The S transform, which is introduced in this correspondence, is an extension of the ideas of the continuous wavelet transform (CWT), and is based on a moving and scalable localizing Gaussian window. It is shown here to have some desirable characteristics that are absent in the continuous wavelet transform. The S transform is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages of the S transform are due to the fact that the modulating sinusoids are £xed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates.

The S-transform is an invertible time-frequency spectral localization technique which combines elements of wavelet transforms and short-time Fourier transforms. In previous usage, the frequency dependence of the analyzing window of the S-transform has been through horizontal and vertical dilations of a basic functional form, usually a Gaussian. In this paper, we present a generalized S-transform in which two prescribed functions of frequency control the scale and the shape of the analyzing window, and apply it to determining P-wave arrival time in a noisy seismogram. The S-transform is also used as a time-frequency filter; this helps in determining the sign of the P arrival.

Steketee's Elasticity Theory of Dislocations is generalized to real Earth models. Taken into account are; (i) self-gravitation, (ii) radial variation of elastic properties, density and gravity, (iii) initial hydrostatic stress, (iv) the presence of the liquid core. Volterra's formula for the displacement field is found to hold in the more general circumstances for slip faults.The dilemma, previously pointed out by Jeffreys and Vicente, which arises when the Adams and Williamson condition is assumed not to hold everywhere perfectly in the core, is resolved. This result also bears on the theory of Earth tides and tidal loading.Changes in the inertia tensor are shown to arise only from spheroidal displacement fields of degree zero and two. These fields have virtually no attenuation with distance from the fault. In the one example in which a direct comparison can be made, the present theory gives a factor of 7.5 increase over a mapped half-space theory and a factor of 2.9 increase over the result for a uniform, spherical Earth, in the contribution to secular polar shift and excitation of Chandler wobble. Calculated and observed levels appear now to be in agreement.

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