Geoscience Reference
In-Depth Information
constant amount. Scargle (1982) showed that this particular choice of the
of set ˄ has the consequence that the solution for
P
x
(ˉ) is identical to a least-
squares i t of sine and cosine functions to the data series
y
(
t
):
h e least-squares i t of harmonic functions to data series in conjunction
with spectral analysis had previously been investigated by Lomb (1976), and
hence the method is called the normalized Lomb-Scargle Fourier transform.
h e term
normalized
refers to the factor
s
2
in the dominator of the equation
for the periodogram.
Scargle (1982) has shown that the Lomb-Scargle periodogram has an
exponential probability distribution with a mean equal to one, assuming that
the noise is Gaussian distributed. h e probability that
P
x
(ˉ) will be between
some positive quantity
z
and
z
+
dz
is exp(-
z
)
dz
. If we scan
M
independent
frequencies, the probability of none of them having a value larger than
z
is
(1-exp(-
z
))
M
. We can therefore compute the false-alarm probability of the
null hypothesis (i.e., the probability that a given peak in the periodogram is
not signii cant) using
Press et al. (1992) suggested using the Nyquist criterion (Section 5.2) to
determine the number of independent frequencies
M
, assuming that the
data were evenly spaced. In this case, the appropriate value for the number of
independent frequencies is
M
=2
N
, where
N
is the length of the time series.
More detailed discussions of the Lomb-Scargle method are given in Scargle
(1989) and Press et al. (1992). An excellent summary of the method and a
TURBO PASCAL program to compute the normalized Lomb-Scargle power
spectrum of paleoclimatic data have been published by Schulz and Stattegger
(1998). A convenient MATLAB algorithm
lombscargle
for computing the
Lomb-Scargle periodogram has been published by Brett Shoelson (h e
MathWorks, Inc.) and can be downloaded from
File Exchange
at
http://www.mathworks.de/matlabcentral/fileexchange/993-lombscargle-m
h e following MATLAB code is based on the original FORTRAN code
published by Scargle (1989). Signii cance testing uses the methods proposed
by Press et al. (1992) explained above.
We i rst load the synthetic data that were generated to illustrate the use
of the evolutionary or windowed power spectrum method in Section 5.6.
