Article,
Polarization analysis and polarization filtering of three-component signals with the time-frequency S transform
Geophysical Journal International, 165 (2): 596--606 (May 2006)
DOI: 10.1111/j.1365-246X.2006.02937.x
Abstract
From basic Fourier theory, a one-component signal can be expressed
as a superposition of sinusoidal oscillations in time, with the Fourier
amplitude and phase spectra describing the contribution of each sinusoid
to the total signal. By extension, three-component signals can be
thought of as superpositions of sinusoids oscillating in the x-,
y-, and z-directions, which, when considered one frequency at a time,
trace out elliptical motion in three-space. Thus the total three-component
signal can be thought of as a superposition of ellipses. The information
contained in the Fourier spectra of the x-, y-, and z-components
of the signal can then be re-expressed as Fourier spectra of the
elements of these ellipses, namely: the lengths of their semi-major
and semi-minor axes, the strike and dip of each ellipse plane, the
pitch of the major axis, and the phase of the particle motion at
each frequency. The same type of reasoning can be used with windowed
Fourier transforms (such as the S transform), to give time-varying
spectra of the elliptical elements. These can be used to design signal-adaptive
polarization filters that reject signal components with specific
polarization properties. Filters of this type are not restricted
to reducing the whole amplitude of any particular ellipse; for example,
the 'linear' part of the ellipse can be retained while the 'circular'
part is rejected. This paper describes the mathematics behind this
technique, and presents three examples: an earthquake seismogram
that is first separated into linear and circular parts, and is later
filtered specifically to remove the Rayleigh wave; and two shot gathers,
to which similar Rayleigh-wave filters have been applied on a trace-by-trace
basis.
