1

Connecting near and farfield earthquake triggering to dynamic strain 1

Nicholas J. van der Elst 2

Emily E. Brodsky 3

Department of Earth and Planetary Science, University of California, Santa Cruz 4

5

Abstract: Any earthquake can trigger more earthquakes. This triggering occurs in both 6

the classical aftershock zone, as well as in the farfield. These near and distant 7

populations of triggered earthquakes may or may not be distinct in terms of triggering 8

mechanism. Here we look for a distinction between the two populations by examining 9

how the observed intensity of triggering scales with the amplitude of the triggering strain 10

in each. To do so, we apply a new statistical metric based on earthquake interevent times 11

to a large dataset and measure earthquake triggering as a function of dynamic strain 12

amplitude, where strain is estimated from empirical ground motion regressions. This 13

method allows us to identify triggering at strain amplitudes down to 3×10-9, which is 14

orders of magnitude smaller than previously reported. This threshold appears to be an 15

observational limit, and suggests that extremely small strains can trigger faults that are 16

sufficiently near failure. We also discover that faults respond in the same way to 17

dynamic strains in both the near and farfield. Surprisingly, there is no need for an 18

additional triggering component to explain the nearfield seismicity rates. Statistical 19

seismicity simulations validate the interevent time method, and strongly suggest that the 20

observed number of triggered earthquakes scales linearly with peak dynamic strain. The 21

empirical connection between dynamic strain and earthquake triggering constitutes a 22

short-term earthquake-forecasting model based on observed seismic shaking. 23

2

24

1. Introduction 25

Triggered earthquakes provide a window into the physics of earthquake nucleation 26

because the forces initiating rupture can be inferred. Because the strain at which a fault is 27

triggered is a measure of its strength, it may be possible to gain insight into the 28

distribution of fault strength by studying the statistics of earthquake triggering [Brodsky 29

and Prejean, 2005; Gomberg, 2001]. In this paper, we use changes in earthquake 30

recurrence times to place a new bound on the minimum strain at which a fault can be 31

triggered and determine how the likelihood of triggering scales with the amplitude of the 32

trigger. 33

34

Estimating the amplitude of the trigger is complicated by the existence of two principal 35

candidate mechanisms for triggering, neither of which is perfectly understood [Freed, 36

2005; Hill and Prejean, 2007]. Static triggering is related to the permanent increase or 37

decrease of stress on one fault due to the coseismic relaxation of stress on another [King 38

et al., 1994]. Dynamic triggering is associated with transient strains carried by radiated 39

seismic waves [Hill and Prejean, 2007]. Static strain changes are permanent, but decay 40

in amplitude quickly with distance away from a fault and are thus unlikely to trigger 41

distant earthquakes. Dynamic strains are generally larger, especially at distance, but 42

present a challenge in explaining prolonged triggering [Brodsky, 2006; Gomberg, 2001]. 43

44

The relative contributions of the two triggering mechanisms are expected to differ as a 45

function of distance from the generative earthquake. Remotely triggered earthquakes are 46

3

believed to result entirely from dynamic triggering, both because static strains are 47

negligible at large distances, and because distant triggering often coincides with the 48

arrival of surface waves [Anderson et al., 1994; Brodsky et al., 2000; Gomberg and 49

Johnson, 2005; Hill et al., 1993; Velasco et al., 2008]. On the other hand, short-range 50

triggering has been attributed to either static or dynamic mechanisms by different 51

researchers [Felzer and Brodsky, 2006; Pollitz and Johnston, 2006; Stein et al., 1994]. 52

Static strains must play a role in determining the location and average rates of 53

earthquakes over long times, because faults are ultimately loaded by quasi-static tectonic 54

motion. Nevertheless, dynamic triggering clearly influences the timing and rate of 55

nearfield earthquakes, as demonstrated by highly directional ruptures, which have 56

asymmetric aftershock zones reflective of the asymmetry in radiation pattern [Gomberg 57

et al., 2003; Kilb et al., 2000]. The relative contribution of the two mechanisms in 58

nearfield triggering remains poorly constrained. 59

60

In this study, we exploit the understanding that dynamic strain dominates farfield 61

earthquake triggering in order to constrain the contribution of static triggering in the 62

nearfield. We first determine an empirical relationship between triggering intensity and 63

dynamic strain in the farfield. Then we compare this farfield relationship to nearfield 64

observations and look for an additional component that would indicate a superposition of 65

static strain triggering on top of the dynamic strain relationship. We ultimately find that 66

earthquake triggering scales with dynamic strain, and that this scaling is identical whether 67

we are looking at remotely triggered earthquakes at distances of hundreds of kilometers, 68

or at local aftershocks triggered within a few kilometers of their mainshock. In the 69

4

process, we develop a measure of earthquake triggering that is significantly more 70

sensitive to low triggering rates than previous measures. 71

72

The first several sections of this article concern the development of the triggering metric. 73

First, we define the interevent time statistic and set up the method for transforming 74

interevent time ratios into an estimate of average triggering intensity within a population 75

of earthquakes. Next, we construct populations on the basis of local dynamic strain and 76

describe the data selection and processing. We then apply the method and show that 77

triggering intensity scales with peak dynamic strain, following the same functional 78

relationship in both the near and farfield triggered earthquake populations. We also 79

report a new maximum bound on the triggering threshold in California of 3×10-9 strain, 80

several orders of magnitude lower than previous estimates. A statistical seismicity 81

simulation is then used to help interpret and calibrate the results. The simulation strongly 82

suggests that the total number of triggered earthquakes in the real catalog is directly 83

proportional to peak dynamic strain. Finally, we evaluate the implications and robustness 84

of the results. 85

86

2. Measuring earthquake rate changes using interevent times 87

A comparison of triggering rates in the near and farfield requires a metric that can be 88

applied to both populations of earthquakes. This metric needs to be sensitive enough to 89

detect the very small triggering rates associated with the very small strains common to 90

the farfield. Previously, triggered earthquakes have been identified by inspecting 91

seismicity rates [Harrington and Brodsky, 2006; Hill et al., 1993; Stark and Davis, 1996] 92

5

or by filtering waveforms to emphasize short-period energy within the surface wave 93

trains of large, distant earthquakes [Brodsky et al., 2000; Hill and Prejean, 2007; Velasco 94

et al., 2008]. Quantitative estimates of triggering usually involve calculating the 95

likelihood of observing a number of post-trigger events given the previous seismicity rate 96

[Anderson et al., 1994; Gomberg et al., 2001; Hough, 2005]. If the likelihood of the rate 97

increase occurring by chance is low enough, triggering is inferred. 98

99

Any estimate that computes likelihood of triggering based on counting the number of 100

triggered earthquakes relative to a pre-trigger count, like the β-statistic [Matthews and 101

Reasenberg, 1988], is limited in several ways. First, the pre-trigger seismicity rate must 102

be resolved for comparison, and this is inherently difficult. Because most earthquakes 103

occur as clusters of aftershocks, the seismicity rate is always changing. Background 104

seismicity level should therefore be measured at a time as close to the purported trigger 105

as possible in as short a window as possible. Different areas will permit different length 106

windows depending on their background level of seismicity and thus a constant window 107

for an entire dataset may not sufficiently capture the data. Second, an earthquake count 108

can only resolve an integer increase in the number of earthquakes for any individual 109

sequence. A slight advancement in the timing of subsequent earthquakes will only rarely 110

result in a triggered earthquake within the counting time window so only large levels of 111

triggering can be resolved with statistical robustness. Finally, an earthquake count also 112

includes all secondarily triggered earthquakes, that is, aftershocks of aftershocks. These 113

secondary earthquakes are not strictly problematic, because they should still be produced 114

in proportion to the number of primary triggered earthquakes when averaged over many 115

6

events, but they complicate the relationship between trigger amplitude and number of 116

triggered quakes by introducing variance into the measurements. 117

118

To detect triggering at very low dynamic strain amplitudes, our metric must use an 119

adaptive time window to measure background rates, be sensitive to small increases in 120

seismicity rates, and be insensitive to secondary aftershocks. 121

122

2.1 The R-statistic 123

We meet the above requirements by developing a statistic based only on the interevent 124

times between the last earthquake before the trigger and the first earthquake after. The 125

normalized interevent time R is defined by 126

R ≡t2

t1 + t2, (1)

where t1 and t2 are the waiting times to the first earthquake before and after the putative 127

trigger (Figure 1). The R-statistic was originally developed to study triggered quiescence 128

[Felzer and Brodsky, 2005]; we use it here to look for a triggered rate increase. 129

130

The statistic R is a random variable distributed between 0 and 1. The strategy here is to 131

measure the distribution of R on a population of earthquakes that are subject to similar 132

triggering conditions. For instance, the population can be a drawn from a variety of areas 133

subject to the same strain. If there is no triggering and t2 is on average equal to t1, then R 134

is distributed uniformly with a mean value R = 12 . On the other hand, if triggering does 135

occur, t2 will be on average smaller than t1 and R < 12 . More triggering results in a 136

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smaller R (Figure 2). Therefore, R provides a measure of triggering intensity within a 137

population of earthquakes. 138

139

The R statistic naturally solves the three problems identified with earthquake counting 140

methods by defining an appropriate time window for each event based on the interevent 141

times, utilizing the statistics of large populations, and focusing on the first recorded 142

earthquake rather than the entire triggered sequence. 143

144

One of the unusual features of the R-statistic is that there is no time limit for the inclusion 145

of triggered events. Both immediate and delayed triggering are included in the 146

measurements. This comprehensiveness is desirable because of issues of catalog 147

completeness, as well as the physical implications of delayed triggering. We will return 148

to the issue of delayed triggering at the end of the paper after the statistic has been 149

implemented. 150

151

2.2. Interpreting R as seismicity rate change 152

The mean of R qualitatively captures the intensity of triggering, but as of yet we have no 153

theoretical basis for interpreting R in terms of seismicity rate changes. Interpreting 154

interevent times in terms of the number of expected earthquakes requires the introduction 155

of a probabilistic model for earthquake recurrence. This is equivalent to the common use 156

of probabilistic models to get expected interevent times from earthquake counts 157

[Reasenberg and Jones, 1989]. 158

159

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If earthquake recurrence were perfectly periodic and uniform, Equation 1 could be simply 160

rearranged to solve for the fractional increase in earthquake rate as a function of the 161

measured R . A somewhat more sophisticated model for earthquake recurrence is a step-162

wise homogeneous Poisson process, in which earthquakes occur randomly in time with 163

an average rate λ1 before the trigger and a new average rate λ2 afterward. We could also 164

use an inhomogeneous Poisson process, where the triggered earthquake rate decays with 165

time according to Omori’s law [Reasenberg and Jones, 1989]. 166

167

For any of these models, we can compute the expectation (expected mean) of R given an 168

input rate change. For instance, for the stepwise homogeneous Poisson process, the 169

expectation of R is given by 170

R =1n2

n +1( ) ln n +1( ) − n⎡⎣ ⎤⎦ , (2)

where n is the fractional rate change 171

n =λ2 − λ1λ1

(3)

(See Appendix A, Equation A10). For the inhomogeneous Poisson process, where a 172

finite number of triggered earthquakes are added to the preexisting seismicity rate, the 173

fractional rate change depends on the time over which the count is made and is defined 174

slightly differently (Appendix A). 175

176

The expectation of R for a given fractional rate change n is similar for each of the 177

probabilistic models especially for the small rate changes we expect to identify (Figure 178

3). The interpretation to follow therefore does not depend strongly on the particular 179

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probability model used. This is because we are looking only at the time to the first event, 180

and not the distribution of times to subsequent events, where most of the differences exist 181

between the models. 182

183

For the rest of this paper, we use the step-wise homogeneous Poisson model, because it is 184

the simplest stochastic model, has an analytical solution, and does not require 185

independent calibration of the background seismicity rate. By the method of moments 186

[Casella and Berger, 2002], we set the expectation of R equal to the measured sample 187

mean, R , and solve numerically for n. Because the transformation from R to n is 188

somewhat model dependent, the calculated fractional rate change n is hereafter referred 189

to as triggering intensity and interpreted as a qualitative measure of rate change that 190

illuminates the scaling of triggering intensity within populations. We will ultimately use 191

statistical simulations to calibrate n and get a quantitative estimate of number of 192

earthquakes triggered for a given strain. 193

194

3. Defining Populations 195

The R-statistic measures triggering intensity in a population of earthquakes. Therefore, 196

the first step in applying R to a real dataset is to define reasonable populations so that we 197

can evaluate the different rate changes in each one. 198

199

Because long-range triggering is clearly associated with dynamic strain, we start by 200

constructing sets of earthquakes with a common dynamic strain. Dynamic strain is 201

proportional to the amplitude of seismic waves 202

10

ε ~ AΛ~ VCS

, (4)

where A is displacement amplitude, Λ is wavelength, V is particle velocity and CS is 203

seismic wave (phase) velocity [Love, 1927]. In principle, dynamic strain can therefore be 204

calculated wherever there is a seismogram. Ground motions are converted to strain by 205

dividing by wavelength or wave speed, depending on whether the regression is for 206

displacement or velocity, respectively. In the nearfield, S-waves are the largest motion, 207

and in the farfield, surface waves are larger. Both have a phase velocity CS of ~3.5 km/s. 208

209

Previous work has investigated the accuracy of using ground motions as a proxy for 210

strain at depth by comparing strain estimated from seismometer data to strain measured 211

by strainmeters [Gomberg and Agnew, 1996]. This work found that seismometer data 212

predicted strain amplitudes within ±20% of the strainmeter measurements. 213

214

3.1. Estimating strain with empirical ground motion regressions 215

In order to take full advantage of the large number of earthquakes in the catalog, we must 216

calculate strain at any point on the map for any trigger in the catalog, not only where we 217

have seismic stations and archived waveforms. Fortunately, peak empirical ground 218

motions are well studied in the nearfield for engineering purposes and at regional and 219

teleseismic distances for calibrating magnitude scales [Boatwright et al., 2003; Joyner 220

and Boore, 1981; Lay and Wallace, 1995; Richter, 1935]. We use these regressions to 221

estimate ground motion as a function of distance and magnitude for nearfield and farfield 222

waves, respectively. 223

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224

Specifically, for long-range strain we use the surface wave magnitude relation [Lay and 225

Wallace, 1995], 226

log10 A20 = MS −1.66 log10 Δ − 2 , (5)

where A20 is in µm and Δ is in degrees. This equation is commonly used to assign a 227

catalog magnitude based on a measured amplitude at some distance. We turn the 228

procedure around and use the catalog magnitude to calculate an amplitude. This 229

approach uses the long-period waves (T=20 s) as indicators of the peak strain, implicitly 230

assuming that the short-period body waves are attenuated at large distances. The 231

displacement A is converted to velocity for the 20 second waves as V = 2πA20 T [Aki 232

and Richards, 2002]. Equation 5 was historically calibrated using a similar catalog of 233

global earthquakes to the one we use for potential triggers, and so provides a good 234

measure of average amplitude despite being imperfect for any individual earthquake. 235

The equation is designed for distances on the order of at least 800 km [Lay and Wallace, 236

1995]. This sets the minimum distance for the population of long-range triggers. 237

238

The amplitude of the short-range triggers is calculated from an empirical peak ground 239

velocity regression determined from California ShakeMap data, 240

log10V = 1.06ML − 0.0063r − log10 r +1.35 , (6)

where V is velocity in µm/s, r is distance in km, and ML is Richter magnitude [Boatwright 241

et al., 2003]. The reported uncertainties in the constants are: 1.06±0.02, 0.0063±0.002 242

and 1.35±0.23, corresponding to a variation of about one half order of magnitude in the 243

observed wave amplitude associated with any individual earthquake. Considering the 244

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additional uncertainty associated with estimating strain from seismometer data, we 245

consider our estimates accurate within an order of magnitude. 246

247

3.2. Constructing Populations Over Space 248

The interevent time method only looks at single earthquake pairs bracketing a potential 249

trigger, but a large quake may trigger numerous earthquakes throughout the study area. 250

We therefore split the study region into spatial bins and calculate R for each of these bins 251

(Figure 4). This allows us to get a robust distribution of R-values for each trigger and 252

ensures that the measurements are not dominated by whichever aftershock sequence is 253

most active at the time. Using a spatial bin that is much smaller than the wavelength of 254

the long-range trigger also ensures that measured triggering intensity reflects the strain at 255

that site. 256

257

The higher the number of bins, the higher the number of bracketing pairs for each trigger 258

quake, down to a lower size limit where single earthquakes begin to be isolated. A bin 259

size of 0.1º×0.1º gives the maximum number of bracketing pairs for the whole catalog of 260

potential triggers, but we perform the analysis using several different bin sizes to ensure 261

robustness of the results with respect to parameter choices. 262

263

To measure nearfield triggering, we use a disk centered on the mainshock epicenter. The 264

radius of the disk is set to give the same area as the long-range bins. Because the R 265

statistic is calculated over an area around the mainshock epicenter, a mean peak velocity 266

is calculated over that area. Assuming radial symmetry and neglecting the inelastic 267

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attenuation term (2nd term on right hand side of Equation 6) at these small distances, the 268

average peak velocity is 269

V D( ) = 1πD2 V r( )2πr dr ≈ 2 ⋅101.06ML +1.35

0

D

∫ ⋅D−1 , (7)

where V(r) was defined in Equation 6 and D is the radius of aftershock collection. 270

271

3.3 Earthquake Catalogs 272

Using the R statistic on earthquake populations requires large accurate catalogs for both 273

potential trigger earthquakes and for local seismicity. The trigger catalog is drawn from 274

the global ANSS catalog from 1984 through April 2008. A depth cutoff is imposed at 275

100 km, because deep earthquakes do not generate significant surface waves. Only 276

earthquakes with surface wave amplitude greater than ten micrometers displacement are 277

treated as potential triggers, because preliminary work finds no observable triggering 278

signal at this or lower amplitudes. This minimum corresponds to a MS4.5 earthquake at 279

800 km. We show below that this cutoff is below the observational threshold for long-280

range triggering. 281

282

Both potential long-range triggered quakes and potential nearfield triggers are drawn 283

from the ANSS catalog for the California study region. Other catalogs have considerably 284

smaller location errors than the ANSS catalog, but contain considerably fewer 285

earthquakes. Location error should not be a significant source of error for this study, 286

because the required spatial precision is on the order of the spatial bin size. We therefore 287

choose the catalog with the largest number of earthquakes. The interevent time method 288

should not be sensitive to regional variations in completeness magnitude, because the 289

14

incompleteness should affect the pre and post-trigger catalogs in a consistent way. 290

However, we impose a magnitude threshold of 2.1, based on the roll-off in the Gutenberg 291

Richter distribution for the catalog as a whole, to protect against large swings in 292

completeness level with time. The study area extends from 114° to 124° west, and from 293

32° to 42° north. 294

295

We also look at the scaling of triggering intensity with dynamic strain in Japan. Here we 296

use the JMA catalog from 1997 through March 2006. For consistency with California, 297

we limit the catalog of local events to shallower than 15 km within the land area of the 298

four main islands of Japan. The magnitude of completeness for the JMA catalog of 299

shallow crustal events is below 2.1, but we impose this larger magnitude cutoff for 300

consistent comparison with California. 301

302

3.4. Practicalities of Implementation 303

In order to evaluate the significance of R as an indicator of triggering, we require 304

confidence bounds on R . We use the bootstrap method to generate confidence bounds by 305

randomly resampling the R distribution for a given population to generate a suite of 306

estimates of R [Casella and Berger, 2002]. 307

308

We also take into consideration two potential sources of undesirable bias for realistic sets 309

of earthquakes. One is related to the superposition of Omori’s law on measurements 310

made in aftershock sequences, and the other to the finiteness of the earthquake catalog. 311

312

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For the farfield case, the timing of the distant trigger is usually uncorrelated to the timing 313

of earlier (or non-triggered) quakes in the study region, because of the large spatial 314

separation. However, this is not the case for short-range triggered earthquakes, where an 315

Omori-type rate decay (~t-1) may be superimposed on the timing of both the trigger quake 316

and the subsequent triggered quake. This decay biases the R statistic toward higher 317

values and obscures the triggering signal. We suppress this bias by requiring that the 318

trigger event be much larger than the immediately preceding event, ensuring that the 319

average rate increase due to the trigger will be much larger than the Omori-law rate 320

decrease superimposed on the entire sequence. A higher magnitude difference better 321

insures against bias, but significantly reduces the number of eligible trigger quakes. We 322

find that R is stable for a minimum magnitude difference of one unit. One magnitude 323

unit corresponds to a roughly tenfold increase in total triggering power, and Omori’s law 324

ensures that the difference is in general much greater than this, because the influence of 325

the previous earthquake decays rapidly with time. 326

327

Another potential source of bias is related to the finiteness of the catalog. This is 328

especially problematic in regions where seismicity rates are low. To understand this 329

effect, consider a putative trigger near the beginning of the catalog. The probability that 330

the first prior event occurs before the start of the catalog is much greater than the 331

probability that the first subsequent event occurs beyond the end, simply because of the 332

position of the trigger quake within the catalog. Since we cannot measure t1 for 333

earthquakes that occurred before the start of the catalog, we only calculate R when t1 is 334

unusually small, and therefore obtain disproportionately large values of R. The mean R 335

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can thus be biased by the non-uniform distribution of global trigger times. We determine 336

the bias numerically by measuring R in 1000 simulations using the real trigger times, but 337

with local catalogs consisting of uniformly distributed random times. This calculated 338

bias is subtracted from the values of R measured for the actual catalog. For simplicity, 339

the bias-corrected mean is referred to below as R . 340

341

4. Observed Triggering Intensity as a Function of Dynamic Strain 342

We begin attacking real data by measuring the interevent times for a well-known case of 343

pervasive triggering in order to establish that R and n behave as designed. The 2002 344

magnitude 7.9 Denali earthquake generated peak dynamic strains on the order of 2-3×10-7 345

for the California study area, according to the empirical regressions, and is known to have 346

triggered significant seismicity [Anderson et al., 1994; Gomberg et al., 2004]. For this 347

initial case study, we disregard amplitude variations and define a population consisting of 348

the full study area. The resulting R distribution reflects significant triggering (Figure 5). 349

The sample mean R (with 95% confidence limits) is 0.475 (0.461-0.488), corresponding 350

to a fractional rate change of 0.16 (0.08-0.26) according to Equation 2. A simple 351

earthquake count before and after the Denali earthquake indicates a 22% seismicity rate 352

increase in the following 24 hours. These estimates agree within error. We conclude that 353

R is capable of capturing triggering in a case with known seismicity rate increases. 354

355

The real utility of the method becomes apparent when it is applied to the full ANSS 356

dataset with over 3000 potential farfield triggers, and 12,000 nearfield triggers meeting 357

our criteria. Figure 6 shows measured R distributions for different dynamic strain 358

17

amplitudes, corresponding to various combinations of magnitude and distance for the 359

long-range dataset, and various magnitudes at constant distance for the short-range data. 360

The R-distributions show evidence of both immediate triggering, in the form of large 361

spikes near R=0, and protracted triggering, in the form of a continuing decrease with 362

increasing R. The distributions show larger proportions of small R for higher strain 363

levels, as expected. 364

365

Triggering intensity for the entire dataset, computed according to Equation 2, is plotted in 366

Figure 7. Triggering intensity scales continuously with peak dynamic strain over five 367

orders of magnitude in strain amplitude. Triggering intensity has the same relationship to 368

peak dynamic strain in both long and short-range populations. There is no increase in 369

triggering intensity in the nearfield that requires an additional triggering component. The 370

continuity shows that long-range triggered earthquakes are no more rare or unusual than 371

expected given the amplitude of dynamic strains at distant sites. For very small farfield 372

strain levels, the triggering is rare and difficult to identify, but does not differ in a 373

fundamental way from triggering at higher strain levels in the nearfield. 374

375

The apparent triggering threshold in California is 3x10-9 strain, where the threshold is 376

defined as the strain at which triggering intensity n is non-zero (or equivalently R < 0.5 ) 377

at the 95% confidence level. This threshold is not dependent on the transformation from 378

R to triggering intensity n. For a crustal shear modulus of 30 GPa, this corresponds to a 379

stress of 0.1 kPa. This estimate is several orders of magnitude smaller than previously 380

reported for dynamic triggering [Brodsky and Prejean, 2005; Gomberg and Davis, 1996; 381

18

Gomberg and Johnson, 2005; Stark and Davis, 1996]. Previous estimates have been 382

based on counting statistics. If n>1, the change in seismicity rate is comparable to the 383

background rate, and triggering is easily observable by counting methods. For a Poisson 384

process, the variance is equal to the average rate, so n>1 also roughly corresponds to the 385

threshold for statistical significance using an earthquake count. Therefore, only the 386

seismicity rate increases corresponding to n>1, i.e. strains of nearly 10-5, were easily 387

observable in previous studies. 388

389

The strain threshold is smaller than tidal stresses [Cochran et al., 2004; Scholz, 2003]. 390

This is somewhat puzzling, because tidal stresses might be expected to “clean out” all 391

available nucleation sites on a daily basis and set a lower limit for dynamic triggering. 392

Strain tensors associated with crustal earthquakes are likely oriented with more variety 393

than those due to the tides, however, and may access faults that tidal strains are incapable 394

of triggering. In addition, the forcing at the relatively long periods of the tides may be 395

intrinsically different from the dynamic strains imposed at the short periods of seismic 396

waves [Beeler and Lockner, 2003; Gomberg et al., 1997; Savage and Marone, 2008] 397

398

Triggering intensity in shallow crustal Japan (Figure 7b) is reduced relative to California 399

in both the long and short-range populations, with a higher triggering threshold of 10-6 400

strain. However, where long-range strain is large enough to overlap in amplitude with 401

short-range strain, the triggering intensity in the long and short-range earthquake 402

populations is identical. This overlap further reinforces the continuity observed in Figure 403

7a. The relative absence of long-range triggering in Japan has been documented before 404

19

[Harrington and Brodsky, 2006], but this study shows that the reduced triggering 405

susceptibility extends to the nearfield, as well. This difference in triggerability may 406

reflect the difference in tectonic style (compressive vs. transpressive) between the two 407

study areas. 408

409

Evaluating the probability of triggering as a function of strain (Figure 8) provides an 410

interpretation of the observed triggering threshold. Probability is calculated from the 411

triggering intensity n using the same Poissonian statistical model as before (Appendix B). 412

The probability of triggering an earthquake within its own recurrence interval, as a 413

function of n, is given by 414

P NEQs ≥ 1( ) = 1− exp − n +1( ){ } . (8)

The baseline probability of having an earthquake in the absence of any rate change (n=0) 415

is 63%. A positive n produces a positive probability gain. Figure 8 shows that the 416

probability gain decreases smoothly to zero as n decreases. For the ~10-9 strain bin in 417

California, there are ~105 interevent time measurements (~103 triggers x ~102 local 418

earthquakes) and this number of events is sufficient to establish the statistical significance 419

of the 0.003% probability gain observed. An order of magnitude more observations 420

would be needed to push the observable threshold an order of magnitude lower. This 421

exceeds the size of the earthquake catalog, and we infer that the absence of detected 422

triggering at strains of less than 10-9 reflects an observational limit and not a physical 423

threshold. 424

425

5. Validation and calibration through statistical seismicity simulations 426

20

We have shown that there is a continuous trend in the scaling of triggering intensity with 427

dynamic strain in both near and farfield populations. However, there is a problem 428

interpreting the slope of this trend. Previous work using earthquake counting and 429

carefully declustered seismicity catalogs has shown that the number of local aftershocks 430

following a mainshock of magnitude M goes as 10αM , with α=1 [Felzer et al., 2004; 431

Helmstetter et al., 2005]. This study finds that triggering intensity n varies as 100.5M 432

(Figure 7). We will now show that this discrepancy arises from the application of a 433

probability model derived for isolated earthquake sequences to a catalog containing 434

superimposed triggering cascades. Fortunately, this effect can be quantified and 435

calibrated using statistical simulations. 436

437

As described in Section 2.3, triggering intensity is estimated from inter-event times using 438

a probabilistic model for earthquake recurrence times. None of the models described 439

consider the effect of the earthquake cascade on the estimation of rate change. The 440

transformation from R to n implicitly assumes that we correctly associate earthquakes 441

with their respective triggers, but a real catalog contains numerous superimposed 442

triggering cascades. The first earthquake before and after a prospective trigger may or 443

may not then be causally related. If they belong to a different earthquake sequence, they 444

will introduce R-values sampled from a uniform distribution, and the resulting 445

distribution will be some combination of the two curves illustrated in Figure 2. This has 446

the effect of dampening the observed triggering signal. 447

448

To evaluate whether the presence of superimposed earthquake cascades can explain the 449

21

discrepancy between our recovered slope in Figure 7 and previous work based directly on 450

earthquake counts, we generate an artificial earthquake catalog that follows the usually 451

observed statistics of magnitude, timing, and triggering distributions (Appendix C). A 452

well-established method for generating such a catalog is the epidemic triggered 453

aftershock sequence (ETAS) [Ogata, 1992]. ETAS uses well-known empirical statistical 454

seismicity laws as probability distributions to generate stochastic seismicity catalogs 455

(Appendix C). Numerous researchers have used ETAS models to study the complex 456

statistical repercussions of simple earthquake cascades [Felzer et al., 2002; Felzer et al., 457

2004; Hardebeck et al., 2008; Helmstetter and Sornette, 2003; Holliday et al., 2008]. 458

Here we use ETAS to study the effects of superimposed earthquake triggering sequences 459

on our measure of triggering intensity. 460

461

We first apply the R statistic to a zero-dimensional ETAS catalog. The zero dimensional 462

model simulates earthquakes in time only, disregarding spatial distribution, in order to 463

isolate the effect of the earthquake cascade. The triggering law in the simulation 464

corresponds to the case of the inhomogeneous Poisson process with an Omori decay 465

(Appendix A), and the number of triggered earthquakes scales with α=1. The 466

inhomogeneous Poisson model for interevent times is used to transform R to a triggered 467

rate change. 468

469

Because causality is known in the simulation, we can investigate how the use of the first 470

earthquake before and after the trigger affects the recovered scaling relationship. If 471

interevent times are measured with respect to known branches of the cascade, that is, if 472

22

the causal relationships are known, the transformation from R recovers the scaling law 473

that was put into the model (Figure 9a). If we instead use the first earthquake before and 474

after a putative trigger, a scaling with α=0.5 is recovered, similar to that recovered for the 475

real catalog (Figure 9b). This demonstrates that the discrepancy between our scaling and 476

that found in other studies is due to the inclusion of some non-causally related interevent 477

times in the R distribution. Our interevent time observations are, in fact, consistent with 478

the number of triggered earthquakes being directly proportional to strain, as found in 479

previous studies. Consequently, the triggering intensity calculated by our transformation 480

represents a lower bound on the real fractional rate change. 481

482

Triggering intensity n (Equation 2) should be a good measure of earthquake rate as long 483

as it scales in a consistent manner with R . The zero-dimensional simulation shows that 484

n indeed scales consistently, but this simulation only considered nearfield aftershock 485

triggering, not triggering from distant earthquakes unconnected to the local earthquake 486

cascade. It is not immediately obvious that the effect of unknown parentage will be 487

identical in the near and farfield. Since we interpret the continuity in the scaling from 488

farfield to nearfield as indicative of continuity in the triggering physics, we need to verify 489

that the continuity is robust despite the imperfect transformation. 490

491

To verify the robustness of the continuity of the near and farfield trends, we now apply 492

the R statistic to a full space-time simulated ETAS catalog. ETAS models have generally 493

been used to study near and intermediate-field triggering. We make a key modification 494

to the model in order to introduce far field triggering, replacing the aftershock 495

23

productivity law and the spatial clustering kernel with the simple rule that earthquakes 496

are triggered in direct proportion to dynamic strain. In practice, this means adjusting the 497

parameters of the standard empirical descriptions to match the magnitude and distance 498

components of the empirical ground motion regressions. Suggestively, this requires only 499

a slight modification of the parameters estimated by Felzer [2002] and Hardebeck et al., 500

[2008]. We are then able to generate both normal aftershock triggering and long-range 501

triggering due to distant sources in the same simulated earthquake catalog. The 502

productivity constant for long-range triggers is set to match the proportionality in the 503

nearfield (Appendix C), and comes to about 300 earthquakes km-2 per unit strain. 504

505

Applying the R statistic to the simulated catalog, we recover the continuous trend 506

observed in the real data for several ETAS parameter sets taken from the literature (Table 507

1; Figure 10). The log-slope of the trend (α) is different for different simulation runs, 508

and correlates most strongly with the fraction of triggered earthquakes generated in a 509

particular realization (Figure 10d). We interpret this as meaning that for a higher fraction 510

of triggered events (low background fraction), there is a higher probability that the 511

earthquakes used to compute R are related to some other ongoing sequence rather than 512

the putative trigger, introducing a higher proportion of uniformly distributed samples into 513

the R distribution. As a result, the recovered slope correlates with the proportion of 514

background earthquakes in the final simulated catalog. The precise relationship between 515

the absolute value of n and the other statistics of the catalog is beyond the scope of this 516

study. 517

518

24

Importantly, the continuity between the long and short-range trends is robustly recovered 519

for all simulations. The simulation that best matches the observations (Figure 10a) has a 520

background fraction of ~25%. This is in agreement with the most recent estimates for 521

California seismicity, which put the background fraction at ~18-24% [Hainzl et al., 522

2006; Marsan and Lengline, 2008]. The agreement between the simulation and the 523

observations is remarkable in that the parameters going into the simulation are based on 524

aftershock numbers and rates [Felzer et al., 2002], and are not tuned to match the 525

interevent time observations. The assignment of all aftershock production to dynamic 526

strain is required in order to explain the observed interevent times in the nearfield, given 527

the model assumption that intensity of triggering falls off according to Omori’s law. This 528

suggests that roughly 75% of the earthquakes in the real seismicity catalog are 529

dynamically triggered. The agreement between the observed and simulated data in the 530

farfield is also remarkable, given that the modeled triggering rate is based only on the 531

proportionality observed in the nearfield. 532

533

The ETAS simulation validates the interpretation with respect to the continuity of the 534

trend and strongly suggests that the real data reflect a direct proportionality between 535

fractional rate change and peak dynamic strain, with a productivity of ~300 earthquakes 536

km-2 per unit strain. 537

538

6. Discussion 539

6.1 Implications for dynamic triggering 540

Peak dynamic strain is a good empirical predictor of triggering intensity in both the near 541

25

and farfield (Figure 7). This continuity from the farfield, where we know that all 542

earthquakes are triggered dynamically, into the nearfield, implies that dynamic strains 543

determine the timing and rate of earthquake triggering at all distance. How do dynamic 544

strains, which produce no permanent load change, nonetheless dominate earthquake 545

triggering rates? The low threshold for dynamic triggering suggests that arbitrarily small 546

dynamic strains can trigger earthquakes on nucleation sites that are sufficiently near 547

failure. Without a physical threshold for dynamic triggering, the question becomes one 548

of a balance of timescales -- the timescale over which a nucleation site is loaded to failure 549

quasi-statically vs. the time between dynamic strain events large enough to push the fault 550

the rest of the way. If the dynamic trigger recurrence time is smaller than the quasi-static 551

time to failure, the fault will be triggered dynamically. A fault far from failure is unlikely 552

to be triggered by any but the largest dynamic strain events. However, as the fault nears 553

failure, not only are smaller and smaller dynamic strains required for triggering, but the 554

availability of sufficient triggers increases due to the greater abundance of small 555

earthquakes. 556

557

In fact, a simple scaling argument shows that a fault is just as likely to be triggered by a 558

small strain event as by a large one. We have shown that the number of dynamically 559

triggered earthquakes is linearly proportional to strain. The peak dynamic strain 560

increases as ~10M, so the number of triggered earthquakes for a given strain event also 561

goes as 10M. The Gutenberg-Richter distribution gives that the number of earthquakes 562

with magnitude M goes as ~10-M. Therefore, the total triggering power for each 563

magnitude bin as a whole is constant with respect to the amplitude of the dynamic strain; 564

26

(9)

Small earthquakes with magnitudes below the level of catalog completeness are therefore 565

very important in triggering subsequent earthquakes. The cascade model implies that the 566

duration and total number of earthquakes triggered in any given sequence is therefore 567

strongly dependent on the magnitude of the smallest physically possible earthquake 568

[Sornette and Werner, 2005]. Similar arguments for the importance of small earthquakes 569

have been made previously based on statistical considerations [Felzer et al., 2004; Felzer 570

and Brodsky, 2006; Helmstetter et al., 2005]. 571

572

6.2 How delayed earthquakes can be triggered earthquakes 573

Earthquakes arbitrarily delayed from the mainshock are allowed to contribute to the 574

triggering signal in the measurements made here. We choose not to limit the delay time 575

for triggering for several reasons. First, primary triggered earthquakes may be obscured 576

by the passage of the generative seismic waves. This is especially true for nearfield 577

triggering, where it has been demonstrated that a tremendous number of early aftershocks 578

are usually missing from earthquake catalogs [Kagan, 2004; Peng et al., 2007]. In this 579

case, the first earthquake in the catalog may actually be a secondarily triggered 580

earthquake, that is, an aftershock of an obscured, directly triggered quake [Brodsky, 581

2006]. The timing of the delayed secondary quake provides a lower bound on the rate 582

increase associated with the direct triggering. Furthermore, if the Gutenberg-Richter 583

relationship holds for magnitudes much smaller than the completeness magnitude, many 584

small directly triggered quakes will not make it into the catalog, but may still play an 585

N !( ) = NTriggers "NTriggered

N !( )#10M "10$M= constant

27

important role in prolonging the aftershock sequence, as demonstrated by Equation 9. 586

588

Second, it is possible that dynamic strains trigger earthquakes by inducing a semi-589

permanent change in the properties of the fault patch, rather than only through transiently 590

exceeding the fault strength. Several studies have posited long-lasting changes in the 591

mechanical properties or effective stresses within fault zones related to the passage of 592

high-amplitude seismic waves. [Brodsky et al., 2003; Elkhoury et al., 2006; Johnson and 593

Jia, 2005; Parsons, 2005]. Delayed triggering may then simply reflect the prolonged 594

nature of the triggering process. 595

596

Regardless of whether delayed triggering reflects an incompletely observed earthquake 597

cascade or a prolonged physical perturbation of the fault conditions, uncorrelated (non-598

triggered) events are sampled from a uniform distribution of R values, and their inclusion 599

will mute the triggering signal but not introduce a bias in the mean. 600

601

6.3. Robustness of the observations with regard to parameters 602

The binning of the data and the separation of triggered quakes into farfield and nearfield 603

populations required the introduction of some arbitrary parameters. We want to be 604

certain that the continuity in triggering intensity between the near and farfield populations 605

can be interpreted as a continuity in the triggering mechanism. The success of the 606

statistical simulation in reproducing the observations lends some confidence to this 607

interpretation, but we also check the robustness of the results with respect to the data 608

selection parameters. 609

28

610

The first arbitrary parameter is the spatial bin size. The results shown in Figure 7-8 use a 611

bin size of 0.1º, because this maximizes the number of R values we can calculate. Figure 612

11 confirms that the continuous scaling is not sensitive to the spatial bin size, as long as 613

the number of data points remains high. We show results for bin sizes between ~8 km2 614

and 123 km2 (0.025º - 0.1º on a side). For larger bins, either the reduced quantity of data 615

or the masking of triggered activity by unrelated local aftershocks causes confidence 616

limits to exceed the mean triggering signal for the farfield data. 617

618

The distance cutoff for farfield triggers also does not influence the results. Trials using 619

minimum far-field cutoff distances of 800 km through 3200 km also recover the 620

continuous scaling (Figure 12). The variance in R begins to overwhelm the triggering 621

signal with cutoffs above 3200 km, due to the diminished dataset, and the surface wave 622

magnitude relationship is not valid for distances less than the order of 800 km, as 623

mentioned before. 624

625

Finally, to make sure the long-range triggering signal is not generated entirely by isolated 626

geothermal areas, we plot the contribution of each spatial bin to the total measured 627

triggering intensity, combining all the farfield amplitude bins (Figure 13). Geothermal 628

regions (particularly Long Valley and the Salton Trough) contribute strongly, as 629

expected, but virtually all regions of active seismicity in California contribute to the long-630

range triggering signal. 631

632

29

6.4 Relation to previous work 633

Dividing earthquakes into populations with common strain is a novel way of looking at 634

the scaling of triggering intensity with dynamic strain. Previous work has shown the 635

relevance of dynamic triggering in the nearfield by comparing the falloff in aftershock 636

density away from a mainshock to the falloff of seismic waves at near and intermediate 637

distances [Felzer and Brodsky, 2006; Gomberg and Felzer, 2008]. These correlations 638

cannot be trivially mapped to a particular function of dynamic strain, however, because 639

the decay in triggering intensity is superimposed on the decay of available nucleation 640

sites away from the mainshock. It therefore becomes necessary to carefully analyze the 641

statistics of non-triggered (background) seismicity in order to extract the triggering 642

function. The method defined here does not suffer from this ambiguity, because each 643

value of R that goes into a distribution measures the change in seismicity rate at a single 644

site. Since we use various combinations of magnitude and distance for the farfield case, 645

and various magnitudes but constant distance for the nearfield case, there is not a one-to-646

one relationship between dynamic strain and distance from the mainshock in these 647

populations. We therefore do not need to be concerned about the geometry of the local 648

fault network. 649

650

7. Conclusion 651

The observations presented here have the following implications: (1) extremely small 652

strains can trigger faults if they are sufficiently near failure, down to observed levels of 653

3×10-9 strain. (2) Dynamic strain events are frequent enough and productive enough to 654

control the timing of the majority of aftershocks, generating roughly 75% of the 655

30

earthquakes in the California seismicity catalog. (3) There is no qualitative distinction 656

between long-range triggered earthquakes and aftershocks. 657

658

These observations place strong constraints on mechanisms of earthquake nucleation. 659

Furthermore, the empirical connection between triggering rate and dynamic strain lays 660

the foundation for a probabilistic description of earthquake clustering based on 661

instrumentally observed seismic wave amplitudes. This has major implications for time-662

dependent forecasting of earthquakes. 663

664

Appendix A: Expectation of R for a Poisson process with a step change in intensity 665

To find the expectation of R, we first derive the distribution of R by transforming the 666

probability density functions for the times to the first events before and after the trigger, 667

t1 and t2. 668

669

The interevent times in a Poisson process follow an exponential distribution with the 670

general form 671

f t( ) = λ t( )exp − λ t( )dt

0

t

∫{ } (A1)

where λ(t) is the intensity, or average rate of earthquakes, at time t. The joint distribution 672

of the two independent recurrence times is then the product of two exponential 673

distributions. Defining 674

N t( ) = λ t( )dt

0

t

∫ (A2)

the joint distribution of the interevent times is 675

31

f (t1,t2 ) = λ1 t1( )λ2 t2( )exp −N1 t1( ) − N2 t2( ){ } . (A3)

where subscripts 1 and 2 refer to the rates before and after the trigger respectively. 676

677

We derive the joint distribution of R, where 678

R =t2

t1 + t2, (A4)

and another arbitrary variable, e.g. T = t2 , by substituting the inverse definitions of R and 679

T into Equation A3 and multiplying by the absolute value of the Jacobian of the 680

transformation [Casella and Berger, 2002]. The Jacobian is given by 681

J = det∂t1

∂R∂t1

∂T∂t2

∂R∂t2

∂T

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= −

TR2

1R−1

0 1= −

TR2

. (A5)

The joint distribution for R and T is then 682

f R,T( ) = λ1TR− T⎛

⎝⎜⎞⎠⎟λ2 T( )exp −N1

TR− T⎛

⎝⎜⎞⎠⎟− N2 T( )

⎧⎨⎩

⎫⎬⎭TR2

. (A6)

The marginal distribution of R is obtained by integrating Equation A6 with respect to T. 683

The expectation of R is therefore the double integral 684

R = λ1TR− T⎛

⎝⎜⎞⎠⎟λ2 T( )exp −N1

TR− T⎛

⎝⎜⎞⎠⎟− N2 T( )

⎧⎨⎩

⎫⎬⎭TR0

∞

∫01

∫ dTdR . (A7)

685

For the stepwise homogenous Poisson process with a step change in otherwise constant 686

intensity λi, the solution to Equation A7 is 687

32

R =λ1λ2

λ2 − λ1( )2λ1λ2

+ ln λ2λ1

⎛⎝⎜

⎞⎠⎟−1

⎛

⎝⎜⎞

⎠⎟. (A8)

Equation A8 can be expressed as a function of the fractional rate change, 688

n ≡N2 t( ) − N1 t( )

N1 t( )=λ2 − λ1λ1

, (A9)

Note that the second equality in Equation A9 only holds for the cases where λ1 and λ2 are 689

independent of time. 690

691

Substituting Equation A9 into Equation A8 yields, 692

R =1n2

n +1( ) ln n +1( ) − n⎡⎣ ⎤⎦ . (A10)

Equation A10 is Equation 2 in the main text with the parameter n identified as triggering 693

intensity. For a measured sample mean R , triggering intensity is found numerically by 694

identifying R with R and solving Equation A10 for n. 695

696

For the inhomogeneous Poisson process with Omori-law decay in the triggered rate 697

change, the expectation of R is calculated according to Equation A7, with 698

λ2 t( ) = λ1 +k

t + c( )p . (A11)

In this case, the fractional rate change depends on the time over which it is measured: 699

n =N2 t( ) − N1 t( )

N1 t( ) =λ1t + k t + c( )

0

t

∫− p

dt − λ1t

λ1t. (A12)

For comparison between models (Figure 3), we choose to calculate n at time t = λ1−1 , the 700

33

expected time to the first event given the background rate λ1. Equation A12 then 701

simplifies to 702

n = k t + c( )− p dt

0

1λ1∫ . (A13)

703

Appendix B: Calculation of Earthquake Probability from R 704

Appendix A shows how to transform the distribution of R into fractional earthquake rate 705

change n, assuming Poisson distributed interevent times. The same statistical model can 706

be used to calculate the probability of triggering an earthquake given the estimated rate 707

change. For a homogeneous Poisson process, the probability of observing exactly m 708

events in time period t, given the average event rate λ, is given by 709

P n = m | λt( ) =

λt( )m

e−λt

m!. (B1)

The probability of observing one or more events is equal to one minus the probability of 710

observing zero events. 711

P n > 0 | λt( ) = 1− P n = 0 | λt( ) = 1− exp −λt{ } . (B2)

Over one recurrence interval (time t = λ1−1 ), the probability of observing at least one 712

event given fractional rate change n is 713

P N EQs ≥ 1| n( ) = 1− exp − n +1( ){ } . (B3)

That is, if the average seismicity rate were one earthquake per day, the probability of 714

seeing an earthquake on any given day, in the absence of triggering, is about 63%. 715

Equation B3 then gives the adjusted probability of seeing that “daily” earthquake given 716

the increase in seismicity rate measured by n. Equation B3 is Equation 8 in the main text. 717

34

718

Appendix C: Modified ETAS simulation 719

The Epidemic-Type Aftershock Sequence (ETAS) model uses empirical probability 720

distributions to stochastically generate realistically clustered earthquake catalogs [Ogata, 721

1998]. We briefly summarize the governing equations here and direct the interested 722

reader to the studies cited in the main text for more information. 723

724

(1) Earthquake magnitudes are assigned from a Gutenberg-Richter probability 725

distribution, 726

N(M ) = 10a−bM , (C1)

where N is the number of earthquakes with magnitude greater than or equal to M, and a 727

and b are constants, with b typically around 1. 728

729

(2) The temporal decay of aftershock sequences is governed by the Modified Omori’s 730

law, which states that aftershock rate decreases approximately as 1 over the time since 731

the mainshock. 732

R(t)∝ c + t( )− p , (C2)

where R is the instantaneous aftershock rate, c is a constant that effectively keeps the rate 733

finite at zero time, and p is the decay exponent. 734

735

(3) An aftershock productivity law is necessary to close the equations in time and 736

magnitude. Previous work shows that number of aftershocks scales as a power law with 737

mainshock magnitude [Felzer et al., 2004; Helmstetter et al., 2005]. 738

35

NAS ∝10αM , (C3)

where α is a constant near 1 and M is mainshock magnitude. Equations C2 and C3 are 739

related, in that the integral of R(t) over the duration of the aftershock sequence equals NAS, 740

and we define a productivity constant A such that 741

NAS =A ⋅10α M −Mmin( )

c + t( )p0

∞

∫ dt , (C4)

where Mmin is the minimum magnitude in the simulation. For p > 1 , NAS is finite. For 742

p ≤ 1 , NAS must be calculated over a finite time period. Equation C4 is calibrated to 743

reproduce Bath’s Law (with α=1), which states that the largest aftershock of a sequence 744

is on average ~1 magnitude unit below the mainshock magnitude. 745

746

(4) A full space-time simulation also requires a law describing the spatial clustering of 747

aftershocks. For example, Felzer and Brodsky [Felzer and Brodsky, 2006] give 748

ρ r( )∝ r−γ , (C5)

where ρ is linear aftershock density at distance r from the mainshock, and γ is a constant. 749

750

We replace rules (3) and (4) with an equivalent rule that also reproduces Bath’s Law and 751

a power-law decrease in linear aftershock density, but is based on dynamic strain rather 752

than magnitude. The equivalent rule specifies that the number of aftershocks per unit 753

area scales linearly with peak dynamic strain. 754

NAS =κεdyn . (C6)

The constant of proportionality κ is found by dividing the number of aftershocks 755

36

predicted by Equation C4 by the peak dynamic strain integrated over the aftershock zone. 756

For simplicity, the constant of proportionality between strain and magnitude in Equation 757

6 (main text) is rounded from to 1. Anelastic attenuation is also neglected for the 758

calibration. This gives 759

κ =A ⋅10− Mmin +1.35( )c 1− p( )γCS

2π p −1( ) Dγmax − D

γmin( ) , (C7)

where CS is the shear wave speed, and Dmax and Dmin are the maximum and minimum 760

bounds of the local aftershock zone, imposed to make the simulation numerically 761

tractable. 762

763

Remotely triggered earthquakes are generated in proportion to κ times their dynamic 764

strain amplitude. In this way, triggering associated with both local earthquakes and the 765

surface waves of distant earthquakes is simulated simultaneously in a self-consistent 766

manner. 767

768

We use a version of Felzer and Felzer’s matlab code, modified to include a separate 769

catalog of global triggers (http://pasadena.wr.usgs.gov/office/kfelzer/AftSimulator.html) 770

[Felzer et al., 2002]. This code was also used by Hardebeck et al., [2008]. An estimate of 771

the spatially varying California background seismicity rate is included with the code 772

(Figure 13b), and several sets of parameters previously determined for California are 773

taken from the literature (Table 1). The dimensions of the aftershock zone, specified for 774

computational efficiency and to keep the number of aftershocks finite, are left at the 775

default values of Dmin=0.001 km and Dmax=500 km, respectively. We run 30 simulations 776

37

for each parameter set. 777

779

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942

Figure Captions 943

Table 1. Three sets of parameters used in the ETAS simulation. All three sets have been 944

derived for California data and are taken from the literature. A is the productivity 945

constant that controls the number of aftershocks per mainshock, c is the time offset in the 946

Modified Omori’s law, p is the time decay of aftershock rate in Omori’s law, and κ is the 947

productivity constant as a function of strain, calculated according to Equation C7. 948

949

Figure 1. Cartoon timeline illustrating the variables contributing to the R statistic. The 950

time t1 is the time since the last earthquake before the trigger, and t2 is the waiting time to 951

the first earthquake after the trigger. 952

953

Figure 2. Schematic cartoon illustrating the distinction between the distribution of R in a 954

case with no triggering (black) and a simulated case of strong triggering (red). The 955

integral of the probability density is 1 in both cases. The mean value of R is 0.50 in the 956

non-triggered case and 0.46 in the triggered example. 957

958

Figure 3. (a) Expectation of R (Equation A7), i.e. predicted , as a function of rate 959

change for three probabilistic models for earthquake recurrence. Fractional rate change is 960

the normalized triggered earthquake rate (Equation 3). The curves are all very similar. 961

R

45

Using one model or another to transform from to rate change will not affect the 962

interpretation of thresholds or scaling with trigger amplitude. The curves are especially 963

similar for small rate changes, like those sought in this study. (b) The difference between 964

the predicted R and its value in the absense of triggering (ΔR = 0.5 − R ) plotted on a log 965

scale to highlight the similarity between curves for small rate changes. 966

967

Figure 4. Cartoon illustrating the construction of earthquake populations for analysis 968

with the R statistic. For the long-range case, various combinations of magnitude and 969

distance are combined to create populations of earthquakes bracketing potential triggers 970

of common dynamic strain amplitude at the site of the triggered quakes. The study area 971

is gridded, and one R measurement is made in each bin for each trigger. Zones of 972

common strain form arcs within the study zone. For the short-range case, populations are 973

constructed by combining all earthquakes within some small radius of potential triggers 974

of common magnitude. 975

976

Figure 5. Distribution of R for spatial bins of any dynamic strain amplitude associated 977

with the 2002 M7.9 Denali earthquake. The high proportion of small R values 978

demonstrates triggering in a large proportion of the 0.1º×0.1º bins. The mean of R is 979

0.475. Data are smoothed by a 0.09 unit cosine filter for clarity. 980

981

Figure 6. Empirical probability densities for R. (a) Long-range California data (trigger 982

distance > 800 km) for four strain increments. (b) Short-range data, for magnitudes 3.1 to 983

5.1 in five increments. Curves have been smoothed for clarity using a cosine weighted 984

R

46

running average with a window width of 0.09 units. The oscillations at this wavelength 985

are therefore a spurious effect of the smoothing. Curves are truncated at the limits to 986

avoid plotting edge effects of the smoothing. 987

988

Figure 7. Triggering intensity (n in Equation 2) as a function of peak dynamic strain. 989

(a) California, (b) Japan. Long-range (>800 km) triggers are in red and short-range 990

triggers (<6 km) are in blue. Error bars are 95% confidence limits. The green point in 991

panel (a) corresponds to the Denali earthquake. The horizontal bar at bottom right shows 992

the 2σ uncertainty associated with the nearfield strain estimate. Open symbols are used 993

to denote points below the threshold for which data are consistently significant. The 994

black dashed line in both panels shows the weighted least squares fit to the combined 995

long and short-range data in California, for comparison with Japan. This line goes as 996

ε0.52+/- 0.03. 997

998

Figure 8. Probability of having an earthquake within the pre-trigger recurrence interval 999

(Equation 2), as a function of peak dynamic strain. (a) California, (b) Japan. Colors and 1000

error bars are as in Figure 7. The horizontal dotted line shows the baseline probability of 1001

having an earthquake within its own recurrence interval in the absence of triggering 1002

(~63%). The black dashed line is the best-fit line from Figure 7 transformed using 1003

Equation 8, and shows that the rapid increase in probability is consistent with a smooth 1004

increase in triggering intensity. Again, exactly the same curve is used in both (a) and (b) 1005

to facilitate comparison between Japan and California. 1006

1007

47

Figure 9. Testing the effect of the earthquake cascade on the transformation of to 1008

fractional rate change n. A zero-dimensional (time only) ETAS model is used to generate 1009

a simulated seismicity catalog in which the triggering law and causal relationships are 1010

known. If R is calculated using the first causally related earthquake before and after the 1011

trigger, the transformation recovers the imposed triggering law (blue curve). If the R 1012

calculation is not restricted to known causally related earthquakes, the method recovers a 1013

reduced scaling exponent (red curve). Error bars are 95% confidence limits. The black 1014

dashed line is the input scaling relationship. The blue dashed line is the best fit to the 1015

data using unknown causality and has a slope of 0.5. 1016

1017

Figure 10. Triggering intensity (n from Equation 2), as a function of peak dynamic 1018

strain for simulated seismicity catalogs, using three different sets of ETAS parameters 1019

estimated for California (Table 1). The parameters are from (a) [Hardebeck et al., 2008], 1020

(b) [Console et al., 2007], (c) [Reasenberg and Jones, 1989]. Panel (a) includes the 1021

observed data from the ANSS catalog as filled symbols (Figure 7a). The recovery of a 1022

continuous trend between near and far-field data in the simulated catalogs demonstrates 1023

that the continuity recovered for the real data is robust. In each case, triggering is 1024

simulated to be a linear function of dynamic strain, but a power law slope of less than 1 is 1025

recovered by the interevent time method. Panel (d) shows that the recovered slope scales 1026

with the proportion of background quakes in the catalog. The circled point marks a 1027

simulation with a recovered slope similar to that obtained for the real data. 1028

1029

Figure 11. Triggering intensity as a function of strain for different bin sizes. Curves on 1030

R

48

the left are farfield data; curves on the right are nearfield. The legend gives the farfield 1031

spatial bin dimension in degrees for each curve. The nearfield aftershock radius is scaled 1032

to cover the same area as the farfield bins. For clarity of presentation, data are plotted 1033

against the peak dynamic strain integrated over the bin, rather than the averaged strain, 1034

and are therefore offset. The slopes of the curves and the absolute values of the 1035

triggering intensity change slightly with bin size, but the continuity between trends is 1036

robust. 1037

1038

Figure 12. Sensitivity to long-range trigger cutoff distance for the California dataset. 1039

The legend gives the minimum distance for potential farfield trigger earthquakes. Results 1040

are not sensitive to distance cutoffs above 800 km, although uncertainties grow larger 1041

because of the reduced catalog size for larger cutoffs. The data in red are the same as the 1042

long-range points shown in Figure 7a. The surface wave magnitude relation (Equation 5) 1043

is not applicable for distance cutoffs less than 800 km. 1044

1045

Figure 13. Geographical distribution of triggering susceptibility in California partially 1046

reflects the background activity rate. a) Triggering intensity in 0.1º spatial bins for 1047

strains above 10-9. b) Background seismicity rate [Hardebeck et al., 2008], expressed in 1048

terms of the number of magnitude 4 and greater earthquakes per year in each ~0.5º bin. 1049

The plots are qualitatively similar, implying that all regions of active seismicity are 1050

triggerable, and no single region dominates the triggering signal. Both maps are 1051

smoothed by a 0.3º Gaussian kernel. 1052

1053

1054

49

1054

A c (days) p κ

(km-2 strain-1)

[Hardebeck et al., 2008] 0.008 0.095 1.34 308

[Console et al., 2007] 0.00288 0.00437 1.146 257

[Reasenberg and Jones, 1989] 0.0035 0.05 1.08 327

Table 1. ETAS Parameters 1055

t1 t2

1stquakebefore

(poten1al)Triggerquake

1stquakea7er

Figure 1.

R = 12R < 1

2

Figure 2.

0 0.2 0.4 0.6 0.8 10

1

3

4

Prob

abilityDen

sity

R

2

10‐3 10‐2 10‐1 10‐0 101 102 1030

0.1

0.2

0.3

0.4

0.5

Expe

cted

R

Frac1onalRateChangen

10‐3 10‐2 10‐1 10‐0 101 102 10310‐4

10‐3

10‐2

10‐1

Expe

cted

ΔR

Frac1onalRateChangen

PeriodicPoissonOmori

Figure 3.

a

b

PeriodicPoissonOmori

Nearfieldtrigger

Farfieldtrigger

Zonesofcommonpeakdynamicstrain

Figure 4.

Figure 5.

0 0.2 0.4 0.6 0.8 10

0.8

1.2

1.8

Prob

abilityDen

sity

R

1

1.6

1.4

2

0 0.2 0.4 0.6 0.8 10

0.95

1.1

Prob

abilityDen

sity

R

1

1.05

Dynam

icStrain

10‐8

10‐7

10‐9

6

Figure 6.

0 0.2 0.4 0.6 0.8 10

1

Prob

abilityDen

sity

R

2

5

Dynam

icStrain

10‐5

10‐4

3

4

a

b

Figure 7.

PeakDynamicStrain

10‐1010‐3

TriggeringIntensity

b 10‐9 10‐8 10‐7 10‐6 10‐5 10‐4 10‐3 10‐2

10‐2

10‐1

100

101

102FarfieldtriggeringNearfieldtriggering

PeakDynamicStrain10‐10

10‐3

TriggeringIntensity

a 10‐9 10‐8 10‐7 10‐6 10‐5 10‐4 10‐3 10‐2

10‐2

10‐1

100

101

102FarfieldtriggeringNearfieldtriggering

Figure 8.

PeakDynamicStrain10‐10

b 10‐9 10‐8 10‐7 10‐6 10‐5 10‐4 10‐3 10‐2

FarfieldtriggeringNearfieldtriggering

PeakDynamicStrain10‐10

Prob

abilityofTriggering

a 10‐9 10‐8 10‐7 10‐6 10‐5 10‐4 10‐3 10‐2

0.6

FarfieldtriggeringNearfieldtriggering

0.7

0.8

0.9

1

‐0.1

Prob

abilityGain

0

0.1

0.2

0.3

Prob

abilityofTriggering

0.6

0.7

0.8

0.9

1

‐0.1

Prob

abilityGain

0

0.1

0.2

0.3

Figure 9.

Magnitude2

TriggeringIntensity

3 4 5 6 710‐2

10‐1

100

103

104

101

102

8

PeakDynamicStrain10‐10

10‐3

TriggeringIntensity

a 10‐9 10‐8 10‐7 10‐6 10‐5 10‐4 10‐3 10‐2

10‐2

10‐1

100

101

102

PeakDynamicStrain10‐10

10‐3

TriggeringIntensity

b

10‐9 10‐8 10‐7 10‐6 10‐5 10‐4 10‐3 10‐2

10‐2

10‐1

100

101

102

PeakDynamicStrain10‐10

10‐3

TriggeringIntensity

c 10‐9 10‐8 10‐7 10‐6 10‐5 10‐4 10‐3 10‐2

10‐2

10‐1

100

101

102

BackgroundFrac1oninCatalog0.1

0.4

RecoveredSlop

e

d

Simula1on:Hardebeck Simula1on:Console

Simula1on:Reasenberg

0.5

0.6

0.7

0.8

0.9

1

0.2 0.3 0.4 0.5 0.6 0.7

Figure 10.

TotalDynamicStrain

10‐3

TriggeringIntensity

10‐9 10‐8 10‐7 10‐6 10‐5 10‐4 10‐3 10‐2

10‐2

10‐1

100

101

102

10‐1 100

Figure 12.

PeakDynamicStrain

10‐3

TriggeringIntensity

10‐2

10‐1

100

10‐10 10‐8 10‐7 10‐6 10‐510‐9

Figure 11.

800km1600km3200km

0.025°0.05°0.1°

TriggeringIntensity

0.03

Figure 13.

0.02

0.01

0

0.035

0.025

0.015

0.005

‐0.005

b

a

Backgrou

ndAc1vity(M

≥4quakes/yr)0.7

0.8

0.6

0.5

0.4

0.3

0.2

0.1

0