Symmetrically normed residuals (SNR) are then defined as: 



SNR 



^ ' ^+,m ' if Rm > 



•^ / <^-,m ' if Rm < 



(38) 



Thus, the symmetrically normed residuals are ratios of the residuals to 

 the rms positive or negative residuals, as the case may be. 



Originally, the analysis was made with rms residual, disregarding 

 whether the residuals were positive or negative. This, however, led to 

 ridiculous results particularly in the subsequent use in simulation. The 

 use of different norming divisors for negative and positive residuals 

 avoided these peculiarities completely. 



The residuals and the corresponding SNR's are plotted in Figures 13 

 through 24. The symmetrical normalization seems to very adequately pro- 

 duce a uniform cloud of points. There does not seem to be any tendency 

 for the SNR's to be systematically large or small at any particular 

 frequency. Generally about 60 percent of the SNR's fall below zero. 



As a check against sequential dependence among the SNR's, neighboring 

 pairs of SNR's were plotted on a scatter diagram in two-dimensional space. 

 The first member of the pair of SNR's was the x-coordinate while the 

 second member was the y-coordinate for the plotted point. Any tendency fo 

 big values to follow big values (or the reverse) would show up as a clus- 

 tering tendency on such a plot. Complete independence, on the other hand, 

 would show up as a uniform cloud of points. 



The plots of this type for the 12 Hurricane Carla data sets are given 

 in Figures 25 through 36. The point scatter is really quite uniform for 

 all of the records with no obvious dependencies showing up. However, it 

 should be emphasized that this type of examination only reveals overall 

 average dependencies. There may be dependencies between neighboring SNR's 

 at certain frequencies which are counteracted by opposing dependencies 

 at other frequencies. However, the earlier graphs (Figs. 13 through 24) 

 would show any strong dependencies tied to frequencies if they were 

 present . 



Everything considered, the analyses strongly support the conclusion 

 that the spectral fluctuations have been successfully decomposed into a 

 smoothed spectrum plus a constant (either a+ or a_) times independent 

 random noise. The smoothed spectrum and the constants are frequency 

 dependent. The noise apparently does not depend on frequency. In symbols, 

 this decomposition can be written: 



P(fm) = B(fm) + c • (SNR) , (39) 



25 



