For a real wave element we have, where S(£,m,f) = S(-Jl,-m,-f ) , see 

 Equation (3.2). R(X,Y,T) =Ex|, (i tl, (|X * V, Y"^ f T) 5 (W-fjifld^ Ji* 



+ t*j>(-iiir(9x+>»Y-*^T))5(-fi->i.-l)«JlUMJlf 



which is simply the sum of the covariance functions of two complex 

 wave elements which are conjugate pairs. It also follows that 

 R(X,Y,T) is real valued. 



Reverting to the complex wave element form, and noting that the 

 expected ensemble value of cross products between different wave ele- 

 ments is zero in a manner similar to the case of cross products shown 

 above, we obtain the composite general relationship 



(3.4) 



We have demonstrated, but not rigorously proven, that the covari- 

 ance function R(X,Y,T) is the three-dimensional Fourier transform of 

 the directional power spectrum S(J^,m,f). 



We cannot hope to be able to estimate R(X,Y,T) for continuous 

 values of X, Y, and T. However, there is a way around this problem, 

 we can write the above as ^ .^- .. \ ^__ 



which is in the form of a single dimension (variable f) Fourier trans- 

 form of the term in brackets [ ]'s. Note this term is not a function 

 of T. It depends only on the value of (X,Y) . Further, by Fourier 

 transform pairs we can write this expression as 





(3.6) 



-OP 



In general, assuming that the term in [ ]'s is complex, we can write 



[C(xxo-iQ(xxn: = 





(3.7) 



11 



