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 Shape-Slope Relation in Observed Gamma Raindrop Size Distributions: Statistical Error or Useful Information?*
February 28, 2008
*Zhang, G., J. Vivekanandan and E.A. Brandes, 2003. JAS, 20, 1106-1119.
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.
n(D) = No Dμe(- Λ D)
Λ= slope of droplets
μ= shape of droplets (=0 for M-P)
No= number of droplets
No vs. μ
No vs. Λ
μ vs. Λ
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
Fit line for this paper
Λ =0.0365 μ 2+ 0.735 μ + 1.935 (2)
GOAL: To see if μ-Λ relationship is due to natural phenomena or if it only results from statistical error.
Can calculate the three parameters from any three moment estimators
Done here for 2nd, 4th and 6th moments
Where the ratio of moments is
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
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
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
-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
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.
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.
This is why μ-Λrelationship is useful!
Which can be solved by either
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.
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.