10. An Overall Check of the Theory. 



During a two- week period in February 1953, the Weather Bureau 

 observers on the various coast guard cutters of the Atlantic Weather 

 Patrol made visual wave observations which were to be used as a check 

 of the forecasting procedures given by Pierson, Neumann, and James 

 (1955). A check for internal consistency of the data showed that the 

 observations had been carefully made. 



The height observation for a particular report consisted of the 

 tabulation of 50 individual crest- to- trough heights. A considerable 

 range of height values was reported for each observation. For a given 

 observation, the reported height values would range from 10 feet to 

 30 feet, which appeared to agree with the theory that the wave heights 

 were distributed according to equation (1). The average of the 50 

 reported heights was computed, and the significant height was found 

 from the average height by multiplying the average height by 1.416/0.886 

 (or 1.60), which is the ratio of the significant height to the average 

 height. 



Twelve of the reported observations were selected because of the 

 high waves that were present. Forecasts based on the theory of the 

 manual by Pierson, Neumann, and James (1955) were prepared without 

 knowledge of the observed values. The results of the comparison of 

 the forecasted values with the observed significant heights computed 

 as described above are presented in table 7 under the heading. Observed 

 Significant Height (no correction) and Forecast Significant Height. 



The results of the comparison of the forecast and observed values 

 were most disappointing. Errors as big as 17 feet resulted. The average 

 forecast error was 8.6 feet. There was also a definite bias in that all 

 but two of the forecast values were less than the observed values. The 

 column labeled Error (uncorrected observations) shows these results. 

 There was evidently something wrong! 



At that time, none of the work on confidence limits or on truncated 

 distributions as discussed above had been applied to the data although 

 the theory was a standard part of statistical texts as it is given, for 

 example, by Cramer (1946) and Wilks (1951). The theories discussed 

 above were then developed and applied to the observational data. 



A closer look at the reported height values shows that the heights 

 did not follow the distribution given by doubling the coefficients in 

 table 1. The low waves predicted by the probability distribution 

 function were either missing or reported in far too small a proportion. 



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