John L. Sabo and David M. Post. 2008. Quantifying periodic, stochastic, and catastrophic environmental variation. Ecological Monographs 78:19–40.

Appendix B. Fourier tutorial.

Estimation of periodic and stochastic sources of variation from time series of daily average discharge measurements

Note: This document contains some information presented in the main body of the manuscript for the purpose of clarity and continuity

We estimated seasonal and inter-annual variability in daily flow data as the amplitude and noise of the hydrograph signal (A_rms and N_rms, respectively) in the frequency domain by transforming normalized and detrended daily time series (d_t) using a Fourier transform:

(B.1)

Here, i is the sqrt(-1), t is the time step (in years, or t = days/365), T is the Nyquist frequency (equivalent to ½ the span of the number of daily observations, n) and D is the discrete Fourier transform of the time series (d) evaluated at the frequency v_k , where

, for k = 0, 1,. . . .T-1

(B.2)

Thus, v_k gives the characteristic frequencies of signals in the time series whose periods are . Prior to transformation we tested all time series for the presence of significant linear (non-periodic) trends via linear regression (of normalized discharge, d_t vs. t) and detrended time series with significant slopes by using the residual values from (d_t vs. t).

The power (i.e., squared amplitude) and phase of each signal were calculated (respectively) as:

	(B.3a)
and
	(B.3b)

where, I and R are the imaginary and real components of D(k) and is expressed in radians.

Extracting seasonal signal

The sample power spectrum, , can be noisy, even for a long time series. To reduce variance and associated problems with aliasing, we tapered the spectra using the hanning function (Kottegoda 1980). We then tested the significance of peaks in the smoothed spectrum at frequencies of 1, 2, 3, 4, 6 and 12 (or periods of 1 year, ad 6, 4, 3, 2 and 1 months). We searched for peaks at these fixed frequencies because they correspond to temporal scales of seasonality. By searching for fixed frequency scales we also significantly reduce the potential for the detection of false signals as a result of aliasing. Significance at each frequency was determined by assuming that a spectral density (power) estimate was a chi-squared random variable with 2.67L degrees of freedom (Kottegoda 1980, Shumway 1988). Here we set the bandwidth at 5 (L = n/M; M = n/5) following (Kottegoda 1980). To control for type-II error inflation associated with multiple significance testing of peaks (6 peaks tested in each power spectrum), we adjusted the significance level using the sequential Bonferroni correction. Specifically, the six peaks were ranked from high to low power and tested with Type-I error rates of = 0.05/rank ~ 0.05, 0.025, 0.0134, 0.0125, 0.01, and 0.0083, respectively. In this way, significance testing provided the characteristic frequencies, (and their associated amplitudes and phases) of the seasonal trend in the data. The frequency, phase, and amplitude of significant spectral densities were then used to calculate the seasonal trend in the daily hydrograph.

Quantification of seasonal and inter-annual variation in daily discharge

We calculated the average seasonal signal as the root mean squared amplitude of the characteristic frequencies:

(B.4)

where f is the number of characteristic frequencies detected (above) and A_k is the amplitude (i.e., ) of the signal at one of these frequencies. We expressed the average inter-annual variation as the root mean squared noise, or RMS amplitude of all non-characteristic frequencies:

.

(B.5)

Summing the power of the characteristic frequencies and leftover noise may provide a more realistic comparison of the relative magnitude of these two sources of variation. We use RMS values here to circumvent problems with comparing values across streams with different amounts of data (variable time series length). Moreover, RMS values for these parameters are more standard in other fields. For example, the signal to noise ratio is calculated using RMS values, and can be expressed on a linear (decibel) scale as:

(B.6)

such that SNR values greater than one indicate strong seasonal relative to inter-annual variation and SNR < 1 indicates high inter-annual relative to seasonal variation in daily discharge. We defined hydrographs with no significant characteristic frequencies as “aseasonal” (i.e., A_rms = 0). In this case, SNR is undefined (log (0) = -). Thus for subsequent analyses using SNR, we used hydrographs only if they have significant seasonal signal(s).

LITERATURE CITED

Kottegoda, N. T. 1980. Stochastic water resources technology. John Wiley and Sons, New York, New York, USA.

Shumway, R. H. 1988. Applied statistical time series analysis. Prentice Hall, Englewood Cliffs, New Jersey, USA.