Microphysical Properties of Raindrops
2.4 Drop-Size Distributions
2.4.1 Historical Review
The study of the drop-size distributions (DSDs) has been the subject of theoretical and practical interest by meteorologists and radio engineers for over half a century. DSDs provide one of the most comprehensive descriptions of rain and together with its moments form the basis of the defin ition and computation of parameters in a wide array of applications, most appropriately in this study of rain attenuation for microwave and
millimetre-waves. The scattering and attenuation due to rain is highly dependent upon the chosen DSD. DSDs exhibit significant spatial and temporal variability.
Consequently the average DSDs for several climatic regions and rainfall types will be examined and discussed in this section. These DSD functions will be used as input parameters in the ensuing chapters.
Detailed documentation of DSD measurements date back as early as Weisner (1895, cited Campos 1999). The DSD measurements were undertaken over the tropics using an absorbent filter paper method. Sheets of filter paper dusted with a water soluble-dye were exposed to rain for a brief time interval. After impacting with the sheet each raindrop would cause a spot marked permanently by the dye. A known empirical relation was then used to convert the size of the spots to the actual raindrop size. The simplicity of this method led to its subsequent usage by many investigators most notably Marshall and Palmer (1948). However, this method has a crucial flaw, namely the spot diameter size is a function of not only the raindrop size but also the fall velocity. Furthermore, large raindrops would splatter upon impact with the sheet and this making it difficult to determine their exact size.
Another common method was the flour method of Laws and Parsons (1943). Their pioneering measurements for different rain types in Washington DC are still used as a standard today. In the flour method a pan containing a thin layer of fine flour is exposed to rain for a few seconds. After a while, each raindrop forms a dry, hard dough pellet. A known relationship between the size or mass of the dough pellets and raindrop size was then used. It was noted that for rain events, the DSD would vary appreciably even when the rain rate was the same. Hence averaged distributions for different rain rates ranging from 0.25 mm h-I to 150 mm h-I were computed. Over 60 years later, the Laws and Parsons DSD measurements are still considered typical of the average DSD for both widespread rain (lower rain rates) and convective rain (higher rain rates). Marshall and
Palmer (1948) showed that both their measurements in Ottawa and those of Laws and Parsons (1943) can be modelled using a negative exponential relation (Marshall and Palmer 1948). Both Marshall and Palmer (1948) and Laws and Parsons (1943) are the most commonly used DSDs for the evaluation of the scattering and attenuation of electromagnetic waves by rain and will be discussed in greater detail later in this chapter.
Other techniques include the camera method (e.g. lones 1959, 1992; Cataneo and Stout 1968) and the oil method (Ugai et al. 1977). In the camera method, two perpendicularly aligned cameras take synchronous pictures of the raindrops. However, the marginal increase in accuracy did not warrant the considerable calculations which this method entails. Hence to date, camera methods have been used predominantly for drop shape measurements, most notably Pruppacher and Pitter (1971). In the oil method a pan of castor oil is exposed to rain and the size of the raindrops floating in the oil is then measured. The viscosity of the oil can be adjusted so even raindrops with radius as small as 0.025 mm can be measured.
The techniques mentioned so far are very laborious and time consummg. More sophisticated measurement techniques have been implemented with the development of automated recording devices. These devices have made use of electromechanical, optical and even electrostatic sensors. The most widely used of theses devices is the distrometer of loss and Waldvogel (1967). It contains an electromechanical sensor to measure the momentum of each raindrop and subsequently the DSD. Using this instrument, loss et al. (1968) found that DSD varied significantly for different rain types. The parameters for the average exponential distributions of 3 different rain types were obtained. They include the "drizzle" distribution (ID), for very light widespread rain, the "thunderstorm" distribution (JT), for extreme convective rain and the
"widespread" distribution (JW), which is nearly identical to Marshall and Palmer (1948) distribution (MP). These DSDs will be discussed in further detail later in this section.
The simplicity of the single-parameter exponential distributions (MP, JD, JW and JT) has led to its application by many investigators. However it has been shown on several occasions that it fails to accurately model measurements taken over a short period of time (Joss and Gori 1978; Zawadzki and de Agostinho Antonio 1988). The number of raindrops with very small and large diameters tends to be overestimated. Even if two- parameter exponential distributions are adopted this problem is not alleviated. Another problem with the traditional exponential distribution is that they may not necessarily be consistent with the rain-rate integral equation, (e.g. Olsen et al. 1978; de Wolf 2001;
Matzler 2002). Minor modifications and normalisations may be required as will be discussed later in this section.
To prevent such shortfalls and improve accuracy more complicated multi-parameter distributions were necessary. This was made possible with the introduction of sophisticated remote sensing techniques. By measuring more than one remotely sensed variable simultaneously, the parameters of multi-parameter distributions were evaluated promptly without the need for intricate empirical relations. This was shown by Atlas and Ulbrich (1974), by measuring the radar reflectivity and microwave attenuation, to evaluate a two parameter gamma model. An alternative method was adopted by Seliga and Bringi (1976) where a dual-polarization radar was used to concurrently measure both the vertical and horizontal polarized reflectivity factors.
To date many investigators have used advanced techniques such as those discussed above and have proposed DSD functions providing greater accuracy and flexibility. The most noteworthy are the modified gamma distribution (e.g. Ulbrich 1983; Willis, 1984;
de Wolf, 2001) and the lognormal distribution (e.g. Feingold and Levin 1986; Ajayi and Olsen 1986; Adimula and Ajayi 1996). The gamma distribution is most practical since it
has a single curvature parameter with which to model deviations from the exponential distribution. Furthermore it reduces to the traditional exponential distribution when this curvature parameter tends to zero. The lognormal distributions are equally advantageous since higher moments of such distributions are also lognormally distributed. The overview of the history and the evolution of OSO measurements and modelling is now completed. In the next subsection the basic definitions and formulas will be given. This will be followed by the mathematics and implementations of the prominent OSDs.