FISHERY BULLETIN: VOL. 69. NO. I 



the surface water in March above that in Feb- 

 ruary. This concept was used in studies of the 

 Hawaiian oceanographic climate (Seckel, 1962, 

 1969) and has been ajipHed to Hawaiian fishery 

 problems (Seckel and Waldron, I960; Seckel, 

 1963). 



Rigorously, the theory of distribution of pro- 

 perties in the sea states that the change of sea- 

 surface temperature during a time interval, say 

 from the first day of one month to the first day 

 of the next month, is equal to the integral of 

 all meteorological and oceanographic processes 

 affecting the temperature during the time in- 

 terval: 1^ 



8 h — S II = f (all processes) J/. 



e I, is the temperature at the beginning and di, 

 is the temperature at the end of the interval. 

 In application, the choice of e „ and 0/, presents 

 the following problems: The difference in the 

 observed temperatures at times a and b also 

 reflects the effect of short-term variability 

 ("noise") that is not of interest in monitoring 

 the large-scale events. If one uses monthly 

 mean temperatures in the heat budget equation 

 that include observations made 15 days before 

 and after times a and b, then the change of 

 temperature incorporates the effect of processes 

 that lie outside the interval of interest. Al- 

 though mean values usually provide an adequate 

 measure of the temperature change during given 

 time intervals, the true change of temperature 

 can be obscured. One can overcome the problems 

 caused by the two unsatisfactory methods of 

 obtaining measures of the temjierature change 

 by finding suitable functions that filter out un- 

 desirable short-term variability without obscur- 

 ing the basic temperature and salinity trends. 



Techniques that can be used in the smoothing 

 of time series data have been reviewed by Hol- 

 loway (1958) and usually involve moving aver- 

 ages of the data to which weighting factors have 

 been assigned. 



Curve fitting provides another method of aji- 

 proach. A useful technique that has been used 

 in this report, is to obtain an analytic expression 

 for the temperature and salinity as a function 

 of time by Fourier analysis. The Fourier series 

 is efficiently, and therefore inexpensively, de- 



rived by computer. Efficiency is furthered in 

 that graphs can be produced by automatic plot- 

 ter. The Fourier series provides a least-squares 

 fit of the observed values. It permits filtering 

 of undesired variability, facilitates statistical 

 evaluation of the data, and — within limits — pro- 

 vides insight into the properties of the distri- 

 bution. 



These advantages will become apparent in the 

 following sections of this report. The results 

 of the analyses for each year of observation 

 are presented in the appendi.x in both tabular 

 and graphical form. 



THE FOURIER METHOD 



Fourier series are well known, widely applied, 

 and adequately described in texts of advanced 

 calculus. A good description can be found in 

 Sokolnikoff (1939) where the derivation of the 

 Fourier coefficients by least-squares method is 

 also presented. 



The temperature or salinity is expressed as 

 a function of time, t, in the Fourier series: 



.S„ (/) = h L (/l„cos/;a)' + B,is\nnu>i), 



where w 



2tt 



1,2,3, 



, and T is the fundamental 



period. For example, if harmonic analysis is 

 to be performed on data collected for a dura- 

 tion of 1 year, T would be 365 days. 



The Fourier series contains the coefficients 

 .4(1, A„, and B„ that are given by the Fourier 

 integrals 

 2 



An = 



and 



«,, = 



i=T 





T ^0 



F{!)cos{nui)ili. n 



0.1,2, 



F(thin{nui)dr. n = 1,2.3, 



The coefficient Aq is the special case of A^i with 

 n — 0. In our application F(i) is the temper- 

 ature or salinity at the time t. Of course, the 

 functional relationship between temperature and 

 time or salinity and time is not known so that 



182 



