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 t o 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.Analytically continued seismic spectra Consider the displacement response of the Earth u(r, t ) to an earthquake (Gilbert & Dziewonski 1975,2.1 .I 1 ;hereafter M).where sk is the displacement of the kth free oscillation (normalized to a total energy of unity), .)k is the source excitation time function, cdk the frequency, and a k = uk/2Qk the damping parameter. The real part cdk and imaginary part (Yk of the kth complex normal mode eigenfrequency, (7k = cdk + i(Yk, are functionals of the mechanical structure of the Earth. There is considerable motivation to make accurate observations of numerous (Tk, and in particular to improve the observational accuracy of the (Yk's. In this paper we exploit the fact that u(r, t ) is available only for te [O, T] where T is the duration of the seismic

Deep Earth Structure: Q of the Earth from Crust to Core

“…As noted in a previous review of this topic (Romanowicz and Durek, 2000), early studies based on normal mode and surface-wave data developed measurement and inversion methodologies and established the main features of the variation of Q m with depth (e.g., Anderson and Archambeau, 1966;Buland and Gilbert, 1978;Deschamps, 1977;Geller and Stein, 1978;Gilbert and Dziewonski, 1975;Jobert and Roult, 1976;Kanamori, 1970;Roult, 1975;Sailor and Dziewonski, 1978). These studies found that 1. shear-wave attenuation is low in the lithosphere; 2. there is a high-attenuation zone roughly corresponding to the low-velocity zone generally associated with the asthenosphere; 3. below 200 km depth, Q m increases with depth with a sharp gradient across the transition zone; and 4.…”

The frequency dependence of Q in the Earth and implications for mantle rheology and Chandler wobble

“…from about 0.2 to 0.4. The mode lSll was recently retrieved from a normal mode spectrum by Buland & Gilbert (1978). This mode has a period of 426 s and samples the mantle and core in roughly the same way as oSz.…”

Stacking for the frequencies and Qs of 0S0 and 1S0

“…The frequencies of oSo and ,So were first measured using the moment ratio method as described in Buland & Gilbert (1978). In this technique we divide the first moment of the power with respect to frequency by the power integrated across the same frequency band.…”

“…With the frequency determined by the moment ratio method we can next measure the Q using the minimum width method, also described by Buland & Gilbert (1978). Here the time series is multipiied by a growing exponential exp(air) which will in effect cancel at least some of the exp(-cud) decay of the mode of interest.…”