Earthquake source parameters from inversion of S-wave spectra.
| copyright: | 2011-2022 Claudio Satriano <satriano@ipgp.fr> |
|---|---|
| license: | CeCILL Free Software License Agreement, Version 2.1 (http://www.cecill.info/index.en.html) |
source_spec inverts the S-wave displacement spectra from
station recordings of a single event.
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):
S(f) =
\frac{1}{r}
\times
\frac{2 R_{\Theta\Phi}}
{4 \pi \rho_h^{1/2} \rho_r^{1/2} \beta_h^{5/2} \beta_r^{1/2}}
\times
M_O
\times
\frac{1}{1+\left(\frac{f}{f_c}\right)^2}
\times
\exp \left( \frac{-\pi r f}{Q_O V_S} \right)
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:
M(f) \equiv
r \times
\frac{4 \pi \rho_h^{1/2} \rho_r^{1/2} \beta_h^{5/2} \beta_r^{1/2}}
{2 R_{\Theta\Phi}}
\times S(f) =
M_O \times
\frac{1}{1+\left(\frac{f}{f_c}\right)^2}
\times
\exp \left( \frac{-\pi r f}{Q_O V_S} \right)
Then the spectrum is converted in unities of magnitude (the Y_{data} (f) vector used in the inversion):
Y_{data}(f) \equiv
\frac{2}{3} \times
\left( \log_{10} M(f) - 9.1 \right)
The data vector is compared to the teoretical model:
Y_{data}(f) =
\frac{2}{3}
\left[ \log_{10} \left(
M_O \times
\frac{1}{1+\left(\frac{f}{f_c}\right)^2}
\times
\exp \left( \frac{-\pi r f}{Q_O V_S} \right)
\right) - 9.1 \right] =
=
\frac{2}{3} (\log_{10} M_0 - 9.1) +
\frac{2}{3} \left[ \log_{10} \left(
\frac{1}{1+\left(\frac{f}{f_c}\right)^2} \right) +
\log_{10} \left(
\exp \left( \frac{-\pi r f}{Q_O V_S} \right) \right)
\right]
Finally coming to the following model used for the inversion:
Y_{data}(f) =
M_w +
\frac{2}{3} \left[ - \log_{10} \left(
1+\left(\frac{f}{f_c}\right)^2 \right) -
\pi \, f t^* \log_{10} e
\right]
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^*.