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):

\[S(f) = M_O \times \frac{2 R_{\Theta\Phi}}{4 \pi \rho \beta^3} \times \frac{1}{1+\left(\frac{f}{f_c}\right)^2} \times \left[ \exp \left( \frac{-\pi r f}{Q_O V_S} \right) \frac{1}{r} \right]\]

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:

\[r \times \frac{4 \pi \rho \beta^3}{2 R_{\Theta\Phi}} \times S(f) = M_O \times \frac{1}{1+\left(\frac{f}{f_c}\right)^2} \times \exp \left( \frac{-\pi r f}{Q_O V_S} \right)\]

Then the spectrum is converted in unities of magnitude (the \(Y_{data}\) vector used in the inversion):

\[ \begin{align}\begin{aligned}Y_{data} = \frac{2}{3} \times \left[ \log_{10} \left( r \times \frac{4 \pi \rho \beta^3}{2 R_{\Theta\Phi}} \times S(f) \right) - 9.1 \right]\\ Y_{data} = \frac{2}{3} \left[ \log_{10} \left( M_O \times \frac{1}{1+\left(\frac{f}{f_c}\right)^2} \times \exp \left( \frac{-\pi r f}{Q_O V_S} \right) \right) - 9.1 \right]\\ Y_{data} = \frac{2}{3} (\log_{10} M_0 - 9.1) + \frac{2}{3} \left[ \log_{10} \left( \frac{1}{1+\left(\frac{f}{f_c}\right)^2} \right) + \log_{10} \left( \exp \left( \frac{-\pi r f}{Q_O V_S} \right) \right) \right]\end{aligned}\end{align} \]

Finally coming to the following model used for the inversion:

\[Y_{data} = M_w + \frac{2}{3} \left[ - \log_{10} \left( 1+\left(\frac{f}{f_c}\right)^2 \right) - \pi \, f t^* \log_{10} e \right]\]

Where \(Mw \equiv \frac{2}{3} (\log_{10} M_0 - 9.1)\) and \(t^* \equiv \frac{r}{Q_O V_S}\)