1. Introduction
The microstructure of liquid precipitation is described statistically by the raindrop size distribution (DSD), the concentration in air of falling raindrops per drop size. Bulk rainfall properties are interrelated by the DSD and can be calculated using its weighted statistical moments (e.g., Ulbrich 1983; Testud et al. 2001). The DSD is measured in situ by disdrometers, which have limits on the sizes of raindrops they can detect. While upper size limits are usually larger than raindrops expected in nature, lower size limits lead to truncations on the measured DSD, and instrumental inaccuracies in measurements of small drops are common. The inability of commonly used disdrometers to accurately measure very small raindrops affects the rainfall properties derived from these instruments’ recordings. In this paper we explore this problem and propose a method to reconstruct concentrations of small drops from such truncated or inaccurate DSD measurements.
Accurate measurements of the DSD are important for many studies. The DSD describes rainfall properties at raindrop scale but provides information that is relevant to larger-scale rainfall processes (e.g., Uijlenhoet and Sempere Torres 2006), including the effects of rain on soil (e.g., van Dijk et al. 2002), rainfall microphysics (e.g., Rosenfeld and Ulbrich 2003), rainfall interactions with telecommunication links (e.g., Crane 1971; Messer et al. 2006; Leijnse et al. 2007; Schleiss and Berne 2010), numerical weather prediction (e.g., Baldauf et al. 2011), and measurement of rain by weather radar (e.g., Marshall et al. 1947; Seliga and Bringi 1976; Bringi and Chandrasekar 2001; Berne and Krajewski 2013). The DSD is highly variable (e.g., Bringi et al. 2003; Jameson et al. 2015b), and most studies of its variability rely on measurements from networks of disdrometers (e.g., Miriovsky et al. 2004; Lee et al. 2009; Tapiador et al. 2010; Tokay and Bashor 2010; Jaffrain et al. 2011; Jameson et al. 2015a; Raupach and Berne 2016; Hachani et al. 2017; Tokay et al. 2017). A problem is that disdrometers provide measurements that may be truncated or inaccurate at the small-drop end of the DSD.
Raindrops are typically sized between 0.1 and 6 mm in equivolume (equivalent-volume sphere) diameter (Uijlenhoet and Sempere Torres 2006). Larger raindrops are rare (Gatlin et al. 2015). There exists a physical maximum to raindrop size, since drops with diameter larger than about 10 mm are unstable and will break up even in laminar airflow (Pruppacher and Klett 2000). The range of detectable raindrop sizes for commonly used disdrometers is typically from a minimum of 0.2–0.3 mm to a maximum larger than 15 mm (e.g., Löffler-Mang and Joss 2000; Schönhuber et al. 2008; Tokay et al. 2013), implying that many empirical DSDs are truncated at the small-drop end. In addition to such truncation of the DSD, comparison studies show that it is likely that most commonly used disdrometers measure small drops with high uncertainty.
Several studies have performed intercomparisons of different disdrometers (e.g., Tokay et al. 2001; Miriovsky et al. 2004; Thurai et al. 2011; Tokay et al. 2013, 2014; Raupach and Berne 2015; Park et al. 2017). Generally speaking, instrumental differences appear more pronounced for the extremes of the DSD, which is to say for very small and very large drops. The 2D video disdrometer (2DVD; Schönhuber et al. 2008) is accepted as a reference instrument for the large end of the DSD (Gatlin et al. 2015; Thurai and Bringi 2018), but has been shown to underestimate concentrations of small drops (Tokay et al. 2001, 2013). Tokay et al. (2001) noted that for rain rates under 20 mm h−1, 2DVD measurements lacked an expected exponential increase in concentrations as drop diameters approached zero. Tokay et al. (2013) concluded that 2DVD measurements under a drop diameter limit of 0.3 mm were unreliable. Sampling issues are a concern for disdrometers (e.g., Jameson and Kostinski 2001; Smith and Kliche 2005; Tapiador et al. 2017), in particular for large drops, since DSD variability increases with drop size (e.g., Jameson et al. 2015b). Instrumental differences at the small-drop end of the DSD are therefore less likely to be related to sampling effects than differences at the large end. Small drops are abundant, so low-order moments may be well sampled, but limited instrumental sensitivity to small drops may result in underestimated measurements. High-order moments, on the other hand, may be more accurately measured with an instrument such as the 2DVD, but samples may not be representative because of the scarcity of large drops.
Recently, Thurai et al. (2017) combined measurements from collocated 2DVD and Meteorological Particle Spectrometer (MPS; Baumgardner et al. 2002) instruments to measure more complete DSD spectra. The MPS provided high drop-size resolution (0.05 mm) for small drops, while the 2DVD provided good resolution (about 0.17 mm) for larger drops. The instruments’ overlap region (0.7–1.2 mm) was found to be in good agreement. The combined spectra exhibited a large number of small drops in a “drizzle mode” (for diameters less than about 0.5–0.7 mm), and a precipitation mode for larger drops, often separated by a shoulder (or plateau) region. These modes were previously identified in aircraft imaging probe data collected in oceanic warm-rain clouds by Abel and Boutle (2012), and resemble “equilibrium” DSDs modeled using simulated raindrop collisions (e.g., McFarquhar 2004; Straub et al. 2010). Thurai et al. (2017) noted that previous conclusions regarding DSD truncation effects (e.g., Willis 1984; Ulbrich 1985; Vivekanandan et al. 2004) may be affected by their findings. The drizzle mode and shoulder region are likely not well captured by commonly used ground-based disdrometers. For example, Prat et al. (2008) used a radar profiler to initialize the top DSDs in a 1D vertical rain shaft model, and studied the DSD evolution to the surface. At the surface, the model predicted many small drops that were not captured by a Joss–Waldwogel disdrometer, and they concluded that the disdrometer did not adequately measure the small-drop end of the DSD.
Some bulk rainfall variables, such as the rain rate or radar reflectivity, are not strongly affected by the concentrations of small drops, but low-order DSD moments such as the total drop concentration are highly sensitive to these numbers. Concentrations of small drops are important for numerical modeling of collision and coalescence processes in rainfall microphysics (e.g., Meyers et al. 1997), since it has been shown that collision energy that leads to raindrop oscillation is greatest when larger drops collide with much smaller drops (Johnson and Beard 1984). Small-drop concentrations are also important for determination of the shape parameter in the popular gamma model of the DSD (Ulbrich 1983). Jameson et al. (2016) found that the spatial patterns of bulk rainfall parameters over the sampling area of a 2DVD could be attributed mainly to spatial patterns in the total number concentration of raindrops over the sampling area, emphasizing the importance of the zeroth moment of the DSD. Thurai et al. (2017) showed that the inclusion of the drizzle mode reduces the DSD-derived mass-weighted mean drop diameter Dm (mm) and increases DSD spectral width, especially in low rain rates, and this affects relationships between Dm and radar variables that may be used to predict characteristic drop diameters (e.g., Liao et al. 2014). Thurai et al. (2017), and references therein, provide more discussion on the importance of accurate, high-resolution measurements of small drops.
DSD normalization refers to a technique in which variability in the DSD is explained via variability in one or more of its statistical moments, and the shape of the DSD is described by a generic “shape function.” Many normalization techniques have been proposed (e.g., Sekhon and Srivastava 1971; Willis 1984; Sempere-Torres et al. 1994; Testud et al. 2001; Illingworth and Blackman 2002; Lee et al. 2004; Yu et al. 2014). Lee et al. (2004) generalized and unified the techniques of Sempere-Torres et al. (1994) and Testud et al. (2001), and provided a double-moment normalization method in which the DSD is represented as a combination of two of its moments of arbitrary order and a double-moment normalized DSD that describes the shape of the DSD. This double-moment normalized DSD, used with DSD moment orders three and four, was shown by Thurai and Bringi (2018) to be stable across two example cases (stratiform and convective rain), as well as in rainfall produced by the outer bands of Hurricane Irma. For a variety of choices of input moment orders, the double-moment normalized DSD has been shown to have low scatter through time series (Lee et al. 2007), and to be stable enough across spatial displacement for the assumption of its invariance to be acceptable for practical use in stratiform rain, with some caveats (Raupach and Berne 2017a). The assumption of an invariant double-moment normalized DSD was used by Raupach and Berne (2017b) in a technique to retrieve DSD estimates from polarimetric radar data.
Double-moment normalization was applied to DSDs that include the drizzle mode by Thurai and Bringi (2018), who used the same MPS and 2DVD instrumental setup as Thurai et al. (2017). They used illustrative cases to show that taking moment orders three and four and a generalized gamma model of the double-moment normalized DSD, the full DSD including the drizzle mode can be modeled properly when the shape parameter is allowed to go negative. They also demonstrated that the generalized gamma model provides better fits on these DSDs than a standard gamma (Ulbrich 1983) model. In this paper we use the same instrumental setup to show how the full DSD including the drizzle mode can be reconstructed from truncated or inaccurate measurements made by common disdrometers. We use the double-moment normalization method of Lee et al. (2004), two moments that are well measured by common disdrometers, and a generalized gamma DSD model to represent the DSD with the drizzle mode. This technique allows for more accurate estimations of the numbers of small raindrops to be made, from measurements in which the numbers of small drops are known to be unreliable.
The rest of this article is structured as follows. Section 2 contains a brief overview of the theory used in this work. In section 3, we describe the data and data processing used. The diameter up to which MPS measurements are used is discussed in section 4. In section 5, we determine which two DSD moment orders measured by a standard disdrometer are best to use with the double-moment normalization model. The best-fitting parameters for a generalized gamma model fitted to double-moment normalized DSDs that include the drizzle mode are found in section 6. In section 7, performance statistics for reconstructed DSDs are discussed. Finally, in section 8, we compare the reconstruction method to a previous routine for correcting measured DSDs. Conclusions are drawn in section 9.
2. Theory
The DSD is the number of raindrops per equivolume diameter D (mm) interval (from 
a. Bulk rainfall parameters
b. Double-moment normalization of the DSD
c. Performance statistics
3. Data
We used the same instruments that were used previously in Thurai et al. (2017) and Thurai and Bringi (2018) in their Greeley, Colorado, and Huntsville, Alabama, campaigns, and details of the instruments, measurement campaigns, and the events listed below (except Tropical Storm Nate) are shown in these two references. At each site, an MPS and 2DVD were collocated to provide measurements of the DSD from 0.1 to 10 mm. The two sites are climatologically different, with Huntsville being lower altitude, wetter, and warmer than Greeley (Thurai et al. 2017): Greeley has a cold semiarid (Köppen Bsk) and Huntsville has a humid subtropical (Cfa) climate (Peel et al. 2007).
The data used here encompass several interesting events. These include, for Greeley, a long and variable event on 17 April 2015 and a multicell storm with large drops on 23 May 2015. At Huntsville, the events include a developing squall line on 11 April 2016 and a convective line with trailing stratiform rain on 30 November 2016. On 8 October 2017, rain produced by Tropical Storm Nate was recorded at Huntsville (Thurai et al. 2018). Some characteristics of the main dataset used in this study are shown in Table 1. Additionally and separately, we tested the results of our technique on data from 11 to 12 September 2017 at Huntsville, when more than eight hours of rainfall produced by the outer bands of Hurricane Irma were recorded. These data are treated separately since some hours of rain from this event have been shown to exhibit unusual characteristics (Thurai and Bringi 2018). We observed unusual combinations of high Nt and low Dm during certain hours on 12 September 2017. While other rain sampled within the hurricane event was more regular, we treated the whole hurricane event as one separate dataset. We call the records from 11 to 12 September 2017 the “hurricane data” and use them only in section 7a.
The earliest and latest time steps (end of integration period) and number of 1-min records (Num), per station, in the MPS and 2DVD data used in this study. The data used did not cover the entire time between first and last times, and only times for which both instruments recorded drops were included. This table does not include the Hurricane Irma data from 11 and 12 Sep 2017 at Huntsville.
a. MPS processing
MPS data were processed as described in the appendix of Thurai et al. (2017), with minor differences that are listed here. The velocities of small drops were calculated from the particle diameter using a relationship provided by the MPS manufacturer, which itself is a least squares fit to the data of Gunn and Kinzer (1949). This relationship was used for all MPS classes, and we note it differs from Eq. (A5) in Thurai et al. (2017). Bringi et al. (2018) showed that MPS measurements of fall speeds of small drops are consistent with a fit to the Gunn and Kinzer data (that of Foote and Toit 1969), when the velocities are adjusted for instrument altitude. We adjusted velocities for altitude using the method of Beard (1985) and air densities calculated using an assumed sea level temperature of 20°C and an assumed humidity of 0.95 at instrument altitude (note that R was calculated using the same humidity value, but for sea level). The depth of field, used in the calculation of the effective sampling area, was calculated as per Eq. (A2) in Thurai et al. (2017), but depth of field values greater than 0.2 m were set to 0.2 m, since this is the distance between the arms of the MPS instrument. The smallest two drop size classes in the MPS data were considered unreliable and were not used, so the smallest MPS drop class we considered was 0.1–0.15 mm. MPS DSDs were calculated at 1-min temporal resolution. Time steps for which there was not a corresponding 2DVD measurement were discarded. The MPS data were then resampled to various lower temporal resolutions assuming that missing times contained no drops.
b. 2DVD processing
The standard procedure for calculating DSDs from 2DVD data (Schönhuber et al. 2008) was used. Since the instruments used here have previously been shown to perform well (Thurai et al. 2017), no filtering based on particle size versus velocity was used, and the 2DVD data were not screened for a recently observed minor anomaly (Larsen and Schönhuber 2018). One unrealistic raindrop was removed from the 2DVD data for having D ≥ 10 mm. 2DVD DSDs were calculated at 1-min temporal resolution using a constant equivolume diameter class width of 0.25 mm, with class diameter intervals closed on the left and open on the right. 2DVD measurements for which there were no corresponding MPS measurement were discarded. Then, 1-min data were resampled to lower temporal resolutions, assuming that drop concentrations for missing times were zero.
4. Combining MPS and 2DVD measurements
When combining measurements from MPS and 2DVD, Thurai et al. (2017) used MPS measurements up to an equivolume drop diameter of 0.7 mm, while Thurai and Bringi (2018) used a cutoff point of 1.2 mm. To determine a cutoff point to use here, we compared concentrations from 2DVD classes from 0.25 to 1.75 mm to MPS concentrations. 2DVD classes had a width of 0.25 mm, so each one was compared to the mean concentration across five MPS classes. Comparisons for each class were only made when both instruments recorded drops for that class. To ensure the comparisons were not affected by any small timing differences, they were made at 1-min, 3-min, 5-min, 10-min, 30-min, and 1-h resolutions. Figure 1 shows the distributions of relative bias of 2DVD concentrations when compared to the MPS as a reference, by diameter class and station.
Relative bias of 2DVD concentrations using the MPS concentrations as a reference, for selected 2DVD equivolume diameter classes per time resolution and station. For display purposes, only three of the tested time resolutions are shown. Crosses show medians; bars show IQRs.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
To find the best match, we took the median of all relative bias values across all time resolutions and both stations. The 2DVD class with the best median bias and therefore the best match was 2DVD class three, for D in the interval 
MPS and 2DVD DSDs were combined for 1-min time steps for which both instruments measured drops. MPS drop concentrations were used for 0.05-mm-width diameter classes up to 0.75-mm equivolume diameter, thus replacing 2DVD concentrations in the first three 2DVD classes. 2DVD drop concentrations were used for 0.25-mm-width classes for larger diameters. The combined 1-min DSDs were resampled to 3-min, 5-min, 10-min, 30-min, 1-h, and 1-day resolutions by taking the mean DSD across time intervals and assuming missing time steps contained no rain. After resampling, combined DSDs that contained raindrops in only one diameter class or had a rain rate less than 0.1 mm h−1 were discarded (at 1-min resolution this resulted in discarding 2.2% and 17.9% of the combined DSDs, respectively). Figure 2 shows example MPS, 2DVD, and combined DSDs from 17 April 2015 at Greeley, for three different temporal resolutions.
Example DSDs, showing MPS measurements (red diamonds), 2DVD measurements (black crosses), and the complete DSD (black line) for three time steps of different resolutions on 17 Apr 2015 at the Greeley site. The log10 scales are used to help discriminate small-drop classes. Rain rates (mm h−1) computed from complete DSDs are shown in the titles, and classes for which N(D) = 0 are not shown. The vertical black line shows 0.75 mm, the point up to which MPS drop concentrations are used in the combined DSDs.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Although the MPS and 2DVD combined DSDs remain truncated, they provide coverage of a more complete drop diameter range than either instrument individually. As shown by Thurai et al. (2017) the combined DSDs provide measurements of small drops that include the drizzle mode, unlike the 2DVD instrument alone. We refer to the combined DSDs as “complete” DSDs and those measured by standard disdrometers as “incomplete” DSDs. Our aim in this paper is to reconstruct the complete DSD as accurately as possible, given only an incomplete DSD.
5. Best-measured DSD moments
Since standard disdrometers tend not to measure the smaller drops in the DSD spectrum very well, they are able to capture some DSD moment orders more accurately than others. The double-moment normalization technique of Lee et al. (2004) is able to represent a DSD based on two of its moments. In this section we show which DSD moments are best represented by incomplete DSDs, with the aim of choosing two DSD moments to use with double-moment normalization.
Taking complete DSDs as the reference, we analyzed the performance of collocated incomplete DSDs by comparing moments of orders zero to seven, and bulk variables R and Dm, for all time resolutions. Figure 3 shows scatterplot comparisons, and statistics are shown in Table 2. As expected, the large number of small drops measured by the MPS means that the complete DSDs have larger lower-order moments than the incomplete DSDs. Higher-order moments are less affected, but differences remain: the addition of the MPS records for small drops does, on occasion, produce significant differences in these moments. Correlation coefficients are highest for moments three to seven and lowest for moment zero.
Density scatterplot comparisons of variables calculated from complete and incomplete DSDs, with all temporal resolutions combined; log10 scales are used to better display the full data range. The 1:1 line is shown in black.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Comparison of variables (Var) between complete DSDs (reference) and incomplete DSDs across all time resolutions. Bias and RMSE have the units of the corresponding variable [R in mm h−1, Dm in mm, and DSD moment of order n (Mn) in mmn m−3]. RB stands for relative bias as a percentage, with 
Raupach and Berne (2017a) showed that the double-moment normalization method of Lee et al. (2004) represents the full spectrum of DSD moments better when the two moments used in the normalization are not both high or both low order. For use with double-moment normalization, then, we chose the two moments in Table 2 with the lowest absolute value of median relative bias, with a constraint that they both have r2 values above 0.98 and that the two moment orders are separated by at least two integer moment orders. Moment seven was not considered, to try to avoid sampling uncertainty caused by the rarity of large drops. The DSD moment orders that we selected as best measured in incomplete DSDs, and best suitable for use with double-moment normalization, were therefore i = 3 and j = 6.
6. Modeling the complete double-moment normalized DSD
The double-moment normalized DSD h(x) is expected to remain relatively invariant over space and time (Lee et al. 2004; Raupach and Berne 2017a; Thurai and Bringi 2018). Finding a representative model for the double-moment normalized DSD is a nontrivial task, however. The generalized gamma model has been used (Lee et al. 2004; Raupach and Berne 2017a,b), including for DSDs with a drizzle mode (Thurai and Bringi 2018). To provide a good fit of Eq. (5) to double-moment normalized DSDs in which there is a drizzle mode, the shape parameter μ must be allowed to go negative (Thurai and Bringi 2018). It should be noted that a negative μ violates the assumption made in Lee et al. (2004) [their Eq. (36)] that the generalized gamma distribution used is a probability density function [pdf; h(x) itself is not a pdf]. Allowing a negative μ provides better numerical fits to empirical double-moment normalized DSDs than when μ is forced to be positive or when an ordinary gamma model is used, however (Thurai and Bringi 2018). For practical purposes, we require a model of h(x) that provides the best possible results when DSDs are reconstructed. To that end, we allow μ to go negative, and in this section we examine different methods for fitting the generalized gamma model to the double-moment normalized DSD.
When the generalized gamma model of the double-moment normalized DSD is used, the best-fitting values of parameters μ and c in Eq. (5) are required. Previous studies used parameter optimization to minimize the difference between measured 
To h(x) from individual DSDs with no weighting, so that the distributions of parameters can be examined.
To median values of h(x) per class of x, with each class weighted by the number of points in the class raised to a power p. Tests were run using no weights (p = 0) and various class widths (dx of 0.02, 0.05, 0.1, and 0.2), with p = 1 and dx = 0.2 (as in Raupach and Berne 2017a), and with p = 4 and dx = 0.2 (as in Raupach and Berne 2017b).
To mean h(x) spectra using classes of x of size dx = 0.2, weighted by the number of points in the class raised to power p, using no weights (p = 0) (as in Lee et al. 2004), and weighted by the number of points in each class (p = 1).
The empirical double-moment normalized DSD h(x) was found for each individual complete DSD, using moments three and six. Then, for each fitting technique and each time resolution, parameters that minimized the difference between 
Figure 4 shows densities for distributions of c and μ fitted to individual 1-min spectra, and properties of these distributions are shown in Table 3. There are well-defined peaks in the densities, for μ at about −0.1, and for c at about 3.3. As shown by the values in Table 3, both distributions are positively skewed, so to select a representative single value for each of the parameters we used their medians. Table 4 shows parameter values found using all tested fitting techniques for 1-min data, including the median parameter values from these individual fits; it is evident from these values that the fitting method used affects the retrieved values of μ and c. The top panel of Fig. 5 shows occurrences of double-moment normalized DSD [empirical values of h(x)] and the fitted models overplotted as lines, for complete DSDs at 1-min resolution. The plot demonstrates the skewed nature of distributions of h(x) for a some classes of x, especially for large normalized diameters, for which the median value of h(x) is zero and there are outlier nonzero values. This skewness affects the reliability of models fitted using mean values of h(x).
Densities for values of μ and log10c fitted to 1-min complete DSDs. For display purposes, the x axes are cropped at 4; 4.4% of μ values were above this limit and are not shown here; μ and c are unitless.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Properties of the distributions of μ and c fitted to complete 1-min DSDs; μ and c are unitless. P10, P25, P75, and P90 stand for the 10th, 25th, 75th, and 90th percentiles respectively.
Generalized gamma model parameters μ and c, for 1-min DSDs, fitted using different fitting techniques. The values of p and class width of x (dx) used during the fit are shown in parentheses.
Density occurrence plot of double-moment normalized complete DSDs at 1-min and 1-h resolution and fitted generalized gamma functions. Crosses show median values of h(x) in classes of x with width 0.02 (for display purposes, not every class is shown), and error bars show the IQR for each class. Occurrence counts of h(x) = 0 are not shown, while zeros in the IQR bars and medians are plotted on the x-axis line. Fitting methods are defined as for Table 4.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Since integration of DSDs in time involves taking the average of the concentrations of drops per diameter class, distributions of h(x) become less skewed as the temporal resolution reduces. This effect can be seen in the bottom panel Fig. 5, in which double-moment normalized DSD densities and fitted models are shown at 1-h resolution. Medians of h(x) are nonzero for larger values of x than in the 1-min data, because the larger the integration time the more large drops are likely to be sampled. This sampling effect means that the best-performing model of the double-moment normalized DSD depends on the temporal resolution of the data.
To test the model performances, each fitted model was used to reconstruct all DSDs using only moments three and six from the complete DSDs, and for the same classes of diameter as the original DSDs. For each reconstructed DSD, DSD moments from zero to seven, R, and Dm were calculated, and compared to the same properties of the original complete DSDs, to compute performance statistics for each model when the input moments are perfectly known. Figure 6 shows the performance of each fitted model on 1-min and 1-h data, using distributions of relative bias for these variables. Reconstruction of moments three and six have low (but nonzero) reconstruction error, because they are the input moments. The greatest error occurs on moment orders zero to two, and it is for these moments that differences are most visible between the different model fits.
Distributions of relative bias for reconstructed DSDs compared to complete DSDs at 1-min and 1-h resolution, using various fits of a generalized gamma model to the double-moment normalized DSD. Crosses show median relative bias, bars show the IQR on relative bias. Here, reconstructions were made using moments from complete DSDs, to show model performance.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Table 5 shows which fitting method performed best for each time resolution tested. To determine the best-performing model, we found the average absolute value of the median relative bias across all variables, and selected the model with the lowest average. Other methods, such as choosing the model with the lowest maximum relative bias or the best minimum r2 value, could be used and would return different results; we chose this method because it prioritized low overall bias. In all time resolutions, the chosen best model was that fitted to the median of h(x), with x divided into classes of width 0.05 or 0.1. Table 5 shows the generalized gamma model parameters for the best-fitting model, the worst median relative bias for that model, and the variable for which that bias was observed. The median relative bias is worst for moments of order either zero or one, which is coherent with our choice of moments three and six to normalize the DSDs (Raupach and Berne 2017a). With decreasing temporal resolution, the best-fitting μ and c values tend to become smaller. The best-fitting models are plotted in Fig. 7. The change in parameter values with decreasing temporal resolution has the effect of lengthening the tail of the double-moment normalized DSD, which is representative of the greater possibility of sampling large drops at larger integration times. That DSD shape depends on the sample size (i.e., integration time) was noted by Joss and Gori (1978), who found that combining many “instant” and highly variable DSDs led to an exponential average DSD. More recently, this idea was also discussed by Bumke and Seltmann (2012).
Generalized gamma model parameters μ and c, for the best-performing fitting method for each time resolution. The values of p and class width dx used during the fit are shown in brackets. M.RB indicates the worst median relative bias for any variable, when the best fitting model is used to reconstruct the DSDs, and Variable shows the variable for which this bias was produced.
The best-fitting models of the double-moment normalized DSD, for each tested time resolution (Res).
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
7. Reconstructed DSD performance
We now have the required ingredients to reconstruct the drizzle mode of the DSD. These ingredients are measurements of DSD moments three and six, which are well captured even when the drizzle mode is not well measured, and the parameters c and μ of a generalized gamma model that describes the double-moment normalized DSD with the correct shape for the complete DSD. The idea now is that, given an instrument that does not capture the drizzle mode, we may use the two DSD moments measured by this instrument and reconstruct the entire DSD including the drizzle mode. In this section we report on the performance of this technique.
For each temporal resolution, DSDs were reconstructed using moments three and six of the incomplete DSD and Eqs. (4) and (5), for class-center drop diameters from the complete DSD diameter classes. The parameters μ and c used were those from Table 5, specific to each temporal resolution. Figure 8 shows several examples of measured complete and incomplete DSDs, along with the reconstructed DSD based on two moments from the incomplete DSD. The figure contains randomly selected individual DSDs for light and medium rainfall rates and covers three different temporal resolutions. Visually, the reconstructed DSDs show slight variations from the complete DSD visible for very large and small drops. Several bulk rainfall statistics are shown for the complete and reconstructed DSDs. Terms R, Dm, and Z are in reasonable agreement, while more variation is shown in Nt, which is expected given the model performances shown in section 6.
Examples of incomplete, complete, and reconstructed DSDs from the dataset. For three time resolutions, DSDs with (top) 1 ≤ R < 2 mm h−1 and (bottom) with R ≥ 5 mm h−1 were randomly chosen to display (based on R calculated from complete DSDs). The reconstructed DSD using the double-moment technique and generalized gamma model is shown in blue, black crosses show the complete DSD, and red diamonds show the incomplete DSD. Axes are both in log10 scale to highlight differences between concentrations of drops, and points for which N(D) = 0 are not shown. Bulk variable values for the complete measured DSD (black) and the reconstructed DSD (blue) are shown for comparison.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Figure 9 compares the distributions of relative bias of incomplete DSDs and reconstructed DSDs, when compared to the complete DSDs as the reference across all time resolutions. To further break down the sources of error, the performances of reconstructed DSDs that used moments from the complete DSDs are also shown in this figure. Errors on DSDs reconstructed from complete DSDs show error in the generalized gamma model only, whereas DSDs reconstructed from incomplete DSDs show slightly worse performance because of the error in incomplete DSD moments three and six. The relative bias performance of the reconstructed DSDs shows an improvement over the performance of the incomplete DSDs themselves, especially for moments zero, one, and two of the DSD.
Distributions of relative bias for incomplete and reconstructed DSDs compared to complete DSDs, across all temporal resolutions. Crosses show median relative bias; bars show relative bias IQR.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Table 6 shows the difference in performance made by the reconstruction, comparing the performance of raw incomplete DSDs to reconstructed DSDs based on incomplete DSD moments. The results show that the reconstruction does not significantly damage the performance of high-order DSD moments, with the worst degradation made on any DSD moment’s median relative bias being 4 percentage points on M4. Meanwhile, the low-order moments’ median relative bias values are improved in the reconstructions, by up to 66 percentage points (on M0). The median relative bias of Dm is improved by 3 percentage points, while that of R is made 5 percentage points worse by the reconstruction. While relative bias is generally improved or kept similar in the reconstructed DSDs, IQR on relative bias is increased by the reconstruction, especially for the low-order moments. This increased uncertainty likely reflects residual variability in the double-moment normalized DSD at the small and large-drop ends of the size distribution, and therefore greater variability in the values of h(x) for those drop sizes. Also, DSD moments close to the input moment orders (three and six) are better constrained by the double-moment normalized DSD (Raupach and Berne 2017a), and the third-order DSD moment is slightly affected by measurement errors on small drops. These reasons are likely why the values of r2 are slightly worse for moments zero and one of the reconstructed DSDs, while staying about the same for other moment orders. Figure 10 shows scatterplots for the reconstructed DSDs, compared to the complete DSDs. Comparing these to Fig. 3, the improvement in low-order moments can be seen, since the points are closer to the 1:1 line, while Dm is slightly improved, and the scatter on the higher-order moments and R is only slightly greater.
Performance statistics for reconstructed DSDs, found by comparing reconstructed to complete DSDs. Differences are shown between the performance of reconstructed DSDs and incomplete DSDs (Table 2), with negative values indicating that the reconstructed DSDs improved the comparison with the complete DSDs. Column labels are as in Table 2.
Density scatterplots comparing variables calculated from complete and reconstructed DSDs. Scales and symbols are as in Fig. 3.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Hurricane Irma
We tested the flexibility of our approach by applying it to data from 11 to 12 September 2017, when the outer rainbands of Hurricane Irma were sampled at the Huntsville site. These data were not used in the training of the method, and sampled rain that is, at least at times, significantly different from other rain at the Huntsville site: the rain sampled in the hurricane data contains number-controlled precipitation for some hours, in which greater rain rates are associated with greater numbers of drops instead of greater drop sizes (Thurai and Bringi 2018).
Figure 11 shows examples of incomplete, complete, and reconstructed DSDs from the hurricane data at three temporal resolutions. For these examples, the drop concentrations are higher than for the DSDs shown in Fig. 8. Visually the reconstructed DSDs look reasonable. There is more difference between measured and reconstructed values of Nt than the other variables, as was the tendency in the training data examples; R values are underestimated. A comparison of relative bias between incomplete DSDs and reconstructed DSDs in the hurricane data is shown in Fig. 12, and performance statistics comparing incomplete to reconstructed DSDs are shown in Table 7. The results are similar to those for the nonhurricane data, with the reconstruction improving the relative bias of DSD moments zero, one, and two and not significantly damaging the high-order moments. Similar to the nonhurricane data, the reconstruction increased the relative bias IQRs for low-order moments, and degraded the median relative bias of R (by 5 percentage points), but in this case r2 values for low-order moments are not much affected by the reconstruction.
Examples of incomplete, complete, and reconstructed DSDs (blue line) from the hurricane data recorded on 11–12 Sep 2017 at the Huntsville site, for the minute, hour, and day with the maximum rain rates on these days. Symbols are as for Fig. 8.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Distributions of relative bias for incomplete DSDs and reconstructed DSDs for the hurricane dataset, across all temporal resolutions. Symbols are as for Fig. 9.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Performance statistics for reconstructed DSDs, with complete DSDs as the reference, on the hurricane data across all time resolutions. Differences are shown between the performance of reconstructed DSDs and incomplete DSDs (incomplete DSD performance not shown) with negative values indicating that the reconstructed DSDs improved the comparison with the complete DSDs. Column labels are as in Table 2.
In the hurricane results, there are larger ranges of relative bias than in the training set, implying greater variability in h(x). The differences between the results of reconstructions using incomplete and complete DSDs are also greater, indicating more error in moments three and six derived from the incomplete DSDs. It is interesting to note that the relative bias distributions for mid- to high-order moments are negatively skewed for incomplete DSDs and DSDs reconstructed from incomplete DSD moments. We infer from this that the difference in the number of small drops between incomplete and complete DSDs in the hurricane data was sometimes of such a magnitude that it significantly affected even large-order moments, which aligns with the observations of number-controlled rainfall during parts of the hurricane event.
8. Comparison to previous correction routine
The results shown in Thurai et al. (2017) and this study have implications for previous disdrometer “correction” routines. Raupach and Berne (2015) provided a method to adjust disdrometer data to match a reference 2DVD, using a correction of recorded velocities, then application of a multiplicative correction factor (this factor was similar to the bias correction suggested by Miriovsky et al. 2004). In Raupach and Berne (2015), the concentrations of small drops recorded by the Parsivels were reduced, to better match the 2DVD. Given the results of Thurai et al. (2017) and this study, it is likely that the 2DVD did not capture well the small-drop end of the spectrum and therefore formed an imperfect reference instrument with underestimated concentrations of small drops. In this section we use data from a network of Parsivel disdrometers and collocated tipping-bucket rain gauges to compare the results of the previous correction routine to DSDs in which the drizzle mode has been reconstructed using our new method.
The instrument network was deployed in Ardèche, France, during the 2013 special observation period (from September to November 2013) of the Hydrological Cycle in the Mediterranean Experiment (HyMeX; Drobinski et al. 2014; Ducrocq et al. 2014; Nord et al. 2017) and is described in Raupach and Berne (2015). The rain gauges used were Précis Méchanique 3039 (and one 3029/1) instruments with a tipping volume of 0.1 mm. Quality control and maintenance of the model 3039 gauge network is described in Nord et al. (2017). Rain gauges can suffer from measurement errors at high temporal resolutions, particularly in light rain (e.g., Habib et al. 2001; Ciach 2003); we therefore use 1-h temporal resolution when testing in light rain conditions. Stations at Villeneuve are not included in this work because of timing inconsistencies between the disdrometers and rain gauges, and stations at Pradel-Vignes and Montbrun are not included here since, respectively, the collocated rain gauge was unreliable and there was no directly collocated rain gauge. To match Raupach and Berne (2015), DSD-based R was calculated assuming a sea level temperature of 15°C and a relative humidity of 0.95 at sea level. Parsivel records were subset to times for which no solid precipitation or hardware errors were detected by the instrument, the Parsivel-derived rain rate was greater than 0.025 mm h−1, and DSD-derived rain rate was greater than or equal to 0.1 mm h−1. The DSD correction factors used were retrained with reprocessed input datasets,1 and the rain gauge data used here are an updated data version.
We applied the reconstruction technique to incomplete DSDs recorded in France, using the double-moment normalized DSD model parameters derived earlier in this article using complete DSDs recorded in the United States. We thus assume large-scale invariance of the double-moment normalized DSD h(x). Figure 13 shows the distributions of h(x) recorded by the Parsivels (France) and the combined 2DVD and MPS data (United States). The differences at the small-drop end due to the drizzle mode are obvious, but the curves are similar for the large-drop end. This aligns with previous studies that showed enough stability in the double-moment normalized DSD for practical use (e.g., Raupach and Berne 2017a).
Comparison of h(x) distributions for HyMeX Parsivel data recorded in France (Parsivel) and complete DSDs recorded in the United States (2DVD+MPS), at 5-min and 1-h resolution. Here, h(x) was grouped in classes of x of width 0.02; center lines show class medians, while the colored bands show the class IQRs. The y axis has a log10 scale and h(x) = 0 is plotted on the x-axis line.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Figure 14 shows the mean and median DSDs across all stations and 5-min integrations for which R ≥ 0.1 mm h−1, for three data treatments. The first treatment was the “basic filter,” in which only drops with unlikely diameter and velocity combinations were removed (the “non-physical filter” in Raupach and Berne 2015). The second treatment was the full “DSD correction” method of Raupach and Berne (2015). The third treatment was “reconstructed,” the drizzle mode reconstruction of this paper, using two reference moments derived from the basic filter set. The same diameter classes as used for the complete DSDs above were used for these reconstructed DSDs. The plot clearly shows both the addition of small drops by the reconstruction of the drizzle mode, and the reduction in small-drop concentrations made by the previous DSD correction method.
Mean and median DSDs for the basic filter, the previous DSD correction, and reconstructed datasets at 5-min temporal resolution. The first two drop classes for Parsivel data are always zero and are not shown; for display purposes values less than 0.0001 mm−1 m−3 are set to zero, and values where N(D) = 0 are plotted on the x-axis line.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Comparisons of rain rates to nearby rain gauges were made at 5-min and 1-h resolutions. At 1-h (5 min) resolution, the minimum rain rate that could be captured by the rain gauges was 0.1 (1.2) mm h−1. Comparisons were made using only those times for which both the rain gauge and the disdrometer reported rainfall. To see the effect of different rain regimes, two extra comparisons were made at 1-h resolution, with thresholds on the rain gauge rain intensities. Rain labeled “not light” was when the gauge measured R ≥ 1.2 mm h−1, and “light” rain was when the gauge measured 0.1 ≥ R ≥ 1.2 mm h−1. Figure 15 shows distributions of relative bias for each station in the network, for the three treatment methods and these thresholds. Comparison statistics for the techniques at 5-min resolution are shown in Table 8.
Relative bias distributions on R by station and treatment method, and thresholded by rain gauge R, for the 2013 HyMeX special observation period data. Crosses show medians; bars show IQRs.
Citation: Journal of Applied Meteorology and Climatology 58, 1; 10.1175/JAMC-D-18-0156.1
Comparison statistics for three DSD treatment methods, comparing rain rate derived from DSDs to those measured by collocated rain gauges, at 5-min resolution and for gauge and Parsivel-based R ≥ 0.1 mm h−1. Dist is the approximate distance from the disdrometer to the reference rain gauge. M.RB is median relative bias. AB shows the relative bias on total rain amount.
The techniques were tested for 10 disdrometer/rain gauge pairs. At 1-h (5-min) resolution, across all records, the reconstructed DSDs produced total rain amounts that were closer to the reference amounts than the previous DSD correction at 6 (7) stations. In terms of median relative bias for all rain rates at 1-h (5 min) resolution, the previous DSD correction improved performance over the basic filter at 4 (3) stations, while the drizzle mode reconstruction performed better than the basic filter at 8 (6) of them, and the reconstruction performed better than the previous correction at 7 (9) stations. In the “not light” comparisons, the previous correction performed better than the basic filter at 3 stations, while performance of the drizzle mode reconstruction was better than the basic filter at 9 of them, and the reconstruction outperformed the previous correction at 7 stations.
As noted by Raupach and Berne (2015), the previous DSD correction produced DSDs that underestimated rain rates in light rain (less than 1.2 mm h−1). The reconstruction technique does not appear to suffer from this issue. For light rain at 1-h resolution, the previous DSD correction outperformed the basic filter at only 3 of stations, yet the reconstruction was better than the basic filter at 9 stations, and the reconstruction had lower median relative bias than the previous DSD correction at 7 stations.
The drizzle mode reconstruction improved the median relative bias on rain intensity over the basic filter and the previous correction routine for at least half the tested stations in all tested scenarios. It also has other advantages over the previous technique: the previous technique relied on a reference instrument that was specific to a time period and location, whereas the new reconstruction method uses a double-moment normalized DSD that can more reasonably be assumed to be constant. The reconstruction technique is more flexible, given that it can use any two well-measured moments of the DSD and can reconstruct DSD concentrations at any desired equivolume drop diameter. Reconstructed DSDs that contain the drizzle mode are also expected to be more representative of the full drop size spectra.
9. Conclusions
Commonly used disdrometers are generally unable to accurately measure concentrations of small drops in the DSD because of uncertainties in small-drop measurements, truncation of measured drop sizes, or both. Thurai et al. (2017) showed that by using collocated MPS and 2DVD disdrometers, more complete DSD spectra can be measured. These “complete” DSDs often exhibit a drizzle mode separated from the larger drop spectrum by a shoulder region. In this article, we have shown how, by using DSD moments that are well measured by standard disdrometers, a more complete DSD can be reconstructed.
The technique shown in this paper uses the double-moment DSD normalization method of Lee et al. (2004) to describe the DSD as a combination of two of its moments and a double-moment normalized DSD that is assumed to be invariant and describes the general shape of the distribution. We showed first that incomplete DSDs that do not include the drizzle mode still record higher-order moments of the DSD with acceptable accuracy, and chose DSD moments three and six to use with the reconstruction technique. A variety of methods for determining the best-fitting parameters of a generalized gamma model were tested, with the conclusion that fitting to median values of double-moment normalized DSD h(x), with small classes of x, produced the models with the least overall relative bias. The models varied with time resolution because of sampling effects. Performance of the reconstruction technique was tested by comparing reconstructed to complete DSDs. The reconstructed DSDs provided less biased low-order DSD moments than the incomplete DSDs, while maintaining the performance of higher-order DSD moments. Naturally, the technique introduces uncertainty compared to the “true” DSD, especially on low-order moments. We hypothesize that this uncertainty is caused by a combination of residual variability in the double-moment normalized DSD, the fact that moments of very different order from the input moment orders are expected to be less well captured by the double-moment normalization (Raupach and Berne 2017a), and uncertainty in the derivation of input moments from incomplete DSDs.
Reconstructions of DSDs in independent and distinctive “hurricane data” showed reasonable and similar performance, which gives confidence in the general applicability of the method. The reconstruction technique was tested against the previous “DSD correction” technique of Raupach and Berne (2015) by comparing DSD-derived rain rates to collocated rain gauges. The reconstruction was shown to improve on the performance of the previous technique for light rain, and to improve on both the previous correction and on noncorrected (basic filter only) DSD performance, for at least half the tested locations in each comparison. It also offers greater flexibility and a more representative estimate of the full DSD spectra than the previous technique. The DSD reconstruction produced reasonable rain-rate estimates for DSDs in France, even though it was trained using data from the United States, which provides further evidence of the stability of the double-moment normalized DSD of Lee et al. (2004).
There are many avenues for future work. For the sake of consistency, it would be desirable to find a parametric probability density function that fits well in the double-moment normalized DSD formulation. Yu et al. (2014) provide a normalization method that insists on a pdf shape function, but the problem of finding a pdf that can represent the drizzle mode’s high number of small drops and plateau region remains. Fitting of a model to the empirical double-normalized DSD is nontrivial and other methods should be investigated (e.g., Cohard and Pinty 2000; Delanoë et al. 2014). The choice of input moments to use for the technique can be freely adjusted according to the performance of the input instrument, and given the large variability and rareness of large drops (e.g., Jameson and Kostinski 2001; Gatlin et al. 2015; Jameson et al. 2015b; Tapiador et al. 2017) it would be useful to consider sampling effects (e.g., Smith and Kliche 2005) in relation to the suitability of very-high-order input moments. We found that the parameters of the best-fitting model of the double-moment DSD depend on the temporal resolution. The links between temporal and spatial sampling effects should be further investigated with regards to the parameters of the double-moment normalized DSD, so that the best parameters can be estimated for varying spatial resolutions, such as for radar volumes and satellite footprints. Residual variability in the double-moment normalized DSD should be investigated. Finally and importantly, more studies with combined MPS and 2DVD (or similar setups) are required, to study the properties and test the ubiquity of the DSD drizzle mode. The best diameter up to which to use MPS measurements is not always easy to determine and should be considered on a case-by-case basis.
Thurai et al. (2017) concluded that DSD retrieval algorithms that assume a constant value of μ may require reevaluation since common disdrometer measurements are likely to provide biased estimates of DSD mass spectrum width and characteristic drop diameter. These algorithms include the Global Precipitation Measurement mission dual-frequency phased-array precipitation radar (GPM DPR; e.g., Hou et al. 2014; Liao et al. 2014; Grecu et al. 2016). We have provided a method for “reconstructing” the drizzle mode in DSDs recorded using ordinary disdrometers. This method could be used to simulate large quantities of DSDs with drizzle modes and may be useful in reevaluation or testing of such algorithms. Thurai et al. (2017) call for studies using larger datasets, which this reconstruction technique may facilitate.
Acknowledgments
Parsivel and rain gauge data were obtained from the HyMeX program, sponsored by Grants MISTRALS/HyMeX, ANR-2011-BS56-027 FLOODSCALE project, Laboratoire de Meteorologie Physique, Domaine Olivier de Serres, CERMOSEM, OHMCV (Cevennes-Vivarais Mediterranean Hydrometeorological Observatory), and LTE-EPFL. HyMeX data URLs are listed in Nord et al. (2017). Other data may be requested from the authors. M. Thurai and V. N. Bringi acknowledge support by the U.S. National Science Foundation under Grant AGS-1431127. The authors thank two anonymous reviewers for their constructive reviews.
REFERENCES
Abel, S. J., and I. A. Boutle, 2012: An improved representation of the raindrop size distribution for single-moment microphysics schemes. Quart. J. Roy. Meteor. Soc., 138, 2151–2162, https://doi.org/10.1002/qj.1949.
Amoroso, L., 1925: Ricerche intorno alla curva dei redditi. Ann. Mat. Pura Appl., 2, 123–159, https://doi.org/10.1007/BF02409935.
Auf der Maur, A. N., 2001: Statistical tools for drop size distributions: Moments and generalized gamma. J. Atmos. Sci., 58, 407–418, https://doi.org/10.1175/1520-0469(2001)058<0407:STFDSD>2.0.CO;2.
Baldauf, M., A. Seifert, J. Förstner, D. Majewski, M. Raschendorfer, and T. Reinhardt, 2011: Operational convective-scale numerical weather prediction with the COSMO model: Description and sensitivities. Mon. Wea. Rev., 139, 3887–3905, https://doi.org/10.1175/MWR-D-10-05013.1.
Baumgardner, D., G. Kok, W. Dawson, D. O’Connor, and R. Newton, 2002: A new groundbased precipitation spectrometer: The meteorological particle sensor (MPS). 11th Conf. on Cloud Physics, Ogden, UT, Amer. Meteor. Soc., 8.6, https://ams.confex.com/ams/pdfpapers/41834.pdf.
Beard, K. V., 1976: Terminal velocity and shape of cloud and precipitation drops aloft. J. Atmos. Sci., 33, 851–864, https://doi.org/10.1175/1520-0469(1976)033<0851:TVASOC>2.0.CO;2.
Beard, K. V., 1985: Simple altitude adjustments to raindrop velocities for Doppler radar analysis. J. Atmos. Oceanic Technol., 2, 468–471, https://doi.org/10.1175/1520-0426(1985)002<0468:SAATRV>2.0.CO;2.
Berne, A., and W. Krajewski, 2013: Radar for hydrology: Unfulfilled promise or unrecognized potential? Adv. Water Resour., 51, 357–366, https://doi.org/10.1016/j.advwatres.2012.05.005.
Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar. Cambridge University Press, 662 pp.
Bringi, V. N., V. Chandrasekar, J. Hubbert, E. Gorgucci, W. L. Randeu, and M. Schoenhuber, 2003: Raindrop size distribution in different climatic regimes from disdrometer and dual-polarized radar analysis. J. Atmos. Sci., 60, 354–365, https://doi.org/10.1175/1520-0469(2003)060<0354:RSDIDC>2.0.CO;2.
Bringi, V. N., M. Thurai, and D. Baumgardner, 2018: Raindrop fall velocities from an optical array probe and 2-D video disdrometer. Atmos. Meas. Tech., 11, 1377–1384, https://doi.org/10.5194/amt-11-1377-2018.
Bumke, K., and J. Seltmann, 2012: Analysis of measured drop size spectra over land and sea. ISRN Meteor., 2012, 296575, https://doi.org/10.5402/2012/296575.
Ciach, G. J., 2003: Local random errors in tipping-bucket rain gauge measurements. J. Atmos. Oceanic Technol., 20, 752–759, https://doi.org/10.1175/1520-0426(2003)20<752:LREITB>2.0.CO;2.
Cohard, J., and J. Pinty, 2000: A comprehensive two-moment warm microphysical bulk scheme. I: Description and tests. Quart. J. Roy. Meteor. Soc., 126, 1815–1842, https://doi.org/10.1256/smsqj.56613.
Crane, R. K., 1971: Propagation phenomena affecting satellite communication systems operating in the centimeter and millimeter wavelength bands. Proc. IEEE, 59, 173–188, https://doi.org/10.1109/PROC.1971.8123.
Delanoë, J. M. E., A. J. Heymsfield, A. Protat, A. Bansemer, and R. J. Hogan, 2014: Normalized particle size distribution for remote sensing application. J. Geophys. Res. Atmos., 119, 4204–4227, https://doi.org/10.1002/2013JD020700.
Drobinski, P., and Coauthors, 2014: HyMeX: A 10-year multidisciplinary program on the Mediterranean water cycle. Bull. Amer. Meteor. Soc., 95, 1063–1082, https://doi.org/10.1175/BAMS-D-12-00242.1.
Ducrocq, V., and Coauthors, 2014: HyMeX-SOP1: The field campaign dedicated to heavy precipitation and flash flooding in the northwestern Mediterranean. Bull. Amer. Meteor. Soc., 95, 1083–1100, https://doi.org/10.1175/BAMS-D-12-00244.1.
Field, P. R., R. J. Hogan, P. R. A. Brown, A. J. Illingworth, T. W. Choularton, and R. J. Cotton, 2005: Parametrization of ice-particle size distributions for mid-latitude stratiform cloud. Quart. J. Roy. Meteor. Soc., 131, 1997–2017, https://doi.org/10.1256/qj.04.134.
Foote, G. B., and P. S. D. Toit, 1969: Terminal velocity of raindrops aloft. J. Appl. Meteor., 8, 249–253, https://doi.org/10.1175/1520-0450(1969)008<0249:TVORA>2.0.CO;2.
Gatlin, P. N., M. Thurai, V. N. Bringi, W. Petersen, D. Wolff, A. Tokay, L. Carey, and M. Wingo, 2015: Searching for large raindrops: A global summary of two-dimensional video disdrometer observations. J. Appl. Meteor. Climatol., 54, 1069–1089, https://doi.org/10.1175/JAMC-D-14-0089.1.
Grecu, M., W. S. Olson, S. J. Munchak, S. Ringerud, L. Liao, Z. Haddad, B. L. Kelley, and S. F. McLaughlin, 2016: The GPM combined algorithm. J. Atmos. Oceanic Technol., 33, 2225–2245, https://doi.org/10.1175/JTECH-D-16-0019.1.
Gunn, R., and G. D. Kinzer, 1949: The terminal velocity of fall for water droplets in stagnant air. J. Meteor., 6, 243–248, https://doi.org/10.1175/1520-0469(1949)006<0243:TTVOFF>2.0.CO;2.
Habib, E., W. F. Krajewski, and A. Kruger, 2001: Sampling errors of tipping-bucket rain gauge measurements. J. Hydrol. Eng., 6, 159–166, https://doi.org/10.1061/(ASCE)1084-0699(2001)6:2(159).
Hachani, S., B. Boudevillain, G. Delrieu, and Z. Bargaoui, 2017: Drop size distribution climatology in Cévennes-Vivarais region, France. Atmosphere, 8, https://doi.org/10.3390/atmos8120233.
Hou, A. Y., and Coauthors, 2014: The Global Precipitation Measurement mission. Bull. Amer. Meteor. Soc., 95, 701–722, https://doi.org/10.1175/BAMS-D-13-00164.1.
Illingworth, A. J., and T. M. Blackman, 2002: The need to represent raindrop size spectra as normalized gamma distributions for the interpretation of polarization radar observations. J. Appl. Meteor., 41, 286–297, https://doi.org/10.1175/1520-0450(2002)041<0286:TNTRRS>2.0.CO;2.
Jaffrain, J., A. Studzinski, and A. Berne, 2011: A network of disdrometers to quantify the small-scale variability of the raindrop size distribution. Water Resour. Res., 47, W00H06, https://doi.org/10.1029/2010WR009872.
Jameson, A. R., and A. B. Kostinski, 2001: What is a raindrop size distribution? Bull. Amer. Meteor. Soc., 82, 1169–1177, https://doi.org/10.1175/1520-0477(2001)082<1169:WIARSD>2.3.CO;2.
Jameson, A. R., M. L. Larsen, and A. B. Kostinski, 2015a: Disdrometer network observations of fine scale spatial–temporal clustering in rain. J. Atmos. Sci., 72, 1648–1666, https://doi.org/10.1175/JAS-D-14-0136.1.
Jameson, A. R., M. L. Larsen, and A. B. Kostinski, 2015b: On the variability of drop size distributions over areas. J. Atmos. Sci., 72, 1386–1397, https://doi.org/10.1175/JAS-D-14-0258.1.
Jameson, A. R., M. L. Larsen, and A. B. Kostinski, 2016: An example of persistent microstructure in a long rain event. J. Hydrometeor., 17, 1661–1673, https://doi.org/10.1175/JHM-D-15-0180.1.
Johnson, D. B., and K. V. Beard, 1984: Oscillation energies of colliding raindrops. J. Atmos. Sci., 41, 1235–1241, https://doi.org/10.1175/1520-0469(1984)041<1235:OEOCR>2.0.CO;2.
Joss, J., and E. G. Gori, 1978: Shapes of raindrop size distributions. J. Appl. Meteor., 17, 1054–1061, https://doi.org/10.1175/1520-0450(1978)017<1054:SORSD>2.0.CO;2.
Larsen, M. L., and M. Schönhuber, 2018: Identification and characterization of an anomaly in two-dimensional video disdrometer data. Atmosphere, 9, https://doi.org/10.3390/atmos9080315.
Lee, C. K., G. W. Lee, I. Zawadzki, and K.-E. Kim, 2009: A preliminary analysis of spatial variability of raindrop size distributions during stratiform rain events. J. Appl. Meteor. Climatol., 48, 270–283, https://doi.org/10.1175/2008JAMC1877.1.
Lee, G. W., I. Zawadzki, W. Szyrmer, D. Sempere-Torres, and R. Uijlenhoet, 2004: A general approach to double-moment normalization of drop size distributions. J. Appl. Meteor., 43, 264–281, https://doi.org/10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2.
Lee, G. W., A. W. Seed, and I. Zawadzki, 2007: Modeling the variability of drop size distributions in space and time. J. Appl. Meteor. Climatol., 46, 742–756, https://doi.org/10.1175/JAM2505.1.
Leijnse, H., R. Uijlenhoet, and J. N. M. Stricker, 2007: Rainfall measurement using radio links from cellular communication networks. Water Resour. Res., 43, W03201, https://doi.org/10.1029/2006WR005631.
Liao, L., R. Meneghini, and A. Tokay, 2014: Uncertainties of GPM DPR rain estimates caused by DSD parameterizations. J. Appl. Meteor. Climatol., 53, 2524–2537, https://doi.org/10.1175/JAMC-D-14-0003.1.
Löffler-Mang, M., and J. Joss, 2000: An optical disdrometer for measuring size and velocity of hydrometeors. J. Atmos. Oceanic Technol., 17, 130–139, https://doi.org/10.1175/1520-0426(2000)017<0130:AODFMS>2.0.CO;2.
Marshall, J. S., R. C. Langille, and W. M. Palmer, 1947: Measurement of rainfall by radar. J. Meteor., 4, 186–192, https://doi.org/10.1175/1520-0469(1947)004<0186:MORBR>2.0.CO;2.
McFarquhar, G. M., 2004: A new representation of collision-induced breakup of raindrops and its implications for the shapes of raindrop size distributions. J. Atmos. Sci., 61, 777–794, https://doi.org/10.1175/1520-0469(2004)061<0777:ANROCB>2.0.CO;2.
Messer, H., A. Zinevich, and P. Alpert, 2006: Environmental monitoring by wireless communication networks. Science, 312, 713–713, https://doi.org/10.1126/science.1120034.
Meyers, M. P., R. L. Walko, J. Y. Harrington, and W. R. Cotton, 1997: New RAMS cloud microphysics parameterization. Part II: The two-moment scheme. Atmos. Res., 45, 3–39, https://doi.org/10.1016/S0169-8095(97)00018-5.
Miriovsky, B. J., and Coauthors, 2004: An experimental study of small-scale variability of radar reflectivity using disdrometer observations. J. Appl. Meteor., 43, 106–118, https://doi.org/10.1175/1520-0450(2004)043<0106:AESOSV>2.0.CO;2.
Nord, G., and Coauthors, 2017: A high space–time resolution dataset linking meteorological forcing and hydro-sedimentary response in a mesoscale Mediterranean catchment (Auzon) of the Ardèche region, France. Earth Syst. Sci. Data, 9, 221–249, https://doi.org/10.5194/essd-9-221-2017.
Park, S.-G., H.-L. Kim, Y.-W. Ham, and S.-H. Jung, 2017: Comparative evaluation of the OTT PARSIVEL2 using a collocated two-dimensional video disdrometer. J. Atmos. Oceanic Technol., 34, 2059–2082, https://doi.org/10.1175/JTECH-D-16-0256.1.
Peel, M. C., B. L. Finlayson, and T. A. McMahon, 2007: Updated world map of the Köppen–Geiger climate classification. Hydrol. Earth Syst. Sci., 11, 1633–1644, https://doi.org/10.5194/hess-11-1633-2007.
Petty, G. W., and W. Huang, 2011: The modified gamma size distribution applied to inhomogeneous and nonspherical particles: Key relationships and conversions. J. Atmos. Sci., 68, 1460–1473, https://doi.org/10.1175/2011JAS3645.1.
Prat, O. P., A. P. Barros, and C. R. Williams, 2008: An intercomparison of model simulations and VPR estimates of the vertical structure of warm stratiform rainfall during TWP-ICE. J. Appl. Meteor. Climatol., 47, 2797–2815, https://doi.org/10.1175/2008JAMC1801.1.
Pruppacher, H. R., and J. D. Klett, 2000: Microphysics of Clouds and Precipitation. 2nd ed. Kluwer Academic Publishers, 954 pp.
Raupach, T. H., and A. Berne, 2015: Correction of raindrop size distributions measured by Parsivel disdrometers, using a two-dimensional video disdrometer as a reference. Atmos. Meas. Tech., 8, 343–365, https://doi.org/10.5194/amt-8-343-2015; Corrigendum, https://doi.org/10.5194/amt-8-343-2015-corrigendum.
Raupach, T. H., and A. Berne, 2016: Small-scale variability of the raindrop size distribution and its effect on areal rainfall retrieval. J. Hydrometeor., 17, 2077–2104, https://doi.org/10.1175/JHM-D-15-0214.1.
Raupach, T. H., and A. Berne, 2017a: Invariance of the double-moment normalized raindrop size distribution through 3D spatial displacement in stratiform rain. J. Appl. Meteor. Climatol., 56, 1663–1680, https://doi.org/10.1175/JAMC-D-16-0316.1.
Raupach, T. H., and A. Berne, 2017b: Retrieval of the raindrop size distribution from polarimetric radar data using double-moment normalisation. Atmos. Meas. Tech., 10, 2573–2594, https://doi.org/10.5194/amt-10-2573-2017.
Rosenfeld, D., and C. W. Ulbrich, 2003: Cloud microphysical properties, processes, and rainfall estimation opportunities. Radar and Atmospheric Science: A Collection of Essays in Honor of David Atlas, Meteor. Monogr., No. 52, Amer. Meteor. Soc., 237–258, https://doi.org/10.1007/978-1-878220-36-3_10.
Schleiss, M., and A. Berne, 2010: Identification of dry and rainy periods using telecommunication microwave links. IEEE Geosci. Remote Sens. Lett., 7, 611–615, https://doi.org/10.1109/LGRS.2010.2043052.
Schönhuber, M., G. Lammer, and W. L. Randeu, 2008: The 2D-video-distrometer. Precipitation: Advances in Measurement, Estimation and Prediction, S. Michaelides, Ed., Springer, 3–31, https://doi.org/10.1007/978-3-540-77655-0_1.
Sekhon, R. S., and R. Srivastava, 1971: Doppler radar observations of drop-size distributions in a thunderstorm. J. Atmos. Sci., 28, 983–994, https://doi.org/10.1175/1520-0469(1971)028<0983:DROODS>2.0.CO;2.
Seliga, T., and V. Bringi, 1976: Potential use of radar differential reflectivity measurements at orthogonal polarizations for measuring precipitation. J. Appl. Meteor., 15, 69–76, https://doi.org/10.1175/1520-0450(1976)015<0069:PUORDR>2.0.CO;2.
Sempere-Torres, D., J. M. Porrà, and J.-D. Creutin, 1994: A general formulation for raindrop size distribution. J. Appl. Meteor., 33, 1494–1502, https://doi.org/10.1175/1520-0450(1994)033<1494:AGFFRS>2.0.CO;2.
Smith, P. L., and D. V. Kliche, 2005: The bias in moment estimators for parameters of drop size distribution functions: Sampling from exponential distributions. J. Appl. Meteor., 44, 1195–1205, https://doi.org/10.1175/JAM2258.1.
Stacy, E. W., 1962: A generalization of the gamma distribution. Ann. Math. Stat., 33, 1187–1192, https://doi.org/10.1214/aoms/1177704481.
Straub, W., K. D. Beheng, A. Seifert, J. Schlottke, and B. Weigand, 2010: Numerical investigation of collision-induced breakup of raindrops. Part II: Parameterizations of coalescence efficiencies and fragment size distributions. J. Atmos. Sci., 67, 576–588, https://doi.org/10.1175/2009JAS3175.1.
Tapiador, F. J., R. Checa, and M. De Castro, 2010: An experiment to measure the spatial variability of rain drop size distribution using sixteen laser disdrometers. Geophys. Res. Lett., 37, https://doi.org/10.1029/2010GL044120.
Tapiador, F. J., and Coauthors, 2017: On the optimal measuring area for pointwise rainfall estimation: A dedicated experiment with 14 laser disdrometers. J. Hydrometeor., 18, 753–760, https://doi.org/10.1175/JHM-D-16-0127.1.
Testud, J., S. Oury, R. A. Black, P. Amayenc, and X. Dou, 2001: The concept of “normalized” distribution to describe raindrop spectra: A tool for cloud physics and cloud remote sensing. J. Appl. Meteor., 40, 1118–1140, https://doi.org/10.1175/1520-0450(2001)040<1118:TCONDT>2.0.CO;2.
Thurai, M., and V. N. Bringi, 2018: Application of the generalized gamma model to represent the full rain drop size distribution spectra. J. Appl. Meteor. Climatol., 57, 1197–1210, https://doi.org/10.1175/jamc-d-17-0235.1.
Thurai, M., W. Petersen, A. Tokay, C. Schultz, and P. Gatlin, 2011: Drop size distribution comparisons between Parsivel and 2-D video disdrometers. Adv. Geosci., 30, 3–9, https://doi.org/10.5194/adgeo-30-3-2011.
Thurai, M., P. Gatlin, V. N. Bringi, W. Petersen, P. Kennedy, B. Notaroš, and L. Carey, 2017: Toward completing the raindrop size spectrum: Case studies involving 2D-video disdrometer, droplet spectrometer, and polarimetric radar measurements. J. Appl. Meteor. Climatol., 56, 877–896, https://doi.org/10.1175/JAMC-D-16-0304.1.
Thurai, M., V. Bringi, P. Gatlin, W. Petersen, and M. Wingo, 2018: Why the generalized gamma?—An answer based on measurements with meteorological particle spectrometer and 2D video disdrometer. ERAD2018: 10th European Conf. on Radar in Meteorology and Hydrology, Ede-Wageningen, Netherlands, KNMI and Wageningen University, 051, https://doi.org/10.18174/454537.
Tokay, A., and P. G. Bashor, 2010: An experimental study of small-scale variability of raindrop size distribution. J. Appl. Meteor. Climatol., 49, 2348–2365, https://doi.org/10.1175/2010JAMC2269.1.
Tokay, A., A. Kruger, and W. Krajewski, 2001: Comparison of drop size distribution measurements by impact and optical disdrometers. J. Appl. Meteor., 40, 2083–2097, https://doi.org/10.1175/1520-0450(2001)040<2083:CODSDM>2.0.CO;2.
Tokay, A., W. A. Petersen, P. Gatlin, and M.