Graphics Programs Reference
In-Depth Information
Such trends are common features in earth sciences. As an example, consider
the glacial-interglacial cycles observed in marine oxygen isotope records
overlain by a long-term cooling trend during the last six million years.
yt = y + 0.005 * t;
plot(t,y,'b-',t,yt,'r-'), axis([0 200 -4 4])
In reality, more complex trends exist, such as higher-order trends or trends
characterized by changing slopes. In practice, it is recommended to elimi-
nate such a trend by fi tting polynomials to the data and to subtract the the
corresponding values. This synthetic time series now contains many charac-
teristics of a typical data set in the earth sciences. It can be used to illustrate
the use of spectral analysis tools that are introduced in the next chapter.
5.3 Autospectral Analysis
Autospectral analysis
aims to describe the distribution of variance contained
in one single signal
x
(
t
) over frequency or wavelength. A simple way to
describe the variance in a signal over a time lag
k
is the autocovariance. An
unbiased estimator of the autocovariance
cov
xx
of the signal
x
(
t
) with
N
data
points sampled at constant time intervals ¨
t
is
The autocovariance series clearly depends on the amplitude of
x
(
t
).
Normalizing the covariance by the variance
σ
2
of
x
(
t
) yields the autocor-
relation sequence. Autocorrelation involves correlating a series of data with
itself, depending on a time lag
k
.
The most popular method to compute power spectra in earth sciences is the
method introduced by Blackman and Tukey (1958). The
Blackman-Tukey
method
estimates the
power-spectral density
by calculating the complex
Fourier transform
X
(
f
) of the autocorrelation sequence
corr
xx
(
k
).
