source_spec¶
Earthquake source parameters from inversion of S-wave spectra.
copyright: | 2012 Claudio Satriano <satriano@ipgp.fr>
2015-2020 Claudio Satriano <satriano@ipgp.fr> |
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license: | CeCILL Free Software License Agreement, Version 2.1 (http://www.cecill.info/index.en.html) |
Overview¶
source_spec
inverts the S-wave displacement spectra from
station recordings of a single event.
Spectral model¶
The Fourier spectrum of the S-wave displacement in far field can be modelled as the product of a source term (Brune model) and a propagation term (geometric and anelastic attenuation of body waves):
where \(r\) is the hypocentral distance; \(R_{\Theta\Phi}\) is the radiation pattern coefficient for S-waves; \(\rho_h\) and \(\rho_r\) are the medium densities at the hypocenter and at the receiver, respectively; \(\beta_h\) and \(\beta_r\) are the S-wave velocities at the hypocenter and at the receiver, respectively; \(M_O\) is the seismic moment; \(f\) is the frequency; \(f_c\) is the corner frequency; \(V_S\) is the average S-wave velocity along the wave propagation path; \(Q_O\) is the quality factor.
In source_spec
, the observed spectra \(S(f)\) are converted in
moment magnitude \(M_w\).
The first step is to multiply the spectrum for the hypocentral distance and convert them to seismic moment units:
Then the spectrum is converted in unities of magnitude (the \(Y_{data} (f)\) vector used in the inversion):
The data vector is compared to the teoretical model:
Finally coming to the following model used for the inversion:
Where \(M_w \equiv \frac{2}{3} (\log_{10} M_0 - 9.1)\) and \(t^* \equiv \frac{r}{Q_O V_S}\).
The parameters to determine are \(M_w\), \(f_c\) and \(t^*\).