Improved resolution of complex eigenfrequencies in analytically continued seismic spectra

Abstract

The response of the Earth to an earthquake is a transient that is effectively zero several days after the event. A recording of the event, of finite duration in time, has a Fourier spectrum that is an entire, or integral, analytic function of frequency. We present a very simple procedure for computing the Fourier spectrum as a function of complex frequency; the analytically continued spectrum. By investigating the properties of the analytically continued spectrum we show how to extract high-Q modes, how to estimate Q either from the amplitude or from the width of a resonance function, and how to improve the resolution of splitting to the theoretical maximum. Examples of these procedures, using observed data, are presented.