224 



TROPOSPHERIC PROPAGATION AND RADIO METEOROLOGY 



become appreciable only when the wavelength is 

 below a certain maximum value and when the drops 

 exceed a certain critical size. Rain echoes are rarely 

 observed at longer waves than S band, but they are 

 common at S band and become very important at 

 the shorter microwaves. 



For a time it was thought that clouds could 

 produce microwave echoes, but more thorough inves- 

 tigations have now established the fact that the 

 droplets in clouds are too small "to produce appreci- 

 able scattering. Only drops that are large enough to 

 constitute genuine rain are seen by a radar, and, 

 especially at S band, light rains will often escape 

 detection. The term "storm echo," invented at a 

 time when the origin of these echoes was not yet 

 clearly understood, should be avoided, and the terms 

 "rain echo" or "precipitation echo" should be used 

 instead. A rain seen by the radar is not necessarily 

 recorded by an observer at the ground, as the rain 

 may be confined to the free atmosphere and never 

 reach the earth. This occurs either when the rain 

 falls in an ascending stratum of air where the air 

 rises more rapidly than the drops fall or when the 

 raindrops evaporate again before reaching the 

 ground. Both cases occur quite commonly in the 

 atmosphere, especially under convective conditions 

 such as are indicated by cumulus clouds and thunder- 

 storms. Snow may also be seen on microwave scopes 

 provided the snowfall is sufficiently heavy. 



While clouds themselves do not produce microwave 

 echoes, they may contain falling rain of one of the 

 forms just indicated. Visual appearances are deceiv- 

 ing, and an imposing looking cumulus cloud might be 

 entirely invisible on the scope, whereas a cloud that 

 is inconspicuous to the eye but contains falling 

 raindrops might give a pronounced echo. 



The question of "shadow" cast by a storm echo 

 is of some operational interest. A shadow is formed 

 when the absorption that accompanies scattering by 

 the raindrops becomes so strong that the remaining 

 radiation no longer suffices to produce visible echoes 

 from targets behind the rain area. This effect is 

 pronounced on X band, and even more on K band, 

 and is often quite conspicuous with airborne equip- 

 ment where it may happen that a rain storm blanks 

 out a sector of the sweep. On S band the absorption 

 is usually much weaker and targets can often be seen 

 behind a rain echo. 



The usefulness of rain echoes for aerial navigation, 

 particularly in the tropics, is now so generally known 

 that the subject need not be discussed further. 



17.4 



SNELL'S LAW 



The ordinary law of refraction known as Snell's 

 law may be expressed as 



Wo sin ft = iii sin ft , 



where ft and ft are the angles which the ray makes 

 with the perpendicular to the boundary. Here it is 

 more convenient to take the angle a between the 

 ray and the boundary surface. Snell's law then reads 



n cos a = n x cos ai . 



The refraction at a sharp boundary is shown in 

 Figure 39 A. If there are several boundaries it is 



r CENTER OF EARTH 



Figure 39. Application of Snell's law of refraction. 



readily seen that Snell's law generalizes (Figure 

 39B) to 



n cos a = n\ cos ai = n 2 cos a 2 = • • • , 



and for a continuously variable layer it becomes 



n cos a = n cos ao , 



where n and a are continuous variables which are 

 functions of the height and the index designates an 

 arbitrary reference level. 



Snell's law for a curved earth may be derived from 

 Figure 39C. For successive boundaries it is found : 



