Assessing the dependency between the magnitudes of earthquakes and the magnitudes of their aftershocks

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1 Research Article Received: 3 September 2013, Revised: 30 January 2014, Accepted: 31 January 2014, Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: /env.2268 Assessing the dependency between the magnitudes of earthquakes and the magnitudes of their aftershocks Kevin Nichols a * and Frederic Paik Schoenberg b Current assumptions on the level of inter-dependency between the magnitudes of earthquakes and the magnitudes of their aftershocks vary significantly, ranging from assumed independence for models like epidemic type aftershock-sequence (ETAS) to assumed positive correlation, at least for the events leading up to the largest earthquakes, for models like accelerated moment release. Recently, a method was developed by Marsan and Lengline for model independent stochastic de-clustering (MISD) of a multidimensional Hawkes self-exciting process, or as is the case here, earthquake activity. After expanding the algorithm to allow for a spatially dependent background rate (henceforth SDMISD), repeated application of SDMISD to a global earthquake catalog allows for estimates with confidence bounds for the average magnitude of an aftershock conditioned upon the magnitude(s) of the earthquake(s) that caused the aftershock. 15 years of data on M 5.3 earthquakes from the global Centroid Moment Tensor earthquake catalog indicate that the magnitudes of aftershocks are dependent on the magnitudes of the earthquakes that cause them, and indeed, larger than average earthquakes are more likely to trigger larger than average aftershocks. Copyright 2014 John Wiley & Sons, Ltd. Keywords: earthquake magnitude; mainshock; aftershock; stochastic de-clustering; ETAS; MISD: SDMISD 1. INTRODUCTION The epidemic type aftershock-sequence (ETAS) model (Ogata 1988, 1998) parametrically estimates the rates of earthquake occurrences over a spatial temporal support. The ETAS model identifies two sources for earthquakes. They are either mainshocks, with rate determined by a spatially dependent background rate, or they are aftershocks, with rate determined by the spatial and temporal coordinates, as well as the magnitudes, of previously occurring earthquakes. Assumedly, both mainshocks and aftershocks are capable of triggering additional aftershocks. Ogata (1998) established the following extension of a Hawkes self-exciting point-process model (Hawkes, 1971) to model the spatial temporal conditional rate of earthquake occurrence:.t; x; y; H t / D.x; y/ C X g.t t i ;x x i ;y y i I M i / (1) iwt i <t where H t represents the history of earthquakes occurring prior to time.t/,.x; y/ represents the background rate of a mainshock earthquake occurring over the spatial temporal support, and P g.t t i ;x x i ;y y i I M i / represents the rate at which aftershock earthquakes occur. Parameterization of g.:/ is often difficult when considering that it is difficult to determine which earthquakes are aftershocks. Once de-clustered, nonparametric methods for estimating g.:/ will prove helpful in determining the level of dependency between earthquake magnitudes and the magnitudes of their aftershocks. One key assumption of most earthquake models is that the magnitudes of earthquakes roughly follow the exponential distribution (f ) (Gutenberg and Richter, 1944), meaning that Z 1 M 0 f.m/dm D e ˇM 0 (2) Parametrization of this density f varies from catalog to catalog but generally is estimated using expectation maximization techniques. * Correspondence to: Kevin Nichols, McCarthy Hall 154, Department of Mathematics, California State University, 800 N. State College Blvd., Fullerton, CA 92831, U.S.A. knichols@fullerton.edu a b McCarthy Hall 154, Department of Mathematics, California State University, 800 N. State College Blvd., Fullerton, CA 92831, U.S.A. Department of Statistics, University of California, Los Angeles, CA, U.S.A. (2014) Copyright 2014 John Wiley & Sons, Ltd.

2 K. NICHOLS AND F. P. SCHOENBERG A key feature of ETAS models, see Schoenberg (2003), is the assumption of separability between the magnitude of an earthquake and the magnitudes of any aftershocks produced by that earthquake. An extension of spatial ETAS models (1) to incorporate the magnitude of potential earthquakes is, 0 1.t; x; y; M; H t / D y/ C X g.t t i ;x x i ;y y i I M i / A (3) iwt i <t which inherently assumes that the magnitude of an aftershock earthquake is independent of the magnitude of the earthquake that produced it (Schoenberg, 2003). In contrast to the assumptions carried by ETAS models for earthquake occurrence, consider now the accelerated moment release model (AMR), for references see Bowman et al., (1998), Bowman and King (2001), Jaumé and Sykes (1999). AMR theory assumes that the potential energy, or more specifically the rate of seismic moment release in moderate earthquakes, increases prior to the occurrence of larger earthquakes (Jaumé and Sykes, 1999). Specifically, the cumulative Benioff strain.t/, which is a measure of seismic energy or seismic potential, can be fitted by a relation of the form: N.t/ X.t/ D E i.t/ 1 2 id1 (4) where N.t/ represents the number of earthquakes occurring prior to time t within a set spatial window, usually restricted to the range of earthquakes associated with the same fault, of the present location, and E i represents the energy of each previously occurring event (Bowman and King, 2001). Most studies supporting AMR theory focus on the earthquakes that precede the very largest earthquakes in historical record (Bowman et al., 1998; Bowman and King 2001; Jaume and Sykes 1999). The inherent assumption being that the magnitudes of the earthquakes preceding the largest earthquakes are in fact inseparable from each other, that is, there is some degree of dependence or positive correlation between the magnitudes of earthquakes that occur in clusters preceding the largest earthquakes. While this contradicts the assumption of ETAS models, the contradiction frames the debate in recent literature as to whether or not the magnitudes of earthquakes and the magnitudes of their aftershocks are correlated. The research performed by Lippiello et al., (2007), Lippiello et al., (2008), Lippiello et al., (2009) indicates there is a positive correlation between earthquakes and the earthquakes that occur after them. However, the research of (Davidsen and Green, 2011) concludes the hypothesis that earthquakes are correlated in magnitude with the earthquakes that follow them cannot be rejected, and any positive correlation that may have been noticed is mainly a result of short-term aftershock incompleteness (STAI). There may be a simple reason as to why the conclusions on the subject are so varied. To date, work on the subject has focused on correlations between one earthquake and the next earthquake(s) to follow it. This does not address the question of interest, namely are the magnitudes earthquakes correlated with the magnitudes of their aftershocks, but rather addresses the question of whether or not the magnitudes of earthquakes are correlated with the magnitudes of earthquakes that occur after them. Perhaps the reason why research has not focused on interdependency between earthquakes and their aftershocks is because it is difficult to determine which earthquakes are main shocks, which earthquakes are aftershocks, and which previously occurring earthquake triggered an aftershock. There have been several proposed methodologies for separating an earthquake catalog into background or primary events and secondary or aftershock events. Most of these methodologies essentially remove the earthquakes in a small spatial temporal window around larger events and assign remaining events as mainshocks; for examples, see Gardener and Knopoff (1990), and Utsu (1995). The obvious problems with this approach are that main shock earthquakes that occur near large earthquakes are misclassified as aftershocks and that smaller earthquakes are not considered self-exciting. In fact, these methods for stochastically de-clustering earthquake catalogs leads to the common misuse of the term aftershock as any earthquake occurring after a large earthquake and misuse of the term foreshock an any earthquake occurring before a large earthquake. Alternative to these window based methodologies are methods that consider the space-time distance between events and then parametrically determine if an event is a mainshock or an aftershock depending on its proximity in both space and time to preceding earthquakes (e.g. Reasenberg, 1985; Davis and Frohlich, 1991). These types of methods are still used today in determining spatial temporal windows for aftershock occurrence; for example, see how aftershocks are defined in (Lippiello et al., 2009). Unfortunately, these methods require parameterization for defining the shape of the aftershock windows or for determining the significance of the link between two events and ultimately underestimate the probability that smaller earthquakes trigger aftershocks. Marsan and Lengline (2008) developed a model independent stochastic de-clustering (MISD) data algorithm for nonparametrically estimating background rate and triggering function g.t t i ;x x i ;y y i I M i /. Consequently, MISD allows for probabilistic assignment of each event as a mainshock or an aftershock, with individual probabilities assigned to each previously occurring earthquake as the trigger for the aftershock. In this paper, we expand the capability of the algorithm to allow for a spatially dependent background rate.x; y/, hence forth spatially dependent model independent stochastic de-clustering (SDMISD). Model independent stochastic de-clustering is an expectation-maximization algorithm. It should be noted that some work has been carried out on improving the MISD algorithm, most notably through Maximized Penalized Likelihood Estimation to increase the accuracy of the algorithm (Lewis and Mohler, 2011). MISD can readily be applied to global earthquake catalogs. MISD does not allow for the error-free classification of events as either mainshocks or aftershocks. However, repeatedly assigning events as mainshocks or aftershocks, and in the case of aftershocks determining the earthquake that triggered the aftershock, will produce a set of pairings between earthquakes and their aftershocks in which the correct lineage will have higher probability and therefor greater representation than any incorrect pairing. This provides an opportunity to observe the dependency of earthquake magnitudes and their aftershock magnitudes. wileyonlinelibrary.com/journal/environmetrics Copyright 2014 John Wiley & Sons, Ltd. (2014)

3 MAGNITUDE DEPENDENCE 2. DATA Fifteen years of earthquake data, ranging from 1996 to 2010, was obtained from the Centroid Moment Tensor (CMT) global catalog (gcmt). The data is freely available online and can be found in NDK format at NDK format includes the following information of interest: latitudes, longitudes, dates, times, and scalar moments M sc, which can in turn easily be converted into moment magnitudes M using the US Geological Survery (USGS) earthquake magnitude policy conversion formula: M D log 10.M sc / 10:7 (5) 1:5 Figure 1 is a global plot of all earthquakes contained within the temporal bounds set for the catalog. There are several different measures of magnitude when it comes to earthquakes, including the Richter scale, measures of fault geometry and seismic moment, measures of energy released, and measures of wavelength such as surface wave length magnitudes and body wave length magnitudes. However, the reason that moment magnitudes are ideal when compared with other choices is that moment magnitudes do not saturate for and after larger events. The consequences of saturation are that typically other measures of earthquake magnitude tend to underestimate the magnitudes of the larger events. As such, for our measure of earthquake moment, we chose moment magnitudes though it is important to note that the forthcoming methods could easily accommodate a separate choice of measure for earthquake magnitude. Most earthquake catalogs are believed to be incomplete for smaller earthquakes, meaning that small earthquakes are often unnoticed or unrecorded in seismology data. Additionally, following the largest earthquakes, there is a period of STAI as seismic instruments for detecting earthquakes are temporarily saturated (Kagan 2004, Zhuang et al., 2004, Helmstetter et al., 2006, Lippiello et al., 2007, Lippiello et al., 2008, Davidsen and Green 2011). As a result, small earthquakes occurring during this period of saturation go undetected by seismic instruments. To address these issues with incompleteness of the catalog, the minimum magnitude of completeness M c is approximated by computing the low-magnitude breakpoint using median based analysis of the segment slope (MBASS) (Amorese, 2007). MBASS is a bootstrap method that essentially assesses at what point the slope of the log-survivor function is linear (as it should be for a complete realization of an Figure 1. Map with spatial occurrences of CMT global catalog earthquakes occurring between 1996 and 2010 Figure 2. The median based analysis of segment slope using a survivor function. Data appears approximately exponentially distributed for events > 5.3 (2014) Copyright 2014 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/environmetrics

4 K. NICHOLS AND F. P. SCHOENBERG exponentially distributed random variable like earthquake magnitudes). The average (and median) low-magnitude breakpoint after 1,000 bootstrap iterations of MBASS for the catalog used in this analysis is M c D 5:3. It is fair to assume that the catalog is relatively complete for moment magnitudes (M > 5:3). Figure 2 is a plot of magnitude against the log frequency of earthquakes greater than that magnitude and helps to visually confirm completeness of the catalog for events with magnitude greater than 5.3 as M c D 5:3 is the lowest magnitude cut-off such that the log-survivor function plot of all data above M c is linear (Wiemer and Wyss, 2000). As incompleteness of events smaller than 5.3 in the catalog may contribute to spurious results any events with magnitude less than 5.3 have been deleted from the catalog in this analysis. Additionally, in consideration of STAI, the relationship between the minimum magnitude of completeness M c.t; m/ and the time t (in days) following an earthquake of magnitude m is M c.t; m/ D m 4:5 0:75log 10.t/ (Helmstetter et al., 2006). In this analysis, to consider earthquakes during periods in which M c is less than the overall minimum magnitude of completeness can lead to spurious results. As such, any earthquakes occurring during a time period in which M c.t; m/ < 5:3 were deleted from the analysis. It should be noted that the impact of removing earthquakes because of STAI was minimal as the minimum magnitude of completeness for the gcmt is relatively high compared to other catalogs. For more information on the gcmt, please see (Ekstrom et al., 2012). The result is a 15-year period in which 11,535.M > 5:3/ earthquakes are used for the forthcoming analyses. 3. METHODS The goal of the proposed methodology is to observe any trends of dependency between the magnitudes of aftershock earthquakes and the earthquakes that trigger them. This first requires stochastic de-clustering of the data. Once the catalog has been de-clustered, conditional averages for aftershock magnitude can be computed by grouping the magnitudes of aftershocks that come from earthquakes of comparable magnitude. While SDMISD allows us to achieve the desired classification of each earthquake as a mainshock or as an aftershock triggered by a previously occurring earthquake, admittedly there will be potential nonsystematic misclassifications. However, by repeatedly applying SDMISD to the same catalog, and assuming that catalog has a reasonably large number of events, on average the SDMISD will correctly compute the lineage between earthquakes and their aftershocks more often than any misclassification. By repeating the process, it is possible to compute the conditional average of aftershock magnitudes conditioned upon the magnitudes of the earthquakes that triggered the aftershock. Additionally, repeating the process allows for an examination on the distribution of conditional means. Using these distributions of conditional aftershock means, confidence intervals are computed for the average aftershock magnitude caused by earthquakes with similar magnitudes. Forthcoming, this section shall be split into three parts. The first is dedicated to an examination of MISD algorithm and how we can implement kernel smoothing to estimate a nonconstant background rate, thus developing SDMISD. In addition to developing SDMISD itself, we implement a methodology for examining results for SDMISD under two competing hypotheses, namely the null hypothesis in which we assume that earthquake magnitudes are independent of each other and the alternative hypothesis in which we assume earthquake magnitudes are dependent on each other. To simulate under the null hypothesis, we can simply re-create our catalog and impose independence by randomly shuffling the magnitudes of the earthquakes in the catalog. To simulate under various versions of the alternative hypothesis will require the creation of synthetic catalogs. The second part of this section is dedicated toward the creation of realistic multidimensional synthetic catalogs. The advantage of illustrating the method using synthetic catalogs before using the global CMT catalog is that we can obtain a descriptive sense of results that are typical for various levels of earthquake magnitude correlation. Finally, in the third part, a methodology is described for repeatedly using SDMISD to create estimates and confidence intervals for the conditional means of aftershock magnitudes depending on the magnitudes of the earthquakes that trigger them MISD and SDMISD The following is a summary of the methodology for probabilistically determining connectivity between earthquakes. For greater detail, please see (Marsan and Lengline, 2008). The methodology begins with a simplified version of (1):.x; y; t/ D 0 C X t i <t i.x;y;t;m/ (6) where.x; y; t/ represents the rate at which earthquakes occur in space time and is the sum of a constant background rate for mainshock occurrence, 0, and the rates for all previously occurring earthquakes to trigger an aftershock, i.x; y; t/. It is important to note the contribution of a previously occurring earthquake at location (x i,y i ), time (t i ), and with magnitude (M i ) to the rate of earthquake occurrence at location.x; y/ and time.t/ is a nonparametric function of distances j.x; y/.x i ;y i /j, t t i and magnitude M i. Model independent stochastic declustering for determining 0 and i.x; y; t/ begins with a set of distances between any pairing of events in the data as well as an aprioriestimate for.j.x; y/j; t;m/and for 0. The algorithm begins by estimating triggering weights w i;j = j.j.x j ;y j /.x i ;y i /j;t j t i ;M i / if t i <t j, w i;j D 0; otherwise, and background weights w 0;j D j 0. j is a normalizing coefficient such that P.w i;j / C w 0;j D 1 for events i occurring prior to event j. The rate contributions of previously occurring earthquakes are then updated as.j.x; y/j; t;m/d 1 N M ıt S.j.x; y/j;ır/ X i;j2a w i;j (7) wileyonlinelibrary.com/journal/environmetrics Copyright 2014 John Wiley & Sons, Ltd. (2014)

5 MAGNITUDE DEPENDENCE where A consists of all pairs such that j.x j ;y j /.x i ;y i /j2j.x; y/j ır, t j t i 2 t ıt and M i 2 M ım (ır;ıt;ım are discretization parameters), N M is the number of earthquakes such that M i 2 M ım,ands.j.x; y/j/ is the surface covered by the disk with radii jı.x; y/j ır. The updated background rate is 0 D 1 NX w 0;j T S j D1 (8) where T and S represent the length of the temporal support and the surface area of the spatial support, respectively. After an initialization of 0 and..x; y/; t; M /, the process of updating weights w 0;j and w i;j and then updating 0 and..x; y/; t; M / continues until convergence. Once MISD has converged the j th event is identified as a mainshock with probability w 0;j P wi;j C w 0;j (9) and as an aftershock triggered by the ith event with probability w i;j P wi;j C w 0;j (10) While MISD is desirable in the sense that it allows for nonparametric definition of the Poissonian (lambda) rate space triggering function g, the requirement for constant background rate is sub-optimal. Typically, to nonparametrically estimate the rate of earthquake occurrence, we would consider kernel smoothing a density function. Literature will show that often it is not the choice of the kernel that matters all that much for kernel smoothing but rather the choice of smoothing parameter. Similar to (Helmstetter et al., 2006), we use an isotropic kernel: K.d/ D C.d/ r 2 C d 2 1:5 (11) where C(d) is normalizing factor and d is the bandwidth determined on a per event basis using cross-validation; see (Helmstetter et al., 2006) for greater detail. Unfortunately, the desired result is a background rate for all earthquakes, when we desire a background rate for mainshock earthquakes. To account for this, we consider the following adaptation to (8). Let 0;j be the background rate at the j th catalog event, K j be the estimated kernel density at the location of the j th catalog event, and KN be the average kernel density across the locations of all events. We substitute for (8) the following: K j NX 0;j D w T S KN 0;j j D1 (12) and then update background rate w 0;j D j 0;j. Note that the mean of 0;j from (12) is equal to 0 from (8) Synthetic catalogs Multiple dimensional earthquake catalogs were simulated under various levels of inter-dependency between primary events and secondary events. Catalogs were simulated by first creating a spatially dependent background rate and randomly generating primary events over a uniform temporal domain. This was performed by kernel smoothing points tracing line segments that represent synthetic fault lines, then sampling coordinates from the kernel support. Magnitudes for the primary events were simulated in accordance with Gutenberg Richter laws by using inversion techniques with an exponential distribution in which the minimum magnitude of completeness set to 5.3, which is the same as our global catalog. The parameter of the exponential distribution used to sample primary event magnitudes was selected so that the expectation of earthquake magnitude in our synthetic catalog was equal to the average earthquake in our global catalog. After generating synthetic primary events, first generation secondary events were generated. The expected number of secondary events (N a ft.m/) over the temporal domain.t / for primary event with magnitude.m/, following a brief period of STAI.c/ was determined in accordance with generalized Omori law (see Helmstetter et al., 2005 for greater detail): N af t.m/ D K m ˇm d / T.1 p/ c.1 p/ 1 p (13) where p D 0:9, K 0 = , D ˇ D 1. Using this expectation as the parameter of a Poisson distribution, the number of secondary events associated with each primary event was generated. The temporal coordinates of secondary events were determined using generalized (2014) Copyright 2014 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/environmetrics

6 K. NICHOLS AND F. P. SCHOENBERG Omori Law by sampling from (13) using the inversion method. The spatial coordinates of secondary events were generated by randomly sampling a uniform direction (), and a distance.d/ from an adapted version of (13): f.d/d K 010. m ˇm d /.d/ p (14) The magnitudes of secondary events triggered M af t.m i / triggered by event m i were generated using the following equation: M af t.m i / D.1 a/m r C am i (15) where.m r / is a new randomly generated primary event. As (a) increases, so does interdependency between primary events and secondary events. After first generation aftershocks are generated from mainshocks, second generation aftershocks are generated from first generation aftershocks and so forth until randomly no further aftershocks are generated Estimates and confidence intervals Repeated probabilistic assignment of earthquakes as either mainshocks or aftershocks creates a large set of earthquake pairings for us to examine. From these pairings, and for each probabilistic assignment of all the earthquakes in a catalog, we can easily compute the average magnitude of aftershocks caused by earthquakes of equal magnitude. After 1,000 such assignments, we can create frequency based confidence intervals for the average aftershock resulting from a mainshock of size.m /. After 1000 of such classifications, the 5th and 95th percentiles for the average aftershock caused by an earthquake of magnitude M serve as realistic 90% confidence bounds on the expected aftershock caused by an earthquake of magnitude M (Efron and Tibshirani, 1986). The conditional expected aftershock caused by an earthquake of size M can then be used inversely to estimate the parameter of a conditional exponential distribution f for earthquake aftershocks caused by an earthquake of size M. Average Aftershock (90% confidence) Average Aftershock (90% confidence) Magnitude of Parent Shock Magnitude of Parent Shock Average Aftershock (90% confidence) Average Aftershock (90% confidence) Magnitude of Parent Shock Magnitude of Parent Shock Figure 3. Simulation based 90% confidence intervals for the average aftershock magnitude (y-axis) corresponding to the magnitude the previously occurring earthquake that excited the aftershock (x-axis). Results were simulated for self-exciting earthquake and aftershock magnitudes are independent (Figure 3(a), top left, 0:0), weakly positively correlated (Figure 3(b), top right, 0:3), positively correlated (Figure 3(c), bottom left, 0:55), and strongly positively correlated (Figure 3(d), bottom right, 0:75) wileyonlinelibrary.com/journal/environmetrics Copyright 2014 John Wiley & Sons, Ltd. (2014)

7 MAGNITUDE DEPENDENCE 4. RESULTS Figure 3 is a plot of the conditional means with confidence intervals of aftershock magnitude as a function of the magnitudes of the earthquakes that come from for the four synthetic catalogs. Important features for Figure 3 include the increase in variability for the conditional averages as the magnitudes of the triggering earthquakes increase. Figure 4 is a display of the average magnitude of aftershocks with confidence intervals conditioned upon the magnitude of the earthquakes they come from. Figure 5 is the same display, only this time the described analyses were performed after randomly shuffling the magnitudes of the earthquakes. Contrasting Figure 4 both with the results of the synthetic catalogs and with the results from imposing independence on the magnitudes of earthquakes and their aftershocks through random shuffling yields the following observations. Table 1 contains the estimated 90% confidence bounds for various earthquake magnitude ranges. Table 1 indicated there is very strong evidence that the magnitudes of earthquakes are correlated with the magnitudes of their aftershocks. The correlation between the magnitudes of earthquakes and their aftershocks is moderate to strong (compare with results from synthetic catalogs in which the correlation was 0.55 or 0.75). It may appear that positive trend does not seem to hold for the largest earthquakes (M > 7). Note, however, that there were very few earthquakes with such magnitudes and with so little data SDMISD struggles to define the triggering function. The result is severe variation in the confidence interval estimates. These are significant results that indicate that the local history of earthquakes are indeed powerful components in modeling not only the rate of future earthquake occurrences but also the magnitudes of future earthquakes. Figure 4. Ninety percent confidence intervals for the average aftershock magnitude (y-axis) associated with varying foreshock magnitudes (M) such that M > 5:3 (x-axis). Results are based on 1000 applications of model independent stochastic declustering (MISD) to the CMT catalog. Horizontal line represents the average magnitude of all aftershocks Figure 5. Ninety percent confidence intervals for the average aftershock magnitude (y-axis) associated with varying foreshock magnitudes (M) such that M > 5:3 (x-axis). Results are based on 1000 applications of model independent stochastic declustering (MISD) to the CMT catalog with randomly shuffled magnitudes. Horizontal line represents the average magnitude of all aftershocks (2014) Copyright 2014 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/environmetrics

8 K. NICHOLS AND F. P. SCHOENBERG Table 1. Ninety percent confidence interval bounds for the average aftershock magnitude associated with varying foreshock magnitudes (M) such that M > 5:3 M Lower bound Upper bound M Lower bound Upper bound DISCUSSION The methodology described in this article has yielded results that potentially could influence the ways in which popular models like ETAS model earthquake rates. An immediate impact could be the inclusion of a conditional term for the distribution of earthquake magnitude depending on whether or not the earthquake is a mainshock or an aftershock. Updating (3) to reflect the findings of this article could potentially yield a new form of the ETAS model in which.t; x; y; M; H t / D f 1.M /.x; y/ C X iwt i <t f 2.M jm i / g.t t i ;x x i ;y y i I M i / (16) where f 1.M / and f 2.M jm i / represent the distribution of mainshock earthquake magnitudes and the distribution of aftershock earthquake magnitudes conditioned upon their triggering earthquake magnitude, respectively. Note that both f 1 and f 2 could easily be nonparametrically estimated using the methodology prescribed in this article and a reasonably sized catalog. REFERENCES Amorese D Applying a change-point detection method on frequency-magnitude distributions. Bulletin of the Seismological Society of America 97: , DOI: / Bowman D, King G Accelarating seismicity and stress accumulation before large earthquakes. Geophysical Research Letters 28(21): Bowman D, Ouillon G, Sammis CG, Sornette D, Sornette A An observational test of the critical earthquake concept. Journal of Geophysics 103(24): Davidsen J, Green A Are earthquake magnitudes clustered? Physical Review Letters 106: , DOI: /PhysRevLett Davis SD, Frohlich C Single-link cluster analysis, synthetic earthquake catalogs, and aftershock identification. Geophysical Journal International 104: Efron B, Tibshirani R Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science 1(1): Ekstrom G, Nettles M, Dziewonski A The global CMT project : centroid-moment tensors for 13,017 earthquakes. Physics of the Earth and Planetary Interiors 200:1 9, DOI: /j.pepi Gardner J, Knopoff L Is the sequence of earthquakes in southern California, with aftershock removed, Poissonian? Bulletin of the Seismological Society of America 64: Gutenberg B, Richter C Frequency of earthquakes in California. Bulletin of the Seismological Society of America 34: Hawkes AG Spectra of some self-exciting and mutually exciting point processes. Biometrika 58: Helmstetter A, Kagan Y, Jackson D Importance of small earthquakes for stress transfers and earthquake triggering. Journal of Geophysical Research 110(B05S08):1 13, DOI: /2004JB Helmstetter A, Kagan Y, Jackson D Comparison of short-term and time-independent earthquake forecast models for southern California. Bulletin of the Seismological Society of America 96: wileyonlinelibrary.com/journal/environmetrics Copyright 2014 John Wiley & Sons, Ltd. (2014)

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