Geology Reference
In-Depth Information
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Time sequence and spectral analysis
New discoveries in Earth dynamics can only be made through the comparison of
theory with observations. Often we are looking for signals close to or below the
noise level; otherwise, they would have already been observed. Thus, the analysis
of observations in both time and frequency domains is of crucial importance.
For several decades now, observations in the time domain have been represented
by discrete samples. The samples may be equally spaced along the time axis or
unequally spaced. Unequally spaced samples may result from inherent properties
of the measurement technique, or from fundamental restrictions such as the vis-
ibility of sources at particular times. Unequally spaced samples may also be the
result of digitiser failure or other instrument problems, leaving gaps in otherwise
equally spaced time sequences. We include the analysis of unequally spaced time
sequences and the application of singular value decomposition to their study. Most
sequences of interest were originally continuous physical signals. Thus, we exam-
ine the e
ects of the sampling process itself on the results of the analysis.
Often observations are made at several locations and it is desired to bring out
common features of the records from di
ff
erent observatories. For this purpose, we
describe in detail, the product spectrum . This may be regarded as a kind of general-
isation of the cross spectrum between two records. As is the case in any spectral
analysis, the estimation of confidence intervals is of prime importance in establish-
ing the significance of the results. We establish methods of estimating confidence
intervals both for the product spectrum and for conventional spectral estimates.
All real observational records are of finite length. It is therefore important to
consider the e
ff
ects of finite record length on any analysis. Often a number of seg-
ments of the record are used to estimate spectral densities, reducing the variance of
the estimate by averaging over estimates on the individual segments. To make more
e
ff
cient use of available data, we employ the Welch overlapping segment analysis
(WOSA), and obtain an asymptotic formula for the variance inflation arising from
the fact that the overlapping segments are not statistically independent.
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