9.5 Spectral and Statistical Analysis of Time Series
329
The de-trended time series (n now becomes:
K
(n(nAt) = (n(nAt) - L bk(nAt)k.
(9.49)
k=O
Filtering of data prior to detailed analysis is desirable for various reasons.
If we are particularly interested in wind-induced waves, the swell component
should be filtered out. On the other hand, if we concentrate only on the
most energetic part of the wave spectrum, the high frequency components with
negligible energy should be removed by filtering.
Digital filtering can be performed in either the time domain or the frequency
domain. A detailed description of those filtering procedures can be found elsewhere, so they are not repeated here (Otnes and Enochson, 1972).
9.5.3 Calculation of Frequency Spectra
Spectral analysis is used to determine the partition of the variance of a time
series as a function of frequency. For stochastic wind wave time series, contributions from the different frequency components are expressed in terms of
the frequency spectral density. In practice, the term spectrum is applied to all
spectral functions such as autospectrum for one time-series, or cross-spectrum
for the two time-series. Frequency spectra are usually estimated by either of
two methods. The first is based on the Wiener-Khinchine theorem and is called
the Blackman-Tukey procedure. The Wiener-Khinchine relations link variance
functions in the time domain to those in the frequency domain (Massel, 1996a;
Emery and Thomson, 1997). In the second method, called the Cooley-Tukey
method, the direct Fast Fourier Transformation is used (Massel, 1996a). In the
following, we briefly discuss both methods.
Blackman-Tukey Method. This procedure requires calculations in the following steps:
1. Subtracting the mean value from digital data {(n}, trend removal and
filtering (if necessary).
2. Calculating the autocorrelation function:
1 N-r"
K(rAt) = - - L (n (n+r, r = 0, 1,2, ... , m,
N - r n=l
(9.50)
where r is called the lag number and m is the maximum lag number
(m < N). Selection of the m value, which provides the optimum estimate
for the autocorrelation function, will be discussed later. A finite value of
m implies that surface elevations ((t), at times t > mAt are uncorrelated.
A typical normalized autocorrelation function, K (T ) /0'2, for surface waves
is shown in Fig. 9.5a.
329
The de-trended time series (n now becomes:
K
(n(nAt) = (n(nAt) - L bk(nAt)k.
(9.49)
k=O
Filtering of data prior to detailed analysis is desirable for various reasons.
If we are particularly interested in wind-induced waves, the swell component
should be filtered out. On the other hand, if we concentrate only on the
most energetic part of the wave spectrum, the high frequency components with
negligible energy should be removed by filtering.
Digital filtering can be performed in either the time domain or the frequency
domain. A detailed description of those filtering procedures can be found elsewhere, so they are not repeated here (Otnes and Enochson, 1972).
9.5.3 Calculation of Frequency Spectra
Spectral analysis is used to determine the partition of the variance of a time
series as a function of frequency. For stochastic wind wave time series, contributions from the different frequency components are expressed in terms of
the frequency spectral density. In practice, the term spectrum is applied to all
spectral functions such as autospectrum for one time-series, or cross-spectrum
for the two time-series. Frequency spectra are usually estimated by either of
two methods. The first is based on the Wiener-Khinchine theorem and is called
the Blackman-Tukey procedure. The Wiener-Khinchine relations link variance
functions in the time domain to those in the frequency domain (Massel, 1996a;
Emery and Thomson, 1997). In the second method, called the Cooley-Tukey
method, the direct Fast Fourier Transformation is used (Massel, 1996a). In the
following, we briefly discuss both methods.
Blackman-Tukey Method. This procedure requires calculations in the following steps:
1. Subtracting the mean value from digital data {(n}, trend removal and
filtering (if necessary).
2. Calculating the autocorrelation function:
1 N-r"
K(rAt) = - - L (n (n+r, r = 0, 1,2, ... , m,
N - r n=l
(9.50)
where r is called the lag number and m is the maximum lag number
(m < N). Selection of the m value, which provides the optimum estimate
for the autocorrelation function, will be discussed later. A finite value of
m implies that surface elevations ((t), at times t > mAt are uncorrelated.
A typical normalized autocorrelation function, K (T ) /0'2, for surface waves
is shown in Fig. 9.5a.