Introduction

Normalization of drop size distributions (DSDs) was used in the past mainly for the purpose of a compact representation of DSDs. Only recently has normalization become a tool to study the variability of DSDs in a systematic manner. Sempere-Torres et al. (1994, 1998; hereinafter referred to together as ST) described a normalization procedure based on the basic notion of scaling functions. In essence, this procedure states that if drop size is scaled by a factor, its number concentration is also scaled in a predetermined manner. This concept is related to the occurrence of power-law relationships between properties of the scaling functions. A general expression for DSDs is proposed by ST in which any DSD is written as a function of D and a reference variable [the ith moment of DSD, M_i = ∫ N(D)Dⁱ dD] in the following way:

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where the scaling or normalization exponents α and β are constant and do not have any functional dependence on M_i. The normalized function g(x₁) = N(D)M^−α_i is called the general distribution function and is independent of the value of 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(x₁) 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(x₁) and to a much lower scatter around these functions. This implies that at least a second moment, characterizing the shape of g(x₁), 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 (D₀) and a number density (M₃/D⁴₀). However, they assume a specific shape (exponential and gamma) of DSDs and impose power relationships between M₃ or D₀ 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

As suggested in the appendix of Sempere-Torres et al. (1998) and fully derived in Porrà et al. (1998), the scaling law (1) can be obtained starting from the statement that any DSD can be expressed as the product of the expected concentration of drops 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

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where the characteristic diameter 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. Thus, the DSD will read as

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Hence, any DSD can be expressed using two parameters (the characteristic diameter and characteristic number density) and a dimensionless pdf 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 (and, in general, any pair of variables obtained from moments of the DSD) should be highly correlated with each other.

This hypothesis leads to an expression for 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_nC_1,nM^γ(n)_i(4)

where 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_cC_1,j+1C_1,jM^β_i(5)

with 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 nth moment, we obtain

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where C_n is a constant given by

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with x = D/D_c. From (6) and (4), it follows that f(M_i) = c′M^α_i, where 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

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where x₁ = DM^−β_i, and 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 nth moment of the normalized DSD is thus defined as

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where C_1,n is a constant that depends on the choice of the reference moment M_i.

Now, from (8) and M_n = ∫ N(D)Dⁿ dD, we obtain

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which shows that the existence of power-law relationships between the moments is a necessary and sufficient condition for scaling DSDs. The exponent of the power law is a function of two normalization parameters: γ(n) = α + (n + 1)β. In addition, taking n = i, a self-consistency constraint is obtained from (10):

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For example, if rainfall rate R is chosen as the reference variable (approximated by the 3.67th moment), we get

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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 relationship¹—while the coefficients in these power laws are determined by moments of the g(x₁) 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 + 2.33β. 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(x₁) 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(x₁), using another moment as reference variable.

A generalization of the scaling normalization

Consider a first normalization of DSDs in which precipitation events are stratified according to the dominant microphysical process that is associated with a value of β and a shape of g(x₁), as in Sempere-Torres et al. (1999, 2000). In the second normalization, the general function g(x₁) is renormalized by introducing an additional reference variable, a moment of g(x₁), M_1,j = ∫ g(x₁)x^j₁ dx₁. We follow the same procedure as in the single-moment normalization. A form of the second-normalized DSD is [see (1) and (8)]

gx₁M^δ_1,jhx₁M^−ε_1,j(14)

where h is the “second normalized” DSD. Here, ε and δ are the new scaling or normalization exponents. The second-normalized diameter is defined as x₂ = x₁M^−ε_1,j. Then, we obtain the following general form of double-moment normalization (see appendix A for detailed derivation):

NDM^{(j+1)/(j−i)}_iM^{(i+1)/(i−j)}_jhx₂

with

x₂DM^1/(j−i)_iM^−1/(j−i)_j(15)

In addition, we obtain the general multiple power-law relationship among moments of DSDs,

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with the following self-consistencies:

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where the coefficient of multiple power C_2,n is defined by the nth moment of the second normalized DSD h(x₂) [C_2,n = ∫ h(x₂)xⁿ₂ dx₂]. The newly introduced reference variable M_1,j disappears, and the jth 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(x₂) 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,nM^1+(n−i)β_i], and the coefficient is the nth moment of g(x₁) [C_1,n = ∫ g(x₁)xⁿ₁ dx₁]. In a similar way, in the double-moment normalization, the coefficient of the multiple power law in (16) is now the nth moment of h(x₂) instead of g(x₁). 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)

Equations (18) and (19) are identical. Hence, when i = 3 and j = 4, h(x₂) is the same as F(D/D_m); apart from the constant C_T = 4⁴/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.

We have thus shown the connection between the double-moment scaling normalization and the one proposed by Testud et al. (2001) by proving that the latter is a particular case of the former. We will show now that the inverse path can be followed as well. For this discussion, we generalize the Testud et al. (2001) normalization using the generalized characteristic number density N^′₀ and the generalized characteristic diameter D^′_m that can be obtained from any combination of two moments M_i and M_j, which are not necessarily consecutive. Similar to (18), we can express DSDs in the following form:

NDN^′₀FDD^′_m(20)

The intrinsic shape of the DSD F(x) does not depend on N^′₀ and D^′_m. From this equation, the nth moment of the DSD is written as

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where x = D/D^′_m. The constant C_T,n is the nth moment of F(x)[C_T,n = ∫ xⁿF(x) dx] that depends on the moment order n. Then, the ith and jth moments of DSDs can be expressed as a function of N^′₀ and D^′_m,

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By solving (22), we can obtain N^′₀ and D^′_m in terms of M_i and M_j:

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Using these generalized equations, (21) reads exactly as (16), with 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)

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By putting (24) into (20), we obtain the general form of the normalization of Testud et al. (2001),

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This general form is identical to (15). In other words, h(x₂) = F(x). If i = 3 and j = 4, N^′₀ = M⁵₃M⁻⁴₄ ∝ N^*₀ and D^′_m = M₄/M₃ = D_m. That is, the N^*₀ and D_m of Testud et al. (2001) are a particular case of N^′₀ and D^′_m when the parameters of the double-moment scaling normalization are the third and fourth moments. Another way of this generalization is shown in appendix B.

For completeness, we can obtain the single-moment scaling normalization from (25) by imposing a power law between any two moments, M_n = C_1,nM^α+(n+1)β_i, as in (10). Then, (25) can be rewritten in the following manner:

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Hence, if the moments of the DSDs are related by a power law, F′ is identical to g(x₁) in (8).

It is worthwhile to reconsider the meaning of N^′₀ (or N^*₀) as a moment in the notation of the double-moment normalization. Now, we derive N^′₀ by taking n = −1 in (16) and from the definition of N^′₀ in (24),

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As a particular case (i = 3 and j = 4), we can derive N^*₀:

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This equation illustrates that N^*₀ ∝ ∫ (1/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 from the −1th moment is extremely sensitive to the number density at small diameters that are problematic because of instrumental limitations. For the DSD model, such as the often used inverse exponential, the value of the −1th moment is undetermined. It is worthwhile to recall that Marshall and Palmer (1948; M–P) derived the intercept parameter N₀ from measured DSDs by extrapolating the linear portion of DSDs in the logN(D)-versus-D diagram to D = 0 mm. They found a constant N₀ 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 (or N^*₀) is derived from any combination of two moments using (24). The choice of two moments depends on the portion of DSDs that is of greater interest. Often, in cloud microphysics we are particularly interested in mass and volume-weighted mean diameter. Then the third and fourth moments are good choices. In this case, the obtained intercept parameter N^′₀ is similar to that from a graphical extrapolation at medium-sized drops because there the third and fourth moments have large weights. When we choose two higher moments (i.e., fifth and sixth moments), N^′₀ from (24) has a larger weight at bigger diameters and its value is similar to that from an extrapolation to zero diameter of the portion of DSD at large-sized drops.

Some consequences

Equation (15) contains all of the relationships between any measurable parameters of the DSD and the reference moments. These relationships are solely determined by the moments of h(x₂), as shown in (16). When we choose M_i = C_uR (∼3.67th moment of the DSD) and M_n = Z (sixth moment of the DSD), the following relationship Z(M_j, R) is obtained:

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where 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(x₁) and the exponent b is related to the normalization exponent β by b = 1 + 2.33β. 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_uM^−0.63₀ and b = 1.63; with j = 3, a = C_2,6C^4.48_uM^−3.48₃ and 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_2,6C^{(j−6)/(j−3.67)}_uC^{2.33/(j−3.67)}R).

Now, we derive M_k(N^′₀, M_l) from the definition of the moment of DSDs. From (21), we obtain the kth and lth moment of DSDs:

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Using this equation, we derive the relationship between M_k and M_l:

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By taking M_l = C_uR (∼3.67th moment of the DSD) and M_k = Z (sixth moment of the DSD), we obtain Z(N^′₀, R):

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As a particular case, we can derive Z(N^*₀, R) from the above equation by taking N^′₀ = M⁵₃M⁻⁴₄ = N^*₀/C_T:

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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,nR^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)}_iM^{(i+1)/(i−j)}_j] and D^′_m [=(M_j/M_i)^1/(j−i)] are calculated, and then N(D) and D are normalized with calculated N^′₀ and D^′_m, respectively. Results for (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.

We now show a quantitative comparison of the single- and double-moment normalization in terms of the compact representation of DSDs. In Fig. 3, we calculate the uncertainty in moment estimation due to the scatter with respect to g(x₁) or h(x₂). From g(x₁) and h(x₂) in Figs. 1b and 2b, DSDs are estimated with R in the single-moment normalization and with M₃ and M₄ 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 nth moment is calculated for both normalizations using the following equation:

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where 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(x₁) and h(x₂) are adjusted to satisfy the self-consistencies], SDFE is zero at M_3.67 for the single-moment normalization (solid line) and at M₃ and M₄ 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 = M₆) 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(x₂) do. Therefore, the applicability of “the optimization study” represented by Fig. 4 to radar remote sensing measurements remains to be explored.

The functions g(x₁) and h(x₂) 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(x₁) and h(x₂). 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(x₁) or h(x₂) 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).

A random variable D ≥ 0 with probability density function

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is said to have a generalized gamma distribution with parameters μ, c, and λ (Stacy 1962). The parameters c, μ, and λ have to be positive. The nth moments m_n are given by

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In section 2, we have shown that DSDs can be expressed as the product of the expected concentration of drops M₀ (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:

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The nth moment of the DSD M_n becomes

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Then, the ith and jth moments of DSDs are

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We can express the parameter λ and M₀ in terms of the two moments, M_i and M_j:

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Thus, the parameter λ has the form of the characteristic diameter D^′_m in the double-moment scaling law [see (24)], except for a constant that depends on the two shape parameters μ and c. In addition, the zeroth moment is defined by a multiple power law. Then, combining the expressions of λ and M₀ into M_n, we find the relationship between M_n and two moments (M_i and M_j):

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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).

For completeness, we now show the scaling form of the generalized gamma DSD. From (38), (41), and the definition of N^′₀ [=M^{(j+1)/(j−i)}_iM^{(i+1)/(i−j)}_j] and D^′_m [=(M_j/M_i)^1/(j−i)] in (24), we obtain

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where x₂ = D/D^′_m. As in (A2), the nth moment of h_{GG,(i,j,μ,c)}(x₂) 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 are held fixed.

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:

1. an exponential DSD model with μ = 1 and c = 1:

   

   

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   for example, when i = 3 and j = 4,

   

   

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2. a gamma DSD model with c = 1:

   

   

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   for example, when i = 3 and j = 4,

   

   

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   where x₂ = D/D^′_m, N^′₀ = M^{(j+1)/(j−i)}_iM^{(i+1)/(i−j)}_j, and 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(x₂) in Figs. 2b and 2d. In this least squares fitting, the sum of the squared difference between logh_GG and logh is minimized by searching the best 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(x₂). The fitted h(x₂) has an “S shape” with slight deviations from the exponential h(x₂), that is, a relative abundance of drops when x₂ < 0.3 and a relative absence when x₂ > 1.5. No particular general significance should be assigned to these values of μ and c because the database used here is very limited.

Similar to the double-moment normalization, the form of the single-moment normalization can be derived from (36) and (37), imposing a simple power law, M_n = C_1,nM^α+(n+1)β₁ as in (26). With the self-consistency constraint α + (i + 1)β = 1, we obtain the normalized form of the generalized gamma DSDs with a single moment,

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where x₁ = DM^−β_i and C_1,0 is a constant defined as M₀/M^α+β_i or the zeroth moment of g_GG,(i,μ,c)(x₁) as in (9). Similar to h_{GG,(i,j,μ,c)}(x₂), the shape of the normalized DSD g_GG,(i,μ,c)(x₁) is not unique but depends on the shape parameters μ and c. Unlike h_{GG,(i,j,μ,c)}(x₂), there is an extra constant C_1,0 that adjusts units. Hence, the unit of C^1/i_1,0M^−β_i should be inverse millimeters. For two simple DSD models, we obtain

1. an exponential DSD model with μ = 1 and c = 1:

   

   

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2. a gamma DSD model with c = 1:

   

   

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The subscripts 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:

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For this case, β = 0.214. With λ = 4.1R^−0.214 and the terminal fall velocity of drops υ(D) = 3.778D^0.67 (m s⁻¹), we obtain a similar value of N₀ = 7.0 × 10³ (m⁻³ mm⁻¹) from (49). To satisfy N₀ = 8.0 × 10³ (m⁻³ mm⁻¹), we expect λ = 4.23R^−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(x₂). This point is illustrated by the functional dependence of h(x₂) on the shape parameters (μ and c) of the generalized gamma DSD in (43). Hence, various h(x₂) 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 M₀ because it represents the total number of particles (a “conservative” quantity in warm nonprecipitating clouds); M₂ because, combined with M₃, it defines the “effective radius;” M₃ because it represents the LWC; or M₄ because, combined with M₃, it defines the mean volume diameter D_m. For the physics of precipitation, the interest of the combination (M₃, M₄) 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 and D^′_m that was a good indication of two rain regimes in Testud et al. (2001) is poor in our dataset (Fig. 8) and the exponent of the power law (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 or D^′_m, as seen in (32) and (34).