Recall that for a single real wave of frequency f^ > and known 

 direction 6^, C^jCfQ), Q^^ (f q) are known (see Equation (5.3)). These 

 values give 



S/K..e..O = S/k.,§i. -■n],-l) = -J . 



When a single, well-directed swell is expected, it is reasonable to 

 assume a single wave for a given frequency exists, and to select 9^ and 

 Aq = a2/4 such that the least square error between the theoretical 

 cross spectral matrix for a single wave (see Equations (5.1) and (5.3)) 

 and an observed cross spectral matrix is a minimum. Accordingly, using 

 the expressions of Equation (5.3) and an observed cross spectral matrix 

 for a given frequency, fg, we can form the squared error 



H = (C - A,)2 



. if ifcj- A. cos(£1i(l. K,^+ v...Y;i)M' 



NOTf. A„=A. ; q,-J|-ZC;i ^^g^ 



Expanding and collecting terms we get 





or using results in Equation (7.7) 



(7.9) 



31 



