With some caution, this estimate is used for each location. The 95 percent 

 significance levels are displayed as the dashed lines in figures 3, 4, and 

 5. The limits are adjusted for the particular series length involved and diverge 

 with increasing lag, since the corresponding sample size decreases. The many 

 oscillations in the autocorrelation functions beyond 100 to 200 days' lag are not 

 significant for the lower curves of figures 3, 4 and 5. The oscillations are 

 merely characteristics of the particular samples of time-series available, and it is 

 useless to subject them to any additional correlation or spectral analysis. 



Of interest is the comparison of the above standard deviation with that 

 resulting from a sometimes used inequality concerning the true variance of the 

 autocorrelation coefficient.^ This inequality is 



T Jo 



r(r)dr 



where pir) is the true autocorrelation function, and T is the sample length. If the 

 Scripps Pier empirical autocorrelation function is numerically integrated out to a 

 lag of 350 days, and if the function is assumed to be zero beyond that lag, an 

 estimate of the inequality a^^ € 0.004424 is obtained. Correspondingly, 

 CTf. < 0.0665. The value ct„ = 0.0293 easily satisfies the inequality. 



REGRESSION ANALYSIS 



The preceding autocorrelation analysis indicates that the surface-temper- 

 ature time-series contains a prominent oscillatory term with a period of 12 months, 

 and that four of the time-series contain an additional oscillatory term with a 

 period of 6 months. 



A general model containing oscillatory functions is 

 k 

 T'= /3o + N «,- sin [27r((D-0,)/365] + e, or expanding, (1) 



ft 

 = l3o+ y [^2j-i sin (27riD/365) + ^,i cos (277fD/365)] 



where D is time measured in days from some arbitrary origin, and T ' is the fitted 

 value of the surface temperature. Fitting the function of equation (2) to the 

 observed surface temperatures T using the method of least squares yields esti- 

 mates of the regression coefficients /S and an estimate of the variance off. The 



(2) 



18 



