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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. Background: Disdrometer.

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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.

background disdrometer
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 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.

background polarimetric radar
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)

  • 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 (μest) and expected values (μ)
background marshall palmer
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
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 Dμe(- Λ D)

Λ= slope of droplets

μ= shape of droplets (=0 for M-P)

No= number of droplets

a parameter problem
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
Zhang et. al (2001)
  • Using disdrometer observations from east-central FL
  • Best results come from μ- Λ correlation

No vs. μ

No vs. Λ

μ vs. Λ

zhang et al 20019
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
Zhang et. al (2001)RR>5 mm/hr
  • Good correlation
  • Correlation only OK

Fit line for this paper

Λ =0.0365 μ 2+ 0.735 μ + 1.935 (2)

zhang et al 200111
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
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 200313
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
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 propagation15
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 propagation16
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 andΛ est are highly correlated (less error)

Standard deviations of est. parameters vs. relative standard error of moment estimators

  • Fixed correlation coeffs
  • Std of μestand Λest 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 andΛ 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 μest and Λestreduces errors in Rest and Do est


Putting in (10) gives

Minimizes standard deviation of Do est

The artifact linear relation between μest and

Λestis the requirement of unbiased moments and it leads to minimum error in rain parameters

3 numerical simulations
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

Difference between lines due to approximation in (11)

One input point: (μ,Λ)=(0,1.935)0)

Errors in moments are small, but errors in of μest and Λest are large and highly correlated- fortunately these do not cause large errors in Rest and Do est

5% std induced, 5.21% std outcome

Errors of moment estimators are correlated, still same 5% relative std

 Correlated moment errors cause smaller errors in estimated DSD parameter and have less effect on the μ– Λ relationship than uncorrelated errors

Standard errors of μ est and Λ est are reduced


In the previous figure, there is a high correlation between μest and Λest 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 μand 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 pairs
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 628
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
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 μest and Λest 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 μest and Λest 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 μest and Λest remains unknown

  • The μ-Λrelation 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-Λrelationship 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
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 μest and Λest 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
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 μis 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-Λrelationship 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

  • μ-Λrelationship
  • μ =2

μ-Λrelationship 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-Λrelationship 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
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
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 discussion40
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 discussion41
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.