1. Introduction

The copolar correlation coefficient ρ_hv between horizontal (H) and vertical (V) polarization radar signals is a measure of the variety of hydrometeor shapes in a pulse volume. It is therefore useful for applications such as identifying the melting layer (Caylor and Illingworth 1989; Brandes and Ikeda 2004; Tabary et al. 2006; Giangrande et al. 2008), ground clutter (e.g., Tang et al. 2014), rain–hail mixtures (Balakrishnan and Zrnić 1990) and interpreting polarimetric signatures in ice (e.g., Andrić et al. 2013), and potentially the retrieval of the drop size distribution (DSD). The standard deviations of differential reflectivity Z_DR and differential phase shift ϕ_dp are both functions of ρ_hv (Bringi and Chandrasekar 2001). Therefore, ρ_hv dictates both the quality of dual-polarization measurements and their weighting in hydrometeor classification schemes (Park et al. 2009). In rainfall, ρ_hv is typically 0.98–1. Giangrande et al. (2008) use data where ρ_hv < 0.97 to identify the melting layer. For hail, ρ_hv can be much lower because of Mie scattering. At present, quantitative use of ρ_hv is hampered by a lack of rigorous confidence intervals accompanying the ρ_hv estimates. Error estimates are available by adopting an empirical approach (Illingworth and Caylor 1991) or a linear perturbation technique (Liu et al. 1994; Torlaschi and Gingras 2003), both of which implicitly assume a Gaussian probability distribution for the ρ_hv samples. We will show that the distribution of ρ_hv samples is in fact non-Gaussian and highly negatively skewed.

Natural raindrop size distributions can be described by a gamma distribution (Ulbrich 1983):

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where D is the equivalent spherical drop diameter, is the intercept parameter, is the median volume drop diameter, and μ is the dispersion parameter (a measure of the drop size spectrum shape). If μ = 0, by exploiting the relationship between drop diameter and drop axis ratio, can be estimated using (Seliga and Bringi 1976). Higher μ corresponds to more monodisperse drop size distributions. Since ρ_hv is sensitive to variations in drop shape, it can in principle be used to estimate μ (Jameson 1987), knowledge of which could improve dual-polarization and dual-frequency (e.g., the Global Precipitation Measurement satellite) rain-rate estimates. Figure 1 shows rain rate R per unit radar reflectivity Z as a function of for simulated gamma distributions with μ = −1, 0, 2, 4, 8, 12, and 16. The rain rate is sensitive to variability in the shape of the drop size spectrum; uncertainty in μ alone could introduce an error in the retrieved rain rate of up to 2.5 dB (almost a factor of 2) for a given pair of Z and observations.

Rain rate (dB, referenced to 1 mm h⁻¹) per unit radar reflectivity as a function of Z_DR computed using Gans theory for gamma distributions of μ = −1, 0, 2, 4, 8, 12, and 16. The rain rate can vary by as much as 2.5 dB for a given pair of Z and Z_DR observations as a result of drop spectrum shape variability.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Rain rate (dB, referenced to 1 mm h⁻¹) per unit radar reflectivity as a function of Z_DR computed using Gans theory for gamma distributions of μ = −1, 0, 2, 4, 8, 12, and 16. The rain rate can vary by as much as 2.5 dB for a given pair of Z and Z_DR observations as a result of drop spectrum shape variability.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Rain rate (dB, referenced to 1 mm h⁻¹) per unit radar reflectivity as a function of Z_DR computed using Gans theory for gamma distributions of μ = −1, 0, 2, 4, 8, 12, and 16. The rain rate can vary by as much as 2.5 dB for a given pair of Z and Z_DR observations as a result of drop spectrum shape variability.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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It is difficult to obtain reliable estimates of μ from observations. Disdrometers suffer from undersampling of large drops, which cause μ values that are derived from the third, fourth, and sixth moments of the drop size distribution to be biased high (Johnson et al. 2014). Furthermore, disdrometers also undercount the number of drops < 0.5 mm (Tokay et al. 2001), which can also introduce a bias in estimates of μ. Estimating DSD parameters using radar is therefore preferable, because of the very large number of drops being sampled. Wilson et al. (1997) made radar observations dwelling in rain at elevation angles above 20° and reported that the difference in the mean Doppler velocity at H and V polarizations provides an estimate of μ, which were in the range of 1 to 11, and, once exceeded 0.5 dB, all the values were above 4. Doppler spectra of rain at vertical incidence with multiple wavelength radars, including wind profiler frequencies that respond to the clear-air motion, have been utilized to estimate μ (Williams 2002; Schafer et al. 2002). These experiments find μ ranges between 0 and 18, but is typically between 0 and 6. Unal (2015) fits the observed Doppler spectra to theoretical drop spectra at S band and retrieves μ in the range from −1 to 5. The disadvantage of these techniques is that they use high-elevation angles; for operational monitoring of surface rainfall, measurements at low-elevation angles are preferable. This motivates the use of ρ_hv to derive μ in rainfall. Illingworth and Caylor (1991) and Thurai et al. (2008) inferred μ from the decrease in ρ_hv as increases. The difficulty here is that any mismatches in the H and V beams will introduce an uncorrelated noise component, so that even for perfectly spherical drizzle droplets, where the true ρ_hv is unity, the radar will always detect a value less than one (we will call this maximum obtainable level of ρ_hv “”; see section 5). From measurements in rain at short range, Illingworth and Caylor (1991) inferred μ values, which if corrected with an estimate of were in the range 0–2, but even for long dwells the estimated errors in μ were quite large. Thurai et al. (2008) analyzed ρ_hv measurements from an operational radar and obtained estimates of μ in the range of 1–3; however, their approach relies on empirically derived relationships between ρ_hv and DSD widths from two-dimensional video disdrometer (2DVD) measurements. Furthermore, the technique is only valid for intense rain Z_DR ≥ 2 dB and ρ_hv < 0.98).

The aim of this paper is to define a new variable, L = −log₁₀(1 − ρ_hv), that has Gaussian error statistics with a width predictable from the number of independent radar pulses. This can be readily estimated by using the observed Doppler spectral width σ_υ. We will then present measurements of L in rainfall as a function of Z_DR, and retrieve estimates of μ by comparing these with predicted L and Z_DR for various three-parameter gamma distributions. The possibility of using this technique to retrieve μ using operational radars is then discussed.

2. The copolar correlation coefficient ()

The quantity ρ_hv is defined as (Doviak and Zrnić 2006)

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where and are the copolar elements of the backscattering matrix averaged over an ensemble of scatterers for the H and V polarizations, respectively, and the asterisk indicates the complex conjugate. It can be estimated by correlating successive power or complex (I and Q) measurements. Examples of power time series in drizzle and heavier rainfall from the 3-GHz Chilbolton Advanced Meteorological Radar (CAMRa) are shown in Figs. 2a and 2b, respectively. The radar is a coherent-on-receive magnetron system, transmitting and receiving alternate H- and V-polarized pulses with a pulse repetition frequency (PRF) of 610 Hz. A cubic polynomial interpolation is used to estimate the H power at the V pulse timing and the V power at the H pulse timing. Its narrow one-way half-power beamwidth (0.28°) makes it capable of very high-resolution measurements. The full capabilities of this radar are discussed in Goddard et al. (1994). The observed fluctuating signals in Fig. 2 are caused by the superposition of the backscattered waves from each drop in the sample volume; the rate of fluctuation is determined by the Doppler spectral width. For drizzle, since the drops are spherical, Z_DR = 0 dB, and the H and V signals are almost perfectly correlated: ρ_hv = 0.995. For heavier rainfall, a systematically lower V power is measured (Z_DR = 1.1 dB), and the signals are visibly less correlated (ρ_hv = 0.987), as a result of the broader axis ratio distributions in the sample volume.

Example time series (0.5 s) for single 75-m gates from 1.5°-elevation dwells in (a) drizzle (Z_DR = 0 dB) at 1203 UTC 6 Feb 2014 and (b) heavier rainfall (Z_DR = 1.1 dB) at 1706 UTC 31 Jan 2014. For both examples, SNR > 40 dB. For drizzle, the H and V echo time series vary in unison as the drops are all spherical. In heavier rainfall, the broader axis ratio distribution causes the H and V time series to be less correlated. The rate of fluctuation of the signals is determined by the Doppler spectral width.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Example time series (0.5 s) for single 75-m gates from 1.5°-elevation dwells in (a) drizzle (Z_DR = 0 dB) at 1203 UTC 6 Feb 2014 and (b) heavier rainfall (Z_DR = 1.1 dB) at 1706 UTC 31 Jan 2014. For both examples, SNR > 40 dB. For drizzle, the H and V echo time series vary in unison as the drops are all spherical. In heavier rainfall, the broader axis ratio distribution causes the H and V time series to be less correlated. The rate of fluctuation of the signals is determined by the Doppler spectral width.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Example time series (0.5 s) for single 75-m gates from 1.5°-elevation dwells in (a) drizzle (Z_DR = 0 dB) at 1203 UTC 6 Feb 2014 and (b) heavier rainfall (Z_DR = 1.1 dB) at 1706 UTC 31 Jan 2014. For both examples, SNR > 40 dB. For drizzle, the H and V echo time series vary in unison as the drops are all spherical. In heavier rainfall, the broader axis ratio distribution causes the H and V time series to be less correlated. The rate of fluctuation of the signals is determined by the Doppler spectral width.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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These estimates of ρ_hv are derived from a finite number of reshufflings, and therefore there is some uncertainty in them. In what follows, we quantify this uncertainty.

3. Theoretical measurement error in estimated correlation of time series

Figure 3a shows the distribution of estimates of the correlation coefficient, (calculated from a finite length time series), as distinct from the true copolar correlation coefficient, (which would be measured for a time series of infinite length). The data were collected during a 1.5°-elevation dwell in drizzle (Z_DR < 0.1 dB), with very high signal-to-noise ratio (SNR; > 40 dB) on 6 February 2014. Each is calculated from 64 H and V pulse pairs (0.21-s dwell) from 75-m range gates with = 1.1 ± 0.1 m s⁻¹. The distribution of has a peak that is close to (which is < 1; see section 5d), but exhibits a very long tail at lower , while there are no data with > 1. Clearly, this distribution is not Gaussian and the negative skewness will negatively bias the mean of many samples compared to the true value of .

The frequency distribution of (a) calculated from a total of 1159 time series (0.21 s; 75-m gates) in drizzle (Z_DR < 0.1 dB) and of (b) = −log₁₀(1 − ). The data were collected at 1203 UTC 6 Feb 2014 during a 1.5°-elevation dwell and have very high SNR (>40 dB). The values for these data range between 0.9 and 1.3 m s⁻¹. Overplotted on is a Gaussian curve with the same mean and standard deviation as in the measured distribution.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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The frequency distribution of (a) calculated from a total of 1159 time series (0.21 s; 75-m gates) in drizzle (Z_DR < 0.1 dB) and of (b) = −log₁₀(1 − ). The data were collected at 1203 UTC 6 Feb 2014 during a 1.5°-elevation dwell and have very high SNR (>40 dB). The values for these data range between 0.9 and 1.3 m s⁻¹. Overplotted on is a Gaussian curve with the same mean and standard deviation as in the measured distribution.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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The frequency distribution of (a) calculated from a total of 1159 time series (0.21 s; 75-m gates) in drizzle (Z_DR < 0.1 dB) and of (b) = −log₁₀(1 − ). The data were collected at 1203 UTC 6 Feb 2014 during a 1.5°-elevation dwell and have very high SNR (>40 dB). The values for these data range between 0.9 and 1.3 m s⁻¹. Overplotted on is a Gaussian curve with the same mean and standard deviation as in the measured distribution.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Fisher (1915) states that sample correlation coefficients () of a true correlation coefficient () calculated from a finite number of Gaussian random variables are skewed for 0. However, the variable

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is Gaussian, with a mean of

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and standard error of

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where N is the number of independent samples used to calculate .

This is directly applicable to estimates of the radar copolar correlation coefficient, by realizing that the I and Q samples that are used to estimate are Gaussian random variables (Doviak and Zrnić 2006). Noting that in meteorological targets is always close to unity so that fractional changes in (1 − ) are always much greater than (1 + ), Eq. (3) can be written as

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Since is normally distributed, the quantity

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is also normally distributed, with a mean of

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and standard deviation of

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for ≫ 3, where is the number of independent I and Q samples used to calculate . Despite having similar characteristics, L is preferred over the use of F as it has the convenient property that = 0.9, 0.99, and 0.999 correspond to L = 1, 2, and 3, respectively, and therefore is more intuitive. Illingworth and Caylor (1991) plotted their data as and their histograms also appear to be Gaussian in shape. Figure 3b illustrates the effect of the transform on the distribution in Fig. 3a. The histogram is now symmetrical and bell shaped. A Gaussian curve with an equal mean and standard deviation to the PDF is overplotted and is an excellent fit to the data, showing that the distributions are indeed Gaussian (and quantile–quantile plots, not shown here for brevity, confirm this).

To determine the number of independent I and Q samples, N_IQ, we consider the autocorrelation function for I and Q samples given by Doviak and Zrnić (2006):

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where is the time spacing between pulses of the same polarization and is the total time lag. Following the definition of Papoulis (1965), the time to independence for I and Q samples for large N_IQ can be shown to be

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where λ is the radar wavelength and is the Doppler spectral width. This is a factor of smaller than the more often used time to independence for reflectivity samples. The number of independent I and Q pulses per sample can therefore be estimated by

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where is the dwell time.

The result [Eq. (9)] is significant as it shows that a confidence interval for any measurement of can be calculated solely in terms of the number of independent I and Q samples used to estimate it, which in turn can be readily estimated using the observed Doppler spectral width and Eq. (12). Furthermore, when multiple samples of are averaged, no bias is introduced to estimates of because of the nonlinear transform. We expand this point in section 4.

To estimate confidence intervals for measurements of , one must

• apply the transform ,

• calculate the standard deviation of using Eq. (9), and

• apply the inverse transform to obtain upper and lower confidence intervals (where will contain the true value 68% of the time and 2 will contain it 98% of the time).

More conveniently, one can simply transform data into and use this for any subsequent analysis, with confidence intervals of ± . This is the approach we follow in the rest of this paper. Although we are focusing on data with very high SNRs in this paper, the theory above should also be valid for weak SNR data, providing that noise introduced is also Gaussian in the I and Q samples.

This theoretical prediction was tested by comparing estimates of using data collected in homogeneous drizzle ( < 0.1 dB) with very good SNRs (> 40 dB). In drizzle, is constant since the drops are spherical and, therefore, any variation is due to the finite . Pulse-to-pulse H and V powers were recorded, and time series of various lengths between 0.2 and 30 s were constructed from these data and used to compute the corresponding and values. Data were binned by , and the standard deviation was computed for each bin. Figure 4 shows how decreases as is increased over more than two orders of magnitude. For ≈ 10, is slightly overestimated, and the data are in excellent agreement with the results predicted by Eq. (9) for > 30.

The quantity as a function of the number of independent I and Q samples used to estimate L for high SNR measurements in drizzle (Z_DR < 0.1 dB; SNR > 40 dB) at 1203 UTC 6 Feb. Different markers correspond to different Doppler spectral widths.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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The quantity as a function of the number of independent I and Q samples used to estimate L for high SNR measurements in drizzle (Z_DR < 0.1 dB; SNR > 40 dB) at 1203 UTC 6 Feb. Different markers correspond to different Doppler spectral widths.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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The quantity as a function of the number of independent I and Q samples used to estimate L for high SNR measurements in drizzle (Z_DR < 0.1 dB; SNR > 40 dB) at 1203 UTC 6 Feb. Different markers correspond to different Doppler spectral widths.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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4. Comparison with existing error statistics

We now compare these new error statistics with existing methods in the literature. From observations of in rain, the bright band, and ice, Illingworth and Caylor (1991) derived empirically the relationship between their mean estimates and their standard deviation:

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where n is the number of 0.2-s time series they used to estimate the mean . Using a linear perturbation technique, Torlaschi and Gingras (2003) derive the following equation for the standard deviation of a measurement:

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where is the number of independent radar reflectivity samples used for its estimation. Note that in Eq. (14) is the true correlation coefficient one is attempting to measure (rather than the measured value, ). This equation represents the standard deviation for infinite SNR conditions and is valid for simultaneous or accurately interpolated H and V sampling. Neither one of these techniques is ideal, relying on either knowing a priori the true correlation coefficient one is attempting to measure (Torlaschi and Gingras 2003) or a number of time series (Illingworth and Caylor 1991) and not the number of independent pulses. It is not possible to compare the method of Illingworth and Caylor (1991) with our proposed method because for their data is unknown, and therefore the number of independent pulses in their time series cannot be quantified. Figure 5a shows the errors on calculated using our new method compared to those calculated using the linear perturbation method of Torlaschi and Gingras (2003) as a function of in rain (). The magnitudes of the upper confidence bounds are largely similar; however, for all the lower confidence interval is higher for Torlaschi and Gingras (2003) (i.e., smaller deviations from are predicted), because of the asymmetric nature of the new confidence intervals on . The largest difference is for small . As increases, both the upper and lower confidence intervals for each method converge. Although Fig. 5a serves as a useful illustration of the difference between the methods, they are not strictly comparable in practice: the error calculation of Torlaschi and Gingras (2003) relies on knowledge of , which in reality is unknown. Conversely, the new method requires no a priori knowledge of and, so, is of much greater practical use.

(a) A comparison of the confidence intervals calculated using the new method and that of Torlaschi and Gingras (2003) in rain () and (b) the bias introduced by averaging instead of , as a function of N_IQ. For all N_IQ, the lower confidence interval is higher for the Torlaschi and Gingras (2003) method, particularly for lower N_IQ, because of the asymmetric nature of the confidence intervals on using the new method. Averaging and not for small N_IQ can lead to a large bias.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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(a) A comparison of the confidence intervals calculated using the new method and that of Torlaschi and Gingras (2003) in rain () and (b) the bias introduced by averaging instead of , as a function of N_IQ. For all N_IQ, the lower confidence interval is higher for the Torlaschi and Gingras (2003) method, particularly for lower N_IQ, because of the asymmetric nature of the confidence intervals on using the new method. Averaging and not for small N_IQ can lead to a large bias.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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(a) A comparison of the confidence intervals calculated using the new method and that of Torlaschi and Gingras (2003) in rain () and (b) the bias introduced by averaging instead of , as a function of N_IQ. For all N_IQ, the lower confidence interval is higher for the Torlaschi and Gingras (2003) method, particularly for lower N_IQ, because of the asymmetric nature of the confidence intervals on using the new method. Averaging and not for small N_IQ can lead to a large bias.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Figure 5b illustrates the theoretical bias introduced by averaging many short samples of , rather than , in rain (). This bias is significant for small . For example, when = 10, the bias on is 0.1, which is significant for the purpose of estimating μ in rainfall; this bias in L could lead to an underestimate of μ of approximately 8 at = 2 dB (see Fig. 8). It is not important whether spatial or temporal averaging is used to increase the number of independent I and Q samples, as long as does not vary substantially over the scales considered.

In summary, confidence intervals that rely on the linear perturbation method overestimate the precision of measurements, and require knowledge of the true one is attempting to measure. Fundamentally, failure to use the transform L when averaging short time series will lead to significant biases in correlation coefficient estimates. This is particularly important for operational applications that typically use very short dwell times (discussed in section 8a), and would lead to a significant bias in retrievals of μ in rain.

5. Practical measurement of

To fully exploit our new error estimates, and retrieve rain DSDs, some practical considerations for the measurement of must first be considered.

b. Signal-to-noise ratio

The addition of noise to the received signals acts to reduce the correlation between the H and V time series. The reduction factor f has been shown (Bringi et al. 1983) to vary predictably as

View Expanded

for simultaneous (or accurately interpolated) H and V sampling, where SNR_H and SNR_V are the signal-to-noise ratios for the H and V polarizations, respectively. This was verified by Illingworth and Caylor (1991) with measurements of in drizzle. While it is in principle possible to correct for the presence of noise using this equation, because of the high degree of precision required in this work, only data with SNR > 34 dB are included in our analysis, which corresponds to a maximum achievable measurement of 0.9996. However, instrumental effects (described in section 5d below) will have the same effect of adding uncorrelated noise, and so in practice this maximum value is never reached.

6. Using L and to estimate μ in rainfall

We now attempt to use our high-precision measurements of L to retrieve μ estimates in rainfall. The independence of (D₀, μ) and (L, ) on the drop number concentration means that a single L and observation pair corresponds to a unique D₀ and μ value. To forward model L and for various gamma distributions, we must first assume an appropriate drop shape model.

a. Mean drop shapes

There are numerous drop shape parameterizations in the literature. Here, we examine drop axis ratios and diameters from the recent experiments of Thurai and Bringi (2005), Szakáll et al. (2008), and the fourth-order polynomial fit to many experiments given by Brandes et al. (2002). Figure 6a shows the mean axis ratio as a function of drop diameter, for each of these models. The Thurai and Bringi (2005) data suggest that mean drop shapes are slightly prolate for D < 1 mm, although it is in the margin of measurement error that the drops are spherical (Beard et al. 2010). Since it is known that drops become spherical as their diameter tends to 0 mm because of surface tension, our fit to the data is adapted so that drops < 1 mm are precisely spherical.

(a) Comparison of mean drop axis ratios as a function of equivalent drop diameter D from recent experiments of Thurai and Bringi (2005), Szakáll et al. (2008), and the fourth-order polynomial fit of older experimental data constructed by Brandes et al. (2002). The model of Thurai and Bringi (2005) has been adapted so that drops are spherical for D < 1 mm. Disdrometer Z_DR is calculated using the drop shape models of (b) Thurai and Bringi (2005), (c) Szakáll et al. (2008), and (d) Brandes et al. (2002) from a 5-h dwell over a nearby Joss–Waldvogel RD-80 impact disdrometer (approximately 7 km away) in a frontal rainband on 25 Apr 2014. The time resolution of the radar measurements was decreased to 30 s to match the integration time of the disdrometer. At a 1.5°-elevation angle, the radar was sampling rain at a height of approximately 183 m above the disdrometer. The dashed line is a 1:1 line. The smallest biases are achieved with the Thurai and Bringi (2005) model, especially for smaller Z_DR, suggesting that these shapes best represent those of natural raindrops. Therefore, this model is chosen for the analysis.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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(a) Comparison of mean drop axis ratios as a function of equivalent drop diameter D from recent experiments of Thurai and Bringi (2005), Szakáll et al. (2008), and the fourth-order polynomial fit of older experimental data constructed by Brandes et al. (2002). The model of Thurai and Bringi (2005) has been adapted so that drops are spherical for D < 1 mm. Disdrometer Z_DR is calculated using the drop shape models of (b) Thurai and Bringi (2005), (c) Szakáll et al. (2008), and (d) Brandes et al. (2002) from a 5-h dwell over a nearby Joss–Waldvogel RD-80 impact disdrometer (approximately 7 km away) in a frontal rainband on 25 Apr 2014. The time resolution of the radar measurements was decreased to 30 s to match the integration time of the disdrometer. At a 1.5°-elevation angle, the radar was sampling rain at a height of approximately 183 m above the disdrometer. The dashed line is a 1:1 line. The smallest biases are achieved with the Thurai and Bringi (2005) model, especially for smaller Z_DR, suggesting that these shapes best represent those of natural raindrops. Therefore, this model is chosen for the analysis.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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(a) Comparison of mean drop axis ratios as a function of equivalent drop diameter D from recent experiments of Thurai and Bringi (2005), Szakáll et al. (2008), and the fourth-order polynomial fit of older experimental data constructed by Brandes et al. (2002). The model of Thurai and Bringi (2005) has been adapted so that drops are spherical for D < 1 mm. Disdrometer Z_DR is calculated using the drop shape models of (b) Thurai and Bringi (2005), (c) Szakáll et al. (2008), and (d) Brandes et al. (2002) from a 5-h dwell over a nearby Joss–Waldvogel RD-80 impact disdrometer (approximately 7 km away) in a frontal rainband on 25 Apr 2014. The time resolution of the radar measurements was decreased to 30 s to match the integration time of the disdrometer. At a 1.5°-elevation angle, the radar was sampling rain at a height of approximately 183 m above the disdrometer. The dashed line is a 1:1 line. The smallest biases are achieved with the Thurai and Bringi (2005) model, especially for smaller Z_DR, suggesting that these shapes best represent those of natural raindrops. Therefore, this model is chosen for the analysis.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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To choose the best mean drop shape model, a 5-h dwell was made with CAMRa at a 1.5°-elevation angle over a nearby Joss–Waldvogel RD-80 impact disdrometer (approximately 7 km away) in a frontal rainband on 25 April 2014. The disdrometer measures drop sizes in 127 size bins from 0.3 to 5.0 mm. The instrument is regularly calibrated by the manufacturer and rain rates estimated with this instrument agree very well with those from a collocated rain gauge. Radar measurements of are calibrated regularly (to within ±0.1 dB) by making observations of drizzle (low Z), which we know to have a value of 0 dB. The range resolution of the radar measurements is 75 m, and is averaged to 30 s to match the integration time used by the disdrometer to estimate the DSD parameters. At this elevation angle, the radar was sampling rain at a height of 183 m above the disdrometer. Figures 6b–d show the observed radar measurement from the closest gate to the disdrometer, as well as the corresponding disdrometer values calculated using the Thurai and Bringi (2005), Szakáll et al. (2008), and Brandes et al. (2002) drop shape models, respectively. The Szakáll et al. (2008) axis ratios are systematically smaller compared to both of the other models for almost all D. Using this model makes the disdrometer estimates of always larger than the radar estimates. Thurai and Bringi (2005) and Brandes et al. (2002) agree for D = 2–7 mm, after which the axis ratios of Thurai and Bringi (2005) are closer to those of Szakáll et al. (2008). Therefore, the radar and disdrometer values for the Thurai and Bringi (2005) and Brandes et al. (2002) models largely agree, apart from 0.4 dB. The largest difference between these models occurs for D < 2 mm. Here, Szakáll et al. (2008) and Brandes et al. (2002) predict more oblate drops than do Thurai and Bringi (2005).

The Szakáll et al. (2008) model produces the largest radar–disdrometer overall bias of ≈0.23 dB. The biases from Brandes et al. (2002) for bins of 0.2, 0.4, and 0.6 dB (±0.1-dB bin width) are 0.09, 0.16, and 0.13 dB, respectively. For the Thurai and Bringi (2005) model, they are only 0.04, 0.08, and 0.09 dB, respectively, and are very similar to results found by Brandes et al. (2002) at higher . These reduced biases at low suggest that the experimental results of Thurai and Bringi (2005) best represent natural raindrop shapes. We therefore chose this model in our analysis. It is unclear why the very small residual difference between radar and disdrometer estimates of using the Thurai and Bringi (2005) shape model is observed. Some possible explanations are that the radar calibration is slightly off causing a systematic underestimation, the small sampling volume of the disdrometer could be biasing , or there could be residual error in the mean drop shape model. However, this very small difference is unimportant for the retrievals that follow.

b. Drop oscillations

Drop oscillations increase the variety of shapes within a radar pulse volume at any given time. This means that the we are attempting to estimate will be lower than that predicted by modeling only the mean drop axis ratios for drops of a given size. To account for this, we must parameterize these drop oscillations. In the Thurai and Bringi (2005) experiment, artificial raindrops were created from a hose and allowed to fall 80 m from a bridge before drop axis ratio and counts were measured with a 2DVD on the valley floor. This fall distance is more than sufficient to allow the drops to achieve steady-state oscillations, and so the standard deviations of axis ratios measured in this experiment are interpreted as drop oscillation amplitudes. However, the large standard deviations of the axis ratios for D < 2 mm are likely artificial, caused by the finite resolution of the 2DVD instrument (Beard et al. 2010). Since drop oscillations are thought to originate from vortex shedding (Beard et al. 2010), which increases as a function of drop size, the magnitude of the oscillations should decrease eventually to zero as the drop diameter tends to 0 mm. Beard and Kubesh (1991) suggest that resonant drop oscillations occur for drop sizes between 1.1 and 1.6 mm; however, more recent measurements from the University of Mainz wind tunnel show that amplitudes of the axis ratios for these drop sizes were less than 0.025 (Szakáll et al. 2010). For this reason, the polynomial fit to oscillation amplitude data from the Mainz wind tunnel (Szakáll et al. 2010) is used for D < 2 mm, which has the desired reduction in oscillation amplitude for small drops.¹ For D > 2 mm, we revert to the more statistically robust drop oscillations from Thurai and Bringi (2005). Since the oscillations are aerodynamically induced, with an amplitude only a function of the drop size, they should not vary with environmental conditions. In our analysis, the oscillations were included by integrating over Gaussian PDFs of axis ratios (Thurai and Bringi 2005) in our Gans theory computations. Figure 7 shows the effect of oscillations on computed L and for values of μ = −1 (solid lines) and μ = 16 (dashed lines). Including drop oscillations for the purpose of estimating μ becomes increasingly important with increasing ; the difference between L at μ = 16 computed with and without oscillations is as large as an equivalent change in μ of ≈ 8. We find that the modification of the oscillation magnitudes for drop diameters < 2 mm has a relatively small impact (<0.01) on predicted L for larger than 0.8 dB where we attempt retrievals of μ. However, we find that the use of Szakáll et al. (2010) oscillations for all drop diameters has a large impact on predicted L values (for μ = −1, L is approximately 0.1 lower). This is potentially important for retrievals of μ.

Predicted L and Z_DR values for gamma distributions of μ = −1 (solid) and 16 (dashed) with no oscillations (gray) and including oscillations (black). The inclusion of drop oscillations is crucial to the interpretation of the L and Z_DR measurements. The is assumed to be 0.9963 to match the case study in section 7.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Predicted L and Z_DR values for gamma distributions of μ = −1 (solid) and 16 (dashed) with no oscillations (gray) and including oscillations (black). The inclusion of drop oscillations is crucial to the interpretation of the L and Z_DR measurements. The is assumed to be 0.9963 to match the case study in section 7.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Predicted L and Z_DR values for gamma distributions of μ = −1 (solid) and 16 (dashed) with no oscillations (gray) and including oscillations (black). The inclusion of drop oscillations is crucial to the interpretation of the L and Z_DR measurements. The is assumed to be 0.9963 to match the case study in section 7.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Comparatively large-amplitude (but short lived, lasting less than approximately 0.4 s) collision-induced oscillations can also occur (Szakáll et al. 2014). Rogers (1989) estimate that the collision rate for an average raindrop in a 55-dBZ rain column is approximately 1 min⁻¹. This would imply that raindrops (even in very heavy rainfall) spend an almost negligible fraction of time (approximately 0.5%) affected by collision-induced oscillations. Raindrop clustering increases the likelihood of these collisions (Jameson and Kostinski 1998). For rain rates of around 10 mm h⁻¹ (comparable to those presented in the following case studies), McFarquhar (2004) estimate the collision rate to be about 5 min⁻¹, implying drops are affected only 3% of the time. For very large rain rates (100 mm h⁻¹), this fraction increases to 6% as the collision rate approximately doubles to 10 min⁻¹. Consequently, their impact on L measurements is likely to be small and can be ignored, other than for exceptional rain rates (Thurai et al. 2013).

Figure 8 shows how L varies as a function of for gamma distributions with μ = −1, 0, 2, 4, 8, 12, and 16 computed using Gans theory with the drop shape and oscillation model discussed above. Note that lines of constant μ diverge with increasing . For 0.5 dB, it becomes possible to distinguish μ, given the typical error on an L measurement (shown in Fig. 9).

Theoretical L and Z_DR results computed using Gans theory for gamma distributions with μ = −1, 0, 2, 4, 8, 12, and 16, using Thurai and Bringi (2005) mean drop axis ratios and the oscillation model described in section 6b. The precision of L required to estimate μ decreases as Z_DR increases. The is assumed to be 0.9963 to match the case study in section 7.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Theoretical L and Z_DR results computed using Gans theory for gamma distributions with μ = −1, 0, 2, 4, 8, 12, and 16, using Thurai and Bringi (2005) mean drop axis ratios and the oscillation model described in section 6b. The precision of L required to estimate μ decreases as Z_DR increases. The is assumed to be 0.9963 to match the case study in section 7.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Theoretical L and Z_DR results computed using Gans theory for gamma distributions with μ = −1, 0, 2, 4, 8, 12, and 16, using Thurai and Bringi (2005) mean drop axis ratios and the oscillation model described in section 6b. The precision of L required to estimate μ decreases as Z_DR increases. The is assumed to be 0.9963 to match the case study in section 7.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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(a) The 2D PDF of L and Z_DR observations, and (b) the normalized 2D PDF such that the distribution equals 1 for each Z_DR bin for observations of L and Z_DR collected from dwells on 25 Nov 2014. Here, L is binned every 0.02, and Z_DR is binned every 0.05 dB. Overplotted are theoretical L and Z_DR values computed using Gans theory for gamma distributions of μ = −1, 0, 2, 4, 8, 12, and 16. Typical errors on L and Z_DR are shown as error bars; the error on Z_DR is very small. The gray dashed line indicates the predicted L and Z_DR observations using DSD parameters from the power-law fit to disdrometer measurements in Williams et al. (2014). The black dashed line indicates the predicted L and Z_DR observations using the relationship of Cao et al. (2008). The for this day is measured to be 0.9963.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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(a) The 2D PDF of L and Z_DR observations, and (b) the normalized 2D PDF such that the distribution equals 1 for each Z_DR bin for observations of L and Z_DR collected from dwells on 25 Nov 2014. Here, L is binned every 0.02, and Z_DR is binned every 0.05 dB. Overplotted are theoretical L and Z_DR values computed using Gans theory for gamma distributions of μ = −1, 0, 2, 4, 8, 12, and 16. Typical errors on L and Z_DR are shown as error bars; the error on Z_DR is very small. The gray dashed line indicates the predicted L and Z_DR observations using DSD parameters from the power-law fit to disdrometer measurements in Williams et al. (2014). The black dashed line indicates the predicted L and Z_DR observations using the relationship of Cao et al. (2008). The for this day is measured to be 0.9963.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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(a) The 2D PDF of L and Z_DR observations, and (b) the normalized 2D PDF such that the distribution equals 1 for each Z_DR bin for observations of L and Z_DR collected from dwells on 25 Nov 2014. Here, L is binned every 0.02, and Z_DR is binned every 0.05 dB. Overplotted are theoretical L and Z_DR values computed using Gans theory for gamma distributions of μ = −1, 0, 2, 4, 8, 12, and 16. Typical errors on L and Z_DR are shown as error bars; the error on Z_DR is very small. The gray dashed line indicates the predicted L and Z_DR observations using DSD parameters from the power-law fit to disdrometer measurements in Williams et al. (2014). The black dashed line indicates the predicted L and Z_DR observations using the relationship of Cao et al. (2008). The for this day is measured to be 0.9963.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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7. Case studies of μ retrieval

We now estimate μ using measurements of L and for stratiform rain case studies on 31 January, 25 April, and 25 November 2014, and a convective case study on 22 May 2014. Typical rain rates for each of these case studies can be found in Table 1. Dwells were made at an elevation angle of 1.5°. Strict data quality filters were applied: SNR > 34 dB, linear depolarization ratio (LDR) < −27 dB (close to the limit of cross-polar isolation) to ensure no melting particle contamination or ground clutter, and range > 5 km to avoid near-field effects. Theoretical L and were computed using Gans theory using the drop shape and oscillation model discussed in section 6 (see Fig. 8). Observations were averaged from 10 to 30 s and from range gates of 75 to 300 m to increase the measurement precision of L. At each gate, the most likely pair of μ and given the observed L and values was obtained by selecting the closest point in a lookup table of gamma DSD calculations. Figure 9a shows the observed L binned every 0.02 and binned every 0.05 dB for the example of 25 November 2015. Overlaid are lines of constant μ = −1, 0, 2, 4, 8, 12, and 16. Figure 9b is the same distribution normalized to sum to 1 for each bin. The on this day was calculated to be 0.9963 (see section 5d). The observations of L and are generally well contained within the expected range. The median error on is ≈ 0.025, and is shown as a representative error bar in Fig. 9. A comparison of these data with disdrometer measurements from Williams et al. (2014) is included. In this experiment, the mass spectrum mean diameter and mass spectrum standard deviation were measured using a 2DVD. A fit was derived from 18 969 measurements of 1 min drop spectra (which can readily be converted to a fit). This was in turn used to predict an relationship, shown by the gray dashed line. Both L and were also predicted using the proposed μ–Λ relationship of Cao et al. (2008), which is also derived from a 2DVD, where

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This is shown by the black dashed line.

Table 1.

Typical rain rates for each of the case studies, calculated from disdrometer measurements (April) and radar-retrieved N₀, D₀, and μ values (January, May, and November).

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The median and interquartile range of retrieved μ per bin for this day are shown in Fig. 10. The median retrieved μ is 5 at = 0.8 dB, increasing to 8 for = 1.6 dB. There is significant spread in retrieved μ values, containing contributions from measurement uncertainty on L, as well as true microphysical variability. The impact of changes in L on retrieved μ is nonlinearly related to μ; contributes more to retrieved μ variability for more monodispersed (higher μ) DSDs than for more polydispersed (lower μ) DSDs. Conversely, the contribution of to retrieved μ variability decreases as increases, as the dual-polarization signature is larger and μ is more easily distinguishable (see Fig. 8). To estimate the contribution that the uncertainty on L measurements makes to this observed variability, μ was retrieved using the median L ± the representative uncertainty depicted in Fig. 9. This was then compared with the interquartile range of the retrieved μ for each bin. For bins of 0.8, 1, 1.2, 1.4, and 1.6 dB, we estimate that 88%, 66%, 32%, 31%, and 27% of the variability, respectively, can be attributed to . For > 1 dB, most of the variability seen in Fig. 10 can be attributed to true microphysical variability.

Box plot of retrieved μ as a function of Z_DR for Z_DR bins of 0.2 dB on 25 Nov 2014, showing the median and interquartile range of the data.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Box plot of retrieved μ as a function of Z_DR for Z_DR bins of 0.2 dB on 25 Nov 2014, showing the median and interquartile range of the data.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Box plot of retrieved μ as a function of Z_DR for Z_DR bins of 0.2 dB on 25 Nov 2014, showing the median and interquartile range of the data.

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Figure 11 shows a comparison with retrieved μ for all of the case studies collected. Each of the dwells in January, April, and November was made in stratiform rain, whereas the May case study contains dwells from convective rain. Overlain are predicted mean μ values (solid gray) and upper and lower bounds that contain 55% of the measurements (dashed gray) of Williams et al. (2014) as a function of from the disdrometer measurements. The solid black line shows the predicted relationship using the μ–Λ relationship of Cao et al. (2008). There is a large spread in the radar-retrieved median μ values from case to case. Each median μ estimate is from a very large number of retrieved μ estimates, such that the standard error is smaller than the markers themselves, and so is not shown. The values of retrieved μ in January are approximately 0, close to an exponential DSD for all smaller than 1.1 dB. This is below the value predicted by Williams et al. (2014) but agrees well with μ predicted by Cao et al. (2008). Interestingly, the case studies of April and November show μ increasing with between 0.5 and 1.5 dB, compared to the trend seen by Williams et al. (2014) and Cao et al. (2008) toward an exponential DSD. The retrieved median μ values from the May case study, although agreeing with the decreasing trend with , are significantly above the Cao et al. (2008) predictions and the upper bound of μ from Williams et al. (2014). Our retrieval suggests that, in this case, the rain rate would be overestimated by almost 2 dB if an exponential DSD or the fit of Cao et al. (2008) is assumed. Whereas the μ values are not outside the full range of data measured by Williams et al. (2014), the use of the proposed relationship would cause an overestimate of about 1 dB (see Fig. 1).

Median retrieved μ as a function of Z_DR for Z_DR bins of 0.1 dB for case studies of 31 Jan, 25 Apr, 22 May, and 25 Nov 2014. The solid gray line is the predicted μ as a function of Z_DR from the power-law fit to the disdrometer measurements of Williams et al. (2014), and corresponds to the dashed gray upper and lower bounds that contain 55% of the data. The solid black line shows the predicted relationship using the μ–Λ relationship of Cao et al. (2008).

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Median retrieved μ as a function of Z_DR for Z_DR bins of 0.1 dB for case studies of 31 Jan, 25 Apr, 22 May, and 25 Nov 2014. The solid gray line is the predicted μ as a function of Z_DR from the power-law fit to the disdrometer measurements of Williams et al. (2014), and corresponds to the dashed gray upper and lower bounds that contain 55% of the data. The solid black line shows the predicted relationship using the μ–Λ relationship of Cao et al. (2008).

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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Median retrieved μ as a function of Z_DR for Z_DR bins of 0.1 dB for case studies of 31 Jan, 25 Apr, 22 May, and 25 Nov 2014. The solid gray line is the predicted μ as a function of Z_DR from the power-law fit to the disdrometer measurements of Williams et al. (2014), and corresponds to the dashed gray upper and lower bounds that contain 55% of the data. The solid black line shows the predicted relationship using the μ–Λ relationship of Cao et al. (2008).

Citation: Journal of Applied Meteorology and Climatology 55, 7; 10.1175/JAMC-D-15-0272.1

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8. Discussion

Our retrievals of μ made using and are typically larger than the radar estimates of μ of between 1 and 3 by Thurai et al. (2008) and of between 0 and 2 by Illingworth and Caylor (1991). Perhaps this is not surprising, given that the imperfect collocation of the H and V sample volumes was unaccounted for, and their would have been biased low as a result of averaging rather than L, both of which are accounted for in our data. Furthermore, Illingworth and Caylor (1991) do not include drop oscillations in their retrievals, which will have led to a significant underestimate of μ. Whereas there is some agreement of the magnitudes of μ for < 1 dB with predicted Williams et al. (2014) and Cao et al. (2008) values, the apparent opposite trend toward more monodisperse distributions is consistent among three of the four case studies. For the retrieved μ to agree with the trend predicted by Williams et al. (2014) or Cao et al. (2008), a reduction in the drop oscillation amplitudes for smaller drops would be required so that predicted L values are higher. However, this would not explain the difference between the May retrieval results and the predicted μ from disdrometer measurements; we estimate that it would require oscillations that are at least an order of magnitude larger to bring these median μ estimates into agreement with Williams et al. (2014) or Cao et al. (2008). An incorrect parameterization of the drop oscillations alone is unlikely to be able to account for the disagreement with Williams et al. (2014) and Cao et al. (2008); however, to better establish the accuracy of the technique, a better quantification of raindrop oscillations is desirable.

The μ estimates derived using radar are sensitive to higher moments of the DSD, whereas disdrometer estimates tend to use lower moments of the DSD (Cao and Zhang 2009). This could be partly responsible for the differences between the radar- and disdrometer-estimated μ values. If the DSD shape is not perfectly described by Eq. (1), the “effective” μ that is derived may be different even if the underlying DSD shape is the same. It is also possible that what we have captured is simply the natural variability of the DSD in different types of rainfall (i.e., convective and stratiform), and there is not a universal relationship. More case studies are needed to gather a statistical understanding of the behavior of μ using this retrieval method.

9. Conclusions

In this paper, a new variable, , is defined that is Gaussian distributed with a width predictable by the number of independent I and Q samples, which in turn can be estimated using the Doppler spectral width. This allows, for the first time, the construction of rigorous confidence intervals on each measurement. The predicted errors using this new method were verified using high-quality measurements in drizzle from the Chilbolton Advanced Meteorological Radar.

The proposed method is of much greater practical use than the linear perturbation error estimation method, as it does not require knowledge of the unknown “true” that one is trying to estimate. The method works for both simultaneous or accurately interpolated alternate sampling. However, it does not work for alternate estimators, which rely on the Gaussian autocorrelation function to estimate the zero-lag correlation between H and V pulses (Sachidananda and Zrnić 1989), where estimates can be >1.

A new technique to account for the imperfect collocation of H and V sampling volumes on measurements is presented. The impact of drop oscillations on the observed L measurements was shown to be significant; omitting oscillations from our Gans simulations leads to an underestimate of retrieved μ of approximately 8. We further show that failure to use L over measurements when averaging can lead to a significant bias low in estimates (and consequently μ), particularly for very short dwell times such as those used operationally.

High-precision measurements of L and in rainfall are then used to estimate μ in the gamma DSD for four case studies. We find that our estimates of μ in stratiform rain somewhat agree in magnitude with those from disdrometer studies for small , but there appears to be a tendency toward more monodisperse DSDs between = 0.8 and 1.5 dB, unlike the trend toward an exponential distribution suggested by disdrometer measurements. The convective case study does display this trend toward lower μ as increases, but the magnitude of μ remains much larger than predicted by disdrometer measurements. If true, this would lead to overestimates of retrieved rain rate by approximately 1 dB if the relationship of Williams et al. (2014) is used, or 2 dB if an exponential distribution or the relationship of Cao et al. (2008) is used. We find that the μ retrieval exhibits sensitivity to the choice of the drop oscillation model. A better understanding of raindrop oscillations would be useful to fully establish the accuracy of our retrieval technique.

The variability in our radar-retrieved μ could simply be the natural variability of the DSD between convective and stratiform rainfall; there may not be a universal relationship. More case studies are desirable to investigate this further.

The μ retrieval technique employed here offers improvements over the radar estimates of Illingworth and Caylor (1991) and Thurai et al. (2008). Illingworth and Caylor (1991) did not take into account the imperfect collocation of the H and V sample volumes on measurements of , the effect of drop oscillations, or the fact their estimates would be biased low by averaging short time series. Each of these effects would cause μ to be underestimated. The same is true of Thurai et al. (2008); however, drop shapes measured by 2DVD measurements include oscillations, and so are included in their μ estimates.

The new error statistics of presented here could aid operational applications that require uncertainty on to be quantified, or use averages of . The use of L operationally to retrieve μ is limited by the use of rapid scan rates and the corresponding few independent I and Q samples. However, assuming that μ is a smoothly varying parameter, averaging L could help improve rain-rate retrievals; the uncertainty on operationally retrieved rain rates using the retrieval technique presented here is estimated to be approximately ±12.5%. Practically, retrieved rain rates are less affected by changes in higher values of μ than by changes in lower values. Therefore, operationally, simply being able to distinguish between regions of “high” and “low” μ with L could be sufficient to provide an improvement over existing retrieval techniques.