Edward A Brandes

Guifu Zhang

J Vivekanandan

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P1A.1 
Drop-Size Distribution Retrieval from Polarimetric Radar Measurements 
Using the Constrained-Gamma Model: Verification and Application 
 
Edward A. Brandes*, Guifu Zhang, and J. Vivekanandan 
 
National Center for Atmospheric Research 
Boulder, Colorado 
 
1. Introduction 
 
Dual-polarization radars typically transmit 
horizontally and vertically polarized electro-
magnetic waves and receive backscattered signals 
that differ for the two polarizations.  Changes in 
signal properties as the waves propagate respond 
to hydrometeor size, shape, and orientation.  The 
measurements can be used to retrieve the 
governing parameters of assumed drop-size 
distribution (DSD) models.  The retrieval model 
examined here is an adaptation of that described 
by Zhang et al. (2001).  The theoretical basis for 
the method is discussed in an accompanying paper 
(Zhang et al. 2003).  Here the method is briefly 
described, and retrieved DSD parameters are 
verified by comparison with video disdrometer 
observations.  The model is then applied to study 
the spatial distribution of DSDs within storms. 
 
2. The constrained-gamma DSD model 
 
It is assumed that raindrops can be represented 
by the gamma distribution (Ulbrich 1983) 
 
N(D)=N0D
µ exp (-ΛD)   , 
 
where N0 is a number concentration parameter, µ is
a distribution shape term, and Λ is a slope 
parameter.  Retrieval of the three DSD governing 
parameters requires three measurements or 
relationships.  The constrained-gamma method of 
Zhang et al. (2001) uses the measurements and 
definitions of radar reflectivity at horizontal and 
vertical polarization 
4
2
, ,24
0
max4
( ) ( )
H V a b
w
D
Z N D f D dD
K
λ
π
= ∫    , 
 
___________ 
*Corresponding author address: Dr. Edward A. 
Brandes, National Center for Atmospheric Research, 
P.O. Box 3000, Boulder CO 80307.  
Email:brandes@ucar.edu 
where fa,b are the backscattering amplitudes along 
the major (a) and minor (b) drop axes;  the 
differential reflectivity 
H
DR
V
Z
Z
Z
=   ; 
and an empirical relation between µ and Λ that was 
derived from disdrometer observations (see also 
Brandes et al. 2003) 
 
Λ = 1.935 + 0.735µ + 0.0365µ2   .   (1) 
 
The DSD can be found by iterating between 
ZDR and the µ–Λ relation to find µ and Λ and then 
using ZH to find N0.  Once the DSD is known, 
physical parameters, such as the total drop 
concentration (NT, m
-3), drop median volume 
diameter (D0, mm), rainwater content (W, g m
-3), 
and rain rate (R, mm h-1), can be computed directly 
from integrals of the DSD.  An alternative is to 
derive a simple set of estimators using the 
constrained-gamma model.  Radar and rain 
parameters were calculated for Λ in the range 0.5 
to 13 and for a fixed value of N0.  Ratios of the 
parameters NT, W, and R with ZH are independent 
of N0 and functions of µ or Λ determined by ZDR 
alone for the constrained-gamma model.  Taking 
logarithms of the ratios and fitting them with 
polynomial relations yields  
2(0.728 2.066 )2.085 10 DR DRZ ZT HN Z
−= ×  
2(0.223 1.124 )45.589 10 10 DR DRZ ZHW Z
−−= × ×  
2(0.165 0.897 )0.00760 10 DR DRZ ZHR Z
−= ×    . 
The units of ZH and ZDR are linear  (mm
6 m-3) and 
dB, respectively.  Similarly, we obtain 
 
3 2
0 0.171 0.725 1.479 0.717DR DR DRD Z Z Z= − + +    . 
 
These relations are unique for the particular axis 
ratio relation used (Brandes et al. 2002) and the 
constrained-gamma model [Eq. (1)].  They are 
valid for ZDR < 2.5 dB. 
 
3. Retrieval comparison with disdrometer 
observations 
 
 Figure 1 shows radar reflectivity 
associated with two short lines of moderate 
thunderstorms observed in east central Florida 
during the PRECIP98 field program.  The southern 
line passed over a video disdrometer located 38 
km from NCAR’s S-Pol radar.  The storms moved 
westward (to the left). 
 
 
 
Fig. 1:  Radar reflectivity measurements of 
moderate thunderstorms, as seen with the 
Melbourne WSR-88 at 1448 UTC on 21 August 
1998. 
 
A time series of ZH, ZDR, NT, and D0 as 
measured or retrieved from radar with the above 
polynomial relations and computed from 
disdrometer observations is given in Fig. 2.  Traces 
for ZH and ZDR are well matched.  The range of 
values with the disdrometer is a little larger, 
probably due to the smaller sampling volume.  
Radar-derived drop concentrations are a little 
smaller than that with the disdrometer.  No bias is 
evident in the D0 retrieval.  Trends agree nicely for 
all parameters.  Additional radar–disdrometer 
comparisons are given by Zhang et al. (2003) and 
Brandes et al. (2003).  A DSD retrieval 
comparison with a wind profiler is described by 
Ellis et al. (2003).   It is clear that the constrained-
gamma method provides useful information 
regarding DSDs. 
 
 
Fig. 2:  Reflectivity (ZH, panel a), differential 
reflectivity (ZDR), total drop concentration (NT), 
and drop median volume diameter (D0) as 
measured or retrieved with the S-Pol radar 
(pluses) and computed from disdrometer 
observations (circles).  The radar antenna 
elevation was 0.5o. 
 
4. Moderate thunderstorm examples 
 
The wind field (storm relative) for the 
thunderstorms depicted in Fig. 1 is presented in 
Fig. 3.  At 0.5 km the prevailing flow is from the 
storm’s left rear (east-southeast).  The flow 
accelerates downwind indicating that the storms 
are dissipating.  Winds throughout much of the 
southern band veer (become more southerly) with 
height (e.g., 2 km).  Winds in the northern band 
become northeasterly at 2 km.  The convergence 
sustains elevated updrafts that dominate above 2 
km (not shown).  Reflectivity maxima in the 
southern band slope southward with height, while 
those in the northern band slope eastward. 
Retrieved DSD physical parameters are shown 
in Fig. 4.  Inspection reveals that peak drop 
concentrations in the southern convective line are 
3000–5000 m−3.  Maximum drop concentrations in 
the northern line are about 3000 m−3.  Drop 
concentrations at the edge of the storms are as 
small as 30–100 m−3.  Estimated drop median 
volume diameters in the core of the southern storm 
(the region with reflectivity >40 dBZ) are fairly 
uniform between 1.4 and 1.8 mm.  The storm area 
with D0 > 1.6 mm is displaced slightly from the 
region >40 dBZ.  This could be a size sorting 
effect where the largest drops are the first to fall  
 
Fig. 3:  Storm-relative dual-Doppler wind field analyses at 0.5 and 2 km height for storms observed on 14 
August 1998.  Reflectivity measurements from the Melbourne, Florida WSR-88D (30 and 40 dBZ) are 
shown (heavy solid contours).  Thin solid (dashed) contours show updrafts (downdrafts).  Contour intervals 
are 0.5 m s-1 at 0.5 km and 1 m s-1 at 2 km 
 
 
 
from the elevated core.  
Rainwater contents with the southern 
convective band are mostly 0.5–3 g m−3 and 
rainfall rates are all <60 mm h−1.  An interesting 
feature is that the 1 g m−3 rainwater content 
contour nearly coincides with the 40-dBZ 
contour.  This is also true for the 20-mm h−1 rain 
rate contour.  The relationship is altered slightly 
in the northern band.  The correspondence 
suggests a reduced dependence on differential 
reflectivity in this case.  
 
5. Summary and conclusions 
 
An overview of the constrained-gamma 
method for retrieving DSD information was 
given, retrievals were verified with disdrometer 
observations, and the model was applied to an 
event with moderate thunderstorms.  The method 
uses radar reflectivity, differential reflectivity, 
and an empirical relation between the shape and 
slope parameters of the gamma DSD. 
Overall, good agreement was found between 
the retrievals and the disdrometer observations.  
Additional radar–disdrometer comparisons are 
 
 
 
 
 
 
 
 
Acknowledgment: This work was supported by 
funds from the National Science Foundation that 
have been designated for the U.S. Weather 
Research Program at the National Center for 
Atmospheric Research. 
needed, preferably at short radar ranges to reduce 
sampling differences.   The differential propagation 
phase measurement is not currently used for 
retrieval purposes.  In theory, the measurement is 
sensitive to total rain content and drop shape.  
Perhaps the measurement can be used as an 
additional constraint to reduce bias by adjusting the 
axis ratio relation (Zhang et al. 2003).  Also, 
retrievals at the leading edge of some convective 
storms often indicate drop concentrations that seem 
too high (NT>10
4.5 m−3).  The problem is believed to 
arise from DSDs that are dominated by small 
numbers of very large drops and not well 
represented by the constrained-gamma model.  
Regardless, this study shows that useful DSD 
information can be deduced from polarimetric radar 
measurements. 
 
 
 
REFERENCES 
 
Brandes, E. A., G. Zhang, and J. Vivekanandan, 
2002: Experiments in rainfall estimation with 
a polarimetric radar in a subtropical 
environment.  J. Appl. Meteor., 41, 674–685. 
          ,           , and           , 2003: An evaluation of 
a drop distribution-based polarimetric radar 
rainfall estimator.  J. Appl. Meteor., 42, 652–
660. 
Ellis, S. M., C. R. Williams, E. A. Brandes, and G. 
Zhang, 2003: Comparison of rain drop size 
distribution retrievals from polarization 
diversity radar and profiling radar  
 
 
using video disdrometer measurements.  (This 
volume) 
Ulbrich, C. W., 1983: Natural variation in the 
analytical form of the raindrop size 
distribution. J. Climate Appl. Meteor., 22, 
1764–1775. 
Zhang, G., J. Vivekanandan, and E. Brandes, 
2001: A method for estimating rain rate and 
drop size distribution from polarimetric radar 
measurements.  IEEE Trans. Geosci. Remote 
Sens., 39, 830–841. 
          ,           , and           , 2003: Constrained 
gamma drop size model for polarimetric radar 
rain estimation: Justification and 
development.  (This volume) 
2
2
2
2
2
2
2.5
2.5
2.5
2.5
2.5
2.5
3
3
3
3
3
3
3
3.5
3.
5
Total drop concentration (logN
T
) [m−3]
D
is
ta
nc
e 
(k
m
)
DSD Parameter Retreival
21 August 1998, 1448 UTC
−40 −35 −30 −25 −20 −15
15
20
25
30
35
40
45
1.2
1.2
1.2
1.2
1.2
1.
4
1.4
1.4
1.4
1.4
1.4
1.
6
1.6
1.6
1.6
1.6
1.6
1.
8
1.8
Drop median volume diameter [mm]
0.5 km
−40 −35 −30 −25 −20 −15
15
20
25
30
35
40
45
0.5
0.5
0.5
0.5
0.5
0.5
1
1
1
1
1
1
2
2
2
Rain water content [g m−3]
D
is
ta
nc
e 
(k
m
)
Distance (km)
−40 −35 −30 −25 −20 −15
15
20
25
30
35
40
45
5
5
5
5
5
5
5
10
10
10
1010
10
10
20
20
20
20
20
20
40
40
40
Rainfall rate [mm h−1]
Distance (km)
−40 −35 −30 −25 −20 −15
15
20
25
30
35
40
45
Fig. 4:  Retrieved total drop concentration, drop median volume diameter, rain water content, and rain rate at 
0.5 km for the storms in Fig. 3.  Light (dark) shading shows radar reflectivity ≥30 dBZ  (≥40 dBZ) as 
measured with the S-Pol radar.  The disdrometer site is shown by a dot. 


P1A.1 Drop-Size Distribution Retrieval from Polarimetric Radar Measurements Using the Constrained-Gamma Model: Verification and Application

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P1A.1 Drop-Size Distribution Retrieval from Polarimetric Radar Measurements Using the Constrained-Gamma Model: Verification and Application

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