Spectral analysis

The statistical technique used to find cyclic components in a time series is known as spectral analysis (Jenkins & Watts 1968; Yevjevich 1972; Bras & Rodriguez-Iturbe 1985). The signal component represents the structured part of the time series, made up of a small number of embedded

Fig. 3. Representative time fluctuations of piezometric levels in the aquifer showing an increase of the irregular data (a-d). Levels are in metres above sea level (masl).

periodicities or cycles repeated over a long time. The noise is a random component; it may be white noise, but more often will be red noise. A time series can be represented by a finite number of measurements. In the present case, a piezometric time series is represented by a succession of water-head data measured at regular or irregular time intervals. When a cycling component is added to another cyclic component of a longer period than the length of the time series, it will give an apparent trend. This, together with possible real trends and other factors, gives rise to noise in the low frequencies, known as red noise.

Harmonic analysis is another name used to denote the estimation of cyclic components in the time series. The time series is supposed to be a linear combination of sinusoidal functions of known periods but of unknown amplitude and phase. The modulus of the amplitude is related to the variance of the time series, explained by the oscillation at each frequency. The representation of the square of the modulus versus frequency is known as the power spectrum. There are a number of methods that can be used to infer the power spectrum: the periodogram (Papoulis 1984), the Blackman-Tukey approach (Blackman & Tukey 1958), maximum entropy (Burg 1972), and Thomson multitaper (Thomson 1982), among others. Each methodology has advantages and disadvantages, for which reason a good strategy is to use various methods and compare the results. This was done with the time series of the piezometric head; nevertheless we can affirm that the indirect method of Blackman-Tukey is a robust approach which gives acceptable results with our data sets. As an example of this comparative analysis, Figure 4 shows the results obtained with the methods of (a) maximum entropy; (b) Thomson multitaper; and (c) Blackman-Tukey. The results obtained were similar, yet we consider the Black-man-Tukey approach to be more robust and therefore more adequate for the analysis of time series.

The power spectrum is calculated from the covariance function by:

Ï FREQUENCY

FREQUENCY

Fig. 4. Power spectra of piezometric data represented in Figure 3c using (a) maximum entropy (b) Thomson multitaper method and (c) Blackman-Tukey methods. In (c) dashed lines represent, starting from the bottom, the underlying power spectrum and the 90%, 95% and 99% confidence levels.

FREQUENCY

Fig. 4. Power spectra of piezometric data represented in Figure 3c using (a) maximum entropy (b) Thomson multitaper method and (c) Blackman-Tukey methods. In (c) dashed lines represent, starting from the bottom, the underlying power spectrum and the 90%, 95% and 99% confidence levels.

where S(v) is estimated power spectrum for frequency v, C(k) is estimated covariance function for the kth lag and l(k) is weighting function, known as a lag-window, which is used to give less weight to the covariance estimates as the lag increases. For large lags, the estimated covariance function is less reliable. The lag-window used was the Tukey window:

where M is maximum number of lags for the covariance function used in the spectral estimation. The maximum number of lags is N — l, with N being the number of experimental data; however, with large values of M a great number of peaks will be seen in the estimated power spectrum, most representing spurious cycles. On the other hand, if M is very small, significant cycles would not be seen in the estimated power spectrum. For this reason we used a number of M = N/2 in order to resolve peaks, and a value of M = N/4 in order to find out which are the most significant peaks.

In addition to using a small value for N, confidence levels were estimated for the inferred power spectrum. Our approach consists of fitting a background power spectrum with no cyclic component, but rather a smooth continuous spectrum, which is done by fitting the spectrum of an autoregressive process of order one, i.e. AR(l). The parameter of this process is estimated from the experimental data. Then, we take into account the known result for the one-side confidence band of power spectrum estimator:

where P is probability operator, S(v) is power spectrum estimate for frequency v, S(v) is underlying power spectrum for frequency v, v is number of degrees of freedom - for the Blackman-Tukey estimate with a Tukey lag-window, the number of degrees of freedom is 2.67N/M, xVa is the a quan-tile of a chi-squared distribution with v degrees of freedom and a is significance level. For this study, we established confidence levels of 90%, 95% and 99%.

0 0

Post a comment