99 



As the order is increased, there are more free parameters, and care must be taken 

 to include sufficient points in each window. As discussed above, for fairly low orders 

 and an array of measurements, the limiting factor is the minimum number of points 

 needed to define the curvature in the window, rather than to specify the system of 

 equations. The above examples were computed from an array of three points, and 

 so required a minimum of only one sample to specify the system at first and second 

 order, and two points for up to order 8. When attempted, the solution would not 

 converge with only one sample. With two samples, the optimization algorithm found 

 a reasonable solution, but this small number of samples would not define the shape of 

 the water surface adequately if it were part of an irregular record, and so three points 

 were used in each window for all orders. In general, a minimum of three points should 

 be used, and more may be necessary to accommodate a highly irregular profile, or 

 the inevitable measurement error in a field record. 



In the case of theoretically generated records the need for more sampling of points 

 poses no problem, but with field records, there are limitations as to the spacing of 

 the sampled points. In order to free up the spacing of points for the LFI method, 

 points are sampled from a cubic spline interpolation of the actual record. This allows 

 the points to be sampled anywhere within the window. While computationally it is 

 possible to sample as many points as necessary in a small window, if that window is, 

 in fact, defined by only a couple of actual data points, it is not appropriate to try 

 to fit a high order solution to a segment defined by only a few observational points. 

 In order to include sufficient actual data points to justify the increased order, the 

 window must be increased in size. While increasing the size of the window permits a 

 higher order solution, it also compromises the local nature of the method. The goal 

 of the LFI method is for the solution to be as local as possible, which is achieved by 

 selecting as small a window as possible at fairly low order. 



Shallow Water A window near a crest of the shallow water steady wave given 

 in 5.5 has been computed at orders 1 through 5. The results are given in Fig. 5.14 

 through 5.23. These figures are analogous to those previously discussed, but on a 

 shallow water wave. 



