A General Approach to Double-Moment Normalization ... - AMS Journals

264

JOURNAL OF APPLIED METEOROLOGY

VOLUME 43

A General Approach to Double-Moment Normalization of Drop Size Distributions GYUWON LEE, ISZTAR ZAWADZKI,

AND

WANDA SZYRMER

J. S. Marshall Radar Observatory, Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

DANIEL SEMPERE-TORRES Grup de Recerca Aplicada en Hidrometeorologia, Universitat Politecnica de Catalunya, Barcelona, Spain

REMKO UIJLENHOET Hydrology and Quantitative Water Management Group, Department of Environmental Sciences, Wageningen University, Wageningen, Netherlands (Manuscript received 10 January 2003, in final form 16 June 2003) ABSTRACT Normalization of drop size distributions (DSDs) is reexamined here. First, an extension of the scaling normalization that uses one moment of the DSD as a scaling parameter to a more general scaling normalization that uses two moments as scaling parameters of the normalization is presented. In addition, the proposed formulation includes all two-parameter normalizations recently introduced in the literature. Thus, a unified vision of the question of DSD normalization and a good model representation of DSDs are given. Data analysis of some convective and stratiform DSDs shows that, from the point of view of the compact representation of DSDs, the double-moment normalization is preferred. However, in terms of physical interpretation, the scaling exponent of the single-moment normalization clearly indicates two different rain regimes, whereas in the double-moment normalization the two populations are not readily separated. It is also shown that DSD analytical models (exponential, gamma, and generalized gamma DSD) have the same scaling properties, indicating that the scaling formalism of DSDs is a very general way of describing DSDs.

1. 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 5 # N(D)D i dD] in the following way: N(D) 5 N(D, M i ) 5 M ai g(DM 2i b ) 5 M ai g(x1 ),

(1)

Corresponding author address: GyuWon Lee, J. S. Marshall Radar Observatory, McGill University, P. O. Box 198, Macdonald Campus, Ste-Anne de Bellevue, QC H9X 3V9, Canada. E-mail: [email protected]

q 2004 American Meteorological Society

where the scaling or normalization exponents a and b are constant and do not have any functional dependence on M i . The normalized function g(x1 ) 5 N(D)M 2i a 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

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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 (D 0 ) and a number density (M 3 /D 40). However, they assume a specific shape (exponential and gamma) of DSDs and impose power relationships between M 3 or D 0 and R, that is, reducing their normalization to a single-moment case [see (30) in Sekhon and Srivastava (1971)]. SempereTorres 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 singlemoment 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. 2. Limitations of the scaling normalization

1 2

NT D pˆ , Dc Dc

N(D) 5

(2)

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

1 2

Mk D pˆ . k11 Dc Dc

(3)

Hence, any DSD can be expressed using two parameters (the characteristic diameter and characteristic number density) and a dimensionless pdf pˆ, which may depend on a number of dimensionless parameters. Note that if k 5 0 then (2) is derived and when k 5 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 k11 (and, in c 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 k11 5 f (M i ). By the hypothesis, the relation c between two DSD moments can be written as a power law M n 5 C1,n M gi (n) ,

(4)

where C1,n is a constant that adjusts the units (the subscript 1 denotes the single-moment scaling normalization framework). The g (n) is the exponent of the power law and depends on the moment order n. Taking n 5 j and j 1 1, the characteristic diameter can be expressed as a function of the reference variable as D c 5 (C1, j11 /C1, j )M bi ,

(5)

with b 5 g ( j 1 1) 2 g ( j). Furthermore, introducing M k /D k11 5 f (M i ) and (3) and (5) in the definition of c the DSD’s nth moment, we obtain Mn 5

As suggested in the appendix of Sempere-Torres et al. (1998) and fully derived in Porra` 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 23 ) (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 1 dD (mm 21 ). If a characteristic diameter D c is used to render the pdf dimensionless, then N(D) 5

the quotient of any two consecutive moments, that is; D c 5 M i11 /M i . In addition, the characteristic number density (m 23 mm 21 ) can be expressed in general as N c 5 M k /D k11 c . Thus, the DSD will read as

5

E

D n N(D) dD

1 2 C1, j11 C1, j

(n11)

C n f (M i )M i(n11)b ,

(6)

where C n is a constant given by Cn 5

E

x n pˆ (x) dx,

(7)

with x 5 D/D c . From (6) and (4), it follows that f (M i ) 5 c9M ai , where c9 5 (C1,n/Cn)(C1, j11/C1, j)2n21, and a 5 g (n) 2 (n 1 1)b. Substituting f (M i ) and D c into (3) we get N(D) 5 N(D, M i ) 5 c9M ai pˆ (c0DM 2i b ) 5 M ai g(DM 2i b ) 5 M ai g(x1 ),

(8)

where x1 5 DM 2i b, and c0 5 (C1, j11/C1, j)21. This is the

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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 M1, n 5

E

g(x1 )x 1n dx1 5 C1, n ,

(9)

where C1,n is a constant that depends on the choice of the reference moment M i . Now, from (8) and M n 5 # N(D)D n dD, we obtain Mn 5

E

D n M ai g(x1 ) dD 5 C1,n M ai 1(n11)b ,

(10)

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: g (n) 5 a 1 (n 1 1)b. In addition, taking n 5 i, a selfconsistency constraint is obtained from (10):

a 1 (i 1 1)b 5 1 and C1,i 5 1 5

E

g(x1 )x1i dx1 .

(11)

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

a 1 4.67b 5 1 and N(D) 5 R

(124.67b)

C1,3.67 5 1 5

E

2b

g(DR ),

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 b 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. 3. 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 b 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 ), M1, j 5 # g(x1 )x 1j dx1 . We follow the same procedure as in the single-moment normalization. A form of the second-normalized DSD is [see (1) and (8)]

(12)

g(x1 ) 5 M 1,d j h(x1 M 2« 1, j ),

(13)

where h is the ‘‘second normalized’’ DSD. Here, « and d are the new scaling or normalization exponents. The second-normalized diameter is defined as x 2 5 x1M 2« 1, j . Then, we obtain the following general form of doublemoment normalization (see appendix A for detailed derivation):

with

g(x1 )x13.67 dx1 .

VOLUME 43

Thus, all of the exponents of the power-law relationships between pairs of DSD moments are linear functions of b—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 5 aR b , the coefficient a is given by the sixth moment (C1,6 ), and the exponent is b 5 1 1 2.33b. 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 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.

(14)

N(D) 5 M i( j11)/( j2i )M j(i11)/(i2j ) h(x 2 ), with j2i ) j2i ) x 2 5 DM 1/( M 21/( . i j

(15)

In addition, we obtain the general multiple power-law relationship among moments of DSDs, M n 5 C2, n M j(n2i )/( j2i ) M i( j2n)/( j2i ) ,

(16)

with the following self-consistencies:

d 1 ( j 1 1)« 5 1, C2,i 5 1 5 C2, j 5 1 5

«5

E E

1 , j2i

h(x2 )x 2i dx2 , h(x2 )x 2j dx2 ,

and (17)

where the coefficient of multiple power C 2,n is defined by the nth moment of the second normalized DSD h(x 2 )

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[C 2,n 5 # h(x 2 )x n2 dx 2 ]. The newly introduced reference variable M1, j disappears, and the jth moment of the original DSD M j is introduced. As a result, the four scaling exponents (a, b, d, and «) 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 )b scaling exponent b [M n 5 C1,n M 11(n2i ], and the coi efficient is the nth moment of g(x1 ) [C1,n 5 # g(x1 )x1n 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(x 2 ) instead of g(x1 ). However, the exponent is purely determined by the orders of two reference moments. Therefore, in this approach the role of b 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.

Testud et al. (2001) proposed the following normalization: (18)

where D m is the volume-weighted mean diameter, a particular characteristic diameter that is derived by D m 5 M 4 /M 3 , and N*0 is defined as N*0 5 C T M 35M 24 4 , where C T is an arbitrary constant that is chosen as 4 4 /G(4). Parameter N*0 can be interpreted as the intercept parameter of an exponential DSD that has the same liquid water content (LWC) and volume-weighted mean diameter as an observed DSD. Testud et al. (2001) showed that F(D/D m ) is remarkably stable and is independent of different rain types. If in (15) we take i 5 3 and j 5 4, we obtain the following: 21 N(D) 5 M 35 M 24 4 h(DM 3 M 4 ).

N(D) 5 N9F(D/D9 0 m ).

(20)

The intrinsic shape of the DSD F(x) does not depend on N90 and D9m. From this equation, the nth moment of the DSD is written as Mn 5

E

n11 N(D)D n dD 5 C T,n N9(D9 , 0 m)

(21)

where x 5 D/D9m. The constant C T,n is the nth moment of F(x)[C T,n 5 # x n F(x) dx] that depends on the moment order n. Then, the ith and jth moments of DSDs can be expressed as a function of N90 and D9m, i11 M i 5 C T,i N9(D9 0 m)

and

j11 M j 5 C T, j N9(D9 . 0 m)

(22)

By solving (22), we can obtain N90 and D9m in terms of M i and M j :

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

N(D) 5 N*F(D/D 0 m ),

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 N90 and the generalized characteristic diameter D9m 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:

(19)

Equations (18) and (19) are identical. Hence, when i 5 3 and j 5 4, h(x 2 ) is the same as F(D/D m ); apart from the constant C T 5 4 4 /G(4). Thus, the normalization of Testud et al. (2001) is a particular case of the doublemoment scaling normalization in which the moments of

1 2

Mi N90 5 C T,i D9m 5

( j11)/( j2i )

1 2 Mj C T, j

1 2 1 2 C T,i C T, j

1/( j2i )

Mj Mi

(i11)/(i2j )

and

1/( j2i )

.

(23)

Using these generalized equations, (21) reads exactly as (16), with C 2,n 5 C T,n (C T,i ) (n2j )/( j2i ) (C T,j ) (n2i )/(i2j ) . By imposing the self-consistency equations (C T,i 5 1 and C T,j 5 1), (23) becomes (see appendix B for the self-consistency) N90 5 M i( j11)/( j2i ) M j(i11)/(i2j ) and D9m 5 (M j /M i )1/( j2i ) .

(24)

By putting (24) into (20), we obtain the general form of the normalization of Testud et al. (2001), j2i ) ) N(D) 5 M i( j11)/( j2i ) M j(i11)/(i2j ) F [DM 1/( M 1/(i2j ]. i j

(25)

This general form is identical to (15). In other words, h(x 2 ) 5 F(x). If i 5 3 and j 5 4, N90 5 M 35M 24 } 4 N*0 and D9m 5 M 4 /M 3 5 D m . That is, the N*0 and D m of Testud et al. (2001) are a particular case of N90 and D9m when the parameters of the double-moment scaling

268


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 5 C1,n M ai 1(n11)b , as in (10). Then, (25) can be rewritten in the following manner: N(D) 5 M i( j11)/( j2i ) [C1, j M ai 1( j11)b ] (i11)/(i2j ) j2i ) 3 F{DM 1/( [C1, j M ai 1( j11)b ]1/(i2j )} i ) ) 5 C 1,(i11)/(i2j M ai F [DC 1/(i2j M 2i b ] j 1, j

5 M ai F9(DM 2i b ).

(26)

Hence, if the moments of the DSDs are related by a power law, F9 is identical to g(x1 ) in (8). It is worthwhile to reconsider the meaning of N90 (or N*0 ) as a moment in the notation of the double-moment normalization. Now, we derive N90 by taking n 5 21 in (16) and from the definition of N90 in (24), N90 5

1 M . C2,21 21

(27)

As a particular case (i 5 3 and j 5 4), we can derive N*0 : N*0 [ C T M 35 M 24 5 4

CT M . C2,21 21

(28)

This equation illustrates that N*0 } # (1/D)N(D) dD 5 M21 , the 21th moment of the DSD, apart from the constant, C T /C 2,21 . However, particular caution is necessary in interpreting this result. A direct calculation of N*0 from the 21th 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 21th moment is undetermined. It is worthwhile to recall that Marshall and Palmer (1948; M–P) derived the intercept parameter N 0 from measured DSDs by extrapolating the linear portion of DSDs in the logN(D)versus-D diagram to D 5 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, N90 (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 N90 is similar to that from a graphical extrapolation at mediumsized drops because there the third and fourth moments have large weights. When we choose two higher moments (i.e., fifth and sixth moments), N90 from (24) has a larger weight at bigger diameters and its value is sim-

VOLUME 43

ilar to that from an extrapolation to zero diameter of the portion of DSD at large-sized drops. 5. 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(x 2 ), as shown in (16). When we choose M i 5 C u R (;3.67th moment of the DSD) and M n 5 Z (sixth moment of the DSD), the following relationship Z(M j , R) is obtained: j23.67) Z 5 [C2,6 C u( j26)/( j23.67) M 2.33/( ]R ( j26)/( j23.67) , j

(29)

where C u is a constant that adjusts the units. In the single-moment scaling normalization, the coefficient a of Z 5 aR b is the sixth moment of g(x1 ) and the exponent b is related to the normalization exponent b by b 5 1 1 2.33b. 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 5 0, a 5 C 2,6 20.63 C 1.63 and b 5 1.63; with j 5 3, a 5 u M0 C 2,6C u4.48M 23.48 and b 5 4.48. As a first approximation, 3 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 5 CR), as expected in the equilibrium process (Zawadzki and Antonio 1988), we obtain a linear R–Z relationship (Z 5 C 2,6C u( j26)/( j23.67) C 2.33/( j23.67) R). Now, we derive M k (N90 , M l ) from the definition of the moment of DSDs. From (21), we obtain the kth and lth moment of DSDs: k11 M k 5 C2, k N9(D9 0 m)

l11 and M l 5 C2,l N9(D9 . (30) 0 m)

Using this equation, we derive the relationship between M k and M l :

1 2

Mk Ml 5 C2, k C 2(k11)/(l11) 2,l N90 N90

(k11)/(l11)

or

(l2k)/(l11) M k 5 C2, k C 2(k11)/(l11) (N9) M l(k11)/(l11) . 2,l 0

(31)

By taking M l 5 C u R (;3.67th moment of the DSD) and M k 5 Z (sixth moment of the DSD), we obtain Z(N90, R):

1 2

Z R 1.5 5 C2,6 C 21.5 2,3.67 C u N90 N90

1.5

or

1.5 20.5 1.5 Z 5 C2,6 C 21.5 R . 2,3.67 C u (N9) 0

(32)

As a particular case, we can derive Z(N*0 , R) from the above equation by taking N90 5 M 35M 24 4 5 N* 0 /C T : 21.5 0.5 20.5 1.5 Z 5 C2,6 C 2,3.67 C 1.5 R . u C T (N*) 0

(33)

The explicit exponent on R, b 5 1.5, in (32) and (33) is close to the climatological value (Z 5 210R1.47 in

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269

Montreal, Quebec, Canada, e.g.). This implies that the correlation between N90 and 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 and R. As already discussed by Testud et al. (2001), the actual exponent in the R–Z relationship depends on the correlation between N90 and 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 (}M21 ) is proportional to R. For M–P DSDs for which N90 is constant (8 3 10 3 m 23 mm 21 ), we expect Z } R1.5 . Similarly, we can derive Z(D9m, R): 2.33 Z 5 C 2,6 C 21 R. 2,3.67 C u (D9 m)

(34)

For the equilibrium process, the characteristic diameter D9m is constant, giving a linear R–Z relationship. For M–P DSDs (D9m } R 0.21 ), the exponent of the R–Z relationship is 1.5. 6. 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). a. 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 g (n) of M n 5 C1,n R g (n) is derived using weighted total least squares fitting (WTLS; Amemiya 1997) in log–log coordinates. The scaling exponent b is derived from WTLS between the calculated exponent g (n) and the moment of order n 1 1 [g (n) 5 a 1 (n 1 1)b]. The other scaling exponent a is derived from the self-consistency constraint in (12). Then, N(D) and D are normalized with R a and R b . The scaling exponent b is slightly smaller than the value of M–P DSDs, indicating that the exponent of Z 5 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

FIG. 1. Single-moment normalization on data from Sempere-Torres et al. (1999, 2000). (a) Scattergram of all normalized DSDs with M i 5 R. An exponential adjustment is shown as a dashed line. (b) The average g(x1 ) of the data points in (a) with bars (dark solid line) indicating std dev; g(x1 ) is adjusted to statisfy the self-consistency by multiplying a factor that imposes C1,3.67 5 1.

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 90 [5M i( j11)/( j2i )M j(i11)/(i2j )] and D9m [5(M j /M i )1/( j2i ) ] are calculated, and then N(D) and D are normalized with calculated N90 and D9m, respectively. Results for (i 5 3, j 5 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-mo-

270


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FIG. 2. Double-moment normalization on data from Sempere-Torres et al. (1999, 2000). (a) Scattergram of all normalized DSDs with M i 5 M 3 and M j 5 M 4 . An exponential adjustment is shown as a dashed line. (b) The average h(x 2 ) of the data points in (a) with dark vertical bars indicating std dev. Similar to g(x1 ) , h(x 2 ) is also adjusted to satisfy the self-consistency. The std dev due to the statistical fluctuation is shown as the less dark vertical bars. (c), (d) Same as in (a) and (b), except for M i 5 M 3 and M j 5 M 6 .

ment normalization, illustrating an advantage of doublemoment normalization in terms of a compact representation of DSDs. By normalizing with (i 5 3, j 5 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 singleand 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(x 2 ) . From g(x1 ) and h(x 2 ) in Figs. 1b and 2b, DSDs are estimated with R in the singlemoment 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 nth moment is calculated for both normalizations using the following equation:

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FIG. 3. SDFE in moment estimation using g(x1 ) and h(x 2 ) . Doublemoment normalization shows an error of less than 30% at the overall range of n.

[ O1

1 SDFE 5 k

2

M n 2 M n,est Mn

]

2 1/ 2

,

(35)

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(x 2 ) are adjusted to satisfy the self-consistencies], SDFE is zero at 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 5 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

FIG. 4. (a) SDFE in the estimate of the nth moment from the average normalized drop size distribution h(x 2 ) when the indicated two consecutive moments are used for the normalization. (b) Same as (a), but when reflectivity and any other moment are used for the normalization.

(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 5 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,

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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. b. 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 nth moment of normalized DSD h(x 2 ) in Fig. 2d. The consistency of both lines indicates that the double-moment scaling law is satisfied well when 2.5 # 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. c. 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 5 3 and j 5 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 g (n) of the power-law relationship [M n 5 C1,n R g (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

FIG. 5. Parameters of multiple power law Mn 5 aM bi M cj from direct least squares fitting (dashed line) of data and from double-moment normalization (solid line) for i 5 3 and j 5 6 (Figs. 2c, d). In the doublemoment normalization, the two exponents are determined from (16) and the coefficient is derived from the nth moment of h(x 2 ) in Fig. 2d.

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FIG. 6. (a) The R–Z WTLS regressions for stratiforms and convective rain. (b) Exponent g (n) of M n 5 C1,n R g(n) as a function of n. The scaling exponent b is determined by the slope in g (n) vs n [g (n) 5 a 1 (n 1 1)b].

defines the scaling exponent b 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 b (determining the exponents of the power laws) and the moments of the g(x1 ) function (determining the coefficients of the power laws). For example, in Z 5 aR b the exponent b is given by b 5 1 1 2.33b and the coefficient a can be obtained by the sixth moment of g(x1 ). In the dataset considered here, b 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, N90 and D9m, can jointly determine the coefficients and

FIG. 7. (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 ) (solid line) is shown with the standard deviation (vertical bars) and exponential adjustment (dashed line). (c) Comparison of h(x 2 ) from different rain regimes. The normalized exponential DSD in (45) is also shown as the long dashed line. Note the remarkably stable shape.

274


the exponents of the power-law relationships and separate the convective and stratiform DSDs of our dataset. 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 ) in Fig. 7c are remarkably stable and independent of rain regimes; h(x 2 ) is slightly different from the normalized exponential DSDs. These DSD shapes are consistent with those of Testud et al. (2001). The consistent forms of h(x 2 ) for the two populations indicate that, in this case, stratiform and convective rain do not generate distinctive shapes of DSDs. We have no concrete explanation for this result. Instead of the broad classification of stratiform and convective rain, a more specific classification based on the dominant physical processes and following research on the physical origin of the changes in h(x 2 ) are needed to explore it. The next question is how well two moments, or N90 and D9m, jointly contain information on the scaling exponent b that nicely separates the two rain regimes in the single-moment normalization. Testud et al. (2001) show that N90 and D9m from the two regimes of tropical rain are well separated so that they are good indicators for the classification. In Fig. 8, we see the correlation between N90 and D9m, as well as the relationship of these two parameters to rain rate. The values of the correlation parameters and relationship are given in Table 1. The table also gives the exponents of Z 5 aR b obtained by direct WTLS, from the single-moment normalization (b 5 1 1 2.33b), and from the relationships between D9m and R [(34)], as well as N90 and R [(31)]. Unlike the result of Testud et al. (2001), Fig. 8a shows that the separation of the two types of precipitation in the (N90, D9m) space is poor. Convective rain shows no correlation (determination coefficient r 2 5 0.01 in Table 1) and has a wide distribution with an upper limit of N90 at 3 3 10 2 m 23 mm 21 . Some points from convective precipitation are mixed with those from stratiform rain. However, stratiform rain has a good correlation (r 2 5 0.66); that is, N90 decreases as D9m increases. Testud et al. (2001) classified two rain regimes by rain intensities at a given DSD spectrum and 10 adjacent spectra along an air plane track. If all of these values are less than 10 mm h 21 , 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 21 . 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 D9m and R is very good (r 2 5 0.54) but is totally nonsignificant between N90 and R (r 2 5 0.04). For the convective rain, the cor-

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FIG. 8. (a) Distribution of points in the (N90, D9m) space and the relationship of (b) N90 and (c) D9m to rain rate for the two types of precipitation.

relation between N90 and R is somewhat better (r 2 5 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 (D9m , R) lead to exponents con-

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TABLE 1. Regression parameters and the determination coefficient (r 2 ) between various parameters of the single-moment (R) and doublemoment (i 5 3 and j 5 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. N09 5 a(D9m) b All Stratiform Convective

N90 5 cR d

b in Z 5 aRb

R 5 e(D9m) f

a

b

r2

c

d

r2

e

f

r2

N90 –R

D9m–R

Single moment

Fitting

116 160 490

23.19 23.23 23.35

0.14 0.66 0.01

45 88 22

0.70 21.16 0.60

0.10 0.04 0.30

0.37 0.60 0.31

4.57 2.78 5.55

0.58 0.54 0.64

1.15 2.08 1.20

1.51 1.84 1.42

1.44 1.65 1.35

1.46 1.75 1.35

sistent with those from direct fitting, whereas the exponents from the relationship (N90 , R) show significant deviation. 7. The functions g(x1 ) 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(x1 ) 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(x1 ) 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 threeparameter 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 p(D) 5

cl (lD) cm21 exp[2(lD) c ] G(m)

(36)

is said to have a generalized gamma distribution with parameters m, c, and l (Stacy 1962). The parameters c, m, and l have to be positive. The nth moments m n are given by mn [

E

D n p(D) dD 5 l2n G(m 1 n/c)/G(m).

(37)

In section 2, we have shown that DSDs can be expressed as the product of the expected concentration of drops M 0 (m 23 ) (the zeroth moment of the DSD) and the probability density function p (mm 21 ). If this probabilistic concept is applied to particle size distributions (Auf der Maur 2001), we have the following form: N(D) 5 M0 p(D) 5 M0

cl (lD) cm21 exp[2(lD) c ]. G(m)

(38)

The nth moment of the DSD M n becomes Mn [

E

D n N(D) dD 5 M0 m n

5 M0 l2n G(m 1 n/c)/G(m).

(39)

Then, the ith and jth moments of DSDs are M i 5 M0 l2i G(m 1 i/c)/G(m) M j 5 M0 l2j G(m 1 j/c)/G(m).

and (40)

We can express the parameter l and M 0 in terms of the two moments, M i and M j :

276

JOURNAL OF APPLIED METEOROLOGY 21/(i2j )

1 2

Mi l5 Mj M0 5 M

2j/(i2j ) i

[1

M

[

]

G(m 1 j/c) G(m 1 i/c)

3 G m1

21/(i2j )

and

[ 1 2] 2]

i/(i2j ) j

VOLUME 43

i G m1 c

j(i2j )

2i/(i2j )

j c

G(m).

(41)

Thus, the parameter l has the form of the characteristic diameter D9m in the double-moment scaling law [see (24)], except for a constant that depends on the two shape parameters m and c. In addition, the zeroth moment is defined by a multiple power law. Then, combining the expressions of l and M 0 into M n , we find the relationship between M n and two moments (M i and M j ): M n 5 CGG,2,n M i(n2j )/(i2j ) M j(i2n)/(i2j ) ,

with

CGG,2,n 5 [G(m 1 i/c)] ( j2n)/(i2j ) [G(m 1 j/c)] (n2i )/(i2j ) 3 G(m 1 n/c).

(42)

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 (m and c) of DSDs. The self-consistency constraints of the double-moment scaling are also satisfied (CGG,2,i 5 1 and CGG,2, j 5 1). For completeness, we now show the scaling form of the generalized gamma DSD. From (38), (41), and the definition of N90 [5M i( j11)/( j2i )M j(i11)/(i2j )] and D9m [5(M j / M i )1/( j2i ) ] in (24), we obtain N(D) 5 N9h 0 GG,(i, j, m, c) (D/D9 m ), hGG,(i, j, m, c) (x2 ) 5 cG

( j1cm)/(i2j ) i

G

(2i2cm)/(i2j ) j

with x 2cm21

[12 ]

G 3 exp 2 i Gj G i 5 G(m 1 i/c),

c/(i2j )

x 2c ,

and G j 5 G(m 1 j/c), (43)

where x 2 5 D/D9m. As in (A2), the nth moment of hGG,(i, j, m, c) (x 2 ) provides the coefficient of the multiple power law CGG,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 D9m 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 reason-

FIG. 9. 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 23 and mean diameter D9m of 1.5 mm.

ably 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 m 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 m 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 m 5 1 and c 5 1:

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TABLE 2. Parameters of h GG,(i, j,m,c) (x 2 ) in (43) adjusted to h(x 2 ) in Figs. 2b and 2d.

i 5 3, i 5 3,

j54 j56

N(D) 5 N90

c

m

h GG,(i, j,m,c) (x 2 )

2.55 2.70

20.20 20.25

1.81x 21.51 exp(20.72x 2.55 2 2 ) 2.25x 21.67 exp(20.91x 2.70 2 2 )

(m 1 3)m13 m21 x exp[2(m 1 3)x2 ], G(m 1 3) 2

(47)

where x 2 5 D/D9m, N90 5 M i( j11)/( j2i )M j(i11)/(i2j ), and D9m 5 (M j /M i )1/( j2i ) . These two examples, when i 5 3 and j 5 4, are similar to those from Testud et al. (2001) and Illingworth and Blackman (2002).

FIG. 10. Adjustment of h(x 2 ) in Figs. 2b and 2d with the form of the generalized gamma (43). The average normalized DSD h(x 2 ) and exponential form in (45) are also shown.

N(D) 5 N9[G(1 1 i )] ( j11)/(i2j ) [G(1 1 j )] (2i21)/(i2j ) 0

75

[G(1 1 i )] 3 exp 2 [G(1 1 j )]

6

1/(i2j )

8

x2 ,

(44)

44 exp(24x2 ); G(4)

and

(45)

N(D) 5 N9[G( m 1 i )] ( j1m)/(i2j ) [G(m 1 j )] (2i2m)/(i2j ) 0

75

[G(m 1 i )] [G(m 1 j )]

6

1/(i2j )

8

x2 , (46)

for example, when i 5 3 and j 5 4,

gGG,(i,m,c) (x1 ) 5

with

cm1i cL GG x cm21 exp[2(LGG x1 ) c ] G(m 1 i/c) 1

[

2) a gamma DSD model with c 5 1:

3 x m2 21 exp 2

N(D) 5 M ai gGG,(i,m, c) (DM 2i b ),

LGG 5 C1,0

for example, when i 5 3 and j 5 4, N(D) 5 N90

To find the functional form of normalized DSDs in our dataset, m and c are determined by a Monte Carlo least squares fitting of (43) to average normalized DSD h(x 2 ) in Figs. 2b and 2d. In this least squares fitting, the sum of the squared difference between loghGG and logh is minimized by searching the best m 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 ) . The fitted 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 m 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 5 C1,nM 1a1(n11)b as in (26). With the self-consistency constraint a 1 (i 1 1)b 5 1, we obtain the normalized form of the generalized gamma DSDs with a single moment,

]

G(m 1 i/c) G(m)

and

1/i

,

(48)

where x1 5 DM 2i b and C1,0 is a constant defined as M 0 / M ai 1b or the zeroth moment of gGG,(i,m,c) (x1 ) as in (9). Similar to hGG,(i, j, m, c) (x 2 ), the shape of the normalized DSD gGG,(i,m,c) (x1 ) is not unique but depends on the shape parameters m and c. Unlike hGG,(i, j, m, c) (x 2 ), there is an extra constant C1,0 that adjusts units. Hence, the unit of 2b C 1/i should be inverse millimeters. For two simple 1,0 M i DSD models, we obtain 1) an exponential DSD model with m 5 1 and c 5 1:

278


N(D) 5 M ai

L Ei11 exp(2L E DM 2i b ) G(1 1 i )

L E 5 [C1,0 G(1 1 i )]1/i ,

with

and

(49)

2) a gamma DSD model with c 5 1: N(D) 5 M ai

LmG1i (DM 2i b )m21 exp(2L G DM 2i b ) L(m 1 i )

with

[

L G 5 C1,0

]

G(m 1 i ) G(m)

1/i

.

(50)

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: N(D) 5 N0 exp(2lD),

with

N0 5 8.0 3 10 3 (m23 mm21 )

and

l 5 4.1R 20.21 (mm21 ).

(51)

For this case, b 5 0.214. With l 5 4.1R and the terminal fall velocity of drops y(D) 5 3.778D 0.67 (m s 21 ), we obtain a similar value of N 0 5 7.0 3 10 3 (m 23 mm 21 ) from (49). To satisfy N 0 5 8.0 3 10 3 (m 23 mm 21 ), we expect l 5 4.23R 20.214 . 20.214

8. 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 (m and c) of the generalized gamma DSD in (43). Hence, various h(x 2 ) are expected from the Sshaped curve of the equilibrium DSDs, the inverse exponential often observed in rain, the gamma shape re-

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sulting 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 KDP ) 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 b is a clear indicator of the processes. However, in the double-moment normalization, the separation of N90 and D9m 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 5 aR b ) is not well determined, especially with the N90–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 b measures the ex-

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ponents 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 doublemoment normalization the two moments jointly contain the same information. In other words, the information on the exponents and the coefficients is mixed within N90 or D9m, as seen in (32) and (34). 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. APPENDIX A

E

h(x2 )x 2n dx2 5 C2, n .

(A4)

The equation (A4) provides a self-consistency constraint in the second normalization by choosing n 5 j:

d 1 ( j 1 1)« 5 1, and C2, j 5 1 5

E

h(x2 )x 2j dx2 .

(A5)

We can see that any moment of g(x1 ) can be expressed as a function of two moments of the original DSDs: M1, n 5

E

g(x1 )x 1n dx1 5

5 M 2i a2(n11)b

E

E

N(D)M 2i a (DM 2i b ) n M 2i b dD

N(D)D n dD

5 M 2i a2(n11)b M n .

(A6)

The reference variable M1, j can be obtained by applying n 5 j. Replacing (A6) into (A4), we obtain the general double-moment relationship between moments of DSDs: M n 5 C2,n M dj 1(n11)« 3 M {i a2d [a1( j11)b ]}1(n11){b2«[a1( j11)b ]} .

A Generalization of the Normalization of Sempere-Torres et al. (1994, 1998) In this appendix, we generalize the single-moment normalization of ST to double-moment normalization. In the double-moment normalization, the general function g(x1 ) is renormalized by introducing an additional reference variable. We follow the same procedure as before: in the second normalization the reference variable is a jth moment of g(x1 ), M1, j 5 # g(x1 )x 1j dx1 . A form of the second normalized DSD is [see (1) and (8)] g(x1 ) 5 M 1,d j h(x1 M 2« (A1) 1, j ), where h is the second-normalized DSD, and « and d are the new normalization exponents. The second-normalized diameter is defined as x 2 5 x1M 2« 1, j . The moments of h are given by M2, n 5

M1,n 5 C 2,n M 1,d1(n11)« . j

(A7)

One interesting point is that the reference variable M1, j of the second normalization does not appear in the equation; instead, (A7) is expressed in terms of the jth moment of the DSD only. Furthermore, we can express x 2 and h(x 2 ) as functions of the ith and jth moments of DSD instead of moments of g(x1 ): x2 5 x1 M 1,2«j 5 DM 2i b [M 2i a2( j11)b M j ]2« 5 DM ai «2[12( j11)«]b M 2« j

and

(A8)

h(x2 ) 5 g(x1 )M 1,2dj 5 N(D)M 2i a [M 2i a2( j11)b M j ]2d 5 N(D)M i(d21)a1( j11)bd M 2j d .

(A9)

Using (A8) and (A9), we can rewrite the second normalization given in (A1) as a function of M i and M j : d21)a2( j11)bd N(D) 5 M 2( i

(A2)

3 M dj h{DM ai «2[12( j11)«]b M 2« j }.

(A10)

Here, the subscript 2 denotes the second scaling normalization and C 2,n are again constants that depend on the shape of h, the choice of M1, j , and the value of n. By combining (A1) and (A2), we obtain the power law between M1, j and M1,n :

The expression above can be further simplified by using the self-consistency constraints (11) and (A8);

M2, n 5

and (A7) can be written as

E

5M

h(x2 )x 2n dx2 5 2d2(n11)« 1, j

E

E

g(x1 )M 21, dj (x1 M 1,2«j ) n M 2« 1, j dx1

j11)« N(D) 5 M i( j11)[(«2b)1( j2i )b«] M 12( j

3 h[DM i(«2b)1( j2i )b« M 2« j ], )« M n 5 C2, n M 11(n2j M i(n2j )[b2«1(i2j )«b] . j

g(x1 )x dx1 n 1

5 M1, n M 21, dj 2(n11)« 5 C2, n , where

(A3)

(A11)

(A12)

Therefore, the double-moment relationship is a function of two parameters, b and «. Furthermore, an additional self-consistency constraint is obtained by setting n 5 i in the above equation. This leads to

280


E

(A13)

Mn 5

Thus, « is determined by the order of the two moments of DSDs used as the reference variables. For example, when R (;3.67th moment) and Z (sixth moment) are used, « ; 0.43. Combining (A13) and (A12) we get

5

« 5 1/( j 2 i )

and C2, i 5 1 5

h(x2 )x 2i dx2 .

M n 5 C 2,n M j(n2i )/( j2i ) M i( j2n)/( j2i ) .

(A14)

Furthermore, we can obtain a simplified form of x 2 and h(x 2 ) as functions of the ith and jth moments of DSD; j2i ) j2i ) x2 5 DM 1/( M 21/( i j

a 1 (i 1 1)b 5 1 and « 5 1/( j 2 i ) and (A15) h(x2 ) 5 N(D)M i( j11)/(i2j ) M j(i11)/( j2i ) with self-consistencies:

d 1 ( j 1 1)« 5 1 and « 5 1/( j 2 i ).

(A16)

Using (A15) and (A16), we can further simplify (A11): N(D) 5 M

M

h[DM

(i11)/(i2j ) j

1/( j2i ) i

M

N(D)D n dD T )D n dD M ai T M dj T F(DM 2i b T M 2« j

5 M ai T 1(n11)b T M dj T 1(n11)« T

E

x n F(x) dx.

21/( j2i ) j

]. (A17)

This equation is the final form of the DSD obtained by the double-moment scaling normalization. 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.

(B3)

The factor C T,n is the nth 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 5 i and n 5 j:

a T 1 (i 1 1)b T 5 1,

d T 1 (i 1 1)« T 5 0,

C T,i 5 1,

a T 1 ( j 1 1)b T 5 0,

d T 1 ( j 1 1)« T 5 1, and

C T, j 5 1.

(B4)

Then, the four exponents of the normalization can be solved as functions of i and j,

a T 5 ( j 1 1)/( j 2 i ),

b T 5 21/( j 2 i ),

d T 5 2(i 1 1)/( j 2 i ), and

« T 5 1/( j 2 i ).

(B5)

Substituting (B5) in (B1) yields the general form of the Testud et al. (2001) normalization, N(D) 5 M i( j11)/( j2i ) M j(i11)/(i2j ) j2i ) ) 3 F [DM 1/( M 1/(i2j ]. i j

APPENDIX B

(B6)

APPENDIX C

A Generalization of the Normalization of Testud et al. (2001) We generalize the Testud et al. (2001) normalization using any two moments, 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) 5 M ai T M dj T F(x)

Statistical Uncertainty in Normalized DSDs Due to Undersampling Let us assume we sample a number of drops Ntot (D i ) within the diameter interval DD i (mm) centered in D i (mm) from the homogeneous rain. The size of each sampling volume V(D i ) (m 3 s 21 ) is identical. If the numbers of drops counted follow the Poisson distribution, we obtain the following characteristics:

s 2 {Ntot (D i )} 5 Ntot (D i ) 5 N(D i )V(D i )DD i ,

with T, x 5 DM 2i b T M 2« j

(B2)

Because F(x) is independent of any M i and M j , the nth 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 nth moment and two reference variables: M n 5 C T,n M ai T 1(n11)b T M dj T 1(n11)« T .

with self-consistencies:

( j11)/( j2i ) i

E E

VOLUME 43

(B1)

where a T , b T , d T , and « 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 nth moment of the DSD can be derived;

23

(C1)

21

where N(D i ) (m mm ) is the average number density at a diameter interval DD i and s 2 { } indicates the variance. From this, we derive the variance in the estimate of the number density s 2{N(D i )}:

s 2 {N(D i )V(D i )DD i } 5 {V(D i )DD i } 2 s 2 {N(D i )} s 2 {N(D i )} 5 N(D i )/V(D i )DD i .

(C2)

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LEE ET AL.

For the normalized DSD h(x 2 ) 5 N(D i )/N90 , the standard deviation of the normalized DSD is

s 2 {N(D i )} 5 s 2 {h(x2 )N9} 0 2 s{h(x2 )} 5 {N(D i )/[(N9) V(D i )DD i ]}1/ 2 0 1/ 2 5 {h(x2 )/[N9V(D . 0 i )DD i ]}

(C3)

The relative standard deviation of the normalized DSD is 21/ 2 s{h(x2 )}/h(x2 ) 5 [N9V(D . 0 i )h(x2 )DD i ]

(C4)

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