to be needed in the long, flat troughs of shallow water waves, or near zero crossings 

 in the record. The additional data provided by the array of measurements, and the 

 fact that the elevation of the water surface is measured, provides for a much more ro- 

 bust optimization than with the subsurface pressure measurement. As a result, these 

 adjustments are necessary less often with an array of water surface measurements. 



Locating the Water Surface at the Array Center 



Once the solution is found, the potential function, and thus the complete kine- 

 matics in the immediate neighborhood of the array, are defined. The solution is likely 

 to be most accurate at the center of the array, and it is often convenient to have a 

 solution at a single point, so the water surface at the center of the array must be 

 found. This is accomplished by setting up another optimization problem with the 

 elevation of the water surface at a few nodes in time throughout the window as the 

 unknowns. 



The free surface boundary conditions (Eqs. 5.4 and 5.3) are applied at the hori- 

 zontal location of the center of the array, at M points in time throughout the solution 

 window. At each point in time, the only unknown is the elevation of the water sur- 

 face, and the two boundary conditions provide two independent equations. The water 

 surface is defined as that location that results in the least error in the FSBCs, in the 

 least squared sense. Each point in time is independent, but the system can be set up 

 to solve for a number of points at once. Enough points should be found to specify 

 the shape of the water surface throughout the window. For the results given here, 

 six points were used in each window (see Figs. 5.6 through 5.23). The mean of the 

 measured water surface at the nodes provides a good first estimate. This system 

 consistently and rapidly converges to a solution. 



