32 



In each window, a third order polynomial is fit to the record in the least squares 

 sense. The second derivative throughout the window can then be computed from this 

 polynomial. By using more than four points in each window, any noise in the record 

 is smoothed out by the least squares fit. This approach proved to be consistently 

 reliable for artificially generated noisy records, resulting in reasonable estimates for 

 the value of the second derivative throughout the window. A set of frequencies is 

 computed from the estimate of the second derivative at each of the considered nodes. 

 The mean of these frequencies is used as a first estimate for the local frequency of the 

 record. 



Amplitude and Phase Once the frequency is known, the amplitude and spatial 

 phase [kx) of a particular record can be found by rearranging the equation as a linear 

 least squares problem by separating the cosine and sine components: 



p^- = a cos {kx — Loti) = b cos uti + c sin uti 



(3.11) 



where a — y/P + c^ and kx — arctan(c/6). This results in the following matrix 

 equation in the unknown amplitudes, b and c. 



coscj^i smujti 

 cosijjt2 sin 07^2 



Pd.2 



(3.12) 



cos ut I sin ut J 



The system is determined ii pd{t) is defined at two points. If / > 2, the system is 

 over determined and can be solved in the least squared sense by common algorithms 

 in numerical linear algebra libraries. This method provides a first estimate for the 

 parameters: a^ kx, and u. 



Refining the Linear Estimates These estimates for the parameters can be quite 

 poor, as they are all dependent on the initial estimate for the second time derivative 



