Geoscience Reference
In-Depth Information
If the time interval between any two successive observations
x
(
t
k
) and
x
(
t
k+1
)
is constant, the time series is said to be equally spaced and the sampling
interval is
h e sampling frequency
f
s
is the inverse of the sampling interval ʔ
t
. We
generally try to sample at regular time intervals or constant sampling
frequencies, but in many earth science examples this is not possible. As an
example, imagine deep-sea sediments sampled at i ve-centimeter intervals
along a sediment core. Radiometric age determinations at certain levels in
the sediment core revealed signii cant l uctuations in the sedimentation
rates. Despite the samples being evenly spaced along the sediment core they
are not equally spaced on the time axis. Here, the quantity
where
T
is the full length of the time series and
N
is the number of data
points, represents only an average sampling interval. In general, a time series
x
(
t
k
) can be represented as the linear sum of a periodic component
x
p
(
t
k
), a
long-term component or trend
x
tr
(
t
k
), and random noise
x
n
(
t
k
).
h e long-term component is a linear or higher-degree trend that can be
extracted by i tting a polynomial of a certain degree and subtracting the
values of this polynomial from the data (see Chapter 4). Noise removal will
be described in Chapter 6. h e periodic - or cyclic in a mathematically less
rigorous sense - component can be approximated by a linear combination of
sine (or cosine) waves that have dif erent amplitudes
A
i
, frequencies
f
i
, and
phase angles ˈ
i
.
h e phase angle ˈ helps to detect temporal shit s between signals of the same
frequency. Two signals
x
and
y
with the same period are out of phase unless
the dif erence between ˈ
x
and ˈ
y
is equal to zero (Fig. 5.2).
h e frequency
f
of a periodic signal is the inverse of the period ˄. h e
Nyquist frequency
f
nyq
is half the sampling frequency
f
s
and represents the
maximum frequency the data can produce. As an example audio compact
