For example, when the sample autocorrelation function goes to zero by 

 the third lag, the ratio of N/2 or N/3 can be used as the number of indepen- 

 dent observations n. The function R(k) is calculated under the assumption 

 that all the elements in the-sample were selected at equally spaced time in- 

 tervals, but some of the scheduled BOMEX rawinsondes were aborted, and the 

 sample data used in this study are therefore not uniformly spaced in time. 

 Because the author was unable to find a discussion in the literature dealing 

 with autocorrelations and missing data, an experiment was designed to test 

 whether the missing data would materially affect the autocorrelation function. 

 First a series of random numbers were generated, and then a new series with a 

 known persistence was developed by using a three-term running sum as follows: 



X(N) = y(N) + y(N+l) + y(N+2), N = 1,...,101, 



where y is the generated random series. The autocorrelation function of X 

 was then calculated. To simulate the missing data, the uniform random number 

 generator was used to generate a random number R^ . When this number was less 

 than 0.204, the corresponding number X- was eliminated from X. In this manner 

 18 observations were removed, and the new X' series contained values that were 

 no longer uniformly spaced. Values of z' equal to zero were then used to re- 

 place the deleted data. The autocorrelation function for X' was calculated 

 by use of equation (1) and the l/(n-k) term was reduced by one each time one 

 of the missing observations appeared in the cross product of ( (zj^) (zj^+j^) ) . 

 Figure 7 shows that the lag at which R(k) goes to zero is not materially 

 changed when 20 percent of the data are missing. Chiu (1960) indicates that 

 the shape of the spectra of temperatures and winds at a point are not changed 

 materially when 10 percent of upper air data are replaced with a "best guess" 

 by a researcher working from an analyzed chart. 



The null hypothesis was tested in all cases at the a = 0.05 level of 

 significance. As these were checks of equality, the two-tailed tests were 

 used. Tabled values for the t and F distributions given by Yamane (1967) were 

 applied in calculating critical values. 



When the null hypothesis is not rejected by the F test, i.e., the samples 

 are from populations with similar variances, the null hypothesis that the 

 sample means of the two samples are equally good estimates of the same popu- 

 lation mean (or mean of the environment) can be tested by use of the Student's 

 t distribution: 



t = 



(xi - Ui) - (X2 - y2) -, /nin2 Cni+n2-2) 



1/n^- + n2S2 



2 2 / ni + no 



with nj^ + n2 - 2 representing effective degrees of freedom. These n's are the 

 adjusted lag coefficient values obtained from N]^/k and N2/k. Since, according 

 to the null hypothesis, yj = U2> ^he equation becomes 



1^1X12 (n]^+n2-2) 



XI - ^2 



t = , / ^ ., 1/ ni + n^ 



+ n2S2- ^ 



V nisi^ 



