the shape–slope relation in observed gamma raindrop size ...8908/datastream/p… · taneous’’...

1106 VOLUME 20J O U R N A L O F A T M O S P H E R I C A N D O C E A N I C T E C H N O L O G Y

q 2003 American Meteorological Society

The Shape–Slope Relation in Observed Gamma Raindrop Size Distributions: StatisticalError or Useful Information?

GUIFU ZHANG, J. VIVEKANANDAN, AND EDWARD A. BRANDES

National Center for Atmospheric Research, Boulder, Colorado

ROBERT MENEGHINI

NASA Goddard Space Flight Center, Greenbelt, Maryland

TOSHIAKI KOZU

Department of Electronic and Control Systems Engineering, Shimane University, Matsue, Shimane, Japan

(Manuscript received 30 April 2002, in final form 15 January 2003)

ABSTRACT

The three-parameter gamma distribution n(D ) 5 N0Dm exp(2LD) is often used to characterize a raindropsize distribution (DSD). The parameters m and L correspond to the shape and slope of the DSD. If m and Lare related to one another, as recent disdrometer measurements suggest, the gamma DSD model is simplified,which facilitates retrieval of rain parameters from remote measurements. It is important to determine whetherthe m–L relation arises from errors in estimated DSD moments, or from natural rain processes, or from acombination of both statistical error and rain physics.

In this paper, the error propagation from moment estimators to rain DSD parameter estimators is studied. Thestandard errors and correlation coefficient are derived through systematic error analysis. Using numerical sim-ulations, errors in estimated DSD parameters are quantified. The analysis shows that errors in moment estimatorsdo cause correlations among the estimated DSD parameters and cause a linear relation between estimators mand . However, the slope and intercept of the error-induced relation depend on the expected values m and L,Land it differs from the m–L relation derived from disdrometer measurements. Further, the mean values of theDSD parameter estimators are unbiased. Consequently, the derived m–L relation is believed to contain usefulinformation in that it describes the mean behavior of the DSD parameters and reflects a characteristic of actualraindrop size distributions. The m–L relation improves retrievals of rain parameters from a pair of remotemeasurements such as reflectivity and differential reflectivity or attenuation, and it reduces the bias and standarderror in retrieved rain parameters.

1. Introduction

Accurate characterization of raindrop size distribution(DSD) and the estimation of DSD parameters using re-mote measurements are needed for inferring rain mi-crophysics. Because various factors contribute to theformation and evolution of rain DSDs, a single explicitfunctional form has not been found. Hence, simple func-tions have been used to model a rain DSD.

Historically, an exponential distribution with two pa-rameters was used to characterize rain DSD. Specialcases of exponential DSDs were determined by Marshalland Palmer (1948) and Laws and Parsons (1943). How-

Corresponding author address: Dr. Guifu Zhang, NCAR/RAP,3450 Mitchell Lane, Building 2, Boulder, CO 80307.E-mail: [email protected]

ever, subsequent DSD measurements have shown thatthe exponential distribution does not capture ‘‘instan-taneous’’ rain DSDs and a more general function isnecessary.

Ulbrich (1983) suggested the use of the gamma dis-tribution for representing rain DSD as

mn(D) 5 N D exp(2LD).0 (1)

The gamma DSD with three parameters (N0, m, and L)is capable of describing a broader range of raindrop sizedistributions than an exponential distribution (a specialcase of the gamma distribution with m 5 0). The threeparameters of the gamma DSD can be obtained fromthree estimated moments. It was shown that the threeparameters are not mutually independent (Ulbrich 1983;Chandrasekar and Bringi 1987; Kozu 1991; Haddad etal. 1997). Hence attempts were made to derive rain

AUGUST 2003 1107Z H A N G E T A L .

FIG. 1. Scatterplots of m–L values obtained using the momentmethod: (a) without filtering and (b) with filtering of rain rate R .5 mm h21 and total counts CT . 1000.

FIG. 2. Median volume diameter (D0) and std dev of massdistribution (sm) as a function of shape parameter (m).

DSDs from a set of mutually independent parameters(Haddad et al. 1997). However, correlations among DSDparameters, if real, may be useful in reducing the num-ber of unknowns and enable the retrieval of the DSDfrom a pair of independent remote measurements suchas reflectivity and attenuation, as in the case of the dual-wavelength radar technique, or the reflectivities at hor-izontal and vertical polarization, as in the case of po-larimetric radar. An approach to find relationship amongthe DSD parameters had been proposed by Kozu et al.(1999). An N0–m relation was found and used by Ulbrich(1983) for retrieving the three DSD parameters fromreflectivity and attenuation. The derived relation waslater attributed to statistical error (Chandrasekar andBringi 1987) and is unstable depending on the methodof fitting procedure. Moreover, fluctuations in N0–mrange over several orders of magnitude; hence, the util-ity of the relation is limited.

Analysis of DSD data collected in Florida during thesummer of 1998 revealed a high correlation between mand L, suggesting that a useful m–L relation could bederived (Zhang et al. 2001). The resulting relation wasused to retrieve rain DSDs from S-band polarizationradar measurements of reflectivity (Z) and differentialreflectivity (ZDR). An independent analysis of DSD ob-servations collected in Oklahoma also indicated the ex-istence of a m–L relation similar to that observed inFlorida (Brandes et al. 2003). Figure 1a shows a scat-terplot between m and L for the Florida dataset. Figure1b presents the relation for the cases with rain .5 mmh21 and large number of counts (CT . 1000). The re-vised relation is

2L 5 0.0365m 1 0.735m 1 1.935. (2)

The relation also holds for the estimated DSD param-eters using the truncated moment method (Ulbrich andAtlas 1998; Vivekanandan et al. 2003). It is noted thatthe scatterplot between m and L for the filtered raincases in Fig. 1b has a higher correlation than that withoutfiltering in Fig. 1a. Therefore, the relation (2) derivedfrom the quality controlled dataset in Fig. 1b shouldhave smaller error.

The median volume diameter (D0) and the standarddeviation of mass distribution (sm) are related to m andL (Ulbrich 1983). If the relation (2) holds, D0 and sm

depend only on m as shown in Fig. 2. Both D0 and sm

decrease as m increases, which means that rain withlarge (small) D0 corresponds to a wide (narrow) distri-bution of rain DSD. As m changes from 10 to 22, D0

increases from 1.1 to 2.7 mm and sm increases from0.3 to 2.4 mm, a range that includes most heavy andmedium rain-rate cases. Thus, the m–L relation suggeststhat a characteristic size parameter such as D0 and theshape of a raindrop spectrum are also related. The re-lation simplifies the three-parameter gamma distributionto a two-parameter constrained gamma DSD model.


However, since the m–L relation was reported, there hasbeen concern as to whether the relation arises purelyfrom statistical error or represents a physical propertyof rain.

In this paper, we describe a detailed analysis of errorpropagation from DSD moment estimators to DSD pa-rameter estimators and evaluate the improvement inphysical parameter retrievals using the m–L relationover existing methods. It is not our intention to identifythe source of the moment errors as studied previously(e.g., Wong and Chidambaram 1985). We address thefollowing issues: 1) what is the origin of the m–L re-lation, and 2) why is it useful for retrieving rain param-eters from remote measurements. In section 2, we derivethe standard errors of estimated DSD parameters ana-lytically. The effects of moment estimator errors on theestimated DSD parameters are then studied using nu-merical simulations in section 3. In section 4, the use-fulness of the m–L relation is examined by comparingDSD retrievals using the m–L relation with those ob-tained using a fixed value of m. Finally, we summarizethe work and discuss the findings in section 5.

2. Theoretical analysis of error propagation

Disdrometer measurements of rain DSD are usuallyprocessed by calculating the statistical moments. Theestimated moments of disdrometer observations are of-ten used to estimate the DSD governing parameters suchas N0, m, and L in (1), and then the DSD parametersare used to calculate physical parameters such as rainrate, characteristic size, and radar measureables. It isimportant to study how the errors associated with DSDmeasurements, the estimated DSD moments, and theDSD model propagate to the estimated DSD parametersand physical parameters. It is also worth pointing outthat remote measurements of rain, such as reflectivityfactor, attenuation, and phase shift at various wave-lengths and polarizations, can be approximated as mo-ments of rain DSD in a sampling volume. Errors in thedetermination of the moments occur both in the radarmeasurement and in the estimation of the moments fromthe radar data.

a. Error propagation from DSD moments to DSDparameters

For the gamma DSD, the three parameters can beestimated from any three moment estimators, Mn withmeans of Mn 5 N0L2(m1n11) G(m 1 n 1 1). As an ex-ample, given the 2d, 4th, and 6th (M2, M4, and M6)moments, the DSD parameters (N0, m, L) can be writtenas

2 1/2(7 2 11h) 2 (h 1 14h 1 1)m 5 , (3)

2(h 2 1)1/2

M G(m 1 5)2L 5 , and (4)[ ]M G(m 1 3)4

m1n11M LnN 5 , (n 5 2, 4, or 6), (5)0 G(m 1 n 1 1)

where the ratio of the moments is h 5 /(M2M6).2M 4

Obviously, the DSD parameters are nonlinear functionsof the DSD moments.

Since the moment estimators (M2, M4, and M6) con-tain measurement errors due to system noise or finitesampling, the estimated gamma DSD parameters (N0,

, ) also have error. Even if the moment estimatorsˆm Lwere determined precisely, the estimated DSD param-eters would fluctuate due to the fact that natural rainDSDs may not exactly follow the assumed gamma dis-tribution.

To understand the error propagation from the momentestimators to the retrieved DSD parameters, a detailederror analysis is described in appendix A, which is basedon the first-order approximation (Papoulis 1965, section7-3). The first-order theory is valid for small fractionalmoment errors (,20%) that are representative of qual-ity-controlled datasets with sufficient drop counts(.1000) (Wong and Chidambaram 1985). Both the mo-ment estimators and the DSD parameter estimators arewritten as sums of their means and fluctuations. Thevariances and covariance of DSD parameter estimators

and are represented asˆm L

2 2 2 2dm ]h ]h ]h ]h ]hˆ ˆ ˆ ˆ ˆvar(m) 5 var(M ) 1 var(M ) 1 var(M ) 1 2 cov(M , M )2 4 6 2 41 2 1 2 1 2 1 2[dh ]M ]M ]M ]M ]M2 4 6 2 4

]h ]h ]h ]hˆ ˆ ˆ ˆ1 2 cov(M , M ) 1 2 cov(M , M ) , (6)2 6 4 6 ]]M ]M ]M ]M2 6 4 6

2]L ]L ]L ]L dm ]h ]L ]L ]L dm ]hˆ ˆ ˆvar(L) 5 var(m) 1 1 2 var(M ) 1 1 2 var(M )2 41 2 1 2 1 2]m ]M ]M ]m dh ]M ]M ]M ]m dh ]M2 2 2 4 4 4

]L ]L ]L dm ]L ]h ]L ]h ]L dm ]L ]hˆ ˆ ˆ ˆ1 2 1 1 cov(M , M ) 1 2 cov(M , M )2 4 2 61 2[ ]]M ]M ]m dh ]M ]M ]M ]M ]m dh ]M ]M2 4 2 4 4 2 2 6


]L dm ]L ]h ˆ ˆ1 2 cov(M , M ), and (7)4 6]m dh ]M ]M4 6

dm ]h ]L ]L dm ]h dm ]h ]L ]L dm ]hˆ ˆ ˆcov(m, L) 5 1 var(M ) 1 1 var(M )2 41 2 1 2dh ]M ]M ]m dh ]M dh ]M ]M ]m dh ]M2 2 2 4 4 4

2 2]L dm ]m dm ]L ]h ]L ]h ]L dm ]h ]hˆ ˆ ˆ1 var(M ) 1 1 1 2 cov(M , M )6 2 41 2 1 2 1 2]m dh ]M dh ]M ]M ]M ]M ]m dh ]M ]M6 2 4 4 2 2 4

dm ]L ]h ]L dm ]h ]h dm ]L ]h ]L dm ]h ]hˆ ˆ ˆ ˆ1 1 2 cov(M , M ) 1 1 2 cov(M , M ),2 6 4 61 2 1 2dh ]M ]M ]m dh ]M ]M dh ]M ]M ]m dh ]M ]M2 6 2 6 4 6 4 6

(8)

FIG. 3. Analytical results of the standard errors of DSD param-eter estimators as a function of the relative error of the momentestimators for fixed correlations among moment errors of r(M 2 ,M 4 ) 5 0.5, r(M 4 , M 6 ) 5 0.5, and r(M 2 , M 6 ) 5 0.25: (a) std( )mand (b) std( ).L

where the derivative and partial derivatives are givenin appendix A.

The covariance terms among the moment estimatorsare included in (6)–(8). However, the correlationsamong errors in the moment estimators depend on whattype of error source is dominant in the dataset. For theDSD measurements, sampling errors among the momentestimates tend to be correlated. The closer the two mo-ments, the higher the correlation. A high-order momentestimator (e.g., M6) may have little correlation with alower-order moment estimator (M2). For remote mea-surements, the errors in reflectivity measurements at twodifferent frequency channels are uncorrelated, whilethose at a dual polarization are partially correlated de-pending on how the signals are collected. The errorsdue to system noise are uncorrelated. Detailed study ofthe correlations among the moment errors is beyond thescope of the present study but has been studied in detailby Chandrasekar and Bringi (1987). Consequently, anumber of correlation coefficients such as 0, 0.5, and0.8 are assumed for this study.

The square roots of (6) and (7) give the standarddeviations of and , which are denoted as std( ) andˆm L mstd( ), respectively. The standard deviations are plottedLfor various parameters in Figs. 3 and 4. Figure 3 showsthe std( )and std( ) as a function of the relative stan-ˆm Ldard error of moment estimators, std(Mn)/Mn. The cor-relation coefficients among the moment estimators usedfor the calculations are fixed at r(M2, M4) 5 0.5, r(M4,M6) 5 0.5, and r(M2, M6) 5 0.25. As expected, thestandard deviations of and increase as the errorsˆm Lin moments increase. It is noted that the standard errorsalso depend on their expected values. The errors forlarge values of m and L can be many times larger thanthose for small values of m and L. This might be thereason for the large scatter on the right side of Fig. 1a.

Figure 4 shows the standard errors and correlationcoefficient of and as a function of the correlationˆm Lcoefficients among the moment errors for fixed relativeerrors in moment estimators of 5%. For convenience inshowing the results, the correlation coefficients among


FIG. 4. Analytical results of standard error of DSD parameter es-timators as a function of the correlation coefficients among the mo-ment estimators for a fixed relative moment error of 5%: (a) std( ),m(b) std( ), and (c) r( , ).ˆ ˆL m L

the moment estimators are chosen such that r(M2, M4)5 r(M4, M6) and r(M2, M6) 5 r(M2, M4)r(M4, M6).As shown in Figs. 4a and 4b, the standard errors in mand estimates decrease as the correlations among mo-Lment estimators increase, a consequence of the fact thatcorrelated moment errors tend to cancel each other inthe retrieval of and values. Such error cancellationˆm Lis obvious in the ratio 5 /(M2M6). The standard2ˆh M4

error of follows the same trend as that of . TheL mcorrelation coefficient between and , r 5 cov( ,ˆm L m

)/[var( ) var( )]1/2, calculated from (6)–(8), is shownˆ ˆL m Lin Fig. 4c and is very high (.0.9).

Further analysis shows that the estimated is highlymsensitive to the change in due to errors in the momenthestimators; as a result, the variance of is large. Formthe same reason, the first term in (7), the variance of

, is the dominant term in the variance of . Therefore,ˆm Land are highly correlated. The high correlation leadsˆm L

to a linear relation between the standard deviations ofand , std( ) ø ]L/]m std( ). This approximateˆ ˆm L L m

relation between the estimation errors is given by

]L L(m 1 3.5)dL ø dm 5 dm. (9)

]m (m 1 4)(m 1 3)

A simple way to study the relation between the error ofestimators and is to start from L 5 (m 1 3.67)/ˆm LD0. Taking the differential gives dL 5 (1/D0) dm 2 (m1 3.67/ )dD0. Since the error in D0 tends to be small2D0

(any good fitting procedure would have small error inestimating physical parameters), neglecting the secondterm yields

1 LdL ø dm 5 dm. (10)

D (m 1 3.67)0

It can be seen that (9) and (10) are very similar. Wereplace the errors (dm, dL) in (10) with the differencesof their estimators ( , ) and expected values (m, L)ˆm Lin (10), respectively, and obtain an artifact linear rela-tion between and estimators, as given byˆm L

LL ø (m 2 m) 1 L. (11)

(m 1 3.67)

The – relation (11) looks like the m–L relation (2)ˆm Las they both have a positive roughly similar slope. How-ever, they are essentially different in terms of both phys-ics and functional relation, although we acknowledgethat statistical errors and physical variations are difficultto separate. Equation (2) is a general relation for themean m and L values, whereas (11) is an artificial re-lation between the estimators and for a pair ofˆm Lexpected values (m, L) because of the introduced errors.The slope and intercept of (11) depend on the expectedvalues (m, L), and the overall relation between m andL remains unspecified. The difference will be studiedfurther through simulations in the next section.


b. Error propagation from DSD parameters tophysical parameters

When gamma rain DSD parameters (N0, m, and L)are known, physical characteristics of rain, such as rainrate (R) and median volume diameter (D0), can be easilycalculated with

24 2(m14.67)R 5 7.125 3 10 N L G(m 1 4.67) and (12)0

m 1 3.67D 5 . (13)0 L

Since the DSD parameters contain errors inherited fromthe moment estimates or remote measurements, the rain-rate estimators (R) and median volume diameter esti-mators (D0) also have error components. The variancesof R and D0 are derived in appendix B as

22 2R G9(m 1 4.67) (m 1 4.67)

2 2ˆ ˆ ˆvar(R) 5 var(N ) 1 2 ln(L) R var(m) 1 R var(L)02 2[ ]N G(m 1 4.67) L0

2 2R G9(m 1 4.67) R (m 1 4.67)ˆ ˆ ˆ1 2 2 ln(L) cov(N , m) 2 2 cov(N , L)0 0[ ]N G(m 1 4.67) N L0 0

2R (m 1 4.67) G9(m 1 4.67) ˆ2 2 2 ln(L) cov(m, L) and (14)[ ]L G(m 1 4.67)

21 D D0 0ˆ ˆ ˆvar(D ) 5 var(m) 1 var(L) 2 2 cov(m, L). (15)0 2 2 2L L L

It is noted that there is a negative sign in front of the last covariance term, cov( , ), in (14) and (15).ˆm LTherefore, the positive correlation between the estimation errors of and reduces the errors in R and D 0

ˆm Lestimators for given errors in DSD parameter estimators.

If the approximate relation (10) is used in the derivation for the variances of R and D 0 , approximateexpressions are obtained as

22R G9(m 1 4.67) m 1 4.67

2ˆ ˆvar(R) ø var(N ) 1 2 ln(L) 2 R var(m)02 [ ]N G(m 1 4.67) m 1 3.670

2R G9(m 1 4.67) m 1 4.67 ˆ1 2 2 ln(L) 2 cov(N , m) and (16)0[ ]N G(m 1 4.67) m 1 3.670

21 (m 1 3.67)ˆvar(D ) ø 1 2 var(m) 5 0. (17)0 2 [ ]L LD0

In this case, the standard error of rain-rate estimates isfurther reduced as compared with that by uncorrelatederrors in DSD parameters, and the standard deviationof the estimator D0 is minimized. It is also noted thatthe mean values of retrieved physical parameters shouldnot be biased by the fluctuation errors in the DSD pa-rameters. In other words, the artifact linear relation be-tween estimators and is the requirement of unbiasedˆm Lmoments and it leads to minimum error in rain param-eters. This becomes even clearer in the simulation re-sults in the next section.

3. Numerical simulations

In support of the error analysis described in the pre-vious section, a numerical simulation was performed to

study the standard errors in the estimates of the DSDparameters and . The steps involved in the simu-ˆm Llation are as follows.

1) Assign gamma DSD parameters with a set of specificvalues (inputs) for N0, m, and L.

2) Calculate the expected moments (M2, M4, and M6)for the specified gamma DSD parameters.

3) Randomize the moments with given standard devi-ations as Mn 5 Mn 1 dMn (n 5 2, 4, 6). The fluc-tuations dMn of the moments represent error and areassumed to be uniformly distributed with a zeromean (a Gaussian error distribution makes almost nodifference in the result). Calculate the estimated DSDparameters (N0, , ) from the randomized momentsˆm Lusing Eqs. (3)–(5).


FIG. 5. Numerical simulations of DSD parameters determinedfrom randomized moments with 5% relative errors for a set of inputparameters: N0 5 8000, m 5 0.0, and L 5 1.935. (a) Independenterrors in moment estimates and (b) correlated errors in moment es-timates with the correlation coefficients of r(M2, M4) 5 0.8, r(M4,M6) 5 0.8, and r(M2, M6) 5 0.64.

4) Calculate the statistics of the estimated DSD param-eters and physical parameters.

The simulation results are shown as scatterplots betweenand in Fig. 5. A relative standard deviation of 5%ˆm L

(0.21-dB error) for each of the moments is used in thesimulations. Figure 5a shows the result for independentrandom errors introduced into the moment estimators.Both the fitted line and the approximate linear relation(11) are also shown. The small difference between theslope of the fitted line and that of (11) is due to theapproximations made in deriving (11). The input (m, L)5 (0, 1.935) is represented by a red circle. Even withsuch small errors in moments, the standard errors of mand are large and highly correlated. The standard de-Lviations of std( ) 5 0.588 and std( ) 5 0.272 and theˆm Lcorrelation coefficient of r( , ) 5 0.992 agree with theˆm Lanalytical results shown in Figs. 4a–c as indicated by thedotted line values at r 5 0, which shows the validity ofthe first-order approximation in the theoretical analysis.Fortunately and importantly, the correlated errors in theDSD parameter estimators and do not cause largeˆm L

errors in the estimated physical parameters R and D0.Rather, the standard errors of R and D0 are very small.The relative standard deviation of std(R)/R 5 5.21% isclose to the introduced errors in moments of 5% and thatof std(D0)/D0 5 2.64% is even smaller than the relativemoment errors. This shows that the moment fitting pro-cedure does not significantly amplify the moment errors.More importantly, the means of the estimated DSD pa-rameters, ^N0& 5 8395.4, ^ & 5 0.071, and ^ & 5 1.967ˆm Lare very close to the input values. The mean rain rateand median volume diameter are ^R& 5 38.58 mm h21

and ^D0& 5 1.897 mm and are essentially identical to theinput values of 38.62 mm h21 and 1.897 mm.

Figure 5b shows the simulation result when the errorsof moment estimators are taken to be correlated but withthe same relative standard deviation of 5% as that inFig. 5a. The correlation coefficients used for the sim-ulation are r(M2, M4) 5 0.8, r(M4, M6) 5 0.8, andr(M2, M6) 5 0.64. As shown in Fig. 5b, the standarderrors of and are reduced considerably from thoseˆm Lwhen the moment errors are not correlated (Fig. 5a).This shows that correlated moment errors cause smallererrors in estimated gamma DSD parameters and haveless effect on the m–L relation than uncorrelated errorsas shown in Figs. 4a and 4b. Again, the means of theestimated DSD parameters, ^N0&, ^ &, and ^ &, as wellˆm Las the means of rain rate (^R&) and median volume di-ameter (^D0&), are almost the same as the input values.The relative standard errors of R and D0 are also verysmall as std(R)/R 5 5.13% and std(D0)/D0 5 1.32%,respectively.

The estimates and scatter mainly along a lineˆm Lcentered at the input m 5 0.0 and L 5 1.935. There isa high correlation between and due to the addedˆm Lerrors in the estimated moments. The correlation leadsto an artifact linear relation between and as shownˆm Lin (11). Note, however, that the slope and intercept ofthe relation depend on the input values m and L, thatis, the means of the estimated moments rather than thefluctuation errors. Also, the scatterplot in Fig. 5 is forone pair of (m, L) 5 (0.0, 1.935) rather than a datasetconsisting of various pairs of (m, L) values. Therefore,the artifact relation in Fig. 5 is different from the m–Lrelation (2) derived from the measurements with quality-controlled data in Fig. 1b. Instead, the mean relationsamong the gamma DSD parameters should have a phys-ical cause and not due to purely statistical error (Chan-drasekar and Bringi 1987).

Realizing the difficulty of separating statistical errorsand physical variations in measurements (e.g., Fig. 1),simulations with various pairs of (m, L) were performed.Pairs of (m, L) were generated both with a randomnumber generator and the constrained relation (2). Re-sults are shown in Fig. 6. Relative standard errors of5% were introduced in the moment estimates with thecorrelation coefficients of r(M2, M4) 5 0.5, r(M4, M6)5 0.5, and r(M2, M6) 5 0.25.

First, a hundred pairs of (m, L) were randomly gen-


FIG. 6. Numerical simulations of DSD parameters determined fromrandomized moments for various pairs of m and L inputs. Relativestandard errors of 5% were introduced in the moment estimates withthe correlation coefficients of r(M2, M4) 5 0.5, r(M4, M6) 5 0.5,and r(M2, M6) 5 0.25: (a) 100 random pairs of m and L, (b) randompairs of m and L with a threshold of 1.0 , D0 , 3.0 mm, and (c)m–L relation pairs.

erated with m between 22 and 10 and L between 0 and15. The relative random errors are added to each set ofmoments to generate 50 sets of moment and DSD pa-rameter estimates. The simulation results are shown inFig. 6a. The estimates and are scattered along aˆm Lline centered on the input m and L (red circles). The

estimated and values cover the whole domain. Theˆm Lscattered points show little correlation between andm

, even when errors are added in the moment estimators.LIt is noted that for heavy and medium rains D0 is typ-ically in the range from 1.0 to 3.0 mm. Applying suchthresholds, we obtain Fig. 6b. The estimated and ˆm Lare now in a confined region because of the threshold1.0 , D0 , 3.0 mm. This shows that physical con-straints (not only errors) determine the pattern of esti-mated and . The scatter increases as m and L in-ˆm Lcrease, which is similar to that in Fig. 1a containingmeasurement errors. But, it is not the same as that shownin Fig. 1b leading to the m–L relation (2).

Second, pairs of m and L values were generated withm varying between 22 and 15 with steps of 1, and Lcalculated from (2) for each m. The simulation resultsare shown in Fig. 6c. The estimated and lie alongˆm La line for each pair of input m and L, and are centeredat the inputs with scatter. For the reason discussed inthe previous section, the larger the input values of mand L, the broader the variation in the estimated andm

. This feature is similar to that in Fig. 1a. However,Lthe means of and depend on the input values of mˆm Land L rather than the added errors in the moment es-timates. The moment errors have little effect on esti-mates and for the small values of m and L (heavyˆm Lrain cases). In contrast, the threshold data in Fig. 1b donot exhibit variations in and that increase as theirˆm Lmean values increase. Therefore, the relation in Fig. 1bis believed to represent the actual physical nature of rainDSD rather than purely statistical error.

The m–L relation (2) derived from measurementsshould not be confused with the linear relationshipshown in Fig. 5, which is a result of statistical errors.It is true that the estimates and exhibit a highˆm Lcorrelation and have a linear relation when moment es-timators contain random errors for a pair of m and L.However, the mean values of estimated DSD parametersand physical parameters are not biased by fluctuationerrors in moment estimators as shown in Figs. 5 and 6.Furthermore, since each pair of m and L has its ownerror-induced linear relation, the overall relation be-tween and remains unknown. Consequently, the m–ˆm LL relation (2), obtained by a second-order polynomialfitting of estimated m and L for a quality-controlleddataset, is believed to be related to the physical natureof rain DSD and contains useful information for thefollowing reasons:

(i) The pattern of the scatter shown in Fig. 6 dependsmainly on the expected values (m, L) rather thanon statistical error alone. The slope and interceptof a linear relation associated with moment errordepend on the expected values (m, L), while themean values of the DSD parameters and physicalparameters are not biased by fluctuation errors inthe moments. Furthermore, relation (2) exhibits a


TABLE 1. DSD parameter retrieval from the 5th and 6th moments.

Param-eters Input

Method 1: m–L relation

Mean Std

Method 2: fixed m 5 2.0

Mean Std

N0

mLRD0

80000.01.935

38.621.897

8926.50.0121.947

38.671.899

4243.90.3330.2464.500.050

8350.72.02.582

35.802.207

5142.80.00.1824.7830.155

quadratic rather than the linear form associatedwith characteristics of the moment errors.

(ii) The moment errors have very little effect on the m–L relation when the rain rate is greater than 5 mmh21 and when the total drop count is large (.1000).For fixed relative errors in moment estimators, theerrors in the estimated and values are large forˆm Llarge m and L values (light rains), while the errorsare small for small m and L values (heavy rains).This is shown in Fig. 6, which is similar to measureddata with sampling errors shown in Fig. 1a. The qual-ity-controlled dataset shown in Fig. 1b, which wasused to derive relation (2), does not exhibit suchincreased spreading with expected values (m, L).

(iii) The relation (2) predicts that the raindrop spectrumis wide when large drops are present (Fig. 2). Thisis consistent with raindrop spectra observed by vid-eo disdrometer. It is important to note that the re-lation (2) and the correlation between m and L arenot a consequence of the relation LD0 5 m 1 3.67since it is theoretically possible for m and L to beuncorrelated in particular and any two of the threeparameters uncorrelated. Even though, in practice,m and L are somewhat correlated because D0 usu-ally varies in a limited range between 1 and 3.0mm for most heavy rain events, the correlationbetween m and L shown in Fig. 6b does not leadto the relation (2). Therefore, we contend that Eq.(2) is partially a consequence of the physical natureof the rain DSDs and not a consequence of the LD0

5 m 1 3.67 relation.(iv) Even though the m–L relation (2) is influenced to

some extent by the errors in moment estimates, itis a useful relation in that it reduces the errors inrain parameter retrieval. The relation (2) simplifiesthe gamma DSD model and enables rain DSD re-trievals from two independent remote measure-ments. Nevertheless, remote measurements, whichcorrespond approximately to moments of the DSDin the radar sampling volume, contain measurementerror. As such, the retrieved and values fromˆm Lremote measurements will contain some spuriouscorrelation. Nevertheless, there is almost no biasin the mean values of the DSD parameters and inphysical parameters such as rain rate ^R& and ^D0&.

4. Retrieval of DSD parameters from twomoments

In a real remote measurement, the number of inde-pendent measurables is generally limited. For example,in a dual-wavelength or dual-polarization radar tech-nique, only two statistical moments are measured ratherthan the three required to determine the three gammaDSD parameters. In practice, the problem is how to re-trieve unbiased physical parameters, such as rain rate andmedian volume diameter with two remote measurables.

Some DSD retrieval algorithms assume one of the

DSD parameters when only two independent remoteobservations are available. For example, m is fixed sothat L and N0 can be retrieved from reflectivity andattenuation, such as in the Tropical Rainfall MeasuringMission (TRMM) algorithm (Kozu and Nakamura 1991;Iguchi et al. 2000; Meneghini et al. 2001). The scatterin Fig. 1 would seem to preclude such an approach.When a m–L relation (2) is used, then two parameterscan be determined from two measurements such as re-flectivity and differential reflectivity (Zhang et al. 2001).In this section, a comparison between the rain retrievalusing the m–L relation and that with fixed m is presentedusing moment pairs that correspond to S-band polari-zation radar measurements. With the dual-polarizationand dual-wavelength radar techniques, the moment pairmight correspond to the 5th and 6th or 3d and 6th mo-ments, respectively. The DSD retrievals are evaluatedbased on numerical simulations and measurements.

a. Gamma DSD parameter retrieval from the 5th and6th moments

Dual-polarization radar measures reflectivity at hor-izontal and vertical polarizations. The 5th and 6th mo-ments are chosen because they are close to the reflec-tivity at vertical polarization (ZV ; M5) and at horizontalpolarization (ZH ; M6). The moment estimators are gen-erated using the procedure described in the previoussection for a set of input DSD parameters having arelative standard error of 5% (0.21 dB) and withoutcorrelation among the moments. We have

M (m 1 6)5L 5 and (18)M6

m17M L6N 5 . (19)0 G(m 1 7)

Expressions (18) and (19) can be solved in two ways:using a m–L relation such as (2) or assuming a fixedm. With the first method, (18) is solved jointly with (2)for m and L from the estimated moments M5 and M6.Then N0 is found from (19). With the second method,m is fixed at 2. Then L and N0 are solved from (18)and (19). The means and standard deviations of the DSDparameter estimators and rain rate as well as medianvolume diameter were calculated and are listed in Table1. The means of the retrieved parameters with the m–


TABLE 2. DSD parameter retrieval from the 3d and 6th moments.

Param-eters Input

Method 1: m–L relation

Mean Std

Method 2: fixed m 5 2.0

Mean Std

N0

mLRD0

80000.01.935

38.621.897

8363.70.0391.964

38.521.889

1106.10.1180.0871.850.024

11 966.62.02.729

40.622.079

2275.80.00.0681.600.052

FIG. 7. Retrieval of rain parameters from S-band polarimetric radarmeasurements: reflectivity and differential reflectivity using the m–L relation and fixed m methods: (a) rain rate and (b) median volumediameter.

L relation are in better agreement with expected valuesthan those with fixed m. It is true that the bias of N0

and L depend on the m bias. But, the bias of rain pa-rameters should be comparable, which are smaller whenthe m–L relation is used. The standard deviations arealso smaller except for m, which was specified.

b. Gamma DSD parameter retrieval from the 3d and6th moments

Dual-wavelength radar measures reflectivity and at-tenuation using reflectivity measurements where at leastone of the channels is attenuating. Because the atten-uation coefficient is proportional to the 3d moment forRayleigh scattering, the DSD parameter retrieval fromthe 3d and 6th moments is studied. Using the momentestimators (M3 and M6) and the gamma DSD model,the DSD parameter can be written as a function of theestimated moments

1/3M (m 1 6)(m 1 5)(m 1 4)3L 5 . (20)[ ]M6

As in the previous section, two methods are used toretrieve the DSD parameters. In the first method, Eq.(20) is jointly solved in conjunction with (2) for m andL from M3 and M6. The estimate N0 is then found from(19). In the second method, m is fixed at 2; L and N0

are then solved from (20) and (19) accordingly. Themeans and standard deviations of the DSD parameterestimators and rain rate, as well as median volume di-ameter are listed in Table 2. Again, the retrieved pa-rameters are much less biased when the m–L relationis used than that when m is fixed. The standard errorsare comparable. However, in the case of a fixed m ap-proach, the actual standard error is a function of m andthe error could be larger and retrieved parameters couldbe biased significantly. In contrast, rain parameters (R andD0) are almost unbiased when the m–L relation is used.

c. Rain parameter retrieval from the S-Polmeasurements

As mentioned previously, polarization radar measuresreflectivity (Z) and differential reflectivity (ZDR), whichcan be used for retrieving rain DSD parameters. Themeasured reflectivity at S band is not exactly the 6thmoment of DSD when large raindrops are present in the

sampling volume. The scattering amplitudes are nu-merically calculated using the T-matrix method (Oguchi1983; Vivekanandan et al. 1991), and then the reflec-tivity and differential reflectivity are computed for as-sumed rain DSDs. Following the procedure in Zhang etal. (2001), Vivekanandan et al. (2003), and Brandes etal. (2003), m and L are determined from ZDR and them–L relation. Here N0 is estimated from Z. The DSDparameters are also retrieved for a fixed m. Rain rateand median volume diameter are calculated.

Data used for this study were collected on 17 Septem-ber 1998 in Florida during the PRECIP98 project. Mea-surements were available from NCAR’s S-Pol radar anda disdrometer operated by University of Iowa (Brandeset al. 2002). The results of disdrometer measurementsand the radar-retrieved values using (i) the m–L relationand (ii) a fixed m are shown in Fig. 7. The fixed m methodoverestimates the rain (almost by a factor of 2 for m 5


0, see Fig. 7a), the radar retrieved value with the m–Lrelation agrees well with the disdrometer measurement.By using the m–L relation, the retrievals of median vol-ume diameter (Fig. 7b) agree more closely with the dis-drometer measurements than by using a fixed m.

5. Summary and discussion

In this paper, detailed analyses of error propagationfrom moment estimators to the estimated gamma DSDparameters were performed. A mathematical approachwas used to quantify the effects of errors in momentson DSD parameters and on R and D0 retrievals. Theretrievals using the m–L relation were compared withthe fixed m approach. The m–L relation is believed tocapture a mean physical characteristic of raindrop spec-tra and is useful for retrieving unbiased DSD parameterswhen only two independent remote measurements areavailable such as Z and ZDR or attenuation.

Theoretical analyses and numerical simulations con-firm that errors in moment measurements (estimates)can cause high correlations in gamma DSD parameterssuch as that observed between and for a single pairˆm Lof expected values in Fig. 5. This error effect, however,should not be equated to the m–L relation (2) derivedfrom a quality-controlled dataset of rain DSD measure-ments. The slope and intercept of the linear relationassociated with moment error depend on the particularvalues of m and L, whereas the mean values of retrievedDSD and physical parameters are not biased by fluc-tuation errors in the moments. The moment errors havelittle effect on the m–L relation for rain rates that con-tribute most to rain accumulations. The m–L relation isconsistent with observation whereby heavy rain is rep-resented with large drops having a broad distribution.Compared to the gamma distribution with a fixed m, theconstrained gamma distribution with the m–L relationis more flexible in representing a wide range of instan-taneous DSD shapes.

Recognizing the difficulty of separating statistical er-rors and physical variations, we believe the errors inDSD parameter estimators should not be consideredmeaningless; rather they should be studied further forthe following reasons.

1) The errors in the estimated DSD parameters arelinked to the functional relations between DSD pa-rameters and moments. The correlations among theestimated gamma DSD parameters due to momenterrors are a result of DSD fitting (moment method),and a requirement of unbiased moments and physicalparameters.

2) Natural rain DSD may not be the same as the math-ematically modeled gamma distribution. In the mod-el we have used here, the difference between actualDSD and assumed gamma distribution can be attri-buted to errors in the moment estimators.

3) ‘‘Fluctuation’’ is a more appropriate description than‘‘error’’ in characterizing the differences of DSD pa-rameters or moments from their expected valuessince each realization could be a real physical event.The DSD parameters should be allowed to vary asin Zhang et al. (2002). It is very difficult to separatethe physical variations from statistical errors.

4) Nevertheless, measurements always contain errorsand as a result the correlation between and mayˆm Lbe strengthened. If such a correlation can improveretrievals such that the bias and standard error inphysical parameters are minimized, it can be a valu-able addition to the retrieval process.

It has been shown that rain DSD retrievals from radarmeasurements that use the m–L relation agree with thein situ measurements better than those obtained withfixed m. The relation is thought to capture the physicalnature of rain DSDs and should provide a way to im-prove DSD estimation by dual-parameter radar retrievaltechniques.

We derived the m–L relation (2) from video disdrom-eter measurements in Florida during the summer of 1998for moderate and heavy rain case (R . 5 mm h21) tominimize the sampling error effect. The relation (2)should be extendable to rain rates smaller than 5 mmh21. The relation is also valid for the observations col-lected in Oklahoma. It is possible that the m–L relationchanges depending on climatology and rain type. If thatis true, a tuned m–L relation based on local DSD ob-servations should be derived and used for accurate rainDSD estimation.

Acknowledgments. The authors appreciate the helpfuldiscussions/communications with Drs. M. K. Politovich,R. J. Doviak, T. Oguchi, and C. Ulbrich. The authorswish to thank Drs. Witold F. Krajewski and Anton Kru-ger from the University of Iowa for making the videodisdrometer data available. The study was partly sup-ported by funds from the National Science Foundationthat have been designated for the U.S. Weather ResearchProgram at the National Center for Atmospheric Re-search (NCAR) and by the National Aeronautics andSpace Administration TRMM Project Office underGrant NAG5-9663, Supplement 3.

APPENDIX A

Derivation of Variances of Estimated DSD Parameters due to Moment ErrorsTo understand the error propagation in the estimation of DSD parameters based on the moment method, the

moments and the DSD parameters can be expressed as sums of their respective means and fluctuations:


M 5 M 1 dM and (A1)n n n

ˆ ˆN 5 N 1 dN ; m 5 m 1 dm; L 5 L 1 dL. (A2)0 0 n

Here we analyze the errors of DSD parameters as determined from the three (2d, 4th, and 6th) moment estimators[Eqs. (5)–(7)]. The analysis of error propagation gives relations between the errors of the estimated DSD parametersand those of the moment estimators as

dm ]h ]h ]hdm 5 dM 1 dM 1 dM , (A3)2 4 61 2dh ]M ]M ]M2 4 6

]L ]L ]LdL 5 dm 1 dM 1 dM , (A4)2 41 2]m ]M ]M2 4

]N ]N ]N0 0 0dN 5 dM 1 dL 1 dm . (A5)0 n1 2]M ]L ]mn

The variance of a DSD parameter estimator is simply an ensemble average of the square of the fluctuation errorbecause the mean of the fluctuation is zero. Hence the variances of and areˆm L

2dm ]h ]h ]h2var(m) 5 ^dm & 5 dM 1 dM 1 dM (A6)2 4 67 1 2 8[ ]dh ]M ]M ]M2 4 6

2]L ]L ]L

2ˆvar(L) 5 ^dL & 5 dm 1 dM 1 dM . (A7)2 471 2 8]m ]M ]M2 4

Similarly, their covariance is

dm ]h ]h ]h ]L ]L ]Lˆcov(m, L) 5 ^dmdL& 5 dM 1 dM 1 dM dm 1 dM 1 dM . (A8)2 4 6 2 47 1 21 28dh ]M ]M ]M ]m ]M ]M2 4 6 2 4

Performing the algebra in (A6)–(A8) and expressing them in terms of the variances and covariance of the momentestimator errors, we obtain expressions for the variances and covariance of and , asˆm L

2 2 2 2dm ]h ]h ]h ]h ]hˆ ˆ ˆ ˆ ˆvar(m) 5 var(M ) 1 var(M ) 1 var(M ) 1 2 cov(M , M )2 4 6 2 41 2 1 2 1 2 1 2[dh ]M ]M ]M ]M ]M2 4 6 2 4

]h ]h ]h ]hˆ ˆ ˆ ˆ1 2 cov(M , M ) 1 2 cov(M , M ) , (A9)2 6 4 6 ]]M ]M ]M ]M2 6 4 6

2]L ]L ]L ]L dm ]h ]L ]L ]L dm ]hˆ ˆ ˆvar(L) 5 var(m) 1 1 2 var(M ) 1 1 2 var(M )2 41 2 1 2 1 2]m ]M ]M ]m dh ]M ]M ]M ]m dh ]M2 2 2 4 4 4

]L ]L ]L dm ]L ]h ]L ]h ]L dm ]L ]hˆ ˆ ˆ ˆ1 2 1 1 cov(M , M ) 1 2 cov(M , M )2 4 2 61 2[ ]]M ]M ]m dh ]M ]M ]M ]M ]m dh ]M ]M2 4 2 4 4 2 2 6

]L dm ]L ]h ˆ ˆ1 2 cov(M , M ), and (A10)4 6]m dh ]M ]M4 6

dm ]h ]L ]L dm ]h dm ]h ]L ]L dm ]hˆ ˆ ˆcov(m, L) 5 1 var(M ) 1 1 var(M )2 41 2 1 2dh ]M ]M ]m dh ]M dh ]M ]M ]m dh ]M2 2 2 4 4 4

2 2]L dm ]m dm ]L ]h ]L ]h ]L dm ]h ]hˆ ˆ ˆ1 var(M ) 1 1 1 2 cov(M , M )6 2 41 2 1 2 1 2]m dh ]M dh ]M ]M ]M ]M ]m dh ]M ]M6 2 4 4 2 2 4

dm ]L ]h ]L dm ]h ]h dm ]L ]h ]L dm ]h ]hˆ ˆ ˆ ˆ1 1 2 cov(M , M ) 1 1 2 cov(M , M ).2 6 4 61 2 1 2dh ]M ]M ]m dh ]M ]M dh ]M ]M ]m dh ]M ]M2 6 2 6 4 6 4 6

(A11)


Hence, (A9)–(A11) constitute the error analysis for the DSD estimated parameters and . The derivative andˆm Lpartial derivatives in the above expressions are obtained from the functional relations (3) and (4) as well as thedefinition of the moment ratio h, given by

dm h 1 725 211 2 2(h 2 1) 2 2[(7 2 11h) 2 c] [2(h 2 1)] , (A12)51 2 6@dh c

where c is

2 1/2c 5 (h 1 14h 1 1) , (A13)2M (m 1 4)(m 1 3)4h 5 5 , (A14)

M M (m 1 7)(m 1 6)2 6

2]h M h45 2 5 2 , (A15a)2]M M M M2 2 6 2

]h 2M 2h45 5 , (A15b)]M M M M4 2 6 4

2]h M h45 2 5 2 , (A15c)2]M M M M6 2 6 6

1/22]L 1 M (2m 1 7) L(m 1 3.5)25 5 , (A16a)[ ]]m 2 M (m 1 4)(m 1 3) (m 1 4)(m 1 3)4

1/2]L 1 (m 1 4)(m 1 3) L

5 5 , (A16b)[ ]]M 2 M M 2M2 2 4 2

1/2]L 1 M (m 1 4)(m 1 3) L25 2 5 2 . (A16c)

3[ ]]M 2 M 2M4 4 4

APPENDIX B

Derivation of Variances of Estimated Rain Physical Parameters due to Errors in DSD Parameters

Rain physical parameters, rain rate (R), and median volume diameter (D0) are related to gamma rain DSDparameters by (9) and (10). The analysis of error propagation gives relations between the errors of the estimatedrain parameters and those of DSD parameters. We have the estimate error in rain rate as expressed by that of mand L as

]R ]R ]RdR 5 dN 1 dm 1 dL. (B1)0]N ]m ]L0

After performing the partial derivatives of rain rate (12) and substituting them in (B1), we have

R G9(m 1 4.67) (m 1 4.67)dR 5 dN 1 2 ln(L) Rdm 2 RdL. (B2)0 [ ]N G(m 1 4.67) L0

The variance of rain estimates is an ensemble average of the square of the estimation error as

2R G9(m 1 4.67) m 1 4.67ˆvar(R) 5 dN 1 2 ln(L) Rdm 2 RdL075 6 8[ ]N G(m 1 4.67) L0

22 2R G9(m 1 4.67) (m 1 4.67)

2 2ˆ ˆ5 var(N ) 1 2 ln(L) R var(m) 1 R var(L)02 2[ ]N G(m 1 4.67) L0


2 2R G9(m 1 4.67) R (m 1 4.67)ˆ ˆ ˆ1 2 2 ln(L) cov(N , m) 2 2 cov(N , L)0 0[ ]N G(m 1 4.67) N L0 0

2R (m 1 4.67) G9(m 1 4.67) ˆ2 2 2 ln(L) cov(m, L). (B3)[ ]L G(m 1 4.67)

For the error in estimated median volume diameter (13), we perform the analysis of error propagation and have

]D ]D 1 D0 0 0dD 5 dm 1 dL 5 dm 2 dL. (B4)0 ]m ]L L L

Hence, its variance is

2 21 D 1 D D0 0 02ˆ ˆ ˆvar(D ) 5 ^dD & 5 dm 2 dL 5 var(m) 1 var(L) 2 2 cov(m, L). (B5)0 0 2 2 271 2 8L L L L L

Equations (B3) and (B5) constitute the error analysis for rain rate and median volume diameter estimates due tothe errors in estimated gamma DSD parameters.

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