7 Approach to the double scaling
normalization of DSD.
7.1
Introduction
After Sempere-Torres et al. 1994 proposed a general formulation of the DSD in terms of a
scaling law (see also, chapter 4), many studies (Sempere-Torres et al. 1998, Salles et al. 1999)
showed that the simple normalization did not collapse when spectra of different precipitation
processes were normalized together.
An important step was introduced by Sempere-Torres et al. 1999 Sempere-Torres et al. 2000
that showed that the parameter and g(x) were different with the type of rain (that is, with the
associated microphysical process responsible of the raindrop formation). This preclassification reduced the scatter around the mean normalized DSD.
This suggests to many authors (Testud et al. 2001, Lee et al. 2004) the need to introduce a
second reference variable to better capture of the natural variability of the DSD. In this
chapter, the first section is devoted to the presentation of the theory of this new approach and
its implications. Then, a methodology to fit the scaling normalization of the DSD from
experimental spectra is presented. Section 7.5 analyzes the application of the theory on the
weather radar measurements. Conclusions are given finally in the last section.
Study of the variability of the rainfall microstructure. A comparison using multi-sensor measurements.
95
Chapter 7: Approach to the double scaling normalization of DSD.
7.2 Double-moment normalization of the DSD. Implication of the
formultaion
Lee et al. 2004 proposed a general expression for the theory of the double normalization in
which the DSD is modelled as a function of the drop diameter (D) and two reference variables
Mi, M j where i, j are the order of two selected moments. (Appendix C gives additional
information fo the presented model).
(
N(D, M i , M j ) = M i M j h DM i M j
)
(7.1)
A number of implications follow from the double-moment normalization
7.2.1 Relations between integral moments of DSD
The first one is that the reference variables Mi, Mj and any other integral moments of the DSD
are related through a multiple power law. Substituting (7.1) into the definition of an arbitrary
integral moment (4.3), and writing x 2 = DM i M i , one obtains:
Mn =
(M
i
)(
M j h( x 2 ) x 2 M i M j
n
) (M
i
)
M j dx 2 =
= a2,n M i (n +1)+ M j (n +1)+
(7.2)
where
a2,n =
h(x )x
2
n
2
dx 2
(7.3)
7.2.2 Self-consistency relationships
A second implication comes from imposing consistency in equation (7.2) (that is, n=i or n=j).
This leads to constraints on the exponents ,,, and the general function h(x2). This leads to
the following self-consistency relationships.
+ (i + 1) = 1
+ ( j + 1) = 0
(7.4)
+ ( j + 1) = 1
+ (i + 1) = 0
a2,i =
2
i
2
2
=1
a2, j
2
j
2
2
=1
h(x )x dx
= h(x )x dx
(7.5)
Interestingly equations (7.4) lead to express the normalizing parameters as a function of the
order of the selected moments (that is, as a function of i and j). Thus, equation can be rewritten
as follows:
N(D, M i , M j ) = M i
j+1
j i
Mj
i+1
i j
1
1
j i
j i
h DM i M j Multi-sensor measurements of Raindrop Size Distribution
(7.6)
96
Chapter 7: Approach to the double scaling normalization of DSD.
In the same way, equation (7.2) could be rewritten as:
M n = a2,n M i
n i
j i
Mj
j n
j i
(7.7)
Equation (7.5) reduce by two the parameters of the general function h(x2).
Once selected the reference, the empirical h(x2), can be derived without any fitting procedure
to determine the scaling exponent.
7.3
Methodology to fit a DSD from experimental spectra
The first step consists on choosing the orders i and j of the reference variables. Any integral
rainfall variable can play this role. They should be selected considering the most appropriate in
each application.
The relationship (4.10) is used to compute both referent moments for each sample DSD during
the storm.
As said, in this case it is not longer necessary to fit scaling parameters and h(x2) can be directly
obtained plotting the scaled diameter versus the scaled experimental concentration of drops,
that is:
• Scaled drop diameter: x 2 = Dp ( M i,t )
1
j i
( M j,t )
1
j i
• Scaled concentration of drops: N 2 = N(Dp , M i , M j ) ( M i,t )
p=1,...m
j+1
j i
( M j,t )
i+1
i j
t=1,....k
Therefore, we have m·k data (being m the number of the diameter bins of the disdrometer and
k the number of time intervals of the rain event)
According to the shape of the normalized DSD, an analytical model can be selected and their
parameters should be fitted verifying the self-consistency equations (7.5). Several functions
(exponential, gamma, lognormal, Weibull) have been used in the past (Lee et al. 2004) as
models to describe the DSD.
For example the generalized gamma function, which verifies the self-consistency equations
could be written as:
hGG = ci
j +cµ
i j
i cµ
i j
j
a2,n = (i )
x2
j n
i j
cµ 1
c (i j ) c
exp i x2 j ( j )
ni
i j
n
i = (µ + i c ) ; j = (µ + j c ) ; n = (µ + n c )
(7.8)
(7.9)
(7.10)
The exponential model for the DSD (7.8) is a particular case of the generalized gamma model
with c = 1 and µ = 1; similarly, the gamma model is obtained with c=1.
Multi-sensor measurements of Raindrop Size Distribution
97
Chapter 7: Approach to the double scaling normalization of DSD.
It is important noting from (7.8) that the exponential formulation does not need to fit any
variable and it is only dependend of the order of the selected reference variables i, j. Figure 7.1
shows the double normalization for event #27 for three different instruments. A good collapse
in a single shape can be observed in the whole spectrum despite of the high scatter present for
the small drops. Without the requirement of any fitting process (in the case of the exponential
function) we obtain a good model for the DSD except for the small drops.
D (mm)
0
10
2
4
4
10
6
4
JW1
10
10
10
0
N2
10
2
-2
-4
0.0
10
10
10
0.5
1.0
1.5
2.0
2.5
3.0
2
0
-2
-4
0.0
0.5
1.0
x2
10
1.5
2.0
x2
4
POSS1
10
N2
N2
10
Parsivel2
10
10
10
2
0
-2
-4
0.0
0.5
1.0
1.5
2.0
2.5
3.0
x2
Figure 7.1.-Double normalization of the DSD for the event #27. The line is the
fitted h(x2) for this case (an exponential model has been chosen).
Multi-sensor measurements of Raindrop Size Distribution
2.5
3.0
Chapter 7: Approach to the double scaling normalization of DSD.
7.4
98
Radar application of the double normalization
The study of the DSD is particularly important to transform the measurements of the
meteorological radar into rainfall intensities.
Although most of meteorological radar only measures one rainfall product (reflectivity),
polarimetric radars also measure a second product (KDP proportional to the 4.6th moment, Lee
et al. 2004). The measure of a second moment is essential to apply the double normalization,
which requires two products of the rainfall to obtain another integral variable, such as the rain
rate.
But still most of the radar needs a Z-R transformation to obtain the rainfall field. In these
cases, simple normalization is still required.
7.5
Conclusions
In this chapter an approach to the double normalization of the DSD is presented. It consists on
using two moments as scaling parameters of the normalization.
The double-moment scaling normalization has been observed quite effective in collapsing
theDSDs around a mean shape, providing an excellent model of DSD variability. We have
also shown that analytical models have the same scaling properties and only it is needed to fit
parameters of h(x2).
Double normalization is not useful on most of weather radar applications, which usually
measures one integral product of the DSD.
Multi-sensor measurements of Raindrop Size Distribution