SECKEL and VONG; HARMONIC FUNCTIONS 



in 1959 and 1965 when it is 0.054/rr, and 

 0.045%f, respectively. 



At Christmas Island (Table 3), the average 

 standard error of estimate at n = 4 (resolution 

 of 1 month) is near 0.33° C and therefore about 

 60 % higher than that for the Koko Head 

 temperatures. As previously mentioned, high 

 temperature variability is to be expected at the 

 Christmas Island sampling site. 



A standard error of estimate based on all 

 samples used to obtain a function obscures the 

 month-to-month changes in variability that may 

 have occurred. At Koko Head the month-to- 

 month changes in temperature variability as re- 

 flected by the standard error of estimate for 

 each month ranges from 0.05° to 0.45° C, the 

 same values for the Koko Head salinities range 

 from 0.006%<. to 0.136%f , and those for Christmas 

 Island temperatures range from 0.17° to 0.66° 

 C. Assuming that sampling error remains 

 constant, the range of variability reflects changes 

 in oceanographic conditions. 



The standard error of estimate computed from 

 the temperature and salinity observations of 

 each month also reflects sampling quality in 

 that low values indicate the residual variability 

 in the ocean plus sampling error. For the Koko 

 Head temperature, low values of the monthly 

 standard error of estimate are near 0.1° C 

 and for the Koko Head salinity they are near 

 0.02%c. The sampling error is therefore with- 

 in ±0.1° C for the temperature and ±0.02^0 

 for the salinity. These are the limits to be ex- 

 pected when bucket sampling of the temperature 

 and salinity is carefully done. 



Finally, how is the quality of fit affected by 

 sampling frequency and how reliable are the 

 expected values that may be obtained from the 

 harmonic functions? The constraint imposed 

 by the sampling frequency on the resolution that 

 may be attained by harmonic analysis has al- 

 ready been discussed. The present question con- 

 cerns improvement of fit when the sampling 

 frequency is increased above the minimum re- 

 quirements. 



At Koko Head the sampling frequency was 

 increased from once to twice weekly in 1961. 

 No significant change can be seen in the stand- 



ard errors of estimate listed in Tables 1 and 

 2 as a result of doubling the sampling frequen- 

 cy. This observation is consistent with results 

 obtained from oceanographic data collected 

 at Ocean Weather Station "P" in the Gulf 

 of Alaska. Tabata (1964: Table 8) lists the 

 monthly mean value and the standard deviation 

 of the temperature at 10-m depth based on data 

 obtained twice daily, data obtained every second, 

 third, fourth, fifth, sixth, and seventh day of 

 July 1959 and May 1961. For July 1959 the 

 mean temperatures range from 10.70° to 10.81° 

 C and the standard deviations range from 0.60° 

 to 0.76° C. For May 1961 the mean temper- 

 atures range from 5.84° to 5.90° C and the 

 standard deviations range from 0.39° to 0.46° C. 



In May 1961 Koko Head temperatures and 

 salinities were sampled on 25 days. The mean 

 of all temperature observations was 24.67° C 

 with standard deviation 0.27° C. The mean of 

 temperatures taken every fifth day was 24.58° 

 C with standard deviation 0.39° C. The mean 

 of all salinity observations was 34.759%o with 

 standard deviation 0.051%r. The mean of 

 salinities taken every fifth day was 34.772%o 

 with standard deviation 0.058%^. The temper- 

 ature results from Koko Head are comparable 

 to those from Ocean Weather Station "P" in 

 that mean values and standard deviations based 

 on diff'erent sampling frequencies fall within ap- 

 proximately the same range. The standard er- 

 rors of estimate for the May 1961 Koko Head 

 temperatures and salinities, based on the har- 

 monic functions with resolution of 1 month, are 

 lower than the standard deviations, namely, 

 0.25° C and 0.02T/,c, respectively. The stand- 

 ard errors of estimate as well as the standard 

 deviations do not change significantly when the 

 sampling frequency is increased above the re- 

 quired minimum to attain a desired resolution 

 by harmonic analysis. 



Increasing the sampling frequency does, how- 

 ever, improve the confidence limits of a mean 

 value or the expected value of a harmonic func- 

 tion. A good measure of the confidence limits 

 of a mean value is the standard error of the 

 mean (the standard deviation divided by the 

 square root of the number of samples) . Return- 



189 



