1. Introduction
Building on the successful Tropical Rainfall Measuring Mission (TRMM), the Global Precipitation Measurement (GPM) mission aims to use multiple satellites to estimate surface rainfall with a 3-h resolution between 65°S and 65°N (Hou et al. 2008). The core GPM satellite will observe precipitation with a cross-track scanning dual-frequency precipitation radar (DPR) and a conically scanning multiple-frequency radiometer. The constellation of GPM satellites will observe precipitation with passive microwave sensors (Huffman et al. 2007).
Algorithms will estimate surface rainfall by using different combinations of GPM observations. “Radar only” algorithms will use DPR observations (e.g., Grecu et al. 2011), “radiometer only” algorithms will use passive microwave observations (e.g., Kummerow et al. 2011), and “combined” algorithms will use both radar and radiometer observations (e.g., Munchak and Kummerow 2011). Algorithms will use probabilistic frameworks that seek to reproduce the observed reflectivities and/or radiances with physically realistic raindrop size distributions (DSDs) following either Bayesian theory (Haddad et al. 2006) or optimal estimation theory (Munchak and Kummerow 2011).
To estimate surface rainfall, retrieval algorithms often assume that the DSD follows a gamma-shaped distribution with three parameters (e.g., Rose and Chandrasekar 2006; Iguchi et al. 2009; Kozu et al. 2009; Grecu et al. 2011; Munchak and Kummerow 2011; Seto and Iguchi 2011; Seto et al. 2013). In the ideal case, three measurements are needed to constrain three unknowns. When only two measurements are available, as in DPR observations (e.g., absolute reflectivity at two radar operating wavelengths), assumptions are needed to constrain the third DSD parameter.
There is concern that μ–Λ relationships similar to Eq. (2) result from mathematical artifacts due to correlations between the three mathematical parameters in Eq. (1) (Chandrasekar and Bringi 1987; Moisseev and Chandrasekar 2007). Another concern is that surface disdrometer observations used in developing μ–Λ relationships may underestimate the number of small raindrops in rain because of wind blowing small raindrops around the instrument inlet or low instrument sensitivity to detecting small raindrops (Moisseev and Chandrasekar 2007). The limited detection of small raindrops causes truncated raindrop spectra that lead to narrower spectra and biased μ–Λ relationships. Even if μ–Λ relationships contain mathematical artifacts, Zhang et al. (2003) have argued that μ–Λ relationships contain physical meaning and lead to improved rain-rate estimates [as later documented by Cao and Zhang (2009)].
Even without concerns over mathematical artifacts, single-value μ–Λ relationships as in Eq. (2) cannot be used in probabilistic rainfall retrieval algorithms because they only provide the expected (or initial) value of a DSD constraint. Probabilistic algorithms need the expected value plus a range of acceptable values to converge to a final solution (Haddad et al. 2006; Munchak and Kummerow 2011). The National Aeronautics and Space Administration (NASA) Precipitation Measurement Missions (PMM) DSD Working Group1 is investigating whether the DSD constraints, or assumptions, used in rainfall retrieval algorithms are observed in field campaign raindrop spectra and whether new constraints can be constructed that fit probabilistic algorithm logic. This study focuses on developing probabilistic DSD constraints through analysis of disdrometer observations. Developing probabilistic algorithms will be described elsewhere (e.g., Munchak and Kummerow 2011).
The DSD Working Group is investigating new DSD constraints by rephrasing the problem in two key ways. First, to avoid mathematical artifacts, relationships between directly measurable physical attributes of the DSD are investigated, and relationships between fitted mathematical parameters of a gamma function are not investigated. Since gamma parameters are not statistically independent, mathematical artifacts will appear in relationships once a DSD is assumed to follow a gamma mathematical model. Second, the relationships between physical DSD attributes are expressed in terms of joint probability distribution functions (joint PDFs) and not only as a best-fit line. The problem is now rephrased as, Given an algorithm estimate of one DSD physical attribute, what is the expected value and range of another DSD physical attribute? After determining joint PDFs of statistically independent DSD physical attributes, joint PDFs of gamma model parameters are constructed so that physically based constraints can be used in probabilistic rainfall retrieval algorithms that are formulated using gamma-shaped DSDs.
This study has the following structure: After defining a normalized gamma DSD, a simple dual-frequency radar rain-rate algorithm is described to highlight how a constraint, or assumption, is needed in the algorithm to solve the three-parameter DSD when only two measurements are available. Without an assumption of a gamma-shaped DSD a priori, section 3 uses the raindrop mass spectrum mean diameter Dm and standard deviation σm to describe the DSD shape. Surface disdrometer observations are introduced in section 4, and in section 5 a power-law relationship between estimated Dm and σm is removed to construct a new mass spectrum standard deviation estimate 
2. Gamma-shaped DSD and a simple DPR algorithm
Dual-frequency ratio (DFR) (dBZ) vs Dm calculated at 13.6- (Ku band) and 35.5-GHz (Ka band) radar frequencies using 
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Dual-frequency ratio (DFR) (dBZ) vs Dm calculated at 13.6- (Ku band) and 35.5-GHz (Ka band) radar frequencies using
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Dual-frequency ratio (DFR) (dBZ) vs Dm calculated at 13.6- (Ku band) and 35.5-GHz (Ka band) radar frequencies using
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
3. Attributes of the raindrop mass spectrum
In the previous section, the DSD was modeled with a modified gamma function using parameters Nw, Dm, and μ. In this section, the DSD is not assumed to have any a priori shape but is expressed as a raindrop number concentration N(D) observed by a surface disdrometer with discrete diameter size bins. By expressing the DSDs as raindrop mass spectra, the shape of the discrete distribution can be described by two attributes: the mass-weighted mean diameter and the mass spectrum standard deviation.
a. Mass spectrum mean diameter and standard deviation
b. Relationship between σm and Dm for simulated mass spectra
To illustrate how σm can increase as Dm increases, Fig. 2a shows three simulated “top hat” mass spectra with constant amplitude for D = 0–2 (squares), D = 0–3 (circles), and D = 0–4 (triangles) mm. The amplitudes were arbitrarily set to ½, ⅓, and ¼, respectively, to help to visualize the three distributions. From Eqs. (14) and (15), Dm and σm for these three distributions were 1.0, 1.5, and 2.0 mm and 0.58, 0.87, and 1.16 mm, respectively. Figure 2b shows these three pairs of Dm and σm values along with a line indicating a linear relationship between Dm and σm for mass spectra having a top-hat shape. Thus, for a general mass spectrum shape, the mass spectrum breadth increases as the mean diameter increases.
Demonstration that σm increases as Dm increases: (a) Simulated top-hat mass spectra with Dm = 1.0 (squares), Dm = 1.5 (circles), and Dm = 2.0 (triangles) mm. Abscissa is raindrop diameter. (b) Calculated σm vs. Dm for simulated top-hat mass spectra. The symbols represent curves shown in (a), and the solid line represents the general relationship for all top-hat mass spectra with various Dm values. The abscissa is mass spectrum mean diameter Dm.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Demonstration that σm increases as Dm increases: (a) Simulated top-hat mass spectra with Dm = 1.0 (squares), Dm = 1.5 (circles), and Dm = 2.0 (triangles) mm. Abscissa is raindrop diameter. (b) Calculated σm vs. Dm for simulated top-hat mass spectra. The symbols represent curves shown in (a), and the solid line represents the general relationship for all top-hat mass spectra with various Dm values. The abscissa is mass spectrum mean diameter Dm.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Demonstration that σm increases as Dm increases: (a) Simulated top-hat mass spectra with Dm = 1.0 (squares), Dm = 1.5 (circles), and Dm = 2.0 (triangles) mm. Abscissa is raindrop diameter. (b) Calculated σm vs. Dm for simulated top-hat mass spectra. The symbols represent curves shown in (a), and the solid line represents the general relationship for all top-hat mass spectra with various Dm values. The abscissa is mass spectrum mean diameter Dm.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
4. Disdrometer observations
To examine Dm and σm relationships in real data, an analysis of spectra collected using low-profile two-dimensional video disdrometers (2DVDs), manufactured by Joanneum Research FgmbH (Graz, Austria; Schönhuber et al. 2007), was undertaken. The diameter resolution was 0.2 mm, with 50 uniformly spaced diameter bins from 0.1 to 9.9 mm. After manually verifying with ancillary observations that precipitation was rain and not snow, the first quality-control stage for each 1-min raindrop spectra consisted of retaining spectra with 1) at least 50 raindrops in at least 3 different diameter bins, 2) reflectivity factor greater than 10 dBZ, and 3) rain rate greater than 0.1 mm h−1. The rain estimates were not divided by rain regime. A total of 29 705 min of raindrop spectra passed these criteria from three disdrometers deployed near Huntsville, Alabama, over an 18-month period from December 2009 to October 2011. After secondary filtering (discussed below), the number of raindrop spectra decreased to 24 872.
a. Disdrometer instrument limitations
Disdrometers count the number of raindrops passing through or hitting a surface. Because of their limited sample volume, disdrometers underestimate the number of small and large drops passing through the sample volume (Ulbrich and Atlas 1998; Kruger and Krajewski 2002). Also, wind can advect small raindrops around the instrument opening, causing the instrument to underestimate further the number of small raindrops. Nešpor et al. (2000) showed wind effects using an early version of the 2DVD, which prompted the development of the low-profile 2DVDs that were used in this study. Underestimating the small and large raindrops has an impact on estimated rain parameters (Wong and Chidambaram 1985; Chandrasekar and Bringi 1987; Smith et al. 1993, 2009; Smith and Kliche 2005) and will artificially narrow mass spectra, leading to underestimated σm.
Using 29 705 min of quality controlled observations from three side-by-side disdrometers located near Huntsville, Fig. 3a shows %Δσm for each 0.1-mm interval of Dm. The squares indicate the mean, and the lines show ±1 standard deviation. For Dm greater than 1 mm, the mean %Δσm is less than 10%, indicating that truncation will have a small impact on σm. For Dm less than 1 mm, however, the mean %Δσm has a very large magnitude, indicating that truncation significantly narrows the spectra causing σm to decrease. This sensitivity to small-drop truncation when Dm is less than 1 mm indicates that the disdrometers are observing some small raindrops. But without independent observations, it is difficult to determine whether wind effects and instrument limitations are reducing the number of detected small drops relative to the unknown true population. To avoid using potentially biased σm estimates in power-law calculations in section 5, all power-law calculations are performed using only estimates with Dm > 1 mm. The power-law relations are then extrapolated into the Dm ≤ 1 mm range.
Sensitivity of estimated σm to truncated spectra: (a) Percent change in σm between the observed and truncated spectra estimated for 0.1-mm intervals of Dm. Squares represent the mean, and lines span over the mean ± one standard deviation (STD). All raindrops 0.5 mm and smaller are removed from spectra before calculating truncated σm. (b) The mean (squares) and mean ± STD (lines) of normalized maximum observed diameter Xmax = Dmax/Dm for 0.1-mm intervals of Dm. The dashed line indicates the threshold used to filter narrow spectra. (c) Percent accumulation of observations as a function of Xmax. Approximately 16% of the observations had Xmax < 1.5, leaving 84% (24 872 min) of observations available for further analysis.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Sensitivity of estimated σm to truncated spectra: (a) Percent change in σm between the observed and truncated spectra estimated for 0.1-mm intervals of Dm. Squares represent the mean, and lines span over the mean ± one standard deviation (STD). All raindrops 0.5 mm and smaller are removed from spectra before calculating truncated σm. (b) The mean (squares) and mean ± STD (lines) of normalized maximum observed diameter Xmax = Dmax/Dm for 0.1-mm intervals of Dm. The dashed line indicates the threshold used to filter narrow spectra. (c) Percent accumulation of observations as a function of Xmax. Approximately 16% of the observations had Xmax < 1.5, leaving 84% (24 872 min) of observations available for further analysis.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Sensitivity of estimated σm to truncated spectra: (a) Percent change in σm between the observed and truncated spectra estimated for 0.1-mm intervals of Dm. Squares represent the mean, and lines span over the mean ± one standard deviation (STD). All raindrops 0.5 mm and smaller are removed from spectra before calculating truncated σm. (b) The mean (squares) and mean ± STD (lines) of normalized maximum observed diameter Xmax = Dmax/Dm for 0.1-mm intervals of Dm. The dashed line indicates the threshold used to filter narrow spectra. (c) Percent accumulation of observations as a function of Xmax. Approximately 16% of the observations had Xmax < 1.5, leaving 84% (24 872 min) of observations available for further analysis.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Figure 3b shows the normalized maximum diameter Xmax = Dmax/Dm for each 0.1-mm interval of Dm. Since our Moisseev and Chandrasekar (2007) method simulations indicated that σm biases decrease as Xmax increases (not shown), all observations with Xmax ≤ 1.5 were filtered from the dataset. Approximately 84% (24 872 min) of the original Huntsville observations had Xmax > 1.5 and were used for further analysis.
b. Observed 2D distributions
Using 24 872 min of filtered Huntsville raindrop spectra (see previous section for the filtering procedure), Fig. 4 shows the frequency of occurrence of reflectivity factor Z (dBZ), rain rate as 10 log10(R) (dBR), and σm as a function of Dm. The pixel with the most occurrences in each panel is normalized to have 0 dB. Each 50% decrease in occurrence has a 3-dB decrease on the logarithmic color scale. Table 1 lists the Pearson correlation coefficients between reflectivity Ze (mm6 m−3), rain rate R (mm h−1), Dm, and σm.
Demonstration that Z, R, and σm are correlated with Dm: frequency of occurrence from 24 872 min of 2DVD observations from Huntsville of (a) Z (dBZ) (b) R (dBR), and (c) σm vs Dm (color tiles; scale is logarithmic defined such that the pixel with the most occurrences has 0 dB and each 50% decrease in occurrence has a 3-dB decrease on the color scale). The solid black lines in (a)–(c) are power-law curves given by the equations in the labels.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Demonstration that Z, R, and σm are correlated with Dm: frequency of occurrence from 24 872 min of 2DVD observations from Huntsville of (a) Z (dBZ) (b) R (dBR), and (c) σm vs Dm (color tiles; scale is logarithmic defined such that the pixel with the most occurrences has 0 dB and each 50% decrease in occurrence has a 3-dB decrease on the color scale). The solid black lines in (a)–(c) are power-law curves given by the equations in the labels.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Demonstration that Z, R, and σm are correlated with Dm: frequency of occurrence from 24 872 min of 2DVD observations from Huntsville of (a) Z (dBZ) (b) R (dBR), and (c) σm vs Dm (color tiles; scale is logarithmic defined such that the pixel with the most occurrences has 0 dB and each 50% decrease in occurrence has a 3-dB decrease on the color scale). The solid black lines in (a)–(c) are power-law curves given by the equations in the labels.
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Pearson correlation coefficients between rain parameters estimated from 24 872 min of filtered Huntsville disdrometer observations. Correlation coefficients are between reflectivity Ze (mm6 m−3), rain rate R (mm h−1), mean mass spectrum diameter Dm (mm), mass spectrum standard deviation σm (mm), and normalized mass spectrum standard deviation 
Using observations with Dm > 1.0 mm (a total of 18 969 observations), power-law curves are estimated with the form 
c. Dm–σm–μ relationships for gamma-shaped DSDs
The σm and Dm estimates in Fig. 4c were calculated directly from the disdrometer spectra using Eqs. (14) and (15) and do not assume a gamma-shaped DSD. As discussed in the introduction, there are mathematical relationships between Dm, σm, and μ for gamma-shaped DSDs (Ulbrich 1983; Ulbrich and Atlas 1998; Bringi and Chandrasekar 2001). These mathematical relationships are derived in this section to define a mapping from DSD physical attributes (Dm and σm) to gamma function parameters (Dm and μ).
Demonstration that weighting σm with 
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Demonstration that weighting σm with
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Demonstration that weighting σm with
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
There are three important points to glean from Fig. 5a. First, since σm and Dm were estimated without assuming a gamma-shaped DSD and the gamma DSD σm,gamma function with μ ranging from 0 to 10 bounds the observed distribution of σm versus Dm, we can conclude that a family of gamma functions can describe the shape of the observed DSDs. Second, the Zhang et al. (2003) μ–Λ relationship passes through the σm–Dm distribution for Dm < 2.0 mm. Third, the σm PDF shown in Fig. 5b is asymmetric and indicates it would be difficult to use σm directly in probabilistic retrieval algorithms that assume that parameters are Gaussian distributed.
Note that Eq. (18) is a simple mathematical relationship between Dm, σm, and μ and was determined after “mathematically” forcing raindrop diameters to extend from Dmin = 0 to Dmax = ∞. Equation (18) does not account for any small-drop truncation in observed disdrometer raindrop spectra as discussed in section 4a. Also, Eq. (18) does not account for finite maximum diameter, Dmax < ∞. Both topics need to be addressed in future work.
5. Statistically independent DSD shape attributes
Figure 4 shows frequency of occurrence of Z, R, and σm as a function of Dm. The largest correlation coefficient is between Dm and σm (see Table 1) and indicates that these two DSD shape attributes are not independent. Developing constraints using Dm and σm will be subject to mathematical artifacts similar to μ–Λ relationships. To avoid potential mathematical artifacts, DSD relationships need to be developed using statistically independent shape attributes. This section uses the method described by Haddad et al. (1996) to construct a new mass spectrum standard deviation that is statistically independent of Dm.
a. Statistically independent DSD shape attributes
Using the 18 969 disdrometer observations with Dm > 1.0 mm, a zero correlation coefficient occurred when bm = 1.36. This is the exponent shown in the power-law curves in Fig. 4c. Using σm and Dm estimated for each disdrometer observation, 
The mean 
One way to interpret Fig. 5c is to consider it as joint PDF plots of Dm and 
Mapping from (Dm, σm) space to (Dm, μ) space: (a) Frequency of occurrence of σm vs Dm (same as Figs. 4c and 5a); black solid line is power-law fit (same as Figs. 4c and 5a), and blue dashed and red dash–dotted lines represent upper and lower 
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Mapping from (Dm, σm) space to (Dm, μ) space: (a) Frequency of occurrence of σm vs Dm (same as Figs. 4c and 5a); black solid line is power-law fit (same as Figs. 4c and 5a), and blue dashed and red dash–dotted lines represent upper and lower
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Mapping from (Dm, σm) space to (Dm, μ) space: (a) Frequency of occurrence of σm vs Dm (same as Figs. 4c and 5a); black solid line is power-law fit (same as Figs. 4c and 5a), and blue dashed and red dash–dotted lines represent upper and lower
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
b. Transformation from physical attributes to gamma parameters
6. Estimated rain rate using μ constraints
This section uses the simple DPR rain-rate retrieval algorithm developed in section 2 to evaluate whether more accurate rain rates occur if μ is held constant or if it is described as a function of Dm. Six different rain-rate estimates are produced using the same observed disdrometer reflectivity 
Demonstration that the 
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Demonstration that the
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Demonstration that the
Citation: Journal of Applied Meteorology and Climatology 53, 5; 10.1175/JAMC-D-13-076.1
Except for the μ = 0 model, the FSE for all models is nearly the same, with an average of 20% ± 4% (the μ = 0 model average is 32%). Table 2 lists the MNB and FSE for the six models at Dm of 1.0, 1.5, and 2.0 mm. The large FSE for all models reflects the simplicity of the rain-rate retrieval algorithm. All models represent the DSD with just one μ value for each Dm. An algorithm that varies μ on the basis of additional information will reduce the FSE.
MNB and FSE for modeled rain-rate estimates at selected Dm and six different values of μ. The number of samples in each 0.1-mm interval is labeled as n.
7. Conclusions
The dual-frequency precipitation radar planned for the core satellite of the Global Precipitation Measurement mission will provide dual-frequency reflectivity measurements of precipitation. Rainfall retrieval algorithms will assume a gamma raindrop size distribution with three mathematical parameters Nw, Dm, and μ. One challenge for the DPR retrieval algorithm is to estimate rainfall that is modeled with three DSD parameters using only two radar measurements. This underconstrained problem requires the algorithm to assume that one parameter is a constant or a function of another parameter. Since GPM rainfall algorithms will use either optimal estimation theory (Munchak and Kummerow 2011) or Bayesian theory (Haddad et al. 2006) to form probabilistic algorithms, the DSD constraint needs to have an initial (or expected) value plus an acceptable range of values. The acceptable range of values allows the algorithm to deviate from the expected value as dictated by the observations.
One option is to constrain the DSD parameters with a μ–Λ constraint (Zhang et al. 2003), but μ–Λ constraints only provide an expected value and do not provide a range of values allowing probabilistic algorithms to deviate from the expected value. Also, since μ and Λ represent mathematical parameters of a gamma function they are highly correlated and thus μ–Λ relationships may contain mathematical artifacts (Chandrasekar and Bringi 1987; Moisseev and Chandrasekar 2007). To avoid these mathematical artifacts, relationships need to be developed before assuming the DSD follows any particular mathematical shape.
This study analyzed over 20 000 minutes of surface disdrometer raindrop mass spectra and found that the mean diameter Dm and mass spectrum standard deviation σm were highly correlated (r2 = 0.91). This high correlation may lead to mathematical artifacts in DSD constraints based on σm–Dm relationships. To avoid mathematical artifacts, a new breadth variable 
Since 
This analysis used disdrometer observations to develop DSD constraints that provide initial values and ranges of acceptable values for underconstrained probabilistic rainfall algorithms. Without any other information, an algorithm can start at the initial value and then use observations and algorithm logic to deviate from this initial value. This analysis provides a statistical representation of DSD parameter assumptions that can be incorporated into algorithm logic. For completeness, note that power-law DSD constraints developed in this analysis should not be used to estimate DSD parameters in disdrometer datasets. The DSD constraints are statistical representations of DSD physical attributes or DSD parameters and do not represent instantaneous values estimated from individual DSD spectra.
There are topics of this study that need further research. First of all, if surface disdrometers underestimate the number of small raindrops, then the mass spectrum will be too narrow and σm will be negatively biased. The σm bias that is due to small raindrop truncation is a function of Dm, with the biases decreasing with increasing Dm. To avoid potential σm biases, this study used σm when Dm was greater than 1.0 mm. Since this work focused on developing joint PDFs of Dm and σm, the impacts of small-drop truncation may be within the upper and lower bounds (or other statistics) of the joint PDFs. Future work needs to address if and how often surface disdrometers underestimate the number of small raindrops in rain, including the raindrops that are advected around the instrument. Future work should also aim to understand how undercounting the number of small raindrops affects joint PDFs derived from observed physical attributes of the DSD.
The maximum observed raindrop diameter Dmax affects the calculated mass spectrum breadth and also has an impact on the mapping of physical attributes in (Dm, 
Another topic that needs further investigation is the site-to-site and rain regime-to-regime variability of the power-law relationship that causes Dm and 
Shifts in 
Acknowledgments
Support for this work was provided by Ramesh Kakar under the NASA Precipitation Measurement Missions (PMM) and NASA Global Precipitation Measurement (GPM) mission, including Grants NNX13AI94G, NNX10AM54G, NNX13AF89G, NNX12AD03A, NNX10AP84G, NNX13AF86G, NNX13AJ55G, NNX13AI89G, and NNX10AH66G. The authors thank Dr. Merhala Thurai for her insightful discussions.
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The DSD Working Group is composed of NASA PMM Science Team members and includes GPM algorithm developers and observational scientists.
