## 1. Introduction

*μ*value is fixed for a given type of rain event. Another approach would be to use a functional dependence between parameters of the DSD distribution function. Ulbrich (1983) and Ulbrich and Atlas (1985) have proposed a relation between the

*N*

_{0}and

*μ*parameters of the DSD function (1). Recently a

*μ*–Λ relation for drop size distribution parameters (Zhang et al. 2001, 2003) was proposed.

Disdrometer observations of drop size distributions are affected by a small sampling volume. As a result of the limited sampling volume, raindrops with large diameters often are not present in the observations (Ulbrich and Atlas 1998). Several studies were dedicated to the analysis of errors in and influence of sampling variability on retrievals of DSD function parameters (Wong and Chidambaram 1985; Chandrasekar and Bringi 1987; Smith et al. 1993; Smith and Kliche 2005). Furthermore, Chandrasekar and Bringi (1987) have shown that the *N*_{0}–*μ* relation (Ulbrich and Atlas 1985) can be attributed to statistical uncertainties and not to a physical relation between these two parameters. Therefore, it is important to understand whether observed relations between DSD parameters are caused by retrieval errors or can be attributed to a physical relation between the parameters.

In this study we use simulations of disdrometer observations of rainfall to investigate the effect of method of moments and data filtering on the *μ*–Λ relation (Zhang et al. 2001, 2003; Seifert 2005). The paper is structured as follows. In section 2 of this paper simulation of disdrometer measurements is discussed. Results of the simulations are discussed in the section 3. In this section we also discuss the influence of filtering disdrometer measurements on reflectivity, rain rate and drop count. In the final section we discuss the *μ*–Λ relation proposed by Zhang et al. (2001, 2003).

## 2. Simulation of disdrometer observations

The simulation of disdrometer rainfall observations is based on two assumptions. First, the number of drops in any given volume is assumed to follow Poisson statistics. Second, diameters of drops in a given volume are determined from a gamma distribution. The simulation procedure we have used for this study is the same as was described in Chandrasekar and Bringi (1987), Smith et al. (1993), and Smith and Kliche (2005).

### a. Formulation

*N*is the intercept parameter,

_{w}*D*

_{0}is the median volume diameter, and

*μ*is the shape parameter of the DSD function.

### b. Simulation approach

The goal of this study is to investigate whether retrieval errors of DSD parameters could induce the *μ*–Λ relation. For this purpose we have made sure that at the input into the simulation the DSD parameters are uncorrelated and vary over the wide range of values. Concretely, log*N _{w}* is uniformly distributed in the range from 2 to 5,

*D*

_{0}is randomly varied in the range from 0.5 to 3.5 mm, and the shape parameter

*μ*is varied in the range from −0.99 to 5. Furthermore, we add physical constraints such that

*Z*< 55 dBZ,

_{h}*R*< 300 mm h

^{−1}, and

*N*≤ 10

_{t}^{4}. Even though the DSD parameters might be partially correlated as discussed by Berne and Uijlenhoet (2005a, b), the purpose of this study is to investigate an effect of data filtering on the relations. Therefore, we intentionally kept all input DSD parameters uncorrelated. This way any correlation that might occur can be traced back to effects of data filtering.

For these simulations we have assumed that measurements are carried out with a video disdrometer, hence the sensor area is 100 cm^{2}. Given a typical observation time of 30 s and a mean raindrop velocity of 5 m s^{−1}, the sampling volume *V _{s}* is equal to 1.5 m

^{3}. We also have found that use of the diameter-dependent sampling volume (Chandrasekar and Bringi 1987) does not influence the results of this study as compared to the results obtained using the same sampling volume for all diameters. Therefore, for simplicity’s sake, the same volume simulation procedure is described here.

The simulation starts by randomly selecting *N _{w}*,

*D*

_{0}, and

*μ*values from the above specified ranges. After that, using (3) and (4) and the sampling volume

*V*, an expected number of drops, 〈

_{s}*C*〉, in a given sample is computed. Then, by using a Poisson deviate (Wong and Chidambaram 1985), the actual number of raindrops

_{t}*C*in the sample volume is calculated, and by drawing

_{t}*C*diameter values from a gamma probability density function (PDF; Chandrasekar and Bringi 1987), the raindrop sizes are found. Then prior to calculating DSD moments the raindrops were subdivided into size categories of 0.2 mm as in (Zhang et al. 2001). In Fig. 1 the distribution of the DSD parameters at the input into simulation is shown. Figure 2 shows the resulting rain-rate distribution that is calculated from input DSD parameters.

_{t}It should be noted that video disdrometers like that used in the work of Zhang et al. (2001, 2003) do not provide accurate measurements of raindrops with diameters less than 0.5 mm. Therefore, we also have investigated the effect of the minimum diameter filtering on the retrieved DSD parameters. To obtain representative statistics 20 000 simulations are performed.

### c. Estimation of DSD parameters

*M*

_{2},

*M*

_{4},

*M*

_{6}, respectively), the DSD parameters can be estimated as follows:

*μ*–Λ relation, and therefore the expressions in (6) can be used to estimate the DSD parameters. Zhang et al. (2003) have studied the effect of random errors in moment estimates on the

*μ*–Λ relation. That study uses error structures from Chandrasekar and Bringi (1987). This analysis, however, does not include all measurement details, such as filtering of data on rain rate and on the number of raindrop counts, or the effect of thresholding on minimum diameter due to instrument limitations. It should also be noted that moment estimators are biased, as has been shown by Haddad et al. (1997) and Smith and Kliche (2005).

## 3. Results

The variability of *μ* versus Λ is limited since these two parameters are not independent. The scatterplot of *μ* versus Λ is bounded. The lower bound is determined by the minimum median volume diameter; in our study it is equal to 0.5 mm. The upper bound is given by the maximum *D*_{0} value, which is 3.5 mm in this study. This effect can be seen in Fig. 3.

### a. Effect of data filtering

Zhang et al. (2003) mentions that prior to curve fitting in the *μ*–Λ scatterplot the data were filtered such that the rain rate *R* is larger than 5 mm h^{−1} and the drop count *C _{t}* is larger than 1000. In Fig. 1 one can see the effect of such filtering on the

*D*

_{0}and

*μ*at the input and output of the simulation. First, we should note that histograms of

*D*

_{0}values at the input and at the output of the simulation are similar. That is not the case for the histograms of

*μ*. The output histogram is shifted toward large values of

*μ*, which can be explained by the bias in the moment estimates (Smith and Kliche 2005; Smith et al. 2005). If we now consider data filtering, one can see that keeping only data where

*R*> 5 mm h

^{−1}reduces number of measurements with small

*D*

_{0}values. The requirement of

*C*being larger than 1000 reduces the number of measurements with large

_{t}*D*

_{0}values. It should also be noted that the result of this filtering is

*μ*being dependent for large Λ values. It can be seen from (4) that for a given

*N*

_{0}and Λ the total drop concentration reduces when

*μ*increases, and therefore for a fixed observation volume the drop count also decreases. This effect is clearly seen in Fig. 3. As a result, the distributions of

*D*

_{0}and

*μ*become narrower; therefore, the spread of the

*μ*–Λ scatterplot also has been reduced, as can be observed in Fig. 3. The error analysis in Zhang et al. (2003) has used the error structure from Chandrasekar and Bringi (1987). The error analysis of Chandrasekar and Bringi (1987) does not include the effect of data filtering; therefore, the spread of the

*μ*–Λ scatterplot that is attributed to the errors in the retrieved DSD parameters is overstated (see Fig. 6b of Zhang et al. 2003). Furthermore, one can observe that Fig. 6b of Zhang et al. (2003) is similar to the unfiltered scatterplot of the retrieved parameters as shown in Fig. 3 of this paper. From the results presented in Fig. 3 we can conclude that data filtering on rain rate and raindrop counts reduces the scatter and the resulting scatterplot is centered on the

*μ*–Λ curves of Zhang et al. (2001, 2003).

### b. Effect of minimum diameter filtering

Video disdrometers are not able to observe small raindrops; the minimum observable diameter is generally accepted to be 0.5 mm. Therefore, observed drop size distributions are not only truncated on the side of large raindrop diameters (Ulbrich and Atlas 1998) but also on the small diameter side. As a result, the retrieved *D*_{0} histogram tends to be shifted toward larger diameters, as can be seen in Fig. 4. By comparing the retrieved and input *D*_{0} histograms shown in Fig. 4, one can see that the former miss values smaller than roughly 0.75 mm. This can also be clearly seen in Fig. 5, where the retrieved *μ*–Λ scatterplot is no longer bounded by the *μ* = 0.5Λ − 3.67 line. If these data are also filtered on rain rate and drop count, we obtain a very narrow scatterplot around the *μ*–Λ curve. This effect can also be observed on the *μ*–*D*_{0} scatterplot shown in Fig. 6. It should also be noted that since *μ* and Λ are not independent parameters, the spread in Fig. 5 is smaller than in Fig. 6. Therefore, part of the observed correlation between *μ* and Λ can be attributed to the fact that the *μ*–Λ distribution is bounded, as can be seen in Fig. 5.

### c. Effect of D_{0} distribution

In the study presented here the *D*_{0} values are varied uniformly over a large range of values, from 0.5 to 3.5 mm. It was observed, however, that *D*_{0} values have typically a narrower distribution, with maximum values lower than 2.5 mm. In this section an effect of the maximum *D*_{0} value on the resulting *μ*–Λ scatterplot is studied. For this investigation we have simulated four scenarios, where input *D*_{0} values have being randomly selected from following ranges: 0.5–3.0, 0.5–2.5, 0.5–2.0, and 0.5–1.5 mm. In Fig. 7 the retrieved *μ*–Λ scatterplots are shown. It can be observed that smaller maximum *D*_{0} value corresponds to a narrower *μ*–Λ scatterplot. Based on this study we can expect a less-defined *μ*–Λ relation for rain events where *D*_{0} values vary over a large range.

### d. Observations using three disdrometers

In this work we have shown that by including filtering of the disdrometer observations and the effect of small diameter filtering one can increase the correlation between *μ* and Λ parameters. Therefore, it is interesting to see whether this artificially induced relation between *μ* and Λ parameters would hold when three moments of the DSD would be estimated from three independent measurements. Chandrasekar and Gori (1991) have shown that observations made using multiple disdrometers can be used to remove correlations between DSD parameters induced by a limited sampling volume. In Fig. 8 the scatterplots of *μ* versus Λ are shown for the case in which three moments are estimated from three independent disdrometer measurements. For this simulation it is assumed that three disdrometers are observing the same rain event described by a gamma drop size distribution with the same parameters. The raindrop counts and diameters of observed raindrop are sampled independently, for these three observations, from Poisson and gamma probability density functions, respectively. In this case errors in the estimated moment become uncorrelated. One can see that in this case there is no longer a correlation between *μ* and Λ.

## 4. Conclusions

In this work we have extended the analysis of Chandrasekar and Bringi (1987) to the case of video disdrometer measurements with data filtering on rain rate and drop count. We have shown that data filtering reduces scatter between *μ* and Λ. Also it is shown that since video disdrometers are not able to correctly measure the concentration of drops with diameters smaller than 0.5 mm, a further reduction in the scatter between the retrieved *μ* and Λ parameters can be observed. From this study we can conclude that the analysis based on work of Chandrasekar and Bringi (1987) is not complete for evaluating the *μ*–Λ relation described in the literature, because it does not include the effects of data filtering, such as minimum diameter filtering due to instrument limitations or limiting the number of drops and rain rate. Nonetheless, simulations using a wide range of input DSD parameters have shown that data filtering that is reported in the literature can have a strong influence on the *μ*–Λ scatterplot. Therefore, extreme care needs to be exercised in interpreting the relation between DSD parameters obtained from disdrometer measurements.

The reported effect of observation errors and data filtering on the *μ*–Λ relation shows that the debate on the validity of this relation is not over. This paper shows that data filtering and minimum diameter filtering can result in a spurious *μ*–Λ relation. The recently reported modeling study (Seifert 2005), however, has shown that there can be a physically based *μ*–Λ relation. Therefore, a part of our investigation was dedicated to finding a measurement setup that can prove or disprove the existence of the *μ*–Λ relation. It was shown that by using three independent disdrometers to determine the DSD moments, and eliminating the inherent correlations among sample moments obtained from a single disdrometer, one would be able to check for the existence of the relationship between the DSD parameters.

## Acknowledgments

The research was supported by the National Science Foundation (ATM-0313881).

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Histogram of the simulated rain-rate values. The figure is truncated at the rain rate of 75 mm h^{−1}. The tail of distribution, nonetheless, extends to rain rates of 300 mm h^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Histogram of the simulated rain-rate values. The figure is truncated at the rain rate of 75 mm h^{−1}. The tail of distribution, nonetheless, extends to rain rates of 300 mm h^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Histogram of the simulated rain-rate values. The figure is truncated at the rain rate of 75 mm h^{−1}. The tail of distribution, nonetheless, extends to rain rates of 300 mm h^{−1}.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplots of *μ*–Λ. (left) Scatterplots of simulation input parameters. (right) Scatterplots of the retrieved DSD parameters. The light gray crosses represent unfiltered data. The dark gray circles represent data filtered such that *R* > 5 mm h^{−1}, *R* < 300 mm h^{−1}, *N _{t}* ≤ 10

^{4}, and

*C*> 1000. The thick solid lines represent borders of scatterplots that correspond to

_{t}*D*

_{0}=0.5 mm and

*D*

_{0}= 3.5 mm. The thin lines correspond to

*μ*–Λ relations from Zhang et al. (2001, 2003).

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplots of *μ*–Λ. (left) Scatterplots of simulation input parameters. (right) Scatterplots of the retrieved DSD parameters. The light gray crosses represent unfiltered data. The dark gray circles represent data filtered such that *R* > 5 mm h^{−1}, *R* < 300 mm h^{−1}, *N _{t}* ≤ 10

^{4}, and

*C*> 1000. The thick solid lines represent borders of scatterplots that correspond to

_{t}*D*

_{0}=0.5 mm and

*D*

_{0}= 3.5 mm. The thin lines correspond to

*μ*–Λ relations from Zhang et al. (2001, 2003).

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplots of *μ*–Λ. (left) Scatterplots of simulation input parameters. (right) Scatterplots of the retrieved DSD parameters. The light gray crosses represent unfiltered data. The dark gray circles represent data filtered such that *R* > 5 mm h^{−1}, *R* < 300 mm h^{−1}, *N _{t}* ≤ 10

^{4}, and

*C*> 1000. The thick solid lines represent borders of scatterplots that correspond to

_{t}*D*

_{0}=0.5 mm and

*D*

_{0}= 3.5 mm. The thin lines correspond to

*μ*–Λ relations from Zhang et al. (2001, 2003).

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 1, but minimum observable diameter is 0.5 mm.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 1, but minimum observable diameter is 0.5 mm.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 1, but minimum observable diameter is 0.5 mm.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 3, but minimum observable diameter is *D*_{min} = 0.5.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 3, but minimum observable diameter is *D*_{min} = 0.5.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 3, but minimum observable diameter is *D*_{min} = 0.5.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplot of retrieved *D*_{0} vs *μ*. Solid line represents *D*_{0}–*μ* relation obtained from *μ*–Λ relation (Zhang et al. 2003). The dark gray circles represent filtered data on rain rate and *C _{t}* and light gray crosses show unfiltered data.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplot of retrieved *D*_{0} vs *μ*. Solid line represents *D*_{0}–*μ* relation obtained from *μ*–Λ relation (Zhang et al. 2003). The dark gray circles represent filtered data on rain rate and *C _{t}* and light gray crosses show unfiltered data.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplot of retrieved *D*_{0} vs *μ*. Solid line represents *D*_{0}–*μ* relation obtained from *μ*–Λ relation (Zhang et al. 2003). The dark gray circles represent filtered data on rain rate and *C _{t}* and light gray crosses show unfiltered data.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplots of *μ*–Λ for different maximum *D*_{0} values. The sampling volume is 1.5 m^{3}. The dashed and solid lines represent *μ*–Λ curves after Zhang et al. (2001, 2003), respectively.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplots of *μ*–Λ for different maximum *D*_{0} values. The sampling volume is 1.5 m^{3}. The dashed and solid lines represent *μ*–Λ curves after Zhang et al. (2001, 2003), respectively.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Scatterplots of *μ*–Λ for different maximum *D*_{0} values. The sampling volume is 1.5 m^{3}. The dashed and solid lines represent *μ*–Λ curves after Zhang et al. (2001, 2003), respectively.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 5, but retrieved from three independent disdrometer observations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 5, but retrieved from three independent disdrometer observations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1

Same as in Fig. 5, but retrieved from three independent disdrometer observations.

Citation: Journal of Atmospheric and Oceanic Technology 24, 5; 10.1175/JTECH2010.1