77 



gauge 1 gauge N 



Figure 5.1: Schematic of system of equations in a window 



(2^1, 2ky, 2a;, 2q, and A1...A1: 10 at first order, 14 at second order, 20 at third 

 order, and 28 at fourth order). The solution is specified when 2MN > 2Y^j +8. 

 This system results in the following least squares optimization. 



M N 



minimizeO(X) = J^ ^/^„(X; x„,j/„,r7^,„,i™)^ + /^_„(X; a:„,y„, ?;„,„, ^™)2 (5.6) 



771=1 n—1 



where: 



X = {k:^,ky,uj,a,Aj,... , Aj) 



for the one wave case, and: 



X = ikx,i,ky^i,ui,ai,kx^2,ky^2,^2-,oi2,Ai,. . . , A/) 



for the two wave case. x„ and y„ are the coordinates of the nth gauge, and rim,n is 

 the measured elevation of the water surface at time tm and gauge n. 



As with the analysis of a pressure record, overspecification can be helpful in accom- 

 modating the measurement errors in field records. More than the minimum number 

 of time samples in the window may be required to define the shape of the water 

 surface in each window. This is less likely to be necessary with two waves than with 

 one, as the number of unknown parameters is much larger. It should be kept in 

 mind, however, as it would be a factor for low order solutions with a large number of 

 measurement locations. 



