Three-parameter raindrop size distribution modelling at a tropical location A. Maitra Three-parameter distribution functions, gamma and lognormal, are used to describe the sizes of raindrops in Calcutta. The three parameters for a drop size distribution are obtained by solving three nonlinear equations formed with three measurements, namely rain rate and rain attenuations at millimetre and optical wavelengths. The two distribution models are compared by employing rainfall parameter estimation methods.
Introduction: The modelling of the drop size distribution (DSD) in tropical rain has been the subject of continued interest in view of varied climatological conditions and the inadequacy of experimental observations in the tropics. Three-parameter distribution models are more flexible than two-parameter models in accounting for a wide variability in raindrop sizes. The three-parameter lognormal function has been considered for tropical rain DSD [1]. The two-parameter distributions, such as the negative exponential and gamma distributions [2], are more widely used particularly for temperate climatic zones. The case of three-parameter gamma distributions has not been adequately examined, at least in the open literature. In this Letter, the rain DSD is obtained in terms of gamma and lognormal distributions, both considered to be three-parameter functions, from ground based rain attenuation measurements. The two distributions are compared by indicating their appropriateness to describe drop sizes and also by estimating integral parameters of rainfall such as the liquid water content and radar reflectivity factor, with these distributions.
where D is the drop diameter in millimetres. Three distribution parameters NT (or N0), Λ and n are considered to depend on the rainfall rate. Γ(n + 1) is the complete gamma function. The lognormal distribution is expressed as
where NT, µ and σ2 are rain rate dependent distribution parameters. The three distribution parameters of a particular distribution are obtained by numerically solving three nonlinear equations by a modified Powell hybrid method [3]. The three equations are formed with colocated measurements of three rain related quantities, namely rain attenuations at millimetre and optical wavelengths (γmm and γopt) over a line-ofsight (LOS) path and rain rate (R), as follows:
where the Q(D)s [mm2] are the extinction coefficients and v(D) [m/s] is the raindrop terminal velocity.
Fig. 1 Comparison between gamma and lognormal distribution of raindrop sizes
Modelling with experimental data: The experimental measurements, first reported in previous literature [4], were obtained at Calcutta (22° 34’ N, 88° 29’ E, elevation 50m), a tropical location in rain climatic zone N, during the monsoon period from July to September 1994. Measurements of rain attenuations at 94GHz and 0.63 µm were carried out over a 110m LOS link and that of rain rate by a fast response rain gauge with integration time 10s at the receiver end. The following power-law curves are fitted between the attenuation and rain rate measurements:
——— gamma ·········· lognormal
The attenuation measurements at different rain rates are considered from the above power-law relations to obtain the solution of eqns. 3 – 5 for DSD parameters. The attenuation values are obtained at rain rates from 5 to 80mm/h, the range of experimental observations, at an interval of 1mm/h. The solutions obtained at different rain rates are best-fitted to the following relations:
Fig. 2 Variation of liquid water content and radar reflectivity factor with rain rate for gamma and lognormal drop-size distributions ——— gamma ··········· lognormal
Background: The raindrop number density N(D) in terms of the gamma distribution is given by
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Comparison between the two distributions: The three-dimensional plots of raindrop size distributions obtained in terms of gamma and lognormal functions are shown in Fig. 1. The two distributions agree well in the drop size range 0.5–2.5mm. The major discrepancy between the two models occurs at drop diameters of < 0.5mm, the gamma model overestimating the number density, which may not be very realistic, compared to the lognormal distribution. The lognormal distribution gives a somewhat higher number density above a drop diameter of 2.5mm than the gamma distribution. To make further comparisons, two rainfall integral parameters, namely the liquid water content W (proportional to the third moment of DSD) and radar reflectivity factor Z (sixth moment of DSD), obtained at different rain rates using the two distribu-
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tions are shown in Fig. 2. There is close agreement, within 10%, between the rainfall parameters estimated with the two distributions up to a rain rate of 50mm/h, above which the gamma distribution gives higher rainfall parameter estimates than the lognormal distribution.
© IEE 2000 Electronics Letters Online No: 20000667 DOI: 10.1049/el:20000667
7 February 2000
A. Maitra (Institute of Radio Physics and Electronics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Calcutta, India) E-mail:
[email protected]
Conclusions: In this Letter we have presented the path-averaged threeparameter rain DSD in terms of gamma and lognormal functions obtained from rain attenuation measurements at two wavelengths at a tropical location. The lognormal distribution gives lower densities for smaller drops, which is more acceptable than the gamma distribution. The rainfall integral parameters obtained with the two distributions agree well up to a rain rate of 50mm/h. Long term drop-size data supported by additional rainfall parameter measurements are needed to indicate a definite preference in terms of a particular three-parameter model of drop-sizes in tropical rain.
References 1 2 3 4
Acknowledgments: A major part of the experimental facilities used for this work was funded by the Department of Electronics, Government of India.
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AJAYI, G.O., and OLSEN, R.L.: ‘Modelling of a tropical raindrop size distribution for microwave and millimeter wave applications’, Radio Sci., 1985, 20, pp. 193–202 ULBRICH, C.A.: ‘Natural variations in the analytical form of the raindrop size distribution’, J. Clim. Appl. Meteorol., 1983, 22, pp. 1764–1775 MAITRA, A., and GIBBINS, C.J.: ‘Modelling of raindrop size distribution from multiwavelength rain attenuation measurements’, Radio Sci., 1999, 34, pp. 657–666 MAITRA, A., DAN, M., BHATTACHARYYA, S., and SEN, A.K.: ‘Modelling of raindrop size distribution from measurements of rain rate and attenuations at millimeter and optical wavelengths at Calcutta’. Proc. 7th URSI Comm. F Open Symp., Ahmedabad, India, November 1995, pp. 50–54
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