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.