83 



in a system of linear equations that can be expressed as the following matrix equation: 



Xq — Xi 

 Xc — X2 



Vc-yi 

 Vc - y-i 



VcA 



Vc,2 



dVc,i 



'dVc.2 



dx 



dx 



dvc,\ 



Sj^cZ 



dy 



dy 







m.i 



m,2 ■ 



■ Vl,M 



VcM 





V2,l 







dVc.M 

 dx 



dy J 





JlN,l 





■ Vnm. 



1 Xc- xpf yc - Vn 



where x„ and y„ are the coordinates of the nth gauge in the array, rjc 



9tic 



(5.11] 



dric. 



"di 



and 



.^''^ are the water surface and gradients of the water surface at the center of the 



dy o 



array at time, i^, and /;„,„ is the water surface elevation at gauge n and time tm- If 

 there are three gauges, the gradients are uniquely specified. If there are more, the 

 system is solved in the least squared sense. 



The water surface is traveling either toward or away from the direction of the 

 water surface slope depending on whether it is moving up or down at the time. The 

 direction is thus determined by the spatial gradient of the water surface, and the sign 

 of the time gradient, yielding a set complex direction vectors: 



Dm = sign 



dr]m 

 dt 



drjm .drjrn 

 dx dx 



(5.12) 



A cubic spline of the water surface at the center of the array (77c) would provide 

 an explicit piecewise polynomial expression that could be directly differentiated to 

 obtain the gradients in time of the water surface. These estimates would be very 

 sensitive to measurement errors in the record. To obtain more robust estimates for 

 the time gradients, a smoothing cubic smoothing spline (de Boor 1978) is employed 

 instead. This algorithm provides a smoothing parameter, p, that can be set at any 

 value between and 1, where p = results in the linear regression fit, and p = 1 

 results in the "natural" cubic spline. The smoothing parameter may be varied to 

 accommodate varying levels of noise in the record. For the examples in this chapter, 

 p = 0.9 was found to be satisfactory. 



