## Introduction

*D*and a reference variable [the

*i*th moment of DSD,

*M*

_{i}= ∫

*N*(

*D*)

*D*

^{i}

*dD*] in the following way:

*α*and

*β*are constant and do not have any functional dependence on

*M*

_{i}. The normalized function

*g*(

*x*

_{1}) =

*N*(

*D*)

*M*

^{−α}

_{i}

*M*

_{i}.

This general expression summarizes all of the previously suggested analytical expressions of DSDs and clarifies the relation between these expressions and the power-law relationships between moments of the distribution that are generally used. However, as shown in their original papers, the normalization is not very effective in collapsing all of the experimental data in a single *g*(*x*_{1}) when observed DSDs taken in a variety of situations are normalized all at once. That is, the scatter of observation points around the normalized DSD is not reduced appreciably by the single-moment scaling normalization. This scatter represents the limitation of the single-moment description of DSD variability.

Later on, Sempere-Torres et al. (1999, 2000) have shown that when DSDs are stratified according to rain type (convective, stratiform, and the transition between the two) the single-moment normalization of each type of DSD leads to different functions of *g*(*x*_{1}) and to a much lower scatter around these functions. This implies that at least a second moment, characterizing the shape of *g*(*x*_{1}), is necessary to capture some of the natural variability of DSDs.

Sekhon and Srivastava (1971) and Willis (1984) show the potential of the normalization of DSDs with two parameters, the median volume diameter (*D*_{0}) and a number density (*M*_{3}/*D*^{4}_{0}*M*_{3} or *D*_{0} and *R,* that is, reducing their normalization to a single-moment case [see (30) in Sekhon and Srivastava (1971)]. Sempere-Torres et al. (1994) used these previous works to propose their general single-moment scaling law of DSDs and their normalization methodology.

Testud et al. (2001) recently expanded their idea without any assumption on the functional form of DSDs and have proposed a double-moment normalization that leads to a more compact representation of all DSDs. Testud et al. point out that the remaining scatter around the normalized function is below the noise level of the disdrometric data, suggesting that their two parameters, the third and fourth moments of the DSDs, are sufficient to capture all of the discernable variability.

The purpose of this paper is to extend the ST single-moment scaling normalization to a double-moment scaling normalization and to establish an explicit relationship between the Testud et al. (2001) and ST approaches. The ST scaling normalization is rewritten in section 2 to demonstrate clearly the essential hypothesis and limitations underlying this scaling law. In section 3 a generalization of this scaling law is proposed, and it is used in section 4 to establish a connection between the Testud et al. (2001) and ST approaches. Some consequences and a simple data analysis of double-moment normalization are shown in sections 5 and 6. In section 7, we show that the DSD models (exponential, gamma, and generalized gamma DSDs) also have the scaling properties.

## Limitations of the scaling normalization

*N*

_{T}(m

^{−3}) (the 0th moment of the DSD) and the probability density function (pdf)

*p,*which gives the probability of finding a drop with a diameter interval between

*D*and

*D*+

*dD*(mm

^{−1}). If a characteristic diameter

*D*

_{c}is used to render the pdf dimensionless, then

*D*

_{c}can be the mean or the volume-weighted mean diameters, or, in general, any weighted diameter expressed in a general way as the quotient of any two consecutive moments, that is;

*D*

_{c}=

*M*

_{i+1}/

*M*

_{i}. In addition, the characteristic number density (m

^{−3}mm

^{−1}) can be expressed in general as

*N*

_{c}=

*M*

_{k}/

*D*

^{k+1}

_{c}

*p̂,*which may depend on a number of dimensionless parameters. Note that if

*k*= 0 then (2) is derived and when

*k*= 3 the normalization of Sekhon and Srivastava (1971) is obtained.

The hypothesis used by ST to obtain a general scaling law for DSDs with one moment *M*_{i} of the DSD as a reference variable is that the moments of the DSDs are related by power laws. Thus, *D*_{c} and *M*_{k}/*D*^{k+1}_{c}

*M*

_{k}as a function of the reference moment

*M*

_{i}. From the definition of

*D*

_{c}, it is a function of

*M*

_{i}, and, therefore, so is

*M*

_{k}/

*D*

^{k+1}

_{c}

*f*(

*M*

_{i}). By the hypothesis, the relation between two DSD moments can be written as a power law

*M*

_{n}

*C*

_{1,n}

*M*

^{γ(n)}

_{i}

*C*

_{1,n}is a constant that adjusts the units (the subscript 1 denotes the single-moment scaling normalization framework). The

*γ*(

*n*) is the exponent of the power law and depends on the moment order

*n*. Taking

*n*=

*j*and

*j*+ 1, the characteristic diameter can be expressed as a function of the reference variable as

*D*

_{c}

*C*

_{1,j+1}

*C*

_{1,j}

*M*

^{β}

_{i}

*β*=

*γ*(

*j*+ 1) −

*γ*(

*j*). Furthermore, introducing

*M*

_{k}/

*D*

^{k+1}

_{c}

*f*(

*M*

_{i}) and (3) and (5) in the definition of the DSD's

*n*th moment, we obtain

*C*

_{n}is a constant given by

*x*=

*D*/

*D*

_{c}. From (6) and (4), it follows that

*f*(

*M*

_{i}) =

*c*′

*M*

^{α}

_{i}

*c*′ = (C

_{1,n}/C

_{n})(C

_{1,j+1}/C

_{1,j})

^{−n−1}, and

*α*=

*γ*(

*n*) − (

*n*+ 1)

*β*. Substituting

*f*(

*M*

_{i}) and

*D*

_{c}into (3) we get

*x*

_{1}=

*DM*

^{−β}

_{i}

*c*″ = (C

_{1,j+1}/C

_{1,j})

^{−1}. This is the general expression proposed by ST, as expressed in (1), where

*g*(

*x*) is the pdf of the DSD affected by some constants taking into account the units. The

*n*th moment of the normalized DSD is thus defined as

*C*

_{1,n}is a constant that depends on the choice of the reference moment

*M*

_{i}.

*M*

_{n}= ∫

*N*(

*D*)

*D*

^{n}

*dD,*we obtain

*γ*(

*n*) =

*α*+ (

*n*+ 1)

*β.*In addition, taking

*n*=

*i,*a self-consistency constraint is obtained from (10):

*R*is chosen as the reference variable (approximated by the 3.67th moment), we get

Thus, all of the exponents of the power-law relationships between pairs of DSD moments are linear functions of *β*—in particular the exponent of the *R*–*Z* relationship^{1}—while the coefficients in these power laws are determined by moments of the *g*(*x*_{1}) as given by (9). For the *R*–*Z* relationship *Z* = *aR*^{b}, the coefficient *a* is given by the sixth moment (*C*_{1,6}), and the exponent is *b* = 1 + 2.33*β.* A series of publications (Uijlenhoet 1999; Salles et al. 2002; Uijlenhoet et al. 2003a, b) have further discussed these concepts.

From the above, we see that the single-moment scaling normalization will be effective only in situations in which power laws between moments are well defined. However, experience shows that, in general, power laws between any two moments of DSDs are statistical, and not deterministic, relationships, with much scatter around the best-fit power law. Thus, it is not surprising that the scatter of observation points around the average normalized DSD is not reduced appreciably by the single-moment scaling normalization.

An important step was added when Sempere-Torres et al. (1999, 2000) have shown that a preclassification of radar images into convective, stratiform, and transition regions of a storm leads to a better stratification by the scaling normalization, with much reduced scatter around the mean normalized DSD, at the same time as the power-law relationships between moments became more deterministic. It was shown that the parameter *β* and *g*(*x*_{1}) change with the type of rain, and, as a consequence, with the associated microphysical process leading to the formation of the DSDs (see also Uijlenhoet et al. 2003a,b). This suggests a generalization of (8) by a second normalization of *g*(*x*_{1}), using another moment as reference variable.

## A generalization of the scaling normalization

*β*and a shape of

*g*(

*x*

_{1}), as in Sempere-Torres et al. (1999, 2000). In the second normalization, the general function

*g*(

*x*

_{1}) is renormalized by introducing an additional reference variable, a moment of

*g*(

*x*

_{1}),

*M*

_{1,j}= ∫

*g*(

*x*

_{1})

*x*

^{j}

_{1}

*dx*

_{1}. We follow the same procedure as in the single-moment normalization. A form of the second-normalized DSD is [see (1) and (8)]

*g*

*x*

_{1}

*M*

^{δ}

_{1,j}

*h*

*x*

_{1}

*M*

^{−ε}

_{1,j}

*h*is the “second normalized” DSD. Here, ε and

*δ*are the new scaling or normalization exponents. The second-normalized diameter is defined as

*x*

_{2}=

*x*

_{1}

*M*

^{−ε}

_{1,j}

*N*

*D*

*M*

^{(j+1)/(j−i)}

_{i}

*M*

^{(i+1)/(i−j)}

_{j}

*h*

*x*

_{2}

*x*

_{2}

*DM*

^{1/(j−i)}

_{i}

*M*

^{−1/(j−i)}

_{j}

*C*

_{2,n}is defined by the

*n*th moment of the second normalized DSD

*h*(

*x*

_{2}) [

*C*

_{2,n}= ∫

*h*(

*x*

_{2})

*x*

^{n}

_{2}

*dx*

_{2}]. The newly introduced reference variable

*M*

_{1,j}disappears, and the

*j*th moment of the original DSD

*M*

_{j}is introduced. As a result, the four scaling exponents (

*α,*

*β,*

*δ,*and ε) disappear and only the orders (

*i*and

*j*) of the two DSD moments used in the normalization remain in (15). If we know two moments, the general form of normalized DSDs

*h*(

*x*

_{2}) is derived without any fitting procedure to decide the scaling exponent, as in the case of the single-moment scaling normalization. In addition, we have not assumed any functional form of the shape of normalized DSDs—that remains free and is to be determined from observations.

In the single-moment normalization, a simple power law between any two moments is assumed [see (4) or (10)]. The exponent of this power law is a function of scaling exponent *β* [*M*_{n} = *C*_{1,n}*M*^{1+(n−i)β}_{i}*n*th moment of *g*(*x*_{1}) [*C*_{1,n} = ∫ *g*(*x*_{1})*x*^{n}_{1}*dx*_{1}]. In a similar way, in the double-moment normalization, the coefficient of the multiple power law in (16) is now the *n*th moment of *h*(*x*_{2}) instead of *g*(*x*_{1}). However, the exponent is purely determined by the orders of two reference moments. Therefore, in this approach the role of *β* and the normalized general function *g,* which have been shown to be related to types of rainfall stratified from the characteristics of the radar echoes (Sempere-Torres et al 1999, 2000; Uijlenhoet et al. 2003a), is now played by the two moments of the original DSD used in the normalization. Thus, the two moments should jointly contain all of the information for the stratification of DSDs.

## Comparison with the normalization of Testud et al. (2001)

*N*

*D*

*N*

^{*}

_{0}

*F*

*D*

*D*

_{m}

*D*

_{m}is the volume-weighted mean diameter, a particular characteristic diameter that is derived by

*D*

_{m}=

*M*

_{4}/

*M*

_{3}, and

*N*

^{*}

_{0}

*N*

^{*}

_{0}

*C*

_{T}

*M*

^{5}

_{3}

*M*

^{−4}

_{4}

*C*

_{T}is an arbitrary constant that is chosen as 4

^{4}/Γ(4). Parameter

*N*

^{*}

_{0}

*F*(

*D*/

*D*

_{m}) is remarkably stable and is independent of different rain types.

Equations (18) and (19) are identical. Hence, when *i* = 3 and *j* = 4, *h*(*x*_{2}) is the same as *F*(*D*/*D*_{m}); apart from the constant *C*_{T} = 4^{4}/Γ(4). Thus, the normalization of Testud et al. (2001) is a particular case of the double-moment scaling normalization in which the moments of order 3 and 4 have been selected as the reference variables.

*N*

^{′}

_{0}

*D*

^{′}

_{m}

*M*

_{i}and

*M*

_{j}, which are not necessarily consecutive. Similar to (18), we can express DSDs in the following form:

*N*

*D*

*N*

^{′}

_{0}

*F*

*D*

*D*

^{′}

_{m}

*F*(

*x*) does not depend on

*N*

^{′}

_{0}

*D*

^{′}

_{m}

*n*th moment of the DSD is written as

*x*=

*D*/

*D*

^{′}

_{m}

*C*

_{T,n}is the

*n*th moment of

*F*(

*x*)[

*C*

_{T,n}= ∫

*x*

^{n}

*F*(

*x*)

*dx*] that depends on the moment order

*n.*Then, the

*i*th and

*j*th moments of DSDs can be expressed as a function of

*N*

^{′}

_{0}

*D*

^{′}

_{m}

*N*

^{′}

_{0}

*D*

^{′}

_{m}

*M*

_{i}and

*M*

_{j}:

*C*

_{2,n}=

*C*

_{T,n}(

*C*

_{T,i})

^{(n−j)/(j−i)}(

*C*

_{T,j})

^{(n−i)/(i−j)}. By imposing the self-consistency equations (

*C*

_{T,i}= 1 and

*C*

_{T,j}= 1), (23) becomes (see appendix B for the self-consistency)

*h*(

*x*

_{2}) =

*F*(

*x*). If

*i*= 3 and

*j*= 4,

*N*

^{′}

_{0}

*M*

^{5}

_{3}

*M*

^{−4}

_{4}

*N*

^{*}

_{0}

*D*

^{′}

_{m}

*M*

_{4}/

*M*

_{3}=

*D*

_{m}. That is, the

*N*

^{*}

_{0}

*D*

_{m}of Testud et al. (2001) are a particular case of

*N*

^{′}

_{0}

*D*

^{′}

_{m}

*M*

_{n}=

*C*

_{1,n}

*M*

^{α+(n+1)β}

_{i}

*F*′ is identical to

*g*(

*x*

_{1}) in (8).

*N*

^{′}

_{0}

*N*

^{*}

_{0}

*N*

^{′}

_{0}

*n*= −1 in (16) and from the definition of

*N*

^{′}

_{0}

*i*= 3 and

*j*= 4), we can derive

*N*

^{*}

_{0}

*N*

^{*}

_{0}

*D*)

*N*(

*D*)

*dD*=

*M*

_{−1}, the −1th moment of the DSD, apart from the constant,

*C*

_{T}/

*C*

_{2,−1}. However, particular caution is necessary in interpreting this result. A direct calculation of

*N*

^{*}

_{0}

*N*

_{0}from measured DSDs by extrapolating the linear portion of DSDs in the log

*N*(

*D*)-versus-

*D*diagram to

*D*= 0 mm. They found a constant

*N*

_{0}that is independent of rain intensities. However, the current work provides a mathematical way of deriving the intercept parameter using any two moments that are of interest. In other words, instead of a graphical extrapolation,

*N*

^{′}

_{0}

*N*

^{*}

_{0}

*N*

^{′}

_{0}

*N*

^{′}

_{0}

## Some consequences

*h*(

*x*

_{2}), as shown in (16). When we choose

*M*

_{i}=

*C*

_{u}

*R*(∼3.67th moment of the DSD) and

*M*

_{n}=

*Z*(sixth moment of the DSD), the following relationship

*Z*(

*M*

_{j},

*R*) is obtained:

*C*

_{u}is a constant that adjusts the units. In the single-moment scaling normalization, the coefficient

*a*of

*Z*=

*aR*

^{b}is the sixth moment of

*g*(

*x*

_{1}) and the exponent

*b*is related to the normalization exponent

*β*by

*b*= 1 + 2.33

*β.*In (29), the exponent of the

*R*–

*Z*relationship depends on the choice of

*j.*The coefficient is also a function of the second reference variable

*M*

_{j}and the moment order

*j.*For example, with

*j*= 0,

*a*=

*C*

_{2,6}

*C*

^{1.63}

_{u}

*M*

^{−0.63}

_{0}

*b*= 1.63; with

*j*= 3,

*a*=

*C*

_{2,6}

*C*

^{4.48}

_{u}

*M*

^{−3.48}

_{3}

*b*= 4.48. As a first approximation, all pairs of moments are related by a power law. Therefore, the

*R*–

*Z*relationship will be adjusted by the relationship between

*M*

_{j}and

*R.*For example, when

*M*

_{j}is proportional to

*R*(

*M*

_{j}=

*CR*), as expected in the equilibrium process (Zawadzki and Antonio 1988), we obtain a linear

*R*–

*Z*relationship (

*Z*=

*C*

_{2,6}

*C*

^{(j−6)/(j−3.67)}

_{u}

*C*

^{2.33/(j−3.67)}

*R*).

*M*

_{k}(

*N*

^{′}

_{0}

*M*

_{l}) from the definition of the moment of DSDs. From (21), we obtain the

*k*th and

*l*th moment of DSDs:

*M*

_{k}and

*M*

_{l}:

*M*

_{l}=

*C*

_{u}

*R*(∼3.67th moment of the DSD) and

*M*

_{k}=

*Z*(sixth moment of the DSD), we obtain

*Z*(

*N*

^{′}

_{0}

*R*):

*Z*(

*N*

^{*}

_{0}

*R*) from the above equation by taking

*N*

^{′}

_{0}

*M*

^{5}

_{3}

*M*

^{−4}

_{4}

*N*

^{*}

_{0}

*C*

_{T}:

*R,*

*b*= 1.5, in (32) and (33) is close to the climatological value (

*Z*= 210

*R*

^{1.47}in Montreal, Quebec, Canada, e.g.). This implies that the correlation between

*N*

^{′}

_{0}

*R*is low when a set of data is taken from a climatological variety of situations, as shown by Testud et al. (2001) for

*N*

^{*}

_{0}

*R.*As already discussed by Testud et al. (2001), the actual exponent in the

*R*–

*Z*relationship depends on the correlation between

*N*

^{′}

_{0}

*R.*For example, for the equilibrium process, the exponent is equal to unity because any moments of the DSD are linearly related—that is,

*N*

^{*}

_{0}

*M*

_{−1}) is proportional to

*R.*For

*M*–

*P*DSDs for which

*N*

^{′}

_{0}

^{3}m

^{−3}mm

^{−1}), we expect

*Z*∝

*R*

^{1.5}. Similarly, we can derive

*Z*(

*D*

^{′}

_{m}

*R*):

*Z*

*C*

_{2,6}

*C*

^{−1}

_{2,3.67}

*C*

_{u}

*D*

^{′}

_{m}

^{2.33}

*R.*

*D*

^{′}

_{m}

*R*–

*Z*relationship. For

*M*–

*P*DSDs (

*D*

^{′}

_{m}

*R*

^{0.21}), the exponent of the

*R*–

*Z*relationship is 1.5.

## Data analysis

Some data analysis will help in adding perspective to the question of double-moment normalization. The data used here are the same as in Sempere-Torres et al. (1999, 2000) and are composed of 1208 one-minute DSDs (over 20 h) measured by the optical spectropluviometer (Salles et al. 1998). DSDs are divided into convective and stratiform rain using the presence of a bright band (BB) and a horizontal gradient of reflectivity, obtained from a nearby scanning radar. Details of the stratification procedure are given in Sempere-Torres et al. (2000).

### Compact representation of DSDs

The normalization of the set of these data is shown in Fig. 1 for the single moment (*R*) and Fig. 2 for the double moment (*M*_{i} and *M*_{j}). In the single-moment normalization, we follow the procedure described by Sempere-Torres et al. (1998), *R* and *M*_{n} (2 ≤ *n* ≤ 6) are calculated, and then the exponent *γ*(*n*) of *M*_{n} = *C*_{1,n}*R*^{γ(n)} is derived using weighted total least squares fitting (WTLS; Amemiya 1997) in log–log coordinates. The scaling exponent *β* is derived from WTLS between the calculated exponent *γ*(*n*) and the moment of order *n* + 1 [*γ*(*n*) = *α* + (*n* + 1)*β*]. The other scaling exponent *α* is derived from the self-consistency constraint in (12). Then, *N*(*D*) and *D* are normalized with *R*^{α} and *R*^{β}.

The scaling exponent *β* is slightly smaller than the value of *M*–*P* DSDs, indicating that the exponent of *Z* = *aR*^{b} is less than 1.5. The scatter of normalized DSDs in Fig. 1a and the standard deviation in Fig. 1b (vertical bars) are large. This result shows the limitation of single-moment scaling normalization in terms of compact representation of DSDs. When all DSDs from different physical processes are normalized together, they do not collapse onto one normalized curve. In other words, all the DSD variability cannot be explained by a single parameter.

We now show results from the double-moment scaling normalization. From 1-min DSDs, *N*^{′}_{0}*M*^{(j+1)/(j−i)}_{i}*M*^{(i+1)/(i−j)}_{j}*D*^{′}_{m}*M*_{j}/*M*_{i})^{1/(j−i)}] are calculated, and then *N*(*D*) and *D* are normalized with calculated *N*^{′}_{0}*D*^{′}_{m}*i* = 3, *j* = 4) in Figs. 2a and 2b are very similar to the ones reported by Testud et al. (2001). In general, the scatter drastically decreases as compared with the single-moment normalization, illustrating an advantage of double-moment normalization in terms of a compact representation of DSDs. By normalizing with (*i* = 3, *j* = 6) in Figs. 2c and 2d, the scatter at smaller normalized diameters (<0.5) slightly increases and vice versa at bigger diameters (>1.5).

The standard deviation (SD: thick vertical bars in Figs. 2b and 2d) from both analyses is still larger than that from the statistical noise (lighter vertical bars next to SD) derived by assuming Poisson fluctuations due to undersampling (see appendix C). This result can be explained by two facts: 1) the possible physical variability that cannot be described by this normalization and 2) the underestimation of the statistical noise by the Poisson process. The statistical fluctuation based on the Poisson statistics usually assumes uniform rain for sampling time (60 s). Jameson and Kostinski (2001) showed that the statistical fluctuation in DSDs can be significantly larger than expected from Poisson statistics when the correlation of rain in time is considered. In addition, the “observational noise” due to the drop sorting adds to variability of observed DSDs.

*g*(

*x*

_{1})

*h*(

*x*

_{2})

*g*(

*x*

_{1})

*h*(

*x*

_{2})

*R*in the single-moment normalization and with

*M*

_{3}and

*M*

_{4}in the double-moment normalization. Then, the moments of DSDs are calculated from the estimated and the original measured DSDs. Last, the standard deviation of fractional error (SDFE) in the

*n*th moment is calculated for both normalizations using the following equation:

*k*is the total number of data and the subscript est indicates the estimated amount. Instead of the uncertainty in the moments, the uncertainty in

*N*(

*D*) can also be obtained by comparing the estimated and the original measured DSDs (not shown here). Because of the self-consistency [

*g*(

*x*

_{1})

*h*(

*x*

_{2})

*M*

_{3.67}for the single-moment normalization (solid line) and at

*M*

_{3}and

*M*

_{4}for the double-moment normalization. In the single-moment normalization, SDFE drastically increases at low and high moments because of large scatter at smaller and bigger normalized diameters in Fig. 1a. This result shows again that the DSD variability cannot be fully described by a single moment (

*R*). However, an error with the double-moment normalization is less than 30% at the overall range of

*n.*These results show the superiority of the double-moment normalization to describe the DSD variability.

A similar error analysis is performed for various combinations of two moments used for the normalization (Fig. 4). When two consecutive moments are used, the error is almost zero for moments close to the ones used for the normalization because of the self-consistency constraints. When the order of the two moments used for the normalization is lower (higher), the error is smaller at lower (higher) moments. The minimum is broader when the order is higher. This fact simply indicates that the slope of the DSDs has less variability at the larger drop sizes.

When the reflectivity factor (*M*_{j} = *M*_{6}) and another moment (*M*_{i}) are used for the normalization, the standard deviation of the fractional error of Fig. 4b is obtained. Again, not surprising, there are two minima (zero) in the error. Because the order *i* is lower, the error at lower (higher) moments decreases (increases). When the order of two moments is far from each other, the overall error is much lower and the error between two moments slightly increases. However, *R* (*n* = 3.67) is estimated always with a precision better than 10%. Because the reflectivity factor is directly measured from radar, in the application to radar remote sensing we prefer to fix the one moment as the reflectivity factor. As mentioned, disdrometric measurements are affected by the statistical uncertainty from the small sampling volume. However, with the sampling volume of the radar, the statistical uncertainty does not play much of a role, but the physical fluctuations of *h*(*x*_{2}) do. Therefore, the applicability of “the optimization study” represented by Fig. 4 to radar remote sensing measurements remains to be explored.

### Consistency of the scaling law for observed DSDs

We now explore how well the observed DSDs at the ground follow a scaling law. To satisfy the double-moment scaling law, observed DSDs should obey (16).

Figure 5 shows the parameters of a multiple power law derived from the scaling formalism (solid line) in (16) and from direct least squares fitting (dashed line). From (16), the exponents should be a linear function of moment order *n,* and the coefficient is the *n*th moment of normalized DSD *h*(*x*_{2})*n* ≤ 8. This result illustrates that observed DSDs can be described reasonably well by the scaling formalism. However, the scaling law is not followed well at small-sized drops, indicated by the discrepancy of both lines at lower moments (*n* ≤ 2). Because lower moments (small-sized drops) are more severely affected by instrumental limitation, drop sorting, and evaporation, we are not certain whether the discrepancy illustrates the failure of the scaling law at a smaller size or the observational limitations.

### Connection between scaling normalizations and physical processes

We now compare the single-moment normalization with the double-moment normalization on the data stratified according to precipitation types (stratiform and convective rain). This comparison provides an idea of the feasibility of both normalizations to identify different precipitation types. From the entire dataset of Sempere-Torres et al. (1999, 2000) we select only those that were identified as stratiform (precipitation with a clearly defined bright band) and those that were classified as convective (no bright band and strong horizontal gradients). For stratiform rain, only the last period (0240–0430 UTC 15 October 1996) is taken, because this period shows the most clearly identified intense bright band. The transition periods, being more ambiguous, are not discussed here. We show results for *i* = 3 and *j* = 4, although the results are similar for other pairs of moments.

Figure 6a shows the *R*–*Z* regression for the two types of precipitation. Although the points are weakly separated, the difference in the two regressions is statistically significant. Note the significantly different exponent. Figure 6b shows the exponent *γ*(*n*) of the power-law relationship [*M*_{n} = *C*_{1,n}*R*^{γ(n)}] between *R* and all other moments of the indicated order. Again, the two regression lines are clearly distinctive for the convective and stratiform rain. The slope of these two regression lines defines the scaling exponent *β* of single-moment normalization for the two populations.

We have said that in the single-moment normalization the parameters of the relationships between moments of the distribution are determined by *β* (determining the exponents of the power laws) and the moments of the *g*(*x*_{1}) function (determining the coefficients of the power laws). For example, in *Z* = *aR*^{b} the exponent *b* is given by *b* = 1 + 2.33*β* and the coefficient *a* can be obtained by the sixth moment of *g*(*x*_{1}). In the dataset considered here, *β* separates well the convective and stratiform DSDs. In the double-moment normalization, the same information is shared between the two moments. The question arises then as to how well the two parameters, *N*^{′}_{0}*D*^{′}_{m}

In Figs. 7a and 7b, the double-moment normalization for stratiform and convective rain shows similar characteristics, although the scatter is slightly less for convective rain. The average normalized DSDs *h*(*x*_{2})*h*(*x*_{2})*h*(*x*_{2})*h*(*x*_{2})

The next question is how well two moments, or *N*^{′}_{0}*D*^{′}_{m}*β* that nicely separates the two rain regimes in the single-moment normalization. Testud et al. (2001) show that *N*^{′}_{0}*D*^{′}_{m}*N*^{′}_{0}*D*^{′}_{m}*Z* = *aR*^{b} obtained by direct WTLS, from the single-moment normalization (*b* = 1 + 2.33*β*), and from the relationships between *D*^{′}_{m}*R* [(34)], as well as *N*^{′}_{0}*R* [(31)]. Unlike the result of Testud et al. (2001), Fig. 8a shows that the separation of the two types of precipitation in the (*N*^{′}_{0}*D*^{′}_{m}*r*^{2} = 0.01 in Table 1) and has a wide distribution with an upper limit of *N*^{′}_{0}^{2} m^{−3} mm^{−1}. Some points from convective precipitation are mixed with those from stratiform rain. However, stratiform rain has a good correlation (*r*^{2} = 0.66); that is, *N*^{′}_{0}*D*^{′}_{m}^{−1}, then the given spectrum is considered as stratiform; otherwise, it is considered to be convective rain. Hence, in their classification, stratiform rain may include weak convection that has rain less than 10 mm h^{−1}. If we include rain at this range as stratiform rain, the separation of the two types of precipitation becomes obviously more evident than before. However, although they had weak precipitation, they clearly showed no BB and a strong horizontal gradient, satisfying convective rain.

The regression relationship between these parameters and *R* is different for the two types of rain. For stratiform precipitation, the correlation between *D*^{′}_{m}*R* is very good (*r*^{2} = 0.54) but is totally nonsignificant between *N*^{′}_{0}*R* (*r*^{2} = 0.04). For the convective rain, the correlation between *N*^{′}_{0}*R* is somewhat better (*r*^{2} = 0.30). In addition, the single-moment normalization provides similar exponents of *R*–*Z* relationships as that from the direct fitting. In the double-moment normalization, the relationships (*D*^{′}_{m}*R*) lead to exponents consistent with those from direct fitting, whereas the exponents from the relationship (*N*^{′}_{0}*R*) show significant deviation.

## The functions *g*(*x*_{1}) and *h*(*x*_{2}) and a scaling model distribution

In the previous sections we have shown that the scaling properties can be studied in observed DSDs without any assumption on the form of the generic distributions *g*(*x*_{1}) and *h*(*x*_{2}). Decades of experience with DSD observations indicate that these display a variety of forms: the quasi-exponential distribution in stratiform rain, the Gaussian-shaped near-monodisperse maritime rain, the S-shaped equilibrium DSDs, the gamma form of evaporating rain, and so on. Our limited data analysis does not intend to represent all of the richness found in nature.

Scaling normalization collapses individual observations into a single *g*(*x*_{1}) or *h*(*x*_{2}) function by displacement (scaling number concentration) and pivoting (scaling size). In the single-moment normalization, the displacement and pivoting are deterministically related; in the double-moment normalization, there is a degree of independence between the two. The scaling normalizations cannot change the shape of the distribution for all of the forms mentioned above to fit them all into one single function at once.

We have shown that the description of DSDs in terms of a double-moment scaling law leads to multiple power laws among moments of DSDs [see (16)]. As intrinsic properties of scaling formalism of DSDs, the exponents of this power law are purely determined by the order of the reference moments and the coefficients depend on the shape of scaling DSDs. We will show in this section that a functional model distribution can contain all the observed forms and, at the same time, include the scaling properties.

Several distribution functions (exponential, lognormal, and gamma) have been used as models to describe naturally occurring DSDs. For the larger drops, the exponential distribution nicely describes climatological averages of DSDs in the lower rain intensities (Marshall and Palmer 1948). Deviations from the exponential form of individual DSDs can be accounted for by the three-parameter gamma DSD. However, the gamma distribution also shows some limitations to describe naturally occurring DSDs, such as the S-shaped equilibrium DSDs. The generalized gamma distribution has more flexibility than the gamma distribution. Several authors illustrate this flexibility and show that observed DSDs can be described better by the generalized gamma distribution (Amoroso 1925; Suzuki 1964; Uijlenhoet 1999; Auf der Maur 2001).

*D*≥ 0 with probability density function

*μ,*

*c,*and

*λ*(Stacy 1962). The parameters

*c,*

*μ,*and

*λ*have to be positive. The

*n*th moments

*m*

_{n}are given by

*M*

_{0}(m

^{−3}) (the zeroth moment of the DSD) and the probability density function

*p*(mm

^{−1}). If this probabilistic concept is applied to particle size distributions (Auf der Maur 2001), we have the following form:

*n*th moment of the DSD

*M*

_{n}becomes

*i*th and

*j*th moments of DSDs are

*λ*and

*M*

_{0}in terms of the two moments,

*M*

_{i}and

*M*

_{j}:

*λ*has the form of the characteristic diameter

*D*

^{′}

_{m}

*μ*and

*c.*In addition, the zeroth moment is defined by a multiple power law. Then, combining the expressions of

*λ*and

*M*

_{0}into

*M*

_{n}, we find the relationship between

*M*

_{n}and two moments (

*M*

_{i}and

*M*

_{j}):

The subscript GG indicates the generalized gamma DSD. This form is similar to the multiple power law in (16); that is, the two exponents are deterministic and do not depend on the shape parameters (*μ* and *c*) of DSDs. The self-consistency constraints of the double-moment scaling are also satisfied (*C*_{GG,2,i} = 1 and *C*_{GG,2,j} = 1).

*N*

^{′}

_{0}

*M*

^{(j+1)/(j−i)}

_{i}

*M*

^{(i+1)/(i−j)}

_{j}

*D*

^{′}

_{m}

*M*

_{j}/

*M*

_{i})

^{1/(j−i)}] in (24), we obtain

*x*

_{2}=

*D*/

*D*

^{′}

_{m}

*n*th moment of

*h*

_{GG,(i,j,μ,c)}(

*x*

_{2}) provides the coefficient of the multiple power law

*C*

_{GG,2,n}. The flexibility of (43) to describe all the observed shapes of DSDs is illustrated in Fig. 9, where the two moments that determine LWC and

*D*

^{′}

_{m}

Scaling properties are prevalent in natural phenomena, and this observation led to the idea of the scaling normalization of DSDs described in sections 2 and 3. The results in (42) and (43) show that the generalized gamma DSD also satisfies scaling properties. Because all naturally occurring DSDs can be described reasonably well by the generalized gamma DSD, it suggests a very general description of all types of DSDs within the scaling framework. It also illustrates well the limitations of the double-moment scaling normalization. In the derivation of the scaling formalism given in sections 3 and 4, no assumption on the shape is imposed. On the other hand, (43) explicitly shows the shape of normalized DSDs. This shape is not unique but depends on the parameters *μ* and *c.* In other words, when original DSDs that have distinctive shapes are normalized with two moments, the normalized DSDs cannot lead to a unique shape. Different physical processes can lead to distinctive shapes of DSDs and will require different values of *μ* and *c* to adjust their shapes. Therefore, we expect a certain degree of scatter in the normalized DSDs because of the physical variability when all DSDs from different physical processes, associated with different DSD shapes, are normalized together.

The exponential and gamma DSD model (particular cases of the generalized gamma DSD) also follow the scaling law. From (43), we show a form of the general double-moment scaling normalized DSDs:

- an exponential DSD model with
*μ*= 1 and*c*= 1:for example, when*i*= 3 and*j*= 4, - a gamma DSD model with
*c*= 1:for example, when*i*= 3 and*j*= 4,where*x*_{2}=*D*/ ,*D*^{′}_{m} =*N*^{′}_{0}*M*^{(j+1)/(j−i)}_{i} , and*M*^{(i+1)/(i−j)}_{j} = (*D*^{′}_{m}*M*_{j}/*M*_{i})^{1/(j−i)}. These two examples, when*i*= 3 and*j*= 4, are similar to those from Testud et al. (2001) and Illingworth and Blackman (2002).

To find the functional form of normalized DSDs in our dataset, *μ* and *c* are determined by a Monte Carlo least squares fitting of (43) to average normalized DSD *h*(*x*_{2})*h*_{GG} and log*h**μ* and *c.* Results are shown in Fig. 10 and Table 2, together with the normalized exponential DSD in (45). A similar curve is obtained from the fitting with overall data points in Figs. 2a and 2c (not shown). The normalized general gamma DSD fits well to *h*(*x*_{2})*h*(*x*_{2}) has an “S shape” with slight deviations from the exponential *h*(*x*_{2}), that is, a relative abundance of drops when *x*_{2} < 0.3 and a relative absence when *x*_{2} > 1.5. No particular general significance should be assigned to these values of *μ* and *c* because the database used here is very limited.

*M*

_{n}=

*C*

_{1,n}

*M*

^{α+(n+1)β}

_{1}

*α*+ (

*i*+ 1)

*β*= 1, we obtain the normalized form of the generalized gamma DSDs with a single moment,

*x*

_{1}=

*DM*

^{−β}

_{i}

*C*

_{1,0}is a constant defined as

*M*

_{0}/

*M*

^{α+β}

_{i}

*g*

_{GG,(i,μ,c)}(

*x*

_{1}) as in (9). Similar to

*h*

_{GG,(i,j,μ,c)}(

*x*

_{2}), the shape of the normalized DSD

*g*

_{GG,(i,μ,c)}(

*x*

_{1}) is not unique but depends on the shape parameters

*μ*and

*c.*Unlike

*h*

_{GG,(i,j,μ,c)}(

*x*

_{2}), there is an extra constant

*C*

_{1,0}that adjusts units. Hence, the unit of

*C*

^{1/i}

_{1,0}

*M*

^{−β}

_{i}

- an exponential DSD model with
*μ*= 1 and*c*= 1: - a gamma DSD model with
*c*= 1:

*E*and

*G*indicate the exponential and gamma DSD, respectively. As an example of the exponential DSDs, Marshall and Palmer (1948) obtained the following exponential DSD from experimental data, although their equation is not perfectly self-consistent:

*β*= 0.214. With

*λ*= 4.1

*R*

^{−0.214}and the terminal fall velocity of drops

*υ*(

*D*) = 3.778

*D*

^{0.67}(m s

^{−1}), we obtain a similar value of

*N*

_{0}= 7.0 × 10

^{3}(m

^{−3}mm

^{−1}) from (49). To satisfy

*N*

_{0}= 8.0 × 10

^{3}(m

^{−3}mm

^{−1}), we expect

*λ*= 4.23

*R*

^{−0.214}.

## Discussion

We have shown here in some detail that ST's and Testud et al.'s (2001) formulations of normalized DSDs are particular cases of the general scaling normalization. No functional form of DSDs is imposed in the normalization. Therefore, the general scaling normalization can reveal any stable shape of normalized DSDs. We must emphasize first that the description of the DSDs with the double-moment normalization captures the pivoting and the displacement of DSDs with changing rain intensity. The former is given by the scaling of *D* with the characteristic diameter, and the latter is given by the scaling of *N*(*D*) with the characteristic number density. Therefore, DSDs that have different slopes and intercept parameters can be collapsed onto a unique normalized DSD. The normalization cannot change the various shapes (especially different curvatures) of DSDs that result from the complex physical processes shaping the distribution, however. Thus, when DSDs that have distinctive curvatures originating from various physical processes are normalized together, they cannot lead to a unique normalized DSD *h*(*x*_{2}). This point is illustrated by the functional dependence of *h*(*x*_{2}) on the shape parameters (*μ* and *c*) of the generalized gamma DSD in (43). Hence, various *h*(*x*_{2}) are expected from the S-shaped curve of the equilibrium DSDs, the inverse exponential often observed in rain, the gamma shape resulting from evaporation, and the quasi-monodisperse DSD sometimes observed in drizzle. The differences in these shapes may be small enough after normalization, but they are nevertheless present. In fact, because of the usually small sample volume of disdrometers and other observational limitations, the physical variation of the shape is often masked by the measurement noise, as indicated by the data analysis in Fig. 2. Therefore, a more sophisticated data analysis is necessary to reveal the connection between physical processes and the shape of DSDs.

Both the single- and double-moment scaling normalization capture DSD pivoting and displacement. The difference is that, while in the single-moment normalization [(8)] the pivoting and displacement are intrinsically related (consequence of power-law relationship between any two moments), the double-moment normalization [(15)] allows partial independence between the two.

The data analysis shows that the double-moment scaling normalization is remarkably effective in collapsing all DSDs around a mean shape (Figs. 2 and 3). The remaining variability can probably be neglected in most applications, and the double-moment normalization provides an excellent model of DSD variability.

The question remains as to whether the choice of the third and fourth moments for the normalization is the best. If the double-moment normalization is used as a tool for cloud physics, it seems that the selected combination of moments should have some clear meaning referring to the problem at hand. Depending on the physical problem to be addressed, it is imaginable to use *M*_{0} because it represents the total number of particles (a “conservative” quantity in warm nonprecipitating clouds); *M*_{2} because, combined with *M*_{3}, it defines the “effective radius;” *M*_{3} because it represents the LWC; or *M*_{4} because, combined with *M*_{3}, it defines the mean volume diameter *D*_{m}. For the physics of precipitation, the interest of the combination (*M*_{3}, *M*_{4}) appears clearly. However, in the application to radar remote sensing, radar measurables such as the sixth moment (*Z*) and 4.6th moment (propotional to *K*_{DP}) is more practical. Furthermore, from the point of view of radar data assimilation, the measurable moments are also preferred.

The single-moment scaling normalization applied after a stratification of DSDs according to a likely dominance of a given microphysical process (Fig. 6) shows that the scaling exponent *β* is a clear indicator of the processes. However, in the double-moment normalization, the separation of *N*^{′}_{0}*D*^{′}_{m}*Z* = *aR*^{b}) is not well determined, especially with the *N*^{′}_{0}*R* relationship (Table 1). Thus, for studies of the relationship between microphysics and DSDs, the single-moment scaling normalization seems preferable. In relation to this point, let us note again that, while in the single-moment normalization *β* measures the exponents of all power laws between DSD moments and the moments of *g*(*x*) give the coefficients (i.e., the information on the two is nicely separated), in the double-moment normalization the two moments jointly contain the same information. In other words, the information on the exponents and the coefficients is mixed within *N*^{′}_{0}*D*^{′}_{m}

## Acknowledgments

This work was triggered by discussions during the stay of the second and fifth authors as visiting professors at the Universitat Politecnica de Catalunya, but the bulk of the work was done by the first author as part of his Ph.D. thesis. We are indebted to the Generalitat de Catalunya for supporting their stay. This work is also partly supported by the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS). The fifth author is supported by the Netherlands Organization for Scientific Research (NWO) through Grant 016.021.003. The second author was greatly stimulated by discussions with Dr. Jacques Testud on this subject. The comments of one of the anonymous reviewers were critical in shaping the final form of this paper.

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## APPENDIX A

### A Generalization of the Normalization of Sempere-Torres et al. (1994, 1998)

*g*(

*x*

_{1}) is renormalized by introducing an additional reference variable. We follow the same procedure as before: in the second normalization the reference variable is a

*j*th moment of

*g*(

*x*

_{1}),

*M*

_{1,j}= ∫

*g*(

*x*

_{1})

*x*

^{j}

_{1}

*dx*

_{1}. A form of the second normalized DSD is [see (1) and (8)]

*g*

*x*

_{1}

*M*

^{δ}

_{1,j}

*h*

*x*

_{1}

*M*

^{−ε}

_{1,j}

*h*is the second-normalized DSD, and ε and

*δ*are the new normalization exponents. The second-normalized diameter is defined as

*x*

_{2}=

*x*

_{1}

*M*

^{−ε}

_{1,j}

*h*are given by

*C*

_{2,n}are again constants that depend on the shape of

*h,*the choice of

*M*

_{1,j}, and the value of

*n.*By combining (A1) and (A2), we obtain the power law between

*M*

_{1,j}and

*M*

_{1,n}:

*M*

_{1,n}

*C*

_{2,n}

*M*

^{δ+(n+1)ε}

_{1,j}

*n*=

*j*:

*g*(

*x*

_{1}) can be expressed as a function of two moments of the original DSDs:

*M*

_{1,j}can be obtained by applying

*n*=

*j.*Replacing (A6) into (A4), we obtain the general double-moment relationship between moments of DSDs:

*M*

_{1,j}of the second normalization does not appear in the equation; instead, (A7) is expressed in terms of the

*j*th moment of the DSD only. Furthermore, we can express

*x*

_{2}and

*h*(

*x*

_{2}) as functions of the

*i*th and

*j*th moments of DSD instead of moments of

*g*(

*x*

_{1}):

*M*

_{i}and

*M*

_{j}:

*β*and ε. Furthermore, an additional self-consistency constraint is obtained by setting

*n*=

*i*in the above equation. This leads to

*R*(∼3.67th moment) and

*Z*(sixth moment) are used, ε ∼ 0.43. Combining (A13) and (A12) we get

*M*

_{n}

*C*

_{2,n}

*M*

^{(n−i)/(j−i)}

_{j}

*M*

^{(j−n)/(j−i)}

_{i}

*x*

_{2}and

*h*(

*x*

_{2}) as functions of the

*i*th and

*j*th moments of DSD;

Last, note that if (A14) is taken as the starting hypothesis it is easy to show that (A17) follows. We have taken here the long route in our derivation to clearly establish the relationship between ST's single-moment normalization and the double-moment normalizations given here.

## APPENDIX B

### A Generalization of the Normalization of Testud et al. (2001)

*M*

_{i}and

*M*

_{j}, where

*i*and

*j*are not necessarily consecutive. A general form of the normalized DSD with any two moments can be defined in the following manner:

*N*

*D*

*M*

^{αT}

_{i}

*M*

^{δT}

_{j}

*F*

*x*

*x*

*DM*

^{−βT}

_{i}

*M*

^{−εT}

_{j}

*α*

_{T},

*β*

_{T},

*δ*

_{T}, and ε

_{T}are newly introduced normalization exponents. Here

*F*(

*x*) is the intrinsic shape of the DSD and depends on the two moments chosen as reference variables. From (B1), the

*n*th moment of the DSD can be derived;

*F*(

*x*) is independent of any

*M*

_{i}and

*M*

_{j}, the

*n*th moment of

*F*(

*x*) is a constant

*C*

_{T,n}that depends on the moment order

*n*and the two reference moments

*M*

_{i}and

*M*

_{j}. Then, (B2) can be rewritten as the double-moment relationship between the

*n*th moment and two reference variables:

*C*

_{T,n}is the

*n*th moment of

*F*(

*x*), and the exponents depend on the normalization parameters. In the above equation, the following self-consistencies are obtained by applying

*n*=

*i*and

*n*=

*j*:

*i*and

*j,*

## APPENDIX C

### Statistical Uncertainty in Normalized DSDs Due to Undersampling

*N*

_{tot}(

*D*

_{i}) within the diameter interval Δ

*D*

_{i}(mm) centered in

*D*

_{i}(mm) from the homogeneous rain. The size of each sampling volume

*V*(

*D*

_{i}) (m

^{3}s

^{−1}) is identical. If the numbers of drops counted follow the Poisson distribution, we obtain the following characteristics:

*N*(

*D*

_{i})

^{−3}mm

^{−1}) is the average number density at a diameter interval Δ

*D*

_{i}and

*σ*

^{2}{ } indicates the variance. From this, we derive the variance in the estimate of the number density

*σ*

^{2}{

*N*(

*D*

_{i})}:

*h*(

*x*

_{2}) =

*N*(

*D*

_{i})/

*N*

^{′}

_{0}

Double-moment normalization on data from Sempere-Torres et al. (1999, 2000). (a) Scattergram of all normalized DSDs with *M*_{i} = *M*_{3} and *M*_{j} = *M*_{4}. An exponential adjustment is shown as a dashed line. (b) The average *h*(*x*_{2})*g*(*x*_{1})*h*(*x*_{2})*M*_{i} = *M*_{3} and *M*_{j} = *M*_{6}.

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Double-moment normalization on data from Sempere-Torres et al. (1999, 2000). (a) Scattergram of all normalized DSDs with *M*_{i} = *M*_{3} and *M*_{j} = *M*_{4}. An exponential adjustment is shown as a dashed line. (b) The average *h*(*x*_{2})*g*(*x*_{1})*h*(*x*_{2})*M*_{i} = *M*_{3} and *M*_{j} = *M*_{6}.

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Double-moment normalization on data from Sempere-Torres et al. (1999, 2000). (a) Scattergram of all normalized DSDs with *M*_{i} = *M*_{3} and *M*_{j} = *M*_{4}. An exponential adjustment is shown as a dashed line. (b) The average *h*(*x*_{2})*g*(*x*_{1})*h*(*x*_{2})*M*_{i} = *M*_{3} and *M*_{j} = *M*_{6}.

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

SDFE in moment estimation using *g*(*x*_{1})*h*(*x*_{2})*n*

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

SDFE in moment estimation using *g*(*x*_{1})*h*(*x*_{2})*n*

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

SDFE in moment estimation using *g*(*x*_{1})*h*(*x*_{2})*n*

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) SDFE in the estimate of the *n*th moment from the average normalized drop size distribution *h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) SDFE in the estimate of the *n*th moment from the average normalized drop size distribution *h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) SDFE in the estimate of the *n*th moment from the average normalized drop size distribution *h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Parameters of multiple power law *M*_{n} = *aM*^{b}_{i}*M*^{c}_{j}*i* = 3 and *j* = 6 (Figs. 2c, d). In the double-moment normalization, the two exponents are determined from (16) and the coefficient is derived from the *n*th moment of *h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Parameters of multiple power law *M*_{n} = *aM*^{b}_{i}*M*^{c}_{j}*i* = 3 and *j* = 6 (Figs. 2c, d). In the double-moment normalization, the two exponents are determined from (16) and the coefficient is derived from the *n*th moment of *h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Parameters of multiple power law *M*_{n} = *aM*^{b}_{i}*M*^{c}_{j}*i* = 3 and *j* = 6 (Figs. 2c, d). In the double-moment normalization, the two exponents are determined from (16) and the coefficient is derived from the *n*th moment of *h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) The *R*–*Z* WTLS regressions for stratiforms and convective rain. (b) Exponent *γ*(*n*) of *M*_{n} = *C*_{1,n}*R*^{γ(n)} as a function of *n.* The scaling exponent *β* is determined by the slope in *γ*(*n*) vs *n* [*γ*(*n*) = *α* + (*n* + 1)*β*]

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) The *R*–*Z* WTLS regressions for stratiforms and convective rain. (b) Exponent *γ*(*n*) of *M*_{n} = *C*_{1,n}*R*^{γ(n)} as a function of *n.* The scaling exponent *β* is determined by the slope in *γ*(*n*) vs *n* [*γ*(*n*) = *α* + (*n* + 1)*β*]

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) The *R*–*Z* WTLS regressions for stratiforms and convective rain. (b) Exponent *γ*(*n*) of *M*_{n} = *C*_{1,n}*R*^{γ(n)} as a function of *n.* The scaling exponent *β* is determined by the slope in *γ*(*n*) vs *n* [*γ*(*n*) = *α* + (*n* + 1)*β*]

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a), (b) Double-moment normalization for the two types of rain separated by the presence of a bright band and a horizontal gradient of reflectivity. The average normalized DSD *h*(*x*_{2})*h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a), (b) Double-moment normalization for the two types of rain separated by the presence of a bright band and a horizontal gradient of reflectivity. The average normalized DSD *h*(*x*_{2})*h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a), (b) Double-moment normalization for the two types of rain separated by the presence of a bright band and a horizontal gradient of reflectivity. The average normalized DSD *h*(*x*_{2})*h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) Distribution of points in the (*N*^{′}_{0}*D*^{′}_{m}*N*^{′}_{0}*D*^{′}_{m}

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) Distribution of points in the (*N*^{′}_{0}*D*^{′}_{m}*N*^{′}_{0}*D*^{′}_{m}

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

(a) Distribution of points in the (*N*^{′}_{0}*D*^{′}_{m}*N*^{′}_{0}*D*^{′}_{m}

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

A sample of possible shapes that can be represented by the generalized gamma function. The curves shown are computed for an LWC of 0.5 g m^{−3} and mean diameter *D*^{′}_{m}

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

A sample of possible shapes that can be represented by the generalized gamma function. The curves shown are computed for an LWC of 0.5 g m^{−3} and mean diameter *D*^{′}_{m}

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

A sample of possible shapes that can be represented by the generalized gamma function. The curves shown are computed for an LWC of 0.5 g m^{−3} and mean diameter *D*^{′}_{m}

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Adjustment of *h*(*x*_{2})*h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Adjustment of *h*(*x*_{2})*h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Adjustment of *h*(*x*_{2})*h*(*x*_{2})

Citation: Journal of Applied Meteorology 43, 2; 10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2

Regression parameters and the determination coefficient (*r*^{ 2} ) between various parameters of the single-moment (*R*) and double-moment (*i* = 3 and *j* = 4) normalizations for the relationships indicated in the upper row. The last column gives the exponent of the *R*–*Z* relationship obtained from direct WTLS. Boldface indicates statistical significance

Parameters of *h* _{GG, (i , j , μ, c )} (*x*_{ 2} ) in (43) adjusted to *h* (*x*_{ 2} )

^{1}

We express the relationship in the conventional *Z*–*R* form but call it an *R*–*Z* relationship because we consider rain rate to be the dependent variable.