34 



Nonlinear Optimization 



Once there is a reasonable first estimate for all the unknown values in the potential 

 function, standard nonlinear optimization routines are adequate for this system. For 

 the results given, the Levenberg-Marquardt algorithm was used as implemented by 

 the Matlab Optimization Toolbox (Grace 1992). 



If the optimization routine successfully finds a minimum, the solution is checked 

 to see if a clearly spurious solution is found. Spurious solutions are identified by the 

 following criteria: 



• Very large or highly variable errors. 



• First order amplitude smaller than higher order amplitudes. 



• Unrealistically large or small frequency or wave number. 



• Large discontinuity between windows in the predicted water surface. 



It is unusual for the optimization routine to converge to a spurious solution. It is far 

 more common for the routine not to converge at all. 



If no solution is found, or a spurious solution is found, it is necessary to revise the 

 parameters of the optimization. For the next attempt, the window width is increased 

 by a factor of 1.5 (1.5ro), and the procedure is repeated. If this is not successful, 

 the window width is increased once more to twice the primary width (2to). When 

 increasing the window width is not successful, the order of the potential function is 

 decreased until a solution is found. If none of these adjustments result in an acceptable 

 solution, the window is skipped, and future analysis must be interpolated through 

 that point. These adjustments are most likely to be needed in the long, flat trough 

 of a shallow water wave, where the window needs to be expanded to include some 

 curvature to indicate the frequency. 



Difficulties may occur in addition near zero crossings, where there is also little 

 curvature in the record. Here the effects of amplitude and frequency may not be 

 independent, as: 



lim asm(ujt) = acot (3.17) 



t-»-o ^ ' 



