IV -43 



We could just substitute the real and imaginary parts of iIji as given by (IV- 36), 

 but would find that the resulting integrals are very difficult to evaluate. In the 

 high frequency (k a > > 1) case, a general evaluation of the integrals has been 

 performed, and may be found in Chernov (Ref. IV-2, following p. 66) where he 

 considers rigorously the problem of a plane wave incident on an inhomogeneous 

 half- space as shown in Figure II- 3. The significant approximation to be made in 

 the high frequency case is the result of the high directivity of the scattering from 

 any small scattering volume. The signal received by an observer located at a 

 point X (see Figure IV- 13) comes principally from those scatterers which are 

 located in a narrow cone with the observer at its apex. The distance between an 

 element of scattering volume and the observer may therefore be approximated 

 according to: 



r= ^(xi -?i)^ + P'' -(xi - ^i) + l ^^ +.... (iV-91) 



where p^ = (X3 - ?af + (^ " ?3) 



The definition of p may be seen from Figure IV- 13. If we substitute this approxi- 

 mation in (IV- 36) and take the real and imaginary parts we obtain 



Sl = 



Xi °° 



k^ f ^^ f f^r ^P sinrkp^/2(xi - ?i)1 



^ '^^M ^^ ^^ ^ — — —^ ^ <i> (IV- 92a) 



Xi °° 



O -0° 



d?3 d?3 cos [kpV2(x. -g.) I ^ ^^^ ^j^_^2^^ 



xi - ?1 



These expressions can be used to obtain the averages given in (IV-90) in a fairly 

 simple form. Since all averages involve taking an average of the product of the 

 index of refraction at two different points, all of these averages will ultimately 

 be in terms of the correlation function describing the inhomogeneity of the medium. 

 The computation of the averages remain a formidable exercise in integration, and 

 may also be found in Chernov. The resulting integrals can be approximated depend- 

 ing on whether the observer distance L is in the focusing or in the interference range. 

 In other words, the nature of the approximation depends on whether the wave param- 



4L 

 eter D = — ^3- is much less or much greater than unity. Following Chernov, we 



£rtliur a.littlcJnt. 

 s-7001-0307 



