Rows 1 and K in equation (A-9) are then switched to give 



I K^ I X^X^ '•' I Xj^Xj(._^ I Xj^X^ I Xj^Xj^^^ "'I 



I ^^y 



I X2XK I 



'"I '^2^K-1 I ^2^1 ^ ^2''K+1 •" I ^2^P ^ ^2^ 



I x^x^ I x^x^ ...VxjX^_^ I xl l^i^K+l •••I^i'^p I ^ly 



I XpXj^ I XpX. 



I X]^y I x^y 



^ p 1 



u. 



i- '"p'"K-l ^ '"p'^i i^ "p"K+l 



^ P 



I ^py 



I XpY I y^ 



(A-IO) 



Techniques are available for partially solving equation (A-10) to obtain 

 d^ in equation (A-2) (e.g., Aubert, Lund, and Thomasell, 1959). The techniques 

 require manipulation of equation (A-10) to convert the first element in the 

 first column to one and other elements to zero. The remaining columns are then 

 orthogonalized with respect to the first column. Correlations between the 

 remaining columns and the time series are computed to identify the most impor- 

 tant remaining constituent. Suppose it is in column 1. Columns and rows are 

 then switched to position the selected constituent in the second row and column. 

 The remaining columns (p - 1) are again orthogonalized with respect to the first 

 two columns. This procedure actually gives a recomputed d^ as well as solving 

 for d^. 



The procedure is repeated for as many steps as desired up to a maximum of 

 p steps. At each step another x^ is selected and the values of the d 

 coefficient are computed for the newly selected Xj and all previously selected 

 x^ so as to explain the maximum amount of variance in the field data time 

 series. 



Because of the quasi-periodic nature of ocean waves, it is desirable to 

 choose periodic functions for the Xj^. Thus, it is convenient to choose 



^2i-i = cos(a)^nAt) 

 x^-j^ = sin(a)^nAt) 



(A-11) 



where m^ is the set of selected frequencies. Substitution of equation (A-11) 

 into equation (A-2) gives 



p/2 

 yg(nAt) = I [d^i.^ cos(a)^nAt) + d^^ sin(w^nAt)] 



i=l 



(A-12) 



or, defining 



= d 



21 



-1» "i = ^ 



2i 



p/2 



yg(nAt) = I [3^ cos(a)^nAt) + a^^ sin(a)^nAt)] 



(A-13) 



1=1 



81 



