the center, and to push the zero contour more realistically to the left along 

 the -q axis. 



It would not be too difficult to remove the effect of the swell from the co- 

 variance surface and to obtain an estimated covariance surface for the sea. How- 

 ever, as Tukey and Hamming [1949] have pointed out the covariance estimates 

 are subject to even more erratic sajnpling variation than the smoothed spec- 

 tral estimates and this sampling variation is not well understood. 



For examples Tukey [1951] has shown covariance functions computed 

 from portions of the same time series. They were markedly different and 

 yet the spectra computed from the different covariance functions were very 

 similar. 



For many types of problems in which knowledge of the covariance function 

 is needed, it has been found that reinverting the smoothed spectral estimates 

 will yield a more reliable covariance surface. Also for simpler problems the 

 simplified spectrumt given above which is symmetrical about 9 = would 

 give a more tractable covariance surface. 



Alternate procedures for determining directional spectra 



A number of alternate procedures for determining directional properties 

 of waves have been proposed and attempted. 



The methods used by Barber [1954], essentially directional antenna 



arrays, are by far the simplest and most economical if fixed positions for the 



wave poles can be maintained. The effects of refraction and perhaps bottom 



friction and percolation, however, make it difficult to generalize to open sea 



conditions and study the full range of components in the spectra. 



246 



