Apparent clustering and apparent background earthquakes biased by undetected seismicity

In models of triggered seismicity and in their estimation from empirical data, the detection threshold m(d) is commonly equated to the magnitude m(0) of the smallest triggering earthquake. This unjustified assumption neglects the possibility of shocks below the detection threshold triggering observable events. We introduce a formalism that distinguishes between the detection threshold md and the minimum triggering earthquake m(0) <= md. By considering the branching structure of one complete cascade of triggered events, we derive the apparent branching ratio n(a) (which is the apparent fraction of aftershocks in a given catalog) and the apparent background source S-a observed when only the structure above the detection threshold md is known due to the presence of smaller undetected events capable of triggering larger events. If, as several recent analyses have shown, earthquake triggering is controlled in large part by the smallest magnitudes, this implies that previous estimates of the clustering parameters may significantly underestimate the true values: For instance, an observed fraction of 55% of aftershocks is renormalized into a true value of 75% of triggered events.