## 1. Introduction

[2] There is much evidence that a seismic event can have a significant effect on the pattern of subsequent seismicity, most obvious in aftershocks of large events. More recently, an important extension of the concept of earthquake interactions has emerged: “triggered seismicity”, in which the usual distinction that foreshocks are precursors of larger main shocks, which in turn trigger smaller aftershocks, becomes blurred. An efficient description of seismicity does not seem to require the division between foreshocks, main shocks, and aftershocks, as they appear indistinguishable in many of their physical and statistical properties [*Helmstetter and Sornette*, 2003a]. An important logical consequence is that cascades of triggered seismicity (aftershocks, aftershocks of aftershocks,…) may play an important role in the overall seismicity budget [*Helmstetter and Sornette*, 2003b; *Felzer et al.*, 2002].

[3] There is a growing interest in phenomenological models of triggered seismicity, which use the Omori law as a coarse-grained proxy for modeling the complex and multifaceted interactions between earthquakes, together with other robust descriptions of seismicity (clustering in space, the Gutenberg-Richter (GR) earthquake size distribution and aftershock productivity laws). The class of epidemic-type aftershock sequences (ETAS) models introduced by *Kagan and Knopoff* [1981] and *Ogata* [1988] offers a parsimonious approach that replaces the classification of foreshocks, main shocks and aftershocks by the concept of earthquake triggering: Earthquakes may trigger other earthquakes through a variety of physical mechanisms without attempting to identify the particular mechanisms.

[4] The questions suggested by this approach include the following: What is the fraction of triggered versus uncorrelated earthquakes? (This is linked to the problem of clustering and a partial answer is given by *Helmstetter and Sornette* [2003b].) How can one use this modeling approach to forecast future seismicity? What are the limits of predictability? (A partial answer using only time-dependent information is given by *Helmstetter and Sornette* [2003c].) How sensitive are forecasts to catalog completeness and type of tectonic deformation? In general, to tackle any such question, one needs to estimate key parameters of the models of triggered seismicity in one way or another. Our present paper shows that there is a nontrivial and important impact of catalog incompleteness on the determination of the parameters quantifying earthquake triggering, with its expected impact on all the above questions.

[5] The most promising approach is in general to use the maximum likelihood method to estimate the model parameters from a catalog of seismicity (time, location, and magnitude) [see, e.g., *Ogata*, 1988; *Kagan*, 1991]. The calculation of the likelihood function requires evaluating the theoretical rate of seismicity at time *t* induced by all past events at times *t*_{i} < *t*. The maximization of the likelihood with respect to the parameters of the model, given the data, then provides an estimate of the parameters. All previous studies have considered that small earthquakes, below the detection threshold, are negligible. Thus the rate of seismicity is calculated as if triggered only by earthquakes above the detection threshold. However, this method is not correct because it does not take into account events below the detection threshold, which may have an important role in the triggering of seismicity. Indeed, small earthquakes have a significant contribution in earthquake triggering because they are much more numerous than larger earthquakes [*Felzer et al.*, 2002; *Helmstetter*, 2003; *Helmstetter et al.*, 2005]. This can simply be seen from the competition between the productivity law ∼10^{αM} giving the number of events triggered by a main shock of magnitude *M* and the relative abundance ∼10^{−bM} of such main shocks given by the Gutenberg-Richter (GR) law: The contribution of earthquakes of magnitude *M* to the overall seismic rate is thus ∼10^{−(b−α)M}, which is dominated by small *M* for α < *b* [*Helmstetter*, 2003] or equally contributed by each magnitude class for α = *b* [*Felzer et al.*, 2002; *Helmstetter et al.*, 2005]. Therefore one needs to take into account small events that are not observed in order to calibrate correctly models of seismicity and obtain reliable answers to our questions stated above. This is an essential bottleneck for the development of earthquake forecasts based on such models.

[6] The purpose of this note is to present a general theoretical treatment of the impact of unobserved seismicity within the framework of models of triggered seismicity. We show by analyzing the branching structure of a complete cascade (cluster) triggered by an independent background event that the unobserved seismicity has the effect of decreasing the real branching ratio *n* and of increasing the number of independent background events *S* into apparent quantities *n*_{a} and *S*_{a}. The bias persists in a catalog of an arbitrary number of clusters (see Appendix B) and may be very significant. We therefore claim that previous work should be reanalyzed from the new perspective of our approach. This leads also to important consequences for the methods presently used to forecast future seismicity based only on incomplete catalogs.

[7] The closely related study by *Sornette and Werner* [2005] also considered the effects of undetected seismicity in models of triggered seismicity. They found that a magnitude cutoff *m*_{0} below which earthquakes do not trigger other events is necessary to make these models convergent and well defined. If each magnitude unit of quakes collectively contributes a comparable amount of triggered events (of any magnitude) to the overall budget, then a lower cutoff *m*_{0} must exist to ensure finite seismicity. *Sornette and Werner* [2005] showed that this cutoff has observable consequences and can thus be estimated from parameters estimated from fits to the statistics of aftershock sequences and from Bth's law. They arrived at four different estimates of *m*_{0} that we employ below to quantify the effects of undetected seismicity on the measured fraction of triggered events in a seismic catalog. *Sornette and Werner* [2005] also discussed possible scenarios for this break in self-similarity. In this article, we continue to explore the effects of undetected earthquakes on the observed seismicity.