The Use of Wavelets in Seismics and Geophysics
by Nico Temme
Wavelets are wave patterns of small size that can be used to analyze
rapid changes in a signal, sharp contrasts in an image, and structures
at different scales. The wavelet method, developed in the eighties and
now provided with a sound mathematical basis, has become a powerful tool
in the field of data compression, noise suppression and processing of images
and signals. At CWI wavelets are studied in a project financed by the Technology
Foundation STW which focuses on the application of wavelets in seismology
and geophysics.
A wavelet is a wave-like function with short extension: its graph oscillates
only over a short distance, or damps very fast; its mean value over the
whole domain equals zero. A wavelet is localized both in frequency and
position. From this 'mother wavelet' a whole family of other wavelets is
derived by displacement and scaling. In this family every scale is represented
for every wanted position. Wavelet analysis operates as a microscope: as
a wavelet family contains 'building blocks' of arbitrary small scale, we
may zoom in at any time on any detail of the signal. Wavelets are well
suited to detect the presence of fractal components in observational data.
As we observe images and acoustic signals mainly by contrasts, wavelets
can be very effective in storing and transmitting such images and signals
by data compression. Wavelets are also successfully applied for, eg, removing
noise in old sound recordings and restoring images from MRI and CT scans.
Research on wavelets and its applications in The Netherlands
takes place for instance at the Royal Dutch Meteorological Institute KNMI
(structure of the clouds), Delft University of Technology (soil structures,
for oil exploration), Groningen University (image processing in medicine)
and CWI (analysis of geophysical/seismic signals). CWI's research is focused
on the development and use of directional time-scale analysis methods.
An important research topic is the combination of the Radon transform
(also called X-ray transform - a well-known technique for reconstructing
a 3D object from a number of cross-sections) and the wavelet transform.
A rigid mathematical formulation of this 'Wavelet X-ray Transform', its
basic properties and its discretization have been obtained. A fast algorithm
for this combined transform is also under development. The computer code
has been established in a Matlab environment and the first results are
promising.
The 'Wavelet X-ray Transform' is applied to the filtering of two-dimensional
seismic data sets. These images contain information about the subsurface
of the earth such as the localization of geological interfaces. However,
they also contain irrelevant parts, for example contributions from waves
coming directly from the explosion (that is used to generate the signals)
to the detection point. CWI's wavelet method of filtering away the irrelevant
parts prior to extraction of the relevant parameters is based on the idea
that relevant and disturbing parts in an image can be separated based on
distinction in position (or time), scale (or frequency) and direction.
Secondly, research is conducted jointly with KNMI on the problems of
polarization in seismic signals, where an estimation of the spectrum is
made by preprocessing the signal data. In a seismogram, the waves coming
from the source and arriving at the Earth's surface at different arrival
times appear as so-called phases. These phases, which can be seen as the
ground motion due to the arrival of a particular wave coming from the source,
are all measured during a short time period. A research aim is to find
the time periods in a seismogram, when they appear. Furthermore we want
to analyze these phases accurately.
Analysis of a short segment in a longer record by spectral estimation,
ie, studying the frequency components of the Fourier spectrum, is hampered
by the fact that its spectrum cannot be determined exactly if the data
process is stochastic or a noisy deterministic signal.
A seismogram is a non-stationary data process and hence, instead of
considering the Fourier spectrum, a direct estimate of the time-varying
spectrum is more appropriate. For this we used the wavelet transform. Since
at higher scales the wavelet transform too cannot deal with the uncertainty
introduced by the short segment, we first preprocessed the segment with
a tapering algorithm, which smoothes the data by multiplying the data segments
with some window function, and then took the wavelet transform of the new
data. The first mathematical results of our research contain error and
convergence estimates of the new algorithm. Successful experiments with
synthetic data will be followed by tests with real seismic data in collaboration
with KNMI.
Other recently started research concerns the design of an algorithm
based on the wavelet transform for the detection of time points, where
the several phases appear in the seismogram. This is relevant for, eg,
distinguishing a (nuclear) explosion from an earthquake and locating the
source of the seismic event. Together with the seismology department of
KNMI, physical properties of the seismic signals, especially their scaling
behaviour, are used to refine the algorithm. See also http://www.cwi.nl/cwi/projects/wavelets.html
Please contact:
Nico Temme - CWI
Tel: +31 20 592 4240
E-mail: Nico.Temme@cwi.nl
