*w- 6 -> = — — — : ( 9) 



.=i j=\ 



where a is a factor of order 1 that is used to satisfy Equation 8, / is the number 

 of gauges, the M- l {f n ) are elements of the inverse of the dimensionless cross- 

 spectral matrix defined by Equation 1, *„(6 m ) is the wave number vector, and x t 

 and x are coordinate position vectors of gauges i and /' , respectively. The wave 

 number vector *„(9 m ) is 



* n (6 m ) = k n cosQ m e x + k n sind m i y (10) 



where e and i are spatial coordinate unit vectors in the x - and y -directions, 

 respectively, and k n is wave number vector magnitude, which is related with grav- 

 itational acceleration g to frequency f n and water depth d through the linear 

 wave dispersion relation 



M?f n = gk n tanhk n d (H) 



As used in this report, horizontal coordinates are such that x increases to the 

 north, and v increases to the west. 



An IMLE result is achieved by iterating through several computational steps. 

 At the r" 1 iteration, an estimate T M tJ {f n ) of the observed cross-spectral matrix 

 M {f ) is computed from the previous directional distribution function estimate 

 />,.,(/■„. B.) by 



r W> - 1 D^^eje-^-^^dQ (12) 



m = l 



A new intermediate directional distribution function estimate D r (f n ,Q m ) is com- 

 puted using the cross-spectral matrix of Equation 12 in the expression 



^o-.e.) = — — — : ri3) 



/=i ;=i 



where a r is adjusted so that Equation 8 is satisfied for D r [f n , Q m ) . A correction 

 is found for !>/(/„, 6 m ) by first computing 



Chapter 3 Primary Data Analysis 



11 



