From equation (20) it is clear that the function y^nAt) actually represents a 

 sum of three sinusoidal waves vd.th frequencies differing by one frequency 

 increment 2iin/N. Thus, the partitioning of wave energy in a very narrow band 

 around the true frequency must be less clearly refined when the cosine bell 

 data window is used. 



b. Application of Fourier Transform Techniques to Field Data . A computer 

 program based on the FFT algorithm was used to estimate A2 and A in 

 equation (13) from field data. The program was applied in the preliminary 

 study of the South Haven data to a 1,024-second record both with and without 

 the cosine bell data window. 



The preliminary South Haven analysis indicated that a resolution of at 

 least 0.001 hertz would be needed to resolve detailed concentrations of 

 spectral energy with confidence. It also indicated that detailed spectral 

 characteristics can change considerably over time intervals of less than 5 

 minutes. The frequency resolution of the FFT analysis is limited by record 

 length, as shown in equation (8), so it is impossible to identify frequencies 

 corresponding to major energy concentrations to an accuracy of 0.001 hertz 

 with a 5-minute record using a direct application of the FFT. An alternative 

 analysis approach is needed. 



Two approaches for artificially increasing the length of a short record 

 were explored. Although the actual frequency resolution must depend on the 

 length of the data time series , it appeared that the nominally greater reso- 

 lution produced with an artificially lengthened record might permit a better 

 definition of major peak frequencies. One approach is to pad the record sym- 

 metrically with zeros to create a 1,024-second record which contains a much 

 shorter length of actual data. 



y(nAt) = 



M - N M + N 



, 1 ^ n < , < n < M 



2 2 



(21) 

 M - N M + N 

 y(nAt) , < ^ < 



where M is the total number of points in padded time series; in this case, 

 M = 1,024. In equation (21), n can range from 1 to M where M is 

 greater than N. 



Another approach is to periodically repeat the short data record as many 

 times as necessary to create a 1,024-second record. 



y(nAt) = y[(n - IN) At] , IN < n < (I + 1) N (22) 



where I is an integer. Care was taken in this approach to reasonably match 

 the elevation, y(At) and y(NAt), and slope, dy(At)/dt and dy(NAt)/dt, at both 

 ends of the short record. However, both the approaches for artificially 

 lengthening the record are capable of generating potentially troublesome 

 spurious peaks in the spectrum and were abandoned. 



Another approach to this problem is to compute spectra from partially 

 overlapped time series defined as 



30 



