where R (f) is the coherence between time series r and s, and 0j.g(f) is 

 the phase of -^glt) minus the phase of 17^ ( t ) at frequency f. Thus if 773 (t) 

 = A cos 2 7rft and 77^ (t) = B sin 2 7rft, the coherence Ri.s(^) = ^ ^"'^ *^s 

 phase angle ^^gCf) = 90°. 



Although the maximum value of Rj-gCf ) = 1* this value would not be 

 anticipated in nature. Instead, a lower value than one is usually obtained. 

 According to Munk et al. (1959) > the value above zero at which Rj.g(f) becomes 

 meaningful can be expressed by: 



955^ confidence limits of nL.g(f)l2_ k * (lO) 



"- -' DF 



where DF = degrees of freedom = (N - r-) / - 



N = number of observations 

 M = number of lags 



c. Corrections to the Data 



Coherences computed from equation (8) are a function of only the 

 frequency of the measured waves and are not dependent upon the relative change 

 of signal amplitude at a given frequency, owing to the constant frequency 

 response factor of each instrument. Also, the phase angle is not dependent 

 upon this factor but only upon the frequencies and, of course, the phase dif- 

 ference between the frequencies. Thus, it has not been necessary to apply any 

 correction to the data for those instrument response characteristics that affeC 

 only the amplitude of the recorded signal. Each instrument produces a phase f 

 shift of the recorded signal as a function of frequency. This delay is differ- 

 ent for the two gravimeters used and must be considered when studying the phase 

 relationships between two incoming signals. 



The phase shift due to each instrument was calculated according to 

 equation (11) (Richter, 1958) 



tan b = f^l (11) 



T -T 



where b = the phase shift in radians 



h = the damping constant 



T = the natural period of the instrxjment (sec) 



T = the wave period (sec) 



The relative phase shift between the gravimeters was also computed. This 

 value was then subtracted from the phase lag computed by the cross-correlation 

 program. 



^Equation (lO) is an approximation derived from a complicated equation given 

 in Munk et al. (1959). Since the writing of this paper, it has been learned 

 from Dr. R. A. Haubrich, a colleague of Prof. Munk, that a better approxi- 

 mation is now in use: 95^ confidence limits of rRj.g(f)p= 6/df. This increases 

 the 95^0 confidence limit from 0.^4 to O.53. Examination of figures 8b, 9b, 

 and 22, where this confidence limit is used, shows that the majority of those 

 coherence peaks previously considered significant are still significant, and 

 the conclusions based on these values still appear to be valid. 



