Two-Parameter Gamma Drop Size Distribution Models for Singapore

IEEE Transactions on Antennas and Propagation, 2010

A model that is less sensitive to errors in the extreme small and large drop diameters, the gamma model with central moments (3, 4 and 6), is proposed to model the rain drop size distribution of Singapore. This is because, the rain rate estimated using measured drop size distribution shows that the contributions of lower drop diameters are small as compared to the central drop diameters. This is expected since the sensitivity of the Joss distrometer degrades for small drop diameters. The lower drop diameters are therefore removed from the drop size data and the gamma model is redesigned for its moments. The effects of the removal of a particular rain drop size diameter on the specific rain attenuation (in dB) and the slant-path rain attenuation calculations with forward scattering coefficients for vertical polarization are analyzed at Ku-band, Ka-band and Q-band frequencies. It is concluded that the sensitivity of the Joss distrometer although affects the rain rate estimation at low rain rates, does not affect the slant path rain attenuation on microwave links. Therefore, the small drop diameters can be ignored completely for slant path rain attenuation calculations in the tropical region of Singapore.

Quarterly Journal of the Royal Meteorological Society, 2013

The gamma family of probability densities, which includes the exponential family as a special case, has recently been used to model raindrop size data. The traditional approach of using method of moments to estimate the gamma distribution parameters, however, is known to be biased and can have substantial errors. Methods superior to the method of moments approach include maximum likelihood. In particular, maximum likelihood estimates have been shown to outperform method of moments estimators both in the case in which the full range of drop sizes are observed as well as the case in which small drop sizes fail to be observed because of the inability of disdrometers to record observations below a threshold. The foregoing comments concern the situation in which drop sizes are measured on a continuous scale. In this work we consider drop sizes from gamma distributions which are classified into broad size bins, as would be the case with data obtained from many disdrometers; we do also allow for the possibility of drop sizes below a threshold or above another threshold not being observed. Maximum likelihood performance in this case is investigated through simulation of sampling from gamma distributions with known parameters. In particular, we compare the performance of the maximum likelihood estimates with those of method of moments and a recently developed weighted least squares procedure. The simulation process, which relies in part on numerical optimization as the maximum likelihood estimates are not expressible in closed-form, is conducted using the R statistical package (http://www.r-project.org/). Slight modifications to this code allow parameter estimation with experimental data.

Journal of Applied Meteorology and Climatology, 2009

A generalized method is presented to derive a ''two scale'' raindrop size distribution (DSD) model over a spatial or temporal domain in which a statistical rain parameter relation exists. The two-scale model is generally defined as a model in which one DSD parameter is allowed to vary rapidly and the other is constant over a certain space or time domain. The existence of a rain parameter relation such as the radar reflectivityrainfall rate (Z-R) relation over a spatial or temporal domain is an example of such a two-scale DSD model. A procedure is described that employs a statistical rain parameter relation with an assumption of the gamma DSD model. An example using Z-R relations obtained at Kototabang, West Sumatra, is presented. The result shows that the resulting two-scale DSD model expressed by conventional DSD parameters depends on the assumed value of parameter m while rain parameter relations such as k-Z e relations from those models using different m values are very close to each other, indicating the stability of the model against the variation of m and the validity of this method. The result is applied to the DSD model for the Tropical Rainfall Measuring Mission (TRMM) precipitation radar 2A25 (versions 5 and 6) algorithm. The derivation procedure of the 2A25 DSD model is described. Through the application of this model, it has become possible to make a logically well-organized rain profiling algorithm and reasonable rain attenuation correction and rainfall estimates, as described in an earlier paper by Iguchi et al.

2020

The Drop Size Distributions (DSDs) measured in northern Benin (West Africa) are analyzed with N(D) and R(D) functions. The N(D) function represents the number of raindrops per unit of volume and by interval of diameters, while the R(D) function represents the rainfall rate per range of diameters. Meteorological variables that are formulated with the N(D) function can also be formulated with the R(D) function. In addition, the R(D) function has the advantage of being independent of the falling drop speed which is not measured well enough by the disdrometers. In this paper, the averages of DSDs per rainfall class are modeled by the gamma and lognormal laws. The parameters of these models are adjusted according to the rain rate. We thus obtained DSD parameterized with the rate of rain. Evaluating the efficiency of this parameterization, we noticed that the R(D) function estimates its useful moments better than the N(D) function. Moreover, we have noted that DSD modeling by the gamma la...

Journal of Atmospheric and Solar-Terrestrial Physics, 2016

This paper investigates the variability of raindrop size distribution (DSD) and rain integral parameters at Ahmedabad, a tropical location, in relation to the radar estimation of rainfall. Rain DSDs for the years 2006-2007 at Ahmedabad (23°04′N, 72°38′E) have been measured using a disdrometer. Variability of DSD is evaluated for different seasons and its effect on the integral rain parameters like radar reflectivity, rainfall intensity and attenuation are examined. A percentage contribution of different drop diameters on rain integral parameters is studied to understand the seasonal behaviour of rain attenuation and radar reflectivity. It is observed that drops with diameter around 3 mm contribute maximum to the radar reflectivity while drops having a diameter around 2 mm contribute the maximum to the rainfall intensity for the present location. The critical diameter range responsible for the maximum contribution in rain attenuation found to shift towards large drops with an increase in rain rate for a fixed frequency. Linear and non-linear regression analysis between radar reflectivity and rainfall intensity show significant variations in different seasons but does not differ much for different regression techniques. Results point to the necessity of considering the seasonal variability of rain DSD in radar remote sensing and will be helpful for better characterizing of rain parameters from radar measurements.

Journal of Atmospheric and Oceanic Technology, 2003

The three-parameter gamma distribution n(D) ϭ N 0 D exp(Ϫ⌳D) is often used to characterize a raindrop size distribution (DSD). The parameters and ⌳ correspond to the shape and slope of the DSD. If and ⌳ are 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 whether the-⌳ relation arises from errors in estimated DSD moments, or from natural rain processes, or from a combination of both statistical error and rain physics. In this paper, the error propagation from moment estimators to rain DSD parameter estimators is studied. The standard errors and correlation coefficient are derived through systematic error analysis. Using numerical simulations, errors in estimated DSD parameters are quantified. The analysis shows that errors in moment estimators do cause correlations among the estimated DSD parameters and cause a linear relation between estimators and. However, the slope and intercept of the error-induced relation depend on the expected values and ⌳, ⌳ and it differs from the-⌳ relation derived from disdrometer measurements. Further, the mean values of the DSD parameter estimators are unbiased. Consequently, the derived-⌳ relation is believed to contain useful information in that it describes the mean behavior of the DSD parameters and reflects a characteristic of actual raindrop size distributions. The-⌳ relation improves retrievals of rain parameters from a pair of remote measurements such as reflectivity and differential reflectivity or attenuation, and it reduces the bias and standard error in retrieved rain parameters.

Advances in Space Research, 2009

A Joss-Waldvogel impact type disdrometer was installed at four different locations in the Indian peninsula during various periods from 2001 till date. The data are analysed to study the nature of rain drop size distribution (DSD) in this region. Out of the three well known distributions that describe DSD, namely, the Marshall-Palmer, Gamma and Lognormal, it has been found that Lognormal distribution fits the DSD in this region better than the other ones. Lognormal distributions for different rain rates were then derived by fitting the lognormal function to the data using a curve fitting software. Then the variation of fit parameters with rain rate was evaluated. Incorporating these variations, into the Lognormal distribution, an empirical equation that describes the DSD in this region for different rain rates was derived. Then this equation was tested with sample data from each of these stations. The data used for validation were not used for fitting lognormal equation to derive the fit parameters. The correlation between the DSD measured and derived using the empirical model was found to be quite good (0.9) except in some cases where the coefficient dropped to 0.75. The empirical model can be updated when more data are available.

2003

Accurate rain estimation from radar measurements has been a difficult task due to the variation of raindrop size distribution (DSD), lack of accurate axis ratio model, measurement error, clutter, and so forth. Previously, rain estimation from weather radars has been largely dependent upon empirical relations such as R-Z relations. The development of polarimetric radar makes accurate rain DSD retrieval and rain rate estimation possible. Polarimetric radar observables: ZDR and KDP depend on the shape of raindrops while the raindrop shape is directly related to drop size, and hence contain the information about rain DSD, and are used to retrieve rain DSDs and improve rain rate estimation. Recently, the constrained Gamma rain DSD retrieval was developed [Brandes et al., 2002&2003a; Vivekanandan et al., 2003; Zhang et al., 2001]. Since KDP has large measurement error and poor range resolution, the constrained Gamma DSD method uses radar measured Z, ZDR, and an observed relation between t...

IEEE Transactions on Geoscience and Remote Sensing, 1997

This paper revisits the problem of nding a parametric form for the rain drop size distribution (DSD) which 1) is an appropriate model for tropical rainfall, and 2) involves statistically independent parameters. Using TOGA/COARE data, we derive a parametrization which meets these criteria. This new parametrization is an improvement on the one that was derived in [3], using TRMM ground truth data from Darwin, Australia. The new COARE data allows us to verify that the spatial variability of the two \shape" parameters is relatively small, thus conrming that this parametrization should be particularly useful for remote sensing applications. We also derive new DSD-based radar-reectivity{rain-rate power laws, whose coecients are directly related to the shape parameters of the DSD. Perhaps most important, since the coecients are independent of the rain-rate itself, and vary little spatially, the relations are ideally suited for rain retrieval algorithms. It should also prove straightforward to extend this method to the problems of estimating cloud hydrometeors from remote-sensing measurements.

Advances in Meteorology, 2015

This work investigates the physical characteristics of raindrop size distribution (DSD) in an equatorial heavy rain region based on three years of disdrometer observations carried out at Universiti Teknologi Malaysia’s (UTM’s) campus in Kuala Lumpur, Malaysia. The natural characteristics of DSD are deduced, and the statistical results are found to be in accordance with the findings obtained from others disdrometer measurements. Moreover, the parameters of the Gamma distribution and the normalized Gamma model are also derived by means of method of moment (MoM) and maximum likelihood estimation (MLE). Their performances are subsequently validated using the rain rate estimation accuracy: the normalized Gamma model with the MLE-generated shape parameterµwas found to provide better accuracy in terms of long-term rainfall rate statistics, which reflects the peculiarities of the local climatology in this heavy rain region. These results not only offer a better understanding of the microphy...