a. Formulation

In this simulation we have used the normalized gamma distribution (Testud et al. 2001; Illingworth and Blackman 2002; Bringi and Chandrasekar 2001)

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where N_w is the intercept parameter, D₀ is the median volume diameter, and μ is the shape parameter of the DSD function.

The change from expression (1) to expression (2) is done to remove dependence of the intercept parameter on μ. By comparing (1) and (2) we can see that N₀ and N_w are related as follows:

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The total drop concentration N_t can be expressed as

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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, logN_w is uniformly distributed in the range from 2 to 5, D₀ 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_h < 55 dBZ, R < 300 mm h⁻¹, and N_t ≤ 10⁴. 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 cm². 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 m³. 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, D₀, and μ values from the above specified ranges. After that, using (3) and (4) and the sampling volume V_s, an expected number of drops, 〈C_t〉, in a given sample is computed. Then, by using a Poisson deviate (Wong and Chidambaram 1985), the actual number of raindrops C_t in the sample volume is calculated, and by drawing C_t 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.

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

Following the approach described by Zhang et al. (2003), we estimate DSD function parameters using a three-moment estimator. Using the second, fourth, and sixth moments (M₂, M₄, M₆, respectively), the DSD parameters can be estimated as follows:

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Ulbrich and Atlas (1998) have proposed a procedure that allows for mitigation of the large drop truncation effect. In Zhang et al. (2003) it was found that large drop truncation does not influence the 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).

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 D₀ and μ at the input and output of the simulation. First, we should note that histograms of D₀ 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 D₀ values. The requirement of C_t being larger than 1000 reduces the number of measurements with large D₀ 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 N₀ 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 D₀ 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 D₀ histogram tends to be shifted toward larger diameters, as can be seen in Fig. 4. By comparing the retrieved and input D₀ 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–D₀ 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 D₀ distribution

In the study presented here the D₀ values are varied uniformly over a large range of values, from 0.5 to 3.5 mm. It was observed, however, that D₀ values have typically a narrower distribution, with maximum values lower than 2.5 mm. In this section an effect of the maximum D₀ value on the resulting μ–Λ scatterplot is studied. For this investigation we have simulated four scenarios, where input D₀ 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 D₀ 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 D₀ 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.