Comparison of ETAS parameter estimates across1

different global tectonic zones2

Chu, A.1, Schoenberg, F.P.2, Bird, P.3, Jackson, D.D.3, and Kagan, Y.Y.33

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1 The Institute of Statistical Mathematics5

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2 Department of Statistics, University of California, Los Angeles7

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3 Department of Earth and Space Sciences, University of California, Los Angeles9

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Email: chu@ism.ac.jp11

Postal address: 10-3 Midori-cho,12

Tachikawa, Tokyo 190-8562, Japan13

1

Abstract14

Branching point process models such as the ETAS (Epidemic-Type Aftershock Se-15

quence) models introduced by Ogata (1988, 1998) are often used in the description,16

characterization, simulation, and declustering of modern earthquake catalogs. The17

present work investigates how the parameters in these models vary across different tec-18

tonic zones. After considering divisions of the surface of the Earth into several zones19

based on the plate boundary model of Bird (2003), ETAS models are fit to the occur-20

rence times and locations of shallow earthquakes within each zone. Computationally,21

the EM-type algorithm of Veen and Schoenberg (2008) is employed for the purpose of22

model fitting. The fits and variations in parameter estimates for distinct zones are com-23

pared. Seismological explanations for the differences between the parameter estimates24

for the various zones are considered, and implications for seismic hazard estimation25

and earthquake forecasting are discussed.26

Keywords: branching processes, earthquakes, epidemic-type aftershock sequence model,27

space-time point process models, maximum likelihood, global earthquake prediction, plate28

tectonics.29

30

1 Introduction31

Researchers have recently defined global tectonic zones according to geophysical character-32

istics such as crust types and relative plate velocities using plate boundary models (Bird,33

2003; Bird and Kagan, 2004; Kagan et al., 2010). The primary purpose of this study is34

2

to investigate the different properties and patterns of seismicity in these different zones by35

exploring the fit of space-time branching point process models to the data within each zone.36

Similarities and differences between the background rates and clustering behavior of earth-37

quake occurrences in different tectonic zones may shed light on the interplay between tectonic38

forces and observed seismicity rates.39

Similar investigations were performed by Kagan et al. (2010), using the branching point40

process model developed by Kagan (1991). Here, we explore the fit of Epidemic-Type Af-41

tershock Sequence (ETAS) models of Ogata (1998) within each tectonic zone. Such models,42

which have become widely used in the study of the properties of earthquake clustering and43

in the forecasting of seismicity, are based on the notion that earthquakes occur at some44

background rate, those background events trigger aftershocks, and those aftershocks trigger45

subsequent aftershocks, etc. ETAS models have been used extensively in the analysis of local46

catalogs (e.g. Ogata, 1998; Schoenberg, 2003; Zhuang et al., 2005; Ogata and Zhuang, 2006;47

Helmstetter et al., 2007), and of global catalogs (Lombardi and Marzocchi, 2007; Marzocchi48

and Lombardi, 2008), but we are unaware of their prior use in the analysis of global tectonic49

zones such as those delineated by Bird (2003).50

This article is organized as follows. We first present a brief description of the delineation51

of tectonic zones and global earthquake data in Section 2. Section 3 describes the ETAS52

model of Ogata (1998), as well as model diagnostics and the computational implementation53

used. Section 4 discusses the results of estimating ETAS models within each tectonic zone,54

including the maximum likelihood estimates (MLEs) of the parameters in the model, their55

standard errors, fitted triggering functions, estimates of the Gutenberg-Richter constant and56

branching ratio. Section 5 presents statistical diagnostics based on the estimated conditional57

3

intensity, and diagnostics using comparisons between the empirical aftershock rates and fitted58

triggering functions. Concluding remarks and discussion of future work follow in Section 6.59

2 Tectonic zones and earthquake catalog60

2.1 Tectonic zones61

Tectonic zones are related to the 7 plate boundary classes defined in global plate model62

PB2002 (Bird, 2003), but zones are defined differently so as to allow for earthquake clas-63

sification even if location, depth, and focal mechanism are poorly known. As displayed in64

Figures 1, there are four major zones (Kagan et al., 2010):65

1. Active continent (including continental parts of all orogens of PB2002, plus continental66

plate boundaries of PB2002).67

2. Slow-spreading ridges (oceanic crust, spreading rate < 40 mm/yr, including trans-68

forms).69

3. Fast-spreading ridges (oceanic crust, spreading rate ≥ 40 mm/yr, including trans-70

forms).71

4. Trenches (including incipient subduction, and earthquakes in outer rise or upper plate).72

In what follows, we refer to the zone consisting of the plate interiors, i.e. the rest of the73

Earth’s surface occupied by the non-colored areas in Figure 1, as zone 0.74

It is reasonable to anticipate different branching behaviors between the zones, and our75

present definition of zones is designed with consideration for practical matters that will76

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permit such testing to be conducted relatively easily. Zones are defined here as collections77

of polygons delineating the surface areas into which epicenters of shallow earthquakes may78

fall. The zones are not divided based on the depths of the earthquakes because these depths79

are usually poorly estimated unless there is local station control, so we follow Bird et al.80

(2002) in limiting our analysis simply to shallow earthquakes of depth less than 70 km. The81

resulting partition produces five major zones, and each zone contains a sufficient number of82

earthquakes within the time period considered (1973-2006) to enable branching models to83

be accurately and reliably estimated.84

Our tectonic zones are defined by objective rules implemented in a program because this85

is reproducible, easy to explain, easy to revise, and not very subject to procedural error.86

Our program assigns tectonic zones to grid points rather than attempting to draw boundary87

curves about each zone. It loops through latitudes (in 0.1o steps) and also loops through88

longitudes (in 0.1o steps) to create a grid of zone index integers which identify the zone at89

each grid point. The decision algorithm for each point is detailed in the Appendix of Kagan90

et al. (2010). The necessary data are available in digital form: elevation from ETOPO591

(Anonymous, 1988), age of seafloor from Mueller et al. (1997), plate boundaries and Euler92

poles from Bird (2003).93

The output is a relatively compact (6 to 12 MB) representation, which is shown in Figure94

1. The gridded zone integers may be downloaded from http://bemlar.ism.ac.jp/wiki/95

index.php/Bird’s_Zones.96

Note that some tectonic zones are the unions of non-continuous patches and may include97

quite different types of earthquakes. For instance, strike-slip and normal-faulting earth-98

quakes are contained in the same mid-ocean spreading ridges and continental rift zones,99

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and similarly, along trenches, subduction-related earthquakes are analyzed collectively along100

with back-arc-spreading earthquakes.101

2.2 Earthquake catalog102

The data used for our analysis were retrieved from the PDE (Preliminary Determination103

of Epicenters) database (USGS, 2008). The time window we study is January 1, 1973 to104

December 31, 2006. Only shallow earthquakes (0 to 70 kilometers in depth) are considered.105

The shallow events constitute approximately 80% of all events in the catalog. Summaries of106

catalog size and area are shown in Table 1. For each zone’s analysis, a lower magnitude cutoff107

threshold of mt 5.0 is used, following the example of Kagan et al. (2010). The PDE dataset108

with lower magnitude threshold 5.0 consists of mostly (97%) body wave (55%), surface wave109

(8%), moment magnitude (31%), and small percentage of local magnitude (3%).110

Table 1 displays each zone’s surface area, in proportion to that of the globe, and the pro-111

portion of all observed events occurring in each zone. Among these zones, zone 4 (trenches)112

has the largest number of events (64.5% out of total 40504 events) while only occupying113

2.98% of the global surface area. Zone 1 (active continent) also contains a disproportion-114

ate number of events, with 16.6% of the events in the global catalog and only 6.58% of the115

global surface area. Zones 1 and 4 are unions of irregularly shaped polygons, with one or two116

larger polygons corresponding to the orogens, and several small and strip-shaped polygons117

corresponding to the plate boundaries. Zones 2 (slow-spreading ridges) and 3 (fast-spreading118

ridges) are similar to one another in number and shape. They each have about 7% of the119

total events (7.47% for zone 2 and 6.35% for zone 3), and occupy a small portion of the120

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global surface area (3.09% for zone 2 and 1.65% for zone 3). Zones 2 and 3 have very long,121

narrow shapes completely constituted by plate boundaries. Lastly, zone 0 (plate-interior),122

the regime excluding zones 1, 2, 3 and 4, occupies 43.8% of the whole global surface and123

contains 5.06% of the global events.124

3 Model and methods125

3.1 Model126

The ETAS model is a type of Hawkes point process model; such models are also sometimes127

called branching or self-exciting point processes. For a temporal Hawkes process, the con-128

ditional rate of events at time t, given information Ht on all events prior to time t, has the129

form130

λ(t|Ht) = µ+∑i:ti<t

g(t− ti), (1)

where µ > 0 is the background rate, g(u) ≥ 0 is the triggering function dictating the rate131

of aftershock activity associated with a prior event, and∫∞0 g(u)du < 1 in order to ensure132

stationarity (Hawkes, 1971).133

These models were called epidemic by Ogata (1988) because of their natural branching134

structure: an earthquake can produce aftershocks, and these aftershocks produce their own135

aftershocks, etc. An example is the time-magnitude ETAS model of Ogata (1988), which136

has magnitude-dependent triggering function137

g(ui;mi) =K0

(ui + c)pea(mi−mt), (2)

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where ui = t − ti is the time elapsed since earthquake i, K0 > 0 is a normalizing constant138

governing the expected number of direct aftershocks triggered by earthquake i, mi denotes139

the recorded magnitude of earthquake i, and mt is the lower cutoff magnitude for the earth-140

quake catalog. The term K0/(ui + c)p describing the temporal distribution of aftershocks is141

known as the modified Omori-Utsu law (Utsu et al., 1995).142

Ogata (1998) extended the ETAS model to describe the space-time-magnitude distribu-143

tion of earthquake occurrences, introducing circular or elliptical spatial functions into the144

triggering function so that the squared distance between an aftershock and its triggering145

event follows a Pareto distribution. For instance, a form for the conditional rate proposed146

in Ogata (1998) is147

λ(t, x, y|Ht) = µ+∑i:ti<t

g(t− ti, x− xi, y − yi,mi),148

where149

g(t− ti, x− xi, y − yi,mi) = K0(t− ti + c)−p((x− xi)2 + (y − yi)2

ea(mi−mt)+ d)

−q

, (3)

and where (xi, yi) represent the Cartesian coordinates of earthquake i.150

With the triggering function (3), the spatial distribution of aftershocks interacts dramat-151

ically with mainshock magnitude. This interaction is sometimes referred to as magnitude152

scaling, since a typical feature of the model and of earthquake catalogs is a gradual widening153

of the spatial-temporal aftershock distribution as the magnitude of the mainshock increases154

(Utsu et al., 1995).155

Typically, in local catalogs, one calculates the squared epicentral distance in the Cartesian156

plane, via D2i = (x − xi)2 + (y − yi)2 as in (3), but for global seismicity it is important to157

8

account for the sphericity of the Earth. Hence in this analysis we replace this term in (3)158

with the squared great circle distance between (x, y) and (xi, yi).159

Ogata (1998) and others have extended the model (1) to include a spatially inhomoge-160

neous background rate µ(x, y). Note that in our analysis, we fit a homogeneous background161

rate in each zone, since our primary goal is to examine the differences in clustering behavior162

and overall rates between the various zones, rather than the spatial heterogeneity within163

each zone. Further, ETAS models are fit to the data within each zone individually; no trig-164

gering between zones is assumed. Implications and extensions to more complex models are165

discussed in Section 6.166

3.2 Methods167

The parameters in the ETAS model may be estimated by maximizing the log-likelihood:168

logL(θ) =∑i

logλ(ti, xi, yi|Hti)−∫ T

0

∫ y1

y0

∫ x1

x0

λ(t, xi, yi)dxdydt (4)

where θ = (µ, a, K0, c, p, d, q) is the vector of parameters to be estimated and [x0, x1] ×169

[y0, y1] × [0, T] is the space-time window where points in the marked point process dataset170

(xi, yi, ti, mi) are observed (Ogata, 1998; Daley and Vere-Jones, 2003, ch. 7). Maximum171

likelihood estimates of the parameters in the ETAS model are obtained based on the EM172

(Expectation-Maximization type algorithm) proposed in Veen and Schoenberg (2008), with173

modification to accommodate the magnitude scaling in (1) and the use of great circle distance174

mentioned in Section 3.1. The algorithm of Veen and Schoenberg (2008) is similar to the175

original, EM algorithm of Dempster, Laird and Rubin (1977), which is now widely used to176

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obtain estimates of parameters in the presence of unobserved random variables influencing177

the likelihood. The EM algorithm is based on an expectation (E) step where the expectation178

of the likelihood given the unobserved variables is computed and a maximization (M) step179

where the parameters maximizing the expected likelihood computed in the E step are found.180

The above steps are typically iterated until a certain tolerance is reached. In the EM-type181

estimation procedure proposed by Veen and Schoenberg (2008), the branching structure182

dictating which earthquake triggered which aftershocks comprises the unobserved random183

variables and is estimated according to the ETAS model in each E step. This assignment184

changes in each iteration as the parameters in the ETAS model change following each M185

step. The idea is similar to the stochastic reconstruction idea of Zhuang, Ogata and Vera-186

Jones (2002, 2004), in which the ETAS model is similarly used to assign a random branching187

structure to an observed earthquake catalog.188

Note that the ETAS model allows assignments where a triggered event has larger magni-189

tude than its parent event and that despite the fact that the branching structure estimated190

in each iteration of the EM algorithm will often be quite erroneous, the resulting estimation191

procedure nevertheless produces quite stable and accurate parameter estimates, as shown by192

Veen and Schoenberg (2008).193

For zones 1, 2, 3, and 0, the time window 1973 to 2001 was used for computation of194

MLEs (maximum likelihood estimates). Due to concerns associated with computer memory195

limitations, the time window 1999 to 2006 was used to obtain the MLEs of zone 4, the196

trench zone; we have ensured that in all of these cases, the MLEs appear already to have197

converged and stabilized within these shortened time windows. The standard error for each198

parameter estimate may be estimated using the square root of the diagonal elements of the199

10

inverse of the Hessian of the log-likelihood (Ogata, 1978). As noted in Wang et al. (2010)200

and Kagan et al. (2010), such asymptotic standard error estimates may be unreliable in201

the presence of strong correlations between parameter estimates, making the statistical sig-202

nificance of differences between parameter estimates for different tectonic zones difficult to203

assess. One method for obtaining more accurate standard errors would be to simulate the204

ETAS model repeatedly, after fitting its parameters by MLE (maximum likelihood estima-205

tion), and subsequently re-estimating the model’s parameters for each simulation. One may206

then use the standard deviation of these estimates as an approximation of the standard error.207

Such repeated simulation and re-estimation of the models is extraordinarily computationally208

intensive, however, and approximate standard errors for each parameter estimate obtained209

individually by investigating the square root of the inverse Hessian when optimizing each210

parameter individually are used as an approximation.211

In addition to studying the behavior of ETAS parameter estimates in the different tectonic212

zones, one may also investigate other properties of the models, such as their information213

gain relative to the stationary Poisson model. The information gain is defined as (Kagan214

and Knopoff, 1977; Kagan, 1991):215

I =`− `0N

(5)216

where ` is the logarithm of the likelihood for the ETAS model, `0 is the logarithm of the217

likelihood for the stationary Poisson process, and N is the catalog size.218

Since the ETAS model contains the Poisson process as a special case when the triggering219

function is 0, the information gain must necessarily be non-negative, and since the difference220

11

between ETAS and the stationary Poisson model is that the former is capable of modeling221

intense clustering behavior, higher values of the information score in this case generally222

signify increased clustering in the earthquake catalog.223

Another comparison of the triggering properties in different zones can be made by in-224

vestigating the branching ratio. Letting m = mi −mt, then G, the total expected number225

of directly triggered aftershocks per earthquake of magnitude mi within each zone, can be226

defined as the integral of (3) over time and space:227

G(m) =∫ ∞0

∫ −∞−∞

∫ −∞−∞

g(t, x, y|Ht) dxdydt = πK0(p− 1)−1c1−p(q − 1)−1d1−qeam. (6)228

In the subcritical case, the expected value E(G) =∫G(m)f(m)dm coincides with the229

branching ratio (Veen, 2006), and estimates of this branching ratio for each zone are discussed230

in Section 4.2.231

In addition to the aforementioned assessment methods, we compare the estimated condi-232

tional intensity at a grid of locations, integrated over time, with the corresponding observed233

rate of earthquake occurrences within a small rectangle around each location in the grid.234

Letting (0, T ) denote the observed time window,235

∫ T

0λ(t, x, y)dt = µT +K0

∑i

∫ T−ti

0(t+ c)−p(Ri

2/ea(mi−mt) + d)−qdt (7)

= µT +K0

∑i

(Ri2/ea(mi−mt) + d)−q(cp−1 − (T − ti + c)p−1)/(p− 1) (8)

where Ri is the distance between the location (x, y) and a previous earthquake’s location,236

(xi, yi); particularly, the great circle distance between (x, y) and (xi, yi) is used in the global237

12

case. Comparison of the integrals in (8) with observed seismicity rates for various locations238

(x, y) may be assessed directly, or via the various standardizations of these differences, as239

discussed in Baddeley et al. (2005). As a further diagnostic, we compare the estimated240

triggering function g in (3), with the empirical, observed aftershock rate. The procedure241

to obtain the empirical aftershock rate is the following: we select earthquakes in a given242

fixed magnitude range and consider their productivity of direct aftershocks by inspecting243

subsequent earthquakes within some small time-space window within the catalog. Note that244

since the background rate is much smaller than the triggering of the conditional intensity in245

(1) at such locations and times, the empirical triggering function and g should be similar in246

such regions if the model is appropriate.247

4 Results248

4.1 Maximum likelihood estimates of ETAS parameters249

The MLEs (maximum likelihood estimates), their standard errors, and the log-likelihoods250

of the ETAS model and the reference model (stationary Poisson) are shown in Table 2.251

The information gains are all larger than 1, showing apparent improvement in fit due to the252

ETAS model compared to the stationary Poisson model, and indicating significant clustering253

in each zone. Zone 0 (plate-interior) has the largest information gain, 4.20, due to the254

more pronounced spatial and temporal heterogeneity of seismicity within this zone. The255

information gains in zones 1 (active continent) and 4 (trenches) are also somewhat large256

(2.62 for zone 1 and 2.24 for zone 4), due to the high level of clustering in these two zones.257

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Zones 2 (slow-spreading ridges) and 3 (fast-spreading ridges) have the smallest information258

gains (1.79 for zone 2 and 1.20 for zone 3), as the seismicity within these zones is relatively259

homogeneous and less clustered than the other zones, so the gain due to the ETAS model,260

relative to a stationary Poisson model, is comparatively small. Note that the analyses in261

Kagan et al. (2010) use the same tectonic zone partitions and calculate the information262

gain relative to an inhomogeneous Poisson model. Kagan et al. (2010) also use different263

parametrization of their critical branching model. Here we apply homogeneous models and264

compute the information gain relative to a homogeneous Poisson model.265

The estimated background rate, µ, differs substantially from one zone to another. Not266

surprisingly, the estimate is largest in zone 4, which has by far the highest seismicity rate,267

as seen in Table 1. Zone 3 has the second largest estimated rate of background seismicity,268

but the estimate of this rate in zone 3 is only one fourth of the corresponding estimated rate269

in zone 4. The estimated background rates in zones 1 and 2 are somewhat similar to one270

another. The fact that the estimated background seismicity rate in zone 4 is approximately271

500 times that in zone 0 demonstrates the tremendous variability in overall seismicity from272

one tectonic zone to another, with the vast majority of background events concentrating in273

the trench zone.274

The MLEs of other parameters can similarly be used to summarize patterns of seismicity,275

including levels of clustering, swarms, and spatial triggering behavior, within the different276

tectonic zones. The exponential term in (2) is known as the productivity law in seismology,277

in which the parameter a governs the relationship between the magnitude of an earthquake278

and its expected number of generated offspring and is also useful in characterizing earth-279

quake sequences quantitatively in relation to their classification into seismic types (Mogi,280

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1963; Utsu, 1970; Ogata, 1988). Smaller estimates of a in zones 2 and 3 are characteristic281

of earthquake swarms, so the relatively low estimates of a in zones 2 and 3 appear to indi-282

cate a correspondingly high incidence of swarming in these tectonic zones. Conversely, the283

estimates of a are largest in zones 1 and 4, indicating a large disparity in aftershock produc-284

tivity between smaller and larger events. Lastly, a is moderately sized in zone 0, indicating285

relatively moderate distinction between large and small events in terms of the productivity286

of subsequent aftershocks in this zone.287

The parameters c and p govern the temporal decay of aftershock activity. The modified288

Omori formula describes the frequency of aftershocks per unit time interval since a preceding289

event as n(t) = K0/(t + c)p (Utsu, 1961). As seen in Table 2, the values of p are similar290

across the zones; this stability indicates the seismic activity in the different tectonic zones is291

somewhat similar in terms of the rate at which the aftershock behavior after an earthquake292

decreases over time. Note that while p seems similar across the zones, the estimate of c in zone293

0 is somewhat higher than that in the other zones. Larger values of c indicate a more gradual294

decaying rate of seismicity following an event, according to the modified Omori formula. The295

estimates of c are much smaller in Zones 2, 3, and 4 than the other zones, indicating that296

these ridges, oceanic transform zones and trenches appear to have sharp temporal decays297

in aftershock activity following earthquakes. Conversely, the apparent aftershock activity298

decays much more slowly and gradually in active continent zones and especially in interplate299

regions, where little aftershock activity is observed even in the close temporal proximity of300

an event.301

The parameter q governs the spatial distribution of aftershocks. In general, smaller q302

corresponds to a more gradual spatial decay of aftershock productivity (long range decay)303

15

while larger q indicates that the triggering effect is mainly confined to a relatively small304

area around an earthquake (short range decay). Our results indicate that, among the major305

zones, zones 1 and 4 have relatively far-reaching triggering effects, and conversely, zones306

2 and 3 have triggering effects that reach a smaller area. Additionally, the value of q for307

zone 0 is much smaller than the corresponding estimates in the major zones, indicating that308

aftershock activity is much more gradually spatially decaying in interplate regions, according309

to the fitted ETAS model. This is likely due in large part to faulting being attributed to310

clustering in the ETAS model, as explained further below.311

Along with q, the square root of d indicates the scale, in km, of the spatial decay in the312

generation of aftershocks. That is, if at a certain time after a particular event of magnitude313

mi, the rate of aftershocks at distance R = 0 from the event is some value s, then at distance314

√dea(mi−mt)/2 the rate decreases to 2−qs, i.e. the rate decreases by a factor of 2−q. As seen315

from Table 2, among the major zones, the estimate of d is smallest in the active continental316

earthquake zone and highest in the ridges and oceanic transform zones. Note that, for two317

models with similar values of q, a higher value of d indicates more gradual spatial decay in318

clustering as one moves further from the mainshock. However, such interpretations can be319

misleading without simultaneously regarding the corresponding estimates of q; zone 3, for320

instance, has the largest estimate of d, indicating more gradual spatial decay with distance,321

but also the largest estimate of q, which indicates sharper spatial decay. The cumulative322

effects of these two parameters on the spatial decay function are explored below.323

K0 is a normalizing constant governing the total expected number of aftershocks per324

earthquake (Ogata 1988, 1998). Estimates of K0 vary dramatically between the various325

tectonic zones, with the highest observed triggering rates in zones 2 and 3. This is quite326

16

counterintuitive, since one would expect higher productivity of aftershocks in the more active327

zones. Additionally, the estimate K0 in zone 0 is approximately the same as that in zone328

1 and zone 4. As noted in Veen (2006), the variance of K0 may be very large due to the329

flatness of (1) and thus large changes in K0 can correspond to very small improvements in330

the log-likelihood (4). In addition, the temporal and spatial components of the triggering331

function g are densities and integrate to unity when integrated over infinite time and space,332

but when integrated over relatively small, thin zones, such as zones 2 and 3 in particular, the333

integral of the spatial component will often deviate substantially from one. This complicates334

the interpretation of K0, since the expected number of aftershocks observed within the given335

zone will deviate markedly from K0 in such cases. Hence it may be preferable to interpret the336

model’s triggering density in total rather than interpret each parameter estimate individually.337

This is explained further below.338

Figure 2 shows both the relationship between the estimated triggering function g and339

∆R or (∆R)2, where ∆R denotes the distance between a mainshock and its aftershock. The340

triggering component is much more pronounced in the active zone (zone 1), as expected,341

compared to the other zones, but this triggering decays vary rapidly as the distance from342

the mainshock increases. Zone 0’s estimated triggering function is noticeably larger than343

that of the other zones. A plausible explanation for the strangely high degree of triggering344

in zone 0 lies in the extreme inhomogeneity of the Earth’s seismicity. Recall that the fitted345

ETAS model has a homogeneous background rate, µ. Within zones 1-4, the seismicity is346

relatively homogeneous, but in zone 0, the ETAS model with homogeneous background rate347

tends to assign a low background rate in order to accommodate areas in zone 0 with no348

seismicity at all, and as a result, the clusters of seismicity that one observes along faults gets349

17

assigned to the triggering portion of the model, although many of the earthquakes along350

these faults might more reasonably be thought of as background events.351

We now return to our concern of the relationship between parameter estimates for a given352

fitted model in a given tectonic zone. Since c and p are both time-related, and d and q are353

both space-related, one might expect strong correlations for these pairs of parameters. As354

noted from simulation studies in Schoenberg et al. (2010), such estimates can be strongly355

correlated. In Figure 3, we observe significant correlations among the pairs (d, q), (d, K0),356

(K0, q), and (K0, information gain), while significant correlation is not observed for some357

other pairs (e.g. for a, Figure 4).358

4.2 Gutenberg-Richter constant, branching ratio359

Maximum likelihood estimates of the Gutenberg-Richter constant β and its corresponding360

b-value differ between tectonic zones, as shown in Table 3. The probability density function361

of m = mi − mt, derived from the Gutenberg-Richter law is (Gutenberg and Richter 1944):362

f(m) =βe−βm

1− e−β(MmaxGR −mt)

(9)

where MmaxGR is defined as the pre-specified maximum magnitude, and the exponent β relates363

to the b-value as β = bln(10) ≈ 2.3b (Veen, 2006). Larger values of β in the above truncated364

exponential density of m indicate more rapid decay in the density function as magnitude365

increases, implying a larger proportion of smaller events. This is the case in zone 2. Zone 3,366

by contrast, has a smaller estimate of β, indicating a smaller proportion of small events.367

Note that a < β for all the tectonic zones, and the difference β − a is largest in zones 2,368

18

3 and 0, implying that the smallest earthquakes contribute most to the overall seismic rate369

(Helmstetter 2003, Sornette and Werner 2005a, 2005b), especially in zones 2, 3 and 0. This370

agrees with the observation in Section 4.1 that zones 2 and 3 tend to contain more swarms371

rather than clusters of clearly distinct mainshocks and aftershocks.372

Table 3 displays the estimated branching ratio (expected number of triggered events per373

mainshock) for the fitted models. Zone 0 has the largest estimated number of aftershocks374

per mainshock. Zone 3 appears to have the fewest aftershocks per mainshock, in agreement375

with the results in Kagan et al. (2010). Among the major zones of primary interest, zone 4376

has the highest estimated branching ratio. This is not surprising because zone 4 is the most377

seismically active regime.378

4.3 Triggering of aftershocks379

Figure 5 shows how the estimated branching ratio, G, based on equation (6), varies as a380

function of mainshock magnitude for different tectonic zones. The curves for zones 2 and381

3 are very similar; the curves corresponding to zones 1 and 4 are also similar. Zone 0382

exhibits a phenomenon between these two types. For zones 1 and 4, a typical M6 event383

would have about 2 expected aftershocks of magnitude ≥ 5.0; a typical M7 event would have384

about 3 expected aftershocks of this size; an M8 event would have about 6 or 7 expected385

aftershocks with M ≥ 5.0; and an M9 event has about 18 or 19 expected aftershocks with386

magnitude ≥ 5.0 within these zones. The number of expected aftershocks increases much387

more dramatically with mainshock magnitude in zones 1 and 4, compared to zones 2, 3388

and 0. For zones 2 and 3, one would expect a typical M6 or M7 event to have only about389

19

one aftershock greater than magnitude 5.0; a M8-9 event would have about two expected390

aftershocks of magnitude ≥ 5.0 in this zone. There is no dramatic change in the number391

of expected aftershocks as the magnitude increases. For zone 0, a typical M6 or M7 event392

would have about 1 or 2 expected aftershocks of magnitude ≥ 5.0, which is somewhat similar393

to zones 2 and 3; a typical M8 or M9 event would have about 3 or 4 direct aftershocks of394

this size. In summary, the aftershock rates of zones 2, 3 and 0 are much more gradual than395

the other zones. This observation coincides with the implication mentioned in Sections 4.1396

and 4.2 that among the major zones, zones 2 and 3 have primarily swarm-type seismicities,397

and zones 1 and 4 have relatively clear distinctions between mainshocks and aftershocks.398

The spatial and temporal ranges of a typical event’s direct (first-generation) aftershocks are399

shown in Figure 5. The top right panel in Figure 5 shows the time (in years) to cover a400

certain proportion, ρ, of direct aftershocks. All the curves are quite similar for ρ < 75%; in401

all zones, at least 75% of the expected direct aftershocks would be expected to occur within 1402

year of the corresponding triggering earthquake. The zones differ, however, for larger values403

of ρ. For example, we expect earthquakes in zone 4 to have 85% of their direct aftershocks404

within 1 or 2 years; an event in zone 1 would have 85% of its expected aftershocks within 5405

years; for zones 0 and 2, 85% of the direct aftershocks are expected to occur within 10 years406

of the mainshock; and for zone 3, one expects it to take nearly 10 years for merely 80% of407

the direct aftershocks to occur. Among the major zones, zones 2 and 3 have less strongly408

clustered seismicity compared to the other zones.409

Figure 5 also shows the spatial range expected to cover a certain proportion of direct410

aftershocks for a typical M6 event. The curves of zones 1, 2, 3 and 4 show that up to 90%411

of the direct aftershocks are within 100 km of their triggering earthquakes, while almost all412

20

aftershocks are within 200 km of their corresponding mainshocks. Zone 0 behaves similarly413

to the other zones except at the upper tail of the percentage of aftershocks to be contained.414

To contain more than 95% of the direct aftershocks, the radius around a mainshock would415

have to be larger than 200 km. Among the major zones, the aftershock radii in zone 2 are the416

smallest, and zone 4 has the largest spatial radii. This is an indication that zone 4 exhibits417

the strongest triggering, so that mainshocks in zone 4 can trigger events to somewhat further418

distances than earthquakes in the other zones. Note that the radii for zone 0 are even larger419

than those for zone 4, which may largely be attributable to the large area of zone 0.420

The bottom right panel in Figure 5 shows the radius (in km) required to contain 95% of421

the expected direct aftershocks for a typical event. The curves of zones 2 and 3 are similar;422

an M7, M8 or M9 event would have 95% of its expected aftershocks within approximately423

100 km. For zones 1 and 4, an M7 event would have 95% of its expected aftershocks within424

150 km; an M8 event would have 95% of its expected aftershocks within 150 to 200 km; and425

in the case of zone 4, we see somewhat larger radii. For zone 1, an M9 event would have426

95% of its expected aftershocks within 300 to 350 km; and an M9 event in zone 4 would427

have 95% of its expected aftershocks within 400 km. Zone 0 is somewhat distinct from the428

other zones for events of M5, M6 and M7 (95% radii are approximately 100 km, 150 km and429

200 km, respectively), but similar to zones 1 and 4 for events of M8 and M9 (95% radii are430

approximately 250 km and 300 km, respectively). Our estimates of the sizes of aftershock431

zones are considerably wider than those estimated in previous research (Abercrombie 1995,432

Kagan et al. 2010). Below (Section 6) we try to explain this ETAS result and its difference433

from other estimates of a focal zone size.434

21

5 Model diagnostics435

5.1 Estimated conditional intensity in space436

The fitted models in each zone may most directly be inspected by examining spatial plots437

of the integrated conditional intensity. In Figure 6, we present an example for each zone:438

1. a sub-region of zone 1 (active continent): Southern California, (−122, −114) in longi-439

tude by (32, 37) in latitude.440

2. a sub-region of zone 2 (slow-spreading ridges): oceanic ridge in the North Atlantic441

Ocean, (−47, −42) in longitude by (20, 30) in latitude.442

3. a sub-region of zone 3 (fast-spreading ridges): oceanic ridge to the south of Australia,443

(145, 150) in longitude by (−60 , −55) in latitude.444

4. a sub-region of zone 4 (trenches): east of Honshu, Japan, (140 , 145) in longitude by445

(30, 40) in latitude.446

The approximation 1o ≈ 111 km, also used in Bird (2003), is employed here.447

One sees in Figure 6 that areas of high estimated intensity correspond to areas of high448

seismicity, and the spatial distribution of the triggering functions appear to fit the data well.449

5.2 Triggering function versus empirical aftershock rate450

Another basic diagnostic tool is to compare the triggering function to the empirical aftershock451

rate. Figure 7 shows the spatial and temporal distribution of all events of magnitude at least452

5.0 occurring within a given space-time window around any event with magnitude in a given453

22

range. For example, for zone 1, we consider in Figure 7 all events occurring after an event454

of estimated magnitude approximately 6.0.455

Note that the estimated background rates for all the zones depicted are small compared456

with the contributions from the triggering functions (see Table 2), so most of the events457

depicted in Figure 7 are attributed by the ETAS model to aftershocks. One can see the458

steep temporal decrease in seismicity following an earthquake, in agreement with the Omori459

law. Spatially, one sees that zone 1 and zone 2 have most of their aftershocks within a 50460

to 100 km radius of the mainshock, but in zone 4, radii of up to 200 km are required to461

contain most of the aftershocks. The spatial plots in Figure 7 do not indicate any strong462

departures from isotropy, at least not in the aggregate over all earthquakes within a zone,463

although in zone 1, aftershocks in the North-South direction appear to be somewhat more464

prevalent. Note that locally, the spatial patterns of aftershocks may be highly anisotropic,465

but that in the aggregate over an entire global zone, when the aftershocks are plotted relative466

to North-South and West-East directions rather than relative to local fault directions, they467

appear nearly isotropic in Figure 7.468

The empirical densities shown in Figure 7 may be compared to the expected densities469

according to the fitted ETAS model, and this comparison is shown in Figure 8. We observe470

that the empirical and expected aftershock density curves coincide very well in each of the471

zones.472

23

6 Conclusions473

We have found that the ETAS models appear to fit rather well to each of the major tectonic474

zones, despite the fact that the estimates of parameters in the ETAS models differ across the475

zones. Most of these differences in parameter estimates can be attributed to the differences476

in overall seismic rates for different zones. Zones 1 (active continent) and 4 (trenches) for477

instance, have the highest overall rates of seismicity and also appear to have the most intense478

estimated triggering functions. Zones 2 (slow-spreading ridges) and 3 (fast-spreading ridges),479

by contrast, appear to have lower background rates, have estimated triggering functions that480

decay more sharply spatially, and generally have parameter estimates consistent with swarms481

rather than traditional mainshock-aftershock clustering.482

As pointed out in Schoenberg et al. (2010), maximum likelihood estimates of ETAS483

parameters may be significantly biased due to the lower cutoff mt. Although the same484

cutoff magnitude of 5.0 is applied to each zone, the bias resulting from this choice of cutoff485

may affect different zones differently due to the varying sparsity of the catalog within each486

zone. A similar sort of truncation bias may be introduced by the irregularly shaped spatial487

windows delineating the different tectonic zones. This may have contributed to biases in the488

parameter estimates, especially for the spatial parameters d and q. Further study is needed489

in order to determine the biases introduced by spatial windows for parameters in ETAS490

models. The time windows selected here, by contrast, appear not to be a major source of491

bias in our parameter estimates; we have tested different time windows of various lengths492

and obtained very similar results for all of the parameters, including the temporal aftershock493

decay parameters c and p. However, the correlations between parameter estimates appear494

24

to be significant, and these correlations can also bias parameter estimates and estimates of495

standard errors.496

Another major source of error in the present analysis is due to the use of homogeneous497

background rates. We have elected to use a homogeneous background rate model because498

of our focus on the triggering behavior in the different tectonic zones. Furthermore, vari-499

ous different methods for estimating inhomogeneous background rates have been proposed,500

including kernel methods and Bayesian smoothing methods, and each may provide substan-501

tially different estimates; we desired our conclusions not to depend critically on our decision502

to use a particular choice of background rate estimation technique. However, our results es-503

pecially for zone 0 appear to depend critically on our choice of homogeneous background rate,504

and a similar study using inhomogeneous ETAS models may be an important direction for505

future work. In addition, one may consider larger models allowing triggering between zones.506

The study of such models with interactions is an important concern for future research. As507

noted in Bird (2003), very small plates within the orogens are likely to be identified in the508

near future, so further modeling of seismicity within single orogens may yield increasingly509

accurate estimates and forecasts of seismicity in the future.510

25

Data and Resources511

All data used in this paper came from published sources listed in the references.512

26

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boundary width, beta, corner magnitude, coupled lithosphere thickness, and coupling527

in seven tectonic settings, Bull. Seismol. Soc. Amer., 94(6), 2380-2399.528

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31

Tables601

32

Zone Area Proportion No. events No. events

(x 10 7 km2) of area in zone in zone / total

1. Active continent 3.36 6.58% 6728 16.6%

2. Slow-spreading ridges 1.58 3.09% 3028 7.47%

3. Fast-spreading ridges 0.844 1.65% 2574 6.35%

4. Trenches 1.52 2.98% 26125 64.5%

0. Plate-interior 43.8 85.8% 2049 5.06%

Total 51.0 100% 40504 100%

Table 1: Areas and PDE catalog sizes for each tectonic zone. Time window is January 1, 1973

to December 31, 2006. Only shallow events (≤ 70 km in depth) with reported magnitudes

of at least 5.0 are considered.

602

33

Estimates Zone 1 Zone 2 Zone 3 Zone 4 Zone 0

Act-Con. Slow Fast Trench Inter.

a 0.980 0.180 0.324 0.942 0.439

(1/magnitude) (0.0179) (0.0543) (0.0530) (0.0181) (0.0466)

c 0.0781 0.0319 0.0307 0.0408 0.497

(days) (0.00341) (0.00282) (0.00320) (0.00195) (0.0396)

d 69.3 227 327 141 69.0

(km2) (0.958) (4.35) (6.91) (2.06) (2.60)

K0 0.539 23.3 77.6 1.33 0.169

(events/day/km2) (0.00963) (0.664) (2.78) (0.0233) (0.00469)

p 1.19 1.16 1.15 1.25 1.21

(0.00309) (0.00414) (0.00488) (0.00430) (0.00440)

q 1.91 2.33 2.49 1.95 1.54

(0.00322) (0.00462) (0.00542) (0.00296) (0.00449)

µ x 10−9 7.63 8.43 15.6 61.7 0.131

(events/day/km2) (0.132) (0.239) (0.398) (0.775) (0.00542)

Log-lik.(ETAS) -109803 -52071 -44562 -380005 -37876

Log-lik.(Poisson) -127451 -57488 -47674 -438746 -46508

Info. gain 2.62 1.79 1.20 2.24 4.20

Table 2: Maximum likelihood estimates of ETAS parameters. Standard errors (diagonal

terms in the Hessian matrix) are reported in parentheses. ’Log-lik (ETAS)’ and ’Log-like

(Poisson)’ refer to the log-likelihood for the ETAS and homogeneous Poisson models, respec-

tively.

34

Zone a β b Branching ratio

1. Active continent 0.980 2.73 1.18 0.521

2. Slow-spreading ridges 0.180 3.15 1.37 0.481

3. Fast-spreading ridges 0.324 2.42 1.05 0.386

4. Trenches 0.942 2.69 1.17 0.559

0. Plate-interior 0.439 3.07 1.33 0.690

Table 3: Estimates of parameters a and β governing the Gutenberg-Richter distribution of

magnitudes, as well as the corresponding b-value (b = β/ln(10)) and expected branching

ratio.

35

Figure Captions603

36

Figure 1: Tectonic zones plotted with world map (Mercator projection). Blue: zone 1 (active

continent), green: zone 2 (slow-spreading ridges), yellow: zone 3 (fast-spreading ridges), red:

zone 4 (trenches), white: zone 0 (plate-interior).

Figure 2: Estimated triggering function g (triggered events / day / km2), fit by maximum

likelihood, as a function of (a) spatial radius or of (b) squared radius. Zone 1: active

continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone 4: trenches,

zone 0: plate-interior.

Figure 3: Scatterplots of estimated ETAS parameters as a function of estimates of K0,

for each zone. The numbers in the plots indicate the corresponding zone numbers. The

estimates for zones 0 and 1 nearly overlap in the plot of the estimate of d versus the estimate

of K0. Zone 1: active continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges,

zone 4: trenches, zone 0: plate-interior.

37

Figure 4: Scatterplots of estimated ETAS parameters as a function of estimates of a, for each

zone. The numbers in the plots indicate the corresponding zone numbers. Zone 1: active

continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone 4: trenches,

zone 0: plate-interior.

Figure 5: Summaries of the estimated triggering functions for each zone. (a) The expected

number of direct (1st generation) aftershocks of an earthquake of a given magnitude, as a

function of that magnitude, according to the fitted ETAS model for each zone. (b) The

expected time required to contain a given proportion of direct aftershocks, according to the

fitted ETAS model for each zone. (c) The expected spatial range (in km) required to contain

a given proportion of direct aftershocks of a typical M6 event, according to the fitted ETAS

models. (d) The radius (in km) of a range expected to contain 95% of the direct aftershocks

of a typical earthquake of a given magnitude, as a function of that magnitude, according

to the fitted ETAS model for each zone. Zone 1: active continent, zone 2: slow-spreading

ridges, zone 3: fast-spreading ridges, zone 4: trenches, zone 0: plate-interior.

Figure 6: Estimated conditional intensities according to ETAS (in color, on logarithmic

scale) and recorded earthquakes (black dots) for particular regions. The events are shallow

events (≤ 70 km deep) of magnitude ≥ 5.0. (a) A sub-region of zone 1 (active continent) in

California with 122 events in 1973-2006. (b) A sub-region of zone 2 (slow-spreading ridges),

an oceanic ridge in the North Atlantic Ocean, with 113 events in 1973-2006. (c) A sub-region

of zone 3 (fast-spreading ridges), an oceanic ridge South of Australia, containing 86 events

in 1973-2006. (d) A sub-region of zone 4 (trenches), East of Honshu, Japan, containing 185

events in 1999-2006.

38

Figure 7: Histograms (left) and scatterplots (right) of events within 30 days and occurring

within the given great circle distance (GCD) of a given event of recorded magnitude. (a)

and (b): Magnitude 6.0 for zone 1, the active continent. (c) and (d): Magnitude 5.3 for zone

2, the slow-spreading ridge zone. (e) and (f): Magnitude 6.0 for zone 4, the trench zone.

Figure 8: Estimated triggering density g (solid curves) and empirical rate (dotted curves) of

events occurring within 30 days and within a given spatial radius, following an earthquake

of a given magnitude. (a) mainshocks of M6.0 in zone 1 (active continent). (b) mainshocks

of M5.3 in zone 2 (slow-spreading ridges). (c) mainshocks of M5.3 in zone 3 (fast-spreading

ridges). (d) mainshocks of M6.0 in zone 4 (trenches). (e) Mainshocks of M5.3 in zone 0

(plate-interior).

39

Figures604

40

Fig

ure

1:T

ecto

nic

zones

plo

tted

wit

hw

orld

map

(Mer

cato

rpro

ject

ion).

Blu

e:zo

ne

1(a

ctiv

eco

nti

nen

t),

gree

n:

zone

2

(slo

w-s

pre

adin

gri

dge

s),

yellow

:zo

ne

3(f

ast-

spre

adin

gri

dge

s),

red:

zone

4(t

rench

es),

whit

e:zo

ne

0(p

late

-inte

rior

).

41

(a) .

5 10 15 200.0e

+00

5.0e

−06

1.0e

−05

1.5e

−05

2.0e

−05

radius(km)

g

Zone 1Zone 2Zone 3Zone 4Zone 0

(b) .

0 100 200 300 4000.0e

+00

5.0e

−06

1.0e

−05

1.5e

−05

2.0e

−05

radius^2(km^2)

g

Zone 1Zone 2Zone 3Zone 4Zone 0

Figure 2: Estimated triggering function g (triggered events / day / km2), fit by maximum

likelihood, as a function of (a) spatial radius or of (b) squared radius. Zone 1: active

continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone 4: trenches,

zone 0: plate-interior.

42

0 20 40 60 80

0.2

0.4

0.6

0.8

1.0

K0

a

0

1

2

3

4

0 20 40 60 80

0.1

0.3

0.5

K0

c

0

12 34

0 20 40 60 80

100

200

300

K0

d

01

2

3

4

0 20 40 60 80

1.16

1.20

1.24

K0

p

0

1

23

4

0 20 40 60 80

1.6

2.0

2.4

K0

q

0

1

2

3

4

0 20 40 60 80

0e+0

03e

−08

6e−0

8

K0

mu

01 2

3

4

0 20 40 60 80

1.5

2.5

3.5

K0

info

rmat

ion

gain

0

1

2

3

4

Figure 3: Scatterplots of estimated ETAS parameters as a function of estimates of K0, for

each zone. The numbers in the plots indicate the corresponding zone numbers. The estimates

for zones 0 and 1 nearly overlap in the plot of the estimate of d versus the estimate of K0.

Zone 1: active continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone

4: trenches, zone 0: plate-interior.

43

0.2 0.4 0.6 0.8 1.0

0.1

0.3

0.5

a

c

0

12 3 4

0.2 0.4 0.6 0.8 1.0

100

200

300

ad

0 1

2

3

4

0.2 0.4 0.6 0.8 1.0

020

4060

80

a

K0

0 1

2

3

4

0.2 0.4 0.6 0.8 1.0

1.16

1.20

1.24

a

p

0

1

23

4

0.2 0.4 0.6 0.8 1.0

1.6

2.0

2.4

a

q

0

1

2

3

4

0.2 0.4 0.6 0.8 1.0

0e+0

03e

−08

6e−0

8

am

u

012

3

4

0.2 0.4 0.6 0.8 1.0

1.5

2.5

3.5

a

info

rmat

ion

gain

0

1

2

3

4

Figure 4: Scatterplots of estimated ETAS parameters as a function of estimates of a, for each

zone. The numbers in the plots indicate the corresponding zone numbers. Zone 1: active

continent, zone 2: slow-spreading ridges, zone 3: fast-spreading ridges, zone 4: trenches,

zone 0: plate-interior.

44

(a) .

5 6 7 8 9 10

05

1015

2025

magnitude

no. o

f dire

ct a

fters

hock

s

Zone 1Zone 2Zone 3Zone 4Zone 0

(b) .

0 2 4 6 8 10

0.70

0.75

0.80

0.85

0.90

0.95

1.00

time (years)

prop

ortio

n of

dire

ct a

fters

hock

s

Zone 1Zone 2Zone 3Zone 4Zone 0

(c) .

0 200 400 600 800

0.70

0.75

0.80

0.85

0.90

0.95

1.00

distance (km)

prop

ortio

n of

dire

ct a

fters

hock

s

Zone 1Zone 2Zone 3Zone 4Zone 0

(d) .

5.0 5.5 6.0 6.5 7.0 7.5 8.0

5010

015

020

025

0

magnitude

km fo

r 95

% o

f dire

ct a

fters

hock

s

Zone 1Zone 2Zone 3Zone 4Zone 0

Figure 5: Summaries of the estimated triggering functions for each zone. (a) The expected

number of direct (1st generation) aftershocks of an earthquake of a given magnitude, as a

function of that magnitude, according to the fitted ETAS model for each zone. (b) The

expected time required to contain a given proportion of direct aftershocks, according to the

fitted ETAS model for each zone. (c) The expected spatial range (in km) required to contain

a given proportion of direct aftershocks of a typical M6 event, according to the fitted ETAS

models. (d) The radius (in km) of a range expected to contain 95% of the direct aftershocks

of a typical earthquake of a given magnitude, as a function of that magnitude, according

to the fitted ETAS model for each zone. Zone 1: active continent, zone 2: slow-spreading

ridges, zone 3: fast-spreading ridges, zone 4: trenches, zone 0: plate-interior.

45

(a) .

−122 −120 −118 −116

3233

3435

36

longitude

latit

ude

−9

−8

−7

−6

(b) .

−47 −46 −45 −44 −43

2022

2426

28

longitude

latit

ude

−9.0

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

−5.5

(c) .

145 146 147 148 149

−60

−59

−58

−57

−56

longitude

latit

ude

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

(d) .

140 141 142 143 144

3032

3436

38

longitude

latit

ude

−8.5

−8.0

−7.5

−7.0

−6.5

−6.0

Figure 6: Estimated conditional intensities according to ETAS (in color, on logarithmic

scale) and recorded earthquakes (black dots) for particular regions. The events are shallow

events (≤ 70 km deep) of magnitude ≥ 5.0. (a) A sub-region of zone 1 (active continent) in

California with 122 events in 1973-2006. (b) A sub-region of zone 2 (slow-spreading ridges),

an oceanic ridge in the North Atlantic Ocean, with 113 events in 1973-2006. (c) A sub-region

of zone 3 (fast-spreading ridges), an oceanic ridge South of Australia, containing 86 events

in 1973-2006. (d) A sub-region of zone 4 (trenches), East of Honshu, Japan, containing 185

events in 1999-2006.

46

(a) .

radius(GCD in km)

Fre

quen

cy

0 50 100 150 200 250

010

2030

(b) .

−200 −100 0 100 200

−20

00

100

200

E−W direction (km)

N−S

dire

ctio

n (k

m)

(c) .

radius(GCD in km)

Fre

quen

cy

0 50 100 150 200 250

010

2030

40

(d) .

−200 −100 0 100 200

−20

00

100

200

E−W direction (km)

N−S

dire

ctio

n (k

m)

(e) .

radius(GCD in km)

Fre

quen

cy

0 50 100 150 200 250

010

2030

4050

60

(f) .

−200 −100 0 100 200

−20

00

100

200

E−W direction (km)

N−S

dire

ctio

n (k

m)

Figure 7: Histograms (left) and scatterplots (right) of events within 30 days and occurring

within the given great circle distance (GCD) of a given event of recorded magnitude. (a)

and (b): Magnitude 6.0 for zone 1, the active continent. (c) and (d): Magnitude 5.3 for zone

2, the slow-spreading ridge zone. (e) and (f): Magnitude 6.0 for zone 4, the trench zone.

47

(a) .

0 50 100 150 200 250

0.00

0.02

0.04

0.06

0.08

0.10

radius (km)

pdf

(b) .

0 50 100 150 200 250

0.00

0.04

0.08

radius (km)

pdf

(c) .

0 50 100 150 200 250

0.00

0.02

0.04

0.06

0.08

radius (km)

pdf

(d) .

0 50 100 150 200 250

0.00

0.02

0.04

0.06

radius (km)

pdf

(e) .

0 50 100 150 200 250

0.00

0.04

0.08

0.12

radius (km)

pdf

Figure 8: Estimated triggering density g (solid curves) and empirical rate (dotted curves) of

events occurring within 30 days and within a given spatial radius, following an earthquake

of a given magnitude. (a) mainshocks of M6.0 in zone 1 (active continent). (b) mainshocks

of M5.3 in zone 2 (slow-spreading ridges). (c) mainshocks of M5.3 in zone 3 (fast-spreading

ridges). (d) mainshocks of M6.0 in zone 4 (trenches). (e) Mainshocks of M5.3 in zone 0

(plate-interior).

48