The ShapeSlope Relation in Observed Gamma Raindrop Size Distributions: Statistical Error or Useful I

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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.
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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.

• 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 ?, 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.thiesclima.com/disdrometer.htm
http//www.arm.gov/instruments/instrument.php?idd
isdrometer
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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.htmlQ5

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Terminology

• DSD parameters
• ? slope of droplets
• µ shape of droplets
• No number of droplets
• Rain parameters/physical parameters
• R rain rate
• Do median volume diameter
• Errors (dµ), estimators (µ est) and expected
  values (µ)

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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(-?D)
• But research has shown that this does not capture
  instantaneous rain DSDs

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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(- ? D)

? slope of droplets µ shape of droplets (0 for
M-P) No number of droplets
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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 ?, µ, and No so we can
  use the gamma distribution

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Zhang et. al (2001)

• Using disdrometer observations from east-central
  FL
• Best results come from µ- ? correlation

µ vs. ?
No vs. µ
No vs. ?
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Zhang et. al (2001)
Little correlation between R and either
parameter Large values of µ and ? (gt15)
correspond to low rain rate (lt 5
mm/hr) Polarimetric measurements are more
sensitive to heavy rain than light rain
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Zhang et. al (2001)RRgt5 mm/hr

• Good correlation

• Correlation only OK

Fit line for this paper ? 0.0365 µ 2 0.735 µ
1.935 (2)
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Zhang et. al (2001)

• So now we have a relationship for µ- ? that
  allows us to retrieve 2 parameters from the radar
  and find the third so we can use the gamma dist
  to get DSDs

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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)

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Zhang et. al (2003)

• The µ-? relationship suggests that a
  characteristic size parameter and the shape of
  the raindrop spectrum are related
• GOAL To see if µ-? relationship is due to
  natural phenomena or if it only results from
  statistical error.

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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
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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
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2. Theoretical analysis of error propagation

• They look at var (µ est), var (? est), and cov (µ
  est, ? est)
• Conclusions are that var (µest) is the dominant
  term in var (? est), due to the sensitivity of µ
  to changes in ? due to errors in the moment
  estimators ? µ est and ? est are highly
  correlated (less error)

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• Standard deviations of est. parameters vs.
  relative standard error of moment estimators
• Fixed correlation coeffs
• Std of µ est and ? est increase as moment errors
  increase
• Errors for large values of µ and ? can be many
  times larger than errors for small values (reason
  for more scatter in Fig. 1a)

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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)
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• High correlation between µ est and ? est leads
  to a linear relation between their std
• The approximate relation between the estimation
  errors is
• Start with ? (µ 3.67)/Do, differentiate and
  neglect Do since errors are small to get

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• Replace errors of µ and ? (dµ, d?) in (10) with
  the differences of their estimators (µ est, ?
  est) and expected values (µ, ? )to get an
  artifact linear relationship between µ est and ?
  est
• There are differences between (11) and (2)

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• 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 ? est reduces
errors in Rest and Do est
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• Putting in (10) gives
• Minimizes standard deviation of Do est
• ?The artifact linear relation between µ est and
• ? est is the requirement of unbiased moments
  and it leads to minimum error in rain parameters

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3. Numerical Simulations
Goal To study the standard errors in the
estimates of µ est and ? est
Adding back on a random deviation, then
recalculate estimated DSD from randomized moments
Look at agreements
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Difference between lines due to approximation in
(11)
Errors in moments are small, but errors in of µ
est and ? est are large and highly correlated-
fortunately these do not cause large errors in
Rest and Do est
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• In the previous figure, there is a high
  correlation between µ est and ? 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 ? 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

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So they test 100 random (µ,?) pairs

• -2lt µ est lt10
• 0lt ? est lt15
• Relative random errors are added to each set of
  moments to generate 50 sets of moment and DSD
  parameters

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Figure 6
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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.

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Fig. 6c

• 6c Generated pairs of µ-? in steps. The larger
  the input values, the broader the variation in
  estimated parameters. This means that µ est and ?
  est depend on the input values of µ and ? rather
  than the added errors in the moment estimators.
• The moment errors have little effect on the
  estimates µ est and ? 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.

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• Each pair has its own error-induced linear
  relation, so the overall relation between µ est
  and ? est remains unknown
• The µ-? 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-? relationship at RRgt 5 mm/hr and when gt1000
  drops (seen in high values in Fig. 6)
• (2) does not exhibit increased spreading at high
  values

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More on why (2) is good

• It predicts a wide raindrop spectrum when large
  drops are present (agrees with disdrometer)
• In practice, µ and ? are somewhat correlated due
  to small range in naturally occurring Do (1
  mmltDolt3 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 ? 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

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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 parameters- need a
  third DSD parameter to use gamma distribution
• Sometimes µ is fixed so ? and No can be retrieved
  from reflectivity and attenuation, but scatter in
  Fig. 1 seems to rule this out
• ?This is why µ-? relationship is useful!

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Dual-Polarization

• Moment pair of 5th (close to vertical
  polarization) and 6th (close to horizontal
  polarization)
• We have
• Which can be solved by either
• µ-? relationship
• µ 2

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µ-? relationship is better, but fixed µ results
are OK It is true that the bias of No and ?
depend on µ bias. But the bias of rain parameter
should be comparable, and they are smaller when
µ-? relationship is used.
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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-? relationship and
  estimate No from (19)
• µ 2, solve for ? and No from (19) and (20)

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Again µ-? 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-? relationship is used.
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FL rain- 9/17/98

• Using process in Zhang (2001) paper, µ and ? are
  determined from ZDR, ZHH and the µ-? relation
• Comparison of disdrometer vs. µ-? relation vs.
  fixed µ
• Fixed µ overestimates rain (by a factor of 2 for
  µ0)
• µ-? relationship derived from radar retrieval
  agrees well with disdrometer

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5. Summary and discussion

• The µ-? 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-? relation
  for most rain events.
• Compared to a fixed µ, the µ-? 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.

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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

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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 ? est
  may be strengthened. This could reduce bias and
  std and improve retrieval process.