Modeling rainfall drop size distribution in southern England using a Gaussian Mixture Model

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. DSD Modeling Standard Statistical ModelsRaindrop size distribution, denoted N(D) and expressed in mm À1 m À3 , is defined as the number of raindrops per unit volume per unit diameter, centered on D (in mm). Thus, N(D)dD, expressed in m À3 , is the number of such drops per unit volume having diameters in the infinitesimal range (D À 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 PUBLICATIONS

Advancing Opportunistic Sensing in Hydrology: A Novel Approach to Measuring Rainfall With Ordinary Surveillance Cameras

“…The DSD can be represented by the product of the density of raindrops within the sampling volume (denoted as n c , m −3 ) and the probability distribution of raindrop size (i.e., D ). The probability distribution of D can be an exponential, gamma, or Weibull distribution (Ekerete et al, ); however, the gamma distribution is the most common choice for practical applications. The gamma distribution has the following probability density function: where a and b are the shape and scale parameters (both are positive) of the distribution, respectively; and Γ( a ) is the gamma function evaluated at a .…”

A Critical Evaluation of the Adequacy of the Gamma Model for Representing Raindrop Size Distributions

“…However, the gamma distribution is not a perfect model and several studies have questioned its adequacy (Kliche et al 2008;Ekerete et al 2015;Cugerone and De Michele 2015). Its acceptance mainly comes from the fact that it is relatively versatile yet simple enough to be useful in practice.…”

“…It is more flexible than the exponential (Marshall and Palmer 1948) and provides a ''reasonably good fit'' to measured DSDs. In addition to the conventional distributions like gamma, more complex models have also been proposed in the literature (Ignaccolo and De Michele 2014;Ekerete et al 2015;Cugerone and De Michele 2015;Thurai and Bringi 2018). Although they are better at representing real DSDs, they are more difficult to use in practice due to their large number of parameters that cannot be retrieved using remote sensing measurements.…”

“…They showed that the gamma distribution has the lowest rejection rate, while Weibull is the most frequently rejected. Ekerete et al (2015) used the chi-square goodness-offit test for testing several candidate models against the observations and concluded that DSDs are somewhere between the bimodal and the gamma shape, suggesting that gamma or lognormal distributions are not fully adequate. Their recommendation is to use a Gaussian mixture model with three centers.…”

“…That diagram is one of the various graphical methods for determining visually whether sample data conform to a reference distribution. Among them the most commonly used graphical tools are the quantile-to-quantile plots or Q-Q plots (Yakubu et al 2014;Watanabe and Ingram 2016) and the density plots (Ekerete et al 2015;Adirosi et al 2016) as well.…”