PUBLICATIONS. Radio Science. Modeling rainfall drop size distribution in southern England using a Gaussian Mixture Model

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1 PUBLICATIONS RESEARCH ARTICLE Key Point: A Gaussian Mixture Model may fit the rainfall DSD better than standard models Modeling rainfall drop size distribution in southern England using a Gaussian Mixture Model K ufre-mfon E. Ekerete 1, Francis H. Hunt 1, Judith L. Jeffery 2, and Ifiok E. Otung 1 1 Mobile and Satellite Communications Research Group, University of South Wales, Pontypridd, UK, 2 STFC Rutherford Appleton Laboratory, Harwell Oxford, UK Correspondence to: F. H. Hunt, Francis.Hunt@southwales.ac.uk Citation: Ekerete, K. u.-m. E., F. H. Hunt, J. L. Jeffery, and I. E. Otung (2015), Modeling rainfall drop size distribution in southern England using a Gaussian Mixture Model, Radio Sci., 50, , doi: / 2015RS Received 10 FEB 2015 Accepted 4 AUG 2015 Accepted article online 6 AUG 2015 Published online 23 SEP 2015 Abstract Understanding and modeling the rainfall drop size distribution is important in a number of applications, in particular predicting and mitigating attenuation of satellite signals in the millimeter band. Various standard statistical distributions have been proposed as suitable models, the first widely accepted being the exponential distribution. Subsequently, gamma and lognormal distributions have been shown to provide better rainfall rate computations. Some empirical studies have revealed bimodal distributions under some circumstances. A natural question to ask therefore is how often gamma and lognormal distributions fit the empirical data. In this paper we fit lognormal and gamma distributions to 1 min slices of rainfall drop size distributions taken from 7 year data from the Chilbolton Observatory in southern England. The chi-square goodness of fit of the models against the data is calculated, and it is found that failure to fit is greater than would normally be expected. This failure to fit is broken down and examined against seasonal variations, different rain rates, atmospheric temperature, and wind speed. Possible reasons for the lack of fit are explored, and alternative fits using models based on Gaussian Mixture Models are developed and found to be an improvement. 1. Introduction Due to the enormous demand for telecommunication services, there is an increasing need to utilize more transmission bandwidth. Providers increasingly use frequencies above 10 GHz since lower-frequency bands are now congested. The transmission of signals at frequencies above 10 GHz is however much more susceptible to attenuation due to precipitation. The precipitation leads to degradation in the desired quality of service and link availability. Raindrops, in particular, absorb and scatter radio wave energy [Åsen and Tjelta, 2003; Moupfouma, 1984]. Rain attenuation affects both Earth to satellite links and terrestrial links, although it is a particular problem for services such as high-definition satellite TV. There is the need to properly estimate the attenuation due to rainfall, as overestimating is wasteful of resources, while underestimating may lead to system outages. In order for engineers to design reliable systems, there is a need to reliably predict how precipitation, and rain in particular, attenuates transmitted signals. Modeling the rainfall drop size distribution (DSD) is a key ingredient in the prediction of rain attenuation, thereby allowing providers to design mitigation techniques to counter attenuation due to these rain events. This work describes a study of rainfall drop size distributions based on data collected between 2003 and 2009 from a RD-69 Joss-Waldgovel Impact Disdrometer located at the Chilbolton Observatory in southern England. Analysis of the DSD is done by fitting standard statistical distributions to the data and then assessing the goodness of fit of each of these distributions to the collected data. Some of the data cannot be accurately modeled by the standard unimodal statistical distributions as they display multimodal behavior. An attempt is made to fit the data using a Gaussian Mixture Model (GMM) American Geophysical Union. All Rights Reserved. 2. DSD Modeling 2.1. Standard Statistical Models Raindrop size distribution, denoted N(D) and expressed in mm 1 m 3,isdefined as the number of raindrops per unit volume per unit diameter, centered on D (in mm). Thus, N(D)dD, expressed in m 3, is the number of such drops per unit volume having diameters in the infinitesimal range (D dd/2, D + dd/2) of size dd centered on D. Various standard classical statistical distributions have been proposed in literature as models EKERETE ET AL. MODELING RAINFALL DSD WITH A GMM 876

2 for N(D), subject to the constraints that the drop size is always nonnegative and varies continuously. Marshall and Palmer [1948] proposed the relationship ND ð Þ ¼ N 0 expð ΛDÞ 0 < D D max (1) where D max is the maximum drop diameter. They note for their data that the coefficient Λ (in mm 1 )isa function of the rainfall rate R (mm/h), according to the relationship Λ ¼ αr β (2) In equation (1), N 0 = 8000 mm 1 m 3, and in equation (2), α = 4.1 and β = Marshall and Palmer [1948] state that the relationship does not hold for small diameters (D < 1.5 mm). However, their method involved measuring the diameter of stains left on dyed filter paper that was briefly exposed to rain, so it is highly likely that some small stains could have been covered over by larger stains which would make the data for small diameters unreliable. So the validity of fitting the suggested exponential distribution to the small diameter data would be questionable. Marshall and Palmer s [1948] data were collected over the summer of 1946 in Canada [Marshall et al., 1947]. The smaller diameter raindrops do nonetheless contribute to rain attenuation in millimeter and submillimeter radio wave transmissions [Jiang et al., 1997], so some other DSD models are needed that accurately represent the relative count of small drops. The lognormal distribution is a well-known positive unimodal statistical distribution. In general, a random variable X is lognormally distributed if log(x θ) is normally distributed with mean μ g and standard deviation σ g for some θ [Johnson et al., 1994]. For use as a model of raindrop distribution θ = 0 is often used, and the distribution is multiplied by a scaling parameter. Kolmogorov [1941] proposed that breaking droplets are lognormally distributed. Other later researchers also modeled rainfall data using the lognormal distribution [Levine, 1954; Mueller, 1966; Markowitz, 1976; Feingold and Levin, 1986; Owolawi, 2011]. The lognormal distribution for the number of drops in a given volume is given in the general form N T 6 lnðd θþ μ g 7 ND ð Þ ¼ pffiffiffiffiffi exp4 5 (3) 2π σgd ð θþ 2σ 2 g A much more detailed treatment of the lognormal distribution is given in Johnson et al. [1994], where different variants of the lognormal distributions are discussed and compared. Ulbrich and Atlas [1984] showed that a gamma distribution yielded better rainfall rate computations when combined with radar data. They used a gamma distribution expressed in the form ND ð Þ ¼ N T D μ expð ΛDÞ 0 D D max (4) with Λ, μ, and N T as the slope, shape, and scaling parameters, respectively, and these allow for the characterization of a wide range of rainfall scenarios, although in the paper they use μ = 2. Note that the exponential distribution is a special case of the gamma distribution with μ = 0. Ulbrich and Atlas [1984] do not actually claim that the DSD is a gamma distribution but simply that a gamma distribution yields more accurate rainfall rate computations. They accept that other distributions might serve equally well. The physical meaning of the parameters in this representation is problematic, since they are not related to any particular scaling, and the physical dimensions of N T vary with μ. Testud et al. [2001] propose scaling the drop size, D, in relation to the so-called volume-weighted mean diameter, or mean volume diameter, D m and putting N(D) in the form ND ð Þ ¼ N 0 FD=D ð m Þ (5) where the description of the shape of the distribution is confined to F and N * 0 is determined only by the drop size and the liquid water content. This normalization has been adopted by many later researchers [Bringi et al., 2003; Montopoli et al., 2008; Islam et al., 2012]; in particular, the gamma distribution under Testud et al. s [2001] normalization takes the form ND ð Þ ¼ N w f ðμþ D μ exp ð4 þ μþ D D m D m (6) EKERETE ET AL. MODELING RAINFALL DSD WITH A GMM 877

3 where N w, μ, and D m are the scaled intercept, shape, and mass-weighted mean diameter parameters, respectively. f(μ), a function of the shape parameter, μ, is defined as fðμþ ¼ 64þ ð μþμþ4 4 4 Γðμ þ 4Þ (7) where Γ is the complete gamma function. Testud et al. s [2001] normalization is useful when comparing the shapes of DSDs. However, it should be noted that it does not in any way change the expressiveness of the distribution, just the parameters used. Although exponential, gamma, and lognormal distributions are the most commonly used models of raindrop distribution in the literature, other distributions have been tried, for example, the Weibull distribution [Sekine and Lind, 1982]; however, this has not been widely adopted and is not used further in this paper. It has been suggested that different models may be needed for different rain types [Adimula and Ajayi, 1996;Maitra, 2004] and regions [Ulbrich and Atlas, 1984;Ajayi and Olsen, 1985; Maciel and Assis, 1990] Fitting Standard Statistical Models to Data Two common general methods of estimating population parameters are the method of moments and maximum likelihood estimation (MLE). The n th moment of the DSD at time instant t is defined as m n ðþ¼ t 0 Dn ND; ð tþdd (8) If N(D,t; D m, N w, μ) is a gamma distribution, then D m is by definition and N w can be calculated as D m ¼ m 4 m 3 (9) N w ¼ 44 m 5 3 6:m 4 4 (10) and μ can similarly be expressed in terms of the moments [Ulbrich and Atlas, 1998]. These expressions can be used to estimate the parameters using a raindrop sample. If N(D,t) is a lognormal distribution, then its parameters can be estimated, using standard methods of estimating parameters of a normal distribution, but applied to a sample of log(d) values. By contrast, MLE selects the parameters for the distribution that make the data most likely. Since the expression for the likelihood is unwieldy, the maximization is often done iteratively with computer assistance, as is the case in this paper. The estimation of parameters is complicated by the nature of the data collection. Notably, an impact disdrometer, as used for data collection in this paper, does not measure drops above or below a certain size, so the sample is truncated. Mallet and Barthes [2009] and Johnson et al. [2011] both provide methods to take this into account when using MLE for estimating gamma distribution parameters. Johnson et al. [2014] further point out that the sample is discretized into various bins, and taking into account this discretization enables a more accurate estimate of the gamma distribution parameters to be made. For the purpose simply of testing goodness of fit of a distribution, the aim of this paper, it is easier to work with the nonnormalized form of the gamma distribution. Specifically a truncated gamma probability density can be written in the form gðd; k; θ; L; UÞ ¼ 1 c Dk 1 e D=θ (11) where k is the shape parameter and θ is the scale parameter for L D U and zero otherwise. Here C is a normalizing constant given by C ¼ U L Dk 1 e D=θ dd ¼ θ k γ k; U γ k; L (12) θ θ where γ(k,s) is the lower incomplete gamma function S 0 xk 1 dx. As explained later in section 3, it is the log of expression (11) that is used in the MLE calculations for this paper. EKERETE ET AL. MODELING RAINFALL DSD WITH A GMM 878

4 2.3. Drop Size Distribution This study used a RD-69 Joss-Waldgovel Impact Disdrometer to collect raindrop distribution data. This disdrometer measures the distribution of raindrops falling on a surface, accumulated into particular drop size ranges, called channels. The RD-69 disdrometer allows for measurement of 127 different channels. Since big raindrops fall faster, the volume distribution of drops can be estimated as [Montopoli et al., 2008; Islam et al., 2012] n i ðþ t N m ðd i ; tþ ¼ (13) A:Δt :v i :ΔD i where at a discrete time instant t, D i is the central drop diameter of the i th channel, n i (t) is the total drop count in that channel, A is the exposed area of the disdrometer s sensor, Δt is the time interval, ΔD i is the width of the bin, and v i is the terminal fall velocity of the raindrops (a function of the raindrop s diameter), given as [Ulbrich, 1983] v i ¼ 3:78 D 0:67 i (14) The assumption is made that the DSDs are spatially homogenous and stationary over short periods, e.g., 1 min; hence, it is reasonable to integrate readings over a 1 min interval and use Δt =60s in expression (13). Although exponential, lognormal, and gamma distributions are popular models for DSDs, the empirical data do not always appear to fit these.radhakrishna and Rao [2009] exhibit data showing multipeak rainfall drop size distributions. They attribute this to the transition between convective and stratiform rain types. Montopoli et al. [2008] found that the MLE-fitted gamma distribution did not fit the tails of the observed DSD well and commented that optimization of the analytical function form was worth further investigation Goodness of Fit Tests In the literature, goodness of fit is usually assessed informally by eye or via the accuracy of the rainfall or other predictions derived from the DSD [Marshall and Palmer, 1948; Ulbrich and Atlas, 1984; Willis, 1984; Adimula and Ajayi, 1996; Fox, 2004; Montopoli et al., 2008; Jassal et al., 2011; Vidyarthi et al., 2012]. A number of researchers use some variant of the sum of square errors (SSEs) [Feingold and Levin, 1986; Adimula and Ajayi, 1996]. Typically, this is a tool to assess relative goodness of fit of two models, the one with the lower SSE being deemed the better fit. Owolawi [2011] does test the absolute rather than relative fit, using the Kolmogorov-Smirnov test due to small sample size. In this paper we also test the absolute fit, whether the proposed model fits the data, using the standard Pearson chi-square test. The Pearson chi-square test can be used to test the fit of a distribution, by using the distribution to calculate the probability of a value falling into a particular class, the distribution of the counts in the classes then being multinomially distributed. If E i is the expected count in the i th class, and O i is the observed count, then with n classes, and estimating k parameters to fit the distribution, the statistic χ 2 ¼ X n ðo i E i Þ 2 (15) i¼1 E i is distributed according to a chi-square distribution with n k 1 degrees of freedom [Lapin, 1990]. A condition commonly given for test validity is that the expected counts must be at least 5 in 80% of the classes and at least 1 in all the classes [Yates et al., 2005; McHugh, 2013]. The calculated statistic is then compared against a confidence threshold and hypothesis that the distribution that fits the data is rejected if it exceeds the threshold. In this work, the volumetric DSD was used to calculate expected drop counts at the disdrometer. The channels were grouped together into classes to ensure counts of at least 5 in all classes, since this condition was easier to implement automatically in a program. In fitting a gamma distribution of the form given in equation (4), the scaling parameter N T was determined by the condition that the total predicted drop count matches the total expected drop count, and the two other parameters, Λ and μ, were estimated. The statistic was then tested against the 95% threshold of a chi-square distribution with n 3 degrees of freedom. EKERETE ET AL. MODELING RAINFALL DSD WITH A GMM 879

5 3. Data and Procedure Used in This Study 3.1. Data Collection This work utilized data captured by the RD-69 Joss-Waldgovel Impact Disdrometer connected to an ADA90 analyzer at the Chilbolton Observatory in southern England ( N, W) between April 2003 and December Data were not captured from July 2005 to May 2006 and for 73 other days in this period. The disdrometer works by converting the vertical momentum of an impacting raindrop into an electrical pulse and estimating the diameter of the raindrop from the amplitude of the pulse. The disdrometer has a surface area of 50 cm 2 and measures raindrop diameters from 0.3 mm to 5.0 mm in 127 gradations, or bins, sampling at 10 s intervals. The 127 size classes are distributed more or less exponentially over the range of drop diameters, and the accuracy rate of the readings is ±5% of measured drop diameter [Distromet, 2002]. It has been shown that the manufacturer calibration of the RD-69 bin sizes can create peaks in the number density of drops observed by the disdrometer. Therefore, for this work we use two possible calibrations of the bin sizes: the original calibration provided by the manufacturer and the ETH calibration of the RD-69 bin sizes [McFarquhar and List, 1993]. Another issue is dead time of the disdrometer due to ringing of the styrofoam cup. This causes the number of small drops detected by the disdrometer to be depressed. Sheppard and Joe [1994] provide a formula for estimating and correcting for this effect. However, for the data used in this paper, fitting gamma or lognormal distributions suggested that there were usually too many drops in the lower bins to fit the model. Therefore, rather than apply a dead time correction, this paper simply considers the case that the counts in the lower 15 bins are unreliable and examines the fit to the data using the ETH calibration but with these bins ignored. In summary, the three versions of the data set considered are (i) using the default manufacturer calibration (MFR), (ii) using the ETH calibration (ETH_ALL), and (iii) using the ETH calibration but ignoring the counts in the lower 15 bins (ETH_TRUNC). To increase confidence in the reliability of the disdrometer data, the disdrometer-derived rain rate was compared with that of a rain gauge in the same vicinity, and it was discovered that the daily disdrometer-calculated rainfalls were about 15% higher on the average than the rain gauge, and this is consistent with other works [Wang et al., 2008; Islam et al., 2012]. This study aggregated six 10 s samples into a 1 min sample to achieve a larger sample. This implicitly assumes that the underlying distribution is approximately stationary over a 1 min time scale. This is the approach taken by previous workers [Montopoli et al., 2008; Townsend and Watson, 2011; Islam et al., 2012]. The wind measurements were made using a cup and vane anemometer at a height of 10 m above ground level, 3 m above a cabin roof. They are located at a distance of approximately 50 m from the disdrometer Data Fitting and Testing For samples corresponding to rain rates of at least 0.1 mm/h, the 1 min sample of drop counts at the disdrometer was converted to a volumetric drop count. Distribution parameters for three distributions were estimated using the methods described in section 2.2 above: a lognormal distribution was fitted to the data by calculating the mean and variance of the logs of the drop diameters, a gamma distribution was fitted using the method of moments, and a truncated gamma distribution was also fitted using maximum likelihood estimation to estimate the two parameters of a gamma probability density function. From each of these distributions, the expected drop counts at the disdrometer were calculated. These were merged into classes ensuring that the drop counts were all at least 5. If this resulted in at least four classes, then the chi-square statistics were calculated and compared with the chi-square 95% confidence thresholds, and a decision reached as to whether the distribution was a good fit to the data. 4. Results and Analyses 4.1. Fitting the Standard Models Using the data and procedure described in section 3, distributions were fitted to a total of 92,100 1 min samples and tested using the Pearson chi-square test. Figure 1 shows a typical 1 min sample of data to which three distributions have been fitted: a lognormal, a gamma using the method of moments, and a truncated gamma using a maximum likelihood fit. The bar chart shows the distribution of the raindrop sizes. EKERETE ET AL. MODELING RAINFALL DSD WITH A GMM 880

6 Drops per m 3 per mm Drop Densities Lognormal Gamma (MoM) Gamma (MLE) Drop diameter (mm) Figure 1. Example of 1 min drop size distributions using ETH_ALL data set. The percentage of fits for each model that were not rejected by the chi-square test is shown in Table 1, in the row entitled overall. (The column labelled GMM is discussed in a later section.) It is noticeable that the gamma MLE fit improves with the use of the ETH calibration and the removal of the lower size bins but perhaps surprising that standard models fit at best only 56% of the time. (To check that this was not simply an implementation error of the chi-square test, pseudo samples of raindrops were generated with parameters matching those calculated, and these were tested and were not rejected around 90% of the time.) Possible reasons for this are discussed in a later section, but this result is consistent with Montopoli et al. [2008], who found that MLE gamma fits appeared to fit the peak of the observed data distribution well but not the tail, and with Radhakrishna and Rao [2009], who observed bimodal DSDs that were certainly not well fitted by standard distributions. It is interesting to see how the percentage of distribution fits varies with parameters such as the season of the year, the rain rate, and the wind speed. Table 1 shows the percentage of nonrejects for spring (March May), summer (June August), autumn (September November), and winter (December February). It is noticeable that all the standard models are worse at modeling the summer rainfall. The variation of the percentage of distribution fits with rain rate is shown in Table 2. If the rainfall is classified as light (less than 2 mm/h), moderate (2 mm/h to 10 mm/h), heavy (10 mm/h to 50 mm/h), and very heavy (more than 50 mm/h) then it is noticeable that the standard models are worse at modeling the higher rain rates. However, it should also be noted that lower rain rates correspond to smaller samples, so provide less data with which to reject an ill-fitting model. It should also be noted however that the percentages at high rain rates may be unreliable since there were few samples in these categories. The variation of the percentage of distribution fits with wind speed is shown in Table 3. Unlike with the rain rate, the fit of standard models seems to improve with increased wind speed. It should again be noted however that the percentages at high wind speeds may be unreliable since there were few samples in these categories. The fundamental result, however, is that a large portion of the data samples did not fit any of the standard distributions. Table 1. Seasonal Variation of DSD Fits (MFR/ETH_ALL/ETH_TRUNC) Season Lognormal Gamma (Method of Moments (MOM)) Gamma (MLE) GMM Spring 33%/31%/33% 22%/21%/27% 48%/53%/60% 51%/54%/50% Summer 24%/22%/26% 15%/14%/20% 36%/38%/47% 47%/46%/46% Autumn 30%/29%/31% 19%/18%/23% 43%/47%/53% 50%/52%/47% Winter 33%/31%/34% 22%/21%/27% 50%/54%/61% 53%/56%/51% Overall 30%/29%/31% 20%/19%/25% 45%/49%/56% 50%/52%/49% EKERETE ET AL. MODELING RAINFALL DSD WITH A GMM 881

7 Table 2. Rain Rate Variation of DSD Fits (MFR/ETH_ALL/ETH_TRUNC) Rain Type Rain Rate (mm/h) Lognormal Gamma (MOM) Gamma (MLE) GMM Light <2 36%/34%/36% 24%/21%/27% 51%/54%/59% 52%/53%/49% Moderate %/19%/23% 10%/14%/20% 32%/40%/52% 48%/53%/51% Heavy %/12%/13% 4%/9%/11% 10%/21%/27% 33%/31%/25% Very heavy >50 8%/17%/16% 1%/6%/6% 10%/24%/24% 39%/37%/25% 4.2. Fitting Nonstandard Models: A Gaussian Mixture Model The results from the preceding section prompt various questions. Is the lack of fit real or just a consequence of noise in the data? And if the lack of fit is real, which distributions might fit better? Examining the data shows that the lack of fit isdefinitely real in some cases: Figure 2 shows the raindrop densities for 26 July 2003 at 16:46. This is clearly a multimodal distribution, and neither the lognormal nor gamma distributions are appropriate models. (Note that this multimodal distribution occurs in the 10 s samples too; it is not the result of aggregation. It is present between 16:45 and 16:48. The day as a whole is characterized by a series of light rain events, all below 5 mm/h, and this bimodal shape is not typical.) As noted earlier, Radhakrishna and Rao [2009] state that this has been observed to occur and explained as being due to a transition from convective to stratiform rain. Jones [1992] also reports secondary peaks in observed drop distributions. Note that this paper is not claiming that the stratiform convective transition is the only possible explanation for multimodality or that multimodality is the only explanation for the lack of fit. It is making the point that multimodal distributions clearly cannot be modeled accurately with unimodal distributions, and between the multimodal and the gamma, there are likely also be a large number of distributions, some of them unimodal, that are also not accurately modeled with a gamma distribution. One distribution that allows multiple modes (and has built in support in MATLAB ) is the Gaussian Mixture Model (GMM) [Bishop, 2006]. Since a Gaussian random variable can assume any real value, and the drop diameter is always positive, the GMM was fitted in the log domain. This is not actually principled; in that, this implies that the distributions multiply in the nonlog domain, something which seems physically implausible. However, the ready availability of GMM tools made this a sensible first thing to attempt. The probability density function for a GMM is of the form pd ð Þ ¼ X " # k w i¼1 i 1 ðln D exp ð Þ μ iþ 2 (16) D where X μ i and σ i represent the mean and standard deviation of the i th cluster and the weights w i satisfy k w i¼1 i ¼ 1. This is multiplied by a scaling parameter to match the actual drop distribution in the log domain. The GMM estimates three parameters for each Gaussian in the mixture: its mean, its variance, and its weight. However, the weights must add to 1; hence, for k Gaussians there are 3k 1 free parameters. These are estimated using the iterative expectation-maximization algorithm [Bishop, 2006]. Fits with 2, 3, and 4 Gaussians were tested here. Obviously, the more parameters used, the better the fit that can be achieved. However the chi-square statistic also reduces as the number of parameters estimated increases and the number of degrees 2σ 2 i Table 3. Wind Speed Variation of DSD Fits (MFR/ETH_ALL/ETH_TRUNC) Wind Type Wind Speed (m/s) Lognormal Gamma (MOM) Gamma (MLE) GMM Calm <1 20%/17%/24% 14%/11%/17% 32%/33%/46% 40%/40%/43% Light air %/20%/26% 15%/12%/19% 36%/38%/50% 46%/45%/46% Light breeze %/16%/23% 14%/11%/18% 38%/39%/50% 46%/45%/47% Gentle breeze %/18%/77% 15%/13%/19% 40%/42%/50% 48%/47%/46% Moderate breeze %/27%/29% 20%/18%/24% 47%/51%/57% 52%/55%/49% Fresh breeze %/43%/41% 26%/26%/31% 51%/58%/62% 56%/61%/54% Strong breeze %/60%/58% 32%/36%/40% 56%/65%/68% 52%/60%/56% Near gale %/70%/69% 36%/49%/53% 56%/70%/71% 41%/45%/43% Gale %/76%/78% 35%/57%/58% 54%/77%/77% 33%/28%/31% Severe gale %/100%/100% 71%/100%/100% 86%/100%/100% 29%/17%/14% EKERETE ET AL. MODELING RAINFALL DSD WITH A GMM 882

8 Drops per m 3 per mm Drop Densities Lognormal Gamma (MoM) Gamma (MLE) Drop diameter (mm) Figure 2. Drop densities with bimodal distribution using ETH_ALL data set. of freedom decreases. The optimal number of centers to use is unclear a priori. Figure 3 shows the fit to the 1 min sampled data in Figure 2 above. This GMM with 3 Gaussians is not rejected by the chi-square test. Experimentally, GMMs using 3 Gaussians provided the best fit to the data, and as can be seen from Table 4, they provide a better fit than any of the standard distributions. 5. Interpretations, Further Work, and Conclusions This paper has shown that neither the gamma nor the lognormal distributions fit the observed DSDs particularly well (one would expect around 95% nonreject rates). The gamma MLE fits better than the others. The fits seem worse at higher rain rates (as shown in Table 2) but better with high wind speeds (as shown in Table 3). The winds may be breaking up the larger drops, making the distributions more like gamma or lognormal distributions. The best fits were in spring, while the worst were in summer. Two possible explanations are 1. The actual volumetric DSDs are well modeled by gamma or lognormal distributions, but the volumetric DSDs estimated from the disdrometer data are not accurate. This might be due to noise in the disdrometer measurements or the imperfect assumption about drop velocities [see, for example, Larsen et al., 2014]. 2. The actual volumetric DSDs are not well modeled by gamma or lognormal distributions A Gaussian Mixture Model with 3 centres 16:46) Pseudo drop counts mu1: mu2: mu3: 0.08 Overall Log drop diameter (mm) Figure 3. GMM with three clusters for 26 July 2003 at 16:46 using MFR data set. EKERETE ET AL. MODELING RAINFALL DSD WITH A GMM 883

9 Table 4. Results for Chi-Square Goodness of Fit for Different Clusters (MFR Data Set) No. of Clusters Parameters Estimated % Nonreject Samples Not Rejected Total Fitted % 43,874 91, % 46,016 91, % 39,606 91,258 The existence of bimodal distributions in the data strongly suggests that the second explanation is sometimes true, and it is reasonable to assume that there are distributions intermediate between the bimodal shape and the gamma shape that will also not be well modeled by gamma or lognormal distributions. The use of a Gaussian Mixture Model with three clusters in the log domain fits better than standard models if the lower bin counts are assumed to be reliable, but there is still room for improvement. Further work is needed to better understand the reasons for the failure of the standard models to fit well and from that to propose better fitting distributions. This should also lead to guidance as to which distributions are best in which particular circumstances. 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