on rainfall retrievals because of the excellent agreement between the profiler retrieved rain rates and the rain gauge trace.

Figure 7 is a time series plot of retrieved rain rates at 1.9 km with full corrections (solid lines) and without considering the vertical air motions (dotted lines) in the high and low rain-rate regions. Note that the instantaneous difference in rain rates can differ by 100% or more when the vertical air motion is large (top panel) and that even when the vertical velocity is small, such as in the low rain-rate regions, the differences can be as large as 25%. The total rainfall accumulation estimate has been reduced from 151 to about 101 mm for the time period of 0500–1700 UTC, and the correlation coefficient between the retrieved and gauge rain rate is 0.72.

Figure 8 is a scatterplot of the relative percentage difference [defined as 100 × (test retrieval − truth)/truth] of rain rates at 1.9 km with full corrections and with spectral width corrections alone as a function of the magnitude of the vertical air motion velocity for the period of 0500–1700 UTC. This plot clearly illustrates the almost linear dependence of the retrieval accuracy with the strength of the mean vertical velocity when the velocity is less than about 1 m s⁻¹. For larger velocities, the uncorrected updrafts produce larger errors. This is at least partly a result of the D⁶ dependence on reflectivity. For an uncorrected updraft, the spectrum is shifted toward smaller velocities and hence a very small drop diameter regime. Consequently, many more small drops are needed to produce the same reflectivity. However, for downdrafts, the spectrum is shifted to higher fall speeds, and hence larger diameters, but the integrated effect is less. The shift in DSDs is shown in Fig. 9. Note that ∫ D⁶N(D) must remain constant, so that ∫ N(D) must increase for an uncorrected updraft. The rain rate is approximately related to D^3.67N(D) (Ulbrich 1983), so it too increases.

It is important to note the effects of improperly corrected mean vertical velocity on different parts of the drop size spectrum. The terminal fall speed of raindrops is related to their diameters, but the speed changes little for diameters greater than about 4–5 mm. Thus, a small change in velocity results in a large (small) change in the diameter in the large (small) diameter regime. For this reason, uncorrected vertical velocities result in more changes to the large drop diameter part of the spectrum.

Figure 10a shows a time series of the median diameter (top panel) and percentage difference (defined as before) in median diameters (middle panel) at 1.9 km with and without corrections for vertical motions along with the vertical air velocities (bottom panel). These differences are largest in the high rain-rate regime, where the vertical velocities are largest with the diameters biased downward in an updraft evident around 1100 UTC. Figure 10b shows a scatterplot of the percentage difference in median diameter against the mean vertical air velocity for the period of observations. There is an almost linear dependence of the relative accuracy of median diameter with the magnitude of the mean vertical velocity. It is notable, however, that these errors are only about 20%, even for a vertical velocity of 1.3 m s⁻¹ with a peak error of about 50% when the vertical velocity is 2.3 m s⁻¹. The theoretical results (thick solid lines) of Atlas et al. (1973) are shown for comparison. They are in reasonable agreement for small velocities but tend to be larger than our results for large vertical velocities. This may be a result of their assumption of the Marshall–Palmer drop size distribution.

In addition to errors induced by the shift of the median diameter, there is an additional problem in fitting a reflectivity-weighted fall speed spectrum that is illustrated in Fig. 11. Drop fall speeds lie between velocity limits of 0.0 and 9.6 m s⁻¹ for small and large drops at mean sea level pressure. If the DSD is in an uncorrected (or improperly corrected) updraft, part of the drop spectrum lies at “fall speeds” greater than 0 m s⁻¹, which is physically unrealistic. The fitting procedure used here does not allow for such errors, and poor fits to the observed radar spectrum result. Similarly, if there is an uncorrected downdraft, part of the spectrum may lie at velocities exceeding physically allowable fall speeds, and again the fitting procedure breaks down. Similar errors may also arise if the spectral broadening due to turbulence and antenna beamwidth is very large and uncorrected, so that the broadened fall speed spectrum has significant amplitude at physically unrealistic velocities. Note that these errors have the opposite sign to that of the mean velocity correction errors. This effect is also one of the reasons for the total errors in D₀ being less than the theoretical calculations of Atlas et al. (1973) (Fig. 10b) for large uncorrected vertical motions.

These comparisons have relevance to the full retrievals because errors in the vertical motion will produce errors similar to the cases where we have neglected the vertical motion. Also note that the difference in sampling between the profilers will induce errors associated with rapid (nonlinear) changes in vertical motion as well as statistical errors in the measurement.

d. Experiment 3: The effect of spectral width corrections

As noted above, there are two components of the vertical air motion information that are used in the retrieval process: the mean vertical velocity and spectral width contributions [see Eqs. (3) and (5)]. The mean vertical air motion correction is obviously important, as shown in the previous section, but the impact of the spectral width is less clear. This effect will now be examined.

Figure 12 is a time series of retrieved rain rates at 1.9 km during the period of relatively intense rain with and without spectral width corrections. The dotted line represents the retrieved rain rates without spectral width corrections, which is compared with the retrieved rain rates (solid lines) using full corrections (spectral width and mean vertical air velocity). Clearly, the spectral width has a minimal effect on the retrieval compared to the mean vertical air motion.

Figure 13 is a plot of the relative percentage difference in retrieved rain rate with and without considering the vertical air motion smearing effect. In most cases, the errors are negative, which indicates that the retrieved rain rates are higher when the spectral width is not included. This effect also can be seen in Fig. 12. The percentage difference in most cases is 10% or less. An almost linear increase in the percentage difference of rain rates with spectral width is clearly seen when the spectral widths are small (<1 m s⁻¹).

To further examine the apparent weak effects of the spectral width, the median volume diameter is investigated. Figure 14 shows a time series of the percentage difference in the median volume diameter at 1.9 km with and without vertical air motion spectral width corrections for an 8-h period. Note that there are variations (∼15%) in the median volume diameter in the high rain-rate region with the uncorrected data biased toward small median diameters. This result is because when the vertical air motion widths are large the precipitation peak is spread and more energy is put into velocities corresponding to small and large diameters. However, the D⁶ reflectivity dependence means that this spread produces many more small drops in the retrieval (and only a small number of large drops) so that the median diameter is biased toward smaller drops. Small variations can be seen in the low rain-rate region (∼5%), but the random errors are comparable. Thus the variation of the median volume diameter seems to be larger if the spectral width is not corrected compared to the variation of rain rates (∼10%). Overall, this variation in the median volume diameter is small but is a definite bias.

e. Experiment 4: The effect of beam mismatching

The precipitation spectral width is equal to the square root of the sum of the air motion width squared and the precipitation width in still air squared. Because the corrections due to the vertical air motions spectral width are small, this indicates that the UHF precipitation width is significantly wider than the vertical air motion width measured by the VHF radar. Now, the question arises whether the VHF vertical air motion spectral width can be taken as the representation of the UHF vertical air motion spectral width in the retrieval process. The beamwidth of the VHF profiler is about 3°, whereas the beamwidth of the UHF profiler used here is about 9°. This difference means that the spectral broadening characteristics due to the beamwidth will not be the same for both profilers. Horizontal velocity information can be used to calculate beam broadening, but the velocity during strong convection is highly variable. An alternative way is to compare the UHF precipitation width with the VHF precipitation width or UHF vertical air motion width with the VHF vertical air motion width.

In the present dataset the UHF vertical air motion echoes were so weak that they were not visible at the height of interest. However, there are many cases, mainly in light rain conditions, in which two distinct spectral peaks are visible in the VHF dataset—one for the vertical air motion and another for the precipitation. Thus, it is possible to compare (statistically) the precipitation components of the UHF spectra with the precipitation component of the VHF spectra in order to assess beam broadening effects. In the analysis procedure, the VHF precipitation peak has been separated from the vertical air motion peak and the spectral width of the precipitation echo is obtained by fitting a Gaussian curve to the precipitation component.

Figure 15 is a scatterplot of the UHF precipitation width versus the VHF precipitation width. The scatter points are clustering above the solid (y = x) line, which indicates that the UHF width is significantly wider than the VHF width. Note that the signal-to-noise ratio of the VHF is quite low, so that much of this scatter is associated with random statistical errors (Doviak and Zernic 1984). The median position of the points in Fig. 15 indicates that the UHF precipitation spectral width is about 10% higher than the VHF precipitation spectral width. Note that in the high rain-rate periods, this difference is likely to be less because convective turbulence will be much higher and will be a greater contributor to the spectral width than during stratiform conditions.

We can estimate a correction factor using the results of Fig. 15. In still air, the precipitation spectral width will be determined only by the DSD. We denote this width as σ_p. We seek an estimate of the 920-MHz vertical air motion width denoted by σ₉₂₀ and can measure the 50-MHz air motion width σ₅₀. The observed 920-MHz spectral width σ_920ob is dominated by precipitation, so we have σ²_920ob = σ²₉₂₀ + σ²_p.

Figure 16 shows the retrievals with the modified width correction. The effect is small, so that in practice the errors induced by the beam mismatch will not be important. In cases where the vertical air motion spectral width becomes wider, the potential for error will increase. However, in strong convection, where turbulence will be intense and beam broadening less important, then σ₉₂₀ → σ₅₀.