source_spec¶
Overview¶
source_spec.py
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 \(f\) is the freqeuncy, \(r\) is the hypocentral distance, \(M_O\) is the seismic moment, \(f_c\) is the corner frequency; \(R_{\Theta\Phi}\) is the radiation pattern coefficient for S-waves, \(\rho\) is the average density of the medium, \(\beta\) and \(V_S\) are the S-wave speed at the source and the average S-wave speed along the wave propagation path, respectively; finally, \(Q_O\) is the quality factor.
In source_spec, the observed spectra \(S(f)\) are converted in moment magnitude \(Mw\).
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 \(Y_{data}\) vector used in the inversion):
Finally coming to the following model used for the inversion:
Where \(Mw \equiv \frac{2}{3} (\log_{10} M_0 - 9.1)\) and \(t^* \equiv \frac{r}{Q_O V_S}\)