The techniques used in the above derivation are a standard part of 

 statistical theory. For another example of how^ they may be applied, 

 see Wilks (1951), pp. 195 through 201. 



Exactly the same factors multiply the average height values or the 



significant height values, since both are simply constants times ^/K 



m. 



For example, if 25 values of ^ were obtained from the artificial 

 >vave record given in figure 3 under the assumptions which were listed, 

 and if the observed mean wave amplitude was 10 feet, then the true 

 wave amplitude as computed from a much larger sample from the same 

 population would be between 8.5 feet and 12.1 feet for 90% of such 

 experiments. If on the other hand the value was based on 100 amplitudes, 

 the true value would be between 9.2 feet and 10.9 feet for 90% of such 

 experiments. 



What has the above derivation to do with actual ocean waves since 

 the properties assumed in the derivation were shown not to be properties 

 of actual waves? The answer is that it appears that the values given 

 are the narrowest bounds possible and that the effect of correlation 

 is always to make the bounds even wider. That is, if the above theory 

 says that the bounds are between, say, 7 feet and 13 feet for a given 

 estimate, then the effects of correlation in the heights make the true 

 bounds even greater, say, from 5 feet to 15 feet. 



The true confidence limits of such an estimate will probably not be 

 known until the study of time series has advanced in this theoretical 

 direction. The range of the theoretical bounds as given above is an 

 underestimate of the range of the true bounds. The true upper bound 

 is greater than the theoretical upper bound, and the true lower bound 

 is less than the theoretical lower bound. 



The model that was made up applies fairly realistically to a series 

 of observations of wave heights as would be made in an actual visual 

 observation. The correlation of unity between a crest amplitude and 

 a succeeding trough amplitude makes each wave height an independent 

 observation instead of the sum of two independent observations which 

 is realistic in the sense that wave heights have been observed to be 

 distributed according to equation (1). Thus the average of N wave heights 

 should be considered to be the average of N independent observations, 

 and table 3 would apply to the computed values. 



The assumption of independence for the individual height vedues is 

 more to be questioned. As stated above, the effect of correlation is 



17 



