Geoscience Reference
In-Depth Information
Signal processing methods are ot en applied to remove a major part of
the noise, although many i ltering methods make arbitrary assumptions
concerning the signal-to-noise ratio. Moreover, i ltering introduces artifacts
and statistical dependencies into the data, which may have a profound
inl uence on the resulting power spectra.
Finally, we introduce a linear long-term trend to the data by adding a
straight line with a slope of 0.005 and an intercept with the y -axis of zero (Fig.
5.3 c). Such trends are common in earth sciences. As an example, consider
the glacial-interglacial cycles observed in marine oxygen isotope records,
overprinted on a long-term cooling trend over the last six million years.
xt = x + 0.005*t;
plot(t,x,'b-',t,xt,'r-'), axis([0 200 -4 4])
In reality, more complex trends exist, such as higher-order trends or trends
characterized by variations in gradient. In practice, it is recommended that
such trends be eliminated by i tting polynomials to the data and subtracting
the corresponding values. Our synthetic time series now contains many
characteristics of a typical earth science data set. It can be used to illustrate
the use of the spectral analysis tools that are introduced in the next section.
Audio
5.2
5.3 Auto-Spectral and Cross-Spectral Analysis
Auto-spectral analysis aims to describe the distribution of variance contained
in a single signal x ( t ) as a function of frequency or wavelength. A simple
way to describe the variance in a signal over a time lag k is by means of
the autocovariance. An unbiased estimator of the autocovariance cov xx of the
signal x ( t ) with N data points sampled at constant time intervals ʔ t is
h e autocovariance series clearly depends on the amplitude of x ( t ).
Normalizing the covariance by the variance ˃ 2 of x ( t ) yields the autocorrelation
sequence. Autocorrelation involves correlating a series of data with itself as
a function of a time lag k .
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