SourceSpec

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)

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):

\[S(f) = \frac{1}{r} \times \frac{2 R_{\Theta\Phi}} {4 \pi \rho_h^{1/2} \rho_r^{1/2} \beta_h^{5/2} \beta_r^{1/2}} \times M_O \times \frac{1}{1+\left(\frac{f}{f_c}\right)^2} \times \exp \left( \frac{-\pi r f}{Q_O V_S} \right)\]

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:

\[ \begin{align}\begin{aligned}Y_{data}(f) = \frac{2}{3} \left[ \log_{10} \left( M_O \times \frac{1}{1+\left(\frac{f}{f_c}\right)^2} \times \exp \left( \frac{-\pi r f}{Q_O V_S} \right) \right) - 9.1 \right] =\\ = \frac{2}{3} (\log_{10} M_0 - 9.1) + \frac{2}{3} \left[ \log_{10} \left( \frac{1}{1+\left(\frac{f}{f_c}\right)^2} \right) + \log_{10} \left( \exp \left( \frac{-\pi r f}{Q_O V_S} \right) \right) \right]\end{aligned}\end{align} \]

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 \(t^*\).