Bias in magnitude for earthquakes with unknown focal mechanism

This study investigates errors induced by the use of average double-couple radiation coefficients for the computation of earthquake magnitude. Indeed, when an earthquake focal mechanism cannot be anticipated before calculating its magnitude, we show that the induced error depends on several factors such as event location, monitoring network aperture, type of seismic phase involved in the calculation (P/S waves) and focal mechanism. Energy radiated at lower amplitude is less likely to be observed, so, applying an absolute lower bound on radiation equal to 0.1, we show that errors in magnitude M are expected to range within the [-0.5; 0.2] interval. This indicates that errors have a stronger tendency towards under-estimation than over-estimation. We also present simulations of the spatial distribution of such errors for two double-couple mechanisms and one tensile opening, which reveal strong spatial dependencies on: i) the radiation pattern; and ii) the coverage of the focal sphere by the monitoring network. We show that the variability of errors significantly decreases with increasing network aperture. We apply this procedure to a field dataset, namely the monitoring of hydraulic fracturing in the Cotton Valley formation of Texas. In this case, if event focal mechanisms were not accounted for, errors in magnitude would generally fall within the above-mentioned interval; except for a few events presenting a larger under-estimation (error would reach M=-0.4 when the magnitude is computed using P waves only). A clear spatial pattern of errors is also observed at the Cotton Valley project, characterized by a west-to-east change in error polarity, but presenting a non-varying pattern in depth. Finally, such errors in magnitude have an insignificant impact on frequency-size distribution of events. b-value estimates based on such frequency-size distributions are within the uncertainty domain of estimates based on corrected frequency-size distributions.