Spectral representation of the seismic source

Seismic source spectra in terms of displacement density[4] can be described in terms of a low pass filter, i. e., with a flat part at low frequencies, an intermediate part where the spectrum starts to decrease, and the high frequency part where we note a steep decay, often proportional fg, where the spectral roll off g is a real number (Fig. 3a). The low frequency part can be directly related to the seismic moment using eq. 2.6. The intermediate part is characterized by the so-called corner frequency f0. It is determined by the intersections of the tangents applied to the flat part and the high frequency part of the spectrum. The spectral roll off g in the high frequency part of the source spectrum typically has a value of 2 in a wide range. The shape of the source spectrum has induced the development of the stochastic source model which forms the base of SHAKYGROUND. Redesigning the spectrum sketched in Fig. 3a in terms of acceleration density (Fig. 3b), one notes a band-limited white spectrum with its lower bound formed by the corner frequency f0. The upper limiting frequency, fmax, is necessary in order to obey the law of energy conservation. For this reason the radiation of seismic waves by the source must have an upper frequency limitation. Furthermore, the attenuation of the waves on their path from the source to the receiver causes additional band limiting effects.

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The acceleration density spectrum shown in Fig. 3b can be interpreted as the spectrum of a band-limited white noise. In fact, this description fits well to the highly irregular and complicated waveforms of observed acceleration seismograms which are considered to be a consequence of heterogeneities (“Barriers” or “Asperities”) within the seismic source. Sometimes the heterogeneities are large enough to form “sub-events”, which can be localized separately. In most cases, however, the heterogeneities cannot be identified explicitly, but contribute anyway to the high frequency radiation. The enriched high frequency radiation is particularly visible in acceleration seismograms (see, e. g., Papageorgiu and Aki, 1983, Brüstle ,1985). The stochastic model, which outline below, accounts for the irregular and complicated waveforms of acceleration seismograms in a straightforward way.

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[4] We are considering the far field contribution of seismic radiation. This is justified since, for the frequency range of technical interest, the near field terms can be neglected.