could be adjusted so that E was very small, then the associated solution would provide a 

 good fit to the complete formulation at these 41 points, and computations have shown that 

 the fit or other phase angles would be comparably good. The problem therefore has evolved 

 into one of minimizing the total error E. The procedure used is a least -squares procedure, 

 which requires formally that 



|i - (111) 



(The parameter i//„ is not determined by the least -squares procedure , but is selected such 

 that the mean water level is not changed by the other variables selected. This will be 

 discussed later.) Examination of Equations (I-ll) and (1-12) further will indicate that the 

 usual least -squares procedure is not appUcable, because the error is not defined as a 

 quadratic function of the unknowns. This problem then falls in the category of a nonUnear 

 least -squares problem. 



The problem was Unearized as follows. Suppose that at the k* iteration, a trial solution 

 is available. The objective is to select changes in the unknowns such that the errors will be 

 reduced. If this were a linear least -squares problem, only one iteration would be required. 

 Expressing the quantity Q in terms of small changes in the unknowns (to be determined at 

 the k*'' iteration). 



k+1 „k 



NN ^Q^ 9Qk 



where 



= S ^ Ji axfel^^^-) -^ 3ir^^ <i-i3> 



3X(n) 3r) 9X(n) 3u 3X (n) 3w 9X (n) ^^"^*^ 



SQ^9Q9r^_l_9Q9u3Q3w9Q3C 

 dL 3n 9L 3u 3L 3w 3L 3C 9L 



(1-15) 



where the 3Q/3rj, 9Q/9u are obtained from Equation (14) and the 977/9X(n), 9u/9X(n), 

 etc., are obtained from Equations, (1-7), (1-8), etc. 



101 



