Polarization analysis and polarization filtering of three-component signals with the time-frequency S transform

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.