related to the effects of equation (141. However, this will he over- 

 whelmed by the averaging against the other "within sequence" lag products 

 provided water level observations separated by more than kjj, time incre- 

 ments are essentially independent of each other and kj^ is small. 



The periodicity introduced in equation (14) causes a corresponding 

 periodicity in C, . 



C,^ = C.]^ = C^_-^ for N/2^k£N . 



(15) 



An approximation to equation (12) would be 



N/2 

 Pf^m^ = H ^k ^°s (2TTfj^kAt) At . 

 k = 2^1 



(16) 



With the introduction of equation (15) and the definition of fj^ (equa- 

 tion 6) , this becomes : 



N-1 N-1 



p(fni) = At Y^ Ck cos (2tTj^ k At] = At V" C^ cos (2iTmk/N) 



(17) 



k=0 



k=0 



The last equation is algebraically equivalent to equation (5) although 

 it appears somewhat different. The equivalence can be seen after equation 



(13) is substituted for Cj^ and the transform is shifted to exponential 



form. Thus, 



N-1 N-1 



k=0 



n=0 



N-1 N-1 



At 



N 



Y" y^ n exp (+i2TTmn/N) m exp (-i2TTm(n+k)/N 



k=0 n=0 



N-1 

 At \p 

 N 2^ 



n=0 



n exp (i2TTmn/N) 



N-1 



E 



k=0 



T]-. exp (-i2TTmk/n) 



(18) 



