3.2 Calculation of stress drop from earthquake source spectra
To calculate the source spectra, source parameters and the empirical
correction spectra, we assume that the earthquake far-field displacement
spectrum can be described by a Brune-type source model (Brune, 1970):
\begin{equation}
s\left(f\right)=\frac{M_{0}}{1+\left(\frac{f}{f_{c}}\right)^{n}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\nonumber \\
\end{equation}where fc is the corner frequency, and n is
high-frequency fall-off rate, which we set to 2 (\(\omega^{-2}\)model). Some studies have allowed the fall-off rate to vary, but found
that it can tradeoff with the corner frequencies (e.g., Shearer et al.,
2019; Trugman & Shearer, 2017; Ye et al., 2013) and so we choose to fix
it here.
Assuming simple circular rupture, the corner frequency
(fc ) can be used to calculate the source radius
(Brune, 1970; Madariaga, 1976):
\begin{equation}
f_{c}=k\frac{\beta}{r}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)\nonumber \\
\end{equation}where \(\beta\) represents the shear velocity, and k is a constant that
depends on model assumptions, such as the source geometry, and rupture
velocity (Kaneko & Shearer, 2014, 2015). We choose k=0.32 for P waves
from Madariaga (1976), which is consistent with AS2007, and Kaneko &
Shearer (2015). The dependence on \(\beta\) introduces a dependence on
depth, since \(\beta\) is depth dependent. For example, if rupture
velocity is assumed to be a constant fraction of \(\beta\) then depth
varying velocity should be used. If a constant \(\beta\) is assumed in
equation (4) for all depths, then this can also introduce an artificial
dependence of source parameters on depth (e.g., Allmann & Shearer,
2007).
The earthquake stress drop (\(\sigma\)) can then be calculated from the
seismic moment (M0) and the source radius (r) following
Eshelby (1957):
\begin{equation}
\sigma=\frac{7}{16}\left(\frac{M_{0}}{r^{3}}\right)=M_{0}\left(\frac{f_{c}}{0.42\beta}\right)^{3}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)\nonumber \\
\end{equation}The stress drop derived from spectral fitting must be considered an
approximation. Theoretically it is related to the dynamic properties of
the earthquake based on a circular rupture model assumption (Brune 1970;
Madariaga, 1976), differing from the “static stress drop” derived from
finite slip source parameters (e.g., Noda et al., 2013). In practice, it
may be closer to a static stress drop since it is essentially the ratio
of the slip to an approximation of the source dimension.