Power Spectrum of Depth Values 



The third method of representing variability is by the 



power spectrum 



13, 14, 15, 16 



The power spectrum U(h) is given 



by the Fourier transform of the autocorrelation, R x . It is the 

 energy (1/2 amplitude squared) per unit bandwidth* and thus 

 emphasizes the bandwidths in which the dominant frequencies 

 occur. The smoothed power spectrum values were obtained as 

 follows: 



U(h) 



\= n-\ 



R(0)+ V R(X) (1 + cos— ) cos 



n 



where h= 0, 1, 2, 3 



A= 0, 1, 2, 3 



n index number of frequency 

 (actual frequencies are given by 

 ft/(2At) cycles/min, At- 1/2 min), 

 and 



n is the lag number 



The power spectra computed from half-minute readings of 

 isotherm depth are given (Appendix D). Each plot is a spectrum 

 of the variations in depth of single isotherms listed (table 2). 



One example of the computed power spectrum (fig. 27) is 

 based on the depth of the 19°C isotherm of section Q (fig. 15). 

 The importance of the power spectrum lies in the curve peaks 

 that indicate frequencies (or periods) in the original data which 

 may have been obscured by background noise. It is significant 

 that this example of power spectrum has several peaks or peak 



*The units for U(h)used here are feet 2 /cycles per min. and Ufa) 

 might better be designated as variance. 



54 



