1. Introduction
Accurate estimates of the particle size distribution (PSD) of hydrometeors are fundamental for many hydrological studies at basin-scale resolution. For instance, for soil erosion analyses (Winder and Paulson 2012; Salles et al. 2002; Petan et al. 2010; Lim et al. 2015), water resources assessment (Lee and Zawadzki 2006; Ten Veldhuis et al. 2014), and remote sensing of precipitation from space (Bringi et al. 2003; Gorgucci et al. 2001; Thurai et al. 2010; Liao et al. 2014), the PSD is important. Estimates of the PSD are also required to objectively characterize the microphysics of rain and thus to parameterize those in numerical models. Such parameterizations are a fundamental component of numerical weather prediction (NWP) models, regional climate models (RCMs), and global circulation/climate models (GCMs), which model precipitation to provide input or are coupled with hydrological models.
Enhancing the atmospheric component of hydrological models via improved parameterizations is therefore beneficial for applications such as water resources assessment and hydropower operations. Within this goal, precise characterization of rain microphysics is key for climate and weather modeling and thus for hydrometeorological studies. Such characterization can be achieved by analyzing the evolution in time of the drop size distribution (DSD) of hydrometeors, which can be measured at ground level using disdrometers.
The empirical modeling of the PSD is based on the choice of the PSD form and parameters. In the case of rain (DSD), two approaches compete: the use of the number concentration of drops NT decoupled from the shape of the distribution (the [NT, E(D), E(D2)] or {NT, E(D), E[log(D)]} models), and the (N0, Λ, μ) model, which embeds in N0 both the shape of the distribution and the number concentration of drops (see Table 1 for a definition of the parameters).
Definitions and method of calculating the PSD variables and the radiometric quantities.
Here we use a comprehensive dataset of disdrometer measurements to show that the NT-based approaches allow a more precise characterization of the DSD and also a physically based modeling of the microphysical processes of rain. Parameter NT is analytically independent of the shape of the PSD {parameterized by E(D) and either E(D2) or E[log(D)]} and can therefore be used as an independent variable in the equations {the other free parameter being either E(D) and E(D2), or E(D) and E[log(D)]}.
Since Chandrasekar and Bringi (1988), several authors have continued to argue that decoupling NT in the PSD is required in order to keep the modeling physical (Jameson et al. 1999; Tapiador et al. 2014). While the topic is not critical for remote sensing, as the retrieval algorithms depend on iterative, empirical optimization algorithms, the nonphysical character of the modeling is indeed an issue in the case of the parameterizations of the microphysics used in numerical models (NWPs, RCMs, and GCMs). Those models attempt to follow a mechanistic, first-principles-based approach to characterize the dynamics of precipitation, so it is desirable to keep all the many components of such complex tools as consistent as possible.
2. Data
Laser disdrometers (from “distribution of drops meters”) measure the PSD of liquid and solid hydrometeors using a collimated, flat laser beam of negligible width and small cross-sectorial area. The actual sampling size is in the order of tens of square centimeters. For instance, Parsivel-1 disdrometers have a 180 mm × 30 mm area (54 cm2), Thies’ disdrometer (Frasson et al. 2011) has 228 mm × 20 mm (46 cm2), and the optical spectro-pluviometer (OSP; Hauser et al. 1984) has 250 mm × 40 mm (100 cm2). Another type of disdrometer, the Joss–Waldvogel disdrometer (JWD; Löffler-Mang and Joss 2000) has an area of about 50 cm2, whereas two-dimensional video disdrometers (2DVDs; Kruger and Krajewski 2002) are in the range of 100 cm2. Infrared-LED-based ODM 470 disdrometers (Lempio et al. 2007) have a 26.4 cm2 cross-section area.
Apart from the limited area, disdrometers present several problems and limitations, namely, uncertainties in measuring very small or very large drops because of noise and a small catching area, respectively, or the existence of “margin fallers” (cf. Lee and Zawadzki 2005a; Jaffrain and Berne 2011; Raupach and Berne 2015; Yuter et al. 2006; You and Lee 2015). Measuring solid precipitation with disdrometers, on the other hand, presents its own challenges (Battaglia et al. 2010).
Notwithstanding those issues, measurements of the DSD from disdrometers are still considered as the most direct estimates of the DSD (Tapiador et al. 2017): the instrument has proven crucial for a detailed understanding of many aspects of hydrological science, such as analyzing the microstructure of rain events (Jameson et al. 2016; Ignaccolo et al. 2009) and the spatiotemporal structure of those in three dimensions (Gires et al. 2014, 2015), analyzing the variability of the DSD across different measurement scales (Lee and Zawadzki 2005b, Raupach and Berne 2016, Tapiador et al. 2010), characterizing the DSD in flood events (Friedrich et al. 2016), delineating flood areas (Beauchamp et al. 2015), assisting radars to estimate the spatial correlation of the DSD (Bringi et al. 2015, Thurai et al. 2012), developing new algorithms (Anagnostou et al. 2013), verifying quantitative precipitation estimates from weather radars (Bringi et al. 2011, Gourley et al. 2009), performing hydrological validation of nowcastings (Berenguer et al. 2005), and carrying out error analysis at catchment scale (Jordan et al. 2003).
a. Instrumental setup
Data from two networks of first-generation OTT Parsivel (Löffler-Mang and Joss 2000) disdrometers were used in this work. The first [the Hydrological Cycle in Mediterranean Experiment (HyMeX) dataset] was deployed in Ardèche, France, in the autumns of 2012 and 2013, as part of HyMeX (Ducrocq et al. 2014; Drobinski et al. 2014; Nord et al. 2017). There were seven (nine) instruments deployed in 2012 (2013) over a region of about 8 × 13 km2. This study region is subject to high-intensity Mediterranean rainfall, especially in the autumn months (e.g., Ricard et al. 2012).
The second network (the Payerne dataset) was deployed on the Swiss Plateau near Payerne, Switzerland, from February to July 2014. Five Parsivels were used to collect DSDs with a maximum interinstrument distance of 23 km.
Overall, the database contains 29 861 five-minute rainfall measurements, of which approximately 39% were in Payerne and the remaining 61% in HyMeX. On average, each station recorded 178 h of rainfall. The climatological differences between the two regions are briefly discussed in Wolfensberger et al. (2016). The DSDs collected in these campaigns represent primarily stratiform rain (Raupach and Berne 2017). Figure 1 gathers the time series of the episodes, including an indication of the differences in daily totals due to the spatial variability of rain, as the disdrometers are separated by several kilometers.
Time series of the disdrometer data used for this study: the HyMeX 2012 and 2013 and the Payerne 2014 campaigns. The graphs show the daily rain accumulations in each of the datasets. The dots show the mean of nonzero accumulations, and the bars are the minimum and maximum nonzero accumulations across the different stations in each network. Note that the dispersion is due to the spatial variability of the rain, as the disdrometers are kilometers apart.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0154.1
b. Data processing
The Parsivel measurement integration time was 30 s. The data were resampled to 1-, 5-, and 10-min temporal resolutions by finding the average DSDs over these periods. The DSDs were corrected using the method of Raupach and Berne (2015), with updated correction factors derived using reprocessed disdrometer data from the HyMeX campaign. To ensure that the data were the best quality possible, they were subset to times for which the estimated rain rate was greater than 0.1 mm h−1, no errors were reported by the instrument, and no solid precipitation was recorded.
DSD parameters were calculated for each 5-min corrected DSD measured by the Parsivel disdrometers. Table 1 summarizes the equations used. For rain rates, the terminal velocities of Beard (1976) were used, together with station altitudes and latitudes and an assumed sea level temperature of 15°C and sea level humidity of 0.95. Integral properties were calculated over truncated classes of equivolume diameters, from 0.2495 to 7 mm (Raupach and Berne 2015).
3. Methods
We compared three approaches to characterize the PSD and thus the microphysics of rain: the maximum entropy (MaxEnt) model in (10), the MaxEnt–gamma model in (12), and the traditional N0 model in (1).
a. PSD modeling: The MaxEnt model
In Tapiador et al. (2014), it was shown that changes in (m, σ2), that is, 
The traditional distributions (gamma, normal, lognormal, beta, Burr, Erlang, Pearson’s, etc.) are just particular cases of (8) for a given choice of fi(D), but there is no need to assume any parametric form in the modeling if the fi(D) are too complex to derive an analytical form.
Whereas higher moments such as E(D3) (skewness) or E(D4) (kurtosis), or effective means given as quotients [such as E(D4)/E(D3)], are less sensitive to outliers, they cannot be easily related with microphysical processes, which should be well captured in the numerical modeling. In addition, capturing dominant microphysical processes is a key component to explore the precipitation processes.
Moreover, being less sensitive is not necessarily an asset when parameterizing the microphysics of precipitation in a numerical model. In this particular field, the methods and needs are different than those in measuring drop size distributions with disdrometers or estimating the DSDs with satellites and radars. In the latter case, there are empirical uncertainties and limitations that are absent in the former, the development of codes for parameterizing the microphysics of rain.
It should be noted here that in (10) the zeroth Lagrange parameter is analytically related with the other two by λ0 = (λ2 − 1)logλ1 + log[Γ(1 − λ2)] (see appendix). Therefore, all three methods under comparison have the same number of parameters.
b. PSD modeling: The MaxEnt–gamma model
c. Parameter estimation
1) Parameter estimation for the MaxEnt–gamma PSD model
The second MaxEnt model assumes that we can get a robust estimate of the arithmetic and logarithm means of the drop diameters. In this case, there is also an analytical technique to derive the parameters.
This approach for parameter estimation was first described by Hory and Martin (2002) for 
2) Parameter estimation for the MaxEnt PSD model
Instead of the central moments, this model uses the first two raw moments (Papoulis 1984) of the distribution, namely, E(D) and E(D2). Qualitatively, E(D) (the mean) is close to the peak of the distribution, while E(D2) tells us about the spread of the drop sizes and is related to both the second central moment (the variance σ2) and to the mean E(D) by E(D2) = σ2 + E(D)2. Therefore, this model is conceptually equivalent to that used in Tapiador et al. (2014).
In contrast to the parameter estimation for the gamma modeling in the previous section, here there is not an analytical form to calculate the parameters, so it is necessary to resort to numerical methods. Thus, we used a Newton–Raphson approach to find the Lagrangian multipliers in (10), following the method described in Zellner and Highfield (1988) and implemented by Woodbury (2004). These results are tagged as MaxEnt or as the “[NT, E(D), E(D2)] model” hereafter.
3) Parameter estimation for the MLE model
The estimation of the parameters in the (N0, Λ, μ)-based model was done by numerical methods by fitting a gamma distribution to each DSD using the maximum likelihood estimation (MLE) method and (modified) R code of Johnson et al. (2014), which allows for truncated and classed DSDs. This is the standard model used today, so the MaxEnt-based models are compared against it.
d. Calculation of the radiometric quantities
Scattering quantities were calculated from the DSDs using the T-matrix code of Mishchenko and Travis (1998), radar frequencies of 13.6 GHz (Ku band) and 35.55 GHz (Ka band), incidence angles of 90° (vertical) and 0° (horizontal), an assumed sea level temperature of 15°C, and temperatures at station altitudes using a standard atmospheric lapse rate of 6.5°C km−1. Raindrop axes were calculated using the formula of Thurai et al. (2007), and drops were assumed to have canting angles described by a normal distribution with a mean of zero and standard deviation of 6°.
The relationship between the reflectivity Z and rainfall rate R is the basis of rainfall estimation using single-polarization radar. The a prefactor and the b exponent in the Z = aRb formulas are derived by a log–log regression of coincident estimates of Z and R over a period (Battan 1973; Marshall et al. 1955). For dual polarimetric instruments, the PSD is still required, as the differential reflectivity Zdr and, to a lesser extent, the specific differential phase Kdp still depend on the PSD, and assumptions on the shape of the distribution have to be made (Gorgucci et al. 2002; Chandrasekar and Bringi 1988; Xiao and Chandrasekar 1997).
4. Results
a. Parameter independence
The [NT, E(D), E(D2)] (MaxEnt), {NT, E(D), E[log(D)]} (MaxEnt–gamma), and the (N0, Λ, μ) models were compared to determine the independence of the shape and location parameters with the number concentration in the case of real rainfall. As discussed above, NT is not mathematically correlated with either E(D), E(D2), or E[log(D)], so any relationship between any of these variables should arise from physical processes occurring in nature. The same does not apply to the (N0, Λ, μ). In this case, as in the equivalent (Nw, Dm, μ) model, there is an embedded correlation in N0 (i.e., the parameters are intrinsically not independent).
Figure 2 clearly exposes the issue. The figure gathers all the data in the HyMeX and Payerne disdrometer database. Log(N0) is highly correlated with Λ (r2 = 0.99) and with μ (r2 = 0.94). This means that the number concentration of drops and the shape of the distribution (i.e., the number of drops per size category) are inextricably linked, as one can expect from the modeling above [(5)]. The only reason the correlations are not perfect is because of truncation errors and limitations in the measurements and parameter fitting.
Cross correlations between the parameters in the (N0, Dm, μ), {NT, E(D), E[log(D)]}, and [NT, E(D), E(D2)] models, for 5-min integrated PSD. Note that the E(D) vs log(NT) scatterplot is repeated for the sake of consistency.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0154.1
The high correlation is also apparent in the Λ versus μ scatterplot, with yields a 0.95 r2 correlation. However, in this case there is nothing in the mathematical definition of both parameters that would anticipate such a result. Having found a high correlation between both parameters means that the spread of the PSD depends on the mean, an observation that everyone working with disdrometers is familiar with: the larger the tail of the distribution is (i.e., the larger the biggest drops are), the larger the mean drop size is (i.e., a longer tail pushes the mean toward the right).
In the case of the [NT, E(D), E(D2)] model, Fig. 2 shows that E(D) and E(D2) are also almost perfectly correlated in observed rainfall (r2 = 0.99 for a 5-min integration). However, and crucially, NT is now clearly uncorrelated with both E(D) and E(D2) (r2 = −0.19 and −0.17, respectively, for a 5-min integration). This is logically consistent since one can observe exactly the same shape of the PDF for vastly different numbers of drops. The implication for the retrieval and modeling of the PSD is that two parameters, NT and either E(D) or E(D2), suffice to determine the PSD, at least for the cases in the database. One is simply the number concentration of drops, thus directly related to the number of drops, and the other is related to the shape and location of the distribution of drops per diameter (DSD), which is a proper PDF.
Figure 2 also shows an analysis of the cross-correlation between the three variables in the (Nw, Dm, μ) model. Here, there is also a strong correlation between Dm and μ (r2 = −0.80 for a 5-min integration). But there is also a significant correlation between Nw and both Dm and μ (r2 = −0.44 and 0.31, respectively, for a 5-min integration). That confirms that the number of drops in the (Nw, Dm, μ) modeling is empirically dependent on the shape of the distribution [not surprising considering (6)], as happens in the (N0, Λ, μ) model. Such dependence is not a desirable feature either for the retrieval of the PSD from remote sensing measurements or for parameterizations in the microphysical modules of weather and climate models. This observation is relevant since the dynamics of the microphysics of rainfall were first described in terms of N0 and Λ parameters [by Marshall and Palmer (1948); later, Ulbrich (1983, 1985) developed a formulation consistent with the 1948 work].
A major problem of characterizing the microphysics of rain in such a way is that N0 is not physical at all, so the result can be intuitively correct but not suitable for implementation in NWP, RCMs, and GCMs unless one is satisfied with having nonphysical variables in a numerical model. Actually, N0 was a fixed value in Marshall and Palmer (1948), which did not have any microphysical interpretation. Moreover, and as described above, N0 is correlated with variables that are supposed to be independent, and that fact plagues the code of any microphysical parameterization (mp) following that approach.
b. Validation using the disdrometer data
There are also compelling practical reasons to favor the [NT, E(D), E(D2)] or {NT, E(D), log[E(D)]} over the (N0, Λ, μ) approach in the parameterization of models. Figure 3 shows the validation of the three approaches with disdrometer measurements (1-min integrations). The white, leftmost box plot in each panel corresponds with the actual estimates of total water content from the disdrometers (ground truth). The standard (N0, Λ, μ)-based modeling, which is parameterized using the MLE method, featured as the green box plots, and the two alternatives within the NT approach, the MaxEnt {with [NT, E(D), E(D2)]} and the MaxEnt–gamma (with {NT, E(D), E[log(D)]}) models are shown in the red and blue box plots, respectively.
Comparison of disdrometer measurements (white, leftmost boxplot) of total water content with the standard (N0, Λ, μ)-based MLE modeling (green) and two alternatives of the [NT, E(D), E(D2)] modeling: a two-moment maximum entropy modeling assuming a gamma distribution PDF (MaxEnt–gamma, blue boxplot) and a two-moment maximum entropy estimate using two crude moments (MaxEnt, red boxplot). Each individual plot in a panel represents a disdrometer station (location) in the DSD database. Both maximum entropy parameterizations are consistently closer to the actual measurements given by the disdrometers than the (N0, Λ, μ)-based approach. The legend of the individual plots is as in Fig. 5.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0154.1
Each individual plot in the panels of Fig. 3 represents a location from the database and a period of observation. The figure shows that the [NT, E(D), E(D2)] and {NT, E(D), E[log(D)]} parameterizations are consistently closer to the actual measurements than those from the (N0, Λ, μ)-based approach.
Figure 4 depicts a particular case, a single 1-min PSD, to illustrate the differences that can be expected. There, it is clear that the (N0, Λ, μ) approach generates a smoother, less precise fit to the actual measurements than the NT-based models. The MaxEnt models correctly identify the mode and do not overestimate the tail of the distribution, resulting in a better fit. The [NT, E(D), E(D2)] model gets the mode right whereas the other two do not perform as well.
An example of how the (N0, Λ, μ) MLE modeling (green line) and the maximum entropy models (red and blue lines) compare with actual disdrometer estimates of the PSD (black line, nominal truth) for a particular data sample (random) from the database. The [NT, E(D), E(D2)] model correctly identifies the mode and does not overestimate the tail of the distribution.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0154.1
Figure 4 is for just one case, but Fig. 5 confirms that the NT-based models compare better with observations than the traditional model for all the cases. The figure gathers the global statistics for all the empirical databases. It is clear then that the (N0, Λ, μ) model consistently overestimates the hydrological variable of interest (total water content W here). The NT-based models, on the other hand, behave better in the large drops section of the spectrum, resulting in the statistics of MaxEnt–gamma model being almost indistinguishable from the reference measurements (truth) in terms of mean, median, quartiles, and extreme values. Note that the extreme values of the (N0, Λ, μ) model are more than 15% off from the actual estimates given by the disdrometers. The fit for reflectivity Z of the MaxEnt–gamma model is also better in terms of mean, median, and quartiles.
As in Fig. 3 for all the cases in the database. (top) The (N0, Λ, μ) MLE modeling consistently overestimates the hydrological variable of interest (water content W). The two maximum entropy models, on the other hand, behave better in the large drops section of the spectrum, resulting in the statistics of MaxEnt being undistinguishable from the reference measurements (truth) in terms of mean, median, quartiles, and extreme values. The MaxEnt–gamma is as accurate as the nonparametric, numerical MaxEnt, and both are better than the N0-based modeling. Note that the extreme values of the (N0, Λ, μ) model are more than 15% off from the actual estimates whereas for MaxEnt the differences are negligible. (bottom) The fit for reflectivity Z of the MaxEnt–gamma model is also better in terms of mean, median, and quartiles.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0154.1
c. Objective characterization of the microphysics of rain
A direct application of the NT-based modeling is in the field of microphysics parameterizations for NWP models, RCMs, and GCMs. The modeling provides a neat account of the main processes operating in clouds. Thus, leaving aside the advection of drops dNT/dt, the PSD varies if drops (NT) are created or destroyed, if the mean size of the drops E(D) increases or decreases, and if the spread in the size distribution of drops E(D2) {or E[log(D)]} widens or narrows. Therefore, in principle it is possible to define 23 possible changes in the PSD (Table 2). The eight possible cases cover all the processes currently found in the parameterizations of rain microphysics models. Note that the processes in Table 2 are consistent with those suggested in Tapiador et al. (2014), but a further refinement is now provided by considering the changes in NT.
Objective characterization of the microphysics in terms of changes in the [NT, E(D), E(D2)] space (the same applies to the {NT, E(D), E[log(D)]} space). The characterization is consistent and extends that in Tapiador et al. (2014) by now accounting for changes in NT. Bold typeface indicates the prevalent microphysical processes as observed in the HyMeX and Payerne databases (see Fig. 7).
Take for instance the 
A slightly more complex case is when 
In evaporation drops are destroyed so all three parameters decrease (case 5). In the event of evaporation affecting only small drops, then 
The succinct description above covers all the possible cases. It should be noted here that to isolate breakup and coalescence processes, changes in PSD parameters need to be analyzed with regard to liquid water content to ensure that changing PSD parameters are not due to evaporation (Williams 2016). In the modeling presented here, this is achieved by considering variations both in NT and in the shape of the PDF.
This objective characterization of microphysics allows us to investigate the evolution of the PSD. Figure 6 shows a few examples. There, the different microphysical processes are color coded, providing a sort of “microphysical sequencing plot” that can be useful to evaluate the parameterizations.
Characterization and evolution of the eight main microphysical processes, as defined in Table 2, for some random sequences extracted from the HyMeX and Payerne databases. Samples are for 10-min spectra. The microphysical sequence plot is useful to track the evolution of the microphysics as observed by the disdrometers.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0154.1
Not all eight processes appear with the same frequency in the observations. The histogram of actual occurrences of the microphysical processes in the database (Fig. 7) shows that coalescence is the main process creating drops, and evaporation is the prevailing process of destruction. The high correlation between E(D) and E(D2) {or E[log(D)]} features as the dominance of those microphysics that implies a joint evolution of the two variables, an observation that is important to evaluate the fidelity of the codes parameterizing the microphysics of rain.
Actual occurrence in the DSD database of the microphysical processes described in Table 2.
Citation: Journal of Hydrometeorology 19, 6; 10.1175/JHM-D-17-0154.1
5. Conclusions and further work
The results show that the use of [NT, E(D), E(D2)] or {NT, E(D), E[log(D)]} instead of (N0, Λ, μ) [or (Nw, Dm, μ)] improves the characterization of the PSD of rainfall. Using a comprehensive and extended database, it has been shown that decoupling NT from the shape of the PSD improves the standard parameterization, as illustrated in all the observational cases of the HyMeX and Payerne databases.
It has been also shown that selecting E(D) and E(D2) to characterize the shape of the PSD does not imply assuming a gamma shape for the PSD. The analytical model (MaxEnt) using raw moments of the distribution does not need a parametric shape. However, if the gamma shape is desired {which implicitly means that the observables are E(D) and E[log(D)]} then the maximum entropy approach improves performance over the traditional modeling.
Regarding numerical models, employing [NT, E(D), E(D2)] or {NT, E(D), E[log(D)]} instead of (N0, Λ and μ) for the microphysics of rainfall has the added advantage of eliminating nonphysical variables in the modeling. Most of the existing parameterizations still include N0 as a variable that has to be prescribed without taking into account that N0 depends on NT, which is predicted by the model. The resulting inconsistencies are then dealt with by resorting to ad hoc assumptions that hinder the full potential of those parameterizations.
Convenient though the use of N0 might be because of legacy reasons, the effort of translating the codes to fully physical variables is worth the trouble. Since E(D) and E(D2) {or E[log(D)]} are correlated in real rainfall, the microphysics only need to predict two independent variables: NT and E(D), instead of assuming N0 and then advancing NT and Λ, which is a modeling that does not always adequately represent the physics of rain as observed in nature.
Further work using the maximum entropy models presented here also includes their use in the retrieval of microphysics by radar and radiometer, including the GPM DPR and GMI instruments. Radiometric observables can be readily calculated from the PSD through radiative transfer calculations, so it is possible to build a database of n-tuples of {NT, E(D), E[log(D)]} on the one hand, and single polarization with double-frequency (or single-frequency double polarization) radiometric variables on the other hand, and then attempting the retrieval of these through a procedure that minimizes a suitable metric.
Acknowledgments
Funding from projects CGL2013-48367-P, CGL2016-80609-R (Ministerio de Economía y Competitividad), UNCM08-1E-086 (Ministerio de Ciencia e Innovacion), and CYTEMA (UCLM) is gratefully acknowledged. HyMeX data were provided by the HyMeX program, sponsored by Grants MISTRALS/HyMeX, ANR-2011-BS56-027 FLOODSCALE project, EPFL-LTE, and OHMCV. The authors also wish to acknowledge fruitful discussions in the Particle Size Distribution Working Group (PSDWG) of the Global Precipitation Measurement (GPM) mission and in the International Precipitation Working Group (IPWG). Haddad's contribution was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
APPENDIX
Singh's Parameter Estimation for the PSD
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