Apparent clustering and apparent background earthquakes biased by undetected seismicity


  • Didier Sornette,

    1. Department of Earth and Space Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California, USA
    2. Laboratoire de Physique de la Matière Condensée, CNRS UMR6622, Université de Nice-Sophia Antipolis, Nice, France
    Search for more papers by this author
  • Maximillian J. Werner

    1. Department of Earth and Space Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California, USA
    Search for more papers by this author


[1] In models of triggered seismicity and in their estimation from empirical data, the detection threshold md is commonly equated to the magnitude m0 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 m0md. By considering the branching structure of one complete cascade of triggered events, we derive the apparent branching ratio na (which is the apparent fraction of aftershocks in a given catalog) and the apparent background source Sa 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.

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 ti < 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 ∼10αM giving the number of events triggered by a main shock of magnitude M and the relative abundance ∼10bM 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 na and Sa. 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 m0 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 m0 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 m0 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.

2. ETAS Model and Smallest Triggering Earthquake

2.1. Definition of the ETAS Model

[8] To make this discussion precise, let us consider the epidemic-type aftershock sequence (ETAS) model, in which any earthquake may trigger other earthquakes, which in turn may trigger more, and so on. Introduced in slightly different forms by Kagan and Knopoff [1981] and Ogata [1988], the model describes statistically the spatiotemporal clustering of seismicity. We choose the ETAS model because of its increasing popularity for the statistical description of earthquake interaction [Kagan and Knopoff, 1981; Ogata, 1988; Console et al., 2003; Zhuang et al., 2004], its establishment as a powerful null hypothesis for forecasting [Helmstetter and Sornette, 2003c; A. Helmstetter et al., Comparison of short-term and long-term earthquake forecast models for southern California, submitted to Bulletin of the Seismological Society of America, 2005; D. Schorlemmer et al., Earthquake likelihood model testing, SCEC preprint, 2005], its simplicity, and its explanatory power of features in catalogs including apparent Gutenberg-Richter b value variations and Omori law exponent variations [Helmstetter and Sornette, 2002], foreshocks [Helmstetter and Sornette, 2003a], and apparent aftershock diffusion [Helmstetter et al., 2003].

[9] The triggering process may be caused by various mechanisms that either compete or combine, such as pore pressure changes due to pore fluid flows coupled with stress variations, slow redistribution of stress by aseismic creep, rate- and state-dependent friction within faults, coupling between the viscoelastic lower crust and the brittle upper crust, stress-assisted microcrack corrosion, and more. The ETAS formulation amounts to a two-scale description: These above physical processes controlling earthquake interactions enter in the determination of effective triggering laws in a first step and the overall seismicity is then seen to result from the cascade of triggering of events triggering other events triggering other events and so on [Helmstetter and Sornette, 2002].

[10] The ETAS model consists of three laws about the nature of seismicity viewed as a marked point process. We restrict this study to the temporal domain only, summing over the whole spatial domain of interest. First, the magnitude of any earthquake, regardless of time, location, or magnitude of the mother shock, is drawn randomly from the exponential Gutenberg-Richter (GR) law. Its normalized probability density function (PDF) is expressed as

display math

where the exponent b is typically close to one, and the cutoffs m0 and mmax serve to normalize the PDF. The upper cutoff mmax is introduced to avoid unphysical, infinitely large earthquakes. Its value was estimated to be in the range 8–9.5 [Kagan, 1999]. As the impact of a finite mmax is quite weak in the calculations below, replacing the abrupt cutoff mmax by a smooth taper would introduce negligible corrections to our results.

[11] Second, the model assumes that direct aftershocks are distributed in time according to the modified “direct” Omori law [see Utsu et al., 1995, and references therein]. Assuming θ > 0, the normalized PDF of the Omori law can be written as

display math

[12] Third, the number of direct aftershocks of an event of magnitude m is assumed to follow the productivity law:

display math

Note that the productivity law (3) is zero below the cutoff m0, i.e., earthquakes smaller than m0 do not trigger other earthquakes, as is typically assumed in studies using the ETAS model. The existence of the small magnitude cutoff m0 is necessary to ensure the convergence of these types of models of triggered seismicity (in the statistical physics of phase transitions and in particle physics, this is called an “ultraviolet” cutoff which is often necessary to make the theory convergent). In a closely related paper, Sornette and Werner [2005] showed that the existence of the cutoff m0 has observable consequences which constrain its physical value. They also discuss possible scenarios for this break in self-similarity, such as a transition from fracture to friction dominated earthquakes [Richardson and Jordan, 2002] or a minimum earthquake size as predicted by rate-and-state friction [Dieterich, 1992; Ben-Zion, 2003].

[13] The key parameter of the ETAS model is defined as the number n of direct aftershocks per earthquake, averaged over all magnitudes. Here, we must distinguish between the two cases α = b and α ≠ b:

display math

for the general case α ≠ b. The special case α = b gives

display math

[14] Three regimes can be distinguished based on the value of n. The case n < 1 corresponds to the subcritical, stationary regime, where aftershock sequences die out with probability one. The case n > 1 describes unbounded, exponentially growing seismicity [Helmstetter and Sornette, 2002]. In addition, the case b < α leads to explosive seismicity with finite time singularities [Sornette and Helmstetter, 2002]. The critical case n = 1 separates the two regimes n < 1 and n > 1. Helmstetter and Sornette [2003b] showed that the branching ratio n is also equal to the fraction of triggered events in a seismic catalog. We consider the case n < 1 which describes stationary seismicity. The branching ratio n measures the distance to the critical state of the crust (n = 1) which may have important implications for the self-organization of the crust.

[15] The fact that we use the same value for the productivity cutoff and the Gutenberg-Richter (GR) cutoff is not a restriction as long as the real cutoff for the Gutenberg-Richter law is smaller than or equal to the cutoff for the productivity law. In that case, truncating the GR law at the productivity cutoff just means that all smaller earthquakes, which do not trigger any events, do not participate in the cascade of triggered events. This should not be confused with the standard incorrect procedure in many previous studies of triggered seismicity of simply replacing the GR and productivity cutoff m0 with the detection threshold md in equations (1) and (3) [see, e.g., Ogata, 1988; Kagan, 1991; Ogata, 1998; Console et al., 2003; Zhuang et al., 2004]. The assumption that md = m0 may lead to a bias in the estimated parameters. Helmstetter et al. [2005, Figure 1] show that events of magnitude 2 trigger their own aftershock sequences. We thus expect m0 to be smaller than md.

[16] Without loss of generality, we consider one independent branch (cluster or cascade of aftershocks set off by a background event) of the ETAS model. We generalize to a seismic catalog of an arbitrary number of clusters in the appendix. Let an independent background event of magnitude M1 occur at some origin of time. The main shock will trigger direct aftershocks according to the productivity law (3). Each of the direct aftershocks will trigger their own aftershocks, which in turn produce their own, and so on. Averaged over all magnitudes, an aftershock produces n direct offspring according to (4). Thus, integrating over time, we can write the average of the total number Ntotal of direct and indirect aftershocks of the initial main shock as an infinite sum over terms of (3) multiplied by n to the power of the generation [Helmstetter and Sornette, 2003b], which can be expressed for n < 1 as

display math

However, since we can only detect events above the detection threshold md > m0, the total number of observed aftershocks Nobs of the sequence is simply Ntotal multiplied by the fraction of events above the detection threshold, given by

display math

according to the GR distribution. The observed number of events in the sequence is therefore

display math

Equation (8) predicts the average observed number of direct and indirect aftershocks of a main shock of magnitude M1 > md. Sornette and Werner [2005] showed that m0 may be estimated using fits of Nobs given by (8) to observed aftershock sequences and Båth's law. The essential parameter needed to constrain m0 is the branching ratio n. As we demonstrate below, typical estimates of n in the literature obtained from a catalog neglect undetected seismicity and therefore cannot be used directly to constrain m0.

[17] Naturally, there is no justification for assuming that md should equal m0, as is done routinely in inversions of catalogs for the parameters of the ETAS model [see, e.g., Ogata, 1988; Kagan, 1991; Ogata, 1998; Console et al., 2003; Zhuang et al., 2004]. First, detection thresholds change over time as instruments and network coverage become better, while the physical mechanisms in the Earth presumably remain the same. No significant deviation from the Gutenberg-Richter distribution or the productivity law has been recorded as the detection threshold md decreased over time [see, e.g., Ouillon and Sornette, 2005, Figure 3]. Second, studies of earthquake occurrence at small magnitude levels below the regional network cutoffs show that earthquakes follow the same Gutenberg-Richter law (for a recent study of mining-induced seismicity, see, e.g., Sellers et al. [2003]), while acoustic emission experiments have shown the relevance of the Omori law at small scales [see, e.g., Nechad et al., 2005, and references therein]. Within the assumption of self-similarity, i.e., a continuation of the GR and productivity laws down to a cutoff, evidence thus points toward a magnitude of the smallest triggering earthquake and a Gutenberg-Richter cutoff that lie below the detection threshold and are thus not directly observable.

[18] The effect of undetected seismicity below the detection threshold is fundamentally different from the effect of earthquakes outside the space-time study window that may contribute to the seismicity budget inside the region. The event incompleteness below the magnitude detection threshold md cannot be treated in analogy to the time and space detection threshold as a finite size boundary effect problem. While events from outside the study area have a decreasing influence on the inside in time according to the Omori law and in space according to a spatial decay function (e.g., Gaussian or power law), the influence of the many events below the detection threshold inside the study area may be very significant because each magnitude range collectively triggers a roughly equal amount of events of any size. The magnitude detection threshold is thus of a different nature than boundary effects and must be addressed.

2.2. Two Interpretations of the ETAS Model

[19] The ETAS model may be viewed in two mathematically equivalent ways that differ in their interpretation. In this section, we develop both views to underline that our results apply in both cases and to stress the equivalence of these two views. The first describes the model as a simple branching model without loops [Kagan, 1991]: The independent background events, due to tectonic loading, may each independently trigger direct aftershocks, each of which may in turn trigger secondary shocks, which in turn may trigger more. Because every triggered event, excluding of course the nontriggered background events, has exactly one main shock (mother), but the mother may have many direct aftershocks (children), the model can be thought of as a simple branching model without loops. The background events are assumed to be a stationary Poisson process with a constant rate. The rate of the aftershocks of a background event is a nonstationary Poisson process that is updated every time another aftershock occurs until the cascade dies out. The intensity is thus conditioned on the specific history of earthquakes. The expectation of the conditional intensity is an average over an ensemble of histories. The predicted number of aftershocks of an independent background event of magnitude M1 as in expression (8) is thus averaged over the ensemble of possible realizations of the aftershock sequence, and it is also averaged over all possible magnitudes of the aftershocks. The branching ratio n is therefore an average not only over magnitudes but also over an ensemble of realizations of the nonstationary Poisson process. In summary, the model consists of statistically independent Poisson clusters of events, which are, however, dependent within one cluster.

[20] The second view of the ETAS model does not allow a unique identification of the mother or trigger of an earthquake. Rather, each aftershock was triggered collectively by all previous earthquakes, each of which contributes a weight determined by the magnitude-dependent productivity law ρ(m) that decays in time according to the Omori law ψ(t) and in space according to a spatial function R(r), often chosen to be an exponential or a power law centered on the event. The instantaneous conditional intensity rate at some time t at location r is given by

display math

where the sum runs over all previous events i with magnitude mi at time ti at location ri. Thus the triggering contribution of a previous event to a later event at time t is given by its own weight (its specific entry in the sum) divided by the total seismicity rate, including the background rate. A nonzero background rate then contributes evenly to all events and corresponds to an omnipresent loading contribution. In this way, earthquakes are seen to be the result of all previous activity including the background rate. This corresponds to a branching model in which every earthquake links to all subsequent earthquakes weighted according to the contribution to triggering. A branching ratio can then be interpreted as a contribution of a past earthquake to a future earthquake, averaged over an ensemble of realizations and all magnitudes. In contrast to the independent background events considered due solely to tectonic loading that exist in the first interpretation, all earthquakes are due to a combination of the background loading and the effect of previous events. This second view becomes the only possible one for nonlinear models whose triggering functions depend nonlinearly on previous events (see, e.g., the recently introduced multifractal earthquake triggering model [Ouillon and Sornette, 2005; Sornette and Ouillon, 2005] and references therein).

[21] These two views are equivalent because the linear formulation of the seismic rate of the ETAS model together with the exponential Poisson process ensures that the statistical properties of the resulting earthquake catalogs are the same. The linear sum over the individual contributions and the Poisson process formulation are the key ingredients that allow the model to be viewed as a simple branching model.

[22] This duality of thinking about the ETAS model is reflected in the existence of two simulation codes in the community, each inspired by one of the two views. A program written by K. Felzer and Y. Gu (personal communication) calculates the background events as a stationary Poisson process and then simulates each cascade independently of the other branches as a nonstationary process. The second code by Ogata [1998], on the other hand, calculates the overall seismicity at each point in time by summing over all previous activity. The latter code is significantly slower because the independence between cascades is not used, and the entire catalog is modeled as the sum of a stationary and a nonstationary process. Despite the different approach, both resulting earthquake catalogs share the same statistical properties and are thus equally acceptable.

[23] While the simulation or forward problem is straightforward when adopting the view of the ETAS model as a branching model with one assigned trigger for any aftershock, the inverse problem of reconstructing the branching structure from a given catalog can at best be probabilistic. Because aftershocks of one mother cannot be distinguished from those of another mother except by spatiotemporal distance, we have no way of choosing which previous earthquake triggered a particular event, or whether it is a background event. Rather, we must resort to calculating the probability of an event at time t to be triggered by any previous event according to the contribution that the previous event has at time t compared to the overall intensity at time t. This probability is of course equal to the weight or triggering contribution that a previous event has on a subsequent event when adopting the collective-triggering view. However, the interpretation remains different since the probability specifies a unique mother in a fraction of many realizations.

[24] Having determined from catalogs a branching structure weighted according to the probability of triggering, one may of course choose to always pick as source of an event the most probable contributor, be that a previous event or the background rate. Another option is to choose randomly according to the probability distribution and thus reconstruct one possible branching structure among the ensemble of many other possible ones. The latter approach has been used by Zhuang et al. [2004] and labeled stochastic reconstruction.

[25] The key point is that equating the detection threshold with the smallest triggering earthquake will most likely lead to a bias in the recovered parameters of a maximum likelihood analysis as performed by Zhuang et al. [2004] and in many other studies. Therefore the weights or probabilities of previous events triggering subsequent events were calculated from biased parameters.

[26] In the following, we show that the branching ratio and the background source events are significantly biased when they are estimated from the apparent branching structure observed above the detection threshold md instead of the complete tree structure down to m0. We adopt the view of the simple branching model to make the derivations more illuminating but all results can be reinterpreted as contributions in the collective-triggering view.

3. Apparent Branching Structure of the ETAS Model

3.1. Apparent Branching Ratio na

[27] Seismic catalogs are usually considered complete above a threshold md, which varies as a function of technology and location. For instance, md ≈ 2 for modern southern California catalogs (and for earthquakes not too close in time to a large main shock [Kagan, 2003]). The analysis of the statistics of the Omori and inverse Omori laws for earthquakes of magnitude down to 3 [Helmstetter, 2003; Helmstetter and Sornette, 2003a] suggests that m0 is smaller than the completeness magnitude md and is thus not directly observable. Thus m0 is the size of the smallest triggering earthquake, which most likely differs significantly in size from the current detection threshold md. By considering the branching structure of the model, we derive the apparent branching ratio and the apparent background source that are found if only the observed (detected) part of the ETAS model is analyzed.

[28] Since aftershock clusters are independent of each other, averages of one cluster are equal to ensemble averages, as nothing but the inherent stochasticity of the model determines the properties of the clusters. One cluster consists of one independent background event (source) and its direct and indirect aftershocks (see Figure 1). However, if not all events of the sequence are detected, then there will appear to be less direct (and indirect) aftershocks, i.e., the branching ratio will appear different. Furthermore, some observed events will be triggered by mother earthquakes below the detection threshold, resulting in apparently independent background events (see Figure 2).

Figure 1.

Schematic representation of the branching structure of the real ETAS model. An independent background earthquake triggers direct aftershocks, which in turn trigger second generation aftershocks, and so on. The structure is complete down to the magnitude of the smallest triggering earthquake m0.

Figure 2.

Schematic representation of the branching structure of the apparent ETAS model. The initial main shock is circled. Only events above the detection threshold md are observed. The apparent branching ratio does not take into account unobserved triggered events (dashed lines). An observed event triggered by a mother below md appears as an untriggered background source event (circled).

[29] This view leads to the conclusion that the average number of direct aftershocks that are observed will be less than the real branching ratio, since some of the triggered events of an observed shock will fall below md and hence not be included in the count. Only the fraction fobs from equation (7) above md of the total direct aftershocks ρ(m) will be observed. Moreover, the PDF P(mmmd) of mother events conditioned on being larger than md is zero for m < md and equal to P(m)/fobs for mmax > mmd. We can thus define the apparent branching ratio as

display math

for the case α ≠ b. The special case α = b gives

display math

Using equation (4) and eliminating k, we have na in terms of n

display math

when α ≠ b, and

display math

when α = b. According to expression (12), nan, where the equality holds for md equal to m0. In principle, equation (12) also holds for n > 1, but we restrict this study to the regime n < 1 for mathematical convenience and because this gives rise to statistically stationary seismic sequences. Figure 3 shows na as a function of n for a range of values of m0 for the case α = b. It demonstrates that the apparent (measurable) fraction of aftershocks may significantly underestimate the true fraction of aftershocks even for m0 not very small. For example, m0 = −5 roughly translates a real branching ratio of n = 0.9 into an apparent branching ratio na = 0.3. Decreasing α below b places more importance on the triggering from small earthquakes and therefore strongly amplifies this effect.

Figure 3.

Apparent fraction of aftershocks (apparent branching ratio) na which varies linearly with the real fraction of aftershocks (real branching ratio) n with a slope fixed by the smallest triggering earthquake m0. As m0 decreases, the apparent fraction of aftershocks significantly underestimates the real fraction. As examples, we chose m0 = md = 3 (solid), i.e., na = n and no events are missed; m0 = 0 (dashed); m0 = −5 (dotted); and m0 = −10 (dash-dotted). We further assumed parameters md = 3, mmax = 8, b = α = 1. A small value of α amplifies this effect (see Figure 4).

[30] In Figure 4, we plot the ratio na/n as a function of the unknown m0. As expected, when m0 = md, the ratio is one because there is no unobserved seismicity. As m0 goes to minus infinity, na approaches zero since almost all seismicity occurs below the threshold. We see clearly that unobserved seismicity results in a drastic underestimate of the fraction of aftershocks.

Figure 4.

Ratio of the apparent fraction of aftershocks (apparent branching ratio) na over the real fraction of aftershocks (real branching ratio) n which varies as a function of the smallest triggering earthquake m0. For m0 = md, na = n and all events are detected above the threshold. For a small value of m0, the ratio becomes small, indicating that na significantly underestimates n. Decreasing α amplifies this effect. We used parameters md = 3 (vertical reference line), mmax = 8, and b = 1. We varied α = 0.5 (dash-dotted), α = 0.8 (dashed), α = 1.0 (solid).

[31] Given an estimate of the magnitude of the smallest triggering earthquake m0 [see Sornette and Werner, 2005, and references therein], one can calculate the true branching ratio from the apparent branching ratio. In fact, Sornette and Werner [2005] obtained four estimates of m0 as a function of n by comparing the ETAS model prediction of the number of observed aftershocks (8) from fits to observed aftershock sequences and from the empirical Båth's law. Their equations (10), (13), (16), and (18) are the estimates of m0 as a function of n and a number of known constants specific to the fits to observed aftershock sequences. We can use these relations of m0 as a function of n to eliminate m0 from equation (12) to obtain direct estimates of n as a function of the measurable na. For simplicity, we restrict the use of their findings to the case α = b. The estimate resulting from the fits performed by Helmstetter et al. [2005] yielded

display math

with the values mmax = 8.5, md = 3, θ = 0.1, c = 0.001, b = 1 and Kfit = 0.008. The study by Felzer et al. [2002] provided another estimate

display math

where mmax = 8.5, md = 3, θT = 0.08, AT = 0.116 image b = α = 1, c = 0.014 and M1 = 6.04. Using the declustering performed by Reasenberg and Jones [1989], Sornette and Werner [2005] obtained

display math

where mmax = 8.5, md = 3, θ = 0.08, a = −1.67, c = 0.05 and b = 1. Finally, using Båth's law, Sornette and Werner [2005] found

display math

where M1ma = 1.2 according to the law, b = 1, mmax = 8.5, and md = 3.

[32] Substituting these four estimates of m0 from equations (14), (15), (16), and (17) into equation (12) for na provides four estimates of na versus n all in terms of known constants. These four estimates of n as a function of na can be used to find the correct fraction of aftershocks from the measurable apparent fraction of aftershocks. Figure 5 shows these four estimates with the above constants.

Figure 5.

Fraction of aftershocks (branching ratio) n that can be estimated from the apparent fraction of aftershocks (apparent branching ratio) na by using four estimates of the smallest triggering earthquake m0 as a function of n as determined by Sornette and Werner [2005] (see text). The estimates of m0 as a function of n were obtained from comparisons of the ETAS model prediction of the number of observed aftershocks and fits to observed aftershock sequences performed by Helmstetter et al. [2005] (solid), Felzer et al. [2002] (dash-dotted), Reasenberg and Jones [1989] (dotted), and Båth's law (dashed). The additional diagonal solid line na = n corresponds to m0 = md (no undetected events). Along any of the four lines, m0 varies from minus infinity to mmax. Given that we can rule out m0md, we can restrict the physical range to the left side of the diagonal na = n.

[33] Figure 5 can be used to find the real fraction of aftershocks from the measured apparent fraction by assuming one of the four estimates of m0 as a function of n. For example, Helmstetter et al. [2005] find that 55 percent of all earthquakes are aftershocks above md = 3. Using their values to estimate m0 as a function of n, we can determine that the real fraction of aftershocks is closer to 75 percent. Thus the size of this effect is significant. Furthermore, having determined a point on the line estimating n from na for all values of m0 fixes the slope of n(na) and therefore m0. Using their values, we find m0 ≃ 1.2. Similar estimates can be made using the apparent fraction of aftershock values found by Felzer et al. [2002] and Reasenberg and Jones [1989]. The uncertainty of the parameters estimated in these studies affects the estimates of m0. Sornette and Werner [2005] analyzed the error propagation and found that the estimates of m0 are most likely order of magnitude calculations.

[34] Assuming that current maximum likelihood estimation methods of the ETAS model parameters, which assume m0 = md, determine a branching ratio that corresponds to the present apparent branching ratio, we can similarly correct these values to find the true fraction of aftershocks using Figure 5. For example, Zhuang et al. [2004] find a “criticality parameter” of about 45 percent, which we take as a proxy for na. Figure 5 shows that the true branching ratio then lies between 0.45 and 0.80, depending on which estimate (among the four models (14), (15), (16), and (17)) of m0 as a function of n is chosen. These calculations suggest that previous estimates of the fraction of aftershocks obtained by various declustering methods significantly underestimated its value.

3.2. Determination of Apparent Background Events Sa of Uncorrelated Seismicity

[35] In order to derive the number of shocks within one cascade that are not triggered by a mother above the threshold and thus appear as independent background events, we need to distinguish between the case where the initial (main) shock of magnitude M1 is observable (i.e., M1md) and the case where it is undetected (i.e., M1 < md).

[36] If M1md, then the initial background event produces ρ(M1) fobs observed direct aftershocks. On average, these will in turn collectively produce ρ(M1)fobsna observed second generation aftershocks. We specifically do not consider events above md triggered from below md, which we deal with below in the definition of the apparent background sources. By continuing this “above water” or “above sea level” cascade for all generations of aftershocks, we can calculate the number of triggered events that are in direct lineage above the threshold back to the main shock as the infinite sum of terms of ρ(M1) fobs multiplied by the apparent branching ratio na to the power of the generation. If, on the other hand, the initial background event is below md, then no such direct above water cascade will be seen. Any observed shock will be triggered by an event below the water. Thus, for the two cases, the above water sequence is expressed as

display math

Furthermore, since in the ETAS model, a small earthquake may trigger large earthquakes, an event below md may produce an observed event above md. An inversion method that reconstructs the entire branching structure of the model from an earthquake catalog will identify these shocks as background events. However, since in reality these events were triggered by earthquakes below the detection threshold, we will refer to them as apparent background events. These events can of course trigger their own cascades. We thus define the apparent background sources Sa as the number of observed events above md that are apparently not triggered, i.e., have “mothers” below md. Again, we distinguish between the cases where the background event magnitude is M1md and M1 < md. For the first, Sa is given by the total number of aftershocks below the threshold multiplied by the average number r of direct aftershocks they trigger above the threshold. For the second case, we must also include the direct aftershocks of the initial background event that are observed:

display math

Now, the number r of observable directly triggered shocks above md averaged over unobserved mothers between m0 and md is given by the following conditional branching ratio:

display math
display math

where we have used that P(mm < md) = P(m)/(1 − fobs) for m < md and zero otherwise. Substituting (21) into the expression for the apparent source (19) and rearranging using (8), we obtain

display math

Equation (22) shows that for each genuine background event of magnitude M1, a perfect inversion method would count Sa apparent background events. Figure 6 plots the number of apparent background events Sa as a function of the branching ratio n for an example aftershock cascade set off by a magnitude m = 5 initial shock. Figure 6 shows that for one cascade, i.e., one independent background event, hundreds of earthquakes appear as apparent background events when m0 < md.

Figure 6.

Number of apparent background events Sa in an aftershock cascade due to a single background event of magnitude M1 = 5 as a function of the fraction of aftershocks (branching ratio) n for several values of the smallest triggering earthquake. For m0 = md, no events are missed. Therefore the number of apparent background events is zero. As m0 decreases, events below the detection threshold trigger events above the threshold, and hence the number of apparent background events increases. We vary m0 = md = 3 (solid, coinciding with x axis), m0 = 0 (dash-dotted), m0 = −5 (dashed), and m0 = −10 (upper solid curve). We used parameters md = 3, mmax = 8, b = 1, and α = 1.0. For very small m0 and n close to 1, almost all events above the detection threshold are triggered from below, and thus Sa becomes very large (see Figure 7). This effect is amplified for decreasing α (not shown).

[37] In Figure 7, we investigate the relative importance of the apparent background events with respect to the observed number of aftershocks of one cascade. According to equation (22)

display math

i.e., a significant fraction nna of events of the actually observed triggered events are falsely identified as background events (since all events are really triggered from a single main shock in our example). For md = m0, the ratio is zero, since no events trigger below the detection threshold. However, as md increases above m0 and more and more events fall below md and become unobserved, the fraction increases until na goes to zero and the ratio approaches n. This effect increases with decreasing α. Small values of α generally place more importance on the cumulative triggering of small earthquakes.

Figure 7.

Ratio of the number of apparent background events Sa over the total observed number Nobs of aftershocks of one cascade varies as nna. Here we show the ratio as a function of the branching ratio n by assuming a particular value of m0. For m0 = md (solid, coinciding with x axis), there are no apparent background sources. For m0 less than md, the ratio increases as more and more of the observed events are triggered by unobserved events. As examples, we show the ratio Sa/Nobs for m0 = 0 (upper solid line), m0 = −5 (dashed), and m0 = −10 (dash-dotted) as a function of the branching ratio n (average number of aftershocks per earthquake also equal to the fraction of aftershocks in a catalog) for parameters md = 3, mmax = 8, b = 1, and α = 1.0. For very small m0, na approaches zero, and the ratio Sa/Nobs approaches its limiting value n, meaning that almost all observed earthquakes were triggered by events below the detection threshold md. The effect of unobserved events triggering observed quakes resulting in an apparent background source rate is further amplified by smaller values of α (not shown).

[38] Expressions (10) and (22) show that analyzing the tree structure of triggered seismicity only above the detection threshold leads to the introduction of an apparent source Sa and an apparent branching ratio na. It is important to realize that both are renormalized simultaneously by using catalogs with md > m0. An unbiased inversion method for the parameters of this averaged, deterministic approximation of the fully stochastic ETAS model would retrieve our analytical results (10) and (22). We conjecture that our time- and space-integrated, magnitude-averaged and clustered version approximates the full ETAS model (equation (9)) well enough so that this bias persists for inversions of parameters of the full model. Accordingly, the value of the background source would be overestimated and the branching ratio underestimated. In fact, one single true sequence will appear as many different sequences, each apparently set off by an apparent background event. Finally, it can be shown (see Appendix A) that the sum of the above water cascade and the cascades due to the apparent background events equal the total number of observed earthquakes, demonstrating the consistency of our decomposition.

[39] Furthermore, we can extend the present approach to a whole catalog that consists of many clusters, each of which is analyzed from the same point of view as above. The calculations are presented in Appendix B and consist simply of summing over all clusters, each of which has been decomposed into a possible real observed source, its resulting above water cascade, the apparent sources and their cascades. We also show the consistency of this decomposition.

[40] We now come back to examine the assumptions made in this work. First, we assumed that events could be clustered into distinct sets that are set off by a real or an apparent source. Second, we approximated the number of aftershocks of each real or apparent source by averaging over the magnitudes of the triggered events. Third, we integrated over time and space so that we could concentrate on pure numbers of events only. In other words, we have removed all stochasticity of the model. Under these assumptions, we have shown that in introducing a detection threshold md, one renormalizes the ETAS model onto itself but with effective parameters Sa and na. The functional form of the model remains the same. However, we have not proved that the instantaneous and stochastic ETAS model as described by equation (9) can be renormalized exactly onto itself with effective parameters for md > m0. For this, we would have to check that all fractional moments (that exist) and all distributions describing the stochastic seismic rates are the same (1) in the catalogs generated by ETAS with md > m0 and (2) in the catalogs generated by the effective ETAS with minimum magnitude of triggering taken equal to md and with the corresponding adequate values of the effective parameters. Our present paper has just shown the already nontrivial result that the first moment (average) of seismic rates of the catalogs of 1 and 2 are identical for the choice of the apparent parameters (10), (11) and (19).

4. Conclusions

[41] We have shown that unbiased estimates of the fraction of aftershocks and the number of independent background events are simultaneously renormalized to apparent values when the smallest triggering earthquake m0 is smaller than the detection threshold md. In summary, main shocks above the threshold will appear to have fewer aftershocks, resulting in a smaller apparent branching ratio. Meanwhile, unobserved events can trigger events above the threshold giving rise to apparently independent background events that seem to increase the constant background rate to an apparent rate. Assuming that current techniques which are used to invert for the parameters of the ETAS model (for example, the maximum likelihood method) under the assumption md = m0 are unbiased estimators of na and Sa, then the obtained values for the fraction of aftershocks and the background source rate correspond to renormalized values because of the assumption that the detection threshold md equals the smallest triggering earthquake m0. We predict that n will be drastically underestimated and S strongly overestimated for m0 much smaller than md.

Appendix A:: Consistency Check: Nobs as the Sum of Above Water Cascades Triggered by the Main Shock and by the Apparent Background Events

[42] To complete the calculations and show consistency of the results, we demonstrate that the observed cascades set off by the apparent background events, when added to the original above water cascade, add up to the total observed number of aftershocks of the whole sequence. Each apparent source event will trigger its own cascade above the threshold md with branching ratio na. The total number of events due to the apparent background events and their cascades above the threshold is

display math

Substituting expression (22) and using (8) gives

display math

Combining the direct above water cascade (18) with the apparent source cascades (A2) gives the total amount of apparent events observed after the initial event

display math

where Nobs is given by (8). The last equality confirms the consistency of our decomposition into apparently triggered earthquakes and apparent sources.

Appendix B:: Generalization to a Catalog of an Arbitrary Number of Clusters

[43] In this section, we generalize our analysis of the apparent branching structure of one cluster to that of a whole catalog consisting of an arbitrary number of clusters. The reasoning developed in section 3.1 can be directly applied as follows.

[44] We begin by writing the instantaneous seismicity rate at time t:

display math

We integrate this expression over time to obtain the total number N of earthquakes in the catalog. We restrict this demonstration to the temporal domain. In order to have a finite catalog, we assume that the integral over the background source rate μ is finite so that the total number s of background events is finite. Stated differently, the integration could also be over a finite but very long period T so that the sources s = μT are finite but the Omori law decays have effectively ended. We thus obtain

display math

where the index i runs over all events.

[45] We now express the total number of earthquakes by grouping each event into one of the s distinct clusters and by averaging over the magnitudes of the indirectly triggered events of the initial background event that set off the cascade. Now the total number is simply the s background events plus the triggered events in their kth cluster, which are averaged over the aftershock magnitudes in the same way as for one cluster in equation (6):

display math

where the index k now only runs over the background events.

[46] The observed number of shocks is expression (B3) multiplied by the fraction of observed events:

display math

[47] Now we can apply to each cluster the same arguments as we did in section 3.1. Let us denote the number of unobserved background events below the detection threshold md by u = 1,… U, so that U = s (1 − fobs), while we call the number of observed background events l = 1, …, L, i.e L = sfobs. Then, the total number of events from all the L above water sequences due to the observed real sources is

display math

where the index l runs over all observed real sources.

[48] The number of apparent sources for each cluster is given by equation (22) in section 3.1. For the whole catalog, the total number of apparent sources is thus the sum of observed real sources, the apparent sources in clusters due to observed sources and the apparent sources from clusters of unobserved sources:

display math

This expression shows that the apparent branching structure renormalizes L = sfobs observed background events into Satot apparent background events. Together with the apparent branching ratio, this completely determines the renormalization of the model when going from m0 to md.

[49] As for the one cluster case, we can check the consistency of our decomposition by testing whether the apparent number of events Na is equal to the observed number of events Nobstot given by (B4). From our decomposition, Na is given by

display math

Substituting the relevant expressions, one can easily show that Na = Nobstot and that our decomposition is consistent. Note that we have to subtract the real observed sources L from the apparent sources Satot because the cascades they set off have already been taken into account in the above water cascade Nabovetot.

[50] In summary, we have generalized the approach to the case of many cascades. We have shown that analyzing the branching structure above the detection threshold of a complete catalog leads to a renormalized ETAS model (for the averaged rates) with an apparent branching ratio na and an apparent number of sources Satot.


[51] We acknowledge useful discussions with A. Helmstetter and J. Zhuang and thank the Associate Editor Frederik Simons, Ian Main, and an anonymous referee for their constructive suggestions. This work is partially supported by NSF-EAR02-30429 and by the Southern California Earthquake Center (SCEC). SCEC is funded by NSF Cooperative Agreement EAR-0106924 and USGS Cooperative Agreement 02HQAG0008. M.J.W. gratefully acknowledges financial support from a NASA Earth System Science Graduate Student Fellowship. SCEC contribution 860.