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A new model to explain the frequency-dependent decay of Fourier acceleration spectra at high frequencies

John G. Anderson, Annabel Haendel, Marco Pilz, & Fabrice Cotton

Published August 13, 2020, SCEC Contribution #10484, 2020 SCEC Annual Meeting Poster #217

Fourier amplitudes of acceleration rapidly decrease at high frequencies beyond the source corner frequency. This spectral decay is commonly modeled by a single parameter kappa introduced by Anderson and Hough (1984), which we will call the „kappa model“. Main features of the kappa model can be derived from three assumptions about earthquake source physics and wave propagation:

(1) The source spectrum is flat above the source corner frequency of the earthquake.
(2) Measurements are taken well above the predominant frequency of the site, i.e. the smoothed site spectrum can be approximated to be flat at high frequencies.
(3) Kappa can be explained by attenuation characterized by a frequency-independent quality factor Q.

Recent observations have found that the high-frequency slope of the acceleration spectrum is not decaying exactly linearly in log-linear space but rather is curved in a concave-upwards manner, resulting in estimated values of kappa that strongly depend on the chosen frequency band of analysis. Moreover, contrary to the third assumption, many seismological studies of high-frequency attenuation have found that Q is, in general, a function of frequency, i.e. Q(f). This study is motivated by the following question: “If the third assumption is replaced with a frequency-dependent Q as inferred in the seismological studies, will the predicted spectral shape better match the observations?”

We thus explore the implications of a frequency-dependent Q model, where Q has a power law dependence on the frequency, to explain the frequency-dependence of kappa. The spectral slope in log-linear space is then described by two parameters, opposed to the single parameter kappa. We refer to this new approach as the “zeta model“ for the high frequency spectral shape. Equivalent to the kappa model, one of the model parameters can be split into a path-dependent and a path-independent part to account for different contributions from the path and site.

We apply the zeta model to vertical array data of the Euroseistest site in Greece. The classical kappa model and the updated zeta models deliver consistent results if the same frequency band of analysis is considered. Yet, the zeta model is better able to capture the concave upwards spectral shape of the data at high frequencies in terms of misfit reduction and a better visual match between the data and the model.

Key Words
earthquake spectrum, kappa, zeta model, fmax, attenuation, Q

Anderson, J. G., Haendel, A., Pilz, M., & Cotton, F. (2020, 08). A new model to explain the frequency-dependent decay of Fourier acceleration spectra at high frequencies. Poster Presentation at 2020 SCEC Annual Meeting.

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