## 1. Introduction

*γ*= 1. The two-parameter exponential distribution and power-law distribution are obtained, respectively, by further setting

*μ*= 0 or Λ = 0.

The exponential distribution and the three-parameter gamma distribution have long dominated the literature on rain and snow PSDs (Marshall and Palmer 1948; Sekhon and Srivastava 1970, hereafter SS70; Hansen and Travis 1974; Ulbrich 1983). The power-law distribution has often been used as a first-order representation of aerosol PSDs, starting with Junge (1955).

The full four-parameter modified (or generalized) gamma distribution (hereafter simply MGD) is used less frequently and is most often encountered in descriptions of haze, fog, and cloud droplet PSDs (Deirmendjian 1969; Tomasi et al. 1975; Hess et al. 1998; Vivekanandan et al. 1999). A major premise of this paper is that the MGD is more fundamentally relevant to theoretical and observational PSD work, especially with respect to inhomogeneous or irregular solid particles such as snow and cloud ice, than is often appreciated.

Depending on the context, the preferred particle size descriptor used in a given PSD formulation may be the mass *m*, the actual geometric diameter *D _{g}*, the volume-equivalent (solid or liquid) spherical diameter

*D*, the actual or volume-equivalent radius (

_{e}*r*or

_{g}*r*), the projected area

_{e}*A*, or the area-equivalent diameter

*D*. For microphysical, chemical, and radiative transfer calculations, it is often necessary to convert from one representation to another, especially when comparing or utilizing distribution parameters obtained from a variety of sources.

_{a}While the derivation of these conversions is mathematically straightforward, it entails some effort and risk of error. This is especially true in the case of complex nonspherical particles for which *m* is found to be proportional to *D ^{b}*, with

*b*< 3 (Locatelli and Hobbs 1974; Mitchell et al. 1996; Vivekanandan et al. 1999; Mitchell 2002; Heymsfield et al. 2002; Delanoë et al. 2007; Schmitt and Heymsfield 2010; Heymsfield et al. 2010), and the two-dimensional

*A*is proportional to

*D*with

^{β}*β*< 2 (Mitchell 1996; Schmitt and Heymsfield 2010).

Noteworthy for the purposes of this paper is that for such particles, even a simple exponential distribution expressed in terms of *D _{g}* becomes an MGD when expressed in terms of melted-equivalent diameter

*D*, and vice versa. For example, if the particle mass is proportional to

_{e}*n*(

_{g}*D*) =

_{g}*N*

_{0g}exp(−Λ

*), then the PSD for the melted-equivalent diameters is*

_{g}D_{g}A major objective of this paper is to provide a single-source reference for the relationships and conversions most likely to be of interest to those working with atmospheric PSDs in the MGD family, especially when particles of nonconstant effective density are involved. Apart from their potential use in quantitative calculations, one practical benefit is the ability to transform published PSDs from different sources to a common basis for ease of comparison.

We begin by briefly reviewing the relevant terminology and key mathematical relationship, starting with generic PSDs (next section) and then focusing specifically on the MGD in section 3. In section 4 we discuss the implications of various choices for representing particle size, followed by a tabulation of general relationships for converting between these representations in section 5 and implications for estimating model parameters in section 6. In section 7 we briefly discuss the rescaling or normalizing of PSDs (Testud et al. 2001; Lee et al. 2004; Delanoë et al. 2005; Field et al. 2007, hereafter F07) in the specific context of MGDs. In section 8, we illustrate the use of the aforementioned conversions by directly comparing three PSD models (SS70; Braham 1990, hereafter B90; F07) published using incompatible representations of particle size.

## 2. General properties of size distributions

*n*(

*x*), where

*x*is some nonnegative variable (e.g., mass or diameter) that uniquely characterizes a particle’s size, is generally defined such that

*n*(

*x*), one may immediately compute several aggregate or bulk quantities from the distribution. The total number concentration of particles of all sizes is

*f*(

*x*) that describes the contribution by a single particle of size

*x*and then integrating over all sizes:

*f*(

*x*) that describes the single-particle contribution to an aggregate property of a distribution can be adequately modeled, at least statistically if not deterministically, as a power law; that is,

*M*is the

_{δ}*δ*th moment of

*n*(

*x*).

For any physically reasonable and therefore finite *f*(*x*), the corresponding aggregate property *F* must also be finite. This requirement restricts the set of admissible size distribution models *n*(*x*). It is not difficult to construct hypothetical size distributions [e.g., *n*(*x*) ∝ 1/(1/*x* + *x*^{4})] for which low-order moments (e.g., total particle count) are well defined while higher-order moments (e.g., total particle volume) are infinite, or vice versa.

## 3. The modified gamma distribution

### a. Definition

*n*(

*x*), with

*μ*, Λ, and

*γ*controlling the shape of the distribution and

*N*

_{0}controlling the overall scaling. If

*x*has dimensions of length (the most common case), then

*N*

_{0}has dimensions

*L*

^{−(μ}

^{+4)}and Λ has dimensions

*L*

^{−γ}. Because of the dependence of the dimensions of

*N*

_{0}and Λ on the values of

*μ*and

*γ*, respectively, one cannot meaningfully compare values of the first two parameters for different distributions if

*μ*and/or

*γ*differ as well.

*x*

_{0}is the modal value of

*x*.

*C*is a normalization constant that depends on distribution parameters

*γ*,

*c*≡ Λ

^{1/γ}, and

*u*≡ (

*μ*+ 1)/

*γ*, with the full PSD then given by

*n*(

*x*) =

*N*

_{tot}

*p*(

*x*).

Occasionally one encounters the term “modified gamma distribution” applied to a three-parameter gamma distribution (Schneider and Stephens 1995; Babb et al. 1999; Miles et al. 2000; Maki et al. 2001), but this usage does not appear to be widespread.

### b. Relationship to other models

#### 1) Exponential distribution

*μ*= 0 and

*γ*= 1, we obtain the widely used exponential distribution

#### 2) Power-law distribution

*x*

_{min},

*x*

_{max}], in which case

The expressions presented below for various moments of MGDs are undefined for Λ = 0 and are in any case applicable only when *n*(*x*) is continuous for all nonnegative *x*. For this reason, we will not discuss power-law PSDs further in this paper, though the conversions given in section 5 may still be used.

#### 3) Gamma distribution

*γ*= 1, is in wide use for precipitation (Ulbrich 1983, 1985; Kozu and Nakamura 1991; Meneghini et al. 1992; Jameson 1993; Iguchi et al. 2000; Maki et al. 2001; Grecu et al. 2004; Rose and Chandrasekar 2006), as well as for cloud droplets and cirrus ice particles (Ackerman and Stephens 1987; Schneider and Stephens 1995; Mitchell et al. 1996; Babb et al. 1999; Miles et al. 2000; Mitchell 2002; Heymsfield et al. 2004; Sun et al. 2006):

*N*

_{0}and Λ are often known respectively as the intercept and slope parameters;

*μ*is often referred to as the shape parameter.

*r*

_{eff}is the effective radius of the distribution, defined in this context

^{1}as

*μ*= 0 (exponential PSD), it follows that

*υ*

_{eff}= ⅓ and Λ = 3/

*r*

_{eff}.

#### 4) Other PSD models

PSD models in occasional use that are not simply special cases of (6) include the normal distribution and the lognormal distribution. For example, Tian et al. (2010) found that some observed cirrus ice particle spectra are better fit by a lognormal distribution than by exponential or gamma distributions (they did not, however, investigate the fit by an MGD).

While many of the general considerations outlined in this paper are valid for lognormal distributions as well, the details of the mathematics require a separate treatment and will not be discussed in this paper. Some general properties of lognormal distributions are summarized by Straka (2009).

In addition, one sometimes encounters composite PSD models based on the sum of two simpler underlying PSDs. For example, Kuo et al. (2004) proposed a bimodal PSD model constructed from the sum of two MGDs. The moments of such a PSD are the sums of the moments of the individual MGDs. Other characteristics, such as the median mass particle size, cannot be derived analytically for a composite PSD.

Also, F07 published empirically derived size distributions for ice particles expressed as rescaled (nondimensional) size distributions Φ_{23}(*x*), where *x* is a nondimensional (scaled) diameter. The fitted Φ_{23}(*x*) models are sums of pure exponentials and three-parameter gamma distributions with fixed parameters. We will return to the Field et al. PSD, as well as their rescaling procedure applied to an MGD, in sections 7 and 8.

### c. Moments and modes of the MGD

#### 1) Analytic expressions

*k*th moment of the MGD is given by

*x*) is the generalization of the integer factorial function (

*x*− 1)! to continuous

*x*.

*n*(

*x*), or

*N*

_{tot}to be finite, we require

*k*th mode

*x*of the distribution, defined here as the value of

_{k}*x*coinciding with the maximum of the function

*x*(

^{k}n*x*):

A more comprehensive discussion of the analytic properties of modified gamma distributions is given by Straka (2009).

#### 2) Parameter estimation from real data

The analytical expressions for PSD moments given above depend on the MGD model being valid for all nonnegative *x*. When working with real data rather than models, it is often necessary to estimate moments and/or parameters of the PSD from observations that are only reliable within a restricted size range [*x*_{min}, *x*_{max}]. In general, parameter estimation from PSD observations is fraught with difficulties related to instrument limitations and statistical sampling, many aspects of which are discussed by Auf der Maur (2001), Smith and Kliche (2005), Brandes et al. (2007), Brawn and Upton (2007), Mallet and Barthes (2009), and Tian et al. (2010), among many others.

This paper does not attempt to address the problems associated with statistical sampling or truncation of observed spectra but rather focuses exclusively on the mathematical conversion between related MGDs, assuming that the parameters of the MGD are known. However, in section 6 we comment briefly on the potential for mathematical inconsistencies when fitting models to data.

### d. Median mass particle size

*x*

_{med}is defined by the following relationship:

*m*(

*x*) is the particle mass, and the total mass concentration of the PSD is

## 4. Choice of size descriptor

In the previous sections, we used *x* as a generic particle size without specifying its precise meaning. The choice of the size variable used in the mathematical representation of a PSD is very flexible, yet the choice affects both the values and the physical dimensions of the parameters *N*_{0}, *μ*, Λ, and *γ* in an MGD.

If one allows the term “size” to encompass all reasonable descriptions of how “big” a particle is, including mass or any of a number of possible measures of linear dimension, area, or volume, one finds that, in the case of irregular particles at least, only the particle mass has an unambiguous meaning that does not depend either on context or on arbitrary definitions. For example, is the “diameter” of a particle its absolute maximum dimension or rather that recorded by a 2D imaging probe? Is the particle “volume” defined as the volume of its circumscribing sphere or by some other measure of the spatial distribution of mass (Petty and Huang 2010)? (The definition of particle volume is of course also inseparable from the definition of particle density.)

Despite the potential ambiguities, there is not only no universal standard way to describe particle size but rather a number of distinct conventions, many in wide use, that lead to superficially incompatible descriptions of a PSD. One major purpose of the present paper is to summarize in convenient form the conversions from one size representation to another. A second purpose is to demonstrate that a simple exponential or gamma PSD referenced to one size descriptor (e.g., liquid equivalent spherical diameter) will often require representation by an MGD for almost any other choice of size descriptor (e.g., geometric diameter).

### a. Mass m

*m*. Although it is not common to do so, we may elect to represent a PSD in terms of the distribution of particle masses. The four parameters of the MGD are supplied with subscripts to make clear their association with this variable, and the PSD thus takes the following form:

*m*

_{med}is then defined by the following relationship:

*m*

_{med}must be expressed in terms of the incomplete gamma function

*γ*(

*a*,

*x*); however, a numerical representation of reasonable accuracy is

### b. Geometric diameter D_{g}

*D*of the particle. One may then specify the size distribution as an MGD of the form

_{g}*D*is crucial. As a practical matter, it is often taken to be the maximum dimension of the particle as projected onto a two-dimensional image plane. For nonspherical randomly oriented particles, it is understood that this maximum dimension bears only a statistical relationship to the true maximum dimension of the particle itself. For example, in the worst-case scenario of a randomly oriented slender needle of length

_{g}*L*, the projected maximum dimension

*D*obeys a rather broad PDF,

_{g}*D*〉 ≈ 0.64

_{g}*L*. Moreover, there may exist only a rough statistical relationship between the true maximum dimension of a stochastically aggregated snowflake, for example, and any other physical property of the aggregate, such as mass. Notwithstanding the likelihood of two layers of scatter in the relationship between observed

*D*and particle

_{g}*m*, it is common to assume a power-law relationship between the two:

*ρ*is independent of size, then we have

*α*=

*ρπ*/6 and

*b*= 3. More generally, however, it is often observed that density decreases with

*D*, in which case

_{g}*b*< 3 and

*a*> 0 are empirically determined coefficients (Mitchell et al. 1996; Vivekanandan et al. 1999; Mitchell 2002; Heymsfield et al. 2002; Delanoë et al. 2007; Heymsfield et al. 2010).

*ρ*

_{0}. If the fraction of solid material is

*f*, and if we neglect the density of air, then

*ρ*=

*fρ*

_{0}. But because

*f*cannot exceed unity, there is a lower limit

*D*

_{g}_{,min}to the size for which (29) can be valid when

*b*< 3 (Brown and Francis 1995; Delanoë et al. 2007). In particular, if the particles are spherical, then

*n*(

_{g}*D*) for which

_{g}*b*< 3, one should assess whether significant numerical errors might arise from assuming (30) for particles smaller than

*D*

_{g}_{,min}(Heymsfield et al. 2010). If so, then numerical integration is in order. Otherwise, we may derive analytical expressions based on (17) to obtain bulk properties of the distribution. For example, the mass concentration is given by

*D*

_{g,med}is defined by the relationship

*γ*= 1).

_{g}### c. Liquid- or solid-equivalent spherical diameter D_{e}

Sometimes it is convenient to express the size of a nonspherical or tenuous particle in terms of the diameter of a mass-equivalent homogeneous sphere of some standard density *ρ*_{0}. For example, the liquid-equivalent diameter of a snowflake is the diameter of the spherical water droplet that results from melting the snowflake.

*D*is tantamount to taking

_{e}*b*= 3 in (30) and

*a*equal to

*D*satisfies

_{e}*D*as follows:

_{e}*D*

_{e,med}is defined by the relationship

*Z*, which is defined as the sixth moment of the distribution of liquid-equivalent diameters:

### d. Area A and area-equivalent diameter D_{a}

In many areas of research, some convenient measure of a particle’s projected area may be of primary interest, not only because it can be directly observed in the field, but also because of its relevance for radiative transfer, microphysical growth processes, and/or particle fall speeds (Mitchell and Arnott 1994; Mitchell 1996; Heymsfield and Miloshevich 2003; Mitchell and Heymsfield 2005). The projected area in question may either be that of a randomly oriented particle or else that of a particle that has a predictable alignment with respect to either an optical instrument or the direction of fall. For atmospheric chemical processes, as well as some radiative properties, it may be the total surface area of the particles rather than the projected area that matters. For fractal-like particles, any value given for either the surface area or projected area is almost certainly resolution dependent.

*A*is the projected area and the particle is a sphere or circular disk oriented perpendicular to the line of sight, then

*α*=

*π*/4 and

*β*= 2. But for an irregular or fractal particle, typically

*α*<

*π*/4 and/or

*β*< 2 (Mitchell 1996; Schmitt and Heymsfield 2010).

*A*is the so-called area ratio

*A*(Heymsfield and Miloshevich 2003; Schmitt and Heymsfield 2009), defined as

_{r}*α*and

*β*to characterize the area–diameter relationship.

If *A* is instead taken to be the total surface area of the particle, then for a uniform sphere, *α* = *π* and *β* = 2. Nonspherical or irregular particles may have surface areas that are either larger or smaller than those of spheres having the same *D _{g}*, so there are few obvious constraints on possible values for

*α*or

*β*.

*A*, one may choose to define a size distribution in terms of

*A*, in which case the MGD form is

*n*(

_{g}*D*), can be directly estimated from 2D images of the particles without the need for ancillary assumptions.

*A*of a particle, one may also define

*D*such that

_{a}*A*represents total surface area, then a preferable choice might be

*α*

_{0}=

*π*. Either way, we have the possibility for yet another PSD model,

### e. Radius

*r*rather than the diameter

*D*. Regardless of whether it is the geometric or equivalent diameter and radius we are speaking of, we require

## 5. Conversion between representations

*x*is to be transformed into PSD expressed in terms of size descriptor

*y*, then we require

*x*and

*y*are related by a power law, then substituting appropriate expressions for

*x*(

*y*) and its derivative into the above allows one to solve for the new parameters

*N*

_{0y},

*μ*, Λ

_{y}*, and*

_{y}*γ*in terms of the old parameters

_{y}*N*

_{0x},

*μ*, Λ

_{x}*, and*

_{x}*γ*. The required expressions are tabulated in Table 1.

_{x}Complete set of conversions from one size descriptor (column heading) to another (row heading). Expressions in boldface are definitional; all others follow from these.

Based on (54), Table 2 lists the complete set of conversions between parameters of MGDs referenced to *m*, *D _{g}*,

*D*,

_{e}*D*, or

_{a}*A*as the independent variable. For conversions between PSD representations using radius and diameter, see (51).

Complete set of conversions between the parameters of MGDs referenced to various particle size descriptors discussed in this paper. Row headings indicate the size descriptor used by the source MGD; column headings indicate the parameters of the target MGD.

## 6. Implications for model fitting

The conversion relationships obtained between different representations of a PSD reveal some potential traps for the unwary when estimating model parameters from real data. To take a simple example, an imaging probe can yield simultaneous estimates of both *A* and *D _{g}* for all of the sampled particles, which in turn can be used to separately obtain (e.g., via moment matching) the four MGD parameters (each) of

*n*(

_{A}*A*) and

*n*(

_{g}*D*). Values of

_{g}*α*and

*β*can then be determined in one of at least two ways: 1) by independently estimating these coefficients via a power-law fit to the sample of

*A*and

*D*, or 2) by solving for

_{g}*α*and

*β*from the MGD parameters determined separately for

*n*(

_{A}*A*) and

*n*(

_{g}*D*).

_{g}But from the conversions given in Table 2, we see that four model parameters (e.g., *N*_{0A}, *μ _{A}*, Λ

*, and*

_{A}*γ*) can be computed exactly from the remaining six (e.g.,

_{A}*N*

_{0g},

*μ*, Λ

_{g}*,*

_{g}*γ*,

_{g}*α*, and

*β*). In other words, the 10 model parameters that could in principle be directly estimated from particle data collected by an imaging probe are not independent and in fact represent only 6 degrees of freedom. Unless care is taken, one is likely to obtain estimates of the 10 parameters that are not mutually consistent. Indeed, even if one simply computes

*α*and

*β*from the other parameters [e.g.,

*A*and

*D*while enforcing perfect mathematical self-consistency. Another question is whether the resulting fits would also be more robust in other measurable ways.

_{g}## 7. Rescaled size distribution

A fairly recent innovation is the use of normalizing or rescaling procedures to obtain a nondimensional PSD expressed as a function of a nondimensional size defined in terms of the actual geometric size and various moments of the original PSD (Testud et al. 2001; Lee et al. 2004; Westbrook et al. 2004; Delanoë et al. 2005; F07; Tian et al. 2010). One motivation is to “explore the underlying shape of the PSD without imposing any a priori expectation for any specific analytic form” (F07). Another is to capture (or parameterize) a spectrum of PSDs in terms of one or two reasonably generic properties such as the low-order moments of the distribution.

*M*is the

_{k}*k*th moment of the PSD,

*x*is a nondimensional particle size, and Φ

_{23}(

*x*) is the nondimensional size distribution obtained used the second and third moments.

*μ*and

_{g}*γ*. The parameters

_{g}*N*

_{0g}and Λ

*play no role in the rescaled MGD, the second of these parameters having been absorbed into the definition of*

_{g}*x*. Moreover, the parameters

*μ*and

*γ*are unchanged by the rescaling. It follows that any pure exponential PSD (

*μ*= 0,

*γ*= 1) rescales to the fixed form (F07):

Because the values of two parameters of the original MGD are lost in the rescaling, its reconstruction from Φ_{23}(*x*) requires the additional specification of any two independent properties (e.g., moments) of the original PSD.

## 8. Example applications

A major purpose for deriving conversions between various equivalent representations of PSDs is to facilitate convenient and direct comparison of PSDs from different sources regardless of the manner in which they were originally specified. Here we consider three published PSDs for snow particles:

a representative measured PSD from Braham (1990, hereafter B90) expressed in terms of

*D*of the snowflakes observed in lake effect snow;_{g}the snow PSD of SS70, which is a function of liquid-equivalent precipitation rate

*R*and expressed in terms of liquid-equivalent diameter*D*; and_{e}the PSD obtained for “snow” in midlatitude stratiform ice cloud by F07, which is given in the nondimensional (rescaled) form described in the previous section and with

*x*as the independent variable.

Note that the snow observed at altitude by F07 is from a markedly different environment than the near-surface snowfall measured by B90 and SS70, and we can therefore expect, a priori, that the three PSDs will not be similar. Our premise, however, is that the PSDs cannot be compared even qualitatively until recast into a common framework.

For the sake of the present illustration, and following F07, we assume the snow particle mass–size relationship of Wilson and Ballard (1999) [*a* = 0.069 and *b* = 2.0 in (29)], which is fairly typical of empirically derived relationships in showing an approximate *D*^{2} dependence of particle mass on geometric diameter. The precise relationship used here is unimportant, since our focus is on the methods and not on the quantitative results.

To be comparable, we require all three PSDs to yield identical snow/ice water content *W*. While SS70 and F07 are parametric and can be varied to yield arbitrary values of *W*, B90 give a series of exponential fits to discrete observations of PSDs in lake effect snow. We therefore choose one representative example from B90 and adjust the other two PSDs to match. Details are given below.

### a. B90

^{−3}to obtain a snow water content of 0.118 g m

^{−3}. We integrated it using

*W*= 0.306 g m

^{−3}. While the (unknowable) true snow water content for this case is not pertinent to the present illustration, the divergence between these two estimates of

*W*underscores the sensitivity of snow water calculations to particle density assumptions.

### b. SS70

### c. F07

*x*defined in (55).

*M*

_{2}in particular is proportional to snow water content by virtue of the assumed

*D*

^{2}dependence of particle mass. Assuming a temperature of −10°C, we find that

*M*

_{2}= 0.004 452 (SI units) yields the target snow water content of 0.306 g m

^{−3}, so that

### d. Conversions

For the above three PSDs, the parameters *μ* = 0 and *γ* = 0 unless otherwise noted. With all four parameters of the MGD specified for a given published distribution (or, in the case of F07, for each term in the composite distribution), we can apply the appropriate conversions in Table 2 to find alternate representations in terms of *D _{g}* or

*D*. The parameter values for these alternate forms are given in Table 3.

_{e}MGD coefficients (SI units) for three published snow PSDs (see text), all yielding a snow water content of 0.306 g m^{−3}.

Note that as a consequence of assuming *b* < 3 in (29), both B90 and F07 translate into MGDs when expressed as *n _{e}*(

*D*), and SS70 becomes an MGD when expressed as

_{e}*n*(

_{g}*D*).

_{g}### e. Comparisons

Recalling that all three PSDs described above yield the same snow/ice water content, they may now be directly compared by plotting the appropriate forms as functions of *D _{e}* (Fig. 1, left column) or

*D*(right column). The differences between the three distributions are striking. In terms of both number concentration (top row) and mass contribution (bottom row), the SS70 distribution is notable for yielding numerous large particles greatly exceeding 2 mm in melted diameter and 4 mm in geometric diameter. These dimensions are consistent with fairly large snow aggregates. At the other extreme, the F07 distribution yields very few particles larger than 1-mm melted diameter or 2-mm geometric diameter.

_{g}The shapes of the PSDs are also distinct, with SS70 being concave-upward in Fig. 1b, while F07 is concave-downward over most of the range. The F07 distribution expressed as mass contribution versus *D _{g}* (Fig. 1d) is also notable for have a second mode at small sizes. In terms of both size and shape, B90 falls between SS70 and F07.

The corresponding median mass melted diameters are 0.503 (F07; determined numerically), 0.755 (B90), and 1.67 mm (SS70), more than a factor of 3 range for the same snow water content. A back-of-the-envelope calculation of equivalent radar reflectivity factors using the sixth moment of the distributions (i.e., assuming the validity of the Rayleigh approximation applied to the volume-equivalent diameters) yields results ranging from 13.6 dB*Z _{e}* for F07 to 31.6 dB

*Z*for SS70, an 18-dB

_{e}*Z*difference.

That the above differences are significant is not unexpected in view of the very different conditions under which the PSDs were obtained. The major point relevant to this paper is that these PSDs *could* be directly compared in multiple frameworks with so little effort, once the necessary conversions had been derived and tabulated in Table 2 along with the analytic expression for the moments given in (17).

## 9. Conclusions

Our review and mathematical analysis of the general PSD model known as the modified gamma distribution highlights the fundamental relevance of the MGD when working with nonspherical atmospheric particles whose area and mass scale with *b* < 3. In particular, a PSD that is a pure exponential or three-parameter gamma distribution when referenced to one size descriptor almost inevitably becomes a four-parameter MGD when referenced to another.

Our findings are consistent with Auf der Maur (2001) but we provide, in Table 2, explicit and convenient rules for the conversion between MGDs expressed in terms of a variety of size representations, including geometric diameter *D _{g}*, volume-equivalent diameter

*D*, area-equivalent diameter

_{e}*D*, mass

_{a}*m*, and surface or projected area

*A*.

By way of illustration, we employed our conversions to transform three published PSDs for “snow”—those of SS70, B90, and F07—into common frames of reference. While large differences in these PSDs were expected owing to differences in the methods and circumstances of the data collection, our conversion of the PSDs to common bases made it straightforward to compare them in more quantitative terms, confirming that the dissimilarities are quite large—a factor of 3 difference in median mass diameter and a roughly 18-dB*Z* difference in radar reflectivity for the same snow/ice water content of 0.306 g m^{−3}. We emphasize that the specific results of these comparisons are tied to both the mass–size relationship used and to the fixed snow water content assumed. Our focus here is on the methods and on the central role of the MGD, not on the quantitative results for specific published PSD models.

The tabulated relationships given herein are intended not only for the purpose of comparisons like the above, but also to facilitate routine work with the MGD family of particle size distribution models, including common exponential and gamma distributions as size descriptor-dependent special cases.

## Acknowledgments

This work was supported by NASA Grants NNX07AE29G and NNX10AGAH69G in support of the Precipitation Measurement Mission (PMM). We thank Dr. Mark Kulie for helpful discussions and two anonymous reviewers for suggestions that led to major improvements to the paper.

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^{1}

The conventional definition of “effective radius” is appropriate to spherical particles. See McFarquhar and Heymsfield (1998) for a variety of definitions applicable to nonspherical ice particles.