ATMOSPHERIC ABSORPTION AND SCATTERING 



163 



and liydc's'- exact attenuation \alucs are available. 

 The attenuation formula in a rain, as given by equa- 

 tion (54), can be transformed easily to another form. 

 If pft denotes the partial precipitation rate of the drops 

 of k cm diameter in a given rain of total precipitation 

 rate p, then clearly. 



P 



= Z.Vk, 



(55) 



s being the diameter of the largest drops in this rain. 

 Now 



Vk = 3.6 X 10'^ Vk Vk Nk mm per hour, (56) 



where V^ is the volume of a raindrop of fc cm diam- 

 eter, Vfc is its terminal velocity in meters per second 

 and iVfc is their number per cubic centimeter. The 

 attenuation of a rain of total precipitation rate p is, 

 then, according to equation (54), 



y , . 0.4343 y VkQt,k 

 a\,p = Z^ otxipk) = „ „ Z^ Iff 



4 = 



3.6 



k = 



VkVk 



db per kilometer, (57) 



after substituting Njc from equation (56) into (54). 

 For a given wavelength X, the ratio Qt.k/yjci'ii is a 

 constant characteristic of drops whose diameter is Ic 

 cm. This ratio will be denoted by qu. The attenuation 

 formula then becomes, finallv. 



0.126 2 



Pkqk, 



(58) 



which shows that the attenuation in rains of a total 

 precipitation rate of p mm per hr depends linearly on 

 the individual precipitation rates pi^ of all the drop 

 groups fc which build up this rain. The attenuation 

 does not depend directly on the total precipitation 

 rate p. The points representing the experimental oli- 

 servations in the coordinate plane (a,p) should cover 

 a certain region of this plane, but no single curve 

 a(/j) exists, since there is no direct relationship be- 

 tween a and p. A curve drawn in this plane is sig- 

 nificant only in so far as it pcririits one to predict a 

 possible attenuation value in any rain of given pre- 

 cipitation rate or vice versa. 



It is, however, possible to draw in the («,;>) co- 

 ordinate plane a straight line which, at a given wave- 

 length, will represent the theoretical upper limit for 

 the attenuation. Indeed, using Table 6 for the attenua- 

 tion in fictitious rains with a distribution of one drop 

 per cubic centimeter, and Table 9, giving the precipi- 



tation associated with the same fictitious rains, one 

 may compute the ratio cnu/pic for any such rain formed 

 by a single group of drops of diameter /■; cm and the 

 precipitation rate pu of the same rain. This ratio for 

 a given wavelength A of the radiation varies with k, 

 the diameter of the drops, and in the diameter range 

 to 0.55 cm this ratio takes on an optimum value for 

 a certain diameter D. This, then, is the slope of the 

 straight line in the (a,p) plane which determines the 

 theoretical upper limit a^^^ of the attenuation in any 

 rain of total rainfall p. 



Table S. Precipitation rates p/N in fictitious rains 

 with a concentration of one drop per cubic centimeter. 



Drop diameter D, cm 



p/N mm/hr 



0.05 

 0.10 

 0.15 

 0.20 

 0.25 

 0.30 

 0.35 

 0.40 

 0.45 

 0.50 

 0.55 



4.99 X 102 

 7.34 X 103 

 3.34 X 10^ 

 9.6 X 10^ 

 2.14 X 105 

 4.08 X 105 

 6.76 X 105 

 1.05 X 106 

 1.54 X 106 

 2.17 X 106 

 2.92 X 106 



The difierent steps taken in computing the total 

 attenuation equation (58) in a rain of total rate of 

 fall of p mm per hour appear in Figure 10 where the 

 drop size distribution and the partial attenuations 

 due to the dift'erent drop groups of a 23.6-mm per 



400 



TOTAL K-BAND ATTENUATION 

 TOTAL X-BANO ATTENUATION 



«((= 2.40 OB/KM 

 !K J = 0.728 DB/KM 



<»Cl,/p= 0,I06DB/KM/MM/HR 

 c.rk(2.5)/p (2.5) =0.102 DB/KM/MM/HR 

 o'-n/f -O.OiZl db/km/mm/hr 

 '^x (2.5)/p 12.5)= 0.0332 db/km/mm/hr 



5 100 - 



0.9 



oa 



0.7 



as - 



0.4 



0.3 1 



0.1 



0.15 0.20 0.25 0.30 0.35 

 DIAMETER OF DROPS IN CM 





 0.40 



Figure 10. Drop size distribution and attenuation in 

 a 22.6-mm per hr rain. Unlabeled curve represents N)- 

 values ; number of raindrops per cu m = Nu.. 



