source_spec.rst

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):

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]

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}

.. automodule:: source_spec
   :members: