Ogilvie 



shown by the open circles in Fig. 2. More points are available than were avail- 

 able in the spectral estimates. The black circles are those points that corre- 

 spond to the diamonds as far as the horizontal axis of the figure is concerned. 

 This approximation to the spectral window yields values for the black dots that 

 compare favorably with the values of the black diamonds that were obtained di- 

 rectly from the spectral computations. The results show that both the shapes of 

 the cross spectra and the location of the zeros in the cross spectra are lost due 

 to poor resolution. The rapid variation in the values indicated by the black dots 

 when convolved with a broad triangular weighting function results in a decrease 

 of the peaks of the estimates for the cross spectra and shifts the zeros to erro- 

 neous values. In Fig. 3 the coherencies as computed from the open circles are 

 shown in order to compare them with the black diamonds. The loss of coherency 

 caused by the shape of the window is confirmed. 



The above computations were based on the assumption that the spectrum as 

 smoothed in Fig. 1 was the true spectrum. It is in fact, an estimate of the true 

 spectrum that also has been convolved with a window of roughly the same shape. 

 The original spectrum cannot be recovered and hence the computations do not 

 completely describe the full effect. An attempt was made to determine what the 

 effect of the convolution is by convolving the true spectral estimate once more 

 and then computing the coherencies that would be obtained by using this new 

 spectrum and the spectra and cross spectra obtained from the open circles and 

 the filled circles of Fig. 2. The result of the computation is shown by the 

 crosses in Fig. 3. Some of the rapid variations in the circles have been re- 

 moved but the same general trend is evident. 



There are two ways to avoid the low coherencies that were obtained in this 

 example. The first requires that a considerably longer record be obtained and 

 that the resolution of the analysis be anywhere from five to ten times greater 

 than that used in this example. The rule is, of course, that the convolution op- 

 erator, which spreads over four frequency intervals, must operate on a portion 

 of a curve that is slowly varying. The work of Dr. Yamanouchi is important in 

 this connection. The computations illustrated by Pier son and Dalzell (1960) 

 show that when the resolution is increased in such spectral and cross spectral 

 estimates the peaks become much higher and the coherencies improve. A sec- 

 ond procedure is to require that, for the same resolution, the estimates of the 

 cross spectra be less rapidly varying. This can be achieved by a "false" time 

 shift of one record with reference to the other. For example, the cross variance 

 function obtained by computing all lag products of t7(x, t) and r)(x + L, t + r) for 

 L fixed yields a function that has a maximum at a certain value of r, say, r^. 

 If the covariance function is then considered to be time shifted so that the value 

 of Tj is zero, one achieves a very nearly even function about this new time 

 origin. The co-spectrum and quadrature spectrum computed from this new 

 time origin in general have the property that the co- spectrum is quite large and 

 the quadrature spectrum is near zero. The coherency between these cross 

 spectra and the original spectrum is then quite large. Pierson and Dalzell have 

 illustrated these ideas by studying both higher resolution spectra analyzed in the 

 same way and by studying time-shifted covariance estimates. Coherencies that 

 fell from 0.9 to 0.5 over a range of six or seven ordinate values were raised by 

 these techniques to from 0.8 to 0.9 by higher resolution and to values greater 

 than 0.9 by a time shift. 



92 



