III-105 



Suppose that we have such a representation for the scattered wave, 

 i.e., we have found a $(X,u) such that (III- 123) is true. Then 



00 00 



p' (x,y)-p (x,y,o)= e'^^^^""^^^^ i{\,u)dXdui (III-124) 



gives the scattered wave in the plane z = . (The coordinate axes are assumed 

 to be oriented so that the plane z =0 lies completely below the surface.) If we 

 use the notation I to denote the Fourier transform of $, then (III-124) also states 



that p' = § 



The autocorrelation function of Pg^ is given by 



00 00 



i ! 



Y (x, y) = (p^c * p;p (x, y) = \ \ p^^ (§ , ri) p^^(§+ X, ri + y) d? dr] (III- 125) 



Let A(^,u)= Y(^. u) be the Fourier transform of ¥(x, y), i.e.. 



CO CX) , ■ 



A(X,u)=j^ \ \ e"^^^^ + ^yN(x,y) dxdy (III-126) 



-OO -00 



Since the Fourier transform of the autocorrelation function (p' * p' ) is the 



^sc ^sc' 

 absolute square of the transform of p , it follows also that 



A(^. U) = |p' (X,^)|^ (III-127) 



1 . We are using * to denote the convolution of two functions, and a superscript 

 bar to indicate complex conjugate . 



^artliur Sl.HittleJnir. 



S-7001-0307 



