76 



the free surface boundary conditions. An array of measurements provides information 

 about the spatial evolution of the signal, helping to better define the fundamental 

 wave number. This usually results in faster and more robust convergence of the 

 optimization. 



5.2 Formulation of the Optimization 



The formulation of the optimization for the LFI method as applied to the in- 

 terpretation of an array of water surface measurements has a great deal in common 

 with the formulation for a subsurface pressure record (chapter 3) and a point water 

 surface measurement (Sobey 1992). Less detail is presented here than in the previous 

 chapter, but the framework will be presented, with an emphasis on the additional 

 information necessary for applying the method to an array of measurements. 



Observational Equations The governing equations presented in the previous sec- 

 tion represent a free surface potential flow, with one or two components propagating 

 in an arbitrary direction. The observational equations are the equations in the system 

 that force the solution to fit the given measured record. As the location of the water 

 surface has been measured, these are the free surface boundary conditions (Eqs. 5.3 

 and 5.4), applied at the horizontal location of each of the nodes in the array. Sufficient 

 independent equations are defined by applying the boundary conditions at a number 

 of times along the measured records, within the window in time considered (Fig. 5.1). 

 The solution is the set of parameters in the potential function that result in the least 

 error in the FSBCs. 



In order to specify the solution, there must be at least as many independent 

 equations as there are unknown parameters in the system of equations. The free 

 surface boundary conditions (/^„ and /^.n) are applied at each of the A'' measured 

 locations and at M time samples in the window, resulting in 2MN independent 

 equations. In the single wave case, there are 4 + J unknown parameters sought in 

 Eq. 5.1 {kj:, ky, Lj, Q, and Ai . . .Aj) in each window, so that if 2MN > 4 + J, the 

 solution is specified. In the two wave case, there are 2^j/ +8 unknowns in Eq. 5.2 



