a. Step-by-step description of the methodology

Smyth and Illingworth (1998) suggested using a “Z_DR ≈ 0 constraint” in the stratiform region behind convective cells to estimate the PIDA and correct radar measurements for differential attenuation. The stratiform region is defined as a low-reflectivity region where the differential phase plateaus. This constraint cannot be used for an operational algorithm because it is not always possible to find such a region for each ray. This limitation is not a problem for the present work since we are essentially interested in characterizing the γ_DP variability and we can afford not to process the unsuitable rays. The application of the “Smyth and Illingworth constraint” (hereinafter referred to as the S&I constraint) is done as follows:

• Select all rays that have at least 30° of differential phase shift (ΔΦ_DP). This threshold value is justified in the following.

• Correct the polarimetric measurements for low signal-to-noise ratio (SNR) biases, nonmeteorological echoes, calibration biases, azimuthal interferences, and differential phase aliasing and offsets (Gourley et al. 2006a).

• Filter Φ_DP and estimate K_DP. In the present analysis, we simply use a running, centered, median filter of length 1.68 km, corresponding to seven 240-m gates. At least 50% of the Φ_DP measurements have to be available (i.e., classified as precipitation) within the filtering window to validate the K_DP estimate. Otherwise, the K_DP estimate is set to a missing value. The rather short path over which K_DP is estimated is related to the fact that K_DP will essentially be used in convective rain.

• Correct Z_H for PIA using Φ_DP assuming a given γ_H:

  

  

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  Actually, as explained below, the choice of γ_H is coupled with the γ_DP estimation.

• For each selected ray, search for light, stratiform precipitation in regions behind convective cells from the radar’s vantage point. A so-called stratiform region is defined in this study as a series of at least 20 consecutive 240-m gates below the freezing level with (corrected) horizontal reflectivity less than 45 dBZ and no differential phase shift. The freezing-level height is retrieved from the brightband identification technique proposed by Tabary et al. (2006). The standard deviation of the differential phase (Φ_DP) over the 20 gates also has to be less than 5° [i.e., slightly more than the typical gate-by-gate noise found with the Trappes radar differential phase measurements in Gourley et al. (2006a)]. That value was determined subjectively by examining a large number of profiles. A stratiform region is more simply identified as a smooth plateau on the Φ_DP profile.

• If the intrinsic horizontal reflectivity (Z_H,intrinsic) is known, then the intrinsic Z_DR,intrinsic can be obtained using an empirical relationship. For instance, Bringi et al. (2001) have proposed the following formula:

  

  

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  Figure 1 helps to assess the relevance of (4) and (5). The data have been obtained from a large number of close-range, nonattenuated, nonshielded, high-SNR data in rain collected by the French polarimetric operational Trappes radar [see Gourley et al. (2006a) for a thorough evaluation of its quality]. The mean Z_DR and its standard deviation have been computed for each 1-dBZ class of Z_H ranging from 0 up to 45 dBZ. The Bringi et al. (2001) model [(3) and (4) above] has been superimposed (diamonds). The agreement is remarkable even though the model tends to overestimate the intrinsic Z_DR by around 0.1 dB for Z_H between 15 and 20 dBZ and to underestimate it by the same amount (0.1 dB) for Z_H between 40 and 45 dBZ. For the present analysis, a linear-by-part model was fitted to the empirical data. It has been represented by a light straight line on Fig. 1. Here are the corresponding relationships:

  

  

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  Figure 1 also helps in setting the minimum value ΔΦ_DP required in the first step of the processing (first item above). The scatter of the intrinsic Z_DR (Z_DR,intrinsic) around the linear-by-part fitted model is typically ±0.2 dB. If we assume a typical γ_DP value of 0.03 dB °⁻¹, then a 30° differential phase shift (which is the value that was retained) corresponds to a PIDA of about 1 dB, which is 5 times larger than the uncertainty on the intrinsic Z_DR. As a consequence, setting ΔΦ_DP to 30° guarantees that the γ_DP estimation has a precision better than 20%.

• Once the intrinsic differential reflectivity (Z_DR,intrinsic) is estimated for each of the 20 gates, then the simple arithmetic mean is taken (Z_{DR,intrinsic,mean}) and the slope γ_DP can be computed as follows:

  

  

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  If the γ_H value assumed initially to correct horizontal reflectivity measurements was not adequate, then the intrinsic Z_DR,intrinsic and, subsequently, the γ_DP estimation may be biased. In the appendix, however, we show that given the typical range of variation of γ_H (between 0.07 and 0.13 dB °⁻¹) the relative error on the estimated γ_DP is around 5%. Even though that amount of error may be considered to be fairly acceptable, a refinement has been introduced to solve for γ_H and γ_DP simultaneously. The approach relies on the assumption that γ_H and γ_DP are proportional in rain. Using numerical simulations, Vulpiani et al. (2008) have shown that the following relationship (see their Fig. 1),

  

  

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  was fairly stable for a large range of temperatures, N_W, μ, and D₀. Later we will present an analysis of 4 yr of observed raindrop spectra that supports this relationship. Similarly, G06 have empirically found ratios γ_DP/γ_H equal to 0.37, 0.27, 0.5, 0.34, 0.42, 0.46, and 0.56 (their Table 1). Those results reveal some variability but part of this variability is due to the uncertainty in the estimated γ_DP and even more γ_H and also to the fact that hail is likely to be present in several of the situations analyzed by G06. The proposed iterative approach assumes an initial value for γ_H (say γ_H0), corrects Z_H for attenuation using (3), estimates the intrinsic Z_DR,intrinsic via (6)–(10), retrieves γ_DP with (11), and finally computes a new γ_H with (12). Convergence is achieved when two successive values of γ_H differ by less than 0.01 dB °⁻¹. Because of the nonlinear form of the (Z_H, Z_DR) relationship [(6)–(10)], the solution for γ_H and γ_DP cannot be obtained in one step. In the coupled retrieval, which relies on the important assumption that precipitation is pure rain, unusually large retrieved γ_DP will be translated into unusually large γ_H. In this example, if the coupling is deactivated and a climatological γ_H value (γ_H0 = 0.1 dB °⁻¹) is used, then Z_H and subsequently Z_DR [(6)–(10) values in the stratiform region will be underestimated and the γ_DP estimation will be biased negatively. The impact of using or not using a coupling between γ_H and γ_DP will be presented and discussed in section 3.

• Once the optimal γ_H and γ_DP have been retrieved, all Z_H and Z_DR measurements along the ray between the radar and the stratiform region are corrected for attenuation using filtered Φ_DP.

• Finally, the characteristics of the attenuating cells are extracted using weighted averages of the various relevant parameters (Z_H, Z_DR, ρ_HV, K_DP, and T), the weights being the attenuation-corrected Z_H values (expressed in linear units) at each gate:

  

  

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  In (13)–(17), z_H is equal to 10^Z_H/10 and Z_H is the attenuation-corrected horizontal reflectivity (dBZ). The summation symbols in (13)–(17) (i.e., Σ_{radar→stratiform_region}) simply mean integration over all gates along the ray between the radar and the stratiform region through the convective cells. To extract the characteristics of the attenuating cells, the proper weights to use in (13)–(17) should be the specific differential attenuation (A_DP) or the specific attenuation (A_H) both expressed in decibels per kilogram. Those two quantities, however, are not known so we use linear horizontal reflectivity (mm⁶ m⁻³) as a proxy [as in Ryzhkov et al. (2007)]. Notice that in rain, the relationship between linear horizontal reflectivity and specific attenuation is close to linear (see Table 1 of Testud et al. 2000). The temperature profile has been reconstructed under the assumption of a constant −6.5° km⁻¹ lapse rate from the freezing-level height retrieved by the brightband identification algorithm (Tabary et al. 2006). Even though this is rarely alluded to, one has to concede that it is extremely difficult to obtain a very accurate estimation of the droplets’ temperature inside the attenuating cells because it is highly variable in space and time and may differ significantly from ambient air temperatures. The space–time resolution of the radiosonde data is clearly not sufficient while mesoscale model forecasts of the wet-bulb temperature fields may not be relevant for the inner part of the convective cells, where significant latent heat release is expected. Jameson (1992) has suggested that the γ_H and γ_DP parameters are typically multiplied by a factor of 1.5 when the temperature decreases from 20° down to 0°C.

  In addition to characterizing the attenuating cells in terms of Z_H, Z_DR, temperature, K_DP, and ρ_HV, we also considered the vertical profile of reflectivity in the most intense part of the attenuating cells, as this is widely used to indicate the presence of hail at ground level. Several indicators [Z_H > 55 dBZ, vertically integrated liquid content, probability of hail (POH), etc.] have been proposed and validated to identify the presence of hail within convective cells (Waldvogel et al. 1979; Eilts et al. 1996; Witt et al. 1998; Joe et al. 2004; Delobbe and Holleman 2006). While it is recognized that those conventional methods typically do not attempt to estimate the height or the size of the hailstones, they have proven to be efficient enough to be used by many operational services to diagnose the presence of hail at ground level. As will be seen later, a central question for the present work and more generally for all attenuation correction methods at C band (Ryzhkov et al. 2007) is whether the attenuating cells contain hail, a rain/hail mixture, or pure rain.

The scan strategy for the Trappes radar for the years 2005 and 2006 is presented in Fig. 2. The elevation angles are scanned on a 15-min basis and range from 1.5° to 9° (9.5° for the year 2006). Two additional low-elevation angles (0.4° and 0.8°) are actually revisited every 5 min, but they were considered to be too much affected by beam blockage and radome interferences for use in the present analysis. Because of this limited number of elevation angles, the vertical structure could only be recovered for ranges between 40 and 80 km. To reconstruct the vertical profile, horizontal reflectivity data were synchronized using a standard cross-correlation advection field at the end of the 15-min period and corrected for attenuation using filtered Φ_DP and the appropriate γ_H (i.e., the one retrieved following the approach described previously). The vertical profile was then extracted at the range of maximum Z_H.

b. Examples

Figure 3 shows an example of the application of the method on a particular ray. Data were taken at 1.5° during a convective situation (1630 UTC 23 June 2005). The raw Z_H and Z_DR profiles are represented by thin lines and their attenuation-corrected counterparts by thick lines. Profiles of raw ϕ_DP (noisy thin line), seven-gate, median-filtered ϕ_DP (thick line), K_DP, and ρ_HV are also plotted in Fig. 3 and show that the total differential phase shift is more than 100°. The PIA and PIDA are in this case respectively equal to 10 and 3 dB, and γ_DP was estimated at 0.025 dB °⁻¹, a typical value (Carey et al. 2000; G06; Ryzhkov et al. 2007). In the most intense part of the attenuating cell (in terms of Z_H, see the vertical bar), Z_H reaches 51.5 dBZ, Z_DR reaches 2.78 dB, and K_DP of reaches 3.9° km⁻¹. It is noteworthy that ρ_HV decreases down to 0.93; ρ_HV normally decreases in convective cells when hydrometeors with a large variety of shapes are present. This occurs in heavy rain when large and small drops are both present (e.g., Keenan et al. 2001, their Fig. 10) and when there are rain/hail mixtures, particularly when the hail is large enough to Mie scatter. In addition, low values of ρ_HV can occur when there are high gradients of reflectivity within the resolution volume (e.g., Ryzhkov 2007). The vertical reflectivity profile steadily decreases from 53 dBZ at the lower levels to 33 dBZ at 10 km. The 45-dBZ level is reached approximately at the height (H_45dBZ) of 7 km. The 0°C isotherm height (H_0°Cisotherm) for that day was at 3.5 km above sea level. Using Delobbe and Holleman’s (2006) formula to assess the probability of hail (POH),

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we get a POH of 0.78. Even though (18) has not been tuned specifically for the Trappes region, one can say that there is presumption of hail in that cell. We recall here that comparisons with surface reports have shown that the conventional POH estimator is fairly successful at predicting hail at ground level. Since in this work we are mainly using data from low-elevation angles at short distances from the radar (i.e., close to the ground), it is fair to assess the presence of hail in the plan position indicator (PPI) using the conventional POH estimator.

Figure 4 is taken from the same episode as Fig. 3 (23 June 2005) but at a slightly different time (1600 UTC) and in another azimuth. The amount of total differential phase shift is about the same as in Fig. 3 (a little more than 100°). Yet in that case, the attenuation is much more severe; the PIA and PIDA values are respectively equal to 20 and 5 dB. The estimated γ_DP is equal to 0.056 dB °⁻¹ (i.e., more than twice as large as in the previous example). Figures 3 and 4 clearly illustrate the fact that the coefficient of proportionality between A_DP and K_DP in (2) does not depend solely upon temperature. The central question is then to know whether the attenuating cells have significantly different characteristics in the two cases. The ρ_HV pattern is rather similar in both cases. A careful examination though reveals a tendency in the second case (the one with high γ_DP) to have slightly lower values of ρ_HV, with some spikes even below 0.9. Likewise, the K_DP profiles are comparable, though higher in the high γ_DP case, in terms of maximum (∼4° km⁻¹) and range extension (20 km). On the other hand, there is a very sharp contrast between the Z_H and even more between the Z_DR profiles. In the high γ_DP case, Z_H reaches 58.5 dBZ and Z_DR 4 dB over almost 10 km. The vertical profile of reflectivity is almost vertical and beyond 60 dBZ up to 6 km. Given the limited vertical coverage, the top of the cell cannot be determined with a high degree of accuracy but one can expect that the POH would probably be beyond 1.

Before moving on to the analysis of all the observations, Fig. 5 illustrates one limitation of the present approach. In this situation, two attenuating cells are clearly visible on the profile: the first one produces a differential phase shift of 20° (maximum Z_H of 50 dBZ) and the other produces 100° (maximum Z_H of 58 dBZ). A stratiform region was identified between 50 and 55 km and the properties of the attenuating cells were automatically extracted using (13)–(16). Given the linear Z_H weighting used in the equations, basically only the second cell was considered when retrieving Z_H, Z_DR, temperature, K_DP, and ρ_HV. A better approach would have consisted of splitting the ray into two parts, analyzing the first part, retrieving the appropriate γ_H and γ_DP, correcting Z_H and Z_DR for attenuation, and then processing the second part. This refinement has not been introduced mainly because a subjective overview of all the profiles showed that, in the vast majority of the cases, attenuation was due to a well-defined single attenuating cell.

a. Introduction

The method proposed in the previous section has been applied to more than 30 convective and mixed cases observed during 2005 and 2006. All tilts between 1.5° and 9° were considered. The freezing-level height varies between 1 and 3.5 km above mean sea level (MSL), the radar being located at 191 m MSL. However, most of the attenuating cells correspond to summer events with freezing-level heights typically at 3 km MSL. For each suitable ray, the following parameters were computed and stored: PIA (dB), PIDA (dB), γ_DP (dB °⁻¹), γ_H (dB °⁻¹), T ^{attenuating_cells} (°C), , r^{attenuating_cells} (km), system differential phase (Φ_DP0), and standard deviation of the differential phase in the stratiform region (std devΦ_DP). Overall, the dataset comprises 216 519 profiles. We have calculated that the S&I constraint could only be applied to 20% of all attenuated rays (defined by Φ_DP > 30°). For 80% of the rays having a significant differential phase shift (Φ_DP > 30°), no suitable stratiform region could be identified, either because a plateau of Φ_DP could not be found in the rain region, the standard deviation of Φ_DP was too high (>5°), Z_H was too high, or the SNR too low. This means that the S&I constraint does not qualify for operational attenuation correction of dual-polarization measurements. Also stored in the database are the vertical profiles of horizontal reflectivity (Z_H) for the attenuating cells located between 40 and 80 km. This information was only available for 10% of all the attenuating cells present in the database.

Figure 6 introduces the quantitative analysis. It simply represents the mean (for visibility purposes) of the standard deviation of the PIDA (dB) as a function of the total differential phase shift (Δϕ_DP; °) for a narrow range of temperature (12.5° < T < 17.5°C) and for different values of (i.e., the Z_DR value of the attenuating cells). All curves in Fig. 6 are approximately linear, which establishes the relevance of (2). Also noteworthy is the fact that their slopes (i.e., the γ_DP parameter) seem to be increasing with , consistent with previous works (Carey et al. 2000; Bringi and Chandrasekar 2001; G06). The subsequent analyses aim at precisely assessing the relationship between γ_DP and the polarimetric values of the attenuating cells.

b. Stratification of the observed attenuating cell properties with γ_DP

Figure 7 presents a stratification of the results according to the estimated γ_DP, with and without the coupling between γ_H and γ_DP being activated in the retrieval in (12). For clarity, no standard deviation bars are presented. We shall first comment on results obtained with the coupling activated. The basic idea of that first stratification is to characterize the properties of the attenuating cells leading to high γ_DP values. First of all, Fig. 7a presents the overall and day-by-day occurrence frequency of the estimated γ_DP. The vertical scale is logarithmic. The maximum frequency is centered at 0.025 dB °⁻¹, which is a value that compares quite well to previously reported results. The distribution is rather broad and values beyond 0.06 dB °⁻¹ occur for 15%—a nonnegligible figure—of the rays presenting a significant differential phase shift (>30°). Several episodes lead to such high γ_DP values. This is a convincing demonstration that the variability of the γ_DP coefficient has to be accounted for in any operational attenuation correction procedure. More recently, Ryzhkov et al. (2007), using independent data from the two SIDPOL C-band polarimetric radars in Alabama and in Canada and a rather different methodology to estimate γ_DP, also found a large variability of γ_DP (see their Table 1). The typical median values they obtained for Alabama were 0.01 (tropical rain), 0.02 (rain with hail), and 0.01 (tornado) dB °⁻¹. In Canada, the typical median values of γ_DP were 0.01–0.04 (small hail), 0.01 (rain), 0.08 (hail), and 0.03–0.06 (rain with hail) dB °⁻¹.

Figures 7b–e show respectively the mean Z_DR, Z_H, ρ_HV, and K_DP values of the attenuating cells as a function of γ_DP. To avoid confusion and competing effects between temperature and DSD characteristics and shapes, these plots are for data all having similar temperatures (T ^{attenuating_cells} between 12.5° and 17.5°C). Therefore, one can say that temperature plays a negligible role in the trends observed in Figs. 7b–e. There is a clear linear relationship between γ_DP and each of the four above-mentioned variables . Larger γ_DP values statistically correspond to larger Z_DR, larger Z_H, smaller ρ_HV, and larger K_DP. In particular, the unusually high γ_DP values (say beyond 0.06 dB °⁻¹) that we will show are very unlikely in rain are caused by cells with Z_DR beyond 5 dB, Z_H beyond 60 dBZ, ρ_HV below 0.94, and K_DP above 4° km⁻¹. Those cells are likely to contain frozen particles (hail or graupel) mixed with rain since such low values of ρ_HV are unlikely in rain.

Interestingly, the “high” γ_DP regime (above 0.06 dB °⁻¹) seems simply to be the linear extrapolation of the “low” regime (below 0.06 dB °⁻¹). Among all the storms that were included in the database, some are known to have been hail producing (Vulpiani et al. 2008). The presence of ice leads to an expected increase in values of Z_H, but also more surprisingly, increases in Z_DR and K_DP. This seriously questions the usability of K_DP to retrieve rain rate in the presence of ice. Finally, Fig. 7f presents the relationship between T ^{attenuating_cells} and γ_DP. Similarly to what was done to remove any temperature impact on Figs. 7b–e, Fig. 7f was built from data having all the same Z_DR ( between 1 and 2 dB). Figure 7f does not reveal any clear relationship between γ_DP and T ^{attenuating_cells}. This may be explained by the fact that the range of T ^{attenuating_cells} present in the database is rather modest (say between 10° and 20°C). Climatologically speaking, the majority of attenuating precipitation systems occur in France at the same season in about the same temperature conditions (15°C). Operational attenuation correction algorithms should try to take into account the temperature effect but it is clearly second order compared with the impact of the microphysical characteristics of the attenuating cells.

The presence of ice in the attenuating cells calls into question the application of a coupling between γ_H and γ_DP. Indeed, the 0.3 factor between γ_H and γ_DP was established using scattering simulations and disdrometer observations in pure rain and little is known about the evolution of the proportionality factor in the presence of ice. This is the reason why a “no coupling” analysis has been carried out where γ_H is set to a climatological value (0.1 dB °⁻¹). Results of this experiment are indicated by crosses on Fig. 7. As expected, the variable that is mostly affected is . In the “coupling” experiment, it reaches the unrealistic value of 72 dBZ (for γ_DP equal 0.1 dB °⁻¹) while in the no coupling experiment, it peaks at 60 dBZ, still an extremely high value that would be consistent with the presence of wet ice. Because it is very likely that unusually high γ_DP leads to unusually high γ_H (in a proportion that is unknown), it is fair to admit that the no coupling curve stands as the minimum bound of while the coupling curve can be considered as its maximum bound. The curve (Fig. 7b) changes a little bit toward slightly lower values for extremely high γ_DP, which is the logical consequence of lower γ_H, hence lower Z_H and Z_DR values in the stratiform region and lower estimated γ_DP. The (Fig. 7d) and (Fig. 7e) are also slightly altered because of the change of the attenuation-corrected Z_H values that are used as weights in the extraction of the properties of the attenuating cells [(13)–(17)]. Overall, the coupling and no coupling curves can be regarded as uncertainty bounds attached to the estimated polarimetric variables of the attenuating cells. The conclusions regarding the surprisingly very high values of γ_DP, K_DP, and Z_DR and the likelihood of (wet) ice remain whether the coupling is considered or not.

c. Comparison with simulations based on Chilbolton disdrometer data

Figure 8 is another way to look statistically at the results. The stratification of the radar observations is done this time according to the properties of the attenuating cells and for various temperature ranges. In addition, on the left-hand side of Fig. 8, we present computations of these variables using raindrop size spectra obtained with a Joss–Waldogel disdrometer over the period June 2003–June 2008 (i.e., 4 yr) sited at Chilbolton (Hampshire, United Kingdom). Chilbolton is located less than 400 km northwest of Trappes. Chilbolton is the closest site to Trappes where long time series of reliable disdrometer data are available. Both sites have a maritime climate with similar total annual rainfall distributed throughout the year but with a very slight tendency to more convective precipitation at Trappes. Table 1 gives the seasonal statistics of rainfall accumulations computed over the period 1971–2000 for Paris and London. It is thus fair to assume that the rain systems affecting both sites have, on average, similar microphysical properties. Because we are interested in events with unusually high attenuation, extreme care is needed to ensure that the disdrometer is operating reliably during the heaviest rain, which occurs only for brief periods. Analysis of the data reveals occasional very short periods of apparently heavy rain composed of only very large drops; closer examination of other gauges showed no rain was falling and that the large drops were probably spurious and caused by crows attacking the Styrofoam cover of the disdrometer. To remove such spurious data, the disdrometer data were compared with three gauges each recording the individual drops (equivalent to 0.004 mm of rain) forming at the base of the gauge funnel; each 30-s disdrometer spectrum was only accepted when two of the three drop counting gauges recorded some rainfall. The disdrometer calibration was monitored by comparing weekly rainfall totals with those from a conventional gauge and any periods of questionable calibration were rejected. After this rigorous quality control, a total of 109 551 spectra were accepted.

The next stage of the disdrometer analysis is to compute values of Z_H, Z_DR, K_DP, ρ_HV, γ_H, and γ_DP at 0°C using the 𝗧-matrix solution technique (Mishchenko 2000) to account for any Mie scattering and resonance at C band and the raindrop axis ratio relationship proposed by Brandes et al. (2002), which has been confirmed as appropriate for rain observed in northern France (Gourley et al. 2009). The 52 500 spectra with a Z_H above 25 dBZ (rain rates above about 1 mm h⁻¹) were selected and the values of Z_DR, Z_H, ρ_HV, and K_DP plotted as a function of γ_DP in Figs. 8e–h, respectively, where the individual points are in gray and the mean values are crosses. Analysis for values of Z_H above 45 dBZ (not shown) showed a linear relationship between γ_H and γ_DP with a very high correlation and a slope of 0.3, thus justifying our fundamental assumption in (12).

In the heaviest rain, the upper limit of 5 mm for the disdrometer may be truncating the spectra and eliminating the largest drops, which may be contributing significantly to the attenuation and the polarization parameters. To account for the truncation, we consider the counts in the last bin (bin no. 127), which provides the total number of drops above 5 mm. Because of saturation of the instrument, drops with a diameter beyond that threshold cannot be accurately sized. The last bin count was nonzero for 201 spectra and reached 23 in the heaviest rainfall cases. To extrapolate the spectra to include the larger drops we fitted our observed truncated spectra to a normalized gamma function of the form N(D) = N_W(D/D₀)^μ exp[−(3.67 + μ)D/D₀] using the method of moments as described by Kozu and Nakamura (1991). The fitted values obtained are very reasonable. The mean value of log(N_W) is 3.8 mm⁻¹ m⁻³ with a standard deviation of 5 dB, D₀ peaks at 1.1 mm with a tail extending to 4 mm, and the μ distribution is quite skewed with a median of 5 and a long tail of positive values. To recreate the missing large drops the full untruncated spectrum was computed from the fits, and the number of drops >5 mm that the disdrometer would have classified in its last bin (bin no.127) was calculated. A scatterplot of the number of predicted drops above 5 mm with those observed was very encouraging. For 218 spectra the extrapolated fit predicted one or more drops >5 mm, and the maximum number of predicted large drops >5 mm was 22. On average the extrapolated predicted count was within 3 counts of the observed number.

The values of Z_H, Z_DR, ρ_HV, K_DP, A_H, and A_DP were computed from the 52 500 spectra fitted to a gamma function and the mean values of Z_DR, Z_H, ρ_HV, and K_DP plotted in Figs. 8e–h as a function of γ_DP for 0° and 20°C. The impact of the extrapolated large drops is, as expected, really only significant for Z_DR, where for Z_DR > 3 dB the inclusion of the extrapolated large drops increases Z_DR by about 1 dB. This change does, however, lead to a much better agreement with the radar observations in Fig. 8a. Note also that the computations using the disdrometer spectra predict a very slight increase in attenuation at the lower temperature, also in agreement with the radar observations. A maximum size of 10 mm was assumed for the extrapolated spectra in Fig. 8, but reducing this to 8 mm did not change the plots significantly; most of the fitted spectra had a positive μ, which introduces a natural truncation of the spectrum. The plot of Z_DR against γ_DP at 20°C for Z_DR > 5 dB is shown dotted in Fig. 8e because there are only three data points. We note that the highest value of γ_DP in nearly all naturally occurring rainfall is 0.06 dB °⁻¹ and the minimum value of ρ_HV derived from the spectra is 0.96, which may be reduced by radar imperfections to 0.95. In their analysis of a large dataset of tropical raindrop spectra, Keenan et al. (2001) found a similar minimum (their Fig. 10). In their Fig. 9, the vast majority of spectra had a γ_DP below 0.06 dB °⁻¹, with just a very few tropical spectra above this value when a maximum drop size of 2.5 D₀ was assumed but not for an 8-mm maximum drop size. This suggests that values of ρ_HV lower than 0.95 and γ_DP above 0.06 dB °⁻¹ found in hot spots in northwest Europe are only rarely due to pure rain but usually indicate targets that include some wet ice.

We now consider the implications of Fig. 8. Figure 8a (mean γ_DP value as a function of ) shows that is an excellent predictor of the γ_DP value. The corresponding disdrometer calculations (Fig. 8e) reproduce the Z_DR dependency very well but only up to 5 dB and γ_DP of 0.06 dB °⁻¹ indicating that values above these limits are likely to be due to partially frozen particles.

Figures 8b and 8f reveal an excellent agreement between the observed and simulated relationships, with both predicting, for example, values of γ_DP of 0.02 and 0.04 dB °⁻¹, for a mean Z_H of 50 and 60 dBZ, respectively. There again, however, the disdrometer predicts no values of γ_DP above 0.06 dB °⁻¹. Figures 8c and 8g show ρ_HV plotted as a function of γ_DP. Lower values of ρ_HV are clearly associated with higher values of γ_DP both on the observations and on the simulations. However, simulations do not yield ρ_HV values below 0.96 (corresponding to γ_DP equal to 0.04 dB °⁻¹). This can be interpreted as further evidence that the attenuating hydrometeors are most unlikely to be pure rain in the “low ρ_HV–high γ_DP” regime. We note that the melting model of hail presented by Rasmussen and Heymsfield (1987), whereby a torus of water forms around the equator of a melting hailstone leading to a stable nontumbling fall mode, is consistent with the “hot spots” having very high Z_H and Z_DR, low ρ_HV, and very large total and differential attenuation. Finally, observed and simulated curves (Figs. 8d,h) both show the same increasing trend, but the disdrometer curves are rather flatter with K_DP values approximately halved. The radar observations tend to show a much clearer increase of γ_DP with increasing , consistent with Fig. 7e. We recall here that observed K_DP have been estimated using a very short path (1.68 km) and should therefore be very close to intrinsic K_DP. The disagreement between observations and simulations can be interpreted as further evidence that ice is contributing to the high values of K_DP and to the attenuation in the hot spots.

d. Vertical profile analysis

To better assess the potential presence of frozen particles in the cells causing unusually high γ_DP, an analysis of the vertical profiles of horizontal reflectivity has been carried out. Figure 9 presents an analysis of those vertical profiles in two contrasting cases: γ_DP < 0.06 dB °⁻¹ (Fig. 9a) and γ_DP > 0.06 dB °⁻¹ (Fig. 9b). Two profiles are presented on each graph: the one retrieved with coupling between γ_H and γ_DP (thick line) and the one retrieved without coupling (thin line). In Fig. 9a, the coupling and the no coupling curves overlap completely and are hardly discernible. In Fig. 9b, the no coupling profile is about 5 dB smaller than the coupling profile. In all cases, the mean Z_H profiles are typical of convective cells. The profiles are almost vertical in the lowest layers (say up to 4 km), then decrease at the rate of −1.5 dB km⁻¹. The high γ_DP profile, however, is shifted by about 5(10) dB in the no coupling (coupling) case toward higher Z_H than the low γ_DP profile. The low γ_DP profile is less than 55 dBZ at all heights whereas the high γ_DP profile exceeds that typical hail threshold up to 4(6) km MSL in the no coupling (coupling) case. The 45-dBZ contour is reached at about 4 km in the low γ_DP case and at some height above 7 km in the high γ_DP cases (with and without coupling). Assuming a 0°C isotherm height at 3 km, Eq. (18) gives values of POH of 0.45 (low γ_DP case) and a value above 0.85 (high γ_DP cases). This strongly supports that hail, possibly combined with rain, is present in the high γ_DP case.