82 



an entire wave from crest to crest or trough to trough, to get a general sense of the 

 directional trend of the wave field. This global solution can then be refined to fit a 

 small segment very closely. This approach is used to establish the initial estimate for 

 the optimization procedure in each window. 



For the initial estimate to the global solution, it is assumed that the water surface 

 records can be approximated with linear wave theory: 



77 = a cos [k:,Xn + kyXJn - Ujtm + Cx) (5.8) 



for the single wave method, and 



■q — aiCos{k-^^iXn + ky,iyn—uJitm+ai} 



(5.9) 



+ G2 cos {kx,2^n + ky,2yn " <^2^m + Ct?) 



for the two wave method, where a„ = AnUj/g, and An is the amplitude of the linear 

 potential function of the nth wave. 



Directional Trend The first step is to determine the directional trend of the wave 

 field. This determination is accomplished by examining the gradients of the water 

 surface throughout the segment considered. Estimates for the spatial gradients and 

 the elevation of the water surface at the center of the array are computed by finite 

 difference approximations. The water surface is expanded in a two dimensional Taylor 

 series about the center of the array: 



r]{x, y) = ri{xc, yc) + (x - x^)-^ + (y - t/c)"^ + • • ■ (5.10) 



where Xc and y^ are the coordinates of the center of the array. This expansion is 

 written for each of the A'^ measured gauges, at each of the M points in time, resulting 



