FISHKRY BLXLETIN: VOL. 69. NO. 1 



a minimum of four or, better, six samples is 

 required to achieve good resolution. For ex- 

 ample, sea-surface temperatures are to be mon- 

 itored and the fundamental period of observa- 

 tions is to be 12 months. Resolution of a 1- 

 month cycle (?i = 12) , requires four samples per 

 month, or sampling once per week. 



APPLICATION OF THE 

 FOURIER METHOD 



In practice, the Fourier method described 

 above must be adapted to each specific applica- 

 tion. In addition to the minimum number of 

 samples necessary in order to attain a desired 

 resolution another restriction applies to vari- 

 ations in the sampling interval. Although the 

 computer program used to obtain the results of 

 this paper allows a varying sampling interval, 

 thus accepting a sequence with missing obser- 

 vations, the sampling interval can be allowed 

 to vary only within limits. For example, at 

 least four samples per month are necessary to 

 resolve a monthly cycle. This cycle will, how- 

 ever, not be resolved if the samples are taken 

 on four consecutive days, rather than being 

 evenly distributed throughout the month. It is 

 also possible to aid the hai-monic analysis in 

 rapid convergence to its best fit with the ob- 

 served values by adjusting the fundamental 

 period of analysis and by performing some pre- 

 liminary operations which are described below. 



APPLICATION TO KOKO HEAD SEA- 

 SURFACE TEMPERATURES AND SALINITIES 



The sampling station is located near Koko 

 Head at the exposed, eastern shore of Oahu so 

 that the sea-surface temperatures and salinities 

 measured there reflect open-ocean conditions. 

 The salinities appear to be affected by runofl' 

 only on rare occasions of heavy rainfall. Both 

 the tempei'atures and salinities are based on 

 bucket samples. The salinity is determined in 

 the Hawaii Area Fishery Research Center, 

 Honolulu. 



Before 1961 samples were collected at weekly 

 intervals and subsequently twice weekly, usually 

 on Tuesday and Friday mornings. Occasionally 



sampling has been missed. The computer pro- 

 gram must therefore accept data with an ir- 

 regular sampling interval. 



The basic period for analysis has been chosen 

 to be 1 year. Harmonic analysis began with the 

 first sample and ended with the last sample of 

 the year. The sampling time, in days and 

 months, was converted to days of the year be- 

 ginning with the first of the year. 



Owing to a longer term trend, the value of a 

 property at the beginning is not necessarily the 

 same as at the end of an annual cycle. In the 

 case of Koko Head salinities and Christmas 

 Island temperatures, it will be seen later that 

 an annual cycle is, in fact, not always apparent. 

 The noncyclic trend during the analysis period 

 can be obtained by linear approximation. Rapid 

 convergence to the best fitting function can then 

 be achieved by performing the harmonic anal- 

 ysis on the residuals of the observed values from 

 a linear fit. 



In our application the first observed value, 

 F(to), and the last observed value, F(ti), for 

 the period were used to obtain the linear equation 



S' = F(?o) + bi 



FUi) - FUo) 

 where b = • 



'/ - '0 



The residuals, /?„, = F(t,n) — [F(to) + bt^}, 



m — 0.1,2, ... /, were used to obtain the Fourier 



coefficients. The Koko Head temperatures and 



salinities for each year are then expressed by 



the function 



k 

 S = K + bt + ^ C„ co<iu(ni — a„) 



« = 1 



where K = FUq) + — and w = —  

 2 T 



The phase angles and coeflicients for each of 

 the years 19.56-69 of the sea-surface tempera- 

 tures are listed in appendix A Table 1, and of 

 the sea-surface salinity are listed in appendix 

 A Table 2. 



The functions for each year together with the 

 observed values of the sea-surface temperature 

 and salinity have been drawn by automatic plot- 

 ter and are presented in appendix B. 



184 



