31 



Initial Estimate 



The first step in each window is to establish an initial estimate for the optimization 

 procedure. Linear wave theory can be used to produce estimates for the parameters 

 of correct magnitudes. The linear subsurface dynamic pressure is: 



Pd = a cos {kx — ut) 



, cosh k(h + z^) (3-^) 



a = Apu \ 



cosh kn 



where A is the amplitude of the linear potential function. The wave number, fc, and 



frequency, a;, are related through the linear dispersion relation: 



{u - kUf = gk tanh kh (3.9) 



Frequency Nielsen (1986, 1989) established a method for determining the param- 

 eters of a local linear approximation to waves from a pressure record. His method 

 determined a local frequency and wave number that could be used to find the location 

 of the water surface. A similar method is used here to determine the first estimate 

 for the local frequency and wave number. Frequency of a sinusoidal signal of the form 

 Pd = a cos (kx — ut) is available from the second derivative: 



^ ^'P' (3.10) 



Pd dt^ 



This approach requires an estimate for the value of the second time derivative of 

 the record throughout each window. An estimate for the second derivative is directly 

 available from the cubic spline of the measured points. This estimate, however, is 

 very sensitive to random error in the measurements. Nielsen suggests estimating the 

 value of the derivative through the use of divided differences. This solution works 

 well for a smooth, sinusoidal record, but is also very sensitive to noise in the record, 

 particularly in areas of small curvature; estimating a second derivative from a small 

 segment of a noisy record can result in large errors. In order to accommodate the 

 inevitable noise in an actual record, a different approach is taken here. 



