TM No. 377 



of the covariance function and the covariance spectra, the higher order 

 errors car. become of lower order s depending upon their degree of non-random- 

 ness. This is caused "by the phenomenon of correlation, in which relatively- 

 small deviations of the data pairs (which tend to be in or out of phase) are 

 highly magnified in the process of taking the products of time sequence pairs. 

 Thus, errors in the individual factors can greatly bias the covariance function. 

 Because of this great sensitivity of the covariance function and the covariance 

 spectra, great care must be taken to insure simultaneity of the two variable 

 records . This requires keeping the data pairs on the same time base after they 

 are abstracted from the two-channel tape record. 



As will be demonstrated in chapter V, any artificial phase lead or lag 

 introduced into one time series with respect to the other can cause large 

 changes in the covariance. An inaccurate covariance estimate can, in turn, 

 cause misinterpretation of the fundamental processes. 



Biasing Errors -= Another processing stage that is critical to the final 

 data output is the linear interpolation. The question naturally arises % how 

 much is the original data distorted by linear interpolation? The justification 

 of linear interpolation is based on the assumption that one has sampled a vari° 

 able closely enough so that linear interpolation does not distort the pattern 

 protrayed by the original raw data points. 



Figure III-8 shows plots of both the uninterpolated w velocity points 

 (lower curve) and the interpolated data points (upper curve). This velocity 

 record is taken from (057A) BEELS-11 at a depth of 0.5 meter. The interpola- 

 tion time interval is 0.2 second. The calibration curves of all the wave meters 

 indicate that, for flow speeds greater than about 25 cm sec" 1 , the period 

 between voltage' pulses is less than the interpolated sampling interval. Thus, 

 for the upper range of speeds nominally peaking at 60-80 cm sec"- (occurring in 

 moderately siaed waves and with the sensor close to the surface), no resolution 

 is lost by interpolation. Examination of the points in figure III-8 indicates 

 little distortion in the interpolated velocity pattern,, On this basis it was 

 assumed that a linear interpolation having a 0.2 second time interval gives a 

 reasonable portrayal of the velocity pattern. 



The linear interpolation for most of the wave meter data was made at 0^ 

 second intervals,, In general, the value of 0.2 second was well chosen, since 

 the wave motions within the upper 3~5 meters of depth supplied data points of 

 the order of time spacing of 0.2 second or less,. Note from figure 111=5 that 

 the repetition rate of the impeller at speeds of about 0„2 second is equivalent 

 to a. velocity of about 25 cm see 3 ",, For records of velocity with relatively 

 low variances (e.g., records below 4-5 meters depth, with average surface wave 

 conditions) , the spacing of the data points becomes, on the average, greater 

 than 0.2 second. In view of this, a few early wave records were interpolated 

 at 0.3, 0.5, and 1.0 second intervals in an attempt to be more consistent with 

 the data point density. The longer time spacing, however, made negligible 

 difference in the calculation of the variance and spectra when compared with 

 statistics produced from the same records using a 0.2 second interpolation time 

 spacingo The lengthening of the sampling time produces a shortening of the 



