The Shape-Slope Relation in
Observed Gamma Raindrop Size
Distributions: Statistical Error or
Useful Information?*
Shaunna Donaher
MPO 531
February 28, 2008
*Zhang, G., J. Vivekanandan and E.A. Brandes, 2003. JAS, 20, 1106-1119.
Background: Disdrometer
•
The disdrometer detects and discriminates
the different types of precipitation as
drizzle, rain, hail, snow, snow grains,
graupel (small hail / snow pellets), and ice
pellets with its Laser optic.
The disdrometer calculates the intensity
(rain rate), volume and the spectrum of the
different kinds of precipitation.
•
http://www.thiesclima.com/disdrometer.htm
The main purpose of the disdrometer is to
measure drop size distribution, which it
captures over 20 size classes from 0.3mm to
5.4mm, and to determine rain rate.
Disdrometer results can also be used to infer
several properties including drop number
density, radar reflectivity, liquid water
content, and energy flux. Two coefficients, N0
and Λ, are routinely calculated from an
exponential fit between drop diameter and
drop number density.
Rain that falls on the disdrometer sensor
moves a plunger on a vertical axis. The
disdrometer transforms the plunger motion
into electrical impulses whose strength is
proportional to drop diameter. Data are
collected once a minute.
http://www.arm.gov/instruments/instrument.php?id=disdrometer
Background: Polarimetric Radar
•
Most weather radars, such as the National Weather Service NEXRAD radar,
transmit radio wave pulses that have a horizontal orientation.
•
Polarimetric radars (also referred to as dual-polarization radars), transmit radio
wave pulses that have both horizontal and vertical orientations. The horizontal
pulses essentially give a measure of the horizontal dimension of cloud (cloud
water and cloud ice) and precipitation (snow, ice pellets, hail, and rain) particles
while the vertical pulses essentially give a measure of the vertical dimension.
•
Since the power returned to the radar is a complicated function of each particles
size, shape, and ice density, this additional information results in improved
estimates of rain and snow rates, better detection of large hail location in summer
storms, and improved identification of rain/snow transition regions in winter
storms.
•
Principle of measurement is based on drops being oblate (bigger the drop= more
oblate)
http://www.cimms.ou.edu/~schuur/radar.html#Q5
Terminology
• DSD parameters
– Λ: slope of droplets
– μ: shape of droplets
– No: number of droplets
• Rain parameters/physical parameters
– R: rain rate
– Do: median volume diameter
• Errors (δμ), estimators (μ est) and expected values (μ)
Background: Marshall-Palmer
• Need an accurate mapping of DSD to get rain rate
• Previously thought an exponential distribution
with two parameters was enough to characterize
rain DSD
n(D)= Noe(-ΛD)
• But research has shown that this does not
capture instantaneous rain DSDs
Gamma distribution
• Ulbrich (1983) suggested using the gamma
function with 3 parameters which is capable
of describing a broader range of DSDs
• Each parameter can be derived from three
estimated moments of a radar retrieval
n(D) = No Dμ e(- Λ D)
Λ= slope of droplets
μ= shape of droplets (=0 for M-P)
No= number of droplets
A parameter problem
• But… the problem is the radar only measures
reflectivity (ZHH) and differential reflectivity
(ZDR) at each gate, so we only have two
parameters
• We need a relation between Λ, μ, and No so
we can use the gamma distribution
Zhang et. al (2001)
• Using disdrometer observations from eastcentral FL
• Best results come from μ- Λ correlation
No vs. μ
No vs. Λ
μ vs. Λ
Zhang et. al (2001)
Little correlation between R
and either parameter
Large values of μ and Λ (>15)
correspond to low rain rate (<
5 mm/hr)
*Polarimetric measurements are more
sensitive to heavy rain than light rain
Zhang et. al (2001)
RR>5 mm/hr
• Correlation only OK
• Good correlation
Fit line for this paper
Λ =0.0365 μ 2 + 0.735 μ + 1.935 (2)
Zhang et. al (2001)
• So now we have a relationship for μ- Λ that
allows us to retrieve 2 parameters from the
radar and find the third so we can use the
gamma dist to get DSDs
Zhang et. al (2003)
• Results from Florida
retrieved from S-Pol radar
• Retrieved using 2nd, 4th
and 6th radar moment
• Fit curve similar to that
observed in Oklahoma
and Australia (varies
slightly for season and
location)
Zhang et. al (2003)
• The μ-Λ relationship suggests that a
characteristic size parameter and the shape of
the raindrop spectrum are related
GOAL: To see if μ-Λ relationship is due to natural
phenomena or if it only results from statistical
error.
2. Theoretical analysis of error propagation
Can calculate the three parameters from any
three moment estimators
Done here for 2nd,
4th and 6th
moments
Where the ratio of moments is
2. Theoretical analysis of error propagation
Moment estimators
have measurement errors due to noise or
finite sampling, so estimated gamma
parameters
will also have errors
Even if moment estimators were precise, parameter estimates would have error
since DSDs do not exactly follow gamma distribution
2. Theoretical analysis of error propagation
• They look at var (μ est), var (Λ est), and cov (μ est, Λ est)
• Conclusions are that var (μest) is the dominant
term in var (Λ est), due to the sensitivity of μ to
changes in η due to errors in the moment
estimators  μ est and Λ est are highly
correlated (less error)
• Standard deviations of est.
parameters vs. relative
standard error of moment
estimators
• Fixed correlation coeffs
• Std of μ est and Λ est increase
as moment errors increase
• Errors for large values of μ
and Λ can be many times
larger than errors for small
values (reason for more
scatter in Fig. 1a)
Standard errors in parameter
estimators decrease as correlation
between moment estimators
increases, due to the fact that
correlated moment errors tend to
cancel each other out in the retrieval
process.
Still have more error in higher
values (low rain rates)
High correlation between μ est and Λ est leads to
a linear relation between their std
The approximate relation between the
estimation errors is
Start with Λ =(μ + 3.67)/Do, differentiate and
neglect Do since errors are small to get
Replace errors of μ and Λ (δμ, δΛ) in (10) with
the differences of their estimators (μ est, Λ est)
and expected values (μ, Λ )to get an artifact
linear relationship between μ est and Λ est
There are differences between (11) and (2)
Once the three parameters are known, rain rate and
median volume diameter can easily be calculated with:
But errors in DSD parameters from moment estimators lead to errors in Rest and Do est
So they look at variance of each estimator. The last term is negative, which means that a
positive correlation between μ est and Λ est reduces errors in Rest and Do est
Putting in (10) gives
Minimizes standard deviation of Do est
The artifact linear relation between μ est and
Λ est is the requirement of unbiased moments
and it leads to minimum error in rain
parameters
3. Numerical Simulations
Goal: To study the standard errors in the estimates of μ est and Λ est
Adding back on a
random deviation,
then recalculate
estimated DSD from
randomized moments
Look at agreements
Independent random errors
introduced into moment
estimators
One input point: (μ,Λ)=(0,1.935)0)
5% std induced, 5.21% std
outcome
Errors of moment
estimators are correlated,
still same 5% relative std
Standard errors of μ est and
Λ est are reduced
Difference between lines due
to approximation in (11)
Errors in moments are
small, but errors in of μ est
and Λ est are large and
highly correlatedfortunately these do not
cause large errors in Rest
and Do est
 Correlated moment
errors cause smaller errors in
estimated DSD parameter
and have less effect on the
μ– Λ relationship than
uncorrelated errors
• In the previous figure, there is a high correlation
between μ est and Λ est due to the added errors in the
estimated moments. This leads to an artifact linear
relationship as seen in (11). This is not the same as
derived relationship between μ and Λ in (2).
• Slope and intercept of line depends on input values.
They only used one point rather than a dataset of many
pairs.
• This is why relation in Fig.5 is different than (2) derived
from quality controlled data
So they test 100 random (μ,Λ) pairs
-2< μ est <10
0< Λ est <15
Relative random errors are added to each set of
moments to generate 50 sets of moment and
DSD parameters
Figure 6
Figure 6
6a: The scattered points show little correlation
between estimators, even when errors are
added to moment estimators.
6b: Using a threshold, estimators are in a
confined region. This shoes that physical
constraints (not only errors) determine the
pattern of estimated DSD parameters. Still
scatter at large values.
Fig. 6c
6c: Generated pairs of μ-Λ in steps. The larger the input
values, the broader the variation in estimated
parameters. This means that μ est and Λ est depend on
the input values of μ and Λ rather than the added
errors in the moment estimators.
The moment errors have little effect on the estimates μ
est and Λ est for heavy rains. This is different from Fig. 1b
which did not have variations in that increased as the
mean values increased.
 The relation in Fig. 1b is believed to represent the
actual physical nature of the rain DSD rather than pure
statistical error.
• Each pair has its own error-induced linear
relation, so the overall relation between μ est and
Λ est remains unknown
• The μ-Λ relation derived in (2) represents the
actual physical nature of rain DSD rather than
purely statistical error
– (2) is quadratic rather than linear
– Moment errors are linear and have little effect on μ-Λ
relationship at RR> 5 mm/hr and when >1000 drops
(seen in high values in Fig. 6)
– (2) does not exhibit increased spreading at high values
More on why (2) is “good”
• It predicts a wide raindrop spectrum when large drops
are present (agrees with disdrometer)
• In practice, μ and Λ are somewhat correlated due to
small range in naturally occurring Do (1 mm<Do<3 mm)
in heavy rain events, the correlation in Fig. 6b does not
lead to (2) therefore (2) is partially due to physical
nature of rain DSDs
• Retrieved μ est and Λ est from remote measurements will
contain some spurious correlation (instrument bias),
but produce almost no bias in mean values of DSD
parameters or in rain rate or median volume diameter
4. Retrieval of DSD parameters from
two moments
• Traditionally only two statistical moments are
measured in remote sensing, so the problem is
how to retrieve unbiased physical parametersneed a third DSD parameter to use gamma
distribution
• Sometimes μ is fixed so Λ and No can be retrieved
from reflectivity and attenuation, but scatter in
Fig. 1 seems to rule this out
This is why μ-Λ relationship is useful!
Dual-Polarization
• Moment pair of 5th (close to vertical polarization)
and 6th (close to horizontal polarization)
We have
Which can be solved by either
1) μ-Λ relationship
2) μ =2
μ-Λ relationship is better, but fixed μ results are OK
It is true that the bias of No and Λ depend on μ bias. But
the bias of rain parameter should be comparable, and
they are smaller when μ-Λ relationship is used.
Dual-Wavelength
• Moment pair of 3rd (attenuation coefficient is
proportional to the 3rd moment for Rayleigh scattering)
and 6th
• Again write DSD parameter as a function of the
estimated moments
• Still using 2 methods:
– Solve with (2) to use μ-Λ relationship and estimate No from
(19)
– μ =2, solve for Λ and No from (19) and (20)
Again μ-Λ relationship is better
Since standard errors are a function of μ, the error could
be larger and retrieved parameters could be biased
significantly. In contrast, rain parameters are almost
unbiased when μ-Λ relationship is used.
FL rain- 9/17/98
• Using process in Zhang (2001)
paper, μ and Λ are
determined from ZDR, ZHH and
the μ-Λ relation
• Comparison of disdrometer
vs. μ-Λ relation vs. fixed μ
• Fixed μ overestimates rain (by
a factor of 2 for μ=0)
• μ-Λ relationship derived from
radar retrieval agrees well
with disdrometer
5. Summary and discussion
• The μ-Λ relationship captures a mean physical
characteristic of raindrop spectra and is useful for retrieving
unbiased rain and DSD parameters when only two remote
measurements exist.
• Moment errors have little effect on μ-Λ relation for most
rain events.
• Compared to a fixed μ, the μ-Λ relationship is more flexible
at representing a wide range of DSD shapes observed from
an in-situ disdrometer.
• This relation should be extendable to smaller rain rates, but
may vary slightly depending on climatology and rain type.
5. Summary and discussion
• It is difficult to separate statistical errors and
physical variations, so the errors in DSD
parameter estimates should not be considered
meaningless.
• They should be studied further
– Linked to functional relations between DSD
parameters and moments
– Natural rain DSD may not follow gamma dist
5. Summary and discussion
• “Fluctuation” is a better term than “error”
since it is difficult to separate nature from
statistical errors
• Measurements always contain errors and as a
results the correlation between μ est and Λ est
may be strengthened. This could reduce bias
and std and improve retrieval process.