The Evaluation of Coherent Scatterng in Rain Radar
4. Rain Drop Size Distributions
4.1 Exponential Distribution At the last conference (1) the power ratio of Marshall - Palmer's rain drop size distribution 5 was presented. The M-P distribution, which is given by an exponential pattern having a simple from is used widdely in the field of f\radar meteorology, In the exponential distribution (3) the number of rain drop N (D) is given by
N (D) N o exp (A D) (1)
where D is drop diameter, N (D) dD is the number of drops of diameter between D and D +dD per unit volume of space, N is the value of N (D) for D = 0 and A = AR a where R is rainfall intensity, A is 41 ad a is 0.21 for M-P distribution.
By integrating Eq. (1) in regard to rain drops from zero to infinite, the total number N of the drops per unit volume is given as
N = N o / A (2)
The total number N changes as a function of N and A However, it is necessary to take notice that the total number of drops will be actually accountedin the volume or space in which the rain drops contributing to the amplitudes of the same frequency component exist rather chan that of per unit volume. WhEN tHE RAINFALLIS OBSERVED BY RADAR> BY THIS REASON> The tatal number of the rain drops used in simulation was fixed to 4096 which corresponds to those of the model distributions discussed in Section 3.
Based on this fact, the comparison of the power ratio between M-P distribution and the model distributions can be made in the same criteria, A plot of M-P distribution is shown by an arrow in Figure 1, which is also that of the model exponential disstribution.
Many scientists use also the exponential distribution to obtain their own rain dropsize distributions by ehanging the parameters N and A (for example. Imai et, al (4)) However as discussed in te foregoing section, if the rain dropsize distribution is givben by the exponential mode the resultant values of the power ratio will be always given by 3.3 dB regardless of rain types or the rainfall intensities in either case, because the statistical index of the exponential distribution gives only one value in spite of any values of the parameter. A
4.2 Weibull Distribution
There are many investigations as for the rain drop -size distribution, though Marshall Palmer's distribution is one of the most famous models, In this section, the power rations due to the drop -size distribution expressed by Weibull function which is more likely applicable for thundershowers bringing the internse rainfall intensity was investigated. The number of drops is given by Sekine (7) as follows:
N(D) = No C\D (D\b) c-1 exp(-(D\b)c) (3)
where
N o = 0.001 cm
-3
c = 0.95R
0.14
b = 0.26R
0.42
The characteristic of this nodel is that the form of the distribution varies with the rainfall intensity R, since the factor c is a function of R. instance, ifR is 1 mm/r. the sttistical index (a / root) statisical index (a /root of) the effective value. 0.725 Is slightly larger than that or the exponential mode. If R is mm/hr. The index is 0.552 whter is positioned in between the right angled triangle and quadrant mode. And if R is mm hr htee index is 0.51 which is below that of the quadrant mode .
The values of the power ratio by welbull distribution are alse shown by circles in Figure is The abscissa is commonly given the statistical idex. The cresponding rainfall intenstties are 1.2.4.8, 16 32 and 64 mm/hr . Only the numbers of 1,8 and 64 mm/hr are dedicated near the circles. Furthermore z-R relation of Weibull function is given by Z=286 R (7)
According to the increase of rainfall intensity level, the corresponding value of the power ratio decreases from 3.2 dB are R = 1mm/hr because of the patter variation inherent to Weibull function. It is important take notice here that the patterns of Weibulll distributie in the intense rainfall intensit level tends to be a sinbgle pea distribution having a maximum value of the number of rain dro in larger drop sizes but not the drop diameter D =0 Accorde ingly, the discrepancy of the power ratio between Weibul distributions and the mode distri butions are expressed by the monotonusly decreasing function.
5. Conclusions
In order to make it clear the power ratio of conherent scattering to incoherent scattering six types of model distributions are investigated to compare with Marshall Palmer drop size distri bution and Weibull distribution used in the field of radar meteorology. Then it was found that each mode of the distribution has the respective inherent statistical index giviven by the ratio of standard deviation to mean value or the squre root of the efective value, and the values of those power ratios are clearly given as a function of the indexes of the distributions. In the case of the exponential distribution which is used for the M-P rain drop size distribution the power ratio is 3.3 dB which is almost the maximum value expected by the monotonusly decreasing function.
On the other hand, in the case of Weibull distribution which was proposed as more realistic rain drop size distribution 7 the index varies with the rainfall intensity. In the case of weaker rainfall intensity such as 1 mm/hr, the index is nearly equal to that of the exponential distribution. According to the increase of the rainfall intensity. The distribution tends to be a single peaked one which has the maximum number of rain drops in larger drop-size regions but not D=0 The power ratio varies from 3.2 dB at 1mm/hr to 4.1 dB at 64 mm/hr.
Accordingly, if Weibull distribution is valid more for representating the actual rain drop size distribution, 4 dB of the power ratio are especially taken into account for the brought by unumaelsiofms Finally, it was found that the ambiguity of about 1 dB of the power ratio have to be considered depending on the rainfall intensity. In another word it can be said that 3.6 dB of the mean power ratio is expected with the deviation of 0.5 dB in the rainfall measurement by radar.
References
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- I Imai M. Fujiwara, I Ichimura and Z. Yoshihara, 1955: Radar reflectivity and the drop size distributuon of rain J. Meteorao Tokyo 7 (7) 422-433.
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