FREQUENCY RESPONSE OF A MARINE ECOSYSTEM 27 



Spectral-Analysis Methods 



Spectral analysis may be considered a statistical method for 

 partitioning the variance of a time series of data among a range of 

 frequencies of oscillation. Spectral analysis performs a function 

 analogous to that of a prism breaking a beam of light into its 

 component fractions at various wavelengths (colors) (see Fig. 3). For 

 the interested reader without a strong mathematical background, we 

 suggest several relatively nontechnical papers on spectral analysis 

 (e.g., Gunnerson, 1966; Wastler, 1969; Piatt and Denman, 1975). In 

 our presentation here we minimize the mathematics underlying 

 spectral analysis. Unless otherwise noted, all techniques, including 

 equations incorporated into FORTRAN IV computer programs 

 (Weisberg, 1974), are taken directly from Bendat and Piersol (1971). 



Basically, spectral analysis takes time-series data sampled at 

 regular intervals from a repeating periodic process, filters it to 

 remove unwanted contaminating signals, removes aiiy linear trend, 

 tapers the two ends of the data smoothly to zero (to avoid placing a 

 spurious artifact in the frequency spectrum), and computes the 

 coefficients of a discrete Fourier series with one of several available 

 numerical algorithms (see Piatt and Denman, 1975). We used a Fast 

 Fourier Transform (FFT), which yields a series of Fourier coef- 

 ficients equal in length to one-half the number of data points. These 

 coefficients yield the variance spectrum for the time series directly. 



A confidence interval about the height of each spectral estimate 

 can be calculated by a moving average of several adjacent spectral 

 estimates. However, decreasing the width of the confidence interval 

 necessitates broadening the frequency bandwidth resolution of the 

 analysis. Thus, typically, an investigator must decide between 

 estimating the heights of a few spectral peaks with great confidence 

 or estimating more peak heights with a corresponding decrease in 

 confidence. 



Once calculated in this manner, spectral estimates for two time 

 series connected by a causal relationship can be analyzed using 

 cross-spectral techniques. A detailed derivation is beyond the scope 

 of this paper, but several resulting parameters are of great interest. 

 Coherence squared (7^ ) can be considered as a correlation coefficient 

 between the two time series, varying from 0.0 to 1.0; i.e., as an 

 estimate at each frequency of the proportion of the variance in one 

 time series which can be explained in terms of variance in the other. 

 When a significance test of the squared coherence between two time 

 series can be shown to be statistically different from zero over a 

 range of frequencies (Amos and Koopmans, 1963), a cause— effect or 

 input— response relationship exists between the two variables at the 



