127 



the potential function. Sufficient independent equations are defined by applying the 

 Bernoulli equation at a number of points in time throughout the considered window. 

 In order to specify the solution to a system of equations, there must be at least 

 as many independent equations as unknown parameters in the system. In order 

 to specify the solution to the LFI formulation for a pressure array, the boundary 

 conditions (/^ and f^ ) are applied at each of the water surface nodes, yielding 2M 

 equations, and the Bernoulli equation (/^j) is applied at / times on each of the A'^ 

 measured pressure records within the local window, yielding NI equations. In the 

 single wave case, there are 4 + J + M unknown parameters sought in Eq. 6.1 {kx-, ky^ 

 a;, Q, Ai . . . Aj, and t]i . . . tjm) in each window, so that if (2M + NI) > (4 + J + M) 

 the solution is specified. In the two wave case, there are 2 ^ j + 8 + M unknowns in 

 Eq. 6.2 {2kx, 2ky, 2w, 2q, Ai . . . Aj, and 771 ... 77m: 10 + M at first order, 14 + M at 

 second order, 20 + M at third order, and 28 + M at fourth order). The solution is 

 specified when (2M + NI) > (2^ j + 8 + M). This system results in the following 

 nonlinear least squares optimization. 



N I 



minimizeO(X) = ^^f^^^{X;Xn,yn,Zn,p,U)'^ + 



m=l 



where: 



X = {kx,ky,u:,a,Aj,... ,Aj,rii ...tjm) 



for the one wave case, and: 



X = {kx,i,ky^i,uJi,ai,kx,2-ky^2,i^2,0!2,Ai,... .Aj^rji ...tjm) 



for the two wave case. a;„, y„, and Zn,p are the coordinates of the nth gauge, Xc and j/c 

 are the coordinates of the center of the array, and rjm is the elevation of water surface 

 node m. 



As with the previous analysis, overspecification is helpful in accommodating the 

 measurement errors in field records, as well as being required to define the shape of 

 measured records and water surface in each window. 



