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\begin{document}

Given the displacement across the fault plane, can define seismic
moment:

\( M_{0}=\mu _{shearmodulus}DS \) 

\begin{itemize}

\item D is displacement 

\item S is area of fault \~{}\( L^{2} \) where L is the length of the fault

\item \( \tau \sim \frac{L}{v} \) for \( v=.7v_{s} \) and \( c_{s}=v_{s}=(\frac{\mu _{shearmodulus}}{\rho })^{\frac{1}{2}}\sim 3km/s \). Compressive waves are somewhat faster. \( \tau \sim (\frac{L}{km})(\frac{v}{km/s})^{-1} \) Big cracks
take awhile to develop. \( c_{p}\sim 1.7c_{s} \) = compressive wave speed.

\item far field displacement, \( \zeta  \), \( r>>c_{s}\tau \sim L \) has \( \zeta \sim (\frac{v}{c_{s}})^{4}D\frac{L}{r}\sim (\frac{v}{c_{s}})^{4}\frac{M_{0}}{\mu Lr} \) \( \sim (\frac{v}{c_{s}})^{4}\frac{M_{0}}{\mu c_{s}\tau r} \)\( \sim (\frac{v}{c_{s}})^{4}\frac{M_{0}}{\rho c^{3}_{s}\tau r} \)

\item \( M_{0} \) is observationally determined.

\end{itemize}

\end{document}