Equations (A-5) can be solved by forming the augmented matrix 



y X X y x^ 



... y X X y x y 



^ 2 P ^ 2 



l_I Xpx 



1 I XpX2 ... I x2 y Xpy 



(A-6) 



The solution is further simplified by adding a (p + l)'th row to equation 

 (A-6) so that the matrix is square and symmetric. 



I x2 I x^i 

 I X2X^ I x2 



I x^Xp I x^y 

 I x^Xp I x^y 



l^'. 



I ^py 



I Xpy y y2 



(A-7) 



I x^y y x^y 

 The solution is obtained by inverting equation (A-7) . 



In using a relationship of the form of equation (A-2) to approximate an 

 ocean wave record, it can be expected that some of the functions x- will be 

 important constituents of the field record and other x^ will be of little 

 significance. The goal of the analysis is to identify and quantify only the 

 major constituents. The most important constituent, x^ , in equation (A-2) 

 is defined as the one which has the highest correlation with the time series 



ly " y x2 y y^ 



(A-8) 



where R^ . „ is the square of the correlation between x. and y. If x^ is 

 the constituent for which equation (A-8) is largest, columns 1 and K in equa- 

 tion (A-7) are switched to give 



y x^x^ y x^x^ ... y x^x^_^ y xj y x^xj^^^ ... y x^Xp y x^y 

 y x^xj^ y x2 ... y x^xj^.^ y x^x^ y x^xj^^^ ... y x^Xp y x^y 



y Xj^ I Xj^x^ 



I Xp^K I V2 •• 



y xj^y y X2y .. 



I XpXj^_^ y Xpx^ y XpXj^^^ 



y xj^_^y y x^y y xj^+^y 



I x^ y Xpy 

 I Xpy y y2 



(A-9) 



80 



