Simple GMPE for underground mines

A simple ground motion prediction equation (GMPE) is developed for peak ground velocity, PGV, and for cumulative absolute displacement, CAD, for underground mines. Assuming the ground velocity at source, ${\text{PGV}}_0=0.63v_S\Delta ϵ$ where $v_S$ is S-wave velocity and $\Delta ϵ$ is the average strain change at seismic sources (Brune in J Geophys Res 75(26):4997–5009, 1970; Kanamori in Phys Earth Planet Inter 5:426–434, 1972), is independent of seismic potency, P, then $\overline{\text{PGV}}(P,R)={\text{PGV}}_0\cdot B$, where $B={\left[c_L{P}^{1/3}/(R+c_L{P}^{1/3})\right]}^{c_R}$ and R is distance. Assuming after Eshelby that at source ${\text{CAD}}_0=q_0$$\Delta {ϵ}^{2/3}{P}^{1/3}$, then $\overline{\text{CAD}}(P,R)={\text{CAD}}_0\cdot B$, where $q_0=0.828494$. The S-wave velocity and the strain drop are strongly constrained by the type of rock and can be assumed, therefore both GMPE have only two parameters to be inverted from ground motion data: $c_L$ and $c_R$. There is no provision made for site effect since in mines almost all sensors are placed in boreholes away from excavations. The basic outcome of ground motion hazard analysis for a given site is a seismic hazard curve that shows the annual rate, or probability, at which a specific ground motion level will be exceeded. It is expected that CAD that includes both the peak and the duration of ground motion may be a better indicator of damage potential than PGV alone, being a single measurement over the whole waveform. Two simple applications are presented. (1) A graphical trigger for damage inspection when the PGV predicted for an event at selected sites exceeds a predetermined level. (2) The cumulative CAD plot that may be a useful tool to monitor the consumption of the deformation capacity of the support due to seismicity.